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 big-bang
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
big-bang 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 late-time 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], large-scale 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 satisfies the condition
(here
and
are the pressure and the
energy density of matter, respectively). In fact, we further require some component of negative pressure,
with
, to realize the acceleration of the universe. The cosmological constant
is the simplest
candidate of dark energy, which corresponds to
. 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
consistent with
observations. Since the accelerated expansion in the very early universe needs to end to connect to the
radiation-dominated universe, the pure cosmological constant is not responsible for inflation. A scalar
field
with a slowly varying potential can be a candidate for inflation as well as for dark
energy.
Although many scalar-field 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 .
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
(
) [140, 365].
While scalar-field models of inflation and dark energy correspond to a modification of the
energy-momentum 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 , where
is the Ricci scalar
and
is the cosmological constant (corresponding to the equation of state
). The
presence of
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
-Cold Dark Matter (
CDM) model (
) fits a number of observational
data well [367
, 368], there is also a possibility for the time-varying 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 is an
arbitrary function of
[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
. In this
formalism the affine connection
depends on
. Note that we will consider here and in the
remaining sections only torsion-free theories. The second is the Palatini formalism [481
] in
which
and
are treated as independent variables when we vary the action. These two
approaches give rise to different field equations for a non-linear Lagrangian density in
,
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 (
) can lead to the accelerated expansion of the Universe
because of the presence of the
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 scalar-field 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 late-time acceleration.
The model with a Lagrangian density (
) 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 matter-dominated epoch because of a large coupling between dark energy and dark
matter [28
, 29
]. These results show how non-trivial 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 Brans–Dicke (BD) theory [100]
with a BD parameter
[467
, 579
, 152
]. Unlike original BD theory [100
], there exists a potential
for a scalar-field 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
, it is not possible to
satisfy local gravity constraints due to the appearance of a long-range 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 “thin-shell” around its surface, the field is
nearly frozen in most regions inside the body. Then the effective coupling between the field and
non-relativistic 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 space-time. 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 high-curvature regime [266
]. For constant density stars, however, a thin-shell 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 . 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
between the field and non-relativistic matter.
This coupling is related to the BD parameter via the relation
[343
, 596
]. One can
recover GR by taking the limit
, i.e.,
. The f (R) gravity in the metric formalism
corresponds to
[28
], i.e.,
. For large coupling models with
it is
possible to design scalar-field 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 late-time
cosmic acceleration. This generalized BD theory also leaves a number of interesting observational and
experimental signatures [596
].
In addition to the Ricci scalar , one can construct other scalar quantities such as
and
from the Ricci tensor
and Riemann tensor
[142
]. For the Gauss–Bonnet (GB)
curvature invariant defined by
, it is known that one can avoid the
appearance of spurious spin-2 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
, i.e.,
or (ii) the Lagrangian density
is a
function of
, i.e.,
. The GB coupling in the case (i) appears in low-energy 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
[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 large-scale 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 Gauss–Bonnet 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
fourth-order 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 , where
is the speed of light,
is reduced
Planck’s constant, and
is Boltzmann’s constant. We define
, where
is the gravitational constant,
is the Planck mass with a reduced value
. Throughout this review, we use a dot for the derivative with respect
to cosmic time
and “
” for the partial derivative with respect to the variable
, e.g.,
and
. We use the metric signature
. The Greek indices
and
run from 0 to 3, whereas the Latin indices
and
run from 1 to 3 (spatial
components).
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