# Generating large primordial fluctuations in single-field inflation for PBH
formation

Jason Kristiano  and Jun’ichi Yokoyama

###### Abstract

In order to produce appreciable amount of primordial black holes (PBHs), the
square amplitude of curvature perturbation must take a large value of
$\mathcal{O}(0.01)$, namely, seven digits larger than the value observed by
cosmic microwave background radiation (CMB) on large scales. Such a large
fluctuation can be achieved by violating the slow-roll (SR) condition within a
short duration. The best known of such possibilities is the ultraslow-roll
(USR) inflation. We calculate the power spectrum of curvature perturbation in
a simple single-field inflation model which evolves through the SR-USR-SR
regimes so that both large-amplitude small-scale fluctuation for PBH formation
and small-amplitude large-scale fluctuation as observed by CMB are realized.
We further calculate the bispectrum and one-loop correction to the power
spectrum induced by the third-order action of curvature perturbation as the
beginning of precision cosmology on small scales. As a result, we show that
single-field inflation model realizing PBH formation can be constrained by the
quantum correction.

##  1 Introduction

PBHs are formed when an overdense region with a magnitude of density
fluctuation close to unity enters the Hubble horizon during radiation
domination [1, 2, 3, 4]. The most popular formation mechanism of such a large
perturbation on small scales relevant to PBHs is quantum effects during cosmic
inflation in the very early universe [5, 6, 7] (see [8] for a review). In this
chapter, we focus on single-field inflation models that can produce large
fluctuations on small scales relevant to PBH formation with particular
attention to our recent finding [9, 10] that higher-order quantum effects can
restrict the feasibility of such models, even though they are observationally
possible because there are no stringent observational constraints on
primordial fluctuations on small scales [11, 12, 13, 14, 15, 16, 17, 18, 19]
unlike on large scales where observations of CMB tightly constrain the
fluctuation spectrum [20, 21, 22].

The simplest inflation model consistent with CMB observation is canonical SR
inflation, where the scalar field which drives the inflation, called inflaton,
is described by a canonical kinetic term and a potential. At a certain range,
the potential has to be slightly tilted to realize the SR condition, so it can
explain CMB observations. Beyond such a range, one can construct a theory that
violates the SR condition, which usually results in the amplification of
fluctuation on small scales [23, 24] due to rapid slowdown of the inflaton’s
velocity. The simplest example of a violation of the SR condition is the USR
condition [25, 26, 27, 28, 29], which can be generalized to the constant-roll
condition [30, 31, 32]. It occurs when the inflaton enters an extremely flat
region in the potential [33], where the first derivative of the potential
vanishes. Peaks of the enhanced fluctuation can collapse into PBH after such a
fluctuation reenters the horizon. An example of potential that realizes the
formation of PBHs while satisfying the observational constraint on large
scales is shown in Fig. 1.

In this chapter, we explain how violation of the SR approximation can lead to
amplification of primordial fluctuation on certain scales. Although we choose
to focus on USR inflation, the method presented in this chapter can be easily
generalized to other mechanism that yields rapid slowdown of the inflaton’s
velocity. In Sec. 2, we introduce two-point functions or the power spectrum of
the fluctuations in SR inflation, USR inflation, and USR to SR transition. In
Sec. 3, we introduce three-point functions of fluctuations in the same setup
as in Sec. 2. In Sec. 4, we work out an inflation model with SR-USR-SR
transitions. In this model, fluctuations are amplified on scales corresponding
to the time of the USR phase. In Sec. 5, we discuss other PBH formation
mechanisms within a single-field inflation framework and summarize our review.

##  2 Two-point functions

The action of canonical inflation is given by

$S=\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}\left[M_{\mathrm{pl}}^{2}R-(\partial_{\mu}\phi)^{2}-2V(\phi)\right],$
(2.1)

where $M_{\mathrm{pl}}$ is reduced Planck scale,
$g=\mathrm{det}~{}g_{\mu\nu}$, $g_{\mu\nu}$ and $R$ are metric tensor and its
Ricci scalar. Consider a spatially flat, homogeneous, and isotropic
background,

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+a^{2}(t)\mathrm{d}\mathbf{x}^{2}=a^{2}(\tau)(-\mathrm{d}\tau^{2}+\mathrm{d}\mathbf{x}^{2}),$
(2.2)

where $\tau$ is conformal time. Equations of motion for the scale factor
$a(t)$ and the homogeneous part of the inflaton $\phi(t)$ are the Friedmann
equations

$H^{2}=\frac{1}{3M_{\mathrm{pl}}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right),~{}\dot{H}=-\frac{\dot{\phi}^{2}}{2M_{\mathrm{pl}}^{2}},$
(2.3)

with $H=\dot{a}/{a}$ being the Hubble parameter, and the Klein-Gordon equation

$\ddot{\phi}+3H\dot{\phi}+\frac{\mathrm{d}V}{\mathrm{d}\phi}=0.$ (2.4)

Here, a dot denotes time derivative.

During inflation, the evolution of the Hubble parameter is parameterized by
the slow-roll (SR) parameters. The SR parameters are defined as

$\epsilon_{n+1}=\frac{\dot{\epsilon}_{n}}{\epsilon_{n}H},~{}\epsilon_{1}\equiv-\frac{\dot{H}}{H^{2}},$
(2.5)

where $n$ is a positive integer. In case the Hubble parameter $H$ is almost a
constant, we have $\epsilon_{1}\ll 1$. This condition is called the quasi-de
Sitter (quasi-dS) approximation. Thus, the scale factor can be approximated as

$a\approx-\frac{1}{H\tau}\propto e^{Ht}.$ (2.6)

Another stronger condition is called the SR approximation, which is defined as
$|\epsilon_{n}|\ll 1$ for every $n$. Therefore, the SR approximation implies a
quasi-dS approximation, while the converse statement is not true. Violation of
the SR approximation does not imply a breakdown of the quasi-dS approximation.

Figure 1:  Schematic picture of the inflaton potential realizing PBH formation
from USR condition. When the inflaton is around $\phi_{\mathrm{CMB}}$, scales
probed by CMB observations leave the horizon and it is in the SR regime. It
enters an extremely flat region at $t=t_{s}$ undergoing an USR period. It
enters the SR period again at $t=t_{e}$ until $\phi_{\mathrm{end}}$, the end
of inflation.

Small perturbation from the homogeneous part, $\phi(t)$, of the inflaton
$\phi(\bm{x},t)$ and metric can be expressed as

$\displaystyle\phi(\bm{x},t)=\phi(t)+\delta\phi(\bm{x},t),$
$\displaystyle\mathrm{d}s^{2}=-N^{2}\mathrm{d}t^{2}+\gamma_{ij}(\mathrm{d}x^{i}+N^{i}\mathrm{d}t)(\mathrm{d}x^{j}+N^{j}\mathrm{d}t),$
(2.7)

where $\gamma_{ij}$ is the three-dimensional metric on slices of constant $t$,
$N$ is the lapse function, and $N^{i}$ is the shift vector. We choose comoving
gauge condition

$\delta\phi(\bm{x},t)=0,~{}\gamma_{ij}(\bm{x},t)=a^{2}(t)e^{2\zeta(\bm{x},t)}\delta_{ij},$
(2.8)

where $\zeta(\bm{x},t)$ is comoving curvature perturbation. $N$ and $N^{i}$
are obtained by solving the constraint equations. In this review, we do not
consider the tensor perturbation.

Expanding the action (2.1) up to the second order of the curvature
perturbation yields

$S^{(2)}[\zeta]=M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}a^{3}\epsilon_{1}\left[\dot{\zeta}^{2}-\frac{1}{a^{2}}(\partial_{i}\zeta)^{2}\right].$
(2.9)

Introducing the Mukhanov-Sasaki variable by $v=zM_{\mathrm{pl}}\zeta$, where
$z=a\sqrt{2\epsilon_{1}}$, the action becomes canonically normalized

$S^{(2)}_{\mathrm{can}}[v]=\frac{1}{2}\int\mathrm{d}\tau~{}\mathrm{d}^{3}x\left[(v^{\prime})^{2}-(\partial_{i}v)^{2}+\frac{z^{\prime\prime}}{z}v^{2}\right],$
(2.10)

where a prime denotes derivative with respect to $\tau$. We can see that the
quantity $z^{\prime\prime}/z$ behaves as an effective time-dependent mass
squared. In momentum space, quantization is performed by promoting the
Mukhanov-Sasaki variable to an operator

$\hat{v}_{\bm{k}}(\tau)=M_{\mathrm{pl}}z(\tau)\hat{\zeta}_{\bm{k}}(\tau)=v_{k}(\tau)\hat{a}_{\bm{k}}+v^{*}_{k}(\tau)\hat{a}_{-\bm{k}}^{\dagger},$
(2.11)

where mode function $v_{k}(\tau)$ satisfies the Mukhanov-Sasaki equation

$v_{k}^{\prime\prime}+\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0.$
(2.12)

In terms of the SR parameters, the effective mass term can be written as

$\frac{z^{\prime\prime}}{z}=(aH)^{2}\left(2-\epsilon_{1}+\frac{3}{2}\epsilon_{2}-\frac{1}{2}\epsilon_{1}\epsilon_{2}+\frac{1}{4}\epsilon_{2}^{2}+\frac{1}{2}\epsilon_{2}\epsilon_{3}\right).$
(2.13)

From (2.10) we find that $\pi^{v}=v^{\prime}$ is momentum conjugate to $v$, so
canonical quantization imposes a commutator

$[\hat{v}_{\bm{k}}(\tau),\hat{\pi}_{\bm{k}^{\prime}}^{v}(\tau)]=i\hbar\delta(\bm{k}-\bm{k}^{\prime}),$
(2.14)

and the operators satisfy
$[\hat{a}_{\bm{k}},\hat{a}_{\bm{k}^{\prime}}^{\dagger}]=(2\pi)^{3}\delta(\bm{k}-\bm{k}^{\prime})$
under the normalization condition

$v_{k}^{\prime*}v_{k}-v_{k}^{\prime}v_{k}^{*}=i\hbar.$ (2.15)

The two-point functions of curvature perturbation and power spectrum can be
written as

$\displaystyle\left\langle\zeta_{\bm{k}}(\tau)\zeta_{\bm{k}^{\prime}}(\tau)\right\rangle\equiv(2\pi)^{3}\delta^{3}(\bm{k}+\bm{k}^{\prime})\left\langle\\!\left\langle\zeta_{\bm{k}}(\tau)\zeta_{-\bm{k}}(\tau)\right\rangle\\!\right\rangle,$
(2.16)
$\displaystyle\Delta^{2}_{s}(k,\tau)\equiv\frac{k^{3}}{2\pi^{2}}\langle\\!\langle\zeta_{\bm{k}}(\tau)\zeta_{-\bm{k}}(\tau)\rangle\\!\rangle,$
(2.17)

the bracket $\langle\cdots\rangle=\bra{0}\cdots\ket{0}$ denotes the vacuum
expectation value (VEV), and $\Delta^{2}_{s}(k,\tau)$ is the power spectrum
multiplied by the phase space density.

