Astrophysics and Space Science

, Volume 342, Issue 1, pp 155–228

Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests

Authors

    • Kobayashi-Maskawa Institute for the Origin of Particles and the UniverseNagoya University
  • Salvatore Capozziello
    • Dipartimento di Scienze FisicheUniversità di Napoli “Federico II”
    • INFN Sez. di NapoliCompl. Univ. di Monte S. Angelo
  • Shin’ichi Nojiri
    • Kobayashi-Maskawa Institute for the Origin of Particles and the UniverseNagoya University
    • Department of PhysicsNagoya University
  • Sergei D. Odintsov
    • Instituciò Catalana de Recerca i Estudis Avançats (ICREA)
    • Instituto de Ciencias del Espacio (CSIC) and Institut d Estudis Espacials de Catalunya (IEEC-CSIC)Facultat de Ciencies
    • Tomsk State Pedagogical University
    • Eurasian International Center for Theoretical Physics and Department of General & Theoretical PhysicsEurasian National University
Invited Review

DOI: 10.1007/s10509-012-1181-8

Cite this article as:
Bamba, K., Capozziello, S., Nojiri, S. et al. Astrophys Space Sci (2012) 342: 155. doi:10.1007/s10509-012-1181-8

Abstract

We review different dark energy cosmologies. In particular, we present the ΛCDM cosmology, Little Rip and Pseudo-Rip universes, the phantom and quintessence cosmologies with Type I, II, III and IV finite-time future singularities and non-singular dark energy universes. In the first part, we explain the ΛCDM model and well-established observational tests which constrain the current cosmic acceleration. After that, we investigate the dark fluid universe where a fluid has quite general equation of state (EoS) [including inhomogeneous or imperfect EoS]. All the above dark energy cosmologies for different fluids are explicitly realized, and their properties are also explored. It is shown that all the above dark energy universes may mimic the ΛCDM model currently, consistent with the recent observational data. Furthermore, special attention is paid to the equivalence of different dark energy models. We consider single and multiple scalar field theories, tachyon scalar theory and holographic dark energy as models for current acceleration with the features of quintessence/phantom cosmology, and demonstrate their equivalence to the corresponding fluid descriptions. In the second part, we study another equivalent class of dark energy models which includes F(R) gravity as well as F(R) Hořava-Lifshitz gravity and the teleparallel f(T) gravity. The cosmology of such models representing the ΛCDM-like universe or the accelerating expansion with the quintessence/phantom nature is described. Finally, we approach the problem of testing dark energy and alternative gravity models to general relativity by cosmography. We show that degeneration among parameters can be removed by accurate data analysis of large data samples and also present the examples.

Keywords

Modified theories of gravityDark energyCosmology

1 Introduction

Cosmic observations from Supernovae Ia (SNe Ia) (Perlmutter et al. 1999; Riess et al. 1998), cosmic microwave background (CMB) radiation (Spergel et al. 2003, 2007; Komatsu et al. 2009, 2011), large scale structure (LSS) (Tegmark et al. 2004; Seljak et al. 2005), baryon acoustic oscillations (BAO) (Eisenstein et al. 2005), and weak lensing (Jain and Taylor 2003) have implied that the expansion of the universe is accelerating at the present stage. Approaches to account for the late time cosmic acceleration fall into two representative categories: One is to introduce “dark energy” in the right-hand side of the Einstein equation in the framework of general relativity (for recent reviews on dark energy, see Caldwell and Kamionkowski 2009; Amendola and Tsujikawa 2010; Li et al. 2011c; Kunz 2012). The other is to modify the left-hand side of the Einstein equation, called as a modified gravitational theory, e.g., F(R) gravity (for recent reviews, see Nojiri and Odintsov 2011, 2006a; Capozziello and Faraoni 2010; Clifton et al. 2012; Capozziello and De Laurentis 2011; Harko and Lobo 2012; Capozziello et al. 2012).

The various cosmological observational data supports the Λ cold dark matter (ΛCDM) model, in which the cosmological constant Λ plays a role of dark energy in general relativity. At the current stage, the ΛCDM model is considered to be a standard cosmological model. However, the theoretical origin of the cosmological constant Λ has not been understood yet (Weinberg 1989). A number of models for dark energy to explain the late-time cosmic acceleration without the cosmological constant has been proposed. For example, a canonical scalar field, so-called quintessence (Chiba et al. 1997; Caldwell et al. 1998; Fujii 1982), a non-canonical scalar field such as phantom (Caldwell 2002), tachyon scalar field motivated by string theories (Padmanabhan 2002), and a fluid with a special equation of state (EoS) called as Chaplygin gas (Kamenshchik et al. 2001; Bento et al. 2002; Bilic et al. 2002, 2009). There also exists a proposal of holographic dark energy (Li 2004; Elizalde et al. 2005; Nojiri and Odintsov 2006b).

One of the most important quantity to describe the features of dark energy models is the equation of state (EoS) wDE, which is the ratio of the pressure P to the energy density ρDE of dark energy, defined as wDEPDE/ρDE. We suppose that in the background level, the universe is homogeneous and isotropic and hence assume the Friedmann-Lemaître-Robertson-Walker (FLRW) space-time. There are two ways to describe dark energy models. One is a fluid description (Nojiri et al. 2005b; Nojiri and Odintsov 2005a; Stefancic 2005) and the other is to describe the action of a scalar field theory. In the former fluid description, we express the pressure as a function of ρ (in more general, and other background quantities such as the Hubble parameter H). On the other hand, in the latter scalar field theory we derive the expressions of the energy density and pressure of the scalar field from the action. In both descriptions, we can write the gravitational field equations, so that we can describe various cosmologies, e.g., the ΛCDM model, in which wDE is a constant and exactly equal to −1, quintessence model, where wDE is a dynamical quantity and −1<wDE<−1/3, and phantom model, where wDE also varies in time and wDE<−1. This means that one cosmology may be described equivalently by different model descriptions.

In this review, we explicitly show that one cosmology can be described by not only a fluid description, but also by the description of a scalar field theory. In other words, the main subject of this work is to demonstrate that one dark energy model may be expressed as the other dark energy models, so that such a resultant unified picture of dark energy models could be applied to any specific cosmology.

This review consists of two parts. In the first part, various dark energy models in the framework of general relativity are presented. First, we introduce the ΛCDM model and the recent cosmological observations. At the current stage, the ΛCDM model is consistent with the observational data. We then explain a fluid description of dark energy and the action representing a scalar field theory. In both descriptions of a fluid and a scalar field theory, we reconstruct representative cosmologies such as the ΛCDM, quintessence and phantom models. Through these procedures, we show the equivalence between a fluid description and a scalar field theory. We also consider a tachyon scalar field theory. Furthermore, we extend the investigations to multiple scalar field theories. In addition, we explore holographic dark energy scenarios. On the other hand, in the second part, modified gravity models, in particular, F(R) gravity as well as F(R) Hořava-Lifshitz gravity and f(T) gravity with T being a torsion scalar (Hehl et al. 1976; Hayashi and Shirafuji 1979; Flanagan and Rosenthal 2007; Garecki 2010; Bengochea and Ferraro 2009; Linder 2010), i.e., pictures of geometrical dark energy, are given. It is illustrated that by making a conformal transformation, an F(R) theory in the Jordan frame can be moved to a corresponding scalar field theory in the Einstein frame. It is also important to remark that as another modified gravitational theory to account for dark energy and the late-time cosmic acceleration, \(F(R, \mathcal{T})\) theory has been proposed in Harko et al. (2011), where \(\mathcal{T}\) is the trace of the stress-energy tensor. We use units of kB=c=ħ=1 and denote the gravitational constant 8πG by κ2≡8π/MPl2 with the Planck mass of MPl=G−1/2=1.2×1019 GeV. Throughout this paper, the subscriptions “DE”, “m”, and “r” represent the quantities of dark energy, non-relativistic matter (i.e., cold dark matter and baryons), and relativistic matter (e.g., radiation and neutrinos) respectively.

The review is organized as follows. In the first part, in Sect. 2 we explain the ΛCDM model. We also present the recent cosmological observational data, in particular, in terms of SNe Ia, BAO and CMB radiation, by defining the related cosmological quantities. These data are consistent with the ΛCDM model.

In Sect. 3, we investigate a description of dark fluid universe. We represent basic formulations for the EoS of dark energy. We also introduce the four types of the finite-time future singularities as well as the energy conditions and give examples of fluid descriptions for the ΛCDM model, the GCG model and a model of coupled dark energy with dark matter. Next, we explore various phantom cosmologies such as a coupled phantom scenario, Little Rip scenario and Pseudo-Rip model. Furthermore, we show that the fluid description of the EoS of dark energy can yield all the four types of the finite-time future singularities. A fluid description with realizing asymptotically de Sitter phantom universe is also examined. In addition, we investigate the inhomogeneous (imperfect) dark fluid universe. We study the inhomogeneous EoS of dark energy and its cosmological effects on the structure of the finite-time future singularities. Moreover, its generalization of the implicit inhomogeneous EoS is presented.

In Sect. 4, we explore scalar field theories in general relativity. We explicitly demonstrate the equivalence of fluid descriptions to scalar field theories. In particular, we concretely reconstruct scalar field theories describing the ΛCDM model, the quintessence cosmology, the phantom cosmology and a unified scenario of inflation and late-time cosmic acceleration. We also consider scalar field models with realizing the crossing the phantom divide and its stability problem.

In Sect. 5, we examine a tachyon scalar field theory. We explain the origin, the model action and its stability conditions.

In Sect. 6, we describe multiple scalar field theories. First, we examine two scalar field theories. We investigate the standard type of two scalar field theories and the stability of the system. We then introduce a new type of two scalar field theories, in which the crossing of the phantom divide can happen, and also explore its stability conditions. Next, we extend the considerations for two scalar field theories to multiple scalar field theories which consist of more scalars. and clearly illustrate those equivalence to fluid descriptions.

In Sect. 7, we study holographic dark energy. We explain a model of holographic dark energy as well as its generalized scenario. In addition, we examine the Hubble entropy in the holographic principle.

In the second part, in Sect. 8, we consider accelerating cosmology in F(R) gravity. First, by using a conformal transformation, we investigate the relations between a scalar field theory in the Einstein frame and an F(R) theory in the Jordan frame. Next, we explore the reconstruction method of F(R) gravity. We explicitly reconstruct the forms of F(R) with realizing the ΛCDM, quintessence and phantom cosmologies. In addition, we study dark energy cosmology in F(R) Hořava-Lifshitz gravity. We first present the model action and then reconstruct the F(R) forms with performing the ΛCDM model and the phantom cosmology. Furthermore, we explain \(F(R, \mathcal{T})\) gravity.

In Sect. 9, we describe f(T) gravity. To begin with, we give fundamental formalism and basic equations. We reconstruct a form of f(T) in which the finite-time future singularities can occur. We also discuss the removal way of those singularities. Furthermore, we represent the reconstructed f(T) models in which inflation in the early universe, the ΛCDM model, the Little Rip scenario and the Pseudo-Rip cosmology are realized. In addition, as one of the most important theoretical touch stones to examine whether f(T) gravity can be an alternative gravitational theory to general relativity, we explore thermodynamics in f(T) gravity. We show the first law of thermodynamics and then discuss the second law of thermodynamics, and derive the condition for the second law to be satisfied.

Next, in the following sections, we develop the observational investigations on dark energy and modified gravity. In Sect. 10, we discuss the basic ideas and concepts of cosmography in order to compare concurring models with the Hubble series expansion coming from the scale factor. In particular, we derive cosmographic parameters without choosing any cosmological model a priori.

In Sect. 11, we examine how it is possible to connect F(R) gravity by cosmography and it is possible to reproduce by it the most popular dark energy models as so called Chevallier-Polarski-Linder (CPL) (Chevallier and Polarski 2001; Linder 2003), which is a parameterization of the EoS for dark energy, or the ΛCDM.

In Sect. 12, we show, as examples, how it is possible to constrain F(R) models theoretically. However, the approach works for any dark energy or alternative gravity model.

In Sect. 13, we discuss constraints coming from observational data. It is clear that the quality and the richness of data play a fundamental role in this context.

Finally, conclusions with the summary of this review are presented in Sect. 14.

2 The Λ cold dark matter (ΛCDM) model

The action of the ΛCDM model in general relativity is described as
$$ S = \int d^4 x \sqrt{-g} \frac{1}{2\kappa^2} ( R - 2\varLambda ) + \int d^4 x {\mathcal{L}}_{\mathrm{M}} ( g_{\mu\nu}, { \varPsi}_{\mathrm{M}} ), $$
(1)
where R is the Ricci scalar, g is the determinant of the metric tensor gμν, Λ is the cosmological constant, and \({\mathcal{L}}_{\mathrm{M}}\) with a matter field ΨM is the matter Lagrangian. The EoS of the cosmological constant, which is the ratio of the pressure PΛ to the energy density ρΛ of the cosmological constant, is given by
$$ w_{\varLambda} \equiv\frac{P_{\varLambda}}{\rho_{\varLambda}} = -1. $$
(2)
We assume the 4-dimensional FLRW metric which describes the homogeneous and isotropic universe
$$ ds^2 = - dt^2 + a^2(t) \biggl[ \frac{dr^2}{1-Kr^2} + r^2 d \varOmega^2 \biggr], $$
(3)
where a(t) is the scale factor, K is the cosmic curvature (K=+1,0,−1 denotes closed, flat, and open universe, respectively), and 2 is the metric of 2-dimensional sphere with unit radius. The redshift z is defined as za0/a−1 with a0=1 being the current value of the scale factor.
In the FLRW background (3), the Einstein equations are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ4_HTML.gif
(4)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ5_HTML.gif
(5)
where \(H=\dot{a}/a\) is the Hubble parameter and the dot denotes the time derivative of /∂t. The fractional densities of dark energy, non-relativistic matter, radiation and the density parameter of the curvature are defined as
$$ \begin{array} {@{}l} \displaystyle\varOmega_{\mathrm{DE}} \equiv\frac{\rho_{\mathrm{DE}}}{\rho_{\mathrm{crit}}^{(0)}}, \qquad \varOmega_{\mathrm{m}} \equiv\frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{crit}}^{(0)}}, \\[4mm] \displaystyle \varOmega_{\mathrm{r}} \equiv \frac{\rho_{\mathrm{r}}}{\rho_{\mathrm{crit}}^{(0)}}, \qquad \varOmega_{K} \equiv - \frac{K}{ ( aH )^2}, \end{array} $$
(6)
where \(\rho_{\mathrm{crit}}^{(0)} = 3H_{0}^{2}/\kappa^{2} \) is the critical density with H0 being the current Hubble parameter. By combining Eq. (4) with the quantities in (6), we find
$$ \varOmega_{\mathrm{DE}} + \varOmega_{\mathrm{m}} + \varOmega_{\mathrm{r}} + \varOmega_{K} = 1. $$
(7)

If there only exists cosmological constant Λ, i.e., ρM=0 and PM=0, from Eq. (5) we have H=Hc=constant, so that de Sitter expansion can be realized. We also note that by comparing Eq. (4) with Eq. (2), we obtain ρΛ=Λ/κ2=−PΛ.

In addition, the scale factor is expressed as
$$ a = a_\mathrm{c} \mathrm{e}^{H_\mathrm{c} t}, $$
(8)
where ac(>0) is a positive constant.

In this section, we present the observational data of SNe Ia, BAO and CMB radiation, which supports the ΛCDM model (for the way of an analysis of observational data, see, e.g., Li et al. 2010).

2.1 Type Ia Supernovae (SNe Ia)

With SNe Ia observations, we find the luminosity distance dL as a function of the redshift z. We define the theoretical distance modulus as μth(zi)≡5log10DL(zi)+μ0. Here, DLH0dL is the Hubble-free luminosity distance and \(d_{L} \equiv\sqrt{ L_{s}/(4\pi\mathcal{F} )}\), where Ls and \(\mathcal{F}\) are the absolute luminosity of a source and an observed flux, respectively, is the luminosity distance. Moreover, μ0≡42.38−5log10h, where hH0/100/[km s−1 Mpc−1] (Kolb and Turner 1990). We express DL as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ9_HTML.gif
(9)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ10_HTML.gif
(10)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ11_HTML.gif
(11)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ12_HTML.gif
(12)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ13_HTML.gif
(13)
where \(\varOmega_{\mathrm{r}}^{(0)}=\varOmega_{\gamma}^{(0)} (1+0.2271N_{\mathrm{eff}})\) with \(\varOmega_{\gamma}^{(0)}\) being the present fractional photon energy density and Neff=3.04 the effective number of neutrino species (Komatsu et al. 2011). In what follows, the superscription “(0)” represents the values at the present time. Moreover, in deriving Eq. (13) we have used the continuity equations
$$ \dot{\rho_{j}}+ 3H ( \rho_{j} + P_{j} ) = 0, $$
(14)
where j= “DE”, “m” and “r”, and Pm=0. Furthermore, \(f_{K} (\mathcal{Y})\) in Eq. (10) is described by
$$ f_K (\mathcal{Y}) = \begin{cases} \sin\mathcal{Y} &\mbox{for $K=+1$,} \\ \mathcal{Y} &\mbox{for $K=0$,} \\ \sinh\mathcal{Y} &\mbox{for $K=-1$.} \end{cases} $$
(15)
By using Eqs. (9), (13) and (15), for the flat universe, we have \(D_{L}(z) =(1+z) \int_{0}^{z} dz'/E(z')\), where E(z) in Eq. (13) with \(\varOmega_{K}^{(0)} = 0\). It follows from this relation, we find H(z)={(d/dz)[dL(z)/(1+z)]}−1. Accordingly, for \(z < \mathcal{O}(1)\), the cosmic expansion history can be obtained through the measurement of the luminosity distance dL(z). We expand Eq. (9) around z=0 as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ16_HTML.gif
(16)
where in deriving the second equality we have used Eq. (13). It is clearly seen from Eq. (16) that suppose there exists dark energy, \(\varOmega_{\mathrm{DE}}^{(0)} > 0\) and wDE<0, and hence the luminosity distance becomes large. We note that since the universe is very close to flat as \(-0.0179 < \varOmega_{K}^{(0)} < 0.0081\) (95 % confidence level (CL)) (Komatsu et al. 2009), even though the universe is open (\(\varOmega_{K}^{(0)} > 0\)), the change of the luminosity distance is small.

By applying the Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations data (Komatsu et al. 2011), the latest distance measurements from the BAO in the distribution of galaxies, and the Hubble constant measurement, for a flat universe, the current value of a constant EoS for dark energy has been estimated as wDE=−1.10±0.14 (68 % CL) in Komatsu et al. (2011). Moreover, as an example of a time-dependent EoS for dark energy, for a linear form wDE(a)=wDE0+wDEa(1−a) (Chevallier and Polarski 2001; Linder 2003), where wDE0 and wDEa are the current value of wDE and its derivative, respectively, by using the WMAP data, the BAO data and the Hubble constant measurement and the high-redshift SNe Ia data, wDE0 and wDEa have been analyzed as wDE0=−0.93±0.13 and \(w_{\mathrm{DE}a} = -0.41^{+0.72}_{-0.71}\) (68 % CL). This form is called as the CPL model (Chevallier and Polarski 2001; Linder 2003). Consequently, for the flat universe, the various recent observational data are consistent with the cosmological constant, i.e., wDE=−1.

We also mention the way of analyzing the χ2 of the SNe Ia data, given by \(\chi_{\mathrm{SN}}^{2}=\sum_{i} [\mu_{\mathrm{obs}}(z_{i})- \mu_{\mathrm{th}}(z_{i})]^{2}/\sigma_{i}^{2}\) with μobs being the observed distance modulus. In the following, the subscripts “th” and “obs” mean the theoretical and observational values, respectively. We expand \(\chi_{\mathrm{SN}}^{2}\) as (Perivolaropoulos 2005; Nesseris and Perivolaropoulos 2005)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ17_HTML.gif
(17)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ19_HTML.gif
(19)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ20_HTML.gif
(20)
Since we do not know the absolute magnitude of SNe Ia, we should minimize \(\chi_{\mathrm{SN}}^{2}\) with respect to μ0 related to the absolute magnitude. We describe the minimum of \(\chi_{\mathrm{SN}}^{2}\) with respect to μ0 as \(\tilde{\chi}_{\mathrm{SN}}^{2}= A_{\mathrm{SN}} - B_{\mathrm{SN}}^{2}/C_{\mathrm{SN}} \) by using, for example, the Supernova Cosmology Project (SCP) Union2 compilation with 557 supernovae, whose redshift range is 0.015≤z≤1.4 (Amanullah et al. 2010).

2.2 Baryon acoustic oscillations (BAO)

Baryons couple to photons strongly until the decoupling era, and therefore we can detect the oscillation of sound waves in baryon perturbations. The BAO is a special pattern in the large-scale correlation function of Sloan Digital Sky Survey (SDSS) luminous red galaxies. Hence, we can use the BAO data to explore the features of dark energy.

We measure the distance ratio dzrs(zd)/DV(z) through the observations of the BAO. The volume-averaged distance DV(z) Eisenstein et al. (2005) and the proper angular diameter distance DA(z) for the flat universe are defined by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ21_HTML.gif
(21)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ22_HTML.gif
(22)
The comoving sound horizon rs(z) is expressed by
$$ r_{s}(z)=-\frac{1}{\sqrt{3}}\int_{0}^{1/ (1+z )} \frac{dz^{\prime}}{H(z^{\prime}) \sqrt{1+ (3\varOmega_{\mathrm{b}}^{(0)}/4\varOmega_{\gamma}^{(0)} )/ (1+z^{\prime} )}}, $$
(23)
with \(\varOmega_{\mathrm{b}}^{(0)} = 2.2765 \times10^{-2} h^{-2}\) and \(\varOmega_{\gamma}^{(0)} = 2.469\times 10^{-5}h^{-2}\) being the current values of baryon and photon density parameters, respectively (Komatsu et al. 2011). The fitting formula of the redshift zd at the drag epoch (Percival et al. 2010), when the sound horizon determines the location of the BAO because baryons are free from the Compton drag of photons, is represented as (Eisenstein and Hu 1998)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ24_HTML.gif
(24)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ25_HTML.gif
(25)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ26_HTML.gif
(26)
For \(\varOmega_{\mathrm{m}}^{(0)}=0.276\) and h=0.705, we have zd≈1021.
By using the BAO data from the Two-Degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey Data Release 7 (SDSS DR7) (Percival et al. 2010), we measure the distance ratio dz at two redshifts z=0.2 and z=0.35 as \(d_{z=0.2}^{\mathrm{obs}}=0.1905\pm0.0061\) and \(d_{z=0.35}^{\mathrm{obs}}=0.1097\pm0.0036\) with the inverse covariance matrix, defined by
$$ {\mathcal{M}}_{\mathrm{BAO}}^{-1} \equiv \left( \begin{array}{@{}c@{\quad}c@{}} 30124 & -17227\\ -17227 & 86977 \end{array} \right). $$
(27)
We express the χ2 of the BAO data as \(\chi_{\mathrm{BAO}}^{2}= (x_{i,\mathrm{BAO}}^{\mathrm{th}}-x_{i,\mathrm{BAO}}^{\mathrm {obs}}) ({\mathcal{M}}_{\mathrm{BAO}}^{-1})_{ij} (x_{j,\mathrm{BAO}}^{\mathrm{th}}-x_{j,\mathrm{BAO}}^{\mathrm{obs}} )\) with xi,BAO≡(d0.2,d0.35).

2.3 Cosmic microwave background (CMB) radiation

By using the CMB data, we can derive the distance to the decoupling epoch z(≃1090) (Komatsu et al. 2009), and hence we constrain the model describing the high-z epoch. Since the expansion history of the universe from the decoupling era to the present time influences on the positions of acoustic peaks in the CMB anisotropies, those are shifted provided that there exists dark energy.

We define the angle for the location of the CMB acoustic peaks as
$$ \theta_A \equiv\frac{r_s (z_{*})}{d_A^{(\mathrm{c})} (z_{*})}, $$
(28)
with \(d_{A}^{(\mathrm{c})} = d_{L}/(1+z)\) being the comoving angular diameter distance. The acoustic scale lA (Hu and Sugiyama 1995, 1996) representing with the CMB multipole corresponding to θA and the shift parameter \(\mathcal{R}\) (Bond et al. 1997; Efstathiou and Bond 1999) are defined as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ29_HTML.gif
(29)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ30_HTML.gif
(30)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ31_HTML.gif
(31)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ32_HTML.gif
(32)
with z being the redshift of the decoupling epoch (Hu and Sugiyama 1996), given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ33_HTML.gif
(33)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ34_HTML.gif
(34)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ35_HTML.gif
(35)
In Eq. (30), ρb and ργ are the energy density of baryons and photons, respectively. Furthermore, \(a_{\mathrm{*}}\) and aeq are the scale factors at the decoupling epoch and the radiation-matter equality time.
It is seen from Eq. (30) that since the change of the cosmic expansion history from the decoupling era to the present time influences on the CMB shift parameter, the multipole lA is shifted. The relation between all peaks and troughs of the observed CMB anisotropies is represented by (Doran and Lilley 2002)
$$ l_n = l_A ( n - \phi_n ), $$
(36)
where n denotes peak numbers and ϕn is the shift of multipoles. For instance, for the first peak n=1 and for the first trough n=1.5. Moreover, the WMAP 5-year results (Komatsu et al. 2009) present the limit on the shift parameter of \(\mathcal{R} = 1.710 \pm0.019\) (68 % CL). In the flat universe, we have \(\mathcal{R}(z_{*}) = \sqrt{\varOmega_{\mathrm{m}}^{(0)}}H_{0} (1+z_{*})D_{A}(z_{*})\). Hence, the smaller \(\varOmega_{\mathrm{m}}^{(0)}\) is, namely, the larger \(\varOmega_{\mathrm{DE}}^{(0)}\) is, the smaller \(\mathcal{R}\) is. For an estimation executed by the WMAP 5-year data analysis (Komatsu et al. 2009) of \(\mathcal{R} = 1.710\), \(\varOmega_{\mathrm{b}}^{(0)} h^{2} = 0.02265\), \(\varOmega_{\mathrm{m}}^{(0)} h^{2} = 0.1369\) and \(\varOmega_{\gamma}^{(0)} h^{2} = 2.469 \times10^{-5}\), with Eq. (30) we obtain lA=299. By plugging this value and ϕ1=0.265 Doran and Lilley (2002) into Eq. (36), we find the first acoustic peak l1=220, which is consistent with the CMB anisotropies observation.
The data from Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations (Komatsu et al. 2011) on CMB can be used. The χ2 of the CMB data is expressed as \(\chi_{\mathrm{CMB}}^{2}=(x_{i,\mathrm{CMB}}^{\mathrm {th}}-x_{i,\mathrm{CMB}}^{\mathrm{obs}}) ({\mathcal{M}}_{\mathrm{CMB}}^{-1})_{ij} (x_{j,\mathrm{CMB}}^{\mathrm{th}}-x_{j,\mathrm{CMB}}^{\mathrm{obs}})\) with \(x_{i,\mathrm{CMB}}\equiv(l_{A}(z_{*}), \mathcal{R}(z_{*}), z_{*})\) and \({\mathcal{M}}_{\mathrm{CMB}}^{-1}\) being the inverse covariance matrix. It follows from the WMAP7 data analysis Komatsu et al. (2011) that lA(z)=302.09, \(\mathcal{R}(z_{*})=1.725\), z=1091.3, and
$$ {\mathcal{M}}_{\mathrm{CMB}}^{-1}=\left( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} 2.305 & 29.698 & -1.333\\ 29.698 & 6825.27 & -113.180\\ -1.333 & -113.180 & 3.414 \end{array} \right). $$
(37)

As a result, we find the \(\chi_{\mathrm{total}}^{2}\) of all the observational data \(\chi_{\mathrm{total}}^{2}=\tilde{\chi}_{\mathrm{SN}}^{2}+\chi _{\mathrm {BAO}}^{2}+\chi_{\mathrm{CMB}}^{2}\). It is known that there exists a fitting procedure called the Markov chain Monte Carlo (MCMC) approach, e.g., CosmoMC (Lewis and Bridle 2002).

Finally, we mention an issue of the origin of the cosmological constant Λ, which is one of the most possible candidates of dark energy because its presence is consistent with a number of observations. In fact, however, the origin is not well understood yet (Weinberg 1989). The vacuum energy density ρv is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ38_HTML.gif
(38)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ39_HTML.gif
(39)
In deriving Eq. (38), we have used the zero-point energy of a field with its mass m, momentum k and frequency ω, \(E = \omega/2 = \sqrt{k^{2} + m^{2}}/2\), and kc(≪m) is a cut-off scale. The vacuum energy density is ρv≃1074 GeV4, provided that kc=MPl. On the other hand, the current observed value of the energy density of dark energy is ρDE≃10−47 GeV4. Thus, the discrepancy between the theoretical estimation and observed value of the current energy density of dark energy is as large as 10121. This is one of the most difficult problems in terms of the cosmological constant. There also exists another problem why the “present” value of the energy density of vacuum in our universe is extremely small compared with its theoretical prediction.

3 Cosmology of dark fluid universe

In this section, we present a description of dark fluid universe (Nojiri et al. 2005a, 2005b; Stefancic 2005). We concentrate on the case in which there is only single fluid which corresponds to dark energy in general relativity.

3.1 Basic equations

We represent the equation of state (EoS) of dark energy as
$$ w_{\mathrm{DE}} \equiv\frac{P}{\rho} = -1 - \frac{f(\rho)}{\rho}, $$
(40)
with
$$ P=-\rho-f(\rho), $$
(41)
where f(ρ) can be an arbitrary function of ρ. In what follows, for the time being, we will investigate the evolution of the universe at the dark energy dominated stage, so that we can regard ρ=ρDE and P=PDE. For simplicity, the “DE” subscription will be omitted as long as there is no need to mention. We also remark that f(ρ) characterizes a deviation from the ΛCDM model.
We assume the flat FLRW metric ds2=−dt2+a2(t)∑i=1,2,3(dxi)2, which is equivalent to Eq. (3) with K=0. In this background, the Einstein equations are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ42_HTML.gif
(42)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ43_HTML.gif
(43)
Equations (42) and (43) correspond to Eqs. (212) and (213) with ρϕ=ρ, Pϕ=P, ρM=0, and PM=0 (i.e., the case in which the scalar field ϕ is responsible for dark energy and there is no any other matter) shown in Sect. 2.1, respectively. Furthermore, the continuity equation of the fluid is given by
$$ \dot{\rho}+ 3H ( \rho+ P) = 0. $$
(44)
From Eq. (44), we find that the scale factor is described as
$$ a = a_{\mathrm{c}} \exp \biggl( \frac{1}{3} \int\frac{d \rho }{f(\rho)} \biggr), $$
(45)
where ac is a constant. In addition, by combining Eqs. (42) and (44) with Eq. (3), we obtain
$$ t = \int\frac{d \rho}{\kappa\sqrt{3\rho} f(\rho)}. $$
(46)
Equations (42) and (43) lead to
$$ \frac{\ddot{a}}{a} = -\frac{\kappa^2}{6} ( \rho+ 3P ) = \frac{\kappa^2}{6} \bigl( 2 \rho+ 3f(\rho) \bigr). $$
(47)

3.2 EoS of dark energy in various cosmological models

Provided that there exists only single fluid of dark energy, (i) for the ΛCDM model, wDE=−1 (f(ρ)=0), (ii) for a quintessence model, −1<wDE<−1/3 (−2/3<f(ρ)/ρ<0), (iii) for a phantom model, wDE<−1 (f(ρ)/ρ>0). If our universe lies beyond wDE=−1 region, then its future can be really dark. In other words, in finite-time phantom/quintessence universe may enter a future singularity.

3.2.1 Finite-time future singularities and energy conditions

In the FLRW background (3), the effective EoS for the universe is given by (Nojiri and Odintsov 2006a, 2011)
$$ w_{\mathrm{eff}} \equiv\frac{P_{\mathrm{eff}}}{\rho_{\mathrm {eff}}} = -1 - \frac{2\dot{H}}{3H^2}, $$
(48)
where ρeff≡3H2/κ2 and \(P_{\mathrm{eff}} \equiv-(2\dot{H}+3H^{2})/\kappa^{2} \) correspond to the total energy density and pressure of the universe, respectively. When the energy density of dark energy becomes perfectly dominant over that of matter, one obtains wDEweff. For \(\dot{H} < 0\ (>0)\), weff>−1 (<−1), representing the non-phantom [i.e., quintessence] (phantom) phase, whereas weff=−1 for \(\dot{H} = 0\), corresponding to the cosmological constant. It is not clear from the very beginning how our universe evolves, and if it ends up in a singularity. This should always be checked.
The finite-time future singularities can be classified into the following four types (Nojiri et al. 2005b):
  • Type I (“Big Rip” (Elizalde et al. 2005; Caldwell et al. 2003; McInnes 2002; Sahni and Shtanov 2003; Nojiri and Odintsov 2003a, 2003b; Faraoni 2002; Gonzalez-Diaz 2004a; McInnes 2005; Singh et al. 2003; Csaki et al. 2005; Wu and Yu 2005; Nesseris and Perivolaropoulos 2004; Sami and Toporensky 2004; Stefancic 2004; Chimento and Lazkoz 2003; Hao and Li 2005; Babichev et al. 2005; Zhang et al. 2006; Dabrowski and Stachowiak 2006; Lobo 2005; Cai et al. 2005; Aref’eva et al. 2005; Lu et al. 2005; Godlowski and Szydlowski 2005; Sola and Stefancic 2005; Guberina et al. 2005a; Dabrowski et al. 2006; Barbaoza 2006)) singularity: In the limit of tts, all the scale factor, the effective energy density and pressure of the universe diverge as a→∞, ρeff→∞ and |Peff|→∞. This also includes the case that ρeff and Peff asymptotically approach finite values at t=ts.

  • Type II (“sudden” Barrow 2004; Nojiri and Odintsov 2004a, 2004b, 2005a; Cotsakis and Klaoudatou 2005; Dabrowski 2005a, 2005b; Fernandez-Jambrina and Lazkoz 2004, 2009; Barrow and Tsagas 2005; Stefancic 2005; Cattoen and Visser 2005; Tretyakov et al. 2006; Balcerzak and Dabrowski 2006; Sami et al. 2006; Bouhmadi-Lopez et al. 2008; Yurov et al. 2008; Koivisto 2008; Brevik and Gorbunova 2008; Barrow and Lip 2009; Bouhmadi-Lopez et al. 2010b; Barrow et al. 2011) singularity: In the limit of tts, only the effective pressure of the universe becomes infinity as aas, ρeffρs and |Peff|→∞.

  • Type III singularity: In the limit of tts, the effective energy density as well as the pressure of the universe diverge as aas, ρeff→∞ and |Peff|→∞.

  • Type IV: In the limit of tts, all the scale factor, the effective energy density and pressure of the universe do not diverge as aas, ρeff→0 and |Peff|→0. However, higher derivatives of H become infinity. This also includes the case that ρeff and/or |Peff| become finite values at t=ts (Shtanov and Sahni 2002).

Here, ts, as(≠0) and ρs are constants. In Abdalla et al. (2005), Briscese et al. (2007), the finite-time future singularities in F(R) gravity have first been observed. Furthermore, the finite-time future singularities in various modified gravity theories have also been studied in Nojiri and Odintsov (2008a), Bamba et al. (2008c, 2010e). In particular, it has recently been demonstrated that the finite-time future singularities can occur in the framework of non-local gravity (Deser and Woodard 2007; Nojiri and Odintsov 2008b; Nojiri et al. 2011b; Zhang and Sasaki 2012) in Bamba et al. (2012f) as well as in f(T) gravity in Bamba et al. (2012d). Also, various studies on the finite-time future singularities have recently been executed, e.g., in Pavon and Zimdahl (2012), Cotsakis and Kittou (2012).

Moreover, there exist four energy conditions.
  1. (a)
    The null energy condition (NEC):
    $$ \rho+ P \geq0. $$
    (49)
     
  2. (b)
    The dominant energy condition (DEC):
    $$ \rho\geq0, \quad \rho\pm P \geq0. $$
    (50)
     
  3. (c)
    The strong energy condition (SEC):
    $$ \rho+ 3P \geq0, \quad \rho+ P \geq0. $$
    (51)
     
  4. (d)
    The weak energy condition (WEC):
    $$ \rho\geq0, \quad \rho+ P \geq0. $$
    (52)
     
If wDE=P/ρ<−1, the Type I (Big Rip) singularity appears within a finite time. In such a case, all energy conditions are violated. On the other hand, even when the strong energy condition in Eq. (51) is satisfied, the Type II (sudden) singularity can appear (Barrow 2004). We remark that if the Type I (Big Rip) singularity exists, all the four conditions are broken, whereas if there exist the Type II, III or IV singularity, only a part of four conditions is broken. Hence, it is expected that the origin of the finite-time future singularities is somehow related with the violation of all or a part of energy conditions.

We mention that “w” singularity has been studied in Kiefer (2010), Dabrowski and Denkiewicz (2009, 2010) and parallel-propagated (p.p.) curvature singularities (Fernandez-Jambrina and Lazkoz 2006; Fernandez-Jambrina 2010) have earlier been investigated. For the “w” singularity, when tts, aas, ρeff→0, |Peff|→0, and the EoS for the universe becomes infinity.

