WUCG-23-07 YITP-23-69

# Highly asymmetric probability distribution from a finite-width upward step
during inflation

Ryodai<EMAIL_ADDRESS>Tomohiro Fujita2,3 Misao
Sasaki4,5,6 1Department of Physics, Waseda University, 3-4-1 Okubo, Shinjuku,
Tokyo 169-8555, Japan  2Waseda Institute for Advanced Study, Waseda
University, 1-6-1 Nishi-Waseda, Shinjuku, Tokyo 169-8050, Japan  3Research
Center for the Early Universe, The University of Tokyo, Bunkyo, Tokyo
113-0033, Japan  4Kavli Institute for the Physics and Mathematics of the
Universe (WPI), The University of Tokyo Institutes for Advanced Study, The
University of Tokyo, Chiba 277-8583, Japan  5Center for Gravitational Physics
and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto
University, Kyoto 606-8502, Japan  6Leung Center for Cosmology and Particle
Astrophysics, National Taiwan University, Taipei 10617, Taiwan

###### Abstract

We study a single-field inflation model in which the inflaton potential has an
upward step between two slow-roll regimes by taking into account the finite
width of the step. We calculate the probability distribution function (PDF) of
the curvature perturbation $P[{\cal{R}}]$ using the $\delta N$ formalism. The
PDF has an exponential-tail only for positive ${\cal{R}}$ whose slope depends
on the step width. We find that the tail may have a significant impact on the
estimation of the primordial black hole abundance. We also show that the PDF
$P[{\cal{R}}]$ becomes highly asymmetric on a particular scale exiting the
horizon before the step, at which the curvature power spectrum has a dip. This
asymmetric PDF may leave an interesting signature in the large scale structure
such as voids.

##  I Introduction

The inflation theory Starobinsky (1980); Sato (1981); Kazanas (1980); Guth
(1981); Linde (1982); Albrecht and Steinhardt (1982), originally introduced to
resolve the problems of the Big Bang cosmology, is currently accepted as a
part of the standard cosmological model. In particular, slow-roll (SR)
inflation driven by a single scalar field produces primordial fluctuations,
which are nearly scale-invariant and almost Gaussian, consistent with the
observed cosmic microwave background (CMB) temperature anisotropies Aghanim
_et al._ (2020); Akrami  _et al._ (2020). Hence, single field SR inflation is
considered to be a strong candidate as an inflation model. However, the
specific mechanism causing inflation is still unknown, and various inflation
models have been proposed within the framework of SR inflation
(e.g.Starobinsky (1980); Linde (1983); Freese  _et al._ (1990); Bezrukov and
Shaposhnikov (2008); Kallosh  _et al._ (2013)). Even among the canonical
single scalar field models, diverse types of inflaton potentials have been
studied so far. Especially at smaller scales than the CMB scale ($k\sim
0.002\text{Mpc}^{-1}$), there are only a little observational constraints and
many possible inflation scenarios are allowed.

Inflationary models associated with primordial black holes (PBHs) Zel’dovich
and Novikov (1967); Hawking (1971); Carr and Hawking (1974); Carr (1975) have
been intensively studied in recent years (e.g. ultra-slow-roll (USR) models
Garcia-Bellido and Ruiz Morales (2017); Kannike  _et al._ (2017); Germani and
Prokopec (2017); Ezquiaga  _et al._ (2018); Motohashi and Hu (2017)). PBHs are
black holes formed through the gravitational collapse of regions of high
density fluctuations above a threshold in the early universe. They are
hypothetical objects but can play many different roles in the history of the
Universe depending on their mass (see Khlopov (2010); Sasaki  _et al._ (2018);
Carr and Kuhnel (2020); Green and Kavanagh (2021); Carr  _et al._ (2021);
Villanueva-Domingo  _et al._ (2021); Carr and Kuhnel (2022); Escrivà  _et al._
(2022); Karam  _et al._ (2023); Özsoy and Tasinato (2023) for recent reviews).
The density fluctuations originate in the curvature perturbation $\cal{R}$
generated during inflation, and the probability of the realisations of large
curvature perturbation must be amplified by some mechanism for a sufficient
abundance of PBH to be formed. The simplest way to achieve this is to enhance
the power spectrum (i.e. the variance) of the curvature perturbation on a
scale small compared to the CMB scale.

In general, however, the power spectrum alone does not completely fix how
likely the production of large amplitude curvature perturbations is. The
probabilistic characteristics of the random variable are determined solely by
the power spectrum only when it follows a Gaussian distribution. In discussing
deviations from Gaussian distribution in the context of primordial cosmology,
higher-order correlators are often used. In particular, the non-gaussian
parameter $f_{\rm NL}$ associated with the bispectrum has been discussed
extensively Komatsu and Spergel (2001); Bartolo  _et al._ (2002); Maldacena
(2003); Acquaviva  _et al._ (2003); Bartolo  _et al._ (2004); Chen (2010).
While SR inflation produces almost Gaussian curvature perturbations
characterized by $f_{\rm NL}\ll 1$, the inflation models in which the SR
condition is violated can produce relatively large $f_{\rm NL}$ (see e.g.
Namjoo  _et al._ (2013); Chen  _et al._ (2013a, b); Martin  _et al._ (2013);
Cai  _et al._ (2018)). It has been pointed out that the PBH abundance is very
sensitive to non-gaussianity in such cases (e.g. Bullock and Primack (1997);
Ivanov (1998); Yokoyama (1998); Byrnes  _et al._ (2012); Bugaev and Klimai
(2013); Young and Byrnes (2013); Young  _et al._ (2016); Franciolini  _et al._
(2018); Biagetti  _et al._ (2018); Atal and Germani (2019); Passaglia  _et
al._ (2019); De Luca  _et al._ (2019); Taoso and Urbano (2021); Davies  _et
al._ (2022); De Luca and Riotto (2022); Matsubara and Sasaki (2022)).

The large and rare fluctuations that cause PBH formation cannot be precisely
assessed by perturbative methods alone such as the $f_{\rm NL}$ parameter. To
capture their nature, we can use a powerful non-perturbative approach called
the $\delta N$ formalism Salopek and Bond (1990); Sasaki and Stewart (1996);
Starobinsky (1985); Sasaki and Tanaka (1998); Lyth  _et al._ (2005); Lee  _et
al._ (2005); Lyth and Rodriguez (2005); Abolhasani  _et al._ (2019). This
formalism allows a direct conversion from the probability distribution
function (PDF) of the scalar field perturbation $\delta\varphi$ to the PDF of
$\cal{R}$. It is known that in inflationary models leading to PBH formation,
an exponential tail appears in the PDF of $\cal{R}$, which has a significant
impact on the PBH abundance Atal  _et al._ (2019, 2020); Biagetti  _et al._
(2021); Kitajima  _et al._ (2021); Gow  _et al._ (2023); Ferrante  _et al._
(2023); Pi and Sasaki (2022) (see also stochastic approach Pattison  _et al._
(2017); Ezquiaga  _et al._ (2020); Figueroa  _et al._ (2021); Pattison  _et
al._ (2021); Ahmadi  _et al._ (2022); Animali and Vennin (2023)).

This paper focuses on a model where the inflaton potential has an upward step
between two SR regions Inomata  _et al._ (2022); Cai  _et al._ (2022a, b).
Models with such an abrupt change have been studied extensively (e.g. Gallego
Cadavid and Romano (2015); Atal  _et al._ (2019); Mishra and Sahni (2020);
Kefala  _et al._ (2021); Inomata  _et al._ (2021, 2022); Dalianis  _et al._
(2021); Zheng  _et al._ (2022); Cai  _et al._ (2022a, b); Kawaguchi and
Tsujikawa (2023); Gu  _et al._ (2023); Fu and Wang (2023); Cai  _et al._
(2023)) and tend to be consistent with the CMB observations, as the transition
takes less time and does not affect much the SR inflaton dynamics on the CMB
scale. Previous studies of the upward step model calculated the PDF of
$\cal{R}$ by ignoring the finite width of the step and reported that the PDF
has a hard cutoff at a certain value of $\cal{R}$ beyond which the PDF is zero
Cai  _et al._ (2022a, b). In this paper, we take a step width $\Delta\varphi$
into consideration. We introduce a smooth step whose first-order derivative is
continuous, and investigate the probabilistic features of the curvature
perturbation using the classical $\delta N$ formalism. We also discuss the
impact of the finite step width on the PBH abundance222Recently, there are
active debates if the enhancement of the curvature power spectrum on the small
scale for PBH generation could significantly affect the power spectrum on the
CMB scale due to the one-loop correction, since Ref. Kristiano and Yokoyama
(2022) raised this issue (see also Inomata  _et al._ (2023)). There are some
counter arguments and this is still unsettled issue Choudhury  _et al._
(2023a, b, c); Kristiano and Yokoyama (2023); Firouzjahi (2023); Riotto
(2023a, b); Firouzjahi and Riotto (2023); Franciolini  _et al._ (2023);
Tasinato (2023); Motohashi and Tada (2023); Choudhury  _et al._ (2023d). In
particular, it is advocated that the loop correction can be sufficiently
suppressed by considering smooth transition between SR and USR Riotto (2023a,
b); Firouzjahi and Riotto (2023). We do not address this issue in the present
paper, but it gives another motivation for introducing the step width
$\Delta\varphi$ that smooths the transition..

We obtain two main results. The first reveals that a tail dependent on the
step width $\Delta\varphi$ appears in the PDF, and the cutoff feature
disappears. In the zero-width limit $\Delta\varphi\rightarrow 0$, the tail
becomes steeper and eventually the cutoff is reproduced. The complementary
cumulative distribution function is then calculated, showing that the step
width $\Delta\varphi$ has a significant impact on the estimate of PBH
abundance. The second is the discovery of a PDF with very high asymmetry. At
certain scales exiting the Hubble horizon before the step, $\cal{R}$ cannot
take large negative values and the PDF becomes asymmetric. The scale that is
most asymmetric is shown to correspond to the dip scale, where
$\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{2}\rangle$ and
$\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{3}\rangle$ are the smallest
compared to the other scales. By illustrating the spatial distribution
realized with the highly asymmetric PDF, we show that the fraction of low-
density regions is smaller at the dip scale.

This paper is organized as follows. In Sec. II, we explain the details of our
model in which the inflaton potential includes an upward step between two SR
regions. In Sec. III, the background equation of motion is solved, and in Sec.
IV we derive the relation between the curvature perturbation $\cal{R}$ and the
scalar perturbation $\delta\varphi$ using the $\delta N$ formalism. In Sec. V,
we compute the PDF and the CCDF of $\cal{R}$ and show that the exponential
tail, which depends on $\Delta\varphi$, appears. In Sec. VI we show that the
highly asymmetric PDF appears and relate it to the dip scale. We then briefly
discuss the implications of the highly asymmetric PDF for the structure of the
universe. Sec. VII is devoted to conclusions.

##  II Upward-step model

In this section, we describe the setup of our model. Let us consider a single-
field inflationary model with a scalar field $\phi$,

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

where $g$ is a determinant of the metric tensor $g_{\mu\nu}$ and $V(\phi)$ is
a potential of the scalar field. We consider a spatially flat Friedmann-
Lemaître-Robertson-Walker background, ${\rm d}s^{2}=-{\rm
d}t^{2}+a^{2}(t)\delta_{ij}{\rm d}x^{i}{\rm d}x^{j}\,,$ where $a(t)$ is a
time-dependent scale factor. From the above action, we obtain two background
equations of motion

$\displaystyle 3h^{2}=\frac{1}{2}h^{2}\pi^{2}+v\,,$ (2)
$\displaystyle\frac{{\rm d}\pi}{{\rm
d}n}+\frac{v}{h^{2}}\pi+\frac{\partial_{\varphi}v}{h^{2}}=0\,,$ (3)

where we introduced dimensionless quantities, $\varphi\equiv\phi/M_{pl}$,
$\pi\equiv{\rm d}\varphi/{\rm d}n$, $v\equiv V(\phi)/V_{0}$ and $h\equiv
M_{pl}H/\sqrt{V_{0}}$. $V_{0}$ is an arbitrary reference point of the
potential, $H=\dot{a}/a$ is the Hubble expansion rate and $n$ is the number of
e-folds defined by ${\rm d}n=H{\rm d}t$. Without loss of generality, we assume
that $\varphi$ moves from positive to zero along the potential and $\pi$ is
always negative during inflation.

