The Effective Field Theory of nonsingular cosmology: II
aa r X i v : . [ g r- q c ] J un The Effective Field Theory of nonsingular cosmology: II
Yong Cai ∗ , Hai-Guang Li † , Taotao Qiu ‡ , and Yun-Song Piao , § School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China Institute of Astrophysics, Central China Normal University, Wuhan 430079, China and Institute of Theoretical Physics, Chinese Academy of Sciences,P.O. Box 2735, Beijing 100190, China
Abstract
Based on the Effective Field Theory (EFT) of cosmological perturbations, we explicitly clarifythe pathology in nonsingular cubic Galileon models and show how to cure it in EFT with newinsights into this issue. With the least set of EFT operators that are capable to avoid instabilitiesin nonsingular cosmologies, we construct a nonsingular model dubbed the Genesis-inflation model,in which a slowly expanding phase (namely, Genesis) with increasing energy density is followedby slow-roll inflation. The spectrum of the primordial perturbation may be simulated numerically,which shows itself a large-scale cutoff, as the large-scale anomalies in CMB might be a hint for.
PACS numbers: ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] . INTRODUCTION Inflation is still being eulogized for its simplicity and also criticized for its past-incompleteness [1][2]. A complete description of the early universe requires physics otherthan only implementing inflation.To the best of current knowledge, the inflation scenario will be past-complete, only if ithappens after a nonsingular bounce which is preceded by a contraction [3][4][5], or a slowexpansion phase (namely, Genesis phase) with increasing energy density [6][7][8]. These twopossibilities will be called bounce-inflation and Genesis-inflation, respectively. Besides be-ing past-complete, a bounce-inflation or Genesis-inflation scenario may explain the probablelarge-scale anomalies in cosmological microwave background (CMB) [4][9]. The nonsingularQuintom bounce [10][11][12] (see also [13]), the ekpyrotic universe [14][15], the Genesis sce-nario [16][17][18][19], and the slow expansion scenario [20][21][22][23] have acquired intensiveattention. In classical nonsingular (past-complete) cosmologies, the Null Energy Condition(NEC) must be violated for a period.The ghost-free bounce models [24][25][26][27][28] have been obtained in cubic Galileon [29]and full Horndeski theory [30][31][32]. However, recently, it has been proved in Ref. [33] thatthere exists a “no-go” theorem for cubic Galileon, i.e., the gradient instability ( c s < c s being sound speed of the scalar perturbations) is inevitable in the corresponding models.See also [34] for the extension to the full Horndeski theory, and [35] for the attempts toavoid “no-go” in Horndeski theory. Relevant studies can also be found in [36][37][38].Recently, in Ref. [39] (see also [40]), we dealt with this issue in the framework of theEffective Field Theory (EFT) [41][42][43][44], which has proved to be a powerful tool. InEFT, the quadratic action of scalar perturbation could always be written in the form (see[39] for detailed derivations) S (2) ζ = Z d xa c (cid:20) ˙ ζ − c s ( ∂ζ ) a (cid:21) , (1)where we have neglected higher-order spatial derivatives of the scalar perturbation ζ , thesound speed squared of scalar perturbation c s = (cid:18) ˙ c a − c (cid:19) /c , (2)with the coefficients c , c and c being time dependent parameters in general, and c > s ∼ ˙ c /a − c >
