A Equations of motion of perturbations

# Primordial perturbations from dilaton-induced gauge fields

## Abstract

We study the primordial scalar and tensor perturbations in inflation scenario involving a spectator dilaton field. In our setup, the rolling spectator dilaton causes a tachyonic instability of gauge fields, leading to a copious production of gauge fields in the superhorizon regime, which generates additional scalar and tensor perturbations through gravitational interactions. Our prime concern is the possibility to enhance the tensor-to-scalar ratio relative to the standard result, while satisfying the observational constraints. To this end, we allow the dilaton field to be stabilized before the end of inflation, but after the CMB scales exit the horizon. We show that for the inflaton slow roll parameter , the tensor-to-scalar ratio in our setup can be enhanced only by a factor of compared to the standard result. On the other hand, for smaller corresponding to a lower inflation energy scale, a much bigger enhancement can be achieved, so that our setup can give rise to an observably large even when . The tensor perturbation sourced by the spectator dilaton can have a strong scale dependence, and is generically red-tilted. We also discuss a specific model to realize our scenario, and identify the parameter region giving an observably large for relatively low inflation energy scales.

98.80.Cq
56

## I Introduction

The cosmological inflation not only solves the naturalness problems in the standard big bang cosmology, but also provides an appealing mechanism to generate the seed of the large scale structure and the cosmic microwave background temperature anisotropies in the present universe (1). During the inflationary phase, primordial gravitational waves can be generated from the quantum fluctuation of metric. The latest joint analysis of BICEP2/Keck Array and Planck data provides an upper limit on such tensor perturbation, implying that the tensor-to-scalar ratio is bounded as (2),

 r≡PtPζ<0.12(95%CL) (1)

at the pivot scale . In the minimal single field inflation scenario, this can be used to constrain the energy scale of inflation based on the standard relation between the tensor power spectrum and the inflationary Hubble scale (3):

 Pt=2H2π2M2P. (2)

On the other hand, many of the well motivated models of particle physics involve a light scalar field which couples to gauge fields in a way to provide an additional source of perturbations. For instance, if the scalar field evolves appropriately during the inflationary epoch, it can cause a tachyonic instability of gauge fields, leading to a copious production of gauge fields. Then the produced gauge fields may result in a significant amount of additional tensor perturbations, so modify the relation (2(4); (6); (7); (8); (5). A well studied example is an axion-like field which couples to gauge field as (9); (10); (11); (12); (14); (13); (15); (16); (17)

 ΔLaxion=132π2ηfFμν~Fμν, (3)

where is the gauge field strength, is its dual, and is the axion decay constant. Regardless of whether it is an inflaton or just a spectator field, a rolling axion with the coupling (3) can generate additional tensor modes which are highly non-gaussian (18), parity-violating (19); (20); (21); (22); (23); (24), and blue-tilted (25). However, if the coupling (3) is strong enough to enhance the tensor-to-scalar ratio significantly, it can lead a large non-gaussianity in scalar perturbation, which is in danger to be incompatible with the recent Planck results (26).

There is another type of well motivated light scalar field, a dilaton (or moduli field) which couples to gauge fields as

 ΔLdilaton=−I2(σ)4FμνFμν, (4)

where can be identified as the field-dependent gauge coupling. As in the case of axion, a rolling dilaton can produce gauge fields by causing a tachyonic instability. Cosmological implications of rolling dilaton with the coupling (4) have been studied extensively in the context of inflationary magnetogenesis (27); (28); (29); (30). As the produced gauge fields are stretched out the horizon during inflation, it may provide the origin of large scale magnetic fields in the present universe. However this mechanism of magnetogenesis is constrained in several ways. Requiring that the electromagnetic energy density should not exceed the inflaton energy density, either the amplitude of the produced magnetic field should be too small to explain the large scale magnetic field in the present universe (31); (32); (33); (34), or the gauge coupling should run from an extremely large value to  (32). In addition, the produced electromagnetic field contributes to the primordial density perturbations, providing a variety of additional constraints on this mechanism of magnetogenesis (35); (36); (37); (38); (39); (40); (41); (42); (43); (44); (45); (46); (47); (48); (49); (50); (51).

