Abstract
Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity — such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as BransDicke theory and GaussBonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
General Relativity (GR) [225, 226] is widely accepted as a fundamental theory to describe the geometric properties of spacetime. In a homogeneous and isotropic spacetime the Einstein field equations give rise to the Friedmann equations that describe the evolution of the universe. In fact, the standard bigbang cosmology based on radiation and matter dominated epochs can be well described within the framework of General Relativity.
However, the rapid development of observational cosmology which started from 1990s shows that the universe has undergone two phases of cosmic acceleration. The first one is called inflation [564, 339, 291, 524], which is believed to have occurred prior to the radiation domination (see [402, 391, 71] for reviews). This phase is required not only to solve the flatness and horizon problems plagued in bigbang cosmology, but also to explain a nearly flat spectrum of temperature anisotropies observed in Cosmic Microwave Background (CMB) [541]. The second accelerating phase has started after the matter domination. The unknown component giving rise to this latetime cosmic acceleration is called dark energy [310] (see [517, 141, 480, 485, 171, 32] for reviews). The existence of dark energy has been confirmed by a number of observations — such as supernovae Ia (SN Ia) [490, 506, 507], largescale structure (LSS) [577, 578], baryon acoustic oscillations (BAO) [227, 487], and CMB [560, 561, 367].
These two phases of cosmic acceleration cannot be explained by the presence of standard matter whose equation of state w = P/ρ satisfies the condition w ≥ 0 (here P and ρ are the pressure and the energy density of matter, respectively). In fact, we further require some component of negative pressure, with w < −1/3, to realize the acceleration of the universe. The cosmological constant Λ is the simplest candidate of dark energy, which corresponds to w = −1. However, if the cosmological constant originates from a vacuum energy of particle physics, its energy scale is too large to be compatible with the dark energy density [614]. Hence we need to find some mechanism to obtain a small value of Λ consistent with observations. Since the accelerated expansion in the very early universe needs to end to connect to the radiationdominated universe, the pure cosmological constant is not responsible for inflation. A scalar field ϕ with a slowly varying potential can be a candidate for inflation as well as for dark energy.
Although many scalarfield potentials for inflation have been constructed in the framework of string theory and supergravity, the CMB observations still do not show particular evidence to favor one of such models. This situation is also similar in the context of dark energy — there is a degeneracy as for the potential of the scalar field (“quintessence” [111, 634, 267, 263, 615, 503, 257, 155]) due to the observational degeneracy to the dark energy equation of state around w = −1. Moreover it is generally difficult to construct viable quintessence potentials motivated from particle physics because the field mass responsible for cosmic acceleration today is very small (m_{ϕ} ≃ 10^{−33} eV) [140, 365].
While scalarfield models of inflation and dark energy correspond to a modification of the energymomentum tensor in Einstein equations, there is another approach to explain the acceleration of the universe. This corresponds to the modified gravity in which the gravitational theory is modified compared to GR. The Lagrangian density for GR is given by f(R) = R − 2Λ, where R is the Ricci scalar and Λ is the cosmological constant (corresponding to the equation of state w = −1). The presence of Λ gives rise to an exponential expansion of the universe, but we cannot use it for inflation because the inflationary period needs to connect to the radiation era. It is possible to use the cosmological constant for dark energy since the acceleration today does not need to end. However, if the cosmological constant originates from a vacuum energy of particle physics, its energy density would be enormously larger than the today’s dark energy density. While the ΛCold Dark Matter (ΛCDM) model (f(R) = R − 2Λ) fits a number of observational data well [367, 368], there is also a possibility for the timevarying equation of state of dark energy [10, 11, 450, 451, 630].
One of the simplest modifications to GR is the f(R) gravity in which the Lagrangian density f is an arbitrary function of R [77, 512, 102, 106]. There are two formalisms in deriving field equations from the action in f(R) gravity. The first is the standard metric formalism in which the field equations are derived by the variation of the action with respect to the metric tensor g_{μν}. In this formalism the affine connection \(\Gamma _{\beta \gamma}^\alpha\) depends on g_{μν}. Note that we will consider here and in the remaining sections only torsionfree theories. The second is the Palatini formalism [481] in which g_{μν} and \(\Gamma _{\beta \gamma}^\alpha\) are treated as independent variables when we vary the action. These two approaches give rise to different field equations for a nonlinear Lagrangian density in R, while for the GR action they are identical with each other. In this article we mainly review the former approach unless otherwise stated. In Section 9 we discuss the Palatini formalism in detail.
The model with f(R) = R + αR^{2} (α > 0) can lead to the accelerated expansion of the Universe because of the presence of the αR^{2} term. In fact, this is the first model of inflation proposed by Starobinsky in 1980 [564]. As we will see in Section 7, this model is well consistent with the temperature anisotropies observed in CMB and thus it can be a viable alternative to the scalarfield models of inflation. Reheating after inflation proceeds by a gravitational particle production during the oscillating phase of the Ricci scalar [565, 606, 426].
The discovery of dark energy in 1998 also stimulated the idea that cosmic acceleration today may originate from some modification of gravity to GR. Dark energy models based on f(R) theories have been extensively studied as the simplest modified gravity scenario to realize the latetime acceleration. The model with a Lagrangian density f(R) = R − α/R^{n} (α > 0, n > 0) was proposed for dark energy in the metric formalism [113, 120, 114, 143, 456]. However it was shown that this model is plagued by a matter instability [215, 244] as well as by a difficulty to satisfy local gravity constraints [469, 470, 245, 233, 154, 448, 134]. Moreover it does not possess a standard matterdominated epoch because of a large coupling between dark energy and dark matter [28, 29]. These results show how nontrivial it is to obtain a viable f(R) model. Amendola et al. [26] derived conditions for the cosmological viability of f(R) dark energy models. In local regions whose densities are much larger than the homogeneous cosmological density, the models need to be close to GR for consistency with local gravity constraints. A number of viable f(R) models that can satisfy both cosmological and local gravity constraints have been proposed in. [26, 382, 31, 306, 568, 35, 587, 206, 164, 396]. Since the law of gravity gets modified on large distances in f(R) models, this leaves several interesting observational signatures such as the modification to the spectra of galaxy clustering [146, 74, 544, 526, 251, 597, 493], CMB [627, 544, 382, 545], and weak lensing [595, 528]. In this review we will discuss these topics in detail, paying particular attention to the construction of viable f(R) models and resulting observational consequences.
The f(R) gravity in the metric formalism corresponds to generalized BransDicke (BD) theory [100] with a BD parameter ω_{BD} = 0 [467, 579, 152]. Unlike original BD theory [100], there exists a potential for a scalarfield degree of freedom (called “scalaron” [564]) with a gravitational origin. If the mass of the scalaron always remains as light as the present Hubble parameter H_{0}, it is not possible to satisfy local gravity constraints due to the appearance of a longrange fifth force with a coupling of the order of unity. One can design the field potential of f(R) gravity such that the mass of the field is heavy in the region of high density. The viable f(R) models mentioned above have been constructed to satisfy such a condition. Then the interaction range of the fifth force becomes short in the region of high density, which allows the possibility that the models are compatible with local gravity tests. More precisely the existence of a matter coupling, in the Einstein frame, gives rise to an extremum of the effective field potential around which the field can be stabilized. As long as a spherically symmetric body has a “thinshell” around its surface, the field is nearly frozen in most regions inside the body. Then the effective coupling between the field and nonrelativistic matter outside the body can be strongly suppressed through the chameleon mechanism [344, 343]. The experiments for the violation of equivalence principle as well as a number of solar system experiments place tight constraints on dark energy models based on f(R) theories [306, 251, 587, 134, 101].
The spherically symmetric solutions mentioned above have been derived under the weak gravity backgrounds where the background metric is described by a Minkowski spacetime. In strong gravitational backgrounds such as neutron stars and white dwarfs, we need to take into account the backreaction of gravitational potentials to the field equation. The structure of relativistic stars in f(R) gravity has been studied by a number of authors [349, 350, 594, 43, 600, 466, 42, 167]. Originally the difficulty of obtaining relativistic stars was pointed out in [349] in connection to the singularity problem of f(R) dark energy models in the highcurvature regime [266]. For constant density stars, however, a thinshell field profile has been analytically derived in [594] for chameleon models in the Einstein frame. The existence of relativistic stars in f(R) gravity has been also confirmed numerically for the stars with constant [43, 600] and varying [42] densities. In this review we shall also discuss this issue.
It is possible to extend f(R) gravity to generalized BD theory with a field potential and an arbitrary BD parameter ω_{BD}. If we make a conformal transformation to the Einstein frame [213, 609, 408, 611, 249, 268], we can show that BD theory with a field potential corresponds to the coupled quintessence scenario [23] with a coupling Q between the field and nonrelativistic matter. This coupling is related to the BD parameter via the relation 1/(2Q^{2}) = 3 + 2ω_{BD} [343, 596]. One can recover GR by taking the limit Q − 0, i.e., ω_{BD} → ∞. The f(R) gravity in the metric formalism corresponds to \(Q =  1/\sqrt 6\) [28], i.e., ω_{BD} = 0. For large coupling models with \(\left\vert Q \right\vert = \mathcal{O}\left(1 \right)\) it is possible to design scalarfield potentials such that the chameleon mechanism works to reduce the effective matter coupling, while at the same time the field is sufficiently light to be responsible for the latetime cosmic acceleration. This generalized BD theory also leaves a number of interesting observational and experimental signatures [596].
In addition to the Ricci scalar R, one can construct other scalar quantities such as R_{μν}R^{μν} and R_{μνρσ} R^{μνρσ} from the Ricci tensor R_{μν} and Riemann tensor R_{μνρσ} [142]. For the GaussBonnet (GB) curvature invariant defined by \(\mathcal{G} \equiv {R^2}  4{R_{\alpha \beta}}{R^{\alpha \beta}} + {R_{\alpha \beta \gamma \delta}}{R^{\alpha \beta \gamma \delta}}\), it is known that one can avoid the appearance of spurious spin2 ghosts [572, 67, 302] (see also [98, 465, 153, 447, 110, 181, 109]). In order to give rise to some contribution of the GB term to the Friedmann equation, we require that (i) the GB term couples to a scalar field ϕ, i.e., \(F\left(\phi \right)\mathcal{G}\) or (ii) the Lagrangian density f is a function of Q, i.e., \(f\left({\mathcal G} \right)\). The GB coupling in the case (i) appears in lowenergy string effective action [275] and cosmological solutions in such a theory have been studied extensively (see [34, 273, 105, 147, 588, 409, 468] for the construction of nonsingular cosmological solutions and [463, 360, 361, 593, 523, 452, 453, 381, 25] for the application to dark energy). In the case (ii) it is possible to construct viable models that are consistent with both the background cosmological evolution and local gravity constraints [458, 188, 189] (see also [165, 180, 178, 383, 633, 599]). However density perturbations in perfect fluids exhibit negative instabilities during both the radiation and the matter domination, irrespective of the form of \(f\left(\mathcal{G} \right)\) [383, 182]. This growth of perturbations gets stronger on smaller scales, which is difficult to be compatible with the observed galaxy spectrum unless the deviation from GR is very small. We shall review such theories as well as other modified gravity theories.
This review is organized as follows. In Section 2 we present the field equations of f(R) gravity in the metric formalism. In Section 3 we apply f(R) theories to the inflationary universe. Section 4 is devoted to the construction of cosmologically viable f(R) dark energy models. In Section 5 local gravity constraints on viable f(R) dark energy models will be discussed. In Section 6 we provide the equations of linear cosmological perturbations for general modified gravity theories including metric f(R) gravity as a special case. In Section 7 we study the spectra of scalar and tensor metric perturbations generated during inflation based on f(R) theories. In Section 8 we discuss the evolution of matter density perturbations in f(R) dark energy models and place constraints on model parameters from the observations of largescale structure and CMB. Section 9 is devoted to the viability of the Palatini variational approach in f(R) gravity. In Section 10 we construct viable dark energy models based on BD theory with a potential as an extension of f(R) theories. In Section 11 the structure of relativistic stars in f(R) theories will be discussed in detail. In Section 12 we provide a brief review of GaussBonnet gravity and resulting observational and experimental consequences. In Section 13 we discuss a number of other aspects of f(R) gravity and modified gravity. Section 14 is devoted to conclusions.
There are other review articles on f(R) gravity [556, 555, 618] and modified gravity [171, 459, 126, 397, 217]. Compared to those articles, we put more weights on observational and experimental aspects of f(R) theories. This is particularly useful to place constraints on inflation and dark energy models based on f(R) theories. The readers who are interested in the more detailed history of f(R) theories and fourthorder gravity may have a look at the review articles by Schmidt [531] and Sotiriou and Faraoni [556].
In this review we use units such that c = ħ = k_{B} = 1, where c is the speed of light, ħ is reduced Planck’s constant, and k_{B} is Boltzmann’s constant. We define \({\kappa ^2} = 8\pi G = 8\pi/m_{{\rm{pl}}}^2 = 1/M_{{\rm{pl}}}^2\), where G is the gravitational constant, m_{pl} = 1.22 × 10^{19} GeV is the Planck mass with a reduced value \({M_{{\rm{pl}}}} = {m_{{\rm{pl}}}}/\sqrt {8\pi} = 2.44 \times {10^{18}}{\rm{Gev}}\). Throughout this review, we use a dot for the derivative with respect to cosmic time t and “_{X}” for the partial derivative with respect to the variable X, e.g., f_{,R} ≡ ∂f/∂R and f_{,RR} ≡ ∂^{2}f/∂R^{2}. We use the metric signature (−, +, +, +). The Greek indices μ and ν run from 0 to 3, whereas the Latin indices i and j run from 1 to 3 (spatial components).
2 Field Equations in the Metric Formalism
We start with the 4dimensional action in f(R) gravity:
where κ^{2} = 8πG, g is the determinant of the metric g_{μν}, and \({{\mathcal L}_M}\) is a matter Lagrangian^{Footnote 1} that depends on g_{μν} and matter fields Ψ_{M}. The Ricci scalar R is defined by R = g^{μν} R_{μν}, where the Ricci tensor R_{μν} is
In the case of the torsionless metric formalism, the connections \(\Gamma _{\beta \gamma}^\alpha\) are the usual metric connections defined in terms of the metric tensor g_{μν}, as
This follows from the metricity relation, \({\nabla _\lambda}{g_{\mu \nu}} = \partial {g_{\mu \nu}}/\partial {x^\lambda}  {g_{\rho \nu}}\Gamma _{\mu \lambda}^\rho  {g_{\mu \rho}}\Gamma _{\nu \lambda}^\rho = 0\).
2.1 Equations of motion
The field equation can be derived by varying the action (2.1) with respect to g_{μν}:
where F(R) = ∂f/∂R. \(T_{\mu \nu}^{\left(M \right)}\) is the energymomentum tensor of the matter fields defined by the variational derivative of \({{\mathcal L}_M}\) in terms of g^{μν}:
This satisfies the continuity equation
as well as Σ_{μν}, i.e., ∇^{μ}Σ_{μν} = 0.^{Footnote 2} The trace of Eq. (2.4) gives
where \(T = {g^{\mu \nu}}T_{\mu \nu}^{\left(M \right)}\) and \(\Box F = \left({1/\sqrt { g}} \right){\partial _\mu}\left({\sqrt { g} {g^{\mu \nu}}{\partial _\nu}F} \right)\).
Einstein gravity, without the cosmological constant, corresponds to f(R) = R and F(R) = 1, so that the term □F(R) in Eq. (2.7) vanishes. In this case we have R = −κ^{2}T and hence the Ricci scalar R is directly determined by the matter (the trace T). In modified gravity the term □F(R) does not vanish in Eq. (2.7), which means that there is a propagating scalar degree of freedom, φ ≡ F(R). The trace equation (2.7) determines the dynamics of the scalar field φ (dubbed “scalaron” [564]).
The field equation (2.4) can be written in the following form [568]
where G_{μν} ≡ R_{μν} − (1/2)g_{μν}R and
Since ∇^{μ}G_{μν} = 0 and \({\nabla ^\mu}T_{\mu \nu}^{\left(M \right)} = 0\), it follows that
Hence the continuity equation holds, not only for Σ_{μν}, but also for the effective energymomentum tensor \(T_{\mu \nu}^{\left(D \right)}\) defined in Eq. (2.9). This is sometimes convenient when we study the dark energy equation of state [306, 568] as well as the equilibrium description of thermodynamics for the horizon entropy [53].
There exists a de Sitter point that corresponds to a vacuum solution (T = 0) at which the Ricci scalar is constant. Since □F(R) = 0 at this point, we obtain
The model f(R) = αR^{2} satisfies this condition, so that it gives rise to the exact de Sitter solution [564]. In the model f(R) = R + αR^{2}, because of the linear term in R, the inflationary expansion ends when the term αR^{2} becomes smaller than the linear term R (as we will see in Section 3). This is followed by a reheating stage in which the oscillation of R leads to the gravitational particle production. It is also possible to use the de Sitter point given by Eq. (2.11) for dark energy.
We consider the spatially flat FriedmannLemaîtreRobertsonWalker (FLRW) spacetime with a timedependent scale factor a(t) and a metric
where t is cosmic time. For this metric the Ricci scalar R is given by
where H ≡ ȧ/a is the Hubble parameter and a dot stands for a derivative with respect to t. The present value of H is given by
where h = 0.72 ± 0.08 describes the uncertainty of H_{0} [264].
The energymomentum tensor of matter is given by \({T^\mu}_\nu ^{\left(M \right)} = {\rm{diag}}\left({ {\rho _M},\,{P_M},\,{P_M},\,{P_M}} \right)\), where ρ_{M} is the energy density and P_{M} is the pressure. The field equations (2.4) in the flat FLRW background give
where the perfect fluid satisfies the continuity equation
We also introduce the equation of state of matter, w_{M} ≡ P_{M}/ρm. As long as w_{M} is constant, the integration of Eq. (2.17) gives \({\rho _M} \propto {a^{ 3\left({1 + {w_M}} \right)}}\). In Section 4 we shall take into account both nonrelativistic matter (w_{M} = 0) and radiation (w_{r} = 1/3) to discuss cosmological dynamics of f(R) dark energy models.
Note that there are some works about the Einstein static universes in f(R) gravity [91, 532]. Although Einstein static solutions exist for a wide variety of f(R) models in the presence of a barotropic perfect fluid, these solutions have been shown to be unstable against either homogeneous or inhomogeneous perturbations [532].
2.2 Equivalence with BransDicke theory
The f(R) theory in the metric formalism can be cast in the form of BransDicke (BD) theory [100] with a potential for the effective scalarfield degree of freedom (scalaron). Let us consider the following action with a new field χ,
Varying this action with respect to χ, we obtain
Provided f,_{χχ}(χ) ≠ 0 it follows that χ = R. Hence the action (2.18) recovers the action (2.1) in f(R) gravity. If we define
the action (2.18) can be expressed as
where U(φ) is a field potential given by
Meanwhile the action in BD theory [100] with a potential U(φ) is given by
where ω_{BD} is the BD parameter and (∇φ)^{2} ≡ g^{μν}∂_{μ}φ∂_{ν}φ. Comparing Eq. (2.21) with Eq. (2.23), it follows that f(R) theory in the metric formalism is equivalent to BD theory with the parameter ω_{BD} = 0 [467, 579, 152] (in the unit κ^{2} = 1). In Palatini f(R) theory where the metric g_{μν} and the connection \(\Gamma _{\beta \gamma}^\alpha\) are treated as independent variables, the Ricci scalar is different from that in metric f(R) theory. As we will see in Sections 9.1 and 10.1, f(R) theory in the Palatini formalism is equivalent to BD theory with the parameter ω_{BD} = −3/2.
