The Effective Field Theory of nonsingular cosmology: II

Based on the Effective Field Theory (EFT) of cosmological perturbations, we explicitly clarify the pathology in nonsingular cubic Galileon models and show how to cure it in EFT with new insights into this issue. With the least set of EFT operators that are capable to avoid instabilities in 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 followed by 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.


I. INTRODUCTION
Inflation is still being eulogized for its simplicity and also criticized for its pastincompleteness [1] [2]. A complete description of the early universe requires physics other than only implementing inflation.
See also [34] for the extension to the full Horndeski theory, and [35] for the attempts to avoid "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 the Effective Field Theory (EFT) [41] [42] [43] [44], which has proved to be a powerful tool. In EFT, the quadratic action of scalar perturbation could always be written in the form (see [39] for detailed derivations) where we have neglected higher-order spatial derivatives of the scalar perturbation ζ, the sound speed squared of scalar perturbation with the coefficients c 1 , c 2 and c 3 being time dependent parameters in general, and c 1 > 0 is needed to avoid the ghost instability. The condition for avoiding gradient instability is The condition of satisfying the inequality is to have c 3 cross 0, which is hardly possible in models based on the cubic Galileon [33] [34]. However, we found that it can easily be satisfied by applying the EFT operator R (3) δg 00 (with R (3) being the 3-dimensional Ricci scalar on spacelike hypersurface and δg 00 = g 00 + 1), so that the gradient instability can be cured.
Though the integral approach (3) is simple and efficient, some details of curing the pathology might actually be missed. In this paper, based on the EFT, using a "non-integral approach", we revisit the nonsingular cosmologies. We begin straightly with (2)

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 action of tensor perturbation γ ij is (see [39] for the derivation of Eqs. (4) to (8)) where 2 4 are coefficients defined in the EFT action (A4).
The quadratic action of the scalar perturbation ζ is given by Eq. (1) with where M 4 2 , m 3 3 andm 2 4 are coefficients defined in the EFT action (A4) and they could be time dependent in general.
Only if c 1 > 0 and c 2 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 1 > 0 is not hard to obtain, as can be seen from Eq. (5), since the cubic Galileon contributes the m 3 3 (t) 2 δKδg 00 operator in EFT. However, since c 3 is also affected by 2 δKδg 00 through γ, c 2 s < 0 is actually inevitable, as will be demonstrated in the following. Since c 1 > 0, the requirement of c 2 s > 0 equals Here, Q T = Qm 4 is required, which cannot be embodied by the Horndeski theory [30][31] [32].
Thus whether c 2 s > 0 or not is controlled by the parameter set (H, γ, Q T , c 2 T , Qm 4 ). In the following, with condition (9), we will re-prove the "no-go" theorem for the cubic Galileon, and clarify how to cure it in EFT. Different from the proof in [33][39] [40], the re-proof is directly based on the derivative inequality instead of integrating it, which we called "non-integral approach". We assume that after the beginning of the hot "big bang" or inflation, γ = H > 0,γ < 0 and Qm 4 = 1.
In the ekpyrotic and bounce models, initially γ = H < 0. In the Genesis model [16] and slow expansion model [21], H > 0 during the Genesis, but actually γ = H − In the cubic Galileon case, f = Q T = Qm 4 = 1. Around t γ , condition (9) is We see that c 2 s < 0 is inevitable around t γ , sinceγ > 0. Thus the nonsingular models based on cubic Galileon is pathological, as first proved by LMR in [33].
In the EFT case, around t γ , condition (9) requires We might have c 2 s > 0, only if (considering only the case where only one of Q T and Qm 4 is modified while the unmodified one is unity) around γ = 0 or In solution (12), at t γ , γ = 0 suggests Q T = 0. Here, since γ = 0 at t γ , 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 2p and p, n being constants. In Ijjas and Steinhardt's model [35], In the bounce model based on the cubic Galileon, Eq. (8) gives Generally, the NEC is violated whenḢ > 0, while the period of c 2 s < 0 corresponds to the phase with γ ≃ 0 andγ > 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 the pathology. Here, we reconfirmed this point.
We see again the details of Qm 4 crossing 0. In both the Genesis model and the bounce model, initially Qm 4 = 1, so if Qm 4 < 0 around t γ , Qm 4 must cross 0 twice. Thus it seems thatQm 4 γ > c 2 T γ 2 is hard to implement. However, with (2) and (7), one always could solve Qm 4 for any given c 2 s , where Q T = 1.