In this paper, we study systematically the scalar and tensor perturbations sourced by a rolling spectator dilaton which couples to gauge field kinetic terms as (4), while taking into account the known observational constraints. Our prime concern is the possibility to enhance the tensor-to-scalar ratio relative to the standard result of the single field inflation scenario, where is the inflaton slow roll parameter. To this end, we allow the spectator dilaton to be stabilized before the end of inflation, but after the CMB scales exit the horizon. As we will see, this makes it possible that is large enough to be observable in the near future, e.g. , even when the inflation energy scale is relatively low to give .

Although our scheme reduces to the conventional single field inflation after the dilaton is stabilized, the dilaton dynamics which took place before the stabilization leaves an imprint on the primordial power spectrum that exit the horizon while the dilaton field is rolling. Imposing the known observational constraints on the scalar perturbation sourced by rolling dilaton, we find that for the inflaton slow roll parameter , the tensor-to-scalar ratio can be modified only by a factor of compared to the standard result. However, for smaller , which corresponds to a lower inflation energy scale, can be enhanced by a much larger factor. Specifically, the tensor perturbation sourced by rolling dilaton can be large enough to give , while satisfying the observational constraints, even when the inflaton slow roll parameter . We also find that the tensor power spectrum in this case can have a strong scale dependence, which is generically red-tilted.

Compared to the axion case, our dilaton scenario has several distinctive features. For instance, both polarizations of tensor mode are equally produced in the dilaton case, while only a certain polarization state is produced in the axion case. Another difference is in the scale dependence. In case that the axion or dilaton coupling to gauge fields is strong enough to generate a large tensor perturbation, the energy density of dilaton-induced gauge fields continues to be growing over the superhorizon regime, while the energy density of axion-induced gauge fields is diluted soon after the horizon crossing. As a result, for the dilaton case the perturbations are produced dominantly in the superhorizon regime, while for the axion case the production of perturbations is active only around the horizon crossing. This results in a strongly red-tilted tensor spectral index for the dilaton case, which is not suppressed by slow roll parameters. On the other hand, for the axion case the tensor spectral index is suppressed by slow roll parameters, although it can be numerically sizable and blue-tilted (25).

This paper is organized as follows. We describe our setup in Section II, and compute the resulting scalar and tensor perturbations in Section III and IV, respectively. In Section V, we discuss the implications of our result and present a specific model with interesting observational consequences. Section VI is the conclusion.

## Ii Setup

We consider an inflationary cosmology described by

 S = ∫d4x√−g[M2P2R+Linf(ϕ)−12∂μσ∂μσ−V(σ)−I2(σ)4FμνFμν], (5)

where GeV is the reduced Planck mass, is the lagrangian density of the inflaton field , and is the field strength of gauge field which couples to the dilaton field . For simplicity, here we assume that there is no direct coupling of the inflaton to the dilaton and gauge fields. We assume also that the inflaton field satisfies the conventional slow-roll conditions, and the total energy density is dominated by the inflaton energy density over the whole period of inflation. We will use the spacially flat gauge, for which the metric perturbations are parametrized as

 ds2=a2(τ)[−(1+2Φ)dτ2+2∂iBdτdxi+(δij+hij)dxidxj], (6)

where the conformal time coordinate is defined as for the Robertson-Walker time coordinate , and satisfies the traceless/transverse condition, . The inflaton, dilaton, and gauge field are expanded also around a homogeneous background as

 ϕ(τ,x) = ϕ0(τ)+δϕ(τ,x), σ(τ,x) = σ0(τ)+δσ(τ,x), Aμ(τ,x) = δAμ(τ,x). (7)

### ii.1 Gauge field production by a rolling dilaton

As is well known, a rolling dilaton field during inflation can develop a tachyonic instability of gauge field, leading to a copious production of gauge fields in the superhorizon regime. Choosing the gauge condition and , the equation of motion of gauge field is given by

 (∂2τ+2∂τII∂τ−∇2)Ai(τ,x)=0. (8)

After the Fourier expansion

 Ai(τ,x) = ∫d3k(2π)3/2^Ai(τ,k)eik⋅x, ^Ai(τ,k) = ∑λ=±ϵi,λ(^k)[Aλ(τ,k)ak,λ+A∗λ(τ,k)a†−k,λ],

it is convenient to redefine the gauge field mode function as

 Vλ≡I(σ0)Aλ, (9)

where depends only on the background dilaton field , not on the dilaton fluctuation. Then the equation of motion of the canonically normalized mode function is given by

 ∂2τVλ(τ,k)+(k2−∂2τII)Vλ(τ,k) = 0 (10)

Note that both helicity states evolve in the same way, so we can drop the helicity index from now. This is different from the axion case where different helicity state experiences different evolution, which results in parity violating phenomena.

