One-loop Corrections in Power Spectrum in Single Field Inflation

Hassan<EMAIL_ADDRESS>
School of Astronomy, Institute for Research in Fundamental Sciences (IPM)
P. O. Box 19395-5531, Tehran, Iran

###### Abstract

We revisit the one-loop correction in curvature perturbation power spectrum in
models of single field inflation which undergo a phase of ultra slow-roll
(USR) inflation. We include the contributions from both the cubic and quartic
interaction Hamiltonians and calculate the one-loop corrections on the
spectrum of the CMB scale modes from the small scale modes which leave the
horizon during the USR phase. It is shown that the amplitude of one-loop
corrections depends on the sharpness of the transition from the USR phase to
the final slow-roll phase. For an arbitrarily sharp transition, the one-loop
correction becomes arbitrarily large, invalidating the perturbative treatment
of the analysis. We speculate that for a mild transition, the large one-loop
corrections are washed out during the subsequent evolution after the USR
phase. The implications for primordial black holes formation are briefly
reviewed.

##  1 Introduction

The inflationary Universe is a successful paradigm for early universe
cosmology which is well supported by cosmological observations [1, 2]. In the
simplest scenario, inflation is driven by a single scalar field which slowly
rolls on top of a nearly flat potential. Among the basic predictions of models
of inflation are that the primordial perturbations are nearly scale invariant,
Gaussian and adiabatic.

A key prediction of most single field models of inflation is that the
curvature perturbations on superhorizon scales are constant [3, 4, 5]. The
conservation of curvature perturbation on superhorizon scales has the great
advantage that one can read off the fundamental properties of the primordial
inflationary universe at the late time when the corresponding modes reenters
the horizon. Because of this fundamental property, single field models of
inflation are particularly interesting. For example, it was shown by Maldacena
[6] that there is a consistency relation [7] between the amplitude of non-
Gaussianity parameter $f_{NL}$ in the squeezed limit and the spectral index of
curvature perturbation $n_{s}-1$, for a review see [8, 9]. While this
consistency condition applies to a vast majority of single field models,
however it has a well-known counter example, the ultra slow-roll (USR) setup,
in which the consistency condition is violated [10]. USR setup is a model in
which the potential is very flat so the inflaton speed falls off exponentially
during inflation [11, 12, 13, 14]. Consequently the curvature perturbations
grow on superhorizon scale which is the main reason behind the violation of
the consistency condition [15, 16, 17, 18, 19]. In particular, it is shown in
[10] that local-type non-Gaussianity with the amplitude $f_{NL}=\frac{5}{2}$
is generated in USR setup with a canonical kinetic energy. This value can be
enhanced if one considers models of non-attractor inflation with non-trivial
sound speed [15, 20].

There have been significant interests in primordial black holes (PBHs)
formation in the setup of USR inflation [21, 22, 23], for a recent review see
[24] and references therein. In order for the PBHs to comprise a sizeable
fraction of dark matter energy density, one requires that the amplitude of
curvature perturbations to grow by about $10^{7}$ orders of magnitude compared
to its amplitude in CMB scale. This can be achieved easily if one considers a
short period of USR inflation between two extended periods of slow-roll
inflation. However, the viability of this setup for PBHs formation was
questioned recently in [25] and [26]. More specifically, it was argued in [25,
26] that the one-loop corrections induced from small scale perturbations which
leave the horizon during the USR phase will induce large contributions in
curvature perturbation power spectrum, invalidating the perturbation theory.
This is somewhat counterintuitive because one expects the superhorizon modes
to be largely unaffected by small scale modes which leave the horizon only
much later during the USR phase. However, following the example of the
violation of consistency condition, things are a bit non-trivial in USR setup
compared to usual slow-roll model. The would-be decaying mode in USR phase is
actually the growing mode and a large growth of curvature perturbation can
induce a source term for the large scale curvature perturbation which may not
support the usual intuition. In addition, a key role is played by the
mechanism in which the USR phase is terminated to be followed by a final
attractor slow-roll phase. The danger of large one-loop effects may seem to be
more sever for sharp transitions while for mild transitions the large one-loop
corrections may be washed out during the transition from the USR phase to the
final attractor slow-roll phase. In this work we revisit the question of one-
loop correction in power spectrum induced from small scale modes which leave
the horizon during the USR phase. To study the system analytically we consider
a general sharp transition and compare our results with those of [25, 26].

The rest of the paper is organized as follows. In section 2 we present our
setup with emphasis on the mechanism of sharp transition from a USR phase to a
slow-roll phase. In section 3 we construct the cubic and quartic Hamiltonians
using the formalism of effective field theory (EFT) of inflation. As a non-
trivial application, in section 4 we employ the cubic action obtained from our
EFT approach and calculate the bispectrum, confirming the previous results in
literature. The one-loop corrections in power spectrum are presented in
details in section 5 followed by summary and discussions in section 6. Some
technicalities concerning the in-in integrals of cubic Hamiltonian are
relegated to Appendix A.

##  2 USR Inflation Setup

In this section we review our setup and gather the equations necessary for our
follow up analysis.

As reviewed in previous section, the USR setup is among the very few models of
single field inflation which violates the Maldacena consistency condition [6]
relating the amplitude of non-Gaussianity parameter $f_{NL}$ in the squeezed
limit to the spectral index $n_{s}-1$ [10, 15, 20]. The origin of the
violation of the consistency condition is that the curvature perturbation is
not frozen on superhorizon scale. In the simple picture where a USR setup is
abruptly terminated by a slow-roll phase it was shown in [10] that
$f_{NL}=\frac{5}{2}$. However, as demonstrated in [27], the final value of
$f_{NL}$ at the end of inflation depends on the details of transition from the
USR phase to the slow-roll phase. If the transition happens mildly then much
of the value of $f_{NL}$ is washed out during the subsequent evolution of the
curvature perturbation in the final slow-roll phase. Consequently, one obtains
$f_{NL}$ at the order of the slow-roll parameter at the end of inflation,
though the consistency condition is still violated.

Recently the question of one-loop correction in USR setup has attracted
interests. More specifically, it was argued in [25] and [26] that the one-loop
corrections from the modes which leave the Hubble radius during the USR phase
can affect the large CMB scale modes such that the induced power spectrum
become comparable to the tree-level power spectrum. This would inevitably
violate the perturbative nature of the system and causes concern. In addition,
it was shown that this large one-loop correction prohibits the formation of
PBHs in USR setup. These conclusions were criticized in [28, 29]. In
particular, it was argued in [29] that if one considers a physical situation
where the transition from the USR phase to the final slow-roll phase takes
place mildly, then the dangerous one-loop corrections are washed out and the
mechanism of PBHs formation in USR setup is still working. We also comment
that one-loop correction in power spectrum was also studied in [32] in a model
of inflation where perturbations are resonantly amplified because of the
oscillatory feature in potential. It was shown that the one-loop correction
can become large to invalidate the perturbative assumption.

In this work we revisit these issues in more details, building upon the
analysis of [25, 26]. As we see the mechanism of transition from the USR phase
to the final slow-roll phase plays important roles so here we briefly review
some analysis concerning the transition as studied in more details in [27]. To
study the evolution analytically, below we consider a sharp transition from a
USR phase to the slow-roll phase while a mild transition requires further
numerical investigation which is beyond the scope of the current work. Also we
consider a scalar field with a canonical kinetic energy with the sound speed
$c_{s}=1$.

The picture we have in mind is a single field inflation driven by the scalar
field $\phi$ with the potential $V(\phi)$. The potential is constructed such
that we have three distinct phases of inflation. The first phase is the usual
slow-roll inflation in which the inflaton slowly rolls on top of the
potential. We assume that this period is long enough, say about 20-30 e-folds
such that the observed CMB and large scale structure perturbations are
generated in this phase. The curvature perturbation is nearly scale invariant
with an amplitude fixed by the COBE normalization. The second stage is the USR
phase in which the potential becomes exactly flat such that the inflaton
velocity falls off exponentially and the curvature perturbation grows
exponentially. This is the phase which was employed extensively in the
literature to generate PBHs formation from the enhanced curvature
perturbations. Of course, one has to terminate the USR phase so the curvature
perturbation does not grow very large to invalidate the perturbation theory.
Therefore, the final stage of inflation is again a slow-roll phase following
the intermediate USR phase. Typically, one requires the duration of the USR
phase to be about a few e-folds to obtain a sizeable fraction of the dark
matter.

With the FLRW metric

$\displaystyle ds^{2}=-dt^{2}+a(t)^{2}d{\bf x}^{2}\,,$ (1)

the evolution of the system in the USR phase is given by

$\displaystyle\ddot{\phi}(t)+3H\dot{\phi}(t)=0\,,\quad\quad
3M_{P}^{2}H^{2}\simeq V_{0},$ (2)

in which $M_{P}$ is the reduced Planck mass, $H$ is the Hubble expansion rate
during inflation and $V_{0}$ is the value of the potential during the USR
phase. As $V_{0}$ is constant, $H$ is very nearly constant while
$\dot{\phi}\propto\frac{1}{a^{3}}$. To parameterize the slow-roll dynamics of
the inflaton field, let us define the two slow-roll parameters related to $H$
as follows,

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}^{2}}{2M_{P}^{2}H^{2}}\,,\quad\quad\eta\equiv\frac{\dot{\epsilon}}{H\epsilon}\,.$
(3)

During the usual slow-roll phase both $\epsilon$ and $\eta$ are nearly
constant and are small, say at the order $10^{-2}$. However, during the USR
phase, $\epsilon$ falls off like $a^{-6}$ while $\eta\simeq-6$. This is the
hallmark of the USR inflation [11]. Going to conformal time $d\tau=dt/a(t)$ in
which $aH\tau\simeq-1$, we can parameterize $\epsilon(\tau)$ as

$\displaystyle\epsilon(\tau)=\epsilon_{i}\big{(}\frac{\tau}{\tau_{i}}\big{)}^{6}\,,$
(4)

in which $\epsilon_{i}$ is the value of $\epsilon$ at the start of inflation
and prior to USP phase. The USR phase is during the period
$\tau_{i}<\tau<\tau_{e}$ so the value of $\epsilon$ at the end of USR phase is
$\epsilon_{e}=\epsilon_{i}\big{(}\frac{\tau_{e}}{\tau_{i}}\big{)}^{6}$.
Defining the number of e-fold as $dN=Hdt$, the duration of the USR phase is
denoted by $\Delta N\equiv N(\tau_{e})-N(\tau_{i})$ and
$\epsilon_{e}=e^{-6\Delta N}\epsilon_{i}$.

