Curvature and entropy perturbations in generalized gravity

Curvature and entropy perturbations in generalized gravity

Xiangdong Ji1,2,* and Tower Wang1,†

1Center for High-Energy Physics, Peking University, Beijing 100871, China2Maryland Center for Fundamental Physics and Department of Physics, University of Maryland, College Park, Maryland 20742, USA

(Received 10 March 2009; published 22 May 2009)

We investigate the cosmological perturbations in generalized gravity, where the Ricci scalar and a

scalar field are nonminimally coupled via an arbitrary function fð’;RÞ. In the Friedmann-Lemaıtre-

Robertson-Walker background, by studying the linear perturbation theory, we separate the scalar-type

perturbations into the curvature perturbation and the entropy perturbation, whose evolution equations are

derived. Then we apply this framework to inflation. We consider the generalized slow-roll conditions and

the quantization initial condition. Under these conditions, two special examples are studied analytically.

One example is the case with no entropy perturbation. The other example is a model with the entropy

perturbation large initially but decaying significantly after crossing the horizon.

DOI: 10.1103/PhysRevD.79.103525 PACS numbers: 98.80.Cq, 04.50.Kd

I. INTRODUCTION

In the past three decades, huge progress has been madeon our understanding of the early universe, both theoreti-cally and observationally. This is implemented by theinflation theory [1–3] merging the general relativity andquantum field theory in an elegant way. On the one hand,inflation theory has naturally explained the initial conditionof big bang cosmology. On the other hand, it also makesquantitative predictions which can be tested by preciseobservational data [4–6].

So far the prevail inflation models are based on theEinstein gravity coupled minimally to a scalar field (ormore scalar fields) [7–11], whereas considerable investi-gations were also performed on models of modified gravitywith higher derivative corrections or nonminimal coupling.Most of them can be classified into two categories:

(i) fðRÞ models without a scalar field [12,13];(ii) Fð’ÞR scalar-tensor theory.

Each of them has only 1 degree of freedom. This is clear ifone takes a conformal transformation as done by [14–16].

Our intention in this paper is to deal with a general typeof model, namely, fð’;RÞ gravity, which unifies and gen-eralizes the above relatively simpler models. The action ofthis model is of the form1

S ¼Z

d4xffiffiffiffiffiffiffi�g

p �1

2fð’;RÞ � 1

2g��@�’@�’� Vð’Þ

�:

(1)

Here the fð’;RÞ term contains a nonminimal couplingbetween the scalar field ’ and the Ricci scalar R, while

Vð’Þ is the potential of the scalar field. In principle Vð’Þcan be absorbed in fð’;RÞ, but we will keep them separate.In contrast with simpler models, the fð’;RÞ model

usually introduces another degree of freedom. Generallyspeaking, due to the new degree of freedom, there is anonvanishing entropy perturbation in most models basedon fð’;RÞ gravity. In the present work, we will distinguishthe entropy perturbation from curvature perturbation andthen study their evolutions.To make our discussion self-contained and clear in

notations, first of all, we collected some previously knownresults in Sec. II. Our general result is presented in Sec. III,where we extract the curvature perturbation and entropyperturbation as well as their evolution equations. Applyingthis formalism to inflation, we study the generalized slow-roll conditions and the quantized initial condition inSec. IV. In Sec. V, some typical examples are studied underthe slow-roll approximation. One example is the case withno entropy perturbation, including the simpler models with1 degree of freedom we mentioned above. The other ex-ample is to add a gð’ÞR2 correction to Einstein gravity.Specifically, we study the gð’Þ ¼ 1

4�’2, Vð’Þ ¼ 1

2m2’2

model in the limit M2p=’

2 � �m2’2=M2p � 1. Ignoring

the coupling of perturbations inside the Hubble horizon,we find the entropy perturbation is large at horizon cross-ing but decays significantly outside the horizon. Initiallythe correlation between the curvature perturbation and theentropy perturbation has been neglected under our approxi-mation, but at the end of inflation they become moderatelyanticorrelated. We summarize and refer to some openproblems in Sec. VI. For reference and as a support toour canonical quantized initial conditions, in Appendix Awe collect the relevant results of a two-field model which isconformally equivalent to the fð’;RÞ model. In the gen-eralized gravity, since it is difficult to draw a clear border-line between gravity and the matter, there is ambiguity indefining curvature perturbation and entropy perturbation.We present a more traditional (but less tractable) definition

*[email protected]†[email protected] this paper, we employ the reduced Planck mass

Mp ¼ 1=ffiffiffiffiffiffiffiffiffiffi8�G

pand set @ ¼ c ¼ 1.

PHYSICAL REVIEW D 79, 103525 (2009)

1550-7998=2009=79(10)=103525(17) 103525-1 � 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.79.103525

of these perturbations in Appendix B. Complementary toSec. III, details for deriving the evolution equation ofentropy perturbation are relegated to Appendix C.

II. REVIEW OF PREVIOUSLY KNOWN RESULTS

The cosmological perturbations in fð’;RÞ gravity havebeen studied actively by Hwang and Noh in [17–21]. Butmost of the investigations mainly concentrated on theorieswith 1 degree of freedom, including the fðRÞ theory and thescalar-tensor theory. For generalized fð’;RÞ theory, thecomplete evolution equations of the first order perturba-tions were obtained in [21], where the background dynam-ics were also shown. In this section, to make our discussionself-contained and clear in notation, we write down theseresults following the notations of [7], except for that themetric signature is taken to be (�;þ;þ;þ). All of theresults collected here can be found in [21,22]. Throughoutthis paper, we will focus on the scalar-type perturbations,working in the longitudinal gauge. The tensor- type per-turbation has been addressed in [20]. There was a discus-sion of scalar-type perturbations in [19], though, forgeneralized gravity, their full evolution equations wereobtained in [21]. In this section, we summarize theseresults in a self-contained way, and at the same time, setup our convention of notations. In the subsequent sections,we will take further steps to get some new results.

The background is described by the Friedmann-Lemaıtre-Robertson-Walker (FLRW) metric

ds2 ¼ �dt2 þ a2ðtÞd~x2 ¼ a2ð�Þð�d�2 þ d~x2Þ; (2)

where t is the comoving time and � is the conformal time,with respect to which the derivatives will be denoted by adot overhead and a superscript prime, respectively. Lateron, for the sake of convenience, we will also use asuperscript ‘‘�’’ to denote the derivative with respect to t.Then in terms of the Hubble parameterH ¼ _a=a, the Ricci

scalar2 can be expressed as R ¼ 6ð2H2 þ _HÞ. Throughoutthis paper, we will only deal with the flat universe. Theresults for a closed or open universe are expected to besimilar.For succinctness let us introduce the notation F ¼

@@R fð’;RÞ. In this paper we will concentrate on the case

with F > 0, but it is easy to extend our results to the caseF � 0. The fluctuation of the scalar field ’ will be denotedby �’. It is well known that the perturbations of the metriccan be decomposed into three types: the scalar type, thevector type, and the tensor type. In the scalar-driven in-flation, these three types are decoupled with each other ifwe only consider two-point correlation functions. Here weare interested in the linear order scalar-type perturbations,so we can treat them exclusively, not worrying about thevector-type and tensor-type perturbations. Consideringscalar-type perturbations only, the perturbed metric takesthe form

ds2 ¼ �ð1þ 2�Þdt2 þ 2a@iBdtdxi

þ a2½ð1� 2c Þ�ij þ 2@i@jE�dxidxj: (3)

Here �ij is the Kronecker delta function. We will mainly

work in the longitudinal gauge, that is, choosingB ¼ E ¼0. If necessary, one can easily recover all of our results intothe gauge-invariant form with the following dictionary[7,21]:

� ! �ðgiÞ ¼ �þ ½aðB� E0Þ�0a

;

c ! c ðgiÞ ¼ c � aHðB� E0Þ;�’ ! �’ðgiÞ ¼ �’þ a _’ðB� E0Þ;�F ! �FðgiÞ ¼ �Fþ a _FðB� E0Þ:

(4)

Corresponding to action (1), the variation of ’ and g�

gives

�1S ¼Z

d4xD0 þZ

d4x

� ffiffiffiffiffiffiffi�gp �

1

2f;’ � V;’

�þ @ð ffiffiffiffiffiffiffi�g

pg�@�’Þ

��’

þZ

d4x

�1

2

ffiffiffiffiffiffiffi�gp �

�g�

�1

2f� 1

2g��@�’@�’� V

�þ FR� � @�’@’þ @�Fg

�� 1

2ð@�g� þ @�g� � @g��Þ

� @�Fg�� 1

2ð@�g� þ @�g� � @g��Þ

�þ @

�1

4

ffiffiffiffiffiffiffi�gp

@�Fð2g��g� � g��g� � g�g��Þ�g��g�

��g�; (5)

where the total derivative term

D0 ¼ @�ð12ffiffiffiffiffiffiffi�g

pFg��

�Þ � @ð12ffiffiffiffiffiffiffi�g

pFg��

��Þ � @ð ffiffiffiffiffiffiffi�gp

g�@�’�’Þþ @½14

ffiffiffiffiffiffiffi�gp

@�Fð2g�g� � g��g � g�g�Þ�g��: (6)

2When defining the Ricci scalar R ¼ g�R�, we take the convention of Ricci tensor R� ¼ @�� @�

�� þ ��

��

��

with the affine connections �� ¼ 1

2 g�ð@�g þ @g� � @g�Þ.

