# Primordial perturbations from ultra-slow-roll single-field inflation with
quantum loop effects

Shu-Lin Cheng1<EMAIL_ADDRESS>Da-Shin Lee2<EMAIL_ADDRESS>Kin-Wang Ng1,3<EMAIL_ADDRESS>1Institute of Physics, Academia Sinica,
Taipei 11529, Taiwan, R.O.C.  2Department of Physics, National Dong Hwa
University, Hualien 97401, Taiwan, R.O.C.  3Institute of Astronomy and
Astrophysics, Academia Sinica, Taipei 11529, Taiwan, R.O.C.

###### Abstract

It is known that the single-field inflation with a transient ultra-slow-roll
phase can produce a large curvature perturbation at small scales for the
formation of primordial black holes. In our previous work, we have considered
quantum loop corrections to the curvature perturbation and found that the
growth of these small-scale modes would affect the curvature perturbation at
large scales probed by cosmic microwave background observation. In this work,
we will further derive the constraints on the growing modes in the transition
between the slow-roll and the ultra-slow-roll phases under the effect of the
loop corrections. Our results would help clarify the recent controversy on
whether or not the primordial-black-hole formation from the single-field
inflation is ruled out at one-loop level.

##  I Introduction

Inflation provides a solution to the horizon and flatness problems in our
Universe and also furnishes a mechanism for generating scalar (curvature)
quantum fluctuations that serve as the seeds of the primordial fluctuations in
the matter density to form the rich network of cosmic structures we observe
today. The inflationary paradigm that successfully explains the cosmological
data relies on the dynamics of a scalar field, the inflaton, evolving slowly
during the inflationary stage with the dynamics determined by a fairly flat
potential. This simple inflationary scenario is referred to as slow-roll (SR)
inflation. A distinct aspect of inflationary perturbations is that they are
generated by quantum fluctuations of the scalar field(s) that drive inflation.
During the stage of primordial acceleration, vacuum fluctuations of the
inflaton field are amplified to the cosmological scales, giving the
approximately scale invariant spectrum of small density fluctuations on large
scales, which are ultimately responsible for the CMB anisotropies and the
large scale structure of our Universe KT .

The recent detection of gravitational waves from the merging of two 30
$M_{\odot}$ black holes ABB  has aroused the interest of primordial black
holes (PBHs). This coalescence might be due to binary PBHs bird ; clesse ;
sasaki . In addition, PBHs could comprise a considerable fraction of the dark
matter, and thus leave imprints throughout the history of the Universe CAR ;
MES ; CAR1 ; Khlopov10  (see Ref. SAS  for a review). To generate sufficiently
large perturbations in the power spectrum of the relevant short scales that
later collapse into PBHs by gravitational pull, an inflation scenario must
deviate from the SR regime. Ultra-slow-roll (USR) inflation was therefore
proposed, invoking a transient USR phase in the single-field inflation KIN ;
MAR ; CHE ; BYR . In our early works SYU ; CHE2 , we have considered the
quantum loop effects to the primordial perturbations generated by a single
scalar field during the dynamics of the SR-USR-SR inflation in the spatially
flat gauge. In Ref. SYU , we have studied the one-loop effects to the power
spectrum in terms of the proper renormalized field quantities to cure the
ultraviolent divergence by introducing the momentum cutoff. For a general
potential model $V(\phi)$ of a scalar field $\phi$, the one-loop effects not
only involve the first and second derivatives of the potential,
$V^{\prime}(\phi)$ and $V^{\prime\prime}(\phi)$, but also the third and fourth
derivatives, $V^{\prime\prime\prime}(\phi)$ and $V^{[4]}(\phi)$. Our estimates
have indicated significant one-loop corrections around the peak of the density
power spectrum generated during the USR inflation. Subsequently, the work was
extended to the Hartree resummation to account for the loop effects in a self-
consistent manner that certainly is beyond the one-loop approximation CHE2 ,
in which the momentum cutoff is set by the Hubble parameter to account for the
sensitive infrared modes of superhorizon scales only in such a nearly de
Sitter background fordbunch . Apart from involving $V^{\prime}(\phi)$ and
$V^{\prime\prime}(\phi)$, we have also included $V^{\prime\prime\prime}(\phi)$
and $V^{[4]}(\phi)$ to consider the loop effects to the background equation of
motion for the inflaton mean-field and the mode equation of the field
perturbation. Using specific inflation model potentials, we have shown that
the presence of the USR phase may not only trigger a huge growth of the
curvature perturbation at small scales that the seed the formation of PBHs,
but also excessively produce long-wavelength modes that leave the horizon in
earlier times of the inflation, potentially at scales measured by cosmic
microwave background (CMB) experiments. To avoid an overproduction of the
perturbations at CMB scales, it turns out that the amount of the produced
small-scale perturbations may be too small for forming PBHs to become a
significant fraction of the dark matter. Recently, there has been a debate on
whether or not the PBH formation from the single-field USR inflation is ruled
out when the effect of one-loop corrections is taken into account, with an
open-ended conclusion that it may depend on the sharpness or smoothness of the
transition between the SR and USR phases kri ; rio ; cho ; cho2 ; kri2 ; rio2
; cho3 ; fir ; mot ; cho4 ; fir2 ; fra ; tas ; fum . In this work, based on
our early results in Ref. CHE2 , we scrutiny the constraints on the transition
between the SR and USR phases and the duration of the USR phase, which help us
find the conditions for a successful inflation model that can accommodate both
small- and large-scale curvature perturbations for the potential production of
PBHs and the CMB observation.

Our presentation is organized as follows. In the next section, we summarize
the relevant equations and the Hartree resummation scheme. The inflation model
potentials are reexamined to find the constraints on the USR phase in Sec.
III. Concluding remarks and discussions are in Sec. IV.

##  II Single-field inflation model and quantum loop effects

We briefly review the relevant equations from the single-field inflation model
that include the quantum back reaction. The details can be referred to Refs.
SYU ; CHE2 . The action under consideration is given by

$\displaystyle S=$ $\displaystyle S_{g}+S_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int d^{4}x\sqrt{-g}R+\int
d^{4}x\sqrt{-g}\Bigg{[}-\frac{1}{2}\;\partial_{\mu}\phi\,\partial^{\mu}\phi-V(\phi)\Bigg{]}\;,$
(1)

with the unit of the reduced Planck mass $M_{Pl}^{-2}=8\pi G_{N}=1$. In the
course of inflation, the inflaton field can be split into the homogeneous mean
field $\Phi_{0}$ and its quantum fluctuations $\varphi(\vec{x},t)$ as

$\phi(\vec{x},t)=\Phi_{0}(t)+\varphi(\vec{x},t)\;,$ (2)

with

$\langle\varphi(\vec{x},t)\rangle=0\;.$ (3)

The mean value is obtained with the non-equilibrium quantum state that later
will be specified to be the Bunch-Davies state often studied in the
literature. Here the further Hartree factorization approximation, namely
$\varphi^{3}\rightarrow 3\langle\varphi^{2}\rangle\varphi$ and
$\varphi^{4}\rightarrow 6\langle\varphi^{2}\rangle\varphi^{2}$, is implemented
clu ; kaz ; SYU ; CHE2 . The Hartree approximation certainly provides the
resummation method in treating the contribution of $\langle\varphi^{2}\rangle$
from the loop effects in a self-consistent manner and is therefore beyond the
one-loop results boyan3 . This approximation will become exact in the case of
$N$ scalar field models as $N\rightarrow\infty$. Apart from the normal
ultraviolent divergence in the momentum-integral that is cured by a
regularization and a renormalization, in the case of minimally coupled
massless inflaton fluctuations in de Sitter space-time,
$\langle\varphi^{2}\rangle$ also suffers from the infrared divergence
fordbunch  that may give sizable effects to the SR inflation boyan1  as well
as the USR inflation SYU . In this work, we will mainly focus on
$\langle\varphi^{2}\rangle$ that comes from the contributions of the infrared
physical momentum modes $(k_{\rm phys}\leq H)$ while introducing a momentum
cutoff $\Lambda=H$ in the momentum integration CHE2 .

