# The separate-universe approach and sudden transitions during inflation

Joseph H. P. Jackson  Hooshyar Assadullahi  Andrew D. Gow  Kazuya Koyama
Vincent Vennin  David Wands

(today)

###### Abstract

The separate-universe approach gives an intuitive way to understand the
evolution of cosmological perturbations in the long-wavelength limit. It uses
solutions of the spatially-homogeneous equations of motion to model the
evolution of the inhomogeneous universe on large scales. We show that the
separate-universe approach fails on a finite range of super-Hubble scales at a
sudden transition from slow roll to ultra-slow roll during inflation in the
very early universe. Such transitions are a feature of inflation models giving
a large enhancement in the primordial power spectrum on small scales,
necessary to produce primordial black holes after inflation. We show that the
separate-universe approach still works in a piece-wise fashion, before and
after the transition, but spatial gradients on finite scales require a
discontinuity in the homogeneous solution at the transition. We discuss the
implications for the $\delta N$ formalism and stochastic inflation, which
employ the separate-universe approximation.

##  1 Introduction

Inflation is a period of rapid, accelerated expansion in the very early
universe [1, 2, 3, 4, 5, 6], which explains a number of observational
properties of our universe, including the origin of large-scale structure.
Initial quantum field fluctuations, $\delta\phi$, are expanded to from small
to super-Hubble scales by the accelerated expansion [7, 8, 9, 10, 11, 12],
where they give rise to primordial curvature perturbations, $\mathcal{R}$ [13,
14]. The almost scale-invariant power spectrum of large-scale perturbations,
$\mathcal{P}_{\mathcal{R}}$, predicted by canonical slow-roll inflation using
linear perturbation theory is in excellent agreement with observations of the
cosmic microwave background (CMB) [15, 16, 17].

We are yet to directly observe the primordial power spectrum on smaller
scales, which left the Hubble radius closer to the end of inflation. A large
enhancement of the perturbations compared to those observed on CMB scales
would be required in order to produce observable signals [18], such as a
stochastic gravitational-wave background [19, 20] or primordial black holes
(PBHs) [21]. The former has been highlighted by the recent pulsar timing array
detection of a stochastic gravitational-wave background [22, 23, 24], which
could be of primordial origin [25]. The latter is of particular interest to
explain some or all of the dark matter in our universe [26, 27, 28, 29].

To produce a large enhancement in the power spectrum, the usual slow-roll
behaviour needs to be interrupted, typically with a period of ultra-slow-roll
(USR) inflation [30, 31]. As this requires breaking away from the slow-roll
attractor solution, a localised feature in the potential is needed [32], which
induces a sudden transition. The transition produces a large effective mass-
squared for a limited time period in the Sasaki–Mukhanov mode equation [33,
34]. The enhanced power spectrum can also amplify non-linearities in
$\mathcal{R}$. This gives the possibility of loop corrections to the power
spectrum dominating over the tree-level result on large scales [35, 36, 37,
38, 39, 40, 41], potentially leading to a breakdown of the perturbative
framework.

One approach which enables us to go beyond a purely perturbative description
is the $\delta N$ formalism [9, 42, 43, 44, 45]. Here we identify the
primordial curvature perturbation, $\mathcal{R}$, with $\delta N$, the
difference between $N(\phi+\delta\phi)$, the local integrated logarithmic
expansion (or e-folds) from an initial spatially-flat hypersurface to the end
of inflation hypersurface, and its mean value $N(\phi)$, i.e.

$\delta N=N(\phi+\delta\phi)-N(\phi)\,.$ (1.1)

It is possible to treat long-wavelength (typically, super-Hubble scale)
fluctuations as classical perturbations of a homogeneous and isotropic
cosmology – the so-called separate-universe approach [46, 47, 48, 49]. The
homogeneous perturbation can be obtained as the long-wavelength limit in a
gradient expansion [50, 51]. This enables us to find the local integrated
expansion, $N(\phi+\delta\phi)$, using the fully non-linear equations of
General Relativity for a homogeneous and isotropic cosmology, evolving the
local fields and expansion rate, while neglecting spatial gradients. In the
classical $\delta N$ approach we use the classical solutions for $N(\phi)$ to
obtain $\delta N$ from Eq. (1.1), which can give a highly non-Gaussian
probability distribution for $\delta N$ [52, 53, 54, 55], even starting from
Gaussian field perturbations, $\delta\phi$. In the stochastic $\delta N$
approach [56, 57, 58] we also include the cumulative effect of quantum field
fluctuations all along the trajectory, leading to a stochastic description for
$\mathcal{N}(\phi)$, where $\mathcal{N}$ now indicates a stochastic variable.