Figure 1:  (Left panel) Schematic figure of the potential (black solid line)
and the scalar field (orange balls). Two slow-roll (SR) potentials are
connected by a smooth upward step with the width $\Delta\varphi$. The black,
magenta and gray arrows represent the SR, step-climbing and relaxation phases,
respectively. (Right panel) Schematic figure of the background trajectory in
the phase space. The scalar field evolves from right to left in time, and
experiences (i) the first SR stage, (ii) the step stage highlighted by magenta
shading, and (iii) the second SR stage. At the beginning of the second SR
stage, the trajectory deviates from the SR attractor (red dashed line), which
is called the relaxation phase. For reference, we superimposed the shape of
the linear order power spectrum on the right panel and indicated the positions
of its peak and dip (red and blue stars).

In our model, the potential $v(\varphi)$ has an upward step-like feature
between two slow-roll regions,

$v(\varphi)=\begin{cases}v_{\rm sr1}(\varphi)\hskip
16.50264pt(\varphi\geq\varphi_{1})\\\ f_{\rm step}(\varphi)\hskip
12.51927pt(\varphi_{2}<\varphi<\varphi_{1})\\\ v_{\rm sr2}(\varphi)\hskip
16.36024pt(\varphi\leq\varphi_{2})\end{cases}\,,$ (4)

where $v_{\rm sr1}$ and $v_{\rm sr2}$ are slow-roll (SR) potentials, which are
generally different from each other and satisfy $v_{\rm
sr2}(\varphi_{2})>v_{\rm sr1}(\varphi_{1})$ to have an upward step in between.
The width of the step region will be an important parameter

$\Delta\varphi\equiv\varphi_{1}-\varphi_{2}.$ (5)

For the upward step $f_{\rm step}$, we employ quadratic functions

$f_{\rm step}(\varphi)=\begin{cases}A_{1}+B_{1}(\varphi-\varphi_{\rm
min})^{2}\hskip 13.6572pt(\varphi_{c}\leq\varphi<\varphi_{1}:\hskip
5.69046pt\text{S1 region})\\\ A_{2}+B_{2}(\varphi-\varphi_{\rm max})^{2}\hskip
12.51927pt(\varphi_{2}<\varphi<\varphi_{c}:\hskip 5.69046pt\text{S2
region})\end{cases}\,,$ (6)

where $\varphi_{c}\equiv(\varphi_{1}+\varphi_{2})/2$ is the midpoint of the
step region. $\varphi_{\rm min}$ and $\varphi_{\rm max}$ denote the minimum
and maximum of the quadratic potential, respectively. Requiring that the
potential and its first $\varphi$-derivative are continuous at
$\varphi=\varphi_{c}$, $\varphi_{1}$, and $\varphi_{2}$, the six constants
$A_{1},A_{2},B_{1},B_{2},\varphi_{\rm min},\varphi_{\rm max}$ are thereby
determined as shown in Appendix. A.

The left panel of Fig. 1 schematically shows our potential $v(\varphi)$. We
chose $v_{\rm sr1}$ and $v_{\rm sr2}$ so that their slopes are different
before and after the upward step, for example, $\partial_{\varphi}v_{\rm
sr1}(\varphi_{1})>\partial_{\varphi}v_{\rm sr2}(\varphi_{2})$. The right panel
of Fig. 1 schematically shows the background trajectory in the phase space.
Due to the presence of the upward step, there are three distinct stages of the
background evolution: (i) The first stage, $\varphi\geq\varphi_{1}$, is the
first SR stage where the background trajectory is on the SR attractor. (ii)
The second, $\varphi_{2}<\varphi<\varphi_{1}$, is the step stage. In the step
stage, the scalar field rapidly loses its kinetic energy and $|\pi|$ becomes
smaller than before the step. (iii) The third, $\varphi\leq\varphi_{2}$, is
the second SR stage. Since $|\pi|$ is made smaller by the step, the trajectory
in the phase space is not on the SR attractor right after the scalar field
finishes climbing up the step. Therefore, after the step, a relaxation phase
takes place and the scalar field asymptotically approaches to the second SR
attractor. Eventually, the trajectory can be regarded as settling on the SR
attractor, and we call the scalar field value at that time as $\varphi_{f}$.

For later convenience, we conclude this section by defining useful quantities
as follows.

$\displaystyle\eta_{i}\equiv 2(2\epsilon_{Vi}-\eta_{Vi})\,,$ (7)
$\displaystyle g\equiv\frac{\pi_{2}}{\pi_{1}}<1\,,$ (8)
$\displaystyle\kappa\equiv\sqrt{\frac{\epsilon_{V1}}{\epsilon_{V2}}}\,,$ (9)
$\displaystyle\omega_{\rm
s1}\equiv\sqrt{\frac{6B_{1}}{A_{1}}}\simeq\frac{\sqrt{2}~{}|\pi_{1}|}{\Delta\varphi}\,,$
(10) $\displaystyle\omega_{\rm
s2}\equiv\sqrt{\frac{-6B_{2}}{A_{2}}}\simeq\frac{\sqrt{2}~{}|\pi_{1}|}{\Delta\varphi}\,,$
(11)

where $\epsilon_{V}\equiv(\partial_{\varphi}v/v)^{2}/2$ and
$\eta_{V}\equiv\partial_{\varphi\varphi}v/v$ are the potential slow-roll
parameters and the subscript $i$ indicates that a quantity is evaluated at
$\varphi=\varphi_{i}$ ($i=1,2$). The approximations used in Eqs. (10) and (11)
assume $\Delta\varphi\ll|\pi_{1}|$ and $g\ll 1$, and the dependence of
$\omega_{\rm s1}$ and $\omega_{\rm s2}$ on $\Delta\varphi$ will be emphasized
in the later discussion.

##  III Background solution

In this section, we briefly discuss the background behavior of the scalar
field in each of the three stages. In solving the background equation of
motion (3), we use the following three approximations. (I) The kinetic energy
of the scalar field is subdominant compared to the potential energy,
$h^{2}\pi^{2}\ll v$. (II) Assuming the width of the step $\Delta\varphi$ is
sufficiently small, we keep only its leading contributions. Consequently,
$\varphi_{\rm min}$ and $\varphi_{\rm max}$ are identified with $\varphi_{1}$
and $\varphi_{2}$, respectively. (III) The Hubble friction is negligible in
the step stage. The detailed derivations of the solutions and the discussions
on the limitations of the above approximations can be found in Appendices. B
and D.

In the SR stages, a general SR solution is applicable,

$\varphi(n)-\varphi_{i}=\frac{2\sqrt{2\epsilon_{Vi}}}{\eta_{i}}\left(1-e^{\frac{\eta_{i}}{2}(n-n_{i})}\right)+\frac{1}{3}\left(\pi_{i}+\sqrt{2\epsilon_{Vi}}\right)\left(e^{\frac{\eta_{i}}{2}(n-n_{i})}-e^{-(3+\frac{\eta_{i}}{2})(n-n_{i})}\right)\,.$
(12)

Again, we used the shorthand notation $X_{i}\equiv X(\varphi_{i})$. One needs
to know $\pi_{i}$ to fix the boundary condition. In the first SR stage, the
background trajectory is on the SR attractor, i.e.
$\pi_{1}=-\sqrt{2\epsilon_{V1}}$ and hence the second term in Eq. (12)
vanishes. In the second SR stage, however, the trajectory is not on the
attractor in the relaxation phase (see Fig. 1). To find $\pi_{2}$ we use the
approximated energy conservation during the step stage,

$\pi_{2}=-\sqrt{\pi_{1}^{2}+6\log\left(\frac{v(\varphi_{1})}{v(\varphi_{2})}\right)}\,.$
(13)

Here $\pi^{2}/2+3\log(v)$ is conserved, because the background equation of
motion with the above approximations (I) and (III) reduces to ${\rm d}\pi/{\rm
d}n+3\partial_{\varphi}\log(v)=0$ as shown in Eq. (B.1).

The step stage is split into the S1 and S2 region, which has a normal and
inverted harmonic potential, respectively (see Eq. (6)). The background
solutions in these regions are

$\displaystyle\varphi(n)-\varphi_{1}=\frac{\pi_{1}}{\omega_{\rm
s1}}\sin\left(\omega_{\rm
s1}(n-n_{1})\right)\,,\qquad(\varphi_{c}\leq\varphi<\varphi_{1})$ (14)
$\displaystyle\varphi(n)-\varphi_{2}=\frac{\pi_{2}}{\omega_{\rm
s2}}\sinh\left(\omega_{\rm
s2}(n-n_{2})\right)\,.\qquad(\varphi_{2}<\varphi<\varphi_{c})$ (15)

Note that for extremely small $\pi_{2}$, Eq. (15) can be inverted as
$\omega_{s2}(n_{2}-n)\simeq\log(2\omega_{s2}(\varphi(n)-\varphi_{2})/|\pi_{2}|)$.
This implies that it takes logarithmically longer time for the background
inflaton to climb up the upward step for smaller $\pi_{2}$. It will have a
significant impact on the final result.

##  IV Calculation of $\delta N$

We now turn our attention to the the curvature perturbation $\cal{R}$.
According to the $\delta N$ formalism, the curvature perturbation $\cal{R}$
can be calculated as the difference in the number of e-folds between the
perturbed spacetime and the background spacetime Salopek and Bond (1990);
Sasaki and Stewart (1996); Starobinsky (1985); Sasaki and Tanaka (1998); Lyth
_et al._ (2005); Lee  _et al._ (2005); Lyth and Rodriguez (2005),

${\cal{R}}=\delta N\equiv
N(\varphi+\delta\varphi,\pi+\delta\pi;\varphi_{f},\pi_{f})-N(\varphi,\pi;\varphi_{f},\pi_{f}),$
(16)

where $N(\varphi,\pi;\varphi^{\prime},\pi^{\prime})$ is the number of e-folds
for which the inflaton takes to evolve from $(\varphi,\pi)$ to
$(\varphi^{\prime},\pi^{\prime})$ in the phase space. In this paper, we assign
$(\varphi,\pi)$ to the background value at the horizon crossing time for a
scale of interest $k$. Note that the velocity perturbation $\delta\pi$ at the
starting point is assumed to be negligibly small. In what follows, we start
from $\varphi$ on the first SR attractor ($\varphi>\varphi_{1}$), and compute
$\delta N$ generated in all stages 333 The e-folding number taken for the
inflaton to pass through the step stage is only about $\Delta N\sim
1/\omega_{\rm s1}\ll 1$ and the corresponding band of wavenumbers that exit
the Hubble horizon during the step stage, is very narrow. Thus, we do not
consider curvature perturbations on such scales in this paper. The scales
crossing the Hubble horizon after the step stage are not studied in this paper
either, as they are similar to the normal SR case, although the evolution of
$\delta\varphi$ well inside the horizon is affected by the step..

