arX

iv:1

303.

2372

v2 [

astr

o-ph

.CO

] 1

7 Se

p 20

13

Towards Anisotropy-Free and Non-Singular Bounce Cosmology with Scale-invariant

Perturbations

Taotao Qiu,1, 2, ∗ Xian Gao,3, 4, 5, † and Emmanuel N. Saridakis6, 7, ‡

1 Leung Center for Cosmology and Particle Astrophysics National Taiwan University, Taipei 106, Taiwan2 Department of Physics, National Taiwan University, Taipei 10617, Taiwan

3Astroparticule et Cosmologie (APC), UMR 7164-CNRS, Universite Denis Diderot-Paris 7,10 rue Alice Domon et Leonie Duquet, 75205 Paris, France

4Laboratoire de Physique Theorique, Ecole Normale Superieure, 24 rue Lhomond, 75231 Paris, France5Institut d’Astrophysique de Paris (IAP), UMR 7095-CNRS,

Universite Pierre et Marie Curie-Paris 6, 98bis Boulevard Arago, 75014 Paris, France6Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece7Instituto de Fısica, Pontificia Universidad de Catolica de Valparaıso, Casilla 4950, Valparaıso, Chile

We investigate non-singular bounce realizations in the framework of ghost-free generalized Galileoncosmology, which furthermore can be free of the anisotropy problem. Considering an Ekpyrotic-likepotential we can obtain a total Equation-of-State (EoS) larger than one in the contracting phase,which is necessary for the evolution to be stable against small anisotropic fluctuations. Since sucha large EoS forbids the Galileon field to generate the desired form of perturbations, we addition-ally introduce the curvaton field which can in general produce the observed nearly scale-invariantspectrum. In particular, we provide approximate analytical and exact semi-analytical expressionsunder which the bouncing scenario is consistent with observations. Finally, the combined Galileon-curvaton system is free of the Big-Rip after the bounce.

PACS numbers: 98.80.-k, 04.50.Kd, 98.80.Cq

I. INTRODUCTION

Non-singular bouncing cosmology [1] has gained sig-nificant interest in recent studies of the early universe.The main reason for such a research direction is that themost popular paradigm of the early universe, namely in-flation, still suffers from the “Big-Bang singularity” prob-lem, which however can be naturally avoided in non-singular bouncing or cyclic cosmologies. Additionally,these paradigms can also solve the horizon, flatness andmonopole problems, and make compatible observationalpredictions such as nearly scale-invariant power spectrumand moderate non-Gaussianities [2–4]. Therefore, theyare recently considered as good alternatives to inflation.

In order to realize a successful bounce several require-ments must be fulfilled. First of all, the basic condition isto have the Hubble parameter change its sign from nega-tive to positive at the bounce, which implies that duringthe bouncing phase the Null Energy Condition (NEC)must be violated, with the total EoS of the universe go-ing below −1 [5, 6]. The NEC violation in the contextof General Relativity is nontrivial [7], usually leading toghost degree(s) of freedom [8, 9], which would demand ei-ther ghost-elimination mechanisms or an extended anal-ysis to a modified gravity context [10, 11].

Apart from the above basic condition, in order for abounce to be a successful alternative to inflation it should

∗Electronic address: xsjqiu@gmail.com,qiutt@ntu.edu.tw†Electronic address: xgao@apc.univ-paris7.fr‡Electronic address: Emmanuel˙Saridakis@baylor.edu

also solve the other Big-Bang problems, and moreover itshould produce a nearly scale-invariant power spectrumas required by observations [12]. These impose morestringent constraints on the bounce evolution, especiallyin the contracting phase. For instance, the horizon prob-lem can be solved if the quantum fluctuations in the farpast lie deep inside the horizon, while they should exit thehorizon in the contracting phase in order to generate per-turbations compatible with observations, provided thatinflation is absent in bouncing scenario. This requires atotal EoS satisfying w > −1/3 in the contracting phase[13, 14]. However, the scale-invariance of the perturba-tions is even harder to be achieved. In particular, as itwas initially shown in [15], if the perturbations generatedin the contracting phase are purely adiabatic, the EoS ofthe contracting universe should satisfy w ≈ 0 in order toproduce the desired spectrum.

However, although bounce models with total EoS w ≈0 before the bounce, namely the “matter bounce”, couldlead to nearly scale-invariant power spectrum, they gen-erally suffer from the “anisotropy problem” [16] in thecontracting phase. In particular, in 4D General Relativ-ity a tiny amount of anisotropic fluctuation from the sim-ple isotropic Friedmann-Robertson-Walker (FRW) geom-etry in the contracting phase, would increase as a−6,where a is the scale factor. Thus it would finally dom-inate over matter-like background, leading to a Big-Crunch singularity with complete anisotropy instead ofa bounce, unless one impose a strong fine-tuning of themodel parameters and the initial amount of anisotropyin order to obtain a bounce before the domination of theanisotropic term. In that sense the “matter bounce” sce-nario is not stable against cosmological anisotropy (for

2

its similar problem in the presence of radiation see [17]).For this reason, we must construct scenarios with totalEoS larger than 1 in order to prevent the dominance ofanisotropy. However, as we mentioned above, a differ-ent EoS may not be able to provide the scale-invariantpower spectrum, if we insist on applying the simpleadiabatic mechanism of generating primordial perturba-tions1. Therefore, we should resort to alternative mech-anisms, such as adiabatic, entropy and conformal ones[18, 20–24] 2.

In the present work we investigate the bounce realiza-tion in the framework of recently proposed generalizedGalileon cosmology [26, 27] (see also [28, 29] for variousdevelopments). Due to the delicate design of the La-grangian form such a theory, which contains higher-orderderivatives, can keep its equation of motion up to second-order and thus is free of ghosts (this was pioneered by thework by Horndeski [30]), but it can indeed provide extradegree(s) of freedom in order to violate NEC. Recently, in[31], the first ghost-free bounce model based on Galileoncosmology was constructed by one of the present authorsand collaborators (see also [32]), and hence in this articlewe will consummate this class of models by addressingthe problems mentioned above. Note that alternativescenarios addressing the anisotropy problem in Galileonbouncing cosmologies have been presented in [33, 34], ofwhich before contracting with w > 1, the universe canbe dominated by cold matter [35], where scale-invariantperturbations could be generated.

First of all, by introducing an Ekpyrotic-like negativepotential we can easily obtain a very large EoS in thecontracting phase, thus the anisotropy problem will beeliminated. However, as mentioned above, a large EoSforbids the Galileon field to generate the desired formof perturbations, thus as a next step we additionally in-troduce the curvaton field which is suitably coupled tothe Galileon field, such that the nearly scale-invariantspectrum can be produced. Finally, we perform a com-plete analysis of the behavior around the bounce point ofthe full Galileon-curvaton system, making use of the “in-verse” reconstruction procedure [36], showing that with aproper choice of the Lagrangian functional forms a non-singular bounce can be reconstructed, which can connectsmoothly to the matter-domination era and moreover al-leviate the Big-Rip singularity which appears in [31].

The plan of the work is the following: In section IIwe briefly review the anisotropy problem. In section IIIwe present the bouncing background evolution before,

1 We would like to mention that here by “simple adiabatic mecha-nism” we mean that the perturbations generated are purely adi-abatic, and the EoS remains constant. However, the term “adia-batic mechanism” which was first proposed in [18] in “Ekpyrotic”scenarios [19], refers to the mechanism that generates adiabaticperturbations via varying EoS.

2 Note that the stability of isotropic solutions in anisotropic per-turbations has been studied in [25].

during and after the bounce, and we show that the per-turbations are stable and free of ghosts. In section IV weanalyze the curvaton mechanism that produces nearlyscale-invariant perturbations. In section V we performa semi-analytical procedure in order to reconstruct anexact bouncing solution that is not followed by a Big-Rip. Finally, in section VI we summarize and we dis-cuss the obtained results. Throughout the manuscriptwe use the (−,+,+,+) metric signature, and units in

which MPl = 1/√8πG = 1.