It is straightforward to generalize the equation of motion for the inflaton
mean field including the loop corrections as

$\ddot{\Phi}_{0}+3\,H\,\dot{\Phi}_{0}+V^{\prime}(\Phi_{0})+\frac{1}{2}V^{\prime\prime\prime}(\Phi_{0})\,\langle\varphi^{2}(\vec{x},t)\rangle=0\,,$
(4)

where the dot means the time derivative with respect to the cosmic time $t$.
The Friedmann equation just including $\langle\varphi^{2}\rangle$ becomes

$\displaystyle H^{2}$
$\displaystyle=\frac{1}{3}\Big{(}\frac{1}{2}{\dot{\Phi}_{0}}^{2}+V(\Phi_{0})\Big{)}+\frac{1}{3}\bigg{\langle}\frac{1}{2}{\dot{\varphi}}^{2}+\frac{\partial_{i}\varphi\partial_{i}\varphi}{2a^{2}}+\frac{1}{2}V^{\prime\prime}(\Phi_{0})\varphi^{2}\bigg{\rangle}+\cdot\cdot\cdot$
(5) $\displaystyle\equiv H_{0}^{2}+\delta H^{2}\,.$

Additionally, the spatial Fourier transform of the field operator
$\varphi_{k}$ obeys the mode equation, which can be obtained from the
quadratic terms of $\varphi$ in the Lagrangian density (II) and the
contributions from the metric perturbations in the spatially flat gauge at
linear order together with the further Hartree factorization approximation as
bau

$\ddot{\varphi_{k}}+3H\dot{\varphi_{k}}+\left[\frac{k^{2}}{a^{2}}+V^{\prime\prime}(\Phi_{0})-\frac{1}{a^{3}M_{P}^{2}}\frac{d}{dt}\left(\frac{a^{3}\dot{\Phi_{0}}^{2}}{H}\right){+\frac{1}{2}V^{[4]}(\Phi_{0})\,\langle\varphi^{2}\rangle}\right]\varphi_{k}=0.$
(6)

In terms of $\zeta_{k}=\varphi_{k}/(\sqrt{2\epsilon})$, the mode equation
becomes clu ; kaz ; SYU ; CHE2

$\ddot{\zeta_{k}}+\left(3+\eta\right)H\dot{\zeta_{k}}+\left[\frac{k^{2}}{a^{2}}+\Delta_{B}\right]\zeta_{k}=0,$
(7)

with the Hubble flow parameters defined as

$\epsilon=-\frac{\dot{H}}{H^{2}},\quad\eta=\frac{\dot{\epsilon}}{\epsilon
H}\,,$ (8)

and

$\displaystyle\Delta_{B}$ $\displaystyle=$
$\displaystyle\Delta_{B3}+\Delta_{B4}\,,\quad{\rm with}$
$\displaystyle\Delta_{B3}$ $\displaystyle=$ $\displaystyle
V^{\prime\prime}(\Phi_{0})-\frac{1}{a^{3}}\frac{d}{dt}\left(\frac{a^{3}\dot{\Phi}_{0}^{2}}{H}\right)+\frac{H}{2}\dot{\eta}+\frac{H^{2}}{4}\eta^{2}-\frac{H^{2}}{2}\epsilon\eta+\frac{3H^{2}}{2}\eta\,,$
$\displaystyle\Delta_{B4}$ $\displaystyle=$
$\displaystyle{V^{[4]}(\Phi_{0})\,\langle\zeta^{2}\rangle\,\epsilon}\,,$ (9)

where $\Delta_{B}$ can be considered as an effective mass-squared term that
can be negative. The term $\Delta_{B3}$ originates from the cubic coupling
that is included as a back reaction to the equation of motion for $\Phi_{0}$
in Eq. (4). The term $\Delta_{B4}$ which appears in the mode equation (7)
directly comes from the quartic coupling. When ignoring the corrections from
quantum fluctuations $\langle\varphi^{2}\rangle$ or $\langle\zeta^{2}\rangle$,
we have $\Delta_{B3}\rightarrow 0$, recovering known Mukhanov-Sasaki equation
msvar . After using both the mean field equation for $\Phi_{0}$ (4) without
the back reaction term and the Friedmann equation for $H_{0}$ in Eq. (5).
$\Delta_{B4}\rightarrow 0$. We thus obtain the equation of motion for $\zeta$
in Eq. (7) with $\Delta_{B}=0$. Nevertheless, including the quantum
fluctuations will give nonzero $\Delta_{B}$ in Eq. (9). This means that the
$\Delta_{B}$ is an indicator that could show the effects of the quantum
fluctuations to the curvature perturbation, which we will discuss later.

For the choice of the standard initial Bunch-Davies vacuum states at early
times $|k\tau|\gg 1$ with $\tau$ being the conformal time defined by
$d\tau=dt/a$, the mode functions behave the same as free-field modes in the
Minkowski space-time. For small $\epsilon$ and $|\eta|$ with ignorable
$\langle\varphi^{2}\rangle$, the Hubble parameter $H$ is approximately a
constant and thus the Universe is in an era of de Sitter space-time with the
scale factor given by $a=e^{H_{0}t}=-1/(H_{0}\tau)$. As such, Eq. (7) has a
simple analytic solution,

$\zeta_{k}=\frac{\pi^{1/2}H_{0}}{2k^{3/2}}\frac{1}{\sqrt{2\epsilon}}(-k\tau)^{3/2}H^{(1)}_{3/2}(-k\tau)\,.$
(10)

In the slow-roll approximation, the power spectrum for the curvature
perturbation is

$\displaystyle\Delta^{2}_{\zeta_{k}}$
$\displaystyle\equiv\frac{k^{3}}{2\pi^{2}}\left|\zeta_{k}\right|^{2}$ (12)
$\displaystyle=\frac{H^{2}}{8\pi^{2}\epsilon}\,,$

where from Eq. (8) $\epsilon=\dot{\Phi}_{0}^{2}/2H_{0}^{2}$ by using the zero-
order background equations. In the next section, we will study how the back
reaction of $\langle\varphi^{2}\rangle$ affects the curvature perturbation.

##  III Quantum fluctuations in SR-USR-SR inflation

In our previous work CHE2 , we consider the inflation models with an
inflection point that will generate a large spike in the power spectrum as the
source for the production of PBHs. Two models are provided with the Universe
undergoing the SR-USR-SR inflation in which at a tree level the created
curvature perturbations are consistent with the Planck CMB measurements,
namely the amplitude of the power spectrum of the curvature perturbation as
well as the levels of the scalar spectral index, the tensor-to-scalar ratio,
and the scalar spectral index running on large cosmological scales:
$\Delta_{\zeta_{k}}^{2}\simeq 2.3\times 10^{-9}$, $n_{s}\simeq 0.97$,
$r<0.06$, and $|dn_{s}/d\ln k|<0.013$ planckinflation ; bk18 , and also show a
spike at relevant scales as the source for the PBH production. However, when
taking into account the loop effects, we will find below that the two models
exhibit different backreaction effects. Although Model 1 can produce large
enough density perturbations at relatively short-wavelength modes during the
transient USR phase for seeding the PBH formation, the resulting loop effects
give rise to an over-production of long-wavelength modes of relevance to CMB
scales. In contrary, Model 2 can produce the long-wavelength perturbations
consistent with observations as well as a sufficient amount of short-
wavelength perturbations for forming PBHs. In this work, we will provide a
detailed analysis in what conditions of the inflaton potential in particular
with the back reaction included, one can have a successful SR-USR-SR
transition that accommodates the CMB observations at large scales, while
producing small-scale perturbations relevant to the formation of PBHs.

###  III.1 Model 1

The potential $V(\phi)$ of Model 1 is parametrized as HER18

$V(\phi)=V_{0}\left(1+c_{1}\frac{\phi}{c_{\Lambda}}+\frac{c_{2}}{2}\frac{\phi^{2}}{c_{\Lambda}^{2}}+\frac{c_{3}}{3!}\frac{\phi^{3}}{c_{\Lambda}^{3}}+\frac{c_{4}}{4!}\frac{\phi^{4}}{c_{\Lambda}^{4}}+\frac{c_{5}}{5!}\frac{\phi^{5}}{c_{\Lambda}^{5}}\right),$
(13)

with

$\displaystyle c_{\Lambda}=0.3,\quad V_{0}={1.681}\times 10^{-14},$
$\displaystyle c_{1}={-0.7276}\times
10^{-4},\;c_{2}=0,\;c_{3}=-0.52,\;c_{4}=1.0,\;c_{5}=-0.6407.$ (14)

The profile of the $V(\phi)$ is depicted in Fig. 1, where on the x-axis we
have replaced the mean-field $\Phi_{0}(N)$ by its corresponding e-folding
number defined by $N=\int^{t}_{0}H\left(t^{\prime}\right)dt^{\prime}$.