0, which is usually integrated as c (cid:12)(cid:12)(cid:12) t f − c (cid:12)(cid:12)(cid:12) t i > Z t f t i ac dt . (3)The condition of satisfying the inequality is to have c cross 0, which is hardly possible inmodels based on the cubic Galileon [33][34]. However, we found that it can easily be satisfiedby applying the EFT operator R (3) δg (with R (3) being the 3-dimensional Ricci scalar onspacelike hypersurface and δg = g + 1), so that the gradient instability can be cured.Though the integral approach (3) is simple and efficient, some details of curing the pathol-ogy might actually be missed. In this paper, based on the EFT, using a “non-integral ap-proach”, we revisit the nonsingular cosmologies. We begin straightly with (2), and clarifythe origin of pathology and show how to cure it in EFT with new insights into what ishappening (Sec. II). To have practice in this clarification, we build a stable model of theGenesis-inflation scenario by using the R (3) δg operator (Sec. III). As a supplementaryremark, we discuss a dilemma in the Genesis scenario (Sec. IV). II. RE-PROOF OF THE “NO-GO” AND ITS AVOIDANCE IN EFT
The EFT is briefly introduced in Appendix A. In the unitary gauge, the quadratic actionof tensor perturbation γ ij is (see [39] for the derivation of Eqs. (4) to (8)) S (2) γ = M p Z d xa Q T (cid:20) ˙ γ ij − c T ( ∂ k γ ij ) a (cid:21) , (4)where Q T = f + m M p > c T = f /Q T > f and m are coefficients defined in the EFTaction (A4).The quadratic action of the scalar perturbation ζ is given by Eq. (1) with c = Q T γ M p h M p Q T ˙ f H − M p Q T (cid:16) f M p ˙ H + ¨ f M p − M (cid:17) − f M p m + 3 ˙ f M p + 3 m i , (5) c = f M p , (6) c = aM p γ Q T Q ˜ m , (7) γ = HQ T − m M p + 12 ˙ f , Q ˜ m = f + 2 ˜ m M p , (8)3here M , m and ˜ m are coefficients defined in the EFT action (A4) and they could betime dependent in general.Only if c > c s >
0, the model is free from ghost and gradient instabilities,respectively. In nonsingular cosmological models based on the cubic Galileon [26][27][28], c > m ( t )2 δKδg operator in EFT. However, since c is also affected by m ( t )2 δKδg through γ , c s < c >
0, the requirement of c s > Hγ + ˙ Q T Q T γ + ˙ Q ˜ m Q ˜ m γ − c T γ Q ˜ m − ˙ γ ! Q T Q ˜ m γ > . (9)Here, Q T = Q ˜ m is required, which cannot be embodied by the Horndeski theory [30][31][32].Thus whether c s > H , γ , Q T , c T , Q ˜ m ).In the following, with condition (9), we will re-prove the “no-go” theorem for the cubicGalileon, and clarify how to cure it in EFT. Different from the proof in [33][39][40], there-proof is directly based on the derivative inequality instead of integrating it, which wecalled “non-integral approach”. We assume that after the beginning of the hot “big bang”or inflation, γ = H >
0, ˙ γ < Q ˜ m = 1. A. Case I: initially γ < Since initially γ < γ has to cross 0 from γ < γ > t γ . The analysis below isalso applicable for all cases with γ crossing 0 from γ < γ > γ = H <
0. In the Genesis model [16] andslow expansion model [21],
H > γ = H − m M p <
0, asdiscussed in Sec. IV. Both belong to the Case I.
In the cubic Galileon case , f = Q T = Q ˜ m = 1. Around t γ , condition (9) is − ˙ γ > . (10)We see that c s < t γ , since ˙ γ >
