Towards the bounce inflationary gravitational wave

In bounce inflation scenario, the inflation is singularity-free, while the advantages of inflation are reserved. We analytically calculate the power spectrum of its primordial gravitational waves (GWs), and show a universal result including the physics of the bounce phase. The spectrum acquires a cutoff at large scale, while the oscillation around the cutoff scale is quite drastic, which is determined by the details of bounce. Our work highlights that the primordial GWs at large scale may encode the physics of the bounce ever happened at about $\sim 60$ efolds before inflation.

Recently, the Planck collaboration [14] have observed the power deficit of CMB TTmode spectrum at large scale. This inspired theorists to think over the physics of the pre-inflation, about ∼ 60 efolds before the inflation during which the evolutions of largest scale perturbations are involved, e.g. [15]. It is interesting that the pre-inflationary physics suggested by the large-scale power deficit might be relevant with the initial singularity problem. The bounce universe, as the solution to this problem, has a long history of study, see [16], [17] for reviews. In Refs. [6], [7], [8], it has been discovered that in the bounce inflation scenario the large-scale anomalies of CMB TT spectrum may be explained naturally. This achievement makes the bounce inflation acquire increasing attention.
The primordial GWs is also the production of the inflation. Recently, lots of experiments aiming at detecting GWs have been implemented or will be implemented, which will bring us a new epoch to understand the physics of inflation scenario, e.g. [18]. There are also others designs to explain the large-scale power deficit of scalar perturbation spectrum, the suppression is attributed to the rapid rolling of scalar field. However, the primordial GWs is independent of the dynamics of scalar field, which thus may be used to identify the physics of the pre-inflationary background.
Thus the features of the primordial GWs at large scale might tell us if such a bounce has ever happened at about ∼ 60 efolds before inflation. Moreover, it might also help us to speculate the property of cyclic universe [24], [25], [26], [27] with such a nonsingular bounce.
In this paper, we analytically calculate the bounce inflationary GWs. The resulting spectrum is written with the recursive Bogoliubov coefficients including the physics of the bounce phase. We also show that our analytic result is completely consistent with the numerical plotting for a realistic model of bounce inflation.

II. OVERVIEW OF BOUNCE INFLATION SCENARIO
FIG. 1: The evolution of a and H =ȧ/a in bounce inflation scenario, based on the model in Ref. [12], which will be showed in details in Sec.IV.
In bounce inflation scenario, the inflation is singularity-free, while the advantages of inflation are reserved, which simply and naturally explains the universe we live.
The idea of bounce inflation showed itself first in Ref. [6], which explained the large-scale suppression of scalar perturbation spectrum observed by WMAP. Based on the Quintom bounce [28], Ref. [29] investigated the evolution of the primordial perturbations in details, and recently, Qiu and Wang [12], and Wan et.al [13], have constructed a realistic model of bounce inflation without instabilities by applying the higher-derivative operator.
The bounce inflation also has been implemented in the positively curved background [9], or by modifying gravity [7] [10]. The matching of perturbation through the bounce with the modified gravity was discussed in [26].
Recently, Liu et.al [8] have found that the bounce inflation brings not only the largescale power deficit of CMB, but also a large hemispherical power asymmetry, as implied by Planck. Current bounds on the primordial GWs have been used to constrain the bounce inflation [11]. However, Ref. [11] used a step-like parameterisation of the primordial GWs spectrum, based on [6], which is only a rough estimate missing the effect of the bounce.
Recently, the detecting of primordial GWs has been on the road, which will possibly tell us more on the inflation and its origin. Therefore, it is significant to have a full study for the bounce inflationary GWs.

A. The contracting phase
The contracting phase is the evolution with H < 0 andḢ < 0. It ends at η B− wheṅ H = 0. Hereafter,Ḣ > 0, the bounce starts.