###  2.1 Slow-roll approximation

With SR approximation, the effective mass becomes

$\frac{z^{\prime\prime}}{z}=(aH)^{2}\left[2-\epsilon_{1}+\frac{3}{2}\epsilon_{2}+\mathcal{O}(\epsilon^{2})\right].$
(2.18)

Up to first-order correction in SR approximation, $aH$ as function of $\tau$
reads

$aH=-\frac{1}{\tau(1-\epsilon_{1})}.$ (2.19)

Then, the effective mass can be expressed in terms of the conformal time as

$\frac{z^{\prime\prime}}{z}=\frac{1}{\tau^{2}}\left(2+3\epsilon_{1}+\frac{3}{2}\epsilon_{2}\right)=\frac{1}{\tau^{2}}\left(\nu^{2}-\frac{1}{4}\right),$
(2.20)

where $\nu$ is defined as

$\nu=\frac{3}{2}+\delta=\frac{3}{2}+\epsilon_{1}+\frac{1}{2}\epsilon_{2}.$
(2.21)

At early time, $\tau\rightarrow-\infty$, the Mukhanov-Sasaki equation
approaches

$v_{k}^{\prime\prime}+k^{2}v_{k}=0,$ (2.22)

which is the equation of a simple harmonic oscillator. Requiring minimum
energy state and normalization (2.15), solution of the Mukhanov-Sasaki
equation at early time is given by

$v_{k}(\tau\rightarrow-\infty)=\frac{1}{\sqrt{2k}}e^{-ik\tau}.$ (2.23)

This minimum energy state corresponds to the initial vacuum state called
Bunch-Davies vacuum, defined as $\hat{a}_{\bm{k}}\ket{0}=0$.

At general time, solution of the Mukhanov-Sasaki equation is given by

$v_{k}(\tau)=\sqrt{-k\tau}\left[c_{1}H_{\nu}^{(1)}(-k\tau)+c_{2}H_{\nu}^{(2)}(-k\tau)\right],$
(2.24)

where $c_{1}$ and $c_{2}$ are coefficients to be fixed to a boundary
condition. Given the asymptotic limit of Hankel functions

$\lim_{x\rightarrow\infty}H_{\nu}^{(1,2)}(x)=\sqrt{\frac{2}{\pi
x}}\exp\left[\pm i\left(x-\frac{\nu\pi}{2}-\frac{\pi}{4}\right)\right],$
(2.25)

we can match the general solution to the early time limit (2.23) that leads to

$c_{1}=\sqrt{\frac{\pi}{4k}}e^{i\frac{\pi}{4}(2\nu+1)}~{}~{}\mathrm{and}~{}~{}c_{2}=0.$
(2.26)

Then, the mode function of Mukhanov-Sasaki variable becomes

$v_{k}(\tau)=\sqrt{\frac{\pi}{4}}e^{i\frac{\pi}{4}(2\nu+1)}\sqrt{-\tau}H_{\nu}^{(1)}(-k\tau).$
(2.27)

To obtain the mode function of the curvature perturbation, we need to know the
explicit function of $z(\tau)$. Starting from

$\frac{z^{\prime}}{z}=aH\left(1+\frac{1}{2}\epsilon_{2}\right),$ (2.28)

we can substitute (2.19) to obtain

$\displaystyle\int\frac{\mathrm{d}z}{z}$
$\displaystyle=-\int\frac{\mathrm{d}\tau}{\tau}\left(1+\epsilon_{1}+\frac{1}{2}\epsilon_{2}\right)$
$\displaystyle\frac{z(\tau)}{z_{\star}}$
$\displaystyle=\left(\frac{\tau}{\tau_{\star}}\right)^{-\left(1+\epsilon_{1}+\frac{1}{2}\epsilon_{2}\right)},$
(2.29)

where $\tau_{\star}$ is an arbitrary time reference and
$z_{\star}=z(\tau_{\star})$. Choosing $\tau_{\star}$ as horizon crossing time
$k=a(\tau_{\star})H(\tau_{\star})$ so

$\displaystyle\tau_{\star}$
$\displaystyle=-\left[\frac{1}{aH(1-\epsilon_{1})}\right]_{\star}=-\frac{1}{k(1-\epsilon_{1})_{\star}},$
(2.30) $\displaystyle z_{\star}$
$\displaystyle=\left(a\sqrt{2\epsilon_{1}}\right)_{\star}=\left(\frac{k}{H}\sqrt{2\epsilon_{1}}\right)_{\star},$
(2.31)

then the explicit function of $z(\tau)$ is

$z(\tau)=\left(\frac{k}{H}\sqrt{2\epsilon_{1}}\right)_{\star}\left[-k\tau(1-\epsilon_{1})_{\star}\right]^{-\left(1+\epsilon_{1}+\frac{1}{2}\epsilon_{2}\right)}.$
(2.32)

Then, the mode function of curvature perturbation reads

$\zeta_{k}(\tau)=\frac{v_{k}(\tau)}{M_{\mathrm{pl}}z(\tau)}=\left(\frac{H}{M_{\mathrm{pl}}k\sqrt{2\epsilon_{1}}}\right)_{\star}\sqrt{\frac{\pi}{4}}(-k\tau)^{\left(1+\epsilon_{1}+\frac{1}{2}\epsilon_{2}\right)}e^{i\frac{\pi}{4}(2\nu+1)}\sqrt{-\tau}H_{\nu}^{(1)}(-k\tau).$
(2.33)

With another asymptotic limit of Hankel function

$\lim_{x\rightarrow
0}H_{\nu}^{(1)}(x)=-\frac{i}{\pi}\Gamma(\nu)\left(\frac{x}{2}\right)^{-\nu},$
(2.34)

the curvature perturbation at the end of inflation $\tau=\tau_{0}(\rightarrow
0)$, approaches

$\zeta_{k}(\tau_{0})=\left(\frac{iH}{M_{\mathrm{pl}}\sqrt{2k^{3}\epsilon_{1}}}\right)_{\star}\frac{1}{\sqrt{4\pi}}2^{3/2}\Gamma(3/2).$
(2.35)

Also, the power spectrum reads

$\Delta^{2}_{s}(k,\tau_{0})=\left(\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1}}\right)_{\star},$
(2.36)

with a weak $k$-dependence due to the horizon crossing condition manifested in
the spectral tilt

$n_{s}(k,\tau_{0})-1=\frac{\mathrm{d}\log\Delta_{s}^{2}(k,\tau_{0})}{\mathrm{d}\log
k}=(-2\epsilon_{1}-\epsilon_{2})_{\star}~{}.$ (2.37)

Another way to derive power spectrum in SR approximation is by looking at
exact de Sitter case. In exact de Sitter, the effective mass is

$\frac{z^{\prime\prime}}{z}=\frac{2}{\tau^{2}}.$ (2.38)

Solution of the Mukhanov-Sasaki equation with Bunch-Davies initial vacuum is

$v_{k}(\tau)=\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)e^{-ik\tau}.$
(2.39)

When we proceed to derive the curvature perturbation, we run into a trouble
because in exact de Sitter we find $\epsilon_{1}=0$. However, as a first-order
approximation of the SR parameters, we can simply consider the finite value of
$\epsilon_{1}$. Also, the Hubble and SR parameters have a weak time
dependence.

Numerical calculation of $\zeta_{k}(\tau)$ shows that evaluating the
proportionality coefficient $1/z$ to $v_{k}(\tau)$ at the horizon crossing
time $\tau=\tau_{\star}$ yields an excellent fit. So the curvature
perturbation (2.33) can be approximated as

$\zeta_{k}(\tau)=\left(\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}}}\right)_{\star}(1+ik\tau)e^{-ik\tau}.$
(2.40)

Substituting $\tau\rightarrow 0$ leads to the power spectrum (2.36).

The power spectrum (2.36) can be expressed as a function of potential. We can
rewrite the Friedmann equation as

$V=H^{2}M_{\mathrm{pl}}^{2}(3-\epsilon_{1}).$ (2.41)

Taking derivative with respect to $\phi$ yields

$V_{,\phi}\equiv\frac{\mathrm{d}V}{\mathrm{d}\phi}=\frac{1}{\sqrt{2}}H^{2}M_{\mathrm{pl}}(6-2\epsilon_{1}+\epsilon_{2}).$
(2.42)

At leading order in the SR approximation, the first SR parameter can be
written as

$\epsilon_{1}\simeq\frac{M_{\mathrm{pl}}^{2}}{2}\left(\frac{V_{,\phi}}{V}\right)^{2}.$
(2.43)

Therefore, $\epsilon_{1}$ has an implicit time dependence because $\phi$
depends on time.

From (2.36), we can see that the power spectrum at the wavenumber $p$
corresponds to $\epsilon_{1}$ evaluated at $\tau=-1/p$. This implies that
observation of the power spectrum at the wavenumber $p$ constrains
$V_{,\phi}(\phi(\tau))$ at $\tau=-1/p$. To solve the horizon problem, the
fluctuations in the CMB must be around $60$ e-folds before the end of
inflation. Moreover, fluctuations are observed within the finite range of
wavenumber $0.005\mathrm{Mpc}^{-1}<p<0.2\mathrm{Mpc}^{-1}$ by CMB. This means
that CMB observations constrain only a finite region of the potential around
$60$ e-folds before the end of inflation. Beyond such a region, there are much
looser observational constraints on the potential. Therefore, it is possible
to have an inflation model in which the SR approximation is violated after the
horizon-crossing time of the CMB scales and before the end of inflation.

Comparison with observations can only be made if and only if we are convinced
that quantum fluctuations generated during inflation behave as classical
statistical fluctuations to leave observable traces in our universe. The
following intuitive argument can be developed for the classicalization of
curvature fluctuations. In superhorizon regime, $k\ll aH$, the mode function
behaves as

$\zeta_{k}(\tau)=\left(\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}}}\right)_{\star}\left[1+\mathcal{O}\left(\frac{k^{2}}{a^{2}H^{2}}\right)\right],$
(2.44)

which means $\zeta_{k}^{\ast}(\tau)=-\zeta_{k}(\tau)$ to the leading order. So
we find
$\hat{\zeta}_{\bm{k}}(\tau)=\zeta_{k}(\tau)(\hat{a}_{\bm{k}}-\hat{a}_{-\bm{k}}^{\dagger})$
and its conjugate momentum
$\hat{\pi}_{\bm{k}}^{\zeta}(\tau)=M_{\mathrm{pl}}^{2}z^{2}(\tau){\zeta}_{\bm{k}}^{\prime}(\tau)(\hat{a}_{\bm{k}}-\hat{a}_{-\bm{k}}^{\dagger})$
have the same operator dependence, and apparently commute with each other,
behaving classically [34]. This is why long-wave quantum fluctuations behave
as if they are classical statistical fluctuations to provide the origin of
large-scale structures and CMB anisotropy. The more precise statement,
however, is that the quantum commutator

$\left[\hat{\zeta}_{\bm{k}}(\tau),\hat{\pi}_{\bm{k}^{\prime}}^{\zeta}(\tau)\right]=i\hbar\delta(\bm{k}-\bm{k}^{\prime})$
(2.45)

always holds, and it is the commutator

$\left[\hat{\zeta}_{\bm{p}}(\tau),\hat{\zeta}_{\bm{q}}^{\prime}(\tau)\right]=\frac{i\hbar}{2M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau)a^{2}(\tau)}\delta(\bm{p}-\bm{q})$
(2.46)

that decreases exponentially during the standard slow-roll inflation.

Observation of the spectral tilt $n_{s}=0.9649\pm 0.0042$ on the CMB scale
implies that the SR parameters $\epsilon_{1},\absolutevalue{\epsilon_{2}}$ at
$\tau=-1/p$ have a value $\mathcal{O}(0.01)$, based on (2.37). If the first SR
parameter decreases rapidly at a later time $\tau\gg-1/p$, the power spectrum
is amplified at wavenumber $k\gg p$. This can be achieved by violating the SR
approximation at a later time. In the next subsections, we will show some
inflation models that violate the SR approximation.