The details of gauge field production by rolling dilaton depends on the functional form of the dilaton coupling . For canonically normalized dilaton field, a particularly well motivated form of the dilaton coupling is

 I(σ)=eσ/Λ, (11)

where is a constant mass parameter. In this case, the evolution rate of the dilaton coupling (relative to the Hubble expansion rate) is given by

 n≡−˙IHI=−˙σHΛ. (12)

If the spectator dilaton underwent a time evolution satisfying

 |¨σ|≪H|˙σ|, (13)

where the dot denotes the derivative with respect to the Robertson-Walker time coordinate , one finds

 ∣∣∣˙nHn∣∣∣=∣∣∣˙HH2−¨σH˙σ∣∣∣≪1. (14)

This suggests that the evolution rate can be approximated as a constant over a certain duration of the dilaton rolling.

To examine the possibility to enhance the tensor-to-scalar ratio , in this paper we consider a scenario that the spectator dilaton rolls over a period of the e-folding number , under the assumption that both the inflaton and the dilaton began to roll at a similar time. Then, as long as , the evolution rate can be approximated as a constant over the entire period of the dilaton rolling. Note that in our scenario, the dilaton field is stabilized before the end of inflation, and therefore can be significantly smaller than the total e-folding number of inflation. For simplicity, we assume that the transition from the rolling dilaton phase to the stabilized dilaton phase takes place within a short time interval . Then the dilaton-dependent gauge coupling evolves as

 I(τ)≡I(σ0(τ))∝a(τ)−n, (15)

where is a nonzero constant during the rolling phase, but right after the dilaton is stabilized. This might be a rather crude approximation for the real dilaton dynamics, but is sufficient for our purpose to explore the possibility to enhance . The reason to consider a dilaton field stabilized before the end of inflation is that it allows to be enhanced by a large factor while satisfying the observational constraints. If the dilaton field rolls until the end of inflation, whenever is significantly affected, scalar perturbation is dominated by the contribution from the rolling dilaton, which would lead to a too large deviation of the scalar spectral index from the observed value, or a too large non-gaussianity.

The evolution rate in (12) can be either positive or negative. Note that changing the sign of amounts to for the gauge coupling . For a positive , the field-dependent gauge coupling runs from the weak coupling regime to the strong coupling regime. For simplicity, we will focus on the case of positive with , where the production of electric fields dominates over the production of magnetic fields. This choice of opens a possibility that the gauge field in our setup can be identified as one of the standard model gauge fields if after the dilaton is stabilized.

For the dilaton coupling (15), the equation of motion of the gauge field mode takes the form

 ∂2τV+[k2−n(n−1)τ2]V=0. (16)

Imposing the Bunch-Davies initial condition,

 limkτ→−∞V(τ,k)=e−ikτ√2k,

the solution is given by

 V(τ,k)=1√2k√−kτπ2H(1)n−1/2(−kτ), (17)

where is the Hankel function of the first kind. Using the asymptotic form of the Hankel function:

 H(1)ν(z)≃−iΓ(ν)π(2z)ν+1Γ(ν+1)(z2)ν−iΓ(−ν)πcosνπ(z2)νforz≪1, (18)

we find that the gauge field mode in the superhorizon regime with is given by

 V(τ,k)≃−i√2kΓ(n−1/2)√π(2−kτ)n−1, (19)

where the blow up of the amplitude in the superhorizon limit (for ) is due to the tachyonic instability of gauge field caused by the rolling dilaton.

For subsequent discussion, it is convenient to define the electric and magnetic fields as

 Ei(τ,x)=−Ia2∂τAi(τ,x),Bi(τ,x)=Ia2(∇×→A)i, (20)

for which the energy density of the gauge field is given by

 ρU(1)≡TU(1)tt=12(|→E|2+|→B|2). (21)

One can now make the Fourier expansion:

 Ei(τ,x) = ∫d3k(2π)3/2ˆEi(τ,k)eik⋅x, ˆEi(τ,k) = ∑λϵi,λ(^k)[E(τ,k)ak,λ+E∗(τ,k)a†−k,λ], Bi(τ,x) = ∫d3k(2π)3/2ˆBi(τ,k)eik⋅x, ˆBi(τ,k) = ∑λλϵi,λ(^k)[B(τ,k)ak,λ+B∗(τ,k)a†−k,λ],

where the corresponding electric and magnetic mode functions are given by

 E(τ,k) = −1a2√k2√−kτπ2H(1)n+1/2(−kτ) (22) ≃ iΓ(n+1/2)√π√k2(Hτ)2(2−kτ)nfor|kτ|≪1, B(τ,k) = 1a2√k2√−kτπ2H(1)n−1/2(−kτ) (23) ≃ −iΓ(n−1/2)√π√k2(Hτ)2(2−kτ)n−1for|kτ|≪1.

