Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests
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DOI: 10.1007/s10509-012-1181-8
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- Bamba, K., Capozziello, S., Nojiri, S. et al. Astrophys Space Sci (2012) 342: 155. doi:10.1007/s10509-012-1181-8
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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 energyCosmology1 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) w_{DE}, which is the ratio of the pressure P to the energy density ρ_{DE} of dark energy, defined as w_{DE}≡P_{DE}/ρ_{DE}. 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 w_{DE} is a constant and exactly equal to −1, quintessence model, where w_{DE} is a dynamical quantity and −1<w_{DE}<−1/3, and phantom model, where w_{DE} also varies in time and w_{DE}<−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 k_{B}=c=ħ=1 and denote the gravitational constant 8πG by κ^{2}≡8π/M_{Pl}^{2} with the Planck mass of M_{Pl}=G^{−1/2}=1.2×10^{19} 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
If there only exists cosmological constant Λ, i.e., ρ_{M}=0 and P_{M}=0, from Eq. (5) we have H=H_{c}=constant, so that de Sitter expansion can be realized. We also note that by comparing Eq. (4) with Eq. (2), we obtain ρ_{Λ}=Λ/κ^{2}=−P_{Λ}.
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)
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 w_{DE}=−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 w_{DE}(a)=w_{DE0}+w_{DEa}(1−a) (Chevallier and Polarski 2001; Linder 2003), where w_{DE0} and w_{DEa} are the current value of w_{DE} and its derivative, respectively, by using the WMAP data, the BAO data and the Hubble constant measurement and the high-redshift SNe Ia data, w_{DE0} and w_{DEa} have been analyzed as w_{DE0}=−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., w_{DE}=−1.
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.
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.
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).
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
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, w_{DE}=−1 (f(ρ)=0), (ii) for a quintessence model, −1<w_{DE}<−1/3 (−2/3<f(ρ)/ρ<0), (iii) for a phantom model, w_{DE}<−1 (f(ρ)/ρ>0). If our universe lies beyond w_{DE}=−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
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 t→t_{s}, all the scale factor, the effective energy density and pressure of the universe diverge as a→∞, ρ_{eff}→∞ and |P_{eff}|→∞. This also includes the case that ρ_{eff} and P_{eff} asymptotically approach finite values at t=t_{s}.
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 t→t_{s}, only the effective pressure of the universe becomes infinity as a→a_{s}, ρ_{eff}→ρ_{s} and |P_{eff}|→∞.
Type III singularity: In the limit of t→t_{s}, the effective energy density as well as the pressure of the universe diverge as a→a_{s}, ρ_{eff}→∞ and |P_{eff}|→∞.
Type IV: In the limit of t→t_{s}, all the scale factor, the effective energy density and pressure of the universe do not diverge as a→a_{s}, ρ_{eff}→0 and |P_{eff}|→0. However, higher derivatives of H become infinity. This also includes the case that ρ_{eff} and/or |P_{eff}| become finite values at t=t_{s} (Shtanov and Sahni 2002).
Here, t_{s}, a_{s}(≠0) and ρ_{s} 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).
- (a)The null energy condition (NEC):$$ \rho+ P \geq0. $$(49)
- (b)The dominant energy condition (DEC):$$ \rho\geq0, \quad \rho\pm P \geq0. $$(50)
- (c)The strong energy condition (SEC):$$ \rho+ 3P \geq0, \quad \rho+ P \geq0. $$(51)
- (d)The weak energy condition (WEC):$$ \rho\geq0, \quad \rho+ P \geq0. $$(52)
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 t→t_{s}, a→a_{s}, ρ_{eff}→0, |P_{eff}|→0, and the EoS for the universe becomes infinity.
3.2.2 Example of a fluid with behavior very similar to the ΛCDM model
3.2.3 Generalized Chaplygin gas (GCG) 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).
3.3 Phantom scenarios
According to the present observations, there is the possibility that the EoS of dark energy w_{DE} 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 (w_{DE}>−1) to the phantom phase (w_{DE}<−1), namely, crossing of the phantom divide line of w_{DE}=−1 occurs (Alam et al. 2004).