3.2.2 Example of a fluid with behavior very similar to the ΛCDM model

To begin with, we investigate a fluid which behaves very similar to the ΛCDM model
$$ f(\rho) = \rho^q -1, $$
(53)
where q is a constant (|q|≪1). In this case, from Eq. (40) we see that |wDE−1|≪0 and therefore this fluid model can be regarded as almost the ΛCDM model. For this fluid, by substituting Eq. (53) into Eqs. (45) and (46), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ54_HTML.gif
(54)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ55_HTML.gif
(55)
where in deriving the approximate equalities we have used |q|≪1.

3.2.3 Generalized Chaplygin gas (GCG) model

Next, we examine the generalized Chaplygin gas (GCG) model proposed in Kamenshchik et al. (2001), Bento et al. (2002). This model is a proposal to explain the origin of dark energy as well as dark matter through a single fluid.
$$ P = -\frac{\mathcal{A}}{\rho^u}, $$
(56)
where \(\mathcal{A} (>0)\) is a positive constant and u is a constant. For u=1, Eq. (56) is for the original Chaplygin gas model (Kamenshchik et al. 2001).
By combining Eq. (56) and the continuity equation (44), we find
$$ \rho= \biggl[\mathcal{A} + \frac{\mathcal{B}}{a^{3 (1+u )}} \biggr]^{1/ (1+u )}, $$
(57)
where \(\mathcal{B}\) is an integration constant. It follows from Eq. (57) that the asymptotic behaviors in the early universe a≪1 as well as the late universe a≫1 are described as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ58_HTML.gif
(58)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ59_HTML.gif
(59)
In the early universe, the energy density evolves ρa−3 and hence this fluid behaves as non-relativistic matter, i.e., dark matter, whereas in the late universe, ρ approaches a constant value of \(\mathcal{A}^{1/(1+u)}\) and therefore it can correspond to dark energy. As a result, the generalized Chaplygin gas model can account for the origin of both dark matter and dark energy.
It is known that the generalized Chaplygin gas model is very close to the ΛCDM model by analyzing the gauge-invariant matter perturbations δM in Carturan and Finelli (2003), Sandvik et al. (2004), which satisfy (Sandvik et al. 2004)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ60_HTML.gif
(60)
where k is a comoving wavenumber and wDE=P/ρ with Eq. (56) is the EoS of the generalized Chaplygin gas. In addition, cs is the sound speed, defined by
$$ c_{\mathrm{s}}^2 \equiv\frac{d P}{d \rho} = - u w_{\mathrm{DE}}. $$
(61)
When z≫1, both wDE and cs approach to zero. Thus, at the matter-dominated stage cs≪1, and after it cs grows. If u>0(<0), \(c_{\mathrm{s}}^{2} >0 (< 0)\) because wDE<0.
It follows from Eq. (60) that δM grows due to the gravitational instability, provided that
$$ \bigl\vert c_{\mathrm{s}}^2 \bigr\vert < \frac{3}{2} \biggl(\frac{aH}{k} \biggr)^2. $$
(62)
On the other hand, if \(\vert c_{\mathrm{s}}^{2} \vert > (3/2 ) (aH/k )^{2}\), the rapid growth or damped oscillation of δM occurs, depending on the sign of \(c_{\mathrm{s}}^{2}\). Around the present time when \(\vert w_{\mathrm{DE}} \vert = \mathcal{O} (1)\) and thus \(\vert c_{\mathrm{s}}^{2} \vert \sim|u|\). The relation (62) imposes the following constraint on u (Sandvik et al. 2004):
$$ |u| \lesssim10^{-5}. $$
(63)
Since the case of u=0 is equivalent to the ΛCDM model, the viable generalized Chaplygin gas model would be very close to the ΛCDM model.

3.2.4 Coupled dark energy with dark matter

Related to the generalized Chaplygin gas model, in which dark energy and dark matter are interpolated each other, we explain a coupled dark energy with dark matter and its cosmological consequences (for recent discussions, see Nojiri and Odintsov 2010; Balakin and Bochkarev 2011a, 2011b).

The equations corresponding to the conservation law are written as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ64_HTML.gif
(64)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ65_HTML.gif
(65)
where wDEPDE/ρDE. Here, Q describes a coupling between dark energy and dark matter, which is assumed to be a constant. We note that the subscription “m” represents quantities of cold dark matter (CDM), and therefore Pm=0. It is important to emphasize that the continuity equation for the total energy density and pressure of dark energy and dark matter, which is the summation of Eqs. (64) and (65), is satisfied.
The solutions for Eqs. (64) and (65) are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ66_HTML.gif
(66)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ67_HTML.gif
(67)
where ρDE(0) is an integration constant.
In the flat FLRW background (3), from Eqs. (42) and (43) the Einstein equations are expressed as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ68_HTML.gif
(68)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ69_HTML.gif
(69)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ70_HTML.gif
(70)
where in deriving the second equality in Eq. (70) we have used Eq. (66). We can obtain an exact de Sitter solution of Eq. (70) as
$$ a = a_0 \mathrm{e}^{H t}, $$
(71)
with
$$ H = -\frac{Q}{3 ( 1+w_{\mathrm{DE}} )}. $$
(72)
Here, a0 is a constant, and obeys
$$ -\frac{3}{\kappa^2} \biggl[ \frac{Q}{3 ( 1+w_{\mathrm{DE}} )} \biggr]^2 = w_{\mathrm{DE}} \rho_{\mathrm{DE} (0)} a_0^{-3 ( 1+w_{\mathrm{DE}} )}. $$
(73)
If the universe is in the phantom phase, i.e., wDE<−1, H in Eq. (72) is positive. Hence, Eq. (73) has a real solution, so that a0 can be a real number.
On the other hand, when only dark energy exists and therefore there is no direct coupling of dark energy with dark matter, i.e., ρm=0 and Q=0, from Eqs. (64) and (68) we find \(\dot{\rho}_{\mathrm{DE}} + 3H ( 1+w_{\mathrm{DE}} ) \rho_{\mathrm{DE}} = 0\) and (3/κ2)H2=ρDE, respectively. These equations can have an expression of H as
$$ H = -\frac{2/[ 3( 1+w_{\mathrm{DE}} ) ]}{t_{\mathrm{s}} -t}, $$
(74)
which leads to a Big Rip singularity at t=ts. Thus, one cosmological consequence of a coupling between dark energy and dark matter is that a de Sitter solution could be realized, and not a Big Rip singularity.
Another cosmological consequence of a coupling between dark energy and dark matter is to present a solution for the so-called coincidence problem, i.e., the reason why the current energy density of dark matter is almost the same order of that of dark energy. H in Eq. (72) may be taken as the present value of the Hubble parameter, which is given by Hp=2.1h×10−42 GeV (Kolb and Turner 1990) with h=0.7 (Komatsu et al. 2011; Freedman et al. 2001). In this case, from Eqs. (66) and (71) we see that the energy density of dark energy becomes constant
$$ \rho_{\mathrm{DE}} = \rho_{\mathrm{DE} (0)} a_0^{-3( 1+w_{\mathrm{DE}})}. $$
(75)
By substituting Eq. (75) into Eq. (67), we acquire
$$ \rho_{\mathrm{m}} = \rho_{\mathrm{m}(0)} a^{-3} - ( 1+w_{\mathrm{DE}} ) \rho_{\mathrm{DE} (0)} a_0^{-3 ( 1+w_{\mathrm{DE}} )}, $$
(76)
where ρm(0) is an integration constant. By plugging Eq. (76) into Eq. (68) and using Eq. (73), we find that ρm(0)=0. Hence, the energy density of dark matter is also constant as
$$ \rho_{\mathrm{m}} = - ( 1+w_{\mathrm{DE}} ) \rho_{\mathrm{DE} (0)} a_0^{-3 ( 1+w_{\mathrm{DE}} )} = - ( 1+w_{\mathrm{DE}} ) \rho_{\mathrm{DE}}, $$
(77)
where the second equality follows from Eq. (75). Thus, provided that the de Sitter solution in Eq. (71) is an attractor one, by taking
$$ \frac{\rho_{\mathrm{m}}}{\rho_{\mathrm{DE}}} = - ( 1+w_{\mathrm{DE}} ) \sim\frac{1}{3}, $$
(78)
from which we have
$$ w_{\mathrm{DE}} \sim-\frac{4}{3}, $$
(79)
the coincidence problem could be resolved. Of course, this is rather qualitative presentation.

3.3 Phantom scenarios

According to the present observations, there is the possibility that the EoS of dark energy wDE would be less than −1. This is called the “phantom phase”. It is known that if the phantom phase is described by a scalar field theory with the negative kinetic energy, which is the phantom model, the phantom field rolls up the potential due to the negative kinetic energy. For a potential unbounded from above, the energy density becomes infinity and eventually a Big Rip singularity appears.

3.3.1 Phantom phase

In order to illustrate the phantom phase, we present a model in which the universe evolves from the non-phantom (quintessence) phase (wDE>−1) to the phantom phase (wDE<−1), namely, crossing of the phantom divide line of wDE=−1 occurs (Alam et al. 2004).

The scale factor is expressed as
$$ a = a_{\mathrm{c}} \biggl( \frac{t}{t_{\mathrm{s}} -t} \biggr)^n, $$
(80)
where ac is a constant, n(>0) is a positive constant, ts is the time when a finite-time future singularity (a Big Rip singularity) appears, and we consider the period 0<t<ts. In this model, for tts, a(t) behaves as atn and hence wDE=−1+2/(3n)>−1, whereas for tts, wDE=−1−2/(3n)<−1.
It follows from Eq. (80) that the Hubble parameter H is given by
$$ H = n \biggl( \frac{1}{t} + \frac{1}{t_{\mathrm{s}} -t} \biggr). $$
(81)
By combining Eq. (42) with Eq. (81), we find
$$ \rho= \frac{3n^2}{\kappa^2} \biggl( \frac{1}{t} + \frac {1}{t_{\mathrm {s}} -t} \biggr)^2. $$
(82)
H as well as ρ becomes minimum at t=ts/2 and those values are
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ83_HTML.gif
(83)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ84_HTML.gif
(84)
Moreover, \(\dot{\rho}\) is written as
$$ \dot{\rho} = \pm2 \rho\sqrt{ \frac{\kappa^2 \rho}{3n^2} - \frac{4}{n t_{\mathrm{s}}} \sqrt{\frac{\kappa^2 \rho}{3}} }, $$
(85)
where we have removed t by using the relation between ρ and t in Eq. (82). Here, the plus (minus) sign denotes the expression of \(\dot{\rho}\) for t>ts/2 (0<t<ts/2), i.e., the phantom (non-phantom) phase. This implies that the energy density of dark energy increases in the phantom phase. Substituting Eqs. (41) and (85) into the continuity equation of dark energy (44), we obtain
$$ f(\rho) = \pm\frac{2 \rho}{3n} \sqrt{ 1 - \frac{4n}{t_{\mathrm{s}}} \sqrt{\frac{3}{\kappa^2 \rho}} }. $$
(86)

As a consequence, it is necessary for the EoS to be doubled valued, so that the transition from the non-phantom phase to the phantom one can occur. Furthermore, we see that f(ρmin)=0. This means that at the phantom crossing point wDE=−1, both H and ρ have those minima, which can also be understood from the definition of wDE in Eq. (40). In addition, when a Big Rip singularity appears at t=ts, from Eq. (86) we find that f(ρ) evolves as f(ρ)=2ρ/(3n), and by using this relation and Eqs. (41) as well as (40) we see that P=−ρ−2ρ/(3n) and wDE=−1−2/(3n), which is a constant. On the other hand, in the opposite limit of t→0, we also have a constant EoS as wDE=−1+2/(3n).

When the crossing of the phantom divide occurs, f(ρ)=0. In order to realize this situation, the integration part of ∫/f(ρ) on the right-hand side (r.h.s.) of Eq. (45) should be finite. Thus, f(ρ) need to behave as
$$ f(\rho) \sim f_{\mathrm{c}} ( \rho- \rho_{\mathrm{c}} )^{s}, \quad 0 < s < 1, $$
(87)
where we have used the condition f(ρc)=0. In general, it is necessary for f(ρ) to be multi-valued around ρ=ρc because 0<s<1.

3.3.2 Coupled phantom scenario

Next, we explore another phantom scenario in which dark energy is coupled with dark matter. We suppose that dark energy can be regarded as a fluid satisfying Eq. (64). Since the coupling Q can be described by a function of a, H, ρm, \(\dot{\rho}_{\mathrm{m}}\), ρDE and \(\dot{\rho}_{\mathrm{DE}}\), we consider the case in which the ratio of ρm to ρDE defined as rρm/ρDE is a constant, that is, Q is represented by scaling solutions. In this case, ρm does not close to zero asymptotically. By using Eqs. (64) and (65), we obtain
$$ \dot{r} = r \biggl[ Q \biggl( \frac{1}{\rho_{\mathrm{m}}} + \frac{1}{\rho_{\mathrm{DE}}} \biggr) -3H ( w_{\mathrm{m}} - w_{\mathrm{DE}} ) \biggr], $$
(88)
where wmPm/ρm. By taking \(\dot{r} = 0\) in Eq. (88), we acquire Q in the presence of scaling solutions as
$$ Q = 3H ( w_{\mathrm{m}} - w_{\mathrm{DE}} ) \frac{\rho_{\mathrm{DE}} \rho_{\mathrm{m}}}{\rho_{\mathrm{DE}} + \rho_{\mathrm{m}}}. $$
(89)
As a model which can be treated analytically, we examine Q given by
$$ Q = \delta H^2, $$
(90)
with δ being a constant. Here, cold dark matter is regarded as a dust, and hence Pm=0. Moreover, we provide that f(ρDE)=ρDE in Eq. (41), i.e.,
$$ P_{\mathrm{DE}} = -2 \rho_{\mathrm{DE}}. $$
(91)
By combining Eq. (68) with Eqs. (64) and (65), a solution is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ92_HTML.gif
(92)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ93_HTML.gif
(93)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ94_HTML.gif
(94)
with
$$ t_{\mathrm{s}} \equiv\frac{9}{\delta\kappa^2}. $$
(95)
We note that the same solution can be derived if Q is described as
$$ Q = \frac{9H \rho_{\mathrm{DE}} \rho_{\mathrm{m}}}{2 ( \rho_{\mathrm{DE}} + \rho_{\mathrm{m}} )}. $$
(96)
The expression of H in Eq. (92) is the same as that in Eq. (81) with n=2/3, and thus a Big Rip singularity appears at t=ts. Incidentally, in this case ts is not determined. We also mention that Eqs. (93) and (94) lead to r=(tst)/t, which implies that this ratio is not a constant but changes dynamically and eventually becomes zero, and hence the solutions in Eqs. (92)–(94) do not correspond to scaling solutions. This is reasonable because around a Big Rip singularity, the energy density of dark energy becomes dominant over that of dark matter completely.

3.3.3 Little Rip scenario

We study a Little Rip scenario (Frampton et al. 2011, 2012a; Brevik et al. 2011; Nojiri et al. 2011a; Astashenok et al. 2012b; Granda and Loaiza 2012; Ivanov and Toporensky 2011; Ito et al. 2011; Belkacemi et al. 2012; Xi et al. 2012; Makarenko et al. 2012; Liu and Piao 2012), which corresponds to a mild phantom scenario. The Little Rip scenario has been proposed to avoid the finite-time future singularities, in particular, a Big Rip singularity within fluid dark energy. In this scenario, the energy density of dark energy increases in time with wDE being less than −1 and then wDE asymptotically approaches wDE=−1. However, its evolution eventually leads to a dissolution of bound structures at some time in the future. This process is called the “Little Rip”.

A sufficient condition in order to avoid a Big Rip singularity is that a(t) should be a nonsingular function for all t. We suppose a(t) is expressed as
$$ a(t) = \mathrm{e}^{\tilde{f}(t)}, $$
(97)
where \(\tilde{f}(t)\) is a nonsingular function. It follows from Eq. (42) that \(\rho= 3 (\dot{a}/a)^{2} = 3 \dot{\tilde{f}}^{2}\). The condition for ρ to be an increasing function of a is that \(d \rho/d a = (6/\dot{a}) \dot{\tilde{f}} \ddot {\tilde{f}} >0\). This condition can be met provided
$$ \ddot{\tilde{f}} >0. $$
(98)
Thus, in all Little Rip scenarios, a(t) is described by Eq. (42) with \(\tilde{f}\) satisfying Eq. (98).
We derive the expression for ρ as an increasing function of the scale factor a and examine the upper and lower bounds on the growth rate of ρ(a) which can be used to judge whether a Big Rig singularity appears. By defining N≡lna, Eq. (42) is rewritten to (Frampton et al. 2011)
$$ t = \int\sqrt{\frac{3}{\rho(N)}} d N. $$
(99)
The condition for the appearance of a finite-time future (Big Rig) singularity to be avoided is that it takes a Big Rig singularity infinite time to appear. This means that
$$ \int_{N_{\mathrm{c}}}^{\infty} \frac{1}{\sqrt{\rho(N)}} d N \to \infty, $$
(100)
where Nc=lnac. For the case P=−ρ1/2 with A being a constant, we have
$$ \frac{\rho}{\rho_1} = \biggl[ \frac{3A}{2\sqrt{\rho_1}} \ln \biggl( \frac{a}{a_{\mathrm{c}}} \biggr) +1 \biggr]^2, $$
(101)
where ρ1 is a constant and ρ=ρ1 and a=ac at a fixed time t1. In this case, A>0 is required, so that there can be exist the phantom phase as wDE<−1. Moreover, the expression of ρ as a function of t is given by
$$ \frac{\rho}{\rho_1} = \mathrm{e}^{\sqrt{3}A (t-t_1 )}. $$
(102)
Furthermore, by using wDE=−1−−1/2 and Eq. (101) we acquire
$$ w_{\mathrm{DE}} = -1 - \biggl[ \frac{3}{2} \ln \biggl( \frac{a}{a_{\mathrm{c}}} \biggr) + \frac{\rho_1}{A} \biggr]^{-1}, $$
(103)
which can also be derived from the relation (a/ρ)(/da)=−3(1+wDE). In addition, by eliminating ρ from Eqs. (101) and (102) we find
$$ \frac{a}{a_{\mathrm{c}}} = \exp \biggl\{ \frac{2\sqrt{\rho_1}}{3A} \bigl[ \mathrm{e}^{\sqrt{3}A/2 (t-t_1 ) -1} \bigr] \biggr\}. $$
(104)
In this expression, by comparing Eq. (104) with Eq. (97), we obtain
$$ \tilde{f} = 2\sqrt{\rho_1} {3A} \bigl[ \mathrm{e}^{\sqrt{3}A/2 (t-t_1 ) -1} \bigr] + \ln a_{\mathrm{c}}. $$
(105)
In this case, \(\ddot{\tilde{f}} = (\sqrt{\rho_{1}}A/2 ) \mathrm{e}^{ (\sqrt{3}A/2 ) (t-t_{1} )} >0\) because of A>0, and hence the condition in Eq. (98) is satisfied. We mention that the single model parameter A satisfies the observational bounds estimated by using the Supernova Cosmology Project (Amanullah et al. 2010). The best fit value is given by A=3.46×10−3 Gyr−1 and the range for 95 % Confidence Level fit is −2.74×10−3 Gyr−1A≤9.67×10−3 Gyr−1 (Frampton et al. 2011).

From Eqs. (103) and (104), we see that when tt1, aac and wDE<−1, i.e., the universe is in the phantom phase. As the universe evolves, a increases, and when tt1, a becomes very large and thus wDE asymptotically becomes close to −1. However, it takes a as well as ρ infinite time to diverge due to Eq. (100), a Big Rip singularity cannot appear at a finite time in the future. This means that a Big Rip singularity can be avoided.

As further recent observations on Little Rip cosmology, in Brevik et al. (2011) it has been demonstrated that a Little Rip scenario can be realized by viscous fluid. On the other hand, in Frampton et al. (2012a) a new interpretation of a Little Rip scenario by means of an inertial force has been presented. It has been shown that a coupling of dark energy with dark matter can eliminate a little rip singularity and an asymptotic de Sitter space-time can appear. Moreover, a scalar field theory with realizing a little rip scenario has been reconstructed.

We investigate the inertial force Finert on a particle with mass m in the context of the expanding universe. When the distance between two points is l, by using the scale factor a, the relative acceleration between those is described as \(l \ddot{a}/a\). If a particle with mass m exists at each of the points, an inertial force on the other mass would be measured by an observer at one of the masses. Thus, Finert is expressed as (Frampton et al. 2011, 2012a)
$$ F_{\mathrm{inert}} = ml \frac{\ddot{a}}{a} = ml \bigl( \dot{H} + H^2 \bigr), $$
(106)
where in deriving the second equality in Eq. (106) we have used Eqs. (42) and (43). If Finert>0 and Finert>Fb, where Fb is a constant force and bounds the two particles, the two particles becomes free, i.e., unbound, so that the bound structure can be dissociated.
If the Hubble parameter is expressed as
$$ H \sim\frac{h_{\mathrm{s}}}{ ( t_{\mathrm{s}} - t )^{\tilde{q}}}, $$
(107)
where hs(>0) is a positive constant and \(\tilde{q} (\geq1)\) is larger than or equal to unity. In this case, a Big Rip singularity appears in the limit of tts. By substituting Eq. (107) into Eq. (106), we obtain (Bamba et al. 2012d)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ108_HTML.gif
(108)
Since in the limit of tts, H and \(\dot{H}\) diverge, Finert becomes infinity. The important point is that for a Big Rip singularity, Finert diverges in the “finite” future time.
On the other hand, a representation of the Hubble parameter realizing Little Rip scenario is described by (Frampton et al. 2012a)
$$ H = H_{\mathrm{LR}} \exp ( \xi t ). $$
(109)
Here, HLR(>0) and ξ(>0) are positive constants. By plugging Eq. (109) into Eq. (106), we find (Bamba et al. 2012d)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ110_HTML.gif
(110)
In the limit of t→∞, the divergence of H and \(\dot{H}\) leads to the consequence of Finert→∞. Thus, a “Rip” phenomenon happens in both cases of a Big Rip singularity and Little Rip scenario. It is remarkable to note that for Little Rip cosmology, it takes Finert the “infinite” future time to become infinity. This feature is different from that in case of a Big Rip singularity, which is the finite time future singularity.

In Little Rip cosmology, as a demonstration we estimate the value of Finert at the present time \(t_{0} \approx H_{0}^{-1}\), where H0=2.1h×10−42 GeV (Kolb and Turner 1990) with h=0.7 (Komatsu et al. 2011; Freedman et al. 2001) is the present value of the Hubble parameter. The bound force \(F_{\mathrm{b}}^{\mathrm{ES}}\) for the Earth-Sun system is given by \(F_{\mathrm{b}}^{\mathrm{ES}} = G M_{\oplus} M_{\odot} / r_{\oplus-\odot}^{2} = 4.37 \times10^{16}~\mathrm{GeV}^{2}\), where M=3.357×1051 GeV (Kolb and Turner 1990) is the Earth mass and M=1.116×1057 GeV (Kolb and Turner 1990) is the mass of Sun, and r⊕−⊙=1 AU=7.5812×1026 GeV−1 (Kolb and Turner 1990) is the Astronomical unit corresponding to the distance between Earth and Sun. We take ξ=H0, m=M and l=r⊕−⊙. In this case, from Eq. (110) we acquire \(F_{\mathrm{inert}} = 2.545 \times10^{78} e H_{0}^{2} [( H_{\mathrm{LR}}/H_{0} ) + e ( H_{\mathrm{LR}}/H_{0} )^{2}]\), where e=2.71828. In order for the current value of Finert to be larger than or equal to \(F_{\mathrm{b}}^{\mathrm{ES}}\), HLR should be HLR≥4.82×10−30 GeV.

3.3.4 Pseudo-rip cosmology

In addition, as an intermediate cosmology between the ΛCDM model, namely, the cosmological constant, and the Little Rip scenario, more recently the Pseudo-Rip model has been proposed in Frampton et al. (2012b). In this case, in the limit of t→∞ the Hubble parameter tends to a constant asymptotically. In other words, the Pseudo-Rip cosmology is a phantom scenario and has a feature of asymptotically de Sitter universe.

A description realizing Pseudo-Rip cosmology is given by (Bamba et al. 2012d)
$$ H(t) = H_{\mathrm{PR}} \tanh \biggl( \frac{t}{t_0} \biggr). $$
(111)
Here, HPR(>0) is a positive constant. It follows from Eq. (111) that we find
$$ a = a_{\mathrm{PR}} \cosh \biggl( \frac{t}{t_0} \biggr), $$
(112)
with aPR(>0) being a positive constant. In the limit of t→∞, H(t)→HPR<∞ and H(t) increases monotonically in time. Hence, the universe evolves in the phantom phase and eventually goes to de Sitter space-time asymptotically. By combining Eqs. (111) and (40), we obtain
$$ w_{\mathrm{DE}} = -1 -\frac{2}{3 t_0 H_{\mathrm{PR}}} \frac{1}{\sinh^2 ( t/t_0)}. $$
(113)
This implies that since \(\dot{H}(t) = H_{\mathrm{PR}}/ [ t_{0} \cosh^{2} ( t/t_{0} ) ] > 0\), the universe always evolves within the phantom phase as wDE<−1, and eventually, when t→∞, wDE→−1, similarly to that in the Little Rip scenario. By comparing the observational present value of wDE as wDE=−1.10±0.14 (68 % CL) (Komatsu et al. 2011) with wDE in Eq. (113) at \(t = t_{0} \approx H_{0}^{-1}\), we obtain the constraint HPR≥2.96×10−42 GeV.
Furthermore, by using Eq. (106) and (111), we acquire (Bamba et al. 2012d)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ114_HTML.gif
(114)
with
$$ F_{\mathrm{inert},\infty}^{\mathrm{PR}} \equiv ml H_{\mathrm {PR}}^2. $$
(115)
This means that in Pseudo-Rip cosmology, Finert becomes finite asymptotically in the limit of t→∞. This originates from the fact that when t→∞, HHPR and \(\dot{H} \to0\). To realize the Pseudo-Rip scenario, the relation \(F_{\mathrm{inert},\infty}^{\mathrm{PR}} > F_{\mathrm {b}}^{\mathrm{ES}}\) should be satisfied, so that the ES system could be disintegrated much before the universe asymptotically goes to de Sitter space-time. For m=M and l=r⊕−⊙, we find \(H_{\mathrm{PR}} > \sqrt{G M_{\odot} / r_{\oplus-\odot}^{3}} = 1.31 \times 10^{-31}~\mathrm{GeV}\). This constraint is much stronger than that obtained from the present value of wDE as HPR≥2.96×10−42 GeV shown above. In Appendix, it is examined how strong the inertial force can constrain the EoS of dark energy.

3.4 Finite-time future singularities

In the ΛCDM model, f(ρ)=0 in Eq. (40) and hence wDE=−1. In a quintessence model, the type II, III and IV singularities can occur. On the other hand, in a phantom model, the singularities of Type I and type II can appear. We explicitly demonstrate that the EoS in Eq. (40) can lead to all the four types of the finite-time future singularities.

3.4.1 Type I and III singularities

We examine the case in which f(ρ) is given by
$$ f(\rho) = A \rho^{\alpha}, $$
(116)
where A and α are constants. By using Eq. (45), we obtain
$$ a = a_{\mathrm{c}} \exp \biggl[ \frac{\rho^{1-\alpha}}{3 (1-\alpha )A} \biggr]. $$
(117)
For α>1, when ρ→∞, a approaches a finite value, whereas, for α<1, when ρ→∞, if A>0 (A<0), a diverges (vanishes). We note that in this sub-subsection (i.e., Sect. 3.4.1), we concentrate on the limit of ρ→∞.1 We also mention the case in the opposite limit of ρ→0 in Sect. 3.6.2.
Moreover, from Eq. (46) we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ118_HTML.gif
(118)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ119_HTML.gif
(119)
For α≥1/2, ρ can diverge within the finite future or past t=ts. Meanwhile, for α≤1/2, it takes infinite time for ρ to diverge and therefore ρ→∞ in the infinite future or past.
It follows from Eq. (41) with Eq. (116) that P=−ρα. When ρ diverges, P also becomes infinity. By substituting Eq. (116) into Eq. (40), we find
$$ w_{\mathrm{DE}} = \frac{P}{\rho} = -1 -A\rho^{\alpha- 1}. $$
(120)
For α>1, when ρ→∞, if A>0 (A<0), wDE→+∞ (wDE→−∞). On the other hand, for α<1, when ρ→∞, if A>0 (A<0), wDE→−1+0 (wDE→−1−0), i.e., wDE approaches −1.
The above considerations are summarized as follows.
  1. (a)

    For α>1, the Type III singularity can exist. If A>0 (A<0), wDE→+∞ (wDE→−∞).

     
  2. (b)

    For 1/2<α<1, if A>0, there can exist the Type I (Big Rip) singularity. If A<0, ρ→∞, a→0. Such a singularity in the past (future) may be called a Big Bang (Big Crunch) singularity. When ρ→∞, if A>0 (A<0), wDE→−1+0 (wDE→−1−0).

     
  3. (c)

    For 0<α≤1/2, there does not exist any “finite-time” future singularity.

     

3.4.2 Type II singularity

Next, we investigate the following form of f(ρ):
$$ f(\rho) = \frac{C}{(\rho_{\mathrm{c}} -\rho)^{\gamma}}, $$
(121)
where C, ρc and γ(>0) are constants. We note that for ρc=0, (−1)γC=A, γ=−α, the expression in Eq. (121) is equivalent to that in Eq. (116). We concentrate on the case in which ρ<ρc. Since P=−ρC/(ρcρ)γ, when ρρc, P diverges. Hence, R=2κ2(ρ−3P) also becomes infinite. From Eq. (45), we obtain
$$ a = a_{\mathrm{c}} \exp \biggl[ -\frac{ (\rho_{\mathrm{c}} -\rho )^{\gamma+1}}{3C ( \gamma+1 )} \biggr]. $$
(122)
Thus, when ρρc, a is finite as aac and hence \(\dot{a}\) is also finite because \(H = \dot{a}/a \propto\sqrt{\rho}\) given by Eq. (42), whereas \(\ddot{a}\) diverges. Furthermore, by using Eq. (46), we find
$$ t \simeq t_{\mathrm{s}} -\frac{ (\rho_{\mathrm{c}} -\rho )^{\gamma+1}}{\kappa C \sqrt{3\rho_{\mathrm{c}}} (\gamma +1 )}, $$
(123)
where ts is an integration constant. Therefore, tts when ρρc. By combining Eq. (121) into Eq. (40), we have
$$ w_{\mathrm{DE}} = -1 -\frac{C}{\rho (\rho_{\mathrm{c}} -\rho )^{\gamma}}. $$
(124)
When ρρc, wDE→−∞ (wDE→∞) for C>0 (C<0). As a consequence, there can exist the Type II (sudden) singularity if f(ρ) is given by Eq. (121). It is interesting to remark that for C<0, when ρ becomes around ρc, the strong energy condition in Eq. (51) is satisfied. This means that the Type II (sudden) singularity can appear in the quintessence era.
Next, we demonstrate a quintessence model as well as a phantom one in which the Type II singularity occurs (Astashenok et al. 2012a). We consider the case that f(ρ) is given by
$$ f(\rho) = \frac{\zeta^2}{1-\rho/\rho_{\mathrm{s}}}, $$
(125)
with ζ(>0) being a positive constant. In the limit of tts, ρ approaches ρs(>0). It follows from Eq. (40) that if f(ρ)>0 with ρ>0, which is realized for ρ0<ρs with ρ0 being the present energy density of the universe, wDE<−1, and hence the universe is in the phantom phase and ρ increases until the pressure of the universe diverges. While, for ρ>0, if −2/3<f(ρ)/ρ<0, −1<wDE<−1/3 and therefore the universe is in the non-phantom (quintessence) phase. For ρ0>ρs, since f(ρ)<0, this case can correspond to a quintessence model. The energy density ρ becomes small in time and finally a Big crunch happens at ρ=ρs. We estimate how long it takes the Type II singularity to appear from the present time t0. The remaining time is described by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ126_HTML.gif
(126)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ127_HTML.gif
(127)
By using Eq. (45), we acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ128_HTML.gif
(128)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ129_HTML.gif
(129)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ130_HTML.gif
(130)
Here, in Eq. (128) the “+ (−)” sign denotes the case that Θ>1 (Θ<1) describing a quintessence (phantom) model. Moreover, the EoS for dark energy at the present time is expressed as
$$ w_{\mathrm{DE}(0)} = -1-\frac{\xi}{\varTheta ( 1-\varTheta )}. $$
(131)
In addition, from Eqs. (9), the luminosity distance dL is described by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ132_HTML.gif
(132)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ133_HTML.gif
(133)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ134_HTML.gif
(134)
Furthermore, the remaining time from the present time to the appearance of the finite-time future singularity is given by
$$ t_{\mathrm{s}} - t_0 = \frac{1}{H_0} \int_{\bar{u}}^{0} \frac{d \bar{u}'}{ (1+\bar{u}' ) \sqrt{\varOmega_{\mathrm {DE}}^{(0)} (1+z' )^{3} +\varOmega_{\mathrm{DE}}^{(0)} \varrho (\bar{u}')}}, $$
(135)
where \(\bar{u} \equiv a_{0}/a - 1\). This variable changes from \(\bar{u} = 0\) at t=t0 to \(\bar{u} = \exp[- ( 1-\varTheta)^{2}/ ( 6\xi)] - 1\) at the time when P diverges, i.e., in the limit of tts. We note that \(\bar{u}\) is equivalent to the redshift z, and hence for \(\bar{u} = z\), \(\varrho(\bar{u}')\) in Eq. (135) becomes equal to ϱ(z) in Eq. (134). As numerical estimations, for the phantom phase with wDE(0)=−1.10 and Θ=0.95, tst0=1.1 Gyr, whereas for the quintessence phase with wDE(0)=−0.98 and Θ=1.05, tst0=5.79 Gyr and tdt0=5.78 Gyr, where td is the time when the cosmic deceleration starts, i.e., \(\ddot{a}\) becomes zero from its positive value (Astashenok et al. 2012a). Here, \(H_{0}^{-1} =13.6~\mathrm{Gyr}\) has been used.
The deceleration parameter qdec and the jerk parameter j are defined by (Sahni et al. 2003)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ136_HTML.gif
(136)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ137_HTML.gif
(137)
where N≡−ln(1+z) is the number of e-folds and N=0 at the present time t=t0. The values of qdec and j at the present time t=t0 are written as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ138_HTML.gif
(138)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ139_HTML.gif
(139)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ140_HTML.gif
(140)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ141_HTML.gif
(141)
If wDE(0)≈−1, except the limit of Θ→1, j0 is not so different from its value for the ΛCDM model, where j0 is unity. Moreover, in the range −1.05<wDE(0)<−0.95, we find \(q_{\mathrm{dec}(0)}^{(\varLambda)} -0.16 < q_{\mathrm{dec}(0)} < q_{\mathrm{dec}(0)}^{(\varLambda)} + 0.16\) (Astashenok et al. 2012a), where \(q_{\mathrm{dec}(0)}^{(\varLambda)} = - 1\) is the value of qdec(0) for the ΛCDM model. Thus, it can be considered that the model in Eq. (125) describing both the quintessence and phantom phases, where the Big Crunch and the Type II singularity eventually happens, respectively, fits the latest supernova data from the Supernova Cosmology project (Amanullah et al. 2010) well. Similar results have been obtained also in Ghodsi et al. (2011), Balcerzak and Denkiewicz (2012).
We remark that for the case describing the quintessence phase, the dissolution of the bound structure before the appearance of the Big Crunch cannot be realized. The inertial force is given by
$$ F_{\mathrm{inert}} = -\frac{ml}{2} \biggl( w_{\mathrm{DE}} + \frac {1}{3} \biggr) \rho. $$
(142)
For the quintessence phase, the maximum value of Finert in Eq. (142) is \(F_{\mathrm{inert}}^{\mathrm{max}} =ml\rho/3\). The inertial force becomes small in time because the energy density of the universe in the quintessence phase decreases.