###  2.2 Ultraslow-roll inflation

In this subsection, we consider a regime where the potential is constant,
called ultra-low-roll inflation (USR) [25, 26, 27, 28, 29]. Because
$\mathrm{d}V/\mathrm{d}\phi=0$, the Klein-Gordon equation (2.4) becomes
$\ddot{\phi}=-3H\dot{\phi}$, so $\dot{\phi}\propto a^{-3}$. This makes
$\epsilon_{1}$ strongly time-dependent and extremely small as

$\epsilon_{1}=\frac{\dot{\phi}^{2}}{2M_{\mathrm{pl}}^{2}H^{2}}\propto a^{-6},$
(2.47)

and the second SR parameter becomes

$\epsilon_{2}\equiv\frac{\dot{\epsilon}_{1}}{\epsilon_{1}H}=2\epsilon_{1}+2\frac{\ddot{\phi}}{\dot{\phi}H}\simeq-6,$
(2.48)

which breaks SR approximation. The effective mass can be written as

$\frac{z^{\prime\prime}}{z}=(aH)^{2}\left[2+\frac{3}{2}\epsilon_{2}+\frac{1}{4}\epsilon_{2}^{2}+\mathcal{O}(\epsilon_{1})\right]=\frac{2}{\tau^{2}},$
(2.49)

which is the same as in the exact de Sitter case. Thus, solution of the
Mukhanov-Sasaki equation with Bunch-Davies initial vacuum is the same as
(2.39). Then, the curvature perturbation is given by

$\zeta_{k}(\tau)=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau)}}(1+ik\tau)e^{-ik\tau}.$
(2.50)

Here, $\epsilon_{1}$ has a strong dependence on time, which can be expressed
as

$\epsilon_{1}(\tau)=\epsilon_{1}(\tau_{\star})\left(\frac{a(\tau)}{a(\tau_{\star})}\right)^{-6},$
(2.51)

where $\tau_{\star}$ is the horizon-crossing time of a characteristic
wavenumber $k_{\star}$. Therefore, the curvature perturbation becomes

$\zeta_{k}(\tau)=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau_{\star})}}\left(\frac{a(\tau)}{a(\tau_{\star})}\right)^{3}(1+ik\tau)e^{-ik\tau}.$
(2.52)

At the end of inflation, the power spectrum is

$\Delta^{2}_{s}(k,\tau_{0})=\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{\star})}\left(\frac{a(\tau_{0})}{a(\tau_{\star})}\right)^{6}.$
(2.53)

To solve the horizon problem, CMB-scale fluctuations must cross the horizon at
$N_{\mathrm{CMB}}\approx 60$ e-folds before the end of inflation. It means
that power spectrum of CMB fluctuations can be written as

$\Delta^{2}_{s}(p,\tau_{0})=\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{\star})}e^{6N_{\mathrm{CMB}}}.$
(2.54)

To explain the observed $\Delta^{2}_{s}(p,\tau_{0})=2.1\times 10^{-9}$, for a
reasonable value of $\epsilon_{1}(\tau_{\star})$, an extremely small value of
$H/M_{\mathrm{pl}}$ is required. Because $H/M_{\mathrm{pl}}$ characterize the
energy scale of inflation, an extremely small value of $H/M_{\mathrm{pl}}$ can
reach the energy scale of the Big Bang nucleosynthesis. Hence, inflation with
only USR period cannot have the required e-folds number to solve the horizon
problem.

###  2.3 Constant-roll inflation

We can generalize the USR condition to $\epsilon_{2}=\mathrm{constant}$ [30,
31, 32]. By taking reference at time $\tau_{\star}$, the first SR parameter
can be written as

$\epsilon_{1}(\tau)=\epsilon_{1}(\tau_{\star})\left(\frac{a(\tau)}{a(\tau_{\star})}\right)^{\epsilon_{2}},$
(2.55)

Rewriting the second SR parameter as $\epsilon_{2}=-2(3+\alpha)$, the
effective mass can be expressed as

$\frac{z^{\prime\prime}}{z}=\frac{1}{\tau^{2}}(2+\alpha)(1+\alpha)=\frac{1}{\tau^{2}}\left(\nu^{2}-\frac{1}{4}\right),$
(2.56)

where $\nu$ is defined as

$\nu=\absolutevalue{\alpha+\frac{3}{2}}.$ (2.57)

Solution of the Mukhanov-Sasaki equation with Bunch-Davies initial condition
is the same as (2.27). The power spectrum is given by

$\Delta^{2}_{s}(k,\tau_{0})=\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{0})}\frac{1}{2\pi}2^{2\nu}[\Gamma(\nu)]^{2}\left(\frac{k}{aH}\right)^{3-2\nu}.$
(2.58)

We can read that the spectral tilt is $n_{s}-1=3-2\nu$. Therefore, there are
two possible solutions for $\alpha$, given by

$\alpha_{\pm}=-\frac{3}{2}\pm\frac{1}{2}(4-n_{s}).$ (2.59)

However, we have to choose a solution that does not grow outside the horizon.
Recall the second-order action of $\zeta$ that reads

$S^{(2)}[\zeta]=M_{\mathrm{pl}}^{2}\int\mathrm{d}\tau~{}\mathrm{d}^{3}x~{}a^{2}\epsilon_{1}\left[(\zeta^{\prime})^{2}-(\partial_{i}\zeta)^{2}\right].$
(2.60)

The equation of motion is given by

$\zeta_{\bm{k}}^{\prime\prime}+\frac{(a^{2}\epsilon_{1})^{\prime}}{a^{2}\epsilon_{1}}\zeta_{\bm{k}}^{\prime}+k^{2}\zeta_{\bm{k}}=0$
(2.61)

At superhorizon scale, we can neglect the last term in the equation of motion.
Hence, evolution of the curvature perturbation outside the horizon is

$\zeta_{k}(\tau_{0})\rightarrow
A_{k}+B_{k}\int^{\tau_{0}}_{\tau_{\star}}\frac{\mathrm{d}\tau}{a^{2}(\tau)\epsilon_{1}(\tau)},$
(2.62)

where $A_{k}$ and $B_{k}$ are time-independent coefficients. The second term
determines whether $\zeta_{k}(\tau_{0})$ grows or decays. Substituting (2.55),
the integral becomes

$\int^{\tau_{0}}_{\tau_{\star}}\frac{\mathrm{d}\tau}{a^{2}(\tau)\epsilon_{1}(\tau)}\sim\tau_{0}^{\epsilon_{2}+3}.$
(2.63)

It shows that curvature perturbation grows outside the horizon if
$\epsilon_{2}<-3$ or equivalently $\alpha>-3/2$. It was pointed out in [24]
that the curvature perturbation grows outside the horizon if
$3-\epsilon_{1}+\epsilon_{2}<0$ is satisfied. Clearly, USR condition satisfies
this condition since $\epsilon_{2}=-6$.

From observation of spectral tilt $n_{s}=0.9649\pm 0.0042$ on CMB scale, we
obtain $\alpha_{+}\approx 0$ and $\alpha_{-}\approx-3$. Solution
$\alpha_{+}\approx 0$ corresponds to $\epsilon_{2}\approx-6$, which
approximately equal to USR inflation. As explained in the previous subsection,
although it yields an almost scale-invariant power spectrum, this solution
cannot explain the required number of e-folds of inflation because the
curvature perturbation grows outside the horizon. Therefore, we have to choose
$\alpha_{-}$ as the solution because it corresponds to decaying curvature
perturbation. With this solution, constant-roll inflation can explain the
observed CMB fluctuations.

###  2.4 Ultraslow-roll to slow-roll transition

In this subsection, we consider an inflation model in which the second
parameter SR evolves from the USR to the SR period at conformal time
$\tau_{s}$. For $\tau<\tau_{s}$, the inflaton is in the USR period. At time
$\tau\geq\tau_{s}$, the inflaton undergoes a transition from the USR period to
the SR period. In the next part, we consider two different evolutions of the
second SR parameter: smooth and sharp transitions. The former means that the
second SR parameter evolves as a continuous function of time, while in the
latter it evolves as a step function of time. We follow the parameterization
of $\epsilon_{2}(\tau)$ in [35], which is also used in many literatures. The
sketches of $\epsilon_{2}(\tau)$ for both cases are shown by the gray and blue
curves in Fig. 2.

Figure 2:  Sketch of $\epsilon_{2}(\tau)$ for sharp and smooth transition from
USR to SR period are shown by the gray and blue curve, respectively. Brown
curve shows sharp transition from SR-USR-SR period, which will be discussed in
Sec. 4.

####  2.4.1 Smooth transition

At time $\tau>\tau_{s}$, consider evolution of the first and second SR
parameter as

$\displaystyle\epsilon_{1}(\tau)=\epsilon_{1}(\tau_{s})\frac{3\left[\left(\dfrac{\tau_{s}}{\tau}\right)^{s}(s-3)+s+3\right]^{2}}{2s^{2}(s+3)}\left(\frac{\tau}{\tau_{s}}\right)^{s+3},$
(2.64)
$\displaystyle\epsilon_{2}(\tau)=s-3+\frac{2s(s+3)}{\left(\dfrac{\tau_{s}}{\tau}\right)^{s}(s-3)+s+3},$
(2.65)

where $s=\sqrt{9-12\eta_{V}}\simeq 3-2\eta_{V}$ and $\eta_{V}\rightarrow 0$.
The parameter $\eta_{V}$ corresponds to the second derivative of the potential
at the transition point. In this model, the second SR parameter evolves from
USR period $\epsilon_{2}(\tau_{s})=-6$ to SR period
$\epsilon_{2}(\tau_{0})=-2\eta_{V}$. With this parametrization, the effective
mass becomes

$\frac{z^{\prime\prime}}{z}=\frac{1}{\tau^{2}}\left(2-3\eta_{V}\right)=\frac{1}{\tau^{2}}\left(\nu^{2}-\frac{1}{4}\right),$
(2.66)

where

$\nu^{2}=\frac{9}{4}-3\eta_{V}.$ (2.67)

Because $\eta_{V}\rightarrow 0$, solution of the Mukhanov-Sasaki equation with
Bunch-Davies initial vacuum is the same as (2.39). Then, the curvature
perturbation is given by

$\zeta_{k}(\tau)=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau)}}(1+ik\tau)e^{-ik\tau},$
(2.68)

and power spectrum at the end of inflation becomes

$\Delta^{2}_{s}(k,\tau_{0})=\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{0})}.$
(2.69)

####  2.4.2 Sharp transition

Consider evolution of the second SR parameter as a step function of time

$\epsilon_{2}(\tau)=-\Delta\epsilon_{2}\left[1-\theta(\tau-\tau_{s})\right],$
(2.70)

where $\Delta\epsilon_{2}=6$. At time $\tau<\tau_{s}$, the inflaton is in the
USR period with the Bunch-Davies initial vacuum. In this period, the curvature
perturbation and its time derivative are given by

$\displaystyle\zeta_{k}(\tau)$
$\displaystyle=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau)}}e^{-ik\tau}(1+ik\tau)$
$\displaystyle=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau_{s})}}\left(\frac{\tau_{s}}{\tau}\right)^{3}e^{-ik\tau}(1+ik\tau),$
(2.71)
$\zeta_{k}^{\prime}(\tau)=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau_{s})}}\left(\frac{\tau_{s}}{\tau}\right)^{3}e^{-ik\tau}\left(k^{2}\tau-\frac{3}{\tau}-3ik\right).$
(2.72)

At time $\tau\geq\tau_{s}$, the inflaton is in the SR period. In this period,
the curvature perturbation and its time derivative are given by

