# Constant-rate inflation: primordial black holes from conformal weight
transitions

Kin-Wang Ng  and Yi-Peng Wu

###### Abstract

Constant-rate inflation, including ultra-slow-roll inflation as a special
case, has been widely applied to the formation of primordial black holes with
a significant deviation from the standard slow-roll conditions at both the
growing and decaying phases of the power spectrum. We derive analytic
solutions for the curvature perturbations with respect to the late-time
scaling dimensions (conformal weights) constrained by the dilatation symmetry
of the de Sitter background and show that the continuity of conformal weights
across different rolling phases is protected by the adiabatic condition of the
inflaton perturbation. The temporal excitation of subleading states (with the
next-to-lowest conformal weights), recorded as the “steepest growth” of the
power spectrum, is triggered by the entropy production in the transition from
the slow-roll to the constant-rate phases.

##  1 Introduction

Primordial black holes (PBHs) are interesting dark matter candidates that can
be constrained by the ongoing and future detection of gravitational waves even
if they only contribute a small fraction of the total dark matter density (see
[1, 2] for recent reviews and references therein). That enhanced primordial
density fluctuations from inflationary models with large curvature
perturbations on small scales is the mainstream scenario for PBH formation.
PBHs from models of inflation, with targeting masses spanning in the range of
$10^{-17}-10^{9}M_{\odot}$, can be realized either in the single-field [9, 3,
16, 4, 7, 5, 8, 12, 18, 13, 19, 11, 17, 15, 14, 6, 20, 10, 21] or multi-field
[22, 23, 25, 31, 27, 24, 30, 28, 26, 34, 32, 35, 33, 29] framework.

In this work, we focus on the class of PBH models of inflation in the single-
field framework with an approximately constant rate of rolling [39, 38, 40,
36, 37]: $\delta\equiv\ddot{\phi}/(H\dot{\phi})\approx\textrm{constant}$,
where $\phi$ is the inflaton field and $H$ is the Hubble parameter during
inflation. The analytic structure of the power spectrum generated by a
constant-rate inflation has been investigated in [11, 18, 12, 41, 19], showing
an essential departure from the standard slow-roll predictions [9] due to the
significant violation of slow-roll conditions in the background dynamics [8].
Quantum diffusion induced by the off-attractor nature of the constant-rate
rolling can introduce non-negligible corrections on the large-scale spectrum
via the stochastic effect [13, 43, 45, 46, 44, 42].

There are at least two model-independent features residing in the numerical
results of the constant-rate models worthy of further clarifications, namely,
better manifestations with very sharp transitions across different rolling
phases and $\dot{\delta}\approx 0$ in each phase [9, 12, 13, 14, 18]. First of
all, the power spectrum $P_{\mathcal{R}}$ of the gauge invariant curvature
perturbation $\mathcal{R}$ on superhorizon scales exhibits a
$(k/k_{\ast})^{4}$ scaling for $k\lesssim k_{\ast}$, where
$k_{\ast}\approx-1/\eta_{\ast}$ is the horizon-crossing scale at which the
inflation begins with a negative constant rate $\delta<-3/2$ (Phase 2 in
Figure 1). Note that $\delta=-3$ is the so-called ultra-slow-roll limit [39,
38, 36, 37]. Such a $(k/k_{\ast})^{4}$ scaling of the power spectrum has been
recognized in analytic approaches [11, 12, 18] and is an enhancement
independent of the value of $\delta$. Here we refer such a $k^{4}$ enhancement
as the “steepest growth” problem [11, 47]. The $P_{\mathcal{R}}\sim k^{4}$
scaling sourced by sub-leading modes in $\mathcal{R}$ has been recognized in
the ultra-slow-roll limit ($\delta=-3$) [47], and, in this work, we extend the
argument to an arbitrary constant-rate inflation with $\delta\leq-3$. 111A
growth of the power spectrum faster than $k^{4}$ scaling (namely with a
spectral index $n_{s}-1>4$) can be realized in the multi-field inflationary
models [48, 30, 35] or with an intermediate $\delta=-1/2$ phase between the
primary slow-roll and the ultra-slow-roll phases in the single-field inflation
[47].

Second, the decay of the power spectrum from its maximal value exhibits a
fixed scaling of the specific power, $P_{\mathcal{R}}\sim k^{6+2\delta}$, for
a duration much longer than the period with a negative constant rate
$\delta<-3$ (see [9, 12, 13, 14, 18] for examples). Such a continuous scaling
of the $k$-dependence in $P_{\mathcal{R}}$ would imply that the curvature
perturbation $\mathcal{R}$ is not sensitive to the change of the inflaton
potential shape in the post negative-constant-rate phase where $\delta$ may
already become positive (Phase 3 in Figure 1). The fixed scaling
$P_{\mathcal{R}}\sim k^{6+2\delta}$ derived by analytic approaches [11, 18] in
Phase 2 cannot be extrapolated to Phase 3, given that the ending time of Phase
2 is not a free parameter for arbitrary extension but is in fact constrained
by the spectral amplitude of $P_{\mathcal{R}}$ for producing the desired PBH
abundance. The fixed scaling of the power spectrum towards the end of
inflation, regardless the change of $\delta$ from negative to positive values,
is referred as the “continuous scaling” problem.

In this work, we review both the steepest growth and continuous scaling
problems from isometries of the de Sitter spacetime, having in mind that the
scaling dimension of the late-time power spectrum (or the two-point
correlator) is constrained by the dilatation symmetry between a rescaling of
the spatial and temporal coordinates [49, 51, 52, 50]. At a fixed time slice,
these de Sitter invariant transformations project on the spatial hypersurface
as conformal invariant operations, where the late-time correlation function
$\langle\mathcal{R}(\vec{x}_{1})\mathcal{R}(\vec{x}_{2})\rangle\sim|\vec{x}_{1}-\vec{x}_{2}|^{-2\Delta}$
precisely describes a two-point conformal correlator with the scaling
dimension (or conformal weight) $\Delta$ [49, 54, 53].

The scaling dimensions/weights of a scalar field with a mass $m$ in $3+1$
dimensional de Sitter space is well known [55, 56, 57]: It is a pair of real
but non-integer numbers $\Delta_{\pm}=3/2\pm\nu$ if $m/H<3/2$, where
$\nu=\sqrt{9/4-m^{2}/H^{2}}$. In the late-time limit, only the branch led by
the lowest power $\Delta_{-}$ survives. For a heavy field with $m/H>3/2$,
scaling dimensions become non-analytic (imaginary) and both $\Delta_{\pm}$
terms are important in the late-time correlators. Signatures of non-analytic
momentum scaling due to the non-local exchange of heavy fields (including
particles with non-zero spins) in curvature perturbations can be recorded in
non-Gaussian correlators of $\mathcal{R}$ as oscillatory features [53, 54, 59,
60, 67, 61, 64, 65, 66, 62, 63].

On the other hand, the scaling dimensions of the late-time curvature
perturbation in the constant-rate inflation is always analytic (real) but can
be negative non-integer. As we will show that, during the $i$-th phase of
inflation with the rate of rolling controlled by $\delta_{i}$, the leading
power is $\Delta_{i}=3/2-\nu_{i}$ with (neglecting small corrections due to
$\epsilon\equiv-\dot{H}/H^{2}$):

$\displaystyle\nu_{i}=\sqrt{\frac{9}{4}+3\delta_{i}+\delta_{i}^{2}}=\left|\frac{3}{2}+\delta_{i}\right|,$
(1.1)

which gives $\nu_{i}=3/2+\delta_{i}$ for $\delta_{i}>-3/2$ and
$\nu_{i}=-\delta_{i}-3/2$ for $\delta_{i}<-3/2$.

We revisit the steepest growth problem in Section 2 and the continuous scaling
problem in Section 3 on the boundary surface at the transition time and show
that both of them are consequences of the continuity of the scaling
dimensions. Our arguments are supported by the bulk solutions in the de Sitter
spacetime (Section 2.3 and 3.2). We show that the realization of PBH scenarios
with enhanced power spectra led by a negative-constant-rate phase introduces
yet unnoticed constraints on the scaling dimensions (see Section 3.3). In
Section 4, we review the entropy production of inflaton perturbations in PBH
scenarios and clarify its connection with the conformal weight continuity.
Finally, conclusions for the inflationary power spectrum characterizing with a
time-varying conformal weight are given in Section 5.

##  2 The (steepest) growth of power spectrum

Figure 1: A schematic description of the single-field inflation scenario
considered in Section 2 and 3, where $\delta_{i}$ is the $i$-th phase rate-of-
rolling of inflaton and $\Delta_{i}$ is the scaling dimensionality (conformal
weight) of the late-time curvature perturbation $\mathcal{R}$ in each phase.

In Section 2 and 3 we consider a three-stage single-field inflation with sharp
transitions to different phases at $\eta=\eta_{\ast}$ and
$\eta=\eta_{\ast\ast}$, as given by Figure 1, where
$\delta\equiv\ddot{\phi}/(H\dot{\phi})$ is the rate-of-rolling of inflaton
$\phi$, and $\Delta\equiv 3/2-\nu$ is the scaling dimension of the curvature
perturbation $\mathcal{R}$ in the late-time limit 222In this work, the late-
time expansion of the mode functions $\mathcal{R}_{k}$ of the curvature
perturbation is expanding with respect to the dimensionless time parameter
$\tau\equiv-z\equiv-k\eta$ in the limit of $\tau\rightarrow 0$. fixed by the
dilatation symmetry of the four-dimensional de Sitter background [53, 54, 49].
Such a piecewise simplification of the inflationary dynamics can also be found
in [19, 42, 21]. $\nu$ is the dimensionless mass parameter of the mode
functions of $\mathcal{R}$. The first slow-roll parameters in Phase 2 and 3
are set to zero for simplicity (namely, $\epsilon_{2}=\epsilon_{3}=0$).

###  2.1 From slow-roll to negative-constant-rate

Let us start with the background evolution of the first two stages of
inflation, which are described by two sets of slow-roll parameters
$\epsilon_{1},\delta_{1},\cdots$ for the physical time $t_{0}\leq t<t_{\ast}$
and $\epsilon_{2},\delta_{2},\cdots$ for $t_{\ast}\leq t<t_{\ast\ast}$, where
$t_{\ast}$ is the critical time for such a phase transition. We are interested
in the cases where $\epsilon_{1}$ is constrained by CMB observations and
$\epsilon_{2}\ll\epsilon_{1}$ is very tiny and is responsible for the
enhancement of the power spectrum associated with the PBH formation.
Definition of the first slow-roll parameters are

$\displaystyle\epsilon=-\frac{\dot{H}(t)}{H^{2}(t)}=1-\frac{\mathcal{H}^{\prime}(\eta)}{\mathcal{H}^{2}(\eta)}=\left\\{\begin{array}[]{cc}\epsilon_{1},\quad\eta_{0}&\leq\eta<\eta_{\ast},\\\
\epsilon_{2},\quad\eta_{\ast}&\leq\eta<\eta_{\ast\ast},\end{array}\right.$
(2.3)

where $\mathcal{H}\equiv a^{\prime}(\eta)/a(\eta)=a(t)H(t)$ is the conformal
Hubble parameter in terms of the conformal time $d\eta=dt/a(t)$. The second
slow-roll parameters are given by

$\displaystyle\epsilon_{i,2}\equiv\frac{\epsilon_{i}^{\prime}}{\mathcal{H}_{i}\epsilon_{i}}\approx
2\epsilon_{i}+2\delta_{i},$ (2.4)

with $i=1,2$. During the slow-roll phase (Phase 1), the derivative
$\epsilon_{1}^{\prime}\sim\mathcal{O}(\epsilon_{1}^{2})$ so that
$\epsilon_{1}$ is treated as a constant for the background evolution. For
$\eta>\eta_{\ast}$, inflation enters the phase with a constant rate of rolling
$\delta_{2}<-3/2$ and $|\delta_{2}|\gg\epsilon_{2}$ significantly violates the
slow-roll conditions. In general, $\epsilon_{2}$ is a function of $\eta$, yet
for models of PBH formation $\epsilon_{2}$ becomes extremely tiny so that one
can ignore its correction to the Hubble parameter $\mathcal{H}_{2}$. The
solutions of $\mathcal{H}$ in each stage are thus given by