3.4.3 Type IV singularity

We further explore the case in which
$$ f(\rho) = \frac{AB \rho^{\alpha+ \beta}}{A \rho^{\alpha} + B \rho ^{\beta}}, $$
(143)
where A, B, α and β are constants. For α>β, we see that f(ρ)→α when ρ→0, while f(ρ)→β when ρ→∞.
For α≠1 and β≠1, Eq. (45) yields
$$ a = a_{\mathrm{c}} \exp \biggl\{ -\frac{1}{3} \biggl[ \frac{\rho^{-\alpha+1}}{ (\alpha-1 )A} + \frac{\rho^{-\beta+1}}{ (\beta-1 )B} \biggr] \biggr\}. $$
(144)
In addition, for α=2β−1, by using Eq. (46), we acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ145_HTML.gif
(145)
where ts is an integration constant. Equation (145) is available for β≠1, β≠3/4, and β≠1/2. Furthermore, from Eq. (40) with Eq. (143), we have
$$ w_{\mathrm{DE}} = -1 - \frac{AB \rho^{\alpha+ \beta-1}}{A \rho^{\alpha} + B \rho^{\beta }}. $$
(146)
For 0<β<1/2, when ρ→0,
$$ P \to-\rho-B \rho^{\beta}. $$
(147)
Equation (145) implies that when ρ→0, tts. Hence, from Eq. (148) we see that in the limit tts, ρ→0 and P→0. For α=2β−1, from Eq. (143) we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ148_HTML.gif
(148)
with
$$ q \equiv1 -\frac{1}{2\beta- 1}. $$
(149)
Since q>2, aac in the limit tts. Moreover, H and \(\dot{H}\) are also finite. However, unless q is an integer, higher derivatives of H, dnH/dtn where n>−1/(2β−1), diverges. As a result, there can exist the Type IV singularity. In this case, i.e., for α=2β−1 with 0<β<1/2, when ρ→0, from Eq. (146) we see that wDE→∞ (wDE→−∞) for B<0 (B>0).

It has been examined in Nojiri and Odintsov (2005a) that if f(ρ) is expressed as Eq. (143), there can also exist the Type I, II and III singularities. For 3/4<β<1 with A>0, the Type I singularity can appear. For A/B<0 with any value of β or for β<0 irrespective the sign of A/B, the Type II singularity can exist. For β>1, the Type III singularity can occur.

3.5 Asymptotically de Sitter phantom universe

We study an example of a fluid realizing asymptotically de Sitter phantom universe. We present an important model constructed in Astashenok et al. (2012b) in which the observational data consistent with the ΛCDM model are satisfied, but it develops the dissolution of the bound structure.

For convenience, we introduce a new variable \(x \equiv\sqrt{\rho}\). By using x, Eqs. (45) and (46) are rewritten to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ150_HTML.gif
(150)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ151_HTML.gif
(151)
where xc is the value of x at t=0. If at x=xf<∞, the integration in Eq. (151) becomes infinity. Hence, it takes the energy density infinite time to arrive at \(\rho_{\mathrm{f}} \equiv x_{\mathrm{f}}^{2}\). In other words, the cosmic expansion becomes exponential behavior asymptotically. In addition, the energy density approaches a constant value, i.e., the cosmological constant, whereas the EoS w for the universe always evolves within the phantom phase w<−1. We examine a model in which f(x) is given by
$$ f(x) = \beta\sqrt{x} \biggl[1 - \biggl(\frac{x}{x_{\mathrm {f}}} \biggr)^{3/2} \biggr]. $$
(152)
We suppose that the energy density of dark energy varies from zero to \(\rho_{\mathrm{f}} = x_{\mathrm{f}}^{2}\). It follows from Eq. (150) that ρ is expressed as a function of the redshift z
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ153_HTML.gif
(153)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ154_HTML.gif
(154)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ155_HTML.gif
(155)
where ρ0 is the energy density of dark energy at the present time t0. Furthermore, the current EoS for dark energy is given by
$$ w_{\mathrm{DE} (0)} = -1-\frac{2\gamma}{3} \frac{1-\Delta}{\Delta }. $$
(156)
By using a parameter v varying from zero at t=t0 to unity in the limit of t→∞, we can obtain the parametric description of the scale factor as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ157_HTML.gif
(157)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ158_HTML.gif
(158)
where in the limit of v→1 the last term in Eq. (158) becomes dominant. With these parametric descriptions, the scale factor can accurately be expressed as
$$ a(t) = a_{\mathrm{c}} \biggl( \frac{2}{3} \biggr)^{2/ (3\gamma )} \exp \biggl[ \frac{x_{\mathrm{f}} ( t-t_0 )}{\sqrt {3}} \biggr]. $$
(159)
The energy density of non-relativistic matter and baryons behaves as ρm=ρm(0)(1+z)3, where ρm(0) is the current value of the energy density of non-relativistic matter and baryons. Thus, it follows from Eq. (9) that the luminosity distance dL is written as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ160_HTML.gif
(160)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ161_HTML.gif
(161)
where we have neglected the contribution from radiation because it is much smaller than those from non-relativistic matter and baryons and dark energy. In case of the ΛCDM model, \(\varOmega_{\mathrm{DE}}^{(0)} = \varOmega_{\varLambda}\) and \(\tilde{h}(z) = 1\) in Eq. (160), where \(\varOmega_{\varLambda} \equiv\rho_{\varLambda}/\rho_{\mathrm {crit}}^{(0)}\). Provided that γ≈0, where β is very small, and Δ≈0, we find \(\tilde{h}(z) \approx1\). Thus, in this case it is impossible to distinguish this model with the ΛCDM model. From Eq. (156), we also have wDE(0)≈−1. On the other hand, if we take Δ=0.5 and γ=0.075, wDE(0)=−1.05. According to the numerical analysis in Astashenok et al. (2012b), the difference between the distance modulus μ in this model and that in the ΛCDM model is estimated as \(\delta\mu\equiv5\log(d_{L}/d_{L}^{(\varLambda\mathrm{CDM})} )< 0.016\), where \(d_{L}^{(\varLambda\mathrm{CDM})}\) is the luminosity distance in the ΛCDM model, for \(\varOmega_{\mathrm{m}}^{(0)} = 0.28\) and \(\varOmega_{\mathrm{DE}}^{(0)} = \varOmega_{\varLambda} = 0.72\). Since errors of the modulus of SNe are ∼0.15, which are larger than the above estimation of δ, this model can fit the observational data as well as the ΛCDM model.
In this model, the Friedmann equation (42) is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ162_HTML.gif
(162)
By substituting Eq. (162) into Eqs. (136) and (137), we acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ163_HTML.gif
(163)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ164_HTML.gif
(164)
For Δ=0.5, γ=0.075, \(\varOmega_{\mathrm{m}}^{(0)} = 0.28\) and \(\varOmega_{\mathrm{DE}}^{(0)} = \varOmega_{\varLambda} = 0.72\), we obtain qdec(0)=−0.54 and j0=1.11. In the ΛCDM model, where Δ=0, we have qdec(0)=−0.58 and j0=1. As a consequence, the values of qdec(0) and j0 in this model are near to those in the ΛCDM model.
In addition, we explore whether the dissolution of the bound structure can occur. By using Eqs. (42), (43) and (106) Finert is also represented as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ165_HTML.gif
(165)
The dimensionless description for Finert is given by (Frampton et al. 2012b)
$$ \bar{F}_{\mathrm{inert}} \equiv\frac{1}{\rho_{\mathrm{DE}(0)}} \biggl( 2\rho_{\mathrm{DE}} (a) + \frac{d \rho_{\mathrm{DE}} (a)}{d a} a \biggr), $$
(166)
where ρDE(0) is the current energy density of dark energy.
The parametric description of the dimensionless inertial force is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ167_HTML.gif
(167)
and tt0 is written by
$$ t-t_0 = \frac{1}{H_0} \int_u^{0} \frac{du'}{ ( 1+u' ) \sqrt{\varOmega_{\mathrm{m}}^{(0)} (1+u' )^{3} +\varOmega_{\mathrm{DE}}^{(0)} \tilde{h}(u')}}. $$
(168)
Here, we have introduced the new variable ua0/a−1, which varies from 0 at the present time t=t0 to −1 in the limit of t→∞. It follows from Eq. (167) that in the limit of t→∞, we find \(\bar{F}_{\mathrm{inert}} (u) \to2\Delta^{-4/3}\). For the Earth-Sun system, the necessary value of \(\bar{F}_{\mathrm{inert}}\) for the disintegration of the bound structure is estimated as \(\bar{F}_{\mathrm{inert}} \gtrsim10^{23}\) (Astashenok et al. 2012b), which leads to the condition for the dissolution Δ≤Δmin=10−17. Moreover, from Eq. (156) we see that for Δ=Δmin=10−17, γ can change from 0 (in which wDE(0)=−1) to 3.6×10−18 (where wDE(0)=−1.24, which is the observational lowest constraint Komatsu et al. 2011). As a consequence, if Δ<Δmin=10−17 and 0<γ<10−18, this model can be compatible with the observational data of SNe. Accordingly, in this model the dissolution of the bound structure can be realized, although it satisfies the observational data which is consistent with the ΛCDM model.

3.6 Inhomogeneous (imperfect) dark fluid universe

In this subsection, we explain inhomogeneous (imperfect) dark fluid universe by following Nojiri and Odintsov (2005a), Capozziello et al. (2006a). We investigate so-called inhomogeneous EoS of dark energy, which the pressure has the dependence not only on the energy density but also on the Hubble parameter H. This idea comes from, e.g., a time dependent bulk viscosity in the ideal fluid (Brevik and Gorbunova 2005; Cataldo et al. 2005; Ren and Meng 2006; Hu and Meng 2006) or a modification of gravity. For a recent study of imperfect fluids, see Pujolas et al. (2011).

3.6.1 Inhomogeneous EoS

An inhomogeneous expression of the pressure is described by
$$ P=-\rho+ f(\rho) + G(H), $$
(169)
where G(H) is a function of H. We note that generally speaking, G is a function of the Hubble parameter H, its derivatives and the scale factor a. However, for simplicity we consider mainly the case that G depends on only H. By substituting Eq. (169) into Eq. (44), the continuity equation of the inhomogeneous fluid is represented by
$$ 0= \dot{\rho}+ 3H \bigl( f(\rho) + G(H) \bigr). $$
(170)
By plugging \(H = \kappa\sqrt{\rho/3}\), which follows from Eq. (42) for the expanding universe (H≥0), into Eq. (170), we obtain
$$ \dot{\rho} = \mathcal{F} (\rho) \equiv-3\kappa \sqrt{\frac{\rho}{3}} \biggl( f(\rho) + G\biggl(\kappa \sqrt{\frac{\rho}{3}}\biggr) \biggr). $$
(171)
On the other hand, by combining Eq. (169) with Eq. (43), we have a representation of G(H) in terms of f as
$$ G(H) = -f\biggl(\frac{3H^2}{\kappa^2}\biggr) + \frac{2}{\kappa^2} \dot{H}. $$
(172)

3.6.2 Influences on the structure of the finite-time future singularities

First, as a concrete example, we examine the case in which the relation between P and ρ in Eq. (41) is changed as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ173_HTML.gif
(173)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ174_HTML.gif
(174)
where wh is the homogeneous EoS, w1 is a constant, and in deriving Eq. (174) we have used Eq. (42). Thus, for the inhomogeneous fluid, the inhomogeneous EoS wih is shifted from the homogeneous EoS wh(=w) as
$$ w \longrightarrow w_{\mathrm{ih}} \equiv w_{\mathrm{h}} + \frac{\kappa^2 w_1}{3}. $$
(175)
From Eq. (175), we see that if wih>−1, there does not appear a Big Rip singularity, even for wh<−1. In other words, provided that if w1≪−1, the universe can evolve to the phantom phase wih<−1, even though in the beginning, the universe is in the non-phantom (quintessence) phase with wh>−1.
Furthermore, in order to demonstrate how the inhomogeneous modification in EoS influences on the structure of the finite-time future singularities, as another example, we investigate the case that
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ176_HTML.gif
(176)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ177_HTML.gif
(177)
where B and β are constants, and by using Eq. (42) \(\tilde{B}\) is defined as
$$ \tilde{B} \equiv B \biggl( \frac{\kappa^2}{3} \biggr)^{\beta}. $$
(178)
For β>α, when ρ is large, the inhomogeneous part, i.e., the second term, of Eq. (177) becomes dominant as
$$ f_{\mathrm{ih}} \to-\tilde{B} \rho^{\beta}. $$
(179)
On the other hand, for β<α, when ρ→0, the inhomogeneous part of Eq. (177) becomes also dominant as in Eq. (179).

We explicitly describe the cases in which the structure of the finite-time future singularities are changed due to the presence of the inhomogeneous term. First, we examine the limit that ρ diverges. For 1/2<α<1 with A>0, if there only exists the homogeneous term without the in homogeneous term G(H) in Eq. (169), i.e., the case in Eq. (116) in Sect. 3.4.1, there can appear the Type I singularity. However, in the presence of the inhomogeneous term as in Eq. (177), for β>1(>α), in which the situation described by Eq. (179) is realized because β>α, in the limit of ρ→∞, |P| also diverges because \(P \to- \rho- \tilde{B} \rho^{\beta}\). Thus, the Type III singularity appears instead of the Type I singularity. For α=1/2, if β>1(>α), the Type III singularity appears, whereas if (α<)1/2<β<1, the Type I singularity occurs. For 0<α<1/2, if β>1(>α) with \(\tilde{B} (B>0)\), the Type III singularity appears, while if (α<)1/2<β<1 with \(\tilde{B} (B>0)\), the Type I singularity occurs.

Second, we consider the opposite limit that ρ tends to zero. In case of the homogeneous EoS in Eq. (116), for 0<α<1/2, in the limit of tts, aas, ρ→0 and |P|→0. The substitution of Eq. (118) into Eq. (117) leads to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ180_HTML.gif
(180)
The exponent (α−1)/(α−1/2) in terms of |tts| in Eq. (180) is not always an integer. Hence, there is the possibility that the higher derivatives of H diverge, even though a becomes finite. As a result, the Type IV singularity can appear. For α<0, in the limit of tts, aas, ρ→0 and |P|→∞. Thus, the Type II singularity can occur.

We explain the case in presence of the inhomogeneous term as in Eq. (177). For α=1/2, if 0<β<1/2(<α), in which the inhomogeneous term becomes dominant over the homogeneous one as in Eq. (179) due to the relation β<α, the Type IV singularity appears, or if β<0, the Type IV singularity occurs. For 0<α<1/2, if β<0(<α), the Type II singularity appears instead of the Type IV singularity.

3.6.3 Implicit inhomogeneous EoS

Next, in a general case, we suppose that in a proper limit, such as ρ being large or small, \(\mathcal{F} (\rho)\) in Eq. (171) is described by
$$ \mathcal{F} (\rho) \sim\bar{\mathcal{F}} \rho^{\gamma}, $$
(181)
where \(\bar{\mathcal{F}}\) and γ are constants. For γ≠1, we can integrate Eq. (171) as
$$ \bar{\mathcal{F}} ( t - t_{\mathrm{c}} ) \sim\frac{\rho^{1-\gamma}}{1-\gamma}, $$
(182)
which is rewritten to
$$ \rho\sim \bigl[ ( 1-\gamma ) \bar{\mathcal{F}} ( t - t_{\mathrm{c}} ) \bigr]^{1/ ( 1-\gamma )}, $$
(183)
where tc is a constant of the integration. It follows from Eq. (42) that the scale factor is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ184_HTML.gif
(184)
If γ=1, from Eq. (171) we find
$$ \rho= \bar{\rho} \mathrm{e}^{\bar{\mathcal{F}}t}, $$
(185)
where \(\bar{\rho}\) is a constant. By substituting Eq. (185) into Eq. (42), we obtain
$$ a = a_{\mathrm{c}} \exp \biggl[ \pm\frac{2\kappa}{\bar{\mathcal{F}}} \sqrt{\frac{\bar{\rho}}{3}} \mathrm{e}^{ ( \bar{\mathcal{F}}/2 )t} \biggr]. $$
(186)
We propose an implicit inhomogeneous EoS by generalizing the expression of \(\mathcal{F} (\rho)\) in Eq. (171) as
$$ \mathcal{F} (P, \rho, H) = 0. $$
(187)
In order to understand the cosmological consequences of the implicit inhomogeneous EoS, we present the following example:
$$ ( P + \rho )^2 - C_{\mathrm{c}} \rho^2 \biggl( 1 - \frac{H_{\mathrm{c}}}{H} \biggr) = 0, $$
(188)
with Cc(>0) and Hc(>0) are positive constants.
Plugging Eq. (188) into Eq. (43) and using Eq. (42), we acquire
$$ \dot{H}^2 = \frac{9}{4} C_{\mathrm{c}} H^4 \biggl( 1 - \frac {H_{\mathrm{c}}}{H} \biggr). $$
(189)
We can integrate Eq. (190) as
$$ H = \frac{16}{9 C_{\mathrm{c}}^2 H_{\mathrm{c}} ( t - t_- ) ( t_+ - t )}, $$
(190)
with
$$ t_\pm\equiv t_{\mathrm{c}} \pm\frac{4}{3 C_{\mathrm{c}} H_{\mathrm {c}}}, $$
(191)
where tc is a constant of integration. By combining Eq. (188) with Eq. (190) and substituting Eq. (190) into Eq. (42), we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ192_HTML.gif
(192)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ193_HTML.gif
(193)
Thus, by using Eq. (192), we have
$$ w = \frac{P}{\rho} = -1 - \frac{3 C_{\mathrm{c}}^2}{4 H_{\mathrm{c}}} ( t - t_{\mathrm{c}} ). $$
(194)
From Eq. (190), we see that if t<t<t+, H>0 because t<tc<t+. At t=tc=(t+t+)/2, H becomes the minimum of H=Hc. On the other hand, in the limit of tt±, H→∞. This may be interpreted that at t=t, there exists a Big Bang singularity, whereas at t=t, a Big Rip singularity appears. It is clearly seen from Eq. (190) that when t<t<tc, w>−1 (the non-phantom (quintessence) phase, and when tc<t<t+, w<−1 (the phantom phase), namely, at t=tc, there can occur the crossing of phantom divide from the non-phantom phase to the phantom one. This is realized by an inhomogeneous term in the EoS.
We present another example in which de Sitter universe is asymptotically realized.
$$ ( P + \rho )^2 + \frac{16}{\kappa^4 t_{\mathrm{c}}^2} ( h_{\mathrm{c}} - H ) \ln \biggl( \frac{h_{\mathrm{c}} - H}{h_1} \biggr) = 0, $$
(195)
where tc, hc and h1 are constants and hc>h1>0.
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ196_HTML.gif
(196)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ197_HTML.gif
(197)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ198_HTML.gif
(198)
It follows from Eq. (196) that
$$ \dot{H} = \frac{2 h_1 t}{t_{\mathrm{c}}^2} \mathrm{e}^{-t^2/t_{\mathrm{c}}^2}. $$
(199)
By using Eqs. (42), (169) and (170), we obtain Eq. (43). For t<0, \(\dot{H} <0\), whereas for t>0, \(\dot{H} >0\). For t<0, weff>−1, while for t>0, weff<−1. In the limit of t→±∞, the universe approaches to de Sitter one asymptotically, and hence there appears a Big Rip singularity nor a Big Bang singularity.
We note that in principle, the implicit inhomogeneous EoS can be expressed in more general form by including higher derivatives of H as
$$ \mathcal{F} (P, \rho, H, \dot{H}, \ddot{H}, \ldots) = 0. $$
(200)
There is a simple example
$$ P = w_{\mathrm{eff}} \rho - \frac{2}{\kappa^2} \dot{H} - \frac{3 ( 1 + w_{\mathrm{eff}} )}{\kappa^2} H^2 . $$
(201)
From Eqs. (42) and (43), we find
$$ \rho= \frac{3}{\kappa^2} H^2, \quad P = w_{\mathrm{eff}} \rho - \frac{2}{\kappa^2} \dot{H} - \frac{3}{\kappa^2} H^2. $$
(202)
It follows from Eqs. (201) and (202) that Eq. (201) is an identity. This implies that there exists a solution for any cosmology, provided that Eq. (201) is satisfied.
We can also find another example
$$ P = w_{\mathrm{eff}} \rho- G_1 - \frac{2}{\kappa^2} \dot{H} + G_2 \dot{H}^2, $$
(203)
where G1 and G2 are constants, and G1(1+weff)>0 is supposed to be satisfied. For G2(1+weff)>0, we have a solution expressing an oscillating universe
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ204_HTML.gif
(204)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ205_HTML.gif
(205)
with a coefficient hc and a frequency \(\bar{\omega}\), given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ206_HTML.gif
(206)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ207_HTML.gif
(207)
On the other hand, for G2(1+weff)<0,
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ208_HTML.gif
(208)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ209_HTML.gif
(209)
where hc is given by Eq. (208) and a frequency \(\tilde{\omega}\) is described as
$$ \tilde{\omega} = \sqrt{-\frac{3 ( 1 + w_{\mathrm{eff}} )}{G_1 \kappa^2}}. $$
(210)
Hence, the above investigations show that a number of models describing cosmology with an inhomogeneous EoS can be constructed.

Finally, we mention that in Balcerzak and Denkiewicz (2012), cosmological density perturbations around the finite-time future singularities with the scale factor being finite have been examined. At the present stage, it seems that a number of models with a finite-time future singularity are not considered to be distinguishable with the ΛCDM model by using the observational test. As a recent investigation, the cosmological density perturbations in k-essence scenario has been investigated in Bamba et al. (2012c).

4 Scalar field theory as dark energy of the universe

In this section, we explore scalar field theories in general relativity.

4.1 Scalar field theories

The action of scalar field theories in general relativity is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ211_HTML.gif
(211)
where ω(ϕ) is a function of the scalar field ϕ and V(ϕ) is the potential of ϕ. (i) For ω(ϕ)=0 and V(ϕ)=Λ/κ2 with Λ being a cosmological constant, the action in Eq. (211) describes the ΛCDM model in Eq. (1). (ii) For ω(ϕ)=+1, this action corresponds to the one for a quintessence model with a canonical kinetic term. (iii) For ω(ϕ)=−1, this action expresses a phantom model. We consider the case in which the scalar field ϕ is a spatially homogeneous one, i.e., it depends only on time t.
In the FLRW background (3), the Einstein equations are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ212_HTML.gif
(212)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ213_HTML.gif
(213)
where ρϕ and Pϕ are the energy density and pressure of the scalar field ϕ, respectively, given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ214_HTML.gif
(214)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ215_HTML.gif
(215)
In addition, ρM and PM are the energy density and pressure of matter, respectively. Equations (214) and (215) give the way to express scalar dark energy as fluid dark energy, where scalar field equation becomes just the conservation law for such fluid description.
Here, for clear understanding, by using Eqs. (214) and (215), we describe the explicit expression of wDE in the case that there exists only single scalar field of dark energy, i.e., dark energy sufficiently dominates over matter. (i) For the ΛCDM model with ω(ϕ)=0 and V(ϕ)=Λ/κ2, wDE=−1. (ii) For a quintessence model with ω(ϕ)=+1,
$$ w_{\mathrm{DE}} = \frac{\dot{\phi}^2 - 2V(\phi)}{\dot{\phi}^2 + 2V(\phi )}, $$
(216)
it follows from which that −1<wDE<−1/3. (iii) For a phantom model with ω(ϕ)=−1
$$ w_{\mathrm{DE}} = \frac{\dot{\phi}^2 + 2V(\phi)}{\dot{\phi}^2 - 2V(\phi )}, $$
(217)
which leads to wDE<−1.
By using Eqs. (212)–(215), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ218_HTML.gif
(218)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ219_HTML.gif
(219)
If there is no coupling between matter and the scalar field ϕ, the continuity equations for matter and ϕ are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ220_HTML.gif
(220)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ221_HTML.gif
(221)
For a matter with a constant EoS wMPM/ρM, from Eq. (220) we find
$$ \rho_{\mathrm{M}} = \rho_{\mathrm{M} \mathrm{c}} a^{-3 (1+w_{\mathrm{M}} )}, $$
(222)
where ρMc is a constant.
It is the interesting case that ω(ϕ) and V(ϕ) are defined in terms of a single function I(ϕ) as (Capozziello et al. 2006b)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ223_HTML.gif
(223)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ224_HTML.gif
(224)
Here, I(ϕ) is defined as
$$ I(\phi) \equiv\frac{d \mathcal{I}(\phi)}{d \phi}, $$
(225)
with \(\mathcal{I}(\phi)\) and \(\mathcal{I}_{\mathrm{c}}\) being an arbitrary twice differentiable function of ϕ and an integration constant, respectively. Thus, we can acquire the following solutions:
$$ \phi= t, \qquad H = I(t). $$
(226)
For this solution, the equation of motion for ϕ is derived from the variation of the action in Eq. (211) over ϕ as
$$ \omega(\phi) \ddot{\phi} + \frac{1}{2} \frac{\partial\omega (\phi )}{\partial\phi} \dot{\phi}^2 + 3H\omega(\phi) \dot{\phi} + \frac{\partial V (\phi)}{\partial\phi} = 0. $$
(227)
From these solutions in Eq. (226), we have
$$ a(t) = a_{\mathrm{c}} \mathrm{e}^{\mathcal{I}(t)}, \quad a_{\mathrm{c}} = \biggl( \frac{\rho_{\mathrm{M} \mathrm{c}} }{\mathcal {I}_{\mathrm{c}}} \biggr)^{1/ [3 (1+w_{\mathrm{M}} ) ]}. $$
(228)
In what follows, we consider the case in which these solutions are satisfied.
For this case, in the FLRW background (3), the effective EoS for the universe is given by Eq. (48) with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ229_HTML.gif
(229)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ230_HTML.gif
(230)
If we define a new scalar field Φ as
$$ \varPhi\equiv\int^{\phi} d\phi\sqrt{\vert{\omega(\phi )}\vert}, $$
(231)
the action in Eq. (211) can be rewritten to the form
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ232_HTML.gif
(232)
where the sign in front of the kinetic term depends on that of ω(ϕ). If the sign of ω(ϕ) is positive (negative), that of the kinetic term is − (+). In the non-phantom phase, the sign of the kinetic term is always −, and in the phantom one it is always +. In principle, it follows from Eq. (231) that ϕ can be solved with respect to χ as ϕ=ϕ(Φ). Hence, the potential \(\tilde{V}(\varPhi)\) is given by \(\tilde{V}(\varPhi) = V ( \phi(\varPhi))\).

4.2 Equivalence between fluid descriptions and scalar field theories

In this subsection, we show the equivalence between fluid descriptions and scalar field theories. We first take a fluid and then construct a scalar field theory with the same EoS as that in a fluid description. This process leads to constraints on a coefficient function of the kinetic term ω(ϕ) and the potential V(ϕ) of the scalar field ϕ in the action in Eq. (211). Through this procedure, we propose a way of expressing a fluid model as an explicit scalar field theory. In other words, we can obtain the explicit expressions of ω(ϕ) and V(ϕ) in the corresponding scalar field theory for a fluid model.

For simplicity, we suppose the dark energy dominated stage, so that we can neglect matter and therefore wDEweff, namely, ρeffρ=ρϕ in Eq. (229) and ρeffρ=ρϕ in Eq. (230). In a fluid description, from Eq. (40) we find
$$ w_{\mathrm{eff}} = \frac{P}{\rho} = -1 - \frac{f(\rho)}{\rho}. $$
(233)
While, in a scalar field theory with the solutions in (226), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ234_HTML.gif
(234)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ235_HTML.gif
(235)
where the second equalities follow from Eq. (41). Since Eq. (42) presents ρ=3H2/κ2, if H(=I(t)) is given, we acquire the expression of ρ as a function of t(=ϕ) as ρ=ρ(t)=ρ(ϕ). Hence, by substituting this relation into Eqs. (234) and (235), in principle we have ω=ω(ϕ) and V=V(ϕ). Thus, this procedure yields an explicit scalar field theory, which corresponds to an original fluid model. On the other hand, as the opposite direction, provided that we have a scalar field theory described by ω(ϕ) and V(ϕ) in the action in Eq. (211). By using Eqs. (214) and (215) and the solution ϕ=t and H=I(t) in Eq. (226), we get the explicit expression of weff in Eq. (233). Hence, by combining this expression and ρ in Eq. (214) and comparing the resultant expression with the representation of weff in Eq. (40), we acquire f(ρ) in the fluid description. Consequently, it can be interpreted that the considerations on both these directions imply the equivalence between the representation of a scalar field theory and the description of a fluid model.
As concrete examples for a fluid description, we first consider the case of Eq. (53). By substituting Eq. (53) into Eqs. (233)–(235), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ236_HTML.gif
(236)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ237_HTML.gif
(237)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ238_HTML.gif
(238)
Second, for the case of Eq. (56), by combining Eq. (56) and Eqs. (233)–(235) we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ239_HTML.gif
(239)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ240_HTML.gif
(240)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ241_HTML.gif
(241)
As the third case described in Eq. (86), by plugging Eq. (86) into Eqs. (233)–(235) we acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ242_HTML.gif
(242)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ243_HTML.gif
(243)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ244_HTML.gif
(244)
As the last example, we consider a model of Little Rip scenario given in Eq. (109). By using Eqs. (42) and (43), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ245_HTML.gif
(245)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ246_HTML.gif
(246)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ247_HTML.gif
(247)
It is meaningful to remark that if the Hubble parameter H can be represented by t, by using the relations (223) and (224) and the solutions (226) we can find the explicit expressions ω=ω(ϕ) and V=V(ϕ). This is clearly demonstrated in the following Sect. 4.3.

4.3 Cosmological models

In this subsection, we reconstruct scalar field theories corresponding to (i) the ΛCDM model, (ii) quintessence model, (iii) phantom model and (iv) unified scenario of inflation and late-time cosmic acceleration. In addition, (v) scalar field models with realizing the crossing the phantom divide is also considered including its stability issue.

4.3.1 The ΛCDM model

For the ΛCDM model, we have
$$ I(\phi) = H_{\mathrm{c}}, $$
(248)
where Hc is a constant. From Eq. (226), we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ249_HTML.gif
(249)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ250_HTML.gif
(250)
Moreover, by using Eqs. (223) and (224), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ251_HTML.gif
(251)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ252_HTML.gif
(252)
with
$$ \mathcal{I}(\phi) = H_{\mathrm{c}} \phi. $$
(253)
In the ΛCDM model, we acquire
$$ w_{\mathrm{eff}} = -1. $$
(254)

4.3.2 Quintessence model

As an example of a quintessence model, we investigate the following model:
$$ I(\phi) = H_{\mathrm{c}} + \frac{H_1}{\phi^n}, $$
(255)
where Hc and H1(>0) are constants and n(>1) is a positive (constant) integer larger than unity. By using Eq. (226), we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ256_HTML.gif
(256)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ257_HTML.gif
(257)
In addition, from Eqs. (223) and (224) we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ258_HTML.gif
(258)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ259_HTML.gif
(259)
with
$$ \mathcal{I}(\phi) = H_{\mathrm{c}} \phi- \frac{H_1}{ (n-1 ) \phi^{n-1}}. $$
(260)
Using Eq. (48), the effective EoS is written as
$$ w_{\mathrm{eff}} = -1 + \frac{2n H_1 t^{n-1}}{3 ( H_{\mathrm{c}} t^{n} + H_1 )^2}. $$
(261)
From Eq. (261), we see that weff>−1 because H1>0 and n>1. Thus, this model corresponds to a quintessence model.

4.3.3 Phantom model

As an example of a phantom model, we explore the following model:
$$ I(\phi) = \frac{H_2}{t_{\mathrm{s}} - \phi} + \frac{H_3}{\phi^2}, $$
(262)
where H2 and H3 are constants and ts is the time when a Big Rip singularity appears. We examine the range 0<t<ts. From Eq. (226), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ263_HTML.gif
(263)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ264_HTML.gif
(264)
Furthermore, it follows from Eqs. (223) and (224) that
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ265_HTML.gif
(265)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ266_HTML.gif
(266)
Using Eq. (48), the effective EoS is expressed as
$$ w_{\mathrm{eff}} = -1 -\frac{2 [H_2 t^3 -2H_3 (t_{\mathrm{s}} - t )^2 ] t}{3 [ H_2 t^2 + H_3 (t_{\mathrm{s}} - t ) ]^2}. $$
(267)
It follows from Eq. (267) that when the time t approaches ts, the second term on the right-hand side (r.h.s.) of Eq. (267) becomes negative because H2t3−2H3(tst)2>0 and therefore weff<−1. Hence, there exists the phantom phase in this model and this consequence originates from the realization of \(\dot{H} > 0\), i.e., superacceleration. We note that when 0<tts, for \(0 < t \lesssim(2H_{3} t_{\mathrm{s}}^{2}/H_{2} )^{1/3}\), weff>−1 and therefore the non-phantom phase exists before the phantom phase appears.

4.3.4 Unified scenario of inflation and late-time cosmic acceleration

As an example of a unified scenario of inflation and the late-time acceleration of the universe, we study the following model:
$$ I(\phi) = h_{\mathrm{c}}^2 \biggl( \frac{1}{t_{\mathrm{s}}^2 - \phi^2} \biggr) + \frac{1}{t_1^2 + \phi^2}, $$
(268)
where hc is a constant, ts corresponds to the time when a Big Rip singularity appears, and t1 is a time. From Eq. (226), we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ269_HTML.gif
(269)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ270_HTML.gif
(270)
Moreover, by using Eqs. (223) and (224), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ271_HTML.gif
(271)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ272_HTML.gif
(272)
with
$$ \mathcal{I}(\phi) = \frac{h_{\mathrm{c}}^2}{2t_{\mathrm{s}}} \ln \biggl( \frac{t_{\mathrm{s}} + \phi}{t_{\mathrm{s}} - \phi} \biggr) + \frac{h_{\mathrm{c}}^2}{t_1} \arctan \biggl( \frac{\phi}{t_1} \biggr). $$
(273)
Using Eq. (48), the effective EoS is expressed as
$$ w_{\mathrm{eff}} = -1 -\frac{8}{3h_{\mathrm{c}}^2} \frac{t ( t - t_{+} ) ( t - t_{-} )}{t_1^2 + t_{\mathrm{s}}^2}, $$
(274)
where \(t_{\pm} \equiv\pm\sqrt{( t_{\mathrm{s}}^{2} - t_{1}^{2})/2}\). There exist two phantom phases: t<t<0 and t>t+, in which weff<−1. On the other hand, there are also two non-phantom phases: −ts<t<t and 0<t<t+, in which weff>−1.

The history of the universe in this model can be interpreted as follows. The universe is created at t=−ts because the value of the scale factor a(t) in Eq. (270) becomes zero a(−ts)=0. During −ts<t<t, there is the first non-phantom phase. The first phantom phase in t<t<0 corresponds to the inflationary stage. After inflation, the second non-phantom phase in 0<t<t+ becomes the radiation/matter-dominated stages. Then, the second phantom phase in t>t+ plays a role of the dark energy dominated stage, i.e., the late-time accelerated expansion of the universe. Finally, a Big Rip singularity occurs at t=ts. As a result, this model can present a unified scenario of inflation in the early universe and the late-time cosmic acceleration. Incidentally, phantom inflation has been studied in Piao and Zhang (2004), Piao and Zhou (2003).

4.3.5 Scalar field models with the crossing the phantom divide

The instability of a single scalar field theory with the crossing the phantom divide was examined in Vikman (2005). In addition, the stability issue in a single scalar field theory as well as a two scalar field theory in which the crossing the phantom divide can be realized has recently been discussed in Saitou and Nojiri (2012). In this subsection, we first examine the stability of a single scalar field theory when the crossing the phantom divide occurs. The considerations for a two scalar field theory with the crossing the phantom divide are presented in Sect. 6.1.2

We define the new variables \(Z \equiv\dot{\phi}\) and YI(ϕ)/H. With these variables, in the flat FLRW background the Friedmann equation (212) with Eq. (214) and the equation of motion for ϕ (227) are rewritten to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ275_HTML.gif
(275)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ276_HTML.gif
(276)
where I,ϕ(ϕ)≡∂I(ϕ)/∂ϕ and I,ϕϕ(ϕ)≡2I(ϕ)/∂ϕ2, N is the number of e-folds and the scale factor is expressed as \(a = \mathrm{e}^{N-N_{0}}\) with N0 being the current value of N. For the solutions (226), we find Z=1 and Y=1. Hence, we examine the small perturbations δZ and δY around this solution as follows
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ277_HTML.gif
(277)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ278_HTML.gif
(278)
By using the solutions (226), we see that these perturbations obey the equation
$$ \frac{d}{dN} \left( \begin{array}{@{}c@{}} \delta X_{\phi} \\ \delta X_{\chi} \end{array} \right) = \mathcal{M}_1 \left( \begin{array}{@{}c@{}} \delta X_{\phi} \\ \delta X_{\chi} \end{array} \right), $$
(279)
where \(\mathcal{M}_{1}\) is a matrix, given by
$$ \mathcal{M}_1 \equiv \left( \begin{array}{@{}c@{\quad}c@{}} -\frac{\ddot{H}}{\dot{H} H}-3 & -3 \\[2mm] -\frac{\dot{H}}{H^2} & -\frac{\dot{H}}{H^2} \end{array} \right). $$
(280)
Thus, we can obtain the eigenvalues of the matrix \(\mathcal{M}_{1}\) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ281_HTML.gif
(281)
The stability condition for the solutions (226) is that both of the eigenvalues m+ and m are negative. When the crossing of the phantom divide occurs around \(\dot{H} \sim0\), for \(\ddot{H} >0\), the transition from the non-phantom (quintessence) phase (\(\dot{H} <0\)) to the phantom one (\(\dot{H} >0\)) occurs, whereas for \(\ddot{H} <0\), the opposite direction transition from the phantom phase to the non-phantom one happens. In case of the expanding universe (H>0), the term \(\ddot{H}/(\dot{H}H)\) is negative. Thus, around the crossing of the phantom divide, from Eq. (281) we find \(m_{+} \sim- \ddot{H}/(\dot{H}H) > 0\) and m∼0. As a result, at the crossing time when \(\dot{H} =0\)m+ becomes +∞ and therefore the solution in Eq. (226) is unstable when the crossing of the phantom divide occurs. In other words, in a single scalar field theory the crossing of the phantom divide cannot be realized.