2.3 Conformal transformation
The action (2.1) in f(R) gravity corresponds to a nonlinear function f in terms of R. It is possible to derive an action in the Einstein frame under the conformal transformation [213, 609, 408, 611, 249, 268, 410]:
where Ω^{2} is the conformal factor and a tilde represents quantities in the Einstein frame. The Ricci scalars R and \(\tilde R\) in the two frames have the following relation
where
We rewrite the action (2.1) in the form
where
Using Eq. (2.25) and the relation \(\sqrt { g} = {\Omega ^{ 4}}\sqrt { \tilde g}\), the action (2.27) is transformed as
We obtain the Einstein frame action (linear action in \(\tilde R\)) for the choice
This choice is consistent if F > 0. We introduce a new scalar field ϕ defined by
From the definition of ω in Eq. (2.26) we have that \(\omega = \kappa \phi/\sqrt 6\). Using Eq. (2.26), the integral \(\int {{{\rm{d}}^4}x} \sqrt { \tilde g} \tilde \Box \omega\) vanishes on account of the Gauss’s theorem. Then the action in the Einstein frame is
where
Hence the Lagrangian density of the field ϕ is given by \({{\mathcal L}_\phi} =  {1 \over 2}{\tilde g^{\mu \nu}}{\partial _\mu}\phi {\partial _\nu}\phi  V\left(\phi \right)\) with the energymomentum tensor
The conformal factor \({\Omega ^2} = F = \exp \left({\sqrt {2/3} \kappa \phi} \right)\) is fielddependent. From the matter action (2.32) the scalar field ϕ is directly coupled to matter in the Einstein frame. In order to see this more explicitly, we take the variation of the action (2.32) with respect to the field ϕ:
that is
Using Eq. (2.24) and the relations \(\sqrt { \tilde g} = {F^2}\sqrt { g}\) and \({\tilde g^{\mu \nu}} = {F^{ 1}}{g^{\mu \nu}}\), the energymomentum tensor of matter is transformed as
The energymomentum tensor of perfect fluids in the Einstein frame is given by
The derivative of the Lagrangian density \({{\mathcal L}_M} = {{\mathcal L}_M}\left({{g_{\mu \nu}}} \right) = {{\mathcal L}_M}\left({{F^{ 1}}\left(\phi \right){{\tilde g}_{\mu \nu}}} \right)\) with respect to ϕ is
The strength of the coupling between the field and matter can be quantified by the following quantity
which is constant in f(R) gravity [28]. It then follows that
where \(\tilde T = {\tilde g_{\mu \nu}}{\tilde T^{\mu \nu \left(M \right)}} =  {\tilde \rho _M} + 3{\tilde P_M}\). Substituting Eq. (2.41) into Eq. (2.36), we obtain the field equation in the Einstein frame:
This shows that the field ϕ is directly coupled to matter apart from radiation \(\left({\tilde T = 0} \right)\).
Let us consider the flat FLRW spacetime with the metric (2.12) in the Jordan frame. The metric in the Einstein frame is given by
which leads to the following relations (for F > 0)
where
Note that Eq. (2.45) comes from the integration of Eq. (2.40) for constant Q. The field equation (2.42) can be expressed as
where
Defining the energy density \({\tilde \rho _\phi} = {1 \over 2}{\left({{\rm{d}}\phi/{\rm{d}}\tilde t} \right)^2} + V\left(\phi \right)\) and the pressure \({\tilde P_\phi} = {1 \over 2}{\left({{\rm{d}}\phi/{\rm{d}}\tilde t} \right)^2}  V\left(\phi \right)\), Eq. (2.46) can be written as
Under the transformation (2.44) together with \({\rho _M} = {F^2}{\tilde \rho _M},\,{P_M} = {F^2}{\tilde P_M}\), and \(H = {F^{1/2}}[\tilde H  ({\rm{d}}F/{\rm{d}}\tilde t)/2F]\), the continuity equation (2.17) is transformed as
Equations (2.48) and (2.49) show that the field and matter interacts with each other, while the total energy density \({\tilde \rho _T} = {\tilde \rho _\phi} + {\tilde \rho _M}\) and the pressure \({\tilde P_T} = {\tilde P_\phi} + {\tilde P_M}\) satisfy the continuity equation \({{\rm{d}}_{\tilde \rho T}}/{\rm{d}}\tilde t + 3\tilde H\left({{{\tilde \rho}_T} + {{\tilde P}_T}} \right) = 0\). More generally, Eqs. (2.48) and (2.49) can be expressed in terms of the energymomentum tensors defined in Eqs. (2.34) and (2.37):
which correspond to the same equations in coupled quintessence studied in [23] (see also [22]).
In the absence of a field potential V(ϕ) (i.e., massless field) the field mediates a longrange fifth force with a large coupling (∣Q∣ ≃ 0.4), which contradicts with experimental tests in the solar system. In f(R) gravity a field potential with gravitational origin is present, which allows the possibility of compatibility with local gravity tests through the chameleon mechanism [344, 343].
In f(R) gravity the field ϕ is coupled to nonrelativistic matter (dark matter, baryons) with a universal coupling \(Q =  1/\sqrt 6\). We consider the frame in which the baryons obey the standard continuity equation ρ_{m} ℝ a^{−3}, i.e., the Jordan frame, as the “physical” frame in which physical quantities are compared with observations and experiments. It is sometimes convenient to refer the Einstein frame in which a canonical scalar field is coupled to nonrelativistic matter. In both frames we are treating the same physics, but using the different time and length scales gives rise to the apparent difference between the observables in two frames. Our attitude throughout the review is to discuss observables in the Jordan frame. When we transform to the Einstein frame for some convenience, we go back to the Jordan frame to discuss physical quantities.
3 Inflation in f(R) Theories
Most models of inflation in the early universe are based on scalar fields appearing in superstring and supergravity theories. Meanwhile, the first inflation model proposed by Starobinsky [564] is related to the conformal anomaly in quantum gravity^{Footnote 3}. Unlike the models such as “old inflation” [339, 291, 524] this scenario is not plagued by the graceful exit problem — the period of cosmic acceleration is followed by the radiationdominated epoch with a transient matterdominated phase [565, 606, 426]. Moreover it predicts nearly scaleinvariant spectra of gravitational waves and temperature anisotropies consistent with CMB observations [563, 436, 566, 355, 315]. In this section we review the dynamics of inflation and reheating. In Section 7 we will discuss the power spectra of scalar and tensor perturbations generated in f(R) inflation models.
3.1 Inflationary dynamics
We consider the models of the form
which include the Starobinsky’s model [564] as a specific case (n = 2). In the absence of the matter fluid (ρ_{M} = 0), Eq. (2.15) gives
The cosmic acceleration can be realized in the regime F = 1 + nαR^{n−1} ≫ 1. Under the approximation F ≃nαR^{n−1}, we divide Eq. (3.2) by 3nαR^{n−1} to give
During inflation the Hubble parameter H evolves slowly so that one can use the approximation ∣Ḣ/H^{2}∣ ♪ 1 and ∣Ḧ/(HḢ)∣ ♪ 1. Then Eq. (3.3) reduces to
Integrating this equation for ϵ_{1} > 0, we obtain the solution
The cosmic acceleration occurs for ϵ_{1} < 1, i.e., \(n > \left({1 + \sqrt 3} \right)/2\). When n = 2 one has ϵ_{1} = 0, so that H is constant in the regime F ≫ 1. The models with n > 2 lead to super inflation characterized by Ḣ > 0 and \(a \propto {\left\vert {{t_0}  t} \right\vert^{ 1/\left\vert {{\epsilon_1}} \right\vert}}\) (t_{0} is a constant). Hence the standard inflation with decreasing H occurs for \(\left({1 + \sqrt 3} \right)/2 < n < 2\).
In the following let us focus on the Starobinsky’s model given by
where the constant M has a dimension of mass. The presence of the linear term in R eventually causes inflation to end. Without neglecting this linear term, the combination of Eqs. (2.15) and (2.16) gives
During inflation the first two terms in Eq. (3.7) can be neglected relative to others, which gives Ḣ ≃ − M^{2}/6. We then obtain the solution
where H_{i} and a_{i} are the Hubble parameter and the scale factor at the onset of inflation (t = t_{i}), respectively. This inflationary solution is a transient attractor of the dynamical system [407]. The accelerated expansion continues as long as the slowroll parameter
is smaller than the order of unity, i.e., H^{2} ≳ M^{2}. One can also check that the approximate relation 3HṘ + M^{2}R ≃ 0 holds in Eq. (3.8) by using R ≃ 12H^{2}. The end of inflation (at time t = t_{f}) is characterized by the condition ϵ_{f} ≃ 1, i.e., \({H_f} \simeq M/\sqrt 6\). From Eq. (3.11) this corresponds to the epoch at which the Ricci scalar decreases to R ≃ M^{2}. As we will see later, the WMAP normalization of the CMB temperature anisotropies constrains the mass scale to be M ≃ 10^{13} GeV. Note that the phase space analysis for the model (3.6) was carried out in [407, 24, 131].
We define the number of efoldings from t = t_{i} to t = t_{f}:
Since inflation ends at t_{f} ≃ t_{i} + 6H_{i}/M^{2}, it follows that
where we used Eq. (3.12) in the last approximate equality. In order to solve horizon and flatness problems of the big bang cosmology we require that N ≳ 70 [391], i.e., ϵ_{1}(t_{i}) ≲ 7 × 10^{−3}. The CMB temperature anisotropies correspond to the perturbations whose wavelengths crossed the Hubble radius around N = 55–60 before the end of inflation.
3.2 Dynamics in the Einstein frame
Let us consider inflationary dynamics in the Einstein frame for the model (3.6) in the absence of matter fluids \(\left({{\mathcal{L}_M} = 0} \right)\). The action in the Einstein frame corresponds to (2.32) with a field ϕ defined by
Using this relation, the field potential (2.33) reads [408, 61, 63]
In Figure 1 we illustrate the potential (3.16) as a function of ϕ. In the regime κϕ ≫ 1 the potential is nearly constant (V(ϕ) ≃ 3M^{2}/(4κ^{2})), which leads to slowroll inflation. The potential in the regime κϕ ≪ 1 is given by V(ϕ) ≃ (1/2)M^{2}ϕ^{2}, so that the field oscillates around ϕ = 0 with a Hubble damping. The second derivative of V with respect to ϕ is
which changes from negative to positive at \(\phi = {\phi _1} \equiv \sqrt {3/2} \left({\ln \,2} \right)/\kappa \simeq 0.169{m_{{\rm{pl}}}}\).
Since F ≃ 4H^{2}/M^{2} during inflation, the transformation (2.44) gives a relation between the cosmic time \(\tilde t\) in the Einstein frame and that in the Jordan frame:
where t = t_{i} corresponds to \(\tilde t = 0\). The end of inflation (t_{f} ≃ t_{i} + 6H_{i}/M^{2}) corresponds to \({\tilde t_f} = \left({2/M} \right)N\) in the Einstein frame, where N is given in Eq. (3.13). On using Eqs. (3.10) and (3.18), the scale factor \(\tilde a = \sqrt F a\) in the Einstein frame evolves as
where \({\tilde a_i} = 2{H_i}{a_i}/M\). Similarly the evolution of the Hubble parameter \(\tilde H = \left({H/\sqrt F} \right)\left[ {1 + \dot F/\left({2HF} \right)} \right]\) is given by
which decreases with time. Equations (3.19) and (3.20) show that the universe expands quasiexponentially in the Einstein frame as well.
The field equations for the action (2.32) are given by
Using the slowroll approximations \({\left({{\rm{d}}\phi/{\rm{d}}\tilde t} \right)^2} \ll V\left(\phi \right)\) and \(\left\vert {{{\rm{d}}^2}\phi/{\rm{d}}{{\tilde t}^2}} \right\vert \ll \left\vert {\tilde H{\rm{d}}\phi/{\rm{d}}\tilde t} \right\vert\) during inflation, one has \(3{\tilde H^2} \simeq {\kappa ^2}V\left(\phi \right)\) and \(3\tilde H\left({{\rm{d}}\phi/{\rm{d}}\tilde t} \right) + {V_{,\phi}} \simeq 0\). We define the slowroll parameters
for the potential (3.16) it follows that
which are much smaller than 1 during inflation (κϕ ≫ 1). The end of inflation is characterized by the condition \(\left\{{{{\tilde \epsilon}_1},\,\left\vert {{{\tilde \epsilon}_2}} \right\vert} \right\} = \mathcal{O}\left(1 \right)\). Solving \({\tilde \epsilon_1} = 1\), we obtain the field value ϕ_{f} ≃ 0.19m_{pl}.
We define the number of efoldings in the Einstein frame,
where ϕ_{i} is the field value at the onset of inflation. Since \(\tilde H{\rm{d}}\tilde t = H{\rm{d}}t\left[ {1 + \dot F/\left({2HF} \right)} \right]\), it follows that \(\tilde N\) is identical to N in the slowroll limit: ∣Ḟ/(2HF)∣ ≃ ∣Ḣ/H^{2}∣ ≪ 1. Under the condition κϕ_{i} ≫ 1 we have
This shows that ϕ_{i} ≃ 1.11m_{pl} for \(\tilde N = 70\). From Eqs. (3.24) and (3.26) together with the approximate relation \(\tilde H \simeq M/2\), we obtain
where, in the expression of \({\tilde \epsilon _2}\), we have dropped the terms of the order of 1/Ñ^{2}. The results (3.27) will be used to estimate the spectra of density perturbations in Section 7.
3.3 Reheating after inflation
We discuss the dynamics of reheating and the resulting particle production in the Jordan frame for the model (3.6). The inflationary period is followed by a reheating phase in which the second derivative \(\ddot R\) can no longer be neglected in Eq. (3.8). Introducing \(\hat R = {a^{3/2}}R\), we have
Since M^{2} ≫ {H^{2}, ∣Ḣ∣} during reheating, the solution to Eq. (3.28) is given by that of the harmonic oscillator with a frequency M. Hence the Ricci scalar exhibits a damped oscillation around R = 0:
Let us estimate the evolution of the Hubble parameter and the scale factor during reheating in more detail. If we neglect the r.h.s. of Eq. (3.7), we get the solution H(t) = const × cos^{2} (Mt/2). Setting H(t) = f(t)cos^{2}(Mt/2) to derive the solution of Eq. (3.7), we obtain [426]
where t_{os} is the time at the onset of reheating. The constant C is determined by matching Eq. (3.30) with the slowroll inflationary solution Ḣ = −M^{2}/6 at t = t_{os}. Then we get C = 3/M and
Taking the time average of oscillations in the regime M(t − t_{os}) ≫ 1, it follows that 〈H〉 ≃ (2/3)(t − t_{os}) ^{−1}. This corresponds to the cosmic evolution during the matterdominated epoch, i.e., 〈a〉 ∝ (t − t_{os})^{2/3}. The gravitational effect of coherent oscillations of scalarons with mass M is similar to that of a pressureless perfect fluid. During reheating the Ricci scalar is approximately given by R ≃ 6Ḣ, i.e.
In the regime M(t − t_{os}) ≫ 1 this behaves as
In order to study particle production during reheating, we consider a scalar field χ with mass m_{χ}. We also introduce a nonminimal coupling (1/2)ξRχ^{2} between the field χ and the Ricci scalar R [88]. Then the action is given by
where f(R) = R + R^{2}/(6M^{2}). Taking the variation of this action with respect to χ gives
We decompose the quantum field χ in terms of the Heisenberg representation:
where \({{\hat a}_k}\) and \(\hat a_{_k}^\dag\) are annihilation and creation operators, respectively. The field χ can be quantized in curved spacetime by generalizing the basic formalism of quantum field theory in the flat spacetime. See the book [88] for the detail of quantum field theory in curved spacetime. Then each Fourier mode χ_{k}(t) obeys the following equation of motion
where k = ∣k∣ is a comoving wavenumber. Introducing a new field u_{k} = aχ_{k} and conformal time η = ∫ a^{−1}dt, we obtain
where the conformal coupling correspond to ξ = 1/6. This result states that, even though ξ = 0 (that is, the field is minimally coupled to gravity), R still gives a contribution to the effective mass of u_{k}. In the following we first review the reheating scenario based on a minimally coupled massless field (ξ = 0 and m_{χ} = 0). This corresponds to the gravitational particle production in the perturbative regime [565, 606, 426]. We then study the case in which the nonminimal coupling ∣ξ∣ is larger than the order of 1. In this case the nonadiabatic particle production preheating [584, 353, 538, 354] can occur via parametric resonance.
3.3.1 Case: ξ = 0 and m_{χ} = 0
In this case there is no explicit coupling among the fields χ and R. Hence the χ particles are produced only gravitationally. In fact, Eq. (3.38) reduces to
where U = a^{2}R/6. Since U is of the order of (aH)^{2}, one has k^{2} ≫ U for the mode deep inside the Hubble radius. Initially we choose the field in the vacuum state with the positivefrequency solution [88]: \(u_k^{(i)} = {e^{ ik\eta}}/\sqrt {2k}\). The presence of the timedependent term U(η) leads to the creation of the particle χ. We can write the solution of Eq. (3.39) iteratively, as [626]
After the universe enters the radiationdominated epoch, the term U becomes small so that the flatspace solution is recovered. The choice of decomposition of χ into â_{k} and \(\hat a_{_k}^\dag\) is not unique. In curved spacetime it is possible to choose another decomposition in term of new ladder operators \({{\hat {\mathcal A}}_k}\) and \(\hat {\mathcal A}_k^\dag\), which can be written in terms of â_{k} and \(\hat a_{_k}^\dag\), such as \({\hat {\mathcal{A}}_k} = {\alpha _k}{{\hat a}_k} + \beta _k^ \ast \hat a_{ k}^\dagger\). Provided that \(\beta _k^ {\ast} \neq 0\), even though â_{k}∣0〉 ≠ 0, we have \({{\hat {\mathcal A}}_k}\left\vert 0 \right\rangle \neq 0\). Hence the vacuum in one basis is not the vacuum in the new basis, and according to the new basis, the particles are created. The Bogoliubov coefficient describing the particle production is
The typical wavenumber in the ηcoordinate is given by k, whereas in the tcoordinate it is k/a. Then the energy density per unit comoving volume in the ηcoordinate is [426]
where in the last equality we have used the fact that the term U approaches 0 in the early and late times.