However, we will show that a sudden transition from slow-roll to ultra-slow-
roll evolution during inflation leads to non-adiabatic behaviour. This gives
rise to particle production on sub-Hubble scales and non-adiabatic
perturbations on super-Hubble scales, sourced by gradient terms [59]. While
the separate-universe description applies both before and after the
transition, the key role played by gradient terms implies that the naïve
separate-universe approach breaks down at the transition itself. This
necessitates a re-appraisal of how we apply the classical and stochastic
$\delta N$ formalisms.

In this paper we first introduce the mathematical framework to identify the
long-wavelength limit of the perturbed fields, which we refer to as
homogeneous solutions, and then discuss their use in the separate-universe
approach in Sec. 2. In Sec. 3 we study the behaviour of the homogeneous
perturbations in a simple model of an instantaneous transition using the
piece-wise linear potential of Starobinsky [60, 61, 62]. Using analytical and
numerical solutions, we explicitly show that the sudden transition leads to
discontinuities in the homogeneous solutions, sourced by gradient terms at the
transition. We then demonstrate in Sec. 4 that the same behaviour is seen in
an example of a smooth potential with a localised feature, using numerical
methods to find the homogeneous component of the perturbation post-transition.
Finally, we discuss in Sec. 5 how the stochastic $\delta N$ approach might be
modified to account for non-adiabatic effects in homogeneous perturbations at
a transition from slow roll to ultra-slow roll. We conclude in Sec. 6.
Supporting calculations and another example of a sudden transition in a smooth
potential are presented in the appendices. Throughout we work in natural units
in which $c=\hbar=8\pi G=1$.

##  2 Separate-universe approach

###  2.1 Homogeneous background

The background dynamics of inflation driven by a spatially-homogeneous
canonical single scalar field, $\phi(t)$, with potential energy $V(\phi)$, in
a homogeneous and isotropic spacetime with scale factor $a(t)$, are given by
the Klein–Gordon equation

$\ddot{\phi}+3H\dot{\phi}+\frac{{\mathrm{d}}V(\phi)}{{\mathrm{d}}\phi}=0\,,$
(2.1)

where an over-dot denotes a derivative with respect to cosmic time, $t$. The
Hubble expansion rate, $H=\dot{a}/a$, is given by the Friedmann equation

$3H^{2}=V(\phi)+\frac{1}{2}\dot{\phi}^{2}\,.$ (2.2)

The inflationary expansion can be described by the Hubble-flow parameters [63,
64]

$\epsilon_{i+1}=\frac{{\mathrm{d}}\ln\epsilon_{i}}{{\mathrm{d}}N}\,,\quad\epsilon_{0}=\frac{H_{\mathrm{in}}}{H}\,,$
(2.3)

where $H_{\mathrm{in}}$ is the Hubble expansion rate at some initial time.
Inflation occurs whenever $\epsilon_{1}<1$. Slow-roll inflation requires
$|\epsilon_{i}|\ll 1\ \forall i\geq 1$, corresponding to $\ddot{\phi}$ being
sub-dominant in Eq. (2.1). The ultra-slow-roll limit of inflation requires
$\epsilon_{1}\ll 1$ and $\epsilon_{2}\simeq-6$, corresponding to
${\mathrm{d}}V(\phi)/{\mathrm{d}}\phi$ being subdominant in Eq. (2.1) [65].
Typically, ultra-slow roll corresponds to a transient phase, followed by a
return to a slow-roll (or constant-roll) attractor [66]. Since slow roll is
always an attractor whenever it exists, a deviation from the slow-roll
attractor requires a feature in the potential, causing a sudden transition to
ultra-slow roll.

###  2.2 Linear perturbations

Inhomogeneous perturbations about the homogeneous and isotropic background can
be described by the Sasaki–Mukhanov variable [33, 34], $v\equiv a\delta\phi$,
where $\delta\phi(t,\vec{x})$ corresponds to scalar field perturbations in the
spatially-flat gauge [67].

Linear perturbations can be decomposed into independent wavemodes, $v_{k}$,
with comoving wavenumber, $k$, which obey the Sasaki–Mukhanov mode equation
[33, 34]

$v_{k}^{\prime\prime}+\left(k^{2}+\mu^{2}\right)v_{k}=0\,,$ (2.4)

where a prime denotes a derivative with respect to comoving time,
$\eta\equiv\int{\mathrm{d}}t/a$, and $\mu^{2}\equiv-z^{\prime\prime}/z$, where
we define
$z=a\dot{\phi}/H=\mathrm{sign}(\dot{\phi})\sqrt{2a^{2}\epsilon_{1}}$. The
time-dependent mass-squared term in Eq. (2.4) can be written in terms of the
Hubble-flow parameters (2.3) as

$\mu^{2}=(aH)^{2}\left(-2+\epsilon_{1}-\frac{3}{2}\epsilon_{2}+\frac{1}{2}\epsilon_{1}\epsilon_{2}-\frac{1}{4}\epsilon_{2}^{2}-\frac{1}{2}\epsilon_{2}\epsilon_{3}\right)\,.$
(2.5)

If we assume that the scalar field starts in the Bunch–Davies vacuum state on
small scales during inflation ($k^{2}\gg|\mu^{2}|$) then this sets the initial
condition for the mode function

$v_{k}^{\mathrm{BD}}=\frac{e^{-ik\eta}}{\sqrt{2k}}\,.$ (2.6)

In our numerical solutions we will assume that each mode $v_{k}$ is in the
Bunch–Davies vacuum state at an initial time $-k\eta_{\mathrm{in}}=10^{3}$.
The subsequent evolution for a given mode is then given by Eq. (2.4) and
depends solely on the time-dependence of $\mu^{2}$ given by Eq. (2.5).

