A large $|\eta|$ approach to single field inflation

Gianmassimo Tasinato1,2

1 Dipartimento di Fisica e Astronomia, Università di Bologna, Italia

2 Physics Department, Swansea University, SA28PP, United Kingdom
email: g.tasinato2208 at gmail.com

###### Abstract

Single field models of inflation capable to produce primordial black holes
usually require a significant departure from the standard, perturbative slow-
roll regime. In fact, in many of these scenarios, the size of the slow-roll
parameter $|\eta|$ becomes larger than one during a short phase of
inflationary evolution. In order to develop an analytical control on these
systems, we explore the limit of $|\eta|$ large, and promote $1/|\eta|$ to a
small quantity to be used for perturbative expansions. Formulas simplify, and
we obtain analytic expressions for the two and three point functions of
curvature fluctuations, which share some of the features found in realistic
inflationary models generating primordial black holes. We study one-loop
corrections in this framework: we discuss criteria for adsorbing ultraviolet
divergences into the available parameters, leaving log-enhanced infrared
contributions of controllable size.

##  1 Introduction and Conclusions

Identifying the nature of dark matter is one of the most challenging open
problems in cosmology [1]. A fascinating possibility is that dark matter is
made of primordial black holes (PBH) [2, 3, 4, 5], forming from the collapse
of high density fluctuations produced during cosmic inflation: see e.g. [6, 7,
8, 9, 10, 11, 12] for reviews. In order for producing PBH, the size of the
inflationary curvature fluctuation spectrum needs to increase by around seven
orders of magnitude, from large to small scales. This condition is not
possible to achieve within a controlled slow-roll expansion in single-field
inflation [13]: a departure from the standard slow-roll conditions is needed.
In several single-field realizations of PBH scenarios, the size $|\eta|$ of
the second slow-roll parameter becomes larger than one during a brief phase of
non-slow-roll evolution (from now on, NSR). Such brief NSR era should last few
e-folds $\Delta N_{\rm NSR}$ of expansion. Examples are ultra-slow-roll models
[14, 15, 16], where $\eta=-6$, and constant roll models [17, 18, 19], where
$|\eta|$ can be larger or smaller than $6$, depending on the properties of the
inflationary potential. In these cases, the evolution of fluctuations
challenges analytical investigations, since the slow-roll expansion breaks
down. Wands duality [20] can be of help in the ultra-slow-roll case, but still
care is needed in connecting slow-roll to NSR eras. Oftentimes, a numerical
analysis is needed.

In this work, we consider large values for the slow-roll quantity $|\eta|$,
and use the inverse $1/|\eta|$ as expansion parameter. A large value of
$|\eta|$ is not inconceivable to obtain at the price of tunings, for example
in constant roll systems. Here we are not interested in model building, but in
investigating the consequences of a large $|\eta|$ limit for the dynamics of
fluctuations. When working at leading order in $1/|\eta|$ formulas simplify,
and we obtain analytic expressions for the two and three point functions of
curvature fluctuations. These analytic results can be useful to get insights
on the properties of curvature fluctuations in PBH scenarios, as well as
understanding the physical consequences of a rapid growth of the curvature
spectrum from large to small scales.

This idealized, large-$|\eta|$ limit has some intriguing analogy with the
large-$N$ limit of $SU(N)$ QCD, a model introduced by ’t Hooft [21] in a
particle physics context. $N$ being the number of colors, the field-theory
analysis can be carried on using a perturbative $1/N$ expansion, and
simplifies in a large-$N$ limit. Real world QCD has $N=3$ colors only, yet the
results of an $1/N$ expansion catch various important properties of standard
QCD: we refer the reader to chapter 8 of [22] for a pedagogical survey.
Calling $g$ the QCD coupling constant, and $N$ the number of colors, ’t Hooft
finds convenient to take the simultaneous limits $g\to 0$, $N\to\infty$, and
$gN^{2}$ fixed [21]. Analogously, in PBH forming scenarios, it is convenient
to consider the limit of vanishing e-folds of NSR expansion, $\Delta N_{\rm
NSR}\to 0$, and at the same time taking $|\eta|\to\infty$, keeping fixed the
product $\left(\Delta N_{\rm NSR}|\eta|\right)$. As we will learn, this
product is associated with the growth of the spectrum from large to small
scales. Keeping $\left(\Delta N_{\rm NSR}|\eta|\right)$ fixed, and expanding
in $1/|\eta|$, the formulas for the curvature fluctuation $n$-point functions
become easier to deal with.

Having analytical control on a perturbative expansion in $1/|\eta|$ allows us
to address the issue of loop corrections, a topic that recently raised much
attention after the important papers [23, 24] appeared. As pointed out in
[23], the same mechanisms that causes the curvature spectrum growth, as needed
for producing PBH, also amplify the effects of loop corrections to the
curvature power spectrum. Their size can become so large to invalidate a
perturbative loop expansion. Many solutions and new perspectives have recently
pointed out [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. In the framework
of a large $|\eta|$ expansion, we show that loop corrections can be placed
under control, at least at the large scales that can affect CMB physics. We
regularize loop integrals by means of ultraviolet and infrared cut-offs, and
analytically compute the effects of loops in a large $|\eta|$ regime. The
resulting ultraviolet divergences can be adsorbed into physically measurable
quantities corresponding to the amplitude and the large-scale tilt of the
spectrum. We are left with log-enhanced infrared contributions, whose size is
small at large scales.

We hope that the tool of an $1/|\eta|$ expansion, although idealized, can lead
to analytical insights allowing to further investigate properties of the
dynamics of curvature fluctuations in PBH scenarios. It will be interesting to
further apply this method to related topics, as the behaviour of higher order
$n$-point functions, and their corresponding loop corrections in a large
$|\eta|$ limit. Having analytic expressions for the primordial correlators can
also be useful for investigating the actual process of PBH formation in the
post-inflationary universe, as well as the generation of second-order
gravitational waves from enhanced curvature spectra: see respectively e.g.
[37] and [38] for reviews. We leave these topics to future investigations.

##  2 System under consideration

We consider single field models of inflation with canonical kinetic terms.
Around a conformally flat cosmological metric,
$ds^{2}\,=\,a^{2}(\tau)\left(-{\rm d}\tau^{2}+d\vec{x}^{2}\right)$, the
quadratic action for the curvature perturbation in Fourier space reads (we set
the Planck mass to unity)

$\displaystyle S_{\rm quad}\,=\,\frac{1}{2}\int
d\tau\,d^{3}k\,{z^{2}(\tau)}\left[\zeta_{k}^{\prime
2}(\tau)+k^{2}\zeta_{k}^{2}(\tau)\right]\,,$ (2.1)

where the pump field $z(\tau)$ is given by

$z(\tau)\,=\,a(\tau)\,\sqrt{2\epsilon(\tau)}\,.$ (2.2)

The definitions of Hubble and slow-roll parameters are

$\displaystyle H(\tau)\,=\,\frac{a^{\prime}(\tau)}{a^{2}(\tau)}\hskip
19.91684pt;\hskip
19.91684pt\epsilon(\tau)\,=\,-\frac{H^{\prime}(\tau)}{a(\tau)\,H^{2}(\tau)}\hskip
19.91684pt;\hskip
19.91684pt\eta(\tau)\,=\,\frac{\epsilon^{\prime}(\tau)}{a(\tau)\,H(\tau)\,\epsilon(\tau)}\,.$
(2.3)

We assume that the first slow-roll parameter $\epsilon(\tau)$ remains small
during the entire duration of inflation, which takes place for negative
conformal time $\tau\leq\tau_{0}\,=\,0$. We also assume that the second
parameter $\eta(\tau)$ remains small for negative values of $\tau$, a part
from a brief time interval $\tau_{1}\leq\tau\leq\tau_{2}$ during which $\eta$
is negative and its size $|\eta|$ becomes larger than one. (See the brief
discussion in Section 1.) During this short phase, which we call non-slow-roll
(NSR) period, we can not make a perturbative slow-roll expansion in $|\eta|$:
other methods are needed to tackle the evolution of fluctuations. In this
work, we explore the possibility to consider the inverse $1/|\eta|$ as a
convenient expansion parameter for pursuing analytical considerations. But
before discussing the role of the $|\eta|$ parameter, we first examine a
quantity related with the duration of NSR phase. We build a dimensionless
positive parameter $\Delta\tau$, as

$\Delta\tau\,=\,-\frac{\tau_{2}-\tau_{1}}{\tau_{1}}\,,$ (2.4)

and we require that $\Delta\tau\ll 1$. This condition implies that the
duration of the NSR phase is short with respect to the typical time-scales one
encounters in treating the system, as e.g. $|\tau_{1}|$ which controls the
onset of the NSR phase. A short duration of non-slow-roll phase is demanded by
the requirement to avoid excessive stochastic effects [39, 40, 41]. Since we
assume that the slow-roll parameter $\epsilon(\tau)$ remains always small
during inflation, we consider for simplicity the limit of pure de Sitter
expansion, with $a(\tau)\,=\,-1/(H_{0}\tau)$ and $H_{0}$ constant during
inflation. If the interval $\Delta\tau$ of eq (2.4) is small, this parameter
has a physical interpretation in terms of a (small) number $\Delta N_{\rm
NSR}$ of e-folds of NSR evolution:

$\displaystyle\Delta N_{\rm NSR}$ $\displaystyle=$
$\displaystyle\ln{\left(\frac{a(\tau_{2})}{a(\tau_{1})}\right)}\,=\,\ln\left({\frac{\tau_{1}}{\tau_{2}}}\right)\,=\,\ln\left(\frac{1}{1-\Delta\tau}\right)\,\simeq\,\Delta\tau\,,$
(2.5)

where in the next-to-last equality we used the definition (2.4), and in the
last equality we expanded for small $\Delta\tau$.