$\zeta_{k}(\tau)=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau_{s})}}\left[\mathcal{A}_{k}e^{-ik\tau}(1+ik\tau)-\mathcal{B}_{k}e^{ik\tau}(1-ik\tau)\right],$
(2.73)
$\zeta_{k}^{\prime}(\tau)=\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1}(\tau_{s})}}k^{2}\tau\left(\mathcal{A}_{k}e^{-ik\tau}-\mathcal{B}_{k}e^{ik\tau}\right),$
(2.74)

where $\mathcal{A}_{k}$ and $\mathcal{B}_{k}$ are general coefficients to be
determined from boundary conditions [36, 37, 38, 39, 40, 41, 42, 43]. This
time, we have to consider both positive and negative frequency solutions
because the boundary condition at $\tau=\tau_{s}$ does not correspond to
Bunch-Davies vacuum. Solutions of the coefficients by requiring continuity of
$\zeta_{k}(\tau)$ and $\zeta_{k}^{\prime}(\tau)$ at transition $\tau=\tau_{s}$
are

$\displaystyle\mathcal{A}_{k}=1+\frac{3(1+k^{2}\tau_{s}^{2})}{2ik^{3}\tau_{s}^{3}},$
(2.75)
$\displaystyle\mathcal{B}_{k}=\frac{3(1+ik\tau_{s})^{2}}{2ik^{3}\tau_{s}^{3}}e^{-2ik\tau_{s}}.$
(2.76)

At the end of inflation, the power spectrum becomes

$\Delta^{2}_{s}(k,\tau_{0})=\frac{k^{3}}{2\pi^{2}}\absolutevalue{\zeta_{k}(\tau_{0})}^{2}=\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{s})}\absolutevalue{\mathcal{A}_{k}-\mathcal{B}_{k}}^{2}.$
(2.77)

For a very long-wavelength modes, $\absolutevalue{k\tau_{s}}\ll 1$, the power
spectrum approaches

$\Delta^{2}_{s}\left(k\ll\frac{1}{\absolutevalue{\tau_{s}}},\tau_{0}\right)=4\times\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{s})}.$
(2.78)

##  3 Three-point functions

So far, we have considered the second-order action of the curvature
perturbation that leads to two-point functions or the power spectrum at the
tree level in the language of quantum field theory. Within the cosmological
perturbation theory framework, higher-order correlation functions resulting
from higher-order expansion of the action are expected to be small. However,
because we are interested in PBH formation, where the curvature perturbation
is large on certain small scales, higher-order correlation functions can be
quite large.

In this section, we discuss the third-order action that yields three-point
functions of the curvature perturbation. Expanding (2.1) to third-order of
$\zeta$ yields the interaction action [44]

$S^{(3)}[\zeta]=S_{\mathrm{bulk}}[\zeta]+S_{\mathrm{B}}[\zeta]+M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}2f(\zeta)\left(\frac{\delta
L}{\delta\zeta}\right)_{1}.$ (3.1)

The bulk interaction $S_{\mathrm{bulk}}[\zeta]$ reads

$\displaystyle
S_{\mathrm{bulk}}[\zeta]=M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}a^{3}$
$\displaystyle\left[\epsilon_{1}^{2}\dot{\zeta}^{2}\zeta+\frac{1}{a^{2}}\epsilon_{1}^{2}(\partial_{i}\zeta)^{2}\zeta-2\epsilon_{1}\dot{\zeta}\partial_{i}\zeta\partial_{i}\chi\right.$
$\displaystyle\left.-\frac{1}{2}\epsilon_{1}^{3}\dot{\zeta}^{2}\zeta+\frac{1}{2}\epsilon_{1}\zeta(\partial_{i}\partial_{j}\chi)^{2}+\frac{1}{2}\epsilon_{1}\dot{\epsilon}_{2}\dot{\zeta}\zeta^{2}\right],$
(3.2)

where $\chi=\epsilon_{1}\partial^{-2}\dot{\zeta}$. $S_{\mathrm{B}}[\zeta]$ is
the surface action given by the total derivatives [45, 46]. The last term is
interaction proportional to the equation of motion in the lowest order,

$\left(\frac{\delta
L}{\delta\zeta}\right)_{1}=\frac{\mathrm{d}}{\mathrm{d}t}(\epsilon_{1}a^{3}\dot{\zeta})-\epsilon_{1}a\partial^{2}\zeta.$
(3.3)

The function $f(\zeta)$ is explicitly given by

$f(\zeta)=\frac{\epsilon_{2}}{4}\zeta^{2}+\frac{\dot{\zeta}}{H}\zeta+\frac{1}{4a^{2}H^{2}}[-(\partial_{i}\zeta)^{2}+\partial^{-2}\partial_{i}\partial_{j}(\partial_{i}\zeta\partial_{j}\zeta)]+\frac{1}{2H}[\partial_{i}\zeta\partial_{i}\chi-\partial^{-2}\partial_{i}\partial_{j}(\partial_{i}\zeta\partial_{j}\chi)].$
(3.4)

Performing field redefinition $\zeta=\bm{\zeta}+f(\bm{\zeta})$ generates
third-order terms from the second-order action (2.9) that cancel relevant
surface terms containing $\dot{\zeta}$. As a result, we find a useful relation
[44, 45, 46]

$S^{(2)}[\zeta]+S^{(3)}[\zeta]=S^{(2)}[\bm{\zeta}]+S_{\mathrm{bulk}}[\bm{\zeta}].$
(3.5)

In terms of $\bm{\zeta}$, the total action is simply given by the second-order
action (2.9) and bulk interaction (3.2).

Three-point functions of $\zeta$ can be written schematically as

$\langle\zeta_{\bm{k}_{1}}(\tau)\zeta_{\bm{k}_{2}}(\tau)\zeta_{\bm{k}_{3}}(\tau)\rangle=\langle\bm{\zeta}_{\bm{k}_{1}}(\tau)\bm{\zeta}_{\bm{k}_{2}}(\tau)\bm{\zeta}_{\bm{k}_{3}}(\tau)\rangle+\mathrm{redefinition~{}terms},$
(3.6)

where the first term is generated by bulk interaction (3.2) and the second
term is boundary contribution at $\tau$ from field redefinition. Higher-order
correction to the expectation value of an operator $\mathcal{O}(\tau)$ is
calculated by the in-in perturbation theory

$\langle\mathcal{O(\tau)}\rangle=\left\langle\left[\bar{\mathrm{T}}\exp\left(i\int_{-\infty}^{\tau}\mathrm{d}\tau^{\prime}H_{\mathrm{int}}(\tau^{\prime})\right)\right]\mathcal{\hat{O}}(\tau)\left[\mathrm{T}\exp\left(-i\int_{-\infty}^{\tau}\mathrm{d\tau^{\prime}}H_{\mathrm{int}}(\tau^{\prime})\right)\right]\right\rangle,$
(3.7)

where $\mathrm{T}$ and $\bar{\mathrm{T}}$ denote time and antitime ordering.
In this case, the interaction Hamiltonian is simply
$H_{\mathrm{int}}=-\int\mathrm{d}^{3}x\mathcal{L}_{\mathrm{bulk}}$, where
$\mathcal{L}_{\mathrm{bulk}}$ is the integrand of $S_{\mathrm{bulk}}$. For
three-point functions, the operator is
$\bm{\zeta}_{\bm{k}_{1}}(\tau)\bm{\zeta}_{\bm{k}_{2}}(\tau)\bm{\zeta}_{\bm{k}_{3}}(\tau)$.
First-order expansion of (3.7) reads

$\langle\mathcal{O(\tau)}\rangle=2\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}~{}\mathrm{Im}\langle\mathcal{\hat{O}}(\tau)H_{\mathrm{int}}(\tau_{1})\rangle.$
(3.8)

Bispectrum
$\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau)\zeta_{\bm{k}_{2}}(\tau)\zeta_{\bm{k}_{3}}(\tau)\rangle\\!\rangle$
is defined as

$\langle\zeta_{\bm{k}_{1}}(\tau)\zeta_{\bm{k}_{2}}(\tau)\zeta_{\bm{k}_{3}}(\tau)\rangle=(2\pi)^{3}\delta(\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3})\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau)\zeta_{\bm{k}_{2}}(\tau)\zeta_{\bm{k}_{3}}(\tau)\rangle\\!\rangle.$
(3.9)

A bispectrum has local type if it can be expressed as

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=\frac{6}{5}f_{\mathrm{NL}}$
$\displaystyle\left[\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}+\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{3}}(\tau_{0})}^{2}\right.$
$\displaystyle\left.+\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{3}}(\tau_{0})}^{2}\right],$
(3.10)

where $f_{\mathrm{NL}}$ is a coefficient called nonlinear parameter.

###  3.1 Slow-roll approximation

In the SR approximation, the first line in cubic self-interaction (3.2) has
$\mathcal{O}(\epsilon^{2})$ coefficients, while the second line has
$\mathcal{O}(\epsilon^{3})$ coefficients. At the end of inflation, when the
relevant modes are in the superhorizon limit, the leading term in the field
redefinition (3.4) is the first term, because the other terms decay outside
the horizon. Therefore, the third-order action becomes

$S_{\mathrm{bulk}}[\bm{\zeta}]=M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}a^{3}\left[\epsilon_{1}^{2}\dot{\bm{\zeta}}^{2}\bm{\zeta}+\frac{1}{a^{2}}\epsilon_{1}^{2}(\partial_{i}\bm{\zeta})^{2}\zeta-2\epsilon_{1}^{2}\dot{\bm{\zeta}}\partial_{i}\bm{\zeta}\partial_{i}\partial^{-2}\dot{\bm{\zeta}}+\mathcal{O}(\epsilon^{3})\right],$
(3.11)

with $\zeta=\bm{\zeta}+\epsilon_{2}\bm{\zeta}^{2}/4$. With the in-in
perturbation theory (3.8), we obtain bispectrum of $\bm{\zeta}$ as

$\displaystyle\langle\\!\langle\bm{\zeta}_{\bm{k}_{1}}(\tau_{0})\bm{\zeta}_{\bm{k}_{2}}(\tau_{0})\bm{\zeta}_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle_{\mathcal{O}(\epsilon^{2})}=$
$\displaystyle~{}\frac{H^{4}}{32M_{\mathrm{pl}}^{4}\epsilon_{1}^{2}(k_{1}k_{2}k_{3})^{3}}$
(3.12)
$\displaystyle\times\left[-\epsilon_{1\star}\sum_{i}k_{i}^{3}+\epsilon_{1\star}\sum_{i\neq
j}k_{i}k_{j}^{2}+8\epsilon_{1\star}\frac{\sum_{i>j}k_{i}^{2}k_{j}^{2}}{\sum_{i}k_{i}}\right].$

To arrive at bispectrum of $\zeta$, we have to include contribution from the
field redefinition

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=\langle\\!\langle\bm{\zeta}_{\bm{k}_{1}}(\tau_{0})\bm{\zeta}_{\bm{k}_{2}}(\tau_{0})\bm{\zeta}_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle$
$\displaystyle+\frac{1}{4}\epsilon_{2}(\tau_{0})\times
2\left[\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}+\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{3}}(\tau_{0})}^{2}+\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{3}}(\tau_{0})}^{2}\right].$
(3.13)

However, assuming $\epsilon_{3}$ as a constant, $\epsilon_{2}(\tau_{0})$
diverges as $\tau_{0}\rightarrow 0$. The explicit time dependence is given by

