f(R) Theories

8/8/2019 f(R) Theories

1/161

Living Rev. Relativity, 13, (2010), 3http://www.livingreviews.org/lrr-2010-3

L I V I N G R E VI E WS

in relativity

f(R) Theories

Antonio De FeliceDepartment o Physics, Faculty o Science, Tokyo University o Science,

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japanemail: [email protected]

http://sites.google.com/site/adefelic/

Shinji TsujikawaDepartment o Physics, Faculty o Science, Tokyo University o Science,

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japanemail: [email protected]

http://www.rs.kagu.tus.ac.jp/shinji/Tsujikawae.html

Accepted on 17 May 2010Published on 23 June 2010

Abstract

Over the past decade, f(R) theories have been extensively studied as one o the simplestmodications to General Relativity. In this article we review various applications o f(R)theories to cosmology and gravity such as ination, dark energy, local gravity constraints,cosmological perturbations, and spherically symmetric solutions in weak and strong gravita-tional backgrounds. We present a number o ways to distinguish those theories rom GeneralRelativity observationally and experimentally. We also discuss the extension to other modiedgravity theories such as BransDicke theory and GaussBonnet gravity, and address modelsthat can satisy both cosmological and local gravity constraints.

This review is licensed under a Creative CommonsAttribution-Non-Commercial-NoDerivs 3.0 Germany License.http://creativecommons.org/licenses/by-nc-nd/3.0/de/
http://www.livingreviews.org/lrr-2010-3http://sites.google.com/site/adefelic/http://www.rs.kagu.tus.ac.jp/shinji/Tsujikawae.htmlhttp://creativecommons.org/licenses/by-nc-nd/3.0/de/http://creativecommons.org/licenses/by-nc-nd/3.0/de/http://www.rs.kagu.tus.ac.jp/shinji/Tsujikawae.htmlhttp://sites.google.com/site/adefelic/http://www.livingreviews.org/lrr-2010-3


2/161

Imprint / Terms o UseLiving Reviews in Relativity is a peer reviewed open access journal published by the Max PlanckInstitute or Gravitational Physics, Am Muhlenberg 1, 14476 Potsdam, Germany. ISSN 1433-8351.

This review is licensed under a Creative Commons Attribution-Non-Commercial-NoDerivs 3.0Germany License: http://creativecommons.org/licenses/by-nc-nd/3.0/de/

Because a Living Reviews article can evolve over time, we recommend to cite the article as ollows:

Antonio De Felice and Shinji Tsujikawa,

(R) Theories,Living Rev. Relativity, 13, (2010), 3. [Online Article]: cited [],

http://www.livingreviews.org/lrr-2010-3

The date given as then uniquely identies the version o the article you are reerring to.

Article Revisions

Living Reviews supports two ways o keeping its articles up-to-date:

Fast-track revision A ast-track revision provides the author with the opportunity to add shortnotices o current research results, trends and developments, or important publications tothe article. A ast-track revision is reereed by the responsible subject editor. I an articlehas undergone a ast-track revision, a summary o changes will be listed here.

Major update A major update will include substantial changes and additions and is subject toull external reereeing. It is published with a new publication number.

For detailed documentation o an articles evolution, please reer to the history document o thearticles online version at http://www.livingreviews.org/lrr-2010-3.
http://creativecommons.org/licenses/by-nc-nd/3.0/de/http://www.livingreviews.org/lrr-2010-3http://www.livingreviews.org/lrr-2010-3http://www.livingreviews.org/lrr-2010-3http://creativecommons.org/licenses/by-nc-nd/3.0/de/


3/161

Contents

1 Introduction 5

2 Field Equations in the Metric Formalism 9

2.1 Equations o motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Equivalence with BransDicke theory . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Conormal transormation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Ination in f(R) Theories 15

3.1 Inationary dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Dynamics in the Einstein rame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Reheating ater ination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.1 Case: = 0 and = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.2 Case: || 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Dark Energy in f(R) Theories 24

4.1 Dynamical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Viable (R) dark energy models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Equation o state o dark energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Local Gravity Constraints 30

5.1 Linear expansions o perturbations in the spherically symmetric background . . . . 30

5.2 Chameleon mechanism in (R) gravity . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Field prole o the chameleon eld . . . . . . . . . . . . . . . . . . . . . . . 325.2.2 Thin-shell solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.3 Post Newtonian parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.4 Experimental bounds rom the violation o equivalence principle . . . . . . 37

5.2.5 Constraints on model parameters in (R) gravity . . . . . . . . . . . . . . . 38

6 Cosmological Perturbations 40

6.1 Perturbation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.2 Gauge-invariant quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Perturbations Generated During Ination 44

7.1 Curvature perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7.2 Tensor perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.3 The spectra o perturbations in ination based on (R) gravity . . . . . . . . . . . 47

7.3.1 The model () = ( > 0) . . . . . . . . . . . . . . . . . . . . . . . . 48

7.3.2 The model () = + 2/(62) . . . . . . . . . . . . . . . . . . . . . . . 48

7.3.3 The power spectra in the Einstein rame . . . . . . . . . . . . . . . . . . . . 49

7.4 The Lagrangian or cosmological perturbations . . . . . . . . . . . . . . . . . . . . 50

8 Observational Signatures o Dark Energy Models in f(R) Theories 52

8.1 Matter density perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.2 The impact on large-scale structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.3 Non-linear matter perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8.4 Cosmic Microwave Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


4/161

9 Palatini Formalism 649.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.2 Background cosmological dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 669.3 Matter perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689.4 Shortcomings o Palatini (R) gravity . . . . . . . . . . . . . . . . . . . . . . . . . 71

10 Extension to BransDicke Theory 7310.1 BransDicke theory and the equivalence with (R) theories . . . . . . . . . . . . . 7310.2 Cosmological dynamics o dark energy models based on BransDicke theory . . . . 7510.3 Local gravity constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7810.4 Evolution o matter density perturbations . . . . . . . . . . . . . . . . . . . . . . . 79

11 Relativistic Stars in f(R) Gravity and Chameleon Theories 83

11.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8311.2 Constant density star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8511.3 Relativistic stars in metric (R) gravity . . . . . . . . . . . . . . . . . . . . . . . . 88

12 GaussBonnet Gravity 9212.1 Lovelock scalar invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9212.2 Ghosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9412.3 () gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

12.3.1 Cosmology at the background level and viable () models . . . . . . . . . 9512.3.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9612.3.3 Solar system constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9912.3.4 Ghost conditions in the FLRW background . . . . . . . . . . . . . . . . . . 10012.3.5 Viability o (

) gravity in the presence o matter . . . . . . . . . . . . . . 101

12.3.6 The speed o propagation in more general modications o gravity . . . . . 10212.4 GaussBonnet gravity coupled to a scalar eld . . . . . . . . . . . . . . . . . . . . 103

13 Other Aspects o f(R) Theories and Modifed Gravity 10513.1 Weak lensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10513.2 Thermodynamics and horizon entropy . . . . . . . . . . . . . . . . . . . . . . . . . 10813.3 Curing the curvature singularity in (R) dark energy models, unied models o

ination and dark energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11113.4 (R) theories in extra dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11113.5 Vainshtein mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11313.6 DGP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11413.7 Special symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

13.7.1 Noether symmetry on FLRW . . . . . . . . . . . . . . . . . . . . . . . . . . 11613.7.2 Galileon symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

14 Conclusions 120

15 Acknowledgements 123

Reerences 124

List o Tables

1 The critical points o dark energy models. . . . . . . . . . . . . . . . . . . . . . . . 76


5/161

(R) Theories 5

1 Introduction

General Relativity (GR) [225, 226] is widely accepted as a undamental theory to describe thegeometric properties o spacetime. In a homogeneous and isotropic spacetime the Einstein eldequations give rise to the Friedmann equations that describe the evolution o the universe. In act,the standard big-bang cosmology based on radiation and matter dominated epochs can be welldescribed within the ramework o General Relativity.

However, the rapid development o observational cosmology which started rom 1990s showsthat the universe has undergone two phases o cosmic acceleration. The rst one is called ina-tion [564, 339, 291, 524], which is believed to have occurred prior to the radiation domination(see [402, 391, 71] or reviews). This phase is required not only to solve the atness and horizonproblems plagued in big-bang cosmology, but also to explain a nearly at spectrum o temperatureanisotropies observed in Cosmic Microwave Background (CMB) [541]. The second accelerating

phase has started ater 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] or reviews).The existence o dark energy has been conrmed by a number o observations such as super-novae 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 o cosmic acceleration cannot be explained by the presence o standard matterwhose equation o state = / satises the condition 0 (here and are the pressureand the energy density o matter, respectively). In act, we urther require some component onegative pressure, with < 1/3, to realize the acceleration o the universe. The cosmologicalconstant is the simplest candidate o dark energy, which corresponds to = 1. However, i thecosmological constant originates rom a vacuum energy o particle physics, its energy scale is toolarge to be compatible with the dark energy density [614]. Hence we need to nd some mechanism to

obtain a small value o consistent with observations. Since the accelerated expansion in the veryearly universe needs to end to connect to the radiation-dominated universe, the pure cosmologicalconstant is not responsible or ination. A scalar eld with a slowly varying potential can be acandidate or ination as well as or dark energy.

