The Effective Field Theory of nonsingular cosmology

In this paper, we explore the nonsingular cosmology within the framework of the Effective Field Theory(EFT) of cosmological perturbations. Due to the recently proved no-go theorem, any nonsingular cosmological models based on the cubic Galileon suffer from pathologies. We show how the EFT could help us clarify the origin of the no-go theorem, and offer us solutions to break the no-go. Particularly, we point out that the gradient instability can be removed by using some spatial derivative operators in EFT. Based on the EFT description, we obtain a realistic healthy nonsingular cosmological model, and show the perturbation spectrum can be consistent with the observations.


I. INTRODUCTION
In the modern era of Cosmology, theories of Hot Big Bang (HBB) and Inflation have achieved great success, and thus have long been regarded as the standard paradigm of the early universe. However, the inflation still suffers from the cosmological singularity problem [1] [2], unless it was preceded by a bounce [3][4] [5] or a Genesis phase [6][7] [8]. It is exciting to study classical nonsingular cosmology, such as bounce universe models [9][10], Genesis models [11][12] [13], slow expansion models [14][15] [16], since we might get classical nonsingular cosmology without begging the details of the unknown UV-complete gravity theory.
One of the most exciting endeavors in this area is to build nonsingular cosmological models with the field theories which can violate the Null Energy Condition (NEC) [17].
Usually the violation of NEC may lead to the ghost instability [18]. This problem can be solved if one considers the so-called Galileon theory [19] or its generalizations (such as the Horndeski theory [20] [21] and its beyond [22]). Making use of the simplest cubic Galileon, many heuristic nonsingular cosmological models have been constructed, e.g. [11] [15][23] [24].
However, it seems quite difficult to avoid the gradient instability problem, which indicates a negative sound speed squared [5] [7][25] [26] and thus leads to an exponential growth of the perturbation [27] [28].
Recently, Libanov, Mironov and Rubakov (LMR) have proved a no-go theorem, which shows that healthy nonsingular cosmological models based on the cubic Galileon does not exist [29]. Hereafter, it was generalized with an additional scalar in Ref. [30] or with the full Horndeski theory in Ref. [31]. However, Ijjas and Steinhardt claimed that there exists a loophole in the proof of Ref. [31] (which was also noticed by the author of Ref. [31]), and they can even reconstruct a fully stable classical bounce [32] throughout the whole evolution by using the "inverse method" [33]. However, we believe that this relevant issue still needs to be studied further.
Prior to LMR's work, studies were also made along other lines. The danger of c 2 s < 0 is mainly attributed to the exponential growth of the amplitude of short wavelength modes.
In [27] [28], it was argued that the strong coupling scale during c 2 s < 0 is low so that the dangerous short wavelength modes lie outside the range of the validity of the effective theory, thus can be disregarded. However, this argument begs unknown strong coupling physics, which actually makes the "classical nonsingular" bounce loose sense. What is the effective theory of nonsingular cosmology is a significant issue. It is interesting to notice that some spatial covariant operators also help to remove the gradient instability [5][7] [8].
The Effective Field Theory (EFT) of cosmological perturbations is extremely powerful and has been widely used to study inflation [34][35] and dark energy [36][37] [38]. It offers a unifying platform to deal with the cosmological perturbations of all kinds of theories, such as the Horndeski theory and its beyond, the Horava gravity [39], and the spatial covariant gravity [40] [41]. In the following context, we will see that it is also a powerful tool for studying nonsingular cosmology.
In this paper, we will explore how to build healthy nonsingular cosmological models within the framework of EFT. Practically, in Sec.II, based on EFT, we clarify how to understand the no-go theorem and how to avoid it. We find that some effective operators can play significant role in building nonsingular cosmological models without pathologies. In Sec.III, we study the evolution of primordial perturbation in nonsingular models with these corresponding effective operators, and find the perturbation spectrum can be consistent with the observations. In Sec.IV, we present a realistic healthy nonsingular bounce model by introducing an effective operator of R (3) δg 00 . Finally, we conclude in Sec.V.
Note added: After our paper appeared in arXiv, nearly simultaneously Creminelli sent us their draft (the preprint [42]), which overlaps substantially with ours.

II. THE FRAMEWORK OF EFT AND THE NO-GO THEOREM
We consider the metric in the ADM form: where N and N i are the lapse function and shift vector, and h ij is the 3-dimentional spatial metric.