0. Thus the nonsingular models basedon cubic Galileon is pathological, as first proved by LMR in [33].
In the EFT case , around t γ , condition (9) requires ˙ Q T Q T γ + ˙ Q ˜ m Q ˜ m γ − c T γ Q ˜ m − ˙ γ ! Q ˜ m > . (11)4e might have c s >
0, only if (considering only the case where only one of Q T and Q ˜ m ismodified while the unmodified one is unity) around γ = 0˙ Q T Q T γ > ˙ γ, (12)or Q ˜ m < , or ˙ Q ˜ m Q ˜ m γ > ˙ γ + c T γ Q ˜ m (for Q ˜ m > . (13)In solution (12), at t γ , γ = 0 suggests Q T = 0. Here, since γ = 0 at t γ , c ∼ Q T /γ diverges.One possibility of removing this divergence is that γ ∼ ( t − t γ ) p and Q T ∼ ( t − t γ ) n around t γ , with n > p and p , n being constants. In Ijjas and Steinhardt’s model [35], γ ∼ t − t γ while Q T ∼ ( t − t γ ) , which belongs to this case.In the bounce model based on the cubic Galileon, Eq. (8) gives γ = H − m M p = H .Generally, the NEC is violated when ˙ H >
0, while the period of c s < γ ≃ γ >
0, these two phases do not necessarily coincide, see Eq. (8).As pointed out by Ijjas and Steinhardt [35], it is the sign’s change of γ that causes thepathology. Here, we reconfirmed this point.In solution (13), if Q ˜ m >
0, at t γ , γ = 0 suggests Q ˜ m = 0; while if Q ˜ m <
0, since Q ˜ m = 1 eventually, Q ˜ m must cross 0 at t ˜ m (generally t ˜ m = t γ ), at which ˙ Q ˜ m γ > c T γ must be satisfied. In both cases, Q ˜ m = 0 is required, as proposed by Cai et.al [39] andCreminelli et.al [40].We see again the details of Q ˜ m crossing 0. In both the Genesis model and the bouncemodel, initially Q ˜ m = 1, so if Q ˜ m < t γ , Q ˜ m must cross 0 twice. Thus it seemsthat ˙ Q ˜ m γ > c T γ is hard to implement. However, with (2) and (7), one always could solve Q ˜ m for any given c s , Q ˜ m = γaM p Z a (cid:0) c c s + c (cid:1) dt , (14)where Q T = 1. 5 . Case II: γ > throughout Since γ > γ > , otherwise γ will diverge in the infinite past. In the cubic Galileon case , condition (9) is Hγ − γ − ˙ γ > . (15)In the bounce model, H < H ∼ Hγ − γ − ˙ γ < . Thus c s < H ∼ / ( − t ) n with the constant n >
1. Thus˙ γHγ ≃ ˙ HH ∼ ( − t ) n − ≫ , (16)which implies Hγ ≪ ˙ γ . Thus with (15), we see that c s < n = 1, Hγ ≪ ˙ γ might be avoided. However, when n = 1, we have H = p/ ( − t ) and a ∼ / ( − t ) p with constant p , thus a → c s > p = H / ˙ H >
1. Therefore, the universe is singular, or fromanother point of view, it is geodesically incomplete since the affine parameter of the gravitongeodesics R t f t i adt is finite for p > t i → −∞ . In the EFT case , condition (9) requires ˙ Q T Q T γ + ˙ Q ˜ m Q ˜ m γ − c T γ Q ˜ m + Hγ − ˙ γ ! Q ˜ m > . (17)We might have c s >
0, only if (considering only the case where either Q T or Q ˜ m is modified)˙ Q T Q T > c T γ − H + ˙ γγ , (18)or ˙ Q ˜ m Q ˜ m < c T γQ ˜ m − H + ˙ γγ (initially Q ˜ m < . (19) Of course, in Case II, we could also have ˙ γ < γ >
0) where pathologies appear. In the case where γ grows from 0 initially, (15) is also obeyed no more. − Hγ + ˙ γ >
0, as in Genesis model and bounce model. Thus the solution (18)suggests ˙ Q T >