Following the assumption of [27], suppose the potential after the USR phase
are in the form to support a period of slow-roll inflation such that

$\displaystyle
V(\phi)=V(\phi_{e})+\sqrt{2\epsilon_{V}}V(\phi_{e})(\phi-\phi_{e})+\frac{\eta_{V}}{2}V(\phi_{e})(\phi-\phi_{e})^{2}+...\,.$
(5)

Here $2\epsilon_{V}\equiv\big{(}V^{\prime}(\phi_{e})/V(\phi_{e})\big{)}^{2}$
and $\eta_{V}\equiv V^{\prime\prime}(\phi_{e})/V(\phi_{e})$ are the slow-roll
parameters associated to the potential in the third slow-roll phase. From the
above expansion we see that the potential is continuous at $\phi=\phi_{e}$. If
one further requires that the derivative of the potential to be continuous as
well then $\epsilon_{V}=0$ and the transition becomes smooth. On the other
hand, if $\epsilon_{V}\neq 0$, then the derivative of the potential is not
continuous and there is a kink in the potential. Depending on the ratio
$\frac{\epsilon_{V}}{\eta_{V}}$ the transition can be either mild or sharp. As
we are interested in a sharp transition to handle the analysis analytically,
below we consider $\eta_{V}=0$. However, this is not a restrictive assumption
and most of our analysis will be carried out to the case where $\eta_{V}=0$ as
long as $\epsilon_{V}>|\eta_{V}|$ to guarantee a sharp transition.

The background field equation is given by [27]

$\displaystyle\frac{d^{2}\phi}{dN^{2}}+3\frac{d\phi}{dN}+3M_{P}\sqrt{2\epsilon_{V}}\simeq
0\,,\quad\quad 3M_{P}^{2}H^{2}\simeq V(\phi_{e})\,.$ (6)

The above equation can be solved easily. Without loss of generality let us
assume the time of transition corresponds to $N=0$. Imposing the continuity of
$\phi$ and $\frac{d\phi}{dN}$ at $N=0$, we obtain

$\displaystyle
M_{P}^{-1}\phi(N)=\frac{C1}{3}e^{-3N}+\frac{h}{6}\sqrt{2\epsilon_{V}}N+C_{2}\,,$
(7)

in which the constants of integration $C_{1}$ and $C_{2}$ are obtained to be

$\displaystyle C_{1}=\sqrt{2\epsilon_{e}}(1+\frac{h}{6})\,,\quad\quad
C_{2}=M_{P}^{-1}\phi_{e}-\frac{\sqrt{2\epsilon_{e}}}{3}(1+\frac{h}{6})\,.$ (8)

Following [27] we have defined the parameter $h$ as

$\displaystyle
h\equiv\frac{6\sqrt{2\epsilon_{V}}}{\dot{\phi}(t_{e})}=-6\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\,.$
(9)

Note that since we assume that $\phi$ is decreasing during its evolution, then
$\dot{\phi}<0$ so $h<0$. As we shall see, $h$ is the key parameter of our
model controlling the sharpness of the transition from the USR phase to the
third slow-roll phase.

Finally, the slow-roll parameter, as defined via Eq. (3) in the final slow-
roll phase $(N>0)$ are given by

$\displaystyle\epsilon(\tau)=\epsilon_{e}\Big{(}\frac{h}{6}-(1+\frac{h}{6})\big{(}\frac{\tau}{\tau_{e}}\big{)}^{3}\Big{)}^{2}$
(10)

and

$\displaystyle\eta(\tau)=-\frac{6(6+h)}{(6+h)-h\big{(}\frac{\tau_{e}}{\tau}\big{)}^{3}}\,.$
(11)

From the above formulas we see that towards the final stage of inflation,
$\tau\rightarrow 0$, $\epsilon\rightarrow\epsilon_{e}(\frac{h}{6})^{2}$ while
$\eta$ vanishes like $\tau^{3}$. While $\epsilon$ is by construction smooth at
the transition point but $\eta$ has a radical change at $\tau=\tau_{e}$. Right
before the transition (i.e. during the USR phase) $\eta=-6$ while right after
transition $\eta=-6-h$. Therefore, very near the transition we can approximate
$\eta$ as follows [27]

$\displaystyle\eta=-6-h\theta(\tau-\tau_{e})\quad\quad\tau_{e}^{-}<\tau<\tau_{e}^{+}\,.$
(12)

In particular, from the above approximation, we have

$\displaystyle\frac{d\eta}{d\tau}=-h\delta(\tau-\tau_{e})\,,\quad\quad\tau_{e}^{-}<\tau<\tau_{e}^{+}\,.$
(13)

We consider two cases of sharp transition: “natural” sharp transition in which
$\eta$ drops to zero right after transition corresponding to $h=-6$. In this
case $\epsilon$ after the transition is frozen to its value at the end of USR,
$\epsilon_{e}$. This is the limit which was studied in [25, 26]. The other
case is “extreme” sharp transition in which $|h|\gg 1$. In this case,
$\epsilon$ after the transition evolves to a larger value and at the end of
inflation (or when the evolution in final stage has reached its attractor
phase) we have the final value $\epsilon_{0}=\epsilon_{e}(\frac{h}{6})^{2}$.

Figure 1:  The Feynman diagrams for the one-loop correction in curvature
perturbation power spectrum
$\langle{\cal{R}}({\bf{p}}_{1}){\cal{R}}({\bf{p}}_{2})\rangle$. The left and
right panel represent the contribution of the cubic Hamiltonian and quartic
Hamiltonian respectively.

##  3 Cubic and Quartic Action in EFT Approach

We are interested in one-loop corrections in power spectrum
$\langle{\cal{R}}({\bf{p}}_{1}){\cal{R}}({\bf{p}}_{2})\rangle$ in which
${\cal{R}}$ is the comoving curvature perturbation. To calculate the one-loop
corrections one has to consider two different types of Feynman diagram as
illustrated in Fig. 1. The left panel corresponds to corrections induced from
a cubic self interaction Hamiltonian. To obtain the cubic Hamiltonian, one has
to calculate the action of curvature perturbation to third order. This was
performed in details in [6]. However, to calculate the one-loop correction
from the right panel of Fig. 1 one has to consider the quartic self
interaction Hamiltonian. To calculate the quartic Hamiltonian one has to
calculate the action to fourth order for the perturbations. This is a
difficult task and requires extra cares as there are contributions from the
first and higher derivatives of $\eta$ which undergo jumps at $\tau=\tau_{e}$.
It was argued briefly in [25] that the quartic self interaction does not
contribute as they are suppressed by higher powers of $\epsilon$ so the
authors of [25] concentrated only on the one-loop correction from the diagram
in the left panel of Fig. 1. Below, we shall show that this assumption is not
correct and indeed there are significant contributions from the quartic self-
interaction associated to the right panel of Fig. 1.

To calculate the cubic and quartic actions of curvature perturbations, we
employ the effective field theory (EFT) of inflation [30, 31]. This is a
particularly useful approach in our setup where we are only interested in
leading corrections and are not concerned with the sub-leading slow-roll
corrections in interaction Hamiltonians. It is well-known that EFT of
inflation is powerful in the decoupling limit where one neglects the
gravitational back-reactions when only the matter perturbations are the
dominant contributions. Here we follow the method employed in [33] which has
used the method of EFT of inflation to calculate the bispectrum for a general
$P(X,\phi)$-type of non-attractor setup. We comment that the EFT approach was
employed recently [34, 35] as well to calculate the loop corrections in
curvature perturbation power spectrum. However, similar to [25], the authors
of these papers also neglected the contributions associated to the Feynman
diagram in the right panel of Fig. 1.

In a near dS background with a time-dependent inflaton field $\phi(t)$, the
four-dimensional diffeomorphism invariance is spontaneously broken to a three-
dimensional spatial diffeomorphism invariance. In the unitary (or comoving)
gauge where the perturbations of inflaton are turned off one is allowed to
write down all terms in the action which are consistent with the remaining
three-dimensional diffeomorphsim invariance. In doing so, the background
inflation dynamics is controlled via the known Hubble expansion rate $H(t)$
and its derivative $\dot{H}(t)$. After writing the full action consistent with
the three dimensional diffeomorphsim invariance, one can restore the full
four-dimensional diffeomorphsim invariance by introducing a scalar field
fluctuations, $\pi(x^{\mu})$, the Goldstone boson associated with the breaking
of the time diffeomorphsim invariance. One big advantage of the EFT approach
is when one goes to the decoupling limit where the gravitational back-
reactions are neglected. This corresponds to neglecting the slow-roll
suppressed interactions in cubic and quartic actions while keeping only the
leading terms which are relevant for large non-Gaussianities. Indeed, these
interactions are those which induce large corrections in one loop integrals as
well.

The full action is $S_{\rm total}=S_{\rm matter}+S_{\rm EH}$ in which $S_{\rm
matter}$ is the matter part of the action while $S_{\rm EH}$ represents the
usual Einstein-Hilbert action. In the decoupling limit, we do not perturb
gravitational action and all leading perturbations come from the matter sector
(i.e. the inflaton perturbations after restoring the four-dimensional
diffeomorphsim invariance). Assuming a canonical scalar field with a sound
speed $c_{s}=1$, $S_{\rm matter}$ consistent with the FLRW inflationary
background is given by [30]

$\displaystyle S_{\rm
matter}=\int\\!d^{4}x\sqrt{-g}\Big{[}-M^{2}_{P}\dot{H}(t+\pi)\Big{(}\frac{1}{N^{2}}(1+\dot{\pi}-N^{i}\partial_{i}\pi)^{2}-\partial^{i}\pi\partial_{j}\pi\Big{)}$
$\displaystyle-M^{2}_{P}\left(3H^{2}(t+\pi)+\dot{H}(t+\pi)\right)\Big{]}\,,$
(14)

in which $N$ and $N^{i}$ are the lapse and shift function in standard ADM
formalism. We note that the form of the above simple action is fixed by the
background dynamics while all other possible terms are discarded in the
gravitational decoupling limit. These discarded interactions include terms
containing various combination of the extrinsic curvature $K_{ij}$ associated
with the embedding of the three-surface $\phi(t)=constant$. As discussed in
[30] the great advantage of EFT of inflation is in the decoupling limit where
one can set $N=1$ and $N^{i}=0$ in the action while neglecting the
perturbations from the $S_{\rm EH}$ action. In this limit the observational
predictions such as the tilt of scalar perturbation or the non-Gaussianity
parameter $f_{NL}$ are determined to leading order by the matter fluctuations.
This was confirmed in non-attractor setup in [33]. Finally, we comment that
there can be other relevant interactions in action (3) if one allows for a
non-trivial sound speed $c_{s}\neq 1$, as studied in [33].