In the first SR stage, the initial perturbation $\delta\varphi$ induces
$\delta N$ as well as the velocity perturbation at the end of the first stage
$\delta\pi_{1}$. We obtain them by comparing Eq. (12) at $i=1$ to its
perturbed expression with $\varphi\to\varphi+\delta\varphi$ and
$\pi_{1}\to\pi_{1}+\delta\pi_{1}$ as (see Appendix.C.1 for derivation)

$\delta
N^{(1)}\simeq-\frac{\delta\varphi}{\pi}\,,\qquad\delta\pi_{1}\simeq-\frac{\eta_{1}}{2}\left(\frac{\pi}{\pi_{1}}\right)^{\frac{6}{\eta_{1}}}\delta\varphi\,,$
(17)

where $\delta N^{(1)}$ denotes the contribution to total $\delta N$ from the
first SR stage. This approximation is valid for $|\delta N^{(1)}|\ll 1/3$. One
can apparently choose $\varphi$ from any values within the first SR stage.
However, since $6/\eta_{1}$ is large, if we choose $\varphi$ far from the
step, $\delta\pi_{1}$ would be substantially suppressed by the factor
$(\pi/\pi_{1})^{6/\eta_{1}}$. This is a consequence of the fact that the
background trajectory is an attractor in the first SR stage. If
$\delta\pi_{1}$ is strongly suppressed, there would be no significant
contribution to total $\delta N$ from the subsequent evolution in the step
stage. In this paper, therefore, we consider $\varphi$ close to $\varphi_{1}$
and investigate the effect of the upward step on the curvature perturbations,
that exit the horizon slightly before entering the step stage.

In the second SR stage, $\delta N$ is produced by $\delta\pi_{2}$, contrary to
$\delta N^{(1)}$ induced by $\delta\varphi$. Perturbing Eq. (13), we find that
$\delta\pi_{2}$ is given by

$\delta\pi_{2}=\pi_{2}\left(\sqrt{1+\frac{2}{g^{2}}\frac{\delta\pi_{1}}{\pi_{1}}+\frac{1}{g^{2}}\left(\frac{\delta\pi_{1}}{\pi_{1}}\right)^{2}}-1\right)\,,$
(18)

where the Hubble friction in the step stage was ignored. Using Eq. (12) at
$i=2$ in a similar way to Eq. (17) but only with
$\pi_{2}\to\pi_{2}+\delta\pi_{2}$, we obtain the contribution to $\delta N$
from the second SR stage as (see Appendix.C.2)

$\delta N^{(2)}\simeq-\frac{\kappa
g}{3}\frac{\delta\pi_{2}}{\pi_{2}}\simeq\frac{\kappa
g}{3}\left(1-\sqrt{1+\frac{2}{g^{2}}\frac{\delta\pi_{1}}{\pi_{1}}+\frac{1}{g^{2}}\left(\frac{\delta\pi_{1}}{\pi_{1}}\right)^{2}}\right)\,.$
(19)

Note that the first approximate equality in Eq. (19) relies only on
$|1/\eta_{2}|\gg 1$ and we do not assume that $\delta\pi_{2}/\pi_{2}$ is
small. Indeed, we will consider the case with
$|\delta\pi_{2}/\pi_{2}|=\mathcal{O}(1)$ soon below.

The contribution to $\delta N$ from the step stage is divided into two: one
from the S1 region $\delta N^{(\rm s1)}$ and the other from the S2 region
$\delta N^{(\rm s2)}$. The former is highly suppressed by $\Delta\varphi$ and
$(\pi/\pi_{1})^{6/\eta_{1}}$, and negligible compared to $\delta N^{(1)}$ (see
Appendix.C.3). In contrast, $\delta N^{(\rm s2)}$ can be significant. As
discussed below Eq. (15), the background e-folds of the S2 region is given by

$N^{\rm(s2)}\simeq\frac{1}{\omega_{\rm
s2}}\sinh^{-1}\left(\frac{\Delta\varphi}{2|\pi_{2}|}\omega_{\rm
s2}\right)\simeq\frac{1}{\omega_{\rm
s2}}\log\left(\frac{\Delta\varphi}{|\pi_{2}|}\omega_{\rm s2}\right)\,.$ (20)

Perturbing it, we obtain

$\delta N^{\rm(s2)}\simeq-\frac{1}{\omega_{\rm
s2}}\log\left(1+\frac{\delta\pi_{2}}{\pi_{2}}\right)\,.$ (21)

It is important to note that when $\delta\pi_{2}$ is comparable to $-\pi_{2}$,
$\delta N^{\rm(s2)}$ may diverge to infinity. This corresponds to the
perturbed cases in which the inflaton exhausts the most of the kinetic energy
to climb up the step and takes an enormous amount of time to pass through it.
As we will see below, $\delta N^{\rm(s2)}$ makes a significant impact on the
curvature perturbation.

Summing up the contributions to $\delta N$ from the three stages, Eqs.
(17),(19) and (21), we obtain

${\cal{R}}=\beta\delta\varphi+\frac{\kappa
g}{3}\left(1-\sqrt{1+\frac{2\gamma}{g^{2}}\delta\varphi+\frac{\gamma^{2}}{g^{2}}\delta\varphi^{2}}\right)-\frac{1}{2\omega_{\rm
s2}}\log\left(1+\frac{2\gamma}{g^{2}}\delta\varphi+\frac{\gamma^{2}}{g^{2}}\delta\varphi^{2}\right)\,,$
(22)

where $\beta$ and $\gamma$ are given by

$\displaystyle\beta=-\frac{1}{\pi}>0\,,\qquad\gamma=\frac{\eta_{1}\beta}{2}\left(\frac{\pi}{\pi_{1}}\right)^{\frac{6}{\eta_{1}}}\simeq\frac{\eta_{1}\beta}{2}\left(\frac{k}{k_{1}}\right)^{3}\,.$
(23)

The second and third terms in Eq. (22) originate from the upward step. The
third term particularly comes from the finite width of the step. In fact,
vanishing the step width $\Delta\varphi\to 0$ (i.e. $\omega_{\rm
s2}\to\infty$) and dropping the third term would reproduce the results of
previous papers that ignored the width of the step Cai  _et al._ (2022a, b).
Instead, substituting $g=1$ into Eq. (22), we obtain a similar logarithmic
dependence on $\delta\varphi$ to the result in the bump-type model Atal  _et
al._ (2019)444 In the limit $g\rightarrow 1$ while keeping $\Delta\varphi$
finite, our step potential becomes a bump feature, not the featureless SR
potential. The previous paper Atal  _et al._ (2019) computed $\delta N$ only
in the bump region. Hence, it is reasonable that the logarithmic term in Eq.
(22) approaches to Eq. (1.5) in the previous paper Atal  _et al._ (2019) for
$g=1$ and $\varphi=\varphi_{1}$. However, its coefficient is not exactly the
same but differs by $\sqrt{3}$, probably because our assumption (II) i.e.
$\varphi_{\rm max}\simeq\varphi_{2}$ and $\varphi_{\rm min}\simeq\varphi_{1}$,
is broken in the limit $g\to 1$.

##  V Probability distribution function of $\cal{R}$

The second and third terms in Eq. (22) make the curvature perturbation highly
non-Gaussian. To investigate the properties of $\cal{R}$, we calculate its
probability distribution function (PDF). The PDF of $\cal{R}$ is associated
with the PDF of $\delta\varphi$ by probability conservation,
$P[{\cal{R}}]=P[\delta\varphi]|{\rm d}\delta\varphi/{\rm d}{\cal{R}}|$. For
simplicity, we assume that $\delta\varphi$ follows a Gaussian distribution
with variance $\sigma_{\delta\varphi}^{2}$.555We mention that $\delta\varphi$
may have a non-Gaussianity of non-local type, arising from non-trivial higher
order interactions. However, since such a non-Gaussianity is known to be
perturbatively small in comparison with the local type non-Gaussianity for a
canonical scalar field Chen  _et al._ (2008), we ignore it. Considering a
possible non-perturbative effect of a non-local type non-Gaussianity is an
important issue, but it is beyond the scope of the present paper. From Eq.
(22), the PDF of $\cal{R}$ is given by

$P[{\cal{R}}]={\displaystyle\sum_{i}}\frac{\frac{1}{\sqrt{2\pi\sigma_{\delta\varphi}^{2}}}\left(1+\frac{2\gamma}{g^{2}}\delta\varphi_{i}+\frac{\gamma^{2}}{g^{2}}\delta\varphi_{i}^{2}\right)\exp\left(-\frac{\delta\varphi_{i}^{2}}{2\sigma_{\delta\varphi}^{2}}\right)}{\left|\beta\left(1+\frac{2\gamma}{g^{2}}\delta\varphi_{i}+\frac{\gamma^{2}}{g^{2}}\delta\varphi_{i}^{2}\right)-\frac{\gamma}{g^{2}}\left(1+\gamma\delta\varphi_{i}\right)\left(\frac{\kappa
g}{3}\sqrt{1+\frac{2\gamma}{g^{2}}\delta\varphi_{i}+\frac{\gamma^{2}}{g^{2}}\delta\varphi_{i}^{2}}+\frac{1}{\omega_{\rm
s2}}\right)\right|}\,,$ (24)

where $\delta\varphi_{i}$’s ($i=a,b$) are the two solutions of Eq. (22) for a
given $\cal{R}$666Strictly speaking there is the possibility that Eq. (22) has
another solution for a given $\cal{R}$ in a region $\delta\varphi<-2/\gamma$.
In the following, the contribution of this solution to the probability is
ignored as being small.. Numerically computing the above equation, we
illustrate the PDF of $\cal{R}$ in the left panel of Fig. 2. One can clearly
see an asymmetric non-gaussian shape of the full PDF.

Figure 2: The probability distribution function (PDF) of the curvature
perturbation $\cal{R}$ against $\cal{R}-\langle\cal{R}\rangle$ for
$\gamma=1.5$, $\kappa=60$, $\beta=20$, $g=0.02$, and
$\sigma_{\delta\varphi}=2\times 10^{-5}$. The mean value
$\langle{\cal{R}}\rangle=\mathcal{O}(10^{-3})$ is negligibly small. (Left
panel) The solid blue line denotes the full PDF obtained numerically based on
Eq. (24) for $\omega_{\rm s2}=50$. The coloured dotted lines represent the
Gaussian (red), cutoff (green) and exponential tail (black) given in Eq. (25).
(Right panel) The coloured lines denote the full PDFs for $\omega_{\rm
s2}=30,50,100,250,400$ and $1000$ from top to bottom. They agree with the
analytical behavior of the tail $P[{\cal{R}}]\propto\exp(-2\omega_{\rm
s2}{\cal{R}})$. Since $\omega_{\rm s2}\propto 1/\Delta\varphi$, a steeper step
tends to produce a steeper tail.

To understand several features of the obtained PDF, it is useful to compare it
with simplified PDFs, which take into account only one term out of three in
Eq. (22). If one term gives a dominant contribution, an analytic expression of
the PDF is available. Note that for the first term, which is a linear term of
$\delta\varphi$, we linearize the other terms and collect all the
contributions. We find that only with the linearized, second, third term in
Eq. (22), the PDF would be (see Appendix. E for derivation)

$P[{\cal{R}}]\propto\begin{cases}\exp\left[-\frac{{\cal{R}}^{2}}{2\left(\beta-\frac{\kappa\gamma}{3g}-\frac{\gamma}{\omega_{\rm
s2}g^{2}}\right)^{2}\sigma_{\delta\varphi}^{2}}\right]\hskip
125.19194pt:(\text{Gaussian})\\\
\left(1-\frac{3{\cal{R}}}{g\kappa}\right)\exp\left[-\frac{1}{2\sigma_{\delta\varphi}^{2}}\frac{9g^{2}{\cal{R}}^{2}}{\gamma^{2}\kappa^{2}}\left(1-\frac{3}{2}\frac{{\cal{R}}}{g\kappa}\right)^{2}\right]\hskip
66.86414pt:(\text{Cutoff})\\\ \exp\left(-2\omega_{\rm
s2}{\cal{R}}\right)\exp\left[-\frac{1}{2\sigma_{\delta\varphi}^{2}}\frac{g^{2}}{4\gamma^{2}}\Bigl{(}\exp\left(-2\omega_{\rm
s2}{\cal{R}}\right)-1\Bigr{)}^{2}\right]\hskip 18.49411pt:(\text{Exponential
tail})\end{cases}$ (25)

where $g\ll 1$ is assumed for simplicity. We dubbed these three cases as
Gaussian, Cutoff and Exponential tail after their features. The linearized
term yields a Gaussian PDF. The PDF of the second term exhibits a sharp cutoff
at ${\cal{R}}={\cal{R}}_{\rm cutoff}\equiv\kappa g/3$ due to its prefactor.
The PDF of the third term characterized by its exponential tail,
$\exp(-2\omega_{\rm s2}{\cal{R}})$.