In the regime of $\Delta\tau\ll 1$ we can use the results of [42] (reviewed in
the technical Appendix A): we write the solution for the mode function of the
curvature perturbation $\zeta_{\kappa}(\tau)$ in Fourier space during
different epochs in the inflationary evolution. We define the pivot scale

$k_{\star}\,=\,1/|\tau_{1}|\,,$ (2.6)

corresponding to modes leaving the horizon at the onset of the NSR era. We
express our formulas in terms of dimensionless momentum scales, as follows:

$\kappa\equiv-k\tau_{1}\,=\,{k}/{k_{\star}}\,.$ (2.7)

Our expressions simplify with this notation, as we can easily identify modes
with $\kappa\sim 1$ which cross the horizon at epochs corresponding to the NSR
phase. For this reason, we adopt from now on the dimensionless definition
(2.7) when treating momenta.

The mode function $\zeta_{\kappa}(\tau)$ acquires its usual profile matching
the Bunch-Davies vacuum at short distances:

$\zeta_{\kappa}(\tau)\,=\,-i\,\frac{H_{0}\,(-\tau_{1})^{3/2}}{\sqrt{4\epsilon_{1}}\,\kappa^{3/2}}\left(1-\frac{i\tau}{\tau_{1}}\right)\,e^{i\frac{\kappa\tau}{\tau_{1}}}\hskip
22.76228pt;\hskip 22.76228pt\tau\leq\tau_{1}$ (2.8)

for conformal times $\tau\leq\tau_{1}$, since at early times the modes do not
yet experience the NSR evolution. In the previous equation, $H_{0}$ is the
constant Hubble parameter during inflation, and

$\epsilon_{1}\,=\,\epsilon(\tau_{1})$ (2.9)

is the value of the first slow-roll parameter at $\tau=\tau_{1}$.

For later times $\tau_{2}\leq\tau\leq\tau_{0}$ during inflation, instead, the
profile of the mode function is modified by the effects of the NSR era. See
Appendix A, where we include the behaviour of the mode function in the
interval $\tau_{1}\leq\tau\leq\tau_{2}$ that we do not need in the main text.
(Sufficient to say that the mode functions, with their first derivatives, are
continuous at the transition between slow-roll and non-slow-roll eras.) We
find

$\zeta_{k}(\tau)\,=\,-i\,\frac{H_{0}\,(-\tau_{1})^{3/2}}{\sqrt{4\epsilon_{1}}\,\kappa^{3/2}}\left[{\cal
C}_{1}(\kappa)\,\left(1-\frac{i\tau}{\tau_{1}}\right)\,e^{i\frac{\kappa\tau}{\tau_{1}}}+{\cal
C}_{2}(\kappa)\,\left(1+\frac{i\tau}{\tau_{1}}\right)\,e^{-i\frac{\kappa\tau}{\tau_{1}}}\right]\hskip
8.5359pt;\hskip 8.5359pt\tau_{2}\leq\tau\leq\tau_{0}$ (2.10)

with (recall the definition of $\Delta\tau$ in eq (2.4))

$\displaystyle{\cal C}_{1}(\kappa)$ $\displaystyle=$ $\displaystyle
1-\frac{\eta}{8(1-\Delta\tau)^{2}\,\kappa^{2}}\left[1-e^{2i\kappa\Delta\tau}-2\kappa\Delta\tau\left(i-2\kappa(1-\Delta\tau)\right)\right]\,,$
(2.11) $\displaystyle{\cal C}_{2}(\kappa)$ $\displaystyle=$
$\displaystyle\frac{\eta\,e^{2i(1-\Delta\tau)\kappa}}{8(1-\Delta\tau)^{2}\,\kappa^{2}}\left[1-2i\kappa-e^{2i\kappa\Delta\tau}\left(1-2i\kappa(1-\Delta\tau)\right)\right]\,,$
(2.12)

and

$\eta\,=\,\lim_{\tau\to\tau_{1}^{+}}\eta(\tau)\,.$ (2.13)

From now on, the quantity $\eta$ refers to the definition (2.13), i.e. the
value of the time-dependent $\eta(\tau)$ evaluated at the beginning of the NSR
era. Notice that in the limit of negligible $\eta\to 0$, the two mode
functions (2.8) and (2.10) coincide. Instead, if $|\eta|$ is large in size,
the scale dependence of the mode functions (2.8) and (2.10) differs
considerably. This leads to the opportunity of increasing the size of the
curvature spectrum at small scales, as required by primordial black hole
production. In what comes next, we examine this possibility.

##  3 The two-point function of curvature fluctuations

In this section we show how a suitably defined large-$|\eta|$ limit allows us
to analytically capture the scale dependence of the spectrum of curvature
fluctuations. Starting from the mode functions obtained in the previous
Section, we quantize the system starting from the quadratic action (2.1) for
curvature fluctuations. See e.g. [43] for a textbook discussion. We can easily
compute the two-point function
$\langle\zeta_{\kappa}(\tau_{0})\,\zeta_{\kappa}^{*}(\tau_{0})\rangle$ of
curvature perturbations evaluated at the end of inflation, $\tau=\tau_{0}=0$,
and the corresponding power spectrum (recall our definition (2.7) of
dimensionless scale $\kappa$)

${\cal
P}({\kappa})\,\equiv\,\frac{\kappa^{3}}{2\pi^{2}(-\tau_{1})^{3}}\,\langle\zeta_{\kappa}(\tau_{0})\,\zeta_{\kappa}^{*}(\tau_{0})\rangle^{\prime}\,,$
(3.1)

where a prime indicates the two-point function omitting the momentum-
conserving delta functions. At very large scales, $\kappa\to 0$, one finds the
usual expression

${\cal P}_{0}\,=\,\lim_{\kappa\to 0}{\cal
P}_{\kappa}\,=\,\frac{H_{0}^{2}}{8\pi^{2}\epsilon_{1}}\,,$ (3.2)

with the scale of ${\cal P}_{0}$ of order $10^{-9}$ to match CMB
normalization. Since large scale modes leave the horizon much earlier than the
NSR era, they are unaffected by it. It is convenient to compute the
dimensionless ratio $\Pi(\kappa)$ (see [42]) between the power spectrum (3.1)
evaluated at scale $\kappa$, versus the large-scale spectrum ${\cal P}_{0}$ in
eq (3.2). We find

$\Pi(\kappa)\,\equiv\,\frac{{\cal P}_{\kappa}}{\lim_{\kappa\to 0}{\cal
P}_{\kappa}}\,=\,|{\cal C}_{1}(\kappa)+{\cal C}_{2}(\kappa)|^{2}\,,$ (3.3)

with the scale-dependent ${\cal C}_{1,2}(\kappa)$ given in eqs (2.11), (2.12).
Such ratio can be considered as a dimensionless power spectrum evaluated at
the end of inflation, which singles out the overall amplitude ${\cal P}_{0}$
at large scales, and encampsulates the rich scale dependence of the spectrum
evolving from large to small scales. We plot $\Pi(\kappa)$ in Figure 1, left
panel, for a representative choice of parameters capable to enhance the
spectrum at small scales. Physically, the scale dependence of the spectrum is
due to the brief NSR phase of inflationary evolution. The NSR era is able to
excite the would-be decaying mode at superhorizon scales, which starts to
actively participate to the dynamics of curvature fluctuations. See e.g. [12]
for a recent review.

Figure 1:  Left panel: Plot of the dimensionless power spectrum
${\Pi}(\kappa)$ as defined in eq (3.3): we choose the values $|\eta|=10^{4}$
and $\Delta\tau=0.2$ for the free parameters. Right panel: the black line is
the same as left panel. The dashed red line represents the spectrum
$\hat{\Pi}(\kappa)$ of eq (3.7), choosing the value $\Pi_{0}=1250$ for the
single free parameter. See the discussion after eq (3.8). Notice that the
maximal values of the spectrum occur around the onset of non-slow-roll phase,
for $\kappa\sim{\cal O}(1)$.

Notice that the spectrum has a pronounced dip at intermediate scales, due to a
disruptive interference between the growing and decaying modes of the
curvature fluctuation at super-horizon scales. The dip is followed by a steady
growth (with slope $\kappa^{4}$ as first shown in [44]) until it reaches a
maximal amplitude. See also [45] for a detailed analysis of the shape of the
curvature power spectrum in PBH forming scenarios.