Figure 1: Time evolution of the inflaton potential $V(\phi)/V_{i}$ in Eq. (13)
(blue solid line) and Hubble parameter $H$ (red dashed line) with e-folding
number $N$, where $V_{i}=V(\Phi_{0i})=1.6810\times 10^{-14}$ and
$H_{i}=7.4855\times 10^{-8}$. $N_{1}$ and $N_{2}$ denote the beginning and end
of the USR phase, respectively. All dynamical variables in this figure and in
the following figures are rescaled by the reduced Planck mass,
$M_{p}=2.435\times 10^{18}$ GeV.

The initial conditions of the background field are set to be
$\Phi_{0i}=8.664\times 10^{-4}$ and ${\dot{\Phi}}_{0i}=1.860\times 10^{-11}$
and those of the mode functions are the standard Bunch-Davies vacuum states.
Fig. 2 shows the evolution of the Hubble flow parameters. The Universe starts
from the SR inflation, undergoes the USR inflation for a transient period from
$N_{1}=32.6$ to $N_{2}=37.4$ during which $3+\eta<0$, and then evolutes back
to the SR phase till the end of inflation at $N=63.8$.

Figure 2: Evolution of $\ln(\epsilon/\epsilon_{i})$ (blue dashed line) and
$\eta$ (red solid line) against e-folds $N$, where $\epsilon_{i}=3.08771\times
10^{-8}$ is the initial value of $\epsilon$ for Model 1.

Fig. 3 shows the evolution of $|\dot{\zeta}_{k^{*}}|$ affected by the
contribution of the loop correction $\Delta_{B}$ for a $k$-mode which crosses
out the horizon at $N=5$ ($k^{*}/k_{i}=170.75$). Normally, the value of
$|\zeta_{k}|$ should approach to a constant after the $k$-mode crosses out the
horizon ($k<aH$) since $\ddot{\zeta_{k}}$ could be ignored in Eq. (7) during
the SR phase. Meanwhile, the value of $|\dot{\zeta}_{k}|$ would be
continuously decreasing during the inflation because
$|\dot{\zeta_{k}}|\sim\left|\zeta_{k}\right|k^{2}/(3Ha^{2})$. The effect of
the loop correction appears at $N\sim 14$ for the $k^{*}$ mode, where
$\Delta_{B}>(k/a)^{2}$. The relatively large value of $\Delta_{B}$ before the
USR phase, though suppressed by the slow-roll parameters, is resulted from a
combination of the loop correction, the form of the potential, and the
backreaction, as depicted by Eq. (38) in Appendix A. Therefore, after $N\sim
14$ and before the USR phase, we can approximate the increase of
$|\dot{\zeta_{k}}(N)|$ as

$|\dot{\zeta_{k}}(N)|\sim\frac{|\Delta_{B}(N)|}{3H(N)}|\zeta_{k}(N)|\,.$ (15)

Apparently, this increase is significant compared to the case without higher-
order corrections as seen in Fig. 3.

During the USR phase ($3+\eta<0$), the mode function can be approximated by
$\ddot{\zeta}_{k}+(3+\eta)H\dot{\zeta}_{k}\simeq 0$, so the $\dot{\zeta_{k}}$
could be amplified exponentially as

$\displaystyle\dot{\zeta_{k}}(N)\sim f(N)\,\dot{\zeta_{k}}(N_{1})\,,$
$\displaystyle
f(N)=\exp\left[-\int_{N_{1}}^{N}(3+\eta)dN^{\prime}\right]=\exp\left[-\left(3N^{\prime}+\ln\epsilon\right){\Big{|}}_{N_{1}}^{N}\right].$
(16)

It was shown that the continuity of conformal weights across the transition
from the USR phase (with $\eta\simeq\eta_{\rm USR}$) to the second SR phase
(with $\eta\simeq\eta_{\rm 2SR}$) requires that $\eta_{\rm USR}+\eta_{\rm
2SR}=-6$ wu2021 . In Fig. 2, it is evident that $\eta_{\rm USR}\simeq-7$ and
$\eta_{\rm 2SR}\simeq 1$. This implies that the $\dot{\zeta_{k}}$ reaches its
maximum at $N=N_{2}$ and then becomes exponentially damped shortly after
$N=N_{2}$. Therefore, for $N\gtrsim N_{2}$, we find from Eq. (III.1) that

${\zeta_{k}}(N)-{\zeta_{k}}(N_{1})=\dot{\zeta_{k}}(N_{1})\int_{N_{1}}^{N}\frac{f(N^{\prime})}{H(N^{\prime})}dN^{\prime}\sim\frac{\dot{\zeta_{k}}(N_{1})}{H(N_{1})}A\,,$
(17)

where we have made $H(N^{\prime})\sim H(N_{1})$ and $A$ is an amplification
factor estimated as

$A=2\int_{N_{1}}^{N_{2}}f(N^{\prime})dN^{\prime}\,.$ (18)

Normally, this amplification would not affect $|\zeta_{k}|$ for small $k$
modes because the values of their $|\dot{\zeta_{k}}|$ are too small. So, they
could still be ignored even after being amplified. But the effect of the loop
correction can provide these small $k$ modes with larger values of
$|\dot{\zeta}_{k}|$ at the beginning of the USR phase, which are given by Eq.
(15) as

$|\dot{\zeta_{k}}(N_{1})|\sim\frac{|\Delta_{B}(N_{1})|}{3H(N_{1})}|\zeta_{k}(N_{1})|\,.$
(19)

Hence, combining Eqs. (17) and (19) gives

$|{\zeta_{k}}(N)-{\zeta_{k}}(N_{1})|\sim\frac{A|\Delta_{B}(N_{1})|}{3H^{2}(N_{1})}|\zeta_{k}(N_{1})|\,.$
(20)

In order that the power spectrum of the curvature perturbation for small $k$
modes remains intact, they have to meet the condition,
$|{\zeta_{k}}(N)-{\zeta_{k}}(N_{1})|<|\zeta_{k}(N_{1})|$, which gives

$A<\frac{3H^{2}(N_{1})}{|\Delta_{B}(N_{1})|}\,.$ (21)

Figure 3: Evolution of $|\dot{\zeta}_{k^{*}}|$ with loop corrections (red
solid line), without loop corrections ($\Delta_{B}=0$, blue dashed line), and
its approximated value from Eq. (15) (cyan dotted line) against e-folds $N$,
where $k^{*}/k_{i}=170.75$ and $k_{i}=H_{i}$. The value of
$|\dot{\zeta}_{k^{*}}|$ is amplified during the USR interval between $N_{1}$
and $N_{2}$.

Note that the above condition does not depend on the large scale. The reason
is the following. For any small $k$ modes, when $\Delta_{B}<(k/a)^{2}$, the
time evolution of $\dot{\zeta_{k}}$ is given by
$|\dot{\zeta_{k}}(N)|=|\zeta_{k}(N)|k^{2}/(3Ha^{2})$. At horizon crossing
(when $a=k/H$), the mode $\zeta_{k}$ gets frozen due to the Hubble friction,
so from Eq. (12) the frozen value of $\zeta_{k}$ is given by
$|\zeta_{k}|_{c}=(2\pi^{2}\Delta_{\zeta_{k}}^{2})^{1\over 2}/k^{3\over 2}$,
where $\Delta_{\zeta_{k}}^{2}\simeq 2.3\times 10^{-9}$ on large scales. At
this moment, the amplitude of $\dot{\zeta_{k}}$ is
$|\dot{\zeta_{k}}|_{c}=|\zeta_{k}|_{c}H/3$. Afterwards, we can rewrite the
decrease of $\dot{\zeta_{k}}$ as
$|\dot{\zeta_{k}}(N)|=|\dot{\zeta_{k}}|_{c}(k/aH)^{2}$. At the time when
$\Delta_{B}=(k/a)^{2}$, $\dot{\zeta_{k}}$ has decreased to a value of
$|\dot{\zeta_{k}}|_{c}\Delta_{B}/H^{2}$, which equals to
$|\zeta_{k}|_{c}\Delta_{B}/(3H)$. This value is the same as given by Eq. (15).
In other words, the presence of loop corrections provides the superhorizon
modes with an $k$ independent ratio between $|\dot{\zeta_{k}}|$ and
$|\zeta_{k}|$ that equals to $\Delta_{B}/(3H)$ just before the USR phase.

