The backreaction effect of the sound speed resonance in DBI inflation

We examine the backreaction effect of the enhanced small-scale scalar perturbations from the sound speed resonance (SSR) mechanism for primordial black hole formation in Dirac-Born-Infeld (DBI) inflation on background as well as curvature perturbation, which can generate a considerable amount of primordial black holes (PBH) in the radiation-dominated epoch. Within the perturbative regime, the backreaction effect of perturbations on the background dynamics can be described by an effective action after integrating out the perturbation sector. Starting with the effective field theory of a specific DBI inflation model that incorporates SSR, we obtain the one-loop effective action by integrating out the scalar perturbations at the quadratic level. Using the effective Friedmann equations derived from this one-loop effective action, we solve the Hubble parameter with backreaction and the effective perturbation dynamics on this background as well. Our numerical findings reveal that, for a viable parameter space, the backreaction effect results in a relative correction to the Hubble parameter of approximately $10^{-7}$, whereas the relative correction to the slow-roll parameter can vary between $-0.3$ and $0.1$, before gradually converging to $10^{-7}$. Furthermore, our results show that the backreaction effect on SSR sound speed causes a slight reduction in the resonant peak of the curvature power spectrum, and the subsequent PBH formation predicted by the SSR mechanism remains almost unchanged.


I. INTRODUCTION
The conventional perturbative approach to the classical Einstein's field equation relies on the balance of the perturbative orders between geometry and matter sectors, and implicitly assumes the negligible interference among different orders.However, the intrinsic nonlinearity of Einstein's gravity inevitably allows linear cosmological perturbations around a homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) Universe to affect on the background dynamics, and this backreaction effect of linear perturbations arise as second and higher orders in the perturbation expansion.It is well known that gravitational waves (GWs) could affect the dynamics of the background on which they are propagating, and thus one can define the effective energy-momentum tensor of GWs by expanding Einstein's field equation up to the second order and taking a spatial average [1][2][3][4][5].In addition, considerable attention has been given to the backreaction effect of scalar perturbations in relation to whether small-scale inhomogeneities have an impact on the global FLRW evolution, which may give a new solution to cosmological coincidence problem, i.e., the acceleration epoch begins roughly at the same moment that non-linear structures form [6,7].Besides the above-mentioned classical approach to the cosmological backreaction problem, semi-classical and quantum approaches are also investi-gated [7,8].
For the ultra-slow-roll inflation, the backreaction effect was investigated in Ref. [54] under two concrete models using Hartree factorization, and authors found that the backreaction can boost the curvature perturbation on superhorizon scales.The lattice simulation of the axion-U(1) inflation model in Ref. [55] shows that the non-Gaussianity of curvature perturbation is sensitive to the backreaction.Reference [56] considers the backreaction of a spectator field on an inflationary background using Hartree approximation.Moreover, several recent works on the one-loop corrections from curvature perturbations [57][58][59][60][61][62][63][64][65] and tensor perturbations [66,67] show the potential importance of the one-loop corrections in the context of PBH formation.Reference [68] studied the backreaction in the hybrid inflation by perturbatively expanding the action up to the one-loop order and integrating out the perturbation to obtain an effective oneloop action from the path integral formalism.In this work, we investigate the backreaction effect of the sound speed resonance (SSR) mechanism which enhances smallscale curvature perturbations through the parametric resonance triggered by a time-oscillating sound speed of inflaton [32].Reference [37] has shown that the SSR mechanism can be realized in the framework of Dirac-Born-Infeld (DBI) inflation.Following the method presented in Ref. [68], we integrate out scalar perturbations at the quadratic order using the path integral formalism, and derive the modified Friedmann equations that measure the backreaction effect of perturbations on the background dynamics.We will show the backreaction corrections on the Hubble parameter, slow-roll parameter and sound speed, and further investigate backreaction on the enhancement of power spectrum.
This paper is organized as follows.In Sec.II, we briefly review the SSR realized in DBI inflation.In Sec.III A, we calculate the effective Friedmann equations from the one-loop effective action and provide numerical results for background parameters including the Hubble parameter, slow-roll parameter, and also the sound speed of curvature perturbations.In Sec.III C, we use the results from Sec. III A to calculate the curvature power spectrum with backreaction.Finally, we summarize the results in Sec.IV.

