[table]capposition=top capbtabboxtable[][]

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

Li-Yang<EMAIL_ADDRESS>Hongwei<EMAIL_ADDRESS>and
Puxun<EMAIL_ADDRESS>1Department of Physics and Synergetic
Innovation Center for Quantum Effects and Applications, Hunan Normal
University, Changsha, Hunan 410081, China
2Institute of Interdisciplinary Studies, Hunan Normal University, Changsha,
Hunan 410081, China

###### Abstract

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.

##  I Introduction

When the large curvature perturbations produced during inflation re-enter the
Hubble horizon in the radiation- or matter-dominated epoch, the gravitational
collapse of dense regions will result in generation of primordial black holes
(PBHs) Zeldovich ; Hawking ; Carr ; Meszaros ; Carr1975 ; Khlopov ; Ozsoy2023
; Choudhury2023f . The PBHs have been used to explain the gravitational wave
(GW) events generated by merger of massive black holes observed by the
LIGO/Virgo collaboration lg1 ; lg2 ; lg3 ; lg4 , the six ultrashort-timescale
microlensing events in the OGLE data P.Mroz2017 ; H.Niikura2019 , and the
ninth asteroid in the solar system J.Scholtz2020 ; E.Witten . Furthermore, the
asteroid-mass PBHs can make up all dark matter A.Katz2018 ; H.Niikura2019a ;
A.Barnacka2012 ; P.W.Graham2015 . Accompanying the formation of PBHs, the
large scalar perturbations will become an important GW source and lead to the
so-called scalar induced GWs (SIGWs). The SIGWs may be detected by the future
GW projects such as LISA lisa  Taiji taiji , TianQin tianqin  and PTA pta1 ;
pta2 ; pta3 ; pta4 . An observation of such SIGWs would provide evidence for
the existence of PBHs.

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 $\mathcal{O}(10^{-9})$. To generate a sizable
amount of PBHs requires that the amplitude of the power spectrum of the
curvature perturbations reaches about $\mathcal{O}(10^{-2})$. Since the CMB
observations only give a limit on the curvature perturbations at large scales
Aghanim2020  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 $\mathcal{R}$ is given by
$\mathcal{P_{R}}=\frac{H^{2}}{8\pi^{2}\epsilon c_{s}}$ when the mode exits the
horizon during inflation in the standard slow-roll inflation. Here $\epsilon$
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 Yokoyama1998 ; Choudhury2014 ;
Germani2017 ; Motohashi2017 ; H. Di2018 ; Ballesteros2018 ; Dalianis2019 ;
Gao2018 ; Tada2019 ; Mishra2020 ; Atal2020 ; Ragavendra2021 ; Bhaumik2020 ;
Drees2021 ; C.Fu2020 ; Xu2020 ; Lin2020 ; Dalianis2021 ; Yi2021 ; Gao2021 ;
Yi2021b ; TGao2021 ; Solbi2021 ; Gao2021b ; Solbi2021b ; Zheng2021 ;
Teimoori2021a ; Cai2021 ; Wang2021 ; Fuchengjie2019 ; fuchengjie2020 ;
Dalianis2020 ; Teimoori2021 ; Karam2022 ; Heydari2022 ; Heydari2022b ;
Bellido2017 ; Ezquiaga2018 ; Pi2022 ; Choudhury2023cc ; Meng2022 ; Mu2022 ;
Kawaguchi2022 ; Fu2022 ; Chen2022 ; Gu2023 ; Yi2022 ; S.Choudhury2023 ;
Dalianis2023 ; Zhao2023 ; Heydari2023 ; Choudhury2023a ; Firouzjahi2023 ;
Mishra2023 ; Asadi2023 ; Cole2023 ; Choudhury2023b ; Frolovsky2023 ; Chen2023a
; Su2023 ; Cable2023 ; Escriva2023 ; Choudhury2023c ; Choudhury2023e ;
Choudhury2024  or to suppress the sound speed $c_{s}$ G. Ballesteros ;
Kamenshchik ; Gorji2022 ; Romano3 ; Ballesteros2022 ; R. Zhai ; Qiu2022 ;
Zhai2023 . Additionally, some other ways can also predict the formation of
PBHs Choudhury2023d ; Corral2023 ; Kasai2023 ; Stamou2023 ; Ge2023 ;
Bhattacharya2023 ; Ijaz2023 ; Cheung2023 ; Mansoori2023 ; Gow2023 ; Li2023 ;
Tada2023 ; Cai2023y ; Huang2023a ; Ghoshal2023 ; Saburov2023 ; Tada2023a ;
Huang2023 ; Arya2023 ; Cai2023i ; Martin2020a ; Martin2020b . 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 Germani2017 ; Motohashi2017 ; Ezquiaga2018 ; H.
Di2018 ; Ballesteros2018 ; Dalianis2019 ; Gao2018 ; Drees2021 ; C.Fu2020 ;
Xu2020 ; Lin2020 ; Dalianis2021 ; Yi2021 ; Gao2021 ; Yi2021b ; TGao2021 ;
Solbi2021 ; Gao2021b ; Solbi2021b ; Zheng2021 ; Teimoori2021a ; Cai2021 ;
Wang2021 . Recently, it has been shown that the gravitationally enhanced
friction can also slow the rolling speed of the inflaton Fuchengjie2019 ;
fuchengjie2020 ; Dalianis2020 ; Teimoori2021 ; Heydari2022 ; Heydari2022b . 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 yfcai2018 ; yfcai2019 ;
c.chen2019 ; c.chen2020 ; Addazi2022 ; Cai2020 ; bLi2023 . 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 $\phi$ and gravity. The action of the system has the form

$\displaystyle\mathcal{S}=\int
d^{4}x\sqrt{-g}\left[\frac{M_{\mathrm{pl}}^{2}}{2}R-\frac{1}{2}\left(g^{\mu\nu}-\frac{1}{M_{\mathrm{pl}}^{2}}\theta(\phi)G^{\mu\nu}\right)\nabla_{\mu}\phi\nabla_{\nu}\phi-V(\phi)\right].$
(1)

Here $g$ is the determinant of the metric tensor $g_{\mu\nu}$,
$M_{\mathrm{pl}}$ is the reduced Planck mass, $R$ is the Ricci scalar,
$G_{\mu\nu}$ is the Einstein tensor, $\theta(\phi)$ denotes the coupling
function, and $V(\phi)$ 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 Deffayet2011 ; Kobayashi2011 , which can be free of the
ghost and gradient instabilities Kobayashi2011 . The Lagrangian of such
Horndeskis theories has the term
$G_{5}(\phi,X)G_{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi$, where $G_{5}$ is a
generic function of $\phi$ and
$X\equiv-\partial_{\mu}\phi\partial^{\nu}\phi/2$. By choosing the function
$G_{5}=-\kappa^{2}\chi(\phi)/2$, the term containing $\theta$ in Eq. (1) can
be recovered from the Horndeski’s Lagrangian after integration by parts with
$\theta$ being defined as $\theta\equiv d\chi/d\phi$.

In the spatially flat Friedmann-Robertson-Walker background, the background
equations derived from the action (1) are

$\displaystyle
3H^{2}=\frac{1}{M_{\mathrm{pl}}^{2}}\left[\frac{1}{2}\left(1+\frac{9}{M_{\mathrm{pl}}^{2}}\theta(\phi)H^{2}\right)\dot{\phi}^{2}+V(\phi)\right],$
(2)
$\displaystyle-2\dot{H}=\frac{1}{M_{\mathrm{pl}}^{2}}\left[\left(1+\frac{3}{M_{\mathrm{pl}}^{2}}\theta(\phi)H^{2}-\frac{1}{M_{\mathrm{pl}}^{2}}\theta(\phi)\dot{H}\right)\dot{\phi}^{2}-\frac{1}{M_{\mathrm{pl}}^{2}}\theta_{,\phi}(\phi)H\dot{\phi}^{3}-\frac{2}{M_{\mathrm{pl}}^{2}}\theta(\phi)H\dot{\phi}\ddot{\phi}\right],$
(3)
$\displaystyle\left(1+\frac{3}{M_{\mathrm{pl}}^{2}}\theta(\phi)H^{2}\right)\ddot{\phi}+\left[1+\frac{1}{M_{\mathrm{pl}}^{2}}\theta(\phi)\left(2\dot{H}+3H^{2}\right)\right]3H\dot{\phi}+\frac{3}{2M_{\mathrm{pl}}^{2}}\theta_{,\phi}(\phi)H^{2}\dot{\phi}^{2}+V_{,\phi}(\phi)=0.$
(4)

Here the overdot denotes the derivative with respect to the cosmic time $t$,
$\theta_{,\phi}\equiv d\theta/d\phi$ and $V_{,\phi}\equiv dV/d\phi$.