5 Tachyon scalar field theory

In this section, we examine a tachyon scalar by following Amendola and Tsujikawa (2010). The effective 4-dimensional action for the tachyon field which is an unstable mode of D-branes [non-Bobomol’nyi-Prasad-Sommerfield (non-BPS) branes] is given by
$$ S=-\int d^4x V(\phi) \sqrt{-\det ( g_{\mu\nu} + { \partial}_{\mu} \phi{\partial}_{\nu} \phi )}, $$
(282)
where ϕ is a tachyon scalar field and V(ϕ) is a potential of ϕ. From the action in Eq. (282), the energy-momentum tensor of ϕ is derived as
$$ T_{\mu\nu}^{(\phi)} = \frac{V(\phi) {\partial}_{\mu} \phi{\partial}_{\nu} \phi}{ \sqrt{1+g^{\alpha\beta} {\partial}_{\alpha} \phi{\partial }_{\beta}}} -g_{\mu\nu} V(\phi) \sqrt{1+g^{\alpha\beta} {\partial}_{\alpha} \phi {\partial}_{\beta}}. $$
(283)
In the flat FLRW background (3) (with K=0), \(\rho_{\phi} = -T_{0}^{0\,(\phi)}\) and \(P_{\phi} = T_{i}^{i\,(\phi)}\) are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ284_HTML.gif
(284)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ285_HTML.gif
(285)
We remark that through the use of Eqs. (284) and (285), this model may be written as a fluid. By using \(\ddot{a}/{a} = H^{2} + \dot{H}\) and plugging Eqs. (284) and (285) into Eqs. (42) and (43), we find
$$ \frac{\ddot{a}}{a} = \frac{\kappa^2}{3} \frac{V(\phi)}{\sqrt {1-\dot{\phi}^2}} \biggl( 1- \frac{3}{2} \dot{\phi}^2 \biggr). $$
(286)
It follows from Eq. (286) that the condition for the accelerated expansion \(\ddot{a} > 0\) is given by \(\dot{\phi}^{2} <2/3\). Furthermore, the EoS for ϕ is represented by
$$ w_{\phi} \equiv\frac{P_{\phi}}{\rho_{\phi}} = \dot{\phi}^2 - 1. $$
(287)
Hence, from the above condition \(\dot{\phi}^{2} <2/3\) and Eq. (287) we see that the possible range of the value of wϕ with realizing the cosmic acceleration is −1<wϕ<−1/3, which corresponds to the non-phantom (quintessence) phase.
We derive expressions of V(ϕ) and ϕ in terms of H and \(\dot{H}\). By using the relation \(H^{2} = (\kappa^{2}/3) V(\phi)/\sqrt{1-\dot{\phi}^{2}}\), which follows from Eq. (42) with Eq. (284), and Eq. (286), we have \(\dot{H}/H^{2} = - (3/2) \dot{\phi}^{2}\). The combination of these equations leads to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ288_HTML.gif
(288)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ289_HTML.gif
(289)
We suppose that the scale factor is expressed by a power-law expansion as a(t)∝tp with p>1 being a constant larger than unity. By combining this relation with Eqs. (288) and (289), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ290_HTML.gif
(290)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ291_HTML.gif
(291)
where we have taken the integration constant as zero. As a result, we acquire an inverse square power-law tachyon potential V(ϕ)∝ϕ−2.
In the open string theory, the form of a tachyon potential V(ϕ) is given by (Kutasov and Niarchos 2003)
$$ V(\phi) = \frac{V_0}{\cosh ( \phi/\phi_0 )}, $$
(292)
with \(\phi_{0} = \sqrt{2}\) for the non-BPS D-branes in the superstring and ϕ0=2 for the bosonic string. This form has a ground state in the limit ϕ→∞. Here, V0 and ϕ0 are constants and V(ϕ=ϕ0)=V0. Moreover, when a tachyon potential appears as the excitation of massive scalar fields on the anti D-branes, V(ϕ) is given by (Garousi et al. 2004, 2005; Chingangbam and Qureshi 2005)
$$ V(\phi) = V_0 \mathrm{e}^{m^2 \phi^2 / 2}, $$
(293)
where m is the mass of ϕ and there exists a minimum of V(ϕ) at ϕ=0.
We mention that a tachyon scalar field can be generalized to so-called k-essence (Chiba et al. 2000; Armendariz-Picon et al. 2000, 2001), which is a scalar field with non-canonical kinetic terms (for its application to inflation, so-called k-inflation, see Armendariz-Picon et al. 1999; Garriga and Mukhanov 1999), and for a unified scenario between inflation and late-time cosmic acceleration in the framework of k-essence model, see, e.g., Saitou and Nojiri 2011). The action of k-essence is described by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ294_HTML.gif
(294)
where P(ϕ,X) is a function of a scalar field ϕ and its kinetic term X≡−(1/2)gμνμϕ∂νϕ. It is known that the accelerated expansion can be realized by the kinetic energy even without the scalar field potential. An effective 4-dimensional Lagrangian describing a tachyon field is given by (Garousi 2000; Sen 2002; Gibbons 2002)
$$ P = -V(\phi) \sqrt{1-2X}. $$
(295)
Finally, we discuss a stability issue. We now consider the action in Eq. (294). In this case, the energy density and pressure of ϕ are given by ρϕ=2XP,X and Pϕ=P, respectively, where P,X∂P/∂X. The EoS for ϕ is written as
$$ w_{\phi} = \frac{P}{2X P_{,X} -P}. $$
(296)
If the relation |2XP,X|≪P is realized, wϕ can be close to −1.
We investigate the stability conditions for k-essence in the ultra-violet (UV) regime. In the Minkowski background, we write ϕ as ϕ(t,x)=ϕb(t)+δϕ(t,x), where ϕb(t) is the background part and δϕ(t,x) is the perturbed one. By deriving the Lagrangian and Hamiltonian for δϕ(t,x), the second-order Hamiltonian is obtained as (Piazza and Tsujikawa 2004)
$$ \delta\mathcal{H} = \xi_1 \frac{ (\delta\dot{\phi} )^2}{2} + \xi_2 \frac{ (\nabla\delta\phi )^2}{2} + \xi_3 \frac{ (\delta\phi )^2}{2}, $$
(297)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ298_HTML.gif
(298)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ299_HTML.gif
(299)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ300_HTML.gif
(300)
where P,XX2P/∂X2 and P,ϕϕ2P/∂ϕ2. The stability conditions are represented by the positivity of each three terms on the r.h.s. of Eq. (297). The propagation speed of ϕ is defined as
$$ c_{\mathrm{s}}^{2} \equiv\frac{P_{\phi,X}}{\rho_{\phi,X}} = \frac{\xi_2}{\xi_1}. $$
(301)
As long as the stability conditions ξ1≥0 in Eq. (298) and ξ2≥0 in Eq. (299) are satisfied, \(c_{\mathrm{s}}^{2} \geq0\). In addition, the condition that the propagation speed should be sub-luminal is given by (Garriga and Mukhanov 1999)
$$ P_{,XX} >0. $$
(302)
We note that the finite-time future singularities in tachyon cosmology have also been examined in Gorini et al. (2004), Keresztes et al. (2009).

6 Multiple scalar field theories

In this section, we describe multiple scalar field theories.

6.1 Two scalar field theories

To begin with, in this subsection we investigate two scalar field theories. First, we explain the standard type of two scalar field theories. Next, we discuss a new type of two scalar field theories with realizing the crossing of the phantom divide, which has recently been constructed in Saitou and Nojiri (2012).

6.1.1 Standard two scalar field theories

First, we explore the standard type two scalar field theories (Nojiri and Odintsov 2006b; Capozziello et al. 2006c; Elizalde et al. 2008). The action of two scalar field theories in general relativity is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ303_HTML.gif
(303)
where σ(χ) is a functions of the scalar field χ and V(ϕ,χ) is the potential term of ϕ and χ. Here, we concentrate on the scalar-field part of the action and do not take into account the matter part of it. If there does not exist the second scalar field χ, the action in Eq. (303) is the same as the action in Eq. (211) without the matter part of it.
In the FLRW background (3), the Einstein equations are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ304_HTML.gif
(304)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ305_HTML.gif
(305)
where ρt and Pt are the total energy density and pressure of the two scalar fields ϕ and χ, respectively, given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ306_HTML.gif
(306)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ307_HTML.gif
(307)
Moreover, equations of motion for the scalar fields are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ308_HTML.gif
(308)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ309_HTML.gif
(309)
Since it is possible to redefine the scalar fields through a convenient transformation, we take ϕ=χ=t. If a solution H(t)=I(t) is given, where I(t) is a function of t, by plugging Eq. (304) into Eqs. (308) and (309), we obtain
$$ \omega(\phi) = -\frac{2}{\kappa^2} \frac{\partial I(\phi, \chi)}{\partial\phi}, \qquad \sigma(\chi) = -\frac{2}{\kappa^2} \frac{\partial I(\phi, \chi)}{\partial\chi}. $$
(310)
Here, by taking ϕ=χ=t into consideration, we can interpret I(ϕ,χ) as I(t,t)≡I(t). Thus, we define I(ϕ,χ) as
$$ I(\phi, \chi) = -\frac{\kappa^2}{2} \biggl( \int\omega(\phi) d \phi+ \int\sigma( \chi) d \chi \biggr). $$
(311)
We can represent the potential term V(ϕ,χ) by
$$ V(\phi, \chi) = \frac{1}{\kappa^2} \biggl( 3 I^2 (\phi, \chi) + \frac{\partial I(\phi, \chi )}{\partial\phi} + \frac{\partial I(\phi, \chi)}{\partial\chi} \biggr). $$
(312)
Furthermore, Eq. (305) is rewritten to
$$ -\frac{2}{\kappa^2} \frac{d I(t)}{d t} = \omega(t) + \sigma(t). $$
(313)
In addition, we can describe a part of coefficient of the kinetic terms, which is a function of a scalar field, ω(ϕ) and σ(χ), as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ314_HTML.gif
(314)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ315_HTML.gif
(315)
where \(\tilde{g} (\phi)\) is an arbitrary function of ϕ. Thus, by substituting Eq. (311) with Eqs. (314) and (315) into Eq. (312), we acquire (for details, see Elizalde et al. 2008)
$$ V(\phi, \chi) = \frac{1}{\kappa^2} \biggl( 3 I^2 (\phi, \chi) + \frac{d I(\phi)}{d \phi} + \tilde{g} (\phi) - \tilde{g} (\chi) \biggr). $$
(316)
In order to demonstrate an example of two scalar field theories, we examine the following model in Eq. (262):
$$ I(t) = \frac{H_2}{t_{\mathrm{s}} - t} + \frac{H_3}{t^2}. $$
(317)
In this case, by using Eqs. (314) and (317) we find that ω(ϕ) is expressed as
$$ \omega(\phi) = -\frac{2}{\kappa^2} \biggl[ \frac{H_2}{ (t_{\mathrm{s}} - t )^2} - \frac{2 H_3}{\phi^3} + \tilde{g} (\phi) \biggr], $$
(318)
and σ(χ) is described as in Eq. (315). In addition, it follows from Eqs. (311), (312) and (317) that I(ϕ,χ) and V(ϕ,χ) are represented as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ319_HTML.gif
(319)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ320_HTML.gif
(320)
We investigate the stability of the system described by the solution in Eq. (317). We define the following variables:
$$ X_{\phi} \equiv\dot{\phi}, \qquad X_{\chi} \equiv\dot{\chi}, \qquad Y \equiv\frac{I(\phi, \chi)}{H}. $$
(321)
By using the variables in Eq. (321), Eqs. (304), (308) and (309) are rewritten to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ322_HTML.gif
(322)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ323_HTML.gif
(323)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ324_HTML.gif
(324)
where we have used d/dN=(1/H)d/dt. We analyze the perturbations |δXϕ|≪1, |δXχ|≪1 and |δY|≪1 around (Xϕ,Xχ,Y)=(1,1,1) as
$$ X_{\phi} = 1 + \delta X_{\phi}, \qquad X_{\chi} = 1 + \delta X_{\chi}, \qquad Y = 1 + \delta Y. $$
(325)
The perturbations satisfy the equation
$$ \frac{d}{dN} \left( \begin{array}{@{}c@{}} \delta X_{\phi} \\ \delta X_{\chi} \\ \delta Y \end{array} \right) = M \left( \begin{array}{@{}c@{}} \delta X_{\phi} \\ \delta X_{\chi} \\ \delta Y \end{array} \right), $$
(326)
where M is the matrix, defined by
$$ M \equiv \left( \begin{array}{@{}c@{\quad}c@{\quad}c@{}} -\frac{d \omega(\phi) /d \phi}{H \omega(\phi)}-3 & 0 & 3 \\[2mm] 0 & -\frac{d \sigma(\chi) /d \chi}{H \sigma(\chi)}-3 & 3 \\[2mm] \kappa^2 \frac{\omega(\phi)}{2H^2} & \kappa^2 \frac{\sigma(\chi )}{2H^2} & \kappa^2 \frac{\omega(\phi) + \sigma(\chi)}{2H^2} \end{array} \right). $$
(327)
The characteristic equation for M is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ328_HTML.gif
(328)
where λ denotes an eigenvalue of M and E is a unit matrix. We impose the following conditions
$$ \omega(\phi) \neq0, \qquad \sigma(\chi) \neq0, $$
(329)
so that the eigenvalues can be finite without diverging. As a result, if the conditions Eq. (329) are satisfied, the solution in Eq. (317) does not have infinite instability at the crossing of the phantom divide from the non-phantom phase to the phantom one. For an illustration, e.g., we take, \(\tilde{g} (t) = \tilde{g}_{\mathrm{c}}/t^{3}\) with \(\tilde{g}_{\mathrm{c}}\) being a constant and satisfying \(\tilde{g}_{\mathrm{c}} > 2H_{3}\). From Eq. (319), we find
$$ I(\phi, \chi) = \frac{H_2}{t_{\mathrm{s}} - \phi} - \frac{\tilde{g}_{\mathrm{c}} -2 H_3}{2\phi^2} + \frac{\tilde{g}_{\mathrm{c}}}{2\chi^2}. $$
(330)
Moreover, it follows from Eqs. (314) and (315) that
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ331_HTML.gif
(331)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ332_HTML.gif
(332)
Furthermore, by using Eq. (320) we obtain
$$ V(\phi, \chi) = \frac{1}{\kappa^2} \biggl[ 3 I^2 (\phi, \chi) + \frac{H_2}{ ( t_{\mathrm{s}} - \phi )^2} + \frac{\tilde{g}_{\mathrm{c}} -2 H_3}{\phi^3} - \frac{\tilde{g}_{\mathrm{c}}}{\phi^3} \biggr]. $$
(333)
Examples of two scalar field theories are an oscillating quintom model (Feng et al. 2006; Elizalde et al. 2004) or a quintom with two scalar fields (Zhang et al. 2006) in the framework of general relativity (see also Feng et al. 2005; Li et al. 2005; Zhao et al. 2005; Cai et al. 2010).

6.1.2 New type of two scalar field theories

A new type of two scalar field theories is given by (Saitou and Nojiri 2012)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ334_HTML.gif
(334)
where X≡−(1/2)gμνμϕ∂νϕ, U≡−(1/2)gμνμχ∂νχ. We suppose that there is no direct interaction between the scalar fields and matters, so that the continuity equation of matter can be satisfied \(\rho^{\prime}_{\mathrm{M}} (N) + 3 ( \rho_{\mathrm{M}} (N) + P_{\mathrm{M}} (N) ) = 0\), where a prime denotes the partial derivative with respect to N of /∂N. In the flat FLRW background, for the action in Eq. (334) the gravitational field equations are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ335_HTML.gif
(335)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ336_HTML.gif
(336)
Moreover, the Friedmann equation is also represented as H2(N)=(κ2/3)(ρs(N)+ρM(N)), where ρs is the energy density of the scalar fields, namely, we have expressed the right-hand side by dividing the energy density into the contributions from the scalar fields and matter. By using this expression, the gravitational field equations (335) and (336) are rewritten to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ337_HTML.gif
(337)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ338_HTML.gif
(338)
Furthermore, the equation of motion for the scalar fields are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ339_HTML.gif
(339)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ340_HTML.gif
(340)
where the subscription “,ϕ” denotes a partial derivative with respect to ϕ, e.g., V,ϕ(ϕ,χ)≡∂V(ϕ,χ)/∂ϕ and V,χ(ϕ,χ)≡∂V(ϕ,χ)/∂χ. Thus, we acquire the following solutions
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ341_HTML.gif
(341)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ342_HTML.gif
(342)
provided that V(ϕ,χ), ω(ϕ) and η(χ) satisfy the equations
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ343_HTML.gif
(343)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ344_HTML.gif
(344)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ345_HTML.gif
(345)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ346_HTML.gif
(346)
where J(N) is an arbitrary function and \(\bar{m}\) is a constant. We derive the explicit forms of ω(ϕ), η(χ) and V(ϕ,χ) in the action in Eq. (334) so that these should obey Eqs. (343)–(346). As an example, by using an arbitrary function \(\bar{\alpha}\) and J we can express ω(ϕ) and η(χ) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ347_HTML.gif
(347)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ348_HTML.gif
(348)
In addition, we define another function \(\tilde{J} (\phi, \chi)\) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ349_HTML.gif
(349)
It follows from Eq. (349) that \(\tilde{J} (\bar{m} N, \bar{m} N) = J (N) \). With Eq. (349), we determine the form of V(ϕ,χ) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ350_HTML.gif
(350)
Thus, through this procedure, it is possible to reconstruct any expanding history of the universe by using two arbitrary functions J of \(\phi^{\prime}/\bar{m}\) or \(\chi^{\prime}/\bar{m}\) and \(\bar{\alpha}\) of ϕ or χ.
Next, we explore the stability of the solutions (341) and (342). There are two approaches to study the stability. One is the perturbative analysis. Another is to examine the sound speed of the scalar fields. First, we investigate the first order perturbations from the solutions, given by
$$ \begin{array}{@{}l} \phi= \phi_0 + \delta\phi(N), \qquad \chi= \chi_0 + \delta\chi(N),\\[2mm] \dot{\phi} = \dot{\phi}_0 + \delta x (N), \qquad \dot{\chi} = \dot{\chi}_0 + \delta y (N), \end{array} $$
(351)
where we have defined \(\delta x (N) \equiv\delta\dot{\phi} (N)\) and \(\delta y (N) \equiv\delta\dot{\chi} (N)\). From the Friedmann equation, there exists the following constraint between the perturbations
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ352_HTML.gif
(352)
Hence, in this system there are four degrees of freedom \((\phi, \chi, \dot{\phi}, \dot{\chi})\). By combining Eqs. (337)–(340) and(351) and using Eq. (352), we obtain
$$ \frac{d}{dN} \left( \begin{array}{@{}c@{}} \delta\phi\\ \delta\chi\\ \delta x \\ \delta y \end{array} \right) = \mathcal{M}_2 \left( \begin{array}{@{}c@{}} \delta\phi\\ \delta\chi\\ \delta x \\ \delta y \end{array} \right), $$
(353)
where \(\mathcal{M}_{1}\) is a matrix, given by
$$ \mathcal{M}_2 \equiv \left( \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0 & 0 & H^{-1} & 0 \\[2mm] 0 & 0 & 0 & H^{-1} \\[2mm] \mathcal{M}_2^{(31)} & \mathcal{M}_2^{(32)} & \mathcal{M}_2^{(33)} & \mathcal{M}_2^{(34)} \\[2mm] \mathcal{M}_2^{(41)} & \mathcal{M}_2^{(42)} & \mathcal{M}_2^{(43)} & \mathcal{M}_2^{(44)} \end{array} \right), $$
(354)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ355_HTML.gif
(355)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ356_HTML.gif
(356)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ357_HTML.gif
(357)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ358_HTML.gif
(358)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ359_HTML.gif
(359)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ360_HTML.gif
(360)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ361_HTML.gif
(361)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ362_HTML.gif
(362)
The condition for the solutions (341) and (342) to be stable is that the real part of all the eigenvalues of the matrix \(\mathcal{M}_{2}\) (354) should be negative. The characteristic equation for \(\mathcal{M}_{2}\) is given by
$$\det|\mathcal{M}_2 - \lambda E| = \lambda^4 + A_1 \lambda^3 + A_2 \lambda^2 + A_3 \lambda+ A_4 = 0, $$
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equb_HTML.gif
The solutions of this equation are \(\lambda= ( \pm\sqrt{2\varXi-B_{1}} \pm\sqrt{-B_{1} -2\varXi\pm 4\sqrt{\varXi-B_{3}}} )/2 - A_{1}/4\), where Ξ satisfies the cubic equation \(\varXi^{3} - (B_{1}/2 )\varXi^{2} -B_{3} \varXi+B_{1}B_{3}/2 - B_{2}^{2}/8 = 0\). Here, \(B_{1} = -3A_{1}^{2}/8 + A_{2}\), \(B_{2} = A_{1}^{3}/8 - A_{1} A_{2}/\allowbreak 2 + A_{3}\), and \(B_{3} = -3A_{1}^{4}/256 + A_{1}^{2} A_{2}/16 -A_{1} A_{3}/4 + A_{4}\). The solution of this cubic equation is given by \(\varXi= (-\mathcal{P} - \sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3}} )^{1/3} + (-\mathcal{P} + \sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3} } )^{1/3} + B_{1}/6\), with \(\mathcal{P} \equiv-B_{1}^{3}/216 + B_{1} B_{3}/6 - B_{2}^{2}/16\) and \(\mathcal{Q} \equiv-B_{1}^{2}/36 - B_{2}/3\). We explore the real solution of the cubic equation so that the maximum of the real parts of λ, which is described by Reλmax, can be negative. As a result, Ξ is expressed as follows. (i) For \(\mathcal{Q} < 0\) and \(\mathcal{P}^{2} < |\mathcal{Q}^{3}|\), \(\varXi= 2\sqrt{|\mathcal{Q}|} \cos[ (1/3) \arccos(-\mathcal{P} |\mathcal{Q}|^{-3/2}) ] + B_{1}/6 \). (ii) For \(\mathcal{Q} < 0\), \(\mathcal{P}^{2} > |\mathcal{Q}^{3}|\) and \(\mathcal{P} \geq0\), \(\varXi= -(\mathcal{P} +\sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3}} )^{1/3} -(\mathcal{P} - \sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3}} )^{1/3} + B_{1}/6\). (iii) For \(\mathcal{Q} < 0\), \(\mathcal{P}^{2} > |\mathcal{Q}^{3}|\) and \(\mathcal{P} < 0\), \(\varXi=(-\mathcal{P} +\sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3}} )^{1/3} +(-\mathcal{P} - \sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3}})^{1/3} + B_{1}/6\). (iv) For \(\mathcal{Q} > 0\), \(\varXi= -(\mathcal{P} +\sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3}} )^{1/3} +(-\mathcal{P} +\sqrt{\mathcal{P}^{2} + \mathcal{Q}^{3}} )^{1/3} + B_{1}/6\).
Moreover, Reλmax is represented as follows. If Ξ2B3,
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ363_HTML.gif
(363)
On the other hand, if Ξ2<B3,
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ364_HTML.gif
(364)
Thus, if the conditions described by Eqs. (363) and (364) are satisfied, the solutions (341) and (342) can be stable. Since these solutions (341) and (342) correspond to those (226) for a single scalar field theory discussed in Sect. 4, through the investigations in Sect. 4.3.5, the stability of these solutions implies that the crossing of the phantom divide can be realized in the new type of two scalar field theories described by the action in Eq. (334).

In principle, the stability condition from the perturbative analysis can yield constraints on the forms of, e.g., \(\bar{\alpha}^{\prime} (\bar {m} N)\), \(\bar{\alpha} (\bar{m} N)\), J(N) and \(\omega_{, \phi} (\bar{m} N )\). However, it is difficult to derive the explicit analytical expressions of such a constraint on \(\bar{\alpha}^{\prime} (\bar{m} N)\) or J(N). On the other hand, the stability condition obtained by the sound speed of the scalar fields can present the analytical representations of constraints on \(\bar{\alpha} (\bar{m} N)\). Therefore, we explore the sound speed of the scalar fields. Its square has to be positive for the stability of the universe. The sound speed \(c_{\mathrm{s} j}^{2}\), where j=ϕ ,χ, of the scalar fields are defined as \(c_{\mathrm{s} \phi}^{2} \equiv P_{\phi, X}/\rho_{\phi, X} = (1+2\omega X)/(1+6\omega X)\) and \(c_{\mathrm{s} \chi}^{2} \equiv P_{\chi, X}/\rho_{\chi, X} = (-1+2\eta X)/(-1+6\eta X)\). Hence, the stability condition is expressed as \(0 \leq c_{\mathrm{s} j}^{2} (\leq1)\). For the solutions (341) and (342), this condition can lead to constraints on the function \(\bar{\alpha}\) as \(\bar{\alpha} (\phi= \bar{m} N) \geq J^{\prime}/( 3\bar{m}^{4}H^{4} )\) or \(\bar{\alpha} (\phi= \bar{m} N) \leq J^{\prime}/( 3\bar{m}^{4}H^{4} )-1/( \bar{m}^{2}H^{2})\), and \(\bar{\alpha} (\phi= \bar{m} N) \geq0\) or \(\bar{\alpha} (\phi= \bar{m} N) \leq -1/( \bar{m}^{2}H^{2} )\).

It is interesting to mention that in Sect. 4.3.5, we have shown that the crossing of the phantom divide cannot occur in a single scalar field theory represented by the action in Eq. (211) because of the instability of the solutions (226), whereas in the new type of two scalar field theories whose action is given by Eq. (334), the crossing of the phantom divide can happen due to the stability of the solutions (341) and (342). This result can be a proposal of a clue for the searches on the non-equivalence of dark energy models on a theoretical level.

6.2 n(≥2) scalar field theories

We generalize the investigations for two scalar field theories in Sect. 4.2. The action of n scalar field theories in general relativity is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ365_HTML.gif
(365)
where ωi(ϕi) is a function of a scalar field ϕi and V(ϕ1,ϕ2,…,ϕn) is the potential term of n scalar fields ϕ1,ϕ2,…,ϕn. For n=2, i.e., two scalar field theories, this action in Eq. (365) with ϕ1=ϕ, ϕ2=χ, ω1(ϕ1)=ω(ϕ) and ω2(ϕ2)=σ(χ) is equivalent to that in Eq. (303). In the FLRW background (3), the Einstein equations are given by Eqs. (304) and (305) with ρt and Pt being
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ366_HTML.gif
(366)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ367_HTML.gif
(367)
We can apply the same procedure executed in the case of two scalar field theories for the n scalar field ones. From Eq. (313), we analogously find
$$ \sum_{i=1}^n \omega_i (t) = -\frac{2}{\kappa^2} \frac{d I(t)}{d t}. $$
(368)
Thus, from Eqs. (314) and (316), we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ369_HTML.gif
(369)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ370_HTML.gif
(370)
where I(ϕ1,…,ϕn) can be regarded as I(t,…,t)≡I(t). Furthermore, we acquire the following solutions:
$$ \phi_i = t, \quad H(t) = I (t). $$
(371)
In addition, from Eq. (368) we take
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ372_HTML.gif
(372)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ373_HTML.gif
(373)
Hence, there exist n−1 arbitrary functions \(\tilde{g}_{2}, \ldots, \tilde{g}_{n}\) which can reproduce the solution (371), and therefore the reconstruction can be executed through the procedure explained above.
Here, we explicitly demonstrate the equivalence of multiple scalar field theories to fluid descriptions. By combining the consequences in Eqs. (369)–(371) and the investigations in Eqs. (233)–(235) in terms of the equivalence between fluid descriptions and scalar field theories, we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ374_HTML.gif
(374)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ375_HTML.gif
(375)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ376_HTML.gif
(376)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ377_HTML.gif
(377)
where in deriving the second equalities in Eqs. (375) and (376) we have used Eq. (374), and Eq. (377) follows from Eqs. (375) and (376). Accordingly, these results imply that multiple scalar field theories can also be represented by a fluid description, similarly to that for a scalar field theory as shown in Sect. 4.2.

7 Holographic dark energy

In this section, we study holographic dark energy scenario and its generalization by following Nojiri and Odintsov (2006b) through the analogy with anti de Sitter (AdS)/conformal field theory (CFT) correspondence. In particular, we investigate the case in which the infrared (IR) cut-off scale is represented by a combination of the particle and future horizons, the time when a Big Rip singularity appears (namely, the life time of the universe), the Hubble parameter and the length scale coming from the cosmological constant.

7.1 Model of holographic dark energy

First, we make an overview for a model of holographic dark energy (Li 2004). The energy density of holographic dark energy is proposed as
$$ \rho_{\mathrm{h}} \equiv \frac{3 C_{\mathrm{h}}^2}{\kappa^2 L_{\mathrm{h}}^2}, $$
(378)
with Ch being a numerical constant. Here, Lh is the IR cut-off scale with a dimension of length. At the dark energy dominated stage, we suppose that the contribution of matter is negligible. From Eq. (42), we have the following Friedmann equation: 3H2/κ2=ρh. By combining this equation and Eq. (378), we find
$$ H = \frac{C_{\mathrm{h}}}{L_{\mathrm{h}}}, $$
(379)
where Ch(>0) is assumed to be positive in order to describe the expanding universe.
It is necessary for us to discuss how to take the IR cut-off Lh because if the IR cut-off is identified with the Hubble parameter, the cosmic acceleration cannot be realized. We define the particle Lph and future Lfh horizons as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ380_HTML.gif
(380)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ381_HTML.gif
(381)
In the flat FLRW space-time in Eq. (3) with K=0, provided that the IR cut-off scale Lh is identified with the particle horizon Lph or the future horizon Lfh, we obtain
$$ \frac{d }{d t} \biggl( \frac{C_{\mathrm{h}}}{a H} \biggr) = \pm \frac{1}{a}, $$
(382)
where the plus + (minus −) sign corresponds to Lph (Lfh). We can solve Eq. (382) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ383_HTML.gif
(383)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ384_HTML.gif
(384)
Thus, for Lh=Lfh, an accelerated expansion of the universe can be realized, because the power index hc in terms of t in Eq. (384) is larger than unity and hence power-law inflation can occur. On the other hand, for Lh=Lph, the universe will shrink due to hc<0.
We suppose that the theory is invariant under the change of the time direction as t→−t. In addition, by shifting the origin of time appropriately, we have the following expression for a instead of Eq. (383):
$$ a = a_{\mathrm{c}} ( t_{\mathrm{s}} - t )^{h_{\mathrm {c}}}. $$
(385)
Thus, for hc<0, a Big Rip singularity will appear at t=ts because a diverges at that time.
If we change the direction of time, the particle horizon becomes like a future one as
$$ L_{\mathrm{ph}} \rightarrow \tilde{L}_{\mathrm{fh}} \equiv a(t) \int_{t}^{t_{\mathrm{s}}} \frac{d t^{\prime}}{a (t^{\prime})} = a(t) \int_{0}^{\infty} \frac{d a}{H a^2}. $$
(386)
From Eq. (48) with Eq. (383) or Eq. (385), we obtain
$$ w_{\mathrm{eff}} = -1 + \frac{2}{3 h_{\mathrm{c}}}. $$
(387)
We remark that when we take Lh as the future horizon Lfh in Eq. (381), we can acquire a de Sitter solution
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ388_HTML.gif
(388)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ389_HTML.gif
(389)
where l is a constant denoting a length scale and hence HdS is a constant Hubble parameter describing de Sitter space. It follows from Eq. (378) and Lfh=l that \(\rho_{\mathrm{h}} = 3 C_{\mathrm{h}}^{2}/( \kappa^{2} l^{2} )\). For Ch=1, by using Eq. (389) we identically find the Friedmann equation (42) as 3H2/κ2=ρh, whereas for Ch≠1, the de Sitter solution (389) cannot be satisfied. When we choose Lh as the particle horizon, there does not exist the de Sitter solution because the particle horizon in Eq. (380) varies in time as Lph=(1−et/l)/l and not a constant.

7.2 Generalized holographic dark energy

Next, we explain generalized holographic dark energy (Elizalde et al. 2005; Nojiri and Odintsov 2006b). It has been pointed out in Hsu (2004), Enqvist et al. (2005) that if Lh is taken as Lph, the EoS vanishes because Lph behaves as being proportional to a3/2(t), although the value of the energy density is compatible with the observations. Therefore, we examine the generalization of holographic dark energy. In more general, Lh could be represented as a function of both Lph and Lfh. Provided that the life time of the universe is finite, then ts can correspond to an IR cut-off, and Lfh in Eq. (381) is not well-defined because of the finiteness of the cosmic time t. Thus, the future horizon may be re-defined by
$$ L_{\mathrm{fh}} \rightarrow \tilde{L}_{\mathrm{fh}} \equiv a(t) \int_{t}^{t_{\mathrm{s}}} \frac{d t^{\prime}}{a (t^{\prime})} = a(t) \int_{0}^{\infty} \frac{d a}{H a^2}, $$
(390)
as in (386). By analogy with AdS/CFT correspondence, we suppose that Lh may be described by
$$ L_{\mathrm{h}} = L_{\mathrm{h}} (L_{\mathrm{ph}}, \tilde{L}_{\mathrm{fh}}, t_{\mathrm {s}}), $$
(391)
as long as Lph, \(\tilde{L}_{\mathrm{fh}}\) and ts are finite, because there exist a lot of possible choices for the IR cut-off (Granda and Oliveros 2008, 2009a, 2009b). We note that holographic dark energy from the Ricci scalar curvature, the so-called Ricci dark energy has been explored in Gao et al. (2009), Feng (2008), Zhang (2009), Wang et al. (2011a), Chattopadhyay (2011). Moreover, there exist a number of studies on holographic dark energy in theoretical aspects as well as observational constraints (Huang and Gong 2004; Horvat 2004; Huang and Li 2004, 2005; Wang et al. 2005a, 2005b, 2006; Myung 2005a, 2005b; Enqvist and Sloth 2004; Hsu and Zee 2005; Ito 2005; Gonzalez-Diaz 2004b; Gong et al. 2005; Shen et al. 2005, 2012; Medved 2006; Zhang 2005; Gong and Zhang 2005; Padmanabhan 2005; Guberina et al. 2005b; McInnes 2005; Pavon and Zimdahl 2005; Kim et al. 2006; Zhang and Wu 2005, 2007; Chang et al. 2006; Jassal et al. 2005; Gong 2004; Setare 2006a, 2006b, 2007a, 2007b; Hu and Ling 2006; Li et al. 2006; Setare et al. 2007; Jamil et al. 2009, 2010b; Sadjadi and Jamil 2011; Ling and Pan 2012; Huang and Wu 2012; Xu 2012; Johansson et al. 2012; Leigh et al. 2012; Kiritsis and Niarchos 2012). A concrete example is given by (Elizalde et al. 2005)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ392_HTML.gif
(392)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ393_HTML.gif
(393)
where hc>0, and \(B(\bar{p}, \bar{q})\) is a beta-function defined by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ394_HTML.gif
(394)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ395_HTML.gif
(395)
It follows from Eq. (392) that the solution is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ396_HTML.gif
(396)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ397_HTML.gif
(397)
Furthermore, since we have
$$ L_{\mathrm{fh}} + \tilde{L}_{\mathrm{fh}} = a(t) \int_0^{t_{\mathrm{s}}} \frac{dt}{a} = t_{\mathrm{s}} \biggl( \frac{t}{t_{\mathrm{s}} - t} \biggr)^{h_{\mathrm{c}}} B(\bar{p}, \bar{q}), $$
(398)
by substituting Eq. (398) into Eq. (392), we acquire
$$ \frac{C_{\mathrm{h}}}{L_{\mathrm{h}}} = h_{\mathrm{c}} \biggl( \frac{1}{t} + \frac{1}{t_{\mathrm{s}} - t} \biggr) = H, $$
(399)
where the second equality follows from Eq. (397). Clearly, Eq. (399) is equivalent to Eq. (379).

We note that for the solution (397), from Eq. (48) we find weff=−1+2(ts−2t)/(3hcts). When t→0, weff→−1+2/(3hc)>−1, i.e., the universe is in the non-phantom phase. At t=ts/2, weff=−1. After that, in the limit of tts, weff→−1−2/(3hc)<−1, i.e., the universe is in the phantom phase. Consequently, it can occur the crossing of the phantom divide.