With the spirit of the EFT of cosmological perturbation [34][36] [39], we write down the EFT action for nonsingular cosmological models where we turn off the accelerator vectors a i in [39] for simplicity. We assume the matter part is minimally coupled to field so that the expansion or contraction of the background with respect to physical rulers is unambiguous. The first line describes the background of our model, while the rest is for perturbations. One is also allowed to contain terms such as jk , which we don't bother to write them explicitly and just put them into the ellipsis. All the coefficients are allowed to vary with t, with the dimension [m i ] = 1, [λ i ] = 0, so as to make the action dimensionless. Moreover, in this action we define δK µν = K µν − HH µν , δK = K − 3H , with the induced metric H µν ≡ g µν + n µ n ν and the normal vector is defined as n µ ≡ (−N, 0, 0, 0).
It is rather straightforward to fix the relations among the functions f (t), c(t) and Λ(t), which is in the background part. Varying the first line of action Eq.(2) with respect to N and a, one can get the two equations: For the minimal coupling theories where f (t) = 1, these are nothing but the Friedmann equations, thus we have c(t) = −M 2 pḢ and Λ(t) = M 2 p (Ḣ + 3H 2 ). The c(t) and Λ(t) have the same expressions as those in the EFT of inflation, however, to have a non-singular scenario a crucial condition must be satisfied, i.e., the violation of NEC. That means c(t) must be negative at least for a while. Since the NEC will finally be restored in the expanding universe, we conclude for the EFT of nonsingular cosmology, c(t) must be a function that can pass the zero boundary. For the case with non-minimal coupling, f (t) is nontrivial, then a more complicated constraint will be imposed on c(t) and Λ(t).

A. The no-go theorem
It is straightforward to derive the quadratic action of scalar and tensor perturbation from Eq.(2). We give some main steps of the derivation in Appendix A and just write down the results here. Under the unitary gauge, the quadratic action of scalar perturbation is where we have left the expressions of c i in Appendix A since they are complicated (except for the c 2 , which has a quite simple expression as c 2 = M 2 p f (t)). The sound speed squared reads The conditions to avoid the ghost instability and the gradient instability are Moreover, the quadratic action of tensor perturbation from Eq. (2) is where To avoid the ghost and gradient instability for tensor modes, we need Q T > 0 and c 2 T > 0, respectively.
We begin withċ 3 − ac 2 > 0, which indicates This expression is the key inequality to clarify the no-go theorem. This inequality turns out to be remarkablely general since it is correct not only for the Horndeski theory, but also for these theories beyond the Horndeski. As matter of fact, by mapping the cubic Galileon to the EFT [37], Eq.(10) will lead to the key inequality used to prove the LMR no-go theorem [29] (see the following part of this subsection); and by mapping the whole Horndeski theory to the EFT [37], Eq.(10) will produce the key inequality in Kobayashi's paper [31]. Now let's consider the cubic Galileon L 2 + L 3 with f (t) = 1, Eq.(10) reads and according to the Appendix. A, we find where we can tell that c 3 t f will finally be larger than zero, thus c 3 must equal to zero at sometime t with t i < t < t f , making γ blows away. Therefore the gradient instability cannot be avoided.
The remaining case is that c 3 be always positive. However, from and let t i → −∞, we see this is impossible in a similar manner. So we have reformulated the LMR no-go theorem [29] for the cubic Galileon in the framework of EFT, which indicates the pathologies in nonsingular cosmological models based on the cubic Galileon are inevitable.
It is interesting to note that Eq.(8) can be reformulated as after a disformal redefinition of the metric. Here we have defined a E = c 1/2 and We firstly consider the addition of the effective operator R (3) δg 00 to the cubic Galileon.
It gives a contribution with (∂ζ) 2 ∼ k 2 ζ 2 k to the scalar perturbation, while does not change the tensor perturbation at quadratic order. The EFT action is written as: here we have set f (t) = 1, the coefficients c(t), Λ(t), M 4 2 (t) and m 3 3 (t) can be found by requiring that they have the same time-dependent behaviors as in the cubic Galileon L 2 +L 3 .
The existence of the lastm 2 4 (t) term indicates this model Eq. (17) goes beyond the Horndeski. Note, the dynamical equation for the true degree of freedom is still second order, thus thẽ m 2 4 (t) term here, as well as the higher order spacial derivative terms (R (3) ) 2 and ∇ i R (3) ∇ i R (3) used below, does not introduce the Ostrogradski instability (see, e.g., [37]). According to the Appendix. A, we have However, if at that time we have 2m 2 4 /M 2 p cross −1, the c 3 will cross 0 naturally without the divergence of γ. So the no-go behavior can be avoided, and notice for Eq.(17), since m 2 4 (t) = 0 and Q T = 1, the tensor perturbation will be healthy. Generally, we could set the effective operatorm 2 4 R (3) δg 00 /2 to be dominated only when we meet c 2 s < 0, thus it just modifies the sound speed squared during this time, see Sec. IV for details.