During the oscillating phase of the Ricci scalar the timedependence of U is given by \(U = I(\eta)\sin (\int\nolimits_0^\eta {\omega {\rm{d}}\bar \eta})\), where I(η) = ca(η)^{1/2} and ω = Ma (c is a constant). When we evaluate the term dU/dη in Eq. (3.42), the timedependence of I(η) can be neglected. Differentiating Eq. (3.42) in terms of η and taking the limit \(\int\nolimits_0^\eta {\omega {\rm{d}}\bar \eta} \gg 1\), it follows that
where we used the relation lim_{k→∞} sin(kx)/x = πδ(x). Shifting the phase of the oscillating factor by π/2, we obtain
The proper energy density of the field χ is given by ρ_{χ} = (ρ_{η}/a)/a^{3} = ρη/a^{4}. Taking into account g_{*} relativistic degrees of freedom, the total radiation density is
which obeys the following equation
Comparing this with the continuity equation (2.17) we obtain the pressure of the created particles, as
Now the dynamical equations are given by Eqs (2.15) and (2.16) with the energy density (3.45) and the pressure (3.47)
In the regime M(t − t_{os}) ≫ 1 the evolution of the scale factor is given by a ≃ a_{0}(t − t_{os})^{2/3}, and hence
where we have neglected the backreaction of created particles. Meanwhile the integration of Eq (3.45) gives
where we have used the averaged relation 〈R^{2}〉 ≃ 8M^{2}/(t−t_{os})^{2} [which comes from Eq. (3.33)]. The energy density ρ_{M} evolves slowly compared to H^{2} and finally it becomes a dominant contribution to the total energy density \((3{H^2} \simeq 8\pi {\rho _M}/m_{{\rm{pl}}}^2)\) at the time \({t_f} \simeq {t_{{\rm{os}}}} + 40m_{{\rm{pl}}}^2/({g_{\ast}}{M^3})\). In [426] it was found that the transition from the oscillating phase to the radiationdominated epoch occurs slower compared to the estimation given above. Since the epoch of the transient matterdominated era is about one order of magnitude longer than the analytic estimation [426], we take the value \({t_f} \simeq {t_{{\rm{os}}}} + 400m_{{\rm{pl}}}^2/({g_{\ast}}{M^3})\) to estimate the reheating temperature T_{r}. Since the particle energy density ρ_{M}(t_{f}) is converted to the radiation energy density \({\rho _r} = {g_\ast}{\pi ^2}T_r^4/30\), the reheating temperature can be estimated as^{Footnote 4}
As we will see in Section 7, the WMAP normalization of the CMB temperature anisotropies determines the mass scale to be M ≃ 3 × 10^{−6}m_{pl}. Taking the value g_{*} = 100, we have T_{r} ≲ 5 × 10^{9} GeV. For t > t_{f} the universe enters the radiationdominated epoch characterized by a ∝ t^{1/2}, R = 0, and ρ_{r} ∝ t^{−2}.
3.3.2 Case: ∣ξ∣ ≳ 1
If ∣ξ∣ is larger than the order of unity, one can expect the explosive particle production called preheating prior to the perturbative regime discussed above. Originally the dynamics of such gravitational preheating was studied in [70, 592] for a massive chaotic inflation model in Einstein gravity. Later this was extended to the f(R) model (3.6) [591].
Introducing a new field X_{k} = a^{3/2}χ_{k}, Eq. (3.37) reads
As long as ∣ξ∣ is larger than the order of unity, the last two terms in the bracket of Eq. (3.51) can be neglected relative to ξR. Since the Ricci scalar is given by Eq. (3.33) in the regime M(t − t_{os}) ≫ 1, it follows that
The oscillating term gives rise to parametric amplification of the particle χ_{k}. In order to see this we introduce the variable z defined by M(t − t_{os}) =2z ± π/2, where the plus and minus signs correspond to the cases ξ > 0 and ξ < 0 respectively. Then Eq. (3.52) reduces to the Mathieu equation
where
The strength of parametric resonance depends on the parameters A_{k} and q. This can be described by a stabilityinstability chart of the Mathieu equation [419, 353, 591]. In the Minkowski spacetime the parameters A_{k} and q are constant. If A_{k} and q are in an instability band, then the perturbation X_{k} grows exponentially with a growth index μ_{k}, i.e., \({X_k} \propto {e^{{\mu _k}z}}\). In the regime q ≪ 1 the resonance occurs only in narrow bands around A_{k} = ℓ^{2}, where ℓ = 1, 2, …, with the maximum growth index μ_{k} = q/2 [353]. Meanwhile, for large q(≫ 1), a broad resonance can occur for a wide range of parameter space and momentum modes [354].
In the expanding cosmological background both A_{k} and q vary in time. Initially the field X_{k} is in the broad resonance regime (q ≫ 1) for ∣ξ∣ ≫ 1, but it gradually enters the narrow resonance regime (q ≲ 1). Since the field passes many instability and stability bands, the growth index μ_{k} stochastically changes with the cosmic expansion. The nonadiabaticity of the change of the frequency \(\omega _k^2 = {k^2}/{a^2} + m_\chi ^2  4M\xi \sin \{M(t  {t_{{\rm{os}}}})\}/(t  {t_{{\rm{os}}}})\) can be estimated by the quantity
where the nonadiabatic regime corresponds to r_{na} ≳ 1. For small k and m_{χ} we have r_{na} ≫ 1 around M(t − t_{os}) = nπ, where n are positive integers. This corresponds to the time at which the Ricci scalar vanishes. Hence, each time R crosses 0 during its oscillation, the nonadiabatic particle production occurs most efficiently. The presence of the mass term m_{χ} tends to suppress the nonadiabaticity parameter r_{na}, but still it is possible to satisfy the condition r_{na} ≳ 1 around R = 0.
For the model (3.6) it was shown in [591] that massless χ particles are resonantly amplified for ∣ξ∣ ≳ 3. Massive particles with m_{χ} of the order of M can be created for ∣ξ∣ ≳ 10. Note that in the preheating scenario based on the model \(V(\phi, \chi) = (1/2)m_\phi ^2{\phi ^2} + (1/2){g^2}{\phi ^2}{\chi ^2}\) the parameter q decreases more rapidly (q ∝ 1/t^{2}) than that in the model (3.6) [354]. Hence, in our geometric preheating scenario, we do not require very large initial values of q [such as \(q > {\mathcal O}({10^3})\)] to lead to the efficient parametric resonance.
While the above discussion is based on the linear analysis, nonlinear effects (such as the modemode coupling of perturbations) can be important at the late stage of preheating (see, e.g., [354, 342]). Also the energy density of created particles affects the background cosmological dynamics, which works as a backreaction to the Ricci scalar. The process of the subsequent perturbative reheating stage can be affected by the explosive particle production during preheating. It will be of interest to take into account all these effects and study how the thermalization is reached at the end of reheating. This certainly requires the detailed numerical investigation of lattice simulations, as developed in [255, 254].
At the end of this section we should mention a number of interesting works about gravitational baryogenesis based on the interaction \((1/M_{\ast}^2)\int {{{\rm{d}}^4}x\sqrt { g} {J^\mu}{\partial _\mu}R}\) between the baryon number current J^{μ} and the Ricci scalar R (M_{*} is the cutoff scale characterizing the effective theory) [179, 376, 514]. This interaction can give rise to an equilibrium baryon asymmetry which is observationally acceptable, even for the gravitational Lagrangian f(R) =R^{n} with n close to 1. It will be of interest to extend the analysis to more general f(R) gravity models.
4 Dark Energy in f(R) Theories
In this section we apply f(R) theories to dark energy. Our interest is to construct viable f(R) models that can realize the sequence of radiation, matter, and accelerated epochs. In this section we do not attempt to find unified models of inflation and dark energy based on f(R) theories.
Originally the model f(R) = R − α/R^{n} (α > 0, n > 0) was proposed to explain the latetime cosmic acceleration [113, 120, 114, 143] (see also [456, 559, 17, 223, 212, 16, 137, 62] for related works). However, this model suffers from a number of problems such as matter instability [215, 244], the instability of cosmological perturbations [146, 74, 544, 526, 251], the absence of the matter era [28, 29, 239], and the inability to satisfy local gravity constraints [469, 470, 245, 233, 154, 448, 134]. The main reason why this model does not work is that the quantity f_{,RR} ≡ ∂^{2}f/≡R^{2} is negative. As we will see later, the violation of the condition f_{,RR} > 0 gives rise to the negative mass squared M^{2} for the scalaron field. Hence we require that f_{,RR} > 0 to avoid a tachyonic instability. The condition f_{,R} ≡ ∂f/∂R > 0 is also required to avoid the appearance of ghosts (see Section 7.4). Thus viable f(R) dark energy models need to satisfy [568]
where R_{0} is the Ricci scalar today.
In the following we shall derive other conditions for the cosmological viability of f(R) models. This is based on the analysis of [26]. For the matter Lagrangian \({{\mathcal L}_M}\) in Eq. (2.1) we take into account nonrelativistic matter and radiation, whose energy densities ρ_{m} and ρ_{r} satisfy
respectively. From Eqs. (2.15) and (2.16) it follows that
4.1 Dynamical equations
We introduce the following variables
together with the density parameters
It is straightforward to derive the following equations
where N = ln a is the number of efoldings, and
From Eq. (4.68) the Ricci scalar can be expressed by x_{3}/x_{2}. Since m depends on R, this means that m is a function of r, that is, m = m(r). The ΛCDM model, f(R) = R − 2Λ, corresponds to m = 0. Hence the quantity characterizes the deviation of the background dynamics from the ΛCDM model. A number of authors studied cosmological dynamics for specific f(R) models [160, 382, 488, 252, 31, 198, 280, 72, 41, 159, 235, 1, 279, 483, 321, 432].
The effective equation of state of the system is defined by
which is equivalent to w_{eff} = − (2x_{3} − 1)/3. In the absence of radiation (x_{4} = 0) the fixed points for the above dynamical system are
The points P_{5} and P_{6} are on the line m(r) = − r − 1 in the (r, m) plane.
The matterdominated epoch (Ω_{m} ≃ 1 and w_{eff} − 0) can be realized only by the point P_{5} for m close to 0. In the (r, m) plane this point exists around (r, m) = (−1, 0). Either the point P_{1} or P_{6} can be responsible for the latetime cosmic acceleration. The former is a de Sitter point (w_{eff} = −1) with r = −2, in which case the condition (2.11) is satisfied. The point P_{6} can give rise to the accelerated expansion (w_{eff} < −1/3) provided that \(m > (\sqrt 3  1)/2\), or −1/2 < m < 0, or \(m <  (1 + \sqrt 3)/2\).
In order to analyze the stability of the above fixed points it is sufficient to consider only timedependent linear perturbations δx_{i}(t) (i = 1, 2, 3) around them (see [170, 171] for the detail of such analysis). For the point P_{5} the eigenvalues for the 3 × 3 Jacobian matrix of perturbations are
where m_{5} ≡ m(r_{5}) and \(m_5^\prime \equiv {{{\rm{d}}m} \over {{\rm{d}}r}}({r_5})\) with r_{5} ≈ −1. In the limit that ∣m_{5}∣ ≪ 1 the latter two eigenvalues reduce to \( 3/4 \pm \sqrt { 1/{m_5}}\). For the models with m_{5} < 0, the solutions cannot remain for a long time around the point P_{5} because of the divergent behavior of the eigenvalues as m_{5} → −0. The model f(R) = R − α/R^{n} (α > 0, n > 0) falls into this category. On the other hand, if 0 < m_{5} < 0.327, the latter two eigenvalues in Eq. (4.77) are complex with negative real parts. Then, provided that \(m_5^\prime >  1\), the point P_{5} corresponds to a saddle point with a damped oscillation. Hence the solutions can stay around this point for some time and finally leave for the latetime acceleration. Then the condition for the existence of the saddle matter era is
The first condition implies that viable f(R) models need to be close to the ΛCDM model during the matter domination. This is also required for consistency with local gravity constraints, as we will see in Section 5.
The eigenvalues for the Jacobian matrix of perturbations about the point P_{1} are
where m_{1} = m(r = −2). This shows that the condition for the stability of the de Sitter point P_{1} is [440, 243, 250, 26]
The trajectories that start from the saddle matter point P_{5} satisfying the condition (4.78) and then approach the stable de Sitter point P_{1} satisfying the condition (4.80) are, in general, cosmologically viable.
One can also show that P_{6} is stable and accelerated for (a) \(m_6^\prime <  1,\,(\sqrt 3  1)/2 < {m_6} < 1\), (b) \(m_6^\prime >  1,\,{m_6} <  (1 + \sqrt 3)/2\), (c) \(m_6^\prime >  1,\,  1/2 < {m_6} < 0\), (d) \(m_6^\prime >  1,\,{m_6} \geq 1\). Since both P_{5} and P_{6} are on the line m = −r − 1, only the trajectories from \(m_5^\prime >  1\) to \(m_6^\prime <  1\) are allowed (see Figure 2). This means that only the case (a) is viable as a stable and accelerated fixed point P_{6}. In this case the effective equation of state satisfies the condition w_{eff} > −1.
From the above discussion the following two classes of models are cosmologically viable.

Class A: Models that connect P_{5} (r ≃ −1, m ≃ +0) to P_{1} (r = −2, 0 < m ≤ 1)

Class B: Models that connect P_{5} (r ≃ −1, m ≃ +0) to \({P_6}\left({m =  r  1,\,\left({\sqrt 3  1} \right)/2 < m < 1} \right)\)
From Eq. (4.56) the viable f(R) dark energy models need to satisfy the condition m > 0, which is consistent with the above argument.
4.2 Viable f(R) dark energy models
We present a number of viable f(R) models in the (r, m) plane. First we note that the ΛCDM model corresponds to m = 0, in which case the trajectory is the straight line (i) in Figure 2. The trajectory (ii) in Figure 2 represents the model f(R) = (R^{b} − Λ)^{c} [31], which corresponds to the straight line m(r) = [(1 − c)/c]r + b − 1 in the (r, m) plane. The existence of a saddle matter epoch demands the condition c ≥ 1 and bc ≃ 1. The trajectory (iii) represents the model [26, 382]
which corresponds to the curve m = n(1 + r)/r. The trajectory (iv) represents the model m(r) = −C(r + 1)(r^{2} + ar + b), in which case the latetime accelerated attractor is the point P_{6} with \({\left({\sqrt 3  1} \right)/2 < m < 1}\).
In [26] it was shown that m needs to be close to 0 during the radiation domination as well as the matter domination. Hence the viable f(R) models are close to the ΛCDM model in the region R ≫ R_{0}. The Ricci scalar remains positive from the radiation era up to the present epoch, as long as it does not oscillate around R = 0. The model f(R) = R − α/R^{n} (α > 0, n > 0) is not viable because the condition f_{,RR} > 0 is violated.
As we will see in Section 5, the local gravity constraints provide tight bounds on the deviation parameter m in the region of high density (R ≫ R_{0}), e.g., m(R) ≲ 10^{−15} for R = 10^{5}R_{0} [134, 596]. In order to realize a large deviation from the ΛCDM model such as \(m(R) > {\mathcal O}(0.1)\) today (R = R_{0}) we require that the variable m changes rapidly from the past to the present. The f(R) model given in Eq. (4.81), for example, does not allow such a rapid variation, because evolves as m ≃ (−r −1) in the region R ≫ R_{0}. Instead, if the deviation parameter has the dependence
it is possible to lead to the rapid decrease of m as we go back to the past. The models that behave as Eq. (4.82) in the regime R ≫ R_{0} are
The models (A) and (B) have been proposed by Hu and Sawicki [306] and Starobinsky [568], respectively. Note that R_{c} roughly corresponds to the order of R_{0} for \(\mu = O(1)\). This means that p = 2n + 1 for R ≫ R_{0}. In the next section we will show that both the models (A) and (B) are consistent with local gravity constraints for n ≳ 1.
In the model (A) the following relation holds at the de Sitter point:
where x_{d} ≡ R_{1}/R_{c} and R_{1} is the Ricci scalar at the de Sitter point. The stability condition (4.80) gives [587]
The parameter μ has a lower bound determined by the condition (4.86). When n = 1, for example, one has \({x_d} \geq \sqrt 3\) and \(\mu \geq 8\sqrt 3/9\). Under Eq. (4.86) one can show that the conditions (4.56) are also satisfied.
Similarly the model (B) satisfies [568]
with
When n = 1 we have \({x_d} \geq \sqrt 3\) and \(\mu \geq 8\sqrt 3/9\), which is the same as in the model (A). For general n, however, the bounds on μ in the model (B) are not identical to those in the model (A).
Another model that leads to an even faster evolution of m is given by [587]
A similar model was proposed by Appleby and Battye [35]. In the region R ≫ R_{c} the model (4.89) behaves as f(R) ≃ R − μR_{c} [1 − exp(−2R/R_{c})], which may be regarded as a special case of (4.82) in the limit that p ≫ 1^{Footnote 5}. The Ricci scalar at the de Sitter point is determined by μ, as
From the stability condition (4.80) we obtain
The models (A), (B) and (C) are close to the ΛCDM model for R ≫ R_{cs}, but the deviation from it appears when R decreases to the order of R_{c}. This leaves a number of observational signatures such as the phantomlike equation of state of dark energy and the modified evolution of matter density perturbations. In the following we discuss the dark energy equation of state in f(R) models. In Section 8 we study the evolution of density perturbations and resulting observational consequences in detail.
4.3 Equation of state of dark energy
In order to confront viable f(R) models with SN Ia observations, we rewrite Eqs. (4.59) and (4.60) as follows:
where A is some constant and
Defining ρ_{DE} and P_{DE} in the above way, we find that these satisfy the usual continuity equation
Note that this holds as a consequence of the Bianchi identities, as we have already mentioned in the discussion from Eq. (2.8) to Eq. (2.10).
The dark energy equation of state, w_{DE} ≡ P_{DE}/ρ_{DE}, is directly related to the one used in SN Ia observations. From Eqs. (4.92) and (4.93) it is given by
where the last approximate equality is valid in the regime where the radiation density ρ_{r} is negligible relative to the matter density ρ_{m}. The viable f(R) models approach the ΛCDM model in the past, i.e., F → 1 as R → ∞. In order to reproduce the standard matter era (3H^{2} ≃ κ^{2}ρ_{m}) for z ≫ 1, we can choose A = 1 in Eqs. (4.92) and (4.93). Another possible choice is A = F_{0}, where F_{0} is the present value of F. This choice may be suitable if the deviation of F_{0} from 1 is small (as in scalartensor theory with a nearly massless scalar field [583, 93]). In both cases the equation of state w_{DE} can be smaller than −1 before reaching the de Sitter attractor [306, 31, 587, 435], while the effective equation of state w_{eff} is larger than −1. This comes from the fact that the denominator in Eq. (4.97) becomes smaller than 1 in the presence of the matter fluid. Thus f(R) gravity models give rise to the phantom equation of state of dark energy without violating any stability conditions of the system. See [210, 417, 136, 13] for observational constraints on the models (4.83) and (4.84) by using the background expansion history of the universe. Note that as long as the latetime attractor is the de Sitter point the cosmological constant boundary crossing of w_{eff} reported in [52, 50] does not typically occur, apart from small oscillations of w_{eff} around the de Sitter point.
There are some works that try to reconstruct the forms of f(R) by using some desired form for the evolution of the scale factor a(t) or the observational data of SN Ia [117, 130, 442, 191, 621, 252]. We need to caution that the procedure of reconstruction does not in general guarantee the stability of solutions. In scalartensor dark energy models, for example, it is known that a singular behavior sometimes arises at lowredshifts in such a procedure [234, 271]. In addition to the fact that the reconstruction method does not uniquely determine the forms of f(R), the observational data of the background expansion history alone is not yet sufficient to reconstruct f(R) models in high precision.