0, so that we will have Q T = 0 in infinite past. Thus based on (12) and (18),it seems that though the pathology can be cured by applying Q T , Q T = 0 is inevitable. Amodel with (18) has been proposed by Kobayashi [34] ( Q T ∼ − t ) p , p > n > γ ∼ H ∼ / ( − t ) n , n >
1, (17) is (cid:16) ˙ Q T /Q T (cid:17) − ˙ γγ = n/p < . (20)Initially, Q T ∼ − t ) p = 0.In solution (19), Q ˜ m must cross 0 at t ˜ m to Q ˜ m >
0, as pointed out by Cai et.al [39]and Creminelli et.al [40]. Around t ˜ m , ˙ Q ˜ m > c T γ must be satisfied.In (17), if Q ˜ m > Q ˜ m Q ˜ m > c T γQ ˜ m − H + ˙ γγ (21)is obtained. Thus, similar to (18), we have Q ˜ m = 0 (which definitely requires γ = 0) inthe infinite past. In the Genesis model, Q ˜ m ∼ / ( − t ) p and γ ∼ / ( − t ) n with p > n , since˙ Q ˜ m /Q ˜ m > ˙ γ/γ . However, p > n indicates ˙ Q ˜ m < γ in the infinite past ( Q ˜ m = 0), whichviolates the inequality (21). Thus Q ˜ m > Nonsingular cubic Galileon modelsInitially γ < γ > γ ? √ × c s < √ √ Phase with c s < γ > γ ≃ Hγ − ˙ γ < γ Curing pathology in EFTConditions of c s > Q T (12) (18)Applying Q ˜ m (13) (19)TABLE I: Pathology in nonsingular cubic Galileon cosmological models and its cure in EFT byeither Q T or Q ˜ m . II. APPLICATION TO GENESIS-INFLATION
In this section, we will build a nonsingular model with the solution (19), in which theslow-roll inflation is preceded by a Genesis phase. A Genesis phase is a slowly expandingphase originating form the Minkowski vacuum with a drastic violation of NEC, i.e., ǫ ≪ − A. The setup of the model
The action of the model is S = Z d x √− g h M p R + M p g ( φ ) X + g ( φ ) X ✷ φ + g ( φ ) X − M p V ( φ )+ ˜ m ( t )2 R (3) δg i , (22)where X = −∇ µ φ ∇ µ φ/ ✷ φ = ∇ µ ∇ µ φ , and φ is a dimensionless scalar field, so dimension-less are g ( φ ), g ( φ ), g ( φ ) and V ( φ ).Mapped into the EFT action (A4), (22) corresponds to f = 1 , (23)Λ( t ) = M p V − g ˙ φ (cid:16) H ˙ φ + ¨ φ (cid:17) + 14 g ˙ φ , (24) c ( t ) = M p g ˙ φ − g ˙ φ (cid:16) H ˙ φ − ¨ φ (cid:17) + 12 g ,φ ˙ φ + 12 g ˙ φ , (25) M ( t ) = − g ˙ φ (cid:16) H ˙ φ + ¨ φ (cid:17) + 14 g ,φ ˙ φ + 12 g ˙ φ , (26) m ( t ) = − g ˙ φ , (27) m = 0 , (28)˜ m = 0 . (29)8e can get the background equations3 H M p = M p g ˙ φ − g H ˙ φ + 12 g ,φ ˙ φ + 34 g ˙ φ + M p V , (30)˙ HM p = − M p g ˙ φ + 32 g H ˙ φ − g ˙ φ ¨ φ − g ,φ ˙ φ − g ˙ φ , (31)0 = g ¨ φ + 3 g H ˙ φ + 12 g ,φ ˙ φ − g H ˙ φ M p − g ˙ H ˙ φ M p − g H ˙ φ ¨ φM p + 2 g ,φ ˙ φ ¨ φM p + g ,φφ ˙ φ M p + 3 g H ˙ φ M p + 3 g ˙ φ ¨ φM p + 3 g ,φ ˙ φ M p + M p V φ , (32)where “ ,φ = d/dφ ” and “ ,φφ = d /dφ ”.Initially, the universe is slowly expanding (in the Genesis phase), H ≃