We can further add the term −m 2 4 (t) (δK 2 − δK µν δK µν ) into the effective action Eq.(17), then c 3 changes to be Generally, we may find those coefficients with 2m 2 4 /M 2 p crossing −1 and m 2 4 (t) =m 2 4 (t) which goes beyond Horndeski, to make c 3 cross 0 while Q T = 0 and γ won't blow up. The case of Q T /γ crosses 0 is discussed in [32] with L 4 . However, when m 2 4 (t) =m 2 4 (t), such as for the case of Horndeski theory, crosses 0 suggests the no-go behavior must happen unless the integral (10) is convergent or Q 2 T /γ crosses 0. Obviously, this argument also applies to a general L 4 with time-dependent f (t), as has been argued by T. Kobayashi [31] (see also [32]).
Furthermore, let's consider the effective operators (R (3) ) 2 and ∇ i R (3) ∇ i R (3) , which will give contributions to higher order spatial derivatives with k 4 ζ 2 k and k 6 ζ 2 k . As the operator (R (3) ) 2 has been applied to the nonsingular cosmology in [7], here we'd like to take the following nonsingular model with f (t) = 1, and the coefficients c(t), Λ(t), M 4 2 (t) and m 3 3 (t) are taken according to the EFT mapping for the cubic Galileon L 2 + L 3 [37]. Then we have an effective sound speed squared with where z = √ 2a 2 c 1 and c 2 s are given by Eq. (6). From the equation of motion of ζ Eq.(23), we see that if these effective operators with higher order spatial derivatives have not been included, we'll have a solution of ζ ∼ e −i √ c 2 s k∆τ which indicates an exponential growth when c 2 s < 0. However, the growth turns out to be negligible for the perturbation modes with k∆τ ≪ 1, and can be quite dangerous for the modes with k∆τ ≫ 1 [27] [28]. So we may specifyλ(t) to make c 2 s,ef f (k) ∼ c 2 s for the modes with k∆τ ≪ 1, while make c 2 s,ef f (k) modified to be positive for the modes with k∆τ ≫ 1. Then such kind of exponential growth of ζ due to c 2 s < 0 can be removed.

WORK OF EFT
In the last section we have presented how to evade the no-go theorem within framework of EFT, i.e., by adding the effective operators such as , to the original nonsingular cosmological models based on the cubic Galileon. One might ask if the perturbation spectrum will be modified due to these operators. In this section, we study the perturbation evolution in detail, and show that the predictions can be consistent with the observations.
The equation of motion of ζ is where with u = zζ, z = √ 2a 2 c 1 and c 2 s is given by Eq. (6), the prime denotes the derivative with respect to the conformal time τ = dt/a.
To study the evolution of the primordial perturbation concretely, let's consider a bounce inflation background in this section. We can define the "bouncing phase" as the time interval during which the NEC is violated, i.e.,Ḣ > 0. At the beginning time τ B− and the ending time τ B+ of the "bouncing phase" we haveḢ = 0, while before the beginning time and after the ending time, the NEC is restored and thus leads toḢ < 0.
By adding the effective operators like R (3) δg 00 , (R (3) ) 2 or ∇ i R (3) ∇ i R (3) to the original G-bounce models [23] [44], within the framework of EFT the whole Lagrangian tends to be like the ones in Eq. (17) or Eq. (21). To cure the gradient instability problem in these models, we can set the corresponding operators to be dominated only during the duration ∆τ = τ B+ − τ B− of the bouncing phase. Thus Eq.(23) can be written as where n ≥ 1, and the corresponding operators will respectively contribute ∼ k 2 , k 4 , k 6 corrections to the equation of motion. To be rigorous, all the coefficients A 2n of the k 2n terms should be time-dependent. However, here we set A 2n constant for simplicity.