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

$\displaystyle\epsilon$
$\displaystyle=-\frac{\dot{H}}{H^{2}},\quad\delta_{\phi}=\frac{\ddot{\phi}}{H\dot{\phi}},$
$\displaystyle\delta_{X}$
$\displaystyle=\frac{\dot{\phi}^{2}}{2{M_{\mathrm{pl}}^{2}}H^{2}},\quad\delta_{D}=\frac{\theta\left(\phi\right)\dot{\phi}^{2}}{4{M_{\mathrm{pl}}^{4}}}.$
(5)

When $\\{\epsilon,|\delta_{\phi}|,\delta_{X},|\delta_{D}|\\}\ll 1$ are
satisfied, the slow-roll inflation is obtained. Furthermore, we add the
condition: $|\frac{3\theta(\phi)H^{2}}{M_{\mathrm{pl}}^{2}}|\ll 1$. 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 $\mathcal{R}$. Expanding the action given in Eq. (1)
to the second-order, one can obtain the action of $\mathcal{R}$ A.D.Felice2011
; Germani2010 ; S.Tsujikawa2012 ; Kobayashi2011

$\displaystyle S^{(2)}=\int
dtd^{3}xa^{3}Q\left[\dot{\mathcal{R}}^{2}-\frac{c_{s}^{2}}{a^{2}}(\partial\mathcal{R})^{2}\right],$
(6)

where

$\displaystyle Q=\frac{w_{1}\left(4w_{1}w_{3}+9w_{2}^{2}\right)}{3w_{2}^{2}},$
(7)

and

$\displaystyle
c_{s}^{2}=\frac{3\left(2w_{1}^{2}w_{2}H-w_{2}^{2}w_{4}+4w_{1}\dot{w}_{1}w_{2}-2w_{1}^{2}\dot{w}_{2}\right)}{w_{1}\left(4w_{1}w_{3}+9w_{2}^{2}\right)}$
(8)

with

$\displaystyle w_{1}$
$\displaystyle=M_{\mathrm{pl}}^{2}\left(1-2\delta_{D}\right),$ $\displaystyle
w_{2}$ $\displaystyle=2HM_{\mathrm{pl}}^{2}\left(1-6\delta_{D}\right),$
$\displaystyle w_{3}$
$\displaystyle=-3H^{2}M_{\mathrm{pl}}^{2}\left(3-\delta_{X}-36\delta_{D}\right),$
$\displaystyle w_{4}$
$\displaystyle=M_{\mathrm{pl}}^{2}\left(1+2\delta_{D}\right).$ (9)

Defining $z\equiv a\sqrt{2Q}$ and $u_{k}\equiv z\mathcal{R}_{k}$, we find that
$u_{k}$ satisfies the equation

$\displaystyle\ddot{u}_{k}+H\dot{u}_{k}+\left(\frac{c_{s}^{2}k^{2}}{a^{2}}-\frac{\ddot{z}+H\dot{z}}{z}\right)u_{k}=0.$
(10)

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

$\displaystyle\mathcal{P}_{\mathcal{R}}\left(k\right)=\frac{k^{3}}{2\pi^{2}}\left|\frac{u_{k}}{z}\right|^{2}.$
(11)

The CMB observations have implied that at large scales this power spectrum is
a nearly scale-invariant spectrum with the amplitude being about
$\mathcal{O}(10^{-9})$ Aghanim2020 .

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 $\theta(\phi)$ to take an
oscillating form

$\displaystyle\theta(\phi)=w\sin\left(\frac{\phi}{\phi_{c}}\right)\Theta\left(\phi_{s}-\phi\right)\Theta\left(\phi-\phi_{e}\right).$
(12)

Here $w$ is a dimensionless constant, which must satisfy
$\omega\ll\frac{M_{\mathrm{pl}}^{2}}{3H^{2}}$ since
$|\frac{3\theta(\phi)H^{2}}{M_{\mathrm{pl}}^{2}}|\ll 1$, and $\phi_{c}$ is a
quantity with the same dimension as $\phi$ and is set to be much less than
$\phi$. Meanwhile $\phi_{s}$ and $\phi_{e}$ represent the beginning and the
end of the coupling, respectively, and $\Theta$ is the unit Heaviside step
function. The value of $\phi_{s}$ is chosen to be away from that of $\phi$ 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:
$\mathcal{P}_{\mathcal{R}_{0}}=\frac{V^{3}}{12\pi^{2}M_{\mathrm{pl}}^{6}V_{,\phi}^{2}}$.
The spectral index $n_{s}$ and the tensor-scalar-ratio $r$ are $n_{s}\simeq
1-\left(6\epsilon_{V}-2\eta_{V}\right)$ and $r\simeq{16\epsilon_{V}}$,
respectively, where
$\epsilon_{V}=\frac{1}{2}M_{\mathrm{pl}}^{2}\big{(}\frac{V_{,\phi}}{V}\big{)}^{2}\simeq\epsilon$,
and
$\eta_{V}={M_{\mathrm{pl}}^{2}}\frac{V_{,\phi\phi}}{V}\simeq\epsilon+\delta_{\phi}$.
To be consistent with the CMB observations, we choose the potential of the
inflaton field to be the Starobinsky potential Starobinsky1980

$\displaystyle{V}(\phi)=\Lambda^{4}\left[1-\exp\left(-\sqrt{2/3}\phi/M_{\mathrm{pl}}\right)\right]^{2},$
(13)

where $\Lambda$ is a constant.

##  III parametric resonance

Since the parametric resonance occurs deep inside the Hubble horizon
$\left(c_{s}k\gg aH\right)$, the Eq. (10) can be simplified to be

$\displaystyle\ddot{u}_{k}+c_{s}^{2}\frac{k^{2}}{a^{2}}u_{k}\approx 0.$ (14)

Considering the slow-roll conditions given in Eq. (II),
$w\ll\frac{M^{2}_{\mathrm{pl}}}{3H^{2}}$ and $\phi_{c}\ll\phi$ during
inflation, we find that the background equations (Eqs. (3, 4)) can be reduced
to

$\displaystyle\dot{H}\approx\frac{1}{2M_{\mathrm{pl}}^{4}}H\dot{\phi}^{3}{\theta}_{,\phi}(\phi),~{}~{}~{}~{}\ddot{\phi}\approx-\frac{3}{2M_{\mathrm{pl}}^{2}}H^{2}\dot{\phi}^{2}{\theta}_{,\phi}(\phi),$
(15)

and then the sound speed square of the curvature perturbations can be
simplified to be

$\displaystyle c_{s}^{2}\approx 1+\delta
c_{s}=1+\frac{3w^{2}H\dot{\phi}^{3}}{2\phi_{c}M_{\mathrm{pl}}^{6}}\sin{\frac{2\phi}{\phi_{c}}}.$
(16)

We find that $\delta c_{s}$ oscillates around zero and satisfies $|\delta
c_{s}|\ll 1$. 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 Babichev2008 ; Armendariz2001 ;
Mukhanov2006 ; Armendariz1999 ; Babichev2006 ; Kang2007 ; Armendariz2005 .
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. $\theta(\phi)\sim\sin^{2}(\phi)$, is chosen,
we find that the subliminal oscillation of the sound speed square is possible.
Substituting Eq. (16) into Eq. (14), one can obtain

$\displaystyle\ddot{u}_{k}+\left(\frac{k^{2}}{a^{2}}+\frac{3w^{2}H_{s}\dot{\phi}_{s}^{3}k^{2}}{2\phi_{c}M_{\mathrm{pl}}^{6}a^{2}}\sin\frac{2\phi}{\phi_{c}}\right)u_{k}\approx
0.$ (17)