$\displaystyle\mathcal{H}_{1}(\eta)$
$\displaystyle=\frac{1}{1/\mathcal{H}_{0}+(\epsilon_{1}-1)(\eta-\eta_{0})},$
$\displaystyle\quad\eta_{0}\leq\eta<\eta_{\ast},$ (2.5)
$\displaystyle\mathcal{H}_{2}(\eta)$
$\displaystyle=\frac{1}{1/\mathcal{H}_{\ast}-(\eta-\eta_{\ast})},$
$\displaystyle\quad\eta_{\ast}\leq\eta<\eta_{\ast\ast},$ (2.6)

where $\mathcal{H}_{0}\equiv\mathcal{H}(\eta_{0})$ and
$\mathcal{H}_{\ast}\equiv\mathcal{H}(\eta_{\ast})=[1/\mathcal{H}_{0}+(\epsilon_{1}-1)(\eta_{\ast}-\eta_{0})]^{-1}$.
Initial values $\mathcal{H}_{0}$ and $\eta_{0}$ at the beginning of inflation
shall be regularized in such a way that is convenient for computing the
physical observibles. 333 The conventional regularization of the Hubble
parameter is to define $1/\mathcal{H}_{0}\equiv-(1-\epsilon_{1})\eta_{0}$ so
that  $\displaystyle\mathcal{H}_{1}(\eta)$
$\displaystyle=-\frac{1}{(1-\epsilon_{1})\eta},$
$\displaystyle\quad\eta_{0}\leq\eta<\eta_{\ast},$ (2.7)
$\displaystyle\mathcal{H}_{2}(\eta)$
$\displaystyle=\frac{1}{\epsilon_{1}\eta_{\ast}-\eta},$
$\displaystyle\quad\eta_{\ast}\leq\eta<\eta_{\ast\ast}.$ (2.8) One can see
that $\mathcal{H}_{1}$ takes the familiar form for the computation of the
spectral index or the tensor-to-scalar ratio of the power spectrum in the
framework of single-field inflationary models. However, such a regulation with
respect to the slow-roll phase has a non-trivial limitation with the presence
of a second-stage inflation, as one can see that the Hubble parameter (2.8)
may change into a negative value for $\eta>\epsilon_{1}\eta_{\ast}$ before
reaching the asymptotic boundary surface at $\eta=0$.  Here we use a
regularization with respect to the second-stage expansion as
$1/\mathcal{H}_{\ast}=-\eta_{\ast}$. This gives the reduced expression as

$\displaystyle\mathcal{H}_{1}(\eta)$
$\displaystyle=\frac{1}{-\epsilon_{1}\eta_{\ast}-(1-\epsilon_{1})\eta},$
$\displaystyle\quad\eta_{0}\leq\eta<\eta_{\ast},$ (2.9)
$\displaystyle\mathcal{H}_{2}(\eta)$ $\displaystyle=-\frac{1}{\eta},$
$\displaystyle\quad\eta_{\ast}\leq\eta<\eta_{\ast\ast}.$ (2.10)

One can check that $\mathcal{H}$ is positively defined at all time.

As a warm-up exercise, let us solve the mode functions of a massless scalar
field $\phi$, which can apply to an inflaton in (ultra-)slow-roll with
$\delta=0$ (or $\delta=-3$). (The inflaton mode functions for general
constant-rate cases is given in Section 4). The equation of motion for a
$k$-mode perturbation $\delta\phi_{k}$ in the Fourier expansion during
inflation is

$\displaystyle\delta\phi_{k}^{\prime\prime}+2\mathcal{H}\delta\phi_{k}^{\prime}+k^{2}\delta\phi_{k}=0,$
(2.11)

where $\mathcal{H}$ is given by (2.9) and (2.10). To obtain systematical
expressions of the mode functions for different phases, it is convenient to
use the rescaled variable $u_{k}\equiv a\delta\phi_{k}$ with respect to the
dimensionless parameter $z\equiv k\eta$. The equation of motion for $u_{k}$
derived from (2.11) reads

$\displaystyle\frac{\partial^{2}u_{k}}{\partial
z^{2}}+\left(1+\frac{-\mathcal{H}^{2}-\mathcal{H}^{\prime}}{k^{2}}\right)u_{k}=0.$
(2.12)

We can further rewrite the Hubble parameter in the first and the second stages
as $\mathcal{H}_{1}=k/\tau_{1}$ and $\mathcal{H}_{2}=k/\tau_{2}$,
respectively, where we have defined $\tau_{1}=-\epsilon_{1}z_{\ast}-r_{1}z$
with $r_{1}=1-\epsilon_{1}$ and $\tau_{2}=-r_{2}z$ with
$r_{2}=1-\epsilon_{2}$. As a result, the equations for $u_{k}$ become

$\displaystyle
r_{1}^{2}\,\frac{\partial^{2}u_{1}}{\partial\tau_{1}^{2}}+\left(1+\frac{\epsilon_{1}-2}{\tau_{1}^{2}}\right)u_{1}$
$\displaystyle=0,$ $\displaystyle\quad\eta_{0}\leq\eta<\eta_{\ast},$ (2.13)
$\displaystyle
r_{2}^{2}\,\frac{\partial^{2}u_{2}}{\partial\tau_{2}^{2}}+\left(1+\frac{\epsilon_{2}-2}{\tau_{2}^{2}}\right)u_{2}$
$\displaystyle=0,$ $\displaystyle\quad\eta_{\ast}\leq\eta<\eta_{\ast\ast}.$
(2.14)

Here we keep $\epsilon_{2}$ in the equations for a comparison.

The general solution of (2.13) or (2.14) takes the form of
$\sqrt{\tau_{i}}H^{(1,2)}_{\nu_{i}}(\tau_{i}/r_{i})$, where $H^{(1,2)}_{n}(x)$
is the Hankel function of the first or second kind. With respect to the
standard Bunch-Davies vacuum in the UV limit ($-\eta\rightarrow\infty$), the
coefficient for the $H_{\nu_{1}}^{(2)}$ term is set to zero, and the mode
functions are given by

$\displaystyle u_{1}$
$\displaystyle=c_{1}\sqrt{\tau_{1}}H^{(1)}_{\nu_{1}}\left(\frac{\tau_{1}}{r_{1}}\right),$
$\displaystyle\quad\nu_{1}=\frac{1}{r_{1}}\sqrt{2-\epsilon_{1}+\frac{r_{1}^{2}}{4}},$
(2.15) $\displaystyle u_{2}$
$\displaystyle=c_{2}^{(1)}\sqrt{\tau_{2}}H^{(1)}_{\nu_{2}}\left(-z\right)+c_{2}^{(2)}\sqrt{\tau_{2}}H^{(2)}_{\nu_{2}}\left(-z\right),$
$\displaystyle\quad\nu_{2}=\frac{1}{r_{2}}\sqrt{2-\epsilon_{2}+\frac{r_{2}^{2}}{4}}.$
(2.16)

In the limit of $\epsilon_{i}\rightarrow 0$, $\nu_{i}\rightarrow 3/2$ so that
the scaling dimension of a massless field is known as
$\Delta_{i}=3/2-\nu_{i}\rightarrow 0$. Note that $u_{2}$ is described by a
generic (non Bunch-Davies) vacuum state due to the sharp transition from the
slow-roll phase to the negative-constant-rate phase $\eta=\eta_{\ast}$. The
coefficient $c_{1}$ can be fixed by the standard Bunch-Davies vacuum according
to

$\displaystyle\lim_{-z\rightarrow\infty}u_{1}(\tau_{1})\rightarrow
c_{1}\sqrt{-r_{1}z}H^{(1)}_{\nu_{1}}(-z)=-\frac{i}{\sqrt{2k}}e^{-iz}.$ (2.17)

The large $x$ expansion of
$H^{(1)}_{\nu_{1}}(x)=\sqrt{\frac{2}{\pi}}e^{-i\nu_{1}\pi/2-i\pi/4}e^{ix}/\sqrt{x}+\cdots$
gives

$\displaystyle
c_{1}=-\frac{i}{2}\sqrt{\frac{\pi}{r_{1}k}}e^{\frac{i}{2}(\nu_{1}+1/2)\pi}.$
(2.18)

By matching $u(z)$ and $\partial u/\partial z$ at $z=z_{\ast}$, the
coefficients $c_{2}^{(1)}$ and $c_{2}^{(2)}$ can be solved as 444A useful
simplification, $(H^{(1)}_{\nu
2-1}(x)H^{(2)}_{\nu_{2}}(x)-H^{(1)}_{\nu_{2}}(x)H^{(2)}_{\nu_{2}-1}(x))^{-1}=-i\pi
x/4$, has been used in the results of (2.19) and (2.20).

$\displaystyle c_{2}^{(1)}=\frac{i\pi}{4}c_{1}$
$\displaystyle\left[-\tau_{\ast}H^{(1)}_{\nu_{1}-1}\left(\frac{\tau_{\ast}}{r_{1}}\right)H^{(2)}_{\nu_{2}}(\tau_{\ast})\right.$
(2.19)
$\displaystyle+\left.H^{(1)}_{\nu_{1}}\left(\frac{\tau_{\ast}}{r_{1}}\right)\left(\tau_{\ast}H^{(2)}_{\nu_{2}-1}(\tau_{\ast})+(A_{2}-A_{1})H^{(2)}_{\nu_{2}}(\tau_{\ast})\right)\right],$
$\displaystyle c_{2}^{(2)}=\frac{i\pi}{4}c_{1}$
$\displaystyle\left[\tau_{\ast}H^{(1)}_{\nu_{1}-1}\left(\frac{\tau_{\ast}}{r_{1}}\right)H_{\nu_{2}}^{(1)}(\tau_{\ast})\right.$
(2.20)
$\displaystyle+\left.H^{(1)}_{\nu_{1}}\left(\frac{\tau_{\ast}}{r_{1}}\right)\left(-\tau_{\ast}H^{(1)}_{\nu_{2}-1}(\tau_{\ast})+(A_{1}-A_{2})H^{(1)}_{\nu_{2}}(\tau_{\ast})\right)\right],$

where $A_{1}=r_{1}(1-2\nu_{1})/2$ and $A_{2}=r_{2}(1-2\nu_{2})/2$.