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(ρ_{min})=0. This means that at the phantom crossing point w_{DE}=−1, both H and ρ have those minima, which can also be understood from the definition of w_{DE} in Eq. (40). In addition, when a Big Rip singularity appears at t=t_{s}, 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 w_{DE}=−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 w_{DE}=−1+2/(3n).
3.3.2 Coupled phantom scenario
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 w_{DE} being less than −1 and then w_{DE} asymptotically approaches w_{DE}=−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”.
From Eqs. (103) and (104), we see that when t∼t_{1}, a∼a_{c} and w_{DE}<−1, i.e., the universe is in the phantom phase. As the universe evolves, a increases, and when t≪t_{1}, a becomes very large and thus w_{DE} 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.
In Little Rip cosmology, as a demonstration we estimate the value of F_{inert} at the present time \(t_{0} \approx H_{0}^{-1}\), where H_{0}=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_{⊕}=3.357×10^{51} GeV (Kolb and Turner 1990) is the Earth mass and M_{⊙}=1.116×10^{57} GeV (Kolb and Turner 1990) is the mass of Sun, and r_{⊕−⊙}=1 AU=7.5812×10^{26} GeV^{−1} (Kolb and Turner 1990) is the Astronomical unit corresponding to the distance between Earth and Sun. We take ξ=H_{0}, 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 F_{inert} to be larger than or equal to \(F_{\mathrm{b}}^{\mathrm{ES}}\), H_{LR} should be H_{LR}≥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.
3.4 Finite-time future singularities
In the ΛCDM model, f(ρ)=0 in Eq. (40) and hence w_{DE}=−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
- (a)
For α>1, the Type III singularity can exist. If A>0 (A<0), w_{DE}→+∞ (w_{DE}→−∞).
- (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), w_{DE}→−1+0 (w_{DE}→−1−0).
- (c)
For 0<α≤1/2, there does not exist any “finite-time” future singularity.
3.4.2 Type II singularity
3.4.3 Type IV singularity
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.
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
3.6.2 Influences on the structure of the finite-time future singularities
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.
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
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
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.
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
4.3.2 Quintessence model
4.3.3 Phantom model
4.3.4 Unified scenario of inflation and late-time cosmic acceleration
The history of the universe in this model can be interpreted as follows. The universe is created at t=−t_{s} because the value of the scale factor a(t) in Eq. (270) becomes zero a(−t_{s})=0. During −t_{s}<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=t_{s}. 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
5 Tachyon scalar field theory
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
6.1.2 New type of two scalar field theories
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
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
7.2 Generalized holographic dark energy
We note that for the solution (397), from Eq. (48) we find w_{eff}=−1+2(t_{s}−2t)/(3h_{c}t_{s}). When t→0, w_{eff}→−1+2/(3h_{c})>−1, i.e., the universe is in the non-phantom phase. At t=t_{s}/2, w_{eff}=−1. After that, in the limit of t→t_{s}, w_{eff}→−1−2/(3h_{c})<−1, i.e., the universe is in the phantom phase. Consequently, it can occur the crossing of the phantom divide.
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 w_{eff} has been proposed, although in the present case of H in Eq. (396) we have a dynamical w_{eff}. 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.
We mention the case that α_{h} is negative. From the denominator of Eq. (407), we see that at t=−α_{h}l a Big Rip singularity appears because a diverges. Furthermore, L_{h} can be negative, so that when matter is not included, H=C_{h}/L_{h} can also be negative and therefore the universe will be shrink.
7.3 The Hubble entropy in the holographic principle
We remark that S_{H} in Eq. (420) is positive, even though α_{c}<0. Moreover, S_{H} in Eq. (417) is always positive. However, for the case that S_{H} is given by Eq. (419), if κ^{2}ρ_{c}/(3h_{c})>1, S_{H} can be negative. This implies that entropy of the universe should be negative, provided that S_{H} 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).