II. REALIZATION OF SSR IN DBI INFLATION
In this section, we will review the realization of SSR within DBI inflation [37].In the original SSR mechanism [32], the sound speed of inflaton is assumed to be timeoscillating during a few e-folds, where τ is the conformal time, ξ and k * are the small amplitude and frequency of the sound speed oscillation, respectively.ξ < 1/4 is required such that c 2 s is positively defined, and the oscillation begins at τ s , where k * is chosen to be deep inside the Hubble horizon with |k * τ s | ≫ 1.Consequently, the resonance of the curvature perturbations happens inside the Hubble horizon and leads to PBH formation during the later radiationdominated epoch.
The DBI model contains a non-canonical kinetic term, which can be used to realize a non-trivial sound speed.The DBI action is written as where X ≡ − 1 2 g µν ∇ µ ϕ∇ ν ϕ, and f (ϕ) is the so-called warp factor.The DBI action (2) is inspired by string theory with the inflaton field regarded as the radial position of branes moving inside a warped throat [69][70][71], and it has rich cosmological phenomenology [72][73][74].The sound speed in the DBI model can be expressed as where the dot denotes the derivative with respect to the cosmic time.
To realize a time-oscillating sound speed in the form of Eq. ( 1), Ref. [37] introduces an oscillating term into the well-studied anti-de Sitter warp factor f (ϕ) = λ/ϕ 4 with λ a constant, where φ(τ ) is the homogeneous background of inflaton field ϕ, and φs ≡ φ(τ s ) is evaluated at the beginning of sound speed oscillation.The oscillation function C(ϕ) is defined as (5) where H is the Hubble parameter.Θ φ − φs is the Heaviside step function, which divides the inflationary epoch into the non-oscillating (c 2 s = 1 − 2ξ for τ ≤ τ s ) and oscillating (i.e., Eq. ( 1)) phases.The functional form (5) is derived from the background evolution of inflaton φ(τ ), in a quasi de-Sitter spacetime a(τ ) ≃ 1/(ϵ − 1)Hτ where ϵ ≡ − Ḣ/H 2 is the slow-roll parameter.φi is the value taken at the beginning moment of inflation τ i .Although this solution is solved by non-oscillating sound speed, it is considered applicable to both oscillating and nonoscillating phases [37].The theoretical viability of this DBI model to realize SSR is also investigated in Ref. [37].