Although many scalar-eld potentials or ination have been constructed in the ramework ostring theory and supergravity, the CMB observations still do not show particular evidence toavor one o such models. This situation is also similar in the context o dark energy thereis a degeneracy as or the potential o the scalar eld (quintessence [111, 634, 267, 263, 615,503, 257, 155]) due to the observational degeneracy to the dark energy equation o state around = 1. Moreover it is generally dicult to construct viable quintessence potentials motivatedrom particle physics because the eld mass responsible or cosmic acceleration today is very small( 1033 eV) [140, 365].

While scalar-eld models o ination and dark energy correspond to a modication o theenergy-momentum tensor in Einstein equations, there is another approach to explain the accelera-tion o the universe. This corresponds to the modied gravity in which the gravitational theory ismodied compared to GR. The Lagrangian density or GR is given by () = 2, where isthe Ricci scalar and is the cosmological constant (corresponding to the equation o state = 1).The presence o gives rise to an exponential expansion o the universe, but we cannot use it orination because the inationary period needs to connect to the radiation era. It is possible to usethe cosmological constant or dark energy since the acceleration today does not need to end. How-ever, i the cosmological constant originates rom a vacuum energy o particle physics, its energydensity would be enormously larger than the todays dark energy density. While the -Cold DarkMatter (CDM) model (() = 2) ts a number o observational data well [367, 368], thereis also a possibility or the time-varying equation o state o dark energy [10, 11, 450, 451, 630].

One o the simplest modications to GR is the (R) gravity in which the Lagrangian density

Living Reviews in Relativityhttp://www.livingreviews.org/lrr-2010-3
http://www.livingreviews.org/lrr-2010-3http://www.livingreviews.org/lrr-2010-3


6/161

6 Antonio De Felice and Shinji Tsujikawa

is an arbitrary unction o [77, 512, 102, 106]. There are two ormalisms in deriving eldequations rom the action in (R) gravity. The rst is the standard metric ormalism in which theeld equations are derived by the variation o the action with respect to the metric tensor .In this ormalism the ane connection depends on . Note that we will consider here andin the remaining sections only torsion-ree theories. The second is the Palatini ormalism [481]in which and

are treated as independent variables when we vary the action. These two

approaches give rise to diferent eld equations or a non-linear Lagrangian density in , whileor the GR action they are identical with each other. In this article we mainly review the ormerapproach unless otherwise stated. In Section 9 we discuss the Palatini ormalism in detail.

The model with () = + 2 ( > 0) can lead to the accelerated expansion o the Universebecause o the presence o the 2 term. In act, this is the rst model o ination proposedby Starobinsky in 1980 [564]. As we will see in Section 7, this model is well consistent with thetemperature anisotropies observed in CMB and thus it can be a viable alternative to the scalar-

eld models o ination. Reheating ater ination proceeds by a gravitational particle productionduring the oscillating phase o the Ricci scalar [565, 606, 426].

The discovery o dark energy in 1998 also stimulated the idea that cosmic acceleration today mayoriginate rom some modication o gravity to GR. Dark energy models based on (R) theorieshave been extensively studied as the simplest modied gravity scenario to realize the late-timeacceleration. The model with a Lagrangian density () = / ( > 0, > 0) was proposedor dark energy in the metric ormalism [113, 120, 114, 143, 456]. However it was shown thatthis model is plagued by a matter instability [215, 244] as well as by a diculty to satisy localgravity constraints [469, 470, 245, 233, 154, 448, 134]. Moreover it does not possess a standardmatter-dominated epoch because o a large coupling between dark energy and dark matter [28, 29].These results show how non-trivial it is to obtain a viable (R) model. Amendola et al. [26]derived conditions or the cosmological viability o (R) dark energy models. In local regions

whose densities are much larger than the homogeneous cosmological density, the models need tobe close to GR or consistency with local gravity constraints. A number o viable (R) modelsthat can satisy both cosmological and local gravity constraints have been proposed in . [26, 382,31, 306, 568, 35, 587, 206, 164, 396]. Since the law o gravity gets modied on large distances in(R) models, this leaves several interesting observational signatures such as the modication to thespectra o galaxy clustering [146, 74, 544, 526, 251, 597, 493], CMB [627, 544, 382, 545], and weaklensing [595, 528]. In this review we will discuss these topics in detail, paying particular attentionto the construction o viable (R) models and resulting observational consequences.

The (R) gravity in the metric ormalism corresponds to generalized BransDicke (BD) the-ory [100] with a BD parameter BD = 0 [467, 579, 152]. Unlike original BD theory [100], thereexists a potential or a scalar-eld degree o reedom (called scalaron [564]) with a gravitationalorigin. I the mass o the scalaron always remains as light as the present Hubble parameter 0,it is not possible to satisy local gravity constraints due to the appearance o a long-range thorce with a coupling o the order o unity. One can design the eld potential o (R) gravitysuch that the mass o the eld is heavy in the region o high density. The viable (R) modelsmentioned above have been constructed to satisy such a condition. Then the interaction range othe th orce becomes short in the region o high density, which allows the possibility that themodels are compatible with local gravity tests. More precisely the existence o a matter coupling,in the Einstein rame, gives rise to an extremum o the efective eld potential around which theeld can be stabilized. As long as a spherically symmetric body has a thin-shell around itssurace, the eld is nearly rozen in most regions inside the body. Then the efective couplingbetween the eld and non-relativistic matter outside the body can be strongly suppressed throughthe chameleon mechanism [344, 343]. The experiments or the violation o equivalence principle aswell as a number o solar system experiments place tight constraints on dark energy models basedon (R) theories [306, 251, 587, 134, 101].



7/161

(R) Theories 7

The spherically symmetric solutions mentioned above have been derived under the weak gravitybackgrounds where the background metric is described by a Minkowski space-time. In stronggravitational backgrounds such as neutron stars and white dwars, we need to take into accountthe backreaction o gravitational potentials to the eld equation. The structure o relativistic starsin (R) gravity has been studied by a number o authors [349, 350, 594, 43, 600, 466, 42, 167].Originally the diculty o obtaining relativistic stars was pointed out in [ 349] in connection to thesingularity problem o (R) dark energy models in the high-curvature regime [266]. For constantdensity stars, however, a thin-shell eld prole has been analytically derived in [ 594] or chameleonmodels in the Einstein rame. The existence o relativistic stars in (R) gravity has been alsoconrmed numerically or the stars with constant [43, 600] and varying [42] densities. In thisreview we shall also discuss this issue.

It is possible to extend (R) gravity to generalized BD theory with a eld potential and anarbitrary BD parameter BD. I we make a conormal transormation to the Einstein rame [213,

609, 408, 611, 249, 268], we can show that BD theory with a eld potential corresponds to thecoupled quintessence scenario [23] with a coupling between the eld and non-relativistic matter.This coupling is related to the BD parameter via the relation 1/(22) = 3 + 2BD [343, 596]. Onecan recover GR by taking the limit 0, i.e., BD . The (R) gravity in the metric ormalismcorresponds to = 1/6 [28], i.e., BD = 0. For large coupling models with || = (1) it ispossible to design scalar-eld potentials such that the chameleon mechanism works to reduce theefective matter coupling, while at the same time the eld is suciently light to be responsible orthe late-time cosmic acceleration. This generalized BD theory also leaves a number o interestingobservational and experimental signatures [596].

In addition to the Ricci scalar , one can construct other scalar quantities such as

and rom the Ricci tensor and Riemann tensor [142]. For the Gauss

Bonnet (GB) curvature invariant dened by 2 4 + , it is known thatone can avoid the appearance o spurious spin-2 ghosts [572, 67, 302] (see also [98, 465, 153, 447,110, 181, 109]). In order to give rise to some contribution o the GB term to the Friedmannequation, we require that (i) the GB term couples to a scalar eld , i.e., () or (ii) theLagrangian density is a unction o, i.e., (). The GB coupling in the case (i) appears in low-energy string efective action [275] and cosmological solutions in such a theory have been studiedextensively (see [34, 273, 105, 147, 588, 409, 468] or the construction o nonsingular cosmologicalsolutions and [463, 360, 361, 593, 523, 452, 453, 381, 25] or the application to dark energy). Inthe case (ii) it is possible to construct viable models that are consistent with both the backgroundcosmological evolution and local gravity constraints [458, 188, 189] (see also [165, 180, 178, 383,633, 599]). However density perturbations in perect uids exhibit negative instabilities duringboth the radiation and the matter domination, irrespective o the orm o () [383, 182]. Thisgrowth o perturbations gets stronger on smaller scales, which is dicult to be compatible withthe observed galaxy spectrum unless the deviation rom GR is very small. We shall review suchtheories as well as other modied gravity theories.