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,T_{st}) gravity has also been investigated, where T_{st} 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
8.2 Reconstruction method of F(R) gravity
8.3 Reconstructed F(R) forms and its cosmologies
8.3.1 The ΛCDM cosmology
8.3.2 Quintessence cosmology
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.
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
We note that for \(F(\bar{R}) = \bar{R}\), in the flat FLRW background the gravitational field equations are written as H^{2}={κ^{2}/[3(3λ_{HL}−1)]}ρ_{M} and \(\dot{H} = -\{ \kappa^{2}/[2(3\lambda_{\mathrm {HL}}-1 )]\}( \rho_{\mathrm{M}} + P_{\mathrm{M}} )\) with λ_{HL}>1/3 due to the consistency, and for λ_{HL}→1 these equations become the ordinary Einstein equations in general relativity.
8.4.2 Reconstruction of F(R) form
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
8.5.2 Example
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
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
Conditions to produce the finite-time future singularities in the limit of t→t_{s}
Type | q(≠0,−1) | a | H | \(\dot{H}\) | ρ_{DE} | P_{DE} |
---|---|---|---|---|---|---|
I | q≥1 | a→∞ | H→∞ | \(\dot{H} \to\infty\) | J_{1}≠0 | J_{1}≠0 or J_{2}≠0 |
III | 0<q<1 | a→a_{s} | H→∞ | \(\dot{H} \to\infty\) | J_{1}≠0 | J_{1}≠0 |
II | −1<q<0 | a→a_{s} | H→H_{s} | \(\dot{H} \to\infty\) | J_{2}≠0 | |
IV | q<−1 (q≠integer) | a→a_{s} | H→H_{s} | \(\dot{H} \to 0\) (Higher derivatives of H diverge.) |
9.2.2 Reconstruction of an f(T) gravity model
9.2.3 Removing the finite-time future singularities
Type | q(≠0,−1) | Final appearance | f(T)=AT^{α} | f_{c}(T)=BT^{β} |
---|---|---|---|---|
(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
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=h_{inf}/t, | f(T)=AT^{α}, |
a_{inf}>0 | h_{inf}>1 | α<0 or α=1/2 | |
(ii) ΛCDM model or exponential inflation | a=a_{Λ}exp(H_{Λ}t), | \(H =\sqrt{\varLambda/3}=\mathrm{constant}\), | f(T)=T−2Λ, |
a_{Λ}>0 | Λ>0 | Λ>0 | |
(iii) Little Rip scenario [when t→∞] | a=a_{LR}exp[(H_{LR}/ξ)exp(ξt)], | H=H_{LR}exp(ξt), | f(T)=AT^{α}, |
a_{LR}>0 | H_{LR}>0 and ξ>0 | α<0 or α=1/2 | |
(iv) Pseudo-Rip cosmology | a=a_{PR}cosh(t/t_{0}), | H=H_{PR}tanh(t/t_{0}), | \(f(T) = A \sqrt{T}\) |
a_{PR}>0 | H_{PR}>0 |
Expressions of w_{DE}, q_{dec}, j and s at the present time t=t_{0} for the ΛCDM model, Little Rip scenario and Pseudo-Rip cosmology (Bamba et al. 2012d)
Model | w_{DE(0)} | q_{dec(0)} | j_{0} | s_{0} |
---|---|---|---|---|
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 | \(-1 - [2\delta/(3 \tilde{\mathrm{s}}^{2} )]\), | \(-1 + (\delta^{2} \tilde{\mathrm{t}}^{2} - 1)/(\delta^{2} \tilde{\mathrm{t}}^{2})\), | \(1 + (1-\delta^{3} \tilde{\mathrm{t}}^{2})/(\delta^{3} \tilde{ \mathrm{t}}^{2})\), | \([2/(3 \delta)](\delta^{3} \tilde{\mathrm{t}}^{2} -1) (\delta^{2} \tilde{\mathrm{t}}^{2} +2)^{-1}\) |
δ≡H_{0}/H_{PR}≤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
9.4.2 Second law of thermodynamics
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)=d^{n}F/dR^{n} or f^{(n)}(T)=d^{n}f/dT^{n}, 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 μ_{cp}(z) obtained using the fifth order expansion of the scale factor is the same (within the errors) as the one μ_{DE}(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/μ_{DE} 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/μ_{DE} 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.