Note that so far we have placed no restriction on the values that the Hubble-
flow parameters may take in Eq. (2.5). We will consider models of inflation
with a localised feature in the potential causing a sudden transition to
ultra-slow roll, coinciding with a large mass-squared, $\mu^{2}>(aH)^{2}$, in
Eq. (2.5). This behaviour is illustrated in Fig. 1 for three specific
inflation models which we will investigate in this paper. We will define a
sudden transition to be a period of less than one e-fold ($\Delta N\lesssim
1$) where $\mu^{2}$ is large and positive.

In addition to the Sasaki–Mukhanov variable, $v$, we will be interested in the
comoving curvature perturbation, $\mathcal{R}=v/z$ [67]. For general matter
content, with density $\rho$ and pressure $P$, the time-dependence of
$\mathcal{R}$ is given by

$\dot{\mathcal{R}}=\frac{H}{\rho+P}\delta P_{\mathrm{com}}\,,$ (2.7)

where the comoving pressure perturbation can be split into adiabatic and non-
adiabatic components, $\delta
P_{\mathrm{com}}=c_{{}_{\mathrm{S}}}^{2}\delta\rho_{\mathrm{com}}+\delta
P_{\mathrm{nad}}$, and $c_{{}_{\mathrm{S}}}^{2}=\dot{P}/\dot{\rho}$ is the
adiabatic sound speed. The comoving density perturbation is related via the
Einstein constraint equations to the divergence of the Bardeen metric
potential [67] and thus the time-dependence of $\mathcal{R}$ can be written as

$\dot{\mathcal{R}}=\frac{H}{\rho+P}\left(\frac{2c_{{}_{\mathrm{S}}}^{2}}{a^{2}}\nabla^{2}\Psi+\delta
P_{\mathrm{nad}}\right)\,.$ (2.8)

Therefore, in the absence of non-adiabatic pressure perturbations ($\delta
P_{\mathrm{nad}}=0$), the comoving curvature is constant on sufficiently large
scales where we can neglect spatial gradients of the Bardeen potential
$\nabla^{2}\Psi\to 0$ [68]. This makes $\mathcal{R}$ convenient to track the
evolution of adiabatic density perturbations on large scales during inflation,
and through the end of inflation, as the inflaton energy is transferred to the
particles of the standard model, reheating the primordial universe [69, 70].

During single-field inflation the comoving density perturbation in the scalar
field can also be related to the non-adiabatic pressure perturbation [71] and,
noting that the adiabatic sound speed for a single field is given by
$c_{{}_{\mathrm{S}}}^{2}=1-(2V^{\prime}/3H\dot{\phi})$, we find

$\dot{\mathcal{R}}=-\frac{3H^{2}}{2\dot{V}}\delta P_{\mathrm{nad}}\,.$ (2.9)

Thus the time-dependence of $\mathcal{R}$ is directly determined by the non-
adiabatic part of the scalar field’s pressure perturbation on all scales. In
the following we shall show that the non-adiabatic pressure perturbation does
not vanish on all super-Hubble scales after a sudden transition from slow roll
to ultra-slow roll.

Figure 1: The evolution of the effective mass-squared, $\mu^{2}$, in the
Sasaki–Mukhanov mode equation (2.5) relative to the comoving Hubble rate, near
a sudden transition in the three inflation models investigated in this paper.
The piece-wise linear potential is studied in Sec. 3, the Gaussian bump model
in Sec. 4 and the inflection-point model in Appendix B. For each potential the
transition time, $\eta_{\mathrm{T}}$, is defined to be the first time for
which $\epsilon_{2}<-3$. The dotted vertical lines indicate the time when
ultra-slow roll ends, defined to be when $\epsilon_{2}>-3$. The similarity in
the behaviour of $\mu^{2}$ close to its maximum is due to model parameters
being tuned to produce a PBH abundance consistent with PBHs being all of the
dark matter.

###  2.3 Separate-universe approach

One might expect that the evolution of inhomogeneous scalar field
perturbations on sufficiently long wavelengths, where spatial gradients become
negligible, can be described locally by spatially-homogeneous perturbations of
the background Klein–Gordon equation (2.1). This is the separate-universe
approach [44, 46, 45] commonly used to simplify the study of inhomogeneous
perturbations on large scales.