This review is organized as ollows. In Section 2 we present the eld equations o (R) gravityin the metric ormalism. In Section 3 we apply (R) theories to the inationary universe. Section 4is devoted to the construction o cosmologically viable (R) dark energy models. In Section 5 localgravity constraints on viable (R) dark energy models will be discussed. In Section 6 we providethe equations o linear cosmological perturbations or general modied gravity theories includingmetric (R) gravity as a special case. In Section 7 we study the spectra o scalar and tensormetric perturbations generated during ination based on (R) theories. In Section 8 we discussthe evolution o matter density perturbations in (R) dark energy models and place constraintson model parameters rom the observations o large-scale structure and CMB. Section 9 is devotedto the viability o the Palatini variational approach in (R) gravity. In Section 10 we constructviable dark energy models based on BD theory with a potential as an extension o (R) theories.



8/161


In Section 11 the structure o relativistic stars in (R) theories will be discussed in detail. InSection 12 we provide a brie review o GaussBonnet gravity and resulting observational andexperimental consequences. In Section 13 we discuss a number o other aspects o (R) gravityand modied gravity. Section 14 is devoted to conclusions.

There are other review articles on (R) gravity [556, 555, 618] and modied gravity [171, 459,126, 397, 217]. Compared to those articles, we put more weights on observational and experimentalaspects o(R) theories. This is particularly useul to place constraints on ination and dark energymodels based on (R) theories. The readers who are interested in the more detailed history o(R) theories and ourth-order gravity may have a look at the review articles by Schmidt [531] andSotiriou and Faraoni [556].

In this review we use units such that = = = 1, where is the speed o light, is reducedPlancks constant, and is Boltzmanns constant. We dene 2 = 8 = 8/2pl = 1/

2pl,

where is the gravitational constant, pl = 1.22

1019 GeV is the Planck mass with a reduced

value pl = pl/8 = 2.44 1018 GeV. Throughout this review, we use a dot or the derivativewith respect to cosmic time and , or the partial derivative with respect to the variable , e.g.,, / and , 2/2. We use the metric signature (, +, +, +). The Greek indices and run rom 0 to 3, whereas the Latin indices and run rom 1 to 3 (spatial components).



9/161

(R) Theories 9

2 Field Equations in the Metric Formalism

We start with the 4-dimensional action in (R) gravity:

=1

22

d4

() +

d4( , ) , (2.1)

where 2 = 8, is the determinant o the metric , and is a matter Lagrangian1 thatdepends on and matter elds . The Ricci scalar is dened by =

, where theRicci tensor is

= =

+ . (2.2)

In the case o the torsion-less metric ormalism, the connections are the usual metric connec-

tions dened in terms o the metric tensor , as

=1

2

+

. (2.3)

This ollows rom the metricity relation, = / = 0.

2.1 Equations o motion

The eld equation can be derived by varying the action (2.1) with respect to :

()() 12

() () + () = 2() , (2.4)

where () /. () is the energy-momentum tensor o the matter elds dened by thevariational derivative o in terms o :

() = 2

. (2.5)

This satises the continuity equation

() = 0 , (2.6)

as well as , i.e., = 0.2 The trace o Eq. (2.4) gives

3() + () 2() = 2

, (2.7)

where = () and = (1/

)().Einstein gravity, without the cosmological constant, corresponds to () = and () = 1,

so that the term () in Eq. (2.7) vanishes. In this case we have = 2 and hence theRicci scalar is directly determined by the matter (the trace ). In modied gravity the term() does not vanish in Eq. (2.7), which means that there is a propagating scalar degree oreedom, (). The trace equation (2.7) determines the dynamics o the scalar eld (dubbed scalaron [564]).

1 Note that we do not take into account a direct coupling between the Ricci scalar and matter (such as 1())considered in [439, 80, 81, 82, 248].

2 This result is a consequence o the action principle, but it can be derived also by a direct calculation, using theBianchi identities.



10/161


The eld equation (2.4) can be written in the ollowing orm [568]

= 2

() + ()

, (2.8)

where (1/2) and

2() ( )/2 + + (1 ) . (2.9)

Since = 0 and () = 0, it ollows that

() = 0 . (2.10)

Hence the continuity equation holds, not only or , but also or the efective energy-momentum

tensor () dened in Eq. (2.9). This is sometimes convenient when we study the dark energy

equation o state [306, 568] as well as the equilibrium description o thermodynamics or the horizonentropy [53].

There exists a de Sitter point that corresponds to a vacuum solution ( = 0) at which the Ricciscalar is constant. Since () = 0 at this point, we obtain

() 2() = 0 . (2.11)

The model () = 2 satises this condition, so that it gives rise to the exact de Sitter so-lution [564]. In the model () = + 2, because o the linear term in , the inationaryexpansion ends when the term 2 becomes smaller than the linear term (as we will see inSection 3). This is ollowed by a reheating stage in which the oscillation o leads to the gravi-tational particle production. It is also possible to use the de Sitter point given by Eq. (2.11) or

dark energy.We consider the spatially at FriedmannLematreRobertsonWalker (FLRW) spacetime with

a time-dependent scale actor () and a metric

d2 = dd = d2 + 2() d2 , (2.12)

where is cosmic time. For this metric the Ricci scalar is given by

= 6(22 + ) , (2.13)

where / is the Hubble parameter and a dot stands or a derivative with respect to . Thepresent value o is given by

0 = 100 kmsec1

Mpc1

= 2.1332 1042

GeV , (2.14)

where = 0.72 0.08 describes the uncertainty o 0 [264].The energy-momentum tensor o matter is given by (

) = diag (, , , ), where

is the energy density and is the pressure. The eld equations (2.4) in the at FLRWbackground give

3 2 = ( )/2 3 + 2 , (2.15)2 = + 2( + ) , (2.16)

where the perect uid satises the continuity equation

+ 3( + ) = 0 . (2.17)



11/161

(R) Theories 11

We also introduce the equation o state o matter,

/. As long as is constant,the integration o Eq. (2.17) gives 3(1+ ). In Section 4 we shall take into account bothnon-relativistic matter ( = 0) and radiation ( = 1/3) to discuss cosmological dynamics o(R) dark energy models.

Note that there are some works about the Einstein static universes in (R) gravity [91, 532].Although Einstein static solutions exist or a wide variety o (R) models in the presence o abarotropic perect uid, these solutions have been shown to be unstable against either homogeneousor inhomogeneous perturbations [532].

2.2 Equivalence with BransDicke theory

The (R) theory in the metric ormalism can be cast in the orm o BransDicke (BD) theory [100]with a potential or the efective scalar-eld degree o reedom (scalaron). Let us consider the

ollowing action with a new eld ,

=1

22

d4

[() + ,()( )] +

d4( , ) . (2.18)

Varying this action with respect to , we obtain

,()( ) = 0 . (2.19)Provided ,() = 0 it ollows that = . Hence the action (2.18) recovers the action (2.1) in(R) gravity. I we dene

,() , (2.20)the action (2.18) can be expressed as

=

d4

1

22 ()

+

d4( , ) , (2.21)

where () is a eld potential given by

() =() (())

22. (2.22)

Meanwhile the action in BD theory [100] with a potential () is given by

=

d4

1

2 BD

2()2 ()

+

d4( , ) , (2.23)

where BD is the BD parameter and ()2

. Comparing Eq. (2.21) with Eq. (2.23),it ollows that (R) theory in the metric ormalism is equivalent to BD theory with the parameterBD = 0 [467, 579, 152] (in the unit

2 = 1). In Palatini (R) theory where the metric andthe connection are treated as independent variables, the Ricci scalar is diferent rom that inmetric (R) theory. As we will see in Sections 9.1 and 10.1, (R) theory in the Palatini ormalismis equivalent to BD theory with the parameter BD = 3/2.

2.3 Conormal transormation

The action (2.1) in (R) gravity corresponds to a non-linear unction in terms o. It is possibleto derive an action in the Einstein rame under the conormal transormation [213, 609, 408, 611,249, 268, 410]:

= 2 , (2.24)



12/161


where 2 is the conormal actor and a tilde represents quantities in the Einstein rame. The Ricciscalars and in the two rames have the ollowing relation

= 2( + 6 6) , (2.25)

where

ln ,

, 1 (

) . (2.26)

We rewrite the action (2.1) in the orm

=

d4

1

22

+

d4( , ) , (2.27)

where

=

22. (2.28)

Using Eq. (2.25) and the relation = 4, the action (2.27) is transormed as

=

d4

1

222( + 6 6) 4

+

d4(2 , ) .