XIANGDONG JI AND TOWER WANG PHYSICAL REVIEW D 79, 103525 (2009)

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Then the generalized Friedmann equations are simply

12 _’2 þ V � 1

2fþ 3H2Fþ 3 _HF� 3H _F ¼ 0; (7)

_’ 2 þ 2 _HF�H _Fþ €F ¼ 0: (8)

Equation (8) proves to be very useful and we will use itfrequently without mention. The nonminimal couplingfð’;RÞ brings a new term to the equation of motion forthe scalar field

€’þ 3H _’� 12f;’ þ V;’ ¼ 0: (9)

From (5), we get the Einstein equations of linear pertur-bations [21,22] in the longitudinal gauge

3Hð _c þH�Þ � 1

a2r2c ¼ � 1

2M2p

��; (10)

_c þH� ¼ � 1

2M2p

�q; (11)

c �� ¼ �F

F; (12)

€c þ 3HðH�þ _c Þ þH _�þ 2 _H�þ 1

3a2r2ð�� c Þ

¼ 1

2M2p

�p; (13)

in which ��, �q, and �p are defined as

�� ¼ M2p

F

�_’� _’þ 1

2ð�f;’ þ 2V;’Þ�’� 3H� _F

þ�3 _H þ 3H2 þ 1

a2r2

��Fþ ð3H _F� _’2Þ�

þ 3 _FðH�þ _c Þ�;

@ið�qÞ ¼ �M2p

F@ið _’�’þ � _F�H�F� _F�Þ;

�p ¼ M2p

F

�_’� _’þ 1

2ðf;’ � 2V;’Þ�’þ � €Fþ 2H� _F

þ�� _H � 3H2 � 2

3a2r2

��F� _F _�

� ð _’2 þ 2 €Fþ 2H _FÞ�� 2 _FðH�þ _c Þ�: (14)

Equations (10)–(13) follow, respectively, from the G00, G

0i ,

Gijði � jÞ,Gi

i components of Einstein equations. The equa-

tion of motion for �’ gives a redundant relation.Using Eqs. (11) and (12) to cancel �’ and �F, respec-

tively, in (10), one obtains

Fð €�þ €c Þþ�HFþ 3 _F� 2F €’

_’

�ð _�þ _c Þ

þ�ð3H _Fþ 3 €FÞ� 2 €’

_’ðHFþ 2 _FÞ� F

a2r2

��

þ�ð4 _HFþH _F� €FÞ� 2 €’

_’ðHF� _FÞ� F

a2r2

�c ¼ 0:

(15)

Inserting (11) and (12) into theGii component Eq. (13), it

will result in a relation automatically satisfied by thebackground equation (9).Remembering that �F ¼ F;R�Rþ F;’�’ and _F ¼

F;R_Rþ F;’ _’, we can write (12) in another form

Fð�� c Þ ¼ 2F;R

�3 €c þ 3H _�þ 12H _c þ 6ð2H2 þ _HÞ�

þ 1

a2r2ð�� 2c Þ

�� F;’

_’½Fð _�þ _c Þ

þ ðHFþ 2 _FÞ�þ ðHF� _FÞc �: (16)

In minimally coupled model f ¼ M2pR, Eq. (16) reduces to

the familiar relation � ¼ c .Comments are needed here. We start with five equations

(four perturbed Einstein equations and one equation ofmotion for �’) of three variables ð�’;�; c Þ, so thissystem of equations appears to be overdetermined.However, as we have just mentioned, the equation ofmotion for �’ is redundant, which can be derived fromperturbed Einstein equations. In addition, one of theremaining four equations turns out to be automaticallysatisfied by background equations. This is also understand-able if one recalls the Bianchi identity. Consequently, thereare three independent equations with three variable at last.What we have done was just use one of the equations tocancel �’ and arrive at two equations (15) and (16) withtwo variables �, c .

III. CURVATURE PERTURBATION AND ENTROPYPERTURBATION

The perturbation equations (15) and (16) were used in[21,22] to study certain fð’;RÞ models with only 1 degreeof freedom, where the effects of entropy perturbation havebeen neglected all the while. However, in the most generalcase, the entropy perturbation may play an important rolein the early stage and then decay or translate into thecurvature perturbation, leaving some observable signaturesin fluctuations of cosmic microwave background (CMB)and dark matter.3 Disregarding the entropy perturbation

3A mechanism for entropy perturbation decay into curvatureperturbation was realized in the curvaton scenario [23,24]. Infð’;RÞ gravity one may expect a similar story. It is remarkablethat by curvaton mechanism a large non-Gaussianty can begenerated [23–27].

CURVATURE AND ENTROPY PERTURBATIONS IN . . . PHYSICAL REVIEW D 79, 103525 (2009)

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would obscure many interesting phenomenological predic-tions, such as the residual entropy perturbation and thelarge non-Gaussianity. From this point of view, for gener-alized fð’;RÞ gravity, it is important to take the newdegree of freedom into consideration and study the entropyperturbation seriously.

In this section, we will rearrange the perturbation equa-tions (15) and (16) in order to decompose the scalar-typeperturbations into curvature and entropy components andto get their evolution equations. In subsequent sections,applied to inflation, the evolution dynamics of perturba-tions will be studied under the slow-roll approximation.

For our following study, it is essential to notice thatEq. (15) can be recast in the form

_R ¼�ln

_’2

2F _’2 þ 3 _F2

�� 2HFþ _F

2F _’2 þ 3 _F2�s

þ 2HFþ _F

2F _’2 þ 3 _F2

F

a2r2ð�þ c Þ; (17)

with the curvature perturbation

R ¼ 1

2ð�þ c Þ þ 2HFþ _F

2F _’2 þ 3 _F2

��Fð _�þ _c Þ þ 1

2ð2HFþ _FÞð�þ c Þ

�: (18)

The first term on the right-hand side of (17) can be taken asthe entropy perturbation,4 or more exactly, the relativeentropy perturbation [10]

S ¼ _Fð2HFþ _FÞ_’ð2F _’2 þ 3 _F2Þ

ffiffiffiffiffiffi3

2F

s�s; (19)

where

�s ¼ Fð _�þ _c Þ þ 1


þ 2F _’2 þ 3 _F2

2 _Fð�� c Þ: (20)

Strictly speaking, the second term on the right-hand side of(17) also contributes to the total entropy perturbation, but itis suppressed on the superhorizon scale [10]. Throughoutour paper, we focus on the relative entropy perturbationand refer to it as entropy perturbation for simplicity. Theadiabatic (curvature) and isocurvature (entropy) perturba-tions were investigated in [28–31] for a two-fieldLagrangian with a nonstandard kinetic term. TheLagrangian discussed there is equivalent to the fð’;RÞtheory here, up to a conformal transformation [15,18].

One can check that the curvature perturbation defined in(18) is conformally equivalent to what appeared in [28–31]. See also Appendix A. As another check, in the mini-mally coupled limit f ¼ M2

pR, the expression (18) reduces

to the familiar form R ¼ ��Hð _�þH�Þ= _H.Along the line of [32], we give an apparently more

traditional definition of curvature perturbation Reff andentropy perturbation �seff in Appendix B. At first glance,the traditional definition seems more physical. But it de-pends heavily on an artificial separation of gravity andmatter content. The choice given by (18) and (20) ismore convenient in calculation. Moreover, the curvatureand entropy perturbations at the end of inflation are notnecessarily the ones probed by astronomical observations[5,6], because they may evolve significantly after the exitof inflation, depending on the details of reheating. We willtake a pragmatic attitude and prefer the convenient defini-tion (18) and (20) here.One may still raise a question: why do we claim that R

and S defined above correspond to curvature perturbationand entropy perturbation, respectively? This doubt can beresolved by rewriting (18) and (20) into the followingform:

R ¼ c þ 2HFþ _F

2F _’2 þ 3 _F2_’�’þ 3H _F� _’2

2F _’2 þ 3 _F2�F;

�s ¼ _’2

��’

_’� �F

_F

�:

(21)

When deriving these relations, we have used Eqs. (8), (11),and (12).First of all, with (21) at hand, we can apply it to the

special limits mentioned in Sec. I: (i) fðRÞ models withouta scalar field, and (ii) Fð’ÞR scalar-tensor theory. Thesemodels are extensively studied in the past [17–21]. ForfðRÞ models, _’2 ¼ 0, thus the entropy perturbation van-ishes obviously. As for the scalar-tensor theory, since Fbecomes a function exclusively dependent of ’, we have

�F_F

¼ �’

_’; (22)

which guarantees �s ¼ 0. Now it becomes very clear thatthere is no entropy perturbation in these models. This is inagreement with the fact that there is only 1 degree offreedom in these models. On the other hand, curvatureperturbation R reduces to

R ¼(c þ H

_’ �’; for pure fðRÞ gravity without scalar field;

c þ H_F�F; for f ¼ Fð’ÞR: (23)

4Here is an ambiguity in normalizing entropy perturbation. We define the entropy perturbation by (19), but we will abuse it for �sand �~s since they are proportional to S up to background quantities. Through relations (19) and (28) they are easily traded with eachother. Our normalization of (19) is chosen to ensure that PR� ¼ PS� when perturbations cross the Hubble horizon, as given byEq. (72).