Note that the last approximation for and are valid only for . Otherwise the latter two terms in (18) become important. Note also that the electric field always dominates over the magnetic field in the superhorizon regime with . For a given mode, the electric field on superhorizon scale decreases (), remains constant (), and grows (). As we will see in the subsequent two sections, the gauge fields produced by rolling dilaton can significantly affect the scalar and tensor perturbations when .

## Iii Scalar perturbation

In the spacially flat gauge, the curvature perturbation is given by

 R=−Hδqρ+p, (24)

where is the scalar 3-momentum potential defined as for the energy momentum tensor perturbation . In the multi-fluid case, it can be decomposed as

 R=∑α(ρ+p)α(ρ+p)RαforRα≡−Hδqα(ρ+p)α, (25)

where denotes the fluid species. In our scenario, the dilaton and gauge field fluctuations could constitute an important part of the total curvature perturbation during the rolling phase of dilaton. However, after the dilaton is stabilized, the dilaton perturbation becomes a massive field, and gauge fields are no longer produced. Then the dilaton and gauge field contributions to are quickly diluted away as and . If the universe has experienced a sufficient inflationary expansion after the dilaton is stabilized, which is the case of our prime interest, the curvature perturbation at the end of inflation is determined simply by the inflaton perturbation as

 R≃Rϕ=Hδϕ˙ϕ. (26)

In fact, if the dilaton keeps rolling until the end of inflation, whenever tensor perturbation is significantly affected, scalar perturbation is dominated by the contribution sourced by rolling dilaton. Such scenario then yields a too large spectral index and non-gaussianity to be compatible with the observational constraints (36). In the following, we compute the inflaton perturbation at the end of inflation, including the effect of pre-evolution during the period before the dilaton stabilization.

### iii.1 Evolution of the inflaton and dilaton perturbations

The equations of motion for the background inflaton and dilaton fields are given by

 ϕ′′0+2Hϕ′0+a2∂ϕV(ϕ0) = 0, σ′′0+2Hσ′0+a2∂σV(σ0) = a2∂σII⟨|→E|2−|→B|2⟩, (27)

where the prime denotes the derivative with respect to the conformal time coordinate , and . Assuming a slow-roll motion of the background fields, and also neglecting the back-reaction effects, we obtain the equations of motion of perturbations as

 δϕ′′+2Hδϕ′+k2δϕ+a2⎛⎝∂2ϕV−3˙ϕ20M2P⎞⎠δϕ−3a2˙σ0˙ϕ0M2Pδσ = S1(τ,k), (28) δσ′′+2Hδσ′+k2δσ+a2(∂2σV−3˙σ20M2P)δσ−3a2˙σ0˙ϕ0M2Pδϕ = S2(τ,k)+S3(τ,k), (29)

where the source terms () in the momentum space are given by

 S1(τ,k) = a2˙ϕ02M2PH∫d3p(2π)3/2(k−p)ipjk2[ˆEi(τ,p)ˆEj(τ,k−p)+ˆBi(τ,p)ˆBj(τ,k−p)], S2(τ,k) = a2∂σII∫d3p(2π)3/2[ˆEi(τ,p)ˆEi(τ,k−p)+ˆBi(τ,p)ˆBi(τ,k−p)], S3(τ,k) = a2˙σ02M2PH∫d3p(2π)3/2(k−p)ipjk2,[ˆEi(τ,p)ˆEj(τ,k−p)+ˆBi(τ,p)ˆBj(τ,k−p)].

See Appdenix. A for the derivation of the above equations of motion. The source terms and are due to the gravitational interaction between the inflaton/dilaton fluctuation and the gauge fields produced by the rolling background dilaton, while originates from the direct coupling between the dilaton and gauge fields. As can be seen from (28) and (29), even though there is no direct coupling between the inflaton and dilaton, their perturbations can be mixed with each other by gravitational interaction. As a result, the inflaton perturbation can be significantly affected by the dilaton perturbation sourced by and . As we will see later, the inflaton perturbation sourced by gauge fields comes dominantly from the source term .