II. THE ANISOTROPY PROBLEM

The anisotropy problem is a notorious problem thatgenerally exists in bouncing models with w < 1 in con-tracting phase [16]. In General Relativity, if we allow theexistence of a non-zero anisotropy at the beginning of thecontraction it will evolve scaling as a−6(t). In order todemonstrate this more transparently, without loss of gen-erality we consider as an example the simple anisotropicBianchi-IX metric [37]:

ds2 = −dt2 + a2(t)3∑

i=1

e2βi(t)dxi2, (1)

with β1(t) + β2(t) + β3(t) = 0. The Friedmann Equationwrites as:

3H2 = ρu +1

2

(

3∑

i=1

β2i

)

, (2)

where H = a/a the Hubble parameter and with ρu in-corporating all the fluids in the universe. The βi’s satisfythe equations

βi + 3Hβi = 0 , (3)

which provide the solutions βi ∝ a−3(t). Since the secondpart in the right hand side of equation (2) can be con-sidered as an effective anisotropy term χ2, we concludethat

χ2 ≡ 1

2

(

3∑

i=1

β2i

)

∝ a−6(t) , (4)

and thus the anisotropy term corresponds to an effectiveenergy density with EoS w = 1. Although in an expand-ing universe this term is always sub-dominant and thusisotropization can be achieved, in a contracting case, aslong as it is initially non-zero (even arbitrarily small),the anisotropy will grow fast and become dominant overall species with EoS less than 1, leading finally to a col-lapsing anisotropic universe. For this reason, in orderto avoid the domination of a possible anisotropic fluctu-ation, one has to realize a contracting background that

3

evolves even faster, which requires an EoS larger thanunity in the contracting phase 3.

III. THE GALILEON BOUNCE

In the previous section we briefly showed that in orderto realize a bounce we need an effective EoS w > 1 in thecontracting phase, in order to avoid the domination of ananisotropic fluctuation. In this section we formulate thebounce realization in generalized Galileon cosmology.In the generalized Galileon scenario, where the coeffi-

cients of the various action-terms are considered as func-tions of the scalar field, the corresponding action can bewritten as [27]:

L =

5∑

i=2

Li , (5)

where

L2 = K(φ,X)

L3 = −G3(φ,X)�φ

L4 = G4(φ,X)R +G4,X [(�φ)2 − (∇µ∇νφ) (∇µ∇νφ)]

L5 = G5(φ,X)Gµν (∇µ∇νφ)

−1

6G5,X

[

(�φ)3 − 3(�φ) (∇µ∇νφ) (∇µ∇νφ)

+2(∇µ∇αφ) (∇α∇βφ) (∇β∇µφ)]

. (6)

In this action the functions K and Gi (i = 3, 4, 5) de-pend on the scalar field φ and its kinetic energy X ≡− 1

2∇µφ∇µφ, while R is the Ricci scalar and Gµν is theEinstein tensor. Moreover, Gi,X and Gi,φ (i = 3, 4, 5)denote the partial derivatives of Gi with respect to Xand φ, (Gi,X ≡ ∂Gi/∂X and Gi,φ ≡ ∂Gi/∂φ), and thebox operator is constructed from covariant derivatives:�φ ≡ gµν∇µ∇νφ. In the following we focus on the case

K(φ,X) = X − V (φ), G3(φ,X) = g(φ)X,

G4(φ,X) =1

2, G5(φ,X) = 0. (7)

Therefore, the action that we are going to use reads:

S =

∫

d4x√−g[

1

2R− 1

2∇µφ∇µφ− V (φ)

+g

2∇µφ∇µφ�φ

]

. (8)

We now proceed to a detailed investigation of the abovescenario. Firstly, in the following subsection we provide

3 For some Grand Unification Theories where anisotropic stressesand collisionless particles are taken into account, there may stillbe anisotropy problems, see [38] for more details. However, it isnot the case that we’re currently considering. We thank JohnBarrow for pointing it out to us.

approximate analytical solutions at the far past beforethe bounce, around the bouncing regime, and after thebounce, at the background level. Then in the next sub-section, we analyze the perturbation behavior.

A. Background evolution: analytical results

In the following we impose a flat Friedmann-Robertson-Walker (FRW) background metric of the formds2 = −N2(t)dt2 + a2(t)dx2, where t is the cosmic time,xi are the comoving spatial coordinates, N(t) is the lapsefunction, and a(t) is the scale factor. Varying the action(8) with respect to N(t) and a(t) respectively, and settingN = 1, we obtain the Friedmann equations

H2 =1

3ρ , H = −1

2(ρ+ P ) . (9)

Additionally, we have defined the effective energy densityand pressure as:

ρ =1

2φ2 + V (φ) + 3gHφ3 , (10)

P =1

2φ2 − V (φ) − gφ2φ , (11)

and thus the total EoS of the universe is just

w ≡ P

ρ=

φ2 − 2V (φ) − 2gφ2φ

φ2 + 2V (φ) + 6gHφ3. (12)

Finally, variation of (8) with respect to the Galileon fieldprovides its evolution equation:

Dφ+ Γφ+ Vφ = 0 , (13)

where

D = 1 + 6gHφ+3

2g2φ4 , (14)

Γ =3

2

(

1 + 3gHφ)(

2H − gφ3)

. (15)

In the following we will suitably choose the potentialV (φ) in order to obtain a very large positive equation ofstate in relation (12), namely w > 1, so that our modelwill not suffer from the anisotropy problem in the con-tracting phase. Meanwhile, when the Galileon term G�φbecomes more and more important, the EoS becomesnegative and eventually triggers the bounce.As a specific example, we choose the potential to be

V (φ) = −V0ecφ, (16)

with V0, c > 0, namely a negative exponential poten-tial, which is the usual one in Ekpyrotic scenarios [19].Within this choice, when the nonlinear kinetic term isrelatively small we obtain ρ < P and thus w > 1. Onecould ask whether the negative potential would lead to anegative energy density, however the more negative the

4

potential is, the steeper it is, and thus it gives rise tolarger kinetic term, which can compensate the potentialnegativity (this will be verified later on).

Let us proceed to a qualitative investigation of the dy-namics of such scenario. For simplicity we consider thatφ increases from negative to positive during the evolu-tion, therefore at the beginning when φ starts at a largenegative value the potential (16) is in its “slow-varying”region, and the field moves slowly. In this region the non-linear term G�φ will have a small contribution to theaction. Similarly to the Ekpyrotic models, the universewill contract with a very large positive EoS, but as timepasses φ moves towards positive values and its velocityincreases, the effect of the G�φ term is enhanced, and itcan trigger the bounce. However, after the bounce anddue to the large slope of the potential, the field kineticenergy could increase to unacceptably large values whichcould spoil the validity of the effective theory. Therefore,we should need some mechanisms to slow the field mo-tion down, and this will be demonstrated in detail in thenext sections.

In the following we proceed to the quantitative investi-gation of the scenario, extracting approximate analyticalsolutions for the background evolution in the far past be-fore the bounce, around the bouncing point, and in thefar future after the bounce.