Our goal is to expend the EFT action to fourth order in perturbations of
$\pi$. For this purpose, we build upon the analysis of [33] which calculated
the action to cubic order in order to calculate $f_{NL}$.

To obtain the EFT action to fourth order, we have to expand the quantities
$H(t+\pi)$ and $\dot{H}(t+\pi)$ to the corresponding orders as well. To
simplify the notation, let us define the order of differentiation via
$H^{(n)}\equiv\frac{d^{n}}{dt^{n}}H$. So for example,
$\dot{H}=H^{(1)}=-\epsilon H^{2}$ and
$\ddot{H}=H^{(2)}=-\dot{\epsilon}H^{2}-2\epsilon\dot{H}H$ and so on. It is
crucial to note that during the USR phase $\eta=-6$ so one can not neglect the
higher derivatives of $H$. In addition, as shown in Eq. (13), immediately near
the USR phase $\eta$ has a non-trivial behaviour so we should keep the
derivative of $\eta$ in mind as well. On the other hand, as $\epsilon$ is very
small, we only keep the first order contribution of $\epsilon$ in the
interaction Hamiltonians. This is also consistent with our decoupling limit
assumption where all gravitational back-reactions are discarded to leading
order of $\epsilon$.

With the above discussions in mind, the various derivatives of $H$ necessary
to construct the action to fourth order of $\pi$ are given by

$\displaystyle{H^{(2)}}$ $\displaystyle=$
$\displaystyle-\dot{\epsilon}H^{2}-2\epsilon\dot{H}H\,,$ (15) $\displaystyle=$
$\displaystyle-\eta\epsilon H^{3}+2\epsilon^{2}H^{3}=-\eta\epsilon
H^{3}+O(\epsilon^{2}),$

where we have neglected the higher orders of $\epsilon$ in the last
expression. Continuing using this expansion, we have

$\displaystyle{H^{(3)}}$ $\displaystyle=$ $\displaystyle-\eta^{2}\epsilon
H^{4}-\epsilon\frac{d\eta}{dt}H^{3}+O(\epsilon^{2})\,,$ (16)
$\displaystyle{H^{(4)}}$ $\displaystyle=$ $\displaystyle-\eta^{3}\epsilon
H^{5}-3\epsilon\eta\frac{d\eta}{dt}H^{4}-\epsilon\frac{d\eta}{dt}H^{3}+O(\epsilon^{2})\,,$
(17) $\displaystyle{H^{(5)}}$ $\displaystyle=$ $\displaystyle-\eta^{4}\epsilon
H^{6}-3\epsilon\eta^{2}\frac{d\eta}{dt}H^{5}-3(\frac{d\eta}{dt})^{2}H^{4}-4\epsilon\eta\frac{d^{2}\eta}{dt^{2}}H^{4}-\epsilon\frac{d^{3}\eta}{dt^{3}}H^{2}+O(\epsilon^{2})\,.$
(18)

It is important to note that since there is a jump in $\eta$ from the USR
phase to the third slow-roll limit, we can not neglect the derivates of
$\eta$. As it is evident from Eq. (13), $\dot{\eta}$ contains a delta function
which will contribute to the quartic interaction Hamiltonian as we shall see
below.

With the above expansion at hands, we can calculate the quadratic, cubic and
quartic actions. Starting with the quadratic action necessary to quantize the
free theory we have

$\displaystyle S_{2}=M_{P}^{3}\int dtd^{3}xa^{3}\epsilon
H^{2}\big{(}\dot{\pi}^{2}-(\partial_{i}\pi)^{2}\big{)}\,,$ (19)

while the cubic and quartic actions on the superhorizon limit where we neglect
the spatial derivative terms are given by

$\displaystyle S_{\pi^{3}}=-M_{P}^{3}\int
dtd^{3}xa^{3}\Big{[}\big{(}HH^{(3)}+3\dot{H}H^{(2)}+\frac{1}{3}H^{(4)}\big{)}\pi^{3}+H^{(3)}\pi^{2}\dot{\pi}+H^{(2)}\pi\dot{\pi}^{2}\Big{]}\,,$
(20)

and

$\displaystyle S_{\pi^{4}}=-M_{P}^{3}\int
dtd^{3}xa^{3}\Big{[}\big{(}\dot{H}H^{(3)}+\frac{3}{4}{\ddot{H}}^{2}+\frac{1}{4}HH^{(4)}+\frac{1}{12}H^{(5)}\big{)}\pi^{4}+\frac{H^{(4)}}{3}\pi^{3}\dot{\pi}+\frac{H^{(3)}}{2}\pi^{2}\dot{\pi}^{2}\Big{]}\,.$
(21)

The above actions look problematic since in $H^{(5)}$ and $H^{(4)}$ we have
$\ddot{\eta}$ and $\dddot{\eta}$ which contain the first and the second
derivatives of $\delta(\tau-\tau_{e})$ respectively. However, things become
simplified by noting that various terms in $S_{\pi^{3}}$ and $S_{\pi^{4}}$
group themselves into total derivative terms which do not contribute into the
Hamiltonian analysis. More specifically, we have

$\displaystyle
a^{3}\Big{(}H^{(3)}\pi^{2}\dot{\pi}+\big{(}HH^{(3)}+\frac{1}{3}H^{(4)}\big{)}\pi^{3}\Big{)}=\frac{1}{3}\frac{d}{dt}{\Big{(}a^{3}H^{(3)}\pi^{3}\Big{)}}\,.$
(22)

Discarding this total derivative and also neglecting the term containing
$\dot{H}H^{(2)}\sim O(\epsilon^{2})$ and going to conformal time the cubic
action simplifies to

$\displaystyle S_{\pi^{3}}=M_{P}^{2}H^{3}\int d\tau d^{3}x\,\eta\epsilon
a^{2}\,\pi\pi^{\prime 2}\,.$ (23)

Correspondingly, the cubic interaction Hamiltonian is obtained to be

$\displaystyle{\bf H}_{3}=-M_{P}^{2}H^{3}\eta\epsilon a^{2}\,\pi\pi^{\prime
2}\,.$ (24)

Here and below, a prime denotes the derivative with respect to the conformal
time.

Following a similar strategy for the quartic action, we note that

$\displaystyle
a^{3}\big{(}\frac{1}{4}HH^{(4)}+\frac{1}{12}H^{(5)}\big{)}\pi^{4}+\frac{a^{3}}{3}H^{(4)}\pi^{3}\dot{\pi}=\frac{1}{12}\frac{d}{dt}{\Big{(}a^{3}H^{(4)}\pi^{4}\Big{)}}\,.$
(25)

Neglecting the contribution of this total time derivative in the quartic
action and discarding the higher order term $\ddot{H}^{2}\sim
O(\epsilon^{2})$, the quartic action to leading order of $\epsilon$ is
obtained to be

$\displaystyle S_{\pi^{4}}=\frac{M_{P}^{2}}{2}\int d\tau
d^{3}x\,\epsilon\Big{(}\eta^{2}a^{2}H^{4}+a\frac{d\eta}{d\tau}H^{3}\Big{)}\,\pi^{2}\pi^{\prime
2}\,.$ (26)

Note the important contribution from the term $\frac{d\eta}{d\tau}$ which
causes a delta contribution in the interaction Hamiltonian as $\eta$ undergoes
a jump from the USR phase to the third slow-roll phase as seen in Eq. (13).

To calculate the quartic Hamiltonian, we have to be careful as the time
derivative interaction $\pi^{\prime}\pi^{2}$ in ${\bf H}_{3}$ induces a new
term in the quartic Hamiltonian [37, 38]. More specifically, the quartic
Hamiltonian receives a correction of the value
$+M_{P}^{2}H^{4}\eta^{2}\epsilon a^{2}\,\pi^{2}\pi^{\prime 2}$ from the cubic
action, so the total quartic Hamiltonian is obtained to be

$\displaystyle{\bf H}_{4}=\frac{M_{P}^{2}}{2}\Big{(}H^{4}\eta^{2}\epsilon
a^{2}-\eta^{\prime}M_{P}^{2}H^{3}\eta\epsilon a^{2}\Big{)}\pi^{2}\pi^{\prime
2}\,.$ (27)

Equipped with the above interaction Hamiltonians, we can proceed to calculate
the one-loop corrections in curvature perturbation power spectrum.

However, note that we are interested in curvature perturbation on comoving
surface ${\cal{R}}$ while the above Hamiltonians are written in terms of the
variable $\pi$. There are additional contributions from the non-linear
relations between $\pi$ and ${\cal{R}}$. To cubic order in $\pi$, the relation
between ${\cal{R}}$ and $\pi$ is given by [39, 36]

$\displaystyle{\cal{R}}=-H\pi+\Big{(}H\pi\dot{\pi}+\frac{\dot{H}}{2}\pi^{2}\Big{)}+\Big{(}-H\pi\dot{\pi}^{2}-\frac{H}{2}\ddot{\pi}\pi^{2}-\dot{H}\dot{\pi}\pi^{2}-\frac{\ddot{H}}{6}\pi^{3}\Big{)}\,.$
(28)

However, we calculate the two point correlation functions at the end of
inflation $\tau=\tau_{0}\rightarrow 0$ where it is assume that the system is
in the slow-roll regime and the perturbations are frozen on superhorizon
scales, $\dot{\pi}=\ddot{\pi}=0$. Fortunately, in this limit all the non-
linear corrections in ${\cal{R}}$ in Eq. (28) are suppressed and we end up
with the simple linear relation between ${\cal{R}}$ and $\pi$:

$\displaystyle{\cal{R}}=-H\pi\,,\quad\quad\quad(\tau\rightarrow\tau_{0}).$
(29)

Because of this linear relation between $\pi$ and ${\cal{R}}$, we can
interchangeably use $\pi$ and ${\cal{R}}$ in the following in-in integrals
where the mode functions in the integrals are the solutions of the free theory
(in the interaction picture).