The upper bound (21) can be derived in a more intuitive way, though less
accurate. Fig. 4 shows the evolution of $\left|\zeta_{k^{*}}\right|$ and
$|\dot{\zeta}_{k^{*}}|/H$. When the speed is larger than the displacement in a
Hubble time, i.e., $|\dot{\zeta_{k}}|>H\left|\zeta_{k}\right|$,
$\left|\zeta_{k}\right|$ is increased. To prevent the increment of $k$ modes
at CMB scales, $\left|\zeta_{k}\right|$ should not change much from $N_{1}$ to
$N_{2}$. This means that at $N=N_{2}$ the maximum value of $|\dot{\zeta_{k}}|$
should be smaller than $H\left|\zeta_{k}\right|$. Altogether, the following
conditions must be satisfied:

$|\zeta_{k}(N_{1})|\sim|\zeta_{k}(N_{2})|\,,\quad|\dot{\zeta_{k}}(N_{2})|<H(N_{2})\left|\zeta_{k}(N_{2})\right|\,.$
(22)

The generical amplification behavior of $\dot{\zeta_{k}}$ from $N_{1}$ to
$N_{2}$ during the USR inflation due to the large negative value of $\eta$ is
encoded in Eq. (III.1). We also consider the typical overdamped dynamics of
the mode function ${\zeta_{k}}$ just before $N_{1}$, the start of the USR
inflation, giving the relation between $\dot{\zeta_{k}}(N_{1})$ and
${\zeta_{k}}(N_{1})$ in the period of the SR inflation in Eq. (19) for the
mode $k<k_{1}$ (the mode $k_{1}$ leaving horizon at $N_{1}$) where the loop
effect becomes significant for these small $k$ modes. Together with
$H(N_{1})\sim H(N_{2})$, we find that

$f(N_{2})<\frac{3H^{2}(N_{1})}{|\Delta_{B}(N_{1})|}$ (23)

during the USR inflation from e-folding $N_{1}$ to $N_{2}$. Apparently the
above heuristic derivations can be applied to a general model, which derived
constraint to $\Delta_{B}$ is treated in a model independent manner.
Nevertheless, their precise values indeed depend on the model. For Model 1,
making the approximation that $\eta\simeq-7$ in Eq. (III.1), we find that
$A\simeq f(N_{2})/2$. Certainly, Model 1 violates the upper bound (21), as
shown in Fig. 5.

Figure 4: Evolution of $\left|\zeta_{k^{*}}\right|$ and
$|\dot{\zeta}_{k^{*}}|/H$ with loop corrections (denoted by the red solid line
and the red thin solid line, respectively), and those without loop corrections
(denoted by the blue dashed line and the blue thin dashed line, respectively)
against e-folds $N$, where $k^{*}/k_{i}=170.75$ and $k_{i}=H_{i}$. The
$|\dot{\zeta}_{k^{*}}|$ with loop corrections is amplified to a value
comparable with $H\left|\zeta_{k^{*}}\right|$ at $N\sim N_{2}$, resulting in a
shift of the power spectrum for small $k$ modes. Figure 5: Evolution of
$\Delta_{B}$ (red solid line) and $\Delta_{B3}$ (blue dashed line) against
e-folds $N$. Both are drawn with absolute values. The black dot-dashed line
shows the constraint in Eq. (21), $|\Delta_{B}|<3H^{2}A^{-1}$ at $N=N_{1}$,
that the loop correction could affect the power spectrum for small $k$ modes.
In Model 1, this constraint is violated, so it gives rise to a small-$k$
shift.

It is noted that $\Delta_{B}$ defined in Eq. (9) has no $k$-dependence, but
presumably has $N$-dependence. As seen in Fig. 5, $\Delta_{B}$ remains a
constant or increases before the USR phase. As such, the behaviour of
$\dot{\zeta}$ is equally affected by the loop corrections for any small $k$
modes during the SR phase. The subsequent USR phase provides the extra
exponential enhancement of $\dot{\zeta}$. This enhancement of $\dot{\zeta}$ at
large scales during the USR phase also has no $k$-dependence. Therefore, the
condition in Eq. (21) for large-scale modes is scale independent. We will see
a similar behaviour of $\dot{\zeta}$ before the USR phase in Model 2 below.

Notice that the small-$k$ shift in the power spectrum depends on two factors.
The first is the duration and depth of the USR phase that determine the
amplification factor $A$ in Eq. (18). However, the deepest $\eta_{\rm
USR}\lesssim-6$ since after the USR phase the Universe returns to the second
SR inflation with $\eta_{\rm 2SR}\gtrsim 0$ CHE ; wu2021 . The second is the
sharpness of the transition from the SR phase to the USR phase that controls
the value of $\Delta_{B}$ in Eq. (19), which is mostly contributed by the
$H\dot{\eta}/2$ term of $\Delta_{B3}$ in Eq. (9). The reader can refer to
Appendix A for a detailed analytic calculation of $\Delta_{B}$ in terms of the
Hubble flow parameters and the inflation potential, which allows us to
evaluate approximately $\Delta_{B}$ from the zero-order quantities. In
Appendix B, we give a detailed description of the general behavior of the
evolution of the curvature perturbation for each $k$ mode, using the dynamical
equation for the mode function of relevance to each step of the SR-USR-SR
inflation, with a particular attention to the typical feature of the
occurrence of dip and peak. We show that the locations of dip and peak as well
as the slopes connecting them depend on the model under consideration.

On the other hand, one needs a sufficiently large factor $A$ to boost the
perturbations of the short-wavelength modes for a significant production of
PBHs. To obtain its lower bound, let us now focus on the peak value of the
power spectrum of the curvature perturbation. In Fig. 6, the power spectrum
peaks at N=34. The thin black line shows the approximated value of the power
spectrum under the SR condition, given by
$\Delta_{\epsilon}^{2}=H^{2}/(8\pi^{2}\epsilon)$ in Eq. (12), which overlaps
with the power spectrum of the corresponding $k$ mode at horizon crossing. It
could be enhanced by the growth of superhorizon modes during the USR phase,
leading to a higher peak value of the power spectrum leach2001 ; CHE . The
power spectrum is amplified to about $A^{2}$ times (since
$\Delta^{2}_{\zeta_{k}}\propto|\zeta_{k}|^{2}$) from the value of
$\Delta_{\epsilon}^{2}$ for the $k$ modes that cross out the horizon near the
beginning of the USR phase:

$\Delta_{\zeta(peak)}^{2}\sim A^{2}\Delta_{\epsilon}^{2}(N_{1})\,,\quad{\rm
with}\quad\Delta_{\epsilon}^{2}(N_{1})=\frac{H^{2}(N_{1})}{8\pi^{2}\epsilon(N_{1})}\,.$
(24)

This peak value should lie within the range,

$\Delta^{2}_{\zeta(CDM)}<\Delta^{2}_{\zeta(peak)}<\Delta^{2}_{\zeta(PBH)},$
(25)

where $\Delta^{2}_{\zeta(PBH)}$ is the upper limit derived from the
astrophysical and cosmological constraints on the abundance of PBHs (see, for
example, Ref. SAS ). The $\Delta^{2}_{\zeta(CDM)}$ is supposed to be the
lowest value of the power spectrum with which the PBHs can make up a
significant fraction of the cold dark matter. Therefore, for a successful SR-
USR-SR inflation model, the factor $A$ has to satisfy both the constraints
(21) and (25), which can be combined as

$\frac{\Delta^{2}_{\zeta(CDM)}}{\Delta_{\epsilon}^{2}(N_{1})}<A^{2}<{\rm
Min}\left[\frac{\Delta^{2}_{\zeta(PBH)}}{\Delta_{\epsilon}^{2}(N_{1})},\frac{9H^{4}(N_{1})}{\Delta_{B}^{2}(N_{1})}\right]\,.$
(26)

Figure 6: Power spectrum of the curvature perturbation $\Delta_{\zeta_{k}}$
computed by Eq.(12) with the solution of $\zeta_{k}$ without loop corrections
(blue dashed line), with loop corrections (red solid line), and from the
analytic result in Eq. (12) evaluated at the horizon-crossing time for each
$k$-mode (black thin solid line), where $k=He^{N}$.