III. BACKREACTION IN SSR-DBI INFLATION A. Backreaction effect on the background
In order to obtain an effective background Lagrangian with backreaction, a partition functional consisting of a DBI inflaton and an external source J(x) is introduced as where D[ϕ] is the functional integral over all possible inflaton field configurations.Using the semiclassical approach within the path integral formalism [75], one can perturbatively expand Z[J] in Eq. ( 7) with Eq. ( 2) around the classical background φ, up to the quadratic order in terms of the field perturbation ϕ per (i.e., ϕ = φ + ϕ per .The first-order term of Z[J] vanishes due to the classical equation of motion for ϕ per ).Within the framework of the effective field theory (EFT) of inflation [76], the inflation field perturbation ϕ per is "eaten" by graviton (namely the unitary gauge ϕ per = 0), one can write down the most general inflation action merely based on metric which respects the unbroken spatial diffeomorphism.After using the Stückelberg trick, the time diffeomorphism can be restored and the quadratic action for the scalar degree of freedom of gravity is given by [76] where the dot refers the derivative with respect to the cosmic time.π corresponds to the broken time translation symmetry.The explicit forms of the coefficients α(t), β(t) in the above action can be obtained by matching the SSR-DBI model with EFT action (see Appendix A for details).After integrating out the quadratic action (8) in the partition functional (7), one can obtain the effective Lagrangian for background evolution as, which describes the backreaction effect and is calculated as follows (see Appendix B for details), where we have used the subscript "tot/bg" to represent the background with/without backreaction, and the subscript "br" refers to the backreaction term arising from the integrated perturbation sector.For example, the total background inflaton field is written as φtot = φbg + φbr , where φbg follows the evolution (6).In Eq. ( 11), Λ k ≡ c −1 s,bg H tot is a physical cutoff at the sound horizon which is introduced to prevent the divergence of Z[J] as Appendix B shows, and tot − H tot ṡ acts as an effective mass implicated by Eq. (B6), and s ≡ ċs,bg /(H tot c s,bg ) is a dimensionless parameter representing the rate of change of the sound speed during inflation in presence of the backreaction.Λ 2 M is a parameter with the unit of the mass square [68], whose value is taken such that L br vanishes at the initial time of inflation τ i when SSR and its backreaction have not occurred.Here we assume the backreaction correction on background is homogeneous [77], thus above parameters do not depend on the spatial coordinate.
From the total Lagrangian L tot in Eq. ( 9), we obtain the effective Friedman equations as follows, where δ/δ φbg refers to the variation with respective to φbg .
In principle, one can integrate the above equations to give the Hubble parameter with/without backreaction.However, since it involves the complicated integration of Eqs. ( 5) and (6), it is hard to investigate the effect of backreaction analytically.In the remaining part of this paper, we shall adopt the numerical method and analyze the modification of the SSR-DBI inflation background by the backreaction effect.As an illustration, we shall show the evolution of the Hubble parameter, slow-roll parameter, sound speed and power spectrum of curvature perturbations with and without backreaction for comparison.
For simplicity, we use φbg instead of φtot as the temporal parameter, using the relation φbg = 2ξ λ φ2 bg indicated by Eq. ( 6).According to Eqs. ( 12) and ( 13), we obtain the approximation, On the right-hand side, the background term is given by [37] dH bg while the backreaction term is calculated as where c 2 s,bg = 1 − f ( φbg ) φ2 bg based on Eq. (3).Numerically integrating Eq. ( 14) gives the evolution of H tot , which is shown by the red curve in Fig. 1.The ratio ∆ H ≡ H br /H bg is shown in the subplot by the blue curve.The parameters taken in this paper follow Ref. [37]: λ = 2 × 10 9 , ϵ = 10 −3 , ξ = 0.1, k * = 10, τ s = −14, N s = 19.7521and H i /M Pl = 10 −5 at the initial time τ i , then we calculate Λ 2 M /M Pl ≃ 1.7 × 10 −5 .It is clearly seen from Fig. 1 that H tot is fairly close to H bg during the whole inflationary era, which demonstrates a negligible correction on the background expansion from the backreaction.At the non-oscillating stage before SSR occurs ( φbg < φs ), the backreaction effect almost vanishes as expected, due to the tiny curvature perturbations.While during the oscillating stage, the backreaction grows to around ∆ H ∼ 10 −7 as a result of the enhanced curvature perturbations.Hence, we conclude that even if the peak of curvature perturbations are enhanced by around seven orders of magnitude through the SSR mechanism, their backreaction on the background expansion can be neglected.This is physically reasonable since only a narrow range of scalar mode functions around k * is enhanced in SSR, and their contribution to the background energy density comes from only a small fraction of the whole Fourier modes.Fig. 2 shows the numerical result for the slow-roll parameter with/without backreaction defined as ϵ tot = − Ḣtot /H 2 tot and ϵ bg = − Ḣbg /H 2 bg , and also shows the ratio of them defined as where ϵ br ≡ ϵ tot −ϵ bg .The evolution of Ḣbr / Ḣbg is shown in Fig. 3, which oscillates when φbg ≳ φs , and then decays to Ḣbr / Ḣbg ≃ −10 −7 ≪ 1 for larger φ, since Ḣbr / Ḣbg = d dt (∆ H H bg ) / Ḣbg and ∆ H → −10 −7 as shown in Fig. 1.Thus, we obtain which is consistent with the numerical result in Fig. 2, where the backreaction effect on the slow-roll parameter experiences an oscillation at the beginning of the oscillating phase and then converges to zero.Using the above results, we also investigate the backreaction effect on the background energy density ρ . By the decomposition of ρ = ρ bg + ρ br and P = P bg + P br , one writes the backreaction terms as follows, Therefore, the equation of state of the backreaction term is given by Since Ḣbr / Ḣbg ≪ 1 and ∆ H ≪ 1 as discussed above, only considering leading terms, Eq. ( 21) becomes And since at leading order ρ br ≃ 2∆ H ρ bg , where ∆ H ≤ 0 according to the numerical result from Fig. 1, and ρ bg = 3H 2 bg 8πG > 0, thus one has ρ br < 0. Therefore the backreaction effect of perturbation behaves like a negative cosmological constant, which is similar to the case of chaotic inflation [79].