(2.29)We obtain the Einstein rame action (linear action in ) or the choice

2 = . (2.30)

This choice is consistent i > 0. We introduce a new scalar eld dened by

3/2 ln . (2.31)

From the denition o in Eq. (2.26) we have that = /

6. Using Eq. (2.26), the integrald4

vanishes on account o the Gausss theorem. Then the action in the Einstein rameis

=

d4

1

22 1

2 ()

+

d4(1() , ) , (2.32)

where

() =

2=

2

2

2. (2.33)

Hence the Lagrangian density o the eld is given by = 12 () with theenergy-momentum tensor

() = 2

()

=

1

2 + ()

. (2.34)

The conormal actor 2 = = exp(

2/3 ) is eld-dependent. From the matter ac-tion (2.32) the scalar eld is directly coupled to matter in the Einstein rame. In order tosee this more explicitly, we take the variation o the action (2.32) with respect to the eld :

(

)() +

()

+

= 0 , (2.35)



13/161

(R) Theories 13

that is

, + 1

= 0 , where 1 (

) . (2.36)

Using Eq. (2.24) and the relations = 2 and = 1 , the energy-momentum

tensor o matter is transormed as

() = 2

=()

. (2.37)

The energy-momentum tensor o perect uids in the Einstein rame is given by

() = diag(, , , ) = diag(/2, /2, /2, /2) . (2.38)The derivative o the Lagrangian density

=

() =

(

1()) with respect to is

=

=

1

()

(())

=

,

2()

. (2.39)

The strength o the coupling between the eld and matter can be quantied by the ollowingquantity

,2

= 16

, (2.40)

which is constant in (R) gravity [28]. It then ollows that

=

, (2.41)

where = () = + 3. Substituting Eq. (2.41) into Eq. (2.36), we obtain the eldequation in the Einstein rame:

, + = 0 . (2.42)This shows that the eld is directly coupled to matter apart rom radiation ( = 0).

Let us consider the at FLRW spacetime with the metric (2.12) in the Jordan rame. Themetric in the Einstein rame is given by

d2 = 2d2 = (d2 + 2() d2) ,= d2 + 2() d2 , (2.43)

which leads to the ollowing relations (or > 0)

d = d , = , (2.44)where

= 2 . (2.45)

Note that Eq. (2.45) comes rom the integration o Eq. (2.40) or constant . The eld equa-tion (2.42) can be expressed as

d2

d2+ 3

d

d+ , = ( 3) , (2.46)

where

1

d

d=

1

+

2

. (2.47)



14/161


Dening the energy density =1

2

(d/d)2 + () and the pressure =1

2

(d/d)2

(),

Eq. (2.46) can be written as

d

d+ 3( + ) = ( 3) d

d. (2.48)

Under the transormation (2.44) together with = 2, = 2, and = 1/2[

(d/d)/2], the continuity equation (2.17) is transormed as

d

d+ 3( + ) = ( 3) d

d. (2.49)

Equations (2.48) and (2.49) show that the eld and matter interacts with each other, while thetotal energy density = + and the pressure = + satisy the continuity equation

d/d + 3( + ) = 0. More generally, Eqs. (2.48) and (2.49) can be expressed in terms othe energy-momentum tensors dened in Eqs. (2.34) and (2.37):

() = , () = , (2.50)

which correspond to the same equations in coupled quintessence studied in [23] (see also [22]).In the absence o a eld potential () (i.e., massless eld) the eld mediates a long-range

th orce with a large coupling (|| 0.4), which contradicts with experimental tests in the solarsystem. In (R) gravity a eld potential with gravitational origin is present, which allows thepossibility o compatibility with local gravity tests through the chameleon mechanism [344, 343].

In (R) gravity the eld is coupled to non-relativistic matter (dark matter, baryons) with auniversal coupling = 1/6. We consider the rame in which the baryons obey the standardcontinuity equation

3, i.e., the Jordan rame, as the physical rame in which physical

quantities are compared with observations and experiments. It is sometimes convenient to reerthe Einstein rame in which a canonical scalar eld is coupled to non-relativistic matter. In bothrames we are treating the same physics, but using the diferent time and length scales gives riseto the apparent diference between the observables in two rames. Our attitude throughout thereview is to discuss observables in the Jordan rame. When we transorm to the Einstein rame orsome convenience, we go back to the Jordan rame to discuss physical quantities.



15/161

(R) Theories 15

3 Ination inf

(R

) TheoriesMost models o ination in the early universe are based on scalar elds appearing in superstring andsupergravity theories. Meanwhile, the rst ination model proposed by Starobinsky [564] is relatedto the conormal anomaly in quantum gravity3. Unlike the models such as old ination [339, 291,524] this scenario is not plagued by the graceul exit problem the period o cosmic accelerationis ollowed by the radiation-dominated epoch with a transient matter-dominated phase [565, 606,426]. Moreover it predicts nearly scale-invariant spectra o gravitational waves and temperatureanisotropies consistent with CMB observations [563, 436, 566, 355, 315]. In this section we reviewthe dynamics o ination and reheating. In Section 7 we will discuss the power spectra o scalarand tensor perturbations generated in (R) ination models.

3.1 Inationary dynamicsWe consider the models o the orm

() = + , ( > 0, > 0) , (3.1)

which include the Starobinskys model [564] as a specic case ( = 2). In the absence o the matteruid ( = 0), Eq. (2.15) gives

3(1 + 1)2 =1

2( 1) 3( 1)2 . (3.2)

The cosmic acceleration can be realized in the regime = 1 + 1 1. Under the approxi-mation

1, we divide Eq. (3.2) by 31 to give

2 16

6

. (3.3)

During ination the Hubble parameter evolves slowly so that one can use the approximation|/2| 1 and |/()| 1. Then Eq. (3.3) reduces to

2 1, 1 = 2

( 1)(2 1) . (3.4)

Integrating this equation or 1 > 0, we obtain the solution

11

, 1/1 . (3.5)

The cosmic acceleration occurs or 1 < 1, i.e., > (1+

3)/2. When = 2 one has 1 = 0, so that is constant in the regime 1. The models with > 2 lead to super ination characterizedby > 0 and |0 |1/|1| (0 is a constant). Hence the standard ination with decreasing occurs or (1 +

3)/2 < < 2.

In the ollowing let us ocus on the Starobinskys model given by

() = + 2/(62) , (3.6)

3 There are some other works about theoretical constructions o f(R) models based on quantum gravity, super-gravity and extra dimensional theories [341, 345, 537, 406, 163, 287, 288, 518, 519].



16/161


where the constant has a dimension o mass. The presence o the linear term in eventuallycauses ination to end. Without neglecting this linear term, the combination o Eqs. (2.15) and(2.16) gives

2

2+

1

22 = 3 , (3.7)

+ 3 + 2 = 0 . (3.8)

During ination the rst two terms in Eq. (3.7) can be neglected relative to others, which gives 2/6. We then obtain the solution

(2/6)( ) , (3.9)

exp [( ) (

2/12)(

)2] , (3.10)

122 2 , (3.11)

where and are the Hubble parameter and the scale actor at the onset o ination ( = ),respectively. This inationary solution is a transient attractor o the dynamical system [ 407]. Theaccelerated expansion continues as long as the slow-roll parameter

1 = 2

2

62, (3.12)

is smaller than the order o unity, i.e., 2 2. One can also check that the approximate relation3 + 2 0 holds in Eq. (3.8) by using 122. The end o ination (at time = )is characterized by the condition 1, i.e., /

6. From Eq. (3.11) this corresponds

to the epoch at which the Ricci scalar decreases to 2

. As we will see later, the WMAPnormalization o the CMB temperature anisotropies constrains the mass scale to be 1013 GeV.Note that the phase space analysis or the model (3.6) was carried out in [407, 24, 131].

We dene the number o e-oldings rom = to = :

d ( ) 2

12( )2 . (3.13)

Since ination ends at + 6/2, it ollows that

32

2 1

21(), (3.14)

where we used Eq. (3.12) in the last approximate equality. In order to solve horizon and atnessproblems o the big bang cosmology we require that 70 [391], i.e., 1() 7103. The CMBtemperature anisotropies correspond to the perturbations whose wavelengths crossed the Hubbleradius around = 55 60 beore the end o ination.

3.2 Dynamics in the Einstein rame

Let us consider inationary dynamics in the Einstein rame or the model (3.6) in the absence omatter uids ( = 0). The action in the Einstein rame corresponds to (2.32) with a eld dened by

= 3

2

1

ln =

3

2

1

ln1 +

32 . (3.15)



17/161

(R) Theories 17

Figure 1: The eld potential (3.16) in the Einstein rame corresponding to the model (3.6). Ination isrealized in the regime 1.

Using this relation, the eld potential (2.33) reads [408, 61, 63]

() =

32

42

1 2/32

. (3.16)

In Figure 1 we illustrate the potential (3.16) as a unction o . In the regime 1 thepotential is nearly constant (() 32/(42)), which leads to slow-roll ination. The potentialin the regime 1 is given by () (1/2)22, so that the eld oscillates around = 0 witha Hubble damping. The second derivative o with respect to is

, = 22/3

1 2

2/3

, (3.17)

which changes rom negative to positive at = 1

3/2(ln 2)/ 0.169pl.Since 42/2 during ination, the transormation (2.44) gives a relation between the

cosmic time in the Einstein rame and that in the Jordan rame:

= d

2

( ) 2

12 ( )2

, (3.18)

where = corresponds to = 0. The end o ination ( + 6/2) corresponds to = (2/) in the Einstein rame, where is given in Eq. (3.13). On using Eqs. (3.10) and

(3.18), the scale actor =

in the Einstein rame evolves as

()

1 2

122

/2 , (3.19)

where = 2/. Similarly the evolution o the Hubble parameter = (/

)[1+ /(2)]is given by

() 2

1

2

62 1

2

122

2

, (3.20)



18/161


which decreases with time. Equations (3.19) and (3.20) show that the universe expands quasi-exponentially in the Einstein rame as well.