B. The bounce phase
The bounce phase is the evolution withḢ > 0. The Hubble parameter is parameterised as [12][29] with αM 2 P ≪ 1. Thus we have where a = a B at t = t B . Eq. (8) indicates that the bounce phase is actually a superinflation phase with H rapidly increasing. In Ref. [7], it was argued that this phase results in the large-scale power deficit in CMB. However, we will show that in bounce inflation scenario this power deficit is actually attributed to the contraction before the bounce.
The continuities of a and H at η B− and η B+ suggest Its solution is where l ≡ αa 2 B − k 2 .

C. The inflationary phase
The universe will inflate after η B+ . The background is parameterised as Here Thus the solution of Eq. (3) is where

D. The spectrum of primordial GWs
According to (4), if ǫ inf = 0 we find where P inf is the standard result of the slow-roll inflation.
The perturbation γ ij and its time derivative should be continuous through the match surface. This suggests that we could write the coefficients recursively as where the matric M (2,1) is The effects of pre-inflationary phases are encoded in M (3,2) and M (2,1) . Here, we set the slow-roll parameter of inflation ǫ inf ≃ 0, thus P T is only relevant with the parameters, α, △η B , and ǫ c , noting that ǫ c ≥ 3 must be satisfied to avoid the cosmic anisotropy problem [30]. We plot P T in Fig.3 and Fig.4 by altering the values of different parameters. We see that for k > H B+ , P T ∼ k 0 with a damped oscillation, and is blue shift for small k modes.
We may analytically estimate it as follows.
For large k-modes, i.e. k > H B+ , which implies P T is flat and oscillating rapidly with the maximal amplitude at k ≃ H B+ , just as showed in Fig.3 and Fig.4. However, if the bounce phase lasts shortly enough, i.e. ∆η B ∼ 0, (17) will be reduced to For small k-modes, i.e. k < H B+ , which suggests that P T ∼ ( k H B+ ) 2 is strongly blue for ǫ c ≫ 1, and P T ∼ ( k H B+ ) 3 for ǫ c ≃ 3. The result is consistent with that found in [15]. In (19), However, if ∆η B ∼ 0, f (∆η B ) ∼ 1 and (19) will be reduced to

A. Qiu-Wang model
How to implement the bounce before inflation has been still a significant issue. Recently, Qiu and Wang have proposed a realistic bounce inflation model without instabilities [12].
We briefly review it as follows.
The potential is plotted in upper panels of Fig.6.
We plot the evolution of background in Fig.1 and Fig.5. The evolution of φ is plotted in Fig.6. Initially, we require φ ≪ −λ 1 , 1/ √ κ 1 , 1/ √ κ 2 , and φ rolls down along its ekpyroticlike potential and the universe is contracting. When t = t B− , the ekpyrotic contraction ends, and φ climbs up along its potential, which is essential so that the inflation can occur subsequently, as showed in original [6]. When φ arrives at φ ≃ 0, the bounce will occur.
Hereafter, φ continues to climb up to the potential hill, and then rolls slowly, the slow-roll inflation starts. The reheating will occur around φ ≃ 10, where φ oscillates and decays. It is convenient for us to write the perturbation equation with respect to the physical and considering u k = ah k 2 , we have Initially we have which suggest We numerically plot P T in Fig.7 and Fig.11, which is completely consistent with our analytical result (15). Fig.11 tells us that the bounce phase must be short, otherwise the numerical curve will not overlap with the analytical one. However, our (15) is actually universal, which is independent of whether the bounce is short or not.
In addition, it should be mentioned that after the bounce, a period withφ 2 dominated will appear, see Fig.5. This brief period has been often used to argue the suppression of power spectrum, e.g. [13]. However, in bounce inflation scenario, such a period is actually only relevant with the oscillating of spectrum, while the contraction before the bounce results in the suppression of spectrum at large scale, as seen in Eqs. (17) and (19).