###  III.2 Model 2

The second model we propose to study is parametrized as juan17

$V(\phi)=\frac{\lambda}{12}\phi^{2}\nu^{2}\frac{6-4a\frac{\phi}{\nu}+3\frac{\phi^{2}}{\nu^{2}}}{\left(1+b\frac{\phi^{2}}{\nu^{2}}\right)^{2}},$
(27)

where

$\displaystyle{\lambda=1.12085\times 10^{-6},\quad\nu=0.18,\quad
a=0.7071,\quad b=1.5012\,,}$ (28)

and the potential profile is shown in Fig. 7. The initial conditions for the
mean field are chosen to be $\Phi_{0i}=2.74$ and $\dot{\Phi}_{0i}=-1.36\times
10^{-7}$ and with the parameters above in the model the Hubble parameter
$H_{i}=6.34\times 10^{-6}$. The evolution of the Hubble flow parameters is
plotted in Fig. 8, which shows that the Universe undergoes the SR-USR-SR
inflation. The Universe starts from the SR inflation, undergoes the USR
inflation for a transient period from $N_{1}=46.6$ to $N_{2}=49.9$ during
which $3+\eta<0$, and then evolutes back to the SR phase till the end of
inflation at $N=62.3$.

Figure 7: Time evolution of the inflaton potential $V(\phi)/V_{i}$ in Eq. (27)
(blue solid line) and Hubble parameter $H$ (red dashed line) with e-folding
number $N$ for Model 2, where $V_{i}=V(\Phi_{0i})=1.2287\times 10^{-10}$ and
$H_{i}=6.34\times 10^{-6}$. All dynamical variables in this figure and in the
following figures are rescaled by the reduced Planck mass, $M_{p}=2.435\times
10^{18}$ GeV.  Figure 8: Evolution of $\ln(\epsilon/\epsilon_{i})$ (blue
dashed line) and $\eta$ (red solid line) against e-folds $N$, where
$\epsilon_{i}=2.25686\times 10^{-4}$ is the initial value of $\epsilon$ for
Model 2.

Then, one can compute the corresponding factor $A$ defined in Eq. (18). We
find that Model 2 obeys the constraint (21) that prevents an overproduction of
the large-scale perturbations as shown in Fig. 9. This is further verified by
the power spectrum of the curvature perturbation in Fig. 10, which does not
have a small-$k$ shift. Notice that Model 2 does provide a sufficiently large
amount of small-scale perturbations for the formation of PBHs. We find that
the value of $\Delta_{B}$ is mainly determined by the difference between the
term $H\dot{\eta}/2$ with and without the loop correction, so its value is
ultimately related to the sharpness of $\eta$. Fig. 11 shows that the value of
$\dot{\eta}$ in the USR phase in Model 2 is higher than that in Model 1, so
Model 2 has a larger value of $|\Delta_{B}|$ in the USR phase than Model 1, as
seen in Fig. 5 and Fig. 9. However, the loop effect could be ignored in Model
2. It is because the Hubble parameter $H$ has a greater value and the
amplification factor $A$ is not large enough in Model 2. The values of $A$ are
$8.34\times 10^{6}$ and $4.89\times 10^{4}$ for Model 1 and Model 2,
respectively. If the duration of the USR phase is longer, the amplification
factor $A$ will become larger, making the loop effect more manifest. We have
also drawn the time evolution of relevant quantities for Model 2 in Fig. 12
and Fig. 13.

Figure 9: Evolution of $\Delta_{B}$ (red solid line) and $\Delta_{B3}$ (blue
dashed line) against e-folds $N$. Both are drawn with absolute values. The
black dot-dashed line shows the constraint in Eq. (21),
$|\Delta_{B}|<3H^{2}A^{-1}$ at $N=N_{1}$, that the loop correction could
affect the power spectrum for small $k$ modes. Model 2 satisfies this
constraint. Figure 10: Power spectrum of the curvature perturbation
$\Delta_{\zeta_{k}}$ computed by Eq.(12) with the solution of $\zeta_{k}$
without loop corrections (blue dashed line), with loop corrections (red solid
line), and from the analytic result in Eq. (12) evaluated at the horizon-
crossing time for each $k$-mode (black thin solid line) against e-folds $N$
for Model 2, where $k=He^{N}$. The dashed and solid curves overlap with each
other. Figure 11: Evolution of $\dot{\eta}$ against e-folds $N$ for Model 1
(red solid line) and Model 2 (blue dashed line). Both are drawn with absolute
values. Figure 12: Evolution of $|\dot{\zeta}_{k^{*}}|$ with loop corrections
(red solid line), without loop corrections (blue dashed line), and its
approximated value from Eq. (15) (cyan dotted line) against e-folds $N$ for
Model 2, where $k^{*}/k_{i}=151.19$ and $k_{i}=H_{i}$. The value of
$|\dot{\zeta}_{k^{*}}|$ is amplified during the USR interval between $N_{1}$
and $N_{2}$. Figure 13: Evolution of $\left|\zeta_{k^{*}}\right|$ and
$|\dot{\zeta}_{k^{*}}|/H$ with loop corrections (denoted by the red solid line
and the red thin solid line, respectively), and those without loop corrections
(denoted by the blue dashed line and the blue thin dashed line, respectively)
against e-folds $N$ for Model 2, where $k^{*}/k_{i}=151.19$ and $k_{i}=H_{i}$.
The $|\dot{\zeta}_{k^{*}}|$ with loop corrections is amplified but still below
the value of $H\left|\zeta_{k^{*}}\right|$ at $N\sim N_{2}$, and therefore the
loop correction will not affect the power spectrum. Figure 14: Evolution of
$\Delta_{B}$, drawn with absolute values, against e-folds $N$. In Model 1, the
red thick and thin solid lines are from the approximated formula (38) and the
full Eq. (9), respectively. The corresponding blue thick and thin dashed lines
are for Model 2.

##  IV Conclusion

We have scrutinized the loop effects to the generation of the curvature
perturbation in the single-field inflation with an ultra-slow-roll phase. We
have improved and extended our previous numerical calculation CHE  that by
virtue of the Hartree factorization, self-consistently includes the cubic
coupling as a back reaction to the motion of the inflaton mean field and the
quartic coupling as an effective mass term in the perturbation mode equation.
Our calculation is done in the spatially flat gauge that allows us to trace
the time evolution of the inflaton perturbation under an inflation potential
across the transition between the slow-roll and the ultra-slow-roll phases. We
could thus understand the requirements for a successful inflation model that
can provide with small-scale curvature perturbation for the production of
primordial black holes while being consistent with the cosmic microwave
background measurements on the primordial density perturbation. The condition
in Eq. (26) summarizing all the requirements is the main result of the present
work. It gives us the constraints on the sharpness across the transition from
the slow-roll phase to the ultra-slow-roll phase as well as the duration of
the ultra-slow-roll phase. We have also found that the transition from the
ultra-slow-roll phase to the final slow-roll phase, being either a sharp or a
smooth one, is irrelevant to the loop corrections. Although we derive the
condition (26) using specific inflaton potentials, the underlying physics is
transparent such that it should apply to any ultra-slow-roll inflation models.

The loop correction to the generation of the curvature perturbation in the
single-field inflation with an ultra-slow-roll phase has also been considered
in Refs. kri ; rio ; cho ; cho2 ; kri2 ; rio2 ; cho3 ; fir ; mot ; cho4 ; fir2
; fra ; tas ; fum , in which they have mostly adopted the in-in formalism to
compute the loop correction in the comoving gauge. In principle, both methods
of different choice of the gauges should be equivalent and thus the results
should be comparable. According to mad03 , the respective degrees of freedom
for each of the gauges have linear relation under the gauge transformation
(see Eq.(2.26) in mad03 ). So, one can compare our approach in the spatially
flat gauge with others, say in Ref. kri , in the comoving gauge in the sense
of what loop diagrams are involved. Here we consider the cubic and quartic
interactions and adopt the Hartree factorizations boyan3 . In summary, the
factorization of the cubic interaction gives the backreaction to the mean
field equation of the inflaton. The factorization of the quartic interaction
presumably contributes to the one-loop diagrams to the power spectrum given by
the so-called bubble diagram, which is local in time. The adopted self-
consistent approach is to extend the one-loop bubble diagram to the daisy
diagram. However, in Ref. kri  the one loop diagrams resulting from the cubic
interaction terms are given in terms of the second-order of the coupling
constant, whose resulting loop contributions to the power spectrum turn out to
be nonlocal in time. Since two types of the diagrams give the same order-of-
magnitude effects, one should consider replacing the free field correlators in
the computation of the one-loop diagrams in Ref. kri  by the dressed ones
given by the Hartree factorization to study their effects. At last, in the
condition (26), one needs to evaluate the effective mass of the perturbation
mode at the beginning of the ultra-slow-roll phase. We have analytically
derived an approximated formula for the time evolution of the effective mass
for a given inflaton potential that can be used to evaluate the effective mass
from the zero-order quantities at any time during inflation. The final words
are that the achieved conditions in Eq. (26) are based upon the dynamical
equations for the mode functions relevant to the each step of the SR-USR-SR
inflation but the detailed values of the quantities depend on the models. This
may give us a useful way to discriminate between ultra-slow-roll inflation
models for the formation of primordial black holes.