B. Case II: γ > 0 throughout
Since γ > 0 throughout, we must haveγ 0 during some period initially 1 , otherwise γ will diverge in the infinite past.
In the cubic Galileon case, condition (9) is In the bounce model, H < 0 in the contracting phase, and in the Genesis model, H ∼ 0 in the Genesis phase, both suggest Hγ − γ 2 −γ < 0 2 . Thus c 2 s < 0 is inevitable in the corresponding phases, so the nonsingular models based on the cubic Galileon is pathological.
We see the Genesis model in the cubic Galileon version again in detail. During the slow expansion (Genesis phase), H ∼ 1/(−t) n with the constant n > 1. Thuṡ which implies Hγ ≪γ. Thus with (15), we see that c 2 s < 0 is inevitable in the slow expansion phase. It seems that if n = 1, Hγ ≪γ might be avoided. However, when n = 1, we have H = p/(−t) and a ∼ 1/(−t) p with constant p, thus a → 0 in the infinite past. From (16), we see that c 2 s > 0 requires p = H 2 /Ḣ > 1. Therefore, the universe is singular, or from another point of view, it is geodesically incomplete since the affine parameter of the graviton In the EFT case, condition (9) requires We might have c 2 s > 0, only if (considering only the case where either Q T or Qm 4 is modified) 1 Of course, in Case II, we could also haveγ < 0 during some period, but what we focus on is the period (i.e.,γ 0) where pathologies appear. 2 In the case where γ grows from 0 initially, (15) is also obeyed no more.

III. APPLICATION TO GENESIS-INFLATION
In this section, we will build a nonsingular model with the solution (19), in which the slow-roll inflation is preceded by a Genesis phase. A Genesis phase is a slowly expanding phase originating form the Minkowski vacuum with a drastic violation of NEC, i.e., ǫ ≪ −1, thus the energy density is increasing with the expansion of the universe and hence is free from the initial singularity [16][17] (see also [20]). As will be shown below, our model cannot only get rid of the pathology of instability, but also give rise to a flat spectrum with interesting features at large scales.
During inflation, we set g 1 = 1 and g 2 = g 3 = 0, since we require that the inflationary phase is controlled by a simple slow-roll field 3 .
3 The behaviors of these g i 's in the two phases can easily be matched together by making use of some shape functions [5] [46].

B. The primordial perturbation and its spectrum
In the unitary gauge, the quadratic action of the scalar perturbation is presented in the form of Eq. (1). The coefficients c i are (substituting Eqs. (23) to (29) into (5)(6)(7)) where The sound speed squared c 2 s of scalar perturbation is defined in Eq. (2). Here, wheñ m 2 4 ≡ 0 or Qm 4 = 1, the sound speed squared of the scalar perturbation is reduced to It is easy to see that c 2 s0 = 1 for inflation, since g 2 = g 3 = 0, but not for Genesis. However, using the operatorm 2 4 (t) 2 R (3) δg 00 , we could always set c 2 s = 1 in the Genesis phase, which 4f 2 +f 3 . This suggests Qm 4 = − 3f 3 4f 2 +f 3 is a constant at | − t| ≫ 1, which is consistent with the solution (19).

The equation of motion of ζ is
with u = zζ, z = √ 2a 2 c 1 , the prime denotes the derivative with respect to the conformal time τ = dt/a. The initial state is the Minkowski vacuum, thus u = 1 √ 2csk e −icskτ for ζ modes deep inside the horizon. The power spectrum of ζ is In the following, we will analytically estimate the spectrum of the scalar perturbation.
We set c 2 s = 1 throughout for simplicity, which could be implemented by using Qm 4 (t), as will be illustrated by the numerical simulation.
We require that u 1 (τ m ) = u 2 (τ m ) and u ′ 1 (τ m ) = u ′ 2 (τ m ), with τ m approximately corresponding to the beginning time of inflation phase, and we obtain The power spectrum of ζ is given by where P inf 3−2ν 2 is the power spectrum of scalar perturbation modes that exit horizon during inflation. We see that for the perturbation modes exiting horizon in the Genesis phase, −kτ m ≪ 1, |C 21 − C 22 | 2 ≃ (−kτ m ) 2 , thus P R ∼ k 2 is strong bluetilted, while for the perturbation modes exiting horizon in the inflation phase, −kτ m ≫ 1, Tensor perturbation is unaffected by the R (3) δg 00 operator. Its quadratic action is given in Eq. (4) with Q T = 1 and c 2 T = 1. The spectrum of primordial GWs can be calculated similarly, see also Ref. [6]. Since z ′′ T /z T = a ′′ /a, we have where P inf 3−2ν 2 is the power spectrum of tensor perturbation modes that exit horizon during inflation. Thus the spectrum of primordial GWs has a shape similar to that of the scalar perturbation.