In the left panel of Fig. 2, these three simplified PDFs in Eq. (25) are shown
as dotted coloured lines in comparison with the full PDF. The full PDF (blue
solid) matches the Gaussian one (red dotted) only for small
$\cal{R}-\langle\cal{R}\rangle$ and their discrepancy quickly gets
significant. Since we chose large $\kappa=60$ in this figure, the second term
in Eq. (22) is dominant for a wide region, and the full and the cutoff PDF
(green dotted) are well overlapped up to the cutoff position
$\cal{R}-\langle\cal{R}\rangle\simeq{\cal{R}}_{\rm cutoff}=$ 0.4. For larger
$\cal{R}$, the exponential tail from the third term (black dotted) becomes
prominent, and the full PDF follows it as expected. Therefore, the cutoff
feature in the PDF advocated in the previous works Cai  _et al._ (2022a, b) is
covered up by the tail contribution as long as the finite width of the step is
correctly taken into account.

To scrutinize the tail behavior, we present several PDFs with varying
$\omega_{\rm s2}$ in the right panel of Fig. 2. Recalling that $\omega_{\rm
s2}\propto 1/\Delta\varphi$ and the analytic tail PDF
$P[{\cal{R}}]\propto\exp(-2\omega_{\rm s2}{\cal{R}})$, we expect the slope of
the tail becomes steeper as $\Delta\varphi$ decreases (i.e. the step is
steeper). The figure confirms this behavior. For very large $\omega_{\rm s2}$,
the cutoff of the PDF appears to be reproduced. However, it should be stressed
that it is not a hard cutoff but a steep exponential tail. Thus, the
probability density is still non-zero at $\cal{R}\geq{\cal{R}}_{\rm cutoff}$
even for huge $\omega_{\rm s2}$.

Figure 3:  The complementary cumulative distribution function (CCDF)
$\bar{F}[{\cal{R}}_{c}]$ in Eq. (26) normalized by the Gaussian case
$\bar{F}_{G}[{\cal{R}}_{c}]$ in Eq. (27) for $\omega_{\rm
s2}=30,50,100,250,400$ and $1000$ from top to bottom. The other parameters are
the same as Fig. 2. The vertical dotted black line represents the cutoff value
${\cal{R}}_{\rm cutoff}=\kappa g/3=0.4$. Beyond the cutoff, the CCDF highly
sensitive to $\omega_{\rm s2}\propto 1/\Delta\varphi$, because of the
exponential tail contribution. Thus, the PBH abundance strongly depends on
$\omega_{\rm s2}$, if ${\cal{R}}_{c}>{\cal{R}}_{\rm cutoff}$.

With the evaluation of the PBH abundance in mind, we calculate the probability
$\bar{F}[{\cal{R}}_{c}]$ where $\cal{R}$ is greater than or equal to the
threshold value of the PBH formation ${\cal{R}}_{c}$, namely,

$\bar{F}[{\cal{R}}_{c}]=\int^{\infty}_{{\cal{R}}_{c}}P[{\cal{R}}]{\rm
d}{\cal{R}}\,.$ (26)

It is called the complementary cumulative distribution function (CCDF) in the
context of probability theory. We also introduce the CCDF of the Gaussian PDF

$\bar{F}_{G}[{\cal{R}}_{c}]=\frac{1}{2}\text{Erfc}\left(\frac{{\cal{R}}_{c}}{\sqrt{2\sigma^{2}_{\cal{R}}}}\right)\,,$
(27)

where $\text{Erfc}(x)$ is the complementary error function and the variance is
fixed by that of the simplified Gaussian PDF in Eq. (25), namely
$\sigma_{\cal{R}}=|\beta-\kappa\gamma/3g-\gamma/\omega_{\rm
s2}g^{2}|\sigma_{\delta\varphi}$. Their ratio, $\bar{F}/\bar{F}_{G}$, is
useful for understanding how much the non-Gaussianity impacts on the abundance
of PBH.

In Fig. 3, the normalized CCDFs for different $\omega_{\rm s2}$ are shown
against ${\cal{R}}_{c}-\langle\cal{R}\rangle$. The cutoff value
${\cal{R}}_{\rm cutoff}$ is also shown as a vertical black dotted line. It can
be seen that the CCDF dramatically changes depending on the value of
$\omega_{\rm s2}$, if ${\cal{R}}_{c}-\langle\cal{R}\rangle$ is larger than the
cutoff value. For smaller $\omega_{\rm s2}$, the normalized CCDF becomes
larger beyond the cutoff, because the contribution from the exponential tail
is more significant in comparison to the Gaussian case. For huge $\omega_{\rm
s2}$, $\bar{F}$ is abruptly suppressed at the cutoff but stays finite, because
of the tail contribution. These results imply that when the critical value
${\cal{R}}_{c}$ of the PBH formation is larger than the cutoff ${\cal{R}}_{\rm
cutoff}$, it is crucially important to take into account the finite width of
the upward step for estimating the PBH abundance, because otherwise one may
underestimate it.

##  VI Highly asymmetric PDF

Figure 4: Schematic illustrations of Eq. (22) for $\gamma>0$ and
$\beta>\kappa\gamma/3$. The solid black line represents the sum of the all
three terms, and the dashed coloured lines represent each of them in Eq. (22),
i.e., the first term (red), the second term (blue), and the third term
(green). The minimum value of $\cal{R}$ is denoted by the horizontal dotted
line. The boundary, $-g^{2}/2\gamma$, is shown as the vertical dotted line.
The gray shaded region corresponds to the inflaton trajectories which fail to
climb the step and are stuck at a local minimum.

In this section we will show that the PDFs with unusual shapes can be realised
at certain scales exiting the Hubble horizon before the step stage and discuss
its consequences. For the better understanding of this phenomenon, let us
first revisit the relation between $\cal{R}$ and $\delta\varphi$, Eq. (22),
which has been derived in Sec. IV.

The relation (22) is illustrated as a solid black line in Fig. 4 for
$\gamma>0$ and $\beta>\kappa\gamma/3$. In the figure, the contributions of
each term in Eq. (22) are shown as coloured dashed lines. We can see that at
around $\delta\varphi=-g^{2}/2\gamma$, the third term (green dashed line),
which is the contribution from the step width, becomes dominant and causes
${\cal{R}}$ to diverge to infinity. This gives rise to the exponential tail in
the PDF of $\cal{R}$.

It should be noted that there appears a local minimum of $\cal{R}$. For
$\gamma>0$, one can see by drawing a family of trajectories in the phase space
that perturbed trajectories with $\delta\varphi<0$ have smaller velocities
$|\pi|$ at $\varphi_{1}$ than the background trajectory (i.e.
$|\pi_{1}+\delta\pi_{1}|<|\pi_{1}|$). Therefore it takes an enormous amount of
time for some of the trajectories to pass through the step. As a result, they
may lead to large positive $\delta N$ for $\delta\varphi<0$. One can also see
that from Eq. (22), for $\beta>\kappa\gamma/3$, the first term dominates for
sufficiently large positive $\delta\varphi$, and $\delta N$ increases in
proportion to $\delta\varphi$. Combining the above two facts together with the
fact that $\delta N=0$ for $\delta\varphi=0$, we can conclude that a local
minimum of ${\cal{R}}$ exists, which we denote by ${\cal{R}}_{\rm min}$.777
Note that, if $\gamma<0$, $\delta N$ is a monotonically increasing function of
$\delta\varphi$ and there is no ${\cal{R}}_{\rm min}$. Therefore,
${\cal{R}}_{\rm min}$ exists only if $\gamma>0$, i.e. when the scalar field
accelerates in the first SR stage.. In this section, we discuss the effect of
the presence of ${\cal{R}}_{\rm min}$ on the PDF and the local non-Gaussianity
parameter $f_{\rm NL}^{\rm local}$.

Now let us take a closer look at the PDF shape and consider its consequences.
Expanding Eq. (22) with respect to $\delta\varphi$, we obtain

${\cal{R}}=A\delta\varphi+B\delta\varphi^{2}+\mathcal{O}(\delta\varphi^{3})$
(28)

with

$A\equiv\beta-\frac{\kappa\gamma}{3g}-\frac{\gamma}{\omega_{\rm
s2}g^{2}},\hskip 28.45274ptB\equiv\frac{\gamma^{2}}{g^{2}}\left[\frac{\kappa
g}{6}\left(\frac{1}{g^{2}}-1\right)+\frac{1}{2\omega_{\rm
s2}}\left(\frac{2}{g^{2}}-1\right)\right]\,.$ (29)

With Eq. (28), the local non-gaussianity parameter of $\cal{R}$ is defined and
computed as

$f_{\rm NL}^{\rm
local}\equiv\frac{5}{3}\frac{B}{A^{2}}=\frac{5}{2}\frac{\kappa
g(1-g^{2})+\frac{3}{\omega_{\rm s2}}(2-g^{2})}{\left(\frac{3\beta
g^{2}}{\gamma}-\kappa g-\frac{3}{\omega_{\rm s2}}\right)^{2}}\,.$ (30)

If $A$ vanishes (i.e. ${\cal{R}}={\cal{R}}_{\rm min}=0$ at $\delta\varphi=0$),
the linear term in Eq. (28) drops, the denominator of Eq. (30) disappears, and
$f_{\rm NL}^{\rm local}$ diverges 888 This corresponds to the case where
$\delta N$ is always positive for any $|\delta\varphi|$ regardless of its
sign. This means that the background trajectory passes through the step in the
shortest time compared to all other perturbed trajectories.. It happens when
$\gamma=\gamma_{0}\equiv 3\beta g^{2}\omega_{\rm s2}/(3+\kappa g\omega_{\rm
s2})>0$. Since $\gamma$ depends on $\pi/\pi_{1}$ (see Eq. (23)), its value
varies as the scale of the curvature perturbation of interest $k$ changes.
Thus, $f_{\rm NL}^{\rm local}$ may diverge at a particular scale at which
$\gamma=\gamma_{0}$ and $A=0$. Although diverging $f_{\rm NL}^{\rm local}$
does not mean that any physical quantity becomes singular, it is interesting
to explore what kind of dynamics occurs around this particular scale.

When $A$ vanishes, $\cal{R}$ can take only positive values and then the PDF
only has its right-half part. This simple argument based on the expanded
expression (28) actually applies to the fully non-linear result (22). In Fig.
5, we present the full PDF for varying $\gamma$ from a larger value to
$\gamma_{0}=0.3$. One can see that the full PDF deforms from a rather
symmetric shape into a completely asymmetric one as the scale changes.
Although it is not shown there, for $\gamma<\gamma_{0}$ the PDF regains its
left-half part. At the lower end of the PDF of $\cal{R}$, many different
values of $\delta\varphi$ corresponds to the minimum $\cal{R}$ (see Fig. 4),
their probability density $P[\delta\varphi]$ condenses there, and $P[\cal{R}]$
formally diverges. However, its contribution to the total probability is
negligible and harmless. In short, the peculiar behavior seen in Fig. 5 is not
pathological, and it is remarkable to have such a highly asymmetric PDF of
$\cal{R}$.