The eld equations or the action (2.32) are given by

32 = 2

1

2

d

d

2+ ()

, (3.21)

d2

d2+ 3

d

d+ , = 0 . (3.22)

Using the slow-roll approximations (d/d)2 () and |d2/d2| |d/d| during ination,one has 32 2() and 3(d/d) + , 0. We dene the slow-roll parameters

1 d/d2

122

,

2, 2 d

2

/d2

(d/d) 1 ,

32. (3.23)

For the potential (3.16) it ollows that

1 43

(2/3 1)2 , 2 1 +

2

32

2/3(1 2

2/3) , (3.24)

which are much smaller than 1 during ination ( 1). The end o ination is characterized bythe condition {1, |2|} = (1). Solving 1 = 1, we obtain the eld value 0.19 pl.

We dene the number o e-oldings in the Einstein rame,

=

d 2

, d , (3.25)

where is the eld value at the onset o ination. Since d = d[1 + /(2)], it ollowsthat is identical to in the slow-roll limit: | /(2)| |/2| 1. Under the condition 1 we have

34

2/3 . (3.26)

This shows that 1.11pl or = 70. From Eqs. (3.24) and (3.26) together with the approxi-mate relation /2, we obtain

1

3

4

2

, 2

1

, (3.27)

where, in the expression o 2, we have dropped the terms o the order o 1/2. The results (3.27)

will be used to estimate the spectra o density perturbations in Section 7.

3.3 Reheating ater ination

We discuss the dynamics o reheating and the resulting particle production in the Jordan rameor the model (3.6). The inationary period is ollowed by a reheating phase in which the secondderivative can no longer be neglected in Eq. (3.8). Introducing = 3/2, we have

+

2 34

2 32

= 0 . (3.28)



19/161

(R) Theories 19

Since 2

{2,

|

|}during reheating, the solution to Eq. (3.28) is given by that o the harmonic

oscillator with a requency . Hence the Ricci scalar exhibits a damped oscillation around = 0:

3/2 sin() . (3.29)Let us estimate the evolution o the Hubble parameter and the scale actor during reheating in

more detail. I we neglect the r.h.s. o Eq. (3.7), we get the solution () = const cos2( /2).Setting () = ()cos2(/2) to derive the solution o Eq. (3.7), we obtain [426]

() =1

+ (3/4)( os) + 3/(4) sin[( os)] , (3.30)

where os is the time at the onset o reheating. The constant is determined by matching Eq. (3.30)with the slow-roll inationary solution = 2/6 at = os. Then we get = 3/ and

() =

3

+

3

4( os) + 3

4sin ( os)

1cos2

2( os)

. (3.31)

Taking the time average o oscillations in the regime ( os) 1, it ollows that (2/3)( os)1. This corresponds to the cosmic evolution during the matter-dominated epoch,i.e., ( os)2/3. The gravitational efect o coherent oscillations o scalarons with mass issimilar to that o a pressureless perect uid. During reheating the Ricci scalar is approximatelygiven by 6, i.e.

3

3

+

3

4( os) + 3

4sin ( os)

1sin[( os)] . (3.32)

In the regime (

os

)

1 this behaves as

4 os sin[( os)] . (3.33)

In order to study particle production during reheating, we consider a scalar eld with mass. We also introduce a nonminimal coupling (1/2)2 between the eld and the Ricci scalar [88]. Then the action is given by

=

d4

()

22 1

2 1

22

2 12

2

, (3.34)

where () = + 2/(62). Taking the variation o this action with respect to gives

2

= 0 . (3.35)We decompose the quantum eld in terms o the Heisenberg representation:

(,) =1

(2)3/2

d3

()

+ *()

, (3.36)

where and are annihilation and creation operators, respectively. The eld can be quan-

tized in curved spacetime by generalizing the basic ormalism o quantum eld theory in the atspacetime. See the book [88] or the detail o quantum eld theory in curved spacetime. Theneach Fourier mode () obeys the ollowing equation o motion

+ 3 + 2

2+ 2 + = 0 , (3.37)



20/161


where =|

|is a comoving wavenumber. Introducing a new eld = and conormal time

=

1d, we obtain

d2d2

+

2 + 2

2 +

1

6

2

= 0 , (3.38)

where the conormal coupling correspond to = 1/6. This result states that, even though = 0(that is, the eld is minimally coupled to gravity), still gives a contribution to the efectivemass o . In the ollowing we rst review the reheating scenario based on a minimally coupledmassless eld ( = 0 and = 0). This corresponds to the gravitational particle productionin the perturbative regime [565, 606, 426]. We then study the case in which the nonminimalcoupling || is larger than the order o 1. In this case the non-adiabatic particle productionpreheating [584, 353, 538, 354] can occur via parametric resonance.

3.3.1 Case: = 0 and = 0

In this case there is no explicit coupling among the elds and . Hence the particles areproduced only gravitationally. In act, Eq. (3.38) reduces to

d2d2

+ 2 = , (3.39)

where = 2/6. Since is o the order o ()2, one has 2 or the mode deep insidethe Hubble radius. Initially we choose the eld in the vacuum state with the positive-requency

solution [88]: () =

/

2. The presence o the time-dependent term () leads to thecreation o the particle . We can write the solution o Eq. (3.39) iteratively, as [626]

() = () + 1

0

() sin[( )]()d . (3.40)

Ater the universe enters the radiation-dominated epoch, the term becomes small so that theat-space solution is recovered. The choice o decomposition o into and

is not unique. In

curved spacetime it is possible to choose another decomposition in term o new ladder operators and , which can be written in terms o and , such as = + * . Providedthat * = 0, even though |0 = 0, we have |0 = 0. Hence the vacuum in one basis is not thevacuum in the new basis, and according to the new basis, the particles are created. The Bogoliubovcoecient describing the particle production is

= 2

0

()2d . (3.41)

The typical wavenumber in the -coordinate is given by , whereas in the -coordinate it is /.Then the energy density per unit comoving volume in the -coordinate is [426]

=1

(2)3

0

42d ||2

=1

82

0

d ()

0

d()0

d 2()

=1

322

0

dd

d

0

d() , (3.42)

where in the last equality we have used the act that the term approaches 0 in the early andlate times.



21/161

(R) Theories 21

During the oscillating phase o the Ricci scalar the time-dependence o is given by =() sin(

0

d), where () = ()1/2 and = ( is a constant). When we evaluate theterm d/d in Eq. (3.42), the time-dependence o() can be neglected. Diferentiating Eq. (3.42)in terms o and taking the limit

0

d 1, it ollows thatdd

32

2()cos2

0

d

, (3.43)

where we used the relation lim sin()/ = (). Shiting the phase o the oscillating actorby /2, we obtain

dd

2

32=

42

1152. (3.44)

The proper energy density o the eld is given by = (/)/3 = /

4. Taking into account

* relativistic degrees o reedom, the total radiation density is

=*4

=*4

os

42

1152d , (3.45)

which obeys the ollowing equation

+ 4 =*2

1152. (3.46)

Comparing this with the continuity equation (2.17) we obtain the pressure o the created particles,as

=1

3 *

2

3456. (3.47)

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 ( os) 1 the evolution o the scale actor is given by 0( os)2/3,and hence

2 49( os)2 , (3.48)

where we have neglected the backreaction o created particles. Meanwhile the integration oEq. (3.45) gives

*3

240

1

os , (3.49)

where we have used the averaged relation 2 82/(os)2 [which comes rom Eq. (3.33)]. Theenergy density evolves slowly compared to 2 and nally it becomes a dominant contribution

to the total energy density (32 8/2pl) at the time os+402pl/(*3). In [426] it wasound that the transition rom the oscillating phase to the radiation-dominated epoch occurs slowercompared to the estimation given above. Since the epoch o the transient matter-dominated erais about one order o magnitude longer than the analytic estimation [426], we take the value os + 400

2pl/(*

3) to estimate the reheating temperature . Since the particle energy density

() is converted to the radiation energy density = *24 /30, the reheating temperaturecan be estimated as4

3 10171/4*

pl

3/2GeV . (3.50)

4 In [426] the reheating temperature is estimated by taking the maximum value o reached around the tenoscillations o. Meanwhile we estimate at the epoch where becomes a dominant contribution to the totalenergy density (as in [364]).



22/161


As we will see in Section 7, the WMAP normalization o the CMB temperature anisotropiesdetermines the mass scale to be 3 106pl. Taking the value * = 100, we have 5 109 GeV. For > the universe enters the radiation-dominated epoch characterized by 1/2, = 0, and 2.