Considering a small homogeneous perturbation about the background scalar
field, $\phi\to\phi(t)+\delta\phi_{\mathrm{h}}(t)$, and about the background
spacetime (specifically perturbing the Hubble rate, $H\to H+\delta H$ given by
(2.2), and the lapse function relating cosmic time to proper time,
${\mathrm{d}}t\to(1+A){\mathrm{d}}t$), one obtains a linear equation for the
homogeneous scalar field perturbation [72]

$\ddot{\delta\phi}_{\mathrm{h}}+3H\dot{\delta\phi}_{\mathrm{h}}+\left[\frac{{\mathrm{d}}^{2}V}{{\mathrm{d}}\phi^{2}}-a^{-3}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\left(\frac{a^{3}\dot{\phi}^{2}}{H}\right)\right]\delta\phi_{\mathrm{h}}=0\,.$
(2.10)

One can verify [72] that this is equivalent to setting $k\to 0$ in the
Sasaki–Mukhanov mode equation (2.4) for
$v_{\mathrm{h}}=a\delta\phi_{\mathrm{h}}$:

$v_{\mathrm{h}}^{\prime\prime}+\mu^{2}v_{\mathrm{h}}=0\,.$ (2.11)

From the form of the inhomogeneous mode equation (2.4) we would expect this to
be a good approximation on sufficiently large scales ($k^{2}\ll|\mu^{2}|$).

The general solution for corresponding homogeneous curvature perturbation,
$\mathcal{R}_{\mathrm{h}}=v_{\mathrm{h}}/z$, is given by

$\mathcal{R}_{\textrm{h}}=C+D\int^{\eta_{1}}_{\eta}\frac{{\mathrm{d}}\tilde{\eta}}{z^{2}(\tilde{\eta})}\,,$
(2.12)

where $C$ and $D$ are constants of integration. $C$ corresponds to the
curvature perturbation at $\eta=\eta_{1}$, while $D$ determines the amplitude
of the isocurvature perturbation at $\eta_{1}$. More generally $D$ can be
identified with a non-adiabatic homogeneous perturbation since it controls
$\dot{\mathcal{R}}_{\textrm{h}}$, see Eq. (2.9).

The choice of $\eta_{1}$ in (2.12) is arbitrary and is degenerate with $C$. If
we set $\eta_{1}$ to be the end of inflation, which we take to be
$\eta_{1}=0$, then the two terms appearing in Eq. (2.12) can be identified as
the homogeneous growing and decaying modes. The growing mode is in fact
constant, while the decaying mode decreases in slow roll where
$1/z^{2}\propto\eta^{2}$ and increases in ultra-slow roll where
$1/z^{2}\propto\eta^{-4}$. It vanishes at the end of inflation by
construction.

####  2.3.1 $k^{2}$-expansion

One can include systematic corrections to the homogeneous solution by
performing an expansion of the Sasaki–Mukhanov equation (2.4) in $k^{2}$. To
that end, it is convenient to first write down the equation of motion for
$\mathcal{R}_{k}=v_{k}/z$ that follows from Eq. (2.4), namely

$\displaystyle\mathcal{R}_{k}^{\prime\prime}+2\frac{z^{\prime}}{z}\mathcal{R}_{k}^{\prime}+k^{2}\mathcal{R}_{k}=0\,.$
(2.13)

For any finite $k$ the decomposition into growing and decaying modes,
$\mathcal{R}_{k}=\mathcal{G}+\mathcal{D}$, where $\mathcal{G}$ and
$\mathcal{D}$ are themselves solutions of Eq. (2.13), can be carried out
order-by-order in $k^{2}$, by expanding

$\displaystyle\mathcal{G}(\eta)=\sum_{n}\mathcal{G}_{n}(\eta)k^{2n}\quad\text{and}\quad\mathcal{D}(\eta)=\sum_{n}\mathcal{D}_{n}(\eta)k^{2n}\,.$
(2.14)

At leading order in $k^{2}$ for a given wavemode we recover the homogeneous
behaviour (2.12)

$\mathcal{G}_{0}\equiv C_{k}\quad\text{and}\quad\mathcal{D}_{0}(\eta)\equiv
D_{k}\int^{0}_{\eta}\frac{{\mathrm{d}}\tilde{\eta}}{z^{2}(\tilde{\eta})}\,.$
(2.15)

By identifying terms of the same order in $k^{2}$ in Eq. (2.13), one finds

$\mathcal{G}_{n}^{\prime\prime}+2\frac{z^{\prime}}{z}\mathcal{G}_{n}^{\prime}=-\mathcal{G}_{n-1}\,,$
(2.16)

which leads to the recursive solution for the growing mode

$\mathcal{G}_{n}(\eta)=\int_{\eta}^{\eta_{3}}\frac{{\mathrm{d}}\tilde{\eta}}{z^{2}(\tilde{\eta})}\int^{\tilde{\eta}}_{\eta_{2}}{\mathrm{d}}\accentset{\approx}{\eta}\,z^{2}(\accentset{\approx}{\eta})\mathcal{G}_{n-1}(\accentset{\approx}{\eta})\,,$
(2.17)

with a similar expression for the decaying mode, $\mathcal{D}_{n}(\eta)$.