We also investigate the case that there exists matter with its EoS being wmPm/ρm. In what follows, with wm we define hc as hc≡(2/3)/(1+wm). We assume the existence of an interaction between holographic matters (Amendola 2000; Zimdahl et al. 2001; Chimento and Richarte 2012). The equation for matter corresponding to the continuity equation is given by
$$ \dot{\rho}_{\mathrm{m}} + 3H ( \rho_{\mathrm{m}} + P_{\mathrm{m}} ) = H \frac{4 \rho_{\mathrm{c}}}{h_{\mathrm{c}}} \frac{ ( 1 + \mathcal{X} )^3}{3\mathcal{X}^2}, $$
(400)
where ρc is a constant. We also suppose
$$ \frac{L_{\mathrm{h}}}{C_{\mathrm{h}}} = \biggl( 1 - \frac{\kappa^2 \rho_{\mathrm{c}}}{3 h_{\mathrm{c}}^2} \biggr)^{-1/2} \frac{\mathcal{X}}{h_{\mathrm{c}} ( 1+\mathcal{X} )^2}. $$
(401)
In this case, the Friedmann equation (42) becomes 3H2/κ2=ρh+ρm. Thus, we obtain the solution (397) as well as ρm, given by
$$ \rho_{\mathrm{m}} = \rho_{\mathrm{c}} \biggl( \frac{1}{t} + \frac{1}{t_{\mathrm{s}} - t} \biggr)^2. $$
(402)
By using the Friedmann equation 3H2/κ2=ρh+ρm, Eqs. (397) and (402), we find that the ratio of the energy density of holographic dark energy in Eq. (378) to that of matter becomes constant and it is given by
$$ \frac{\rho_{\mathrm{h}}}{\rho_{\mathrm{m}}} = \frac{3 h_{\mathrm{c}}}{\kappa^2 \rho_{\mathrm{c}}} \biggl( 1 - \frac{\kappa^2 \rho_{\mathrm{c}}}{3 h_{\mathrm{c}}} \biggr). $$
(403)
This consequence may be a resolution of coincidence problem between the current energy density of dark energy and that of dark matter.

In Pavon and Zimdahl (2005), a naive model of such an interaction scenario between dark energy and dark matter in (400) realizing a similar result with a constant weff has been proposed, although in the present case of H in Eq. (396) we have a dynamical weff. In addition, it has been examined in Wang et al. (2005b) that in more general case, the ratio of the energy density of dark energy to that of matter is not constant.

In addition, by extending the relation (391), we examine more general cases that Lh depends on the Hubble parameter H and the length scale which originates from the cosmological constant as Λ=12/l2, i.e., Lh is expressed as
$$ L_{\mathrm{h}} = L_{\mathrm{h}} (L_{\mathrm{ph}}, \tilde{L}_{\mathrm{fh}}, t_{\mathrm{s}}, H, l), $$
(404)
or
$$ L_{\mathrm{h}} = L_{\mathrm{h}} (L_{\mathrm{ph}}, L_{\mathrm{fh}}, t_{\mathrm{s}}, H, l). $$
(405)
The proposal of generalized holographic dark energy in the form in Eqs. (404) or (405) has been made in Nojiri and Odintsov (2006b) where instead of H, the scale factor a was used, supposing that such a cut-off may depend on the scale factor and its derivatives (i.e., also from H). Hence, this is the most general proposal for the IR cut-off which eventually covers all known proposals.
An example in such an extended class of generalized holographic dark energy is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ406_HTML.gif
(406)
where αh(>0) is a positive dimensionless parameter. Here, we find Lh>0 due to Ch>0 and αh(>0). This can be seen from the relations \(( \alpha_{\mathrm{h}}^{2} - \alpha_{\mathrm{h}} -1 )^{2} -2 ( \alpha_{\mathrm{h}}^{3} - 2\alpha_{\mathrm{h}}^{2} + \alpha _{\mathrm{h}} + 1 ) = - ( \alpha_{\mathrm{h}}^{2} - 1/2)^{2} - 2\alpha_{\mathrm{h}} -3/4 <0\) and \(\alpha_{\mathrm{h}}^{3} - 2\alpha_{\mathrm{h}}^{2} + \alpha_{\mathrm {h}} + 1 = \alpha_{\mathrm{h}} ( \alpha_{\mathrm{h}} - 1 )^{2} + 1 >0\). In this case, we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ407_HTML.gif
(407)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ408_HTML.gif
(408)
with Lc being an integration constant. The solution in (407) with (408) yields
$$ H = \frac{1 + \alpha_{\mathrm{h}} [ 1 + t/ ( \alpha_{\mathrm {h}} l ) ]^2}{t [ 1 + t/ ( \alpha_{\mathrm{h}} l ) ]}. $$
(409)

We mention the case that αh is negative. From the denominator of Eq. (407), we see that at t=−αhl a Big Rip singularity appears because a diverges. Furthermore, Lh can be negative, so that when matter is not included, H=Ch/Lh can also be negative and therefore the universe will be shrink.

When t is small, i.e., in the limit of t→0, from Eq. (409) we find
$$ H \rightarrow\frac{\alpha_{\mathrm{h}} + 1}{t}. $$
(410)
This means that in this limit, the energy of the universe is dominated by that of a fluid whose Eos is given by wm=−(αh−1)/[3(αh+1)]. While, when t is large, i.e., in the opposite limit of t→∞, from Eq. (409) we see that H becomes a constant as
$$ H \rightarrow\frac{1}{l}. $$
(411)
This implies that the universe asymptotically approaches to de Sitter space. We explore the following model:
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ412_HTML.gif
(412)
Also in this case, the solution (407) with (408) is again satisfied.

7.3 The Hubble entropy in the holographic principle

It follows from Eqs. (42) and (378) that the Friedmann equation is described by \(3H^{2}/\kappa^{2} = 3 C_{\mathrm{h}}^{2}/( \kappa^{2} L_{\mathrm{h}}^{2} ) + \rho_{\mathrm{m}}\). We define the energy of matter Em, the Casimir energy EC and the Hubble entropy SH as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ413_HTML.gif
(413)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ414_HTML.gif
(414)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ415_HTML.gif
(415)
By substituting Eqs. (413)–(415) into the Friedmann equation shown above, we have (Verlinde 2000) the Cardy-Verlinde holographic formula of the Friedmann equation (Verlinde 2000)
$$ S_{\mathrm{H}}^2 = \bigl( \kappa^2 L_{\mathrm{h}}^2 \bigr)^2 E_{\mathrm{C}} ( E_{\mathrm{C}} + E_{\mathrm{m}} ). $$
(416)
Suppose that ρm can be neglected, Eq. (415) is rewritten to
$$ S_{\mathrm{H}} = \frac{18 \pi C_{\mathrm{h}}^2 L_{\mathrm{h}}^2}{ \kappa^2} = \frac{18 \pi C_{\mathrm{h}}^4}{\kappa^2 H^2}, $$
(417)
where in deriving the second and third equalities, we have eliminated H and Lh by using Eq. (380), respectively. It is seen from Eq. (417) that SH depends on time, not a constant as in case of de Sitter universe in Ke and Li (2005), Setare (2004). In the model in Eq. (392) with the solution (396), from Eq. (417) we acquire
$$ S_{\mathrm{H}} = \frac{18 \pi C_{\mathrm{h}}^4 t^2 ( t_{\mathrm {s}} - t )^2}{\kappa^2 h_{\mathrm{c}}^2 t_{\mathrm{s}}^2}. $$
(418)
Therefore, in this model SH=0 at t=0 and t=ts, and SH becomes maximum at t=ts/2.
On the other hand, for a generalized model in Eq. (401) with the interaction represented in Eq. (400), we have
$$ S_{\mathrm{H}} = \biggl( 1 - \frac{\kappa^2 \rho_{\mathrm{c}}}{3 h_{\mathrm{c}}} \biggr)^{-3} \frac{18 \pi C_{\mathrm{h}}^4 t^2 ( t_{\mathrm{s}} - t )^2}{\kappa^2 h_{\mathrm{c}}^2 t_{\mathrm{s}}^2}. $$
(419)
In comparison with Eq. (418), the form in Eq. (419) is multiplied by the first constant factor on the right-hand side. In case of Eq. (406) or the Friedmann equation \(3H^{2}/\kappa^{2} = 3 C_{\mathrm{h}}^{2}/( \kappa^{2} L_{\mathrm{h}}^{2} )+ \rho_{\mathrm{m}}\), we have
$$ S_{\mathrm{H}} = \frac{18 \pi C_{\mathrm{h}}^4 t^2 [ 1 + t/ ( \alpha_{\mathrm {c}} l ) ]^2}{\kappa^2 \{1 + \alpha_{\mathrm{h}} [ 1 + t/ ( \alpha_{\mathrm{h}} l ) ]^2 \}^2}. $$
(420)
Therefore, in this model SH=0 at t=0, whereas, in the limit of t→∞, SH approaches to a constant as follows.
$$ S_{\mathrm{H}} \rightarrow \frac{18 \pi C_{\mathrm{h}}^4 l^2}{\kappa^2}. $$
(421)

We remark that SH in Eq. (420) is positive, even though αc<0. Moreover, SH in Eq. (417) is always positive. However, for the case that SH is given by Eq. (419), if κ2ρc/(3hc)>1, SH can be negative. This implies that entropy of the universe should be negative, provided that SH corresponds to the upper bound on the entropy of the universe. In Brevik et al. (2004), negative entropy in the phantom phase has been observed. While, if the phantom phase is transient in the late time, the entropy of the universe may remain positive (Nojiri and Odintsov 2005a).

To connect the holographic dark energy scenario with the reconstruction of the corresponding scalar field theory in Sect. 4, we explore another model.
$$ \frac{C_{\mathrm{h}}}{L_{\mathrm{h}}} = \frac{\bar{\alpha}}{3} \varUpsilon^3 - \bar{\beta} \varUpsilon + \bar {\gamma}, $$
(422)
where \(\bar{\alpha} (>0)\), \(\bar{\beta} (>0)\) and \(\bar{\gamma} (>0)\) are positive constants and satisfy
$$ \bar{\gamma} > \frac{2 \bar{\beta}}{3} \sqrt{\frac{\bar{\beta }}{\bar {\alpha}}}. $$
(423)
Here, ϒ is defined by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ424_HTML.gif
(424)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ425_HTML.gif
(425)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ426_HTML.gif
(426)
Equation (424) informs us that H becomes zero only once as a function of ϒ. We also have
$$ \bar{\gamma} = -\frac{\bar{\alpha}}{3} \varUpsilon_{\mathrm {c}}^3 + \bar{\beta} \varUpsilon_{\mathrm{c}}. $$
(427)
Here, ϒc(<0) is a negative constant and it is determined that when ϒ=ϒc, H(ϒ=ϒc)=0.
We reconstruct a corresponding scalar field theory in the holographic dark energy scenario. We take a concrete form of I(ϕ) in Eq. (226) as
$$ I(\phi) = \frac{\bar{\alpha}}{3} (\varUpsilon_{\mathrm{c}} + \phi )^3 - \bar{\beta} (\varUpsilon_{\mathrm{c}} + \phi ) + \bar{\gamma }, $$
(428)
where \(\bar{\gamma}\) is given by Eq. (427). In this model, ω(ϕ) and V(ϕ) in the action in Eq. (211) is expressed as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ429_HTML.gif
(429)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ430_HTML.gif
(430)
It follows from Eq. (226) that we have the solution ϕ=t, H=I(t), and hence a is given by
$$ a = a_{\mathrm{c}} \exp \biggl[ \frac{\bar{\alpha}}{12} (\varUpsilon_{\mathrm{c}} + t )^4 - \frac{\bar{\beta}}{2} (\varUpsilon_{\mathrm{c}} + t )^2 + \bar{\gamma} (\varUpsilon_{\mathrm{c}} + t ) \biggr]. $$
(431)
The scale factor a has a minimum because H=0 at t=0. Therefore, for t<0 the universe will shrink, whereas for t>0 it will expand. From \(\dot{H} = \bar{\alpha} (\varUpsilon_{\mathrm{c}} + t)^{2} - \bar{\beta}\), we see that \(\dot{H} =0\) at
$$ t = t_{\pm} \equiv-\varUpsilon_{\mathrm{c}} \pm \sqrt{ \frac{\bar{\beta}}{\bar{\alpha}}} >0. $$
(432)
By using Eqs. (48) and H=I(t) with Eq. (226) and ϕ=t, weff is written as
$$ w_{\mathrm{eff}} = -1 - \frac{2 [ \bar{\alpha} (\varUpsilon_{\mathrm{c}} + t )^2 - \bar{\beta} ]}{\bar{\alpha} (\varUpsilon_{\mathrm {c}} + \phi )^3 - 3\bar{\beta} (\varUpsilon_{\mathrm{c}} + \phi ) + 3\bar{\gamma}}. $$
(433)
Accordingly, it is seen from Eq. (48) that for t<t<t+, weff>−1, i.e., the universe is in the non-phantom phase, whereas for 0<t<t or t+<t, weff<−1, i.e., the universe is in the phantom phase, and that at t=t±, weff=−1. In summary, at t=t+(t), the crossing of the phantom divide can occur from the non-phantom (phantom) phase to the phantom (non-phantom) one.
In the model in Eq. (422), the Hubble entropy in Eq. (415) is given by
$$ S_{\mathrm{H}} = \frac{18 \pi C_{\mathrm{h}}^4}{ \kappa^2 [ (\bar{\alpha}/3 ) (t+\varUpsilon_{\mathrm {c}} )^3 - \bar{\beta} (t+\varUpsilon_{\mathrm{c}} ) + \bar {\gamma} ]^2}. $$
(434)
Thus, SH is always positive. It follows from Eqs. (427) and (434) that at t=0, SH diverges. From Eq. (417), we see that SHH−2. Thus, for 0<t<t (the phantom phase with \(\dot{H} > 0\)), SH decreases. For t<t<t+ (the non-phantom phase with \(\dot{H} < 0\)), SH increases. For t+<t (the phantom phase with \(\dot{H} > 0\)), SH again becomes small. Eventually, in the limit of t→∞, SH→0. As a result, in the non-phantom phase SH grows, whereas in the phantom phase, SH decreases.

8 Accelerating cosmology in F(R) gravity

In this section, we study an accelerating cosmology in F(R) gravity. First, we consider relations between a scalar field theory in the Einstein frame and an F(R) theory in the Jordan frame. Furthermore, we show how to obtain the ΛCDM, phantom-like or quintessence-like cosmologies in F(R) gravity by following Nojiri and Odintsov (2006c, 2007b), Bamba et al. (2009, 2010b, 2011a), Bamba (2009, 2010), Bamba and Geng (2009b). We mention that the reconstruction of F(R,Tst) gravity has also been investigated, where Tst is the trace of the stress-energy tensor, e.g., in Momeni et al. (2012).

8.1 F(R) gravity and a corresponding scalar field theory

The action describing F(R) gravity with matter is given by
$$ S = \int d^4 x \sqrt{-g} \frac{F(R)}{2\kappa^2} + \int d^4 x {\mathcal{L}}_{\mathrm{M}} ( g_{\mu\nu}, {\varPsi}_{\mathrm{M}} ), $$
(435)
where \({\mathcal{L}}_{\mathrm{M}}\) is the Lagrangian of matter. By making a conformal transformation, we move to the Einstein frame:
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ436_HTML.gif
(436)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ437_HTML.gif
(437)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ438_HTML.gif
(438)
where a tilde denotes quantities in the Einstein frame. We define a new scalar field ϕ as
$$ \phi\equiv\sqrt{\frac{3}{2}} \frac{1}{\kappa} \ln F_{,R}. $$
(439)
Moreover, R is represented by using \(\tilde{R}\) as
$$ R = \mathrm{e}^{1/\sqrt{3} \kappa\phi} \biggl[ \tilde{R} + \sqrt{3} \tilde{\Box} ( \kappa\phi ) - \frac{1}{2} \tilde{g}^{\mu\nu} {\partial}_{\mu} ( \kappa\phi ) {\partial}_{\nu} ( \kappa\phi ) \biggr], $$
(440)
with
$$ \tilde{\Box} ( \kappa\phi ) = \frac{1}{\sqrt{-\tilde{g}}} {\partial}_{\mu} \bigl[ \sqrt{-\tilde{g}} \tilde{g}^{\mu\nu} {\partial}_{\nu} ( \kappa\phi ) \bigr]. $$
(441)
As a result, we acquire the action in the Einstein frame (Maeda 1989; Fujii and Maeda 2003)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ442_HTML.gif
(442)
where the potential V(ϕ) is represented by
$$ V(\phi) = \frac{F_{,R}\tilde{R}-F}{2\kappa^2 (F_{,R} )^2}. $$
(443)
We note that an important cosmological application of the relations between F(R) gravity in the Jordan frame and its corresponding scalar field theory in the Einstein frame to the time variation of the fine structure constant in non-minimal Maxwell-F(R) gravity (Bamba and Odintsov 2008; Bamba et al. 2008b; Bamba and Nojiri 2008) has recently been executed in Bamba et al. (2012e) by using a novel consequence of a static domain wall solution (Toyozato et al. 2012) in F(R) gravity. Such non-minimal Maxwell theories with its coupling to a scalar field or the scalar curvature break the conformal invariance of the electromagnetic fields, so that the large-scale magnetic fields from inflation can be generated (Bamba and Yokoyama 2004a, 2004b; Bamba and Sasaki 2007; Bamba 2007a, 2007b; Bamba et al. 2008a, 2008d, 2012a). This would be considered to be a significant cosmological implication of the investigations of the present section.

8.2 Reconstruction method of F(R) gravity

To begin with, we explain the reconstruction method of F(R) gravity (Nojiri and Odintsov 2006c, 2007b). We introduce proper functions P(ϕ) and Q(ϕ) of a scalar field ϕ and rewrite the action in Eq. (435) to the following form
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ444_HTML.gif
(444)
Since the scalar field ϕ does not have the kinetic term, it may be regarded as an auxiliary scalar field. It follows from Eq. (444) that the equation of motion of ϕ reads
$$ 0=\frac{d P(\phi)}{d \phi} R + \frac{d Q(\phi)}{d \phi}. $$
(445)
Hence, by solving Eq. (445) in terms of R in principle we find the expression ϕ=ϕ(R). By combining this expression and the action in Eq. (444), we obtain the representation of F(R) as
$$ F(R) = P\bigl(\phi(R)\bigr) R + Q\bigl(\phi(R)\bigr). $$
(446)
Furthermore, the variation of the action in Eq. (444) leads to the gravitational field equation
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ447_HTML.gif
(447)
Here, ∇μ is the covariant derivative operator associated with gμν and □≡gμνμν is the covariant d’Alembertian for a scalar field. Moreover, \(T^{(\mathrm{M})}_{\mu\nu}\) is the energy-momentum tensor of matter. We take the flat FLRW space-time ds2=−dt2+a2(t)∑i=1,2,3(dxi)2. In this background, the (μ,ν)=(0,0) and (μ,ν)=(i,j) (i,j=1,…,3) components of Eq. (447) become
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ448_HTML.gif
(448)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ449_HTML.gif
(449)
Here, we have expressed the sum of the energy density and pressure of matters with a constant EoS wMi as ρM and PM, respectively, where the subscription “i” denotes a component of matters. By using Eqs. (448) and (449), we eliminate Q(ϕ), and eventually we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ450_HTML.gif
(450)
If we appropriately redefine the scalar field ϕ, it can be taken as ϕ=t. In addition, we describe the form of a(t) by
$$ a(t) = \bar{a} \exp \bigl( \tilde{g}(t) \bigr), $$
(451)
with \(\bar{a}\) being a constant and \(\tilde{g}(t)\) being a proper function of t. In this case, the Hubble parameter is given by \(H= d \tilde{g}(\phi)/(d \phi)\). By using this expression, Eq. (450) can be rewritten to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ452_HTML.gif
(452)
Here, \(\bar{\rho}_{\mathrm{M} i}\) is a constant. In addition, by solving Eq. (448) in terms of Q(ϕ), we acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ453_HTML.gif
(453)
It is significant to emphasize that by redefining the auxiliary scalar field ϕ as ϕ=Φ(φ) with a proper function Φ and defining \(\tilde{P}(\varphi) \equiv P(\varPhi(\varphi))\) and \(\tilde{Q}(\varphi) \equiv Q(\varPhi(\varphi))\), we obtain the new form of the action
$$ S = \int d^4 x \sqrt{-g} \frac{\tilde{F}(R)}{2\kappa^2} + \int d^4 x {\mathcal{L}}_{\mathrm{M}} ( g_{\mu\nu}, {\varPsi}_{\mathrm{M}} ), $$
(454)
where
$$ \tilde{F}(R) \equiv\tilde{P}(\varphi) R + \tilde{Q}(\varphi). $$
(455)
Since \(\tilde{F}(R) = F(R)\), this action in Eq. (455) is equivalent to that in Eq. (444). Furthermore, φ is the inverse function of Φ, and therefore by using ϕ=ϕ(R) φ can be solved with respect to R as φ=φ(R)=Φ−1(ϕ(R)). Accordingly, there exist the choices in ϕ as a gauge symmetry, and hence ϕ can be identified with time t as ϕ=t. This can be considered as a gauge condition which corresponds to the reparameterization of ϕ=ϕ(φ) (Bamba et al. 2009). As a result, if we obtain the solution t=t(R), by solving Eqs. (452) and (453) and substituting these solutions into Eq. (446), the explicit expression of F(R) can be acquired. It should be noted that in a naive model of F(R) gravity the crossing of the phantom divide cannot be realized because F(R) has to be a double-valued function in order that the crossing of the phantom divide can occur. In fact, however, if the action of F(R) gravity is extended to the form of P(ϕ)R+Q(ϕ), the crossing of the phantom divide can happen. We show an explicit example to realize the crossing of the phantom divide in Sect. 8.3.3.

8.3 Reconstructed F(R) forms and its cosmologies

8.3.1 The ΛCDM cosmology

We demonstrate the reconstruction process of F(R) gravity in which the ΛCDM cosmology is realized (Nojiri and Odintsov 2006c) (for other method of the reconstruction of F(R) gravity reproducing the ΛCDM model, see de la Cruz-Dombriz and Dobado (2006)). In the flat FLRW background, if there exist a matter with its EoS wM and the cosmological constant Λ, the Friedmann equation (42) is written as
$$ H^2 = \frac{\kappa^2}{3} \rho_{\mathrm{M} \mathrm{c}} a^{-3 ( 1 + w_{\mathrm{M}} )} + \frac{1}{l_{\varLambda}^2}, $$
(456)
where ρMc is a constant and lΛ is a length scale related to the cosmological constant Λ. From Eq. (456), we find that the scale factor can be expressed by Eq. (451) with \(\tilde{g}(t)\) being
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ457_HTML.gif
(457)
where tΛ corresponds to an integration constant. We suppose that geometrical dark energy originating from F(R) gravity is dominant over matter and therefore matter contribution can be neglected. In this case, by using Eq. (457), Eq. (452) is expressed as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ458_HTML.gif
(458)
This expression can further been rewritten to a Gauss’s hypergeometric differential equation by replacing the variable ϕ with z as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ459_HTML.gif
(459)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ460_HTML.gif
(460)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ461_HTML.gif
(461)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ462_HTML.gif
(462)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ463_HTML.gif
(463)
By using the Gauss’s hypergeometric function, a solution of Eq. (459) is described as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ464_HTML.gif
(464)
where Pc1 and Pc2 are constants and FG is the Gauss’s hypergeometric function, defined by
$$ F_{\mathrm{G}} (\tilde{\alpha}, \tilde{\beta}, \tilde{\gamma}; z) \equiv \frac{\varGamma(\tilde{\gamma})}{\varGamma(\tilde{\alpha}) \varGamma (\tilde{\beta})} \sum_{n=0}^{\infty} \frac{\varGamma(\tilde{\alpha}+n) \varGamma (\tilde {\beta}+n)}{\varGamma(\tilde{\gamma}+n)} \frac{z^n}{n!}, $$
(465)
with Γ being the Γ function. For simplicity in order to obtain the form of F(R), we set Pc2=0, namely, we take only the first linearly independent solution in Eq. (464). It follows from Eq. (453) that we obtain the form of Q as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ466_HTML.gif
(466)
In the limit of t=ϕ→∞, from Eq. (460) we see z→0. Thus, in this limit, by substituting Eqs. (464) and (466) into Eq. (446) we find
$$ F(R) = P\bigl(\phi(R)\bigr) R + Q\bigl(\phi(R)\bigr) \to P_{\mathrm{c}1} \biggl( R - \frac{6}{l_{\varLambda}^2} \biggr). $$
(467)
As a consequence, by comparing Eq. (467) with Eq. (1), we see that if we take
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ468_HTML.gif
(468)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ469_HTML.gif
(469)
the general relativity with the cosmological constant is realized. Moreover, by plugging Eq. (467) with Eqs. (468) and (469) into the action in Eq. (442), we acquire the action describing the ΛCDM cosmology which corresponds to the one in Eq. (1).

8.3.2 Quintessence cosmology

Next, we reconstruct F(R) gravity in which quintessence-like cosmology is produced (Nojiri and Odintsov 2006c). We investigate the case that \(\tilde{g}(t)\) in Eq. (451) is given by
$$ \tilde{g}(\phi) = \bar{h}(\phi) \ln \biggl( \frac{\phi}{\phi_{\mathrm{c}}} \biggr), $$
(470)
where ϕc is a constant and \(\bar{h}(\phi)\) is a function varying slowly in ϕ. Therefore, an adiabatic approximation is applied to \(\bar{h}(\phi)\), so that the derivative of \(\bar{h}(\phi)\) can be neglected, i.e., \(d\bar{h}(\phi)/d\phi\sim d^{2}\bar{h}(\phi)/d\phi^{2} \sim0\). Through the same procedure as in Sect. 8.3.1, we derive the solutions of P(ϕ) and Q(ϕ). The equation of P(ϕ) in (452) is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ471_HTML.gif
(471)
We acquire a solution of Eq. (471) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ472_HTML.gif
(472)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ473_HTML.gif
(473)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ474_HTML.gif
(474)
where P± are arbitrary constants, and wMr=1/3 for radiation and wMm=0 for non-relativistic matter. Furthermore, by using Eq. (453), Q is written by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ475_HTML.gif
(475)
On the other hand, from Eq. (470) we find \(H \sim\bar{h}(t)/t\) and hence \(R=6(\dot{H} + 2H^{2} ) \sim6(-\bar{h}(t) + 2\bar{h}^{2}(t) )/t^{2}\). Here, we provided that in the limit of ϕ→0, \(\bar{h}(\phi) \to \bar{h}_{\mathrm{i}}\), and in the opposite limit of ϕ→∞, \(\bar{h}(\phi) \to\bar{h}_{\mathrm{f}}\). As a form of \(\bar{h}(\phi)\), we take
$$ \bar{h}(\phi) = \frac{\bar{h}_{\mathrm{i}} + \bar{h}_{\mathrm{f}} \vartheta \phi^2}{1+\vartheta\phi^2}, $$
(476)
where ϑ is a small constant enough for \(\bar{h}(\phi)\) to be a function varying slowly in ϕ. The substitution of Eqs. (474) and (475) into Eq. (446) yields
$$ F(R) = P\bigl(\varPhi_{\mathrm{c}} (R)\bigr) R + Q\bigl(\varPhi_{\mathrm{c}} (R)\bigr), $$
(477)
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ478_HTML.gif
(478)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ479_HTML.gif
(479)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ480_HTML.gif
(480)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ481_HTML.gif
(481)
At the dark energy dominated stage, wDEweff because the energy density of matter can be negligible compared with that of dark energy. In the present model in Eq. (476), from Eq. (48) we have \(w_{\mathrm{DE}} = -1+2/( 3\bar{h}_{\mathrm{f}} )\). Thus, for \(0<\bar{h}_{\mathrm{f}}<1\), −1<wDE<−1/3. As a result, this means that in the reconstructed F(R) gravity model, quintessence-like cosmology can be realized. In addition, for the reconstructed F(R) gravity model in Eq. (477), in the late limit of ϕ→∞, which can be regarded as the limit of the present time, the asymptotic behavior is given by a power-law description as \(F(R) \sim R^{\bar{s}}\) with \(\bar{s} \equiv-( \bar {h}_{\mathrm {f}} -5 +\sqrt{\bar{h}_{\mathrm{f}}^{2} +6\bar{h}_{\mathrm{f}} +1} )/4\). By using this expression, we find that in the action of a scalar field theory in Eq. (442), the potential V(ϕ) in Eq. (443) is written as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ482_HTML.gif
(482)
Here, in deriving the second equality, we have used \(R= [ (1/\bar{s}) \mathrm{e}^{ (2/3) \kappa\phi}]^{1(\bar{s}-1)}\), which follows from Eq. (439). This can be interpreted as quintessence potential.

8.3.3 Phantom cosmology

Furthermore, we reconstruct F(R) gravity in which the crossing of the phantom divide is realized (Bamba et al. 2010b, 2011a; Bamba 2009, 2010; Bamba and Geng 2009b) and eventually phantom-like cosmology is produced.

For matter to be neglected because of the dark energy domination, as an example, we examine the case that \(\tilde{g}(t)\) in Eq. (451) is described by
$$ \tilde{g}(\phi) = - 10 \ln \biggl[ \biggl(\frac{\phi}{t_0} \biggr)^{-\bar {\gamma}} - C_{\mathrm{p}} \biggl(\frac{\phi}{t_0} \biggr)^{\bar{\gamma}+1} \biggr], $$
(483)
where \(\bar{\gamma} (> 0)\) and Cp(>0) are positive constants and t0 is the present time. We note that since there occurs a Big Rip singularity at \(\phi= t_{\mathrm{s}} \equiv t_{0} C_{\mathrm{p}}^{-1/(2\bar{\gamma }+1)}\), we investigate the period 0<t<ts in order for \(\tilde{g}(\phi)\) to be a real number. From Eq. (483), the expression of the Hubble parameter \(H(t)= d \tilde{g}(\phi)/d\phi\) is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ484_HTML.gif
(484)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ485_HTML.gif
(485)
where in deriving Eq. (485) we have used the relation \(t_{\mathrm{s}} = t_{0} C_{\mathrm{p}}^{-1/(2\bar{\gamma }+1)}\) and ϕ=t. A solution of Eq. (452) for P(ϕ) is derived as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ486_HTML.gif
(486)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ487_HTML.gif
(487)
with \(\bar{p}_{\pm}\) being arbitrary constants.
It follows from Eq. (48) that wDEweff at the dark energy dominated stage is expressed as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ488_HTML.gif
(488)
In the limit of t→0, namely, t/ts≪1, from Eqs. (484) and (488) we find \(H(t) \sim10\bar{\gamma}/t\) and \(w_{\mathrm{DE}} \sim-1 + 1/ (15\bar{\gamma}) (> -1)\), respectively. This is the non-phantom (quintessence) phase. While, in the opposite limit of tts it follows from Eqs. (484) and (488) that H(t)∼10/(tst), which leads to \(a(t) \sim\bar{a} ( t_{\mathrm{s}} - t)^{-10}\), and wDE∼−16/15(<−1), respectively. Moreover, d(wDE+1)/dt monotonously decreases in time. Hence, first the universe is in the non-phantom phase. As the time passes, when t closes to ts, the universe enters the phantom phase. Thus, the crossing of the phantom divide line of wDE=−1 can occur at \(t = t_{\mathrm{c}} \equiv t_{\mathrm{s}} [-2\bar{\gamma} + \sqrt{4\bar{\gamma}^{2} + \bar{\gamma}/(\bar{\gamma}+1)} ]^{1/( 2\bar{\gamma} + 1 )}\).
In addition, by using Eq. (486) and substituting it into Eq. (453), we acquire the forms of P(t) and Q(t) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ489_HTML.gif
(489)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ490_HTML.gif
(490)
On the other hand, from \(R=6(\dot{H} + 2H^{2} )\), the relation between R and t is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ491_HTML.gif
(491)
Accordingly, in principle, if we obtain the relation t=t(R) by solving Eq. (491) reversely, the substitution of it into Eqs. (489) and (490) and the combination of those with Eq. (446) yield the explicit form of F(R). We show analytically solvable cases below. In the limit of t→0 (t/ts≪1), from Eq. (491) we obtain \(t \sim\sqrt{60\bar{\gamma}( 20\bar{\gamma} -1)/R}\). By using this asymptotic relation, we acquire (Bamba and Geng 2009a)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ492_HTML.gif
(492)
In the opposite limit of tts, it follows from Eq. (491) that \(t \sim t_{\mathrm{s}} - 3\sqrt{140/R}\). With this asymptotic relation, for the large curvature regime as \(t_{\mathrm{s}}^{2} R \gg1\) we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ493_HTML.gif
(493)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ494_HTML.gif
(494)
For the power-law form of F(R) in Eq. (494), the potential V(ϕ) in Eq. (443) of the action for a scalar field theory in Eq. (442) is described as
$$ V(\phi) \sim\frac{5}{49\bar{F} \kappa^2} \biggl[\frac{2}{7}\mathrm{e}^{ (2/3 ) \kappa\phi} \biggr]^{-3/5}. $$
(495)
This can be interpreted as phantom potential.

Using the reconstruction program with auxiliary scalar fields as discussed in this section, or without the use of auxiliary scalar fields, following Nojiri et al. (2009), one can eventually reconstruct any dark energy cosmology studied in this review. For instance, Little Rip cosmology for modified gravity has been presented in Brevik et al. (2011), Nojiri et al. (2011a).

8.4 Dark energy cosmology in F(R) Hořava-Lifshitz gravity

As a candidate for a renormalizable gravitational theory in four dimensions, the Hořava-Lifshitz gravity has been proposed in Horava (2009a) (for a review on the Hořava-Lifshitz cosmology, see, e.g., Mukohyama 2010), although it cannot maintain the Lorentz invariance. In addition, its extension to an F(R) formalism has been executed in Chaichian et al. (2010). In this subsection, we study cosmology for dark energy in F(R) Hořava-Lifshitz gravity (Elizalde et al. 2010).

8.4.1 F(R) Hořava-Lifshitz gravity

The model action describing F(R) Hořava-Lifshitz gravity with the matter action is given by (Chaichian et al. 2010)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ496_HTML.gif
(496)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ497_HTML.gif
(497)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ498_HTML.gif
(498)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ499_HTML.gif
(499)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ500_HTML.gif
(500)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ501_HTML.gif
(501)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ502_HTML.gif
(502)
where, i,j,k,l run over 1,2,3, Nl is the lapse variable in the Arnowitt-Deser-Misner (ADM) decomposition in (3+1) space-time, Ni is the shift 3-vector (Misner et al. 1973; Arnowitt et al. 2004; Gao 2010), \({\mathcal{L}}_{\mathrm{M}}\) is the Lagrangian of matter which is a perfect fluid, Kij is the extrinsic curvature, R(3) is the spatial scalar curvature, nμ is a unit vector perpendicular to a constant time hypersurface, \(\bar{\kappa}\) is the dimensionless gravitational coupling, and λHL and μHL are constants and these break the full diffeomorphism invariance. Furthermore, Gijkl is the inverse of the generalized De Witt metric and it exists only for \(\bar{\lambda}_{\mathrm{HL}} \neq1/3\) because it follows from Eq. (501) that if \(\bar{\lambda}_{\mathrm{HL}} = 1/3\), Gijkl becomes singular. Moreover, Eij is constructed so that the detailed balance principle restricting the number of free model parameters can be met (Horava 2009a). In the Hořava-Lifshitz gravity, there exists the difference of the scaling properties between the space and time coordinates as xibxi and \(t \to b^{\bar{z}} t\), with b being a constant and \(\bar{z}\) being a dynamical critical exponent. Here, if z=3 in (3+1) space-time dimensions, the theory is renormalizable, whereas for \(\bar{z}=1\), it is the general relativity. Such scaling properties give the theory only the foliation preserving diffeomorphisms, given by \(\delta x^{i} = \bar{\zeta} (x^{i}, t)\) and \(\delta t = \bar{\xi} (t)\), where \(\bar{\zeta}\) and \(\bar{\xi}\) are functions of xi as well as t and t, respectively. We also mention that in the Hořava-Lifshitz gravity (Horava 2009a), Nl is supposed to depend only on time in order for the projectability condition to be satisfied and it is set to be unity (Nl=1) by applying the foliation preserving diffeomorphisms. For \(\bar{z} = 2\) and \(\bar{z} = 3\), the expression of \(W [g_{kl}^{(3)}]\) in Eq. (502) are presented in Horava (2009b). In the spatially flat FLRW space-time \(ds^{2}= -N_{\mathrm{l}}^{2} dt^{2} +\sum_{i=1,2,3} (dx^{i})^{2}\), we have
$$ \bar{R} = \frac{3 (1-3\lambda_{\mathrm{HL}} +6\mu_{\mathrm{HL}} )H^ 2}{N_{\mathrm{l}}^2} + \frac{6\mu_{\mathrm{HL}}}{N_{\mathrm{l}}} \frac{d}{dt} \biggl( \frac{H}{N_{\mathrm{l}}} \biggr). $$
(503)
By varying the action in Eq. (496) with respect to Nl and \(g_{ij}^{(3)}\), we acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ504_HTML.gif
(504)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ505_HTML.gif
(505)
where the prime denotes the derivative with respect to \(\bar{R}\), and ρM and PM are the energy density and pressure of a perfect fluid, respectively. In deriving Eq. (504), we have used the projectability condition for Nl to have only the time dependence, and in obtaining Eq. (505), we have taken Nl=1. For λHL=μHL=1, these resultant equations are reduced to those in ordinary F(R) gravity.
By using the continuity equation \(\dot{\rho}_{\mathrm{M}} + 3H( \rho _{\mathrm{M}} + P_{\mathrm{M}}) = 0\) in terms of a perfect fluid and executing the integration of Eq. (504), we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ506_HTML.gif
(506)
with CHL being a constant of integration, which has to be chosen to 0 so that the constraint equation (504) can be satisfied.