B. Backreaction effect on the sound speed
Next, we investigate the backreaction effect on the sound speed c 2 s as the enhancement of the curvature perturbation ζ is sensitive to c 2 s in SSR [32,37].In the next subsection, we shall further show how the power spectrum of ζ in the SSR-DBI inflation is influenced by the backreaction modification of c 2 s .In this paper, we assume the functional form of the warp factor does not receive any correction from the backreaction, and the sound speed with backreaction in SSR-DBI inflation is thus given by where φtot (τ ) is written as The derivative of φtot under the slow-roll approximation (ϵ tot , ϵ bg ) ≪ 1 and up to the first order of ∆ H , is calculated as Notice that in Eq. ( 14), we do not consider the backreaction in φtot when calculating H br .In principle, one can use φtot to replace φbg to obtain a more precise prediction of H br .More detailed research shall be carried out in our future work.
Based on the form of ϕ tot in Eq. ( 24), the oscillation term C (ϕ tot (τ )) in Eq. ( 5) is not affected by the backreaction.Thus the correction on the warp factor f (ϕ tot ) in Eq. ( 4) can be expanded as f ( φtot (τ )) = 1 − 4 φbr (τ ) φbg (τ ) f ( φbg (τ )).Using the above results, the sound speed with backreaction becomes The evolutions of the sound speed c 2 s,tot/bg are shown in Fig. 4. The subplot shows the comparison with the cor-rection manually amplified to show the difference more explicitly.It can be seen that the correction is large at the beginning of the oscillation stage and decays afterward, which is consistent with the behavior of ϵ br shown in Fig. 2. It is also worth noting that the backreaction does not cause the c s to become larger than untiy.

C. Backreaction effect on the curvature perturbations
With the backreaction effects on the background quantities and the sound speed given in previous sections, we shall examine the backreaction on the power spectrum of curvature perturbations ζ, which is given by and can be decomposed as P tot

IV. CONCLUSIONS
In this paper, we investigate the backreaction effect of the enhanced small-scale curvature perturbations in the SSR-DBI inflation.We find that the backreaction effects on both the background and perturbations can be neglected and do not spoil the previous results in the SSR-DBI inflation model in the absence of backreaction.
As a starting point, we review the SSR-DBI inflation model [37] and adopt the methods presented in Ref. [68] to calculate the backreaction effects of perturbations on the background.After integrating out the quadratic curvature perturbation action, we obtain the effective Lagrangian (11) which describes the backreaction effect on the inflationary background dynamics.
Based on this effective Lagrangian, we numerically solve the evolution of the Hubble parameter H, the slowroll parameter ϵ, and the sound speed c 2 s with backreaction.We find that the backreaction effect causes a tiny correction to the Hubble parameter in an order of 10 −7 compared to the original results in Ref. [37], as shown in Fig. 1.For the slow-roll parameter, as displayed in Fig. 2, the correction resulting from the backreaction effect exhibits an oscillation that begins with an amplitude between −0.3 and 0.1, but gradually decreases to a value as small as −2∆ H ∼ 10 −7 .Additionally, the results presented in Fig. 4 tell us that the sound speed is slightly altered by curvature perturbations.Applying the sound speed with/without backreaction to the Mukhanov-Sasaki equation ( 28 where k denotes the physical momentum and we have taken the continuous limit of summation.The explicit form of A can be obtained from Eq. (A4) using integration by parts, where η ≡ π a 2 α −1/2 is a normalized variable, the effective mass term is M 2 = 1 4 (3 − 2s) 2 H 2 tot − H tot ṡ.The operator A can be read by comparing Eq. (B6) with Eq. (B2), Substituting Eq. (B7) to Eq. (B5), one gets where k ′0 ≡ ik 0 and k ′i ≡ c s k i , and the identity is used [68].
To prevent the divergence of the above integration, we introduce a cutoff at the sound horizon Λ k ≡ c −1 s aH and only the modes of |k ′ | < Λ k contribute to the integration, since the resonant mode of curvature perturbation in SSR-DBI inflation reaches its maximum at the sound horizon and the backreaction effect shall be relatively small for |k ′ | > Λ k .So we carry out the integration for |k ′ | < Λ k and obtain Eq. (11).

1 FIG. 2 :
FIG. 2:The evolutions of slow-roll parameters ϵ tot (red solid) and ϵ bg (gray dashed) in terms of φbg .The ratio ∆ ϵ ≡ ϵ br /ϵ bg is displayed in the subplot by the blue curve.

FIG. 4 :
FIG. 4: The numerical results of sound speeds c 2 s,tot (red solid) and c 2 s,bg (gray dashed), and their ratio is shown by the blue curve in the subplot.

FIG. 5 :
FIG. 5: The numerical results of the curvature power spectra P tot ζ (red solid) and P bg ζ (gray dashed).The small subplot shows a zoom-in to the power spectra around the peak.