We assume that the inflaton field evolves from $\phi_{s}$ to $\phi_{e}$ during
a short time, which indicates that during this short time the evolution of
$\phi$ can be expressed approximately as
$\phi\approx\phi_{s}+\dot{\phi}_{s}\left(t-t_{s}\right)$, where $t_{s}$ is the
time when $\phi=\phi_{s}$. Setting
$x=\frac{\phi}{\phi_{c}}-\frac{\pi}{4}=\frac{\dot{\phi}_{s}}{\phi_{c}}t+\frac{(\phi_{s}-\dot{\phi}_{s}t_{s})}{\phi_{c}}-\frac{\pi}{4}$
and $k_{c}=\frac{\left|\dot{\phi}_{s}\right|}{\phi_{c}}$, we find that the Eq.
(17) can be transformed into the form of the Mathieu equation

$\displaystyle\frac{d^{2}u_{k}}{dx^{2}}+\left[A_{k}(x)-2q\cos
2x\right]u_{k}=0,$ (18)

where

$\displaystyle
A_{k}(x)\equiv\frac{k^{2}}{k_{c}^{2}a^{2}},~{}~{}~{}q\equiv-\frac{3w^{2}H_{s}\dot{{\phi}_{s}}\phi_{c}k^{2}}{4M_{\mathrm{pl}}^{6}a^{2}}=2C\frac{k^{2}}{k_{c}^{2}a^{2}}.$
(19)

For the Mathieu equation, the resonant bands are close to narrow regions near
$A_{k}(x)\simeq n^{2}~{}\left(n=1,2,3...\right)$. The width of each resonant
band is $\Delta k\sim q^{n}$. If $0<q\ll 1$, 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 $\mu_{k}$ which describes the
rate of exponential growth has the form

$\displaystyle\mu_{k}(t)=\Re\left(\sqrt{\left(\frac{q}{2}\right)^{2}-\left(\frac{k}{k_{c}a}-1\right)^{2}}\right)=\Re\left(\sqrt{\left(C\frac{k^{2}}{k_{c}^{2}a^{2}}\right)^{2}-\left(\frac{k}{k_{c}a}-1\right)^{2}}\right).$
(20)

Here $\Re$ refers to taking the real part. Then, we find that the resonance
occurs in a narrow band

$\displaystyle k_{-}<\frac{k}{a}<k_{+},$ (21)

where $k_{-}=k_{c}(1-|C|)$ and $k_{+}=k_{c}(1+|C|)$. In obtaining the
expressions of $k_{\pm}$, we have used the condition $|C|\ll 1$ derived from
$0<q\ll 1$. 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 $\phi_{s}$ and $\phi_{e}$, 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 $\Delta k=k_{+}a_{e}-k_{-}a_{s}$, which is determined mainly by
$k_{c}$, $\phi_{e}$ and $\phi_{s}$.

During the parametric resonance, the curvature perturbations will be enhanced
exponentially

$\displaystyle\mathcal{A}(k)\equiv\left|\frac{u_{k}\left(t_{F}\right)}{u_{k}\left(t_{I}\right)}\right|\simeq\exp\left(\int_{t_{I}}^{t_{F}}\mu_{k}(t)k_{c}dt\right).$
(22)

Defining $\mathcal{B}_{k}(t)=\frac{k}{k_{c}a}$, the above integral can be re-
expressed as

$\displaystyle\mathcal{A}_{k}\left(\mathcal{B}_{k}\left(t_{I}\right),\mathcal{B}_{k}\left(t_{F}\right)\right)\simeq\exp\left(-\frac{k_{c}}{H_{s}}\int_{\mathcal{B}_{k}\left(t_{I}\right)}^{\mathcal{B}_{k}\left(t_{F}\right)}\sqrt{\left(C\mathcal{B}_{k}^{2}\right)^{2}-\left(\mathcal{B}_{k}-1\right)^{2}}\frac{d\mathcal{B}_{k}}{\mathcal{B}_{k}}\right).$
(23)

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: $k_{-}a_{s}<k\leq k_{+}a_{s}$,
$k_{+}a_{s}<k<k_{-}a_{e}$, and $k_{-}a_{e}\leq k<k_{+}a_{e}$, respectively.
The corresponding $\mathcal{B}_{k}(t_{I})$ and $\mathcal{B}_{k}(t_{F})$ can be
calculated as

$\displaystyle\mathcal{B}_{k}(t_{I})=\frac{k}{k_{c}a_{s}},\quad\mathcal{B}_{k}(t_{F})=\frac{k_{-}}{k_{c}}\quad\mathrm{for}\quad
k_{-}a_{s}<k\leq k_{+}a_{s},$ (24)
$\displaystyle\mathcal{B}_{k}(t_{I})=\frac{k_{+}}{k_{c}},\quad\mathcal{B}_{k}(t_{F})=\frac{k_{-}}{k_{c}}\quad\mathrm{for}\quad
k_{+}a_{s}<k<k_{-}a_{e},$ (25)
$\displaystyle\mathcal{B}_{k}(t_{I})=\frac{k_{+}}{k_{c}},\quad\mathcal{B}_{k}(t_{F})=\frac{k}{k_{c}a_{e}}\quad\mathrm{for}\quad
k_{-}a_{e}\leq k<k_{+}a_{e}.$ (26)

Apparently, for the second group: $k_{+}a_{s}<k<k_{-}a_{e}$,
$\mathcal{B}_{k}(t_{I})$ and $\mathcal{B}_{k}(t_{F})$ are independent of $k$,
which results in that $\mathcal{A}_{k}$ is independent of $k$ for this group.

The enhanced power spectrum of the curvature perturbations can be expressed
approximately as

$\displaystyle\mathcal{P}_{\mathcal{R}}\left(k\right)\approx\mathcal{A}_{k}^{2}\mathcal{P}_{\mathcal{R}_{0}}\left(k\right).$
(27)

In the region of $k_{+}a_{s}<k<k_{-}a_{e}$, since $\mathcal{A}_{k}$ 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
$\mathcal{P}_{\mathcal{R}}/\mathcal{P}_{\mathcal{R}_{0}}$ 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.

Figure 1:  The evolution of
$\mathcal{P}_{\mathcal{R}}/\mathcal{P}_{\mathcal{R}_{0}}$ 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 $w=2.36\times
10^{8}$, $\Lambda=3.14\times 10^{-3}M_{\mathrm{pl}}$,
$\phi_{s}=4.987M_{\mathrm{pl}}$, $\phi_{e}=4.98M_{\mathrm{pl}}$ and
$\phi_{c}=9.98\times 10^{-8}M_{\mathrm{pl}}$.  Figure 2: The power spectrum of
the curvature perturbations. The green shaded region is excluded from the
current CMB observation Aghanim2020 . The orange- and blue-shaded regions are
excluded by the $\mu$ distortion of CMB D.J.Fixsen  and the effect on the
$n-p$ ratio during big-bang nucleosynthesis (BBN) K.Inomata2016 ,
respectively. Cyan shaded region indicate the limitations of current PTA
observations in the power spectrum K.Inomata2019 .

##  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 relationship with $k$:

$\displaystyle M(k)=\gamma\frac{4\pi M_{\mathrm{pl}}^{2}}{H}\simeq
M_{\odot}\left(\frac{\gamma}{0.2}\right)\left(\frac{g_{*}}{10.75}\right)^{-\frac{1}{6}}\left(\frac{k}{1.9\times
10^{6}~{}\mathrm{Mpc}^{-1}}\right)^{-2}\;.$ (28)

Here $\gamma$ 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 $\gamma\simeq(1/\sqrt{3})^{3}$ Carr1975 . In Eq. (28),
$M_{\odot}$ 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 radiation-dominated period, we can set
$g_{*}=106.75$.