We note that for \(F(\bar{R}) = \bar{R}\), in the flat FLRW background the gravitational field equations are written as H2={κ2/[3(3λHL−1)]}ρM and \(\dot{H} = -\{ \kappa^{2}/[2(3\lambda_{\mathrm {HL}}-1 )]\}( \rho_{\mathrm{M}} + P_{\mathrm{M}} )\) with λHL>1/3 due to the consistency, and for λHL→1 these equations become the ordinary Einstein equations in general relativity.

8.4.2 Reconstruction of F(R) form

We further analyze Eq. (506). It follows from the continuity equation of a perfect fluid with its constant EoS wMPM/ρM that \(\rho_{\mathrm{M}} = \rho_{\mathrm{M} \, \mathrm{c}} a^{-3( 1+w_{\mathrm{M}} )} \mathrm{e}^{-3 (1+w_{\mathrm{M}}) N}\) with ρMc being a constant, where N≡ln(a/ac) with ac being a constant is the number of e-folds. We replace the cosmic time t as a variable with N, so that Eq. (503) can be rewritten to \(\bar{R} = 3(1-3\lambda_{\mathrm{HL}} +6\mu_{\mathrm{HL}}) \bar{G}(N) +3\mu_{\mathrm{HL}} (d\bar{G}(N)/dN) \), where we have defined \(\bar{G}(N) \equiv H^{2}\) and used it in order to analyze Eq. (503) easier. Since this equation can be solved as \(N =N (\bar{R})\) and H can be represented as H=H(N), Eq. (503) can be described as an equation of \(F(\bar{R})\) in terms of \(\bar{R}\). Accordingly, Eq. (503) can be rewritten to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ507_HTML.gif
(507)
First, we reconstruct the form of F(R) with realizing the ΛCDM cosmology, in which the Friedmann equation (42) can be described by \(\bar{G}(N) = H^{2} = H_{\mathrm{c}}^{2} + (\kappa^{2}/3) \rho_{\mathrm{M} \mathrm{c}} a_{\mathrm{c}}^{-3} \mathrm{e}^{-3N}\) with Hc being a constant, where we have used wM=0 because a perfect fluid is considered to a non-relativistic matter. For general relativity, \(H_{\mathrm{c}}^{2} = \varLambda/3\) as seen in Eq. (4). Moreover, we have \(\mathrm{e}^{-3N} = [\bar{R}-3(1-3\lambda_{\mathrm{HL}} +6\mu_{\mathrm{HL}}) H_{\mathrm{c}}^{2} ]/ \{ \kappa^{2} \rho_{\mathrm{M} \mathrm{c}} a_{\mathrm{c}}^{-3} [1+3(\mu_{\mathrm{HL}}-\lambda _{\mathrm {HL}}) ]\}\). With these equations and wM=0, Eq. (507) is represented as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ508_HTML.gif
(508)
The homogeneous part of Eq. (508) is rewritten to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ509_HTML.gif
(509)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ510_HTML.gif
(510)
where
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ511_HTML.gif
(511)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ512_HTML.gif
(512)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ513_HTML.gif
(513)
We describe the complete solution of Eq. (510) with the Gauss’s hypergeometric function FG as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ514_HTML.gif
(514)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ515_HTML.gif
(515)
with Fc1 and Fc2 being constants. Thus, a class of the reconstructed F(R) theories in Eq. (515) can represent the ΛCDM cosmology. We remark that for μHL=λHL−1/3, \(\bar{R}\) becomes a constant, so that the solution in Eq. (515) can be expressed as \(F(\bar{R}) = F(\bar{R})/(3\lambda_{\mathrm{HL}} -1) - 2\varLambda\), where \(\varLambda=(3/2)(3\lambda_{\mathrm{HL}} -1)H_{\mathrm{c}}^{2}\).
Next, we reconstruct an F(R) form describing the phantom cosmology. Since we examine the dark energy dominated stage, for simplicity, non-relativistic matter contributions are neglected. At the dark energy dominated stage, by using the continuity equation in terms of dark energy, the Hubble parameter can be represented as H=Hph/(tst) with Hph≡−1/[3(1+wDE)], where at t=ts, a Big Rip singularity appears. From this expression, we have \(\bar{G}(N) = H^{2} (N) = H_{\mathrm{ph}} \mathrm{e}^{2N/H_{\mathrm{ph}}}\). In this case, the relation of N to \(\bar{R}\) is described by \(\mathrm{e}^{2N/H_{\mathrm{ph}}} = \bar{R}/ [ H_{\mathrm{ph}} ( A_{\mathrm{ph}} H_{\mathrm{ph}} +6\mu _{\mathrm{HL}} )]\) with Aph being a constant. By combining these relations and Eq. (507), we acquire the Euler equation
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ516_HTML.gif
(516)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ517_HTML.gif
(517)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ518_HTML.gif
(518)
A solution of Eq. (516) is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ519_HTML.gif
(519)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ520_HTML.gif
(520)
where Fph+ and Fph− are constants. Hence, a reconstructed form of F(R) in Eq. (519) can describe the phantom cosmology. As a result, it is considered that by using the procedure explained above, in principle, an F(R) form with representing any cosmology could be reconstructed.

8.5 \(F(R, \mathcal{T})\) gravity

In Harko et al. (2011), the formulations of a novel modified gravitational theory, the so-called \(F(R, \mathcal{T})\) gravity with \(\mathcal{T}\) being the trace of the stress-energy tensor, which can explain the late-time cosmic acceleration, have been investigated. In this subsection, we review this latest theory.

8.5.1 Formulations

The action of \(F(R, \mathcal{T})\) gravity is given by (Harko et al. 2011)
$$ S = \int d^4 x \sqrt{-g} \frac{F(R, \mathcal{T})}{16\pi} + \int d^4 x \sqrt{-g} L_{\mathrm{M}}, $$
(521)
where \(\mathcal{T} = g^{\mu\nu} \mathcal{T}_{\mu\nu}\) is the trace of the stress-energy tensor of matter, defined as (Landau and Lifshitz 1998) \(\mathcal{T}_{\mu\nu} \equiv- (2/\sqrt{-g}) \delta( \sqrt{-g} L_{\mathrm{M}} )/\delta g\varTheta _{\mu\nu}\), LM is the Lagrangian density of matter, and \(F(R, \mathcal{T})\) is an arbitrary function of R and T. Here and in this subsection, we use the unit of G=c=1. As past related studies, a theory whose Lagrangian density is described by an arbitrary function of R and the Lagrangian density of matter as F(R,LM) has been explored in Harko and Lobo (2010). Moreover, in Poplawski (2006) a theory in which the cosmological constant is written by a function of the trace of the stress-energy tensor as \(\varLambda(\mathcal{T})\) has been investigated.
From the action in Eq. (521), the gravitational field equation is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ522_HTML.gif
(522)
with \(\varTheta_{\mu\nu} \equiv g^{\alpha\beta} ( \delta\mathcal{T}_{\alpha\beta}/\delta g^{\mu\nu})\), which follows from the relation \(\delta( g^{\alpha\beta} \mathcal{T}_{\alpha\beta}/\delta g^{\mu \nu}) = \mathcal{T}_{\mu\nu} + \varTheta_{\mu\nu}\), and \(F_{R}(R, \mathcal{T}) \equiv\partial F(R, \mathcal{T})/ \partial R\), \(F_{\mathcal{T}} (R, \mathcal{T}) \equiv\partial F(R, \mathcal{T})/ \partial\mathcal{T}\). The contraction of Eq. (522) yields \(F_{R}(R, \mathcal{T}) R + 3\Box F_{R}(R, \mathcal{T}) -2F(R, \mathcal{T}) =(8\pi- F_{\mathcal{T}}(R, \mathcal{T})) \mathcal{T} - F_{\mathcal{T}}(R, \mathcal{T}) \varTheta\) with ΘgμνΘμν. Combining Eq. (522) and the contracted equation and eliminating the \(\Box F_{R}(R, \mathcal{T})\) term from these equations, we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ523_HTML.gif
(523)
On the other hand, the covariant divergence of Eq. (521) as well as the energy-momentum conservation law \(\nabla^{\mu}[ F_{R}(R, \mathcal{T}) - (1/2) F(R, \mathcal{T}) g_{\mu\nu} +(g_{\mu\nu} \Box- \nabla_{\mu}\nabla_{\nu})\times F_{R}(R, \mathcal{T})] \equiv0\), which corresponds to the divergence of the left-hand side of Eq. (521), we acquire the divergence of \(\mathcal{T}_{\mu\nu}\) as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ524_HTML.gif
(524)
In addition, from \(\mathcal{T}_{\mu\nu} = g_{\mu\nu} L_{\mathrm{M}} -2(\partial L_{\mathrm{M}}/ \partial g^{\mu\nu})\) we have
$$ \frac{\delta\mathcal{T}_{\alpha\beta}}{\delta g^{\mu\nu}} = \biggl( \frac{\delta g_{\alpha\beta}}{\delta g^{\mu\nu}} +\frac{1}{2} g_{\alpha\beta} g_{\mu\nu} \biggr) L_{\mathrm{M}} -\frac{1}{2} g_{\alpha\beta} \mathcal{T}_{\mu\nu} -2 \frac{\partial^2 L_{\mathrm{M}}}{\partial g^{\mu \nu} \partial g^{\alpha\beta}}. $$
(525)
Using the relation \(\delta g_{\alpha\beta}/\delta g^{\mu\nu} = -g_{\alpha\rho}g_{\beta\sigma} \delta_{\mu\nu}^{\rho\sigma}\) with \(\delta_{\mu\nu}^{\rho\sigma} = \delta g^{\rho\sigma}/\delta g^{\mu\nu}\), which follows from \(g_{\alpha\rho} g^{\rho\beta} = \delta_{\alpha}^{\beta}\), we obtain
$$ \varTheta_{\mu\nu} = -2 \mathcal{T}_{\mu\nu} + g_{\mu\nu} L_{\mathrm {M}} -2g^{\alpha\beta} \frac{\partial^2 L_{\mathrm{M}}}{\partial g^{\mu \nu} \partial g^{\alpha\beta}}. $$
(526)
Provided that matter is regarded as a perfect fluid, \(\mathcal{T}_{\mu \nu}\) is expressed as \(\mathcal{T}_{\mu\nu} =( \rho_{\mathrm{M}} + P_{\mathrm{M}} ) u_{\mu}u_{\nu}-P_{\mathrm{M}} g_{\mu\nu}\), where uμ being the four velocity satisfying gμνuμuν=−1, and ρM and PM are the energy density and pressure of the perfect fluid, respectively, we acquire \(\varTheta_{\mu\nu} = -2 \mathcal{T}_{\mu\nu} - P_{\mathrm{M}} g_{\mu\nu}\).

8.5.2 Example

As an example, we examine the case that \(F(R, \mathcal{T}) = R + 2F_{1} (\mathcal{T})\) with \(F_{1} (\mathcal{T})\) being an arbitrary function of \(\mathcal{T}\) and matter is a perfect fluid. In this case, Eq. (522) becomes
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ527_HTML.gif
(527)
where Gμν=Rμν−(1/2)Rgμν is the Einstein tensor. In the flat FLRW background, for the matter to be a dust, i.e., PM=0, and \(F_{1} (\mathcal{T}) = \lambda\mathcal{T}\) with λ being a constant, the gravitational field equations are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ528_HTML.gif
(528)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ529_HTML.gif
(529)
From these equations, we have
$$ \dot{H} + \frac{3 (8\pi+ 2\lambda )}{2 (8\pi+ 3\lambda )} H^2 =0. $$
(530)
The solution of this equation is given by \(H= \tilde{p}/t\) with \(\tilde{p} \equiv[2(8\pi+ 3\lambda )]/[3(8\pi+ 2\lambda)]\). Thus, we obtain \(a = t^{\tilde{p}}\) with \(\tilde{p} > 1\), and consequently the accelerated expansion of the universe can be realized.

9 f(T) gravity

In this section, we explore f(T) gravity.2 It is known that as a candidate of an alternative gravitational theory to general relativity, there exists “teleparallelism” in which the Weitzenböck connection is used (Hehl et al. 1976; Hayashi and Shirafuji 1979; Flanagan and Rosenthal 2007; Garecki 2010). In this theory, there is only torsion T and the curvature R defined by the Levi-Civita connection does not exist. Recently, to account for the late time accelerated expansion of the universe as well as inflation in the early universe (Ferraro and Fiorini 2007, 2008), by extending the teleparallel Lagrangian density described by the torsion scalar T to a function of T as f(T) (Bengochea and Ferraro 2009; Linder 2010), various studies in f(T) gravity have been executed. This concept has the same origin as the idea of F(R) gravity. In order to examine whether f(T) gravity can be worthy of being an alternative theory of gravitation to general relativity, recently a number of aspects of f(T) gravity have widely been investigated in the literature (Capozziello et al. 2011a; Wu and Yu 2010a, 2010b, 2011a, 2011b; Myrzakulov 2010a, 2011; Yerzhanov et al. 2010; Tsyba et al. 2011; Chen et al. 2011; Bengochea 2011; Yang 2011a; Dent et al. 2011; Zheng and Huang 2011; Wang 2011; Zhang et al. 2011; Deliduman and Yapiskan 2011; Li et al. 2011a, 2011b, 2011d; Cai et al. 2011; Chattopadhyay and Debnath 2011; Sharif and Rani 2011; Wei et al. 2011a, 2011b, 2012; Meng and Wang 2011; Boehmer et al. 2011, 2012; Hamani Daouda et al. 2011; Daouda et al. 2012a, 2012b, 2012c; Belo et al. 2011; Geng et al. 2011, 2012; Ferraro and Fiorini 2011a; Wei 2011; Wu and Geng 2011; Gonzalez et al. 2011; Fabbri and Vignolo 2012; Xu et al. 2012; Jamil et al. 2012; Maluf and Faria 2012; Liu et al. 2012; Castello-Branco and da Rocha-Neto 2012; Iorio and Saridakis 2012; Dong et al. 2012a, 2012b; Fu et al. 2012; Baez and Wise 2012; Gu et al. 2012; Tamanini and Boehmer 2012; Cardone et al. 2012; Ferraro 2012; Daouda et al. 2012d; Behboodi et al. 2012; Ulhoa and Amorim 2012; Houndjo et al. 2012; Bamba et al. 2011b, 2010a, 2012d; Bamba 2012; Sotiriou et al. 2011; Miao et al. 2011; Yang 2011b; Ferraro and Fiorini 2011b; Karami and Abdolmaleki 2011, 2012; Maluf et al. 2012; Bamba and Geng 2011; Capozziello et al. 2011a; Setare and Houndjo 2012). For example, the local Lorentz invariance (Li et al. 2011b, 2011d; Sotiriou et al. 2011; Miao et al. 2011), non-trivial conformal frames (Yang 2011b; Ferraro and Fiorini 2011b), thermodynamics Karami and Abdolmaleki (2011, 2012), Maluf et al. (2012), Bamba and Geng (2011), and finite-time future singularities (Bamba et al. 2012d; Setare and Houndjo 2012). In this review, we concentrate on the issues on the finite-time future singularities in f(T) gravity and review the results in Bamba et al. (2012d).

9.1 Basic formalism and fundamental equations

We use orthonormal tetrad components eA(xμ) in the teleparallelism. At each point xμ of the manifold, an index A runs over 0,1,2,3 for the tangent space. The relation to the metric gμν is given by \(g_{\mu\nu}=\eta_{A B} e^{A}_{\mu}e^{B}_{\nu}\), where μ and ν are coordinate indices on the manifold and run over 0,1,2,3. Moreover, \(e_{A}^{\mu}\) forms the tangent vector of the manifold. We define the torsion \(T^{\rho}_{\mu\nu}\) and contorsion \(K^{\mu\nu}_{\rho}\) tensors as
$$ T^\rho_{\mu\nu} \equiv e^\rho_A \bigl( \partial_\mu e^A_\nu- \partial_\nu e^A_\mu \bigr) $$
(531)
and
$$ K^{\mu\nu}_{\rho} \equiv -\frac{1}{2} \bigl(T^{\mu\nu}_{\rho} - T^{\nu\mu}_{\rho} - T_\rho^{\mu\nu} \bigr), $$
(532)
respectively. In general relativity, the Lagrangian density is described by the Ricci scalar R, whereas the teleparallel Lagrangian density is represented by the torsion scalar T. With the torsion tensor as well as the contorsion tensor, we first define the following quantity
$$ S_\rho^{\mu\nu} \equiv\frac{1}{2} \bigl(K^{\mu\nu}_{\rho}+ \delta^\mu_\rho T^{\alpha\nu}_{\alpha}- \delta^\nu_\rho T^{\alpha\mu}_{\alpha} \bigr). $$
(533)
By using this quantity in Eq. (533), the torsion scalar T is described as
$$ T \equiv S_\rho^{\mu\nu} T^\rho_{\mu\nu}. $$
(534)
The modified teleparallel action of f(T) gravity with the matter Lagrangian \({\mathcal{L}}_{\mathrm{M}}\) is expressed as (Linder 2010)
$$ I= \int d^4x \vert{e}\vert \biggl[ \frac{f(T)}{2{\kappa}^2} +{ \mathcal{L}}_{\mathrm{M}} \biggr]. $$
(535)
Here, \(\vert{e}\vert= \det(e^{A}_{\mu})=\sqrt{-g}\). By varying the action in Eq. (535) with respect to the vierbein vector field \(e_{A}^{\mu}\), we acquire the gravitational field equation (Bengochea and Ferraro 2009)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ536_HTML.gif
(536)
with \({T^{(\mathrm{M})}}_{\rho}^{\nu}\) being the energy-momentum tensor of all perfect fluids of ordinary matter such as radiation and non-relativistic matter.
We assume the flat FLRW space-time with the metric ds2=dt2a2(t)∑i=1,2,3(dxi)2.3 Therefore, we have gμν=diag(1,−a2,−a2,−a2) and the tetrad components \(e^{A}_{\mu}= (1,a,a,a)\). From these relations, we find T=−6H2. In this flat FLRW universe, we can write the gravitational field equations in the same forms as those in general relativity
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ537_HTML.gif
(537)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ538_HTML.gif
(538)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ539_HTML.gif
(539)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ540_HTML.gif
(540)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ541_HTML.gif
(541)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ542_HTML.gif
(542)
Here, Fdf/dT and F′≡dF/dT. Moreover, we express the energy density and pressure of all perfect fluids of generic matter as ρM and PM, respectively. These perfect fluids satisfy the continuity equation \(\dot{\rho}_{\mathrm{M}}+3H( \rho_{\mathrm{M}} + P_{\mathrm{M}} )=0\). In addition, for the representations of ρDE in Eq. (539) and PDE in Eq. (540), the standard continuity equation can be met as \(\dot{\rho}_{\mathrm{DE}}+3H ( \rho_{\mathrm{DE}} + P_{\mathrm{DE}} )= 0\).

9.2 Reconstruction of f(T) gravity with realizing the finite-time future singularities

9.2.1 Finite-time future singularities in f(T) gravity

We provided that an expression of the Hubble parameter (Nojiri and Odintsov 2008a) realizing the finite-time future singularities and the resultant scale factor obtained from the form of the Hubble parameter are given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ543_HTML.gif
(543)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ544_HTML.gif
(544)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ545_HTML.gif
(545)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ546_HTML.gif
(546)
with hs(>0), Hs(>0) and as(>0) being positive constants, q(≠0,−1) a non-zero constant, and ts the time when the finite-time future singularity appears. We only deal with the period 0<t<ts because H should be a real number. For the expression of H in Eqs. (543) and (544), the finite-time future singularities occur in the case of each range of the value of q. It follows from ρDEρeff=3H2/κ2 in Eq. (539), \(P_{\mathrm{DE}} \approx P_{\mathrm{eff}} = -(2\dot{H} + 3H^{2})/\kappa^{2}\) in Eq. (540), and the relation T=−6H2 that in the limit of tts, if H→∞, both ρDE and PDE diverge; if H becomes finite but \(\dot{H}\) does infinite, PDE diverges, although ρDE does not. We note that J1 in Eq. (541) only depends on T, i.e., H and J2 in Eq. (542) is proportional to \(\dot{H}\). In Table 1, we summarize the conditions to produce the finite-time future singularities in the limit of tts.
Table 1

Conditions to produce the finite-time future singularities in the limit of tts

Type

q(≠0,−1)

a

H

\(\dot{H}\)

ρDE

PDE

I

q≥1

a→∞

H→∞

\(\dot{H} \to\infty\)

J1≠0

J1≠0 or J2≠0

III

0<q<1

aas

H→∞

\(\dot{H} \to\infty\)

J1≠0

J1≠0

II

−1<q<0

aas

HHs

\(\dot{H} \to\infty\)

 

J2≠0

IV

q<−1 (q≠integer)

aas

HHs

\(\dot{H} \to 0\) (Higher derivatives of H diverge.)

  

9.2.2 Reconstruction of an f(T) gravity model

Next, we reconstruct an f(T) gravity model with realizing the finite-time future singularities in the limit of tts. From Eqs. (539) and (540), the EoS of dark energy is given by
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ547_HTML.gif
(547)
This expression can be rewritten to a fluid description explained in Sect. 3.1 as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ548_HTML.gif
(548)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ549_HTML.gif
(549)
Here, \(\mathcal{J}\) corresponds to −f(ρ) in Eq. (41). The comparison of Eq. (548) with \(P_{\mathrm{eff}} = - \rho_{\mathrm{eff}} -2\dot{H}/\kappa^{2}\) leads to \(\dot{H} + ( \kappa^{2}/2) \mathcal{J} (H, \dot{H}) = 0\). By combining Eq. (549) and this equation, we obtain \(\dot{H}( F +2TF^{\prime}) = 0\). This yields the condition F+2TF′=0 because \(\dot{H} \neq 0\) for H in Eqs. (543) and (544).
On the other hand, Eqs. (537) and (538) can be reduced to
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ550_HTML.gif
(550)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ551_HTML.gif
(551)
where we have also used Eqs. (539)–(542) and \(\dot{H} \neq0\) for H in Eqs. (543) and (544). We see that Eq. (551) is equivalent to the above condition and Eq. (550) corresponds to a consistency condition. For a power-law model given by
$$ f(T) = A T^{\alpha}, $$
(552)
with A(≠0) and α(≠0) being non-zero constants, which with A=1 and α=1 corresponds to general relativity, Eq. (551) becomes F+2TF′=A(−6)α−1(2α−1)H2(α−1)=0, and Eq. (550) reads −f+2TF=A(−6)α(2α−1)H2α=0. In the limit of tts, both of these equations have to be satisfied asymptotically. If α=1/2, these equations are always met. Hence, for q>0, α<0, while for q<0, α=1/2. We remark that these are not sufficient but necessary conditions to realize the finite-time future singularities. In fact, if α<0, the Type I singularity occurs faster than the Type III and finally the Type I singularity happens because the speed of the divergence depends on the absolute value of q. From the same reason, if α=1/2, the Type IV singularity appears faster than the Type II singularity and eventually the Type IV singularity occurs. We also note that for an exponential model f(T)=Ceexp(λeT) with Ce(≠0) and λe(≠0) being non-zero constants and an logarithmic model f(T)=Dlln(γlT) with Dl(≠0) being a non-zero constant and γl(>0) being a positive constant, both Eqs. (543) and (544) cannot be simultaneously met, and therefore these models cannot produce the finite-time future singularities.
For “w” singularity, the scale factor is given by (Dabrowski and Denkiewicz 2009)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ553_HTML.gif
(553)
Here, σ and n are arbitrary constants. When tts, both H and \(\dot{H}\) becomes zero, whereas the effective EoS for the universe weff=(1/3)(2qdec−1)→∞ with \(q_{\mathrm{dec}} \equiv-\ddot{a}a/\dot{a}^{2}\) being the deceleration parameter. Thus, in the above limit of tts, we find F+2TF′=0 because \(\dot{H} (t \to t_{\mathrm {s}}) = 0\). For a power-law model in Eq. (552) with A≠0 and α>1, Eq. (551) can asymptotically be met due to \(\dot{H} (t \to t_{\mathrm{s}}) = 0\), and hence “w” singularity can occur.

9.2.3 Removing the finite-time future singularities

We examine the possibility to remove the finite-time future singularities by taking a power-law correction term fc(T), given by
$$ f_{\mathrm{c}} (T) = B T^\beta. $$
(554)
Here, B(≠0) and β(≠0) are non-zero constants. By plugging the total form of f(T)=ATα+BTβ into Eqs. (550) and (551), we obtain −f+2TF=A(2α−1)Tα+B(2β−1)Tβ≠0 and −F−2TF′=−(2α−1)Tα−1(2β−1)Tβ−1≠0. From the investigations in Sect. 9.2.2, the latter inequality is satisfied when the condition that for q>0, β>0, whereas for q<0, β≠1/2, is met. Accordingly, if β>1, in the limit of tts both (537) and (538) cannot be met. Thus, the finite-time future singularities in f(T) gravity can be removed by a power-low correction as Tβ, where β>1. It is remarkable to mention that a T2 term can cure all the four types of the finite-time future singularities in f(T) gravity, similar to that in F(R) gravity (Nojiri et al. 2011a; Nojiri and Odintsov 2007b). In Table 2, we show necessary conditions for a power-law f(T) model in Eq. (552) to produce the finite-time future singularities and those appearance, and necessary conditions for a power-law correction term fc(T)=BTβ in Eq. (554) to remove the finite-time future singularities.
Table 2

Necessary conditions for the appearance of the finite-time future singularities on a power-law f(T) model in Eq. (552) and those for the removal of the finite-time future singularities on a power-law correction term fc(T)=BTβ in Eq. (554)

Type

q(≠0,−1)

Final appearance

f(T)=ATα

fc(T)=BTβ

(A≠0, α≠0)

(B≠0, β≠0)

I

q≥1

Occur

α<0

β>1

III

0<q<1

α<0

β>1

II

−1<q<0

α=1/2

β≠1/2

IV

q<−1 (q≠integer)

Occur

α=1/2

β≠1/2

We also remark that In terms of the “w” singularity, if a power-law correction term in Eq. (554) with B≠0 and β<0 is taken, the gravitational field equations (537) and (538) cannot be met asymptotically. As a consequence, the power-law correction term can cure the “w” singularity.

9.3 Reconstructed f(T) models performing various cosmologies

Furthermore, we describe the reconstructed f(T) models in which the following various cosmologies are realized: (i) inflation, (ii) the ΛCDM model, (iii) Little Rip scenario and (iv) Pseudo-Rip cosmology. We present expressions of a, H and f(T) realizing the above cosmologies in Table 3. Here, ainf(>0) and aLR(>0) are positive constants and hinf(>1) is a constant larger than unity. We note that the form of f(T) and the conditions for it are derived so that the gravitational field equations (537) and (538).
Table 3

Forms of H and f(T) with realizing (i) inflation, (ii) the ΛCDM model, (iii) Little Rip scenario and (iv) Pseudo-Rip cosmology

Cosmology

a

H

f(T)

(i) Power-law inflation [when t→0]

\(a = a_{\mathrm{inf}} t^{h_{\mathrm{inf}}}\)

H=hinf/t,

f(T)=ATα,

ainf>0

hinf>1

α<0 or α=1/2

(ii) ΛCDM model or exponential inflation

a=aΛexp(HΛt),

\(H =\sqrt{\varLambda/3}=\mathrm{constant}\),

f(T)=T−2Λ,

aΛ>0

Λ>0

Λ>0

(iii) Little Rip scenario [when t→∞]

a=aLRexp[(HLR/ξ)exp(ξt)],

H=HLRexp(ξt),

f(T)=ATα,

aLR>0

HLR>0 and ξ>0

α<0 or α=1/2

(iv) Pseudo-Rip cosmology

a=aPRcosh(t/t0),

H=HPRtanh(t/t0),

\(f(T) = A \sqrt{T}\)

aPR>0

HPR>0

As another quantity to show the deviation of a dark energy model from the ΛCDM model, in addition to the EoS wDE of dark energy in Eq. (48), the deceleration parameter qdec in Eq. (136), and the jerk parameter j in Eq. (137), the snark parameter s is used, which defined as (Sahni et al. 2003)
$$ s \equiv\frac{j - 1}{3 ( q_{\mathrm{dec}} -1/2 )}. $$
(555)
For the ΛCDM model, wDE=−1, qdec=−1, j=1 and s=0. Hence, by examining the deviations of (wDE,qdec,j,s) from (−1,−1,1,0) for the ΛCDM model and using these four parameters as a tool of observational tests, we can distinguish a dark energy model from the ΛCDM model. In Table 4, we display the expressions of wDE, qdec, j and s at the present time t0 for the ΛCDM model, Little Rip scenario and Pseudo-Rip cosmology (Bamba et al. 2012d). Here, s0s(t=t0).
Table 4

Expressions of wDE, qdec, j and s at the present time t=t0 for the ΛCDM model, Little Rip scenario and Pseudo-Rip cosmology (Bamba et al. 2012d)

Model

wDE(0)

qdec(0)

j0

s0

ΛCDM model

−1

−1

1

0

Little Rip scenario

\(-1 -(2/3) \tilde{\chi}\),

\(-1 - \tilde{\chi}\)

\(1 + \chi( \tilde{\chi} + 3)\)

\(-[2\tilde{\chi} ( \tilde{\chi} + 3)] [3 ( 2\tilde{\chi} + 3 )]^{-1}\)

\(\tilde{\chi} \equiv H_{0}/( H_{\mathrm{LR}} e)\leq0.36\),

e=2.71828

Pseudo-Rip cosmology

\(-1 - [2\delta/(3 \tilde{\mathrm{s}}^{2} )]\),

\(-1 + (\delta^{2} \tilde{\mathrm{t}}^{2} - 1)/(\delta^{2} \tilde{\mathrm{t}}^{2})\),

\(1 + (1-\delta^{3} \tilde{\mathrm{t}}^{2})/(\delta^{3} \tilde{ \mathrm{t}}^{2})\),

\([2/(3 \delta)](\delta^{3} \tilde{\mathrm{t}}^{2} -1) (\delta^{2} \tilde{\mathrm{t}}^{2} +2)^{-1}\)

δH0/HPR≤0.497

\(\tilde{\mathrm{s}}^{2} \equiv\sinh^{2} 1 = 1.38\)

\(\tilde{\mathrm{t}}^{2} \equiv\tanh^{2} 1 = 0.580\)

Finally, we mention another feature of f(T) gravity. It has been discussed that in the star collapse, the time-dependent matter instability found in F(R) gravity (Arbuzova and Dolgov 2011; Bamba et al. 2011d), which is related to the well-studied matter instability (Dolgov and Kawasaki 2003) leading to the appearance of a singularity in the relativistic star formation process (Kobayashi and Maeda 2008, 2009; Dev et al. 2008), can also happen in the framework of f(T) gravity (Bamba et al. 2012d).

9.4 Thermodynamics in f(T) gravity

In this section, to explore whether f(T) gravity is worthy of an alternative gravitational theory to general relativity, we investigate thermodynamics in f(T) gravity. In particular, the second law of thermodynamics around the finite-time future singularities is studied by applying the procedure proposed in Bamba and Geng (2010, 2011). Black hole thermodynamics (Bardeen et al. 1973; Bekenstein 1973; Hawking 1975; Gibbons and Hawking 1977) suggested the fundamental relation of gravitation to thermodynamics (for recent reviews, see, e.g., Padmanabhan 2009, 2010a, 2010b). With the proportionality of the entropy to the horizon area, in general relativity the Einstein equation was obtained from the Clausius relation in thermodynamics (Jacobson 1995). This consideration has been extended to more general gravitational theories (Eling et al. 2006; Elizalde and Silva 2008; Bamba et al. 2010c; Wu et al. 2010; Yokokura 2011; Brustein and Hadad 2009; Brustein and Medved 2012).

9.4.1 First law of thermodynamics

It is known in Bamba and Geng (2011, 2010), Bamba et al. (2011c, 2010d), that when the continuity equation of dark component is met as \(\dot{\rho}_{\mathrm{DE}}+3H ( \rho_{\mathrm{DE}} + P_{\mathrm{DE}} )= 0\), we can have an equilibrium description of thermodynamics. In the flat FLRW universe with the metric \(d s^{2} = h_{\alpha\beta} d x^{\alpha} d x^{\beta} +\tilde{r}^{2} d \varOmega^{2}\), where \(\tilde{r}=a(t)r\), x0=t and x1=r with the two-dimensional metric hαβ=diag(1,−a2(t)), 2 is the metric of two-dimensional sphere with unit radius. The radius \(\tilde{r}_{A}\) of the apparent horizon is described by \(\tilde{r}_{A}= 1/H\). The relation \(h^{\alpha\beta} \partial_{\alpha} \tilde{r} \partial_{\beta} \tilde{r}=0\) leads to the dynamical apparent horizon. The time derivative of \(\tilde{r}_{A}= 1/H\) yields \(-d\tilde{r}_{A}/\tilde{r}_{A}^{3} =\dot{H}H dt\). Combining the Friedmann equation (537) with this equation presents \([1/(4\pi G)] d\tilde{r}_{A}=\tilde{r}_{A}^{3} H ( \rho_{\mathrm{t}}+P_{\mathrm{t}}) dt\) with ρtρDE+ρM and PtPDE+PM being the total energy density and pressure of the universe, respectively. The Bekenstein-Hawking horizon entropy in general relativity is written by \(S=\mathcal{A}/(4G)\). Here, \(\mathcal{A}=4\pi\tilde{r}_{A}^{2}\) is the area of the apparent horizon (Bardeen et al. 1973; Bekenstein 1973; Hawking 1975; Gibbons and Hawking 1977). Thus, with the horizon entropy as well as the above relation, we obtain \([1/(2\pi\tilde{r}_{A} )]dS= 4\pi\tilde{r}_{A}^{3} H ( \rho_{\mathrm{t}}+P_{\mathrm{t}} ) dt\). The Hawking temperature TH=|κsg|/(2π) is considered to be the associated temperature of the apparent horizon, and the surface gravity κsg is given by (Cai and Kim 2005)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ556_HTML.gif
(556)
Here, h is the determinant of the metric hαβ and wtPt/ρt is the EoS for the total of energy and matter in the universe. From Eq. (556), we see that if wt≤1/3, κsg≤0. The substitution of Eq. (556) into TH=|κsg|/(2π) yields
$$ T_{\mathrm{H}}=\frac{1}{2\pi\tilde{r}_A} \biggl( 1-\frac{\dot{\tilde{r}}_A}{2H\tilde{r}_A} \biggr). $$
(557)
By combining the above relation of the horizon entropy S with Eq. (557), we find \(T_{\mathrm{H}} dS = 4\pi\tilde{r}_{A}^{3} H (\rho_{\mathrm {t}}+P_{\mathrm{t}}) dt -2\pi\tilde{r}_{A}^{2} (\rho_{\mathrm{t}}+P_{\mathrm{t}} ) d\tilde{r}_{A}\). Moreover, the Misner-Sharp energy (Misner and Sharp 1964; Bak and Rey 2000) is expressed by \(E=\tilde{r}_{A}/(2G) = V\rho_{\mathrm{t}}\), where \(V=4\pi\tilde{r}_{A}^{3}/3\) is the volume inside the apparent horizon. From the second equality, we see that E corresponds to the total intrinsic energy. With this equation, we obtain \(dE=-4\pi\tilde{r}_{A}^{3} H(\rho_{\mathrm{t}}+P_{\mathrm{t}} ) dt+4\pi\tilde{r}_{A}^{2} \rho_{\mathrm{t}} d\tilde{r}_{A}\). In addition, the work density (Hayward 1998; Hayward et al. 1999; Cai and Cao 2007) is defined by W≡−(1/2)(T(M)αβhαβ+T(DE)αβhαβ)=(1/2)(ρtPt) with T(M)αβ and T(DE)αβ being the energy-momentum tensor of matter and that of dark components, respectively. We plug the work density W into the relation on dE derived above, so that we can represent the first law of equilibrium thermodynamics as
$$ T_{\mathrm{H}} dS=-dE+W dV. $$
(558)
Thus, we acquire an equilibrium description of thermodynamics. It follows from the gravitational equations (537) and (538) as well as the above relation on dS that \(\dot{S} = 8\pi^{2} H \tilde{r}_{A}^{4} (\rho_{\mathrm{t}}+P_{\mathrm {t}})= ( 6\pi/G ) ( \dot{T}/T^{2} ) = -( 2\pi/G ) [ \dot{H}/ ( 3H^{3} ) ]> 0\). Accordingly, for the expanding universe (H>0), if the null energy condition ρt+Pt≥0 in Eq. (49) is satisfied, namely, \(\dot{H} \leq0\), S always becomes large.