$\epsilon_{2}(\tau_{0})=\epsilon_{2\star}\left(\frac{\tau_{0}}{\tau_{\star}}\right)^{-\epsilon_{3\star}}\simeq\epsilon_{2\star}\left(1-\epsilon_{3\star}\log\frac{\tau_{0}}{\tau_{\star}}\right).$
(3.14)

At leading order of $\epsilon_{2}(\tau_{0})$, the bispectrum of $\zeta$ is

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle_{\mathcal{O}(\epsilon^{2})}=$
$\displaystyle~{}\frac{H^{4}}{32M_{\mathrm{pl}}^{4}\epsilon_{1}^{2}(k_{1}k_{2}k_{3})^{3}}$
(3.15)
$\displaystyle\times\left[(\epsilon_{2}-\epsilon_{1})_{\star}\sum_{i}k_{i}^{3}+\epsilon_{1\star}\sum_{i\neq
j}k_{i}k_{j}^{2}+8\epsilon_{1\star}\frac{\sum_{i>j}k_{i}^{2}k_{j}^{2}}{\sum_{i}k_{i}}\right].$

In the squeezed limit, where one of the wavenumbers, $k_{1}$, is much smaller
than the others, $k_{1}\rightarrow 0$, the bispectrum becomes

$\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{-\bm{k}_{2}}(\tau_{0})\rangle\\!\rangle=(2\epsilon_{1}+4\epsilon_{2})_{\star}\frac{H^{4}}{32M_{\mathrm{pl}}^{4}\epsilon_{1}^{2}(k_{1}k_{2})^{3}}.$
(3.16)

In terms of the mode function and spectral tilt, the squeezed bispectrum can
be written as

$\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{-\bm{k}_{2}}(\tau_{0})\rangle\\!\rangle=-(n_{s}(k_{2},\tau_{0})-1)\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2},$
(3.17)

which is called Maldacena’s theorem. Here, we have shown Maldacena’s theorem
by explicit derivation. It can also be proven by a model-independent method by
considering the evolution of the small-scale perturbation in a rescaled dS
background due to the long-wavelength perturbation [47].

Another important point discussed in [44] is the bispectrum induced by the
last term in (3.2)

$S_{\mathrm{bulk}}[\bm{\zeta}]\supset\frac{1}{2}M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}a^{3}\epsilon_{1}\dot{\epsilon}_{2}\dot{\bm{\zeta}}\bm{\zeta}^{2}.$
(3.18)

Although it is SR-suppressed compared to (3.11), such term induces bispectrum
that evolves outside the horizon.

$\langle\\!\langle\bm{\zeta}_{\bm{k}_{1}}(\tau_{0})\bm{\zeta}_{\bm{k}_{2}}(\tau_{0})\bm{\zeta}_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle_{\mathcal{O}(\epsilon^{3})}=\left(\epsilon_{2\star}\epsilon_{3\star}\log\frac{\tau_{0}}{\tau_{\star}}\right)\frac{H^{4}}{32M_{\mathrm{pl}}^{4}\epsilon_{1}^{2}(k_{1}k_{2}k_{3})^{3}}(k_{1}^{3}+k_{2}^{3}+k_{3}^{3}),$
(3.19)

where $\star$ denotes horizon crossing of the total wavenumber
$k_{1}+k_{2}+k_{3}$. It shows that bispectrum of $\bm{\zeta}$ evolves outside
the horizon due to $\log\tau_{0}$ dependence. However, for bispectrum of
$\zeta$, there is an additional contribution from the field redefinition
(3.13). After summing with the second term in (3.14), the $\log\tau_{0}$
dependence cancels out. Therefore, bispectrum of $\zeta$ does not evolve
outside the horizon.

###  3.2 Ultraslow-roll inflation

During USR period, $\epsilon_{1}$ becomes extremely small because it scales as
$a^{-6}$. Also, $\dot{\epsilon}_{2}=0$ because $\epsilon_{2}$ is a constant.
Therefore, the contribution of the bulk interaction to the bispectrum is
negligible and the leading contribution comes from the field redefinition. The
relevant terms in the field redefinition are [48]

$f(\bm{\zeta})=\frac{\epsilon_{2}}{4}\bm{\zeta}^{2}+\frac{\dot{\bm{\zeta}}}{H}\bm{\zeta},$
(3.20)

which contribute

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=2\left(\frac{\epsilon_{2}}{4}+3\right)$
$\displaystyle\left[\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}+\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{3}}(\tau_{0})}^{2}\right.$
$\displaystyle\left.+\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{3}}(\tau_{0})}^{2}\right].$
(3.21)

Factor $3$ comes from $\dot{\zeta}_{k}(\tau_{0})=3H\zeta_{k}(\tau_{0})$, which
can be derived from the USR mode function (2.52). Hence, we can read the local
non-Gaussianity as

$\frac{6}{5}f_{\mathrm{NL}}=3\Longrightarrow f_{\mathrm{NL}}=\frac{5}{2}.$
(3.22)

In squeezed limit, the bispectrum becomes

$\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{-\bm{k}_{2}}(\tau_{0})\rangle\\!\rangle=6\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2},$
(3.23)

which clearly violates Maldacena’s theorem (3.17). Indeed, for a shift-
symmetric action of the inflaton, Maldacena’s theorem is generalized in [49,
50]. The squeezed bispectrum (3.23) can be reproduced from a generalized
Maldacena’s theorem after substituting the USR condition.

###  3.3 Ultraslow-roll to slow-roll transition

In case there is a transition of the second SR parameter, $\dot{\epsilon_{2}}$
becomes large during the transition. Therefore, the leading term in the third-
order action is [48, 51, 52, 35, 53]

$S_{\mathrm{bulk}}[\bm{\zeta}]=M_{\mathrm{pl}}^{2}\int\mathrm{d}\tau~{}\mathrm{d}^{3}x~{}a^{2}\left[\frac{1}{2}\epsilon_{1}\epsilon_{2}^{\prime}\bm{\zeta}^{\prime}\bm{\zeta}^{2}+\mathcal{O}(\epsilon^{2})\right].$
(3.24)

Because we are interested in transition from USR to SR period, relation
between $\zeta$ and $\bm{\zeta}$ at the end of inflation is
$\zeta=\bm{\zeta}+\epsilon_{2}\bm{\zeta}^{2}/4$. However, because
$\absolutevalue{\epsilon_{2}(\tau_{0})}\ll 1$, contribution from field
redefinition to the bispectrum is much smaller than the bulk interaction
(3.24). Thus, the bispectrum is given by

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=-2M_{\mathrm{pl}}^{2}$
$\displaystyle\int_{-\infty}^{\tau_{0}}\mathrm{d}\tau~{}\epsilon_{1}(\tau)\epsilon_{2}^{\prime}(\tau)a^{2}(\tau)$
(3.25)
$\displaystyle\times\mathrm{Im}\left[\zeta_{k_{1}}(\tau_{0})\zeta_{k_{2}}(\tau_{0})\zeta_{k_{3}}(\tau_{0})\zeta_{k_{1}}^{*}(\tau)\zeta_{k_{2}}^{*}(\tau)\zeta_{k_{3}}^{\prime*}(\tau)+\mathrm{perm}\right].$

In the next part, we will evaluate the time integral for sharp and smooth
transition of $\epsilon_{2}(\tau)$. Similarly as in Sec. 2.4, we follow the
parametrization of $\epsilon_{2}(\tau)$ in [35].

####  3.3.1 Sharp transition

Consider evolution of the second SR parameter as step function, given by
(2.70), its time derivative is Dirac-delta function

$\epsilon_{2}^{\prime}(\tau)=\Delta\epsilon_{2}\delta(\tau-\tau_{s}).$ (3.26)

Substituting it to the time integral in (3.25) leads to

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=$
$\displaystyle-2M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{s})a^{2}(\tau_{s})\Delta\epsilon_{2}$
(3.27)
$\displaystyle\times\mathrm{Im}\left[\zeta_{k_{1}}(\tau_{0})\zeta_{k_{2}}(\tau_{0})\zeta_{k_{3}}(\tau_{0})\zeta_{k_{1}}^{*}(\tau_{s})\zeta_{k_{2}}^{*}(\tau_{s})\zeta_{k_{3}}^{\prime*}(\tau_{s})+\mathrm{perm}\right].$

Then, we can substitute $\zeta_{k}(\tau)$ and $\zeta_{k}^{\prime}(\tau)$ given
in Sec. 2.4.2 to obtain the bispectrum. Although the general form of
bispectrum is complicated, in the limit $k_{1},k_{2},k_{3}\ll k_{s}$ the
bispectrum becomes

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=$
$\displaystyle~{}2M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{s})a^{2}(\tau_{s})\Delta\epsilon_{2}~{}\left(\frac{H^{2}}{4M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{s})}\right)^{3}$
$\displaystyle\times\frac{4}{k_{s}^{2}}\left(\frac{1}{k_{1}^{3}k_{2}^{3}}+\frac{1}{k_{2}^{3}k_{3}^{3}}+\frac{1}{k_{1}^{3}k_{3}^{3}}\right),$
(3.28)

which is a local type bispectrum. Thus, we can read the nonlinear parameter
$f_{\mathrm{NL}}=5/8$.

####  3.3.2 Smooth transition

Consider evolution of the second SR parameter given by (2.65), its time
derivative is

$\epsilon_{2}^{\prime}(\tau)=\frac{-2s^{2}(s-3)(s+3)}{\tau\left[\left(\frac{\tau_{s}}{\tau}\right)^{s}(s-3)+s+3\right]^{2}}\left(\frac{\tau_{s}}{\tau}\right)^{s}.$
(3.29)

Because the second SR parameter is constant for $\tau<\tau_{s}$, the domain of
time integral in (3.25) is simplified to

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=-2M_{\mathrm{pl}}^{2}$
$\displaystyle\int_{\tau_{s}}^{\tau_{0}}\mathrm{d}\tau~{}\epsilon_{1}(\tau)\epsilon_{2}^{\prime}(\tau)a^{2}(\tau)$
(3.30)
$\displaystyle\times\mathrm{Im}\left[\zeta_{k_{1}}(\tau_{0})\zeta_{k_{2}}(\tau_{0})\zeta_{k_{3}}(\tau_{0})\zeta_{k_{1}}^{*}(\tau)\zeta_{k_{2}}^{*}(\tau)\zeta_{k_{3}}^{\prime*}(\tau)+\mathrm{perm}\right].$

Unfortunately, the general form of bispectrum is complicated. However, it
becomes simple in the squeezed limit $k_{1}\ll k_{2}=k_{3}$ for perturbation
modes that cross the horizon much before the transition $k_{1},k_{2},k_{3}\ll
k_{s}$. Substituting (2.64), (3.29), and (2.68) to the time integral leads to

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{-\bm{k}_{2}}(\tau_{0})\rangle\\!\rangle$
$\displaystyle=\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}\int_{\tau_{s}}^{\tau_{0}}\mathrm{d}\tau\sqrt{\frac{\epsilon_{1}(\tau_{0})}{\epsilon_{1}(\tau)}}\epsilon_{2}^{\prime}(\tau)\left(1+\frac{1}{2}\epsilon_{2}(\tau)\right)$
$\displaystyle=-2\eta_{V}\sqrt{\frac{\epsilon_{1}(\tau_{0})}{\epsilon_{1}(\tau_{s})}}\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}.$
(3.31)

Thus, we can read off the nonlinear parameter

$\frac{12}{5}f_{\mathrm{NL}}=-2\eta_{V}\sqrt{\frac{\epsilon_{1}(\tau_{0})}{\epsilon_{1}(\tau_{s})}},$
(3.32)

which is suppressed by SR parameter $\eta_{V}$. This means that the bispectrum
generated by the transition (2.65) is negligible. In [54], it is argued that
suppression in the bispectrum is a result of (2.65) satisfying Wands duality
condition [55], which makes the integrand of (3.24) a total time derivative.
The squeezed bispectrum in both smooth and sharp transition cases can be
reproduced from the generalized Maldacena’s theorem by considering only
background evolution [56] following [47].