##  4 The analysis of Bispectrum

As a prelude to calculate the one-loop corrections in power spectrum, in this
section we perform the bispectrum analysis to calculate $f_{NL}$ in the USR
setup. This is a two-step model in which the first phase is USR inflation
where the large CMB scales are generated followed by a second phase of slow-
roll inflation. So this setup should not be confused with our main three-
phases setup which was reviewed in section 2 which will be employed for the
loop correction in the next section. While the current analysis is new and
interesting by itself, the reader who is only interested in one-loop
corrections can skip directly to the next section.

The analysis of bispectrum for the simple USR setup was performed originally
in [10], see also [15, 20]. This analysis was extended to a general non-
attractor setup with $c_{s}\neq 1$ in [33] using the EFT approach as in
current analysis. However, the analysis in [33] was for the case of extreme
sharp transition $h\rightarrow-\infty$. Here, we extend the analysis of [33]
to the general case where the transition from USR phase to SR phase is
controlled by the parameter $h\leq-6$. We confirm that our result for $f_{NL}$
is in agreement with the results of [27] for a general sharp transition. This
can be viewed as a non-trivial confirmation of the validity of our EFT
approach to calculate the one-loop corrections in power spectrum in the follow
up section.

Going to Fourier space, the comoving curvature perturbations ${\cal{R}}$ is
written as

$\displaystyle{\cal{R}}({\bf x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i{\bf
k}\cdot{\bf x}}\hat{\cal{R}}_{\bf k}(t)\,,$ (30)

in which the operator $\hat{\cal{R}}_{\bf k}(t)$ is written in terms of the
creation and annihilation operators as $\hat{\cal{R}}_{\bf
k}(t)={\cal{R}}_{k}(t)a_{\bf k}+{\cal{R}}^{*}_{k}(t)a_{-\bf k}^{\dagger}$. The
creation and annihilation operators satisfy the usual commutation relations
$[a_{\bf k},a^{\dagger}_{-\bf k^{\prime}}]=(2\pi)^{3}\delta({\bf k}+{\bf
k^{\prime}})$.

The mode function ${\cal{R}}_{k}$ in the USR phase with the Bunch-Davies
initial condition is given by [10]

$\displaystyle{\cal{R}}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon_{i}k^{3}}}\big{(}\frac{\tau_{i}}{\tau}\big{)}^{3}(1+ik\tau)e^{-ik\tau}\,,$
(31)

in which $\epsilon_{i}$ is the value of the slow-roll parameter at the start
of USR phase $\tau=\tau_{i}$. We note that on superhorizon scales where
$k\tau\rightarrow 0$, ${\cal{R}}_{k}$ grows like
${\cal{R}}_{k}\propto\tau^{-3}$ which is the hallmark of the USR phase. After
the transition to the slow-roll phase at $\tau=\tau_{e}$, the mode function
during the final slow-roll phase ($\tau>\tau_{e}$) is given by

$\displaystyle{\cal{R}}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon(\tau)k^{3}}}\Big{[}\alpha_{k}(1+ik\tau)e^{-ik\tau}+\beta_{k}(1-ik\tau)e^{ik\tau}\Big{]}\,,$
(32)

with $\epsilon(\tau)$ given by Eq. (10) while

$\displaystyle\alpha_{k}=1+\frac{ih}{4k^{3}\tau_{e}^{3}}(1+k^{2}\tau_{e}^{2})\,,\quad\quad\beta_{k}=-\frac{ih}{4k^{3}\tau_{e}^{3}}{(1+ik\tau_{e})^{2}}e^{-2ik\tau_{e}}\,.$
(33)

It is instructive to look at the curvature perturbation power spectrum at the
end of inflation $\tau_{0}\rightarrow 0$. From the above expression for
${\cal{R}}(\tau)$ we obtain

$\displaystyle
P_{\cal{R}}(k,\tau_{0})=\big{|}{\cal{R}}_{k}(\tau_{0})\big{|}^{2}=\frac{(h-6)^{2}}{h^{2}}\Big{(}\frac{H^{2}}{4k^{3}M_{P}^{2}\epsilon_{e}}\Big{)}=\Big{(}1+\frac{\epsilon(\tau_{0})}{\epsilon(\tau_{e})}\Big{)}^{2}\Big{(}\frac{H^{2}}{4k^{3}M_{P}^{2}\epsilon_{e}}\Big{)}\,.$
(34)

In the sharp transition with $h=-6$, we obtain
$P_{R}(\tau_{0})=\Big{(}\frac{H^{2}}{k^{3}M_{P}^{2}\epsilon_{e}}\Big{)}=4P_{\cal{R}}(\tau_{e})$.
Curiously, the power at the end of inflation is enhanced by a factor 4
compared to the power at the end of USR phase. This is because the curvature
perturbation is not frozen immediately after the USR phase and it keeps
evolving till reaching the final attractor phase. On the other hand, for the
extreme sharp transition with $h\rightarrow-\infty$ where the mode function
freezes shortly after the USR phase the power at the end of inflation is
approximately the same as the power at the end of USR phase as expected. This
difference in the behaviours of the mode function for $h=-6$ and $|h|\gg 1$
plays important roles in the final value of $f_{NL}$ measured at the end of
inflation.

As the relation between $\pi$ and ${\cal{R}}$ is linear when measured at the
end of inflation (${\cal{R}}=-H\pi)$, the three-point correlation calculated
at the end of inflation is given by

$\displaystyle\big{\langle}{\cal{R}}_{\bf p_{1}}(\tau_{0}){\cal{R}}_{\bf
p_{2}}(\tau_{0}){\cal{R}}_{\bf p_{3}}(\tau_{0})\big{\rangle}$ $\displaystyle=$
$\displaystyle-H^{3}\big{\langle}\pi_{\bf p_{1}}(\tau_{0})\pi_{\bf
p_{2}}(\tau_{0})\pi_{\bf p_{3}}(\tau_{0})\big{\rangle}$ (35) $\displaystyle=$
$\displaystyle 2H^{3}\mathrm{Im}\int d\tau\big{\langle}{\bf
H}_{3}(\tau)\pi_{\bf p_{1}}(\tau_{0})\pi_{\bf p_{2}}(\tau_{0})\pi_{\bf
p_{3}}(\tau_{0})\big{\rangle}\,,$

yielding to

$\displaystyle\big{\langle}{\cal{R}}_{\bf p_{1}}{\cal{R}}_{\bf
p_{2}}{\cal{R}}_{\bf
p_{3}}\big{\rangle}|_{\tau_{0}}=-4M_{P}^{2}\,\mathrm{Im}\int
d\tau\eta(\tau)\epsilon
a^{2}\big{\langle}{\cal{R}}^{*}_{p_{1}}(\tau_{0}){\cal{R}}^{*}_{p_{2}}(\tau_{0}){\cal{R}}^{*}_{p_{3}}(\tau_{0}){\cal{R}}_{p_{1}}(\tau){\cal{R}}^{\prime}_{p_{2}}(\tau){\cal{R}}^{\prime}_{p_{3}}(\tau)\big{\rangle}\,.$
(36)

As the function $\eta(\tau)$ evolves from the USR phase to the final slow-roll
phase we divide the integral above into two regions $\tau_{i}<\tau<\tau_{e}$
with $\eta=-6$ and $\tau_{e}<\tau<\tau_{0}$ with $\eta(\tau)$ given by Eq.
(11). For the first contribution, we obtain

$\displaystyle\big{\langle}{\cal{R}}_{\bf p_{1}}(\tau_{0}){\cal{R}}_{\bf
p_{2}}(\tau_{0}){\cal{R}}_{\bf
p_{3}}(\tau_{0})\big{\rangle}_{(1)}=-\frac{3(h-6)^{2}(h+12)}{16h^{3}}\Big{(}\frac{H^{2}}{M_{P}^{2}\epsilon_{e}}\Big{)}\Big{(}\frac{1}{p_{1}^{3}p_{2}^{3}}+\frac{1}{p_{1}^{3}p_{3}^{3}}+\frac{1}{p_{2}^{3}p_{3}^{3}}\Big{)}\,.$
(37)

On the other hand, from the second contribution for the period
$\tau_{e}<\tau<\tau_{0}$ we obtain

$\displaystyle\big{\langle}{\cal{R}}_{\bf p_{1}}(\tau_{0}){\cal{R}}_{\bf
p_{2}}(\tau_{0}){\cal{R}}_{\bf
p_{3}}(\tau_{0})\big{\rangle}_{(2)}=\frac{3(h-6)^{2}(h+6)}{8h^{3}}\Big{(}\frac{H^{2}}{M_{P}^{2}\epsilon_{e}}\Big{)}\Big{(}\frac{1}{p_{1}^{3}p_{2}^{3}}+\frac{1}{p_{1}^{3}p_{3}^{3}}+\frac{1}{p_{2}^{3}p_{3}^{3}}\Big{)}\,.$
(38)

The non-Gaussianity parameter $f_{NL}$ is defined via

$\displaystyle\big{\langle}{\cal{R}}_{\bf p_{1}}(\tau_{0}){\cal{R}}_{\bf
p_{2}}(\tau_{0}){\cal{R}}_{\bf
p_{3}}(\tau_{0})\big{\rangle}\equiv\frac{5}{6}f_{NL}\Big{(}P_{\cal{R}}(p_{1},\tau_{0})P_{\cal{R}}(p_{2},\tau_{0})+2\,\mathrm{cyclic\,\,permutations}\Big{)}\,.$
(39)

Using the value of the power spectrum given by Eq. (34) and combining the two
contributions Eqs. (37) and (38) we obtain

$\displaystyle f_{NL}=\frac{5h^{2}}{2(h-6)^{2}}\,.$ (40)

The above result matches exactly with the result obtained111As mentioned
earlier, we work in the simplified limit where $\eta_{V}=0$ in the slow-roll
phase while [27] keeps this slow-roll parameter as well. in [27]. In the limit
of extreme sharp transition $h\rightarrow-\infty$, we obtain the original
result in [10] with $f_{NL}=\frac{5}{2}$. On the other hand, for a sharp
transition with $h=-6$ in which $\eta$ jumps to zero at the end of USR with
$\epsilon$ in the second phase held fixed to its value at the end of USR phase
($\epsilon(\tau_{0})=\epsilon(\tau_{e})$), we obtain $f_{NL}=\frac{5}{8}$.