Based on the Press-Schechter theory Tada2019 ; Young2014 , the production rate
of PBHs with mass $M(k)$ is

$\displaystyle\beta(M)=\int_{\delta_{c}}\frac{d\delta}{\sqrt{2\pi\sigma^{2}(M)}}e^{-\frac{\delta^{2}}{2\sigma^{2}(M)}}=\frac{1}{2}\operatorname{erfc}\left(\frac{\delta_{c}}{\sqrt{2\sigma^{2}(M)}}\right)$
(29)

after assuming that the probability distribution function of the disturbance
obeys the Gaussian distribution. Here $\mathrm{erfc}$ is the complementary
error function, and $\delta_{c}$ is the threshold for the relative density
perturbation of the PBH formation, which is chosen to be $\delta_{c}\simeq
0.4$ Musco2013 ; Harada2013  in our calculation of the PBH abundance. The
variance $\sigma^{2}(M)$ represents the coarse-grained density contrast with
the smoothing scale $k$, and it takes the form

$\displaystyle\sigma^{2}(M(k))=\frac{16}{81}\int d\ln
q\;W^{2}\left(qk^{-1}\right)\left(qk^{-1}\right)^{4}\mathcal{P}_{\mathcal{R}}(q)\;.$
(30)

Here $W$ is the window function. We find that the PBH mass spectrum can be
obtained from the following equation

$\displaystyle
f(M)\equiv\frac{1}{\Omega_{\mathrm{DM}}}\frac{d\Omega_{\mathrm{PBH}}}{d\ln
M}\simeq\frac{\beta(M)}{1.84\times
10^{-8}}\left(\frac{\gamma}{0.2}\right)^{\frac{3}{2}}\left(\frac{10.75}{g_{*}}\right)^{\frac{1}{4}}\left(\frac{0.12}{\Omega_{\mathrm{DM}}h^{2}}\right)\left(\frac{M}{M_{\odot}}\right)^{-\frac{1}{2}}.$
(31)

Here $\Omega_{\mathrm{DM}}$ represents the current dark matter density
parameter and $h$ is the reduced Hubble constant. While
$\Omega_{\mathrm{DM}}h^{2}$ is constrained to be
$\Omega_{\mathrm{DM}}h^{2}\simeq 0.12$ by the Planck 2018 observations
Aghanim2020 . 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
$\frac{\Omega_{\mathrm{PBH}}}{\Omega_{\mathrm{DM}}}\simeq 0.99$.

Figure 3:  The mass spectrum of PBHs. The colored regions are ruled out by
observations including the extragalactic gamma ray background
($\mathrm{EG}\gamma$ B.J.Carr2010  ), the galactic center 511 keV gamma-ray
line (INTEGRAL) R.Laha , the white dwarfs explosion (WD) P.W.Graham2015 , the
microlensing events with Subaru HSC (Subaru HSC) H.Niikura2019 , with the
Kepler satellite (Kepler) K.Griest , with EROS/MACHO (EROS/MACHO) P.Tisserand
, and the CMB V. Poulin .

##  V SIGWs

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:

$\displaystyle
h_{ij}^{\prime\prime}+2\mathcal{H}h_{ij}^{\prime}-\nabla^{2}h_{ij}=-4\mathcal{T}_{ij}^{lm}S_{lm},$
(32)

where a prime denotes the derivative with respect to the conformal time,
$\mathcal{H}\equiv a^{\prime}/a$, $\mathcal{T}_{ij}^{lm}$ is the transverse-
traceless projection operator, and

$\displaystyle
S_{ij}=4\Psi\partial_{i}\partial_{j}\Psi+2\partial_{i}\Psi\partial_{j}\Psi-\frac{1}{\mathcal{H}^{2}}\partial_{i}\left(\mathcal{H}\Psi+\Psi^{\prime}\right)\partial_{j}\left(\mathcal{H}\Psi+\Psi^{\prime}\right)$
(33)

is the GW source term Ananda2007 ; Baumann2007 . Here $\Psi$ is the metric
scalar perturbation. In the radiation-dominated era, the evolution of $\Psi$
is
$\Psi_{k}(\eta)=\psi_{k}\frac{9}{(k\eta)^{2}}\left(\frac{\sin(k\eta/\sqrt{3})}{k\eta/\sqrt{3}}-\cos(k\eta/\sqrt{3})\right)$
Baumann2007 , where $\psi_{k}$ is the primordial perturbation, which relates
with the power spectrum of the primordial curvature perturbations through

$\displaystyle\left\langle\psi_{\mathbf{k}}\psi_{\tilde{\mathbf{k}}}\right\rangle=\frac{2\pi^{2}}{k^{3}}\left(\frac{4}{9}\mathcal{P}_{\mathcal{R}}(k)\right)\delta(\mathbf{k}+\tilde{\mathbf{k}}).$
(34)

Solving Eq. (32), one can obtain the GW energy density for each logarithmic
interval $k$ Kohri2018 :

$\displaystyle\Omega_{\mathrm{GW}}\left(\eta_{c},k\right)=$
$\displaystyle\frac{1}{12}\int_{0}^{\infty}dv\int_{|1-v|}^{|1+v|}du\left(\frac{4v^{2}-\left(1+v^{2}-u^{2}\right)^{2}}{4uv}\right)^{2}\mathcal{P}_{\mathcal{R}}(ku)\mathcal{P}_{\mathcal{R}}(kv)$
$\displaystyle\times\left(\frac{3}{4u^{3}v^{3}}\right)^{2}\left(u^{2}+v^{2}-3\right)^{2}\times\bigg{\\{}\left[-4uv+\left(u^{2}+v^{2}-3\right)\ln\left|\frac{3-(u+v)^{2}}{3-(u-v)^{2}}\right|\right]^{2}$
$\displaystyle+\pi^{2}\left(u^{2}+v^{2}-3\right)^{2}\Theta(v+u-\sqrt{3})\bigg{\\}}\,,$
(35)

where $\eta_{c}$ represents the time when $\Omega_{\mathrm{GW}}$ stops to
grow. The current energy density spectrum of SIGWs can be expressed as
Kohri2018 ; K.Inomata2019

$\displaystyle\Omega_{\mathrm{GW},0}h^{2}=0.83\left(\frac{g_{*}}{10.75}\right)^{-1/3}\Omega_{\mathrm{r},0}h^{2}\Omega_{\mathrm{GW}}\left(\eta_{c},k\right),$
(36)

where $\Omega_{\mathrm{r},0}h^{2}$ is the current density parameter of
radiation which is set to be $4.2\times 10^{-5}$. The current frequency $f$ of
SIGWs is related to $k$ through

$\displaystyle f=1.546\times
10^{-15}\frac{k}{1\mathrm{Mpc}^{-1}}\mathrm{~{}Hz}.$ (37)

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.

Figure 4:  The current energy spectrum (solid blue line) of SIGWs. The various
dotted lines represent the sensitive intervals of different gravitational wave
detectors including SKA ska , EPTA epta , TAIJI taiji , TIANQIN tianqin , LISA
lisa , and ALIGO aligo .

##  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 Fuchengjie2019 . 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 fuchengjie2020 , 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 Kristiano2022  and the parametric
resonance from the oscillating potential Inomata2023 . 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.

###### Acknowledgements.

We appreciate very much the insightful comments and helpful suggestions by
anonymous referees. This work is supported by the National Key Research and
Development Program of China Grant No. 2020YFC2201502, and by the National
Natural Science Foundation of China under Grants No. 12275080 and No.
12075084.

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$\eta=\eta_{1}\vee\eta_{2}$ and let $\\{\mu_{W}\\}_{W\in\eta}$ be the
canonical disintegration of $\mu$ with respect to $\eta$. Note that
$\mu_{\eta_{1}(\omega,x)}\ll\nu_{(\omega,x)}$ implies that
$\mu_{\eta(\omega,y)}\ll\nu_{(\omega,x)}$ for every
$\eta(\omega,y)\subset\eta_{1}(\omega,x)$. Therefore, by the uniqueness of the
canonical disintegration, one has
$\mu_{\eta_{2}(\omega,x)}\ll\nu_{(\omega,x)}$ for $\mu$-almost every
$(\omega,x)$.

The statement (1) implies (2) and, (3) implies (1) follow immediately from the
proposition below. ∎

######  Proposition 5.1.

Assuming that $(\Phi,\mu)$ has a positive Lyapunov exponent, then there is a
random stack $\mathcal{S}$ with positive $\mu$-measure, and a measurable
partition $\eta$ on $\widetilde{\mathcal{S}}:=\cup_{n\geq
0}\Phi^{n}\mathcal{S}$ such that

  1. (a)

$\eta$ subordinate to $W^{u}(\omega,x)$,

  2. (b)

$\eta$ is $\Phi$-decreasing,

  3. (c)

for any measurable set $E\subset\Omega\times\mathcal{B}$, the function
$(\omega,x)\mapsto\nu_{(\omega,x)}(E_{\omega}\cap\eta(\omega,x)_{\omega})$ is
finite-valued and measurable on a full $\mu$-measure set, and

  4. (d)

$\vee_{n\geq 0}\Phi^{-n}\eta$ is partition of $\widetilde{\mathcal{S}}$ into
points modulo $\mu$-measure $0$.