Let us divide the inflaton perturbation into four pieces,

 δϕ=δϕ(v)+δϕ(S1)+δϕ(S2)+δϕ(S3), (30)

where represents the piece from vacuum fluctuation, while () represent the parts induced by the source terms . To obtain the solution, it is convenient to rotate the field basis into the propagation eigenbasis. For this, we rewrite (28) and (29) as (52); (15)

 [∂2τ+(k2−2τ2)+1τ2(ΔϕΔΔΔσ)](aδϕaδσ)=a(τ)(S1S2+S3), (31)

where

 Δα=∂2αV−3˙α20/M2PH2−3ϵ≃3(ηα−2ϵα)−3ϵ(α=ϕ,σ),Δ=−3˙ϕ0˙σ0M2PH2≃−6√ϵϕϵσ (32)

for the slow roll parameters

 ϵα≡M2P2(∂αVV)2,ηα≡M2P(∂2αVV),ϵ≡−˙HH2. (33)

In our setup, these slow roll parameters are small and can be approximated as constant over the time scale of our interest. Then the propagation eigenstates () defined as

 (aδϕaδσ)=(cosθsinθ−sinθcosθ)(v1v2), (34)

obey

 [∂2τ+(k2−2τ2)+1τ2(Δ+00Δ−)](v1v2)=a(τ)(S1cosθ−(S2+S3)sinθ(S2+S3)cosθ+S1sinθ), (35)

where the rotation angle is determined as

 sin2θ=−2ΔΔ+−Δ−,cos2θ=Δϕ−ΔσΔ+−Δ−, (36)

with

 Δ±=12(Δϕ+Δσ)±12√(Δϕ−Δσ)2+4Δ2. (37)

One can now split the propagation eigenstates into two pieces:

 v1=v(v)1+v(s)1,v2=v(v)2+v(s)2, (38)

where () denote the piece from vacuum fluctuation, while are the piece sourced by gauge fields. Here we are interested in the sourced part which is given by

 v(s)1(τ,k) = cosθ∫τdτ′a(τ′)Gk(τ,τ′;Δ+)S1−sinθ∫τdτ′a(τ′)Gk(τ,τ′;Δ+)(S2+S3), v(s)2(τ,k) = cosθ∫τdτ′a(τ′)Gk(τ,τ′;Δ−)(S2+S3)+sinθ∫τdτ′a(τ′)Gk(τ,τ′;Δ−)S1,

where the Green function obeys

 [∂2τ+(k2−2−Δ±τ2)]Gk(τ,τ′;Δ±)=δ(τ−τ′). (39)

See Appendix. B for the properties of this Green function up to first order in slow-roll parameters.

After the dilaton field is stabilized, but before the inflation is over, the dilaton fluctuation and the source terms are rapidly diluted away, while leaving the inflaton perturbation frozen to be constant in the superhorizon regime. The inflaton perturbation sourced by gauge fields is determined to be

 a(τ)δϕ(s)(τ,k)=v(s)1cosθ+v(s)2sinθ=aδϕ(S1)+aδϕ(S2)+aδϕ(S3), (40)

where

 a(τ)δϕ(S1) ≃ ∫τdτ′a(τ′)Gk(τ,τ′;0)S1(τ′,k), a(τ)δϕ(S2) ≃ sin2θ2∫τdτ′a(τ′)[Gk(τ,τ′;Δ−)−Gk(τ,τ′;Δ+)]S2(τ′,k), a(τ)δϕ(S3) ≃ sin2θ2∫τdτ′a(τ′)[Gk(τ,τ′;Δ−)−Gk(τ,τ′;Δ+)]S3(τ′,k). (41)

In fact, the three source terms are not equally important. We can estimate their relative importance by tracking their dependence on the gravitational coupling , as well as investigating the coupling structure of the inflaton and dilaton perturbations. (See Fig. 1 for instance.) It is then straightforward to find

 δϕ(S1)H ∼ 1H3(˙ϕ02M2PH)∫d3p(ki−pi)pjk2ˆEi(τ,p)ˆEj(τ,k−p), (42) δϕ(S2)H ∼ 1H3(˙ϕ02M2PH)(˙IHI)∫d3pˆEi(τ,p)ˆEi(τ,k−p), (43) δϕ(S3)H ∼ 1H3(˙ϕ02M2PH)(˙σ20M2PH2)∫d3p(ki−pi)pjk2ˆEi(τ,p)ˆEj(τ,k−p). (44)

Note that we are considering the inflaton perturbation after the dilaton is stabilized. However and in the coefficients of the above estimates correspond to the values while the dilaton is rolling, which were approximated as nonzero constants.