We can compute the power spectrum of the massless scalar field according to
the definition

$\displaystyle\left\langle\delta\phi_{\mathbf{k}}\delta\phi_{\mathbf{p}}\right\rangle=(2\pi)^{3}\delta^{(3)}\left(\mathbf{k}+\mathbf{p}\right)P_{\delta\phi}(k;\eta)\frac{2\pi^{2}}{k^{3}},$
(2.21)

where
$\delta\phi_{\mathbf{k}}=\delta\phi_{k}a_{\mathbf{k}}+\delta\phi_{k}^{\ast}a_{-\mathbf{k}}^{\dagger}$
and $a_{\mathbf{k}}^{\dagger}$ ($a_{\mathbf{k}}$) is the creation
(annihilation) operator of the free vacuum satisfying the commutation relation
$[a_{\mathbf{k}},a_{-\mathbf{p}}^{\dagger}]=(2\pi)^{3}\delta^{(3)}(\mathbf{k}+\mathbf{p})$.
The spectrum of $\delta\phi$ is led by

$\displaystyle P_{\delta\phi}(k;\eta)$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{1}{a^{2}(\eta)}\left|u_{k}(\eta)\right|^{2}.$
(2.22)

Here the scale factor $a(\eta)$ shall be expressed in terms of $a_{k}\equiv
a(\eta_{k})$ at which the $k$-mode exits the horizon. For
$k<k_{\ast}=-1/\eta_{\ast}$ we can solve the scale factor $a(\eta)$ for the
two stages via $\mathcal{H}$ given by (2.9) and (2.10) respectively, which
indicates

$\displaystyle\frac{a_{\ast}}{a_{k}}$
$\displaystyle=\left[-\eta_{\ast}\mathcal{H}_{k}\right]^{-1/r_{1}},$ (2.23)
$\displaystyle\frac{a}{a_{\ast}}$
$\displaystyle=\left(\frac{\eta}{\eta_{\ast}}\right)^{-1},$ (2.24)

where $\mathcal{H}_{k}=\mathcal{H}(\eta_{k})=a_{k}H_{k}=k$. The spectrum is
therefore

$\displaystyle
P_{\delta\phi}(k;\eta)=\frac{k^{3}}{2\pi^{2}}\left(\frac{a_{\ast}}{a(\eta)}\frac{1}{a_{\ast}}\right)^{2}\left|c_{2}^{(1)}\sqrt{-k\eta}H^{(1)}_{\nu_{2}}(-k\eta)+c_{2}^{(2)}\sqrt{-k\eta}H^{(2)}_{\nu_{2}}(-k\eta)\right|^{2}.$
(2.25)

The small $x$ expansion of the Hankel functions are
$H^{(1,2)}_{n}(x)=\mp\frac{i}{\pi}2^{n}\Gamma(n)x^{-n}+\cdots$, and therefore
for $k\ll-1/\eta$, the late-time limit of the power spectrum reads

$\displaystyle P_{\delta\phi}^{\rm
IR}(k;\eta)=\frac{kH_{k}^{2}}{2\pi^{2}}\left(-z_{\ast}\right)^{\frac{2}{r_{1}}-2}\left(-z\right)^{3-2\nu_{2}}\left|\left(c_{2}^{(2)}-c_{2}^{(1)}\right)\frac{i}{\pi}2^{\nu_{2}}\Gamma(\nu_{2})\right|^{2},$
(2.26)

where both $c_{2}^{(1)}$ and $c_{2}^{(2)}$ are functions of
$\tau_{\ast}=-z_{\ast}=k/k_{\ast}$, and

$\displaystyle c_{2}^{(2)}-c_{2}^{(1)}=$
$\displaystyle\frac{i\pi}{2}c_{1}\tau_{\ast}\left[J_{\nu_{2}}(\tau_{\ast})H_{\nu_{1}-1}^{(1)}\left(\frac{\tau_{\ast}}{r_{1}}\right)\right.$
(2.27)
$\displaystyle\qquad\left.+H_{\nu_{1}}^{(1)}\left(\frac{\tau_{\ast}}{r_{1}}\right)\left(\frac{A_{1}-A_{2}}{\tau_{\ast}}J_{\nu_{2}}(\tau_{\ast})-J_{\nu_{2}-1}(\tau_{\ast})\right)\right].$

In the case with $\epsilon_{1}=\epsilon_{2}=0$, $\nu_{1}=\nu_{2}=3/2$ and
$c_{2}^{(2)}-c_{2}^{(1)}=-c_{1}$. One can reproduce the standard result for a
massless scalar as $P_{\delta\phi}^{\rm IR}(k;\eta)=H_{k}^{2}/(4\pi^{2})$.

###  2.2 Boundary arguments for steepest growth

In this section we provide a simple argument to show that the enhancement of
the curvature perturbation from slow-roll (Phase 1) to negative-constant-rate
(Phase 2) is a $\mathcal{R}\sim k^{2}$ growth for a finite range $k_{\rm
min}<k<k_{\ast}$ for modes exit the horizon close to the end of Phase 1 [11,
18]. Such a growth that corresponds to a power spectrum with the spectral
index $n_{s}-1=4$ is recognized as the steepest possible growth beyond the
limitation of the ordinary matter power spectrum [11]. Our arguments only base
on the continuity of matching the late-time curvature perturbations,
$\mathcal{R}_{1}$ and $\mathcal{R}_{2}$, at the boundary surface
$\eta=\eta_{\ast}$ with their leading dimensionality, $\Delta_{1}$ and
$\Delta_{2}$, constrained by the dilatation symmetry of the de Sitter
background [53, 54].

The first step is to find out the correct scaling dimension of $\mathcal{R}$
in each phase. The dynamics of the curvature perturbation $\mathcal{R}$ can be
computed by virtue of the Mukhanov-Sasaki variable $v=-y\mathcal{R}$, where
$y=a\sqrt{\epsilon}$ includes the first slow-roll parameter and each Fourier
mode function $v_{k}$ follows the equation

$\displaystyle
v_{k}^{\prime\prime}+\left(k^{2}-\frac{y^{\prime\prime}}{y}\right)v_{k}=0.$
(2.28)

The ratio
$y^{\prime\prime}/y=\mathcal{H}^{2}(2+2\epsilon+3\delta+\delta^{2}+\cdots)$
features the time-varying mass term for the mode function. To identify the
steepest growth, it is enough to impose
$\epsilon_{1}=\delta_{1}=\epsilon_{2}=0$ for the computation of $v_{k}$, which
gives $y_{1}^{\prime\prime}/y_{1}=2\mathcal{H}_{1}^{2}$ and
$y_{2}^{\prime\prime}/y_{2}=\mathcal{H}_{2}^{2}(2+3\delta_{2}+\delta_{2}^{2})$.
Note that such a simplification also makes
$\mathcal{H}_{1}=-1/\eta=\mathcal{H}_{2}$ and thus (2.28) reads

$\displaystyle\frac{\partial^{2}v_{1}}{\partial\tau_{1}^{2}}+\left(1+\frac{\nu_{1}^{2}-1/4}{\tau_{1}^{2}}\right)v_{1}$
$\displaystyle=0,$ $\displaystyle\quad\eta_{0}\leq\eta<\eta_{\ast},$ (2.29)
$\displaystyle\frac{\partial^{2}v_{2}}{\partial\tau_{2}^{2}}+\left(1+\frac{\nu_{2}^{2}-1/4}{\tau_{2}^{2}}\right)v_{2}$
$\displaystyle=0,$ $\displaystyle\quad\eta_{\ast}\leq\eta<\eta_{\ast\ast},$
(2.30)

where $\tau=-z=-k\eta$. One can observe that $\nu_{1}=3/2$ and
$\nu_{2}=(9/4+3\delta_{2}+\delta_{2}^{2})^{1/2}=|3/2+\delta_{2}|$, which
implies that $\Delta_{1}=0$ and $\Delta_{2}=3/2-\nu_{2}=3+\delta_{2}$ for
$\delta_{2}<-3/2$.

Given that $\Delta_{1}$ and $\Delta_{2}$ are only real numbers, we can expand
the curvature perturbation $\mathcal{R}_{k}=-v_{k}/y$ in the limit of
$\tau=-k\eta\rightarrow 0$ as

$\displaystyle
k^{3/2}\mathcal{R}_{i}\rightarrow\mathcal{C}_{i0}(-k\eta)^{\Delta_{i}}+\mathcal{C}_{i1}(-k\eta)^{\Delta_{i}+2}+\cdots,$
(2.31)

where $\mathcal{C}_{in}$ are late-time coefficients for the power series of
$(-k\eta)^{\Delta_{i}+2n}$. 555The even power $2n$ of this expansion comes
from the power series expression of the Hankel function, where  $\displaystyle
H_{\nu}^{(1)}(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\left(i\csc\pi\nu\right)\left(\frac{x}{2}\right)^{2n}\left[e^{-i\pi\nu}\frac{1}{\Gamma(n+1+\nu)}\left(\frac{x}{2}\right)^{\nu}-\frac{1}{\Gamma(n+1-\nu)}\left(\frac{x}{2}\right)^{-\nu}\right],$
and $H_{\nu}^{(2)}(x)$ is the complex conjugate if both $\nu$ and $x$ are
real.  The coefficients $\mathcal{C}_{in}$ in the definition of (2.31) are
dimensionless but can have $k$-dependence, and the combination
$\mathcal{C}_{in}k^{\Delta_{i}+2n}$ can be taken as primary operators of a
conformal field theory with dimension $\Delta_{i}+2n$. In particular,
$\mathcal{C}_{1n}$ are fixed by $c_{1}$ in (2.18) from the Bunch-Davies
vacuum, and the explicit form of $\mathcal{C}_{2n}$ can be found in the next
section. For the late-time expansion to be valid in Phase 1 the momentum $k$
shall satisfy $\tau_{\ast}\equiv-k\eta_{\ast}=k/k_{\ast}\ll 1$.

There are two boundary conditions come from the continuity of $\mathcal{R}$ at
$\eta=\eta_{\ast}$. The first condition
$\mathcal{R}_{1}(\eta_{\ast})=\mathcal{R}_{2}(\eta_{\ast})$ gives

$\displaystyle\mathcal{C}_{10}\left(\frac{k}{k_{\ast}}\right)^{\Delta_{1}}+\mathcal{C}_{11}\left(\frac{k}{k_{\ast}}\right)^{\Delta_{1}+2}+\cdots=\mathcal{C}_{20}\left(\frac{k}{k_{\ast}}\right)^{\Delta_{2}},$
(2.32)

which implies that the leading contribution to the coefficient
$\mathcal{C}_{20}$ in the limit of $k/k_{\ast}\ll 1$ is given by the term with
the lowest power and thus
$\mathcal{C}_{20}\sim(k/k_{\ast})^{\Delta_{1}-\Delta_{2}}$. As a result, the
late-time curvature perturbation obtained in Phase 2 reads
$k^{3/2}\mathcal{R}_{2}\sim k^{\Delta_{1}}$, which is independent of
$\Delta_{2}$. This asserts that the very large-scale modes (well outside the
horizon during Phase 1) are not affected by a subsequent phase transition at
late times.

However, for the special case $\Delta_{1}\rightarrow 0$, the subleading terms
$\mathcal{C}_{1n}$ with higher powers ($n>0$) can become important when
$k/k_{\ast}\rightarrow 1$ and change the momentum dependence of
$\mathcal{C}_{20}$ . This fact is manifest from the first derivative
$\mathcal{R}_{1}^{\prime}(\eta_{\ast})=\mathcal{R}_{2}^{\prime}(\eta_{\ast})$
as

$\displaystyle
A_{1}\mathcal{C}_{10}\left(\frac{k}{k_{\ast}}\right)^{\Delta_{1}-1}+\left(A_{1}\mathcal{C}_{11}+B_{1}\right)\left(\frac{k}{k_{\ast}}\right)^{\Delta_{1}+1}+\cdots=A_{2}\mathcal{C}_{20}\left(\frac{k}{k_{\ast}}\right)^{\Delta_{2}-1},$
(2.33)

where $A_{i}=(\epsilon_{i,2}/2+\Delta_{i}r_{i})$ contains the second slow-roll
parameter due to the derivative
$y_{i}^{\prime}/y_{i}=\mathcal{H}_{i}(1+\epsilon_{i,2}/2)$.

If Phase 1 is an exact de Sitter inflation with $\epsilon_{1}=\delta_{1}=0$,
then $A_{1}=\Delta_{1}=0$ so that (2.33) gives
$\mathcal{C}_{20}\sim(k/k_{\ast})^{\Delta_{1}-\Delta_{2}+2}$ for
$k/k_{\ast}\sim\mathcal{O}(1)$. Therefore the late-time curvature perturbation
exhibits an unusual scaling as

$\displaystyle
k^{3/2}\mathcal{R}_{2}(\eta)\rightarrow\mathcal{C}_{20}(-k\eta)^{\Delta_{2}}\sim
k^{\Delta_{1}+2},$ (2.34)

for a certain range of $k$. This is nothing but a nearly $n_{s}-1=4$ growth of
the power spectrum ($P_{\mathcal{R}}\sim k^{2\Delta_{1}+4}$) when
$\Delta_{1}\ll 1$. Note that for general cases with $\Delta_{1}\neq 0$, (2.33)
reaches to the same conclusion
$\mathcal{C}_{20}\sim(k/k_{\ast})^{\Delta_{1}-\Delta_{2}}$ as that of (2.32).
If Phase 2 approaches to the slow-roll limit with $A_{2}\rightarrow 0$ and
$\Delta_{2}\rightarrow 0$, then one will find that
$\mathcal{C}_{2n}\rightarrow\mathcal{C}_{1n}$ and the first constraint (2.32)
converges to be the same as (2.33).