It is particularly interesting to evaluate the value of $\Pi(\kappa)$ at very
small scales, $\kappa\to\infty$, which informs us on the total amount of the
growth of the spectrum. See Figure 1, left panel. Plugging into (3.3) the
expressions for ${\cal C}_{1,2}$ of eqs (2.11), (2.12) and taking the small-
scale limit, we find

$\displaystyle\lim_{\kappa\to\infty}\Pi(\kappa)$ $\displaystyle=$
$\displaystyle\left(\frac{1+\left(|\eta|/2-1\right)\,\Delta\tau}{1-\Delta\tau}\right)^{2}\,,$
(3.4) $\displaystyle\equiv$ $\displaystyle(1+\Pi_{0})^{2}\,,$

where in the second line we introduce a constant parameter $\Pi_{0}$
controlling the enhancement of the spectrum from large to small scales
($\Pi_{0}=0$ means no enhancement). We would like a large enhancement of the
spectrum at small scales for producing PBH. Since we are in a regime of small
$\Delta\tau$, as discussed in Section 2, we need to consider large values for
the parameter $|\eta|$ during the NSR period (we make the hypothesis that
$\eta$ is negative, hence the absolute value). In fact, in the limit of
$|\eta|$ large and $\Delta\tau$ small, expression (3.4) simplifies to

$\Pi_{0}\,\simeq\,\frac{|\eta|\Delta\tau}{2}\,.$ (3.5)

The combination (3.5), as well as the considerations above, motivates us to
take the simultaneous limits:

$|\eta|\gg 1\hskip 19.91684pt;\hskip 19.91684pt\Delta\tau\ll 1\hskip
19.91684pt;\hskip 19.91684pt{\rm keep\,\,\,\,}\Pi_{0}\,\,\,\,{\rm fixed\,.}$
(3.6)

This is reminiscent of the ’t Hooft limit one encounters in particle physics
[21], as explained in Section 1. In fact, combining $|\eta|$ and $\Delta\tau$
into the fixed quantity $\Pi_{0}$ allows us to consistently perform expansions
in the small parameter $1/|\eta|$, maintaining at the same time control on the
effects of the NSR through the quantity $\Pi_{0}$. In most PBH scenarios we
aim to a total enhancement of the order $10^{6}-10^{7}$ in eq (3.4). Then the
quantity $\Pi_{0}$ results by itself large, of order $10^{3}-10^{4}$.

Adopting the limits of eq (3.6), the expression for the ratio (3.3)
simplifies. We substitute $\Delta\tau\,=\,2\Pi_{0}/|\eta|$ in eq (3.3), and
expand for large values of $|\eta|$ keeping $\Pi_{0}$ fixed. At leading order
in this expansion, we obtain

$\hat{\Pi}(\kappa)\,=\,1-4\kappa\,\Pi_{0}\,\cos{\kappa}\,j_{1}(\kappa)+4\kappa^{2}\,\Pi_{0}^{2}\,j^{2}_{1}(\kappa)\,,$
(3.7)

where a hat indicates that we only include the leading order in an expansion
in $1/|\eta|$, following the conditions of eq (3.6). The spherical Bessel
function $j_{1}(\kappa)$ is given by

$j_{1}(\kappa)\,=\,\frac{\sin\kappa}{\kappa^{2}}-\frac{\cos\kappa}{\kappa}\hskip
19.91684pt;\hskip 19.91684ptj_{1}(\kappa\ll
1)\,=\,\frac{\kappa}{3}-\frac{\kappa^{3}}{30}+{{\cal O}(\kappa^{5})}\,.$ (3.8)

We represent formula (3.7) in Figure 1, right panel, in comparison with the
result obtained by the more accurate formula (3.3). The latter, plotted in the
left panel of the figure, makes use of a small $\Delta\tau$ limit only,
without the further expansion in $1/|\eta|$ of eq (3.6). The resulting profile
of the spectrum is very similar in both cases, at least in the regime
$\kappa\leq 5$, indicating that the limits of eq (3.6) give trustable results
for the spectrum at least at relatively large scales. It is not difficult to
use eq (3.7) to analytically determine the position of the dip, finding
agreement with other works in the literature [42].

It is remarkable to obtain such a simple formula (3.7) for the scale
dependence of the curvature power spectrum, whose momentum profile shares
features with more realistic PBH models discussed in the literature. This
formula depends on a single free parameter $\Pi_{0}$. Besides parameterizing
the total enhancement of the spectrum, this quantity also governs the scale
dependence of the spectrum at large scales. Expanding (3.7) up to
$\kappa^{2}$:

$\hat{\Pi}(\kappa)\,=\,1-\frac{4\,\Pi_{0}}{3}\kappa^{2}+{\cal
O}(\kappa^{4})\,,$ (3.9)

making manifest the role of $\Pi_{0}$ in controlling the deviations from a
flat spectrum. We can be more precise and analytically compute the spectral
index associated with eq (3.7):

$\displaystyle\hat{n}_{s}(\kappa)-1$ $\displaystyle\equiv$
$\displaystyle\frac{d\ln\hat{\Pi}(\kappa)}{d\ln\kappa}\,,$ (3.11)
$\displaystyle=$
$\displaystyle\frac{2\,\kappa\,\Pi_{0}\left[(1-2\kappa^{2})\sin{(2\kappa)}-2\kappa\cos{(2\kappa)}\right]}{\kappa^{2}+4\kappa\Pi_{0}\cos{\kappa}\left(\kappa\cos{\kappa}-\sin{\kappa}\right)+4\Pi_{0}^{2}\left(\kappa\cos{\kappa}-\sin{\kappa}\right)^{2}}$
$\displaystyle-\frac{\Pi_{0}^{2}\left[4-(4-8\kappa^{2})\cos{(2\kappa)}+4\kappa(\kappa^{2}-2)\sin{(2\kappa)}\right]}{\kappa^{2}+4\kappa\Pi_{0}\cos{\kappa}\left(\kappa\cos{\kappa}-\sin{\kappa}\right)+4\Pi_{0}^{2}\left(\kappa\cos{\kappa}-\sin{\kappa}\right)^{2}}\,.$

The rich dependence in momentum scale of the spectral index in eq (3.11)
reflects the scale dependence of the spectrum in Fig 1. We represent it in Fig
2 for a range of momenta going from the dip position to small scales.
Comparing Figures 1 and 2, we notice that, after the dip position, the maximal
growth slope of the spectrum is $n_{s}-1\leq 4$. This agrees with the more
sophisticated analysis [44] based on complete expressions for the curvature
power spectrum, outside the large $|\eta|$ limit we consider here.

Figure 2:  The spectral index as given in eq (3.11), choosing $\Pi_{0}=1250$.

##  4 The three-point function of curvature fluctuations

We now apply the previous set-up to the study of three-point function of
curvature fluctuations, evaluated at the end of inflation. This quantity
controls the non-Gaussianity of curvature fluctuations in PBH scenarios. We
assume that the slow-roll parameter $\epsilon(\tau)$ remains always small,
while $\eta(\tau)$ experiences a sharp transition between the slow-roll and
non-slow-roll phases, at $\tau=\tau_{1}$ and $\tau=\tau_{2}$. The $n$-point
functions of $\zeta$ can be computed using the in-in formalism [46, 47, 48].
Let ${\cal O}(\tau)$ the operator one wishes to determine (for us, the three-
point function
$\langle\zeta_{\kappa_{1}}(\tau_{0})\zeta_{\kappa_{2}}(\tau_{0})\zeta_{\kappa_{3}}(\tau_{0})\rangle$),
and ${\cal H}_{\rm int}$ the interaction Hamiltonian. We map the time
evolution of the operator from the initial $\big{|}{\rm in}\rangle$ vacuum up
to the time the operator ${\cal O}(\tau)$ is evaluated, and then we map back
to the $\big{|}{\rm in}\rangle$ vacuum again. In formulas: $\langle{\rm
in}\Big{|}{\bar{T}}e^{-i\int{\cal H}_{\rm
int}(\tau^{\prime})d\tau^{\prime}}\,{\cal O}(\tau)\,{T}e^{i\int{\cal H}_{\rm
int}(\tau^{\prime})d\tau^{\prime}}\Big{|}{\rm in}\rangle$. In our case, since
we focus on sudden transitions, there is a single dominant contribution to the
interaction Hamiltonian [23, 24], which can be extracted from the third-order
action of perturbations in single field inflation [46]:

${\cal H}_{\rm int}\,=\,-\frac{1}{2}\int
d^{3}x\,a^{2}(\tau)\epsilon(\tau)\,\eta^{\prime}(\tau)\,\zeta^{2}(\tau,\vec{x})\,\zeta^{\prime}(\tau,\vec{x})\,.$
(4.1)

We assume that $|\eta|$ is negligible during slow-roll evolution
($\tau<\tau_{1}$ and $\tau_{2}<\tau<\tau_{0}$) while it is large during the
intermediate NSR phase, $\tau_{1}\leq\tau\leq\tau_{2}$. We adopt a sharp-
transition Ansatz [24] for the time-derivative of $\eta(\tau)$

$\eta^{\prime}(\tau)\,=\,\Delta\eta\left[-\delta(\tau-\tau_{1})+\delta(\tau-\tau_{2})\right]\,.$
(4.2)

where the times $\tau_{1,2}$ correspond to the onset and end of the NSR phase
during inflation.