##  4 Generation of large fluctuation

In Sec. 2.2, we have shown that violation of the SR approximation, especially
the USR condition, leads to the growth of curvature perturbation outside the
horizon [23, 24]. Although the USR period cannot realize the required e-folds
number of inflation, it can be useful for other purposes. If the inflaton
undergoes USR period only for a short duration, the curvature perturbation
will be enhanced only within that duration. At the end of inflation, it
corresponds to amplification of the curvature perturbation at a range of
wavenumbers that cross the horizon during the USR period.

Assuming sharp transitions, evolution of the second SR parameter corresponding
to the potential in Fig. 1 is shown by the brown curve in Fig. 2. The inflaton
undergoes USR period in the flat region of potential from time $t_{s}$ to
$t_{e}$. In order to explain the observed CMB fluctuations, at early time,
$t\lesssim t_{s}$, the inflation was in SR period with slightly tilted
potential. At late time, $t\gtrsim t_{e}$, there is no observational
constraint on the potential. A potential is viable as long as the curvature
perturbation does not grow during the final period.

In this section, we study the power spectrum and bispectrum of curvature
perturbation in such a setup.

###  4.1 Power spectrum

At early time, $t\lesssim t_{s}$, the inflaton was in SR period with Bunch-
Davies initial vacuum, so the curvature perturbation is given by

$\zeta_{k}(\tau)=\left(\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1s}}}\right)_{\star}e^{-ik\tau}(1+ik\tau),$
(4.1)

where $\epsilon_{1s}$ is the first SR parameter during this SR period.

At $t_{s}\lesssim t\lesssim t_{e}$, the inflaton is in the intermediate USR
period. We define $\tau_{s}$ and $\tau_{e}$ as the conformal time
corresponding to $t_{s}$ and $t_{e}$, respectively. The first SR parameter can
be written as $\epsilon_{1}(\tau)=\epsilon_{1s}(\tau/\tau_{s})^{6}$ based on
proportionality in (2.47). Therefore, the curvature perturbation becomes

$\zeta_{k}(\tau)=\left(\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1s}}}\right)_{\star}\left(\frac{\tau_{s}}{\tau}\right)^{3}\left[\mathcal{A}_{k}e^{-ik\tau}(1+ik\tau)-\mathcal{B}_{k}e^{ik\tau}(1-ik\tau)\right],$
(4.2)

where coefficients $\mathcal{A}_{k}$ and $\mathcal{B}_{k}$ are determined by
matching to the SR solution (4.1) at the boundary. Solutions of the
coefficients by requiring continuity of $\zeta_{k}(\tau)$ and
$\zeta_{k}^{\prime}(\tau)$ at transition $\tau=\tau_{s}$ are

$\displaystyle\mathcal{A}_{k}=1-\frac{3(1+k^{2}\tau_{s}^{2})}{2ik^{3}\tau_{s}^{3}},$
(4.3)
$\displaystyle\mathcal{B}_{k}=-\frac{3(1+ik\tau_{s})^{2}}{2ik^{3}\tau_{s}^{3}}e^{-2ik\tau_{s}}.$
(4.4)

At late time, $t\gtrsim t_{e}$, the inflaton goes back to SR dynamics. The
curvature perturbation can be written as

$\zeta_{k}(\tau)=\left(\frac{iH}{2M_{\mathrm{pl}}\sqrt{k^{3}\epsilon_{1s}}}\right)_{\star}\left(\frac{\tau_{s}}{\tau_{e}}\right)^{3}\left[\mathcal{C}_{k}e^{-ik\tau}(1+ik\tau)-\mathcal{D}_{k}e^{ik\tau}(1-ik\tau)\right],$
(4.5)

where coefficients $\mathcal{C}_{k}$ and $\mathcal{D}_{k}$ are determined by
matching to the USR solution (4.2) at the boundary. Solutions of the
coefficients by requiring continuity of $\zeta_{k}(\tau)$ and
$\zeta_{k}^{\prime}(\tau)$ at transition $\tau=\tau_{e}$ are

$\displaystyle\mathcal{C}_{k}=\frac{-1}{4k^{6}\tau_{s}^{3}\tau_{e}^{3}}$
$\displaystyle\left\\{9(k\tau_{s}-i)^{2}(k\tau_{e}+i)^{2}e^{2ik(\tau_{e}-\tau_{s})}\right.$
$\displaystyle\left.-\left[k^{2}\tau_{s}^{2}(2k\tau_{s}+3i)+3i\right]\left[k^{2}\tau_{e}^{2}(2k\tau_{e}-3i)-3i\right]\right\\},$
(4.6) $\displaystyle\mathcal{D}_{k}=\frac{3}{4k^{6}\tau_{s}^{3}\tau_{e}^{3}}$
$\displaystyle\left\\{e^{-2ik\tau_{s}}[3+k^{2}\tau_{e}^{2}(3-2ik\tau_{e})](k\tau_{s}-i)^{2}\right.$
$\displaystyle\left.+ie^{-2ik\tau_{e}}\left[3i+k^{2}\tau_{s}^{2}(2k\tau_{s}+3i)\right](k\tau_{e}-i)^{2}\right\\}.$
(4.7)

As before, $k_{s}$ and $k_{e}$ are wavenumbers which cross the horizon at
$\tau_{s}$ and $\tau_{e}$, respectively. At the end of inflation,
$\tau_{0}~{}(\rightarrow 0)$, the tree-level power spectrum is

$\Delta^{2}_{s(0)}(k,\tau_{0})=\frac{k^{3}}{2\pi^{2}}\absolutevalue{\zeta_{k}(\tau_{0})}^{2}=\left(\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1s}}\right)_{\star}\left(\frac{k_{e}}{k_{s}}\right)^{6}\absolutevalue{\mathcal{C}_{k}-\mathcal{D}_{k}}^{2}.$
(4.8)

On large scale, the power spectrum approaches an almost scale-invariant limit

$\Delta_{s(\mathrm{SR})}^{2}(k)\equiv\Delta^{2}_{s(0)}(k\ll
k_{s},\tau_{0})=\left(\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1s}}\right)_{\star}\left[1+\mathcal{O}\left(\frac{k^{2}}{k_{s}^{2}}\right)\right],$
(4.9)

which is consistent with CMB observation. Assuming $k_{e}\gg k_{s}$, the next
order approximation is

$\Delta^{2}_{s(0)}(k,\tau_{0})\simeq\left(\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{1s}}\right)_{\star}\left[1-\frac{8}{5}\left(\frac{k_{e}}{k_{s}}\right)^{3}\left(\frac{k}{k_{s}}\right)^{2}+\frac{24}{35}\left(\frac{k_{e}}{k_{s}}\right)^{3}\left(\frac{k}{k_{s}}\right)^{4}+\mathcal{O}\left(\frac{k^{6}}{k_{s}^{6}}\right)\right].$
(4.10)

The minus sign in the coefficient of $(k/k_{s})^{2}$ is important because it
can explain a dip in the power spectrum before amplification start. As $k$
approaches $k_{s}$, the $k^{2}$-term grows until it reaches unity. At this
point, the $k^{4}$-term is not large enough yet, so the power spectrum almost
vanishes. This yields a dip in the power spectrum, which happens at the
wavenumber

$k_{\mathrm{dip}}\simeq
k_{s}\left[\frac{5}{8}\left(\frac{k_{s}}{k_{e}}\right)^{3}\right]^{1/2}.$
(4.11)

It has been argued that this dip may lead to an interesting observational
consequence [57].

At $k_{\mathrm{dip}}<k<k_{s}$, the $k^{4}$-term grows while the $k^{6}$-term
is still small, which explains the apparent $k^{4}$-growth in the power
spectrum [38]. On a small scale with a larger wavenumber, $k_{s}\lesssim
k\lesssim k_{e}$, the power spectrum is oscillating around

$4\times\Delta_{s(\mathrm{PBH})}^{2}\equiv
4\times\Delta_{s(\mathrm{SR})}^{2}(k_{s})\left(\frac{k_{e}}{k_{s}}\right)^{6},$
(4.12)

whose high-density peak may collapse into PBHs. Numerical coefficient 4
appears for the same reason as (2.78). It is amplified by a factor of
$(k_{e}/k_{s})^{6}$ compared to the CMB-scale power spectrum. To generate a
significant abundance of PBHs, typically peak with $\mathcal{O}(0.01)$ on
small-scale power spectrum is required. It means that the power spectrum is
amplified $\mathcal{O}(10^{7})$ compared to CMB-scale. In terms of the number
of e-folds during the USR period
$N_{\mathrm{USR}}=\log[a(\tau_{e})/a(\tau_{s})]$, we need to have

$N_{\mathrm{USR}}=\log\frac{\tau_{s}}{\tau_{e}}=\log\frac{k_{e}}{k_{s}}\simeq\frac{1}{6}\log
10^{7}=2.68,$ (4.13)

to achieve a significant abundance of PBHs.

Figure 3:  Power spectrum of the curvature perturbation. At CMB pivot scale,
$p_{*}=0.05~{}\mathrm{Mpc}^{-1}$, the power spectrum is almost scale invariant
with amplitude $\Delta_{s(\mathrm{SR})}^{2}(p_{*})=2.1\times 10^{-9}$, based
on observational result [21]. At $k_{\mathrm{dip}}<k<k_{s}$, the power
spectrum grows with function $k^{4}$ until it reaches the peak. At
$k_{s}<k<k_{e}$, the power spectrum is oscillating around
$4\times\Delta_{s(\mathrm{PBH})}^{2}$. Similar behavior at $k>k_{e}$ because
SR period is imposed after the USR period.

In this model, we can see that the power spectrum remains amplified even for
$k>k_{e}$, which oscillates around $\Delta_{s(\mathrm{PBH})}^{2}$. This
happens because the inflaton goes directly to the SR period after the USR
period. In case the inflaton undergoes a constant-roll period with
$\epsilon_{2}>0$ after the USR period, the power spectrum at the end of
inflation will decrease for $k>k_{e}$.

In this review, we choose an USR period for the intermediate phase as an
example. In principle, the intermediate phase can be generalized to a
constant-roll period as long as the curvature perturbation grows outside the
horizon. The requirement was discussed below Eq. (2.63). However, for a
general constant-roll period, the mode function becomes the Hankel function,
which is given by (2.24) with (2.57). Technically, it becomes more difficult
to match the coefficient of constant-roll period with the SR period because it
involves derivative of Hankel function. In [40], a method to approximate the
mode function is proposed by utilizing the fact that the duration of the SR
violating regime is extremely short compared to the total duration of
inflation. An inflation model with two phases of SR violating period is also
introduced in [40] that yields $k^{8}$ growth in the power spectrum, which is
much steeper than $k^{4}$ growth.