103525-4

These are exactly the ones that appeared in [21]. Ourdefinition of curvature perturbation R naturally general-izes them in a unified form.

Secondly, in favor of the small dictionary (4), one canprove that the definitions of R and �s are gauge invariant.In other words, the (B� E0) terms cancel outautomatically.

As a further check, since our definition of curvatureperturbation is gauge invariant, by choosing a special(but not longitudinal) gauge

ð2HFþ _FÞ _’�’ðcÞ þ ð3H _F� _’2Þ�FðcÞ ¼ 0; (24)

we get a neat relation

R ¼ c ðcÞ: (25)

In minimally coupled models, the same relation (25) holds

in comoving gauge or uniform density gauge �’ðcÞ ¼ 0.Here the gauge condition (24) generalizes the comovinggauge condition, so we can take it as a generalized comov-ing gauge. In fact, just as in minimal models, relation (25)has a geometric interpretation. The spatial curvature to thefirst order of perturbations is given by

ð3ÞR ¼ 4

a2c ðcÞ: (26)

Therefore, in gauge (24), the adiabatic perturbation R is

proportional to the spatial curvature ð3ÞR. This is why wegive it the name ‘‘curvature perturbation.’’According to our definition in this section, the entropy

perturbation is proportional to �’_’ � �F

_F. This is natural if

one remembers that a new scalar degree of freedom isrelated to F in the present theory. It gets dynamical whenF becomes dynamical. This form is very similar to theentropy perturbation in models with two fields, say ’1, ’2,corresponding to an entropy perturbation proportional to�’1

_’1� �’2

_’2.

We would like to emphasize that the perturbations de-fined in Appendix B make sense only if one views thefð’;RÞ theory effectively as Einstein gravity with exoticmatter contents. This viewpoint is not so reliable since thegravity has been modified in the fð’;RÞ model. Fur-thermore, what we are really interested in is the evolutionof � and c . As we will show below, the choice in thissection is powerful to study their evolution.In Appendix C, we demonstrate that the evolution of

entropy perturbation obeys the equation

€�s ¼��ln

_’2ð2F _’2 þ 3 _F2Þ_F2

�� 3H

�_�sþr2

a2�sþ

��F _’2

_F

�� þ �3 _F €’

_’� _’2 � 3

2€F

�� þ�2 €’

_’� 3 _F

2F

��3 _F €’

_’� _’2 � 3 €F

�

þ�_’2

_Fþ 3 _F

2F

��2Fð2H2 þ _HÞ þ 3 _F €’

_’� _’2 � 3 €F� F2

3F;R

� F _FF;’

2F;R _’

�

þ��

ln_’2ð2F _’2 þ 3 _F2Þ

_F2

�� 3H

��_’2 þ 3

2€F�

�F _’2

_F

�� 3 _F €’

_’

��2 _F

2F _’2 þ 3 _F2�s

þ�2F _’2

3 _Fþ F

�ln

_F2

2F _’2 þ 3 _F2

��r2

a2ð�þ c Þ: (27)

In terms of

�~s ¼ _F

_’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4F _’2 þ 6 _F2

p �s; (28)

this equation can be reexpressed as

�€~sþ 3H� _~s�r2

a2�~s ¼ 1

2

�ln

_’2ð2F _’2 þ 3 _F2Þ_F2

��32ln

_’2ð2F _’2 þ 3 _F2Þ_F2

�� 3H��ln

�_’2ð2F _’2 þ 3 _F2Þ

_F2

��~s

þ��F _’2

_F

�� þ�3 _F €’

_’� _’2 � 3

2€F

�� þ�2 €’

_’� 3 _F

2F

��3 _F €’

_’� _’2 � 3 €F

�

þ�_’2

_Fþ 3 _F

2F

��2Fð2H2 þ _HÞ þ 3 _F €’

_’� _’2 � 3 €F� F2

3F;R

� F _FF;’

2F;R _’

�

þ��

ln_’2ð2F _’2 þ 3 _F2Þ

_F2

�� 3H

��_’2 þ 3

2€F�

�F _’2

_F

�� 3 _F €’

_’

��2 _F

2F _’2 þ 3 _F2�~s

þ�2F _’2

3 _Fþ F

�ln

_F2

2F _’2 þ 3 _F2

�� _F


p r2

a2ð�þ c Þ: (29)


103525-5

At the same time, according to (17), the evolution equationof R is

_R ¼�ln

_’2

2F _’2 þ 3 _F2

�� 2 _’ð2HFþ _FÞ_F

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4F _’2 þ 6 _F2

p �~s

þ 2HFþ _F

2F _’2 þ 3 _F2

F

a2r2ð�þ c Þ: (30)

There are similar equations in nonstandard two-fieldmodels [28–31]. In principle, Eqs. (29) and (30) may beobtained by a conformal transformation from the counter-parts in [28–31]. However, having avoided the intricaciesof conformal transformation at the perturbation level, ourderivation in the fð’;RÞ frame is straightforward.Nonetheless, it is still interesting to compare the resultshere and those in [28–31] via the conformal transformation[15,18]. Such a comparison would confirm the conformalequivalence at the perturbation level.

Until now we have not made any approximation.Equations (29) and (30) determine the dynamics of entropyand curvature perturbations exactly. They are applicable invarious cosmological stages and scenarios of the FLRWuniverse. Given a concrete model, the classical evolution ofperturbations can be numerically followed utilizing theseequations. Towork in the inflation scenario and generate anappropriate spectrum of density fluctuation, we shouldconsider the generalized slow-roll conditions and the initialcondition. This is a task of the next section.

IV. SLOW-ROLL APPROXIMATION ANDQUANTIZATION

In this section we will study the perturbations under thegeneralized slow-roll approximation. First we study theclassical evolution. At the end of this section, we willdiscuss the quantization of perturbations as an initialcondition.

From Eq. (18), we know that the curvature perturbationR is fully determined by�þ c and its time derivative. Sothe dynamics of �þ c informs us the dynamics of R.Inserting (18) into (17), one obtains

1

F

�ln2F _’2 þ 3 _F2

_’2

��sþ ð €�þ €c Þ

þ�ln

aF3

2F _’2 þ 3 _F2

��ð _�þ _c Þ

þ 1

2½lnða2FÞ��

�lnð2HFþ _FÞ22F _’2 þ 3 _F2

��ð�þ c Þ

� r2

a2ð�þ c Þ ¼ 0: (31)

We observe that substituting this equation into (27) willresult in a fourth order differential equation of �þ c . Butthat equation would be rather difficult to solve directly. Inthis section, we rewrite Eqs. (27) and (31) into a tractable

form. In the next section, we will get their analyticalsolutions in some special examples.First, let us define two new variables:

uR ¼ F3=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4F _’2 þ 6 _F2

p ð�þ c Þ; uS ¼ a�~s: (32)

Similar to �þ c , the variable uR tells us the evolution ofcurvature perturbationR. Following relation (31), it obeysthe equation of motion

a _’F1=2

_F

�ln2F _’2 þ 3 _F2

_’2

��uS þ u00R �r2uR þm2

Ra2uR

¼ 0; (33)

with the ‘‘mass squared’’

m2R ¼ F3=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2F _’2 þ 3 _F2p � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2F _’2 þ 3 _F2p

F3=2

��

� 1

2

�ln2F _’2 þ 3 _F2

aF3

��ln2F _’2 þ 3 _F2

F3

��þ 1

2½lnða2FÞ��

�lnð2HFþ _FÞ22F _’2 þ 3 _F2

��: (34)

If we introduce the notation

E ¼ 2HFþ _F

F3=2; (35)

then one can easily prove that

_E ¼ � 2F _’2 þ 3 _F2

2F5=2: (36)