Finally we mention a number of works [115, 118, 119, 265, 319, 515, 542, 90] about the use of metric f(R) gravity as dark matter instead of dark energy. In most of past works the powerlaw f(R) model f = R^{n} has been used to obtain spherically symmetric solutions for galaxy clustering. In [118] it was shown that the theoretical rotation curves of spiral galaxies show good agreement with observational data for n = 1.7, while for broader samples the bestfit value of the power was found to be n = 2.2 [265]. However, these values are not compatible with the bound ∣n − 1∣ < 7.2 × 10^{−19} derived in [62, 160] from a number of other observational constraints. Hence, it is unlikely that f(R) gravity works as the main source for dark matter.
5 Local Gravity Constraints
In this section we discuss the compatibility of f(R) models with local gravity constraints (see [469, 470, 245, 233, 154, 448, 251] for early works, and [31, 306, 134] for experimental constraints on viable f(R) dark energy models, and [101, 210, 330, 332, 471, 628, 149, 625, 329, 45, 511, 277, 534, 133, 445, 309, 89] for other related works). In an environment of high density such as Earth or Sun, the Ricci scalar R is much larger than the background cosmological value R_{0}. If the outside of a spherically symmetric body is a vacuum, the metric can be described by a Schwarzschild exterior solution with R = 0. In the presence of nonrelativistic matter with an energy density ρ_{m}, this gives rise to a contribution to the Ricci scalar R of the order κ^{2}ρ_{m}.
If we consider local perturbations δR on a background characterized by the curvature R_{0}, the validity of the linear approximation demands the condition δR ≪ R_{0}. We first derive the solutions of linear perturbations under the approximation that the background metric \(g_{\mu \nu}^{(0)}\) is described by the Minkowski metric η_{μν}. In the case of Earth and Sun the perturbation δR is much larger than R_{0}, which means that the linear theory is no longer valid. In such a nonlinear regime the effect of the chameleon mechanism [344, 343] becomes important, so that f(R) models can be consistent with local gravity tests.
5.1 Linear expansions of perturbations in the spherically symmetric background
First we decompose the quantities R, F(R), and T_{μν} into the background part and the perturbed part: R = R_{0} + δR, F = F_{0}(1 + δ_{F}), and T_{μν} = ^{(0)}T_{μν} + δT_{μν} about the approximate Minkowski background \((g_{\mu \nu}^{(0)} \approx {\eta _{\mu \nu}})\). In other words, although we consider R close to a meanfield value R_{0}, the metric is still very close to the Minkowski case. The linear expansion of Eq. (2.7) in a timeindependent background gives [470, 250, 154, 448]
where δT ≡ η^{μν}δT_{μν} and
The variable m is defined in Eq. (4.67). Since 0 < m(R_{0}) < 1 for viable f(R) models, it follows that M^{2} > 0 (recall that R_{0} > 0).
We consider a spherically symmetric body with mass M_{c}, constant density ρ (= −δT), radius r_{c}, and vanishing density outside the body. Since δ_{F} is a function of the distance r from the center of the body, Eq. (5.1) reduces to the following form inside the body (r < r_{c}):
whereas the r.h.s. vanishes outside the body (r > r_{c}). The solution of the perturbation δ_{F} for positive M^{2} is given by
where c_{i} (i = 1, 2, 3, 4) are integration constants. The requirement that \({({\delta _F})_{r > {r_c}}} \rightarrow 0\) as r → ∞ gives c_{4} = 0. The regularity condition at r = 0 requires that c_{2} = −c_{1}. We match two solutions (5.4) and (5.5) at r = r_{c} by demanding the regular behavior of δ_{F}(r) and \(\delta _F^{\prime}(r)\). Since δ_{F} ∝ δR, this implies that R is also continuous. If the mass M satisfies the condition Mr_{c} ≪ 1, we obtain the following solutions
As we have seen in Section 2.3, the action (2.1) in f(R) gravity can be transformed to the Einstein frame action by a transformation of the metric. The Einstein frame action is given by a linear action in \(\tilde R\), where \(\tilde R\) is a Ricci scalar in the new frame. The firstorder solution for the perturbation h_{μν} of the metric \({\tilde g_{\mu \nu}} = {F_0}({\eta _{\mu \nu}} + {h_{\mu \nu}})\) follows from the firstorder linearized Einstein equations in the Einstein frame. This leads to the solutions h_{00} = 2 GM_{c}/(F_{0}r) and h_{ij} = 2GM_{c}/(F_{0}r) δ_{ij}. Including the perturbation δ_{F} to the quantity F, the actual metric g_{μν} is given by [448]
Using the solution (5.7) outside the body, the (00) and (ii) components of the metric g_{μν} are
where\(G_{{\rm{eff}}}^{(N)}\) and γ are the effective gravitational coupling and the postNewtonian parameter, respectively, defined by
For the f(R) models whose deviation from the ΛCDM model is small (m ≪ 1), we have M^{2} ≃ R_{0}/[3m(R_{0})] and R ≃ 8πGρ. This gives the following estimate
where \({\Phi _c} = G{M_c}/({F_0}{r_c}) = 4\pi G\rho r_c^2/(3{F_0})\) is the gravitational potential at the surface of the body. The approximation Mr_{c} ≪ 1 used to derive Eqs. (5.6) and (5.7) corresponds to the condition
Since F_{0}δ_{F} = f_{,rr}(R_{0})δR, it follows that
The validity of the linear expansion requires that δR ≪ R_{0}, which translates into δ_{F} ≪ m(R_{0}). Since δ_{F} ≃ 2GM_{c}/(3F_{0}r_{c}) = 2Φ_{c}/3 at r = r_{c}, one has δ_{F} ≪ m(R_{0}) ≪ 1 under the condition (5.12). Hence the linear analysis given above is valid for m(R_{0}) ≫ Φ_{c}.
For the distance r close to r_{c} the post Newtonian parameter in Eq. (5.10) is given by γ≃ 1/2 (i.e., because Mr ≪ 1). The tightest experimental bound on γ is given by [616, 83, 617]:
which comes from the timedelay effect of the Cassini tracking for Sun. This means that f(R) gravity models with the light scalaron mass (Mr_{c} ≪ 1) do not satisfy local gravity constraints [469, 470, 245, 233, 154, 448, 330, 332]. The mean density of Earth or Sun is of the order of ρ ≃ 1–10 g/cm^{3}, which is much larger than the present cosmological density \(\rho _c^{(0)} \simeq {10^{ 29}}g/{\rm{c}}{{\rm{m}}^3}\). In such an environment the condition δR ≪ R_{0} is violated and the field mass M becomes large such that Mr_{c} ≫ 1. The effect of the chameleon mechanism [344, 343] becomes important in this nonlinear regime (δR ≫ R_{0}) [251, 306, 134, 101]. In Section 5.2 we will show that the f(R) models can be consistent with local gravity constraints provided that the chameleon mechanism is at work.
5.2 Chameleon mechanism in f(R) gravity
Let us discuss the chameleon mechanism [344, 343] in metric f(R) gravity. Unlike the linear expansion approach given in Section 5.1, this corresponds to a nonlinear effect arising from a large departure of the Ricci scalar from its background value R_{0}. The mass of an effective scalar field degree of freedom depends on the density of its environment. If the matter density is sufficiently high, the field acquires a heavy mass about the potential minimum. Meanwhile the field has a lighter mass in a lowdensity cosmological environment relevant to dark energy so that it can propagate freely. As long as the spherically symmetric body has a thinshell around its surface, the effective coupling between the field and matter becomes much smaller than the bare coupling ∣Q∣. In the following we shall review the chameleon mechanism for general couplings Q and then proceed to constrain f(R) dark energy models from local gravity tests.
5.2.1 Field profile of the chameleon field
The action (2.1) in f(R) gravity can be transformed to the Einstein frame action (2.32) with the coupling \(Q =  1/\sqrt 6\) between the scalaron field \(\phi = \sqrt {3/(2{\kappa ^2})}\) ln F and nonrelativistic matter. Let us consider a spherically symmetric body with radius \({\tilde r_c}\) in the Einstein frame. We approximate that the background geometry is described by the Minkowski spacetime. Varying the action (2.32) with respect to the field ϕ, we obtain
where \(\tilde r\) is a distance from the center of symmetry that is related to the distance r in the Jordan frame via \(\tilde r = \sqrt F r = {e^{ Q\kappa \phi}}r\). The effective potential V_{eff} is defined by
where ρ* is a conserved quantity in the Einstein frame [343]. Recall that the field potential V(ϕ) is given in Eq. (2.33). The energy density \(\tilde \rho\) in the Einstein frame is related with the energy density ρ in the Jordan frame via the relation \(\tilde \rho = \rho/{F^2} = {e^{4Q\kappa \phi}}\rho\). Since the conformal transformation gives rise to a coupling Q between matter and the field, \(\tilde \rho\) is not a conserved quantity. Instead the quantity \({\rho ^{\ast}} = {e^{3Q\kappa \phi}}\rho = {e^{ Q\kappa \phi}}\tilde \rho\) corresponds to a conserved quantity, which satisfies \({\tilde r^3}{\rho ^{\ast}} = {r^3}\rho\). Note that Eq. (5.15) is consistent with Eq. (2.42).
In the following we assume that a spherically symmetric body has a constant density ρ* = ρ_{A} inside the body \((\tilde r < {\tilde r_c})\) and that the energy density outside the body \((\tilde r > {\tilde r_c})\) is ρ* = ρ_{B} (≪ρ_{A}). The mass M_{c} of the body and the gravitational potential Φ_{c} at the radius \({\tilde r_c}\) are given by \({M_c} = (4\pi/3)\tilde r_c^3{\rho _A}\) and \({\Phi _c} = G{M_c}/{\tilde r_c}\), respectively. The effective potential has minima at the field values ϕ_{A} and ϕ_{B}:
The former corresponds to the region of high density with a heavy mass squared \(m_A^2 \equiv {V_{{\rm{eff,}}\phi \phi}}({\phi _A})\), whereas the latter to a lower density region with a lighter mass squared \(m_B^2 \equiv {V_{{\rm{eff,}}\phi \phi}}({\phi _B})\). In the case of Sun, for example, the field value ϕ_{B} is determined by the homogeneous dark matter/baryon density in our galaxy, i.e., ρ_{B} ≃ 10^{−24} g/cm^{3}.
When Q > 0 the effective potential has a minimum for the models with V_{,ϕ} < 0, which occurs, e.g., for the inverse powerlaw potential V(ϕ) = M^{4+n}ϕ^{−n}. The f(R) gravity corresponds to a negative coupling \((Q =  1/\sqrt 6)\), in which case the effective potential has a minimum for V_{,ϕ} > 0. As an example, let us consider the shape of the effective potential for the models (4.83) and (4.84). In the region R ≫ R_{c} both models behave as
For this functional form it follows that
The r.h.s. of Eq. (5.20) is smaller than 1, so that ϕ < 0. The limit R → ∞ corresponds to ϕ → −0. In the limit ϕ → −0 one has V → μR_{c}/(2κ^{2}) and V_{,ϕ} → ∞. This property can be seen in the upper panel of Figure 3, which shows the potential V(ϕ) for the model (4.84) with parameters n = 1 and μ = 2. Because of the existence of the coupling term \({e^{ \kappa \phi/\sqrt 6}}{\rho ^{\ast}}\), the effective potential V_{eff}(ϕ) has a minimum at
Since R ∼ κ^{2}ρ*≫ R_{c} in the region of high density, the condition ∣κϕ_{M}∣≪ 1 is in fact justified (for n and μ of the order of unity). The field mass m_{ϕ} about the minimum of the effective potential is given by
This shows that, in the regime R ∼ κ^{2}ρ* ≫ R_{c}, m_{ϕ} is much larger than the present Hubble parameter \({H_0}(\sim \sqrt {{R_c}})\). Cosmologically the field evolves along the instantaneous minima characterized by Eq. (5.22) and then it approaches a de Sitter point which appears as a minimum of the potential in the upper panel of Figure 3.
In order to solve the “dynamics” of the field ϕ in Eq. (5.15), we need to consider the inverted effective potential (−V_{eff}). See the lower panel of Figure 3 for illustration [which corresponds to the model (4.84)]. We impose the following boundary conditions:
The boundary condition (5.25) can be also understood as \({\lim\nolimits_{\tilde r \rightarrow \infty}}{\rm{d}}\phi {\rm{/d}}\tilde r = 0\). The field ϕ is at rest at \(\tilde r = 0\) and starts to roll down the potential when the mattercoupling term κQρ_{A}e^{Qκϕ} in Eq. (5.15) becomes important at a radius \({\tilde r_1}\). If the field value at \(\tilde r = 0\) is close to ϕ_{A}, the field stays around ϕ_{A} in the region \(0 < \tilde r < {\tilde r_1}\). The body has a thinshell if \({\tilde r_1}\) is close to the radius \({\tilde r_c}\) of the body.
In the region \(0 < \tilde r < {{\tilde r}_1}\) one can approximate the r.h.s. of Eq. (5.15) as \({\rm{d}}{V_{{\rm{eff}}}}/{\rm{d}}\phi \simeq m_A^2(\phi  {\phi _A})\) around ϕ = ϕ_{A}, where \(m_A^2 = {R_c}{({\kappa ^2}{\rho _A}/{R_c})^{2(n + 1)}}/[6n(n + 1)]\). Hence the solution to Eq. (5.15) is given by \(\phi (\tilde r) = {\phi _A} + A{e^{ {m_A}\tilde r}}/\tilde r + B{e^{{m_A}\tilde r}}/\tilde r\), where A and B are constants. In order to avoid the divergence of ϕ at \(\tilde r = 0\) we demand the condition B = −A, in which case the solution is
In fact, this satisfies the boundary condition (5.24).
In the region \({{\tilde r}_1} < \tilde r < {{\tilde r}_c}\) the field \(\vert\phi (\tilde r)\vert\) evolves toward larger values with the increase of \({\tilde r}\). In the lower panel of Figure 3 the field stays around the potential maximum for \(0 < \tilde r < {{\tilde r}_1}\), but in the regime \({{\tilde r}_1} < \tilde r < {{\tilde r}_c}\) it moves toward the left (largely negative ϕ region). Since ∣V_{,ϕ}∣ ≪ κQρ_{A}e^{Qκϕ}∣ in this regime we have that dV_{eff}/dϕ ≃ κQρ_{A} in Eq. (5.15), where we used the condition Qκϕ ≪ 1. Hence we obtain the following solution
where C and D are constants.
Since the field acquires a sufficient kinetic energy in the region \({{\tilde r}_1} < \tilde r < {{\tilde r}_c}\), the field climbs up the potential hill toward the largely negative ϕ region outside the body \((\tilde r > {{\tilde r}_c})\). The shape of the effective potential changes relative to that inside the body because the density drops from ρ_{A} to ρ_{B}. The kinetic energy of the field dominates over the potential energy, which means that the term dV_{eff}/dϕ in Eq. (5.15) can be neglected. Recall that one has ∣ϕ_{B}∣ ≫ ∣ϕ_{A}∣ under the condition ρ_{A} ≫ ρ_{B} [see Eq. (5.22)]. Taking into account the mass term \(m_B^2 = {R_c}{({k^2}{\rho _B}/{R_c})^{2(n + 1)}}/[6n(n + 1)]\), we have \({\rm{d}}{V_{{\rm{eff}}}}/{\rm{d}}\phi \simeq m_B^2(\phi  {\phi _B})\) on the r.h.s. of Eq. (5.15). Hence we obtain the solution \(\phi (\tilde r) = {\phi _B} + E{e^{ {m_B}(\tilde r  {{\tilde r}_c})}}/\tilde r + F{e^{{m_B}(\tilde r  {{\tilde r}_c})}}/\tilde r\) with constants E and F. Using the boundary condition (5.25), it follows that F = 0 and hence
Three solutions (5.26), (5.27) and (5.28) should be matched at \(\tilde r = {{\tilde r}_1}\) and \(\tilde r = {{\tilde r}_c}\) by imposing continuous conditions for ϕ and \({\rm{d}}\phi {\rm{/d}}\tilde r\). The coefficients A, C, D and E are determined accordingly [575]:
where
if the maxss m_{B} outside the body is small to satisfy the condition \({m_B}{{\tilde r}_c} \ll 1\) and m_{A} ≫ m_{B}, we can neglect the contribution of the m_{B}dependent terms in Eqs. (5.29)–(5.32). Then the field profile is given by [575]
Originally a similar field profile was derived in [344, 343] by assuming that the field is frozen at ϕ = ϕ_{A} in the region \({{\tilde r}_1} < \tilde r < {{\tilde r}_c}\).
The radius r_{1} is determined by the following condition
This translates into
where \({\Phi _c} = {\kappa ^2}{M_c}/(8\pi {{\tilde r}_c}) = {\kappa ^2}{\rho _A}\tilde r_c^2/6\) is the gravitational potential at the surface of the body. Using this relation, the field profile (5.37) outside the body reduces to
If the field value at \(\tilde r = 0\) is away from ϕ_{A}, the field rolls down the potential for \(\tilde r > 0\). This corresponds to taking the limit \({{\tilde r}_1} \to 0\) in Eq. (5.40), in which case the field profile outside the body is given by
This shows that the effective coupling is of the order of Q and hence for \(\vert Q\vert = {\mathcal O}(1)\) local gravity constraints are not satisfied.
5.2.2 Thinshell solutions
Let us consider the case in which \({{\tilde r}_1}\) is close to \({{\tilde r}_c}\), i.e.
This corresponds to the thinshell regime in which the field is stuck inside the star except around its surface. If the field is sufficiently massive inside the star to satisfy the condition \({m_A}{{\tilde r}_c} \gg 1\), Eq. (5.39) gives the following relation
where ϵ_{th} is called the thinshell parameter [344, 343]. Neglecting secondorder terms with respect to \(\Delta {{\tilde r}_c}/{{\tilde r}_c}\) and \(1/({m_A}{{\tilde r}_c})\) in Eq. (5.40), it follows that
where Q_{eff} is the effective coupling given by
Since õ_{th} ≪ 1 under the conditions \(\Delta {{\tilde r}_c}/{{\tilde r}_c} \ll 1\) and \(1/({m_A}{{\tilde r}_c}) \ll 1\), the amplitude of the effective coupling Q_{eff} becomes much smaller than 1. In the original papers of Khoury and Weltman [344, 343] the thinshell solution was derived by assuming that the field is frozen with the value ϕ = ϕ_{A} in the region \(0 < \tilde r < {{\tilde r}_1}\). In this case the thinshell parameter is given by \({\epsilon _{{\rm{th}}}} \simeq \Delta {{\tilde r}_c}/{{\tilde r}_c}\), which is different from Eq. (5.43). However, this difference is not important because the condition \(\Delta {{\tilde r}_c}/{{\tilde r}_c} \gg 1/({m_A}{{\tilde r}_c})\) is satisfied for most of viable models [575].