1. Solution far before the bounce

Far before the bounce, as we assumed, φ begins witha large negative value and moves slowly, while the non-linear term of φ is negligible. The energy density andpressure reduces to

ρ ≃ 1

2φ2 + V (φ) , P ≃ 1

2φ2 − V (φ) , (17)

similarly to a single scalar field with an Ekpyrotic po-tential. Thus, we can choose the model parameters andthe initial conditions of φ in order to acquire scaling so-lutions, namely to obtain a scale factor evolving as

a(t) ∼ (t∗ − t)p ∼ |η∗ − η|p

1−p , p ≡ 2

3(1 + w), (18)

where the subscript ‘∗’ denotes the time where the non-linear term in equations (10)-(13) becomes important.Note that since we are considering the region where t <t∗, we have t∗ − t > 0. For completeness we have alsoexpressed the above solution using the conformal time η,related to the cosmic time t through dt = adη. Similarly,the field φ(t) scales as

φ(t) ≃ −2

cln(t∗ − t) , p =

2

c2. (19)

Moreover, taking the derivatives of the above expressionswe find

φ(t) ≃ 2

c(t∗ − t), (20)

H(t) ≃ p

t− t∗, (21)

and therefore the energy density from expression (10)becomes

ρ(t)t<t∗ = 3H2 ∝ (t− t∗)−2 (22)

Finally, using relations (18) and (19), for the total EoSusing (12) we obtain

w ≃ 1

3c2 − 1 = const. (23)

2. Solution around the bounce

Around the bouncing point tB the nonlinear term be-comes important. From the Friedmann equation (9) wecan express H as:

H =1

2gφ3 ± 1

6

√

9g2φ6 + 6(φ2 − 2V0ecφ) , (24)

in which the nonlinear term becomes important and thelast terms in (10) and (11) can no longer be neglected.Without loss of generality we can keep the minus-signbranch in order to obtain a positive φ(t).4 Therefore,when the universe goes from the contracting (H < 0) to

the expanding phase (H > 0), one has φ >√2V0e

cφ/2

before and φ <√2V0e

cφ/2 after the bounce. It is there-fore natural to consider the solution at the bounce regionto be

φ =√

2V0(αt+ β)ecφ

2 , (25)

with α and β being two parameters satisfying the condi-tions

α < 0 , αtB + β = 1 . (26)

Note that substitution of (25) into the φ-equation of mo-tion (13), provides the necessary value for the coefficientα in order to have self-consistency.Integrating equation (25) leads to

e−cφ02 − e−

cφ

2 =√

2V0

[α

2(t20 − t2) + β(t0 − t)

]

, (27)

4 Note that in general the Hubble parameter could transit fromthe minus-sign branch to the plus-sign branch either before (forφinitial < 0) or after the bounce (for φinitial > 0). Thus, abovewe assume that if such transition exist it will take place after thebounce. However, as we will shortly see below, at late times andunder the curvaton backreaction, expression (24) is not exactlyvalid any more. Hence, we do not examine in detail the relationbetween the two branches.

5

with t0 a boundary value for t, and φ0 the correspondingvalue of φ. The above formula will be simplified if we sett0 in the far future when φ0 becomes very large, and thus

e−cφ02 ≈ 0. In this case the expression for φ(t) becomes

φ = −2

cln{

−√

2V0(t0− t)[α

2(t− t0)+αt0+β

]}

, (28)

which leads to

φ =4(αt+ β)

c(t0 − t)[α(t− t0) + 2(αt0 + β)], (29)

φ =2

c

{

1

(t0 − t)2+

α2

[α(t+ t0) + 2β]2

}

. (30)

It would be useful if we could approximate the aboveexpressions around the bouncing point tB, namely t →tB = (1−β)/α. In this case, and neglecting the constantterms in order to extract the pure scaling behavior, weobtain

φ ≃ 4α(t− tB)

c [(β + αt0)2 − 1],

φ ≃ 4α2[

(β + αt0)2 + 1

]

(t− tB)

c [(β + αt0)2 − 1]2 ,

φ ≃ 8α3[

3(β + αt0)2 + 1

]

(t− tB)

c(β + αt0 − 1)3(β + αt0 + 1)3, (31)

that is φ exhibits a linear behavior around tB . Followingthe same way, one could also insert these expressions intoequation (24) and (12) to straightforwardly obtain theapproximate solutions for H(t) and w(t), respectively.

3. Solution after the bounce

Far after the bounce, when t approaches t0, the solu-tions are still given by (28)-(30). Thus, approximatingthem at t → t0, and neglecting the constant terms inorder to extract the pure scaling behavior, we acquire

φ ≃ −2

cln(t0 − t) , (32)

φ ≃ 2

c

1

t0 − t, (33)

φ ≃ 2

c

1

(t0 − t)2. (34)

In this case, inserting these expressions into (24), wecan extract a simple approximate expression forH(t) too,namely

H ≃ 1

(t0 − t)3, (35)

which leads to a Big-Rip singularity when t approachest0. This can be easily explained since when the non-linear terms become very important the last terms in

(10) and (11) become dominant. Thus, when the en-

ergy density is dominated by the 3gHφ3 term, namely3H2 = ρ ∼ 3gHφ3, we straightforwardly find that theHubble parameter H will be proportional to φ3, and (35)is verified. Finally, the total EoS can also be obtained byinserting these expressions into (12).

4. Numerical verification

-20 -15 -10 -5 0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

t

FIG. 1: The evolution of the Galileon field φ with respect tot. We choose the initial conditions to be φi = −

√3(ln 20)/3

and φi =√3/60, and the parameters to be c = 2

√3, g = 1,

and V0 = 1/12, respectively.

-20 -15 -10 -5 00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

t

d /dt

FIG. 2: The evolution of the speed of the Galileon field φwith respect to t. We choose the initial conditions to be φi =−√3(ln 20)/3 and φi =

√3/60, and the parameters to be c =

2√3, g = 1, and V0 = 1/12, respectively.

We close the background investigation by performingan exact numerical elaboration in order to verify the

6

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

t

H

FIG. 3: The evolution of the Hubble parameter H with respectto t. We choose the initial conditions to be φi = −

√3(ln 20)/3

and φi =√3/60, and the parameters to be c = 2

√3, g = 1,

and V0 = 1/12, respectively.

-20 -15 -10 -5 0-30

-27

-24

-21

-18

-15

-12

-9

-6

-3

0

3

6

9

t

w

FIG. 4: The evolution of the EoS w with respect to t. Wechoose the initial conditions to be φi = −

√3(ln 20)/3 and

φi =√3/60, and the parameters to be c = 2

√3, g = 1, and

V0 = 1/12, respectively.

above approximate expressions in the various regimes. Inparticular, we numerically solve equations (9) and (13),imposing (19) as our initial conditions. In Figures 1, 2,

3 and 4, we respectively present φ(t), φ(t), the Hubbleparameter H(t) as well as the EoS w(t). As we observe,

both φ(t) and φ(t) are monotonically increasing. More-over, before the bounce the universe contracts with ascaling solution with EoS w being constant and largerthan unity (in this specific example w = 3), and thus thescenario is free from the anisotropy problem discussed insection II.

As time passes the nonlinear term becomes important

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0-0.02

0.00

0.02

0.04

0.06

0.08

0.10

t

FIG. 5: The evolution of the total energy density ρ of theGalileon field, with respect to t. We choose the initial con-ditions to be φi = −

√3(ln 20)/3 and φi =

√3/60, and the

parameters to be c = 2√3, g = 1, and V0 = 1/12, respec-

tively.

and triggers the bounce, which forces H(t) to changefrom negative to positive. Note that the numerical re-sults confirm that φ(t) has a positive value during thebouncing period, and thus it justifies our choice of theminus sign in (24). After the bounce, the nonlinear ki-netic term makes the scalar-field energy density increas-ing, leading the universe to a Big-Rip. One can also seethat the total EoS w(t) indeed indicates the bounce fol-lowed by the Big-Rip, verifying the analytical results thathas been obtained in preceding paragraphs.

Finally, in Fig. 5, we present the evolution of the totalenergy density ρ of the Galileon field, calculated through(10). As we observe ρ is always positive, despite the useof a negative potential, due to the increase of the kineticenergy, and it becomes zero only at the bounce point asexpected. Therefore, in the present work we do not needmechanisms that could transit the universe to positivepotential energy [39], however it would be desirable toconsider a mechanism that could smooth the increase ofthe Galileon kinetic energy, by either considering a boundin the potential, or couple φ to other matter fields such isradiation, which could lead to energy transfer away fromit (a procedure that could lead to the universe preheat-ing too). These mechanisms lie beyond the scope of thepresent work and are left for a future investigation.