###  2.3 Bulk solutions

We now perform a detailed calculation to support the boundary arguments in the
previous section. Taking $\mathcal{H}_{i}=k/\tau_{i}$, the equation of motion
(2.28) shares a similar structure as those of (2.29) and (2.30) so that the
solution of $v_{k}$ in each phase takes the form of

$\displaystyle v_{1}$
$\displaystyle=c_{1}\sqrt{\tau_{1}}H^{(1)}_{\nu_{1}}\left(\tau_{1}\right),$
$\displaystyle\quad\nu_{1}=3/2+\mathcal{O}(\epsilon),$ (2.35) $\displaystyle
v_{2}$
$\displaystyle=c_{2}^{(1)}\sqrt{\tau_{2}}H^{(1)}_{\nu_{2}}\left(\tau_{2}\right)+c_{2}^{(2)}\sqrt{\tau_{2}}H^{(2)}_{\nu_{2}}\left(\tau_{2}\right),$
$\displaystyle\quad\nu_{2}=|\delta_{2}+3/2|,$ (2.36)

where $\nu_{2}=-\delta_{2}-3/2$ for $\delta_{2}<-3/2$ and $c_{1}$ is given by
(2.18). Matching the mode functions at $z=z_{\ast}$, $c_{2}^{(1)}$ and
$c_{2}^{(2)}$ are solved by the Cauchy boundary conditions as

$\displaystyle c_{2}^{(1)}=\frac{i\pi}{4}c_{1}$
$\displaystyle\left[-\tau_{\ast}H^{(1)}_{\nu_{1}-1}\left(\tau_{\ast}\right)H^{(2)}_{\nu_{2}}(\tau_{\ast})\right.$
(2.37)
$\displaystyle\;+\left.H^{(1)}_{\nu_{1}}\left(\tau_{\ast}\right)\left(\tau_{\ast}H^{(2)}_{\nu_{2}-1}(\tau_{\ast})+(A_{2}-A_{1})H^{(2)}_{\nu_{2}}(\tau_{\ast})\right)\right],$
$\displaystyle c_{2}^{(2)}=\frac{i\pi}{4}c_{1}$
$\displaystyle\left[\tau_{\ast}H^{(1)}_{\nu_{1}-1}\left(\tau_{\ast}\right)H_{\nu_{2}}^{(1)}(\tau_{\ast})\right.$
(2.38)
$\displaystyle\;+\left.H^{(1)}_{\nu_{1}}\left(\tau_{\ast}\right)\left(-\tau_{\ast}H^{(1)}_{\nu_{2}-1}(\tau_{\ast})+(A_{1}-A_{2})H^{(1)}_{\nu_{2}}(\tau_{\ast})\right)\right],$

with $A_{i}=(\epsilon_{i,2}/2+\Delta_{i}r_{i})$ for $i=1,2$, where
$\epsilon_{1}=\delta_{1}=\epsilon_{2}=0$ gives $A_{1}=0$ and
$A_{2}=3+2\delta_{2}$.

Figure 2: The power spectrum $P_{\mathcal{R}}$ from the slow-roll (Phase 1) to
the negative constant rate (Phase 2) given by (2.3) with $H_{k}=10^{-5}$ in
Planck unit. The dotted lines are IR expansions given by (2.3). The dashed-
vertical line is $k=k_{\ast\ast}$.

The power spectrum of $\mathcal{R}$ during Phase 2,
$\eta_{\ast}<\eta\leq\eta_{\ast\ast}$, is given by

$\displaystyle
P_{\mathcal{R}}(k;\eta)=\frac{k^{3}}{2\pi^{2}}\frac{1}{y^{2}(\eta)}\left|v_{k}(\eta)\right|^{2}=\frac{k^{3}}{2\pi^{2}}\frac{1}{a^{2}(\eta)\epsilon_{2}(\eta)}\left|v_{2}(\eta)\right|^{2},$
(2.39)

where $\epsilon_{2}(\eta)=\epsilon_{\ast}(\eta/\eta_{\ast})^{-2\delta_{2}}$
and $\epsilon_{\ast}=\epsilon_{1}$ is a constant for the slow-roll phase
probed on CMB scales. Denoting that $a_{\ast\ast}\equiv a(\eta_{\ast\ast})$,
we have $a_{\ast}/a_{\ast\ast}=k_{\ast}/k_{\ast\ast}$, and the power spectrum
at $\eta=\eta_{\ast\ast}$ reads

$\displaystyle P_{\mathcal{R}}(k;\eta_{\ast\ast})$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{1}{a^{2}_{k}\epsilon_{\ast}}\left(\frac{k_{\ast}}{k_{\ast\ast}}\right)^{2\delta_{2}}\left(\frac{k}{k_{\ast\ast}}\right)^{2}\left|v_{2}(\eta_{\ast\ast})\right|^{2}$
$\displaystyle=\frac{kH_{k}^{2}}{2\pi^{2}}\frac{1}{\epsilon_{\ast}}\left(\frac{k_{\ast}}{k_{\ast\ast}}\right)^{2+2\delta_{2}}\left(\frac{k}{k_{\ast}}\right)^{2}\left|v_{2}(\eta_{\ast\ast})\right|^{2},$
(2.40)

with $a_{k}=k/H_{k}$ and $k=\mathcal{H}(\eta_{k})=-1/\eta_{k}$. This
expression (2.3) reproduces the numerical result of the theoretical model in
[9] with the analytical structure identified in [18].

To see the analytical structure of (2.3) more explicitly, let us focus on
superhorizon modes, $k\ll k_{\ast\ast}$, by expanding the Hankel functions in
$v_{2}$ to find that

$\displaystyle P_{\mathcal{R}}^{\rm IR}(k;\eta_{\ast\ast})$
$\displaystyle=\frac{kH_{k}^{2}}{2\pi^{2}}\frac{1}{\epsilon_{\ast}}\left(\frac{k_{\ast}}{k_{\ast\ast}}\right)^{2\delta_{2}}\left|c_{1}f^{2}_{\mathcal{R}}(\tau_{\ast})\right|^{2}\left|\frac{i}{\pi}2^{\nu_{2}}\Gamma(\nu_{2})\right|^{2}\left(\frac{k}{k_{\ast\ast}}\right)^{3-2\nu_{2}}$
$\displaystyle=\frac{H_{k}^{2}}{8\pi}\frac{1}{\epsilon_{\ast}}\left|f^{2}_{\mathcal{R}}(\tau_{\ast})\right|^{2}\left|\frac{i}{\pi}2^{\nu_{2}}\Gamma(\nu_{2})\right|^{2}\left(\frac{k_{\ast}}{k_{\ast\ast}}\right)^{6+4\delta_{2}}\tau_{\ast}^{6+2\delta_{2}},$
(2.41)

where in terms of $\tau_{\ast}=-z_{\ast}=k/k_{\ast}$ we can transfer the ratio
to $k/k_{\ast\ast}=\tau_{\ast}k_{\ast}/k_{\ast\ast}$. One can see that
$(k_{\ast}/k_{\ast\ast})^{6+4\delta_{2}}=e^{-(6+4\delta_{2})\Delta N}$ is the
factor of the enhancement due to a negative constant rate $\delta_{2}<-3/2$
for a period of $\Delta N$ $e$-folding numbers [18].

The $k$-dependence of the spectrum is led by functions of $\tau_{\ast}$ where
we have defined

$\displaystyle f^{2}_{\mathcal{R}}(\tau_{\ast})$
$\displaystyle\equiv\frac{c_{2}^{(2)}-c_{2}^{(1)}}{c_{1}}$ (2.42)
$\displaystyle=\frac{i\pi}{2}\tau_{\ast}\left[J_{\nu_{2}}(\tau_{\ast})H_{1/2}^{(1)}\left(\tau_{\ast}\right)+H_{3/2}^{(1)}\left(\tau_{\ast}\right)\left(\frac{-3-2\delta_{2}}{\tau_{\ast}}J_{\nu_{2}}(\tau_{\ast})-J_{\nu_{2}-1}(\tau_{\ast})\right)\right].$

In the limit of $\tau_{\ast}\ll 1$, the leading terms of $f_{\mathcal{R}}$ are

$\displaystyle
f^{2}_{\mathcal{R}}(\tau_{\ast})=-\sqrt{\frac{\pi}{2^{3}}}\left(\frac{\tau_{\ast}}{2}\right)^{\nu_{2}}\left[\frac{3+2\delta_{2}+2\nu_{2}}{\Gamma(1+\nu_{2})}\tau_{\ast}^{-3/2}+\frac{\delta_{2}}{2}\frac{7+2\delta_{2}+2\nu_{2}}{\Gamma(2+\nu_{2})}\tau_{\ast}^{1/2}+\cdots\right],$
(2.43)

where the first term in proportion to $\tau_{\ast}^{\nu_{2}-3/2}$ vanishes
accidentally since $\nu_{2}=-\delta_{2}-3/2$ for $\delta_{2}<-3/2$. Therefore
$f^{2}_{\mathcal{R}}\sim\tau_{\ast}^{\nu_{2}+1/2}$ for $k<k_{\ast}$ and the
power spectrum $P_{\mathcal{R}}^{\rm IR}\sim\tau_{\ast}^{4}$ shows a $k^{4}$
enhancement for $k_{\rm min}<k<k_{\ast}$ independent of the value of
$\delta_{2}$. The minimal scale of growth $k_{\rm
min}/k_{\ast}=(k_{\ast}/k_{\ast\ast})^{\nu_{2}}$ is estimated by the condition
$P_{\mathcal{R}}(k=k_{\ast})=(k_{\ast}/k_{\ast\ast})^{6+2\delta_{2}}P_{\mathcal{R}}(k\ll
k_{\ast})$ [18]. For $k<k_{\rm min}$, the constant modes dominate
$P_{\mathcal{R}}$, thus reproducing the standard $k$-invariant result on very
large scales.

On the other hand, in the limit of $\tau_{\ast}\gg 1$,
$f^{2}_{\mathcal{R}}\rightarrow e^{i\pi(1+2\nu_{2})/4}$ becomes a purely
oscillatory function of $k$ so that the power-law scaling
$P_{\mathcal{R}}\sim\tau_{\ast}^{2\Delta_{2}}=\tau_{\ast}^{6+2\delta_{2}}$ for
$k>k_{\ast}$. This describes the power spectrum for $k$ modes exit the horizon
well in side Phase 2. The maximal of $P_{\mathcal{R}}$ appears at $k=k_{\ast}$
if $\delta_{2}<-3$ and at $k=k_{\ast\ast}$ if $-3<\delta_{2}<-3/2$. In Figure
2 we plot the full expression (2.3) and the IR approximation (2.3) of the
power spectrum solved in this section with different choices of
$\delta_{2}<-3/2$. In the limit of $\tau_{\ast}\rightarrow\infty$,
$v_{2}(\eta_{\ast\ast})$ has no additional power-law dependence on $k$ (see
the discussion of the large $k$ limit in Section 3.4 for more details) so that
$P_{\mathcal{R}}\sim\tau_{\ast}^{2}$, while $P_{\mathcal{R}}^{\rm
IR}\sim\tau_{\ast}^{6+2\delta_{2}}$ holds for the extrapolated regime beyond
$k>k_{\ast\ast}$. For the exact USR case with $\delta_{2}=-3$, one finds that
$P_{\mathcal{R}}^{\rm IR}\sim\tau_{\ast}^{0}$ stands for a scale-invariant
spectrum in the large $k$ limit. The limitation of the continuous scaling of
$P_{\mathcal{R}}$ after the end of Phase 2 is discussed in Section 3.

##  3 Continuous decay of power spectrum

The duration of Phase 2, namely $\Delta N=\ln(k_{\ast\ast}/k_{\ast})$, with a
negative constant rate, $\delta_{2}<-3/2$, is the key factor that determines
the magnitude of enhancement to the power spectrum at the beginning of Phase
2: $P_{\mathcal{R}}(k=k_{\ast})/P_{\mathcal{R}}(k\ll
k_{\ast})\sim(k_{\ast}/k_{\ast\ast})^{6+4\delta_{2}}=e^{-(6+4\delta_{2})\Delta
N}$. This means that the ending time of Phase 2, namely
$\eta=\eta_{\ast\ast}=-1/k_{\ast\ast}$, cannot be arbitrarily extended (to
some number close to zero) or otherwise the power spectrum becomes arbitrarily
large. Thus a realistic power spectrum on the boundary surface at the end of
inflation must include also the solutions of mode functions in the post
negative-constant-rate phase (Phase 3). In particular, to realize a spiky
power spectrum for PBH formation around a certain mass scale, Phase 3 shall
address how the curvature perturbation can be successfully reduced from the
enhancement after Phase 2.