A natural conclusion from the above analysis is that the value of $f_{NL}$
measured at the time of end of inflation depends crucially on the mechanism of
transition. For the case of extreme sharp transition where the mode function
freezes to its final slow-roll value immediately after the USR phase, one can
use the value of $f_{NL}$ calculated right before or right after the USR phase
as employed in [10] and [33]. However, if the mode function keeps evolving
after the USR phase (including the “ natural ” case with $h=-6$), it is
crucial to make the measurement at the end of inflation. As demonstrated in
[27], the maximum value of $f_{NL}$ obtained in this setup is
$f_{NL}=\frac{5}{2}$.

From the above exercise we conclude that our EFT approach is trusted to
calculated the loop corrections in power spectrum since we are interested in
leading corrections from the modes which are growing in the USR phase while
discarding the subleading slow-roll corrections. This is the limit where the
EFT formalism is guaranteed to be trusted.

##  5 One-Loop Correction in Power Spectrum

After a warmup exercise to calculate the bispectrum using our EFT formalism,
we are in a position to calculate the one-loop corrections in power spectrum.
As mentioned earlier, for a consistent treatment we have to include the
corrections from both the cubic Hamiltonian ${\bf H}_{3}$ and the quartic
Hamiltonian ${\bf H}_{4}$. Somehow, it was concluded in [25] that there is no
contribution from the quartic self interaction of curvature perturbation and
the quartic action is suppressed by higher powers of $\epsilon$ [39]. While
there are non-linear relations between ${\cal{R}}$ (or notation $\zeta$ used
in [25] ) but our specific calculation shows that ${\bf H}_{4}$ is actually
non-zero and has the same order of $\epsilon$ as ${\bf H}_{3}$. Looking at the
structure of ${\bf H}_{4}$ given in Eq. (27) we see that it contains two
terms. The first term make contribution in the “ bulk” either in the region
$\tau_{i}<\tau<\tau_{e}$ or $\tau_{e}<\tau<\tau_{0}$ while the second term is
a localized delta term $\delta(\tau-\tau_{e})$ which makes a contribution
right at the transition time $\tau_{e}$ as given in Eq. (13).

We present the analysis for a general value of $h$ for the sharp transitions.
However, the analysis of [25] is for the case $h=-6$ when the value of the
slow-roll parameter in the third slow-roll phase is fixed to its value at the
end of USR phase, $\epsilon(\tau_{0})=\epsilon(\tau_{e})$.

To calculate the loop corrections, we employ the standard in-in formalism [40]
in which the expectation value of the operator $\hat{O}[\pi]$ at the end of
inflation $\tau_{0}$ is given by the perturbative Dyson series,

$\displaystyle\langle\hat{O}[\pi(\tau_{0})]\rangle=\Big{\langle}\Big{[}\bar{\mathrm{T}}\exp\Big{(}i\int_{-\infty}^{\tau_{0}}d\tau^{\prime}H_{in}(\tau^{\prime})\Big{)}\Big{]}\,\hat{O}[\pi(\tau_{0})]\,\Big{[}\mathrm{T}\exp\Big{(}-i\int_{-\infty}^{\tau_{0}}d\tau^{\prime}H_{in}(\tau^{\prime})\Big{)}\Big{]}\Big{\rangle}$
(41)

in which $\mathrm{T}$ and $\bar{\mathrm{T}}$ represents the time ordering and
anti-time ordering respectively while $H_{in}(t)$ represents the interaction
Hamiltonian which in our case is $H_{in}(\tau)={\bf H}_{3}+{\bf H}_{4}$.

To calculate the in-in analysis we need the mode function for $\pi$ (or
${\cal{R}})$ during the USR and the follow up slow-roll phase. Imposing the
Bunch-Davies initial condition during the first slow-roll stage (i.e. for
$\tau<\tau_{i}$) we have

$\displaystyle{\cal{R}}^{(1)}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon_{i}k^{3}}}(1+ik\tau)e^{-ik\tau}\,,\quad\quad(\tau<\tau_{i})\,.$
(42)

in which $\epsilon_{i}$ is the value of the slow-roll parameter at the start
of inflation when the CMB scale mode has left the horizon. The superscript (1)
indicate the first phase. During the USR phase, the mode function is given
formally by an expansion similar to Eq. (32):

$\displaystyle{\cal{R}}^{(2)}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon_{i}k^{3}}}\big{(}\frac{\tau_{i}}{\tau}\big{)}^{3}\Big{[}\alpha^{(2)}_{k}(1+ik\tau)e^{-ik\tau}+\beta^{(2)}_{k}(1-ik\tau)e^{ik\tau}\Big{]}\,,$
(43)

with the coefficients $\alpha^{(2)}_{k}$ and $\beta^{(2)}_{k}$ given by

$\displaystyle\alpha^{(2)}_{k}=1+\frac{3i}{2k^{3}\tau_{i}^{3}}(1+k^{2}\tau_{i}^{2})\,\quad\quad\beta^{(2)}_{k}=-\frac{3i}{2k^{3}\tau_{i}^{3}}{(1+ik\tau_{i})^{2}}e^{-2ik\tau_{i}}\,.$
(44)

Finally, imposing the continuity of the mode function and its time derivative
at $\tau_{e}$, the mode function in the final slow-roll phase, denoted by the
superscript (3), is obtained to be

$\displaystyle{\cal{R}}^{(3)}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon(\tau)k^{3}}}\Big{[}\alpha^{(3)}_{k}(1+ik\tau)e^{-ik\tau}+\beta^{(3)}_{k}(1-ik\tau)e^{ik\tau}\Big{]}\,,$
(45)

with $\epsilon(\tau)$ given by Eq. (10) while the coefficients
$\alpha^{(3)}_{k}$ and $\beta^{(3)}_{k}$ are given by,

$\displaystyle\alpha^{(3)}_{k}=\frac{1}{8k^{6}\tau_{i}^{3}\tau_{e}^{3}}\Big{[}3h(1-ik\tau_{e})^{2}(1+ik\tau_{i})^{2}e^{2ik(\tau_{e}-\tau_{i})}-i(2k^{3}\tau_{i}^{3}+3ik^{2}\tau_{i}^{2}+3i)(4ik^{3}\tau_{e}^{3}-hk^{2}\tau_{e}^{2}-h)\Big{]}$

and

$\displaystyle\beta^{(3)}_{k}=\frac{1}{8k^{6}\tau_{i}^{3}\tau_{e}^{3}}\Big{[}3(1+ik\tau_{i})^{2}(h+hk^{2}\tau_{e}^{2}+4ik^{3}\tau_{e}^{3})e^{-2ik\tau_{i}}+ih(1+ik\tau_{e})^{2}(3i+3ik^{2}\tau_{i}^{2}+2k^{3}\tau_{i}^{3})e^{-2ik\tau_{e}}\Big{]}$

Armed with the above mode functions, we are ready to calculate the one-loop
corrections in power spectrum. To clarify the notation, the momentum for the
CMB modes are denoted by ${\bf p}_{1}$ and ${\bf p}_{2}$ while the momentum
for the small scale modes running in the loop are denoted by ${\bf q}$. Since
we are interested only to the modes which leave the horizon during the USR
phase, we cut the loop integrals in the intervals $q_{i}\leq q<q_{e}$ in which
$q_{i}=-\frac{1}{\tau_{i}}$ and $q_{e}=-\frac{1}{\tau_{e}}$. The number of
e-folds of the USR phase $\Delta N=N(\tau_{e})-N(\tau_{i})$ is related to
$q_{i}$ and $q_{e}$ via

$\displaystyle e^{-\Delta N}=\frac{\tau_{e}}{\tau_{i}}=\frac{q_{i}}{q_{e}}\,.$
(46)

For a rapid growth of the mode function during USR phase we typically require
$\Delta N>1$.

###  5.1 Corrections from Quartic Self Interaction

We start with the loop corrections from the quartic self interaction Eq. (27)
as the in-in integrals are easier to handle.

The one-loop correction for the CMB-scale mode ${\bf p}_{1}$ and
${\bf{p}}_{2}$ induced from ${\bf H}_{4}$ is given by

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle=H^{2}\langle\pi_{{\bf{p}}_{1}}(\tau_{0})\pi_{{\bf{p}}_{2}}(\tau_{0})\rangle=-2H^{2}\mathrm{Im}\int_{-\infty}^{\tau_{0}}d\tau\big{\langle}{\bf
H}_{4}(\tau)\pi_{\bf p_{1}}(\tau_{0})\pi_{\bf
p_{2}}(\tau_{0})\big{\rangle}\,.$ (47)

As the ${\bf H}_{4}$ term has contributions from the bulk
$\tau_{i}<\tau<\tau_{0}$ and from the source term $\delta(\tau-\tau_{e})$, we
have two different types of in-in integrals: in-in performed in the bulk and
in-in performed at $\tau=\tau_{e}$. Let us first start with the bulk
contribution.

After performing all the contractions, we end up with integral over the
momentum ${\bf{q}}$ running in the loop yielding to two different
contributions, which we denote by terms A and B as follows

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle|_{\mathrm{bulk}}\equiv(2\pi)^{3}\delta^{3}({\bf{p}}_{1}+{\bf{p}}_{2})(A+B)\,,$
(48)

with

$\displaystyle A=-2M_{P}^{2}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int
d\tau(\eta^{2}-\frac{\eta^{\prime}}{aH})\epsilon(\tau)a(\tau)^{2}|{\cal{R}}^{\prime}_{q}(\tau)|^{2}\mathrm{Im}\Big{[}{\cal{R}}_{\bf{p}}^{*}(\tau_{0})^{2}{\cal{R}}_{\bf{p}}(\tau)^{2}\Big{]}\,,$
(49)

and

$\displaystyle B=-8M_{P}^{2}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int
d\tau(\eta^{2}-\frac{\eta^{\prime}}{aH})\epsilon(\tau)a(\tau)^{2}\mathrm{Im}\Big{[}{\cal{R}}_{\bf{p}}^{*}(\tau_{0})^{2}{\cal{R}}_{\bf{p}}(\tau){\cal{R}}^{\prime}_{\bf{p}}(\tau){\cal{R}}_{\bf{q}}(\tau){\cal{R}}^{\prime}_{\bf{q}}(\tau)^{*}\Big{]}\,.$
(50)

Note that as $p\ll q$, we have approximated ${\bf{p}}-{\bf{q}}\simeq-{\bf{q}}$
in the above integrals to leading orders in $p$.