###### Proof.

Using the notations in Section 4.4, we first construct a random stack. Let
$r\in(1/2,1)$ and let

$\mathcal{S}^{r}_{\omega}:=\bigcup\\{W^{u}_{\delta,(\omega,y)}\cap\bar{B}(x_{\ast},2rr_{2}):y\in
A_{\ast}\ \text{such that}\ (\omega,y)\in\Gamma_{\ell_{0}}\\},$

and define $\mathcal{S}^{r}:=\bigcup\\{\\{\omega\\}\times
S^{r}_{\omega}:\exists y\in A_{\ast}\ \text{such that}\
(\omega,y)\in\Gamma_{\ell_{0}}\\}$. Let
$\widetilde{\mathcal{S}}:=\widetilde{\mathcal{S}}^{r}=\cup_{n\geq
0}\Phi^{n}\mathcal{S}^{r}$, since $\mu$ is $\Phi$-ergodic one has
$\mu(\widetilde{\mathcal{S}})=1$.

Let $\xi:=\\{\\{\omega\\}\times W_{\omega}:W_{\omega}\in\xi_{\omega}\\}$ be a
measurable partition of $\mathcal{S}^{r}$, where $\xi_{\omega}$ is the
measurable partition of $\mathcal{S}^{r}_{\omega}$ into unstable leaves. For
each $n\geq 0$, let
$\xi^{n}=\\{\Phi^{n}(W):W\in\xi\\}\cup\\{\widetilde{\mathcal{S}}-\Phi^{n}(\mathcal{S}^{r})\\}$
be a measurable partition of $\widetilde{\mathcal{S}}$, and let
$\eta=\vee_{n=0}^{\infty}\xi^{n}$. Note that $\eta$ is a refinement of the
partition $\eta_{0}:=\\{\\{\omega\\}\times\mathcal{B}\\}_{\omega\in\Omega}$.
For every $(\omega,x)\in\widetilde{\mathcal{S}}$, a point
$(\omega,y)\in\eta(\omega,x)$ if and only if

$\left\\{\begin{aligned}
&f^{-n}_{\omega}y\in\xi_{\theta^{-n}\omega}(f^{-n}_{\omega}x)\quad&\text{if}&\
\Phi^{-n}(\omega,x)\in\mathcal{S}^{r};\\\
&\Phi^{-n}(\omega,y)\notin\mathcal{S}^{r}&\text{if}&\
\Phi^{-n}(\omega,x)\notin\mathcal{S}^{r}.\end{aligned}\right.$ (5.1)

Therefore, one has that $\eta$ is $\Phi$-decreasing. This proves item (b).

For each $(\omega,x)\in\widetilde{\mathcal{S}}$, there exists
$N(\omega,x)\in\mathbb{N}$ such that $\Phi^{-N}(\omega,x)\in\mathcal{S}^{r}$.
Then, by (5.1) we have $\eta(\omega,x)_{\omega}\in
f^{N}_{\theta^{-n}(\omega)}W^{u}_{\operatorname{loc}}(\Phi^{-N}(\omega,x))$.
To show $\eta$ subordinate to $W^{u}$, we only requirement that
$\eta(\omega,x)_{\omega}$ contains a neighborhood of $x$ in
$W^{u}_{\operatorname{loc}}(\omega,x)$ for $\mu$-almost every $(\omega,x)$
left.

We now choose $r$ carefully to guarantee that. Let

$\partial\mathcal{S}^{r}_{\omega}:=\bigcup\\{W^{u}_{\delta,(\omega,y)}\cap\\{z:|z-x_{\ast}|=2rr_{2}\\}:y\in
A_{\ast}\ \text{such that}\ (\omega,y)\in\Gamma_{\ell_{0}}\\},$

and let

$\partial\mathcal{S}^{r}:=\bigcup\\{\\{\omega\\}\times\partial\mathcal{S}^{r}_{\omega}:\exists
y\in A_{\ast}\ \text{with}\ (\omega,y)\in\Gamma_{\ell_{0}}\\}.$

For fixed $(\omega,x)\in\widetilde{\mathcal{S}}\cap\Gamma$. Since
$\\{\omega\\}\times\partial(\eta(\omega,x)_{\omega})\subset\cup_{n\geq
0}\Phi^{n}(\partial(\mathcal{S}^{r}))$, we have that for
$0<\varepsilon<{\delta}^{\prime}_{1}$ (${\delta}^{\prime}_{1}$ be as in
Theorem 4.1), if
$f^{-n}_{\omega}W^{u}_{\varepsilon}(\omega,x)\cap(\partial(\mathcal{S}^{r}))_{\theta^{-n}\omega}=\varnothing$
then $W^{u}_{\varepsilon}(\omega,x)\subset\eta(\omega,x)_{\omega}$. To
guarantee that, we needed that for every $y\in W^{u}_{\varepsilon}(\omega,x)$
and every $n\in\mathbb{N}$,

$|f^{-n}_{\omega}y-f^{-n}_{\omega}x|<|(2rr_{2}-|f^{-n}_{\omega}x-x_{\ast}|)|.$

For every $y\in W^{u}_{\varepsilon}(\omega,x)$, by Theorem 4.1 (iii) we have

$3|f^{-n}_{\omega}y-f^{-n}_{\omega}x|\leq|f^{-n}_{\omega}y-f^{-n}_{\omega}x|^{\prime}_{\Phi^{-n}(\omega,x)}\leq|x-y|^{\prime}_{(\omega,x)}e^{\frac{-n\lambda}{2}}.$

Therefore, we need to take $\varepsilon(\omega,x)$ small enough such that

$\varepsilon<\min_{n\in\mathbb{N}}\\{{\delta}^{\prime}_{1},\frac{1}{3}\ell(\omega,x)e^{\frac{n\lambda}{2}}|(2rr_{2}-|f^{-n}_{\omega}x-x_{\ast}|)|\\}.$

Take $r\in(1/2,1)$, then the following subset

$\\{r:\sum_{n=0}^{\infty}\mu(\\{(\omega,x):|(|x-x_{\ast}|-2rr_{2})|<e^{\frac{-n\lambda}{2}}\\})<\infty\\}$
(5.2)

have full Lebesgue measure on the interval $(1/2,1)$ by standard estimations
(see [16, 17] for details). Since there are most countable choice of $r$ such
that $\mu(\partial\mathcal{S}^{r})>0$, one can choose $r\in(1/2,1)$ such that
$r$ belongs to the set (5.2), and $\mu(\partial\mathcal{S}^{r})=0$. Moreover,
by $\mu$ is $\Phi$ invariant, we also have

$\sum_{n=0}^{\infty}\mu(\\{(\omega,x):|(|f^{-n}_{\omega}x-x_{\ast}|-2rr_{2})|<e^{\frac{-n\lambda}{2}}\\})<\infty.$

By a Borel-Cantelli type argument, for $\mu$-almost every $(\omega,x)$ we have

$|(|f^{-n}_{\omega}x-x_{\ast}|-2rr_{2})|\geq e^{\frac{-n\lambda}{2}}$

except for a finite number of integers $n\in\mathbb{N}$. So, one can choose
$\varepsilon(\omega,x)>0$ such that
$W^{u}_{\varepsilon}(\omega,x)\subset\eta(\omega,x)_{\omega}$. Item (a) be
proved.

It follows from Lemma 4.11 the sequence of functions

$(\omega,x)\mapsto\nu_{(\omega,x)}(E_{\omega}\cap(\vee_{i=0}^{n-1}\xi^{i}(\omega,x))_{\omega})\quad\text{for}\
n\in\mathbb{N}$

are measurable on $\cup^{n-1}_{i=0}\Phi^{i}(\Gamma_{\ell_{0}}\cap(\Omega\times
A_{\ast}))$, and decreasing for $n$. Therefore, we have

$(\omega,x)\mapsto\nu_{(\omega,x)}(E_{\omega}\cap\eta(\omega,x)_{\omega})=\lim_{n\rightarrow\infty}\nu_{(\omega,x)}(E_{\omega}\cap(\vee_{i=0}^{n}\eta^{i}(\omega,x))_{\omega})$

is measurable on a full $\mu$-measure set
$\cup^{\infty}_{n=0}\Phi^{i}(\Gamma_{\ell_{0}}\cap(\Omega\times A_{\ast}))$.
Item (c) be proved.