###  3.1 Boundary arguments for continuous scaling

With a careful look at the numerical results of [9, 18, 12], one finds that
$\delta_{3}$, the rate of rolling in the post negative-constant-rate phase, is
still approximately a constant and it can have a positive value as large as
$\mathcal{O}(1)$, which significantly violates the slow-roll conditions. Based
on this observation we construct Phase 3 with a constant rate parameter
$\delta_{3}\geq 0$, and we expect that the late-time scaling of the curvature
perturbation is again led by the conformal weight $\Delta_{3}$ with respect to
the dilatation symmetry as

$\displaystyle
k^{3/2}\mathcal{R}_{3}\rightarrow\mathcal{C}_{30}(-k\eta)^{\Delta_{3}}+\mathcal{C}_{31}(-k\eta)^{\Delta_{3}+2}+\cdots,$
(3.1)

where $\Delta_{3}=3/2-\nu_{3}=-\delta_{3}$ can include the slow-roll case if
$\delta_{3}\ll 1$.

As argued in Section 2.2, the boundary condition
$\mathcal{R}_{2}(\eta_{\ast\ast})=\mathcal{R}_{3}(\eta_{\ast\ast})$ for $k\ll
k_{\ast\ast}$ indicates
$\mathcal{C}_{30}\sim\mathcal{C}_{20}(k/k_{\ast\ast})^{\Delta_{2}-\Delta_{3}}$
so that $k^{3/2}\mathcal{R}_{3}\sim\mathcal{C}_{20}k^{\Delta_{2}}$ shows a
scaling independent of $\Delta_{3}$ in Phase 2. Note that
$\mathcal{C}_{20}\propto c_{1}f_{\mathcal{R}}^{2}$ given by (2.42) becomes
$k$-independent in the limit of $k\gg k_{\ast}$, and therefore
$k^{3/2}\mathcal{R}_{3}\sim k^{\Delta_{2}}$ reproduces the continuous scaling
$P_{\mathcal{R}}\sim k^{2\Delta_{2}}=k^{6+2\delta_{2}}$ in [11, 12, 9, 18] to
Phase 3 where inflaton is no longer rolling with a negative constant rate.

The continuous $k$-scaling of the curvature perturbation from Phase 2 to Phase
3 breaks down when $k/k_{\ast\ast}\gg 1$. Since
$\mathcal{R}_{i}(k\rightarrow\infty)\sim c_{i}^{(1,2)}e^{\pm ik\eta}/y_{i}$,
the boundary condition
$\mathcal{R}_{1}(\eta_{\ast})=\mathcal{R}_{2}(\eta_{\ast})$ in the large $k$
limit indicates that the absolute value $|\mathcal{C}_{20}|$ is
$k$-independent. Similarly, $|\mathcal{C}_{30}|$ is $k$-independent in the
limit of $k\rightarrow\infty$ due to the boundary condition
$\mathcal{R}_{2}(\eta_{\ast\ast})=\mathcal{R}_{3}(\eta_{\ast\ast})$. As a
result, the asymptotic momentum scaling of the power spectrum reads
$P_{\mathcal{R}}\sim k^{2\Delta_{3}}=k^{-2\delta_{3}}$ in the large $k$ limit.

###  3.2 Unified bulk solutions

We now derive the full analytic solutions to be observed on the final boundary
at the end of Phase 3. The mode function of the Mukhanov-Sasaki variable in
Phase 3 takes the form of

$\displaystyle
v_{3}=c_{3}^{(1)}\sqrt{\tau_{3}}H^{(1)}_{\nu_{3}}\left(\tau_{3}\right)+c_{3}^{(2)}\sqrt{\tau_{3}}H^{(2)}_{\nu_{3}}\left(\tau_{3}\right),$
$\displaystyle\quad\nu_{3}=\delta_{3}+3/2,$ (3.2)

where $\tau_{3}=-k\eta$ and $\mathcal{H}_{3}=k/\tau_{3}=-1/\eta$ by taking
$\epsilon_{3}=0$.

Let us define $k_{\rm end}=-1/\eta_{\rm end}$ as the scale at the end of Phase
3 and the duration of Phase 2 and 3 are fixed by the ratios $x_{2}\equiv
k_{\ast}/k_{\ast\ast}$, $x_{3}\equiv k_{\ast\ast}/k_{\rm end}$ which are
always smaller than unity. This gives $z_{\rm end}=-k\eta_{\rm
end}=\tau_{\ast}x_{2}x_{3}$, and the Phase 3 coefficients solved at the
boundary surface $-\eta=k_{\ast\ast}$ are given by

$\displaystyle c_{3}^{(1)}=$
$\displaystyle\frac{i\pi}{4}c_{1}\tau_{\ast}x_{2}\left[\left(c_{2}^{(1)}H_{\nu_{2}}^{(1)}(\tau_{\ast}x_{2})+c_{2}^{(2)}H_{\nu_{2}}^{(2)}(\tau_{\ast}x_{2})\right)H^{(2)}_{\nu_{3}-1}(\tau_{\ast}x_{2})\right.$
(3.3)
$\displaystyle\;+\left.\left(c_{2}^{(1)}H^{(1)}_{\nu_{2}+1}(\tau_{\ast}x_{2})+c_{2}^{(2)}H^{(2)}_{\nu_{2}+1}(\tau_{\ast}\tau_{\ast\ast})\right)H_{\nu_{3}}^{(2)}(\tau_{\ast}x_{2})\right],$
$\displaystyle c_{3}^{(2)}=$
$\displaystyle-\frac{i\pi}{4}c_{1}\tau_{\ast}x_{2}\left[\left(c_{2}^{(1)}H_{\nu_{2}}^{(1)}(\tau_{\ast}x_{2})+c_{2}^{(2)}H_{\nu_{2}}^{(2)}(\tau_{\ast}x_{2})\right)H^{(1)}_{\nu_{3}-1}(\tau_{\ast}x_{2})\right.$
(3.4)
$\displaystyle\;+\left.\left(c_{2}^{(1)}H^{(1)}_{\nu_{2}+1}(\tau_{\ast}x_{2})+c_{2}^{(2)}H^{(2)}_{\nu_{2}+1}(\tau_{\ast}x_{2})\right)H_{\nu_{3}}^{(1)}(\tau_{\ast}x_{2})\right].$

Note that $c_{2}^{(1,2)}$ solved by (2.37) and (2.38) are scale-dependent
functions of $\tau_{\ast}=k/k_{\ast}$.

Since $\delta_{3}>0$ implies an enhancement of $\epsilon_{3}$, we find that
$\epsilon_{\rm end}=\epsilon(\eta_{\rm
end})=\epsilon_{\ast}x_{2}^{-2\delta_{2}}x_{3}^{-2\delta_{3}}$. The final
power spectrum is then

$\displaystyle P_{\mathcal{R}}(k;\eta_{\rm end})$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{1}{a^{2}_{\rm end}\epsilon_{\rm
end}}\left|v_{3}(\eta_{\rm end})\right|^{2}$
$\displaystyle=\frac{kH_{k}^{2}}{2\pi^{2}}\left(\tau_{\ast}x_{2}x_{3}\right)^{2}\frac{x_{2}^{2\delta_{2}}x_{3}^{2\delta_{3}}}{\epsilon_{\ast}}\left|v_{3}(\eta_{\rm
end})\right|^{2},$ (3.5)

and the expansion of $v_{3}$ for $k/k_{\rm end}\ll 1$ gives

$\displaystyle P_{\mathcal{R}}^{\rm IR}(k;\eta_{\rm end})$
$\displaystyle=\frac{H_{k}^{2}}{8\pi}\frac{x_{2}^{2\delta_{2}}x_{3}^{2\delta_{3}}}{\epsilon_{\ast}}\left|f_{\mathcal{R}}^{3}(\tau_{\ast},x_{2})\right|^{2}\left|\frac{i}{\pi}2^{\nu_{3}}\Gamma(\nu_{3})\right|^{2}\left(\tau_{\ast}x_{2}x_{3}\right)^{3-2\nu_{3}}$
(3.6)
$\displaystyle=\frac{H_{k}^{2}}{8\pi}\frac{x_{2}^{2\delta_{2}}}{\epsilon_{\ast}}\left|f_{\mathcal{R}}^{3}(\tau_{\ast},x_{2})\right|^{2}\left|\frac{i}{\pi}2^{3/2+\delta_{3}}\Gamma(3/2+\delta_{3})\right|^{2}\left(\tau_{\ast}x_{2}\right)^{-2\delta_{3}},$
(3.7)

where we have defined
$f_{\mathcal{R}}^{3}\equiv(c_{3}^{(2)}-c_{3}^{(1)})/c_{1}$. We remark that the
general expression in terms of $\nu_{2}$ and $\nu_{3}$ applies to arbitrary
values of $\delta_{2}$ and $\delta_{3}$, while the specific expression using
$\nu_{2}=-3/2-\delta_{2}$ and $\nu_{3}=3/2+\delta_{3}$ is for
$\delta_{2}<-3/2$ and $\delta_{3}\geq 0$.

Figure 3: The power spectrum $P_{\mathcal{R}}$ given by (3.2) with
$H_{k}=10^{-5}$ in Planck unit from the phases of slow-roll (Phase 1) to
negative-constant-rate (Phase 2) with $\delta_{2}=-2$ (left panel) and
$\delta_{2}=-3.5$ (right panel). The dashed-vertical line is $k=k_{\ast\ast}$,
and for $k>k_{\ast\ast}$ a positive-constant-rate $\delta_{3}=1$ is used in
Phase 3. The late-time (IR) expansions of the power spectrum (3.6) in terms of
the dashed lines are in well agreement with the full expression (3.2) in solid
lines.

Figure 4: The power spectrum $P_{\mathcal{R}}$ given by (3.2) with
$H_{k}=10^{-5}$ in Planck unit from the phases of slow-roll (Phase 1) to
negative-constant-rate (Phase 2) with $\delta_{2}=-2$ (left panel) and
$\delta_{2}=-3.5$ (right panel). The dashed-vertical line is $k=k_{\ast\ast}$,
and for $k>k_{\ast\ast}$ a secondary slow-roll phase $\delta_{3}=0$ is used in
Phase 3. The late-time (IR) expansions of the power spectrum (3.6) in terms of
the dashed lines are in well agreement with the full expression (3.2) in solid
lines.

In Figure 3, we plot examples with a positive constant rate $\delta_{3}>0$ in
Phase 3, and in Figure 4, we plot examples with $\delta_{3}=0$, where Phase 3
is effectively a slow-roll inflation. 666Single-field models of inflation for
PBH formation ending with an effective slow-roll phase can be found in [11,
13, 16, 20]. In the left (right) panel of Figures 3 and 4, we use
$-3<\delta_{2}<-3/2$ (and $\delta_{2}<-3$) so that the Phase 2 spectrum
$P_{\mathcal{R}}\sim k^{2\Delta_{2}}=k^{6+2\delta_{2}}$ is a growing
(decaying) phase, respectively. One can see that the steepest growth
$P_{\mathcal{R}}\sim k^{4}$, outlined by the “steepest” dotted lines, starts
from $k_{\rm min}\approx k_{\ast}x_{1}^{\nu_{2}}$ in Phase 1 and continues to
the beginning in Phase 2. Similarly, a continuous scaling of
$P_{\mathcal{R}}\sim k^{2\Delta_{2}}$, outlined by the “continuous” dotted
lines, occurs from Phase 2 to the beginning of Phase 3 for arbitrary choices
of $\delta_{3}$. All examples indicate that the continuous scaling from Phase
2 breaks down in the limit of $k/k_{\ast\ast}\gg 1$ and the asymptotic
behavior is again led by the conformal weight of Phase 3 as
$P_{\mathcal{R}}\sim k^{2\Delta_{3}}$.