9.4.2 Second law of thermodynamics

We then explore the second law of thermodynamics. The Gibbs equation of all the matter and energy fluid is expressed as THdSt=d(ρtV)+PtdV=Vdρt+(ρt+Pt)dV. We can write the second law of thermodynamics as dSsum/dtdS/dt+dSt/dt≥0. Here, SsumS+St with St being the entropy of total energy inside the horizon. If the temperature of the universe is the same as that of the apparent horizon (Gong et al. 2007; Jamil et al. 2010a), this can be represented as
$$ \frac{dS_{\mathrm{sum}}}{dt} = -\frac{6\pi}{G} \biggl( \frac{\dot{T}}{T} \biggr)^2 \frac{1}{4HT + \dot {T}}, $$
(559)
where we have used \(V=4\pi\tilde{r}_{A}^{3}/3\), Eqs. (538), (557) and the relation on \(\dot{S}\) shown in the last part of Sect. 9.4.1. Thus, if the condition that \(Y \equiv-( 4HT + \dot{T} ) = 12H ( 2H^{2} + \dot{H} ) \geq0\) (Bamba and Geng 2011) is met, the second law of thermodynamics can be verified. When tts, for all of the four types of the finite-time future singularities in Table 1, the relation \(2H^{2} + \dot{H} \geq0\) is always satisfied. Since H>0 for the expanding universe, the second law of thermodynamics described by Eq. (559) can be met around the finite-time future singularities including in the phantom phase (\(\dot{H} >0\)), although at the exact time of the appearance of singularity of t=ts this classical description of thermodynamics could not be applicable.

10 Testing dark energy and alternative gravity by cosmography: generalities

Next, we move to the comparison of the theoretical studies on dark energy and modified gravity with the observational data. In this section, we introduce the idea and concept of cosmography to observationally test dark energy and alternative gravitational theory to general relativity.

The observed accelerated expansion of the cosmic fluid can be faced in several equivalent ways. In other words, both dark energy models and modified gravity theories seem to be in agreement with data. As a consequence, unless higher precision probes of the expansion rate and the growth of structure will be available, these two rival approaches could not be discriminated. This confusion about the theoretical background suggests that a more conservative approach to the problem of the cosmic acceleration, relying on as less model dependent quantities as possible, is welcome. A possible solution could be to come back to the cosmography (Weinberg 1972) rather than finding out solutions of the Friedmann equations and testing them. Being only related to the derivatives of the scale factor, the cosmographic parameters make it possible to fit the data on the distance-redshift relation without any a priori assumption on the underlying cosmological model: in this case, the only assumption is that the metric is the FLRW one (and hence not relying on the solution of cosmological equations). Almost eighty years after Hubble’s discovery of the expansion of the universe, we can now extend, in principle, cosmography well beyond the search for the value of the only Hubble constant. The SNeIa Hubble diagram extends up to z=1.7 thus invoking the need for, at least, a fifth order Taylor expansion of the scale factor in order to give a reliable approximation of the distance-redshift relation. As a consequence, it could be, in principle, possible to estimate up to five cosmographic parameters, although the still too small data set available does not allow to get a precise and realistic determination of all of them.

Once these quantities have been determined, one could use them to put constraints on the models. In a sense, we can revert to the usual approach, consisting with deriving the cosmographic parameters as a sort of byproduct of an assumed theory. Here, we follow the other way of expressing the quantities characterizing the model as a function of the cosmographic parameters. Such a program is particularly suited for the study of alternative theories as F(R) or f(T) gravity (Capozziello et al. 2008, 2011a; Bouhmadi-Lopez et al. 2010a) and any equivalent description of dynamics by effective scalar fields. As it is well known, the mathematical difficulties in analyzing the solution of field equations make it quite problematic to find out analytical expressions for the scale factor and hence predict the values of the cosmographic parameters. A key role in F(R) gravity and f(T) gravity is played by the choice of the function. Under quite general hypotheses, it is possible to derive relations between cosmographic parameters and the present time values of F(n)(R)=dnF/dRn or f(n)(T)=dnf/dTn, with n=0,…,3, whatever F(R), f(T) or their equivalent scalar-field descriptions are.

Once the cosmographic parameters are determined, the method allows to investigate the cosmography of alternative theories matching with observational data.

It is worth stressing that the definition of the cosmographic parameters only relies on the assumption of the FLRW metric. As such, it is however difficult to state a priori to what extent the fifth order expansion provides an accurate enough description of the quantities of interest. Actually, the number of cosmographic parameters to be used depends on the problem one is interested in.

To illustrate the method, one can be concerned only with the SNeIa Hubble diagram so that one has to check that the distance modulus μcp(z) obtained using the fifth order expansion of the scale factor is the same (within the errors) as the one μDE(z) of the underlying physical model. Being such a model of course unknown, one can adopt a phenomenological parameterization for the dark energy EoS and look at the percentage deviation Δμ/μDE as a function of the EoS parameters. Note that one can always use a phenomenological dark energy model to get a reliable estimate of the scale factor evolution (see, for example, Cardone et al. 2004).

Here, we will carry out such an approach using the so called CPL model (Chevallier and Polarski 2001; Linder 2003), introduced below, and verified that Δμ/μDE is an increasing function of z (as expected), but still remains smaller than 2 % up to z∼2 over a wide range of the CPL parameter space. On the other hand, halting the Taylor expansion to a lower order may introduce significant deviation for z>1 that can potentially bias the analysis if the measurement errors are as small as those predicted by future observational surveys. However, the fifth order expansion is both sufficient to get an accurate distance modulus over the redshift range probed by SNeIa and necessary to avoid dangerous biases. As shown in Capozziello and Izzo (2008), Capozziello et al. (2011b), the method can highly be improved by adopting BAO and Gamma Ray Bursts (GRBs) as cosmic indicators.

As stated above, the key rule in cosmography is the Taylor series expansion of the scale factor with respect to the cosmic time. To this aim, it is convenient to introduce the following functions:
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ560_HTML.gif
(560)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ561_HTML.gif
(561)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ562_HTML.gif
(562)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ563_HTML.gif
(563)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ564_HTML.gif
(564)
which are usually referred to as the Hubble, deceleration, jerk, snap, and lerk parameters, respectively. (Here, for clear understanding, we have again defined the Hubble, deceleration (as q, which is the same as qdec in Eq. (136)), jerk and snap parameters.) It is then a matter of algebra to demonstrate the following useful relations:
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ565_HTML.gif
(565)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ566_HTML.gif
(566)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ567_HTML.gif
(567)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ568_HTML.gif
(568)
Equations (565)–(568) make it possible to relate the derivative of the Hubble parameter to the other cosmographic parameters. The distance - redshift relation may then be obtained, starting from the Taylor expansion of a(t) along the lines described in Visser (2004), Wang and Mukherjee (2004), Cattoen and Visser (2007).
By these definitions, the series expansion to the 5th order in time of the scale factor is
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ569_HTML.gif
(569)
from which, we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ570_HTML.gif
(570)
It is easy to see that Eq. (570) is the inverse of redshift z, being the redshift defined by
$$1 + z = \frac{a(t_{0})}{a(t)} . $$
The physical distance traveled by a photon that is emitted at time t and absorbed at the current epoch t0 is
$$D = c \int dt = c (t_{0} - t_{*}) , $$
where c is the speed of light. Assuming \(t_{*} = t_{0} - \frac{D}{c}\) and inserting it into Eq. (570), we have
$$ 1 + z = \frac{a(t_{0})}{a(t_{0}-\frac{D}{c})} =\frac{1}{1 - \frac{H_{0}}{c}D - \frac{q_{0}}{2} (\frac{H_{0}}{c} )^{2}D^{2} - \frac{j_{0}}{6} (\frac{H_{0}}{c} )^{3}D^{3} + \frac{s_{0}}{24} (\frac{H_{0}}{c} )^{4}D^{4} - \frac{l_{0}}{120} (\frac{H_{0}}{c} )^{5}D^{5} +O[(\frac{H_{0} D}{c})^{6}]}. $$
(571)
The inverse of this expression becomes
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ572_HTML.gif
(572)
Then, we reverse the series z(D)→D(z) to have the physical distance D expressed as a function of the redshift z
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ573_HTML.gif
(573)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ574_HTML.gif
(574)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ575_HTML.gif
(575)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ576_HTML.gif
(576)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ577_HTML.gif
(577)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ578_HTML.gif
(578)
From this, we obtain
$$ D(z) = \frac{c z}{H_{0}} \bigl\{ \mathcal{D}_{z}^{0} + \mathcal{D}_{z}^{1} z + \mathcal{D}_{z}^{2} z^{2} + \mathcal{D}_{z}^{3} z^{3} + \mathcal{D}_{z}^{4} z^{4} + O\bigl(z^{5} \bigr) \bigr\} $$
(579)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ580_HTML.gif
(580)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ581_HTML.gif
(581)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ582_HTML.gif
(582)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ583_HTML.gif
(583)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ584_HTML.gif
(584)
In standard applications, other quantities can result in become useful
  • the luminosity distance:
    $$ d_{L} = \frac{a(t_{0})}{a(t_{0}-\frac{D}{c})} \bigl(a(t_{0}) r_{0} \bigr), $$
    (585)
  • the angular-diameter distance:
    $$ d_{A} = \frac{a(t_{0}-\frac{D}{c})}{a(t_{0})} \bigl(a(t_{0}) r_{0} \bigr), $$
    (586)
where r0(D) is given by
$$ r_{0}(D) = \left\{ \begin{array}{@{}l@{\quad}l} \sin( \int_{t_{0}- \frac{D}{c}}^{t_{0}} \frac{c \mathrm{d}t}{a(t)}) & K = +1; \\[2mm] \int_{t_{0}- \frac{D}{c}}^{t_{0}} \frac{c \mathrm{d}t}{a(t)} & K = 0; \\[2mm] \sinh( \int_{t_{0}- \frac{D}{c}}^{t_{0}} \frac{c \mathrm{d}t}{a(t)} ) & K = -1. \end{array} \right. $$
(587)
If we consider the expansion for short distances, namely, if we insert the series expansion of a(t) into r0(D), we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ588_HTML.gif
(588)
To convert from physical distance traveled to r coordinate, we have to consider that the Taylor series expansion of sin-sinh functions is
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ589_HTML.gif
(589)
so that Eq. (570) with the spatial curvature K term becomes
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ590_HTML.gif
(590)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ591_HTML.gif
(591)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ592_HTML.gif
(592)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ593_HTML.gif
(593)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ594_HTML.gif
(594)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ595_HTML.gif
(595)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ596_HTML.gif
(596)
Using these definitions for luminosity distance, we acquire
$$ d_{L}(z) = \frac{c z}{H_{0}} \bigl\{ \mathcal{D}_{L}^{0} + \mathcal{D}_{L}^{1} z + \mathcal{D}_{L}^{2} z^{2} + \mathcal{D}_{L}^{3} z^{3} + \mathcal{D}_{L}^{4} z^{4} + O\bigl(z^{5} \bigr) \bigr\} $$
(597)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ598_HTML.gif
(598)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ599_HTML.gif
(599)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ600_HTML.gif
(600)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ601_HTML.gif
(601)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ602_HTML.gif
(602)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ603_HTML.gif
(603)
While, for the angular diameter distance we find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ604_HTML.gif
(604)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ605_HTML.gif
(605)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ606_HTML.gif
(606)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ607_HTML.gif
(607)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ608_HTML.gif
(608)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ609_HTML.gif
(609)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ610_HTML.gif
(610)
We define \(\varOmega_{0} = 1 + \frac{K c^{2}}{H_{0}^{2} a_{0}^{2}}\), which can be considered a purely cosmographic parameter, or \(\varOmega_{0} = 1 - \varOmega_{K}^{(0)} = \varOmega_{\mathrm{m}}^{(0)} + \varOmega _{\mathrm{r}}^{(0)} + \varOmega_{X}^{(0)}\), where \(\varOmega_{X}^{(0)}\) corresponds to the current fractional densities of dark energy, if we explore the dynamics of the universe. With these parameters, we obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ611_HTML.gif
(611)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ612_HTML.gif
(612)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ613_HTML.gif
(613)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ614_HTML.gif
(614)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ615_HTML.gif
(615)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ616_HTML.gif
(616)
and
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ617_HTML.gif
(617)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ618_HTML.gif
(618)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ619_HTML.gif
(619)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ620_HTML.gif
(620)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ621_HTML.gif
(621)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ622_HTML.gif
(622)
Previous relations have been derived for any value of the curvature parameter. To illustrate the method, however, we can assume a spatially flat universe, using the simplified versions for K=0. Now, since we are going to use supernovae data, it will be useful to give as well the Taylor series of the expansion of the luminosity distance at it enters the modulus distance, which is the quantity about which those observational data inform. The final expression for the modulus distance based on the Hubble free luminosity distance, μ(z)=5log10dL(z), is
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ623_HTML.gif
(623)
with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ624_HTML.gif
(624)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ625_HTML.gif
(625)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ626_HTML.gif
(626)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ627_HTML.gif
(627)

11 An example: testing F(R) gravity by cosmography

The cosmographic approach can be used to deal with F(R) gravity (Capozziello et al. 2008). However, similar considerations perfectly hold also for f(T) gravity (Capozziello et al. 2011a) or any scalar-tensor gravity model. In order to construct the cosmographic apparatus, we describe the Friedmann equation (42) in the FLRW space-time in F(R) gravity as
$$ H^2 = \frac{1}{3} \biggl( \frac{\rho_{\mathrm{M}}}{F'(R)} + \rho_{\mathrm{curv}} \biggr), $$
(628)
where the prime denotes the derivative with respect to R, the gravitational coupling is taken as κ2=1 and ρcurv is the energy density of an effective curvature fluid:
$$ \rho_{\mathrm{curv}} = \frac{1}{F'(R)} \biggl[ \frac{1}{2} \bigl( F(R) - R F'(R) \bigr) - 3 H \dot{R} F''(R) \biggr]. $$
(629)
Assuming there is no interaction between the matter and the curvature terms (we are in the Jordan frame), the matter continuity equation gives the usual scaling \(\rho_{\mathrm{M}} = \rho_{\mathrm{M}} (t = t_{0}) a^{-3} = 3 H_{0}^{2} \varOmega_{\mathrm{M}} a^{-3}\), with \(\varOmega_{\mathrm{M}}^{(0)}\) the matter density parameter at the present time. The continuity equation for ρcurv then reads
$$ \dot{\rho}_{\mathrm{curv}} + 3 H (1 + w_{\mathrm{curv}}) \rho_{\mathrm {curv}} = \frac{3 H_0^2 \varOmega_{\mathrm{M}}^{(0)} \dot{R} F''(R)}{ ( F'(R) )^2} a^{-3} $$
(630)
with
$$ w_{\mathrm{curv}} = -1 + \frac{\ddot{R} F''(R) + \dot{R} [ \dot{R} F'''(R) - H F''(R) ]}{ [ F(R) - R F'(R) ]/2 - 3 H \dot{R} F''(R)}, $$
(631)
the barotropic factor of the curvature fluid. It is worth noticing that the curvature fluid quantities ρcurv and wcurv only depend on the form of F(R) and its derivatives up to the third order. As a consequence, considering only those current values (which may naively be obtained by replacing R with R0 everywhere), two F(R) theories sharing the same values of F(R0), F′(R0), F″(R0), F‴(R0) will be degenerate from this point of view. One can argue that this is not strictly true because different F(R) theories will lead to different expansion rates H(t) and hence different current values of R and its derivatives. However, it is likely that two F(R) functions that exactly match with each other up to the third order derivative today will give rise to the same H(t) at least for tt0, so that \((R_{0}, \dot{R}_{0}, \ddot{R}_{0})\) will be almost the same. Combining Eq. (630) with Eq. (628), one finally gets the following master equation for the Hubble parameter
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ632_HTML.gif
(632)
Expressing the scalar curvature R as function of the Hubble parameter as
$$ R = - 6 \bigl( \dot{H} + 2 H^2 \bigr) $$
(633)
and inserting the resultant expression into Eq. (632), one ends with a fourth order nonlinear differential equation for the scale factor a(t) that cannot easily be solved even for the simplest cases (for instance, F(R)∝Rn). Moreover, although technically feasible, a numerical solution of Eq. (632) is plagued by the large uncertainties on the boundary conditions (i.e., the current values of the scale factor and its derivatives up to the third order) that have to be set to find out the scale factor.

Motivated by these difficulties, we now approach the problem from a different viewpoint. Rather than choosing a parameterized expression for F(R) and then numerically solving Eq. (632) for given values of the boundary conditions, we try to relate the current values of its derivatives to the cosmographic parameters (q0,j0,s0,l0) so that constraining them in a model independent way can give us a hint for what kind of F(R) theory is able to fit the observed Hubble diagram. Note that a similar analysis, but in the context of the energy conditions in F(R), has yet been presented in (Perez Bergliaffa 2006). However, in that work, an expression for F(R) is given and then the snap parameter is computed in order for it to be compared to the observed one. On the contrary, our analysis does not depend on any assumed functional expression for F(R).

As a preliminary step, it is worth considering again the constraint equation (633). Differentiating with respect to t, we easily get the following relations:
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ634_HTML.gif
(634)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ635_HTML.gif
(635)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ636_HTML.gif
(636)
Evaluating these at the present time and using Eqs. (565)–(568), we finally obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ637_HTML.gif
(637)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ638_HTML.gif
(638)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ639_HTML.gif
(639)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ640_HTML.gif
(640)
which will turn out to be useful in the following.
We come back to the expansion rate and master equations (628) and (632). Since they have to hold along the full evolutionary history of the universe, they are naively satisfied also at the present time. Accordingly, we may evaluate them in t=t0 and thus we easily obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ641_HTML.gif
(641)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ642_HTML.gif
(642)
Using Eqs. (565)–(568) and (637)–(640), we can rearrange Eqs. (641) and (642) as two relations among the Hubble constant H0 and the cosmographic parameters (q0,j0,s0), on one hand, and the present day values of F(R) and its derivatives up to third order. However, two further relations are needed in order to close the system and determine the four unknown quantities F(R0), F′(R0), F″(R0), F‴(R0). A first one may be easily obtained by noting that, inserting back the physical units, the rate expansion equation reads
$$ H^2 = \frac{8 \pi G}{3 F'(R)} \bigl(\rho_{\mathrm{m}} + \rho_{\mathrm {curv}} F'(R) \bigr), $$
(643)
which clearly shows that, in F(R) gravity, the Newton’s gravitational constant G (restored for the moment) is replaced by an effective (time dependent) Geff=G/F′(R). On the other hand, it is reasonable to assume that the value of Geff at the present time is the same as that of the Newton’s one, so that we can acquire the simple constraint
$$ G_{\mathrm{eff}}(z = 0) = G \rightarrow F'(R_0) = 1. $$
(644)
In order to find the fourth relation we need to close the system, we first differentiate both sides of Eq. (632) with respect to t. We thus obtain
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ645_HTML.gif
(645)
with F(iv)(R)=d4F/dR4. We now suppose that F(R) may be well approximated by its third order Taylor expansion in terms of (RR0), i.e., we set
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ646_HTML.gif
(646)
In such an approximation, we find F(n)(R)=dnF/Rn=0 for n≥4, so that naively F(iv)(R0)=0. Evaluating Eq. (645) at the present time, we acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ647_HTML.gif
(647)
Now, we can schematically proceed as follows. We evaluate Eqs. (565)–(568) at z=0 and plug these relations into the left-hand sides of Eqs. (641), (642) and (647). Then, we insert Eqs. (637)–(640) into the right-hand sides of these same equations, so that only the quantities related to the cosmographic parameters (q0,j0,s0,l0) and the F(R) term can enter both sides of these relations. Finally, we solve them under the constraint (644) with respect to the current values of F(R) and its derivatives up to the third order. After some algebra, we eventually end up with the desired result
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ648_HTML.gif
(648)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ649_HTML.gif
(649)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ650_HTML.gif
(650)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ651_HTML.gif
(651)
where we have defined
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ652_HTML.gif
(652)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ653_HTML.gif
(653)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ654_HTML.gif
(654)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ655_HTML.gif
(655)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ656_HTML.gif
(656)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ657_HTML.gif
(657)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ658_HTML.gif
(658)
Equations (648)–(658) make it possible to estimate the current values of F(R) and its first three derivatives as function of the Hubble constant H0 and the cosmographic parameters (q0,j0,s0,l0) provided a value for the matter density parameter \(\varOmega_{\mathrm{M}}^{(0)}\) is given. This is a somewhat problematic point. Indeed, while the cosmographic parameters may be estimated in a model independent way, the fiducial value for \(\varOmega_{\mathrm{M}}^{(0)}\) is usually the outcome of fitting a given dataset in the framework of an assumed dark energy scenario. However, it is worth noting that all the different models converge on the concordance value \(\varOmega_{\mathrm{M}}^{(0)} \simeq0.25\) which is also in agreement with astrophysical (model independent) estimates from the gas mass fraction in galaxy clusters. On the other hand, it has been proposed that F(R) theories may avoid the need for dark matter in galaxies and galaxy clusters (Capozziello et al. 2004, 2007; Martins and Salucci 2007; Sobouti 2007; Saffari and Sobouti 2007; Mendoza and Rosas-Guevara 2007). In such a case, the total matter content of the universe is essentially equal to the baryonic one. According to the primordial elements abundance and the standard Big bang nucleosynthesis (BBN) scenario, we therefore find \(\varOmega_{\mathrm{M}}^{(0)} \simeq\omega_{b}/h^{2}\) with \(\omega_{b} = \varOmega_{b}^{(0)} h^{2} \simeq0.0214\) (Kirkman et al. 2003) and h the Hubble constant in units of 100 km/s/Mpc. Setting h=0.72 in agreement with the results of the HST Key project (Freedman et al. 2001), we hence obtain \(\varOmega_{\mathrm{M}}^{(0)} = 0.041\) for a universe consisting of baryons only. Thus, in the following we will consider both cases when numerical estimates are needed.

It is worth noticing that H0 only plays the role of a scaling parameter giving the correct physical dimensions to F(R) and its derivatives. As such, it is not surprising that we need four cosmographic parameters, namely (q0,j0,s0,l0), to fix the four F(R) related quantities F(R0), F′(R0), F″(R0), F‴(R0). It is also worth stressing that Eqs. (648)–(651) are linear in the F(R) quantities, so that (q0,j0,s0,l0) can uniquely determine the former ones. On the contrary, inverting them to acquire the cosmographic parameters as a function of the F(R) ones, we do not obtain linear relations. Indeed, the field equations in F(R) theories are nonlinear fourth order differential equations in terms of the scale factor a(t), so that fixing the derivatives of F(R) up to the third order can make it possible to find out a class of solutions, not a single one. Each one of these solutions will be characterized by a different set of cosmographic parameters. This explains why the inversion of Eqs. (648)–(658) does not give a unique result for (q0,j0,s0,l0).

As a final comment, we again investigate the underlying assumptions leading to the above derived relations. While Eqs. (641) and (642) are exact relations deriving from a rigorous application of the field equations, Eq. (647) heavily relies on having approximated F(R) with its third order Taylor expansion (646). If this assumption fails, the system should not be closed because a fifth unknown parameter enters the game, namely F(iv)(R0). Actually, replacing F(R) with its Taylor expansion is not possible for all the class of F(R) theories. As such, the above results only hold in those cases where such an expansion is possible. Moreover, by truncating the expansion to the third order, we are implicitly assuming that higher order terms are negligible over the redshift range probed by the data. That is to say, we are assuming that
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ659_HTML.gif
(659)
over the redshift range probed by the data. Checking the validity of this assumption is not possible without explicitly solving the field equations, but we can guess an order of magnitude estimate considering that, for all the viable models, the background dynamics should not differ too much from the ΛCDM one at least up to z≃2. Using the expression of H(z) for the ΛCDM model, it is easily to see that R/R0 is a quickly increasing function of the redshift, so that, in order Eq. (659) should hold, we have to assume that F(n)(R0)≪F‴(R0) for n≥4. This condition is easier to check for many analytical F(R) models.

Once such a relation is verified, we have to still worry about Eq. (644) relying on the assumption that the cosmological gravitational constant is exactly the same as the local one. Although reasonable, this requirement is not absolutely demonstrated. Actually, the numerical value usually adopted for the Newton’s constant G is obtained from laboratory experiments in settings that can hardly be considered homogeneous and isotropic. Similarly, the space-time metric in such conditions has nothing to do with the cosmological one, so that strictly speaking, matching the two values of G should be an extrapolation. Although commonly accepted and quite reasonable, the condition Glocal=Gcosmo could (at least, in principle) be violated, so that Eq. (644) could be reconsidered. Indeed, as we will see, the condition F′(R0)=1 may not be verified for some popular F(R) models which has recently been proposed in the literature. However, it is reasonable to assume that Geff(z=0)=G(1+ε) with ε≪1. When this be the case, we should repeat the derivation of Eqs. (648)–(651) by using the condition F′(R0)=(1+ε)−1. By executing the Taylor expansion of the results in terms of ε to the first order and comparing with the above derived equations, we can estimate the error induced by our assumption ε=0. The resultant expressions are too lengthy to be reported and depend on the values of the matter density parameter \(\varOmega_{\mathrm{M}}^{(0)}\), the cosmographic parameters (q0,j0,s0,l0) and ε in a complicated way. Nevertheless, we have numerically checked that the error induced on F(R0), F″(R0), F‴(R0) are much lower than 10 % for the value of ε as high as an unrealistic ε∼0.1. However, results are reliable also for these cases (Capozziello et al. 2008).

11.1 The CPL model

A determination of F(R) and its derivatives in terms of the cosmographic parameters need for an estimate of these latter from the data in a model independent way. Unfortunately, even in the nowadays era of precision cosmology, such a program is still too ambitious to give useful constraints on the F(R) derivatives, as we will see later. On the other hand, the cosmographic parameters may also be expressed in terms of the dark energy density and EoS parameters so that we can work out what are the current values of F(R) and its derivatives giving the same (q0,j0,s0,l0) of the given dark energy model. To this aim, it is convenient to adopt a parameterized expression for the dark energy EoS in order to reduce the dependence of the results on any underlying theoretical scenario. Following the prescription of the Dark Energy Task Force (Albrecht et al. 2006), we will use the CPL parameterization for the EoS of dark energy by setting (Chevallier and Polarski 2001; Linder 2003)
$$ w = w_0 + w_a (1 - a) = w_0 + w_a z (1 + z)^{-1}, $$
(660)
so that, in a flat universe filled by dust matter and dark energy, the dimensionless Hubble parameter E(z)=H/H0 in Eq. (13) reads
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ661_HTML.gif
(661)
with \(\varOmega_{X}^{(0)} = 1 - \varOmega_{\mathrm{M}}^{(0)}\) because of the assumption that the universe is flat. Here and in the following, we omit the inferior “DE” of w0 and wa. In order to determine the cosmographic parameters for such a model, we avoid integrating H(z) to get a(t) by noting that d/dt=−(1+z)H(z)d/dz. We can use such a relation to evaluate \((\dot{H}, \ddot{H}, d^{3}H/dt^{3}, d^{4}H/dt^{4})\) and then solve Eqs. (565)–(568), evaluated at z=0, with respect to the parameters of interest. Some algebra finally gives
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ662_HTML.gif
(662)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ663_HTML.gif
(663)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ664_HTML.gif
(664)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ665_HTML.gif
(665)
Inserting Eqs. (662)–(665) into Eqs. (648)–(658), we acquire lengthy expressions (which we do not report here) giving the current values of F(R) and its first three derivatives as a function of \((\varOmega_{\mathrm{M}}^{(0)}, w_{0}, w_{a})\). It is worth noting that the F(R) model obtained is not dynamically equivalent to the starting CPL one. Indeed, the two models have the same cosmographic parameters only today. For instance, the scale factor is the same between the two theories only over the time period during which the fifth order Taylor expansion is a good approximation of the actual a(t). It is also meaningful to stress that such a procedure does not select a unique F(R) model, but rather a class of fourth order theories all sharing the same third order Taylor expansion of F(R).

11.2 The ΛCDM case

With these caveats in mind, it is significant to first examine the ΛCDM model, which is described by setting (w0,wa)=(−1,0) in the above expressions, and hence we have
$$ \begin{cases} q_0 = \displaystyle\frac{1}{2} - \frac{3}{2} (1-\varOmega_{\mathrm{M}}^{(0)} ), \\ j_0 = \displaystyle 1, \\ s_0 = \displaystyle 1 - \frac{9}{2} \varOmega_{\mathrm{M}}^{(0)}, \\ l_0 = \displaystyle1 + 3 \varOmega_{\mathrm{M}}^{(0)} + \frac{27}{2} (\varOmega_{\mathrm{M}}^{(0)} )^2. \end{cases} $$
(666)
When inserted into the expressions for the F(R) quantities, these relations give the remarkable result :
$$ F(R_0) = R_0 + 2\varLambda, \qquad F''(R_0) = F'''(R_0) = 0, $$
(667)
so that we obviously conclude that the only F(R) theory having exactly the same cosmographic parameters as the ΛCDM model is just F(R)∝R, i.e., general relativity (GR). It is important to mention that such a result comes out as a consequence of the values of (q0,j0) in the ΛCDM model. Indeed, should we have left (s0,l0) undetermined and only fixed (q0,j0) to the values in (666), we should have got the same result in (667). Since the ΛCDM model fits a large set of different data well, we do expect that the actual values of (q0,j0,s0,l0) do not differ too much from those in ΛCDM model. Therefore, we plug Eqs. (648)–(658) into the following expressions :
$$ \begin{array}{@{}l} q_0 = q_0^{\varLambda} {\times} (1 + \varepsilon_q), \qquad j_0 = j_0^{\varLambda} {\times} (1 + \varepsilon_j),\\[2mm] s_0 = s_0^{\varLambda} {\times} (1 + \varepsilon_s), \qquad l_0 = l_0^{\varLambda} {\times} (1 + \varepsilon_l), \end{array} $$
(668)
with \((q_{0}^{\varLambda}, j_{0}^{\varLambda}, s_{0}^{\varLambda}, l_{0}^{\varLambda})\) given by Eqs. (666) and (εq,εj,εs,εl) quantify the deviations from the values in the ΛCDM model allowed by the data. A numerical estimate of these quantities may be obtained, e.g., from a Markov chain analysis, but this is outside our aims. Since we are here interested in a theoretical examination, we prefer to study an idealized situation where the four quantities above all share the same value ε≪1. In such a case, we can easily investigate how much the corresponding F(R) deviates from GR by exploring the two ratios F″(R0)/F(R0) and F‴(R0)/F(R0). Inserting the above expressions for the cosmographic parameters into the exact (not reported) formulae for F(R0), F″(R0) and F‴(R0), taking those ratios and then expanding to first order in ε, we finally find
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ669_HTML.gif
(669)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ670_HTML.gif
(670)
which are dimensionless quantities and hence more suitable to estimate the order of magnitudes of the different terms. Inserting our fiducial values into \(\varOmega_{\mathrm{M}}^{(0)}\), we have
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ671_HTML.gif
(671)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ672_HTML.gif
(672)
For values of ε up to 0.1, the above relations show that the second and third derivatives are at most two orders of magnitude smaller than the zeroth order term F(R0). Actually, the values of η30 for a baryon only model (first row) seems to argue in favor of a larger importance of the third order term. However, we have numerically checked that the above relations approximates very well the exact expressions up to ε≃0.1 with an accuracy depending on the value of \(\varOmega_{\mathrm{M}}^{(0)}\), being smaller for smaller matter density parameters. Using the exact expressions for η20 and η30, our conclusion on the negligible effect of the second and third order derivatives are significantly strengthened.

Such a result holds under the hypotheses that the narrower are the constraints on the validity of the ΛCDM model, the smaller are the deviations of the cosmographic parameters from the values in the ΛCDM model. It is possible to show that this indeed the case for the CPL parameterization we are considering. On the other hand, we have also assumed that the deviations (εq,εj,εs,εl) take the same values. Although such hypothesis is somewhat ad hoc, we argue that the main results are not affected by giving it away. In fact, although different from each other, we can still assume that all of them are very small so that Taylor expanding to the first order should lead to additional terms into Eqs. (669)–(670) which are likely of the same order of magnitude. We may thus conclude that, if the observations confirm that the values of the cosmographic parameters agree within ∼10 % with those predicted by the ΛCDM model, we must recognize that the deviations of F(R) from the GR case, F(R)∝R, should be vanishingly small.

It should be emphasized however, that such a conclusion only holds for those F(R) models satisfying the constraint (659). It is indeed possible to work out a model having F(R0)∝R0, F″(R0)=F‴(R0)=0, but F(n)(R0)≠0 for some n. For such a (somewhat ad hoc) model, Eq. (659) is clearly not satisfied so that the cosmographic parameters have to be evaluated from the solution of the field equations. Accordingly, the conclusion above does not hold, so that one cannot exclude that the resultant values of (q0,j0,s0,l0) are within 10 % of those in the ΛCDM model.

11.3 The constant EoS model

Let us now take into account the condition w=−1, but still retains wa=0, thus obtaining the so called quiescence models. In such a case, some problems arise because both the terms (j0q0−2) and \({\mathcal{R}}\) may vanish for some combinations of the two model parameters \((\varOmega_{\mathrm{M}}^{(0)}, w_{0})\). For instance, we find that j0q0−2=0 for w0=(w1,w2) with
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ673_HTML.gif
(673)
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ674_HTML.gif
(674)
On the other hand, the equation \(\mathcal{R}(\varOmega_{\mathrm {M}}^{(0)}, w_{0}) = 0\) may have different real roots for w depending on the adopted value of \(\varOmega_{\mathrm{M}}^{(0)}\). Denoting collectively with wnull the values of w0 that, for a given \(\varOmega_{\mathrm{M}}^{(0)}\), make \((j_{0} - q_{0} -2) {\mathcal{R}}(\varOmega_{\mathrm{M}}^{(0)}, w_{0})\) taking the null value, we individuate a set of quiescence models whose cosmographic parameters give rise to divergent values of F(R0), F″(R0) and F‴(R0). For such models, F(R) is clearly not defined, so that we have to exclude these cases from further consideration. We only note that it is still possible to work out an F(R) theory reproducing the same background dynamics of such models, but a different route has to be used.
Since both q0 and j0 now deviate from those in the ΛCDM model it is not surprising that both F″(R0) and F‴(R0) take finite non null values. However, it is more interesting to study the two quantities η20 and η30 defined above to investigate the deviations of F(R) from the GR case. These are plotted in Figs. 1 and 2 for the two fiducial values of \(\varOmega_{\mathrm{M}}^{(0)}\). Note that the range of w0 in these plots have been chosen in order to avoid divergences, but the lessons we will draw also hold for the other w0 values.
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Fig. 1

The dimensionless ratio between the current values of F″(R) and F(R) as a function of the constant EoS w0 of the corresponding quiescence model. Short dashed and solid lines refer to models with \(\varOmega_{\mathrm{M}}^{(0)} = 0.041\) and 0.250, respectively

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Fig. 2

The dimensionless ratio between the current values of F‴(R) and F(R) as a function of the constant EoS w0 of the corresponding quiescence model. Legend is the same as in Fig. 1

As a general comment, it is clear that, even in this case, F″(R0) and F‴(R0) are from two to three orders of magnitude smaller than the zeroth order term F(R0). Such a result could yet be guessed from the previous discussion for the ΛCDM case. Actually, relaxing the hypothesis w0=−1 is the same as allowing the values of the cosmographic parameters to deviate from those in the ΛCDM model. Although a direct mapping between the two cases cannot be established, it is nonetheless evident that such a relation can be argued and hence make the outcome of the above plots not fully surprising. It is nevertheless worth noting that, while in the ΛCDM case, η20 and η30 always have opposite signs, this is not the case for quiescence models with w>−1. Indeed, depending on the value of \(\varOmega_{\mathrm{M}}^{(0)}\), we can have F(R) theories with both η20 and η30 positive. Moreover, the lower \(\varOmega_{\mathrm{M}}^{(0)}\) is, the higher the ratios η20 and η30 are for a given value of w0. This can be explained qualitatively noticing that, for a lower \(\varOmega_{\mathrm{M}}^{(0)}\), the density parameter of the curvature fluid (playing the role of an effective dark energy) must be larger and thus claim for higher values of the second and third derivatives.

11.4 The general case

Finally, we study evolving dark energy models with wa≠0. Needless to say, varying three parameters allows to get a wide range of models that cannot be discussed in detail. Therefore, we only concentrate on evolving dark energy models with w0=−1 in agreement with some most recent analysis. The results on η20 and η30 are plotted in Figs. 3 and 4 where these quantities are displayed as functions of wa. Note that we are exploring models with positive wa so that w(z) can tend to w0+wa>w0 for z→∞ and the EoS for dark energy can eventually approach the dust value w=0. Actually, this is also the range favored by the data. We have, however, excluded values where η20 or η30 diverge. Considering how they are defined, it is clear that these two quantities diverge when F(R0)=0, so that the values of (w0,wa) making (η20,η30) diverge may be found by solving
$$ \mathcal{P}_0(w_0, w_a) \varOmega_{\mathrm{M}}^{(0)} + \mathcal{Q}_0(w_0, w_a) = 0, $$
(675)
where \({\mathcal{P}}_{0}(w_{0}, w_{a})\) and \({\mathcal{Q}}_{0}(w_{0}, w_{a})\) are obtained by inserting Eqs. (662)–(665) into the definitions (652)–(653). For such CPL models, there is no F(R) model having the same cosmographic parameters and, at the same time, satisfying all the criteria needed for the validity of our procedure. In fact, if F(R0)=0, the condition (659) is likely to be violated so that higher than third order must be included in the Taylor expansion of F(R) thus invalidating the derivation of Eqs. (648)–(651).
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Fig3_HTML.gif
Fig. 3

The dimensionless ratio between the present day values of F″(R) and F(R) as function of the wa parameter for models with w0=−1. Legend is the same as in Fig. 1

https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Fig4_HTML.gif
Fig. 4

The dimensionless ratio between the present day values of F‴(R) and F(R) as function of the wa parameter for models with w0=−1. Legend is the same as in Fig. 1

Under these caveats, Figs. 3 and 4 demonstrate that allowing the EoS for dark energy to evolve does not change significantly our conclusions. Indeed, the second and third derivatives, although being not null, are nevertheless negligible with respect to the zeroth order term, and therefore the consequence is in favor of a GR-like F(R) with only very small corrections. Such a result is, however, not fully unexpected. From Eqs. (662) and (663), we see that, having set w0=−1, the parameter q0 is the same as that in the ΛCDM model, while j0 reads \(j_{0}^{\varLambda} + (3/2)(1 - \varOmega_{\mathrm{M}}^{(0)}) w_{a}\). As we have stressed above, the Einstein-Hilbert Lagrangian F(R)=R+2Λ is recovered when \((q_{0}, j_{0}) = (q_{0}^{\varLambda}, j_{0}^{\varLambda})\) whatever the values of (s0,l0) are. Introducing a wa≠0 makes (s0,l0) differ from those in the ΛCDM model, but the first two cosmographic parameters are only mildly affected. Such deviations are then partially washed out by the complicated way they enter in the determination of the values of F(R) at the present time and its first three derivatives.