###### Acknowledgements.

This work was supported in part by the National Science and Technology Council
(NSTC) of Taiwan, R.O.C. under Grant Nos. 111-2112-M-259-006-MY3 (DSL) and
111-2112-M-001-065 (KWN).

##  Appendix A Differences induced by the loop correction

The loop correction induces small changes in the Hubble flow parameters:

$\epsilon=-\frac{\dot{H}}{H^{2}}=\epsilon_{0}+\delta\epsilon\,$ (29)

with

$\displaystyle\epsilon_{0}$ $\displaystyle=$
$\displaystyle\frac{\dot{\Phi}_{0}^{2}}{2H^{2}}\,,$ (30)
$\displaystyle\delta\epsilon$ $\displaystyle\simeq$
$\displaystyle\frac{\dot{\Phi}_{0}}{6H^{3}}V^{[3]}\left\langle\varphi^{2}\right\rangle\,,$
(31)

and

$\eta=\frac{\dot{\epsilon}}{\epsilon H}=\eta_{0}+\delta\eta$ (32)

with

$\displaystyle\eta_{0}$ $\displaystyle=$
$\displaystyle-6+\frac{\dot{\Phi}_{0}^{2}}{H^{2}}-\frac{\dot{\Phi}_{0}V^{\prime}}{H^{3}\epsilon_{0}}\,,$
(33) $\displaystyle\delta\eta$ $\displaystyle\simeq$
$\displaystyle\frac{1}{6H^{4}\epsilon}\left\\{\frac{1}{2}\left\langle\varphi^{2}\right\rangle\left(V^{[3]}\left[3H\dot{\Phi}_{0}\left(\epsilon_{0}-1\right)-V^{\prime}\right]+\dot{\Phi}_{0}^{2}V^{[4]}\right)\right.$
(34)
$\displaystyle\qquad\quad\left.-V^{[3]^{2}}\left(\frac{1}{2}\left\langle\varphi^{2}\right\rangle\right)^{2}+\dot{\Phi}_{0}^{2}V^{[4]}\left\langle\varphi\dot{\varphi}\right\rangle\right\\}\,,$

and its derivative

$\displaystyle\dot{\eta}_{0}$ $\displaystyle=$
$\displaystyle-12H\epsilon_{0}-\frac{5\dot{\Phi}_{0}V^{\prime}}{H^{2}}+\frac{2\dot{\Phi}_{0}^{2}}{H}\epsilon_{0}+\frac{\dot{\Phi}_{0}V^{\prime}}{H^{2}\epsilon_{0}}\left(3+\eta_{0}\right)+\frac{V^{\prime
2}-\dot{\Phi}_{0}^{2}V^{\prime\prime}}{H^{3}\epsilon_{0}}\,,$ (35)
$\displaystyle\delta\dot{\eta}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{6H^{4}\epsilon}\left\\{\frac{1}{2}\left\langle\varphi^{2}\right\rangle\left(V^{[3]}\left[-6\dot{\Phi}_{0}^{3}+\frac{\dot{\Phi}_{0}^{5}}{H^{2}}-\frac{2V^{\prime
2}}{\dot{\Phi}_{0}}+\frac{3\dot{\Phi}_{0}^{2}V^{\prime}}{H}+3H^{2}\dot{\Phi}_{0}\left(3\epsilon^{2}-6\epsilon+3+\eta\right)\right.\right.\right.$
(36)
$\displaystyle\left.\left.\left.-\dot{\Phi}_{0}V^{\prime\prime}+HV^{\prime}\left(-7\epsilon+3+\eta\right)\right]+V^{[4]}\left[H\dot{\Phi}_{0}^{2}\left(7\epsilon-9-\eta\right)-\dot{\Phi}_{0}V^{\prime}\right]+\dot{\Phi}_{0}^{3}V^{[5]}\right)\right.$
$\displaystyle\left.+\left(\frac{1}{2}\left\langle\varphi^{2}\right\rangle\right)^{2}\left(V^{[3]^{2}}\left[-\frac{2V^{\prime}}{\dot{\Phi}_{0}}+H\left(-7\epsilon+3+\eta\right)\right]-4\dot{\Phi}_{0}V^{[3]}V^{[4]}\right)\right.$
$\displaystyle\left.+\left\langle\varphi\dot{\varphi}\right\rangle\left[V^{[3]}H\dot{\Phi}_{0}\left(7\epsilon-6-\eta\right)+2\dot{\Phi}_{0}^{2}V^{[4]}\right]\right.$
$\displaystyle\left.-3V^{[3]^{2}}\frac{1}{2}\left\langle\varphi^{2}\right\rangle\left\langle\varphi\dot{\varphi}\right\rangle+\dot{\Phi}_{0}V^{[3]}\left(\left\langle\dot{\varphi}^{2}\right\rangle+\left\langle\varphi\ddot{\varphi}\right\rangle\right)\right\\}\,.$

The further expansion of $H$ can be done by writing $H=H_{0}+\delta H$ where
we have found that $\delta H$ is ignorable in terms of the parameters we
adopt. These correction terms are so much smaller than the zero-order terms
that they do not affect the inflation kinematics. In the mode equation of
$\zeta_{k}$, since the sum of the zero-order terms in $\Delta_{B3}$ vanishes,
the loop correction terms in $\Delta_{B3}$ become relevant to the mode
solutions. After performing a lengthy but straightforward calculation, we have
found a fairly well approximation for $\Delta_{B}$, which can be obtained from
the zero-order mode solution $\zeta_{k}$ of Eq. (7) with $\Delta_{B}=0$:

$\displaystyle\Delta_{B}$ $\displaystyle=$ $\displaystyle
V^{\prime\prime}(\Phi_{0})-\frac{1}{a^{3}}\frac{d}{dt}\left(\frac{a^{3}\dot{\Phi}_{0}^{2}}{H}\right)+\frac{H}{2}\dot{\eta}+\frac{H^{2}}{4}\eta^{2}-\frac{H^{2}}{2}\epsilon\eta+\frac{3H^{2}}{2}\eta+V^{[4]}\frac{1}{2}\left\langle\varphi^{2}\right\rangle$
(37) $\displaystyle\simeq$
$\displaystyle\frac{1}{12H^{3}\epsilon_{0}}\left\\{\frac{1}{2}\left\langle\varphi^{2}\right\rangle\left(V^{[3]}\left[H^{2}\dot{\Phi}_{0}\left(10\epsilon_{0}^{2}-12\epsilon_{0}+18+3\eta_{0}\right)-\dot{\Phi}_{0}V^{\prime\prime}\right.\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+HV^{\prime}\left(-6\epsilon_{0}+6+\eta_{0}\right)\right]$
$\displaystyle\qquad\qquad\qquad\quad\left.+V^{[4]}\left[H\dot{\Phi}_{0}^{2}\left(8\epsilon_{0}-6-\eta_{0}\right)-3\dot{\Phi}_{0}V^{\prime}\right]+\dot{\Phi}_{0}^{3}V^{[5]}\right)$
$\displaystyle\quad\qquad\left.+\left\langle\varphi\dot{\varphi}\right\rangle\left(V^{[3]}\left[H\dot{\Phi}_{0}\left(8\epsilon_{0}-9-\eta_{0}\right)-2V^{\prime}\right]+2\dot{\Phi}_{0}^{2}V^{[4]}\right)\right.$
$\displaystyle\quad\qquad\left.+\dot{\Phi}_{0}V^{[3]}\left(\left\langle\dot{\varphi}^{2}\right\rangle+\left\langle\varphi\ddot{\varphi}\right\rangle\right)\right\\}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{12H^{3}\epsilon_{0}}\left\\{\epsilon_{0}\left\langle\zeta^{2}\right\rangle\left(V^{[3]}\left[H^{2}\dot{\Phi}_{0}\left(10\epsilon_{0}^{2}-12\epsilon_{0}+18-6\eta_{0}+7\epsilon_{0}\eta_{0}\right)-\dot{\Phi}_{0}V^{\prime\prime}\right.\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.+HV^{\prime}\left(-6\epsilon_{0}+6-\eta_{0}\right)+H\dot{\Phi}_{0}\dot{\eta_{0}}\right]$
$\displaystyle\qquad\qquad\qquad\quad\left.+V^{[4]}\left[H\dot{\Phi}_{0}^{2}\left(8\epsilon_{0}-6+\eta_{0}\right)-3\dot{\Phi}_{0}V^{\prime}\right]+\dot{\Phi}_{0}^{3}V^{[5]}\right)$
$\displaystyle\quad\qquad\left.+2\epsilon_{0}\left\langle\zeta\dot{\zeta}\right\rangle\left(V^{[3]}\left[H\dot{\Phi}_{0}\left(8\epsilon_{0}-9+\eta_{0}\right)-2V^{\prime}\right]+2\dot{\Phi}_{0}^{2}V^{[4]}\right)\right.$
$\displaystyle\quad\qquad\left.+2\dot{\Phi}_{0}V^{[3]}\epsilon_{0}\left(\left\langle\dot{\zeta}^{2}\right\rangle+\left\langle\zeta\ddot{\zeta}\right\rangle\right)\right\\}\,.$

