Primordial non-Guassianity in ination with gravitationally enhanced friction Li-Yang Chen Hongwei Yuyand Puxun Wuz

2025-05-02 0 0 551.45KB 19 页 10玖币

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Primordial non-Guassianity in inﬂation with gravitationally

enhanced friction

Li-Yang Chen∗, Hongwei Yu†and Puxun Wu‡

Department of Physics and Synergetic Innovation

Center for Quantum Eﬀects and Applications,

Hunan Normal University, Changsha, Hunan 410081, China

Abstract

The gravitationally enhanced friction can reduce the speed of the inﬂaton to realize an ultra-

slow-roll inﬂation, which will amplify the curvature perturbations. The ampliﬁed perturbations

can generate a sizable amount of primordial black holes (PBHs) and induce simultaneously a signif-

icant background gravitational waves (SIGWs). In this paper, we investigate the primordial non-

Gaussianity of the curvature perturbations in the inﬂation with gravitationally enhanced friction.

We ﬁnd that when the gravitationally enhanced friction plays a role in the inﬂationary dynamics,

the non-Gaussianity is noticeably larger than that from the standard slow-roll inﬂation. During

the regime in which the power spectrum of the curvature perturbations is around its peak, the

non-Gaussianity parameter changes from negative to positive. When the power spectrum is at its

maximum, the non-Gaussianity parameter is near zero (∼ O(0.01)). Furthermore, the primordial

non-Gaussianity promotes the formation of PBHs, while its eﬀect on SIGWs is negligible.

∗clyrion@hunnu.edu.cn

†hwyu@hunnu.edu.cn

‡pxwu@hunnu.edu.cn

arXiv:2210.05201v2 [gr-qc] 26 Oct 2022

I. INTRODUCTION

Inﬂation resolves most of the problems, such as the ﬂatness, horizon and monopole prob-

lems, that plague the standard cosmological model [1–4]. During inﬂation the curvature

perturbations are stretched outside the Hubble horizon and then stop propagating with

the amplitudes frozen at certain nonzero values. Inﬂation predicts a nearly scale-invariant

spectrum for the curvature perturbations, which is well consistent with the CMB observa-

tions [5]. The CMB observations indicate that the amplitude PRof the power spectrum of

the curvature perturbations is about 10−9[5]. After inﬂation, these super-horizon perturba-

tions, which will reenter the Hubble radius during the radiation- or matter-dominated era,

result in the formation of large scale cosmic structures and at the same time lead to possi-

ble generation of primordial black holes (PBHs) [6–9]. The possibility is however slim for

the standard slow-roll inﬂation since the amplitude of the power spectrum of the curvature

perturbations is too small (∼10−9).

If a sizable amount of PBHs is formed in the early universe, PBHs with diﬀerent

masses can be used to explain diﬀerent astronomical events. For example, the O(10)M,

O(10−5)Mand O(10−12)MPBHs can explain the gravitational wave events observed by

the LIGO/Virgo collaboration [10–13] and six ultrashort-timescale microlensing events in

the OGLE data [14,15], and make up all dark matter [16–20], respectively, where Mis

the mass of the Sun. To generate abundant PBHs, PRis required to reach the order of

O(10−2). Since the CMB observations have put stringent constraints on PRonly at the

CMB scales, we can realize the production of abundant PBHs by enhancing the amplitude

of the power spectrum of the curvature perturbations about seven orders at small scales. As

PR∝1/ with being the slow roll parameter, a natural way to amplify the curvature per-

turbations is to include an ultra-slow-roll period during inﬂation. Flattening the inﬂationary

potential can reduce the rolling speed of the inﬂaton, which gives rises to an ultra-slow-roll

inﬂation [21–44]. The ultra-slow-roll inﬂation can also be achieved via slowing down the

inﬂaton by gravitationally enhancing friction [45–50]. Moreover, some other mechanisms,

such as parametric resonance [51–56], have also been proposed to amplify the curvature

perturbations.

When the ampliﬁed curvature perturbations reenter the Hubble horizon during the

radiation- or matter-dominated era, they will not only generate the PBHs, but also lead

simultaneously to large scalar metric perturbations, which become an eﬀective source of

background gravitational waves. These gravitational waves, called the scalar induced grav-

itational waves (SIGWs), may be detectable by the future GW projects such as LISA [57],

Taiji [58], TianQin [59] and PTA [60–63].