###  4.2 Bispectrum

Consider the evolution of the second SR parameter shown by the brown curve in
Fig. 2. Its time derivative can be modelled as Dirac-delta function

$\epsilon_{2}^{\prime}(\tau)=\Delta\epsilon_{2}[-\delta(\tau-\tau_{s})+\delta(\tau-\tau_{e})].$
(4.14)

Substituting it to the time integral in (3.25) leads to

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{\bm{k}_{3}}(\tau_{0})\rangle\\!\rangle=2M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{s})a^{2}(\tau_{s})\Delta\epsilon_{2}\mathrm{Im}\left[\zeta_{k_{1}}(\tau_{0})\zeta_{k_{2}}(\tau_{0})\zeta_{k_{3}}(\tau_{0})\zeta_{k_{1}}^{*}(\tau_{s})\zeta_{k_{2}}^{*}(\tau_{s})\zeta_{k_{3}}^{\prime*}(\tau_{s})\right]$
$\displaystyle-2M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{e})a^{2}(\tau_{e})\Delta\epsilon_{2}\mathrm{Im}\left[\zeta_{k_{1}}(\tau_{0})\zeta_{k_{2}}(\tau_{0})\zeta_{k_{3}}(\tau_{0})\zeta_{k_{1}}^{*}(\tau_{e})\zeta_{k_{2}}^{*}(\tau_{e})\zeta_{k_{3}}^{\prime*}(\tau_{e})\right]+\mathrm{perm}.$
(4.15)

In squeezed limit, the bispectrum becomes

$\displaystyle\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{-\bm{k}_{2}}(\tau_{0})\rangle\\!\rangle=-\left\\{4\Delta\epsilon_{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{e})a^{2}(\tau_{e})\mathrm{Im}\left[\frac{\zeta_{k_{2}}^{2}(\tau_{0})}{\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}}\zeta_{k_{2}}^{*}(\tau_{e})\zeta_{k_{2}}^{\prime*}(\tau_{e})\right]\right.$
$\displaystyle\left.-4\Delta\epsilon_{2}M_{\mathrm{pl}}^{2}\epsilon_{1}(\tau_{s})a^{2}(\tau_{s})\mathrm{Im}\left[\frac{\zeta_{k_{2}}^{2}(\tau_{0})}{\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}}\zeta_{k_{2}}^{*}(\tau_{s})\zeta_{k_{2}}^{\prime*}(\tau_{s})\right]\right\\}\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}.$
(4.16)

We define the term inside parenthesis as $C_{0}(k)$, so the bispectrum can be
written as

$\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau_{0})\zeta_{\bm{k}_{2}}(\tau_{0})\zeta_{-\bm{k}_{2}}(\tau_{0})\rangle\\!\rangle=-C_{0}(k_{2})\absolutevalue{\zeta_{k_{1}}(\tau_{0})}^{2}\absolutevalue{\zeta_{k_{2}}(\tau_{0})}^{2}.$
(4.17)

We can show that $C_{0}(k)=n_{s}(k,\tau_{0})-1$ analytically, which confirms
Maldacena’s theorem explicitly [10].

Figure 4:  Plot of $C_{0}(k)$ and $n_{s}(k,\tau_{0})-1$. We choose
$k_{e}/k_{s}=10$ only for illustrative purposes.

At the end of inflation, we have shown that only the bulk interaction (3.24)
contributes to the bispectrum. During the USR period, the bispectrum also
satisfies Maldacena’s theorem, where both the bulk interaction (3.24) and
field redefinition (3.20) are important. Therefore, in general time, the
squeezed bispectrum is

$\langle\\!\langle\zeta_{\bm{k}_{1}}(\tau)\zeta_{\bm{k}_{2}}(\tau)\zeta_{-\bm{k}_{2}}(\tau)\rangle\\!\rangle=-(n_{s}(k_{2},\tau)-1)\absolutevalue{\zeta_{k_{1}}(\tau)}^{2}\absolutevalue{\zeta_{k_{2}}(\tau)}^{2}.$
(4.18)

We can read that spectral tilt of the short-scale perturbation, depends on
time even at superhorizon scale, specifically around the transition time. Plot
of the spectral tilt at the end of USR period and at the end of inflation are
shown in Fig. 5. Thus, in contrast to the case of the SR inflation in Sec. 3.1
where time dependence $\log\tau_{0}$ is cancelled, the squeezed bispectrum
evolves outside the horizon in this model. In case all wavenumbers have
magnitude around the peak scale, the bispectrum is expected to deviate
significantly from Maldacena’s limit.

In the presence of peaks and troughs in the power spectrum, Maldacena’s
theorem is generalized in [58]. In addition, the consistency relation (4.18)
fixes the sign of $f_{\mathrm{NL}}$ at the peak, which has an implication for
the abundances of PBH [59].

Figure 5:  Plot of $n_{s}-1$ at the end of USR period and at the end of
inflation. For a range of wavenumber between $p_{*}$ and $k_{s}$, the spectral
tilt evolves, altough such perturbations already far outside the horizon at
the end of USR period.

###  4.3 Beyond tree-level contribution

So far, we have discussed two-point and three-point functions of the curvature
perturbation at tree-level approximation. We have shown that the sharp
transition of the second SR parameter generates a $\mathcal{O}(1)$
contribution to the nonlinear parameter $f_{\mathrm{NL}}$. It is a large
contribution to the bispectrum, which implies that a higher-order correction
than first-order perturbation theory can also be large.

The second-order expansion of in-in perturbation theory reads

$\displaystyle\langle\mathcal{O(\tau)}\rangle_{(1)}=\langle\mathcal{O(\tau)}\rangle_{(0,2)}^{\dagger}+\langle\mathcal{O(\tau)}\rangle_{(1,1)}+\langle\mathcal{O(\tau)}\rangle_{(0,2)},$
(4.19)
$\displaystyle\langle\mathcal{O(\tau)}\rangle_{(1,1)}=\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\int_{-\infty}^{\tau}\mathrm{d}\tau_{2}\left\langle
H_{\mathrm{int}}(\tau_{1})\mathcal{\hat{O}}(\tau)H_{\mathrm{int}}(\tau_{2})\right\rangle,$
$\displaystyle\langle\mathcal{O(\tau)}\rangle_{(0,2)}=-\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\int_{-\infty}^{\tau_{1}}\mathrm{d}\tau_{2}\left\langle\mathcal{\hat{O}}(\tau)H_{\mathrm{int}}(\tau_{1})H_{\mathrm{int}}(\tau_{2})\right\rangle.$

In the presence of cubic self-interaction in the Hamiltonian, the second-order
perturbation contributes to one-loop correction to the power spectrum and
tree-level contribution to the trispectrum. For an inflation model with sharp
transition of the second SR parameter, a one-loop correction to the large-
scale power spectrum was first derived in [9]. In this case, the operator is
$\zeta_{\bm{p}}(\tau)\zeta_{-\bm{p}}(\tau)$, where $\mathbf{p}$ is the CMB
scale wavevector, evaluated at $\tau=\tau_{0}~{}(\rightarrow 0)$.

At the end of inflation, there is no contribution from the field redefinition
because it is evaluated at the SR period. Only the bulk interaction (3.24)
contributes to the one-loop correction. Substituting the corresponding
Hamiltonian of (3.24) to (4.19) yields

$\displaystyle\langle\zeta_{\bm{p}}(\tau_{0})\zeta_{-\bm{p}}(\tau_{0})\rangle_{(1,1)}=$
$\displaystyle~{}\frac{1}{4}M_{\mathrm{pl}}^{4}\int_{-\infty}^{\tau_{0}}\mathrm{d}\tau_{1}~{}a^{2}(\tau_{1})\epsilon_{1}(\tau_{1})\epsilon_{2}^{\prime}(\tau_{1})\int_{-\infty}^{\tau_{0}}\mathrm{d}\tau_{2}~{}a^{2}(\tau_{2})\epsilon_{1}(\tau_{2})\epsilon_{2}^{\prime}(\tau_{2})$
$\displaystyle\times\int\prod_{a=1}^{6}\left[\frac{\mathrm{d}^{3}k_{a}}{(2\pi)^{3}}\right]\delta(\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3})\delta(\bm{k}_{4}+\bm{k}_{5}+\bm{k}_{6})$
$\displaystyle\times\left\langle\zeta_{\bm{k}_{1}}^{\prime}(\tau_{1})\zeta_{\bm{k}_{2}}(\tau_{1})\zeta_{\bm{k}_{3}}(\tau_{1})\zeta_{\bm{p}}(\tau_{0})\zeta_{-\bm{p}}(\tau_{0})\zeta_{\bm{k}_{4}}^{\prime}(\tau_{2})\zeta_{\bm{k}_{5}}(\tau_{2})\zeta_{\bm{k}_{6}}(\tau_{2})\right\rangle,$
(4.20)
$\displaystyle\langle\zeta_{\bm{p}}(\tau_{0})\zeta_{-\bm{p}}(\tau_{0})\rangle_{(0,2)}=$
$\displaystyle-\frac{1}{4}M_{\mathrm{pl}}^{4}\int_{-\infty}^{\tau_{0}}\mathrm{d}\tau_{1}~{}a^{2}(\tau_{1})\epsilon_{1}(\tau_{1})\epsilon_{2}^{\prime}(\tau_{1})\int_{-\infty}^{\tau_{1}}\mathrm{d}\tau_{2}~{}a^{2}(\tau_{2})\epsilon_{1}(\tau_{2})\epsilon_{2}^{\prime}(\tau_{2})$
$\displaystyle\times\int\prod_{a=1}^{6}\left[\frac{\mathrm{d}^{3}k_{a}}{(2\pi)^{3}}\right]\delta(\bm{k}_{1}+\bm{k}_{2}+\bm{k}_{3})\delta(\bm{k}_{4}+\bm{k}_{5}+\bm{k}_{6})$
$\displaystyle\times\left\langle\zeta_{\bm{p}}(\tau_{0})\zeta_{-\bm{p}}(\tau_{0})\zeta_{\bm{k}_{1}}^{\prime}(\tau_{1})\zeta_{\bm{k}_{2}}(\tau_{1})\zeta_{\bm{k}_{3}}(\tau_{1})\zeta_{\bm{k}_{4}}^{\prime}(\tau_{2})\zeta_{\bm{k}_{5}}(\tau_{2})\zeta_{\bm{k}_{6}}(\tau_{2})\right\rangle.$
(4.21)

Performing Wick contraction and summing both terms lead to

$\displaystyle\langle\\!\langle\zeta_{\bm{p}}(\tau_{0})\zeta_{-\bm{p}}(\tau_{0})\rangle\\!\rangle_{(1)}=\frac{1}{4}M_{\mathrm{pl}}^{4}\epsilon_{1}^{2}(\tau_{e})a^{4}(\tau_{e})(\Delta\epsilon_{2})^{2}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\left\\{\zeta_{p}(\tau_{0})\zeta_{p}^{*}(\tau_{0})\left(4\zeta_{p}^{\prime}\zeta_{p}^{\prime*}\zeta_{q}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}\right.\right.$
$\displaystyle\left.\left.+8\zeta_{p}^{\prime*}\zeta_{p}\zeta_{q}^{\prime}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}+8\zeta_{p}^{\prime}\zeta_{p}^{*}\zeta_{q}^{\prime*}\zeta_{q}\zeta_{k}\zeta_{k}^{*}+16\zeta_{p}\zeta_{p}^{*}\zeta_{q}^{\prime}\zeta_{q}^{\prime*}\zeta_{k}\zeta_{k}^{*}\right)_{\tau=\tau_{e}}-\mathrm{Re}\left[\zeta_{p}(\tau_{0})\zeta_{p}(\tau_{0})\right.\right.$
(4.22)
$\displaystyle\left.\left.\times\left(4\zeta_{p}^{\prime*}\zeta_{p}^{\prime*}\zeta_{q}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}+8\zeta_{p}^{\prime*}\zeta_{p}^{*}\zeta_{q}^{\prime}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}+8\zeta_{p}^{\prime*}\zeta_{p}^{*}\zeta_{q}^{\prime*}\zeta_{q}\zeta_{k}\zeta_{k}^{*}+16\zeta_{p}^{*}\zeta_{p}^{*}\zeta_{q}^{\prime}\zeta_{q}^{\prime*}\zeta_{k}\zeta_{k}^{*}\right)_{\tau=\tau_{e}}\right]\right\\}.$

Here, we only show the leading contribution to the one-loop correction that
comes from integral of Dirac-delta function at $\tau=\tau_{e}$. The
contributions of $\tau=\tau_{s}$ are subleading, which are explicitly
evaluated in [10].

