Resonant amplification of curvature perturbations in inflation model with periodical derivative coupling

In this paper, we introduce a weak, transient and periodical derivative coupling between the inflaton field and gravity, and find that the square of the sound speed of the curvature perturbations becomes a periodic function, which results in that the equation of the curvature perturbations can be transformed into the form of the Mathieu equation in the sub-horizon limit. Thus, the parametric resonance will amplify the curvature perturbations so as to generate a formation of abundant primordial black holes (PBHs). We show that the generated PBHs can make up most of dark matter. Associated with the generation of PBHs, the large scalar perturbations will give rise to the scalar induced gravitational waves which may be detected by future gravitational wave projects.

However, in the standard slow-roll inflation, which predicts nearly scale-invariant curvature perturbations, the possibility of the formation of PBHs is negligible.This is because the cosmic microwave background radiation (CMB) observations have implied a very small amplitude of the power spectrum of the curvature perturbations of about O(10 −9 ).To generate a sizable amount of PBHs requires that the amplitude of the power spectrum of the curvature perturbations reaches about O(10 −2 ).Since the CMB observations only give a limit on the curvature perturbations at large scales [28] and the magnitude of the power spectrum at scales smaller than the CMB one is not restricted strongly by any observations, the formation of an abundant PBHs will be possible if there are some mechanisms to enhance the curvature perturbations at small scales.
It is well known that the amplitude of the power spectrum of the curvature perturbations R is given by P R = H 2 8π 2 ϵcs when the mode exits the horizon during inflation in the standard slow-roll inflation.Here ϵ is the slow-roll parameter, which is proportional to the rolling speed of the inflaton, c s is the sound speed of the curvature perturbations and H is the Hubble parameter.So, a natural way to enhance the curvature perturbations is to reduce the rolling speed of the inflaton  or to suppress the sound speed c s [94][95][96][97][98][99][100][101].Additionally, some other ways can also predict the formation of PBHs .The decrease of the rolling speed of the inflaton can be realized by flattening the potential of the inflaton field.The corresponding inflation model is called the inflection-point inflation .Recently, it has been shown that the gravitationally enhanced friction can also slow the rolling speed of the inflaton [59][60][61][62][63][64].In this mechanism, a derivative coupling between the inflaton field and gravity is invoked.In addition, the growth of curvature perturbations caused by parametric resonance has also been extensively studied [125][126][127][128][129][130][131].The parametric resonance can be obtained by adding a periodic correction to the potential of the inflaton field or considering a periodic sound speed.As the derivative coupling can realize the decrease of the inflaton's rolling speed, can it lead to the parametric resonance to amplify the curvature perturbations?This motivates us to finish the present study.We find that the parametric resonance will occur after introducing a periodical derivative coupling between the inflaton field and gravity, since the equation of the curvature perturbations can be transformed into the form of the Mathieu equation.We demonstrate that the enhanced curvature perturbations will lead to an abundant generation of PBHs, which can make up most of dark matter, and the SIGWs may be detected by the future GW projects.
The rest of this paper is organized as follows: In Sec.II, we will introduce the inflation model with a periodical derivative coupling between the inflaton field and gravity.Sec.III discusses the parametric resonance and Sec.IV describes the formation of PBHs.In Sec.V, we investigate the SIGWs.Finally, we give our conclusions in Sec.VI.