When we assess the abundance of PBHs and the energy density of SIGWs, the curvature

perturbations are assumed usually to be of a Gaussian distribution. This is because the

curvature perturbations generated during the standard slow-roll inﬂation are nearly Gaussian

with negligible non-Gaussianity. However, once the inﬂation departs from the slow-roll

inﬂation or it is driven by the noncanonical ﬁelds, the primordial non-Gaussianity of the

curvature perturbations may no longer be ignored. The primordial non-Gaussianity in the

ultra-slow-roll inﬂation has been studied widely [64–76], becuase the abundance of PBHs is

extremely sensitive to the primordial non-Gaussianity of the curvature perturbations. For

the PBHs generated from inﬂation with gravitationally enhanced friction mechanism [45,

77,78], the primordial non-Gaussianity might be non-negligible too since the inﬂation ﬁeld

couples derivatively with the gravity and the rolling of the inﬂaton is ultra slow. In this paper

we study, in the ultra-slow-roll inﬂation achieved through gravitationally enhanced friction,

the non-Gaussianity of the curvature perturbations and its eﬀect on the PBH abundance

and the energy density of SIGWs.

The paper is organized as follows: In Sec. II, we brieﬂy review the inﬂation model with

the nonminimal derivative coupling between inﬂation ﬁeld and gravity. Sec. III studies the

primordial non-Gaussianity of the curvature perturbations. In Sec. IV, the eﬀect of the

non-Gaussianity of the curvature perturbations on the abundance of PBHs and the energy

density of SIGWs are assessed. Finally, we give our conclusions in Sec. V.

II. INFLATION WITH THE GRAVITATIONALLY ENHANCED FRICTION

To enhance the friction term in the equation of motion of the inﬂaton through the gravity,

we consider a nonminimal derivative coupling between the inﬂaton ﬁeld φand gravity, with

the action given by

S=Zd4x√−g"M2

2R−1

2 gµν −1

θ(φ)Gµν !∇µφ∇νφ−V(φ)#,(1)

where Mpl is the reduced Planck mass, and gis the determinant of the metric tensor gµν ,R

is the Ricci scalar, Gµν is the Einstein tensor, θ(φ) is the coupling function, and V(φ) is the

potential of the scalar inﬂaton ﬁeld.

In the spatially ﬂat Friedmann-Robertson-Walker background

ds2=−dt2+a(t)2dx2(2)

with a(t) being the scale factor, one can obtain, from the action (1), the background equa-

tions

3H2=1

pl "1

2 1 + 9

θ(φ)H2!˙

φ2+V(φ)#,(3)

−2˙

H=1

pl " 1 + 3

θ(φ)H2−1

θ(φ)˙

H!˙

φ2−1

θ,φH˙

φ3−2

θ(φ)H˙

φ¨

φ#,(4)

1 + 3

θ(φ)H2!¨

φ+"1 + 1

θ(φ)2˙

H+ 3H2#3H˙

φ+3

2M2

θ,φH2˙

φ2+V,φ = 0 ,(5)

where an overdot denotes the derivative with respective to the cosmic time t,H=˙a

ais the

Hubble parameter, θ,φ =dθ/dφ, and V,φ =dV/dφ.

To describe the slow-roll inﬂation, we deﬁne the slow-roll parameters

=−˙

H2, δφ=¨

H˙

φ,

δX=˙

φ2

2M2

plH2, δD=θ˙

φ2

4M4

.(6)

When {, |δφ|, δX, δD}  1 are satisﬁed, the slow-roll inﬂation is obtained.

In order to ﬁnd the power spectrum of the curvature perturbations, we need to derive the

quadratic action for the curvature perturbations Rfrom the action given in Eq. (1), which

takes the form [79–81]

S(2) =Zdtd3xa3Q˙

R2−c2

a2(∂R)2,(7)

where

Q=w1(4w1w3+ 9w2

3w2

,(8)

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Primordialnon-GuassianityininationwithgravitationallyenhancedfrictionLi-YangChen,HongweiYuyandPuxunWuzDepartmentofPhysicsandSynergeticInnovationCenterforQuantumEectsandApplications,HunanNormalUniversity,Changsha,Hunan410081,ChinaAbstractThegravitationallyenhancedfrictioncanreducethespeedoftheinato...

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Primordial non-Guassianity in ination with gravitationally enhanced friction Li-Yang Chen Hongwei Yuyand Puxun Wuz

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