For $p\ll k$, the one-loop correction can be simplified to

$\langle\\!\langle\zeta_{\bm{p}}(\tau_{0})\zeta_{-\bm{p}}(\tau_{0})\rangle\\!\rangle_{(1)}=\frac{1}{4}M_{\mathrm{pl}}^{4}(\Delta\epsilon_{2})^{2}\absolutevalue{\zeta_{p}(\tau_{0})}^{2}\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}16\left[\epsilon_{1}^{2}a^{4}\absolutevalue{\zeta_{k}}^{2}\mathrm{Im}(\zeta_{p}\zeta_{p}^{\prime*})\mathrm{Im}(\zeta_{k}\zeta_{k}^{\prime*})\right]_{\tau=\tau_{e}}.$
(4.23)

We find that the one-loop correction is exactly proportional to the quantum
commutator (2.46), which implies the importance of decaying mode even for
superhorizon mode function. If we only consider the constant mode of curvature
perturbation at superhorizon scale, we will obtain an incorrect result
$\left[\hat{\zeta}_{\bm{p}}(\tau),\hat{\zeta}_{\bm{q}}^{\prime}(\tau)\right]=0$.
From commutator (2.46), we find

$\mathrm{Im}(\zeta_{k}(\tau)\zeta_{k}^{\prime*}(\tau))=\frac{1}{4M_{\mathrm{pl}}^{2}\epsilon(\tau)a^{2}(\tau)},$
(4.24)

which does not depend on wavenumber $k$. As we have already seen, in the
standard SR inflation this quantity decreases exponentially as a function of
conformal time. However, during USR regime, since $\epsilon$ decreases much
more rapicly than $a^{-2}$, the right-hand-side of (2.46) and (4.24) increases
exponentially, which makes the one-loop correction to the power spectrum large
even on the large scale probed by CMB.

Then, substituting (4.24) into (4.23), we obtain a one-loop correction to the
power spectrum

$\Delta_{s(1)}^{2}(p,\tau_{0})=\frac{1}{4}(\Delta\epsilon_{2})^{2}\Delta_{s(0)}^{2}(p,\tau_{0})\int_{k_{\mathrm{IR}}}^{k_{\mathrm{UV}}}\frac{\mathrm{d}k}{k}\Delta_{s(0)}^{2}(k,\tau_{e}).$
(4.25)

Because we are interested in the effect of large fluctuation on small scales
related to PBH formation to the fluctuations on large scales observed by CMB,
a natural choice of cutoffs are $k_{\mathrm{IR}}=k_{s}$ and
$k_{\mathrm{UV}}=k_{e}$. For $k_{e}/k_{s}\ll 1$, evaluating the wavenumber
integral with small-scale mode function (4.2) yields

$\Delta_{s(1)}^{2}(p,\tau_{0})=\frac{1}{4}(\Delta\epsilon_{2})^{2}\Delta_{s(0)}^{2}(p,\tau_{0})\Delta_{s(\mathrm{PBH})}^{2}\left(1.1+\log\frac{k_{e}}{k_{s}}\right).$
(4.26)

Requiring the one-loop correction to be much smaller than tree-level
contribution yields an upper-bound on small-scale power spectrum

$\Delta_{s(\mathrm{PBH})}^{2}\ll\frac{1}{(\Delta\epsilon_{2})^{2}}\simeq
0.03,$ (4.27)

where the numerical value is obtained by substituting $\Delta\epsilon_{2}=6$
for transition from USR to SR period. Such a bound suggests that sharp
transition of the second SR parameter cannot generate large fluctuation on
small scales, otherwise perturbation theory breaks down. This result should be
understood under caveat that there are other contributions from quartic
Hamiltonian, both induced by the third-order action (3.24) and fourth-order
action.

##  5 Discussion and summary

###  5.1 Discussion

Following [41], PBH formation models in single-field inflation can be
classified into two categories. The first category is models with features in
the inflationary potential or nonminimally coupled inflaton with potential
defined in the Einstein frame. Models with an extremely flat feature or an
inflection point in the potential fall in this category [33, 60, 61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82,
83]. These models can be derived from high-energy theories such as
supergravity [60, 61, 62, 63, 64, 65, 66], axion monodromy [67, 68], scalaron
in $R^{2}$-gravity [69], $\alpha$-attractor [70, 71, 72, 73], and string
theory [74, 75, 76]. Also, there are models from Higgs inflation which do not
require theories beyond the standard model [77, 78, 79, 80, 81]. Other
examples of features are a tiny bump or dip [84, 85, 86, 87, 88, 89], an
upward or downward step [90, 91, 92, 32, 93, 94, 95], polynomial shape [96,
97, 98, 99], and Coleman-Weinberg potential [100, 24, 101, 102]. In these
examples, modification of potential makes a sharp transition of the second SR
parameter on the inflaton dynamics.

The second category is models with modified gravity or beyond nonminimally
coupled inflaton. Examples for this category are models based on $k$ [103] or
$G$ [104] inflation [105, 106, 107, 108, 109, 110], the effective field theory
of inflation [111, 112, 113], $f(R)$ gravity [114], a nonminimal derivative
coupling [115, 116, 117], Gauss-Bonnet inflation [118, 119], and bumpy axion
inflation [120]. In these examples, amplification of small-scale perturbation
can be realized by a sharp transition of the second SR parameter and/or other
parameters. In [111, 112], amplification of small-scale perturbation is caused
by a sharp transition of the sound speed, $c_{s}$, a quantity that
parametrizes deviation from a canonical kinetic term. In this case, coupling
$\epsilon_{2}^{\prime}$ in cubic self-interaction (3.24) is modified to
$(\epsilon_{2}/c_{s}^{2})^{\prime}$ [121, 122]. For this category, rapid
slowdown of the inflaton’s velocity might not be the only reason of
amplification of the curvature perturbation.

There have been some works following [9, 10], most of them derive and
qualitatively confirm the one-loop correction by other methods, although
quantitatively has not been settled yet in the community. Despite of that, our
claim [9] was criticized by the author of [123] who attempted to evaluate the
same correction with a different method. In case of sharp transition of the
second SR parameter, we have shown in [10] that after more careful and proper
calculation, one can reproduce our result even if the method proposed in [123]
is used. In case the second SR parameter evolves continuously with function
(3.29), the bispectrum is suppressed by a SR parameter $\eta_{V}$ [124]. In
[54], we have argued that suppression on the bispectrum and one-loop
correction occur only for a specific smooth transition that satisfies Wands
duality condition [55].

Indeed, there are issues related to regularization and renormalization of the
divergence. In [9], we show that constraint (4.27) corresponds to the
requirement to perform the renormalization order by order in perturbation
theory. Another method for performing regularization, renormalization, and
resummation of loop corrections is proposed in [125, 126, 127], where the
sound speed of the curvature perturbation is introduced within the effective
field theory of inflation framework. Numerical integration of the one-loop
correction has been performed in [128, 129], confirming our analytical result
in the limit of the sharp transition. Also, Ref. [128] pointed out that a dip
in the power spectrum (4.11) is a tree-level artifact, which disappears after
adding a one-loop correction. One-loop correction to the power spectrum around
the peak scale has been investigated in [130]. Specifically, for an
inflationary potential with oscillatory features, it has been calculated in
[131, 132].

Moreover, the $\delta N$ formalism and stochastic inflation in the presence of
a sharp transition of the second SR parameter have been investigated in [133,
134]. The one-loop correction has also been derived from the $\delta N$
formalism in [135, 136]. The extension from USR to constrant-roll period has
been derived in [137], which yields a constraint involving a constant second
SR parameter and small-scale power spectrum. In the limit
$\absolutevalue{\epsilon_{2}}\rightarrow\infty$, bispectrum and one-loop
correction has been investigated in [138, 139], based on a method that
approximates the curvature perturbation in the presence of a very short period
where the SR approximation is violated [40]. In addition to cubic self-
interaction, quartic self-interaction [140, 141] in the Hamiltonian can also
contribute to the one-loop correction, in first-order perturbation theory. Its
contribution has been investigated, either by direct expansion of the quartic
derivative of the potential [142] or by an effective field theory approach
[143, 144].

However, there are two works that completely disagree with our conclusions
[145, 146]. In their approach, instead of performing field redefinition, they
calculate a one-loop correction directly from third-order action with total
time derivative terms and terms proportional to the equation of motion, as
shown in (3.1). Currently, they have been criticized by [147]. Other
perspectives on the effect of the total time derivative on the correlation
function have been presented in [148, 149]. We hope to solve this discrepancy
by pointing out what is missing in [145, 146]. Beyond the power spectrum, one-
loop correction to the bispectrum has been explored in [150].

While we have been performing computation in the comoving gauge, some authors
do so adopting the flat-slicing gauge. In this gauge, perturbation is
represented by the inflaton’s fluctuation $\delta\phi$, with vanishing
curvature perturbation. Prior to our work, the quantum correction to
$\delta\phi$ was investigated in [151, 152] and followed up in [153], which is
consistent with our claim. The effect of a smooth transition of the second SR
parameter in the flat-slicing gauge has been explored in [154, 155]. However,
there is a result [156] that is inconsistent with our claim due to
inappropriate order counts.

###  5.2 Summary

In this chapter, we have explained the mechanism to generate large primordial
fluctuation in single-field inflation. In order to amplify the curvature
perturbation, violation of the SR approximation is needed. We have discussed
the simplest example of an inflation model that violates the SR approximation,
called USR inflation. We have shown that USR inflation leads to growth of the
curvature perturbation outside the horizon. The consequence is that USR
inflation cannot explain the required e-folds number to solve the horizon
problem.

Nevertheless, one can utilize the properties of the USR period for another
purpose. If the USR period is sandwiched by two SR periods, the curvature
perturbation grows temporary during the USR period. At the end of inflation,
the power spectrum of the curvature perturbation is amplified on small scales
corresponding to the time scale of the USR period, while keeping the small
amplitude on large scale consistent with CMB observations.

Moreover, we have explained the nonlinear evolution of the curvature
perturbation induced by cubic self-interaction, which yields three-point
functions or bispectrum. The cubic self-interaction consists of bulk
interaction and field redefinition. In USR inflation, only the field
redefinition contributes to the bispectrum. When a transition from USR to SR
period exists, the bulk interaction that is proportional to the time
derivative of the second SR parameter becomes important. If the transition is
sharp, the nonlinear parameter of the bispectrum has a $\mathcal{O}(1)$ value.
This means that the bispectrum is large enough, which indicates that the
nonlinear effect cannot be neglected. At the same time, such a nonlinear
effect gives backreaction to the power spectrum as a one-loop correction that
leads to a constraint on the small-scale power spectrum associated with
transition of the second SR parameter.

## Acknowledgments

J. K. acknowledges the support from JSPS KAKENHI Grants No. 22KJ1006 and No.
22J20289. J. K. is also supported by the Global Science Graduate Course (GSGC)
program of The University of Tokyo. J. Y. is supported by JSPS KAKENHI Grant
No. 20H05639.

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