C. Numerical simulation
In the numerical calculation, we set with f 1,2,3 , q 1,2,3,4 , φ 0,1,2 and λ being dimensionless constants. When φ ≪ φ 0 , we have g 1 = −f 1 e 2φ , g 2 = f 2 and g 3 = f 3 , which brings a Genesis phase (36), while φ ≫ φ 0 , we have g 1 = 1 and g 2 = g 3 = 0, the slow-roll inflation will occur with V (φ) ∼ φ 2 . When φ ≪ φ 2 , We do not require φ 0 = φ 2 but φ 0 > φ 2 . We start the simulation at t i ≪ −1, and we seṫ and We show the evolution of φ andφ in Fig. 1, and the evolution of a, H and ǫ in Fig.   2. In Fig. 3(a), c 1 is plotted, and c 1 > 0 is satisfied. In Fig. 3(b), we see that γ does not cross 0, which implies that, in the Genesis phase, c 2 s0 < 0 (see Fig. 4(a)), as proved in Sec. IV. By including the operator R (3) δg 00 , we could have c 2 s > 0, and so cure the gradient instability. The spectrum of the scalar perturbation can be simulated numerically, which is plotted in Fig. 5. The spectrum obtained has a cutoff at large scale k < k * and is nearly scale-invariant for k > k * , as displayed in Eq. (51).

IV. THE DILEMMA OF γ IN THE GENESIS SCENARIO
In the Genesis scenario based on the cubic Galileon [16], (also [21]), we have during the Genesis, where f 2 < 0. In Ref. [16], Thus if a hot "big bang" or inflation (γ = H > 0) starts after the Genesis phase, γ must cross 0 at t γ (c 2 s0 < 0 around t γ , which may be cured by applying Qm 4 ). It is obvious that when γ = 0, c 1 in (1) will be divergent. Though this divergence might not be a problem, it will affect the numerical simulation for perturbations [47] [48], unless Q T /γ 2 is finite at t γ , as in Ijjas and Steinhardt's model [35].
However, after inflation, γ crossing 0 is still inevitable.
In our model, the Genesis is followed by the slow-roll inflation, γ = H > 0 for inflation.
Thus we will have γ > 0 throughout. However, for the cubic Galileon, the expense is during the Genesis. Here, this pathology is cured in EFT by applying (19).

V. 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 of the pathology in nonsingular Galileon models and its cure in EFT.
We clarify the application of the operatorm 2 4 R (3) δg 00 /2 in EFT, which is significant for curing the gradient instability. We show that if Qm 4 < 0 around γ = 0 is adopted to cure the gradient instability, in solution (13) (with γ < 0 and Qm 4 = 1 initially), Qm 4 must cross 0 twice; while in solution (19) (with γ > 0 throughout), initially Qm 4 < 0 must be satisfied, Qm 4 will cross 0 to Qm 4 > 0 at tm 4 , and crosses 0 only once. Thus at a certain time, Qm 4 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 < 0 initially, c 2 s < 0 will occur in the phase with γ ≃ 0 andγ > 0, while the NEC is violated whenḢ > 0 (bounce phase), these two phases do not necessarily coincide. As pointed out by Ijjas and Steinhardt [35], it is the sign's change of γ that causes c 2 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 > 0 throughout) has not been studied before. In Sec. III, we design such a model, in which a slow expansion phase (namely, the Genesis phase) is followed by slow-roll inflation. Under the unitary gauge, sinceγ > 0 and γ > 0 (not crossing 0), the evolution of primordial perturbations can be simulated numerically. The simulation displays that the spectrum acquires a large-scale cutoff, as expected in Ref. [6].
We conclude that, based on EFT, not only a stable nonsingular cosmological scenario may be built without getting involved in unknown physics, but also the phenomenological possibilities of its implementation are far richer than expected (see also recent [50][51] for the higher spatial derivative operators).