3.3.2 Case: || 1I || is larger than the order o unity, one can expect the explosive particle production calledpreheating prior to the perturbative regime discussed above. Originally the dynamics o suchgravitational preheating was studied in [70, 592] or a massive chaotic ination model in Einsteingravity. Later this was extended to the (R) model (3.6) [591].

Introducing a new eld = 3/2, Eq. (3.37) reads

+2

2 + 2 +

9

4 2

3

2

= 0 . (3.51)

As long as || is larger than the order o unity, the last two terms in the bracket o Eq. (3.51) can beneglected relative to . Since the Ricci scalar is given by Eq. (3.33) in the regime ( os) 1,it ollows that

+

2

2+ 2

4

os sin{( os)}

0 . (3.52)

The oscillating term gives rise to parametric amplication o the particle . In order to seethis we introduce the variable dened by ( os) = 2 /2, where the plus and minus signscorrespond to the cases > 0 and < 0 respectively. Then Eq. (3.52) reduces to the Mathieuequation

d2

d2 + [ 2 cos(2)] 0 , (3.53)where

=42

22+

422

, =8||

( os) . (3.54)

The strength o parametric resonance depends on the parameters and . This can be describedby a stability-instability chart o the Mathieu equation [419, 353, 591]. In the Minkowski spacetimethe parameters and are constant. I and are in an instability band, then the perturbation grows exponentially with a growth index , i.e., . In the regime 1 the resonanceoccurs only in narrow bands around = 2, where = 1, 2,..., with the maximum growth index = /2 [353]. Meanwhile, or large ( 1), a broad resonance can occur or a wide range oparameter space and momentum modes [354].

In the expanding cosmological background both and vary in time. Initially the eld is in the broad resonance regime ( 1) or || 1, but it gradually enters the narrow resonanceregime ( 1). Since the eld passes many instability and stability bands, the growth index stochastically changes with the cosmic expansion. The non-adiabaticity o the change o therequency 2 =

2/2 + 2 4 sin{( os)}/( os) can be estimated by the quantity

na 2

= |2/2 + 2 cos{( os)}/( os)||2/2 + 2 4 sin{( os)}/( os)|3/2 , (3.55)

where the non-adiabatic regime corresponds to na 1. For small and we have na 1around ( os) = , where are positive integers. This corresponds to the time at whichthe Ricci scalar vanishes. Hence, each time crosses 0 during its oscillation, the non-adiabaticparticle production occurs most eciently. The presence o the mass term tends to suppress



23/161

(R) Theories 23

the non-adiabaticity parameter na, but still it is possible to satisy the condition na 1 around = 0.

For the model (3.6) it was shown in [591] that massless particles are resonantly amplied or|| 3. Massive particles with o the order o can be created or || 10. Note that inthe preheating scenario based on the model (, ) = (1/2)2

2 + (1/2)222 the parameter

decreases more rapidly ( 1/2) than that in the model (3.6) [354]. Hence, in our geometricpreheating scenario, we do not require very large initial values o [such as > (103)] to lead tothe ecient parametric resonance.

While the above discussion is based on the linear analysis, non-linear efects (such as themode-mode coupling o perturbations) can be important at the late stage o preheating (see,e.g., [354, 342]). Also the energy density o created particles afects the background cosmologicaldynamics, which works as a backreaction to the Ricci scalar. The process o the subsequentperturbative reheating stage can be afected by the explosive particle production during preheating.

It will be o interest to take into account all these efects and study how the thermalization isreached at the end o reheating. This certainly requires the detailed numerical investigation olattice simulations, as developed in [255, 254].

At the end o this section we should mention a number o interesting works about gravita-tional baryogenesis based on the interaction (1/2* )

d4

between the baryon num-ber current and the Ricci scalar (* is the cut-of scale characterizing the efective the-ory) [179, 376, 514]. This interaction can give rise to an equilibrium baryon asymmetry which isobservationally acceptable, even or the gravitational Lagrangian () = with close to 1. Itwill be o interest to extend the analysis to more general (R) gravity models.



24/161


4 Dark Energy inf

(R

) TheoriesIn this section we apply (R) theories to dark energy. Our interest is to construct viable (R)models that can realize the sequence o radiation, matter, and accelerated epochs. In this sectionwe do not attempt to nd unied models o ination and dark energy based on (R) theories.

Originally the model () = / ( > 0, > 0) was proposed to explain the late-timecosmic acceleration [113, 120, 114, 143] (see also [456, 559, 17, 223, 212, 16, 137, 62] or relatedworks). However, this model sufers rom a number o problems such as matter instability [215, 244],the instability o cosmological perturbations [146, 74, 544, 526, 251], the absence o the matterera [28, 29, 239], and the inability to satisy local gravity constraints [469, 470, 245, 233, 154, 448,134]. The main reason why this model does not work is that the quantity , 2/2 isnegative. As we will see later, the violation o the condition , > 0 gives rise to the negativemass squared 2 or the scalaron eld. Hence we require that , > 0 to avoid a tachyonic

instability. The condition , / > 0 is also required to avoid the appearance o ghosts (seeSection 7.4). Thus viable (R) dark energy models need to satisy [568]

, > 0 , , > 0 , or 0 (> 0) , (4.56)where 0 is the Ricci scalar today.

In the ollowing we shall derive other conditions or the cosmological viability o (R) models.This is based on the analysis o [26]. For the matter Lagrangian in Eq. (2.1) we take intoaccount non-relativistic matter and radiation, whose energy densities and satisy

+ 3 = 0 , (4.57)

+ 4 = 0 , (4.58)

respectively. From Eqs. (2.15) and (2.16) it ollows that

3 2 = ( )/2 3 + 2( + ) , (4.59)2 = + 2 [ + (4/3)] . (4.60)

4.1 Dynamical equations

We introduce the ollowing variables

1

, 2 6 2

, 3 62

, 4 2

3 2, (4.61)

together with the density parameters

2

3 2= 1 1 2 3 4, 4 , DE 1 + 2 + 3 . (4.62)

It is straightorward to derive the ollowing equations

d1d

= 1 3 32 + 21 13 + 4 , (4.63)d2d

=13

2(23 4 1) , (4.64)

d3d

= 13

23(3 2) , (4.65)d4d

= 234 + 1 4 , (4.66)



25/161

(R) Theories 25

where = ln is the number o e-oldings, and

d ln d ln

=,

,, (4.67)

d ln d ln

= ,

=32

. (4.68)

From Eq. (4.68) the Ricci scalar can be expressed by 3/2. Since depends on , thismeans that is a unction o , that is, = (). The CDM model, () = 2,corresponds to = 0. Hence the quantity characterizes the deviation o the backgrounddynamics rom the CDM model. A number o authors studied cosmological dynamics or specic(R) models [160, 382, 488, 252, 31, 198, 280, 72, 41, 159, 235, 1, 279, 483, 321, 432].

The efective equation o state o the system is dened by

ef 1 2/(32) , (4.69)

which is equivalent to ef = (23 1)/3. In the absence o radiation (4 = 0) the xed pointsor the above dynamical system are

1 : (1, 2, 3) = (0, 1, 2), = 0, ef = 1 , (4.70)2 : (1, 2, 3) = (1, 0, 0), = 2, ef = 1/3 , (4.71)3 : (1, 2, 3) = (1, 0, 0), = 0, ef = 1/3 , (4.72)

4 : (1, 2, 3) = (4, 5, 0), = 0, ef = 1/3 , (4.73)5 : (1, 2, 3) =

3

1 + , 1 + 4

2(1 + )2,

1 + 4

2(1 + )

, (4.74)

= 1 (7 + 10)2(1 + )2

, ef = 1 +

, (4.75)

6 : (1, 2, 3) =

2(1 )1 + 2

,1 4

(1 + 2), (1 4)(1 + )

(1 + 2)

,

= 0, ef =2 5 62

3(1 + 2). (4.76)

The points 5 and 6 are on the line () = 1 in the (, ) plane.The matter-dominated epoch ( 1 and ef 0) can be realized only by the point 5 or

close to 0. In the (, ) plane this point exists around (, ) = (1, 0). Either the point 1or 6 can be responsible or the late-time cosmic acceleration. The ormer is a de Sitter point(ef =

1) with =

2, in which case the condition (2.11) is satised. The point 6 can give rise

to the accelerated expansion (ef < 1/3) provided that > (3 1)/2, or 1/2 < < 0, or < (1 + 3)/2.