0. We set V = 0, g = − f e φ , g = f and g = f , see e.g. Ref. [7], with f , , being dimensionless constants.Thus with Eq. (30), we have M p g ˙ φ + g ˙ φ = 0, which suggests e φ = 3 f M p f ˙ φ . (33)The solution is ˙ φ = 1( − t ) , t < . (34)Eq. (31) reads ˙ H = f − f M p ˙ φ . Thus we get H = f − f M p − t ) (35)after the integration. In principle, there could be a constant, i.e., H = f − f M p − t ) + const ,however, in that case we will have H ≈ const initially, which is geodesically incomplete, seealso [45].Additionally, from Eq. (35), we have a ( t ) = e R Hdt = exp (cid:18) f − f M p t (cid:19) ≃ (cid:18) f − f M p t (cid:19) , (36)while we set a ( −∞ ) = 1.During inflation, we set g = 1 and g = g = 0, since we require that the inflationaryphase is controlled by a simple slow-roll field . The behaviors of these g i ’s in the two phases can easily be matched together by making use of some shapefunctions [5][46]. . The primordial perturbation and its spectrum In the unitary gauge, the quadratic action of the scalar perturbation is presented in theform of Eq. (1). The coefficients c i are (substituting Eqs. (23) to (29) into (5)(6)(7)) c = ˙ φ M p γ h φ M p ( g ,φ + 2 g ) − g M p (cid:16) H ˙ φ + ¨ φ (cid:17) + 3 g ˙ φ i − ˙ HM p γ , (37) c = M p , (38) c = aM p γ Q ˜ m , (39)where γ = H + g M p ˙ φ , Q ˜ m = 1 + 2 ˜ m M p . (40)The sound speed squared c s of scalar perturbation is defined in Eq. (2). Here, when˜ m ≡ Q ˜ m = 1, the sound speed squared of the scalar perturbation is reduced to c s = 1 + 4 ˙ φ h g M p (cid:16) ¨ φ − H ˙ φ (cid:17) + g ˙ φ + ˙ φ M p ( g ,φ + g ) i HM p + ˙ φ h g M p (cid:16) H ˙ φ + ¨ φ (cid:17) − g ˙ φ − φ M p ( g ,φ + 2 g ) i , (41)It is easy to see that c s = 1 for inflation, since g = g = 0, but not for Genesis. However,using the operator ˜ m ( t )2 R (3) δg , we could always set c s = 1 in the Genesis phase, whichrequires ˜ m = − M p ( f + f )4 f + f . This suggests Q ˜ m = − f f + f is a constant at | − t | ≫
1, whichis consistent with the solution (19).The equation of motion of ζ is u ′′ + (cid:18) c s k − z ′′ z (cid:19) u = 0 , (42)with u = zζ , z = √ a c , the prime denotes the derivative with respect to the conformaltime τ = R dt/a . The initial state is the Minkowski vacuum, thus u = √ c s k e − ic s kτ for ζ modes deep inside the horizon. The power spectrum of ζ is P R = k π (cid:12)(cid:12)(cid:12) uz (cid:12)(cid:12)(cid:12) . (43)In the following, we will analytically estimate the spectrum of the scalar perturbation.We set c s = 1 throughout for simplicity, which could be implemented by using Q ˜ m ( t ), aswill be illustrated by the numerical simulation.10n the Genesis phase, substituting Eqs. (34), (35) into (5), we have c = 108 f M p (4 f + f ) ( − t ) . (44)Thus z = 6 √ f M p f + f ( − t ) · exp (cid:18) f − f M p t (cid:19) . (45)Then it is straightforward to obtain z ′′ z ≈ ( f − f ) M p τ ≈ τ , where τ = R a dt ≈ t . Thus thesolution of Eq. (42) is u = √− πτ h C · H (1)1 / ( − kτ ) + C · H (2)1 / ( − kτ ) i , (46)where C and C are functions of k , H (1) ν and H (2) ν are the first and the second kind Hankelfunctions of ν − th order, respectively. The initial condition u = √ k e − ikτ indicates C = i , C = 0 . (47)In the inflation phase, c = ǫM p , thus z = p ǫa M p . We set ǫ ≪ z ′′ /z ≈ (2 + 3 ǫ ) /τ . The solution of Eq. (42) is u = √− πτ (cid:2) C · H (1) ν ( − kτ ) + C · H (2) ν ( − kτ ) (cid:3) (48)with ν ≈ / ǫ .We require that u ( τ m ) = u ( τ m ) and u ′ ( τ m ) = u ′ ( τ m ), with τ m approximately corre-sponding to the beginning