Figure 5: The PDFs of $\cal{R}$ against $\cal{R}-\langle\cal{R}\rangle$ for
$\kappa=1$, $\beta=20$, $g=0.02$, $\sigma_{\delta\varphi}=5\times 10^{-5}$ and
$\omega_{\rm s2}=50$. $\gamma$, which corresponds to the scale of $\cal{R}$,
varies from $0.5$ (light yellow) to $\gamma_{0}=0.3$ (dark purple) with colour
gradient. The probability density for negative $\cal{R}$ gradually disappears
as the scale changes. The mean value is
$\langle{\cal{R}}\rangle={\cal{O}}(10^{-5})$. Each PDF is normalized so that
the total probability is unity, and the contribution from the apparently
diverging part at the lower end of the PDF is negligibly small.

The most direct observables related to the PDF may be the variance and
skewness of $\cal{R}$, i.e.
$\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{2}\rangle$ and
$\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{3}\rangle$. Using Eq. (28), their
ratio can be computed as

$\frac{\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{3}\rangle}{\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{2}\rangle{}^{2}}=\frac{6A^{2}B\sigma_{\delta\varphi}^{4}+8B^{3}\sigma_{\delta\varphi}^{6}}{A^{4}\sigma_{\delta\varphi}^{4}+4A^{2}B^{2}\sigma_{\delta\varphi}^{6}+4B^{4}\sigma_{\delta\varphi}^{8}}\simeq\begin{cases}\frac{18}{5}f_{\rm
NL}^{\rm local}\hskip 5.69046pt(A\gg B\sigma_{\delta\varphi})\\\
\frac{2}{B\sigma_{\delta\varphi}^{2}}\hskip 18.49411pt(A\ll
B\sigma_{\delta\varphi})\end{cases}$ (31)

For $A\ll B\sigma_{\delta\varphi}$ this ratio is not related to $f_{\rm
NL}^{\rm local}$, while in the other limit they coincide up to an
$\mathcal{O}(1)$ numerical factor. Fig. 6 shows the variance, skewness and
their ratio for $\gamma$ around $\gamma_{0}$. Since $\gamma$ corresponds to
the scale $k$, they are closely connected to the power spectrum and the
bispectrum of the curvature perturbation. Both of them has a dip at
$\gamma=\gamma_{0}=0.3$ where the linear contribution from $\delta\varphi$
disappears (i.e. $A=0$). This corresponds to the dip in the power spectrum
that frequently appears in PBH formation modelsByrnes  _et al._ (2019). In
linear perturbation theory, $\delta\varphi^{2}$ is neglected, ${\cal{R}}$
would vanish from Eq. (28), and the dip is infinitely deep at
$\gamma=\gamma_{0}$.999The dip would never become exactly zero in linear
perturbation theory, which is in contradiction with our result. We suspect
this is due to the neglection of the next leading order terms in gradient
expansion in the $\delta N$ formalism. In any case, since the nonlinear terms
dominate the spectrum at the dip, this inaccuracy does not affect the
conclusion that the deep dip disappears once nonlinear terms are included. In
contrast, our result with the $\delta N$ formalism indicates that the dip
depth is finite due to the non-linear contribution. It has been reported that
the inclusion of a loop correction results in a shallower dip Franciolini  _et
al._ (2023), which is consistent with our result.

Figure 6:  (Left panel) The variance
$\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{2}\rangle$ (red) and the skewness
$\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{3}\rangle$ (blue) of the
curvature perturbation against $\gamma$ representing the corresponding scale
around $\gamma_{0}=0.3$. They have sharp dips at $\gamma=\gamma_{0}$ because
the linear term in Eq. (28) vanishes. (Right panel) The ratio of the these two
$\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{3}\rangle/\langle({\cal{R}}-\langle{\cal{R}}\rangle)^{2}\rangle$
compared to the asymptotic expressions, $2/B\sigma_{\delta\varphi}^{2}$ (gray
dashed) and $18f_{\rm NL}^{\rm local}/5$ (black dashed), given in Eq. (31).
They match at $\gamma=\gamma_{0}$ and at far region from it, respectively, as
expected. The parameter choice is the same as Fig. 5.

Figure 7:  Illustration of the distributions of the curvature perturbation
$\cal{R}$ based on different PDFs. $\cal{R}$ is randomly assigned to each of
$40\times 40$ two-dimensional lattice points according to the Gaussian PDF
(left) and the asymmetric PDF with $\gamma=\gamma_{0}$ (right). Both panels
use the same colour scheme, with red for high and blue for low value of
$\cal{R}$. Identifying $\cal{R}$ with the density fluctuation, one can observe
the spatial distribution of the primordial density contrast. In the Gaussian
distribution, high and low density regions appear in equal proportions,
whereas very low density regions (dark blue) are not found in the highly
asymmetric PDF, which implies the distribution is significantly biased.

In order to intuitively understand the consequence of the highly asymmetric
PDF, we present Fig. 7 in which $\cal{R}$ is regarded as a classical
stochastic quantity and its values are randomly assigned to lattices on a two-
dimensional plane. $\cal{R}$ is probabilistically weighted by the Gaussian PDF
in the left panel and the highly asymmetric PDF for $\gamma=\gamma_{0}$ in the
right panel, respectively. It can be clearly seen that the distribution is
biased to positive value in the highly asymmetric case compared to the
Gaussian case. In the highly asymmetric case, $\cal{R}$ does not appear with
large negative values because of $\cal{R}_{\rm min}$. When one interprets Fig.
7 as the spatial distribution of density fluctuations, this result would imply
that there are fewer voids (low-density regions) than the normal Gaussian
case. It would be interesting to consider such a signature in observation. We
leave it for future study.

We expect that such asymmetric PDFs generically appear in models in which the
inflaton experiences an accelerated SR phase, passes an inflection point, and
subsequently enters a decelerated phase. This is because when we consider
$\delta N$ from the accelerated SR phase, it can become large and positive for
both $\delta\varphi>0$ and $\delta\varphi<0$, which leads to the existence of
${\cal{R}}_{\rm min}$ in the same way as our model.

Assuming the inflaton $\varphi$ is initially slow-rolling and moving toward
the negative direction, the scalar field perturbation $\delta\varphi$ in the
accelerated phase affects the velocity at the inflection point $\pi_{1}$ in
such a way that its absolute value $|\pi_{1}+\delta\pi_{1}|$ becomes larger
for $\delta\varphi>0$. Conversely, $|\pi_{1}+\delta\pi_{1}|$ becomes smaller
than $|\pi_{1}|$ for $\delta\varphi<0$. For such trajectories it takes longer
time to pass through the decelerated phase, which leads to large positive
$\delta N$.

On the other hand, for $\delta\varphi>0$, although the time to pass through
the decelerated phase may become shorter by the increase in the absolute value
of the velocity, depending on the values of the model parameters as well as on
the scale of interest, the number of $e$-folds may eventually increase for
sufficiently large $\delta\varphi$, as the perturbed trajectory starts at a
greater distance from the inflection point. This gives rise to the appearance
of ${\cal{R}}_{\rm min}$. In our model the condition that this happens is
$\beta>\kappa\gamma/3$.

Furthermore, corresponding to the scale at which $\gamma=\gamma_{0}$ in our
model, we expect that there is a scale at which $\delta N$ is non-negative,
regardless of the sign of $\delta\varphi$. This occurs when the term linear in
$\delta\varphi$ vanishes. Apparently, the PDF in this case is highly
asymmetric.

##  VII Conclusion

We studied an inflationary model in which the inflaton potential includes a
finite-width upward step between two slow-roll stages. The inflaton loses its
kinetic energy during the step stage where it climbs up the step, which has a
large effect on the statistical distribution of the curvature perturbation. We
derived the relation between the curvature perturbation $\cal{R}$ and the
scalar field perturbation $\delta\varphi$ and obtained Eq. (22) by using the
$\delta N$ formalism. We found that the step-width $\Delta\varphi$ plays a
important role in the estimation of $\cal{R}$.

In Sec. V, we calculated the PDF of the curvature perturbation $\cal{R}$ and
obtained Eq. (24). The result is shown in Fig 2. For ${\cal{R}}<{\cal{R}}_{\rm
cutoff}=\kappa g/3$ the PDF follows the Cutoff PDF given in Eq. (25), while
for ${\cal{R}}>{\cal{R}}_{\rm cutoff}=\kappa g/3$ the exponential tail
$P[{\cal{R}}]\propto\exp(-2\omega_{\rm s2}{\cal{R}})$ is dominant. The slope
of the tail depends on the step width $\Delta\varphi$ through $\omega_{\rm
s2}\simeq\sqrt{2}~{}|\pi_{1}|/\Delta\varphi$. The CCDF was also calculated,
and the significant impact on the PBH abundance of this exponential tail was
illustrated in Fig. 3. We conclude that the step width should be taken into
account for the accurate estimation of the PBH abundance.

In Sec. VI we discussed the highly asymmetric PDF. As shown in Fig. 4, for
$\gamma>0$ and $\beta>\kappa\gamma/3$, $\cal{R}$ has a minimum value
${\cal{R}}_{\rm min}$. At a particular scale $\gamma=\gamma_{0}$ exiting the
Hubble horizon before the step, $\cal{R}_{\rm min}$ is realised with
$\delta\varphi=0$, which means that $A$ as defined in Eq. (29) vanishes and
that $\cal{R}$ is supported by $\delta\varphi^{2}$ instead of $\delta\varphi$.
We found that this scale $\gamma_{0}$ corresponds to the dip of the curvature
power spectrum known in other models. On this scale, $f_{\rm NL}^{\rm local}$
formally diverges and the highly asymmetric PDF appears as shown in Figs. 5
and 6. The highly asymmetric PDF may result in a lower abundance of low-
density regions such as voids (see Fig. 7) compared to the normal Gaussian
case.

In this paper, we mainly investigated curvature perturbations on the scales
exiting the Hubble horizon just before the background inflaton enters the step
stage. It would also be interesting to calculate the PDF for scales exiting
the Hubble horizon during and after the step stage for the comprehensive
understanding of this model. In doing so, one needs to track the time
evolution of $\delta\varphi$ more precisely, to include the contribution of
$\delta\pi$, and to take into account the intrinsic non-gaussianity of
$\delta\varphi$. As another approach, it is also fascinating to apply the
stochastic-$\delta N$ formalism to this upward step model. We leave these as
future work.

## Acknowledgments

We thank Shinji Tsujikawa for useful discussions and comments. This work is
supported by JSPS KAKENHI grants 18K13537, 20H05854, 23K03424 (TF), 20H05853
(MS), and by the World Premier International Research Center Initiative (WPI
Initiative), MEXT, Japan.

##  Appendix A Parameters in $f_{\rm step}$

The parameters in Eq. (6) are determined by imposing the continuity conditions
on the potential $v(\varphi)$ and its first derivative $\partial_{\varphi}v$
at $\varphi=\varphi_{1},\varphi_{2}$ and $\varphi_{c}$. We then obtain the
parameters of $f_{\rm step}$ as

$\displaystyle A_{1}=v_{\rm
sr1}(\varphi_{1})-\frac{\left(\partial_{\varphi}v_{\rm
sr1}(\varphi_{1})\right)^{2}}{4B_{1}}\,,$ (A.1) $\displaystyle A_{2}=v_{\rm
sr2}(\varphi_{2})-\frac{\left(\partial_{\varphi}v_{\rm
sr2}(\varphi_{2})\right)^{2}}{4B_{2}}\,,$ (A.2) $\displaystyle
B_{1}=\frac{2\Delta v}{(\Delta\varphi)^{2}}+\frac{3\partial_{\varphi}v_{\rm
sr1}(\varphi_{1})+\partial_{\varphi}v_{\rm
sr2}(\varphi_{2})}{2\Delta\varphi}\,,$ (A.3) $\displaystyle
B_{2}=-\frac{2\Delta v}{(\Delta\varphi)^{2}}-\frac{\partial_{\varphi}v_{\rm
sr1}(\varphi_{1})+3\partial_{\varphi}v_{\rm
sr2}(\varphi_{2})}{2\Delta\varphi}\,,$ (A.4) $\displaystyle\varphi_{\rm
min}=\varphi_{1}-\frac{\partial_{\varphi}v_{\rm sr1}(\varphi_{1})}{2B_{1}}\,,$
(A.5) $\displaystyle\varphi_{\rm
max}=\varphi_{2}-\frac{\partial_{\varphi}v_{\rm sr2}(\varphi_{2})}{2B_{2}},$
(A.6)

where $\Delta v\equiv v_{\rm sr2}(\varphi_{2})-v_{\rm sr1}(\varphi_{1})$
corresponds to the height of the step.