During the inflation stage, the Hubble parameter and theeffective energy density are almost constants. In the fol-lowing, we will study perturbation dynamics under theslow-roll approximation. For the convenience of calcula-tion, our definition of slow-roll parameters is

�1 ¼_H

H2; �2 ¼

€H

H _H; 1 ¼ €’

H _’;

2 ¼ ’:::

H €’; �1 ¼

_F

HF; �2 ¼

_E

HE;

�3 ¼€F

H _F; �4 ¼

€E

H _E; �6 ¼ E

:::

H €E;

(37)

while the slow-roll conditions are given by

j�ij � 1; j ij � 1; j�ij � 1: (38)

Be careful with the notation and sign difference betweenthe slow-roll parameters here and those in most literature.Under the slow-roll conditions, the background equations(7)–(9) are significantly simplified,

V � 12fþ 3H2F ’ 0; (39)


103525-6

_’ 2 þ 2 _HF�H _F ’ 0; (40)

3H _’� 12f;’ þ V;’ ’ 0: (41)

Notice that Eq. (41) can be consistently derived from theleading order Friedmann equations (39) and (40).Neglecting higher order terms, we have

_E ¼ � _’2

F3=2; €E ¼ 3 _F _’2 � 4F _’ €’

2F5=2;

�2 ’ �1 � 1

2�1; �4 ’ 2 1 � 3

2�1;

�6 ’ 1 � 5

2�1 þ 3�1�3 � �1 1 � 4 1 2

3�1 � 4 1

:

(42)

Under the slow-roll approximation, the Hubble parame-ter is almost constant during inflation. In terms of slow-rollparameters, the nondimensionalized mass squared for uRis

m2R

H2’ 2�1 � 1: (43)

Therefore, to leading order, the evolution equation of uR is

u00Rk þ k2uRk þm2Ra2uRk þ �uSk ¼ 0 (44)

in the Fourier space. Here we have used the notation

� ’ signð _’ÞaH ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1 � 2�1

p(45)

with signð _’Þ ¼ _’=j _’j. In other words, the sign of � de-pends on the sign of _’.

At the same time, in terms of uS , the evolution equa-tion (29) of entropy perturbation can be written to theleading order as

u00Sk þ k2uSk þm2Sa

2uSk þ �k2uRk ¼ 0; (46)

in which

m2S

H2’ 5

2�1 � 5�1 þ F

3H2F;R

þ_FF;’

2H2F;R _’� 6; (47)

�

aH¼ _F

H _’F1=2

�2 _’2

3 _Fþ

�ln

_F2

2F _’2 þ 3 _F2

��

’ signð _’Þ 23

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1 � 2�1

p: (48)

In the above, we discussed the classical perturbationsand a generalized slow-roll condition. The initial conditionis governed by a quantum theory of fluctuations. That is,we should expand action (1) to the second order withrespect to perturbations, taking the form

�2S ¼Z

d3 ~xd�

�1

2ð@�vRÞ2 þ 1

2ð@�vSÞ2 � 1

2ð@ivRÞ2

� 1

2ð@ivSÞ2 þ CJvR@�vS � VðvR; vSÞ

�: (49)

In this action the covariant variables vR and vS are linearsuperpositions of perturbations. Then the canonical quan-tization leads to the initial condition

vRk ! 1ffiffiffiffiffi2k

p e�ik�eRk; vSk ! 1ffiffiffiffiffi2k

p e�ik�eSk (50)

in the short wavelength limit k=ðaHÞ ! 1. HerefeRk; eSkg is the orthogonal basis

he�k; e�k0 i ¼ ��ðk� k0Þ; �; � ¼ R;S: (51)

From the argument in Appendix A, we believe the canoni-cal variables in fð’;RÞ gravity should be

vR ¼ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2F

2F _’2 þ 3 _F2

s

��_’�’þ 3 _Fc þ Fð _’2 � 3H _FÞ

2HFþ _Fð�þ c Þ

�

¼ a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2F

2F _’2 þ 3 _F2

s2F _’2 þ 3 _F2

2HFþ _FR;

vS ¼ffiffiffi3

paffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2F _’2 þ 3 _F2p ½ _F�’þ F _’ð�� c Þ�

¼ffiffiffi3

pa _F


p �s ¼ ffiffiffi6

puS : (52)

A detailed form of (49) and a closer analysis of canonicalquantization will be presented elsewhere.When solving Eqs. (44) and (46), the initial condition

(50) for vR and vS sets the initial condition of uR and uS.In the short wavelength limit, we find vRk ! 2u0Rk. Thus

we have

uRk ! 1

ð2kÞ3=2 ei½ð�=2Þ�k��eRk; uSk ! 1ffiffiffiffiffiffiffiffi

12kp e�ik�eSk:

(53)

One can check that this solution meets (44) and (46) well inthe limit k=ðaHÞ ! 1.

V. ANALYTICAL EXAMPLES

In Eqs. (44) and (46), it is clear that uRk and uSk arecoupled, while the coupling strength is controlled by � and�. In general the coupled equations are difficult to solveanalytically, thus numerical approaches are needed.Nevertheless, under some circumstances, analytical resultsare available. Let us focus on the case �s ¼ 0 in Sec. VA.In Sec. VB, we discuss the gð’ÞR2 model with a non-vanishing entropy perturbation. Under certain approxima-tion, we solve a special case analytically, resulting innearly scale-invariant power spectra.


103525-7

A. No relative entropy perturbation: �s ¼ 0

This is essentially the case with only 1 degree of free-dom. Therefore, when �s ¼ 0, Eq. (46) is expected to besatisfied automatically, while the dynamics of the survivaldegree of freedom is described by Eq. (44). Although wecannot give a general proof to this claim, we can check it inthe two classes of models mentioned previously: (i) fðRÞmodels without a scalar field and (ii) the Fð’ÞR scalar-tensor theory. For fðRÞ models, this is quite clear becauseuS ¼ 0 and � ¼ 0. For Fð’ÞR scalar-tensor theory, it isless obvious. But looking at Eq. (47), one may notice thatwe have assumed F;R � 0 when writing down (46) and

there is a singularity in m2S. Multiplying Eq. (46) with F;R

to eliminate the illegal singularity in m2S , we will find the

resulting equation is well defined and automaticallysatisfied.

We cannot exclude the possibility that the above claimdoes not hold in some cases, although so far we cannotcome up with such an example. In those cases, Eq. (46)gives uR ¼ 0, which also satisfies (44). But then there willbe no quantum fluctuations. So we are not interested in thispossibility even if it exists.

Hence let us assume Eq. (46) is satisfied more generallyin the cases we are interested with �s ¼ 0. Then we are leftwith one differential equation of uR,

u00Rk þ k2uRk þm2Ra2uRk ¼ 0; (54)

which is obtained by setting uS ¼ 0 in Eq. (44).In accordance with the initial condition (53), we find the

solution for (54) is

uRk ¼ � 1

4k3=2eið�1=2Þð�=2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffi��k�

pHð1Þ

ð�k�Þ

¼ Cffiffiffiz

pHð1Þ

ðzÞ;

with 2 ¼ 1

4�m2

R

H2:

(55)

Here we introduced the notation z ¼ �k�. This form ofsolution is consistent with the minimal inflation model [8].For ’ 1

2 , it gives a nearly scale-invariant power spectrum.

The coefficient C is determined by quantization condition(53). Keep in mind that in the limit z ! 1,

Hð1Þ ðzÞ !

ffiffiffiffiffiffi2

�z

sei½z�ð�=2Þ�ð�=4Þ� / e�ik�: (56)

According to inflation theory, the CMB fluctuations areseeded by primordial quantum fluctuations inside the hori-zon in the early universe. When the primordial fluctuationscrossed the Hubble horizon, they began the classical evo-lution phase. So the initial condition for classical evolutionis dictated by quantization. Here we did not study thequantization condition carefully. It would be important toput this result on firmer ground by quantizing the secondorder action directly.If the scale factor is exponentially growing a� eHt, then

there is a relation

aH ¼ � 1

�¼ k

z: (57)

Straightforward calculations give

u0Rk ¼ �Ck

� ffiffiffiz

pHð1Þ

�1ðzÞ þ1� 2

2ffiffiffiz

p Hð1Þ ðzÞ

�;

u00Rk ¼ �Ck2� ffiffiffi

zp þ 1� 42

4z3=2

�Hð1Þ

ðzÞ:(58)

Notice that a prime denotes the derivative with respect toconformal time �.In the long wavelength limit k� ! 0, for ’ 1

2 , we

obtain to the leading order of slow-roll parameters,

Rk ’ 1

4k3=2

�2HFþ _F

F

�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF

2F _’2 þ 3 _F2

s ��k�

2

�ð1=2Þ�

’ H2

k3=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF

2F _’2 þ 3 _F2

s: (59)

As a result, the power spectrum of curvature perturbation is

P R ¼ k3

2�2jRkj2 ¼ H4F

2�2ð2F _’2 þ 3 _F2Þ ; (60)

and thus the spectral index is

nR � 1 ¼ 4 _H

H2þ _F

HF� ð2F _’2 þ 3 _F2Þ�

Hð2F _’2 þ 3 _F2Þ : (61)

Recalling that in [21] Hwang and Noh gave the corre-sponding result

nR � 1 ’(4 _HH2 þ _F

HF � 2 €FH _F

; for pure fðRÞ gravity without scalar field;4 _HH2 þ _F

HF � ð2F _’2þ3 _F2Þ�Hð2F _’2þ3 _F2Þ ; for f ¼ Fð’ÞR: (62)

Obviously the final results (61) and (62) are perfectly matched. Notice that for the case of pure fðRÞ gravity, we can recover(62) from (61) by setting _’ ¼ 0.