B. Perturbations

In the previous subsection we analyzed the backgroundevolution of the Galileon bounce. Thus, we can now pro-ceed to the investigation of the perturbations, focusingon their stabilities. It proves convenient to foliate theFRW metric in an Arnowitt-Deser-Misner (ADM) form

7

[40]:

ds2 = −N2dt2 + hij(dxi +N idt)(dxj +N jdt) , (36)

where N is the lapse function, N i is the shift vector, andhij is the induced 3-metric. One can then perturb thesefunctions as:

N = 1 +A , Ni = ∂iψ , hij = a2(t)e2ζδij , (37)

where A, ψ and ζ are the scalar metric perturbations. Asusual it is useful to define the (gauge-invariant) comovingcurvature perturbation through

R ≡ ζ +H

φδφ , (38)

and hence in the uniform φ gauge we acquire δφ = 0 andζ = R.Under the above perturbation scheme, the action (8)

perturbed up to second order becomes:

S(2) =

∫

dηd3xa2QR

c2s

[

R′2 − c2s(∂R)2]

, (39)

where η is the conformal time. In the above expressionwe have introduced the sound-speed squared c2s, and thequantity QR related to instabilities, which in our specificscenario read as

c2s = Q−1R [1 + 2g(φ+ 2Hφ)− 1

2g2φ4] , (40)

QR = 1 + 6gHφ+3

2g2φ4 . (41)

Note that according to the definition of D in (14) we

-20 -15 -10 -5 00.0

0.5

1.0

1.5

2.0

2.5

3.0

t

c2s

FIG. 6: The evolution of the squared speed of sound c2swith respect to t. We choose the initial conditions to beφi = −

√3(ln 20)/3 and φi =

√3/60, and the parameters to

be c = 2√3, g = 1, and V0 = 1/12, respectively.

obtain 2QR = D.

The perturbative action (39) could in principle lead toghosts and gradient instabilities, which would be catas-trophic since it is this action that can be written as acanonical form and then be quantized. From its form onecan see that the avoidance of ghosts requires the factorin front of the kinetic term of the perturbation variableR to be positive, namely QR/c

2s > 0, while the absence

of gradient instabilities requires c2s ≥ 0 [29], which meansthe ratio of the factors of spatial and time derivativesmust be positive. In order to show that this is the casefor the exact behavior too, in Figures 6 and 7 we re-spectively depict c2s and QR, arising from the numericalelaboration of the full system.

-20 -15 -10 -5 00

2

4

6

8

10

t

QR

FIG. 7: The evolution of the instability-related quantity QR

with respect to t. We choose the initial conditions to be φi =−√3(ln 20)/3 and φi =

√3/60, and the parameters to be c =

2√3, g = 1, and V0 = 1/12, respectively.

From the above analysis we can see that our model isstable under both ghost and gradient instabilities. How-ever, it generally generates a (deep) blue tilted powerspectrum, which cannot be consistent with observations.Note that in [31] this was shown for w = 1/3, there-fore in our present case where w > 1 the blue tilt ofthe power spectrum is even stronger. This implies thatthe anisotropy-free requirement w > 1 and the scale-invariant perturbation generation cannot be obtained si-multaneously in the current case, as was already men-tioned in the Introduction. For this reason, we shouldintroduce an additional mechanism in order to be ableto generate the nearly invariant perturbation spectrum,without spoiling the solution to the anisotropy problem.This can be performed by the curvaton mechanism, aswe will present in the next section.

IV. THE CURVATON MECHANISM AND THE

SCALE-INVARIANT SPECTRUM

In the previous section we showed that the Galileonscenario under an Ekpyrotic-like potential can exhibit a

8

bouncing solution naturally. In the contracting phase thecorresponding total EoS of the universe satisfies w > 1 inorder for the evolution to be free of the anisotropy prob-lem discussed in section II. However, under this require-ment the corresponding perturbations, although free ofghost and gradient instabilities, cannot give rise to anearly scale-invariant power spectrum, as it is requiredby observational data [12].In [31] it was shown that this problem can be solved by

introducing a curvaton field σ coupled to the Galileon φ.As it is usual for curvaton fields [41], σ does not affect thebouncing background behavior, but it can lead to a powerspectrum in agreement with observations. In particular,through a specific Galileon-curvaton coupling, the curva-ton field lies effectively in a “fake” de-Sitter expansion ormatter-like contraction, and thus it can generate a nearlyscale-invariant power spectrum. This is the idea behindthe “conformal” mechanism [24] and in this section weinvestigate the necessary form of the coupling functions.Let us consider the curvaton action as

Sσ =1

2

∫

d4x√−g[−F(φ)(∂σ)2 − 2G(φ)W (σ)] , (42)

allowing for the most general form of coupling between σand φ. The functions F(φ) and G(φ) depend on φ, whileW (σ) is the potential for σ. Variation of action (42) withrespect to σ gives its background evolution equation as:

σ0 +(a3F)·

a3F σ0 +GFWσ0 = 0 , (43)

where σ0 is the background value of σ and Wσ0 corre-sponds to ∂W/∂σ|σ0 . In the following, and up to the endof this section, we omit the subscript “0”, denoting thebackground by a simple σ, unless explicitly mentioned.Additionally, the energy density and pressure of σ can berespectively written as:

ρσ =1

2F σ2 + GW , Pσ =

1

2F σ2 − GW . (44)

We now proceed to the examination of the perturba-tions generated from the curvaton field. Perturbing it byδσ and defining u ≡ a

√Fδσ, we can extract the corre-

sponding perturbation equation as a second-order differ-ential equation:

u′′ +

(

k2 + a2GFWσσ − z′′

z

)

u = 0 , (45)

where the prime denotes derivative with respect to theconformal time η, Wσσ ≡ ∂2W (σ)/(∂σ)2 is the secondderivative of the potential with respect to σ, and we havedefined z ≡ a

√F . As usual, the power spectrum gener-

ated by δσ is defined as:

Pδσ =k3

2π2

∣

∣

∣

u

z

∣

∣

∣

2

. (46)

Therefore, we deduce that the condition for obtaining ascale-invariant power spectrum of δσ is:

a2GWσσ

F − z′′

z≃ − 2

|η∗ − η|2 . (47)

There are several ways to satisfy the condition (47).The simplest one is to set W (σ) = 0, in which the firstterm in the above condition disappears, and thus we needjust to suitably choose F in order to obtain z′′/z ≃ 2|η∗−η|−2. Alternatively we can incorporate the effects of boththe kinetic and the potential terms of σ. In the followingsubsections we consider these cases separately.

A. W (σ) = 0

UnderW (σ) = 0, the condition (47) of obtaining scale-invariant power spectrum becomes

z ∝ |η∗ − η|2 or |η∗ − η|−1 . (48)

For the case of z ∝ |η∗ − η|2 we obtain

δσ =u

z∼ k

32 , k−

32 |η∗ − η|−3 , (49)

the latter of which dominates over the former. Therefore,using expression (46) we can obtain the power spectrumas

Pδσ ∼ k0|η∗ − η|−6 . (50)

As we observe, the spectrum is indeed scale-invariant butit has an increasing amplitude.On the other hand, for the case of z ∝ |η∗ − η|−1 we

acquire

δσ =u

z∼ k−

32 , k

32 |η∗ − η|3 , (51)

the former of which dominates over the latter. Therefore,using (46) we can obtain the power spectrum as

Pδσ ∼ k0. (52)

In this case the spectrum is scale-invariant and moreoverit is conserved on super-horizon scales.The absence of W (σ) leads to an absence of G(φ) too.