Soon we will discuss a criterium to select the constant $\Delta\eta$. But
first, we apply the aforementioned in-in approach with the interaction
Hamiltonian (4.1), and eq (4.2). The curvature three-point function, evaluated
at the end of inflation $\tau_{0}(=0)$, results [24]

$\displaystyle\langle\zeta_{\kappa_{1}}(\tau_{0})\zeta_{\kappa_{2}}(\tau_{0})\zeta_{\kappa_{3}}(\tau_{0})\rangle^{\prime}=$
$\displaystyle-2\Delta\eta\Big{(}\epsilon(\tau_{2})a^{2}(\tau_{2})\,{\rm
Im}\left[\left(\zeta_{\kappa_{1}}(\tau_{0})\zeta^{*}_{\kappa_{1}}(\tau_{2})\right)\left(\zeta_{\kappa_{2}}(\tau_{0})\zeta^{*}_{\kappa_{2}}(\tau_{2})\right)\left(\zeta_{\kappa_{3}}(\tau_{0})\partial_{\tau_{2}}\zeta^{*}_{\kappa_{3}}(\tau_{2})\right)\right]-(\tau_{2}\to\tau_{1})\Big{)}$
$\displaystyle+{\rm perms\,.}$ (4.3)

where recall that the prime means that we understand the momentum-conserving
delta functions. In the squeezed limit, eq (4.3) reduces to

$\displaystyle\lim_{\kappa_{1}\to
0;\,\kappa_{2}\simeq\kappa_{3}}\langle\zeta_{\kappa_{1}}(\tau_{0})\zeta_{\kappa_{2}}(\tau_{0})\zeta_{\kappa_{3}}(\tau_{0})\rangle^{\prime}$
$\displaystyle=\,-\,4\Delta\eta\,\epsilon(\tau_{2})a^{2}(\tau_{2})\,|\zeta_{\kappa_{1}}(\tau_{0})|^{2}\,|\zeta_{\kappa_{2}}(\tau_{0})|^{2}$
$\displaystyle\times\left\\{{\rm
Im}\left[\frac{\zeta_{\kappa_{2}}^{2}(\tau_{0})}{|\zeta_{\kappa_{2}}(\tau_{0})|^{2}}\zeta_{\kappa_{2}}^{*}(\tau_{2})(\zeta^{\prime}_{\kappa_{2}}(\tau_{2}))^{*}\right]-\frac{\epsilon(\tau_{1})a^{2}(\tau_{1})}{\epsilon(\tau_{2})a^{2}(\tau_{2})}{\rm
Im}\left[\frac{\zeta_{\kappa_{2}}^{2}(\tau_{0})}{|\zeta_{\kappa_{2}}(\tau_{0})|^{2}}\zeta_{\kappa_{2}}^{*}(\tau_{1})(\zeta^{\prime}_{\kappa_{2}}(\tau_{1}))^{*}\right]\right\\}\,.$
(4.4)

The squeezed limit refers to modes with very large momenta $\kappa_{1}$ which
leave the horizon much earlier than the onset of the non-slow-roll (NSR)
phase. When selecting large-scale modes with $\kappa_{2}$ small, also far from
the NSR epoch, we do expect that the standard Maldacena consistency relation
[46] holds. Namely

$\displaystyle\lim_{\kappa_{1}\to
0;\,\kappa_{2}\simeq\kappa_{3}}\langle\zeta_{\kappa_{1}}(\tau_{0})\zeta_{\kappa_{2}}(\tau_{0})\zeta_{\kappa_{3}}(\tau_{0})\rangle^{\prime}\,=\,-(n_{s}(\kappa_{2})-1)\,|\zeta_{\kappa_{1}}(\tau_{0})|^{2}\,|\zeta_{\kappa_{2}}(\tau_{0})|^{2}\,.$
(4.5)

Figure 3:  Left panel Check of Maldacena consistency relation for the squeezed
limit of the three point function. Black line: the quantity $1-n_{s}$. Red
dashed line: the squeezed limit of the three-point function of eq (4.4) (we
omit the factors
$|\zeta_{\kappa_{1}}(\tau_{0})|^{2}\,|\zeta_{\kappa_{2}}(\tau_{0})|^{2}$). We
use the mode functions in eqs (2.10), and choose the values $|\eta|=10^{4.1}$,
$\Delta\tau=0.002$. Right panel: Plot of the scale dependence of the
equilateral three-point function, the quantity $-f_{\rm eq}/\Pi_{0}^{3}$, as
defined in the main text, eq (4.8). The profile is remarkably similar to the
power spectrum of Fig 1.

We substitute our expressions for the mode functions in eq (2.10), and take
the small $\kappa_{2}$ limit. Using the results of Section 3 for computing the
spectral index, the two expressions (4.4) and (4.5) match once we select a
certain value for the parameter $\Delta\eta$ which enters in the Ansatz (4.2).
Neglecting contributions that vanish in the large-$|\eta|$ limit, we find the
requirement

$\Delta\eta\,=\,\frac{|\eta|}{(1+\Pi_{0})}+\frac{\Pi_{0}(12+34\Pi_{0}+25\Pi_{0}^{2})}{2(1+\Pi_{0})^{2}(1+2\Pi_{0})}\,,$
(4.6)

as well as the expected condition 111In fact, we are working in a regime of
large $|\eta|$ and very small $\Delta\tau$, as dictated by relations (3.6).
Hence, we obtain  $\displaystyle\epsilon(\tau_{1})\,a^{2}(\tau_{1})$
$\displaystyle=$
$\displaystyle\epsilon(\tau_{2})\,a^{2}(\tau_{2})\left(1+\tau_{1}\,\frac{\epsilon^{\prime}(\tau_{1})}{\epsilon(\tau_{1})}\,\Delta\tau\right)\left(1+2\tau_{1}\,\frac{a^{\prime}(\tau_{1})}{a(\tau_{1})}\,\Delta\tau\right)\,,$
$\displaystyle=$
$\displaystyle\epsilon(\tau_{2})\,a^{2}(\tau_{2})\left(1+\tau_{1}\,a(\tau_{1})H(\tau_{1})\left(\eta+2\right)\,\Delta\tau\right)\,,$
$\displaystyle=$
$\displaystyle\epsilon(\tau_{2})\,a^{2}(\tau_{2})\left(1-\left(\eta+2\right)\,\Delta\tau\right)\,,$
$\displaystyle=$
$\displaystyle\epsilon(\tau_{2})\,a^{2}(\tau_{2})\left(1+2\Pi_{0}\right)\,,$
in agreement with condition (4.7).

$\displaystyle\frac{\epsilon(\tau_{1})a^{2}(\tau_{1})}{\epsilon(\tau_{2})a^{2}(\tau_{2})}\,=\,1+2\Pi_{0}\,.$
(4.7)

Interestingly, although the constant $\Delta\eta$ has been fixed to satisfy
Maldacena condition in the small-$\kappa_{2}$ limit, the resulting expression
(4.4) for the squeezed three-point function that matches well with single-
field Maldacena consistency relation also for larger scales: see Fig 3, left
panel, which is also in agreement with [49, 50]. The resulting squeezed non-
Gaussianity is strongly scale-dependent [51, 52].

The squeezed limit of the three-point function, as in eq (4.4), is not the
only interesting configuration. From the complete expression for the three-
point function, eq (4.3), we can also consider other shapes. For example, let
us consider the equilateral limit $\kappa_{i}=\kappa$ for $i=1,2,3$. In Fig 3,
right panel, we represent the value for the three-point function as a function
of the dimensionless scale $\kappa$, divided by the square of the large-scale
power spectrum, eq (3.2) (we further divide it by $\Pi_{0}^{3}$). Namely,

$\frac{f_{\rm
eq}(\kappa)}{\Pi_{0}^{3}}\equiv\frac{\langle\zeta_{\kappa}(\tau_{0})\zeta_{\kappa}(\tau_{0})\zeta_{\kappa}(\tau_{0})\rangle^{\prime}}{\Pi_{0}^{3}\,{\cal
P}_{0}^{2}}\,.$ (4.8)

This quantity aims to capture the scale-dependence of the non-Gaussian
equilateral limit [53], analogously to the scale-dependent part of the power
spectrum of eq (3.2). Remarkably, the profile of the scale-dependence for the
equilateral shape (changing its overall sign) is similar to the profile of the
scale-dependent power spectrum: compare Fig 1 with Fig 3, right panel. It
would be interesting to find a physical reason for this result.

The non-Gaussianity of curvature fluctuations in PBH scenarios is an important
observable with several phenomenological ramifications for PBH formation [54,
55, 56, 57, 58]. We refer the reader to [59] for a recent comprehensive
analysis, and further references therein.

##  5 Loop corrections

In this section we apply the previous tools to study loop contributions to the
inflationary power spectrum [60, 61, 62, 63, 64]. In developing our arguments,
we closely follow the clear technical discussion of [24], but we make use of
our large $|\eta|$ expansion, and the corresponding solutions for the mode
functions discussed in Section 2. We are especially interested in examining
the physical implications of large loop corrections in our approach, and the
role of the scale dependence of the spectrum. Moreover, we discuss a proposal
to adsorb quadratic ultraviolet divergences into the available bare parameters
in a large $|\eta|$ limit, at least at large scales relevant for CMB physics.
We are left with log-enhanced, infrared effects whose size is small at large
scales. This is an important step in order to clarify the relation between
loops and physically measurable quantities.