V. DISCUSSION
The bounce inflation is successful in solving the initial singularity problem of inflation and is also competitive for explaining the power deficit in CMB at large scale. This might provide us chance to comprehend the origin of inflation thoroughly. There are also other designs to explain the power deficit, such as [33], [34], [35], [36], but not involve the initial singularity problem. The primordial GWs straightly encodes the evolution property of spacetime, thus it is interesting to have a detailed study for it.
We analytically calculated the bounce inflationary GWs. The resulting spectrum is written with the recursive Bogoliubov coefficients including the physics of the bounce phase. The spectrum acquires a large-scale cutoff due to the contraction, while the oscillation around the cutoff scale is quite drastic, which is determined by the details of bounce. We also show that our analytic result is completely consistent with the numerical plotting for a realistic model of bounce inflation.
In original Ref. [6], the perturbation spectrum was calculated without considering the bounce phase, which is controversial, since conventionally it is thought that the bounce should affect the spectrum. However, we find that if the bounce phase lasts shortly enough, the effect of the bounce on the primordial GWs may be negligible, and the calculation without considering the bounce phase is robust.
We should point out that what we applied is the perturbation equation without modifying gravity, which may be implemented only when the null energy condition is broke. However, the physics of bounce phase is unknown, it is obvious that the high-curvature corrections of gravity may also result in the occurrence of bounce, e.g. the Gauss-Bonnet correction [37], [38], and non-local gravity [39], which will inevitably modify the perturbation equation.
We find that the corresponding corrections will aggravate the oscillating behavior around the cutoff scale H B+ , however, the spectrum in the regime of k ≪ H B− and k ≫ H B+ is hardly affected. We will come back to this issue in upcoming work.
In addition, as has been mentioned, the bounce inflation may also explain a large dipole power asymmetry in CMB at low-l. However, the asymmetry might also appear in CMB B-mode polarization [40], [41]. Moreover, during the bounce, there might be a large parity violation [42]. It is interesting to have a reestimate for the relevant issues.
Series of experiments aiming at detecting GWs have been implemented or will be implemented. Our work suggests that searching primordial GWs at large scale might tell us if the bounce has ever happened before inflation.

Acknowledgments
This work is supported by NSFC, No. 11222546, 11575188, and the Strategic Priority Research Program of Chinese Academy of Sciences, No. XDA04000000.
Appendix A: The effects of u k continuum and h k continuum on spectrum When we do calculations, it is convenience to write h k as u k /2a(η), since the evolution of u k satisfies a Bessel equation. However, the real GWs mode is h k , not u k . Thus which of u k or h k is continuous on the matching surface might be a question needed to be clarified.
Our result (15) is based on the continuum of h k , which we will argue as follows.
We define the phase 'i' and 'i + 1' as the conjoint phases, both are matched at η 0 . The perturbations in different phase satisfy and where the prefactor 2 is neglected. The the continuum of h k andḣ k at the matching surface Thus we have, Finally, Generally, a i+1 = a i at the matching surface. Thus if a ′ i+1 = a ′ i , the continuum of u is equal to that of h.
In Sec.III.D, a ′ i+1 = a ′ i is assured. We plot P T with the continuum of u and h, respectively, in Fig.8, which are completely identical. However, in Appendix B, the bounce phase is neglected, and we have a ′ (η B+ ) ≃ −a ′ (η B− ), which indicates that a ′ is not continuous. We analytically calculate P T with the continuum of u, and plot it in Fig.9. It is found that the analytical result obtained can't match with the numeric curve accurately. In original idea of bounce inflation [6], the primordial GWs spectrum was calculated by straightly jointing the perturbation solution in the contracting phase to that in the expanding phase. It is interesting to estimate if the effect of bounce phase negligible.
We do not consider the bounce phase, which implies ∆η = 0 and H B+ = H B− = H B . The spectrum of primordial GWs is still Eq. (15). However, c 3,1 and c 3,2 should be determined by straightly matching the solutions (7) and (14). The perturbation γ ij and its time derivative should be continuous through the match surface. Thus we have with the matric N (3,1) (1) For large k-modes, i.e. k > H B , which is correspond to (17). While for small k-modes, i.e. k < H B , the result is same with (19).
In Fig.10, we plot P T with (16)  overlap. This indicates that if the bounce lasts shortly enough, the effect of the bounce on the primordial GWs may be negligible. It is noticed in Sec.III that in realistic model of bounce inflation, the period of bounce is actually short enough, thus the result without considering the bounce phase is robust, see Fig.11.