Two arbitrary integration limits have to be introduced, $\eta_{2}$ and
$\eta_{3}$. We will require that the $k^{2}$-corrections vanish at the end of
inflation at all orders in $k$, which sets $\eta_{3}=0$ in $\mathcal{G}_{n}$,
and we further impose that $\eta_{3}=0$ in $\mathcal{D}_{n}$ as well, such
that at the end of inflation the value of $\mathcal{R}_{k}$ corresponds to
$C_{k}$ and the power spectrum of the curvature perturbation reads

$\displaystyle\mathcal{P}_{\mathcal{R}}(k)=\frac{k^{3}}{2\pi^{2}}\left|C_{k}\right|^{2}\,.$
(2.18)

There is no unique prescription for $\eta_{2}$. In practice, one notices that
changing $\eta_{2}$ in Eq. (2.17) adds a contribution proportional to
$\mathcal{D}_{0}$ into $\mathcal{G}_{n}$ and $\mathcal{D}_{n}$. In particular,
the $k^{2}$-correction to the growing mode, $\mathcal{G}_{1}(\eta)$, is only
defined up to terms proportional to the leading-order decaying mode,
$\mathcal{D}_{0}(\eta)$ [59].

#### Solution for massless inflaton field in de Sitter

In order to illustrate how the $k^{2}$-expansion works, let us consider the
simple limit of a massless inflaton field in de Sitter, corresponding to
$\epsilon_{1}\to 0$, such that $H$ is a constant and $a=-1/(H\eta)$, and
$\mu^{2}=\to-2/\eta^{2}$ in Eq. (2.5). This holds in both the slow-roll limit
($\epsilon_{i}\to 0$ for all $i\geq 1$) or the ultra-slow-roll limit
($\epsilon_{1}\to 0$, $\epsilon_{2}\to-6$, and $\epsilon_{i}\to 0$ for all
$i\geq 3$). In this case Eq. (2.13) can be solved analytically

$\mathcal{R}_{k}=\frac{\mathrm{sign}({\dot{\phi}})}{2a\sqrt{k\epsilon_{1}}}\left[\alpha_{k}\left(1-\frac{i}{k\eta}\right)e^{-ik\eta}+\beta_{k}\left(1+\frac{i}{k\eta}\right)e^{ik\eta}\right]\,.$
(2.19)

where the integration constants $\alpha_{k}$ and $\beta_{k}$ are Bogoliubov
coefficients. In the Bunch–Davies vacuum (2.6), $\alpha_{k}=1$ and
$\beta_{k}=0$ but we will keep them general for now.

Let us perform the $k^{2}$-expansion in the slow-roll limit where
$\epsilon_{1}$ can be considered to be a constant. First, from Eq. (2.15), one
has $\mathcal{G}_{0}=C_{k}$ and
$\mathcal{D}_{0}=-D_{k}H^{2}\eta^{3}/(6\epsilon_{1})$. Then, for the growing
mode, one can show that the ansatz $\mathcal{G}_{n}=C_{k}g_{n}\eta^{2n}$
satisfies Eq. (2.17) if

$g_{n}=-\frac{g_{n-1}}{2n(2n-3)}\,.$ (2.20)

Note that in order to get this expression, we must set $\eta_{2}$ in such a
way that the contribution coming from the lower bound in the inner integral of
Eq. (2.17) vanishes, which requires $\eta_{2}=-\infty$ (asymptotic past) for
$n=1$ and $\eta_{2}=0$ (asymptotic future) for $n\geq 2$. Since $g_{0}=1$
(given that $\mathcal{G}_{0}=C_{k}$ by definition), this leads to

$g_{n}=\frac{(-1)^{n}(1-2n)}{(2n)!}\,.$ (2.21)

As a consequence, Eq. (2.14) leads to

$\displaystyle\mathcal{G}(\eta)=C_{k}\sum_{n=0}^{\infty}g_{n}(k\eta)^{2n}=C_{k}\sum_{n=0}^{\infty}(-1)^{n}\frac{1-2n}{(2n)!}(k\eta)^{2n}=C_{k}\left[\cos(k\eta)+k\eta\sin(k\eta)\right]$
(2.22)

where we have recognised the Taylor series of $\cos(k\eta)+k\eta\sin(k\eta)$.