II. INFLATION WITH PERIODICAL DERIVATIVE COUPLING
We consider an inflation model with a non-minimal derivative coupling between the inflaton field ϕ and gravity.The action of the system has the form Here g is the determinant of the metric tensor g µν , M pl is the reduced Planck mass, R is the Ricci scalar, G µν is the Einstein tensor, θ(ϕ) denotes the coupling function, and V (ϕ) is the potential of the inflaton field.This action belongs to a class of the general Horndeski's theories with second-order equations of motion [132,133], which can be free of the ghost and gradient instabilities [133].The Lagrangian of such Horndeskis theories has the term G 5 (ϕ, X)G µν ∇ µ ∇ ν ϕ, where G 5 is a generic function of ϕ and X ≡ −∂ µ ϕ∂ ν ϕ/2.By choosing the function G 5 = −κ 2 χ(ϕ)/2, the term containing θ in Eq. ( 1) can be recovered from the Horndeski's Lagrangian after integration by parts with θ being defined as θ ≡ dχ/dϕ.
In the spatially flat Friedmann-Robertson-Walker background, the background equations derived from the action (1) are Here the overdot denotes the derivative with respect to the cosmic time t, θ ,ϕ ≡ dθ/dϕ and To describe the slow-roll inflation, we define the slow-roll parameters where and with Defining z ≡ a √ 2Q and u k ≡ zR k , we find that u k satisfies the equation Here k is the wave-number and a is the cosmic scale factor.Solving Eq. ( 10) leads to the power spectrum of the curvature perturbations The CMB observations have implied that at large scales this power spectrum is a nearly scale-invariant spectrum with the amplitude being about O(10 −9 ) [28].
To generate a sizable amount of PBHs, we need to enhance the curvature perturbations at scales smaller than the CMB one through the parametric resonance.Thus, we choose the coupling function θ(ϕ) to take an oscillating form Here w is a dimensionless constant, which must satisfy ω ≪ | ≪ 1, and ϕ c is a quantity with the same dimension as ϕ and is set to be much less than ϕ.Meanwhile ϕ s and ϕ e represent the beginning and the end of the coupling, respectively, and Θ is the unit Heaviside step function.The value of ϕ s is chosen to be away from that of ϕ at the beginning of inflation, and thus the derivative coupling does not affect the curvature perturbations at the CMB scale.So, the amplitude of the power spectrum of the curvature perturbations at the CMB scale remains to be of the standard form: . The spectral index n s and the tensor-scalar-ratio r are To be consistent with the CMB observations, we choose the potential of the inflaton field to be the Starobinsky potential [137] where Λ is a constant.
Since the parametric resonance occurs deep inside the Hubble horizon (c s k ≫ aH), the Eq. ( 10) can be simplified to be Considering the slow-roll conditions given in Eq. ( 5), w ≪ 3H 2 and ϕ c ≪ ϕ during inflation, we find that the background equations (Eqs.(3,4)) can be reduced to and then the sound speed square of the curvature perturbations can be simplified to be We find that δc s oscillates around zero and satisfies |δc s | ≪ 1.So, the sound speed could exceed the speed of light.However, it has been found that the superluminal sound speed will not result in the causal paradoxes when the scalar field is non-trivial [138][139][140][141][142][143][144].This suggests that there may be no violation of causality from the superluminal sound speed for the model considered in this paper.If a different coupling function, i.e. θ(ϕ) ∼ sin 2 (ϕ), is chosen, we find that the subliminal oscillation of the sound speed square is possible.
Substituting Eq. ( 16) into Eq.( 14), one can obtain We assume that the inflaton field evolves from ϕ s to ϕ e during a short time, which indicates that during this short time the evolution of ϕ can be expressed approximately as ϕ ≈ and k c = | φs| ϕc , we find that the Eq. ( 17) can be transformed into the form of the Mathieu equation where For the Mathieu equation, the resonant bands are close to narrow regions near A k (x) ≃ n 2 (n = 1, 2, 3...).The width of each resonant band is ∆k ∼ q n .If 0 < q ≪ 1, the resonance in the first resonant band (n = 1) is the most violent.Therefore, we only consider the influence of the first resonant band on u k .
For the first instability, the Floquet index µ k which describes the rate of exponential growth has the form Here ℜ refers to taking the real part.Then, we find that the resonance occurs in a narrow band where In obtaining the expressions of k ± , we have used the condition |C| ≪ 1 derived from 0 < q ≪ 1.Since Eq. ( 21) is time dependent, the duration that the k mode stays in the resonant band is finite, which is given by T in (k) = min(t e , t F ) − max(t s , t I ) with k satisfying k − a s < k < k + a e , where subscripts s and e represent the moment that the inflaton field equals ϕ s and ϕ e , respectively, and t I and t F represent, respectively, the time of k mode entering and leaving the resonant band.Thus, the resonant amplified width of the power spectrum is ∆k = k + a e − k − a s , which is determined mainly by k c , ϕ e and ϕ s .
During the parametric resonance, the curvature perturbations will be enhanced exponentially Defining B k (t) = k kca , the above integral can be re-expressed as The amplified modes can be divided into three groups: (1) the modes entering the band before t s ; (2) the modes entering the band after t s and exiting before t e ; (3) the modes exiting the band after t e .For these three groups, the wavenumber satisfies: Apparently, for the second group: k + a s < k < k − a e , B k (t I ) and B k (t F ) are independent of k, which results in that A k is independent of k for this group.
The enhanced power spectrum of the curvature perturbations can be expressed approximately as In the region of k + a s < k < k − a e , since A k is independent of k, the enhanced power spectrum in this region will have a plateau, which can be seen in Fig. (1), where we plot the evolution of P R /P R 0 with k.In this figure the blue and red lines represent the numerical results from Eq. ( 10) and the approximate ones given in Eq. ( 27), respectively.It is easy to see that the approximate results are consistent well with the numerical ones, and the power spectrum can be enhanced by several orders.These enhanced curvature perturbations can lead to the generation of significant gravitational quadrupole moments during inflation, which will emit GWs.However, this issue is beyond the scope of the present paper and is left to be investigated in the future.
Figure (2) shows the power spectrum of the curvature perturbations from numerical calculation, which indicates clearly that the power spectrum is compliant with the CMB observations at the CMB scale, and it can be amplified to generate a sizable amount of PBHs at scales smaller than the CMB one.