The interaction Hamiltonian that we consider is given in eq (4.1); as in
Section 4, we focus on a sharp transition between slow-roll regimes and an
intermediate non-slow-roll regime for $\tau_{1}\leq\tau\leq\tau_{2}$. We
consider for definiteness the two-point function of curvature fluctuations in
momentum space, evaluated at the scale $p$ (dimensionless in the sense that
the momentum is multiplied by $-\tau_{1}$, as in eq (2.7)). The corresponding
1-loop contributions can be found utilising the in-in formalism. We follow
[24]: loop corrections are conveniently decomposed as

$\displaystyle\langle\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{0})\rangle_{\rm
loop}\,=\,\langle\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{0})\rangle_{(1,1)}+2\,{\rm
Re}\left[\langle\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{0})\rangle_{(2,0)}\right]\,,$
(5.1)

where

$\displaystyle\langle\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{0})\rangle_{(1,1)}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\,\int_{-\infty}^{\tau_{0}}\,d\tau_{a}\,a^{2}(\tau_{a})\,\epsilon(\tau_{a})\,\eta^{\prime}(\tau_{a})\int_{-\infty}^{\tau_{0}}\,d\tau_{b}\,a^{2}(\tau_{b})\,\epsilon(\tau_{b})\,\eta^{\prime}(\tau_{b})$
(5.2)
$\displaystyle\times\int\Pi_{i=1}^{6}\,\frac{d^{3}k_{i}}{(2\pi)^{3}}\,\delta^{3}(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3})\,\delta^{3}(\vec{k}_{4}+\vec{k}_{5}+\vec{k}_{6})$
$\displaystyle\times\langle\zeta^{\prime}_{\vec{k}_{1}}(\tau_{a})\zeta_{\vec{k}_{2}}(\tau_{a})\zeta_{\vec{k}_{3}}(\tau_{a})\zeta_{\vec{p}}(\tau_{0})\zeta_{-\vec{p}}(\tau_{0})\zeta^{\prime}_{\vec{k}_{4}}(\tau_{b})\zeta_{\vec{k}_{5}}(\tau_{b})\zeta_{\vec{k}_{6}}(\tau_{b})\rangle\,,$

and

$\displaystyle\langle\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{0})\rangle_{(2,0)}$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\,\int_{-\infty}^{\tau_{0}}\,d\tau_{a}\,a^{2}(\tau_{a})\,\epsilon(\tau_{a})\,\eta^{\prime}(\tau_{a})\int_{-\infty}^{\tau_{0}}\,d\tau_{b}\,a^{2}(\tau_{b})\,\epsilon(\tau_{b})\,\eta^{\prime}(\tau_{b})$
(5.3)
$\displaystyle\times\int\Pi_{i=1}^{6}\,\frac{d^{3}k_{i}}{(2\pi)^{3}}\,\delta^{3}(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3})\,\delta^{3}(\vec{k}_{4}+\vec{k}_{5}+\vec{k}_{6})$
$\displaystyle\times\langle\zeta_{\vec{p}}(\tau_{0})\zeta_{-\vec{p}}(\tau_{0})\zeta^{\prime}_{\vec{k}_{1}}(\tau_{a})\zeta_{\vec{k}_{2}}(\tau_{a})\zeta_{\vec{k}_{3}}(\tau_{a})\zeta^{\prime}_{\vec{k}_{4}}(\tau_{b})\zeta_{\vec{k}_{5}}(\tau_{b})\zeta_{\vec{k}_{6}}(\tau_{b})\rangle\,.$

From now on, to simplify the calculations we focus on a large-scale regime
where the size of the external momentum $p$ is much smaller than the momenta
$k_{i}$ over which we integrate [23, 24]. This allows to simplify formulas
substituing $k-p\simeq k$, and permits us to obtain analytic results. We will
discuss in due course the limitations we should impose on $p$ for satisfying
this condition, and their physical implications.

Substituting our Ansatz (4.2) in the case a sharp transition at the times
$\tau_{1}$ and $\tau_{2}$ between slow-roll and non-slow-roll phases, the
result acquires the following structure [24]

$\displaystyle\langle\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{0})\rangle_{\rm
loop}$ $\displaystyle=$
$\displaystyle\left(2\epsilon(\tau_{2})a^{2}(\tau_{2})\right)^{2}\Delta\eta^{2}\,|\zeta_{\vec{p}}(\tau_{0})|^{2}\,$
(5.4) $\displaystyle\times$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\Big{[}|\zeta_{\vec{k}}(\tau_{2})|^{2}\,{\rm
Im}(\zeta_{p}(\tau_{2})\zeta^{\prime*}_{p}(\tau_{2}))\,{\rm
Im}(\zeta_{k}(\tau_{2})\zeta^{\prime*}_{k}(\tau_{2}))$
$\displaystyle-4\frac{\epsilon(\tau_{1})a^{2}(\tau_{1})}{\epsilon(\tau_{2})a^{2}(\tau_{2})}\,{\rm
Im}(\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{2}))\,{\rm
Im}(\zeta_{k}^{\prime}(\tau_{2})\zeta_{k}(\tau_{2})\zeta^{*}_{k}(\tau_{1})\zeta^{\prime*}_{k}(\tau_{1}))$
$\displaystyle-2\frac{\epsilon(\tau_{1})a^{2}(\tau_{1})}{\epsilon(\tau_{2})a^{2}(\tau_{2})}\,{\rm
Im}(\zeta_{p}(\tau_{2})\zeta^{\prime*}_{p}(\tau_{2}))\,{\rm
Im}(\zeta_{k}^{2}(\tau_{2})\zeta_{k}^{*}(\tau_{1})\zeta_{k}^{\prime*}(\tau_{1}))$
$\displaystyle+\frac{\epsilon^{2}(\tau_{1})a^{4}(\tau_{1})}{\epsilon^{2}(\tau_{2})a^{4}(\tau_{2})}\,|\zeta_{\vec{k}}(\tau_{1})|^{2}\,{\rm
Im}(\zeta_{p}(\tau_{1})\zeta^{\prime*}_{p}(\tau_{1}))\,{\rm
Im}(\zeta_{k}(\tau_{1})\zeta_{k}^{\prime*}(\tau_{1}))\Big{]}\,.$

Since the integrand functions are rotationally invariant, the three-
dimensional integrals over internal momenta can be decomposed into integrals
over the real line as

$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\left(\dots\right)$
$\displaystyle=$ $\displaystyle\int_{\Lambda_{\rm
IR}/{|\eta|}^{1/2}}^{\mu/|\eta|^{1/2}}\frac{k^{2}\,dk}{2\pi^{2}}\left(\dots\right)+\int^{\Lambda_{\rm
UV}/{|\eta|}^{1/2}}_{\mu/|\eta|^{1/2}}\frac{k^{2}\,dk}{2\pi^{2}}\left(\dots\right)$
(5.5)

with $\Lambda_{\rm IR}$ and $\Lambda_{\rm UV}$ corresponding to a very small
infrared (IR) and a very large ultraviolet (UV) cut-off 222From now on, our
approach is a different from [23], where $\Lambda_{\rm IR,UV}$ are scales of
modes leaving the horizon at the start and end of the NSR era. In our case,
being the NSR epoch very short, we do not make this identification. . They are
dimensionless quantities, obtained multiplying physical momentum scales with
$|\tau_{1}|$, as in eq (2.7). For convenience, as a technical device we
rescale the extrema of integration by $1/|\eta|^{1/2}$, to simplify our
results in a large $|\eta|$ limit. The intermediate dimensionless scale $\mu$
is introduced in order to physically separate the loop corrections in an IR
part (the first integral in eq (5.5)) and a UV part (the second integral). We
can think of $\mu\sim 1$ as a scale where NSR effects take place. This
separation will be essential for our arguments.

We decompose the resulting power spectrum at the end of inflation $\tau_{0}=0$
as a tree level and a loop part

$\displaystyle{\cal P}_{\rm tot}(p)$ $\displaystyle=$
$\displaystyle\frac{p^{3}}{2\pi^{2}(-\tau_{1})^{3}}\,\langle\zeta_{p}(\tau_{0})\zeta^{*}_{p}(\tau_{0})\,=\,{\cal
P}_{0}\,\hat{\Pi}(p)\left[1+{L}^{\rm IR}_{\rm loop}+{L}^{\rm UV}_{\rm
loop}\right]\,,$ (5.6)

with ${\cal P}_{0}$ the amplitude of the large scale tree-level power spectrum
as in eq (3.2), and $\hat{\Pi}(p)$ is the momentum-dependent function of eq
(3.7), controlling the scale-dependent ratio between small-scale and large-
scale spectra. Eq (5.6) contains the quantity ${L}_{\rm loop}$, the loop
contribution (5.4), with the momentum integrals decomposed as in eq (5.5). We
collect as an overall factor the momentum dependent quantity ${\cal
P}_{0}\,\hat{\Pi}(p)$.