In the leading order, we also find nR � 1 ¼ 1� 2. This indicates _Rk=ðHRkÞ ’ 0 for long wavelength perturbations.That is to say, under the generalized slow-roll approximation, the curvature perturbation is almost conserved outsidethe horizon. On the one hand, this extends the previous result in [21] to the general case �s ¼ 0 [and Eq. (46) satisfied


103525-8

automatically]. On the other hand, to get richer phe-nomena, we should take the entropy perturbation intoaccount and solve evolution equations (44) and (46) moregenerally.

B. gð’ÞR2 correction

A relatively simple but nontrivial model with nonvanish-ing entropy perturbation is to consider the R2 correctionwith a ’-dependent coefficient,

fð’;RÞ ¼ M2pRþ gð’ÞR2: (63)

For this class of model, under the slow-roll approxima-tion, Eqs. (39)–(41) become

V ’ 3M2pH

2; _’2 þ 2 _HF�H _F ’ 0;

3H _’ ’ 72g;’H4 � V;’:

(64)

Then we get the following relations:

F ¼ M2p þ 2gR ’ M2

p þ 8gV

M2p

;

V;’ _’ ’ 6M2pH _H ¼ 2HV�1;

V;’ €’þ V;’’ _’2 ’ 6M2pðH €H þ _H2Þ;

3H €’þ 3 _H _’ ’�8g;’V

2

M4p

� V;’

�;’

_’;

3H’:::þ 6 _H €’þ3 €H _’ ’

�8g;’V

2

M4p

� V;’

�;’€’

þ�8g;’V

2

M4p

� V;’

�;’’

_’2:

(65)

The slow-roll parameters (37) can be expressed in terms ofg and V and their derivatives with respect to ’,

�1 ¼_H

H2’ 4g;’V;’

M2p

�M2pV

2;’

2V2;

1 ¼ €’

H _’’ 8g;’’V

M2p

þ12g;’V;’

M2p

�M2pV;’’

VþM2

pV2;’

2V2;

�1 ¼_F

HF’ 16�1VðgVÞ;’V;’ðM4

pþ8gVÞ ; �2 ¼_E

HE’ �1�1

2�1;

�3 ¼€F

H _F¼ 1þ

2�1VðgVÞ;’’V;’ðgVÞ;’ ; �4 ¼

€E

H _E’ 2 1�3

2�1;

�2 ¼€H

H _H’ 1��1þ

2�1VV;’’

V2;’

;

2 ¼ ’:::

H €’’ 1��1� �1 €H

1H _Hþ2�1M

2p

1V;’

�8g;’V

2

M4p

�V;’

�;’’

;

�6 ¼ E:::

HE:::’ 1�5

2�1þ3�1�3��1 1�4 1 2

3�1�4 1

: (66)

Subsequently, the coefficients in Eqs. (44) and (46) are

m2R ’ H2ð2�1 � 1Þ;� ’ signð _’ÞaH ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�1 � 2�1p

;

� ’ signð _’Þ 23aH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1 � 2�1

p;

m2S ’ H2

�5

2�1 � 3�1 þ

M4p

2gV� 2þ 48g;’ðgVÞ;’

gM2p

�:

(67)

The above results are still too complicated to make somesense. Particularly, since there are so many slow-roll pa-rameters, one may even worry about whether the slow-rollconditions (38) can be satisfied simultaneously. However,for the special case5

gð’Þ ¼ 14�’

2; Vð’Þ ¼ 12m

2’2; (68)

the slow-roll parameters take a much simpler form asfollows:

�1 ¼_H

H2’ 2�m2’2

M2p

� 2M2p

’2; 1 ¼ €’

H _’’ 8�m2’2

M2p

;

�1 ¼_F

HF’ 4�1�m

2’4

M4p þ �m2’4

; �2 ¼_E

HE’ �1 � 1

2�1;

�3 ¼€F

H _F¼ 1 þ 3�1; �4 ¼

€E

H _E’ 2 1 � 3

2�1;

�2 ¼€H

H _H’ 1; 2 ¼ ’

:::

H €’’ 1 þ 3�1;

�6 ¼ E:::

H €E’ 3�1 þ 2 1 � 5

2�1 þ �1 1

4 1 � 3�1

: (69)

We have been concentrating on the case F > 0, so in thisexample we limit our attention to the case with � > 0.Obviously the slow-roll conditions (38) are satisfiedwhen M2

p � ’2 � M2p=ð�m2Þ. To meet this condition

we should fine-tune the coupling constants to be very small�m2 � 1. This is the large field inflation. Although thevalue of field’ is larger than the Planck mass, thanks to thesmall coupling constants, its energy density is still less thanthe Planck energy density.In the above special case, if we further assume

M2p=’

2 � �m2’2=M2p � 1 during inflation, then we

will find

�1 ’ 2�m2’2

M2p

; 1 ’ �1 ’ 4�1;

m2R ’ �2�1H

2; � ’ signð _’ÞaH ffiffiffiffiffiffiffiffi2�1

p;

m2S ’ ð31�1 � 2ÞH2; � ’ signð _’Þ 2

3aH

ffiffiffiffiffiffiffiffi2�1

p:

(70)

5As a matter of fact, a model with R2’2 correction wasdiscussed in [33].


103525-9

The model with coefficients (70) is relatively simple. Letus discuss it in some detail.

Again, due to the nonvanishing � and �, the interactionterms form an obstacle to our analytical study. However, itis still interesting to make some rough estimates by ne-glecting these interactions before the horizon crossing/Hubble exit. The problem is akin to the one we met ininflation of a coupled multiple field. In fact, there is anexcellent analysis of coupling effects in [34]. It turned outif the coupling terms are of slow-roll order, then they willgive a correction of slow-roll order compared to the leadingorder contribution. In our case, the coefficients of couplingterms are of orderOð ffiffiffiffiffi

�1p Þ, so we expect their corrections to

the power spectra are suppressed by Oð ffiffiffiffiffi�1

p Þ.Our scheme is taking limitM2

p=’2 � �m2’2=M2

p � 1,

and disregarding the interaction terms related to � and �inside the horizon k=ðaHÞ * 1. In accordance with initialcondition (53), we get an analytical solution to Eqs. (44)and (46),

uRk ¼ � 1

4k3=2eið1�ð1=2ÞÞ�=2 ffiffiffiffiffiffiffiffiffiffiffiffiffi��k�

pHð1Þ

1ð�k�ÞeRk;

with 21 ¼

1

4�m2

R

H2;

uSk ¼ � 1

2ffiffiffiffiffi6k

p eið2�ð3=2ÞÞ�=2 ffiffiffiffiffiffiffiffiffiffiffiffiffi��k�p

Hð1Þ2ð�k�ÞeSk;

with 22 ¼

1

4�m2

S

H2: (71)

From this solution, we find at the time of the horizoncrossing, the power spectra of curvature and entropy per-turbations are nearly scale invariant,

PR� ¼ k3

2�2jRk�j2 ’ H4

4�2 _’2

��’ M2

p

96�2�2m2’4�;

PS� ¼ k3

2�2jSk�j2 ’ H4

4�2 _’2

��’ M2

p

96�2�2m2’4�;

(72)

while their cross correlation is negligible,6

P C� ¼ k3

2�2hRk�;Sk�i ’ 0: (73)

Therefore the spectral indices at the horizon crossing are

nR� � 1 ¼ nS� � 1 ¼ 4�1� � 2 1� ¼ �4�1�: (74)

The variables with an asterisk subscript take their values atthe horizon-crossing time k ¼ aH. Note we have chosenthe normalization of S in (19) so that PR� ¼ PS�.