There are two different contributions in the bulk for both A and B terms:
during the USR phase with $\tau_{i}<\tau<\tau_{e}$ and after the USR phase
$\tau_{e}<\tau<\tau_{0}$. During the first interval, $\eta=-6$ and
$\eta^{\prime}=0$ and we calculate the corresponding in-in integrals in
details below. On the other hand, during the second bulk phase, we
approximately have

$\displaystyle\eta^{2}-\frac{\eta^{\prime}}{aH}\simeq\frac{18(6+h)\tau^{3}}{h\tau_{e}^{3}}+O(\tau^{5})\,.$
(51)

Therefore, we see that this combination falls off rapidly after the USR phase.
Furthermore, as the system reaches the attractor phase rapidly after the
transition, the mode function ${\cal{R}}_{\bf{q}}(t)$ approaches its usual
slow-roll form and we will have additional suppressions from the time
derivatives of the mode function. Correspondingly, one expects the
contribution from the final slow-roll region $\tau_{e}<\tau<\tau_{0}$ to be
much smaller than those coming from the USR phase. We have checked explicitly
that this is indeed the case. Therefore, in the rest of the analysis we
consider the contribution from the first bulk region during the USR phase.

During the USR phase, the superhorizon modes grow like $\tau^{-3}$ and

$\displaystyle|{\cal{R}}^{\prime}_{q}(\tau)|^{2}\simeq\frac{9}{\tau^{2}}|{\cal{R}}_{q}(\tau)|^{2}\simeq\frac{9}{\tau^{2}}\big{(}\frac{\tau_{e}}{\tau}\big{)}^{2}|{\cal{R}}_{q}(\tau_{e})|^{2}\,.$
(52)

On the other hand,

$\displaystyle\mathrm{Im}\Big{(}{\cal{R}}_{\bf{p}}^{*}(\tau_{0})^{2}{\cal{R}}_{\bf{p}}(\tau)^{2})\Big{)}=\frac{1}{24}\frac{H^{2}}{M_{P}^{4}\epsilon_{i}^{2}p^{3}}\frac{\tau_{e}^{3}}{r^{6}\tau^{3}}\big{(}\frac{6-h}{h}\tau^{3}+\tau_{e}^{3}\big{)}\,.$
(53)

Plugging the expressions (52) and (53) in Eq. (49) with $\eta=-6$ we end up
with a time integral of the following form for the A-term:

$\displaystyle\int_{\tau_{i}}^{\tau_{e}}d\tau\Big{(}\frac{6-h}{h}\tau^{-4}+\tau_{e}^{3}\tau^{-7}\Big{)}=\big{(}\frac{1}{6}-\frac{2}{h}\big{)}\tau_{e}^{-3}\,.$
(54)

Plugging all numerical factors together, we finally obtain

$\displaystyle
A=27\big{(}\frac{2}{h}-\frac{1}{6}\big{)}\frac{H^{2}}{M_{P}^{2}\epsilon_{i}p^{3}}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}|{\cal{R}}_{q}(\tau_{e})|^{2}\,.$
(55)

To perform the above loop integral we restrict ourselves to modes which become
superhorizon during the USR phase with $q_{i}<q<q_{e}$. Also, the power
spectrum at the end of USR phase is given by
$P_{\cal{R}}(\tau_{e})=|{\cal{R}}_{q}(\tau_{e})|^{2}=\frac{H^{2}}{4M_{P}^{2}\epsilon_{i}q^{3}}e^{6\Delta
N}$. Performing the remaining loop integral over $q$ yields the factor
$\frac{1}{2\pi^{2}}\ln(\frac{q_{e}}{q_{i}})=\frac{\Delta N}{2\pi^{2}}$.
Combining all results together, we have

$\displaystyle A=\Big{(}\frac{27}{2h}-\frac{9}{8}\Big{)}\big{(}{\Delta
N}{e^{6\Delta
N}}\big{)}\frac{H^{4}}{2\pi^{2}M_{P}^{4}\epsilon_{i}^{2}p^{3}}\,.$ (56)

To calculate the B-term, we note that to leading order in $p$,

$\displaystyle{\cal{R}}_{\bf{p}}^{*}(\tau_{0})^{2}{\cal{R}}_{\bf{p}}(\tau){\cal{R}}^{\prime}_{\bf{p}}(\tau)=-\frac{iH^{4}}{16M_{P}^{4}\epsilon_{i}^{2}p^{3}}\,\frac{\tau_{i}^{6}}{\tau^{4}}\,.$
(57)

As the above combination is imaginary by itself, the remaining factor in the
B-term simplifies to

$\displaystyle{\cal{R}}_{\bf{q}}(\tau){\cal{R}}^{\prime}_{\bf{q}}(\tau)^{*}=-\frac{3}{\tau}|{\cal{R}}_{q}(\tau)|^{2}=-\frac{3}{\tau}\big{(}\frac{\tau_{e}}{\tau}\big{)}^{6}|{\cal{R}}_{q}(\tau_{e})|^{2}\,.$
(58)

Curiously, from the above expressions we realize that there is no explicit
dependence on the parameter $h$ in the B-term. Performing a simple time
integral and the momentum integral over $q$ as in the case of A-term, we
finally obtain

$\displaystyle B=\frac{9}{4}\big{(}{\Delta N}{e^{6\Delta
N}}\big{)}\frac{H^{4}}{2\pi^{2}M_{P}^{4}\epsilon_{i}^{2}p^{3}}\,.$ (59)

After calculating the bulk contributions, now we calculate the localized terms
induced by $\eta^{\prime}$ at $\tau=\tau_{e}$. Again, we have two
contributions from the source terms denoted by terms C and D. The term C is
similar to term A and the term D is similar to term B but with the
understanding that $\eta^{\prime}=-h\delta(\tau-\tau_{e})$. More specifically,

$\displaystyle
C=-2hM_{P}^{2}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\epsilon(\tau_{e})a(\tau_{e})|{\cal{R}}^{\prime}_{q}(\tau_{e})|^{2}\mathrm{Im}\Big{[}{\cal{R}}_{\bf{p}}^{*}(0)^{2}{\cal{R}}_{\bf{p}}(\tau_{e})^{2}\Big{]}\,,$
(60)

and

$\displaystyle
D=-8hM_{P}^{2}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\epsilon(\tau_{e})a(\tau_{e})\mathrm{Im}\Big{[}{\cal{R}}_{\bf{p}}^{*}(\tau_{0})^{2}{\cal{R}}_{\bf{p}}(\tau_{e}){\cal{R}}^{\prime}_{\bf{p}}(\tau_{e}){\cal{R}}_{\bf{q}}(\tau_{e}){\cal{R}}^{\prime}_{\bf{q}}(\tau_{e})^{*}\Big{]}\,.$
(61)

Performing the loop integrals as in the bulk cases and, using Eqs. (53) and
(57) now calculated at $\tau=\tau_{e}$, we obtain

$\displaystyle C=\frac{9}{8}\big{(}{\Delta N}{e^{6\Delta
N}}\big{)}\frac{H^{4}}{2\pi^{2}M_{P}^{4}\epsilon_{i}^{2}p^{3}}\,,$ (62)

and

$\displaystyle D=\frac{3h}{8}\big{(}{\Delta N}{e^{6\Delta
N}}\big{)}\frac{H^{4}}{2\pi^{2}M_{P}^{4}\epsilon_{i}^{2}p^{3}}\,.$ (63)

Curiously, we see that the D-term has an explicit factor of $h$ while this is
not the case for the C-term.

Summing the terms $A+B+C+D$, the total contribution from the quartic
Hamiltonian corresponding to the right panel of Fig. 1 is obtained to be

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{4}}=\frac{3}{8h}\big{(}h^{2}+6h+36\big{)}\big{(}{\Delta N}{e^{6\Delta
N}}\big{)}\frac{H^{4}}{2\pi^{2}M_{P}^{4}\epsilon_{i}^{2}p^{3}}\,(2\pi)^{3}\delta^{3}({\bf{p}}_{1}+{\bf{p}}_{2})\,.$
(64)

We notice that the one-loop correction is a non-linear function of the
parameter $h$ and grows linearly for $|h|\gg 1$.

###  5.2 Corrections from Cubic Self Interaction

Now we calculate the loop correction from the cubic interaction ${\bf H}_{3}$
corresponding to the right panel of Fig. 1. There are noticeable differences
compared to the quartic case. First, one simplification is that there is no
source term as there is no localized term from $\eta^{\prime}$ in ${\bf
H}_{3}$. However, a major complication is that the integrals in the bulk
become nested (i.e. double integrals). We present the detail technicalities of
the nested integrals in Appendix A and here outline the main steps leading to
the final result.

Expanding the Dyson series to second order in ${\bf H}_{3}$ we have

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{3}}=\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(2,0)}+\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}+\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(0,2)}$
(65)

in which

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(2,0)}$
$\displaystyle=$
$\displaystyle-\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{\langle}{\bf
H}_{3}(\tau_{1}){\bf
H}_{3}(\tau_{2}){\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}$
(66) $\displaystyle=$
$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle^{\dagger}_{(0,2)}\,,$

and

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}=-\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{\langle}{\bf
H}_{3}(\tau_{1}){\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0}){\bf
H}_{3}(\tau_{2})\big{\rangle}\,.$ (67)

As in the case of quartic self interaction, the main contributions in the in-
in integrals come from the USR phase and the contributions from the final
slow-roll phase are negligible. As explained before, this is because the mode
functions approach their slow-roll values and their derivatives falls off
exponentially during the slow-roll phase.