Thus, we only need to suitably determine the form ofF(φ) according to the condition (48). Note that far be-fore the bounce we have already assumed the scale-factoransatz (18), and therefore (48) requires just

F ∝ |η∗ − η|2(2−3p)

1−p or |η∗ − η| 2p−1 . (53)

Furthermore, from the ansatz solution (19) for φ and theabove expressions we deduce that the suitable choice ofthe form of F in terms of φ might be

F ∝ e2(3−c2)φ/c or ecφ , (54)

where we have made use of the relation p = 2/c2 (notethat since in contracting phase w ≫ 1, we have 0 < p≪1).We close this subsection by examining the backreac-

tion of σ field on the background evolution, since al-though the curvaton is necessary for the correct per-turbation generation we would not desire it to spoil the

9

background bouncing behavior itself. In the contractionregion where t < t∗, the background energy density ofthe system scales as ρt<t∗ ∼ (t∗ − t)−2, as it was foundin (22). On the other hand, the evolution equation ofthe curvaton field (43) gives σ ∼ a−3F−1, and thus itsenergy density in (44) becomes ρσ ≃ F σ2/2 ∼ a−6F−1.Since we know the time-dependence of the scale factorfrom (18) and the time-dependence of F from (53), westraightforwardly deduce that for the solution branchwhere z ∝ |η∗ − η|2 we obtain ρσ ∼ (t∗ − t)−4, whilefor the solution branch where z ∝ |η∗ − η|−1 we ac-quire ρσ ∼ (t∗ − t)2(1−3p) ∼ (t∗ − t)2 (in the last stepwe used that p ≪ 1). Therefore, our analysis indicatesthat in the solution branch where z ∝ |η∗ − η|2 (with

F ∝ e2(3−c2)φ/c), one has to suitably tune the initialconditions in order for the curvaton not to destroy thebackground bouncing behavior. However, in the secondsolution branch where z ∝ |η∗−η|−1 (with F ∝ ecφ), theenergy density of the curvaton field grows slower thanthat of the background, and thus the background bounc-ing evolution is not altered by the backreaction of thecurvaton.

B. W (σ) 6= 0

We now examine the case where the curvaton poten-tial is non-zero, in order to investigate its effect on theperturbation generation. Without loss of generality andfor simplicity we assume that F is approximately a con-stant (we set F = 1), although extension to general F isstraightforward.In order to see what condition (47) gives in this case,

we recall that at the early stage of the bouncing phase,where the perturbation δσ is generated, the scale factorevolves according to (18), and therefore since z ≡ a

√F

we obtain

z′′

z=a′′

a≃ p

1− p

(

p

1− p− 1

)

1

|η − η∗|2. (55)

Furthermore, introducing a∗ = a(η∗) we can write

a2GFW,σσ = a2∗GW,σσ |η∗ − η|

2p1−p . (56)

Inserting equations (55) and (56) into (47) we deducethat the condition for obtaining a scale-invariant powerspectrum of δσ becomes

a2∗GWσσ =3p− 2

(1− p)2|η∗ − η|−

21−p . (57)

As a specific example we consider the well-studied caseof a quadratic potential, namely W (σ) = m2

σσ2/2, in

which case Wσσ = const. Hence, condition (57), usingalso the φ-evolution from (19), gives

G ∝ ecφ . (58)

Therefore, we extract that the field perturbation δσscales as:

δσ =u

z∼ k−

32 |η∗ − η| 1

p−1 , k32 |η∗ − η|

3p−2p−1 . (59)

We mention that since we assume F = 1 (that is z = a)the “fake” effect is absent, and thus the dominating modeof the perturbations is always the growing mode.We close this subsection by examining the backreaction

of σ field on the background evolution, since we wouldnot want the curvaton to spoil the background bouncingbehavior itself. The σ-evolution equation (43) becomes

σ + 3Hσ +m2σGσ = 0 . (60)

Since according to relations (58) and (19) we have G ∼ecφ ∼ (t∗ − t)−2, we can write G = G0(t∗ − t)−2 withG0 being an arbitrary constant. Therefore, equation (60)accepts the solution

σ ∼ (t∗ − t)12 [1−3p±

√(1−3p)2−4m2

σG0] . (61)

Finally, substituting it into (44) for the curvaton energydensity we obtain

ρσ ∼ (t∗ − t)[1−3p±√

(1−3p)2−4m2σG0]−2 . (62)

Comparing the background energy-density evolution(22) with the curvaton energy-density evolution (62) wededuce that the requirement for the latter to grow slowerthan the former is to have 1−3p±

√

(1− 3p)2 − 4m2σG0 >

0. However, since in order to solve the anisotropy prob-lem we focus on w > 1 (or equivalently p < 1/3), thenprovided G0 > 0 the above requirement is always satis-fied. Therefore, in the scenario at hand the energy den-sity of the curvaton field will never dominate over thebackground evolution, that is the background bouncingbehavior will not be destroyed by its backreaction.We close this section with a comment on the preserva-

tion of the scale invariance across and after the bounce,which is in general a crucial question in bouncing sce-narios, and on the matching conditions we impose. Inthe above analysis we required δσ and δσ′ to be continu-ous across the bounce, which is consistent with Deruelle-Mukhanov matching conditions [42]. Considering theexpanding phase, since it usually contains two modes,namely the constant and the growing/decaying one, theperturbation spectrum may or may not get altered de-pending on whether mode-mixing is realized or not, ordepending on whether the varying modes in contract-ing/expanding phase are growing/decaying, which is de-termined by the background. For instance, in the abovecase where F(φ) ∼ ecφ and W (σ) = 0, the contractingmodes are decaying, thus by using the aforementionedmatching conditions scale-invariance will be maintainedif the expanding mode is growing, which requires thebackground EoS (or the effective EoS, if there is a “fak-ing”) to be no larger than 1, and therefore we may eas-ily preserve the scale-invariance in the expanding phaseby slightly constraining the background evolution. Amore detailed discussion on these will be taken on in afollowing-up paper. Similar results can be found in [33].

10

V. RECONSTRUCTING THE EXACT

SOLUTION AROUND THE BOUNCE

In the previous sections we constructed the Galileonbounce scenario free of the anisotropy problem, in whichwe added the curvaton field in order to obtain a scale-invariant power spectrum of primordial perturbationsgenerated in the contracting phase. Additionally, weshowed that under soft requirements on the choice ofF(φ) or G(φ), the backreaction of the curvaton field willnot alter the background bouncing evolution.However, after the bounce the effect of the curvaton

field can be significant, and in particular it can regu-larize the universe evolution in order not to result to aBig-Rip. In order to examine what classes of couplingfunctions can provide this overall behavior, in this sec-tion we semi-analytically reconstruct them following the“inverse” procedure [36], in which we impose as input thedesired bouncing scale factor, reconstructing suitably thevarious function in order to correspond to a consistentand exact solution of the full system of equations.The complete action of the Galileon-curvaton system,

consisted of both (8) and (42), can be written as:

Stotal =

∫

d4x√−g

[

1

2R− 1

2∇µφ∇µφ− V (φ)

+g

2∇µφ∇µφ�φ−F(φ)(∂σ)2 − 2G(φ)W (σ)

]

.(63)

Note that although matter and radiation could be in-cluded straightforwardly, in the above action we haveneglected them in order to examine the pure effects ofthe Galileon-curvaton evolution. Thus, the cosmologi-cal equations in the FRW metric are the first Friedmannequation:

3H2 =φ2

2+V (φ)+3gHφ3+

1

2F(φ)σ2+G(φ)W (σ) , (64)

and the evolution equations for the two fields, namely

σ + σ

[

3H +∂F(φ)

∂φ

φ

F(φ)

]

+G(φ)F(φ)

∂W (σ)

∂σ= 0 (65)

and

φ

[

1 + 6gHφ+3

2g2φ4

]

− 1

2

∂F(φ)

∂φσ2 +

∂G(φ)∂φ

W (σ)