##  Appendix B Solving background EOM

Here we solve the background equation of motion for the scalar field. In the
following, we assume that the kinetic energy of the scalar field is
subdominant compared to the potential energy, $h^{2}\pi^{2}\ll v$, throughout
the three stages (approximation (I) in Sec. III). This reduces the EOM (3) to

$\frac{{\rm d}\pi}{{\rm d}n}+3\pi+3\frac{\partial_{\varphi}v}{v}=0\,.$ (B.1)

We analytically solve this approximate EOM (B.1) in each of the three stages.

###  B.1 SR stages

In this subsection we discuss the dynamics before and after the step. The step
stage will be addressed in the next subsections. In these stages, the
potential is almost flat and the Taylor expansion is a sufficiently good
approximation for the potential. To solve Eq. (B.1) analytically, we expand
the potential term up to the first order in the vicinity of
$\varphi=\varphi_{i}$. Then, we obtain the equation of motion

$\frac{{\rm d}\pi}{{\rm
d}n}+3\pi+3\sqrt{2\epsilon_{Vi}}-\frac{3}{2}\eta_{i}(\varphi-\varphi_{i})=0\,,$
(B.2)

for $\varphi\geq\varphi_{1}$ ($i=1$) and $\varphi\leq\varphi_{2}$ ($i=2$),
respectively. The solutions are

$\varphi(n)-\varphi_{i}=C_{i}^{+}(\pi_{i})e^{\lambda_{i}^{+}(n-n_{i})}+C_{i}^{-}(\pi_{i})e^{\lambda_{i}^{-}(n-n_{i})}+\frac{2\sqrt{2\epsilon_{Vi}}}{\eta_{i}}\,,$
(B.3)

where $\lambda_{i}^{\pm}$ and $C_{i}^{\pm}$ are constants for a given
trajectory (i.e. for given boundary conditions at $n=n_{i}$),

$\displaystyle\lambda_{i}^{+}=-\frac{3}{2}\left(1+\sqrt{1+\frac{2\eta_{i}}{3}}\right)\simeq-3-\frac{\eta_{i}}{2}\,,$
(B.4)
$\displaystyle\lambda_{i}^{-}=-\frac{3}{2}\left(1-\sqrt{1+\frac{2\eta_{i}}{3}}\right)\simeq\frac{\eta_{i}}{2}\,,$
(B.5) $\displaystyle
C_{i}^{+}(\pi_{i})=\frac{1}{\lambda_{i}^{+}-\lambda_{i}^{-}}\left(\pi_{i}+\lambda_{i}^{-}\frac{2\sqrt{2\epsilon_{Vi}}}{\eta_{i}}\right)\simeq-\frac{1}{3}\left(\pi_{i}+\sqrt{2\epsilon_{Vi}}\right)\,,$
(B.6) $\displaystyle
C_{i}^{-}(\pi_{i})=\frac{1}{\lambda_{i}^{-}-\lambda_{i}^{+}}\left(\pi_{i}+\lambda_{i}^{+}\frac{2\sqrt{2\epsilon_{Vi}}}{\eta_{i}}\right)\simeq\frac{1}{3}\left(-\frac{6\sqrt{2\epsilon_{Vi}}}{\eta_{i}}+\pi_{i}+\sqrt{2\epsilon_{Vi}}\right)\,.$
(B.7)

For later discussion, we show $\pi_{2}$ with the first-order
$\Delta\varphi$-corrections from the effect of the Hubble friction term. From
the energy conservation law which includes the dissipation caused by the
Hubble friction term, we obtain

$\displaystyle\pi_{2}$ $\displaystyle=$
$\displaystyle-\sqrt{\pi_{1}^{2}-6\log\left(\frac{v(\varphi_{2})}{v(\varphi_{1})}\right)-6\int^{n_{2}}_{n_{1}}\pi^{2}(n){\rm
d}n}\,$ (B.8) $\displaystyle\simeq$
$\displaystyle-\sqrt{\pi_{1}^{2}-6\log\left(\frac{v(\varphi_{2})}{v(\varphi_{1})}\right)-3\pi_{1}^{2}N^{(\rm
s1)}+3\pi_{c}\Delta\varphi-3\pi_{2}^{2}N^{(\rm s2)}}\,,$ (B.9)

where we used the zeroth-order solutions (14) and (15) to derive the second
line. In Eq. (B.8), the third term inside the square root corresponds to the
dissipation term, which we ignored in the main context as the approximation
(III) in Sec. III.

Note, however, that the last three terms in Eq. (B.9) are not negligible
compared to the sum of the first two terms for $\omega_{\rm s1}g^{2}<1$.
Therefore, for the purpose of estimating $\pi_{2}$, the approximation (III) is
not a good approximation. However, as we will see later in Appendix. D, the
approximation (III) can be considered as a good approximation for estimating
$\delta\pi_{2}$.

###  B.2 S1 region

Substituting Eq. (6) into Eq. (B.1) yields the equation of motion to be
solved. To solve it analytically, we approximate the denominator of the third
term in Eq. (B.1) to be the constant value $A_{1}$ shown in Eq. (A.1). This
approximation is valid under the assumption that the step height $\Delta v$ is
sufficiently smaller than the value of the potential. Then the EOM is given by

$\frac{{\rm d}\pi}{{\rm d}n}+3\pi+\omega_{\rm s1}^{2}(\varphi-\varphi_{\rm
min})=0\,.$ (B.10)

Assuming $\omega_{\rm s1}^{2}>9/4$ and setting the values at $n=n_{1}$ as
$\varphi=\varphi_{1}$ and $\pi=\pi_{1}$, the solution is

$\varphi(n)-\varphi_{\rm min}=D_{\rm
s1}e^{-\frac{3}{2}(n-n_{1})}\sin\left(\sqrt{\omega_{\rm
s1}^{2}-\frac{9}{4}}(n-n_{1})+\theta_{\rm s1}\right)\,,$ (B.11)

where $D_{\rm s1}$ and $\theta_{\rm s1}$ are constants for a given trajectory,
defined by

$\displaystyle D_{\rm
s1}=-\sqrt{\frac{\pi_{1}^{2}+3\pi_{1}(\varphi_{1}-\varphi_{\rm
min})+\omega_{\rm s1}^{2}(\varphi_{1}-\varphi_{\rm min})^{2}}{\omega_{\rm
s1}^{2}-\frac{9}{4}}}\,,$ (B.12) $\displaystyle\cos\theta_{\rm
s1}=\frac{\pi_{1}+\frac{3}{2}(\varphi_{1}-\varphi_{\rm min})}{D_{\rm
s1}\sqrt{\omega_{\rm s1}^{2}-\frac{9}{4}}}\hskip 14.22636pt\text{and}\hskip
14.22636pt\sin\theta_{\rm s1}=\frac{\varphi_{1}-\varphi_{\rm min}}{D_{\rm
s1}}\,.$ (B.13)

Here we check that if $\Delta\varphi$ is sufficiently small, Eq. (B.11)
coincides with the solution, Eq. (14), obtained from the approximated EOM to
which approximations (II) and (III) are applied. For this purpose, it is
sufficient to compare $\omega_{\rm s1}^{2}(\varphi_{1}-\varphi_{\rm min})^{2}$
to $\pi_{1}^{2}$,

$\frac{\omega_{\rm s1}^{2}(\varphi_{1}-\varphi_{\rm
min})^{2}}{\pi_{1}^{2}}=\frac{3(\partial_{\varphi}v_{\rm
sr1}(\varphi_{1}))^{2}}{2A_{1}B_{1}\pi_{1}^{2}}\simeq\frac{3}{4}\frac{(\partial_{\varphi}v_{\rm
sr1}(\varphi_{1}))^{2}}{v_{\rm
sr1}(\varphi_{1})}\frac{(\Delta\varphi)^{2}}{\pi_{1}^{2}\Delta v}\sim
9\epsilon_{V1}\frac{(\Delta\varphi)^{2}}{\pi_{1}^{4}}={\cal{O}}\left(\frac{1}{\omega_{\rm
s1}^{2}}\right)\,,$ (B.14)

where we used the approximation $\Delta v\sim v_{\rm
sr1}(\varphi_{1})\pi_{1}^{2}/6$, which can be derived by the energy
conservation law (13). From Eq. (B.14), it follows that if $\omega_{\rm s1}$
is much larger than 1, $D_{\rm s1}$, $\theta_{\rm s1}$ and $\varphi_{\rm min}$
are reduced to $\pi_{1}/\omega_{\rm s1}$, 0 and $\varphi_{1}$, respectively,
i.e. Eq. (14) is reproduced from Eq. (B.11).

###  B.3 S2 region

Similarly, for the S2 region, the EOM to be solved is

$\frac{{\rm d}\pi}{{\rm d}n}+3\pi-\omega_{\rm s2}^{2}(\varphi-\varphi_{\rm
max})=0\,,$ (B.15)

where we used the fact that $B_{2}$ is a negative value. Setting the values at
$n=n_{2}$ as $\varphi=\varphi_{2}$ and $\pi=\pi_{2}$, the solution is

$\varphi(n)-\varphi_{\rm max}=D_{\rm
s2}e^{-\frac{3}{2}(n-n_{2})}\sinh\left(\sqrt{\omega_{\rm
s2}^{2}+\frac{9}{4}}(n-n_{2})+\theta_{\rm s2}\right)\,,$ (B.16)

where $D_{\rm s2}$ and $\theta_{\rm s2}$ are constants for a given trajectory,
defined by

$\displaystyle D_{\rm
s2}=-\sqrt{\frac{\pi_{2}^{2}+3\pi_{2}(\varphi_{2}-\varphi_{\rm
max})-\omega_{\rm s2}^{2}(\varphi_{2}-\varphi_{\rm max})^{2}}{\omega_{\rm
s2}^{2}+\frac{9}{4}}}\,,$ (B.17) $\displaystyle\theta_{\rm
s2}=\log\left[\frac{1}{D_{\rm s2}\sqrt{\omega_{\rm
s2}^{2}+\frac{9}{4}}}\left(\pi_{2}+\left(\frac{3}{2}+\sqrt{\omega_{\rm
s2}^{2}+\frac{9}{4}}\right)(\varphi_{2}-\varphi_{\rm max})\right)\right]\,.$
(B.18)

Similar to Eq. (B.14), we compare $\omega_{\rm
s2}^{2}(\varphi_{2}-\varphi_{\rm max})^{2}$ to $\pi_{2}^{2}$ to find a
condition under which Eq. (15) is reproduced. We obtain

$\frac{\omega_{\rm s2}^{2}(\varphi_{2}-\varphi_{\rm
max})^{2}}{\pi_{2}^{2}}\simeq\frac{3}{4}\frac{(\partial_{\varphi}v_{\rm
sr2}(\varphi_{2}))^{2}}{v_{\rm
sr2}(\varphi_{2})}\frac{(\Delta\varphi)^{2}}{\pi_{2}^{2}\Delta v}\sim
9\epsilon_{V2}\frac{(\Delta\varphi)^{2}}{\pi_{1}^{2}\pi_{2}^{2}}={\cal{O}}\left(\frac{1}{\kappa^{2}g^{2}\omega_{\rm
s2}^{2}}\right)\,,$ (B.19)

where we used the approximation $\Delta v\sim v_{\rm
sr2}(\varphi_{2})\pi_{1}^{2}/6$. Thus, if $\kappa g\omega_{\rm s2}\gg 1$ is
satisfied, then $D_{\rm s2}$, $\theta_{\rm s2}$ and $\varphi_{\rm max}$ can be
approximated by $\pi_{2}/\omega_{\rm s2}$, 0 and $\varphi_{2}$ and Eq. (B.16)
reduces to Eq. (15).