Combining all contributions (see Appendix A for further details) we obtain

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{3}}=-8M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}\,\mathrm{Im}\Big{[}X^{*}_{1}(\tau_{1})\big{(}\delta(\tau_{2})+2\beta(\delta\tau_{2})\,\big{)}\Big{]}\,,$
(68)

in which

$\displaystyle\delta(\tau)$ $\displaystyle\equiv$ $\displaystyle 2\epsilon\eta
a^{2}{\cal{R}}^{\prime}_{q}(\tau)^{2}\mathrm{Im}\big{[}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}_{p}(\tau)\big{]}\,,$
$\displaystyle\beta(\tau)$ $\displaystyle\equiv$ $\displaystyle 2\epsilon\eta
a^{2}{\cal{R}}^{\prime}_{q}(\tau){\cal{R}}_{q}(\tau)\mathrm{Im}\big{[}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}^{\prime}_{p}(\tau)\big{]}\,,$
(69)

and

$\displaystyle X_{1}(\tau)\equiv\epsilon\eta
a^{2}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}_{p}(\tau){\cal{R}}^{\prime}_{q}(\tau)^{2}\,.$
(70)

Calculating the above integrals yields the final result as follows

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{3}}=-\frac{27(h-10)}{8h}\big{(}\Delta Ne^{6\Delta
N}\big{)}\frac{H^{4}}{2\pi^{2}M_{P}^{4}\epsilon_{i}^{2}p^{3}}\,(2\pi)^{3}\delta^{3}({\bf{p}}_{1}+{\bf{p}}_{2})\,.$
(71)

As in the quartic case, we see a non-linear dependence on the parameter $h$.

###  5.3 Total one-Loop Corrections

Having calculated the one-loop corrections from the quartic and cubic
Hamiltonians, Eqs. (64) and (71), the total one-loop corrections in power
spectrum is given by (after neglecting the trivial factor
$(2\pi)^{3}\delta^{3}({\bf{p}}_{1}+{\bf{p}}_{2})$)

$\displaystyle\Delta{\cal{P}}({\bf{p}})\equiv\frac{p^{3}}{2\pi^{2}}\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{\mathrm{1-loop}}=\frac{2}{h}\big{(}3h^{2}-9h+378)(\Delta
Ne^{6\Delta N}){\cal{P}}_{\mathrm{CMB}}^{2}\,,$ (72)

in which
${\cal{P}}_{\mathrm{CMB}}=\frac{H^{2}}{8\pi^{2}M_{P}^{2}\epsilon_{i}}\sim
2\times 10^{-9}$ is the power spectrum on the CMB scale.

Eq. (72) is the main result of this paper. As mentioned before, we see a
nonlinear dependence of the one-loop correction to the parameter $h$. Note
that this result obtained in the limit of sharp transition so Eq. (72) is not
applicable for the limit $|h|\ll 1$. For the particular case $h=-6$, where
$\epsilon$ in the final slow-roll stage keeps its value at the end of USR
phase, from Eq. (72) we obtain $\Delta{\cal{P}}=-180(\Delta Ne^{6\Delta
N}){\cal{P}}_{\mathrm{CMB}}^{2}$. This is the limit which we can compare our
result with the results of [25]. In their notation,
$\Delta\eta(\tau_{e})=-h=6$ and up to a model dependent constant they would
obtain $\Delta{\cal{P}}=\frac{\Delta\eta(\tau_{e})^{2}}{4}(\Delta Ne^{6\Delta
N}){\cal{P}}_{\mathrm{CMB}}^{2}=9(\Delta Ne^{6\Delta
N}){\cal{P}}_{\mathrm{CMB}}^{2}$. Therefore, our result is larger than that of
[25] by an additional factor $-20$. We suspect that the assumption in [25]
that the quartic Hamiltonian does not contribute to one-loop correction is
unjustified which causes this discrepancy. Indeed, we see non-trivial
contributions from quartic Hamiltonian as studied in details in section (5.1).
Other than this numerical discrepancy, our analysis confirm the main
conclusion in [25] that the one-loop corrections from small scale modes which
leaves the horizon during the USR phase can induce significant correction to
large scale power spectrum with an amplitude proportional to $e^{6\Delta N}$.

There are some important comments in order regarding our one-loop correction
in power spectrum. The first comment is that in the extreme sharp transition
with $|h|\gg 1$, the one-loop correction in power spectrum becomes arbitrarily
large. Indeed, in this limit we would have
$\frac{\Delta{\cal{P}}}{{\cal{P}}_{\mathrm{CMB}}}\simeq 6h(\Delta Ne^{6\Delta
N}){\cal{P}}_{\mathrm{CMB}}$ so the amplitude of this ratio will easily become
larger than unity. It is more useful to eliminate the parameter $h$ in this
ratio in favour of the final slow-roll parameter
$\epsilon_{0}=\epsilon(\tau_{0})$. Using Eq. (10) (or directly from the
definition of $h$ in Eq. (9) ) we have
$|h|=6\sqrt{\frac{\epsilon_{0}}{\epsilon_{e}}}$ while
$\epsilon_{e}=e^{-6\Delta N}\epsilon_{i}$ so we obtain

$\displaystyle\big{|}\frac{\Delta{\cal{P}}}{{\cal{P}}_{\mathrm{CMB}}}\big{|}=36\sqrt{\frac{\epsilon_{0}}{\epsilon_{i}}}(\Delta
Ne^{3\Delta N}){\cal{P}}_{\mathrm{CMB}}\,.$ (73)

Assuming that there is no large hierarchy between the CMB scale $\epsilon_{i}$
and the final value $\epsilon_{0}$, we see that the above ratio scales like
$36\,e^{3\Delta N}{\cal{P}}_{\mathrm{CMB}}$. For example, with $\Delta N\simeq
6$ the above ratio becomes larger than unity and the perturbative
approximation breaks down. Alternatively, Eq. (72) can be used to put an upper
bound on the duration of the USR period, the number of e-fold $\Delta N$. For
example, for the standard sharp transition with $h=-6$ and
$\epsilon_{0}=\epsilon_{e}$, the ratio becomes order unity for $\Delta N\simeq
2.3$. A longer period of USR phase will amplify the amplitude of curvature
perturbations such that eventually the one-loop corrections become comparable
to the tree level. In either way, i.e. either by taking $|h|$ to be
arbitrarily large or for a fixed value of $|h|$ considering a long enough
period of USR phase, one would invalidate the perturbative theory as the one-
loop contribution surpasses the tree level contribution.

The second comment is the issue concerning PBHs formation as debated in [25,
28, 29]. As argued in [25] in order to generate PBHs as candidate for dark
matter from USR quantum fluctuations, one typically requires that
${\cal{P}}(\tau_{e})\simeq e^{6\Delta N}{\cal{P}}_{\mathrm{CMB}}\sim 10^{-2}$
or so. However, in order to trust the perturbative approximation, we need
$\frac{\Delta{\cal{P}}}{{\cal{P}}_{\mathrm{CMB}}}\ll 1$. Combining this
requirement with Eq. (72) we conclude that for sharp transition
${\cal{P}}(\tau_{e})\ll 10^{-2}$ so the advocated mechanism to generate enough
PBHs as the candidate for dark matter does not work in USR setup with a sharp
transition.

The third comment is that Eq. (72) is obtained in the limit of sharp
transition, corresponding to $h\leq-6$. A more realistic setup is when the
transition from a USR phase to the final slow-roll phase happens mildly. As we
discussed in section 4 for the case of bispectrum, the effects of the
transition on the final value of $f_{NL}$ were studied in details in [27]. It
was demonstrated that the final value of $f_{NL}$ can be washed out if the
transition is mild. The maximum value $f_{NL}=\frac{5}{2}$ happens only for
the case of extreme sharp transition $|h|\gg 1$ while for the standard sharp
transition with $h=-6$, one obtains a smaller value $f_{NL}=\frac{5}{8}$ as we
demonstrated it explicitly in section 4. However, for a mild transition with
small value of $|h|$, the $f_{NL}$ parameter falls off to the order of the
slow-roll parameters (while the Maldacena consistency condition is still
violated). We believe that a similar effects can happen for the one-loop
corrections as well. If the transition takes place in a mild manner, then the
leading order corrections proportional to $e^{6\Delta N}$ will be washed out
by the subsequent evolution of the mode function during the transition from
the USR phase to the final slow-roll phase. This point was advocated in [28]
as well. If so, the USR setup may still be employed as a viable mechanism to
generate PBHs formation as a candidate for dark matter.

##  6 Summary and Discussions

In this work we have revisited the question of one-loop effects on large scale
curvature perturbations power spectrum induced from small scale modes which
become superhorizon during an intermediate USR phase. This is an important
question both theoretically and also for its implications for the PBHs
formation.

Our results largely support the conclusion made in [25, 26] that the induced
one-loop corrections can become large enough to kill the popular mechanism of
PBHs formation in the USR setup. As the one-loop corrections become very large
they invalidate the perturbative assumption before one proceeds to calculate
the PBHs abundance in this setup. Having said this, there is a discrepancy
between our results and those of [25, 26] as our result is larger by a factor
of $20$ compared to that of [25, 26] in the limit where these two calculations
have overlap (i.e. in the special case $h=-6$). We believe this discrepancy is
caused by the assumption in [25, 26] that the quartic self interaction does
not contribute to the one-loop calculation. Our direct calculation based on
EFT formalism shows that there are non-trivial contributions from the quartic
Hamiltonian which yield to a comparable contribution to one-loop power
spectrum as does the quartic Hamiltonian.

Our analytical results were obtained in the limit of sharp transition where
one neglects the relaxation period after the USR phase. An interesting
conclusion from our result is that for an infinitely sharp transition the
induced one-loop correction can be arbitrarily large, invalidating the
perturbative approximation completely. Of course, in a realistic case, one
needs a relaxation period so it is possible that much of the one-loop
corrections are washed out during the subsequent evolution of the mode
function after the USR phase. This was certainly the case in $f_{NL}$ story as
demonstrated explicitly in [27]. If so, this will also support the arguments
in [28] that in a realistic setup with a mild transition, the one-loop
corrections become harmless and one may still employ the USR setup for the
PBHs formation. We would like to come back to this important question and
examine the effects of a mild transition on the one-loop calculations.

As usual in QFT analysis, in performing the loop corrections one has to take
into account the renormalization of the divergences, both UV and IR. In this
work, we were mainly interested in the contributions from the modes in the
interval $q_{i}\leq q\leq q_{e}$ which become superhorizon during the USR
phase. Our main goal was to check if there is a cancellation between the cubic
and quartic contributions for the one-loop corrections. Also our second main
goal was to investigate the roles of the sharpness of the transition. For
these motivations the renormalization of the power spectrum was not an
immediate concern. However, to read off the final physical power spectrum one
has to examine the divergence associated to the modes with momentum larger
than $q_{e}$ and take into account the question of renormalization.