For the decaying mode, likewise, the ansatz
$\mathcal{D}_{n}=-D_{k}H^{2}/(6\epsilon_{1})d_{n}\eta^{3+2n}$ satisfies Eq.
(2.17) if $d_{n}=-d_{n-1}/[2n(2n+3)]$, where this time we have set
$\eta_{2}=0$ for all $n$. Together with $d_{0}=1$, this leads to
$d_{n}=6(-1)^{n}(n+1)/(2n+3)!$, and Eq. (2.14) yields

$\displaystyle\mathcal{D}(\eta)=$ $\displaystyle-
D_{k}\frac{H^{2}}{6\epsilon_{1}k^{3}}\sum_{n=0}^{\infty}d_{n}(k\eta)^{2n+3}=-D_{k}\frac{H^{2}}{2\epsilon_{1}k^{3}}\left[\sin(k\eta)-k\eta\cos(k\eta)\right]\,.$
(2.23)

It is then straightforward to check that Eq. (2.19) can indeed be rewritten as
$\mathcal{R}_{k}=\mathcal{G}+\mathcal{D}$, provided one makes the
identification

$C_{k}=\frac{iH\mathrm{sign}({\dot{\phi}})}{2\sqrt{k^{3}\epsilon_{1}}}(\alpha_{k}-\beta_{k})\quad\text{and}\quad
D_{k}=-\mathrm{sign}({\dot{\phi}})\frac{\sqrt{\epsilon_{1}k^{3}}}{H}(\alpha_{k}+\beta_{k})\,.$
(2.24)

We can thus identify the homogeneous growing and decaying mode solutions
(2.15)

$\mathcal{R}_{\textrm{h}}=\frac{iH\mathrm{sign}({\dot{\phi}})}{2\sqrt{k^{3}\epsilon_{1}}}\left[\left(\alpha_{k}-\beta_{k}\right)-i\left(\alpha_{k}+\beta_{k}\right)\frac{(k\eta)^{3}}{3}\right]\,.$
(2.25)

In the slow-roll limit, $\epsilon_{1}$ is a constant111We note that (2.25)
follows from (2.19) and thus holds more generally for a massless inflaton
field in de Sitter, including both the ultra-slow-roll regime where
$\epsilon_{1}$ is small, but time-dependent. However in the ultra-slow-roll
limit, where $\sqrt{\epsilon_{1}}\propto\eta^{3}$, the identification of the
constant and time-dependent terms in (2.25) is swapped [73]., and the first
term in the square bracket is the constant growing mode. The second $\eta^{3}$
term is the homogeneous decaying mode. Note however that in slow roll at late
times ($\eta\to 0$) the leading $k^{2}$-correction to the growing mode, i.e.
the $n=1$ term in Eq. (2.22), which decays as $\eta^{2}$, dominates over the
homogeneous decaying mode, which decays as $\eta^{3}$.

The most common approximation for the description of perturbations on super-
Hubble scales during inflation is thus to take the homogeneous growing mode
for a massless field starting in the Bunch–Davies vacuum state
($\alpha_{k}=1$, $\beta_{k}=0$) in de Sitter

$\mathcal{R}_{\textrm{h}}=\frac{iH\mathrm{sign}({\dot{\phi}})}{2\sqrt{k^{3}\epsilon_{1}}}\,.$
(2.26)

This is equivalent to setting $\delta\phi_{\textrm{h}}=iH/\sqrt{2k^{3}}$,
leading to the power spectrum for long-wavelength field fluctuations evaluated
at Hubble exit, $\mathcal{P}_{\delta\phi_{h}}=(H/2\pi)^{2}$.

###  2.4 Super-Hubble matching

####  2.4.1 Homogeneous matching

The idea of the separate-universe approach [46] is to approximate the dynamics
of perturbations on large scales by the homogeneous solutions, i.e. to use
only the $0^{\mathrm{th}}$-order terms in the $k^{2}$-expansion (2.14) above a
certain scale. In practice, this can be implemented via a homogeneous-matching
procedure, where we use the solution to the full equation of motion (2.13)
below the matching scale, and match it to its homogeneous counterpart (2.12)
above that scale. In what follows, we will express the matching scale in terms
of the ratio, $\sigma$, of the wavenumber to the Hubble scale, i.e. a wavemode
$k$ crosses the matching scale when $k=\sigma aH$ and we choose $\sigma<1$
corresponding to matching on super-Hubble scales.