5.2.3 Post Newtonian parameter
We derive the bound on the thinshell parameter from experimental tests of the post Newtonian parameter in the solar system. The spherically symmetric metric in the Einstein frame is described by [251]
where \(\tilde {\mathcal A}(\tilde r)\) and \(\tilde {\mathcal B}(\tilde r)\) are functions of \({\tilde r}\) and dΩ^{2} = dθ^{2} + (sin^{2} θ)dϕ^{2}. In the weak gravitational background \((\tilde {\mathcal A}(\tilde r) \ll 1\) and \(\tilde {\mathcal B}(\tilde r) \ll 1)\) the metric outside the spherically symmetric body with mass M_{c} is given by \(\tilde {\mathcal A}(\tilde r) \simeq \tilde {\mathcal B}(\tilde r) \simeq G{M_c}/\tilde r\).
Let us transform the metric (5.46) back to that in the Jordan frame under the inverse of the conformal transformation, \({g_{\mu \nu}} = {e^{2Q\kappa \phi}}{{\tilde g}_{\mu \nu}}\). Then the metric in the Jordan frame, \({\rm{d}}{s^2} = {e^{2Q\kappa \phi}}{\rm{d}}{{\tilde s}^2} = {g_{\mu \nu}}{\rm{d}}{x^\mu}{\rm{d}}{x^\nu}\), is given by
Under the condition ∣Qκϕ∣ ≪ 1 we obtain the following relations
In the following we use the approximation \(r \simeq \tilde r\), which is valid for ∣Qκϕ∣ ≪ 1. Using the thinshell solution (5.44), it follows that
where we have used the approximation ∣ϕ_{B}∣ ≫ ∣ϕ_{A}∣ and hence ϕ_{B} ≃ 6QΦ_{c}ϵ_{th}/κ.
The term Qκϕ_{B} in Eq. (5.48) is smaller than \({\mathcal A}(r) = G{M_c}/r\) under the condition r/r_{c} < (6Q^{2}ϵ_{th})^{−1}. Provided that the field ϕ reaches the value ϕ_{B} with the distance r_{B} satisfying the condition r_{B}/r_{c} < (6Q^{2}∊_{th})^{−1}, the metric \({\mathcal A}(r)\) does not change its sign for r < r_{B}. The postNewtonian parameter γ is given by
The experimental bound (5.14) can be satisfied as long as the thinshell parameter ϵ_{th} is much smaller than 1. If we take the distance r = r_{c}, the constraint (5.14) translates into
where ϵ_{th,⊙} is the thinshell parameter for Sun. In f(R) gravity \((Q =  1/\sqrt 6)\) this corresponds to ϵ_{th,⊙} < 2.3 × 10^{−5}.
5.2.4 Experimental bounds from the violation of equivalence principle
Let us next discuss constraints on the thinshell parameter from the possible violation of equivalence principle (EP). The tightest bound comes from the solar system tests of weak EP using the freefall acceleration of Earth (a_{⊕}) and Moon (a_{Moon}) toward Sun [343]. The experimental bound on the difference of two accelerations is given by [616, 83, 617]
Provided that Earth, Sun, and Moon have thinshells, the field profiles outside the bodies are given by Eq. (5.44) with the replacement of corresponding quantities. The presence of the field ϕ(r) with an effective coupling Q_{eff} gives rise to an extra acceleration, a^{fifth} = ∣Q_{eff}∇ϕ(r)∣. Then the accelerations a_{⊕} and a_{Moon} toward Sun (mass M_{⊙}) are [343]
where ϵ_{th, ⊕} is the thinshell parameter of Earth, and Φ_{⊙} ≃ 2.1 × 10^{−6}, Φ_{⊕} ≃ 7.0 × 10^{−10}, Φ_{Moon} ≃ 3.1 × 10^{−11} are the gravitational potentials of Sun, Earth and Moon, respectively. Hence the condition (5.52) translates into [134, 596]
which corresponds to ϵ_{th,⊕} < 2.2 × 10^{−6} in f(R) gravity. This bound provides a tighter bound on model parameters compared to (5.51).
Since the condition ∣ϕ_{B}∣≫ ∣ϕ_{A}∣ is satisfied for ρ_{A} ≫ ρ_{B}, one has ϵ_{th,⊕} ≃ κϕ_{B}/(6QΦ_{⊕}) from Eq. (5.43). Then the bound (5.55) translates into
5.2.5 Constraints on model parameters in f(R) gravity
We place constraints on the f(R) models given in Eqs. (4.83) and (4.84) by using the experimental bounds discussed above. In the region of high density where is much larger than R_{c}, one can use the asymptotic form (5.19) to discuss local gravity constraints. Inside and outside the spherically symmetric body the effective potential V_{eff} for the model (5.19) has two minima at
The bound (5.56) translates into
where x_{d} ≡ R_{1}/R_{c} and R_{1} is the Ricci scalar at the latetime de Sitter point. In the following we consider the case in which the Lagrangian density is given by (5.19) for R ≥ R_{1}. If we use the models (4.83) and (4.84), then there are some modifications for the estimation of R_{1}. However this change should be insignificant when we place constraints on model parameters.
At the de Sitter point the model (5.19) satisfies the condition \(\mu = x_d^{2n + 1}/[2(x_d^{2n}  n  1)]\). Substituting this relation into Eq. (5.58), we find
For the stability of the de Sitter point we require that m(R_{1}) < 1, which translates into the condition \(x_d^{2n} > 2{n^2} + 3n + 1\). Hence the term \(n/[2(x_d^{2n}  n  1)]\) in Eq. (5.59) is smaller than 0.25 for n > 0.
We use the approximation that R_{1} and ρ_{B} are of the orders of the present cosmological density 10^{−29} g/cm^{3} and the baryonic/dark matter density 10^{−24} g/cm^{3} in our galaxy, respectively. From Eq. (5.59) we obtain the bound [134]
Under this condition one can see an appreciable deviation from the ΛCDM model cosmologically as R decreases to the order of R_{c}.
If we consider the model (4.81), it was shown in [134] that the bound (5.56) gives the constraint n < 3 × 10^{−10}. This means that the deviation from the ΛCDM model is very small. Meanwhile, for the models (4.83) and (4.84), the deviation from the ΛCDM model can be large even for \(n = {\mathcal O}(1)\), while satisfying local gravity constraints. We note that the model (4.89) is also consistent with local gravity constraints.
6 Cosmological Perturbations
The f(R) theories have one extra scalar degree of freedom compared to the ΛCDM model. This feature results in more freedom for the background. As we have seen previously, a viable cosmological sequence of radiation, matter, and accelerated epochs is possible provided some conditions hold for f(R). In principle, however, one can specify any given H = H(a) and solve Eqs. (2.15) and (2.16) for those f(R(a)) compatible with the given H(a).
Therefore the background cosmological evolution is not in general enough to distinguish f(R) theories from other theories. Even worse, for the same H(a), there may be some different forms of f(R) which fulfill the Friedmann equations. Hence other observables are needed in order to distinguish between different theories. In order to achieve this goal, perturbation theory turns out to be of fundamental importance. More than this, perturbations theory in cosmology has become as important as in particle physics, since it gives deep insight into these theories by providing information regarding the number of independent degrees of freedom, their speed of propagation, their timeevolution: all observables to be confronted with different data sets.
The main result of the perturbation analysis in f(R) gravity can be understood in the following way. Since it is possible to express this theory into a form of scalartensor theory, this should correspond to having a scalarfield degree of freedom which propagates with the speed of light. Therefore no extra vector or tensor modes come from the f(R) gravitational sector. Introducing matter fields will in general increase the number of degrees of freedom, e.g., a perfect fluid will only add another propagating scalar mode and a vector mode as well. In this section we shall provide perturbation equations for the general Lagrangian density f(R, ϕ) including metric f(R) gravity as a special case.
6.1 Perturbation equations
We start with a general perturbed metric about the flat FLRW background [57, 352, 231, 232, 437]
where α, β, ψ, γ, are scalar perturbations, S_{i}, F_{i} are vector perturbations, and h_{ij} is the tensor perturbations, respectively. In this review we focus on scalar and tensor perturbations, because vector perturbations are generally unimportant in cosmology [71].
For generality we consider the following action
where f(R, ϕ) is a function of the Ricci scalar R and the scalar field ϕ, ω(ϕ) and V(ϕ) are functions of ϕ, and S_{M} is a matter action. We do not take into account an explicit coupling between the field ϕ and matter. The action (6.2) covers not only f(R) gravity but also other modified gravity theories such as BransDicke theory, scalartensor theories, and dilaton gravity. We define the quantity F(R, ϕ) ≡ ∂, f/∂R. Varying the action (6.2) with respect to g_{μν} and ϕ, we obtain the following field equations
where \(T_{\mu \nu}^{(M)}\) is the energymomentum tensor of matter.
We decompose ϕ and F into homogeneous and perturbed parts, \(\phi = \bar \phi + \delta \phi\) and \(F = \bar F + \delta F\), respectively. In the following we omit the bar for simplicity. The energymomentum tensor of an ideal fluid with perturbations is
where υ characterizes the velocity potential of the fluid. The conservation of the energymomentum tensor (∇^{μ}T_{μν} = 0) holds for the theories with the action (6.2) [357].
For the action (6.2) the background equations (without metric perturbations) are given by
where R is given in Eq. (2.13).
For later convenience, we define the following perturbed quantities
Perturbing Einstein equations at linear order, we obtain the following equations [316, 317] (see also [436, 566, 355, 438, 312, 313, 314, 492, 138, 33, 441, 328])
where δR is given by
We shall solve the above equations in two different contexts: (i) inflation (Section 7), and (ii) the matter dominated epoch followed by the latetime cosmic acceleration (Section 8).
6.2 Gaugeinvariant quantities
Before discussing the detail for the evolution of cosmological perturbations, we construct a number of gaugeinvariant quantities. This is required to avoid the appearance of unphysical modes. Let us consider the gauge transformation
where δt and δx characterize the time slicing and the spatial threading, respectively. Then the scalar metric perturbations α, β, ϕ and E transform as [57, 71, 412]
Matter perturbations such as δϕ and δρ obey the following transformation rule
Note that the quantity δF is also subject to the same transformation: \(\overset \wedge {\delta F} = \delta F  \dot F\delta t\). We express the scalar part of the 3momentum energymomentum tensor \(\delta T_i^0\) as
for the scalar field and the perfect fluid one has \(\delta q =  \dot \phi \delta \phi\) and δq = −(ρ_{M} + P_{M})υ, respectively. This quantity transforms as
One can construct a number of gaugeinvariant quantities unchanged under the transformation (6.20):
Since \(\delta q =  \dot \phi \delta \phi\) for singlefield inflation with a potential V(ϕ), \({\mathcal R}\) is identical to \({{\mathcal R}_{\delta \phi}}\) [where we used \(\rho = {{\dot \phi}^2}/2 + V(\phi)\) and \(P = {{\dot \phi}^2}/2  V(\phi)]\). In f(R) gravity one can introduce a scalar field ϕ as in Eq. (2.31), so that \({{\mathcal R}_{\delta F}} = {{\mathcal R}_{\delta \phi}}\). From the gaugeinvariant quantity (6.31) it is also possible to construct another gaugeinvariant quantity for the matter perturbation of perfect fluids:
where w_{M} = P_{M}/ρ_{M}.
We note that the tensor perturbation h_{ij} is invariant under the gauge transformation [412].
We can choose specific gauge conditions to fix the gauge degree of freedom. After fixing a gauge, two scalar variables δt and δx are determined accordingly. The Longitudinal gauge corresponds to the gauge choice \(\hat \beta = 0\) and \(\hat \gamma = 0\), under which \(\delta t = a(\beta + \alpha \dot \gamma)\) and δx = γ. In this gauge one has \(\hat \Phi = \hat \alpha\) and \(\hat \Psi =  \hat \psi\), so that the line element (without vector and tensor perturbations) is given by
where we omitted the hat for perturbed quantities.
The uniformfield gauge corresponds to \(\overset \wedge {\delta \phi} = 0\) which fixes \(\delta t = \delta \phi/\dot \phi\). The spatial threading δx is fixed by choosing either \(\hat \beta = 0\) or \(\hat \gamma = 0\) (up to an integration constant in the former case). For this gauge choice one has \({{\hat {\mathcal R}}_{\delta \phi}} = \hat \psi\). Since the spatial curvature \(^{(3)}{\mathcal R}\) on the constanttime hypersurface is related to ϕ via the relation \(^{(3)}{\mathcal R} =  4{\nabla ^2}\psi/{a^2}\), the quantity \({\mathcal R}\) is often called the curvature perturbation on the uniformfield hypersurface. We can also choose the gauge condition \(\overset \wedge {\delta q} = 0\) or \(\overset \wedge {\delta F} = 0\).
7 Perturbations Generated During Inflation
Let us consider scalar and tensor perturbations generated during inflation for the theories (6.2) without taking into account the perfect fluid (S_{M} = 0). In f(R) gravity the contribution of the field ϕ such as δϕ is absent in the perturbation equations (6.11)–(6.16). One can choose the gauge condition ϕF = 0, so that \({{\mathcal R}_{\delta F}} = \psi\). In scalartensor theory in which F is the function of ϕ alone (i.e., the coupling of the form F(ϕ)R without a nonlinear term in R), the gauge choice δϕ = 0 leads to \({{\mathcal R}_{\delta \phi}} = \psi\). Since δF = F_{,ϕ}δϕ = 0 in this case, we have \({{\mathcal R}_{\delta F}} = {{\mathcal R}_{\delta \phi}} = \psi\).
We focus on the effective singlefield theory such as f(R) gravity and scalartensor theory with the coupling F(ϕ)R, by choosing the gauge condition δϕ = 0 and δF = 0. We caution that this analysis does not cover the theory such as \({\mathcal L} = \xi (\phi)R + \alpha {R^2}\) [500], because the quantity F depends on both ϕ and R (in other words, δF = F_{,ϕ}δϕ + F_{,R}δR). In the following we write the curvature perturbations \({{\mathcal R}_{\delta F}}\) and \({{\mathcal R}_{\delta \phi}}\) as \({\mathcal R}\).
7.1 Curvature perturbations
Since δϕ = 0 and δF = 0 in Eq. (6.12) we obtain
Plugging Eq. (7.34) into Eq. (6.11), we have
Equation (6.14) gives
where we have used the background equation (6.7). Plugging Eqs. (7.34) and (7.35) into Eq. (7.36), we find that the curvature perturbation satisfies the following simple equation in Fourier space
where k is a comoving wavenumber and
Introducing the variables \({z_s} = a{\sqrt Q _s}\) and \(u = {z_s}{\mathcal R}\), Eq. (7.37) reduces to
where a prime represents a derivative with respect to the conformal time η = ∫ a^{−1}dt.
In General Relativity with a canonical scalar field ϕ one has ω = 1 and F = 1, which corresponds to \({Q_s} = {{\dot \phi}^2}/{H^2}\). Then the perturbation u corresponds to \(u = a[  \delta \phi + (\dot \phi/H)\psi ]\). In the spatially flat gauge (ω = 0) this reduces to u = − aδϕ, which implies that the perturbation u corresponds to a canonical scalar field δχ = aδϕ. In modified gravity theories it is not clear at this stage that the perturbation \(u = a\sqrt {{Q_s}} {\mathcal R}\) corresponds a canonical field that should be quantized, because Eq. (7.37) is unchanged by multiplying a constant term to the quantity Q_{s} defined in Eq. (7.38). As we will see in Section 7.4, this problem is overcome by considering a secondorder perturbed action for the theory (6.2) from the beginning.
In order to derive the spectrum of curvature perturbations generated during inflation, we introduce the following variables [315]
where \(E \equiv F[\omega + 3{{\dot F}^2}/(2{\kappa ^2}{{\dot \phi}^2}F)]\). Then the quantity Q_{s} can be expressed as
if the parameter ϵ_{1} is constant, it follows that η =−1/[(1−ϵ_{1})aH] [573]. If \({{\dot \epsilon}_i} = 0\) (i = 1, 2, 3, 4) one has
then the solution to Eq. (7.39) can be expressed as a linear combination of Hankel functions,
where c_{1} and c_{2} are integration constants.
During inflation one has ∣ϵ_{i}∣ ≪ 1, so that \(z_s^{^{\prime\prime}}/{z_s} \approx {(aH)^2}\). For the modes deep inside the Hubble radius (k ≫ aH, i.e., ∣kη∣ ≫1) the perturbation u satisfies the standard equation of a canonical field in the Minkowski spacetime: u″+ k^{2}u ≃ 0. After the Hubble radius crossing (k = aH) during inflation, the effect of the gravitational term \(z_s^{^{\prime\prime}}/{z_s}\) becomes important. In the superHubble limit (k ≪ aH, i.e., ∣kη≪ 1) the last term on the l.h.s. of Eq. (7.37) can be neglected, giving the following solution
where c_{1} and c_{2} are integration constants. The second term can be identified as a decaying mode, which rapidly decays during inflation (unless the field potential has abrupt features). Hence the curvature perturbation approaches a constant value c_{1} after the Hubble radius crossing (k < aH).
In the asymptotic past (kη → −∞) the solution to Eq. (7.39) is determined by a vacuum state in quantum field theory [88], as \(u \rightarrow {e^{ ik\eta}}/\sqrt {2k}\). This fixes the coefficients to be c_{1} = 1 and c_{2} = 0, giving the following solution
We define the power spectrum of curvature perturbations,
Using the solution (7.45), we obtain the power spectrum [317]
where we have used the relations \(H_\nu ^{(1)}(k\vert \eta \vert) \rightarrow  (i/\pi)\Gamma (\nu){(k\vert \eta \vert/2)^{ \nu}}\) for kη → 0 and \(\Gamma (3/2) = \sqrt \pi/2\). Since the curvature perturbation is frozen after the Hubble radius crossing, the spectrum (7.47) should be evaluated at k = aH. The spectral index of \({\mathcal R}\), which is defined by \({n_{\mathcal R}}  1 = {\rm{d ln}}\,{{\mathcal P}_{\mathcal R}}/{\rm{d}}\,{\rm{ln}}\,k{\vert _{k = aH}}\), is
where \({\nu _{\mathcal R}}\) is given in Eq. (7.42). As long as ∣ϵ_{i}∣(i = 1, 2, 3, 4) are much smaller than 1 during inflation, the spectral index reduces to
where we have ignored those terms higher than the order of ϵ_{i}’s. Provided that ∣ϵ_{i}∣ ≫ 1 the spectrum is close to scaleinvariant \(({n_{\mathcal R}} \simeq 1)\). From Eq. (7.47) the power spectrum of curvature perturbations can be estimated as
A minimally coupled scalar field ϕ in Einstein gravity corresponds to ϵ_{3} = 0, ϵ_{4} = 0 and \({Q_s} = {{\dot \phi}^2}/{H^2}\), in which case we obtain the standard results \({n_{\mathcal R}}  1 \simeq  4{\epsilon _1}  2{\epsilon _2}\) and \({{\mathcal P}_{\mathcal R}} \simeq {H^4}/(4{\pi ^2}{{\dot \phi}^2})\) in slowroll inflation [573, 390].
7.2 Tensor perturbations
Tensor perturbations h_{ij} have two polarization states, which are generally written as λ = +, × [391]. In terms of polarization tensors \(e_{ij}^ +\) and \(e_{ij}^ \times\). they are given by
If the direction of a momentum k is along the zaxis, the nonzero components of polarization tensors are given by \(e_{xx}^ + =  e_{yy}^ + = 1\) and \(e_{xy}^ \times = e_{yx}^ \times = 1\).