There are a number of directions in which the current analysis can be
extended. One interesting direction to proceed is to consider the effects of a
non-trivial sound speed $c_{s}$ for the induced one-loop corrections in power
spectrum. As a related comment we mention that the relation between the one-
loop correction in power spectrum and the amplitude of non-Gaussianity was
studied in [41]. It was concluded that in order for the one-loop corrections
in a model of inflation with small $c_{s}$ to be under control, then the
amplitude of the equilateral-type non-Gaussianity has to be small. The other
interesting question is to calculate the one-loop correction using the
formalism of stochastic inflation. Indeed, this question was already studied
in [42, 43] in the USR setup. It was shown there that
$\Delta{\cal{P}}\propto{\cal{P}}^{2}\propto e^{12\Delta N}$. This is unlike
our current case where $\Delta{\cal{P}}\propto e^{6\Delta N}$. The reason is
that in [42, 43] one has a two-phase model where an early USR phase is
followed by a slow-roll phase and the CMB scale modes leave the horizon during
the USR phase. However, in our current analysis, we have a three-phase setup
of inflation in which the CMB scale modes leave the horizon at the first slow-
roll stage long prior to the USR phase. Finally, the other interesting
question is to investigate the one-loop corrections for the tensor power
spectrum. As there are interactions between two scalars and one tensor
perturbations, then naturally there can be large one-loop corrections in
tensor power spectrum as well. We would like to come back to these interesting
questions in future.

Acknowledgments: I am grateful to Mohammad Hossein Namjoo for many insightful
discussions during the progress of this work. I would like to thank Xingang
Chen, Junichi Yokoyama, Mahdiyar Noorbala, Jason Kristiano and Sina Hooshangi
for useful discussions and comments on the draft. This research is partially
supported by the “Saramadan” Federation of Iran.

##  Appendix A Loop corrections from the cubic Hamiltonian

In this Appendix, we present the details of the analysis for the loop
corrections from the cubic Hamiltonian.

As mentioned in Section 5.2, expanding the Dyson series to second order in
${\bf H}_{3}$ we have

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{3}}=\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(2,0)}+\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}+\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(0,2)}$
(74)

in which

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(2,0)}$
$\displaystyle=$
$\displaystyle-\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{\langle}{\bf
H}_{3}(\tau_{1}){\bf
H}_{3}(\tau_{2}){\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}$
(75) $\displaystyle=$
$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle^{\dagger}_{(0,2)}\,,$

and

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}=-\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{\langle}{\bf
H}_{3}(\tau_{1}){\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0}){\bf
H}_{3}(\tau_{2})\big{\rangle}\,.$ (76)

As explained in the main text, the dominant contributions for the time
integrals above come from the USR phase as the mode functions are growing and
the time derivatives are not slow-roll suppressed. During the USR phase
$\eta=-6$ so it simply can be taken out of the time integrals. With this
discussion in mind, let us start with Eq. (76). After performing all the
contractions and considering the limit that $|{\bf{p}}|\ll|{\bf{q}}|$, we
obtain four different contributions in
$\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}$
denoted by $A_{i}$ for $i=1...4$. For example, $A_{1}$ is given by

$\displaystyle
A_{1}=4\eta^{2}M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}\epsilon
a^{2}\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}\epsilon
a^{2}{\cal{R}}_{p}(\tau_{e}){\cal{R}}^{*}_{p}(\tau_{e}){\cal{R}}_{p}(\tau_{1}){\cal{R}}^{*}_{p}(\tau_{2}){\cal{R}}^{\prime}_{q}(\tau_{1})^{2}{\cal{R}}^{{}^{\prime}*}_{q}(\tau_{2})^{2}\,,$
(77)

with similar expressions for $A_{2},A_{3}$ and $A_{4}$.

Deining

$\displaystyle X_{1}(\tau)\equiv\eta\epsilon
a^{2}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}_{p}(\tau){\cal{R}}^{\prime}_{q}(\tau)^{2}\,,$
(78)

we can express $A_{1}$ as follows

$\displaystyle
A_{1}=4M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{1}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}X_{1}^{*}(\tau_{2})\,.$
(79)

Similarly, defining $X_{2}$ as

$\displaystyle X_{2}(\tau)\equiv\eta\epsilon
a^{2}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}^{\prime}_{p}(\tau){\cal{R}}^{\prime}_{q}(\tau){\cal{R}}_{q}(\tau)\,,$
(80)

with some efforts one can show that the remaining terms $A_{1},A_{2}$ an
$A_{3}$ can be written as follows

$\displaystyle
A_{2}=8M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{1}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}X_{2}^{*}(\tau_{2})\,,$
(81) $\displaystyle
A_{3}=8M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{1}^{*}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}X_{2}(\tau_{2})\,,$
(82)

and

$\displaystyle
A_{4}=16M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{2}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}X_{2}^{*}(\tau_{2})\,.$
(83)

Using the asymptotic form of the mode function at $\tau=\tau_{0}$ one can
check that $X_{1}\propto p^{-3}$ with the subleading correction of order
$p^{-1}$ while $X_{2}\propto p^{-2}$ with the subleading correction of order
$p^{0}$. Since we have the large hierarchy $p\ll q$, we can safely neglect the
contribution of $A_{4}$ in the following analysis. Intuitively speaking,
$X_{2}$ has a time derivative ${\cal{R}}^{\prime}_{p}(\tau)$ compared to
$X_{1}$ which has no corresponding time derivative so naturally $X_{2}$ is
further suppressed compared to $X_{1}$ by an extra factor of $p$.

Now we calculate
$\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(2,0)}$.
Performing all the contractions, we have four different contributions denoted
by $B_{i},i=1,...4$. For example, $B_{1}$ has the following form

$\displaystyle
B_{1}=8\eta^{2}M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}\epsilon
a^{2}\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}\epsilon
a^{2}{\cal{R}}_{p}(\tau_{e}){\cal{R}}^{*}_{p}(\tau_{e}){\cal{R}}_{p}(\tau_{1}){\cal{R}}^{*}_{p}(\tau_{2}){\cal{R}}^{\prime}_{q}(\tau_{1})^{2}{\cal{R}}^{{}^{\prime}*}_{q}(\tau_{2}){\cal{R}}^{*}_{q}(\tau_{2}),$
(84)

with similar expressions for $B_{2},B_{3}$ and $B_{4}$.

To calculate $B_{i}$, we further define the quantities $Z_{1}$ and $Z_{2}$ as
follows:

$\displaystyle Z_{1}(\tau)\equiv\eta\epsilon
a^{2}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}_{p}(\tau){\cal{R}}^{{}^{\prime}*}_{q}(\tau)^{2}\,,$
(85)

and

$\displaystyle Z_{2}(\tau)\equiv\eta\epsilon
a^{2}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}^{\prime}_{p}(\tau){\cal{R}}^{{}^{\prime}*}_{q}(\tau){\cal{R}}^{*}_{q}(\tau)\,.$
(86)

We note that in their dependence on $p$, $Z_{1}$ is similar to $X_{1}$ while
$Z_{2}$ is similar to $X_{2}$. With these definition, one obtains

$\displaystyle
B_{1}=-4M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{1}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}Z_{1}(\tau_{2})\,,$
(87) $\displaystyle
B_{2}=-8M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{1}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}Z_{2}(\tau_{2})\,,$
(88) $\displaystyle
B_{3}=-8M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{2}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}Z_{1}(\tau_{2})\,,$
(89)

and

$\displaystyle
B_{4}=-16M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}X_{2}(\tau_{1})\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}Z_{2}(\tau_{2})\,.$
(90)

To proceed further, we note that

$\displaystyle Z_{1}=X_{1}^{*}+i\delta^{*}$ (91)

in which

$\displaystyle\delta(\tau)\equiv 2\epsilon\eta
a^{2}{\cal{R}}^{\prime}_{q}(\tau)^{2}\mathrm{Im}\big{[}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}_{p}(\tau)\big{]}\,,$
(92)

and

$\displaystyle Z_{2}=X_{2}^{*}+i\beta^{*}$ (93)

in which

$\displaystyle\beta(\tau)\equiv 2\epsilon\eta
a^{2}{\cal{R}}^{\prime}_{q}(\tau){\cal{R}}_{q}(\tau)\mathrm{Im}\big{[}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}^{\prime}_{p}(\tau)\big{]}\,.$
(94)

Combining the results for
$\langle{\cal{R}}_{{\bf{p}}_{1}}{\cal{R}}_{{\bf{p}}_{2}}\rangle_{(2,0)}=\langle{\cal{R}}_{{\bf{p}}_{1}}{\cal{R}}_{{\bf{p}}_{2}}\rangle_{(0,2)}^{\dagger}$
and $\langle{\cal{R}}_{{\bf{p}}_{1}}{\cal{R}}_{{\bf{p}}_{2}}\rangle_{(1,1)}$
and after some algebra one obtains

$\displaystyle\langle{\cal{R}}_{{\bf{p}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{3}}=-8M_{P}^{4}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\int_{\tau_{i}}^{\tau_{e}}d\tau_{1}\int_{\tau_{i}}^{\tau_{1}}d\tau_{2}\,\mathrm{Im}\Big{[}X^{*}_{1}(\tau_{1})\big{(}\delta(\tau_{2})+2\beta(\delta\tau_{2})\,\big{)}\Big{]}\,.$
(95)

Note that in obtaining the above results we have neglected the contributions
from the terms $A_{4}$ and $B_{4}$ which are subleading as mentioned
previously.

We can simplify $\delta$ and $\beta$ in the integral (95) to some extent.
Using the asymptotic form of the mode function, we have:

$\displaystyle\mathrm{Im}\big{[}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}_{p}(\tau)\big{]}=\frac{H^{3}\tau_{i}^{6}}{12M_{P}^{2}\epsilon_{i}h\tau_{e}^{3}\tau^{3}}\big{(}h\tau_{e}^{3}+(6-h)\tau^{3}\big{)}\,,$
(96)

and

$\displaystyle\mathrm{Im}\big{[}{\cal{R}}_{p}^{*}(\tau_{0}){\cal{R}}^{\prime}_{p}(\tau)\big{]}=-\frac{H^{2}\tau_{i}^{6}}{4M_{P}^{2}\epsilon_{i}\tau^{4}}\,.$
(97)

Using these expressions in $\delta$ and $\beta$ and plugging them in Eq. (95)
we obtain the correction from the cubic Hamiltonian as given in Eq. (71). To
perform the integrals we employ the Maple computational software.

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