We would like to pause here and comment on the scaleinvariance of the entropy perturbation. If readers gothrough our calculation carefully, they would find the limitM2

p=’2 � �m2’2=M2

p � 1 is of key importance in mak-

ing PS� scale invariant. In this limit, theM4p=ð2gVÞ term is

small, hence m2S=H

2 ’ �2 and subsequently 2 ’ 3=2. IfM4

p=ð2gVÞ is not small enough, we cannot get such a value

of 2 and then the entropy power spectrum will not be scaleinvariant. Even in the limit we have taken, there are subtle-ties with the M4

p=ð2gVÞ term. If M4p=ð2gVÞ �Oð�1Þ, we

should pick it up when calculating m2S . However, for

simplicity of our following estimation, when writing down(70), we have assumed M4

p=ð2gVÞ � �1.

If one considers interactions outside the horizon, itproves useful to describe the evolution of perturbations interms of a general transfer matrix

RS

� �¼ 1 TRS

0 TSS

� �RS

� ��; (75)

which gives the power spectra of scalar-type perturbationsat the end of inflation

P R ¼ PR� þ T2RSPS�; PS ¼ T2

SSPS�;

PC ¼ TRSTSSPS�:(76)

These are the power spectra probed by astronomical ob-servation [5,6], unless they changed significantly from theend of inflation to the matter-radiation decoupling.Although the cross-correlation power spectrum is negli-

gibly small when the perturbations cross the Hubble hori-zon, it may still be large at the end of inflation. This is

usually evaluated by the cross-correlation coefficient ~� [9][here we use a tilde to distinguish it from the notation thatappeared in Eq. (44)] or the correlation angle � [11]:

~� ¼ cos� ¼ PCffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPRPS

p : (77)

In terms of the entropy-to-curvature ratio

rS ¼ PS

PR; (78)

it is simplified by the transfer relation (76),

cos� ¼ signðTSSÞTRS

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirS�

1þ T2RSrS�

s¼ signðTSSÞTRSffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ T2RS

q :

(79)

Here we have used the notation signðTSSÞ ¼ TSS=jTSSjand the fact rS� ’ 1. Following the line of [34], we expectthe coupling terms inside the horizon would introduceOð ffiffiffiffiffi

�1p Þ corrections to rS�.As a matter of fact, well after the Hubble exit k=ðaHÞ �

1, the interaction term �k2uRk in Eq. (46) is negligible;thus the entropy perturbation evolves independently out-

6This is because we have neglected the coupling terms in (44)and (46) before the Hubble crossing. The fact is, when takingcoupling terms into consideration, we expect the cross correla-tion is not negligible here, PC�=PS� �Oð ffiffiffiffiffi

�1p Þ. But to get the

explicit value of it, one should either perform the higher ordercalculation or take a numerical method.


103525-10

side the horizon. As usually done in the literature, whendealing with the evolution equation of entropy perturba-

tion, one can ignore the second order term (i.e., €Sk term)and terms proportional to k2. Recalling relations (19) and(52)

uS ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFð2F _’2 þ 3 _F2Þpffiffiffi3

p ð2HFþ _FÞ S; (80)

from Eq. (46) we can get the evolution equation of S to theleading order,

_S k ¼ �H

3

�m2

S

H2þ 2þ �1 � 9

4�1 � 3�2 þ 3

2�4

�Sk

¼ �SHSk; (81)

which is a first order differential equation because thesecond order term has been ignored. This equation, to-gether with (17) outside the horizon

_Rk ¼ _’_F

ffiffiffiffiffiffi2F

3

s �ln

_’2

2F _’2 þ 3 _F2

��Sk

¼ signð _’Þ�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð�1 � 2�1Þ

3

s �2 1 � 5

2�1 � �4

�HSk

¼ �RHSk; (82)

dictates the transfer matrix (75). When evaluating m2S in

Eq. (81), we should keep the slow-roll order quantities in(47) because �S is of the slow-roll order. Taking �S and�R as their average values between the horizon exit andthe end of inflation, we can quickly write down the solutionfor (81) and (82):

Sk ¼ Sk� exp�Z t

t��SHdt

�¼ Sk�e�SðN��NÞ;

Rk �Rk� ¼Z t

t��RHSkdt ¼

Z �R

�SSk�de

�R

�SdN

¼ �R

�SSk�½e�S ðN��NÞ � 1�; (83)

in which N ¼ ln½aend=aðtÞ� stands for the e-folding num-ber from time t to the end of inflation. Using this solution

one may check that €Sk=ðH _SkÞ �Oð�1Þ. So it was reason-

able for us to ignore the €Sk term.With the above results and formulas, it is not hard to

obtain the power spectra at the end of inflation:

PR ’ PR� þ PS��2

R

�2S

½e�SðN��NÞ � 1�2;

PS ’ PS�e2�S ðN��NÞ;

PC ’ PS��R

�Se�SðN��NÞ½e�SðN��NÞ � 1�:

(84)

The spectral indices at the end of inflation are

nR � 1 ¼ nS� � 1� 2�2R�Se

�S ðN��NÞ½e�SðN��NÞ � 1��2

S þ�2R½e�SðN��NÞ � 1�2 ;

nS � 1 ¼ nS� � 1� 2�S ;

nC � 1 ¼ nS� � 1��S½2e�SðN��NÞ � 1�e�S ðN��NÞ � 1

:

(85)

In model (68) withM2p=’

2 � �m2’2=M2p � 1, we find

3H _’> 0, so signð _’Þ ¼ 1. Under the slow-roll approxima-tion, we get

�R ’ �ffiffiffiffiffiffiffiffi4�13

s; �S ’ � 29

3�1; rS� ’ 1: (86)

If we estimate �R and �R with their values at the Hubbleexit, the final result will be very simple (taking N ¼ 0 atthe end of inflation):

PR

PS�’ 1þ 12

841�1�ðe�29�1�N�=3 � 1Þ2;

PS

PS�’ e�58�1�N�=3;

PC

PS�’ 2

29

ffiffiffiffiffiffiffi3

�1�

se�29�1�N�=3ðe�29�1�N�=3 � 1Þ;

nR � 1 ¼ �4�1�;

nS � 1 ¼ 46

3�1�;

nC � 1 ¼ 17

3�1�:

(87)

At the end of inflation, the curvature perturbation and theentropy perturbation are moderately anticorrelated, withthe cross-correlation coefficient

~� ¼ cos� ’ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

12

841�1� þ 12

s; (88)

but the entropy-to-curvature ratio is small enough,

rS ’ 841�1�e�58�1�N�=3

841�1� þ 12: (89)

The evolution of the power spectra of the curvature andentropy perturbations and their correlation have beenshown in Fig. 1. We can see the power spectrum of thecurvature perturbation, denoted by the solid blue line, al-most doubled from the horizon crossing (atN� � N ’ 0) tothe end of inflation (at N� � N ’ 60). Although the en-tropy perturbation (denoted by the dashed purple line) waslager than the curvature perturbation at the Hubble cross-ing, it was decaying exponentially with respect to N� � Nin the superhorizon scale. The correlation between thecurvature perturbation and the entropy perturbation is de-picted by the dot-dashed brown line. After crossing out the


103525-11

horizon, it kept a negative value, with its amplitude at firstincreasing and then decreasing.

In Fig. 2, we illustrate the dependence of the correlationcoefficient (top graph), the logarithm of entropy-to-curvature ratio (middle graph), and the tensor-to-scalarratio (bottom graph) on N� � N. It is clear that theentropy-to-curvature ratio dropped down quickly outsidethe horizon. The entropy perturbation and the curvatureperturbation was totally uncorrelated (under the decoupledapproximation) at the Hubble exit but evolved to be mod-erately anticorrelated at the end of inflation.

At the end of inflation, if one assumes nR � 1 ’ �0:04,then rS ’ 4� 10�6. This is well below the upper boundput by the five-year Wilkinson Microwave AnisotropyProbe (WMAP) [5]. Of course, we cannot take thesenumbers seriously, because our above analytical resultsare meaningful only for estimation. The coupling betweenentropy perturbation and curvature perturbation has beensystematically ignored until the horizon crossing. In addi-tion we have estimated �R and �R with their Hubble-exitvalues. Remember that in writing down (70) we haveneglected the M4

p=ð2gVÞ term. This may also have intro-

duced some uncertainties if M4p=ð2gVÞ �Oð�1Þ. To get a

robust conclusion, one should carefully take all of theseeffects into account, resorting to the numerical method.