At the matching scale, we require $\mathcal{R}_{\textrm{h}}=\mathcal{R}_{k*}$
and $\mathcal{R}_{\textrm{h}}^{\prime}=\mathcal{R}_{k*}^{\prime}$, where the
subscript “$*$” refers to quantities evaluated at the matching scale, hence
they depend implicitly on $k$. This leads to the matching conditions

$\displaystyle\hat{C}_{k}=\mathcal{R}_{k*}+z^{2}_{*}\mathcal{R}_{k*}^{\prime}\int_{\eta_{*}}^{0}\frac{\mathrm{d}\tilde{\eta}}{z^{2}(\tilde{\eta})}\quad\text{and}\quad\hat{D}_{k}=-z^{2}_{*}\mathcal{R}_{k*}^{\prime}\,,$
(2.27)

where hats denote quantities estimated by the homogeneous-matching procedure.
In the above expressions, $\mathcal{R}_{k*}$ and $\mathcal{R}_{k*}^{\prime}$
are obtained by solving the full linear dynamics, Eq. (2.13), below the
matching scale, taking initial conditions in the Bunch–Davies state at past
infinity. At late times, $\mathcal{R}_{k}$ approaches $C_{k}$, and thus
$\hat{C}_{k}$ in Eq. (2.27) can be used to approximate the curvature
perturbation at the end of inflation.

####  2.4.2 Matching including $k^{2}$-corrections

In practice, if we match soon after Hubble-crossing, i.e., $\sigma\simeq 1$,
the values of $\mathcal{R}_{k*}$ and $\mathcal{R}_{k*}^{\prime}$, and hence
our estimates $\hat{C}_{k}$ and $\hat{D}_{k}$ in Eq. (2.27), are liable to be
contaminated by gradient terms. One can thus improve our estimate of the
solution on large scales by including the leading order $k^{2}$-corrections.
While this can be done analytically in the slow-roll limit where
$\epsilon_{1}$ is a constant, in general this would be computationally
inefficient, as from Eq. (2.17), a double integration is needed for the
$k^{2}$ correction to the growing mode and a triple integration would be
needed for the decaying-mode correction.

Instead, the freedom in the choice of the upper limit of the integral in Eq.
(2.12) can be used to simplify the numerical computation so that the dominant
$k^{2}$-correction will be the correction to the adiabatic solution. This
suggests we should choose the adiabatic/isocurvature
($\mathcal{A}/\mathcal{I}$) split at the matching time, $\eta_{1}=\eta_{*}$ in
Eq. (2.12). We then have

$\mathcal{R}_{\textrm{h}}=\mathcal{A}_{0}+\mathcal{I}_{0}\int^{\eta_{*}}_{\eta}\frac{{\mathrm{d}}\tilde{\eta}}{z^{2}(\tilde{\eta})}\,,$
(2.28)

and hence we can estimate the leading-order terms in a $k^{2}$-expansion for
the adiabatic and isocurvature modes at the matching scale, where

$\hat{\mathcal{A}}_{0}=\mathcal{R}_{k*}\quad\text{and}\quad\hat{\mathcal{I}}_{0}=-z^{2}_{*}\mathcal{R}_{k*}^{\prime}\,.$
(2.29)

If we further set $\eta_{3}=\eta_{2}=\eta_{*}$ in (2.17) to obtain the
$k^{2}$-correction to the adiabatic mode,

$\mathcal{A}_{1}(\eta)=\mathcal{A}_{0}\int_{\eta}^{\eta_{*}}\frac{{\mathrm{d}}\tilde{\eta}}{z^{2}(\tilde{\eta})}\int^{\tilde{\eta}}_{\eta_{*}}{\mathrm{d}}\accentset{\approx}{\eta}\,z^{2}(\accentset{\approx}{\eta})\,,$
(2.30)

then this correction makes no contribution to $\mathcal{R}_{k}$ and
$\mathcal{R}_{k}^{\prime}$ at the matching time, significantly simplifying the
computation.

This choice gives a straightforward form for describing the behaviour of
$\mathcal{R}_{k}$ for $\eta\geq\eta_{*}$ including the leading-order
$k^{2}$-corrections

$\displaystyle\hat{\mathcal{R}}_{k}(\eta)=\mathcal{R}_{k*}\left[1-k^{2}\int_{\eta_{*}}^{\eta}\frac{{\mathrm{d}}\tilde{\eta}}{z^{2}(\tilde{\eta})}\int^{\tilde{\eta}}_{\eta_{*}}{\mathrm{d}}\accentset{\approx}{\eta}\,z^{2}(\accentset{\approx}{\eta})\right]+z^{2}_{*}\mathcal{R}_{k*}^{\prime}\int_{\eta_{*}}^{\eta}\frac{\mathrm{d}\tilde{\eta}}{z^{2}(\tilde{\eta})}\,.$
(2.31)

It is clear that if one sets $k^{2}=0$, then the homogeneous matching (2.27)
is recovered. This result was obtained in Ref. [59], where the super-Hubble
enhancement of a mode between the matching time and the end of inflation due
to $k^{2}$-corrections was calculated in a model with a transient ultra-slow-
roll phase.