time of inflation phase, and we obtain C = − i e − ikτ m r − π kτ m h kτ m H (2) ν − ( − kτ m ) + (2 ν − − ikτ m ) H (2) ν ( − kτ m ) i , (49) C = i e − ikτ m r − π kτ m h kτ m H (1) ν − ( − kτ m ) + (2 ν − − ikτ m ) H (1) ν ( − kτ m ) i . (50)The power spectrum of ζ is given by P R = P inf R · | C − C | , k ≪ aH , (51)where P inf R = H inf π M p ǫ · (cid:0) kaH (cid:1) − ν is the power spectrum of scalar perturbation modes thatexit horizon during inflation. We see that for the perturbation modes exiting horizon inthe Genesis phase, − kτ m ≪ | C − C | ≃ ( − kτ m ) , thus P R ∼ k is strong blue-tilted, while for the perturbation modes exiting horizon in the inflation phase, − kτ m ≫ | C − C | ≃
1, thus P R ∼ k − ν = k − ǫ is flat with a slightly red tilt.11ensor perturbation is unaffected by the R (3) δg operator. Its quadratic action is givenin Eq. (4) with Q T = 1 and c T = 1. The spectrum of primordial GWs can be calculatedsimilarly, see also Ref. [6]. Since z ′′ T /z T = a ′′ /a , we have P T = P infT · | C − C | , k ≪ aH , (52)where P infT = H inf π M p · (cid:0) kaH (cid:1) − ν is the power spectrum of tensor perturbation modes that exithorizon during inflation. Thus the spectrum of primordial GWs has a shape similar to thatof the scalar perturbation. C. Numerical simulation
In the numerical calculation, we set g ( φ ) = f e φ f e φ tanh h q ( φ − φ ) i , (53) g , ( φ ) = f , − tanh h q , ( φ − φ ) i , (54) V ( φ ) = λ φ − φ ) h q ( φ − φ ) i (55)with f , , , q , , , , φ , , and λ being dimensionless constants. When φ ≪ φ , we have g = − f e φ , g = f and g = f , which brings a Genesis phase (36), while φ ≫ φ , we have g = 1 and g = g = 0, the slow-roll inflation will occur with V ( φ ) ∼ φ . When φ ≪ φ , V ( φ ) ≈
0, while φ ≫ φ , V ( φ ) ≈ λ ( φ − φ ) . We do not require φ = φ but φ > φ .We start the simulation at t i ≪ −
1, and we set˙ φ ( t i ) = 1( − t i ) , φ ( t i ) = 12 ln (cid:20) f f M p − t i ) (cid:21) , (56)and a ( t i ) = 1 , H ( t i ) = f − f M p − t i ) . (57)We show the evolution of φ and ˙ φ in Fig. 1, and the evolution of a , H and ǫ in Fig.2. In Fig. 3(a), c is plotted, and c > γ doesnot cross 0, which implies that, in the Genesis phase, c s < R (3) δg , we could have c s >
0, and so cure the gradient12nstability. The spectrum of the scalar perturbation can be simulated numerically, which isplotted in Fig. 5. The spectrum obtained has a cutoff at large scale k < k ∗ and is nearlyscale-invariant for k > k ∗ , as displayed in Eq. (51). FIG. 1: The evolution of φ and ˙ φ , while we set f = 5, f = − . f = − f , q = 1, q = 0 . q = 0 . q = 2, λ = 4 × − , φ = 7, φ = 22 . φ = 5 . (a) a and H (b) ǫ FIG. 2: The evolution of a , H and ǫ , while we set f = 5, f = − . f = − f , q = 1, q = 0 . q = 0 . q = 2, λ = 4 × − , φ = 7, φ = 22 . φ = 5 . IV. THE DILEMMA OF γ IN THE GENESIS SCENARIO
In the Genesis scenario based on the cubic Galileon [16], (also [21]), we have γ = H + f M p ˙ φ = f + 4 f M p ˙ φ (58)during the Genesis, where f <
0. 13 a) c (b) γ/H FIG. 3: The evolution of c , γ/H and ǫ , while we set f = 5, f = − . f = − f , q = 1, q = 0 . q = 0 . q = 2, λ = 4 × − , φ = 7, φ = 22 . φ = 5 . (a) c s (with ˜ m ≡