For the action (6.2) the Fourier components h_{λ} (λ = +, ×) obey the following equation [314]
This is similar to Eq. (7.37) of curvature perturbations, apart from the difference of the factor F instead of Q_{s}. Defining new variables \({z_t} = a\sqrt F\) and \({u_\lambda} = {z_t}{h_\lambda}/\sqrt {16\pi G}\), it follows that
We have introduced the factor 16πG to relate a dimensionless massless field h_{λ} with a massless scalar field u_{λ} having a unit of mass.
If \({{\dot \epsilon}_i} = 0\), we obtain
We follow the similar procedure to the one given in Section 7.1. Taking into account polarization states, the spectrum of tensor perturbations after the Hubble radius crossing is given by
which should be evaluated at the Hubble radius crossing (k = aH). The spectral index of \({{\mathcal P}_T}\) is
where ν_{t} is given in Eq. (7.54). If ∣ϵ_{i}∣ ≪ 1, this reduces to
then the amplitude of tensor perturbations is given by
We define the tensortoscalar ratio
For a minimally coupled scalar field ϕ in Einstein gravity, it follows that n_{T} ≃ −2ϵ_{1}, \({{\mathcal P}_T} \simeq 16{H^2}/(\pi m_{{\rm{p1}}}^2)\), and r ≃ 16ϵ_{1}.
7.3 The spectra of perturbations in inflation based onf(R) gravity
Let us study the spectra of scalar and tensor perturbations generated during inflation in metric f(R) gravity. Introducing the quantity E = 3Ḟ^{2}/(2κ^{2}), we have \({\epsilon _4} = \ddot F/(H\dot F)\) and
Since the field kinetic term \({{\dot \phi}^2}\) is absent, one has ϵ_{2} = 0 in Eqs. (7.42) and (7.49). Under the conditions ∣ϵ_{i}∣ ≪ 1 (i = 1, 3, 4), the spectral index of curvature perturbations is given by \({n_{\mathcal R}}  1 \simeq  4{\epsilon _1} + 2{\epsilon _3}  2{\epsilon _4}\).
In the absence of the matter fluid, Eq. (2.16) translates into
which gives ϵ_{1}≃ −ϵ_{3} for ∣ϵ_{4}∣ ≪ 1. Hence we obtain [315]
From Eqs. (7.50) and (7.60), the amplitude of \({\mathcal R}\) is estimated as
Using the relation ϵ_{1} ≃ −ϵ_{3}, the spectral index (7.57) of tensor perturbations is given by
which vanishes at firstorder of slowroll approximations. From Eqs. (7.58) and (7.63) we obtain the tensortoscalar ratio
7.3.1 The model f(R) = αR^{n} (n > 0)
Let us consider the inflation model: f(R) = αR^{n} (n > 0). From the discussion given in Section 3.1 the slowroll parameters ϵ_{i} (i = 1, 3, 4) are constants:
In this case one can use the exact results (7.48) and (7.56) with \({\nu _{\mathcal R}}\) and ν_{t} given in Eqs. (7.42) and (7.54) (with ϵ_{2} = 0). Then the spectral indices are
If n = 2 we obtain the scaleinvariant spectra with \({n_{\mathcal R}} = 1\) and n_{T} = 0. Even the slight deviation from n = 2 leads to a rather large deviation from the scaleinvariance. If n = 1.7, for example, one has \({n_{\mathcal R}}  1 = {n_T} =  0.13\), which does not match with the WMAP 5year constraint: \({n_{\mathcal R}} = 0.960 \pm 0.013\) [367].
7.3.2 The model f(R) = R+R^{2}/(6 M^{2})
For the model f(R) = R+R^{2}/(6M^{2}), the spectrum of the curvature perturbation \({\mathcal R}\) shows some deviation from the scaleinvariance. Since inflation occurs in the regime R ≫ M^{2} and ∣Ḣ∣≪ H^{2}, one can approximate F ≃R/(3M^{2}) ≃ 4H^{2}/M^{2}. Then the power spectra (7.63) and (7.58) yield
where we have employed the relation ϵ_{3} ≃ − ϵ_{1}.
Recall that the evolution of the Hubble parameter during inflation is given by Eq. (3.9). As long as the time t_{k} at the Hubble radius crossing (k = aH) satisfies the condition (M^{2}/6)(t_{k} − t_{i}) ≪ H_{i}, one can approximate H(t_{k}) ≪ H_{i}. Using Eq. (3.9), the number of efoldings from t = t_{k} to the end of inflation can be estimated as
then the amplitude of the curvature perturbation is given by
The WMAP 5year normalization corresponds to \({{\mathcal P}_{\mathcal R}} = (2.445 \pm 0.096) \times {10^{ 9}}\) at the scale k = 0.002 Mpc^{—1} [367]. Taking the typical value N_{k} = 55, the mass M is constrained to be
Using the relation F ≪ 4H^{2}/M^{2}, it follows that ϵ_{4} ≃ − ϵ_{1}. Hence the spectral index (7.62) reduces to
for N_{k} = 55 we have \({n_{\mathcal R}} \simeq 0.964\), which is in the allowed region of the WMAP 5year constraint (\({n_{\mathcal R}} = 0.964 \pm 0.013\) at the 68% confidence level [367]). The tensortoscalar ratio (7.65) can be estimated as
which satisfies the current observational bound r < 0.22 [367]. We note that a minimally coupled field with the potential V(ϕ) = m^{2}ϕ^{2}/2 in Einstein gravity (chaotic inflation model [393]) gives rise to a larger tensortoscalar ratio of the order of 0.1. Since future observations such as the Planck satellite are expected to reach the level of \(r = {\mathcal O}({10^{ 2}})\), they will be able to discriminate between the chaotic inflation model and the Starobinsky’s f(R) model.
7.3.3 The power spectra in the Einstein frame
Let us consider the power spectra in the Einstein frame. Under the conformal transformation \({{\tilde g}_{\mu \nu}} = F{g_{\mu \nu}}\), the perturbed metric (6.1) is transformed as
We decompose the conformal factor into the background and perturbed parts, as
In what follows we omit a bar from F. We recall that the background quantities are transformed as Eqs. (2.44) and (2.47). The transformation of scalar metric perturbations is given by
Meanwhile vector and tensor perturbations are invariant under the conformal transformation (\(({{\tilde S}_i} = {S_i},{{\tilde F}_i} = {F_i},{{\tilde h}_{ij}} = {h_{ij}})\)).
Using the above transformation law, one can easily show that the curvature perturbation \({\mathcal R} = \psi  H\delta F/\dot F\) in f(R) gravity is invariant under the conformal transformation:
Since the tensor perturbation is also invariant, the tensortoscalar ratio in the Einstein frame is identical to that in the Jordan frame. For example, let us consider the model f(R) = R + R^{2}/(6M^{2}). Since the action in the Einstein frame is given by Eq. (2.32), the slowroll parameters \({{\tilde \epsilon}_3}\) and \({{\tilde \epsilon}_4}\) vanish in this frame. Using Eqs. (7.49) and (3.27), the spectral index of curvature perturbations is given by
where we have ignored the term of the order of \(1/\tilde N_k^2\). Since \({{\tilde N}_k} \simeq {N_k}\) in the slowroll limit (∣Ḟ/(2HF)∣ ≪ 1), Eq. (7.78) agrees with the result (7.72) in the Jordan frame. Since \({Q_s} = {({\rm{d}}\phi {\rm{/d}}\tilde t)^2}/{H^2}\) in the Einstein frame, Eq. (7.59) gives the tensortoscalar ratio
where the background equations (3.21) and (3.22) are used with slowroll approximations. Equation (7.79) is consistent with the result (7.73) in the Jordan frame.
The equivalence of the curvature perturbation between the Jordan and Einstein frames also holds for scalartensor theory with the Lagrangian \({\mathcal L} = F(\phi)R/(2{\kappa ^2})  (1/2)\omega (\phi){g^{\mu \nu}}{\partial _\mu}\phi {\partial _\nu}\phi  V(\phi)\) [411, 240]. For the nonminimally coupled scalar field with F(ϕ) = 1 − ξk^{2}ϕ^{2} [269, 241] the spectral indices of scalar and tensor perturbations have been derived by using such equivalence [366, 590].
7.4 The Lagrangian for cosmological perturbations
In Section 7.1 we used the fact that the field which should be quantized corresponds to \(u = a{\sqrt Q _s}{\mathcal R}\). This can be justified by writing down the action (6.1) expanded at secondorder in the perturbations [437]. We recall again that we are considering an effective singlefield theory such as f(R) gravity and scalartensor theory with the coupling F(ϕ)R. Carrying out the expansion of the action (6.2) in second order, we find that the action for the curvature perturbation \({\mathcal R}\) (either \({{\mathcal R}_{\delta F}}\) or \({{\mathcal R}_{\delta \phi}}\)) is given by [311]
where Q_{s} is given in Eq. (7.38). In fact, the variation of this action in terms of the field \({\mathcal R}\) gives rise to Eq. (7.37) in Fourier space. We note that there is another approach called the Hamiltonian formalism which is also useful for the quantization of cosmological perturbations. See [237, 209, 208, 127] for this approach in the context of f(R) gravity and modified gravitational theories.
Introducing the quantities \(u = {z_S}{\mathcal R}\) and \({z_S} = a{\sqrt Q _s}\), the action (7.80) can be written as
where a prime represents a derivative with respect to the conformal time η = ∫ a^{−1}dt. The action (7.81) leads to Eq. (7.39) in Fourier space. The transformation of the action (7.80) to (7.81) gives rise to the effective mass^{Footnote 6}
We have seen in Eq. (7.42) that during inflation the quantity \(z_s^{^{\prime\prime}}/{z_s}\) can be estimated as \(z_s^{^{\prime\prime}}/{z_s} \simeq 2{(aH)^2}\) in the slowroll limit, so that \(M_s^2 \simeq  2{H^2}\). For the modes deep inside the Hubble radius (k ∫ aH) the action (7.81) reduces to the one for a canonical scalar field u in the flat spacetime. Hence the quantization should be done for the field \(u = a\sqrt {{Q_s}} {\mathcal R}\), as we have done in Section 7.1.
From the action (7.81) we understand a number of physical properties in f(R) theories and scalartensor theories with the coupling F(ϕ)R listed below.

1.
Having a standard d’Alambertian operator, the mode has speed of propagation equal to the speed of light. This leads to a standard dispersion relation ω = k/a for the highk modes in Fourier space.

2.
The sign of Q_{s} corresponds to the sign of the kinetic energy of \({\mathcal R}\). The negative sign corresponds to a ghost (phantom) scalar field. In f(R) gravity (with \(\dot \phi = 0\)) the ghost appears for F < 0. In BransDicke theory with F(ϕ) = κ^{2}ϕ and ω(ϕ) = ω_{BD}/ϕ [100] (where ϕ > 0) the condition for the appearance of the ghost \((\omega {{\dot \phi}^2} + 3{F^2}/(2{\kappa ^2}F) < 0)\) translates into ω_{BD} < −3/2. In these cases one would encounter serious problems related to vacuum instability [145, 161].

3.
The field u has the effective mass squared given in Eq. (7.82). In f(R) gravity it can be written as
$$M_s^2 =  {{72{F^2}{H^4}} \over {{{(2FH + {f_{,RR}}\dot R)}^2}}} + {1 \over 3}F\left({{{288{H^3}  12HR} \over {2FH + {f_{,RR}}\dot R}} + {1 \over {{f_{,RR}}}}} \right) + {{f_{,RR}^2{{\dot R}^2}} \over {4{F^2}}}  24{H^2} + {7 \over 6}R\,,$$(7.83)where we used the background equation (2.16) to write Ḣ in terms of R and H^{2}. In Fourier space the perturbation u obeys the equation of motion
$$u^{\prime\prime} +({{k^2} + M_s^2{a^2}})\;u = 0\,.$$(7.84)For \({k^2}/{a^2} \gg M_s^2\), the field u propagates with speed of light. For small k satisfying \({k^2}/{a^2} \ll M_s^2\), we require a positive \(M_s^2\) to avoid the tachyonic instability of perturbations. Recall that the viable dark energy models based on f(R) theories need to satisfy Rf_{,RR} ≪ F (i.e., m = Rf_{,RR}/f_{,R} ≪ 1) at early times, in order to have successful cosmological evolution from radiation domination till matter domination. At these epochs the mass squared is approximately given by
$$M_s^2 \simeq {F \over {3{f_{,RR}}}}\,,$$(7.85)which is consistent with the result (5.2) derived by the linear analysis about the Minkowski background. Together with the ghost condition F > 0, this leads to f_{,RR} > 0. Recall that these correspond to the conditions presented in Eq. (4.56).
8 Observational Signatures of Dark Energy Models in f(R) Theories
In this section we discuss a number of observational signatures of dark energy models based on metric f(R) gravity. Our main interest is to distinguish these models from the ΛCDM model observationally. In particular we study the evolution of matter density perturbations as well as the gravitational potential to confront f(R) models with the observations of largescale structure (LSS) and Cosmic Microwave Background (CMB). The effect on weak lensing will be discussed in Section 13.1 in more general modified gravity theories including f(R) gravity.
8.1 Matter density perturbations
Let us consider the perturbations of nonrelativistic matter with the background energy density ρ_{m} and the negligible pressure (P_{m} = 0). In Fourier space Eqs. (6.17) and (6.18) give
where in the second line we have used the continuity equation, \({{\dot \rho}_m} + 3H{\rho _m} = 0\). The density contrast defined in Eq. (6.32), i.e.
obeys the following equation from Eqs. (8.86) and (8.87):
where B ≡ Hυ − ψ and we used the relation \(A = 3(H\alpha  \dot \psi) + ({k^2}/{a^2})\chi\).
In the following we consider the evolution of perturbations in f(R) gravity in the Longitudinal gauge (6.33). Since χ = 0, α = Φ, ψ = −Ψ, and \(A = 3(H\Phi + \dot \Psi)\) in this case, Eqs. (6.11), (6.13), (6.15), and (8.89) give
where B = Hυ + Ψ. In order to derive Eq. (8.92), we have used the mass squared M^{2} = (F/F_{,R} − R)/3 introduced in Eq. (5.2) together with the relation δR = δF/F_{,R}.
Let us consider the wavenumber k deep inside the Hubble radius (k ≫ aH). In order to derive the equation of matter perturbations approximately, we use the quasistatic approximation under which the dominant terms in Eqs. (8.90)–(8.93) correspond to those including k^{2}/a^{2}, δρ_{m} (or δ_{m}) and M^{2}. In General Relativity this approximation was first used by Starobinsky in the presence of a minimally coupled scalar field [567], which was numerically confirmed in [403]. This was further extended to scalartensor theories [93, 171, 586] and f(R) gravity [586, 597]. Precisely speaking, in f(R) gravity, this approximation corresponds to
and
From Eqs. (8.90) and (8.91) it then follows that
Since (k^{2}/a^{2} + M^{2})δF ≃ κ^{2}δρ_{m}/3 from Eq. (8.92), we obtain
We also define the effective gravitational potential
This quantity characterizes the deviation of light rays, which is linked with the Integrated SachsWolfe (ISW) effect in CMB [544] and weak lensing observations [27]. From Eq. (8.97) we have
From Eq. (6.12) the term Hυ is of the order of H^{2}Φ/(κ^{2}ρ_{m}) provided that the deviation from the ΛCDM model is not significant. Using Eq. (8.97) we find that the ratio 3Hυ/(\(3H\upsilon/(\delta {\rho _m}/{\rho _m})\)) is of the order of (aH/k)^{2}, which is much smaller than unity for subhorizon modes. Then the gaugeinvariant perturbation δ_{m} given in Eq. (8.88) can be approximated as δ_{m} ≃ δρ_{m}/ρ_{m}. Neglecting the r.h.s. of Eq. (8.93) relative to the l.h.s. and using Eq. (8.97) with δρ_{m} ≃ ρ_{m}δ_{m}, we get the equation for matter perturbations:
where G_{eff} is the effective (cosmological) gravitational coupling defined by [586, 597]
We recall that viable f(R) dark energy models are constructed to have a large mass M in the region of high density (R ≫ R_{0}). During the radiation and deep matter eras the deviation parameter m = Rf_{,RR}/f_{,R} is much smaller than 1, so that the mass squared satisfies
if m grows to the order of 0.1 by the present epoch, then the mass M today can be of the order of H_{0}. In the regimes M^{2} ≫ k^{2}/a^{2} and M^{2} ≪ k^{2}/a^{2} the effective gravitational coupling has the asymptotic forms G_{eff} ≃ G/F and G_{eff} ≃ 4G/(3F), respectively. The former corresponds to the “General Relativistic (GR) regime” in which the evolution of δ_{m} mimics that in GR, whereas the latter corresponds to the “scalartensor regime” in which the evolution of δ_{m} is nonstandard. For the f(R) models (4.83) and (4.84) the transition from the former regime to the latter regime, which is characterized by the condition M^{2} = k^{2}/a^{2}, can occur during the matter domination for the wavenumbers relevant to the matter power spectrum [306, 568, 587, 270, 589].
In order to derive Eq. (8.100) we used the approximation that the timederivative terms of δF on the l.h.s. of Eq. (8.92) is neglected. In the regime M^{2} ≫ k^{2}/a^{2}, however, the large mass M can induce rapid oscillations of δF. In the following we shall study the evolution of the oscillating mode [568]. For subhorizon perturbations Eq. (8.92) is approximately given by
The solution of this equation is the sum of the matter induce mode δF_{ind} ≃ (κ^{2}/3)δρ_{m}/(k^{2}/a^{2}+ M^{2}) and the oscillating mode δF_{osc} satisfying
As long as the frequency \(\omega = \sqrt {{k^2}/{a^2} + {M^2}}\) satisfies the adiabatic condition \(\vert \dot \omega \vert \, \ll {\omega ^2}\), we obtain the solution of Eq. (8.104) under the WKB approximation:
where c is a constant. Hence the solution of the perturbation δR is expressed by [568, 587]
For viable f(R) models, the scale factor a and the background Ricci scalar R^{(0)} evolve as a ∝ t^{2/3} and R^{(0)} ≃ 4/(3t^{2}) during the matter era. Then the amplitude of δR_{osc} relative to R^{(0)} has the timedependence
The f(R) models (4.83) and (4.84) behave as m(r) = C(−r − 1)^{p} with p = 2n + 1 in the regime R ≫ R_{c}. During the matterdominated epoch the mass M evolves as M ∝ t^{−(p+1)}. In the regime M^{2} ≫ k^{2}/a^{2} one has ∣δR_{osc}∣/R^{(0)} ∝ t^{−(3p+1)/2} and hence the amplitude of the oscillating mode decreases faster than R^{(0)}. However the contribution of the oscillating mode tends to be more important as we go back to the past. In fact, this behavior was confirmed in the numerical simulations of [587, 36]. This property persists in the radiationdominated epoch as well. If the condition ∣δR∣ < R^{(0)} is violated, then R can be negative such that the condition f_{,R} > 0 or f_{,RR} > 0 is violated for the models (4.83) and (4.84). Thus we require that ∣δR∣ is smaller than R^{(0)} at the be ginning of the radiation era. This can be achieved by choosing the constant in Eq. (8.106) to be sufficiently small, which amounts to a fine tuning for these models.