+3

2φ{

2H + gφ[

6H2 − φ2 −F(φ)σ2 − 3gHφ3]}

+∂V (φ)

∂φ= 0 , (66)

respectively.One could solve the above equations fully numerically,

imposing specific ansantzes and initial conditions, how-ever doing so he does not have control of what functions-ansantzes, parameter choices and initial conditions, leadto bouncing solutions. That is why the aforementioned,

semi-analytical, “inverse” procedure, where the scale fac-tor is imposed a priori, is better and more appropriatefor the analysis of this section, allowing for a system-atic control on the conditions of the bounce realization.We mention that since there are more unknown functionsthan equations, one can in general always reconstruct thedesired evolution.Let us impose a desired bouncing scale factor a(t) as

an input, by which H(t) is also known. Furthermore,we consider V (φ) as usual, and we impose φ(t) at willtoo. Thus, only three free functions remain, namelyF(φ), G(φ) andW (σ) which must be derived by the equa-tions, along with the solution for σ(t). Since there arethree independent cosmological equations, namely equa-tions (64), (65) and (66), we must also impose by handone more of the above four functions. We prefer to setW (σ) = 0, since this was one (simpler) case that wasapproximately analyzed in subsection IVA (however onecould easily consider other W (σ) forms too). Such achoice simplifies things since the function G(φ) also dis-appears from the equations, and therefore the cosmo-logical equations (64)-(66) are considered as differentialequations for σ(t) and F(t). Thus, after obtaining the so-lution, and since we know φ(t), we can reconstruct F(φ).Equation (64) can be algebraically solved in order to

obtain σ2 as

σ2(t) =1

F(t)

[

6H(t)2 − 2V (φ(t)) − φ2(t)

−6gH(t)φ3(t)]

. (67)

Substituting this into (65) gives a simple first order dif-ferential equation for F(t) of the form

h(t,F(t), F(t)) = 0 , (68)

which can be easily solved. Thus, from the solution ofF(t) and the known φ(t) we can reconstruct F(φ).We mention here that the above procedure holds for

every input functions, with the only requirement beingthe obtained σ2 from relation (67) to be positive, other-wise there is no solution that can correspond to the inputfunctions. With the above semi-analytical procedure onehas full control on how to choose the model parametersand the initial conditions in order to get a positive σ2 in(67). On the other hand, if one tries to solve fully nu-merically the three equations (64)-(66) simultaneously, itis very hard to determine the model parameters and theinitial conditions in order to get a consistent bouncingsolution.In order to apply explicitly the above reconstructing

procedure, without loss of generality we choose a bounc-ing scale factor of the form

a(t) = aB

(

1 +3

2ωt2)1/3

, (69)

where aB is the scale factor at the bouncing point and ω isa positive parameter which describes how fast the bounce

11

takes place. The above ansatz presents the bouncingbehavior, where t varies in the bounce region, that isbetween the times tB− and tB+, with t = tB = 0 thebouncing point, however one could use at will any otherbouncing ansatz. Straightforwardly we find

H(t) =ωt

(1 + 3ωt2/2), (70)

and thus the universe is free of a Big-Rip after thebounce.

0.454

0.456

0.458

0.460

0.462

0.464

0.466

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0-0.20.00.20.40.60.81.01.21.41.6

t

F

FIG. 8: (Colored online) The solution for the coupling func-tion F(t) and the imposed Galileon field φ(t), under the im-posed bouncing ansatz (69). We choose the parameters asg = 1, V0 = 1/12, c = 2

√3, ω = 1, φI = 0.1, tI = −100, and

tB± = ±1.

0.456 0.457 0.458 0.459 0.460 0.461 0.462 0.463 0.464 0.465-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

F

FIG. 9: (Colored online) The reconstructed coupling functionF(φ) under the imposed bouncing ansatz (69), using Fig. 8.We choose the parameters as g = 1, V0 = 1/12, c = 2

√3,

ω = 1, φI = 0.1, tI = −100, and tB± = ±1.

For the field φ and the potential V (φ), enlightenedby the analysis of subsection IIIA and without loss of

generality, respectively we assume

φ(t) = φI ln(t− tI) (71)

and

V (φ) = −V0ecφ , (72)

while as we mentioned we set W (σ) = 0.We follow the procedure described above, and for the

model parameters we choose g = 1, V0 = 1/12, c = 2√3,

ω = 1, φI = 0.1, tI = −100, and tB± = ±1. In Fig. 8we depict the solution for F(t) and also the known φ(t)from (71), and in Fig. 9 we present the correspondingreconstructed F(φ). Finally, for completeness in Fig. 10we show the solution for σ(t).

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.075

76

77

78

79

80

t

FIG. 10: (Colored online) The solution for the curvaton fieldσ(t), under the imposed bouncing ansatz (69). We choose theparameters as g = 1, V0 = 1/12, c = 2

√3, ω = 1, φI = 0.1,

tI = −100, and tB± = ±1.

We close this section by mentioning that in principleone could think of other reconstructing procedures, forinstance setting F(φ) and reconstruct G(φ) and W (σ),or even setting φ(t) and σ(t) and reconstruct F(φ), G(φ)and W (σ). So there can actually be many possibilitiesto realize the non-singular bounce.

VI. CONCLUSIONS

Bounce cosmology is an interesting paradigm since italleviates the Big-Bang singularity problem. Addition-ally, it can solve the Big-Bang problems, and nearly scale-invariant primordial perturbations can be incorporatedtoo. These features make bouncing cosmologies success-ful alternatives to inflation. However, there are manydetailed issues that should be carefully addressed dur-ing the establishment of bouncing cosmology, and in thepresent work we tried to confront some of them.First of all, the NEC violation, which is required for

the bounce realization, may bring ghost degrees of free-dom. In the above analysis we were based on the Galileon

12

scenario, which is a higher-derivative construction free ofghosts, and thus we obtained a bouncing evolution freeof ghost and gradient instabilities.However, there is a second problem that may disturb

the bounce construction, namely that in the contractingphase even a tiny anisotropic fluctuation from the totallyisotropic FRW geometry will be radically enhanced anddestroy completely the FRW evolution. The solution ofthis “anisotropy problem” requires the total EoS of theuniverse to lie in the regime w > 1 in the contractingphase. Thus, starting from [31] where the anisotropyproblem was present, in this work we were able to solveit and obtain w > 1 by considering an Ekpyrotic-likepotential with negative value. This is one of the maincontributions of the present article.The above solution of the anisotropy problem through

a large EoS has an undesired effect, namely it spoilsthe generation of a nearly scale-invariant power spec-trum, that a scalar with smaller EoS can bring throughadiabatic perturbation. Therefore, in order to still beable to produce a power spectrum in agreement withobservations, we additionally introduced in the scenarioa second, curvaton field, coupled to the Galileon one,which can indeed generate the desired perturbations inan isocurvature way. In our analysis we presented thismechanism in general, and we analyzed explicitly twospecific examples where nearly scale-invariant perturba-tions are generated. Finally, we examined the conditionsunder which the curvaton field does not cause a signifi-cant backreaction on the background bouncing behaviorcaused by the Galileon field.Furthermore, the curvaton field, apart from the gener-

ation of the desired perturbations, has another importantrole, namely after the bounce it can regularize the back-ground evolution in order to avoid a Big-Rip singularity,which is caused by the Galileon field itself. In partic-ular, although the curvaton backreaction is not signifi-cant at the background level before the bounce, duringand after the bounce it becomes important and changesthe background evolution. In order to see this effect weperformed a semi-analytically “inverse” analysis, recon-structing suitably the desired bouncing evolution of thescale factor, which is free of a Big-Rip without any fine-tuning. We mention here that since the region wherethe scale-invariant spectrum is generated lies in the con-

tracting phase, while the region where the Big-Bang isavoided is around and after the bounce point, the con-ditions on the functions that generate scale-invarianceperturbations should hold in the contracting phase whilethose for the Big-Bang avoidance should hold around andafter the bounce. Therefore, one can always match therequired function form of the contracting regime withthe required form of the bounce regime, to obtain bothperturbation scale invariance and Big-Bang avoidance,although not always analytically.