IV. PBHS
When the sufficiently large curvature perturbations re-enter the Hubble horizon during the radiation-dominated period, the gravity of the high-density regions will overcome the radiation pressure and lead to the formation of PBHs.The PBH mass has the following The evolution of P R /P R 0 with k.The blue and red lines represent the numerical results from Eq. ( 10) and the approximate ones given in Eq. ( 27).The parameters are set to be FIG. 2: The power spectrum of the curvature perturbations.The green shaded region is excluded from the current CMB observation [28].The orange-and blue-shaded regions are excluded by the µ distortion of CMB [145] and the effect on the n − p ratio during big-bang nucleosynthesis (BBN) [146], respectively.Cyan shaded region indicate the limitations of current PTA observations in the power spectrum [147].
relationship with k: Here γ is the ratio of the mass of PBH to the total mass of the Hubble horizon when the PBH is formed.It represents the effective collapse rate, and its specific value is related to the details of gravitational collapse.In our analysis we set γ ≃ (1/ √ 3) 3 [5].In Eq. ( 28), M ⊙ represents the solar mass, and g * is the number of degrees of freedom of the relativistic particle at the time of the PBH formation.Assuming that the PBHs form in the radiationdominated period, we can set g * = 106.75.
after assuming that the probability distribution function of the disturbance obeys the Gaussian distribution.Here erfc is the complementary error function, and δ c is the threshold for the relative density perturbation of the PBH formation, which is chosen to be δ c ≃ 0.4 [149,150] in our calculation of the PBH abundance.The variance σ 2 (M ) represents the coarse-grained density contrast with the smoothing scale k, and it takes the form Here W is the window function.We find that the PBH mass spectrum can be obtained from the following equation Here Ω DM represents the current dark matter density parameter and h is the reduced Hubble constant.While Ω DM h 2 is constrained to be Ω DM h 2 ≃ 0.12 by the Planck 2018 observations [28].We show the numerical results of the PBH mass spectrum in Fig. (3) and find that the PBHs can make up most of dark matter since Ω PBH Ω DM ≃ 0.99.
Associated with the formation of PBHs, which are assumed to be generated in the radiation-dominated era, the large metric scalar perturbations become an important GW source and radiate the observable SIGWs.The second-order tensor perturbations h ij satisfy the equation: where a prime denotes the derivative with respect to the conformal time, H ≡ a ′ /a, T lm ij is the transverse-traceless projection operator, and is the GW source term [156,157].Here Ψ is the metric scalar perturbation.In the radiationdominated era, the evolution of 3) [157], where ψ k is the primordial perturbation, which relates with the power spectrum of the primordial curvature perturbations through Solving Eq. ( 32), one can obtain the GW energy density for each logarithmic interval k [158]: where η c represents the time when Ω GW stops to grow.The current energy density spectrum of SIGWs can be expressed as [147,158] Ω GW,0 h 2 = 0.83 g * 10.75 where Ω r,0 h 2 is the current density parameter of radiation which is set to be 4. the sensitive intervals of different gravitational wave detectors including SKA [159], EPTA [160], TAIJI [22], TIANQIN [23], LISA [21], and ALIGO [161].
In Fig. ( 4), we show the current energy spectrum of SIGWs.One can see that the SIGWs possess a multi-peak structure and may be detected by the future GW projects including LISA, Taiji and TianQin.