##  Appendix C Derivation of $\delta N$ in each stage

###  C.1 First SR stage

In the first SR stage the background trajectory is on the SR attractor. From
Eq. (B.3), we have

$\displaystyle\varphi-\varphi_{1}=-\frac{2\sqrt{2\epsilon_{V1}}}{\eta_{1}}e^{-\lambda_{1}^{-}N^{(1)}}+\frac{2\sqrt{2\epsilon_{V1}}}{\eta_{1}}\,,$
(C.1)
$\displaystyle\pi=-\sqrt{2\epsilon_{V1}}+\frac{\eta_{1}}{2}(\varphi-\varphi_{1})\,,$
(C.2)

where $N^{(1)}\equiv N(\varphi,\pi;\varphi_{1},\pi_{1})$.

Now, what we are here to discuss is the deviation from the SR attractor, thus
it is not sufficient just to perturb the background solutions (C.1) and (C.2).
Replacing $\pi_{1}$ with $\pi_{1}+\delta\pi_{1}$ in $C_{1}^{\pm}$, Eq.(B.3)
gives a perturbed trajectory,

$\displaystyle\varphi(n)-\varphi_{1}=C_{1}^{+}(\pi_{1}+\delta\pi_{1})e^{\lambda_{1}^{+}(n-n_{1}^{\prime})}+C_{1}^{-}(\pi_{1}+\delta\pi_{1})e^{\lambda_{1}^{-}(n-n_{1}^{\prime})}+\frac{2\sqrt{2\epsilon_{V1}}}{\eta_{1}}\,,$
(C.3)
$\displaystyle\pi(n)=C_{1}^{+}(\pi_{1}+\delta\pi_{1})\lambda_{1}^{+}e^{\lambda_{1}^{+}(n-n_{1}^{\prime})}+C_{1}^{-}(\pi_{1}+\delta\pi_{1})\lambda_{1}^{-}e^{\lambda_{1}^{-}(n-n_{1}^{\prime})}\,,$
(C.4)

which correspond to a trajectory passing through ($\varphi_{1}$,
$\pi_{1}+\delta\pi_{1}$) at $n=n_{1}^{\prime}$. We assign $\delta\varphi$ so
that, in the phase space, ($\varphi+\delta\varphi$, $\pi$) lies on the
perturbed trajectory given by Eqs. (C.3) and (C.4). Note that $\delta\varphi$
is uniquely determined by $\delta\pi_{1}$. Conversely, $\delta\pi_{1}$ should
not be chosen such that there is no corresponding $\delta\varphi$. This allows
us to substitute $\varphi(n^{\prime})=\varphi+\delta\varphi$ and
$\pi(n^{\prime})=\pi$ into Eqs. (C.3) and (C.4) as values at $n=n^{\prime}$.
Then, solving the system of equations for
$e^{\lambda_{1}^{+}(n^{\prime}-n_{1}^{\prime})}$ and
$e^{\lambda_{1}^{-}(n^{\prime}-n_{1}^{\prime})}$, we obtain

$\displaystyle\delta\pi_{1}\simeq-\lambda_{1}^{-}\delta\varphi
e^{\lambda_{1}^{+}{\cal{N}}^{(1)}}\,,$ (C.5)
$\displaystyle\delta\pi_{1}\simeq\left(-\lambda_{1}^{+}\delta\varphi+\left(1-\frac{\lambda_{1}^{+}}{\lambda_{1}^{-}}\right)\pi_{*}\right)e^{\lambda_{1}^{-}{\cal{N}}^{(1)}}-\left(1-\frac{\lambda_{1}^{+}}{\lambda_{1}^{-}}\right)\pi_{1}\,,$
(C.6)

where ${\cal{N}}^{(1)}\equiv
N(\varphi+\delta\varphi,\pi;\varphi_{1},\pi_{1}+\delta\pi_{1})=n_{1}^{\prime}-n^{\prime}$.
To derive these results, we used the background solution (C.2) and
$\pi_{1}=-\sqrt{2\epsilon_{V1}}$.

What we need to know in particular is the relation to $\delta\varphi$ for each
of $\delta\pi_{1}$ and $\delta N^{(1)}$($\equiv{\cal{N}}^{(1)}-N^{(1)}$). In
principle, it is possible to solve them because there are two equations for
two unknown quantities. However, since this system of equations contains two
exponential functions, it is not possible to obtain an explicit form of the
analytic solution. In order to solve them analytically, we limit our
discussion to considering only the case in which $\delta N^{(1)}$ is
sufficiently small, and perform an approximation that ignores second and
higher orders for $\delta N^{(1)}$ and $\delta\varphi$. That is, the
exponential functions in Eqs. (C.5) and (C.6) are approximately

$e^{\lambda_{1}^{\pm}{\cal{N}}^{(1)}}=e^{\lambda_{1}^{\pm}N^{(1)}}e^{\lambda_{1}^{\pm}\delta
N^{(1)}}\simeq\left(\frac{\pi_{1}}{\pi}\right)^{\frac{\lambda_{1}^{\pm}}{\lambda_{1}^{-}}}\left(1+\lambda_{1}^{\pm}\delta
N^{(1)}\right)\,,$ (C.7)

where we used $e^{\lambda_{1}^{-}N^{(1)}}\simeq\pi_{1}/\pi$, which is derived
from Eqs. (C.1) and (C.2). Notice that this approximation (C.7) is only valid
if $|\delta N^{(1)}|\ll 1/3$ is satisfied. This then makes it possible to
solve the system of equations analytically, and the final results can be
obtained as

$\displaystyle\delta\pi_{1}\simeq-\lambda_{1}^{-}\left(\frac{\pi_{1}}{\pi}\right)^{\frac{\lambda_{1}^{+}}{\lambda_{1}^{-}}}\delta\varphi\simeq-\frac{\eta_{1}}{2}\left(\frac{\pi}{\pi_{1}}\right)^{\frac{6}{\eta_{1}}}\delta\varphi\,,$
(C.8) $\displaystyle\delta
N^{(1)}\simeq\frac{\lambda_{1}^{+}-\lambda_{1}^{-}\left(\frac{\pi}{\pi_{1}}\right)^{1-\frac{\lambda_{1}^{+}}{\lambda_{1}^{-}}}}{\lambda_{1}^{-}-\lambda_{1}^{+}}\frac{\delta\varphi}{\pi}\simeq-\frac{\delta\varphi}{\pi}\,.$
(C.9)

Note that in the above discussion $\delta N^{(1)}$ is limited to being small,
but $N^{(1)}$ is not limited at all.

###  C.2 Second SR stage

By solving Eq. (B.3) for $\varphi\leq\varphi_{2}$ in conjunction with the
equation for $\pi$ derived by differentiating Eq. (B.3) with respect to $n$,
we obtain $N^{(2)}(\equiv
N(\varphi_{2},\pi_{2};\varphi_{f},\pi_{f})=n_{f}-n_{2})$ as

$N^{(2)}\simeq\frac{1}{\lambda_{2}^{-}}\log\left(\frac{\left(1-\frac{\lambda_{2}^{+}}{\lambda_{2}^{-}}\right)\pi_{f}}{\pi_{2}+\frac{\lambda_{2}^{+}}{\lambda_{2}^{-}}\sqrt{2\epsilon_{V2}}}\right)\,,$
(C.10)

where $\pi_{f}$ is the value of $\pi$ at $n=n_{f}$. Replacing $\pi_{2}$ with
$\pi_{2}+\delta\pi_{2}$ in $C_{2}^{\pm}$, Eq.(B.3) gives a perturbed
trajectory. We can immediately obtain

${\cal{N}}^{(2)}\simeq\frac{1}{\lambda_{2}^{-}}\log\left(\frac{\left(1-\frac{\lambda_{2}^{+}}{\lambda_{2}^{-}}\right)(\pi_{f}+\delta\pi_{f})}{\pi_{2}+\delta\pi_{2}+\frac{\lambda_{2}^{+}}{\lambda_{2}^{-}}\sqrt{2\epsilon_{V2}}}\right)\,,$
(C.11)

where ${\cal{N}}^{(2)}\equiv
N(\varphi_{2},\pi_{2}+\delta\pi_{2};\varphi_{f},\pi_{f})$. Recall that we
ignore $\delta\pi_{f}$ by choosing $n_{f}$ as the time later than which the
trajectory converges to the SR attractor. Then $\delta N^{(2)}$ is given by

$\delta
N^{(2)}\equiv{\cal{N}}^{(2)}-N^{(2)}\simeq-\frac{1}{\lambda_{2}^{-}}\log\left(1+\frac{\frac{\delta\pi_{2}}{\pi_{2}}}{1-\frac{\lambda_{2}^{+}}{\lambda_{2}^{-}}\frac{1}{\kappa
g}}\right)\simeq-\frac{\kappa g}{3}\frac{\delta\pi_{2}}{\pi_{2}}\,.$ (C.12)

Note that the last approximate equality in Eq. (C.12) is an approximation due
to the fact that $|\lambda_{2}^{+}/\lambda_{2}^{-}|$ is much larger than 1,
not that $\delta\pi_{2}/\pi_{2}$ is small. Substituting Eq. (18) into Eq.
(C.12) leads to Eq. (19), but note that Eq. (18) ignores the Hubble friction.
In Appendix. D, we will discuss the validity of Eq. (18).

###  C.3 Step stage

First, let us estimate the number of e-folds in the S1 region. Assuming that
$\Delta\varphi$ is sufficiently small, i.e. $\omega_{\rm s1}\gg 1$, Eq. (B.11)
gives

$-\frac{1}{2}\Delta\varphi\simeq\frac{\pi_{1}}{\omega_{\rm
s1}}e^{-\frac{3}{2}N^{(\rm s1)}}\sin{\left(\omega_{\rm s1}N^{(\rm
s1)}\right)}\,,$ (C.13)

where $N^{(\rm s1)}\equiv
N(\varphi_{1},\pi_{1};\varphi_{c},\pi_{c})=n_{c}-n_{1}$. The scalar field
should not oscillate around the local minimum of the potential as it climbs
through the step. This fact tells us that

$N^{(\rm s1)}\leq\frac{\pi}{2\omega_{\rm s1}}\ll 1\,,$ (C.14)

where we used that $\omega_{\rm s1}$ is an inverse power of the order of
$\Delta\varphi$ and is very large compared to 1. In this context, since the
damping factor $\exp(-3N^{(\rm s1)}/2)$ is approximately equal to 1, we have

$N^{(\rm s1)}\simeq\frac{1}{\omega_{\rm
s1}}\arcsin\left(-\frac{\Delta\varphi}{2\pi_{1}}\omega_{\rm s1}\right)\,.$
(C.15)

Then varying $N^{(\rm s1)}$ with respect to $\delta\pi_{1}$ yields

$\delta N^{(\rm
s1)}=-\frac{1}{2}\frac{\Delta\varphi}{\sqrt{\pi_{1}^{2}-\left(\frac{1}{2}\Delta\varphi\omega_{\rm
s1}\right)^{2}}}\frac{\delta\pi_{1}}{\pi_{1}}\simeq-\frac{1}{\omega_{\rm
s1}}\frac{\delta\pi_{1}}{\pi_{1}}\simeq-\frac{\eta_{1}}{2\omega_{\rm
s1}}\left(\frac{\pi}{\pi_{1}}\right)^{1+\frac{6}{\eta_{1}}}\delta N^{(1)}\,.$
(C.16)

Here, $\delta\pi_{1}/\pi_{1}$ is assumed to be much smaller than 1, and
$\delta N^{(\rm s1)}$ is expanded up to the first order of
$\delta\pi_{1}/\pi_{1}$. It can be seen that $\delta N^{(\rm s1)}$ is strongly
suppressed by $\Delta\varphi$ and $(\pi/\pi_{1})^{1+\frac{6}{\eta_{1}}}$
compared to $\delta N^{(1)}$. For this reason, we have ignored $\delta N^{(\rm
s1)}$ in the main part of the paper.