12 Theoretical constraints on the model parameters

In the preceding section, we have worked out a method to estimate F(R0), F″(R0) and F‴(R0) resorting to a model independent parameterization of the EoS for dark energy. However, in the ideal case, the cosmographic parameters are directly estimated from the data so that Eqs. (648)–(658) can be used to infer the values of the quantities related to F(R). These latter can then be used to put constraints on the parameters entering an assumed fourth order theory assigned by an F(R) function characterized by a set of parameters p=(p1,…,pn) provided that the hypotheses underlying the derivation of Eqs. (648)–(658) are indeed satisfied. We show below two interesting cases which clearly highlight the potentiality and the limitations of such an analysis.

12.1 Double power law Lagrangian

As a first interesting example, we take (Nojiri and Odintsov 2003c)
$$ F(R) = R \bigl(1 + \alpha R^{n} + \beta R^{-m} \bigr) $$
(676)
with n and m two positive real numbers. The following expressions are immediately obtained:
$$ \begin{cases} F(R_0) = R_0 (1 + \alpha R_0^{n} + \beta R_0^{-m} ), \\ F'(R_0) = 1 + \alpha(n + 1) R_0^n - \beta(m - 1) R_0^{-m}, \\ F''(R_0) = \alpha n (n + 1) R_0^{n - 1} + \beta m (m - 1) R_0^{-(1 + m)}, \\ F'''(R_0) = \alpha n (n + 1) (n - 1) R_0^{n - 2} \\ \phantom{F'''(R_0) =}{}- \beta m (m + 1) (m - 1) R_0^{-(2 + m)}. \end{cases} $$
(677)
Denoting the values of F(i)(R0) determined through Eqs. (648)–(658) by ϕi (with i=0,…,3), we can solve
$$ \begin{cases} F(R_0) = \phi_0, \\ F'(R_0) = \phi_1, \\ F''(R_0) = \phi_2, \\ F'''(R_0) = \phi_3, \end{cases} $$
(678)
which is a system of four equations in the four unknowns (α,β,n,m) that can analytically be solved proceeding as follows. First, we solve the first and second equation with respect to (α,β) and we obtain
$$ \begin{cases} \alpha= \displaystyle{\frac{1 - m}{n + m} \biggl( 1 - \frac{\phi_0}{R_0} \biggr) R_0^{-n}}, \\ \beta= \displaystyle{- \frac{1 + n}{n + m} \biggl( 1 - \frac{\phi_0}{R_0} \biggr) R_0^{m}}, \end{cases} $$
(679)
while, solving the third and fourth equations, we get
$$ \begin{cases} \alpha= \displaystyle{\frac{\phi_2 R_0^{1 - n} [ 1 + m + (\phi_3/\phi_2) R_0 ]}{n (n + 1) (n + m)}}, \\ \beta= \displaystyle{\frac{\phi_2 R_0^{1 + n} [ 1 - n + (\phi_3/\phi_2) R_0 ]}{m (1 - m) (n + m)}}. \end{cases} $$
(680)
Equating the two solutions, we acquire a system of two equations in the two unknowns (n,m), given by
$$ \begin{cases} \displaystyle{\frac{n (n + 1) (1 - m) ( 1 - \phi_0/R_0 )}{\phi_2 R_0 [ 1 + m + (\phi_3/\phi_2) R_0 ]}} = 1, \\ \displaystyle{\frac{m (n + 1) (m - 1) ( 1 - \phi_0/R_0 )}{\phi_2 R_0 [ 1 - n + (\phi_3/\phi_2) R_0 ]}} = 1. \end{cases} $$
(681)
Solving with respect to m, we find two solutions. The first one is m=−n, which has to be discarded because it makes (α,β) go to infinity. The only acceptable solution is
$$ m = - \bigl[ 1 - n + (\phi_3/\phi_2) R_0 \bigr] $$
(682)
which, inserted back into the above system, leads to a second order polynomial equation for n with solutions
$$ n = \frac{1}{2} \biggl[1 + \frac{\phi_3}{\phi_2} R_0 {\pm} \frac{\sqrt{{\mathcal{N}}(\phi_0, \phi_2, \phi_3)}}{\phi_2 R_0 (1 + \phi_0/R_0)} \biggr]. $$
(683)
Here, we have defined
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ684_HTML.gif
(684)
Depending on the values of (q0,j0,s0,l0), Eq. (683) may lead to one, two or any acceptable solution, i.e. real positive values of n. This solution has to be inserted back into Eq. (682) to determine m and then into Eqs. (679) or (680) to estimate (α,β). If the final values of (α,β,n,m) are physically viable, we can conclude that the model in Eq. (676) is in agreement with the data giving the same cosmographic parameters inferred from the data themselves. Exploring analytically what is the region of parameter space of (q0,j0,s0,l0) which leads to acceptable solutions of (α,β,n,m) is a daunting task far outside the aim of the present work.

12.2 The Hu-Sawicki model

One of the most pressing problems of F(R) theories is the need to escape the severe constraints imposed by the Solar System tests. A successful model has recently been proposed by Hu and Sawicki (Hu and Sawicki 2007) (HS)4
$$ F(R) = R - R_{\mathrm{c}} \frac{\alpha(R/R_{\mathrm{c}})^n}{1 + \beta (R/R_{\mathrm{c}})^n}. $$
(685)
As for the double power law model discussed above, there are four parameters which we can be expressed in terms of the cosmographic parameters (q0,j0,s0,l0).
As a first step, it is trivial to have
$$ \begin{cases} F(R_0) = \displaystyle{R_0 - R_{\mathrm{c}} \frac{\alpha R_{0 \mathrm {c}}^n}{1 + \beta R_{0 \mathrm{c}}^n}}, \\[4mm] F'(R_0) = \displaystyle{1 - \frac{\alpha n R_{\mathrm{c}} R_{0 \mathrm {c}}^{n}}{R_0 (1 + \beta R_{0 \mathrm{c}}^n)^2}}, \\[4mm] F''(R_0) = \displaystyle{\frac{\alpha n R_{\mathrm{c}} R_{0 \mathrm{c}}^n [ (1 - n) + \beta(1 + n) R_{0 \mathrm{c}}^n ]}{R_0^2 (1 + \beta R_{0 \mathrm{c}}^n)^3}}, \\[4mm] F'''(R_0) = \displaystyle{\frac{\alpha n R_{\mathrm{c}} R_{0 \mathrm {c}}^n (A n^2 + B n + C)}{R_0^3 (1 + \beta R_{0 \mathrm{c}}^n)^4}}, \end{cases} $$
(686)
with R0c=R0/Rc and
$$ \begin{cases} A = - \beta^2 R_{0 \mathrm{c}}^{2n} + 4 \beta R_{0 \mathrm{c}}^n - 1, \\ B = 3 (1 - \beta^2 R_{0 \mathrm{c}}^{2n} ), \\ C = -2 (1 - \beta R_{0 \mathrm{c}}^{n} )^2. \end{cases} $$
(687)
Equating equations in (686) to the four quantities (ϕ0,ϕ1,ϕ2,ϕ3) defined as above, we could, in principle, solve this system of four equations in four unknowns to obtain (α,β,Rc,n) in terms of (ϕ0,ϕ1,ϕ2,ϕ3). Then, by using Eqs. (648)–(658) we acquire the expressions of (α,β,Rc,n) as functions of the cosmographic parameters. However, setting ϕ1=1 as required by Eq. (649) gives the only trivial solution αnRc=0 so that the HS model reduces to the Einstein-Hilbert Lagrangian F(R)=R. In order to escape this problem, we can relax the condition F′(R0)=1 to F′(R0)=(1+ε)−1. As we have discussed in Sect. 11, this is the same as assuming that the current effective gravitational constant Geff,0=G/F′(R0) only slightly differs from the usual Newton’s one, which seems to be a quite reasonable assumption. Under this hypothesis, we can analytically solve the equations for (α,β,Rc,n) in terms of (ϕ0,ε,ϕ2,ϕ3). The actual values of (ϕ0,ϕ2,ϕ3) will be no more given by Eqs. (648)–(651), but we have checked that they deviate from those expressions5 much less than 10 % for ε up to 10 % well below any realistic expectation.
With this caveat in mind, we first solve
$$ F(R_0) = \phi_0, \qquad F''(R_0) = (1 + \varepsilon)^{-1}, $$
(688)
to acquire
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Eque_HTML.gif
Inserting these expressions into the equations in (686), it is easy to check that Rc cancels out, so that we can no more determine its value. Such a result is, however, not unexpected. Indeed, Eq. (685) can trivially be rewritten as
$$ F(R) = R - \frac{\tilde{\alpha} R^n}{1 + \tilde{\beta} R^n} $$
(689)
with \(\tilde{\alpha} = \alpha R_{\mathrm{c}}^{1 - n}\) and \(\tilde {\beta } = \beta R_{\mathrm{c}}^{-n}\) which are indeed the quantities determined by the above expressions for (α,β). Reversing the discussion, the current values of F(i)(R) depend on (α,β,Rc) only through the two parameters \((\tilde{\alpha}, \tilde{\beta})\). Accordingly, the use of cosmographic parameters is unable to break this degeneracy. However, since Rc only plays the role of a scaling parameter, we can arbitrarily set its value without loss of generality.
On the other hand, this degeneracy allows us to get a consistency relation to immediately check whether the HS model is viable or not. Indeed, solving the equation F″(R0)=ϕ2, we find
$$ n = \frac{ (\phi_0/R_0 ) + [(1 + \varepsilon)/\varepsilon ](1 - \phi_2 R_0) - (1 - \varepsilon)/(1 + \varepsilon)}{1 - \phi_0/R_0}, $$
(690)
which can then be inserted into the equations F‴(R0)=ϕ3 to obtain a complicated relation among (ϕ0,ϕ2,ϕ3) which we do not report for sake of shortness. Solving such a relation with respect to ϕ3/ϕ0 and executing the Taylor expansion to first order in ε, the resultant constraint reads
$$ \frac{\phi_3}{\phi_0} \simeq- \frac{1 + \varepsilon}{\varepsilon} \frac{\phi_2}{R_0} \biggl[ R_0 \biggl( \frac{\phi_2}{\phi_0} \biggr) + \frac{\varepsilon\phi_0^{-1}}{1 + \varepsilon} \biggl( 1 - \frac{2 \varepsilon}{1 - \phi_0/R_0} \biggr) \biggr]. $$
(691)
If the cosmographic parameters (q0,j0,s0,l0) are known with sufficient accuracy, one could compute the values of (R0,ϕ0,ϕ2.ϕ3) for a given ε (eventually, using the expressions obtained for ε=0) and then check if they satisfied this relation. If this is not the case, one can immediately give off the HS model also without the need of solving the field equations and fitting the data. In fact, given the still large errors on the cosmographic parameters, such a test only remains in the realm of (quite distant) future applications. However, the HS model works for other tests as shown in Hu and Sawicki (2007) and thus a consistent cosmography analysis has to be combined with them.

13 Constraints coming from observational data

Equations (648)–(658) relate the values of F(R) and its first three derivatives at the present time to the cosmographic parameters (q0,j0,s0,l0) and the matter density \(\varOmega_{\mathrm{M}}^{(0)}\). In principle, therefore, a measurement of these latter quantities makes it possible to put constraints on F(i)(R0), with i={0,…,3}, and hence on the parameters of a given fourth order theory through the method shown in the previous section. Actually, the cosmographic parameters are affected by errors which obviously propagate onto the F(R) quantities. Indeed, since the covariance matrix for the cosmographic parameters is not diagonal, one has to also take care of this fact to estimate the final errors on F(i)(R0). A similar discussion also holds for the errors on the dimensionless ratios η20 and η30 introduced above. As a general rule, indicating with \(g(\varOmega_{\mathrm{M}}^{(0)}, \mathbf{p})\) a generic F(R) related quantity depending on \(\varOmega_{\mathrm{M}}^{(0)}\) and the set of cosmographic parameters p, its uncertainty reads
$$ \sigma_{g}^2 = \biggl\vert \frac{\partial g}{\partial\varOmega_{\mathrm {M}}^{(0)}} \biggr \vert^2 \sigma_{M}^2 + \sum _{i = 1}^{i = 4}{ \biggl\vert \frac{\partial g}{\partial p_i} \biggr \vert^2 \sigma_{p_i}^2} + \sum _{i \neq j}{2 \frac{\partial g}{\partial p_i} \frac{\partial g}{\partial p_j} C_{ij}}, $$
(692)
where Cij are the elements of the covariance matrix (being \(C_{ii} = \sigma_{p_{i}}^{2}\)), we have set (p1,p2,p3,p4)=(q0,j0,s0,l0). and assumed that the error σM on \(\varOmega_{\mathrm{M}}^{(0)}\) is uncorrelated with those on p. Note that this latter assumption strictly holds if the matter density parameter is estimated from an astrophysical method (such as estimating the total matter in the universe from the evaluated halo mass function). Alternatively, we will assume that \(\varOmega_{\mathrm{M}}^{(0)}\) is constrained by the CMB radiation related to experiments. Since these latter mainly probes the very high redshift universe (zzlss≃1089), while the cosmographic parameters are concerned with the cosmos at the present time, one can argue that the determination of \(\varOmega_{\mathrm{M}}^{(0)}\) is not affected by the details of the model adopted for describing the late universe. Indeed, we can reasonably assume that, whatever the candidate for dark energy or alternative gravitational theory such as F(R) gravity is, the decoupling epoch, i.e., the era when we can observe through the CMB radiation, is well approximated by the standard GR with a model comprising only dust matter. Hence, we will make the simplifying (but well motivated) assumption that σM may be reduced to very small values and is uncorrelated with the cosmographic parameters.
Under this assumption, the problem of estimating the errors on \(g(\varOmega_{\mathrm{M}}^{(0)}, \mathbf{p})\) reduces to the analysis of the covariance matrix for the cosmographic parameters given the details of the data set used as observational constraints. We address this issue by computing the Fisher information matrix (see, e.g., Tegmark et al. 1997 and references therein) defined as
$$ F_{ij} = \biggl\langle\frac{\partial^2 L}{\partial\theta_i \partial\theta_j} \biggr\rangle, $$
(693)
with \(L = -2 \ln{{\mathcal{L}}(\theta_{1}, \ldots, \theta_{n})}\), where \({\mathcal{L}}(\theta_{1}, \ldots, \theta_{n})\) is the likelihood of the experiment and (θ1,…,θn) is the set of parameters to be constrained, and 〈…〉 denotes the expectation value. In fact, the expectation value is computed by evaluating the Fisher matrix elements for fiducial values of the model parameters (θ1,…,θn), while the covariance matrix C is finally obtained as the inverse of F.
A key ingredient in the computation of F is the definition of the likelihood which depends, of course, of what experimental constraint one is using. To this aim, it is worth remembering that our analysis is based on fifth order Taylor expansion of the scale factor a(t) so that we can only rely on observational tests probing quantities that are well described by this truncated series. Moreover, since we do not assume any particular model, we can only characterize the background evolution of the Universe, but not its dynamics which, being related to the evolution of perturbations, unavoidably need the specification of a physical model. As a result, the SNeIa Hubble diagram is the ideal test to constrain the cosmographic parameters. We therefore defined the likelihood as
https://static-content.springer.com/image/art%3A10.1007%2Fs10509-012-1181-8/MediaObjects/10509_2012_1181_Equ694_HTML.gif
(694)
where the distance modulus to redshift z is described as
$$ \mu_{\mathrm{th}}(z, H_0, \mathbf{p}) = 25 + 5 \log_{10} \biggl(\frac {c}{H_0} \biggr) + 5 \log_{10} {D_L(z, \mathbf{p})}. $$
(695)
Here, DL(z) is the Hubble free luminosity distance and it follows from Eq. (9) that DL(z) in the flat universe is expressed as
$$ D_L(z) = (1 + z) \int_{0}^{z}{\frac{dz}{H(z)/H_0}}. $$
(696)
Using the fifth order Taylor expansion of the scale factor, we find an analytical expression for DL(z,p), so that in the computation of Fij no numerical integration need (this consequence makes the estimate faster). As a last ingredient, we need to specify the details of the SNeIa survey giving the redshift distribution of the sample and the error on each measurement. Following (Kim et al. 2004), we adopt6
$$ \sigma(z) = \sqrt{\sigma_{\mathrm{sys}}^2 + \biggl( \frac {z}{z_{\mathrm{max}}} \biggr)^2 \sigma_m^2} $$
(697)
where zmax is the maximum redshift of the survey, σsys is an irreducible scatter in the SNeIa distance modulus, and σm is to be assigned depending on the photometric accuracy.
In order to run the Fisher matrix calculation, we have to set a fiducial model, according to the predictions in the ΛCDM model for the cosmographic parameters. For \(\varOmega_{\mathrm{M}}^{(0)} = 0.3\) and the Hubble constant h=0.72 Freedman et al. (2001) in units of 100 km/s/Mpc, we acquire
$$ (q_0, j_0, s_0, l_0) = (-0.55, 1.0, -0.35, 3.11). $$
(698)
As a first consistency check, we compute the Fisher matrix for a survey mimicking the recent database in Davis et al. (2007) and thus set \(({\mathcal{N}}_{\mathrm{SNeIa}}, \sigma_{m}) = (192, 0.33)\). After marginalizing over h (which, as well known, is fully degenerate with the SNeIa absolute magnitude \({\mathcal{M}}\)), we find the uncertainties
$$ (\sigma_1, \sigma_2, \sigma_3, \sigma_4) = (0.38, 5.4, 28.1, 74.0), $$
(699)
where we are still using the indexing introduced above for the cosmographic parameters. These values are reasonably well compared with those obtained from a cosmographic fitting of the Gold SNeIa dataset (John 2004; John 2005)7
$$ \begin{array}{@{}l} q_0 = -0.90 {\pm} 0.65, \qquad j_0 = 2.7 {\pm} 6.7,\\[2mm] s_0 = 36.5 {\pm} 52.9, \qquad l_0 = 142.7 {\pm} 320. \end{array} $$
(700)
Because of the Gaussian assumptions we rely on, the Fisher matrix forecasts are known to be lower limits on the determination of a set of parameters, the accuracy to which a given experiment can attain. This is indeed the case with the comparison suggesting that our predictions are quite optimistic. It is worth stressing, however, that the analysis in John (2004, 2005) used the Gold SNeIa dataset which is poorer in high redshift SNeIa than the one in Davis et al. (2007) we are mimicking, so that larger errors on the higher order parameters (s0,l0) could be expected.
Rather than computing the errors on F(R0) and its first three derivatives, it is more interesting to look at the precision attainable on the dimensionless ratios (η20,η30) introduced above because they quantify how much deviations from the linear order exist. For the fiducial model we are considering, both η20 and η30 vanish, while, using the covariance matrix for a survey at the present time and setting \(\sigma_{M}/\varOmega_{\mathrm{M}}^{(0)} \simeq10~\%\), their uncertainties read
$$ (\sigma_{20}, \sigma_{30}) = (0.04, 0.04). $$
(701)
As an application, we can look at Figs. 1 and 2 showing how (η20,η30) depend on the current value of the EoS w0 for F(R) models sharing the same cosmographic parameters of a dark energy model with its constant EoS. As it is clear, also considering only the 1σ range, the full region plotted is allowed by such large constraints on (η20,η30). Thus, this means that the full class of corresponding F(R) theories is viable. As a consequence, we may conclude that the SNeIa data at the present time are unable to discriminate between a Λ dominated universe and this class of fourth order gravity theories.
As a next step, we investigate a SNAP-like survey (Aldering et al. 2004) and therefore take \(({\mathcal{N}}_{\mathrm{SNeIa}}, \sigma_{m}) = (2000, 0.02)\). We use the same redshift distribution in Table 1 of Kim et al. (2004) and add 300 nearby SNeIa in the redshift range (0.03,0.08). The Fisher matrix calculation gives the uncertainties on the cosmographic parameters
$$ (\sigma_1, \sigma_2, \sigma_3, \sigma_4) = (0.08, 1.0, 4.8, 13.7). $$
(702)
The significant improvement of the accuracy in the determination of (q0,j0,s0,l0) is translated into a reduction of the errors on (η20,η30), which is now given by
$$ (\sigma_{20}, \sigma_{30}) = (0.007, 0.008), $$
(703)
where we have supposed that, when SNAP data will be available, the matter density parameter \(\varOmega_{\mathrm{M}}^{(0)}\) would be determined with a precision \(\sigma_{M}/\varOmega_{\mathrm{M}}^{(0)} \sim1~\%\). Looking again at Figs. 1 and 2, it is clear that the situation is improved. Indeed, the constraints on η20 makes it possible to narrow the range of allowed models with low matter content (the dashed line), while models with typical values of \(\varOmega_{\mathrm{M}}^{(0)}\) are still viable for w0 covering almost the full horizontal axis. On the other hand, the constraint on η30 is still too weak, so that almost the full region plotted can be allowed.
Finally, we examine an hypothetical future SNeIa survey working at the same photometric accuracy as SNAP and with the same redshift distribution, but increasing the number of SNeIa up to \({\mathcal{N}}_{\mathrm{SNeIa}} = 6 \times 10^{4}\) as expected from, e.g., DES (Abbott et al. 2005), PanSTARRS (Kaiser 2005), SKYMAPPER (Schmidt et al. 2005), while still larger numbers may potentially be observed by ALPACA (Corasaniti et al. 2006) and LSST (Tyson 2002). Such a survey can achieve
$$ (\sigma_1, \sigma_2, \sigma_3, \sigma_4) = (0.02, 0.2, 0.9, 2.7) $$
(704)
so that, with \(\sigma_{M}/\varOmega_{\mathrm{M}}^{(0)} \sim0.1~\%\), we obtain
$$ (\sigma_{20}, \sigma_{30}) = (0.0015, 0.0016). $$
(705)
Figure 1 shows that, with such a precision on η20, the region of w0 values allowed essentially reduces to the value in the ΛCDM model, while, from Fig. 2, it is clear that the constraint on η30 definitively excludes models with low matter content further reducing the range of values of w0 to quite small deviations from the w0=−1. We can therefore conclude that such a survey will be able to discriminate between the concordance ΛCDM model and all the F(R) theories giving the same cosmographic parameters as quiescence models other than the ΛCDM itself.

A similar discussion may be repeated for F(R) models sharing the same values of (q0,j0,s0,l0) as those in the CPL model even if it is less intuitive to grasp the efficacy of the survey being the parameter space multivalued. For the same reason, we have not explored what is the accuracy on the double power-law or HS models, even if this is technically possible. In fact, one should first estimate the errors on the current values of F(R) and its three time derivatives and then propagate them on the model parameters by using the expressions obtained above.

In conclusion, notwithstanding the common claim that we live in the era of precision cosmology, the constraints on (q0,j0,s0,l0) are still too weak to efficiently apply the program we have outlined above. We have shown how it is possible to establish a link between the popular CPL parameterization of the EoS for dark energy and the derivatives of F(R), imposing that they share the same values of the cosmographic parameters. This analysis has led to the quite interesting conclusion that the only F(R) function, which is able to give the same values of (q0,j0,s0,l0) as those in the ΛCDM model, is indeed F(R)=R+2Λ. A similar conclusion holds also in the case of f(T) gravity (Capozziello et al. 2011a). If future observations will inform us that the cosmographic parameters are those of the ΛCDM model, we can therefore rule out all F(R) theories satisfying the hypotheses underlying our derivation of Eqs. (648)–(651). Actually, such a result should not be considered as a no way out for higher order gravity. Indeed, one could still work out a model with null values of F″(R0) and F‴(R0) as required by the above constraints, but non-vanishing higher order derivatives. One could well argue that such a contrived model could be rejected on the basis of the Occam razor, but nothing prevents from still taking it into account if it turns out to be both in agreement with the data and theoretically well founded.

If new SNeIa surveys will determine the cosmographic parameters with good accuracy, acceptable constraints on the two dimensionless ratios η20F″(R0)/F(R0) and η30F‴(R0)/F(R0) could be obtained, and thus these quantities allow us to discriminate among rival F(R) theories. To investigate whether such a program is feasible, we have pursued a Fisher matrix based forecasts of the accuracy, which future SNeIa surveys can achieve, on the cosmographic parameters and hence on (η20,η30). It turns out that a SNAP-like survey can start giving interesting (yet still weak) constraints allowing us to reject F(R) models with low matter content, while a definitive improvement is achievable with future SNeIa survey observing ∼104 objects and hence makes it possible to discriminate between the ΛCDM model and a large class of fourth order theories. It is worth emphasizing, however, that the measurement of \(\varOmega_{\mathrm{M}}^{(0)}\) should come out as the result of a model independent probe such as the gas mass fraction in galaxy clusters which is, at present, still far from the 1 % requested precision. On the other hand, one can also rely on the \(\varOmega_{\mathrm{M}}^{(0)}\) estimate from the anisotropy and polarization spectra of the CMB radiation even if this comes to the price of assuming that the physics at recombination is strictly described by GR, so that one has to limit its attention to F(R) models reducing to F(R)∝R during that epoch. However, such an assumption is quite common in many F(R) models available in literature and therefore it is not a too restrictive limitation.

A further remark is in order concerning what kind of data can be used to constrain the cosmographic parameters. The use of the fifth order Taylor expansion of the scale factor makes it possible to not specify any underlying physical model by relying on the minimalist assumption that the universe is described by the flat FLRW metric. While useful from a theoretical perspective, such a generality puts severe limitations to the dataset one can use. Actually, we can only resort to observational tests depending only on the background evolution so that the range of astrophysical probes reduces to standard candles (such as SNeIa and possibly GRBs (Capozziello and Izzo 2008)) and standard rods (such as the angular size-redshift relation for compact radiosources). Moreover, pushing the Hubble diagram to z∼2 may rise the question of the impact of gravitational lensing amplification on the apparent magnitude of the adopted standard candle. The magnification probability distribution function depends on the growth of perturbations (Holz and Wald 1998; Holz and Linder 2005; Hui and Greene 2006; Frieman 1997; Cooray et al. 2006), so that one should worry about the underlying physical model in order to estimate whether this effect biases the estimate of the cosmographic parameters. However, it has been shown (Riess et al. 2007; Jonsson et al. 2006; Gunnarsson et al. 2006; Nordin et al. 2008; Sarkar et al. 2008) that the gravitational lensing amplification does not alter the measured distance modulus for z∼1 SNeIa significantly. Although such an analysis has been executed for models based on GR, we can argue that, whatever the F(R) model is, the growth of perturbations finally leads to a distribution of structures along the line of sight that is as similar as possible to the observed one so that the lensing amplification can be approximately the same. We can therefore discuss that the systematic error made by neglecting lensing magnification is lower than the statistical ones expected by the future SNeIa surveys. On the other hand, one can also try reducing this possible bias further by using the method of flux averaging (Wang 2000) even if, in such a case, our Fisher matrix calculation should be repeated accordingly. Furthermore, it is significant to note that the constraints on the cosmographic parameters may be tightened by imposing some physically motivated priors in the parameter space. For instance, we can suppose that the Hubble parameter H(z) always stays positive over the full range probed by the data or that the transition from past deceleration to present acceleration takes place over the range probed by the data (so that we can detect it). Such priors should be included in the likelihood definition so that the Fisher matrix should be recomputed. This is left for a forthcoming work.

Although the data at the present time are still too limited to efficiently discriminate among rival dark energy models, we are confident that an aggressive strategy aiming at a very precise determination of the cosmographic parameters could offer stringent constraints on higher order gravity without the need of solving the field equations or addressing the complicated problems related to the growth of perturbations. Almost 80 years after the pioneering distance-redshift diagram by Hubble, the old cosmographic approach appears nowadays as a precious observational tool to investigate the new developments of cosmology.

14 Conclusion

In summary, we have presented the review of a number of popular dark energy models, such as the ΛCDM model, Little Rip and Pseudo-Rip scenarios, the phantom and quintessence cosmologies with the four types (I, II, III and IV) of the finite-time future singularities and non-singular universes filled with dark energy.

In the first part, we have explained the ΛCDM model and recent various cosmological observations to give the bounds on the late-time acceleration of the universe. Furthermore, we have investigated a fluid description of the universe in which the dark fluid has a general form of the EoS covering the inhomogeneous and imperfect EoS. We have explicitly shown that all the dark energy cosmologies can be realized by different fluids and also considered their properties. It has also been demonstrated that at the current stage the cosmological evolutions of all the dark energy universes may be similar to that of the ΛCDM model, and hence these models are compatible with the cosmological observations. In particular, we have intensively studied the equivalence of different dark energy models. We have described single and multiple scalar field theories, tachyon scalar theory and holographic dark energy, in which the quintessence/phantom cosmology with the current cosmic acceleration can be represented, and eventually verified those equivalence to the corresponding fluid descriptions.

In the second part, as another equivalent class of dark energy models, in which dark energy has its geometrical origins, namely, modifications of gravitational theories, we have examined F(R) gravity including its extension to F(R) Hořava-Lifshitz gravity and f(T) gravity. It has clearly been explored that in these models, the ΛCDM model or the late-time cosmic acceleration with the quintessence/phantom behavior can be performed.

Finally, it is significant to remark that there are a number of various dark energy models which we did not discuss in this review, such as \(F(\mathcal{G})\) gravity (Nojiri and Odintsov 2005b; Nojiri et al. 2006; Li et al. 2007), where \(\mathcal{G} \equiv R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho \sigma }R^{\mu\nu\rho\sigma}\) with Rμν and Rμνξσ being the Ricci tensor Riemann tensors, respectively, is the Gauss-Bonnet invariant, \(F(R,\mathcal{G})\) gravity (Cognola et al. 2006), scalar-Gauss-Bonnet dark energy (Nojiri et al. 2005a; Bamba et al. 2007), k-essence dark energy models (Chiba et al. 2000; Armendariz-Picon et al. 2000, 2001), ghost condensates scenario (Arkani-Hamed et al. 2004a) (for its extension to inflation, see Arkani-Hamed et al. 2004b), viscous dark energy (Brevik and Gorbunova 2005; Cataldo et al. 2005; Ren and Meng 2006; Hu and Meng 2006), non-minimal derivative dark energy models (Capozziello et al. 2000; Daniel and Caldwell 2007; Jimenez and Maroto 2009; Saridakis and Sushkov 2010; Germani and Kehagias 2010; Granda and Escobar 2009; Granda 2011), G-essence dark energy models (Myrzakulov 2010b; Bamba et al. 2012g), non-local gravity (Deser and Woodard 2007; Nojiri and Odintsov 2008b; Nojiri et al. 2011b; Zhang and Sasaki 2012) produce by quantum effect, which is investigated to account for the coincidence problem of dark energy and dark matter, and galileon dark energy models (Nicolis et al. 2009; Deffayet et al. 2009a, 2009b, 2010a, 2010b; Shirai et al. 2012) (for its application to inflation, called G-inflation, which has recently been proposed, see Kobayashi et al. 2010, 2011a, 2011b; Kamada et al. 2011) [as recent reviews on galileon models, see, e.g., Trodden and Hinterbichler 2011; de Rham 2012]. In particular, galileon gravity has recently been studied very extensively in the literature. The most important feature of the Lagrangian for the galileon scalar field is that the equation of motion derived from the Lagrangian is up to the second-order, so that the appearance of an extra degree of freedom with the existence of a ghost can be avoided. The galileon field originates from a brane bending mode in the Dvali-Gabadadze-Porrati (DGP) brane world scenario (Dvali et al. 2000; Deffayet et al. 2002), and therefore galileon gravity might be regarded as an indirect resolution for the issue of a ghost in the self-accelerating branch of the DGP model. Since we have no enough space to describe the details of all these models, we again mention the important procedure of our approach to show the equivalence of dark energy models to represent each cosmology. In all of the above models, it follows from Eqs. (42) and (43) that in the flat FLRW background the gravitational equations can be described as H2=(κ2/3)ρDE and \(\dot{H} = -(\kappa^{2}/2 ) ( \rho_{\mathrm{DE}} + P_{\mathrm{DE}} )\). In each model, the difference is only the forms of the energy density ρDE and pressure PDE of dark energy. Hence, the expression of the Hubble parameter H to describe the concrete cosmology, e.g., the ΛCDM, quintessence and phantom cosmologies, can be reconstructed by using these gravitational field equations. Similarly, by applying ρDE and PDE, the EoS wDE≡DE/ρDE in the fluid description in Eq. (40) with Eq. (41) can also be presented.

Finally, it is worth stressing the role of cosmography in this discussion. As shown, it is a fundamental tool because it allows, in principle, to discriminate among models without a priori assumptions but just laying on constraints coming from data. However, the main criticism to this approach is related to the extension of the Hubble series, the quality and the richness of data samples. In particular, observations cannot be extended at any redshift and, in most of cases, are not suitable to track models up to early epochs. However, the forthcoming observational campaigns should ameliorate the situation removing the degeneration emerging at low redshifts and allowing a deeper insight of models.

Footnotes
1

It has also been examined in Stefancic (2005) that for α<0, when ρ→0, there can appear the Type II singularity.

 
2

For clarity, we use the notation “F(R)” gravity and “f(T)” gravity throughout this review.

 
3

In this section, the metric signature of (+,−,−,−) is adopted.

 
4

Note that such a model does not pass the matter instability test and therefore some viable generalizations (Nojiri and Odintsov 2007c, 2008c; Cognola et al. 2008; Bamba et al. 2012b) have been proposed.

 
5

Note that the correct expressions for (ϕ0,ϕ2,ϕ3) may still formally be written as Eqs. (648)–(651), but the polynomials entering them are now different and also depend on powers of ε.

 
6

Note that, in Kim et al. (2004), the authors assume the data are separated in redshift bins so that the error becomes \(\sigma^{2} = \sigma_{sys}^{2}/{\mathcal{N}}_{bin} + {\mathcal{N}}_{bin} (z/z_{max})^{2} \sigma_{m}^{2}\) with \({\mathcal{N}}_{bin}\) the number of SNeIa in a bin. However, we prefer to not bin the data so that \({\mathcal{N}}_{bin} = 1\).

 
7

Actually, such estimates have been obtained by computing the mean and the standard deviation from the marginalized likelihoods of the cosmographic parameters. Hence, the central values do not represent exactly the best fit model, while the standard deviations do not give a rigorous description of the error because the marginalized likelihoods are manifestly non-Gaussian. Nevertheless, we are mainly interested in an order of magnitude estimate so that we would not care about such statistical details.

 

Acknowledgements

First of all, all of us would like to thank all the collaborators in our works explained in this review: Artyom V. Astashenok, Iver Brevik, Vincenzo F. Cardone, Mariafelicia De Laurentis, Emilio Elizalde, Paul H. Frampton, Chao-Qiang Geng, Diego Sáez-Gómez, Zong-Kuan Guo, Shih-Hao Ho, Yusaku Ito, Win-Fun Kao, Shota Kumekawa, Ruth Lazkoz, Antonio Lopez-Revelles, Kevin J. Ludwick, Chung-Chi Lee, Ling-Wei Luo, Jiro Matsumoto, Ratbay Myrzakulov, Nobuyoshi Ohta, Rio Saitou, Vincenzo Salzano, Misao Sasaki, Robert J. Scherrer, Lorenzo Sebastiani, Norihito Shirai, Yuta Toyozato, Shinji Tsujikawa, Jun’ichi Yokoyama, Artyom V. Yurov and Sergio Zerbini. K.B. and S.D.O. would like to acknowledge the very kind hospitality as well as support at Eurasian National University. S.D.O. also appreciates the Japan Society for the Promotion of Science (JSPS) Short Term Visitor Program S11135 and the very warm hospitality at Nagoya University where the work has progressed. The work is supported in part by Global COE Program of Nagoya University (G07) provided by the Ministry of Education, Culture, Sports, Science & Technology (S.N.); the JSPS Grant-in-Aid for Scientific Research (S) # 22224003 and (C) # 23540296 (S.N.); and MEC (Spain) project FIS2010-15640 and AGAUR (Catalonia) 2009SGR-994 (S.D.O.).