We substitute in the general formulas (5.5) our mode functions (2.10). We
analytically perform both the IR and the UV integrals, which are much
simplified in the large $|\eta|$ regime of eq (3.6), which keeps $\Pi_{0}$
fixed. At leading order in $1/|\eta|$, the dominant contribution to the IR
piece of the loop correction results

$\displaystyle{L}^{\rm IR}_{\rm loop}$ $\displaystyle=$
$\displaystyle-p^{2}\,\frac{{\cal
P}_{0}}{3}\,\frac{\Pi_{0}^{4}}{(1+\Pi_{0})^{2}(1+2\Pi_{0})}\,\ln{\left(\frac{\mu}{\Lambda_{\rm
IR}}\right)}\,,$ (5.7)

where we include only the log-enhanced part. We neglect power-law quadratic
pieces depending on the small quantity $\Lambda_{\rm IR}$, and on $\mu$ which,
being of order one, is suppressed with respect to the logarithm in eq (5.7),
in case of a large ratio ${\mu}/{\Lambda_{\rm IR}}$. This IR contribution can
be interpreted as a secular effects caused by modes crossing the horizon from
the onset of inflation until around the epoch of NSR, controlled respectively
by the scales $\Lambda_{\rm IR}$ and $\mu$. IR contributions are typically
characterized by large logarithms, whose effects might contribute to
observable quantities, if inflation lasts long. In our case, we can estimate
$\ln{\left({\mu}/{\Lambda_{\rm
IR}}\right)}\sim\ln\left[{a(\tau_{1})/a(\tau_{\rm start})}\right]$). Hence, we
can expect the logarithm to be of order say $10^{2}$. See e.g. the clear
discussion in [61].

The dominant contribution to the UV integral is a quadratic divergence in
$\Lambda_{\rm UV}$. We write the result only including the contribution
quadratic in the UV cut-off

$\displaystyle{L}^{\rm UV}_{\rm loop}$ $\displaystyle=$
$\displaystyle-\frac{{\cal P}_{0}\,\Pi_{0}\,\Lambda_{\rm
UV}^{2}}{(1+\Pi_{0})}\left(\frac{5}{6}+\frac{3j_{1}(p)-p}{3\,p}\right)\,,$
(5.8)

and we neglect subleading contributions. The spherical Bessel function
$j_{1}(p)$ defined in eq (3.8). The UV part contains the effects of small-
scale modes, which remain in a thermal vacuum within a subhorizon regime
during the first phase of inflation, until the short NSR phase occurs. These
modes should not participate to the dynamics of the NSR era during inflation,
and the associated UV divergences are expected to be adsorbed into
appropriate, physically measurable quantities (see e.g. [65] for a detailed
analysis within slow-roll models).

We adopt this viewpoint, and assume that the contributions of ${L}^{\rm
UV}_{\rm loop}$ are adsorbed into the available parameters by means of an
appropriate renormalization procedure. We discuss in Appendix B a way to do
so. We are left with the log-enhanced loop contributions of eq (5.7). All our
results are derived under the approximation stated after eq (5.3): to
analytically compute the integrals, we make the hypothesis that the momentum
$p$ at which the two-point function (5.6) is computed is well smaller than the
momentum scales over which integrate, $i.e.$ the lower extremum of the
integral

$p^{2}\,\leq\,\frac{\Lambda_{\rm IR}^{2}}{|\eta|}\,.$ (5.9)

Since we are working at leading order in a $1/|\eta|$ expansion, the previous
condition informs us that we should only focus on the very first terms in a
momentum expansion of our formulas. Using the expression (3.7), we consider eq
(5.6) up to second order in an expansion in momentum $p$, including the IR
loop contributions:

$\displaystyle{\cal P}_{\rm tot}(p)$ $\displaystyle=$ $\displaystyle{\cal
P}_{0}-\frac{4{\cal P}_{0}\,\Pi_{0}}{3}\left[1+\,\frac{{\cal
P}_{0}}{4}\,\frac{\Pi_{0}^{3}}{(1+\Pi_{0})^{2}(1+2\Pi_{0})}\,\ln{\left(\frac{\mu}{\Lambda_{\rm
IR}}\right)}\right]\,{p^{2}}+{\cal O}(p^{4})\,.$ (5.10)

Hence, the log-enhanced IR loop only gives a contribution to the quadratic
term in the expansion. Its size is small, being suppressed by a factor ${\cal
P}_{0}\simeq 10^{-9}$ with respect to the tree level term, so even a large
logarithm is unable to give large effects. The coefficients depending on
$\Pi_{0}$ give order one effects, in the limit of $\Pi_{0}$ large.

The main difference with respect to [23], who focussed on $|\eta|=6$, is that
the physical loop effects are suppressed by a factor $p^{2}$, rendering them
small at large CMB scales $p\ll 1$. It would be interesting to pursue our
large-$|\eta|$ program further, estimating the loop corrections also at
smaller scales of $p$ of order one, understanding whether the associated
divergences can be adsorbed into measurable quantities. A computation of loop
effects valid for larger $p$ was recently carried out in the interesting work
[36] using a numerical procedure, in a framework similar to the one of [23].
Probably, also in the large-$|\eta|$ set-up we discuss here, a proper
computation of the associated loop integrals need to be handled by means of a
numerical approach. It would be important to then investigate at what extend
divergences renormalize the tree-level spectrum, for example using more
sophisticated techniques as the ones pursued by [32] in a related context.
Moreover, it would be interesting to study higher loops, and consequences of
higher order interactions, in a complete framework for handling loop-
divergences in PBH scenarios in a physically controllable set-up based on a
large $|\eta|$ expansion. We leave these investigations for future works.

### Acknowledgments

It is a pleasure to thank Maria Mylova, Ogan Özsoy, and Ivonne Zavala for
useful input. GT is partially funded by the STFC grant ST/T000813/1. For the
purpose of open access, the author has applied a Creative Commons Attribution
licence to any Author Accepted Manuscript version arising.

##  Appendix A Curvature perturbations and the NSR regime

In this technical Appendix, we briefly review the results developed in [42] to
determine analytic solutions for inflationary mode functions during non-slow-
roll regimes, referring the reader to [42] for more details. Starting from the
quadratic action (2.1) of curvature perturbations, it is convenient to
introduce a Mukhanov-Sasaki variable
$v_{k}(\tau)\,=\,\zeta_{k}(\tau)/z(\tau)$, satisfying the equation

$v_{k}^{\prime\prime}(\tau)+\left[k^{2}-\frac{z^{\prime\prime}(\tau)}{z(\tau)}\right]\,v_{k}(\tau)\,=\,0\,,$
(A.1)

in momentum space. In our case, the inflationary evolution for
$\tau\leq\tau_{0}$ undergoes different phases. We have an initial slow-roll
phase for $\tau\leq\tau_{1}$, where both the slow-roll parameters
$\epsilon(\tau)$ and $\eta(\tau)$ are very small. We can approximate this as
pure de Sitter phase. Then, for $\tau_{1}\leq\tau\leq\tau_{2}$ we have non-
slow-roll evolution where $\epsilon(\tau)$ keeps small, while $\eta(\tau)$ is
negative but potentially large in size. We denote with $\epsilon_{1}$ and
$\eta$ the values of the slow-roll parameters evaluated at
$\tau\to\tau_{1}^{+}$. Finally, a slow-roll phase $\tau_{2}<\tau\leq\tau_{0}$
where the slow-roll parameters return to very small values. Again, we
approximate this last phase to pure de Sitter. We assume that the pump field
$z(\tau)$ is continuous at the transitions.

In the de Sitter limit, while $z^{\prime\prime}(\tau)/z(\tau)\,=\,2/\tau^{2}$
for $\tau\leq\tau_{1}$ and $\tau_{2}<\tau\leq\tau_{0}$, the time-profile for
this quantity can be richer. As in [42], we adopt an Ansatz

$\displaystyle v_{k}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{i\,H_{0}\,z(\tau)\,e^{-ik\tau}}{2\sqrt{\epsilon_{1}\,k^{3}}}\,{\cal
C}_{1}(k)\left[1+ik\tau+(ik\tau_{1})^{2}A_{(2)}(\tau)+(ik\tau_{1})^{3}A_{(3)}(\tau)\right]$
$\displaystyle-\frac{i\,H_{0}\,z(\tau)\,e^{ik\tau}}{2\sqrt{\epsilon_{1}\,k^{3}}}\,{\cal
C}_{2}(k)\left[1-ik\tau+(-ik\tau_{1})^{2}A_{(2)}(\tau)+(-ik\tau_{1})^{3}A_{(3)}(\tau)\right]$

for the Mukhanov-Sasaki mode function.