We close this work by mentioning that there could beother possibilities to avoid the Big-Rip singularity. Forinstance, an alternative evolution after the bounce wouldbe to assume that the Galileon and curvaton fields de-cay to standard model particles [43]. In this case thedecaying Galileon energy density cannot trigger the BigRip anymore, and additionally it can produce the mattercontent of the universe. Such a detailed analysis of thepost-bounce evolution, and its relation to the subsequentthermal history of the universe, lies beyond the scope ofthe present work and it is left for a future investigation.

Note added: After completing our manuscript, wecame to know that studies of bounce cosmology aimingto the same issue has been done in [44], in which simi-lar results are obtained for different (conformal) Galileonmodels but with the same (Ekpyrotic-like) potential.

Acknowledgments

T.Q. thanks Robert Brandenberger for his useful com-ments. The work of T.Q. is funded in part by theNational Science Council of R.O.C. under Grant No.NSC99-2112-M-033-005-MY3 and No. NSC99-2811-M-033-008 and by the National Center for Theoretical Sci-ences. X.G. was supported by ANR (Agence Nationalede la Recherche) grant “STR-COSMO” ANR-09-BLAN-0157-01. The research of E.N.S. is implemented withinthe framework of the Action “Supporting PostdoctoralResearchers” of the Operational Program “Educationand Lifelong Learning” (Actions Beneficiary: GeneralSecretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and theGreek State.

[1] M. Novello and S. E. P. Bergliaffa, Phys. Rept. 463, 127(2008) [arXiv:0802.1634 [astro-ph]].

[2] Y. -F. Cai, T. Qiu, R. Brandenberger, Y. -S. Piao andX. Zhang, JCAP 0803, 013 (2008) [arXiv:0711.2187 [hep-th]]; Y. -F. Cai, T. -t. Qiu, J. -Q. Xia and X. Zhang,Phys. Rev. D 79, 021303 (2009) [arXiv:0808.0819[astro-ph]]; Y. -F. Cai, T. -t. Qiu, R. Brandenbergerand X. -m. Zhang, Phys. Rev. D 80, 023511 (2009)[arXiv:0810.4677 [hep-th]].

[3] Y. -F. Cai and X. Zhang, JCAP 0906, 003 (2009)

[arXiv:0808.2551 [astro-ph]]; Y. -F. Cai, W. Xue,R. Brandenberger and X. Zhang, JCAP 0905, 011 (2009)[arXiv:0903.0631 [astro-ph.CO]]; Y. -F. Cai, W. Xue,R. Brandenberger and X. -m. Zhang, JCAP 0906, 037(2009) [arXiv:0903.4938 [hep-th]].

[4] T. Qiu and K. -C. Yang, JCAP 1011, 012 (2010)[arXiv:1007.2571 [astro-ph.CO]].

[5] Y. -F. Cai, T. Qiu, Y. -S. Piao, M. Li and X. Zhang,JHEP 0710, 071 (2007) [arXiv:0704.1090 [gr-qc]].

[6] Y. -F. Cai, E. N. Saridakis, M. R. Setare and J. -Q. Xia,

13

Phys. Rept. 493, 1 (2010) [arXiv:0909.2776 [hep-th]].[7] Y. -F. Cai and E. N. Saridakis, J. Cosmol. 17,

7238 (2011) [arXiv:1108.6052 [gr-qc]]; S. ’i. Nojiri andE. N. Saridakis, arXiv:1301.2686 [hep-th].

[8] S. M. Carroll, M. Hoffman and M. Trodden, Phys. Rev.D 68, 023509 (2003) [astro-ph/0301273].

[9] J. M. Cline, S. Jeon and G. D. Moore, Phys. Rev. D 70,043543 (2004) [hep-ph/0311312].

[10] S. Capozziello and M. De Laurentis, Phys. Rept. 509,167 (2011) [arXiv:1108.6266 [gr-qc]].

[11] T. Biswas, A. Mazumdar and W. Siegel, JCAP 0603,009 (2006) [hep-th/0508194]; T. Biswas, T. Koivisto andA. Mazumdar, JCAP 1011, 008 (2010) [arXiv:1005.0590[hep-th]]; T. Biswas, E. Gerwick, T. Koivisto andA. Mazumdar, Phys. Rev. Lett. 108, 031101 (2012)[arXiv:1110.5249 [gr-qc]]; T. Biswas, A. S. Koshelev,A. Mazumdar and S. Y. .Vernov, JCAP 1208, 024 (2012)[arXiv:1206.6374 [astro-ph.CO]].

[12] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik,G. Hinshaw, N. Odegard, K. M. Smith and R. S. Hill etal., arXiv:1212.5225 [astro-ph.CO]; G. Hinshaw, D. Lar-son, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunk-ley, M. R. Nolta and M. Halpern et al., arXiv:1212.5226[astro-ph.CO].

[13] Y. -S. Piao and Y. -Z. Zhang, Phys. Rev. D 70, 043516(2004) [astro-ph/0403671]; Y. -S. Piao, Phys. Lett. B606, 245 (2005) [hep-th/0404002].

[14] T. Qiu, JCAP 1206, 041 (2012) [arXiv:1204.0189[hep-ph]]; T. Qiu, Phys. Lett. B 718, 475 (2012)[arXiv:1208.4759 [astro-ph.CO]].

[15] F. Finelli and R. Brandenberger, Phys. Rev. D 65,103522 (2002) [hep-th/0112249].

[16] K. E. Kunze and R. Durrer, Class. Quant. Grav. 17, 2597(2000) [arXiv:gr-qc/9912081]; J. K. Erickson, D. H. Wes-ley, P. J. Steinhardt and N. Turok, Phys. Rev. D69, 063514 (2004) [arXiv:hep-th/0312009]; B. Xue andP. J. Steinhardt, Phys. Rev. Lett. 105, 261301 (2010)[arXiv:1007.2875 [hep-th]]; B. Xue and P. J. Steinhardt,Phys. Rev. D 84, 083520 (2011) [arXiv:1106.1416 [hep-th]].

[17] J. Karouby and R. Brandenberger, Phys. Rev. D 82,063532 (2010) [arXiv:1004.4947 [hep-th]]; J. Karouby,T. Qiu and R. Brandenberger, Phys. Rev. D 84, 043505(2011) [arXiv:1104.3193 [hep-th]].

[18] J. Khoury and P. J. Steinhardt, Phys. Rev. Lett.104, 091301 (2010) [arXiv:0910.2230 [hep-th]]; A. Linde,V. Mukhanov and A. Vikman, JCAP 1002, 006 (2010)[arXiv:0912.0944 [hep-th]]; J. Khoury and P. J. Stein-hardt, Phys. Rev. D 83, 123502 (2011) [arXiv:1101.3548[hep-th]].

[19] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,Phys. Rev. D 64, 123522 (2001) [arXiv:hep-th/0103239];J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Stein-hardt and N. Turok, Phys. Rev. D 65, 086007(2002) [arXiv:hep-th/0108187]; J. Khoury, B. A. Ovrut,P. J. Steinhardt and N. Turok, Phys. Rev. D 66, 046005(2002) [arXiv:hep-th/0109050].

[20] F. Finelli, Phys. Lett. B 545, 1 (2002) [hep-th/0206112];A. Notari and A. Riotto, Nucl. Phys. B 644, 371(2002) [hep-th/0205019]; K. Koyama and D. Wands,JCAP 0704, 008 (2007) [hep-th/0703040 [HEP-TH]];K. Koyama, S. Mizuno and D. Wands, Class. Quant.Grav. 24, 3919 (2007) [arXiv:0704.1152 [hep-th]].