Another issue of our analysis is the special limit we havetaken: M2

p=’2 � �m2’2=M2

p � 1. Taking such a limit is

twofold. On the one hand, it enables us to get a nearlyscale-invariant entropy perturbation. On the other hand, wefind 3H _’ ’ m2’ð�m2’4 �M4

pÞ=M4p > 0 in this limit;

hence the scalar field was not rolling down but rolling upthe potential Vð’Þ. Also the Hubble parameter is forced togrow (�1 > 0). The same thing also happens in ‘‘phantom’’inflation (single-field inflation with a kinetic term of the

wrong sign), see, e.g., [35]. In our model, as the inflaton ’grows, the slow-roll parameter �1 tends to order unity andthe inflation is expected to cease then. It is still unclear howto terminate inflation and trigger reheating in this model.We leave it as an open problem for future investigation. Amore interesting problem is to get a more realistic, slow-roll model.According to the result in [21], to the leading order, the

power spectrum of tensor-type perturbation in fð’;RÞgravity is

P T ’ H2

2�2F¼ PS�ð2�1 � 4�1Þ: (90)

10 20 30 40 50 60 70N N

0.5

0.0

0.5

1.0

1.5

2.0

2.5

PR,S,C PS

FIG. 1 (color online). The evolutions of power spectra withrespect to e-folding number N� � N after crossing the horizon.The curvature power spectrum PR is signified by a solid blueline. The entropy power spectrum PS is denoted by a dashedpurple line. The cross-correlation power spectrum PC is depictedby a dot-dashed brown line. All of the power spectra arenormalized by PS�, the entropy power spectrum at the horizoncrossing. The vertical dotted black line corresponds to N� �N ¼ 60.

10 20 30 40 50 60 70N N

1.0

0.8

0.6

0.4

0.2

0.0

cos

10 20 30 40 50 60 70N N

6

4

2

0

log10 rS

0 10 20 30 40 50 60 70 N N0.00

0.01

0.02

0.03

0.04

0.05

rT

FIG. 2 (color online). The evolutions of the correlation coef-ficient cos� (top graph), entropy-to-curvature ratio rS (its loga-rithm, middle graph), and tensor-to-scalar ratio rT (bottomgraph) with respect to e-folding number N� � N after crossingthe horizon. The vertical dotted black lines correspond to N� �N ¼ 60. The horizontal dotted black line corresponds to cos� ¼�1, that is, the totally anticorrelated situation.


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It is conserved after the Hubble exit, with a spectral index

nT ’ 2 _H

H2� _F

HF¼ 2�1 � �1: (91)

For the specific model (68), in the limit M2p=’

2 ��m2’2=M2

p � 1, they are

P T

PS�’ 4�1�; nT ’ �2�1�: (92)

In our case, the tensor spectrum is red tilted, which isdifferent from phantom inflation. One would have noticedthat the entropy-to-scalar ratio is small at the end ofinflation,

rT ¼ P T

PR’ 3364�21�841�1� þ 12

’ 0:02: (93)

Its evolution curve outside the horizon is plotted in thelower graph of Fig. 2.

VI. SUMMARY

We investigated the cosmological perturbations in gen-eralized gravity, where the Ricci scalar and a scalar fieldare coupled nonminimally by an arbitrary function fð’;RÞ,with the Einstein gravity as a special limit. This generalform unifies the usual modified gravity [36] and scalar-tensor theory, but often introduces an additional degree offreedom. In the FLRW background, by studying the firstorder perturbation theory, we decomposed the scalar-typeperturbations into the curvature perturbation and the en-tropy perturbation, whose evolution equations are ob-tained. The effects of entropy perturbation in this class ofmodel were seldom studied in the past.

Then we applied this framework to inflation theory. Theslow-roll conditions and the quantized initial condition arediscussed. The quantization of second order action willappear in a future work, which will put our discussion atthe end of sec. IV on firmer ground.

Analytically we studied two special examples: the �s ¼0 models in Sec. VA and the gð’ÞR2 model in Sec. VB. Inthe former example we unified the previous known resultsin a unique form and extended them to the general case.The latter example is more interesting. In the modelgð’Þ ¼ 1

4�’2, Vð’Þ ¼ 1

2m2’2, we paid attention to the

limit M2p=’

2 � �m2’2=M2p � 1. This limit helps us to

get a nearly scale-invariant entropy perturbation. Althoughthe entropy perturbation was large at the Hubble exit, itdecreased significantly outside the horizon. At the end ofinflation, the ‘‘entropy-to-curvature ratio’’ we defined by(78) was of order 10�6, well below the constraint of five-year WMAP data.

In spite of the above results, there are still some impor-tant problems unsolved. First, the quantization should beperformed to confirm the normalization in (53) directly.Second, to get robust conclusions and richer phenomena,

maybe one has to employ a numerical method to solveevolution equations (44) and (46) more generally. Third,the formalism we developed in Sec. III can be applied toother cosmological stages and scenarios, such as the pre-heating stage and the curvaton scenario, etc. In [37–40],different schemes of modified gravity were investigatedmainly concerning their effects on the late time evolutionof our universe; the possible effect of fð’;RÞ gravity on thelate universe is still waiting for us to study.

ACKNOWLEDGMENTS

We are grateful to Bin Chen and Miao Li for reading apreliminary version of this manuscript. We also thank Yi-Fu Cai, Xian Gao, Yi Wang, Zhi-Bo Xu, and Wei Xue foruseful discussions.

APPENDIX A: TWO-FIELD MODEL ANDCONFORMAL TRANSFORMATION

For comparison, let us collect here the well-knownresults for two-field inflation with a nonstandard kineticterm, which is conformally equivalent to the fð’;RÞ gen-eralized gravity [14–16,21]. Usually the fð’;RÞ formulismis called the Jordan frame while the nonstandard two-fieldformulism is called the Einstein fame. In this Appendix, weuse the super/subscripts J and E to distinguish differentframes. The action two-field inflation with a nonstandardkinetic term in the Einstein fame is

S ¼Z

d4xffiffiffiffiffiffiffiffiffiffi�gE

p �1

16�GRE � 1

2g��E @E��@

E��

� 1

2e2bð�Þg��E @E�’@

E�’� Vð’;�Þ

�: (A1)

Reference [28] has an original and clear discussion on thecurvature and entropy perturbations in this model. Thereaders may get details there. In a further study [10], itwas clarified that the total entropy perturbation includesnot only the relative entropy perturbation, but also anentropy perturbation proportional to k2=a2H2. Althoughit will give a correction at the scale k� aH, its contribu-tion to entropy perturbation is suppressed in a long wave-length, so we do not consider its effects in ourinvestigation.In the longitudinal gauge, the canonical variable in this

frame is given by

v� ¼ aE

��þ @Et �

HE

c E

�;

v’ ¼ aEeb

��’þ @Et ’

HE

c E

�:

(A2)

They are related to the canonical variables for curvatureand entropy perturbations


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vR ¼ @Et �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið@Et �Þ2 þ e2bð@Et ’Þ2p v�

þ eb@Et ’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið@Et �Þ2 þ e2bð@Et ’Þ2p v’

¼ aEffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið@Et �Þ2 þ e2bð@Et ’Þ2

pHE

R;

vS ¼ @Et �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið@Et �Þ2 þ e2bð@Et ’Þ2p v’

� eb@Et ’ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið@Et �Þ2 þ e2bð@Et ’Þ2p v�

¼ aEffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið@Et �Þ2 þ e2bð@Et ’Þ2

pHE

S:

(A3)

Making use of the conformal transformation

gE� ¼ F

M2p

gJ� (A4)

and the identification

�

Mp¼

ffiffiffi3

2

sln

F

M2p

; (A5)

one can prove the following relations [15,18,21]:

aE ¼ffiffiffiffiF

pMp

aJ;

dtE ¼ffiffiffiffiF

pMp

dtJ;

HE ¼ Mpð2HJFþ @Jt FÞ2F3=2

;

c E ¼ 1

2ð�J þ c JÞ;

�� ¼ffiffiffi3

2

sMpðc J ��JÞ:

(A6)

If we take

bð�Þ ¼ ��=ð ffiffiffi6

pMpÞ;

Vð�;�Þ ¼ M4p

2F2½RJFð’;RJÞ � fð’;RJÞ�;

(A7)

then action (1) can be perfectly reproduced from action(A1). For details of derivation, see Ref. [15]. In theseframes, the conformal times are coincident d�E ¼ d�J,so the canonical quantized variables (A3) are exactlyequivalent to (52) by the above conformal transformation.This conformal equivalence is powerful. Given a specificfunction fð’;RJÞ, one can know the detailed form of

Vð’;�Þ using F ¼ @@RJ

fð’;RJÞ and (A5), and then do

our job in the more familiar Einstein frame.7

APPENDIX B: TRADITIONAL DEFINITION OFCURVATURE AND ENTROPY PERTURBATIONS

In all of our discussion, we take the curvature andentropy perturbations as defined in (18) and (20). Butsuch a definition is different from the traditional one. Ifwe regard the fð’;RÞ theory as the Einstein gravity withexotic matter contents induced by the nonminimal cou-pling and follow the spirit of [32], then we arrive at atraditional form of curvature and entropy perturbations.Let us elaborate a little on this point. One should keep inmind that this point of view is different from that inAppendix A. Although we also use the term ‘‘Einsteingravity’’ here, it does not mean the Einstein frame.Formally, Friedmann equations (7) and (8) can be re-

written as

H2 ¼ 1

3M2p

�; _H ¼ � 1

2M2p

ð�þ pÞ; (B1)

with

� ¼ M2p

F

�1

2_’2 þ V þ RF� f

2� 3H _F

�;

p ¼ M2p

F

�1

2_’2 � V � RF� f

2þ €Fþ 2H _F

�:

(B2)

Here in the effective energy density and pressure we havereckoned the contribution of the nonminimal coupling. Ourfinal result seriously depends on this trick. The perturba-tions obey Eqs. (10)–(13). Then the comoving curvatureperturbation is given by

R eff ¼ c � H

�þ p�q ¼ c �H

_Hð _c þH�Þ: (B3)

The curvature perturbation on the uniform density hyper-surface is well defined,

�eff ¼ �c �H

_�� ¼ �c þH

_Hð _c þH�Þ � 1

3 _H

r2

a2c :

(B4)

At the same time, the entropy perturbation �seff is definedby

T�seff ¼ �p� c2s��; c2s ¼ @p

@�¼ _p

_�¼ � 3H _H þ €H

3H _H:

(B5)

It would be interesting to notice that

7We thank the referee for an emphasis on this point.