For $\tau<\tau_{1}$, the mode equation is the same as in a standard slow-roll
era: in order to match with the Bunch-Davies vacuum, we select $A_{(n)}=0$ for
$n\geq 2$, as well as ${\cal C}_{2}=0$ and ${\cal C}_{1}=1$ in eq
(LABEL:ansfu1). For $\tau_{1}\leq\tau\leq\tau_{2}$, we can use the Ansatz
(LABEL:ansfu1) in the evolution equation (A.1), and solve the equation order
by order in powers of $(k\tau_{1})$: see [42]. At each order $n$ in
$(k\tau_{1})^{n}$, the equation can be solved at leading order in an expansion
in the parameter $\Delta\tau$ of eq (2.4) controlling the duration of the non-
slow-roll era. For each $n$, the result depends on powers of the quantity
$d\ln{[z^{2}(\tau)/a^{2}(\tau)]}/d\ln\tau$, evaluated at time $\tau_{1}^{+}$
at the onset of the NSR era. This quantity was dubbed $\alpha$ in [42]: in the
present instance, within single field inflation with canonical kinetic terms
and in a pure de Sitter limit, it corresponds to the quantity $-\eta$ (we use
the definitions (2.3)). After computing each quantity $A_{(n)}$, the resulting
series in eq (LABEL:ansfu1) can be resummed analytically in terms of
exponentials. The result of the resummation is [42]

$\displaystyle v_{k}(\tau)$ $\displaystyle=$
$\displaystyle-\frac{i\,H_{0}\,z(\tau)\,e^{-ik\tau}}{2\sqrt{\epsilon_{1}\,k^{3}}}\left[1+ik\tau+\frac{\eta}{4}\left(1-2ik(\tau-\tau_{1})-e^{2ik(\tau-\tau_{1})}\right)\right]\,,$
(A.3)

valid for $\tau_{1}\leq\tau\leq\tau_{2}$. This mode function continuously
connects, together with its first derivative, with the mode function (and the
Bunch-Davies vacuum) for $\tau\leq\tau_{1}$. We can finally connect the result
of eq (A.3) with de Sitter mode function at later times
$\tau_{2}\leq\tau\leq\tau_{0}$, imposing continuity of the function and its
first derivative at $\tau=\tau_{2}$. The solution corresponds to Ansatz
(LABEL:ansfu1) with $A_{(n)}=0$, and the scale-dependent functions ${\cal
C}_{1}$ and ${\cal C}_{2}$ are collected in eqs (2.11) and (2.12) of the main
text.

##  Appendix B Renormalization of UV divergences

In this Appendix we briefly discuss a method for adsorbing the UV
quadratically-divergent parts (5.8) of the loop contributions into the
available parameters of the system, at least at large scales for $p\ll 1$. The
quantities available for this procedure are the overall amplitude ${\cal
P}_{0}$ defined in eq (3.2), and the factor $\Pi_{0}$ controlling the scale-
dependence of the tree level spectrum (3.7). As stated in the main text, we
can trust our results only on a large-scale, small-$p$ regime. (See discussion
around eq (5.9).) Expanding the total power spectrum (5.6) up to quadratic
order in $p$, and including the UV one-loop contributions given in eq (5.8),
we obtain:

$\displaystyle{\cal P}_{\rm tot}(p)$ $\displaystyle=$ $\displaystyle{\cal
P}_{0}\left(1-\frac{5\,\Pi_{0}\,\Lambda_{\rm UV}^{2}\,{\cal
P}_{0}\,}{6\,(1+\Pi_{0})}\right)-\frac{4{\cal
P}_{0}\,\Pi_{0}}{3}\left(1-\frac{103\,\Lambda_{\rm UV}^{2}\,{\cal
P}_{0}}{120\,(1+\Pi_{0})}\right)\,{p^{2}}+{\cal O}(p^{4})\,.$ (B.1)

The parenthesis contain the UV-divergent loop contributions, suppressed by a
factor ${\cal P}_{0}$ with respect to the tree-level terms. Higher loop
corrections give contributions to eq (B.1) with powers higher than two in
${\cal P}_{0}$. In the present one-loop instance, we can trust our results
only up to quadratic contributions ${\cal P}_{0}^{2}$. We can then adsorb the
UV-divergent parts of eq (B.1) into a redefinition of the bare quantities
${\cal P}_{0}$ and $\Pi_{0}$, which are mapped into measurable quantities
${\cal P}_{\rm ms}$ and $\Pi_{\rm ms}$ at large scales:

$\displaystyle{\cal P}_{0}$ $\displaystyle\to$ $\displaystyle{\cal P}_{\rm
ms}\left(1+\frac{5\,\Lambda_{\rm UV}^{2}\,{\Pi}_{\rm ms}\,}{6\,(1+\Pi_{\rm
ms})}\,{\cal P}_{\rm ms}\right)\,,$ (B.2) $\displaystyle\Pi_{0}$
$\displaystyle\to$ $\displaystyle\Pi_{\rm ms}\left(1+\frac{\Lambda_{\rm
UV}^{2}\,\left(103-100\Pi_{\rm ms}\right)}{120\,(1+\Pi_{\rm ms})}\,{\cal
P}_{\rm ms}\right)\,.$ (B.3)

By means of this definition, we express eq (B.1) as

$\displaystyle{\cal P}_{\rm tot}(p)$ $\displaystyle=$ $\displaystyle{\cal
P}_{\rm ms}-\frac{4{\cal P}_{\rm ms}}{3}\,\Pi_{\rm ms}\,{p^{2}}+{\cal
O}(p^{4})+{\cal O}({\cal P}_{\rm ms}^{2}).$ (B.4)

Hence quadratically divergent, one-loop effects get adsorbed into bare
quantities. The result is expressed in terms of the the measurable amplitude
${\cal P}_{\rm ms}$ of the spectrum, and on the parameter $\Pi_{\rm ms}$
controlling its scale dependence at very large scales (see the discussion
around eq (3.9)).