###  3.3 PBH scenarios

So far we have derived analytic power spectra from instantaneous transitions
of phases with different constant rate-of-rollings. To obtain the unified
formula (3.6), there is no assumption applied to the asymptotic scaling
dimension towards the end of inflation (here we identify as $\Delta_{3}$) so
that $\delta_{3}$ in general can be a free parameter that does not necessarily
relate to the negative constant rate $\delta_{2}$ in Phase 2. This implies
that PBH scenarios with a continuous scaling to the end of inflation should
involve with non-trivial constraints. To see this, let us divide the existing
models into two classes with respect to the scaling behavior of the power
spectrum on scales much smaller than the (steepest) growing phase led by a
negative $\delta<-3$. We refer scenarios with an approximately constant
scaling dimension $|\Delta(N\rightarrow N_{\rm end})|\sim\mathcal{O}(1)$ that
significantly violates the standard slow-roll conditions as the first class
[14, 11, 9, 5, 18, 12, 13, 7], and those with $|\Delta(N\rightarrow N_{\rm
end})|\ll 1$ being essentially a secondary slow-roll phase as the second class
[11, 13, 16, 20].

To demonstrate the PBH scenarios in both secondary slow-rolling (SSR) and
slow-roll-violating (SRV) classes, we adopt the parametrization of the
inflaton potential by a power-law series [12] as

$\displaystyle
V(\phi)=V_{0}\left[\alpha_{0}+\alpha_{1}\frac{\phi}{\Lambda}+\frac{\alpha_{2}}{2}\left(\frac{\phi}{\Lambda}\right)^{2}+\frac{\alpha_{3}}{3!}\left(\frac{\phi}{\Lambda}\right)^{3}+\frac{\alpha_{4}}{4!}\left(\frac{\phi}{\Lambda}\right)^{4}+\frac{\alpha_{5}}{5!}\left(\frac{\phi}{\Lambda}\right)^{5}\right],$
(3.8)

where $\Lambda$ measures the vacuum expectation value of $\phi$. The power
spectrum is solved by the numerical evaluation of $\epsilon(N)$ with respect
to the $e$-folding number $N=\ln a$ as
$P_{\mathcal{R}}(N)=H_{\ast}^{2}/(2\pi^{2}M_{\rm pl}^{2}\epsilon(N))$, where
$H_{\ast}$ is the initial value of the Hubble parameter.

Figure 5: The power spectrum $P_{\mathcal{R}}$ of the slow-roll-violating
(SRV) class with the transition from $\delta<0$ to $\delta>0$.
$H_{\ast}=10^{-8}M_{\rm pl}$ is used in this plot. The evolution of $\delta$
from negative to positive values is constrained by the continuity of the
scaling dimension (conformal weight) $\Delta$. The dotted lines are fitted
power spectra with the spectral index given by $n_{s}-1=-2\Delta_{3}$, where
$\Delta_{3}=\Delta(N=50)$ is used.

For SRV scenarios with the slow-roll violation in the decaying phase of the
power spectrum till the end of inflation ($\Delta>0$ for $N\rightarrow N_{\rm
end}$), we use $\Lambda=0.3M_{\rm pl}$, $V_{0}=3M_{\rm pl}^{2}H_{\ast}^{2}$,
$\alpha_{0}=1$ and $\alpha_{2}=0$. This choice subjects to a kind of small
field inflation potential and therefore the initial value is taken in the
limit of $\phi/\Lambda\ll 1$. In Figure 5, we fix
$\\{\alpha_{1},\alpha_{3}\\}=\\{-0.73\times 10^{-4},-0.52\\}$ and change
$\\{\alpha_{4},\alpha_{5}\\}$ to obtain different values of the rolling rate
$\delta(N)$ for the $e$-folding numbers $N\gtrsim 36$. For SRV 1 (SRV 2) in
Figure 5, $\alpha_{4}=0.95$ ($\alpha_{4}=1$) is used and $\alpha_{5}<0$ must
be accurately tuned such that the inflaton has just enough momentum to roll
away from the inflection point or otherwise a period of stable $\delta<0$
cannot be realized. Such a fine-tuning condition ensures
$\Delta_{2}=3+\delta_{2}\approx\Delta_{3}=-\delta_{3}$, so that the scaling
dimension found in Phase 2 (see Section 2) approximately continues to Phase 3.
For the evolution of $\delta(N)$ with respect to $N$, we have
$\\{\delta(38),\delta(50)\\}=\\{-3.718,0.721\\}$ for SRV 1 and
$\\{\delta(38),\delta(50)\\}=\\{-3.458,0.470\\}$ for SRV 2, which imply nearly
constant conformal weights:$\\{\Delta_{2},\Delta_{3}\\}=\\{-0.718,-0.721\\}$
for SRV 1 and $\\{\Delta_{2},\Delta_{3}\\}=\\{-0.458,-0.470\\}$ for SRV 2.

Figure 6: The power spectrum $P_{\mathcal{R}}$ of the secondary slow-rolling
(SSR) class with the transition from $\delta<0$ to $\delta\approx 0$.
$H_{\ast}=8\times 10^{-6}M_{\rm pl}$ is used in this plot. The dotted lines
are fitted power spectra with the spectral index given by
$n_{s}-1=-2\Delta_{3}$, where $\Delta_{3}=\Delta(N=60)$ is used.

For SSR scenarios where a secondary slow roll follows the growing phase led by
a negative rate ($\Delta\approx 0$ for $N\rightarrow N_{\rm end}$), we use
$\Lambda=M_{\rm pl}$ and $\alpha_{0}=\alpha_{1}=\alpha_{5}=0$. This choice
subjects to a specific potential form of the punctuated inflation [20, 68, 70,
69]. In Figure 6, we use
$\\{\alpha_{2},\alpha_{3},\alpha_{4}\\}=\\{1,-4/\phi_{\ast},6/\phi_{\ast}^{2}\\}$
with $\phi_{\ast}=1.983M_{\rm pl}$, $V_{0}/(M_{\rm
pl}^{2}H_{\ast}^{2})=6.1\times 10^{-4}$ and the initial value
$\phi_{0}=18M_{\rm pl}$ for SSR 1. For SSR 2, we choose
$\\{\alpha_{2},\alpha_{3},\alpha_{4}\\}=\\{1,-4/\phi_{\ast},6/\phi_{\ast}^{2}\\}$
with $\phi_{\ast}=1.985M_{\rm pl}$, $V_{0}/(M_{\rm
pl}^{2}H_{\ast}^{2})=3.9\times 10^{-4}$, and $\phi_{0}=20M_{\rm pl}$. We find
that $\\{\delta(37),\delta(60)\\}=\\{-2.987,-5.86\times 10^{-3}\\}$ for SSR 1
and $\\{\delta(46),\delta(60)\\}=\\{-2.980,-8.89\times 10^{-3}\\}$ for SSR 2.
These results also manifest the approximately continuous scaling dimensions
between the range of
$\\{\Delta_{2},\Delta_{3}\\}=\\{\Delta(37),\Delta(60)\\}=\\{1.27\times
10^{-2},5.86\times 10^{-3}\\}$ for SSR 1 or
$\\{\Delta_{2},\Delta_{3}\\}=\\{\Delta(46),\Delta(60)\\}=\\{1.98\times
10^{-2},8.89\times 10^{-3}\\}$ for SSR 2.

In summary, the asymptotic scaling dimension towards the end of inflation,
$\Delta_{3}$, is in principle a free parameter governed by the inflaton
potential. However, an inflation with a stable period of constant rate
$\delta_{2}<-3/2$ for the PBH formation imposes an additional constraint
across the scaling dimensions. For PBH scenarios that realize a continuous
decay of the power spectrum to the end of inflation (mostly for the generation
of PBHs in a certain mass range), $\Delta_{3}=-\delta_{3}$ is constrained by
the dimension $\Delta_{2}=3+\delta_{2}$ (for $\delta_{2}<-3$) of the negative
rate so that $\delta_{3}\rightarrow-3-\delta_{2}$. On the other hand, if one
intends to realize a secondary slow-roll phase with $\Delta_{3}\approx 0$, the
continuity $\Delta_{3}=\Delta_{2}$ implies that the negative rate always goes
to the ultra-slow-roll limit where $\delta_{2}\rightarrow-3$. 777We thank also
the numerical confirmations of this property from H. V. Ragavendra and J. Silk
in their study [20].

###  3.4 $N$-stage inflation

The unified mode function derived in Section 3.2 can be extended to more
general cases with $N>3$ phases of inflation (see also [11]), as long as the
rolling rates, $\delta_{N}$, in the $N$-th phase of inflation are nearly
constant and the background spacetime is sufficiently close to the de Sitter
phase ($\epsilon_{N}\ll 1$) in each phase. One of the possible application for
the $N$-stage constant-rate inflation is to generate multiple peaks in the
power spectrum for producing PBHs in different mass ranges [71].

Although the $N$-stage extension may largely increase the complexity of the
final mode functions at the end of inflation, there are some generic features
shared among the boundary arguments that shall be outlined in this section:

  1. 1.

The final spectrum. The instantaneous transition of rolling rates to
$\delta_{N}$ is equivalent to altering only the effective mass for the
Mukhanov-Sasaki variable $v_{N}$ in the $N$-th phase. Therefore the solution
of $v_{N}$ takes the similar structure as

$\displaystyle
v_{N}=c_{N}^{(1)}\sqrt{\tau_{N}}H_{\nu_{N}}^{(1)}(\tau_{N})+c_{N}^{(2)}\sqrt{\tau_{N}}H_{\nu_{N}}^{(2)}(\tau_{N}),$
(3.9)

where $\nu_{N}=|3/2+\delta_{N}|$ and $\tau_{N}=-k\eta$. At the end of
inflation, $\tau_{\rm end}=k/k_{\rm end}=\tau_{\ast}x_{2}x_{3}\cdots x_{N}$,
where $x_{N}=k_{\ast(N-1)}/k_{\rm end}\approx a_{\ast(N-1)}/a_{\rm
end}=e^{-\Delta N_{\ast(N-1)}}$ measures the ratio of the horizon scale
between the end of $N-1$-th stage to the end of $N$-th stage. The final power
spectrum for $\tau_{\rm end}\ll 1$ modes is then

$\displaystyle P_{\mathcal{R}}^{\rm IR}(k;\eta_{\rm
end})=\frac{H_{k}^{2}}{8\pi}\frac{1}{\epsilon_{\rm
end}}\left|f_{\mathcal{R}}^{N}(\tau_{\ast},x_{2},\cdots,x_{N})\right|^{2}\left|\frac{i}{\pi}2^{\nu_{N}}\Gamma(\nu_{N})\right|^{2}\left(\frac{k}{k_{\rm
end}}\right)^{2\Delta_{N}},$ (3.10)

where $\Delta_{N}=3/2-\nu_{N}$,
$f_{\mathcal{R}}^{N}\equiv(c_{N}^{(2)}-c_{N}^{(1)})/c_{1}$ and $\epsilon_{\rm
end}=\epsilon_{\ast}x_{2}^{-2\delta_{2}}x_{3}^{-2\delta_{3}}\cdots
x_{N}^{-2\delta_{N}}$.

  2. 2.