The number of e-folds for the S2 region can then be calculated by imposing the
same condition, i.e., $\Delta\varphi$ is small (quantitatively,
$\Delta\varphi\ll\kappa g|\pi_{1}|$). Since $\omega_{\rm s2}$ is considerably
larger than $3/2$, the damping factor $e^{-3(n-n_{2})/2}$ in Eq. (B.16) is
negligible. Under this approximation, Eq. (B.16) can be solved inversely for
$N^{(\rm s2)}\equiv N(\varphi_{c},\pi_{c};\varphi_{2},\pi_{2})=n_{2}-n_{c}$,
and it yields

$N^{(\rm s2)}\simeq\frac{1}{\omega_{\rm
s2}}\sinh^{-1}\left(\frac{\Delta\varphi}{2|\pi_{2}|}\omega_{\rm
s2}\right)\simeq\frac{1}{\omega_{\rm
s2}}\log\left(\frac{\Delta\varphi}{|\pi_{2}|}\omega_{\rm s2}\right)\,.$ (C.17)

The number of e-folds in the perturbed trajectory, ${\cal{N}}^{(\rm s2)}\equiv
N(\varphi_{c},\pi_{c}+\delta\pi_{c};\varphi_{2},\pi_{2}+\delta\pi_{2})$, is
also represented in the same form. As a result, we obtain

$\delta N^{(\rm s2)}\simeq-\frac{1}{\omega_{\rm
s2}}\log\left(1+\frac{\delta\pi_{2}}{\pi_{2}}\right)\,.$ (C.18)

It is important to note that this may not be just a correction term. Certainly
$1/\omega_{\rm s2}$ is of the order of $\Delta\varphi$, however when
$\delta\pi_{2}$ is comparable to $-\pi_{2}$, $\delta N^{(\rm s2)}$ may diverge
to infinity. This is a consequence of the existence of perturbed trajectories
that barely reach the local maximum of the potential and take an enormous
amount of time to pass through the step. Therefore, the contribution to the
total $\delta N$ from $\delta N^{(\rm s2)}$ cannot be ignored and must be
taken into account.

##  Appendix D Hubble friction corrections during step stage

In the main text we imposed the approximation condition (III) and neglected
the Hubble friction during the step stage. However, as we have seen in
Appendix. B.1, the ${\cal{O}}(\Delta\varphi)$ corrections are not necessarily
negligible compared to the zeroth orders in the background solution, since the
first and second terms in Eq. (B.9) largely cancel each other out for small
$g$. In this section we verify that they are negligible at the perturbation
level and that Eq. (18) is a good enough approximation.

From Eq. (B.9), we can obtain

$\displaystyle\pi_{2}+\delta\pi_{2}$ $\displaystyle=$
$\displaystyle\pi_{2}\biggl{[}1+\frac{2}{g^{2}}\frac{\delta\pi_{1}}{\pi_{1}}+\frac{1}{g^{2}}\left(\frac{\delta\pi_{1}}{\pi_{1}}\right)^{2}-\frac{3}{g^{2}}\left(1+\frac{\delta\pi_{1}}{\pi_{1}}\right)^{2}\delta
N^{(\rm
s1)}-\frac{3\delta\pi_{1}}{g^{2}\pi_{1}}\left(2+\frac{\delta\pi_{1}}{\pi_{1}}\right)N^{(\rm
s1)}$ (D.1) $\displaystyle\hskip
28.45274pt+\frac{3\Delta\varphi}{g^{2}}\delta\pi_{c}-3\left(1+\frac{\delta\pi_{2}}{\pi_{2}}\right)^{2}(N^{(\rm
s2)}+\delta N^{(\rm s2)})+3N^{(\rm s2)}\biggr{]}^{\frac{1}{2}}\,.$

In the following, for simplicity, the second and higher orders of
$\delta\pi_{1}/\pi_{1}$ are truncated. This is well justified for $g\ll 1$.
Substituting the already calculated $\Delta\varphi$ zeroth order solutions
(C.15), (C.16), (C.17), (C.18) and (18) into Eq. (D.1), we obtain

$\displaystyle\pi_{2}+\delta\pi_{2}=\pi_{2}\biggl{[}1+\frac{2}{g^{2}}\frac{\delta\pi_{1}}{\pi_{1}}-\frac{3}{g^{2}}\delta
N^{(\rm s1)}-\frac{6\delta\pi_{1}}{g^{2}\pi_{1}}N^{(\rm
s1)}+\frac{3\Delta\varphi}{g^{2}\pi_{c}}\frac{\delta\pi_{1}}{\pi_{1}}-\frac{6\delta\pi_{1}}{g^{2}\pi_{1}}N^{(\rm
s2)}-3\left(1+\frac{\delta\pi_{2}}{\pi_{2}}\right)^{2}\delta N^{(\rm
s2)}\biggr{]}^{\frac{1}{2}}$ $\displaystyle\hskip
8.5359pt\simeq\pi_{2}\biggl{[}1+\frac{2}{g^{2}}\left(1-\frac{3}{2\omega_{\rm
s1}}+\frac{3\Delta\varphi}{2\pi_{c}}-\frac{3}{\omega_{\rm
s2}}\log\left(\frac{\sqrt{2}}{g}\right)\right)\frac{\delta\pi_{1}}{\pi_{1}}+\frac{3}{2\omega_{\rm
s2}}\left(1+\frac{2}{g^{2}}\frac{\delta\pi_{1}}{\pi_{1}}\right)\log\left(1+\frac{2}{g^{2}}\frac{\delta\pi_{1}}{\pi_{1}}\right)\biggr{]}^{\frac{1}{2}}\,.$
(D.2)

Therefore, if the step width is sufficiently small, i.e. $\omega_{\rm
s1},\omega_{\rm s2}\gg 1$ and $\omega_{\rm s2}\gg|3\log g|$, the
$\Delta\varphi$ corrections are subdominant and negligible. In this case, the
Hubble friction can be ignored when calculating $\delta\pi_{2}$ with $g$ as an
input parameter, resulting in Eq. (18) being a good approximation.

##  Appendix E Derivation of PDF for each case

Using Eq. (24), we can numerically obtain the PDF of $\cal{R}$. However, since
it is not possible to solve Eq. (22) analytically, we cannot obtain the
analytically explicit form of the PDF in terms of $\cal{R}$. Instead of
solving Eq. (22) exactly, it may be useful to derive the shape of the PDF in
each case where one term in Eq. (22) makes a larger contribution than others,
in order to understand the specific features of the full PDF in this model.
For simplicity, in the following discussion we assume that $g\ll 1$, otherwise
the PDF in our model becomes close to the Gaussian case.

  1. 1.

The linear perturbative regime
In this case, the PDF is obviously given by a Gaussian distribution with
variance $\sigma_{\cal{R}}^{2}=(\beta-\kappa\gamma/3g-\gamma/\omega_{\rm
s2}g^{2})^{2}\sigma_{\delta\varphi}^{2}$, namely,

$P[{\cal{R}}]=\frac{1}{\sqrt{2\pi\left(\beta-\frac{\kappa\gamma}{3g}-\frac{\gamma}{\omega_{\rm
s2}g^{2}}\right)^{2}\sigma_{\delta\varphi}^{2}}}\exp\left(-\frac{{\cal{R}}^{2}}{2\left(\beta-\frac{\kappa\gamma}{3g}-\frac{\gamma}{\omega_{\rm
s2}g^{2}}\right)^{2}\sigma_{\delta\varphi}^{2}}\right)\,.$ (E.1)

  2. 2.

The second term dominant regime
Based on the assumption $g\ll 1$, we can ignore the third term in the square
root in the second term of Eq. (22), i.e.

${\cal{R}}\simeq\frac{\kappa
g}{3}\left(1-\sqrt{1+\frac{2\gamma}{g^{2}}\delta\varphi}\right)\,.$ (E.2)

The inverse of this equation gives us

$\delta\varphi\simeq\frac{9}{2}\frac{{\cal{R}}^{2}}{\gamma\kappa^{2}}-3\frac{g{\cal{R}}}{\gamma\kappa}\,,$
(E.3)

and the PDF is

$P[{\cal{R}}]=\frac{1}{\sqrt{2\pi\sigma_{\delta\varphi}^{2}}}\frac{3g}{|\gamma|\kappa}\left(1-\frac{3{\cal{R}}}{g\kappa}\right)\exp\left[-\frac{1}{2\sigma_{\delta\varphi}^{2}}\frac{9g^{2}{\cal{R}}^{2}}{\gamma^{2}\kappa^{2}}\left(1-\frac{3}{2}\frac{{\cal{R}}}{g\kappa}\right)^{2}\right]\,.$
(E.4)

We note that in this case there is a cutoff at ${\cal{R}}=\kappa g/3$. It can
be seen that the PDF decreases rapidly towards 0 at ${\cal{R}}=\kappa g/3$ due
to the factor $1-3{\cal{R}}/g\kappa$. Since the cutoff depends not only on
$\kappa$ but also on $g$, the smaller $g$ we choose, the smaller the cutoff
tends to be.

  3. 3.

The third term dominant regime
Around $\delta\varphi\simeq-2\gamma/g^{2}$, the main contribution to $\cal{R}$
is the third term in Eq. (22), namely,

${\cal{R}}\simeq-\frac{1}{2\omega_{\rm
s2}}\log\left(1+\frac{2\gamma}{g^{2}}\delta\varphi\right)\,.$ (E.5)

Thus $\delta\varphi$ can be given by

$\delta\varphi\simeq\frac{g^{2}}{2\gamma}\Bigl{(}\exp\left(-2\omega_{\rm
s2}{\cal{R}}\right)-1\Bigr{)}\,,$ (E.6)

and then the PDF is

$P[{\cal{R}}]=\frac{1}{\sqrt{2\pi\sigma_{\delta\varphi}^{2}}}\frac{g^{2}\omega_{\rm
s2}}{|\gamma|}\exp\left(-2\omega_{\rm
s2}{\cal{R}}\right)\exp\left[-\frac{1}{2\sigma_{\delta\varphi}^{2}}\frac{g^{2}}{4\gamma^{2}}\Bigl{(}\exp\left(-2\omega_{\rm
s2}{\cal{R}}\right)-1\Bigr{)}^{2}\right]\,.$ (E.7)

For a sufficiently large ${\cal{R}}$, this PDF is almost proportional to
$\exp(-2\omega_{\rm s2}{\cal{R}})$, i.e. the tail appears. Recalling that
$\omega_{\rm s2}\propto 1/\Delta\varphi$, we can see that the effect of this
tail is suppressed by $\Delta\varphi$. If we take the case of zero width
limit, $\Delta\varphi=0$, then the tail vanishes and the cutoff described in
Refs. Cai  _et al._ (2022a, b) can be reproduced. However, the PDF is nonzero
even for ${\cal{R}}$ larger than the cutoff value, ${\cal{R}}_{\rm
cutoff}=\kappa g/3$, unless $\Delta\varphi$ is exactly zero. In this sense, we
can conclude that the cutoff is a side effect of ignoring the finite step
width.

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