Item (d) follows from (5.1) and Lemma 4.3 directly. ∎

###  5.2. SRB measures implies entropy formula

###### proof of Theorem A (ergodic case).

Since Ruelle’s inequality [28] (see also [19] for the random case on Banach
spaces) tells us that

$h({\mu}^{\prime}|\mathbb{P})\leq\int\sum_{i}m_{i}\lambda^{+}_{i}d{\mu}^{\prime}$

for any $\Phi$-invariant probability measure ${\mu}^{\prime}$ projection to
$\mathbb{P}$. Then, it suffices to show that if $\mu$ is a random SRB measure
the reverse inequality holds. We assume that $\mu$ is ergodic here. Let $\eta$
be constructed in proposition 5.1. Recall that
$\eta_{0}:=\\{\\{\omega\\}\times\mathcal{B}\\}_{\omega\in\Omega}$. Since
$\eta$ is $\Phi$-decreasing and $\eta_{0}\leq\eta$, we have

$\displaystyle h(\mu|\mathbb{P},\eta)$
$\displaystyle=\lim_{n\rightarrow\infty}\frac{1}{n}H_{\mu}(\vee_{i=0}^{n}\Phi^{-1}\eta|\eta_{0})$
$\displaystyle=\frac{1}{n}H_{\mu}(\Phi^{-n}\eta|\eta)=H_{\mu}(\Phi^{-1}\eta|\eta).$

It suffices to show that

$H_{\mu}(\Phi^{-1}\eta|\eta)=\int\log J^{u}(\omega,x)d\mu(\omega,x)=\sum
m_{i}\lambda_{i}^{+},$

where $J^{u}(\omega,x)=\det(Df_{\omega}(x)|E^{u}(\omega,x))$. Note that $\log
J^{u}\in L^{1}(\mu)$, the second equality follows immediately from the
Birkhoff ergodic theorem and Proposition 3.1.

By Proposition 5.1 (c), we let $\nu$ be the $\sigma$-finite measure defined by

$\nu(K)=\int v_{(\omega,x)}((\eta(\omega,x)\cap K)_{\omega})d\mu(\omega,x).$

By assumption, we have $\mu\ll\nu$. Let $\rho=\frac{d\mu}{d\nu}$, then for
$\mu$-almost every $(\omega,x)$ one has

$\rho(\omega,y)=\frac{d\mu_{\eta(\omega,x)}}{d\nu_{(\omega,x)}}(y)\quad\text{for}\
\nu_{(\omega,x)}\text{-almost every}\ y\in\eta(\omega,x)_{\omega}.$

Recall that the information function $I(\Phi^{-1}\eta|\eta)$ is

$I(\Phi^{-1}\eta|\eta)(\omega,x)=-\log\mu_{\eta(\omega,x)}((\Phi^{-1}\eta)(\omega,x)_{\omega}).$

Let
$L(\omega,x)=\rho(\omega,x)(J^{u}(\omega,x))^{-1}(\rho(\Phi(\omega,x)))^{-1}$.
For every measurable set $E\subset\mathcal{B}$. On the one hand, since $\eta$
is $\Phi$-decreasing, one has

$\displaystyle\mu_{(\Phi^{-1}\eta)(\omega,x)}(E)$
$\displaystyle=\dfrac{\mu_{\eta(\omega,x)}((\Phi^{-1}\eta)(\omega,x)_{\omega}\cap
E)}{\mu_{\eta(\omega,x)}((\Phi^{-1}\eta)(\omega,x)_{\omega})}$
$\displaystyle=e^{I(\Phi^{-1}\eta|\eta)(\omega,x)}\int_{((\Phi^{-1}\eta)(\omega,x))_{\omega}\cap
E}\rho(\omega,y)d\nu_{(\omega,x)}(y).$

On the other hand, by $\mu$ is $\Phi$-invariant, we have for $\mu$-almost
every $(\omega,x)$

$\displaystyle\mu_{(\Phi^{-1}\eta)(\omega,x)}(E)$
$\displaystyle=\mu_{\eta(\Phi(\omega,x))}(f_{\omega}E)$
$\displaystyle=\int_{f_{\omega}(((\Phi^{-1}\eta)(\omega,x))_{\omega}\cap
E)}\rho(\theta(\omega),z)d\nu_{\Phi(\omega,x)}(z)$
$\displaystyle=\int_{f_{\omega}(((\Phi^{-1}\eta)(\omega,x))_{\omega}\cap
E)}\rho(\theta(\omega),z)J^{u}(\omega,f^{-1}_{\theta\omega}(z))d((f_{\omega})_{\ast}\nu_{(\omega,x)})(z)$
$\displaystyle=\int_{((\Phi^{-1}\eta)(\omega,x))_{\omega}\cap
E}\rho(\Phi(\omega,y))J^{u}(\omega,y)d\nu_{(\omega,x)}(y),$

where we use the change of variables formula for induced measures on finite-
dimensional manifolds in the third equality (recall $f^{-1}_{\theta\omega}(y)$
is the unique point such that $f_{\omega}(f^{-1}_{\theta\omega}(y))=y$).
Consequently, by $E$ is arbitrary and
$I(\Phi^{-1}\eta|\eta)(\omega,y)=I(\Phi^{-1}\eta|\eta)(\omega,x)$ for any
$y\in((\Phi^{-1}\eta)(\omega,x))_{\omega}$, we get that for $\mu$-almost every
$(\omega,x)$

$I(\Phi^{-1}\eta|\eta)(\omega,y)=-\log L(\omega,y)\quad\text{for}\
\nu_{(\omega,x)}\text{-almost every}\
y\in((\Phi^{-1}\eta)(\omega,x))_{\omega}.$

Therefore, for $\mu$-almost every $(\omega,x)$ one has

$I(\Phi^{-1}\eta|\eta)(\omega,x)=\log
J^{u}(\omega,x)+\log\dfrac{\rho(\Phi(\omega,x))}{\rho(\omega,x)}.$

Since $\log J^{u}\in L^{1}(\mu)$ and $I(\Phi^{-1}\eta|\eta)(\omega,x)\geq 0$,
one has $\log^{-}\frac{\rho\circ\Phi}{\rho}\in L^{1}(\mu)$. Therefore, by
Lemma 3.1 one has $\int\log\frac{\rho\circ\Phi}{\rho}d\mu=0$. So, we have

$H_{\mu}(\Phi^{-1}\eta|\eta)=\int\log J^{u}(\omega,x)d\mu(\omega,x).$

This completes the proof. ∎

###  5.3. Entropy formula implies SRB measures

We assume that $(\Phi,\mu)$ satisfies (H1)-(H5) and $\mu$ is ergodic. Let the
partition $\eta$ be constructed in Proposition 5.1. We first prove the
following:

######  Lemma 5.2.

Let $(\Phi,\mu)$ with a positive Lyapunov exponent for $\mu$-almost
everywhere, and let $\eta$ be as in Proposition 5.1. Assume

$h(\mu|\mathbb{P},\eta)=\int\log J^{u}d\mu,$

then we have $\mu$ is a Random SRB measure.

###### Proof.

Let $\eta,\xi,\mathcal{S}$ and $\widetilde{\mathcal{S}}$ be as in Proposition
5.1.

For $(\omega,x)\in\mathcal{S}$. Recall the Lipschitz-continuous function
$z\mapsto\Delta^{\omega}(x,z)$ on $\xi(\omega,x)_{\omega}$ which defined by

$z\mapsto\Delta^{\omega}(x,z):=\prod_{n=1}^{\infty}\dfrac{J^{u}(\Phi^{-n}(\omega,x))}{J^{u}(\Phi^{-n}(\omega,z))}.$

And the function $q(\omega,z)$ on $\mathcal{S}$ defined by

$q(\omega,z):=\dfrac{\Delta^{\omega}(x,z)}{\int_{\xi_{\omega}(x)}\Delta^{\omega}(x,y)d\nu_{(\omega,x)}(y)},\quad
z\in\xi(\omega,x)_{\omega}.$

Since $\eta$ is subordinate to $W^{u}(\omega,x)$, for every $(\omega,x)$ there
exists $n\geq 0$ such that
$f^{-n}_{\theta^{-n}\omega}(\eta(\Phi^{-n}(\omega,x))_{\theta^{-n}\omega})$
contained in a leaf of $\mathcal{S}_{\omega}$. By assumption (H1) and Theorem
4.1, $f^{n}_{\theta^{-n}\omega}$ is a $C^{2}$ embedding on each unstable leaf.
Then, we have that for $\mu$-almost every $(\omega,x)$, the map
$z\mapsto\Delta^{\omega}(x,z)$ is Lipschitz-continuous and bounded on
$\eta(\omega,x)_{\omega}$. By Proposition 5.1 (c), for $\mu$-almost every
$(\omega,x)$ we have $\nu_{(\omega,x)}(\eta(\omega,x)_{\omega})>0$. Therefore,
the following function

$p(\omega,z):=\dfrac{\Delta^{\omega}(x,z)}{\int_{\eta(\omega,x)_{\omega}}\Delta^{\omega}(x,y)d\nu_{(\omega,x)}(y)},\
z\in\eta(\omega,x)_{\omega},$

is well defined for $\mu$-almost every $(\omega,x)$.