- 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)
11 An example: testing F(R) gravity by cosmography
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 (q_{0},j_{0},s_{0},l_{0}) 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).
It is worth noticing that H_{0} 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 (q_{0},j_{0},s_{0},l_{0}), to fix the four F(R) related quantities F(R_{0}), F′(R_{0}), F″(R_{0}), F‴(R_{0}). It is also worth stressing that Eqs. (648)–(651) are linear in the F(R) quantities, so that (q_{0},j_{0},s_{0},l_{0}) 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 (q_{0},j_{0},s_{0},l_{0}).
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 G_{local}=G_{cosmo} could (at least, in principle) be violated, so that Eq. (644) could be reconsidered. Indeed, as we will see, the condition F′(R_{0})=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 G_{eff}(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′(R_{0})=(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 (q_{0},j_{0},s_{0},l_{0}) and ε in a complicated way. Nevertheless, we have numerically checked that the error induced on F(R_{0}), F″(R_{0}), F‴(R_{0}) 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
11.2 The ΛCDM case
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 (ε_{q},ε_{j},ε_{s},ε_{l}) 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(R_{0})∝R_{0}, F″(R_{0})=F‴(R_{0})=0, but F^{(n)}(R_{0})≠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 (q_{0},j_{0},s_{0},l_{0}) are within 10 % of those in the ΛCDM model.
11.3 The constant EoS model
As a general comment, it is clear that, even in this case, F″(R_{0}) and F‴(R_{0}) are from two to three orders of magnitude smaller than the zeroth order term F(R_{0}). Such a result could yet be guessed from the previous discussion for the ΛCDM case. Actually, relaxing the hypothesis w_{0}=−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, η_{20} and η_{30} 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 η_{20} and η_{30} positive. Moreover, the lower \(\varOmega_{\mathrm{M}}^{(0)}\) is, the higher the ratios η_{20} and η_{30} are for a given value of w_{0}. 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
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 w_{0}=−1, the parameter q_{0} is the same as that in the ΛCDM model, while j_{0} 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 (s_{0},l_{0}) are. Introducing a w_{a}≠0 makes (s_{0},l_{0}) 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(R_{0}), F″(R_{0}) and F‴(R_{0}) 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=(p_{1},…,p_{n}) 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
12.2 The Hu-Sawicki model
13 Constraints coming from observational data
A similar discussion may be repeated for F(R) models sharing the same values of (q_{0},j_{0},s_{0},l_{0}) 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 (q_{0},j_{0},s_{0},l_{0}) 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 (q_{0},j_{0},s_{0},l_{0}) 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″(R_{0}) and F‴(R_{0}) 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 η_{20}∝F″(R_{0})/F(R_{0}) and η_{30}∝F‴(R_{0})/F(R_{0}) 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 (η_{20},η_{30}). 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 ∼10^{4} 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 R_{μν} and R_{μνξσ} 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 H^{2}=(κ^{2}/3)ρ_{DE} 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 ρ_{DE} and pressure P_{DE} 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 ρ_{DE} and P_{DE}, the EoS w_{DE}≡DE/ρ_{DE} 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.
It has also been examined in Stefancic (2005) that for α<0, when ρ→0, there can appear the Type II singularity.
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.
Note that the correct expressions for (ϕ_{0},ϕ_{2},ϕ_{3}) may still formally be written as Eqs. (648)–(651), but the polynomials entering them are now different and also depend on powers of ε.
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 \({\mathcal{N}}_{bin} = 1\).
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.).