[21] J. -L. Lehners, P. McFadden, N. Turok and P. J. Stein-

hardt, Phys. Rev. D 76, 103501 (2007) [hep-th/0702153[HEP-TH]].

[22] E. I. Buchbinder, J. Khoury and B. A. Ovrut, Phys. Rev.D 76, 123503 (2007) [hep-th/0702154].

[23] J. -L. Lehners and P. J. Steinhardt, Phys. Rev. D 77,063533 (2008) [arXiv:0712.3779 [hep-th]].

[24] K. Hinterbichler and J. Khoury, JCAP 1204, 023 (2012)[arXiv:1106.1428 [hep-th]]; M. Libanov, S. Mironovand V. Rubakov, Phys. Rev. D 84, 083502 (2011)[arXiv:1105.6230 [astro-ph.CO]]; P. Creminelli, Phys.Rev. D 85, 041302 (2012) [arXiv:1108.0874 [hep-th]].

[25] I. Y. .Aref’eva, N. V. Bulatov, L. V. Joukovskayaand S. Y. .Vernov, Phys. Rev. D 80, 083532 (2009)[arXiv:0903.5264 [hep-th]]; E. Elizalde, E. O. Pozdeevaand S. Y. .Vernov, Phys. Rev. D 85, 044002 (2012)[arXiv:1110.5806 [astro-ph.CO]].

[26] A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev.D 79, 064036 (2009) [arXiv:0811.2197 [hep-th]]; C. Def-fayet, G. Esposito-Farese and A. Vikman, Phys. Rev. D79, 084003 (2009) [arXiv:0901.1314 [hep-th]]; A. Nicolis,R. Rattazzi and E. Trincherini, JHEP 1005, 095 (2010)[Erratum-ibid. 1111, 128 (2011)] [arXiv:0912.4258 [hep-th]].

[27] C. Deffayet, S. Deser and G. Esposito-Farese, Phys. Rev.D 80, 064015 (2009) [arXiv:0906.1967 [gr-qc]]; C. Def-fayet, X. Gao, D. A. Steer and G. Zahariade, Phys. Rev.D 84, 064039 (2011) [arXiv:1103.3260 [hep-th]].

[28] F. P. Silva and K. Koyama, Phys. Rev. D 80, 121301(2009) [arXiv:0909.4538 [astro-ph.CO]]; P. Creminelli,A. Nicolis and E. Trincherini, JCAP 1011, 021 (2010)[arXiv:1007.0027 [hep-th]]; C. Deffayet, S. Deser andG. Esposito-Farese, Phys. Rev. D 82, 061501 (2010)[arXiv:1007.5278 [gr-qc]]; C. Deffayet, O. Pujolas,I. Sawicki and A. Vikman, JCAP 1010, 026 (2010)[arXiv:1008.0048 [hep-th]]; T. Kobayashi, M. Yamaguchiand J. ’i. Yokoyama, Phys. Rev. Lett. 105, 231302(2010) [arXiv:1008.0603 [hep-th]]; L. Levasseur Perreault,R. Brandenberger and A. -C. Davis, Phys. Rev. D 84,103512 (2011) [arXiv:1105.5649 [astro-ph.CO]]; Z. G. Liu,J. Zhang and Y. S. Piao, Phys. Rev. D 84, 063508(2011), arXiv:1105.5713 [astro-ph.CO]; T. Kobayashi,M. Yamaguchi and J. ’i. Yokoyama, Prog. Theor. Phys.126, 511 (2011) [arXiv:1105.5723 [hep-th]]; J. Evslin,T. Qiu, JHEP 1111, 032 (2011) [arXiv:1106.0570 [hep-th]]; X. Gao and D. A. Steer, JCAP 1112, 019 (2011)[arXiv:1107.2642 [astro-ph.CO]]; X. Gao, T. Kobayashi,M. Yamaguchi and J. ’i. Yokoyama, Phys. Rev. Lett. 107,211301 (2011) [arXiv:1108.3513 [astro-ph.CO]]; H. Wang,T. Qiu and Y. -S. Piao, Phys. Lett. B 707, 11 (2012)[arXiv:1110.1795 [hep-ph]]; M. Li, T. Qiu, Y. Cai andX. Zhang, JCAP 1204, 003 (2012) [arXiv:1112.4255 [hep-th]]; X. Gao, T. Kobayashi, M. Shiraishi, M. Yamaguchi,J. ’i. Yokoyama and S. Yokoyama, arXiv:1207.0588[astro-ph.CO]; Z. -G. Liu and Y. -S. Piao, Phys.Lett. B 718, 734 (2013) [arXiv:1207.2568 [gr-qc]];G. Leon and E. N. Saridakis, JCAP 1303, 025 (2013)[arXiv:1211.3088 [astro-ph.CO]].

[29] A. De Felice and S. Tsujikawa, JCAP 1202, 007 (2012)[arXiv:1110.3878 [gr-qc]].

[30] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974).[31] T. Qiu, J. Evslin, Y. -F. Cai, M. Li and X. Zhang, JCAP

1110, 036 (2011) [arXiv:1108.0593 [hep-th]].[32] D. A. Easson, I. Sawicki and A. Vikman, JCAP 1111,

021 (2011) [arXiv:1109.1047 [hep-th]].

14

[33] Y. -F. Cai, D. A. Easson and R. Brandenberger, JCAP1208, 020 (2012) [arXiv:1206.2382 [hep-th]].

[34] Y. -F. Cai, R. Brandenberger and P. Peter,arXiv:1301.4703 [gr-qc].

[35] C. Lin, R. H. Brandenberger and L. Levasseur Perreault,JCAP 1104, 019 (2011) [arXiv:1007.2654 [hep-th]].

[36] Y. -F. Cai and E. N. Saridakis, JCAP 0910, 020(2009) [arXiv:0906.1789 [hep-th]]; Y. -F. Cai andE. N. Saridakis, Class. Quant. Grav. 28, 035010 (2011)[arXiv:1007.3204 [astro-ph.CO]]; Y. -F. Cai, S. -H. Chen,J. B. Dent, S. Dutta and E. N. Saridakis, Class. Quant.Grav. 28, 215011 (2011) [arXiv:1104.4349 [astro-ph.CO]];Y. -F. Cai, C. Gao and E. N. Saridakis, JCAP 1210, 048(2012) [arXiv:1207.3786 [astro-ph.CO]].

[37] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravita-tion, San Francisco (1973).

[38] J. DBarrow and K. Yamamoto, Phys. Rev. D 82, 063516(2010) [arXiv:1004.4767 [gr-qc]].

[39] G. N. Felder, A. V. Frolov, L. Kofman and A. D. Linde,Phys. Rev. D 66, 023507 (2002) [hep-th/0202017];J. E. Lidsey and D. J. Mulryne, Phys. Rev. D 73, 083508(2006) [hep-th/0601203]; T. Biswas, T. Koivisto andA. Mazumdar, arXiv:1105.2636 [astro-ph.CO].

[40] R. L. Arnowitt, S. Deser and C. W. Misner,arXiv:gr-qc/0405109.

[41] D. H. Lyth and D. Wands, Phys. Lett. B 524, 5(2002) [arXiv:hep-ph/0110002]; D. H. Lyth, C. Un-garelli and D. Wands, Phys. Rev. D 67, 023503 (2003)[arXiv:astro-ph/0208055].

[42] N. Deruelle and V. F. Mukhanov, Phys. Rev. D 52, 5549(1995) [gr-qc/9503050].

[43] D. Langlois and T. Takahashi, arXiv:1301.3319 [astro-ph.CO]; H. Assadullahi, H. Firouzjahi, M. H. Namjooand D. Wands, arXiv:1301.3439 [hep-th].

[44] M. Osipov and V. Rubakov, arXiv:1303.1221 [hep-th].