0) and c s (with ˜ m ( t )) (b) Q ˜ m and ˜ m (c) ˙ Q ˜ m FIG. 4: The sound speed of the scalar perturbation with and without ˜ m , while we set f = 5, f = − . f = − f , q = 1, q = 0 . q = 0 . q = 2, λ = 4 × − , φ = 7, φ = 22 . φ = 5 . IG. 5: The spectrum P R of the scalar perturbation, while we set f = 5, f = − . f = − f , q = 1, q = 0 . q = 0 . q = 2, λ = 4 × − , φ = 7, φ = 22 . φ = 5 . k ∗ corresponds tothe comoving wave number of the perturbation mode which exits horizon around the beginning ofinflation. In Ref. [16], f = − f , which suggests γ = f M p ˙ φ <
0. Thus if a hot “big bang” orinflation ( γ = H >
0) starts after the Genesis phase, γ must cross 0 at t γ ( c s < t γ , which may be cured by applying Q ˜ m ). It is obvious that when γ = 0, c in (1) willbe divergent. Though this divergence might not be a problem, it will affect the numericalsimulation for perturbations [47][48], unless Q T /γ is finite at t γ , as in Ijjas and Steinhardt’smodel [35].In the model of [7], the Genesis is followed by Galileon inflation [49]. Though f = − f and γ = f M p ˙ φ < γ < g = 0 in (22) during inflation. Thus it seems that γ might not necessarily cross 0.However, after inflation, γ crossing 0 is still inevitable.In our model, the Genesis is followed by the slow-roll inflation, γ = H > γ must satisfy γ >
0. In the Genesis phase, this suggests f > − f .Thus we will have γ > c s = 1 − f + 4 f f < . CONCLUSION Based on the EFT of cosmological perturbations, we revisit the nonsingular cosmologies,using the “non-integral approach”. By doing this, we could have a clearer understanding ofthe pathology in nonsingular Galileon models and its cure in EFT.We clarify the application of the operator ˜ m R (3) δg / Q ˜ m < γ = 0 is adopted to curethe gradient instability, in solution (13) (with γ < Q ˜ m = 1 initially), Q ˜ m must cross0 twice; while in solution (19) (with γ > Q ˜ m < Q ˜ m will cross 0 to Q ˜ m > t ˜ m , and crosses 0 only once. Thus at a certain time, Q ˜ m meeting 0 is required, as pointed out first by Cai et.al [39], and also by Creminelli et.al [40].We also clarify that in the bounce model with γ < c s < γ ≃ γ >
0, while the NEC is violated when ˙
H > γ that causes c s <
0. Here, we verify this point. In Genesis model [16][7],and also [21], the case is similar, as discussed in Sec. IV.The nonsingular model with the solution (19) ( γ > γ > γ >
Acknowledgments
We thank Youping Wan for helpful comments. The work of YSP is supported by NSFC,Nos. 11575188, 11690021, and also supported by the Strategic Priority Research Program ofCAS, Nos. XDA04075000, XDB23010100. The work of T. Q. is supported by NSFC underGrant Nos. 11405069 and 11653002. 16 ppendix A: EFT of cosmological perturbations
With the ADM line element, we have g µν = N k N k − N N j N i h ij , g µν = − N − N j N N i N h ij − N i N j N , (A1)and √− g = N √ h , where N i = h ij N j . The unit one-form tangent vector is defined as n ν = n ( dt/dx µ ) = ( − N, , ,