Obviously is subdominant compared to as it is further suppressed by the slow roll parameter . On the other hand, as we will see, we need to enhance the tensor-to-scalar ratio through a rolling dilaton, so the factor in (43) does not cause an additional suppression of . In fact, from the asymtotic behavior (22) of the electric mode function, one can easily recognize that the momentum integral of (42)–(43) for receives the main contribution from the region near or . We then have schematically

 δϕ(S1)(τ,k)δϕ(S2)(τ,k)∼∫dq[q|^k−q|](3/2−n)∫dq[q|^k−q|](1/2−n), (45)

where is the dimensionless normalized wave vector. This implies that the inflaton perturbation sourced by gauge fields is dominated by for the case with , where the rolling dilaton can enhance the tensor-to-scalar ratio significantly. We will therefore consider only in the following discussion of scalar perturbation sourced by gauge fields.

### iii.2 Scalar power spectrum

The power spectrum of the inflaton perturbation is defined as

 ⟨δϕ(k)δϕ(k′)⟩=2π2k3Pδϕ(k)δ(3)(k+k′). (46)

In our case, the inflaton perturbation consists of the contribution from vacuum fluctuation and the piece sourced by gauge fields during the phase of rolling dilaton. Since the sourced part is dominated by , we have

 δϕ≃δϕ(v)+δϕ(S2). (47)

As and are uncorrelated, the power spectrum of the curvature perturbation after the dilaton is stabilized is given by

 PR(k)=P(v)R(k)+P(s)R(k), (48)

where is the nearly scale invariant power spectrum originating from the vacuum fluctuation of the inflaton field:

 Missing or unrecognized delimiter for \right (49)

and is the sourced power spectrum:

 P(s)R(k)≃(H˙ϕ)2P(S2)δϕ(k). (50)

Let us now evaluate the sourced power spectrum. Using the solution of in (41), we find

 ⟨δϕ(S2)(τ,k)δϕ(S2)(τ,k′)⟩=sin22θ4a2∫τdτ1a(τ1)[Gk(τ,τ1;Δ−)−Gk(τ,τ1;Δ+)] ×∫τdτ2a(τ2)[Gk′(τ,τ2;Δ−)−Gk′(τ,τ2;Δ+)]⟨S2(τ1,k)S2(τ2,k′)⟩. (51)

Ignoring the subdominant magnetic field, we find also

 ⟨S2(τ1,k)S2(τ2,k′)⟩≃2a(τ1)2a(τ2)2(I,σI)2δ(3)(k+k′)×∫d3p(2π)3[1+(^p⋅ˆk−p)2]E(τ1,p)E(τ1,|k−p|)E∗(τ2,p)E∗(τ2,|k−p|), (52)

where is the electric mode function given in (22). Then the power spectrum of the sourced curvature perturbation is obtained to be

 P(s)R(k) ≃ 24n−2n29π4Γ4(n+1/2)(HMP)4{∫d3q(2π)3[1+(^q⋅^q′)2]q−2n+1q′−2n+1 (53) × ∣∣∣∫zℓzdz′z′−2n+3[(23−lnz′z)+z3z′3(lnzz′−23)]∣∣∣2},

where and are the normalized wave vectors, and . Here the integration over is performed from to , where

 zℓ=1max(q,q′) (54)

corresponds to the time when both and become a superhorizon mode.

Regarding to the integration over , we note that the electric field stays constant (), or grows () in the superhorizon limit . As a result, the integration suffers from an infrared divergence when the internal momentum approaches to the poles at . On the other hand, only the scales that exit the horizon after the beginning of inflation are relevant for us. Then a physical infrared cutoff at can be applied to regulate the integral over , where corresponds to the scale that leaves the horizon at the beginning of inflation (42). Around the region where or , can be set as or . In the following, we will focus on the case with

 n−2>O(0.1), (55)

in which the dilaton-induced gauge fields can significantly affect the primordial perturbations. We then find

 P(s)R(k)≃24nn227π6Γ4(n+1/2)(HMP)4q−2n+4in2n−4∣∣∣9z−2n