The large $k$ limit. The mode function (3.9) has the general property
$\sqrt{\tau_{N}}H_{\nu_{N}}^{(1,2)}(\tau_{N})\rightarrow\sqrt{2/\pi}e^{\mp
i\frac{\pi}{2}(\nu_{N}+\frac{1}{2})}e^{\pm i\tau_{N}}$ in the limit of
$k\rightarrow\infty$ where $\tau_{N}\rightarrow\infty$. This leads to the
matching of the boundary conditions at each phase transition, $\eta_{\ast
N}=-1/k_{\ast N}$ with $k/k_{\ast N}\gg\frac{\pi}{2}(\nu_{N}+\frac{1}{2})$,
which is approximately given by the relation

$\displaystyle\left(c_{N}^{(2)}+c_{N}^{(1)}\right)$
$\displaystyle\cos\left(-k/k_{\ast
N}\right)+\left(c_{N}^{(2)}-c_{N}^{(1)}\right)i\sin\left(-k/k_{\ast N}\right)$
(3.11)
$\displaystyle=\left(c_{N+1}^{(2)}+c_{N+1}^{(1)}\right)\cos\left(-k/k_{\ast
N+1}\right)+\left(c_{N+1}^{(2)}-c_{N+1}^{(1)}\right)i\sin\left(-k/k_{\ast
N+1}\right),$

which implies that $c_{N+1}^{(2)}-c_{N+1}^{(1)}\rightarrow
c_{N}^{(2)}-c_{N}^{(1)}$. The coefficient $c_{1}$ of the primary slow-roll
phase fixed by the Bunch-Davies vacuum (2.18) ensures that
$f_{\mathcal{R}}^{2}=(c_{2}^{(2)}-c_{2}^{(1)})/c_{1}$ is a pure oscillatory
function of $k$ (no power-law dependence) in the limit of
$k\rightarrow\infty$. One can repeat the matching process to find that
$f_{\mathcal{R}}^{N}\equiv(c_{N}^{(2)}-c_{N}^{(1)})/c_{1}$ is also a pure
oscillatory function of $k$, and thus the final spectrum (3.10) has the
asymptotic scaling dimension $P_{\mathcal{R}}^{\rm IR}(k\gg
k_{\ast(N-1)};\eta_{\rm end})\sim k^{2\Delta_{N}}$.

  3. 3.

Continuous scaling. For the $N$-th scaling dimension $|\Delta_{N}|\gg 0$, the
boundary argument in Section 2.2 and 3.1 indicate that
$k^{3/2}\mathcal{R}_{N+1}\sim\mathcal{C}_{N0}k^{\Delta_{N}}$ (even with
$|\Delta_{N+1}|\rightarrow 0$) and $k^{3/2}\mathcal{R}_{N+1}\rightarrow
k^{\Delta_{N}}$ when $\tau_{\ast}\rightarrow 0$. This means that there is a
short period of continuous scaling $P_{\mathcal{R}}\sim k^{2\Delta_{N}}$ for
$k\gtrsim k_{\ast N}$ and the continuous scaling breaks down to
$P_{\mathcal{R}}\sim k^{2\Delta_{N+1}}$ when $k\gg k_{\ast N}$ due to the
large $k$ behavior discussed above.

  4. 4.

Steepest growth. For PBH scenarios that allow multiple peaks in the power
spectrum, the $P_{\mathcal{R}}\sim k^{4}$ growth appears whenever a slow-roll
phase ($\Delta_{N-1}\rightarrow 0$) transits to a negative-constant-rate phase
($\Delta_{N}=3+\delta_{N}$ with $\delta_{N}<-3/2$), as argued in Section 2.2.
To reduce the power spectrum from the peak values after enhancement, we need
$\Delta_{N+1}=-\delta_{N+1}$ to be a positive-constant-rate phase with
$\delta_{N+1}>0$. To maintain a stable phase with $\delta_{N}<-3$, the
inflaton potential should satisfy the condition $\delta_{N+1}=-3-\delta_{N}$
as numerically shown in Section 3.3.

##  4 Criterion of conformal weight violation

In the slow-roll ($\delta=0$) and the ultra-slow-roll ($\delta=-3$) phases,
the scaling dimension, $\Delta=3/2-|3/2+\delta|=0$, vanishes identically. A
question arises following this fact is that, why the $P_{\mathcal{R}}\sim
k^{4}$ growth of the power spectrum only occurs when inflation transits from
the slow-roll (SR) phase to the ultra-slow-roll (USR) phase, yet the $k^{4}$
growth does not occur in the inverse (USR $\rightarrow$ SR) transition? (In
fact, $P_{\mathcal{R}}$ keeps scale-invariant for the USR to SR transition,
see [72].) In this section, we show that the $k^{4}$ growth as a conformal-
weight-violating process is due to the entropy production of the inflaton
perturbation driven by the sharp deceleration of the coherent inflaton motion
in Phase 2.

The criterion for realizing an enhanced power spectrum through a decay of the
first slow-roll parameter $\epsilon_{H}\simeq\dot{\phi}^{2}/(2M_{\rm
pl}^{2}H_{\ast}^{2})$ has been given in [74], and the connection with entropy
production for such an enhancement in USR inflation has been pointed out in
[20, 75]. Here, we apply the discussion to the generic constant-rate inflation
to identify the origin of the $k^{4}$ growth. Our formula for the inflaton
perturbation is obtained in the spatially flat slicing of the metric:

$\displaystyle
ds^{2}=(1+\alpha)^{2}dt^{2}+a^{2}h_{ij}\left(dx^{i}+\beta^{i}dt\right)\left(dx^{j}+\beta^{j}dt\right),$
(4.1)

where $\alpha\equiv N-1$, $\beta^{i}\equiv N^{i}$, and $N$ ($N^{i}$) is
nothing but the lapse (shift) function in the standard Arnowitt-Deser-Minser
(ADM) formalism. We simply take $h_{ij}=\delta_{ij}$ since the tensor
perturbation is irrelevant to our discussion. At the linear order, the
Einstein equation for the energy density, pressure, and momentum flux gives
the definition to the inflaton fluid perturbation as

$\displaystyle\delta\rho_{\phi}$
$\displaystyle=\dot{\phi}\left(\delta\dot{\phi}-\alpha\dot{\phi}\right)+V_{\phi}\delta\phi,$
(4.2) $\displaystyle\delta p_{\phi}$
$\displaystyle=\dot{\phi}\left(\delta\dot{\phi}-\alpha\dot{\phi}\right)-V_{\phi}\delta\phi,$
(4.3) $\displaystyle\delta q_{\phi}$ $\displaystyle=-\dot{\phi}\delta\phi,$
(4.4)

where $\delta\rho_{m}\equiv\delta\rho_{\phi}-3H\delta q_{\phi}$ represents the
gauge-invariant total matter perturbation [76]. Together with the vanishing of
the anisotropic stress for the linear inflaton perturbation, one can solve
$\alpha$ and $\beta$ in terms of $\delta\phi$ as

$\displaystyle\alpha=\frac{\dot{\phi}}{2HM_{\rm
pl}^{2}}\delta\phi,\qquad\partial^{2}\beta=\frac{\dot{\phi}^{2}}{2H^{2}M_{\rm
pl}^{2}}\frac{d}{dt}\left(-\frac{H}{\dot{\phi}}\delta\phi\right),$ (4.5)

where $\beta$ is the scalar mode of $\beta^{i}$ with respect to the
decomposition $\beta^{i}=\partial_{i}\beta+\beta_{T}^{i}$ and
$\partial_{i}\beta_{T}^{i}=0$. These are all the quantities we need for the
discussion of entropy production.

Inflaton mode functions. The constant-rate condition illustrated in Figure 1
in fact specifies the first and second field derivatives of the inflaton
potential $V(\phi)$ so that one can solve the mode functions of $\delta\phi$
based on the given parameter $\delta$. Let us derive the analytic solutions of
$\delta\phi_{k}$ before going to the fluid dynamics of the inflaton field.

Recalling that the coherent motion of $\phi$ follows
$\ddot{\phi}+3H\dot{\phi}+V_{\phi}=0$, which can be translated to
$\delta+3=-V_{\phi}/(H\dot{\phi})$ in terms of the rate-of-rolling parameter
$\delta$. Taking the time derivative of $\delta$, one finds

$\displaystyle\frac{\dot{\delta}}{H}=-\frac{V_{\phi\phi}}{H^{2}}+(\epsilon-\delta)(\delta+3),$
(4.6)

where the constant-rate condition $\dot{\delta}=0$ with $\epsilon\ll\delta$
indicates a constant inflaton mass $V_{\phi\phi}=-\delta(\delta+3)H^{2}$.

On the other hand, the equation of motion of $\delta\phi$ in the flat slicing
(4.1) is given by

$\displaystyle\delta\phi^{\prime\prime}+2\mathcal{H}\delta\phi^{\prime}+\left(k^{2}+a^{2}V_{\phi\phi}\right)\delta\phi+2\left(\phi^{\prime\prime}+2\mathcal{H}\phi^{\prime}\right)\alpha+\phi^{\prime}\left(\alpha^{\prime}+a\partial^{2}\beta\right)=0,$
(4.7)

where the prime denotes the derivative with respect to the conformal time. One
can check from the solutions (4.5) to find that those terms involved with
$\alpha$ or $\beta$ are suppressed by $\epsilon_{H}$ in the de Sitter
background. Therefore, using the constant-rate condition (4.6) and dropping
$\epsilon_{H}$ suppressed terms, the leading equation of motion for the mode
function $u_{k}=a\delta\phi_{k}$ reads

$\displaystyle\frac{\partial^{2}u}{\partial
z^{2}}+\left[1-\frac{2+\delta(\delta+3)}{z^{2}}\right]u=0,$ (4.8)

which is nothing but the equation of motion for the mode function $v_{k}$ of
the Mukhanov-Sasaki variable given in (2.29) and (2.30) with
$\nu\equiv|\delta+3/2|$. This is not surprising since we are reproducing the
standard expression for the curvature perturbation in flat slicing [77, 76]:

$\displaystyle\mathcal{R}=-\frac{H}{\dot{\phi}}\delta\phi.$ (4.9)

The solution (4.5) is therefore
$\partial^{2}\beta=\epsilon_{H}\dot{\mathcal{R}}$.

Entropy production. The entropy perturbation in single-field inflation is
usually negligible. However, the excitation of both constants of integration,
$c_{N}^{(1,2)}$, due to the transition into a negative rolling rate in Phase
2, indicates that the late-time curvature perturbation contains both adiabatic
and entropy modes. It is the exponential growth of the entropy mode with
$\delta_{2}<-3/2$ that sources the enhancement of $P_{\mathcal{R}}$ on
superhorizon scales [75]. A numerical experiment to demonstrate the growth of
superhorizon modes can be found in [12]. To see the role of entropy
perturbation in the enhancement of $P_{\mathcal{R}}$, let us recall the gauge-
invariant definition for a single scalar fluid [76] as

$\displaystyle S_{\phi}=H\left(\frac{\delta
p_{\phi}}{\dot{p}_{\phi}}-\frac{\delta\rho_{\phi}}{\dot{\rho}_{\phi}}\right),$
(4.10)

where in constant-rate inflation $\dot{\rho}_{\phi}=-3H\dot{\phi}^{2}$ and
$\dot{p}_{\phi}=(2\delta+3)H\dot{\phi}^{2}$. Taking the perturbations given by
(4.2), one can manipulate the definition to find that

$\displaystyle
S_{\phi}=\frac{2(\delta+3)}{3(2\delta+3)}\frac{1}{\dot{\phi}}\left(\delta\dot{\phi}-\alpha\dot{\phi}-\delta
H\delta\phi\right)=\frac{2(\delta+3)}{3(2\delta+3)}\frac{\delta\rho_{m}}{\dot{\phi}^{2}}.$
(4.11)

Given that the total matter perturbation $\delta\rho_{m}=2HM_{\rm
pl}^{2}\partial^{2}\beta$ in flat slicing, we can connect the time-derivative
of the curvature perturbation (4.9) to the entropy perturbation as

$\displaystyle\dot{\mathcal{R}}=-\frac{H}{\dot{\phi}}\left(\delta\dot{\phi}-\alpha\dot{\phi}-\delta
H\delta\phi\right)\equiv-\frac{3(2\delta+3)}{2(\delta+3)}HS_{\phi}.$ (4.12)

This relation shows the generic suppression of superhorizon evolution of the
curvature perturbation in slow-roll inflation ($\delta\rightarrow 0$) as
$d\mathcal{R}/dN\sim S_{\phi}\sim(k/aH)^{2}\ll 1$.