During the contracting phase τ < τ B− , the background can be parameterized as where ν c = 1/2 − 1 ǫc−1 . The solution of Eq.(25) can be given as where H (1) ν and H (2) ν are the ν-th order Hankel function of the first and the second kind. Initially, the perturbations are deep inside the horizon. The initial condition can be taken During the bouncing phase τ B− < τ < τ B+ , the solution of Eq. (26) is where c 2,1 and c 2,2 are determined by the evolution of the contracting phase. By considering the effective operators, the effective sound speed squared c 2 s,ef f > 0 for short wavelength perturbation modes, thus there won't be any dangerous growths of the curvature perturbation ζ.
During the inflation τ > τ B+ , the background can be parameterized as where ν e = 1/2 − 1 ǫe−1 . The solution of Eq.(25) can be given as The power spectrum is calculated as The information of the evolution history of the universe and the contributions of the EFT operators are encoded in c 3,1 and c 3,2 . Though we work with bounce inflation scenario, actually, our result is also applicable to the bounce scenario, as will be seen.
By requiring the continuity of u and u ′ at the matching surfaces, we obtain where the components of the matrix M (2,1) are and H B+ is the comoving Hubble parameter at τ B+ .
Considering the long wavelength limit, k/H B+ ≪ 1, we have In bounce scenario where the bounce is followed by the Hot Big-Bang expansion, P ζ is given by Eq. (37), since the perturbation modes with k/H B+ > 1 can be hardly produced during the expansion after the bounce. The result is consistent with that in ekpyrotic universe [45] [46] [47].
Thus the spectrum of primordial perturbations in bounce scenario is unaffected by the corresponding spatial derivative operators in Eq.(2).
However, in the bounce inflation scenario, the perturbation modes with k/H B+ > 1 will be produced during the inflation after the bounce. When we take the short wavelength limit, k/H B+ ≫ 1, the |c 3,1 − c 3,2 | 2 acquires drastic oscillation and even diverges when k/H B+ → ∞. Without making qualitative deviation, we have The "1" in right-hand side of Eq.(39) actually stands for the terms ∼ k 0 , such as cos 2 (2dA n k n ). Here, we do not specify it, since it makes no qualitative difference when k/H B+ ≫ 1. In order to satisfy the observations, Eq.(39) should be nearly scale invariant.
Thus the operator R (3) δg 00 ∼ k 2 in EFT Eq.(2) is applicable, but the operators (R (3) ) 2 ∼ k 4 for k/H B+ ≫ 1, which are unacceptable. This result could be general, though the drastic oscillations in Eq.(39) might be attributed to the matching method and the oversimplified approximation we have used.

MODEL
In this section we apply the effective operator R (3) δg 00 to cure the gradient instability faced by the G-bounce inflation model proposed in [5] (see also [26]). The G-bounce inflation background was built by using the cubic Galileon, which can be written in the EFT language as c(t) = 1 2φ  though we still don't know the complete theory of the quantum gravity. However, the popular nonsingular cosmological models based on the cubic Galileon are afflicted by the LMR no-go theorem, which means we have to go beyond the cubic Galileon to construct models without pathologies.
In this paper, we have explored the nonsingular cosmology within the framework of EFT.
We have illustrated how to avoid the no-go theorem in theories beyond Horndeski, and pointed out how could the effective operators, such as R (3) δg 00 , (R (3) ) 2 and ∇ i R (3) ∇ i R (3) , play significant roles in building healthy nonsingular cosmological models. We also have studied the perturbation evolution of these healthy models. We find that the spectrum of the primordial perturbation can be consistent with the observations.
We conclude that based on EFT, a fully healthy nonsingular bounce model can be built without begging any unknown physics. As an application of the EFT, we have presented a realistic healthy bounce inflation model by making use of the operator R (3) δg 00 . The study of classical nonsingular cosmology in the framework of EFT will be helpful for understanding the evolution and the gravity theory in the primordial universe.
during the time while this work was started. YW also wishes to thank T. Qiu and 3-dimensional metric on the hypersurface is H µν = g µν + n µ n ν , thus Moreover, the extrinsic curvature on the hypersurface is where L n is the Lie derivative with respective to n µ , and D µ is the covariant derivative associate with H µν . The Ricci scalar is decomposed as where R (3) is the induced 3-dimensional Ricci scalar associated with H µν . Note that in general, when there is a non-minimal coupling between the scalar field and R, the last term in Eq.(A4) cannot be discarded.
Substituting Eqs.(A10) to (A13) into the action (2) and using the Hamiltonian constraints we find Note that δK = K − 3H contains only scalars up to the quadratic order, as well as the last term in Eq.(A4). Substituting the above results into action (2), we obtain the quadratic action of tensor perturbation, which is displayed in Eq. (8).