0) and n ν = g µν n µ = (1 /N , − N i /N ), which satisfies n µ n µ = − H µν = g µν + n µ n ν , thus H µν = N k N k N j N i h ij , H µν = h ij . (A2)The extrinsic curvature is K µν ≡ L n H µν , where L n is the Lie derivative with respective to n µ . The induced 3-dimensional Ricci scalar R (3) associated with H µν is R (3) = R + K − K µν K µν − ∇ µ ( Kn µ − n ν ∇ ν n µ ) . (A3)Without higher-order spatial derivatives, the EFT reads [39] S = Z d x √− g h M p f ( t ) R − Λ( t ) − c ( t ) g + M ( t )2 ( δg ) − m ( t )2 δKδg − m ( t ) (cid:0) δK − δK µν δK µν (cid:1) + ˜ m ( t )2 R (3) δg i + S m [ g µν , ψ m ] , (A4)where δg = g + 1, δK µν = K µν − H µν H with H being the Hubble parameter. The coeffi-cient set ( f, c, Λ , M , m , m , ˜ m ) specifies different theories and could be time-dependent ingeneral . A particular subset ( m = ˜ m ) of EFT (A4) is the Horndeski theory. S m [ g µν , ψ m ]is the matter part, which is minimally coupled to the metric g µν .To obtain the quadratic actions for scalar and tensor perturbations, we will work in theunitary gauge, thus we set h ij = a e ζ ( e γ ) ij , γ ii = 0 = ∂ i γ ij . (A5) Different conventions of the nomenclatures of these coefficients were adopted during the development ofthe EFT of cosmological perturbations (see e.g., [41][42][43][44]). Here, we follow the convention used inRefs. [43][44]. ζ and tensor pertur-bation γ ij , as exhibited in Eqs. (1) and (4), respectively (see [39] for detailed derivations). [1] A. Borde and A. Vilenkin, Phys. Rev. Lett. , 3305 (1994) [gr-qc/9312022].[2] A. Borde, A. H. Guth and A. Vilenkin, Phys. Rev. Lett. , 151301 (2003) [gr-qc/0110012].[3] Y. -S. Piao, B. Feng and X. -m. Zhang, Phys. Rev. D , 103520 (2004) [hep-th/0310206];Y. -S. Piao, Phys. Rev. D , 087301 (2005) [astro-ph/0502343]; Y. -S. Piao, S. Tsujikawaand X. -m. Zhang, Class. Quant. Grav. , 4455 (2004) [hep-th/0312139].[4] Z. G. Liu, Z. K. Guo and Y. S. Piao, Phys. Rev. D , 063539 (2013) [arXiv:1304.6527[astro-ph.CO]].[5] T. Qiu and Y. T. Wang, JHEP , 130 (2015) [arXiv:1501.03568 [astro-ph.CO]].[6] Z. G. Liu, H. Li and Y. S. Piao, Phys. Rev. D , no. 8, 083521 (2014) [arXiv:1405.1188[astro-ph.CO]].[7] D. Pirtskhalava, L. Santoni, E. Trincherini and P. Uttayarat, JHEP , 151 (2014)[arXiv:1410.0882 [hep-th]].[8] T. Kobayashi, M. Yamaguchi and J. Yokoyama, JCAP , no. 07, 017 (2015)[arXiv:1504.05710 [hep-th]].[9] Z. G. Liu, Z. K. Guo and Y. S. Piao, Eur. Phys. J. C , no. 8, 3006 (2014) [arXiv:1311.1599[astro-ph.CO]].[10] Y. F. Cai, T. Qiu, Y. S. Piao, M. Li and X. Zhang, JHEP , 071 (2007) [arXiv:0704.1090[gr-qc]].[11] Y. F. Cai, T. t. Qiu, R. Brandenberger and X. m. Zhang, Phys. Rev. D , 023511 (2009)[arXiv:0810.4677 [hep-th]].[12] Y. B. Li, J. Quintin, D. G. Wang and Y. F. Cai, JCAP , no. 03, 031 (2017)[arXiv:1612.02036 [hep-th]].[13] M. Roshan and F. Shojai, Phys. Rev. D , no. 4, 044002 (2016) [arXiv:1607.06049 [gr-qc]].[14] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, Phys. Rev. D , 123522 (2001)[hep-th/0103239].[15] J. L. Lehners, Phys. Rept. , 223 (2008) [arXiv:0806.1245 [astro-ph]].
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