103525-14

�eff þReff ¼2M2

p

3ð�þ pÞr2

a2c : (B6)

Finally, one can quickly prove

� _Reff ¼ H_H

�€c þH _�� €H

_Hð _c þH�Þ þ 2 _H�

�; (B7)

and

� _H

H_Reff ¼ 1

2M2p

ð�p� c2s��Þ þ c2sa2

r2c

þ 1

3a2r2ðc ��Þ

¼ 1

2M2p

T�seff þ c2sa2

r2c þ 1

3a2r2ðc ��Þ:

(B8)

Now we see that the curvature perturbation Reff andentropy perturbation �seff are related in the traditionalmanner. However, the full expression of �seff is rathercomplicated, accordingly its evolution equation is evenmore difficult to understand. Therefore, the perturbationspresented in this Appendix are not convenient in studyingthe generalized gravity. Although the definition here ap-pears very natural if one takes fð’;RÞ gravity as an effec-tive Einstein theory, a more convenient choice should be(18) and (20).

APPENDIX C: EVOLUTION OF ENTROPYPERTURBATION

From (18) and (20), we get

Fð _�þ _c Þ ¼ � 1


þ 2F _’2 þ 3 _F2

2HFþ _F

�R� 1

2ð�þ c Þ

�; (C1)

�� c ¼ 2 _F

2F _’2 þ 3 _F2�sþ 2 _F

2HFþ _F

�1

2ð�þ c Þ�R

�:

(C2)

Now Eqs. (15) and (16) can be rewritten in the form

Fð €�þ €c Þ ¼�2F €’

_’�HF� 3 _F

�ð _�þ _c Þ

þ�€’

_’ð2HFþ _FÞ � ð2HFþ _FÞ�

�ð�þ c Þ

þ�3 _F €’

_’� _’2 � 3 €F

�ð�� c Þ

þ F

a2r2ð�þ c Þ; (C3)

Fð €�� €c Þ ¼�4HF� 3 _Fþ 2F €’

_’� F2F;’

3F;R _’

�ð _�þ _c Þ

� 3HFð _�� _c Þ þ�2Fð2H2 þ _HÞ

� ð2HFþ _FÞ� þ ð2HFþ _FÞ�€’

_’� FF;’

6F;R _’

��

� ð�þ c Þ þ�2Fð2H2 þ _HÞ þ 3 _F €’

_’

� _’2 � 3 €F� F2

3F;R

� F _FF;’

2F;R _’

�ð�� c Þ

þ 2F

3a2r2ð�þ c Þ þ F

a2ð�� c Þ: (C4)

Differentiating (20) with respect to time and making use of(C3), one obtains

_�s ¼�2F €’

_’� 3

2_F

�ð _�þ _c Þ þ 2F _’2 þ 3 _F2

2 _Fð _�� _c Þ

þ�€’

_’ð2HFþ _FÞ � 1

2ð2HFþ _FÞ�

�ð�þ c Þ

þ��

F _’2

_F

�� þ 3 _F €’

_’� _’2 � 3

2€F

�ð�� c Þ

þ F

a2r2ð�þ c Þ; (C5)

which gives

2F _’2 þ 3 _F2

2 _Fð _�� _c Þ ¼ _�sþ

�3

2_F� 2F €’

_’

�ð _�þ _c Þ þ

�1

2ð2HFþ _FÞ� � €’

_’ð2HFþ _FÞ

�ð�þ c Þ

þ�_’2 þ 3

2€F�

�F _’2

_F

�� 3 _F €’

_’

�ð�� c Þ � F

a2r2ð�þ c Þ: (C6)

Taking the derivative of (C5) once more, we have


103525-15

€�s ¼�2F €’

_’� 3

2_F

�ð €�þ €c Þ þ 2F _’2 þ 3 _F2

2 _Fð €�� €c Þ þ

��2F €’

_’

�� þ €’

_’ð2HFþ _FÞ � ðHFþ 2 _FÞ�

�ð _�þ _c Þ

þ��

2F _’2

_F

�� þ 3 _F €’

_’� _’2

�ð _�� _c Þ þ

��€’

_’ð2HFþ _FÞ

�� 1

2ð2HFþ _FÞ��

�ð�þ c Þ

þ��

F _’2

_F

�� þ�3 _F €’

_’� _’2 � 3

2€F

��ð�� c Þ þ F

a2r2ð _�þ _c Þ þ ð _F� 2HFÞ r

2

a2ð�þ c Þ: (C7)

Substituting (C3), (C4), and (C6), (C1) and (C2) into (C7) step by step to eliminate €�þ €c and €�� €c , _�� _c , _�þ _c ,and �� c , respectively, one will arrive at a rather scattering equation:

€�s ¼��ln

_’2ð2F _’2 þ 3 _F2Þ_F2

�� 3H

�_�sþr2

a2�sþ

��€’

_’ð2HFþ _FÞ

�� 1

2ð2HFþ _FÞ��

þ�2 €’

_’� 3 _F

2F

��€’

_’ð2HFþ _FÞ � ð2HFþ _FÞ�

�þ

�_’2

_Fþ 3 _F

2F

��2Fð2H2 þ _HÞ � ð2HFþ _FÞ� þ ð2HFþ _FÞ

��€’

_’� FF;’

6F;R _’

��þ

��ln

_’2ð2F _’2 þ 3 _F2Þ_F2

�� 3H

��1

2ð2HFþ _FÞ� � €’

_’ð2HFþ _FÞ

��ð�þ c Þ

þ��2F €’

_’

�� þ €’

_’ð2HFþ _FÞ � ðHFþ 2 _FÞ� þ

�2 €’

_’� 3 _F

2F

��2F €’

_’�HF� 3 _F

�

þ�_’2

_Fþ 3 _F

2F

��4HF� 3 _Fþ 2F €’

_’� F2F;’

3F;R _’

�þ

��ln

_’2ð2F _’2 þ 3 _F2Þ_F2

�� 3H

��3

2_F� 2F €’

_’

��

�� 2HFþ _F

2Fð�þ c Þ �

�_’2

_Fþ 3 _F

2F

�2 _F

2HFþ _F

�1

2ð�þ c Þ �R

��

þ��F _’2

_F

�� þ�3 _F €’

_’� _’2 � 3

2€F

�� þ �2 €’

_’� 3 _F

2F

��3 _F €’

_’� _’2 � 3 €F

�

þ�_’2

_Fþ 3 _F

2F

��2Fð2H2 þ _HÞ þ 3 _F €’

_’� _’2 � 3 €F� F2

3F;R

� F _FF;’

2F;R _’

�þ

��ln

_’2ð2F _’2 þ 3 _F2Þ_F2

�� 3H

�

��_’2 þ 3

2€F�

�F _’2

_F

�� 3 _F €’

_’

��2 _F

2F _’2 þ 3 _F2�sþ 2 _F

2HFþ _F

�1

2ð�þ c Þ �R

��

þ�� 1

2ð2HFþ _FÞ þ _F� 2HFþ 2F €’

_’� 3

2_Fþ 2F _’2

3 _Fþ _F� F

�ln

_’2ð2F _’2 þ 3 _F2Þ_F2

�� þ 3HF

�r2

a2ð�þ c Þ:

(C8)

This equation looks terribly lengthy. However, repeatedly employing the relation (8) or, namely,

ð2HFþ _FÞ� ¼ 3H _F� _’2; (C9)

after careful calculation, we find the coefficients of the 2 _F2HFþ _F

½12 ð�þ c Þ �R� and remaining (�þ c ) terms are exactlyvanished, resulting in a much simpler form (27).

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Documents

Curvature and entropy perturbations in generalized gravity