For the models (4.83) and (4.84) one has F = 1 − 2nμ(R/R_{c})^{−2n−1} in the regime R ≫ R_{c}. Then the field ϕ defined in Eq. (2.31) rapidly approaches 0 as we go back to the past. Recall that in the Einstein frame the effective potential of the field has a potential minimum around ϕ = 0 because of the presence of the matter coupling. Unless the oscillating mode of the field perturbation δϕ is strongly suppressed relative to the background field ϕ^{(0)}, the system can access the curvature singularity at ϕ = 0 [266]. This is associated with the condition ∣δR∣ < R^{(0)} discussed above. This curvature singularity appears in the past, which is not related to the future singularities studied in [461, 54]. The past singularity can be cured by taking into account the R^{2} term [37], as we will see in Section 13.3. We note that the f(R) models proposed in [427] [e.g., f(R) = R− αR_{c} ln(1+R/R_{c})] to cure the singularity problem satisfy neither the local gravity constraints [580] nor observational constraints of largescale structure [194].
As long as the oscillating mode δR_{osc} is negligible relative to the matterinduced mode δR_{ind}, we can estimate the evolution of matter perturbations δ_{m} as well as the effective gravitational potential Φ_{eff}. Note that in [192, 434] the perturbation equations have been derived without neglecting the oscillating mode. As long as the condition ∣δR_{osc}∣ < ∣δR_{ind}δ∣ is satisfied initially, the approximate equation (8.100) is accurate to reproduce the numerical solutions [192, 589]. Equation (8.100) can be written as
where N = lna, w_{eff} = −1 − 2 Ḣ/(3H^{2}), and Ω_{m} = 8πGρ_{m}/(3FH^{2}). The matterdominated epoch corresponds to w_{eff} = 0 and Ω_{m} = 1. In the regime M^{2} ≫ k^{2}/a^{2} the evolution of δ_{m} and Φ_{eff} during the matter dominance is given by
where we used Eq. (8.99). The matterinduced mode δR_{ind} relative to the background Ricci scalar R^{(0)} evolves as ∣δR_{ind}∣/R^{(0)} ∝ t^{2/3} ∝ δ_{m}. At late times the perturbations can enter the regime M^{2} ≪ k^{2}/a^{2}, depending on the wavenumber k and the mass M. When M^{2} ≪ k^{2}/a^{2}, the evolution of δ_{m} and Φ_{eff} during the matter era is [568]
For the model m(r) = C(−r − 1)^{p}, the evolution of the matterinduced mode in the region M^{2} ≪ k^{2}/a^{2} is given by \(\vert \delta {R_{{\rm{ind}}}}\vert/{R^{(0)}} \propto {t^{ 2p +}}(\sqrt {33}  5)/6\). This decreases more slowly relative to the ratio ∣δR_{osc}∣/R^{(0)} [587], so the oscillating mode tends to be unimportant with time.
8.2 The impact on largescale structure
We have shown that the evolution of matter perturbations during the matter dominance is given by δ_{m} ∝ t^{2/3} for M^{2} ≫ k^{2}/a^{2} (GR regime) and \({\delta _m} \propto {t^{(\sqrt {33}  1)/6}}\) for M^{2} ≪ k^{2}/a^{2} (scalartensor regime), respectively. The existence of the latter phase gives rise to the modification to the matter power spectrum [146, 74, 544, 526, 251] (see also [597, 493, 494, 94, 446, 278, 435] for related works). The transition from the GR regime to the scalartensor regime occurs at M^{2} = k^{2}/a^{2}. If it occurs during the matter dominance (R ≃ 3H^{2}), the condition M^{2} = k^{2}/a^{2} translates into [589]
where we have used the relation M^{2} ≃ R/(3m) (valid for m ≪ 1).
We are interested in the wavenumbers k relevant to the linear regime of the galaxy power spectrum [577, 578]:
where h = 0.72 ± 0.08 corresponds to the uncertainty of the Hubble parameter today. Nonlinear effects are important for k ≳ 0.2 h Mpc^{−1}. The current observations on large scales around k ∼ 0.01 h Mpc^{−1} are not so accurate but can be improved in future. The upper bound = 0.2 h Mpc^{−1} corresponds to k ≃ 600a_{0}H_{0}, where the subscript “0” represents quantities today. If the transition from the GR regime to the scalartensor regime occurred by the present epoch (the redshift z = 0) for the mode k = 600a_{0}H_{0}, then the parameter m today is constrained to be
When m(z = 0) ≲ 3 × 10^{−6} the linear perturbations have been always in the GR regime by today, in which case the models are not distinguished from the ΛCDM model. The bound (8.113) is relaxed for nonlinear perturbations with k ≳ 0.2 h Mpc^{−1}, but the linear analysis is not valid in such cases.
If the transition characterized by the condition (8.111) occurs during the deep matter era (z ≫ 1), we can estimate the critical redshift z_{k} at the transition point. In the following let us consider the models (4.83) and (4.84). In addition to the approximations \({H^2} \simeq H_0^2\Omega _m^{(0)}{(1 + z)^3}\) and R ≃ 3H^{2} during the matter dominance, we use the the asymptotic forms m ≃ C(−r − 1)^{2n+1} and r ≃ −1 − μR_{c}/R with C = 2n(2n + 1)/μ^{2n}. Since the dark energy density today can be approximated as \(\rho _{{\rm{DE}}}^{(0)} \approx \mu {R_c}/2\), it follows that \(\mu {R_c} \approx 6H_0^2\Omega _{{\rm{DE}}}^{(0)}\). Then the condition (8.111) translates into the critical redshift [589]
For n = 1, μ = 3, \(\Omega _m^{(0)} = 0.28\), and k = 300a_{0}H_{0} the numerical value of the critical redshift is z_{k} = 4.5, which is in good agreement with the analytic value estimated by (8.114).
The estimation (8.114) shows that, for larger k, the transition occurs earlier. The time t_{k} at the transition has a kdependence: t_{k} ∝ k^{−3/(6n+4)}. For t > t_{k} the matter perturbation evolves as \({\delta _m} \propto {t^{(\sqrt {33}  1)/6}}\) by the time t = t_{Λ} corresponding to the onset of cosmic acceleration (ä = 0). The matter power spectrum \({P_{{\delta _m}}} = \vert {\delta _m}{\vert ^2}\) at the time t_{Λ} shows a difference compared to the case of the ΛCDM model [568]:
We caution that, when z_{k} is close to z_{Λ} (the redshift at t = t_{Λ}), the estimation (8.115) begins to lose its accuracy. The ratio of the two power spectra today, i.e., \({P_{{\delta _m}}}({t_0})/{P_{{\delta _m}}}^{\Lambda {\rm{CDM}}}({t_0})\) is in general different from Eq. (8.115). However, numerical simulations in [587] show that the difference is small for n of the order of unity.
The modified evolution (8.110) of the effective gravitational potential for z < z_{k} leads to the integrated SachsWolfe (ISW) effect in CMB anisotropies [544, 382, 545]. However this is limited to very large scales (low multipoles) in the CMB spectrum. Meanwhile the galaxy power spectrum is directly affected by the nonstandard evolution of matter perturbations. From Eq. (8.115) there should be a difference between the spectral indices of the CMB spectrum and the galaxy power spectrum on the scale (8.112) [568]:
Observationally we do not find any strong signature for the difference of slopes of the two spectra. If we take the mild bound Δn_{s} < 0.05, we obtain the constraint n > 2. Note that in this case the local gravity constraint (5.60) is also satisfied.
In order to estimate the growth rate of matter perturbations, we introduce the growth index γ defined by [484]
where \({{\tilde \Omega}_m} = {\kappa ^2}{\rho _m}/(3{H^2}) = F{\Omega _m}\). This choice of \({{\tilde \Omega}_m}\) comes from writing Eq. (4.59) in the form 3H^{2} = ρ_{DE} + κ^{2}ρ_{m}, where ρ_{DE} ≡ (FR − f)/2 − 3 HḞ + 3H^{2}(1 − F) and we have ignored the contribution of radiation. Since the viable f(R) models are close to the ΛCDM model in the region of high density, the quantity F approaches 1 in the asymptotic past. Defining ρ_{DE} and \({{\tilde \Omega}_m}\) in the above way, the Friedmann equation can be cast in the usual GR form with nonrelativistic matter and dark energy [568, 270, 589].
The growth index in the ΛCDM model corresponds to γ ≃ 0.55 [612, 395], which is nearly constant for 0 < z < 1. In f(R) gravity, if the perturbations are in the GR regime (M^{2} ≫ k^{2}/a^{2}) today, γ is close to the GR value. Meanwhile, if the transition to the scalartensor regime occurred at the redshift z_{k} larger than 1, the growth index becomes smaller than 0.55 [270]. Since \(0 < {{\tilde \Omega}_m} < 1\), the smaller γ implies a larger growth rate.
In Figure 4 we plot the evolution of the growth index γ in the model (4.83) with n = 1 and μ = 1.55 for a number of different wavenumbers. In this case the present value of γ is degenerate around γ_{0} ≃ 0.41 independent of the scales of our interest. For the wavenumbers k = 0.1 h Mpc^{−1} and k = 0.01 h Mpc^{−1} the transition redshifts correspond to z_{k} = 5.2 and z_{k} = 2.7, respectively. Hence these modes have already entered the scalartensor regime by today.
From Eq. (8.114) we find that z_{k} gets smaller for larger n and μ. If the mode k = 0.2 h Mpc^{−1} crossed the transition point at \({z_k} > {\mathcal O}(1)\) and the mode k = 0.01 h Mpc^{−1} has marginally entered (or has not entered) the scalartensor regime by today, then the growth indices should be strongly dispersed. For sufficiently large values of n and μ one can expect that the transition to the regime M^{2} ≪ k^{2}/a^{2} has not occurred by today. The following three cases appear depending on the values of n and μ [589]:

(i)
All modes have the values of γ_{0} close to the ΛCDM value: γ_{0} =. 55, i.e., 0.53 ≲ γ_{0} ≲ 0.55.

(ii)
All modes have the values of γ_{0} close to the value in the range 0.40 ≲ γ_{0} ≲ 0.43.

(iii)
The values of γ_{0} are dispersed in the range 0.40 ≲ γ_{0} ≲ 0.55.
The region (i) corresponds to the opposite of the inequality (8.113), i.e., m(z = 0) ≲ 3 × 10^{−6}, in which case n and μ take large values. The border between (i) and (iii) is characterized by the condition m(z = 0) ≈ 3 × 10^{−6}. The region (ii) corresponds to small values of n and μ (as in the numerical simulation of Figure 4), in which case the mode k = 0. 01 h Mpc^{−1} entered the scalartensor regime for \({z_k} > {\mathcal O}(1)\).
The regions (i), (ii), (iii) can be found numerically by solving the perturbation equations. In Figure 5 we plot those regions for the model (4.84) together with the bounds coming from the local gravity constraints as well as the stability of the latetime de Sitter point. Note that the result in the model (4.83) is also similar to that in the model (4.84). The parameter space for n ≲ 3 and \(\mu = {\mathcal O}(1)\) is dominated by either the region (ii) or the region (iii). While the present observational constraint on γ is quite weak, the unusual converged or dispersed spectra found above can be useful to distinguish metric f(R) gravity from the ΛCDM model in future observations. We also note that for other viable f(R) models such as (4.89) the growth index today can be as small as γ_{0} ≃ 0.4 [589]. If future observations detect such unusually small values of γ_{0}, this can be a smoking gun for f(R) models.
8.3 Nonlinear matter perturbations
So far we have discussed the evolution of linear perturbations relevant to the matter spectrum for the scale k ≲ 0.01–0.2 h Mpc^{−1}. For smaller scale perturbations the effect of nonlinearity becomes important. In GR there are some mapping formulas from the linear power spectrum to the nonlinear power spectrum such as the halo fitting by Smith et al. [540]. In the halo model the nonlinear power spectrum P(k) is defined by the sum of two pieces [169]:
where P_{L}(k) is a linear power spectrum and
Here M is the mass of dark matter halos, \(\rho _m^{(0)}\) is the dark matter density today, dn/dln M is the mass function describing the comoving number density of halos, y(M, k) is the Fourier transform of the halo density profile, and b(M) is the halo bias.
In modified gravity theories, Hu and Sawicki (HS) [307] provided a fitting formula to describe a nonlinear power spectrum based on the halo model. The mass function dn/d ln M and the halo profile ρ depend on the rootmeansquare σ(M) of a linear density field. The ShethTormen mass function [535] and the NavarroFrenkWhite halo profile [449] are usually employed in GR. Replacing σ for σ_{GR} obtained in the GR dark energy model that follows the same expansion history as the modified gravity model, we obtain a nonlinear power spectrum P(k) according to Eq. (8.118). In [307] this nonlinear spectrum is called P_{∞}(k). It is also possible to obtain a nonlinear spectrum P_{0}(k) by applying a usual (halo) mapping formula in GR to modified gravity. This approach is based on the assumption that the growth rate in the linear regime determines the nonlinear spectrum. Hu and Sawicki proposed a parametrized nonlinear spectrum that interpolates between two spectra P_{∞}(k) and P_{0}(k) [307]:
where c_{nl} is a parameter which controls whether P(k) is close to P_{0}(k) or P_{∞}(k). In [307] they have taken the form Σ^{2}(k) = k^{3}P_{L}(k)/(2π^{2}).
The validity of the HS fitting formula (8.120) should be checked with Nbody simulations in modified gravity models. In [478, 479, 529] Nbody simulations were carried out for the f (R) model (4.83) with n = 1/2 (see also [562, 379] for Nbody simulations in other modified gravity models). The chameleon mechanism should be at work on small scales (solarsystem scales) for the consistency with local gravity constraints. In [479] it was found that the chameleon mechanism tends to suppress the enhancement of the power spectrum in the nonlinear regime that corresponds to the recovery of GR. On the other hand, in the post Newtonian intermediate regime, the power spectrum is enhanced compared to the GR case at the measurable level.
Koyama et al. [371] studied the validity of the HS fitting formula by comparing it with the results of Nbody simulations. Note that in this paper the parametrization (8.120) was used as a fitting formula without employing the halo model explicitly. In their notation P_{0} corresponds to “P_{nonGR}” derived without nonlinear interactions responsible for the recovery of GR (i.e., gravity is modified down to small scales in the same manner as in the linear regime), whereas P_{∞} corresponds to “P_{GR}” obtained in the GR dark energy model following the same expansion history as that in the modified gravity model. Note that c_{nl} characterizes how the theory approaches GR by the chameleon mechanism. Choosing Σ as
where P_{L} is the linear power spectrum in the modified gravity model, they showed that, in the f(R) model (4.83) with n = 1/2, the formula (8.120) can fit the solutions in perturbation theory very well by allowing the timedependence of the parameter c_{nl} in terms of the redshift z. In the regime 0 < z < 1 the parameter c_{nl} is approximately given by c_{nl}(z = 0) = 0.085.
In the left panel of Figure 6 the relative difference of the nonlinear power spectrum P(k) from the GR spectrum P_{GR}(k) is plotted as a dashed curve (“no chameleon” case with c_{nl} = 0) and as a solid curve (“chameleon” case with nonzero c_{nl} derived in the perturbative regime). Note that in this simulation the fitting formula by Smith et al. [540] is used to obtain the nonlinear power spectrum from the linear one. The agreement with Nbody simulations is not very good in the nonlinear regime (k > 0.1h Mpc^{−1}). In [371] the power spectrum P_{nonGR} in the no chameleon case (i.e., c_{nl} = 0) was derived by interpolating the Nbody results in [479]. This is plotted as the dashed line in the right panel of Figure 6. Using this spectrum P_{nonGR} for c_{nl} ≠ 0, the power spectrum in Nbody simulations in the chameleon case can be well reproduced by the fitting formula (8.120) for the scale k < 0.5h Mpc^{−1} (see the solid line in Figure 6). Although there is some deviation in the regime k > 0.5h Mpc^{−1}, we caution that Nbody simulations have large errors in this regime. See [530] for clustered abundance constraints on the f(R) model (4.83) derived by the calibration of Nbody simulations.
In the quasi nonlinear regime a normalized skewness, \({S_3} = \langle \delta _m^3\rangle/{\langle \delta _m^2\rangle ^2}\), of matter perturbations can provide a good test for the picture of gravitational instability from Gaussian initial conditions [79]. If largescale structure grows via gravitational instability from Gaussian initial perturbations, the skewness in a universe dominated by pressureless matter is known to S_{3} = 34/7 in GR [484]. In the ΛCDM model the skewness depends weakly on the expansion history of the universe (less than a few percent) [335]. In f(R) dark energy models the difference of the skewness from the ΛCDM model is only less than a few percent [576], even if the growth rate of matter perturbations is significantly different. This is related to the fact that in the Einstein frame dark energy has a universal coupling \(Q =  1/\sqrt 6\) with all nonrelativistic matter, unlike the coupled quintessence scenario with different couplings between dark energy and matter species (dark matter, baryons) [30].
8.4 Cosmic Microwave Background
The effective gravitational potential (8.98) is directly related to the ISW effect in CMB anisotropies. This contributes to the temperature anisotropies today as an integral [308, 214]
where τ is the optical depth, η = ∫a^{−1}dt is the conformal time with the present value η_{0}, and j_{ℓ}[k(η_{0} − η)] is the spherical Bessel function for CMB multipoles ℓ and the wavenumber k. In the limit ℓ ≫ 1 (i.e., smallscale limit) the spherical Bessel function has a dependence j_{ℓ}(x) ≃ (x/ℓ)^{ℓ−1/2}, which is suppressed for large ℓ. Hence the dominant contribution to the ISW effect comes from the low ℓ modes (\(\ell = \mathcal O(1)\)).
In the ΛCDM model the effective gravitational potential is constant during the matter dominance, but it begins to decay after the Universe enters the epoch of cosmic acceleration (see the left panel of Figure 7). This latetime variation of Φ_{eff} leads to the contribution to Θ_{ISW}, which works as the ISW effect.
For viable f(R) dark energy models the evolution of Φ_{eff} during the early stage of the matter era is constant as in the ΛCDM model. After the transition to the scalartensor regime, the effective gravitational potential evolves as \({\Phi _{{\rm{eff}}}} \propto {t^{(\sqrt {33  5)}/6}}\) during the matter dominance [as we have shown in Eq. (8.110)]. The evolution of Φ_{eff} during the accelerated epoch is also subject to change compared to the ΛCDM model. In the left panel of Figure 7 we show the evolution of Φ_{eff} versus the scale factor a for the wavenumber k = 10^{−3} Mpc^{−1} in several different cases. In this simulation the background cosmological evolution is fixed to be the same as that in the ΛCDM model. In order to quantify the difference from the ΛCDM model at the level of perturbations, [628, 544, 545] defined the following quantity
where m = Rf_{,RR}/f_{,R}. If the effective equation of state w_{eff} defined in Eq. (4.69) is constant, it then follows that R = 3H^{2}(1–3 w_{eff}) and hence B = 2 m. The stability of cosmological perturbations requires the condition B > 0 [544, 526]. The left panel of Figure 7 shows that, as we increase the values of B today (= B_{0}), the evolution of Φ_{eff} at late times tends to be significantly different from that in the ΛCDM model. This comes from the fact that, for increasing B, the transition to the scalartensor regime occurs earlier.