Suppose we have that $\mu_{\eta(\omega,x)}\ll\nu_{(\omega,x)}$ for
$\mu$-almost $(\omega,x)$. Then there exist some function $\rho$ such that
$d\mu_{\eta(\omega,x)}=\rho\nu_{(\omega,x)}$ for $\nu_{(\omega,x)}$-almost
every $y\in\eta(\omega,x)_{\omega}$. By the argument in the proof of Theorem
A, the function $\rho$ must satisfy $\int_{\eta(\omega,x)_{\omega}}\rho
d\nu_{(\omega,x)}=1$ and
$\rho(\Phi(\omega,y))J^{u}(\omega,y)\rho(\omega,y)^{-1}$ must be constant on
$(\Phi^{-1}\eta)(\omega,x)_{\omega}$. So, one can guess that $\rho=p$.

The function $p$ is measurable by Lemma 4.11 and Lemma 4.13 with standard
arguments. So, we can define a probability measure $\upsilon$ on
$\widetilde{\mathcal{S}}$ by

$\upsilon(K)=\int(\int_{(K\cap\eta(\omega,x))_{\omega}}p(\omega,y)dv_{(\omega,x)}(y))d\mu(\omega,x)$

for each $K$ measurable. By construction, the measure $\upsilon$ and $\mu$
project to the same measure on the quotient space
$\widetilde{\mathcal{S}}/\eta$, and the conditional measure of $\upsilon$ on
each element of $\eta$, which we denote by $\upsilon_{\eta(\omega,x)}$, is
absolutely continuous with respect to $\nu_{(\omega,x)}$ with the densities
function $p$. It suffices to prove that $\upsilon=\mu$.

For each $n\in\mathbb{N}$, by the construction we have

$\upsilon_{\eta(\omega,x)}((\Phi^{-n}\eta)(\omega,x)_{\omega})=\dfrac{\int_{(\Phi^{-n}\eta)(\omega,x)_{\omega}}\Delta^{\omega}(x,y)d\nu_{(\omega,x)}(y)}{\int_{\eta(\omega,x)_{\omega}}\Delta^{\omega}(x,y)d\nu_{(\omega,x)}(y)}.$

By the change of variables formula (recall $f^{-n}_{\theta^{n}\omega}(y)$ is
the unique point such that $f^{n}_{\omega}(f^{-n}_{\theta^{n}\omega}(y))=y$),

$\displaystyle\int_{(\Phi^{-n}\eta)(\omega,x)_{\omega}}\Delta^{\omega}(x,z)d\nu_{(\omega,x)}(z)$
$\displaystyle=$
$\displaystyle\int_{f^{-n}_{\theta^{n}\omega}(\eta(\Phi^{n}(\omega,x))_{\theta^{n}\omega})}\Delta^{\omega}(x,z)d\nu_{(\omega,x)}(z)$
$\displaystyle=$
$\displaystyle\int_{\eta(\Phi^{n}(\omega,x))_{\theta^{n}\omega}}\Delta^{\omega}(x,f^{-n}_{\theta^{n}\omega}(y))d((f^{n}_{\omega})_{\ast}\nu_{(\omega,x)})(y)$
$\displaystyle=$
$\displaystyle\int_{\eta(\Phi^{n}(\omega,x))_{\theta^{n}\omega}}\dfrac{\Delta^{\omega}(x,f^{-n}_{\theta^{n}\omega}(y))}{\det(Df^{n}_{\omega}(f^{-n}_{\theta^{n}\omega}(y))|E^{u}(\omega,f^{-n}_{\theta^{n}\omega}(y)))}d\nu_{\Phi^{n}(\omega,x)}(y)$
$\displaystyle=$
$\displaystyle\dfrac{1}{\det(Df^{n}_{\omega}(x)|E^{u}(\omega,x))}\int_{\eta(\Phi^{n}(\omega,x))_{\theta^{n}\omega}}\Delta^{\theta^{n}\omega}(f^{n}_{\omega}x,y)d\nu_{\Phi^{n}(\omega,x)}(y).$

Let
$L(\omega,x):=\int_{\eta(\omega,x)_{\omega}}\Delta^{\omega}(x,y)d\nu_{(\omega,x)}(y)$,
then we have

$\upsilon_{\eta(\omega,x)}((\Phi^{-n}\eta)(\omega,x)_{\omega})=\dfrac{L(\Phi^{n}(\omega,x))}{L(\omega,x)}\cdot\dfrac{1}{\det(Df^{n}_{\omega}(x)|E^{u}(\omega,x))}.$

Using the same argument in the proof of Theorem A, for each $n>0$ we have

$\int-\log\upsilon_{\eta(\omega,x)}((\Phi^{-n}\eta)(\omega,x)_{\omega})d\mu(\omega,x)=\int\log\det(Df^{n}_{\omega}(\omega,x)|E^{u}(\omega,x))d\mu(\omega,x).$
(5.3)

Note that the main assumption of the lemma tells us

$\displaystyle\int-\log\mu_{\eta(\omega,x)}((\Phi^{-n}\eta)(\omega,x)_{\omega})$
$\displaystyle d\mu(\omega,x)=H_{\mu}(\Phi^{-n}\eta|\eta)$ (5.4)
$\displaystyle=$
$\displaystyle\int\log\det(Df^{n}_{\omega}(\omega,x)|E^{u}(\omega,x))d\mu(\omega,x).$

Consider the $\widetilde{\mathcal{S}}/(\Phi^{-n}\eta)$-measurable function

$\psi_{n}(\omega,x)=\dfrac{\upsilon_{\eta(\omega,x)}((\Phi^{-n}\eta)(\omega,x)_{\omega})}{\mu_{\eta(\omega,x)}((\Phi^{-n}\eta)(\omega,x)_{\omega})},$

which is defined for $\mu$-almost everywhere. Together with (5.3) and (5.4) we
have $\int\log\psi_{n}d\mu=0$. Let $\mathbb{B}_{n}$ be the
sub-$\sigma$-algebra of measurable subsets that are unions of elements of
$\Phi^{-n}\eta$, and let $\mu_{n}$ and $\upsilon_{n}$ be the restriction of
$\mu$ and $\upsilon$ on $\mathbb{B}_{n}$ respectively. Decompose
$\upsilon_{n}=\upsilon_{n}^{\mu}+\upsilon^{\perp}_{n}$ where
$\upsilon_{n}^{\mu}\ll\mu_{n}$ and $\upsilon^{\perp}_{n}$ is mutually singular
with $\mu_{n}$ ($\upsilon^{\perp}_{n}$ can be strictly positive), then
$\psi_{n}=\frac{d\upsilon_{n}^{\mu}}{d\mu_{n}}$. Thus, we have
$\int\psi_{n}d\mu\leq 1$.

By Jensen’s inequality we have

$0=\int\log\psi_{n}d\mu\leq\log\int\psi_{n}d\mu\leq 0$

and the equality holds if and only if $\psi_{n}$ is constant for $\mu$-almost
everywhere. So, $\psi_{n}\equiv 1$ for $\mu$-almost every $(\omega,x)$. This
implies $\mu$ and $\upsilon$ coincide on $\mathbb{B}_{n}$. Since
$\Phi^{-n}\eta$ partition into points modulo $\mu$-measure 0, one has
$\mathbb{B}_{n}\nearrow\mathcal{F}\otimes\mathcal{B}$, and so $\mu=\upsilon$.
This completes the proof. ∎

###### proof of theorem B (ergodic case).

By Lemma 5.2, it suffices to show that

$h(\mu|\mathbb{P},\eta)=h(\mu|\mathbb{P}).$

We will use the following lemma which needed that $A$ is visible:

######  Lemma 5.3 ([18] Proposition 4.2).

Let $(\Phi,\mu)$ be a random dynamical system where $\mu$ is $\Phi$-invariant
and supported on a random compact set $A$ with $\Phi(A)=A$. For any function
$\tau:A\rightarrow(0,1)$ with $\log\tau\in L^{1}(\mu)$. If $A$ is visible,
then there exists a measurable partition $\mathcal{P}$ of $A$ such that
$h(\mu|\mathbb{P},\mathcal{P})<\infty$ and
$\mathcal{P}(\omega,x)_{\omega}\subset B(x,\tau(\omega,x))$ for $\mu$-almost
every $(\omega,x)$.