## References

  * [1] G. Bertone and D. Hooper, “History of dark matter,” Rev. Mod. Phys. 90 no. 4, (2018) 045002, arXiv:1605.04909 [astro-ph.CO].
  * [2] S. Hawking, “Gravitationally collapsed objects of very low mass,” Mon. Not. Roy. Astron. Soc. 152 (1971) 75.
  * [3] B. J. Carr and S. W. Hawking, “Black holes in the early Universe,” Mon. Not. Roy. Astron. Soc. 168 (1974) 399–415.
  * [4] P. Ivanov, P. Naselsky, and I. Novikov, “Inflation and primordial black holes as dark matter,” Phys. Rev. D 50 (1994) 7173–7178.
  * [5] J. Garcia-Bellido, A. D. Linde, and D. Wands, “Density perturbations and black hole formation in hybrid inflation,” Phys. Rev. D 54 (1996) 6040–6058, arXiv:astro-ph/9605094.
  * [6] M. Y. Khlopov, “Primordial Black Holes,” Res. Astron. Astrophys. 10 (2010) 495–528, arXiv:0801.0116 [astro-ph].
  * [7] J. García-Bellido, “Massive Primordial Black Holes as Dark Matter and their detection with Gravitational Waves,” J. Phys. Conf. Ser. 840 no. 1, (2017) 012032, arXiv:1702.08275 [astro-ph.CO].
  * [8] M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, “Primordial black holes—perspectives in gravitational wave astronomy,” Class. Quant. Grav. 35 no. 6, (2018) 063001, arXiv:1801.05235 [astro-ph.CO].
  * [9] B. Carr and F. Kuhnel, “Primordial Black Holes as Dark Matter: Recent Developments,” Ann. Rev. Nucl. Part. Sci. 70 (2020) 355–394, arXiv:2006.02838 [astro-ph.CO].
  * [10] A. M. Green and B. J. Kavanagh, “Primordial Black Holes as a dark matter candidate,” J. Phys. G 48 no. 4, (2021) 043001, arXiv:2007.10722 [astro-ph.CO].
  * [11] A. Escrivà, F. Kuhnel, and Y. Tada, “Primordial Black Holes,” arXiv:2211.05767 [astro-ph.CO].
  * [12] O. Özsoy and G. Tasinato, “Inflation and Primordial Black Holes,” arXiv:2301.03600 [astro-ph.CO].
  * [13] H. Motohashi and W. Hu, “Primordial Black Holes and Slow-Roll Violation,” Phys. Rev. D 96 no. 6, (2017) 063503, arXiv:1706.06784 [astro-ph.CO].
  * [14] W. H. Kinney, “Horizon crossing and inflation with large eta,” Phys. Rev. D 72 (2005) 023515, arXiv:gr-qc/0503017.
  * [15] J. Martin, H. Motohashi, and T. Suyama, “Ultra Slow-Roll Inflation and the non-Gaussianity Consistency Relation,” Phys. Rev. D 87 no. 2, (2013) 023514, arXiv:1211.0083 [astro-ph.CO].
  * [16] K. Dimopoulos, “Ultra slow-roll inflation demystified,” Phys. Lett. B 775 (2017) 262–265, arXiv:1707.05644 [hep-ph].
  * [17] H. Motohashi, A. A. Starobinsky, and J. Yokoyama, “Inflation with a constant rate of roll,” JCAP 09 (2015) 018, arXiv:1411.5021 [astro-ph.CO].
  * [18] S. Inoue and J. Yokoyama, “Curvature perturbation at the local extremum of the inflaton’s potential,” Phys. Lett. B 524 (2002) 15–20, arXiv:hep-ph/0104083.
  * [19] K. Tzirakis and W. H. Kinney, “Inflation over the hill,” Phys. Rev. D 75 (2007) 123510, arXiv:astro-ph/0701432.
  * [20] D. Wands, “Duality invariance of cosmological perturbation spectra,” Phys. Rev. D 60 (1999) 023507, arXiv:gr-qc/9809062.
  * [21] G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions,” Nucl. Phys. B 72 (1974) 461.
  * [22] S. Coleman, Aspects of Symmetry: Selected Erice Lectures.  Cambridge University Press, Cambridge, U.K., 1985.
  * [23] J. Kristiano and J. Yokoyama, “Ruling Out Primordial Black Hole Formation From Single-Field Inflation,” arXiv:2211.03395 [hep-th].
  * [24] J. Kristiano and J. Yokoyama, “Response to criticism on ”Ruling Out Primordial Black Hole Formation From Single-Field Inflation”: A note on bispectrum and one-loop correction in single-field inflation with primordial black hole formation,” arXiv:2303.00341 [hep-th].
  * [25] A. Riotto, “The Primordial Black Hole Formation from Single-Field Inflation is Not Ruled Out,” arXiv:2301.00599 [astro-ph.CO].
  * [26] S. Choudhury, M. R. Gangopadhyay, and M. Sami, “No-go for the formation of heavy mass Primordial Black Holes in Single Field Inflation,” arXiv:2301.10000 [astro-ph.CO].
  * [27] S. Choudhury, S. Panda, and M. Sami, “No-go for PBH formation in EFT of single field inflation,” arXiv:2302.05655 [astro-ph.CO].
  * [28] A. Riotto, “The Primordial Black Hole Formation from Single-Field Inflation is Still Not Ruled Out,” arXiv:2303.01727 [astro-ph.CO].
  * [29] S. Choudhury, S. Panda, and M. Sami, “Quantum loop effects on the power spectrum and constraints on primordial black holes,” arXiv:2303.06066 [astro-ph.CO].
  * [30] H. Firouzjahi, “One-loop Corrections in Power Spectrum in Single Field Inflation,” arXiv:2303.12025 [astro-ph.CO].
  * [31] H. Motohashi and Y. Tada, “Squeezed bispectrum and one-loop corrections in transient constant-roll inflation,” arXiv:2303.16035 [astro-ph.CO].
  * [32] S. Choudhury, S. Panda, and M. Sami, “Galileon inflation evades the no-go for PBH formation in the single-field framework,” arXiv:2304.04065 [astro-ph.CO].
  * [33] H. Firouzjahi and A. Riotto, “Primordial Black Holes and Loops in Single-Field Inflation,” arXiv:2304.07801 [astro-ph.CO].
  * [34] E. Tomberg, “Stochastic constant-roll inflation and primordial black holes,” arXiv:2304.10903 [astro-ph.CO].
  * [35] H. Firouzjahi, “Loop Corrections in Gravitational Wave Spectrum in Single Field Inflation,” arXiv:2305.01527 [astro-ph.CO].
  * [36] G. Franciolini, A. Iovino, Junior., M. Taoso, and A. Urbano, “One loop to rule them all: Perturbativity in the presence of ultra slow-roll dynamics,” arXiv:2305.03491 [astro-ph.CO].
  * [37] A. Escrivà, “PBH Formation from Spherically Symmetric Hydrodynamical Perturbations: A Review,” Universe 8 no. 2, (2022) 66, arXiv:2111.12693 [gr-qc].
  * [38] G. Domènech, “Scalar Induced Gravitational Waves Review,” Universe 7 no. 11, (2021) 398, arXiv:2109.01398 [gr-qc].
  * [39] C. Pattison, V. Vennin, H. Assadullahi, and D. Wands, “Quantum diffusion during inflation and primordial black holes,” JCAP 10 (2017) 046, arXiv:1707.00537 [hep-th].
  * [40] J. M. Ezquiaga, J. García-Bellido, and V. Vennin, “The exponential tail of inflationary fluctuations: consequences for primordial black holes,” JCAP 03 (2020) 029, arXiv:1912.05399 [astro-ph.CO].
  * [41] D. G. Figueroa, S. Raatikainen, S. Rasanen, and E. Tomberg, “Implications of stochastic effects for primordial black hole production in ultra-slow-roll inflation,” JCAP 05 no. 05, (2022) 027, arXiv:2111.07437 [astro-ph.CO].
  * [42] G. Tasinato, “An analytic approach to non-slow-roll inflation,” Phys. Rev. D 103 no. 2, (2021) 023535, arXiv:2012.02518 [hep-th].
  * [43] A. R. Liddle and D. H. Lyth, Cosmological inflation and large scale structure.  2000\.
  * [44] C. T. Byrnes, P. S. Cole, and S. P. Patil, “Steepest growth of the power spectrum and primordial black holes,” JCAP 06 (2019) 028, arXiv:1811.11158 [astro-ph.CO].
  * [45] O. Özsoy and G. Tasinato, “On the slope of the curvature power spectrum in non-attractor inflation,” JCAP 04 (2020) 048, arXiv:1912.01061 [astro-ph.CO].
  * [46] J. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 05 (2003) 013, arXiv:astro-ph/0210603.
  * [47] X. Chen, M.-x. Huang, S. Kachru, and G. Shiu, “Observational signatures and non-Gaussianities of general single field inflation,” JCAP 01 (2007) 002, arXiv:hep-th/0605045.
  * [48] D. Seery and J. E. Lidsey, “Primordial non-Gaussianities in single field inflation,” JCAP 06 (2005) 003, arXiv:astro-ph/0503692.
  * [49] O. Özsoy and G. Tasinato, “CMB $\mu$T cross correlations as a probe of primordial black hole scenarios,” Phys. Rev. D 104 no. 4, (2021) 043526, arXiv:2104.12792 [astro-ph.CO].
  * [50] O. Özsoy and G. Tasinato, “Consistency conditions and primordial black holes in single field inflation,” Phys. Rev. D 105 no. 2, (2022) 023524, arXiv:2111.02432 [astro-ph.CO].
  * [51] C. T. Byrnes, S. Nurmi, G. Tasinato, and D. Wands, “Scale dependence of local fNL,” JCAP 02 (2010) 034, arXiv:0911.2780 [astro-ph.CO].
  * [52] C. T. Byrnes, M. Gerstenlauer, S. Nurmi, G. Tasinato, and D. Wands, “Scale-dependent non-Gaussianity probes inflationary physics,” JCAP 10 (2010) 004, arXiv:1007.4277 [astro-ph.CO].
  * [53] X. Chen, “Running non-Gaussianities in DBI inflation,” Phys. Rev. D 72 (2005) 123518, arXiv:astro-ph/0507053.
  * [54] C. T. Byrnes, E. J. Copeland, and A. M. Green, “Primordial black holes as a tool for constraining non-Gaussianity,” Phys. Rev. D 86 (2012) 043512, arXiv:1206.4188 [astro-ph.CO].
  * [55] S. Young and C. T. Byrnes, “Primordial black holes in non-Gaussian regimes,” JCAP 08 (2013) 052, arXiv:1307.4995 [astro-ph.CO].
  * [56] S. Passaglia, W. Hu, and H. Motohashi, “Primordial black holes and local non-Gaussianity in canonical inflation,” Phys. Rev. D 99 no. 4, (2019) 043536, arXiv:1812.08243 [astro-ph.CO].
  * [57] M. Biagetti, V. De Luca, G. Franciolini, A. Kehagias, and A. Riotto, “The formation probability of primordial black holes,” Phys. Lett. B 820 (2021) 136602, arXiv:2105.07810 [astro-ph.CO].
  * [58] V. Atal and C. Germani, “The role of non-gaussianities in Primordial Black Hole formation,” Phys. Dark Univ. 24 (2019) 100275, arXiv:1811.07857 [astro-ph.CO].
  * [59] M. Taoso and A. Urbano, “Non-gaussianities for primordial black hole formation,” JCAP 08 (2021) 016, arXiv:2102.03610 [astro-ph.CO].
  * [60] S. Weinberg, “Quantum contributions to cosmological correlations,” Phys. Rev. D 72 (2005) 043514, arXiv:hep-th/0506236.
  * [61] M. S. Sloth, “On the one loop corrections to inflation and the CMB anisotropies,” Nucl. Phys. B 748 (2006) 149–169, arXiv:astro-ph/0604488.
  * [62] D. Seery, “One-loop corrections to the curvature perturbation from inflation,” JCAP 02 (2008) 006, arXiv:0707.3378 [astro-ph].
  * [63] L. Senatore and M. Zaldarriaga, “On Loops in Inflation,” JHEP 12 (2010) 008, arXiv:0912.2734 [hep-th].
  * [64] N. Bartolo, E. Dimastrogiovanni, and A. Vallinotto, “One-Loop Corrections to the Power Spectrum in General Single-Field Inflation,” JCAP 11 (2010) 003, arXiv:1006.0196 [astro-ph.CO].
  * [65] D. Boyanovsky, H. J. de Vega, and N. G. Sanchez, “Quantum corrections to slow roll inflation and new scaling of superhorizon fluctuations,” Nucl. Phys. B 747 (2006) 25–54, arXiv:astro-ph/0503669.
  * [66] B. Delamotte, “A Hint of renormalization,” Am. J. Phys. 72 (2004) 170–184, arXiv:hep-th/0212049.