In order to analyze the stability o the above xed points it is sucient to consider only time-dependent linear perturbations () ( = 1, 2, 3) around them (see [170, 171] or the detail o suchanalysis). For the point 5 the eigenvalues or the 3 3 Jacobian matrix o perturbations are

3(1 + 5),35

5(25635 + 160

25 315 16)

45(5 + 1), (4.77)

where 5 (5) and 5 dd (5) with 5 1. In the limit that |5| 1 the lattertwo eigenvalues reduce to 3/4

1/5. For the models with 5 < 0, the solutions cannot

remain or a long time around the point 5 because o the divergent behavior o the eigenvalues



26/161


as 5

0. The model () =

/ ( > 0, > 0) alls into this category. On the otherhand, i 0 < 5 < 0.327, the latter two eigenvalues in Eq. (4.77) are complex with negative realparts. Then, provided that 5 > 1, the point 5 corresponds to a saddle point with a dampedoscillation. Hence the solutions can stay around this point or some time and nally leave or thelate-time acceleration. Then the condition or the existence o the saddle matter era is

() +0 , dd

> 1 , at = 1 . (4.78)

The rst condition implies that viable (R) models need to be close to the CDM model duringthe matter domination. This is also required or consistency with local gravity constraints, as wewill see in Section 5.

The eigenvalues or the Jacobian matrix o perturbations about the point 1 are

3, 32

25 16/12

, (4.79)

where 1 = ( = 2). This shows that the condition or the stability o the de Sitter point 1is [440, 243, 250, 26]

0 < ( = 2) 1 . (4.80)The trajectories that start rom the saddle matter point 5 satisying the condition (4.78) and thenapproach the stable de Sitter point 1 satisying the condition (4.80) are, in general, cosmologicallyviable.

One can also show that 6 is stable and accelerated or (a) 6 < 1, (

31)/2 < 6 < 1, (b)

6 > 1, 6 < (1 +

3)/2, (c) 6 > 1, 1/2 < 6 < 0, (d) 6 > 1, 6 1. Since both 5and 6 are on the line = 1, only the trajectories rom 5 > 1 to 6 < 1 are allowed

(see Figure 2). This means that only the case (a) is viable as a stable and accelerated xed point6. In this case the efective equation o state satises the condition ef > 1.From the above discussion the ollowing two classes o models are cosmologically viable.

Class A: Models that connect 5 ( 1, +0) to 1 ( = 2, 0 < 1) Class B: Models that connect 5 ( 1, +0) to 6 ( = 1, (

3 1)/2 < < 1)

From Eq. (4.56) the viable (R) dark energy models need to satisy the condition > 0, which isconsistent with the above argument.

4.2 Viable f(R) dark energy models

We present a number o viable (R) models in the (, ) plane. First we note that the CDM

model corresponds to = 0, in which case the trajectory is the straight line (i) in Figure 2. Thetrajectory (ii) in Figure 2 represents the model () = ( ) [31], which corresponds to thestraight line () = [(1 )/] + 1 in the (, ) plane. The existence o a saddle matter epochdemands the condition 1 and 1. The trajectory (iii) represents the model [26, 382]

() = ( > 0, 0 < < 1) , (4.81)which corresponds to the curve = (1 + )/. The trajectory (iv) represents the model () =( + 1)(2 + + ), in which case the late-time accelerated attractor is the point 6 with(

3 1)/2 < < 1.In [26] it was shown that needs to be close to 0 during the radiation domination as well as

the matter domination. Hence the viable (R) models are close to the CDM model in the region

0. The Ricci scalar remains positive rom the radiation era up to the present epoch, as long



27/161

(R) Theories 27

- 0 . 2 0

0 . 0

0 . 2 0

0 . 4 0

0 . 6 0

0 . 8 0

1 . 0

1 . 2

- 2 . 2 - 2 - 1 . 8 - 1 . 6 - 1 . 4 - 1 . 2 - 1 - 0 . 8

m

r

..

..

.m = - r -1

r = - 2

(i)

(ii)

(iii)

(iv)

P5

P6

P1

Figure 2: Four trajectories in the (, ) plane. Each trajectory corresponds to the models: (i) CDM,(ii) () = (), (iii) () = with > 0, 0 < < 1, and (iv) () = ( +1)(2 + + ).From [31].

as it does not oscillate around = 0. The model () = / ( > 0, > 0) is not viablebecause the condition

,> 0 is violated.

As we will see in Section 5, the local gravity constraints provide tight bounds on the deviationparameter in the region o high density ( 0), e.g., () 1015 or = 1050 [134, 596].In order to realize a large deviation rom the CDM model such as () > (0.1) today ( = 0)we require that the variable changes rapidly rom the past to the present. The (R) model givenin Eq. (4.81), or example, does not allow such a rapid variation, because evolves as (1)in the region 0. Instead, i the deviation parameter has the dependence

= ( 1) , > 1 , > 0 , (4.82)

it is possible to lead to the rapid decrease o as we go back to the past. The models that behaveas Eq. (4.82) in the regime 0 are

(A) () = (/)2

(/)2 + 1with ,, > 0 , (4.83)

(B) () =

1 (1 + 2/2) with ,, > 0 . (4.84)The models (A) and (B) have been proposed by Hu and Sawicki [306] and Starobinsky [568],respectively. Note that roughly corresponds to the order o 0 or = (1). This means that = 2 + 1 or 0. In the next section we will show that both the models (A) and (B) areconsistent with local gravity constraints or 1.

In the model (A) the ollowing relation holds at the de Sitter point:

=(1 + 2 )

2

21 (2 + 22

2)

, (4.85)



28/161


where

1/ and 1 is the Ricci scalar at the de Sitter point. The stability condition (4.80)gives [587]

24 (2 1)(2 + 4)2 + (2 1)(2 2) 0 . (4.86)The parameter has a lower bound determined by the condition (4.86). When = 1, or example,one has

3 and 83/9. Under Eq. (4.86) one can show that the conditions (4.56) are

also satised.Similarly the model (B) satises [568]

(1 + 2)+2 1 + ( + 2)2 + ( + 1)(2 + 1)4, (4.87)

with

=(1 +

2)+1

2[(1 + 2)+1

1

( + 1)2]

. (4.88)

When = 1 we have 3 and 83/9, which is the same as in the model (A). For general, however, the bounds on in the model (B) are not identical to those in the model (A).

Another model that leads to an even aster evolution o is given by [587]

(C) () = tanh (/) with , > 0 . (4.89)

A similar model was proposed by Appleby and Battye [35]. In the region the model (4.89)behaves as () [1 exp(2/)], which may be regarded as a special case o (4.82)in the limit that 1 5. The Ricci scalar at the de Sitter point is determined by , as

= cosh

2()

2 sinh() cosh()

. (4.90)

From the stability condition (4.80) we obtain

> 0.905 , > 0.920 . (4.91)

The models (A), (B) and (C) are close to the CDM model or , but the deviationrom it appears when decreases to the order o . This leaves a number o observationalsignatures such as the phantom-like equation o state o dark energy and the modied evolution omatter density perturbations. In the ollowing we discuss the dark energy equation o state in (R)models. In Section 8 we study the evolution o density perturbations and resulting observationalconsequences in detail.

4.3 Equation o state o dark energy

In order to conront viable (R) models with SN Ia observations, we rewrite Eqs. (4.59) and (4.60)as ollows:

32 = 2 ( + + DE) , (4.92)

2 = 2 [ + (4/3) + DE + DE] , (4.93)

where is some constant and

2DE (1/2)( ) 3 + 32( ) , (4.94)2DE + 2 (1/2)( ) (32 + 2)( ) . (4.95)

5 The cosmological dynamics or the model () = [1 exp(2/)] was studied in [396].



29/161

(R) Theories 29

Dening DE and DE in the above way, we nd that these satisy the usual continuity equation

DE + 3(DE + DE) = 0 . (4.96)

Note that this holds as a consequence o the Bianchi identities, as we have already mentioned inthe discussion rom Eq. (2.8) to Eq. (2.10).

The dark energy equation o state, DE DE/DE, is directly related to the one used in SN Iaobservations. From Eqs. (4.92) and (4.93) it is given by

DE = 2+ 32 + 2/3

32 2( + ) ef

1 (/) , (4.97)

where the last approximate equality is valid in the regime where the radiation density is negligiblerelative to the matter density . The viable (R) models approach the CDM model in the past,i.e., 1 as . In order to reproduce the standard matter era (32 2) or 1,we can choose = 1 in Eqs. (4.92) and (4.93). Another possible choice is = 0, where 0 isthe present value o . This choice may be suitable i the deviation o 0 rom 1 is small (asin scalar-tensor theory with a nearly massless scalar eld [583, 93]). In both cases the equationo state DE can be smaller than 1 beore reaching the de Sitter attractor [306, 31, 587, 435],while the efective equation o state ef is larger than 1. This comes rom the act that thedenominator in Eq. (4.97) becomes smaller than 1 in the presence o the matter uid. Thus (R)gravity models give rise to the phantom equation o state o dark energy without violating anystability conditions o the system. See [210, 417, 136, 13] or observational constraints on themodels (4.83) and (4.84) by using the background expansion history o the universe. Note that aslong as the late-time attractor is the de Sitter point the cosmological constant boundary crossingo ef reported in [52, 50] does not typically occur, apart rom small oscillations o ef around

the de Sitter point.There are some works that try to reconstruct the orms o (R) by using some desired orm or

the evolution o the scale actor () or the observational data o SN Ia [117, 130, 442, 191, 621, 252].We need to caution that the procedure o reconstruction does not in general guarantee the stabilityo solutions. In scalar-tensor dark energy models, or example, it is known that a singular behaviorsometimes arises at low-redshits in such a procedure [234, 271]. In addition to the act that thereconstruction method does not uniquely determine the orms o (R), the observational data othe background expansion history alone is not yet sucient to reconstruct (R) models in highprecision.