Let $\tau:\Gamma_{\ell_{0}}\rightarrow\mathbb{N}$ be the first return time to
$\Gamma_{\ell_{0}}$, extend $\tau$ to all of $A$ by setting
$\tau|_{A\setminus\Gamma_{\ell_{0}}}=0$. Let $\rho_{\delta}(\omega,x):=\delta
l^{-2}_{0}e^{2\tau(\omega,x)\varepsilon_{0}}$. Since $\tau\in L^{1}(\mu)$ we
have $\log\rho_{\delta}\in L^{1}(\mu)$. Using Lemma 5.3, there exists a
partition $\mathcal{P}$ of $A$ with finite conditional entropy and
$\mathcal{P}(\omega,x)_{\omega}\subset B(x,\rho_{\delta}(\omega,x))$.

Let $\mathcal{P}^{+}:=\vee_{n\geq 0}\Phi^{n}\mathcal{P}$. It is
straightforward to check that for $\mu$-almost every $(\omega,x)$, for each
$y\in\mathcal{P}^{+}(\omega,x)_{\omega}$ one has

$|f_{\omega}^{-n}y-f_{\omega}^{-n}x|^{\prime}_{\Phi^{-n}(\omega,x)}\leq\delta\ell(\Phi^{-n}(\omega,x))^{-1}.$

Then, by Lemma 4.3 one has $\mathcal{P}^{+}(\omega,x)_{\omega}\subset
W^{u}_{\delta}(\omega,x)$ (note that we assume $E^{c}=\\{0\\}$).

For each $r>0$. Choose a countable partition $\mathcal{P}_{0}$ of $A$ with
finite conditional entropy such that
$h(\mu|\mathbb{P})<h(\mu|\mathbb{P},\mathcal{P}_{0})-r$. Let
$\mathcal{Q}:=\mathcal{P}\vee\\{\widetilde{\mathcal{S}}\cap
A,A-\widetilde{\mathcal{S}}\\}\vee\mathcal{P}_{0}$. Then, we also have
$h(\mu|\mathbb{P})<h(\mu|\mathbb{P},\mathcal{Q})-r$. Let
$\mathcal{Q}^{+}=\vee_{n\geq 0}\Phi^{n}\mathcal{Q}$. Use the fact that
$\mathcal{P}^{+}(\omega,x)_{\omega}\subset W^{u}_{\delta}(\omega,x)$ and the
construction of $\eta$ we can check that
$\eta(\omega,x)_{\omega}\supset\mathcal{Q}^{+}(\omega,x)_{\omega}$ for
$\mu$-almost every $(\omega,x)$. This implies that
$\eta\leq\mathcal{Q}^{+}\vee\eta_{0}$. Then, we have

$\displaystyle h(\mu|\mathbb{P},\eta)$
$\displaystyle=\frac{1}{n}H_{\mu}(\Phi^{-n}\eta|\eta)$
$\displaystyle\geq\frac{1}{n}H_{\mu}(\vee_{j=0}^{n}\Phi^{-j}\mathcal{Q}|\eta)-\frac{1}{n}H_{\mu}(\vee_{j=0}^{n}\Phi^{-j}\mathcal{Q}|\Phi^{-n}\eta)$
$\displaystyle\geq\frac{1}{n}H_{\mu}(\vee_{j=0}^{n}\Phi^{-j}\mathcal{Q}|\mathcal{Q}^{+}\vee\eta_{0})-\frac{1}{n}H_{\mu}(\vee_{j=0}^{n}\Phi^{-j}\mathcal{Q}|\Phi^{-n}\eta)$
$\displaystyle\geq
h(\mu|\mathbb{P},\mathcal{Q})-\frac{1}{n}H_{\mu}(\vee_{j=0}^{n}\Phi^{-j}\mathcal{Q}|\Phi^{-n}\eta),$

the second term in above can be shown to be $<r$ for large $n$ since
$\vee_{n\geq 0}\Phi^{-n}\eta$ partition into points $\mu$-modulo $0$.
Therefore, we have that

$h(\mu|\mathbb{P},\eta)\geq
h(\mu|\mathbb{P},\mathcal{Q})-r>h(\mu|\mathbb{P})-2r.$

By arbitrarily of $r$, we have $h(\mu|\mathbb{P},\eta)=h(\mu|\mathbb{P})$.
This completes the proof of Theorem B in the ergodic case. ∎

##  6\. The non-ergodic case

The case when $\mu$ is not ergodic can be proved in the same way with the
following modifications.

The non-ergodic case of the Multiplicative Ergodic Theorem is as follows:
fixed a measurable function $\lambda_{\alpha}$ with
$0>\lambda_{\alpha}>l_{\alpha}$. Then there is a measurable function
$r:\Gamma\rightarrow\mathbb{N}$ such that for every $(\omega,x)\in\Gamma$,
there are $r(\omega,x)$ Lyapunov exponents

$\lambda_{1}(\omega,x)\geq\cdots\lambda_{r(\omega,x)}(\omega,x)(>\lambda_{\alpha}(\omega,x))$

and a unique splitting

$\mathcal{B}=E_{1}(\omega,x)\oplus\cdots E_{r(\omega,x)}(\omega,x)\oplus
F(\omega,x)$

with respect to the properties (a)-(d) in Theorem 3.1 hold. Moreover,
$\lambda_{i}(\omega,x)$ and $m_{i}(\omega,x):=\dim E_{i}(\omega,x)$ are
measurable functions on $\\{(\omega,x):r(\omega,x)\geq i\\}.$ The limit in
Proposition 3.1 also exists for $\mu$-almost everywhere, but the equality is

$\displaystyle\int\lim_{n\rightarrow\infty}\frac{1}{n}\log\det(Df^{n}_{\omega}(x)|\bigoplus_{j=1}^{k}E_{i_{j}}(\omega,x))d\mu(\omega,x)$
$\displaystyle=\int m_{i}(\omega,x)\lambda_{i_{j}}(\omega,x)d\mu(\omega,x).$

The unstable, center, and stable subspace in Section 4.1 are also defined, and
let $\lambda^{\pm}(\omega,x)$ be as before, which are measurable functions
with positive (negative) values but not be bounded away from $0$. For
$m,n\in\mathbb{N}$ and $p,q\in\mathbb{Z}^{+}$, let

$\displaystyle\Gamma(m,n,p,q):=\\{(\omega,x):$ $\displaystyle\dim
E^{u}(\omega,x)=m,\dim E^{c}(\omega,x)=n;$
$\displaystyle\lambda^{+}(\omega,x)\geq\frac{1}{p},\lambda^{-}(\omega,x)\leq-\frac{1}{q}\\}.$

Then, each $\Gamma(m,n,p,q)$ is either empty, or it is $\Phi$-invariant, and

$\Gamma=\bigcup_{m,n,p,q}\Gamma(m,n,p,q).$

Therefore, the results in Section 3 and Section 4 are also holding on
$\Gamma(m,n,p,q)$. Note that $\Gamma(m,n,p,q)$ is not disjoint, we let

$\bar{\Gamma}(m,n,p,q)=\Gamma(m,n,p,q)\setminus\Gamma(m,n,p-1,q-1).$

Then, $\bar{\Gamma}$ is disjoint, and $\Gamma$ is a countable union of
$\bar{\Gamma}(m,n,p,q)$. We can see that $\bar{\Gamma}$ is an alternative to
ergodic decomposition [8].

In non-ergodic cases, it may happen that the set $\widetilde{\mathcal{S}}$
which is defined in Section 5 is not a full $\mu$-measure set. However, it is
no problem that since there is at most a countable union of disjoint sets of
the form $\widetilde{\mathcal{S}}$ will cover a full $\mu$-measure set.

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