VI. CONCLUSIONS
In order to produce a sizable amount of PBHs, the amplitude of the power spectrum of the small-scale curvature perturbations must be enhanced by about 7 orders of magnitude compared to that at the CMB scale.In inflation models in which the inflaton field couples derivatively with gravity, it has been found that the curvature perturbations can be amplified through the gravitationally enhanced friction mechanism [59].In this paper, we find that, if there is a weak, transient and periodical derivative coupling between the inflaton field and gravity, the sound speed square of the curvature perturbations becomes a periodic function, which results in that the equation of the curvature perturbations in the sub-horizon limit can be transformed into the form of the Mathieu equation.Thus, for some k-modes, the parametric resonance will amplify their fluctuations.These amplified fluctuations are stretched to be super-horizon by the inflation and then the power spectrum will be enhanced at scales smaller than the CMB one.When the enhanced curvature perturbations re-enter the horizon during radiation-dominated era, they will lead to the formation of PBHs, which can explain most of dark matter.Associated with the generation of PBHs, the large scalar perturbations will radiate the observable SIGWs.We demonstrate that the current energy spectrum of the SIGWs has a multi-peak structure, which is different from that in the inflation model with the gravitationally enhanced friction [60], and it can be detected by future GW projects including LISA, Taiji and TianQin.Therefore, the future detection of SIGWs will help us to distinguish different mechanisms of enhancing curvature perturbations at small scales.Finally, it is worth noting that when loop corrections are considered the perturbation theory will be broken in the cases of the amplification of the primordial curvature perturbations due to the decrease of the inflaton's rolling speed [162] and the parametric resonance from the oscillating potential [163].Whether the loop corrections will break the perturbation theory in the scenario considered in the present paper needs to be studied separately since the gravity, which couples derivatively with the inflaton field, is different from the theory of general relativity.

When 2 M 2 pl| ≪ 1 .
{ϵ, |δ ϕ |, δ X , |δ D |} ≪ 1 are satisfied, the slow-roll inflation is obtained.Furthermore, we add the condition: | 3θ(ϕ)H Thus, the background dynamics in the non-minimally derivative coupled inflation model will be almost the same as that in the minimal coupling case During inflation, the quantum fluctuations provide the seed for the formation of large scale cosmic structures.The fluctuations are described by using the curvature perturbations R. Expanding the action given in Eq. (1) to the second-order, one can obtain the action of R[133][134][135][136]

FIG. 3 :
FIG. 3: The mass spectrum of PBHs.The colored regions are ruled out by observations including

2 ×FIG. 4 :
FIG. 4: The current energy spectrum (solid blue line) of SIGWs.The various dotted lines represent and k − a e ≤ k < k + a e , respectively.The corresponding B k (t I ) and B k (t F ) can be calculated as