Finally we mention a number o works [115, 118, 119, 265, 319, 515, 542, 90] about the use ometric (R) gravity as dark matter instead o dark energy. In most o past works the power-law(R) model = has been used to obtain spherically symmetric solutions or galaxy clustering.In [118] it was shown that the theoretical rotation curves o spiral galaxies show good agreementwith observational data or = 1.7, while or broader samples the best-t value o the power wasound to be = 2.2 [265]. However, these values are not compatible with the bound | 1| 0 (recall that 0 > 0).

We consider a spherically symmetric body with mass , constant density (=

), radius

, and vanishing density outside the body. Since is a unction o the distance rom the centero the body, Eq. (5.1) reduces to the ollowing orm inside the body ( < ):

d2

d2 +

2

d

d 2 =

2

30 , (5.3)

whereas the r.h.s. vanishes outside the body ( > ). The solution o the perturbation orpositive 2 is given by

() = 3

+ 4

, (5.5)



31/161

(R) Theories 31

where ( = 1, 2, 3, 4) are integration constants. The requirement that ()>

0 as gives 4 = 0. The regularity condition at = 0 requires that 2 = 1. We match two solutions

(5.4) and (5.5) at = by demanding the regular behavior o () and (). Since ,

this implies that is also continuous. I the mass satises the condition 1, we obtainthe ollowing solutions

() 230

. (5.7)

As we have seen in Section 2.3, the action (2.1) in (R) gravity can be transormed to theEinstein rame action by a transormation o the metric. The Einstein rame action is given bya linear action in , where is a Ricci scalar in the new rame. The rst-order solution orthe perturbation o the metric = 0 ( + ) ollows rom the rst-order linearizedEinstein equations in the Einstein rame. This leads to the solutions 00 = 2/(0) and = 2/(0) . Including the perturbation to the quantity , the actual metric isgiven by [448]

=

+ . (5.8)Using the solution (5.7) outside the body, the (00) and () components o the metric are

00 1 + 2()ef

, 1 + 2

()ef

, (5.9)

where ()ef and are the efective gravitational coupling and the post-Newtonian parameter,

respectively, dened by

()ef

0

1 +

1

3

, 3

3 + . (5.10)

For the (R) models whose deviation rom the CDM model is small ( 1), we have2 0/[3(0)] and 8. This gives the ollowing estimate

()2 2

(0), (5.11)

where = /(0) = 42/(30) is the gravitational potential at the surace o the body.The approximation 1 used to derive Eqs. (5.6) and (5.7) corresponds to the condition

(0) . (5.12)Since 0 = ,(0), it ollows that

=,(0)

,(0) . (5.13)

The validity o the linear expansion requires that 0, which translates into (0).Since 2/(30) = 2/3 at = , one has (0) 1 under the condition(5.12). Hence the linear analysis given above is valid or (0) .

For the distance close to the post Newtonian parameter in Eq. (5.10) is given by 1/2(i.e., because 1). The tightest experimental bound on is given by [616, 83, 617]:

|

1

|< 2.3

105 , (5.14)



32/161


which comes rom the time-delay efect o the Cassini tracking or Sun. This means that (R)gravity models with the light scalaron mass ( 1) do not satisy local gravity constraints [469,470, 245, 233, 154, 448, 330, 332]. The mean density o Earth or Sun is o the order o 1 10 g/cm3, which is much larger than the present cosmological density

(0) 1029 g/cm3. In

such an environment the condition 0 is violated and the eld mass becomes large suchthat 1. The efect o the chameleon mechanism [344, 343] becomes important in this non-linear regime ( 0) [251, 306, 134, 101]. In Section 5.2 we will show that the (R) models canbe 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 (R) gravity. Unlike the linearexpansion approach given in Section 5.1, this corresponds to a non-linear efect arising rom a large

departure o the Ricci scalar rom its background value 0. The mass o an efective scalar elddegree o reedom depends on the density o its environment. I the matter density is sucientlyhigh, the eld acquires a heavy mass about the potential minimum. Meanwhile the eld has alighter mass in a low-density cosmological environment relevant to dark energy so that it canpropagate reely. As long as the spherically symmetric body has a thin-shell around its surace,the efective coupling between the eld and matter becomes much smaller than the bare coupling||. In the ollowing we shall review the chameleon mechanism or general couplings and thenproceed to constrain (R) dark energy models rom local gravity tests.

5.2.1 Field profle o the chameleon feld

The action (2.1) in (R) gravity can be transormed to the Einstein rame action ( 2.32) with thecoupling =

1/

6 between the scalaron eld = 3/(22) ln and non-relativistic matter.

Let us consider a spherically symmetric body with radius in the Einstein rame. We approximatethat the background geometry is described by the Minkowski space-time. Varying the action ( 2.32)with respect to the eld , we obtain

d2

d2+

2

d

d def

d= 0 , (5.15)

where is a distance rom the center o symmetry that is related to the distance in the Jordanrame via =

= . The efective potential ef is dened by

ef() = () + * , (5.16)

where * is a conserved quantity in the Einstein rame [343]. Recall that the eld potential () is

given in Eq. (2.33). The energy density in the Einstein rame is related with the energy density in the Jordan rame via the relation = /2 = 4. Since the conormal transormationgives rise to a coupling between matter and the eld, is not a conserved quantity. Instead thequantity * = 3 = corresponds to a conserved quantity, which satises 3* = 3.Note that Eq. (5.15) is consistent with Eq. (2.42).

In the ollowing we assume that a spherically symmetric body has a constant density * = inside the body ( < ) and that the energy density outside the body ( > ) is * = ( ). The mass o the body and the gravitational potential at the radius are givenby = (4/3)3 and = /, respectively. The efective potential has minima at theeld values and :

,() + = 0 , (5.17)

,() + = 0 . (5.18)



33/161

(R) Theories 33

The ormer corresponds to the region o high density with a heavy mass squared 2

ef,(),whereas the latter to a lower density region with a lighter mass squared 2 ef,(). In thecase o Sun, or example, the eld value is determined by the homogeneous dark matter/baryon

density in our galaxy, i.e., 1024 g/cm3.

0.0365

0.037

0.0375

0.038

0.0385

0.039

0.0395

0.04

-0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

V/(mpl2Rc

)

/mpl

de Sitter

singular

-0.056

-0.054

-0.052

-0.05

-0.048

-0.046

-0.044

-0.042

-0.04

-0.038

-0.036

-0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0

-Veff

/(mpl2Rc

)

/mpl

Figure 3: (Top) The potential () = (

)/(222) versus the eld =

3/(16)pl ln or

the Starobinskys dark energy model (4.84) with = 1 and = 2. (Bottom) The inverted efectivepotential ef or the same model parameters as the top with * = 102pl. The eld value, at whichthe inverted efective potential has a maximum, is diferent depending on the density *, see Eq. (5.22). Inthe upper panel de Sitter corresponds to the minimum o the potential, whereas singular means thatthe curvature diverges at = 0.

When > 0 the efective potential has a minimum or the models with , < 0, which occurs,e.g., or the inverse power-law potential () = 4+. The (R) gravity corresponds to anegative coupling ( = 1/6), in which case the efective potential has a minimum or , > 0.As an example, let us consider the shape o the efective potential or the models ( 4.83) and (4.84).In the region both models behave as

()

1

(/)2 . (5.19)



34/161


For this unctional orm it ollows that

= 26

= 1 2(/)(2+1) , (5.20)

() =22

4

6

1 (2 + 1)

6

22+1

. (5.21)

The r.h.s. o Eq. (5.20) is smaller than 1, so that < 0. The limit corresponds to 0.In the limit 0 one has /(22) and , . This property can be seen in theupper panel o Figure 3, which shows the potential () or the model (4.84) with parameters

= 1 and = 2. Because o the existence o the coupling term /6*, the efective potential

ef() has a minimum at

=

6

2*2+1

. (5.22)

Since 2* in the region o high density, the condition || 1 is in act justied (or and o the order o unity). The eld mass about the minimum o the efective potential isgiven by

2 =1

6( + 1)

2*

2(+1). (5.23)

This shows that, in the regime 2* , is much larger than the present Hubble param-eter 0 (

). Cosmologically the eld evolves along the instantaneous minima characterized

by Eq. (5.22) and then it approaches a de Sitter point which appears as a minimum o the potentialin the upper panel o Figure 3.

In order to solve the dynamics o the eld in Eq. (5.15), we need to consider the invertedefective potential (ef). See the lower panel o Figure 3 or illustration [which corresponds tothe model (4.84)]. We impose the

f(R) Theories

Documents

Transcript of f(R) Theories