To obtain them, we have ignored all high-order terms like
$\left\langle\varphi^{2}\right\rangle^{2}$,
$\left\langle\varphi^{2}\right\rangle\left\langle\varphi\dot{\varphi}\right\rangle$,
and etc. We can further omit all $\epsilon_{0}$ terms because $\epsilon_{0}\ll
1$, giving

$\displaystyle\Delta_{B}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{6H\dot{\Phi}_{0}^{2}}\left\\{\frac{1}{2}\left\langle\varphi^{2}\right\rangle\left(V^{[3]}\left[3H^{2}\dot{\Phi}_{0}\left(6+\eta_{0}\right)-\dot{\Phi}_{0}V^{\prime\prime}+HV^{\prime}\left(6+\eta_{0}\right)\right]\right.\right.$
(38)
$\displaystyle\qquad\qquad\qquad\left.+V^{[4]}\left[H\dot{\Phi}_{0}^{2}\left(-6-\eta_{0}\right)-3\dot{\Phi}_{0}V^{\prime}\right]+\dot{\Phi}_{0}^{3}V^{[5]}\right)$
$\displaystyle\quad\qquad\left.+\left\langle\varphi\dot{\varphi}\right\rangle\left(V^{[3]}\left[H\dot{\Phi}_{0}\left(-9-\eta_{0}\right)-2V^{\prime}\right]+2\dot{\Phi}_{0}^{2}V^{[4]}\right)\right.$
$\displaystyle\quad\qquad\left.+\dot{\Phi}_{0}V^{[3]}\left(\left\langle\dot{\varphi}^{2}\right\rangle+\left\langle\varphi\ddot{\varphi}\right\rangle\right)\right\\}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{6H\dot{\Phi}_{0}^{2}}\left\\{\epsilon_{0}\left\langle\zeta^{2}\right\rangle\left(V^{[3]}\left[6H^{2}\dot{\Phi}_{0}\left(3-\eta_{0}\right)-\dot{\Phi}_{0}V^{\prime\prime}\right.\right.\right.$
$\displaystyle\qquad\qquad\qquad\qquad\quad\left.+HV^{\prime}\left(6-\eta_{0}\right)+H\dot{\Phi}_{0}\dot{\eta_{0}}\right]$
$\displaystyle\qquad\qquad\qquad\left.+V^{[4]}\left[H\dot{\Phi}_{0}^{2}\left(-6+\eta_{0}\right)-3\dot{\Phi}_{0}V^{\prime}\right]+\dot{\Phi}_{0}^{3}V^{[5]}\right)$
$\displaystyle\quad\qquad\left.+2\epsilon_{0}\left\langle\zeta\dot{\zeta}\right\rangle\left(V^{[3]}\left[H\dot{\Phi}_{0}\left(-9+\eta_{0}\right)-2V^{\prime}\right]+2\dot{\Phi}_{0}^{2}V^{[4]}\right)\right.$
$\displaystyle\quad\qquad\left.+2\dot{\Phi}_{0}V^{[3]}\epsilon_{0}\left(\left\langle\dot{\zeta}^{2}\right\rangle+\left\langle\zeta\ddot{\zeta}\right\rangle\right)\right\\}\,.$

This is the final relatively simple form that can be used to evaluate
$\Delta_{B}$ from the zero-order quantities. The results are shown in Fig. 14.

##  Appendix B Dip, peak, and slopes of the power spectrum of curvature
perturbations

The power spectrum of the curvature perturbation can be obtained from the
evolution of the $\zeta_{k}$. The shape of the power spectrum has been largely
discussed. For a summary, the reader may refer to Ref. qin23  and references
therein. In this appendix, we plan to give more elaborated estimates of the
locations of the dip and peak as well as the slopes of the power spectrum,
using the results obtained in the main text.

Apparently, both of the dip and peak start to appear during the USR regime
whereas the flat spectrum is developed in the period of the first SR phase.
So, the dynamics of the modes with $k<k_{1}$ at $N$ before the USR phase,
where the mode $k_{1}$ leaves the horizon at $N_{1}$, is given by

$\displaystyle\left(3+\eta\right)H\dot{\zeta_{k}}+\left(\frac{k^{2}}{a^{2}}+\Delta_{B}\right)\zeta_{k}(t)\simeq
0\,.$ (39)

Thus,

$\dot{\zeta_{k}}(N_{1})\simeq-\frac{H(N_{1})}{3}\left(\frac{k^{2}}{k_{1}^{2}}+\frac{\Delta_{B}(N_{1})}{H^{2}(N_{1})}\right)\zeta_{k}(N_{1})\,,$
(40)

where the value $\eta$ has been ignored in the SR phase. According to Eq.
(III.1), $|\dot{\zeta_{k}}(N)|$ grows by a factor of $A$ in the USR regime,
giving $|{\zeta_{k}}(N)|$ at $N_{2}$ as

$\displaystyle|\zeta_{k}(N_{2})|^{2}\simeq|\zeta_{k}(N_{1})|^{2}\left[1-\frac{A}{3}\left(\frac{k^{2}}{k_{1}^{2}}+\frac{\Delta_{B}(N_{1})}{H^{2}(N_{1})}\right)\right]^{2}.$
(41)

Then, the power spectrum at $N_{2}$ is obtained as

$\displaystyle\Delta_{\zeta_{k}}^{2}(N_{2})$ $\displaystyle=$
$\displaystyle\frac{k^{3}}{2\pi^{2}}\left|\zeta_{k}(N_{2})\right|^{2}$ (42)
$\displaystyle\simeq$
$\displaystyle\frac{k^{3}}{2\pi^{2}}|\zeta_{k}(N_{1})|^{2}\left[1-\frac{A}{3}\left(\frac{k^{2}}{k_{1}^{2}}+\frac{\Delta_{B}(N_{1})}{H^{2}(N_{1})}\right)\right]^{2}$
$\displaystyle\simeq$
$\displaystyle\Delta_{\epsilon}^{2}(N_{k})\left[1-\frac{A}{3}\left(\frac{k^{2}}{k_{1}^{2}}+\frac{\Delta_{B}(N_{1})}{H^{2}(N_{1})}\right)\right]^{2}\,,$

where the power spectrum of the mode with $k<k_{1}$ at $N_{1}$ can be
approximated by
$\Delta_{\zeta_{k}}^{2}(N_{1})\simeq\Delta_{\epsilon}^{2}(N_{k})$, where
$N_{k}$ is the time when the $k$-mode crosses out the horizon. The dip is
located at $k_{\rm dip}$ when $|\Delta_{\zeta_{k}}(N_{2})|^{2}$ reaches its
minimum value, so we obtain

$\displaystyle k_{\rm
dip}=k_{1}\sqrt{\frac{3}{A}-\frac{\Delta_{B}(N_{1})}{H^{2}(N_{1})}}\,.$ (43)

In the models under consideration, we have $\Delta_{B}(N_{1})<0$ as a result
of the loop contributions. Thus, the value of $k_{\rm dip}$ and its
corresponding horizon cross-out $N$ will shift to a larger value as compared
with that in the absence of loop corrections (i.e., $\Delta_{B}(N_{1})=0$).
This can be seen in Fig. 6 of Model 1. Such a shift is not so large for small
values of $\Delta_{B}(N_{1})$ as shown in Fig. 10 of Model 2. Therefore, the
sign of $\Delta_{B}$ at the start of the USR phase may become important to
locate the dip of the power spectrum in a general model. Additionally, the
large value of the amplification factor $A$ developed during the USR phase
drives $k_{\rm dip}$ to a relatively small value as seen in both Fig. 6 and
Fig. 10, and the steepest slope between $k_{\rm dip}$ and $k_{1}$ is about
$k^{4}$. Afterwards, the evolution of the power spectrum in the stage of the
reentry into the SR phase does not significantly change the $k$ dependence.