The above definition of the entropy perturbation can be misleading to a
constant $\mathcal{R}$ in the exact USR inflation with $\delta\rightarrow-3$
where $\dot{\mathcal{R}}=S_{\phi}\rightarrow 0$. However, non-zero entropy has
been created in the primary slow-roll phase (Phase 1) and its change in time
is governed by

$\displaystyle\dot{S}_{\phi}=-\left(3+\epsilon_{H}+2\delta\right)HS_{\phi}+\frac{2(\delta+3)}{3(2\delta+3)}\frac{k^{2}}{a^{2}}\frac{\mathcal{R}}{H},$
(4.13)

where we have applied the constant-rate condition (4.6) to simplify the
expression. For a consecutive phase with $\delta<-3/2$, one finds
$\dot{S}_{\phi}\approx-(3+2\delta)HS_{\phi}$ leading to a rapid growth
$S_{\phi}\sim e^{-(3+2\delta)N}$ on large scales with $k\ll aH$, which allows
the subdominant entropy mode to overcome the adiabatic perturbation.

The relation $\dot{\mathcal{R}}=\partial^{2}\beta/\epsilon_{H}\sim HS_{\phi}$
provides an alternative way to see that the late-time curvature perturbation
contributed by the entropy production has the scaling property
$\mathcal{R}_{S}\sim k^{2}$, where $\mathcal{R}_{S}$ comes from the
integration

$\displaystyle\mathcal{R}\equiv\mathcal{R}_{A}+\mathcal{R}_{S}=\mathcal{R}_{A}+k^{2}\int\frac{\beta(t)}{\epsilon_{H}(t)}dt.$
(4.14)

$\mathcal{R}_{A}$ is nothing but the adiabatic mode of the curvature
perturbation, which is a constant in the slow-roll and the USR phases.
$\mathcal{R}_{S}\geq\mathcal{R}_{A}$ is realized when $\epsilon_{H}$ decays to
a very small value with $\delta<-3/2$, which summarizes the criterion of
superhorizon enhancement given in [74]. Comparing the adiabatic-entropy
decomposition (4.14) with the boundary argument in Section 2.2, we find that
$\mathcal{R}_{S}\sim k^{\Delta+2}$ is the subleading mode with the next-to-
lowest conformal weight. As a result, the temporary domination of
$\mathcal{R}_{S}$ leads to a growing spectrum $P_{\mathcal{R}}\sim k^{4}$ and
breaks the continuity of the leading conformal weight of the system. 888The
$k^{4}$ growth reflects the superhorizon evolution of the curvature
perturbation for modes (in the range of $k_{\rm min}<k<k_{\ast}$) that have
exited the horizon close to the end of Phase 1. This superhorizon evolution is
induced by the deceleration dynamics of inflaton (namely $\delta_{2}<0$) in
Phase 2. The larger the spectral amplitude is enhanced at $k=k_{\ast}$, the
smaller the wavenumber $k_{\rm min}$ needs to be (see Figure 2). For modes
that exit the horizon well inside Phase 2 or in Phase 3 with
$\delta_{3}=-3-\delta_{2}$ (that is for $k_{\ast}\ll k<k_{\rm end}$), the
spectral amplitude at the end of inflation is governed by two factors related
to the Phase 2 parameters as $P_{\mathcal{R}}(k>k_{\ast};\eta_{\rm
end})\sim(k_{\ast}/k_{\ast\ast})^{2\delta_{2}}(k/k_{\ast\ast})^{6+2\delta_{2}}$,
as shown in (3.6) or (2.3). The scale-independent factor
$(k_{\ast}/k_{\ast\ast})^{2\delta_{2}}$ is solely determined by the duration
of Phase 2, namely $\Delta
N=\ln(k_{\ast\ast}/k_{\ast})=\ln(\eta_{\ast}/\eta_{\ast\ast})$, due to the
decay of the first slow-roll parameter
$\epsilon_{2}(\eta_{\ast\ast})=\epsilon_{\ast}(\eta_{\ast}/\eta_{\ast\ast})^{-2\delta_{2}}=\epsilon_{\ast}e^{2\delta_{2}\Delta
N}$. The scale-dependent factor $(k/k_{\ast\ast})^{6+2\delta_{2}}$ describes
the momentum scaling on the final boundary (fixed by the conformal weight:
$6+2\delta_{2}=2\Delta_{2}=2\Delta_{3}$) converted from the superhorizon
(time) evolution of the curvature perturbation with respect to the dilatation
symmetry. For the transient USR case with $\delta_{2}=-3$, the decay of
$\epsilon$ during Phase 2, factorized by
$(k_{\ast}/k_{\ast\ast})^{2\delta_{2}}$, is the only enhancement of the power
spectrum $P_{\mathcal{R}}$.

On the other hand, in a Phase $N$ with a decaying $P_{\mathcal{R}}$ driven by
a positive rate $\delta_{N}>0$, the parameter $\epsilon_{H}$ is growing with
time so that the entropy mode $\mathcal{R}_{S}$ never dominates the adiabatic
perturbation. This is the reason why the conformal weight of $\mathcal{R}$
keeps unchanged from the previous phase ($\Delta_{N}=\Delta_{N-1}$), where the
constraint $\delta_{N}=-3-\delta_{N-1}$ for $\delta_{N-1}\leq-3$ is protected
by the adiabatic condition.

To answer the question at the beginning of this section, let us denote the
conjugate momentum of the curvature perturbation $\mathcal{R}$ in a
dimensionless way based on the general mode function (3.9) as

$\displaystyle\pi_{N}^{\mathcal{R}}\equiv\frac{\partial\mathcal{R}_{N}}{\partial
z}=-A_{N}\frac{\mathcal{R}_{N}}{z}+\frac{\sqrt{-z}}{y_{N}}\left[c_{N}^{(1)}H_{\nu_{N}-1}^{(1)}(-z)+c_{N}^{(2)}H_{\nu_{N}-1}^{(2)}(-z)\right],$
(4.15)

where $A_{N}\equiv\Delta_{N}+\delta_{N}=\epsilon_{N,2}+\Delta_{N}$ is nothing
but the $A_{i}$ appears in (2.33) with $i=N$ and $\epsilon_{i}\rightarrow 0$.
Given that $\Delta_{N}=3/2-|3/2+\delta_{N}|$, one finds two possibilities as

$\displaystyle
A_{N}=\Delta_{N}+\delta_{N}=\left\\{\begin{array}[]{lll}\;0,&\quad\delta_{N}\geq-3/2,&\quad(\textrm{adiabatic})\\\
\;2\delta_{N}+3,&\quad\delta_{N}<-3/2&\quad(\textrm{non-
adiabatic}).\end{array}\right.$ (4.18)

We shall refer the case with $A_{N}\neq 0$ as the “non-adiabatic” phase for
the curvature perturbation with $k^{3/2}\mathcal{R}_{N}\sim
c_{N}^{(1,2)}(-z)^{\Delta_{N}}$ in the late-time limit, where the leading
scaling power of the conjugate momentum is simply
$k^{3/2}\pi_{N}^{\mathcal{R}}\sim k^{3/2}A_{N}\mathcal{R}_{N}/z\sim
c_{N}^{(1,2)}(-z)^{\Delta_{N}-1}$. On the other hand, the case with $A_{N}=0$
is referred as the “adiabatic” phase where $k^{3/2}\pi_{N}^{\mathcal{R}}\sim
c_{N}^{(1,2)}(-z)^{1+\Delta_{N}}$ and the conjugate momentum for the leading
mode in $\mathcal{R}_{N}$ is missing.

It is now clear that the transition from an adiabatic Phase $N$ to a non-
adiabatic Phase $N+1$ has to respect the continuity of the conjugate momentum
at $z=z_{\ast N}$, where $c_{N}^{(1,2)}(-z_{\ast N})^{1+\Delta_{N}}\sim
c_{N+1}^{(1,2)}(-z_{\ast N})^{\Delta_{N+1}-1}$ implies
$c_{N+1}^{(1,2)}\sim(-z_{\ast N})^{2+\Delta_{N}-\Delta_{N+1}}$ in the large
$k$ limit for Phase $N$, see Section 3.4. This gives

$\displaystyle k^{3/2}\mathcal{R}_{N+1}(-z)\sim
c_{N+1}^{(1,2)}(-z)^{\Delta_{N+1}}\sim(k/k_{\ast N})^{2+\Delta_{N}}$ (4.19)

when seeing from a late-time boundary $-z\ll-z_{\ast N}$. As a result, a
$P_{\mathcal{R}}\sim k^{4}$ growth appears with $\Delta_{N}\rightarrow 0$ due
to the entropy production across the transition. This is indeed the case for
the SR to USR transition where $\Delta_{N}=\Delta_{N+1}=0$ but $A_{N}=0$,
$A_{N+1}=-3$.

For a non-adiabatic to adiabatic transition (such as the USR to SR transition
[72]), one can repeat the above process to find that the continuity of the
conjugate momentum is led by $k^{3/2}\mathcal{R}_{N+1}(-z)\sim(k/k_{\ast
N})^{\Delta_{N}-2}$, which in fact describes a decaying mode since the range
holds only for $k/k_{\ast N}\lesssim 1$. In this case the leading term comes
from the matching of $\mathcal{R}$ at the phase transition and the conformal
weight is inherited from the adiabatic mode $\mathcal{R}_{A}$ in the preceding
phase, as argued in Section 3.1 and 3.2. Finally, the continuity of
$\pi_{N+1}^{\mathcal{R}}$ from an adiabatic to adiabatic (or a non-adiabatic
to non-adiabatic) transition gives $c_{N+1}^{(1,2)}\sim(k/k_{\ast
N})^{\Delta_{N}-\Delta_{N+1}}$, which is the same result as that obtained from
the continuity of $\mathcal{R}$ matching.

In summary, the conformal weight violation driven by a temporary entropy
domination only occurs for an adiabatic to non-adiabatic transition.

##  5 Conclusions

In this work, we have shown that the scaling dimensions (or conformal weights)
fixed by the dilatation symmetry of the de Sitter background can be a powerful
tool to understand the final power spectrum at the end of inflation for a
majority classes of PBH scenarios that are involved with enhanced spectral
amplitudes led by a phase with the rolling rate of the inflaton field being
negative and nearly constant. The continuity of the conformal weights across
different scaling dimensionalities imposes a constraint to the inflaton
rolling rate in the PBH scenarios, where the spectral shape on scales towards
the end of inflation can be a power-law decay or scale-invariant. The apparent
violation of the continuity across conformal weights from the (primary) slow-
roll phase to the negative-constant-rate phase can be realized by the
efficient entropy production in the PBH scenarios. Our analytic formalism has
identified the entropy perturbation as the subleading mode carrying the next-
to-lowest conformal weights and its temporary domination triggers the $k^{4}$
growth of the power spectrum, which is the answer to the steepest growth
problem [11]. We have provided a generalized adiabatic condition in terms of
the conformal weight $\Delta$ and the rolling rate $\delta$ for the constant-
rate inflation. Despite that those PBH scenarios in general require fine
tuning the model parameters, it is interesting to consider them as the
realization of conformal field systems that experience the time-varying
conformal weights during inflation.

Further study is required to clarify the necessary conditions between
adiabaticity and the continuity across scaling dimensions. Revisiting the
problem via the quantization of scalar fields in terms of the hyperbolic
coordinates for the $3+1$ de Sitter space [55, 56, 57, 58] is one of the
conceivable working directions. It is also interesting to seek for possible
signatures, due to the time-varying conformal weights, residing in higher-
order or non-Gaussian correlation functions of the curvature perturbation. The
presented analytic formulae could be applied as a zeroth-order result to study
subhorizon corrections to the superhorizon correlators arisen from the off-
attractor nature of the constant-rate inflation [13, 43, 45, 72, 73, 46, 44].

## Acknowledgments

The authors thank Xingang Chen, Kalliopi Petraki, H. V. Ragavendra, and Joseph
Silk for their helpful comments and discussions. K.-W. Ng is supported in part
by the Ministry of Science and Technology (MOST) of Taiwan, R.O.C. under Grant
No. MOST 109-2112-M-001-003. Y.-P. Wu is supported by the the Agence Nationale
de la Recherche (ANR) Accueil de Chercheurs de Haut Niveau (ACHN) 2015 grant
(“TheIntricateDark” project).

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