From the right panel of Figure 7 we find that, as B_{0} increases, the CMB spectrum for low multipoles first decreases and then reaches the minimum around B_{0} = 1.5. This comes from the reduction in the decay rate of Φ_{eff} relative to the ΛCDM model, see the left panel of Figure 7. Around B_{0} = 1.5 the effective gravitational potential is nearly constant, so that the ISW effect is almost absent (i.e., Θ_{ISW} ≈ 0). For B_{0} ≳ 1.5 the evolution of Φ_{eff} turns into growth. This leads to the increase of the largescale CMB spectrum, as B_{0} increases. The spectrum in the case B_{0} = 3.0 is similar to that in the ΛCDM model. The WMAP 3year data rule out B_{0} > 4.3 at the 95% confidence level because of the excessive ISW effect [545].
There is another observational constraint coming from the angular correlation between the CMB temperature field and the galaxy number density field induced by the ISW effect [544]. The f(R) models predict that, for B_{0} ≳ 1, the galaxies are anticorrelated with the CMB because of the sign change of the ISW effect. Since the anticorrelation has not been observed in the observational data of CMB and LSS, this places an upper bound of B_{0} ≳ 1 [545]. This is tighter than the bound B_{0} < 4.3 coming from the CMB angular spectrum discussed above.
Finally we briefly mention stochastic gravitational waves produced in the early universe [421, 172, 122, 123, 174, 173, 196, 20]. For the inflation model f(R) = R + R^{2}/(6M^{2}) the primordial gravitational waves are generated with the tensortoscalar ratio of the order of 10^{−3}, see Eq. (7.73). It is also possible to generate stochastic gravitational waves after inflation under the modification of gravity. Capozziello et al. [122, 123] studied the evolution of tensor perturbations for a toy model f = R^{1+ϵ} in the FLRW universe with the powerlaw evolution of the scale factor. Since the parameter ϵ is constrained to be very small (∣ϵ∣ < 7.2 × 10^{−19}) [62, 160], it is very difficult to detect the signature of f(R) gravity in the stochastic gravitational wave background. This property should hold for viable f(R) dark energy models in general, because the deviation from GR during the radiation and the deep matter era is very small.
9 Palatini Formalism
In this section we discuss f(R) theory in the Palatini formalism [481]. In this approach the action (2.1) is varied with respect to both the metric g_{μν} and the connection \(\Gamma _{\beta \gamma}^\alpha\). Unlike the metric approach, g_{μν} and \(\Gamma _{\beta \gamma}^\alpha\) are treated as independent variables. Variations using the Palatini approach [256, 607, 608, 261, 262, 260] lead to secondorder field equations which are free from the instability associated with negative signs of f_{,RR} [422, 423]. We note that even in the 1930s Lanczos [378] proposed a specific combination of curvaturesquared terms that lead to a secondorder and divergencefree modified Einstein equation.
The background cosmological dynamics of Palatini f(R) gravity has been investigated in [550, 553, 21, 253, 495], which shows that the sequence of radiation, matter, and accelerated epochs can be realized even for the model f(R) = R − a/R^{n} with n > 0 (see also [424, 457, 495]). The equations for matter density perturbations were derived in [359]. Because of a large coupling Q between dark energy and nonrelativistic matter dark energy models based on Palatini f(R) gravity are not compatible with the observations of largescale structure, unless the deviation from the ΛCDM model is very small [356, 386, 385, 597]. Such a large coupling also gives rise to nonperturbative corrections to the matter action, which leads to a conflict with the Standard Model of particle physics [261, 262, 260] (see also [318, 472, 473, 475, 55]).
There are also a number of works [470, 471, 216, 552] about the Newtonian limit in the Palatini formalism (see also [18, 19, 107, 331, 511, 510]). In particular it was shown in [55, 56] that the nondynamical nature of the scalarfield degree of freedom can lead to a divergence of nonvacuum static spherically symmetric solutions at the surface of a compact object for commonlyused polytropic equations of state. Hence Palatini f(R) theory is difficult to be compatible with a number of observations and experiments, as long as the models are constructed to explain the latetime cosmic acceleration. Moreover it is also known that in Palatini gravity the Cauchy problem [609] is not wellformulated due to the presence of higher derivatives of matter fields in field equations [377] (see also [520, 135] for related works). We also note that the matter Lagrangian (such as the Lagrangian of Dirac particles) cannot be simply assumed to be independent of connections. Even in the presence of above mentioned problems it will be useful to review this theory because we can learn the way of modifications of gravity from GR to be consistent with observations and experiments.
9.1 Field equations
Let us derive field equations by treating g_{μν} and \(\Gamma _{\beta \gamma}^\alpha\) as independent variables. Varying the action (2.1) with respect to g_{μν}, we obtain
where F(R) = ∂f/∂R, R_{μν}(Γ) is the Ricci tensor corresponding to the connections \(\Gamma _{\beta \gamma}^\alpha\), and \(T_{\mu \nu}^{(M)}\) is defined in Eq. (2.5). Note that R_{μν}(Γ) is in general different from the Ricci tensor calculated in terms of metric connections R_{μν}(g). The trace of Eq. (9.1) gives
where \(T = {g^{\mu \nu}}T_{\mu \nu}^{(M)}\). Here the Ricci scalar R(T) is directly related to T and it is different from the Ricci scalar R(g) = g^{μν}R_{μν}(g) in the metric formalism. More explicitly we have the following relation [556]
where a prime represents a derivative in terms of R(T). The variation of the action (2.1) with respect to the connection leads to the following equation
In Einstein gravity (f(R) = R − 2Λ and F(R) = 1) the field equations (9.2) and (9.4) are identical to the equations (2.7) and (2.4), respectively. However, the difference appears for the f(R) models which include nonlinear terms in R. While the kinetic term □F is present in Eq. (2.7), such a term is absent in Palatini f(R) gravity. This has the important consequence that the oscillatory mode, which appears in the metric formalism, does not exist in the Palatini formalism. As we will see later on, Palatini f(R) theory corresponds to BransDicke (BD) theory [100] with a parameter ω_{BD} = −3/2 in the presence of a field potential. Such a theory should be treated separately, compared to BD theory with ω_{BD} ≠ −3/2 in which the field kinetic term is present.
As we have derived the action (2.21) from (2.18), the action in Palatini f(R) gravity is equivalent to
where
Since the derivative of U in terms of φ is U_{,φ} = R/(2κ^{2}), we obtain the following relation from Eq. (9.2):
Using the relation (9.3), the action (9.5) can be written as
Comparing this with Eq. (2.23) in the unit κ^{2} = 1, we find that Palatini f(R) gravity is equivalent to BD theory with the parameter ω_{BD} = −3/2 [262, 470, 551]. As we will see in Section 10.1, this equivalence can be also seen by comparing Eqs. (9.1) and (9.4) with those obtained by varying the action (2.23) in BD theory. In the above discussion we have implicitly assumed that \({\mathcal L_M}\) does not explicitly depend on the Christoffel connections \(\Gamma _{\mu \nu}^\lambda\). This is true for a scalar field or a perfect fluid, but it is not necessarily so for other matter Lagrangians such as those describing vector fields.
There is another way for taking the variation of the action, known as the metricaffine formalism [299, 558, 557, 121]. In this formalism the matter action S_{M} depends not only on the metric g_{μν} but also on the connection \(\Gamma _{\mu \nu}^\lambda\). Since the connection is independent of the metric in this approach, one can define the quantity called hypermomentum [299], as \(\Delta _\lambda ^{\mu \nu} \equiv ( 2/\sqrt { g})\delta {\mathcal L_M}/\delta \Gamma _{\mu \nu}^\lambda\). The usual assumption that the connection is symmetric is also dropped, so that the antisymmetric quantity called the Cartan torsion tensor, \(S_{\mu \nu}^\lambda \equiv \Gamma _{[\mu \nu ]}^\lambda\), is defined. The nonvanishing property of \(S_{\mu \nu}^\lambda\) allows the presence of torsion in this theory. If the condition \(\Delta _\lambda ^{[\mu \nu ]} = 0\) holds, it follows that the Cartan torsion tensor vanishes \((S_{\mu \nu}^\lambda = 0)\) [558]. Hence the torsion is induced by matter fields with the antisymmetric hypermomentum. The f(R) Palatini gravity belongs to f(R) theories in the metricaffine formalism with \(\Delta _\lambda ^{\mu \nu} = 0\). In the following we do not discuss further f(R) theory in the metricaffine formalism. Readers who are interested in those theories may refer to the papers [557, 556].
9.2 Background cosmological dynamics
We discuss the background cosmological evolution of dark energy models based on Palatini f(R) gravity. We shall carry out general analysis without specifying the forms of f(R). We take into account nonrelativistic matter and radiation whose energy densities are ρ_{m} and ρ_{r}, respectively. In the flat FLRW background (2.12) we obtain the following equations
together with the continuity equations, \({\dot \rho _m} + 3H{\rho _m} = 0\) and \({\dot \rho _r} + 4H{\rho _r} = 0\). Combing Eqs. (9.9) and (9.10) together with continuity equations, it follows that
where
In order to discuss cosmological dynamics it is convenient to introduce the dimensionless variables:
by which Eq (9.12) can be written as
Differentiating y_{1} and y_{2} with respect to N = ln a, we obtain [253]
where
The following constraint equation also holds
Hence the Ricci scalar R can be expressed in terms of y_{1} and y_{2}.
Differentiating Eq. (9.11) with respect to t, it follows that
from which we get the effective equation of state:
The cosmological dynamics is known by solving Eqs. (9.16) and (9.17) with Eq. (9.18). If C(R) is not constant, then one can use Eq. (9.19) to express R and C(R) in terms of y_{1} and y_{2}.
The fixed points of Eqs. (9.16) and (9.17) can be found by setting dy_{1}/dN = 0 and dy_{2}/dN = 0. Even when C(R) is not constant, except for the cases C(R) = −3 and C(R) = −4, we obtain the following fixed points [253]:

1.
P_{r}: (y_{1},y_{2}) = (0, 1),

2.
P_{m}: (y_{1}, y_{2}) = (0, 0),

3.
P_{d}: (y_{1}, y_{2}) = (1, 0).
The stability of the fixed points can be analyzed by considering linear perturbations about them. As long as dC/dy_{1} and dC/dy_{2} are bounded, the eigenvalues λ_{1} and λ_{2} of the Jacobian matrix of linear perturbations are given by

1.
P_{r}: (λ_{1}, λ_{2}) = (4 + C(R), 1),

2.
P_{m}: (λ_{1}, λ_{2}) = (3 + C(R), −1),

3.
P_{d}: (λ_{1}, λ_{2}) = (−3 − C(R), −4 − C(R)).
In the ΛCDM model (f(R) = R − 2Λ) one has w_{eff} = −y_{1} + y_{2}/3 and C(R) = 0. Then the points P_{r}, P_{m}, and P_{d} correspond to w_{eff} = 1/3, (λ_{1}, λ_{2}) = (4, 1) (radiation domination, unstable), w_{eff} = 0, (λ_{1}, λ_{2}) = (3, −1) (matter domination, saddle), and w_{eff} = −1, (λ_{1}, λ_{2}) = (−3, −4) (de Sitter epoch, stable), respectively. Hence the sequence of radiation, matter, and de Sitter epochs is in fact realized.
Let us next consider the model f(R) = R − β/R^{n} with β > 0 and n > −1. In this case the quantity C(R) is
The constraint equation (9.19) gives
The latetime de Sitter point corresponds to R^{1+n} = (2 + n)β, which exists for n > −2. Since C(R) = 0 in this case, the de Sitter point P_{d} is stable with the eigenvalues (λ_{1}, λ_{2}) = (−3, −4). During the radiation and matter domination we have β/R^{1+n} ≪ 1 (i.e., f(R) ≃ R) and hence C(R) = 3n. P_{r} corresponds to the radiation point (w_{eff} = 1/3) with the eigenvalues (λ_{1}, λ_{2}) = (4 + 3n, 1), whereas P_{m} to the matter point (w_{eff} = 0) with the eigenvalues (λ_{1}, λ_{2}) = (3 + 3n, −1). Provided that n > −1, P_{r} and P_{m} correspond to unstable and saddle points respectively, in which case the sequence of radiation, matter, and de Sitter eras can be realized. For the models f(R) = R + αR^{m} − β/R^{n}, it was shown in [253] that unified models of inflation and dark energy with radiation and matter eras are difficult to be realized.
In Figure 8 we plot the evolution of w_{eff} as well as y_{1} and y_{2} for the model f(R) = R − β/R^{n} with n = 0.02. This shows that the sequence of (P_{r}) radiation domination (w_{eff} = 1/3), (P_{m}) matter domination (w_{eff} = 0), and de Sitter acceleration (w_{eff} = −1) is realized. Recall that in metric f(R) gravity the model f(R) = R − β/R^{n} (β > 0, n > 0) is not viable because f_{,RR} is negative. In Palatini f(R) gravity the sign of f_{,RR} does not matter because there is no propagating degree of freedom with a mass M associated with the second derivative f_{,RR} [554].
In [21, 253] the dark energy model f(R) = R − β/R^{n} was constrained by the combined analysis of independent observational data. From the joint analysis of SuperNova Legacy Survey [39], BAO [227] and the CMB shift parameter [561], the constraints on two parameters n and β are n ∈ [−0.23, 0.42] and β ∈ [2.73, 10.6] at the 95% confidence level (in the unit of H_{0} = 1) [253]. Since the allowed values of n are close to 0, the above model is not particularly favored over the ΛCDM model. See also [116, 148, 522, 46, 47] for observational constraints on f(R) dark energy models based on the Palatini formalism.
9.3 Matter perturbations
We have shown that f(R) theory in the Palatini formalism can give rise to the latetime cosmic acceleration preceded by radiation and matter eras. In this section we study the evolution of matter density perturbations to confront Palatini f(R) gravity with the observations of largescale structure [359, 356, 357, 598, 380, 597]. Let us consider the perturbation δρ_{m} of nonrelativistic matter with a homogeneous energy density ρ_{m}. Koivisto and KurkiSuonio [359] derived perturbation equations in Palatini f(R) gravity. Using the perturbed metric (6.1) with the same variables as those introduced in Section 6, the perturbation equations are given by
where the Ricci scalar R can be understood as R(T).
From Eq. (9.28) the perturbation δF can be expressed by the matter perturbation δρ_{m}, as
where m = RF_{,R}/F. This equation clearly shows that the perturbation δF is sourced by the matter perturbation only, unlike metric f(R) gravity in which the oscillating mode of δF is present. The matter perturbation δρ_{m} and the velocity potential υ obey the same equations as given in Eqs. (8.86) and (8.87), which results in Eq. (8.89) in Fourier space.
Let us consider the perturbation equations in Fourier space. We choose the Longitudinal gauge (χ = 0) with α = Φ and ψ = Ψ. In this case Eq. (9.26) gives
Under the quasistatic approximation on subhorizon scales used in Section 8.1, Eqs. (9.24) and (8.89) reduce to
Combining Eq. (9.30) with Eq. (9.31), we obtain
where
Then the matter perturbation satisfies the following Eq. [597]
The effective gravitational potential defined in Eq. (8.98) obeys
In the above approximation we do not need to worry about the dominance of the oscillating mode of perturbations in the past. Note also that the same approximate equation of δ_{m} as Eq. (9.35) can be derived for different gauge choices [597].
The parameter ζ is a crucial quantity to characterize the evolution of perturbations. This quantity can be estimated as ζ ≈ (k/aH)^{2}m, which is much larger than m for subhorizon modes (k ≫ aH). In the regime ζ ≪ 1 the matter perturbation evolves as δ_{m} ℝ t^{2/3}. Meanwhile the evolution of δ_{m} in the regime ζ ≫ 1 is completely different from that in GR. If the transition characterized by ζ = 1 occurs before today, this gives rise to the modification to the matter spectrum compared to the GR case.
In the regime ζ ≫ 1, let us study the evolution of matter perturbations during the matter dominance. We shall consider the case in which the parameter m (with ∣m∣ ≪ 1) evolves as
where p is a constant. For the model f(R) = R − μR_{c}(R/R_{c})^{n} (n < 1) the power p corresponds to p = 1 + n, whereas for the models (4.83) and (4.84) with n > 0 one has p = 1 + 2n. During the matter dominance the parameter ζ evolves as ζ = ±(t/t_{k})^{2p+2/3}, where the subscript “k” denotes the value at which the perturbation crosses ζ = ±1. Here + and − signs correspond to the cases m > 0 and m < 0, respectively. Then the matter perturbation equation (9.35) reduces to
When m > 0, the growing mode solution to Eq. (9.38) is given by
This shows that the perturbations exhibit violent growth for p > −1/3, which is not compatible with observations of largescale structure. In metric f(R) gravity the growth of matter perturbations is much milder.
When m < 0, the perturbations show a damped oscillation:
where \(x = \sqrt 6 {e^{(3p + 1)(N  {N_k})/2}}/(3p + 1)\), and θ is a constant. The averaged value of the growth rate f_{δ} is given by \({\bar f_\delta} =  (3p + 2)/4\), but it shows a divergence every time x changes by π. These negative values of f_{δ} are also difficult to be compatible with observations.
The f(R) models can be consistent with observations of largescale structure if the universe does not enter the regime ∣ζ∣ > 1 by today. This translates into the condition [597]
Let us consider the wavenumbers 0.01 h Mpc^{−1} ≲ k ≲ 0.2 h Mpc^{−1} that corresponds to the linear regime of the matter power spectrum. Since the wavenumber k = 0.2 h Mpc^{−1} corresponds to k ≈ 600a_{0}H_{0} (where “0” represents present quantities), the condition (9.41) gives the bound ∣m(z = 0)∣ ≲ 3 × 10^{−6}.
If we use the observational constraint of the growth rate, f_{δ} ≲ 1.5 [418, 605, 211], then the deviation parameter m today is constrained to be ∣m(z = 0)∣ ≲ 10^{−5}–10^{−4} for the model f(R) = R − λR_{c}(R/R_{c})^{n} (n < 1) as well as for the models (4.83) and (4.84) [597]. Recall that, in metric f(R) gravity, the deviation parameter can grow to the order of 0.1 by today. Meanwhile f(R) dark energy models based on the Palatini formalism are hardly distinguishable from the ΛCDM model [356, 386, 385, 597]. Note that the bound on m(z = 0) becomes even severer by considering perturbations in nonlinear regime. The above peculiar evolution of matter perturbations is associated with the fact that the coupling between nonrelativistic matter and a scalarfield degree of freedom is very strong (as we will see in Section 10.1).
The above results are based on the fact that dark matter is described by a cold and perfect fluid with no pressure. In [358] it was suggested that the tight bound on the parameter m can be relaxed by considering imperfect dark matter with a shear stress. Although the approach taken in [358] did not aim to explain the origin of a dark matter stress Π that cancels the kdependent term in Eq. (9.35), it will be of interest to further study whether some theoretically motivated choice of Π really allows the possibility that Palatini f(R) dark energy models can be distinguished from the ΛCDM model.