For relatively large $k$ modes with $k_{2}>k>k_{1}$ that cross out the horizon
during the USR regime, the strong $k$ dependence of the power spectrum is
apparent. For such a $k$-mode which the effect of the loop correction (or the
$\Delta_{B}$ term) can be ignored in its dynamical equation (7), we rewrite
the equation in terms of the conformal time $\tau$ as

$\displaystyle\frac{d^{2}\zeta_{k}}{d\tau^{2}}+(2+\eta)aH\frac{d\zeta_{k}}{d\tau}+k^{2}\zeta_{k}=0.$
(44)

The power spectrum for the $k$-mode at $N_{2}$ can be estimated by its
evolution starting from $N_{1}$ to $N_{k}$ when the mode crosses out the
horizon and then further to $N_{2}$ during the USR phase:

$\Delta_{\zeta_{k}}^{2}(N_{2})\sim\Delta_{\zeta_{k}}^{2}(N_{1})\left|e^{\int_{N_{1}}^{N_{k}}\left[-(\eta+2)/2+\sqrt{(\eta+2)^{2}-4k^{2}/(aH)^{2}}/2\right]dN}\right|^{2}\left|\frac{k}{k_{2}}e^{\int_{N_{k}}^{N_{2}}\left[-(\eta+2)/2+|\eta+2|/2\right]dN}\right|^{2}\,,$
(45)

where we have used approximate solutions of Eq. (44) for $\zeta_{k}$ and
$\zeta_{k}^{\prime}\equiv d\zeta_{k}/d\tau$, and $\eta\simeq-7$ which is
almost a constant. We have ignored the $k$ dependence in the evolution from
$N_{k}$ to $N_{2}$ when $\zeta_{k}$ becomes a superhorizon mode with $k/aH<1$.
In addition, the evolution of the mode functions are such that
$|\zeta_{k}(N_{k})|\sim|\dot{\zeta}_{k}(N_{k})|/H<|\dot{\zeta}_{k}(N_{2})|/H\sim|\zeta_{k}(N_{2})|$,
allowing us to estimate the evolution of $|\zeta_{k}|$ during $N_{k}$ and
$N_{2}$ as

$\left|\frac{\zeta_{k}(N_{2})}{\zeta_{k}(N_{k})}\right|\sim\left|\frac{\dot{\zeta}_{k}(N_{2})/H}{\dot{\zeta}_{k}(N_{k})/H}\right|\sim\left|\frac{\zeta_{k}^{\prime}(N_{2})/a(N_{2})H}{\zeta_{k}^{\prime}(N_{k})/a(N_{k})H}\right|\sim\left|\frac{\zeta_{k}^{\prime}(N_{2})/k_{2}}{\zeta_{k}^{\prime}(N_{k})/k}\right|\,.$

In Eq. (45), the $k$ dependence in $\Delta_{\zeta_{k}}^{2}(N_{1})$ can be
extracted by approximating it as

$\displaystyle\Delta_{\zeta_{k}}^{2}(N_{1})=\frac{k^{3}}{2\pi^{2}}\left|\zeta_{k}(N_{1})\right|^{2}$
$\displaystyle\sim$
$\displaystyle\frac{k^{3}}{2\pi^{2}}\left|\frac{1}{2a(N_{1})\sqrt{k\epsilon(N_{1})}}\right|^{2}$
(46) $\displaystyle\sim$
$\displaystyle\Delta_{\epsilon}^{2}(N_{1})\frac{k^{2}}{k_{1}^{2}}\,,$

where we have used the asymptotic solution of Eq. (44),

$\zeta_{k}\rightarrow\frac{1}{a\sqrt{2k}}\frac{e^{-ik\tau}}{\sqrt{2\epsilon}}$
(47)

as $-k\tau\gg 1$ for early times and evaluated it at $N_{1}$. At last, we
arrive at

$\displaystyle\Delta_{\zeta_{k}}^{2}(N_{k})$ $\displaystyle\sim$
$\displaystyle\Delta_{\epsilon}^{2}(N_{1})\frac{k^{2}}{k_{1}^{2}}\left|e^{\int_{N_{1}}^{N_{k}}-(\eta+2)dN}\right|\left|e^{\int_{N_{1}}^{N_{k}}\sqrt{(\eta+2)^{2}-4k^{2}/(aH)^{2}}dN}\right|\frac{k^{2}}{k_{2}^{2}}e^{-\int_{N_{k}}^{N_{2}}(2\eta+4)dN}$
(48) $\displaystyle\sim$
$\displaystyle\Delta_{\epsilon}^{2}(N_{1})\frac{k_{1}^{\eta}}{k_{2}^{2\eta+6}}k^{\eta+6}\left|e^{\int_{N_{1}}^{N_{k}}\sqrt{(\eta+2)^{2}-4e^{-2N}k^{2}/H^{2}}dN}\right|\,.$

With $\eta=-7$ during the USR regime, the power spectrum decreases as
$k^{-1}$, resulting in a peak roughly near $N_{1}$ at the start of the USR
regime. Including the evolution from $N_{1}$ to $N_{k}$ in the exponential
factor shifts the peak location toward a larger $k$ or a larger $N$ that
certainly depends on the details of a given model.

As for the $k$ modes that cross out the horizon after the USR phase with
$k_{2}<k<k_{e}$, where the $k_{e}$ mode just undergoes the horizon crossing at
the end of inflation at $N_{e}$, their mode functions are such that
$|\zeta_{k}|>|\dot{\zeta}_{k}|/H$. The power spectrum at $N_{e}$ can be
estimated by its evolution from $N_{2}$ till $N_{e}$ as

$\displaystyle\Delta_{\zeta_{k}}^{2}(N_{e})$ $\displaystyle\sim$
$\displaystyle\Delta_{\zeta_{k}}^{2}(N_{2})\left|e^{\int_{N_{2}}^{N_{k}}\frac{1}{2}\left[-(\eta+2)\pm\sqrt{(\eta+2)^{2}-4k^{2}/(aH)^{2}}\right]dN}\right|^{2}\left|e^{\int_{N_{k}}^{N_{e}}\left[-(\eta+2)+(\eta+2)\right]dN}\right|$
(49) $\displaystyle\sim$
$\displaystyle\Delta_{\epsilon}^{2}(N_{2})k_{2}^{\eta}k^{-\eta}\left|e^{\int_{N_{2}}^{N_{k}}\sqrt{(\eta+2)^{2}-4e^{-2N}k^{2}/H^{2}}dN}\right|$
$\displaystyle\sim$
$\displaystyle\Delta_{\epsilon}^{2}(N_{k})\left|e^{\int_{N_{2}}^{N_{k}}\sqrt{(\eta+2)^{2}-4e^{-2N}k^{2}/H^{2}}dN}\right|\,,$

where we have used the fact that

$\displaystyle\Delta_{\zeta_{k}}^{2}(N_{2})=\frac{k^{3}}{2\pi^{2}}\left|\zeta_{k}(N_{2})\right|^{2}$
$\displaystyle\sim$
$\displaystyle\frac{k^{3}}{2\pi^{2}}\left|\frac{1}{2a(N_{2})\sqrt{k\epsilon(N_{2})}}\right|^{2}$
(50) $\displaystyle\sim$
$\displaystyle\Delta_{\epsilon}^{2}(N_{2})\frac{k^{2}}{k_{2}^{2}}\,,$

from the solution in Eq. (47). The slope of the power spectrum is about
$k^{-\eta}$ and for $\eta>0$ in the second SR phase the power spectrum
decreases as $k$ increases. The slope of $\Delta_{\zeta_{k}}^{2}(N)$ after
$N_{2}$ shares the same slope of $\Delta_{\epsilon}^{2}(N)\propto k^{-\eta}$
as shown in Fig. 6 and Fig. 10.

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