####  2.4.3 The $\delta N$ formalism

An alternative method to implement the separate-universe approach is the
$\delta N$ formalism [9, 74, 43, 44, 45]. This follows from the fact that, on
large scales, the curvature perturbation can be identified with the
fluctuation in the local expansion $N=\ln(a)$. In particular, the curvature
perturbation on uniform-density hypersurfaces [67], $\zeta$, can be identified
with the perturbed expansion, $\delta N$, between an initial spatially-flat
hypersurface during inflation and a final hypersurface of uniform energy
density at the end of inflation. In the models that we consider, the
perturbations become adiabatic on super-Hubble scales by the end of inflation,
and the curvature perturbation on uniform-density hypersurfaces then coincides
(up to a sign convention) with the comoving curvature perturbation [46, 45]

$\zeta=-\mathcal{R}-\frac{3H}{2\dot{V}}\delta P_{\mathrm{nad}}\,.$ (2.32)

In practice, as in the homogeneous-matching procedure described in Sec. 2.4,
one solves for the full linear dynamics (2.13) below the matching scale, to
find $\mathcal{R}_{k*}$ and $\mathcal{R}_{k*}^{\prime}$, and hence
$\delta\phi_{k*}$ and $\dot{\delta\phi}_{k*}$ in the spatially-flat gauge
(using $\delta\phi=v/a=\dot{\phi}\mathcal{R}/H$ in that gauge). Above the
matching scale, the homogeneous dynamics is described by the function
$N(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$, which returns the number of
e-folds realised using the background equations of motion (2.1) and (2.2) from
the initial field values $(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$ up to
a late-time hypersurface of uniform energy density (which in practice we take
to be the end-of-inflation surface). One then evaluates the fluctuation in the
expansion

$\displaystyle\delta
N_{k}=N\left(\phi_{*}+\delta\phi_{k*},\dot{\phi}_{*}+\dot{\delta\phi}_{k*}\right)-N\left(\phi_{*},\dot{\phi}_{*}\right)$
(2.33)

and at late times, this gives the curvature perturbation $\zeta_{k}=\delta
N_{k}$.

The above formula can be expanded in the field perturbations $\delta\phi_{k}$
and $\dot{\delta\phi}_{k}$, leading to

$\displaystyle\delta N_{k}\simeq\frac{\partial
N}{\partial\phi_{\mathrm{in}}}\left(\phi_{*},\dot{\phi}_{*}\right)\delta\phi_{k*}+\frac{\partial
N}{\partial\dot{\phi}_{\mathrm{in}}}\left(\phi_{*},\dot{\phi}_{*}\right)\dot{\delta\phi}_{k*}\,.$
(2.34)

At linear order, the $\delta N$ formalism is strictly equivalent to the
homogeneous-matching procedure discussed in Sec. 2.4. In Appendix A, we prove
this equivalence for the model discussed in Sec. 3. However, in principle one
can use the fully non-linear background equations (2.1) and (2.2), which
enables one to use Eq. (2.33) beyond linear order. In practice, we will not
consider non-linear effects in what follows, so we will restrict our
discussion to the homogeneous-matching procedure, given that it is equivalent
to the linearised $\delta N$ formalism.

##  3 Piece-wise linear potential

Starobinsky’s piece-wise linear model [60] provides an illustrative example of
a sudden transition, with the advantage of being fully tractable analytically.
It has been extensively studied in the literature, see e.g. Refs. [61, 62,
75].

In this toy model, the potential function has the form

$V(\phi)=\begin{cases}V_{0}+A_{+}(\phi-\phi_{\mathrm{T}})&\text{for
}\phi\geq\phi_{\mathrm{T}}\,,\\\ V_{0}+A_{-}(\phi-\phi_{\mathrm{T}})&\text{for
}\phi<\phi_{\mathrm{T}}\,,\end{cases}$ (3.1)

with the constants $A_{+}>A_{-}>0$. Initial conditions are set such that
$\phi_{\mathrm{in}}>\phi_{\mathrm{T}}$ and the system reaches the slow-roll
attractor in the pre-transition phase of the dynamics. Immediately after the
transition, the field velocity inherited from the $\phi>\phi_{\mathrm{T}}$
region is larger than the slow-roll velocity in the $\phi<\phi_{\mathrm{T}}$
region, so there is a transient phase of slow-roll violation, until a second
slow-roll attractor is reached at late times. In the limit where $A_{+}\gg
A_{-}$, this transient behaviour is an ultra-slow-roll phase.

###  3.1 Homogeneous background

We will consider the case where $V_{0}\gg A_{\pm}|\phi-\phi_{\mathrm{T}}|$,
which implies that $\epsilon_{1}\ll 1$ throughout and that $H$ is effectively
constant. In this de Sitter limit, the Klein–Gordon equation (2.1) can be
solved analytically to give

$\phi(\eta)=\begin{cases}\frac{A_{+}}{3H^{2}}\ln{(-k_{\mathrm{T}}\eta)}+\phi_{\mathrm{T}}&\text{for
}\eta\leq\eta_{\mathrm{T}}\,,\\\
\frac{A_{-}}{3H^{2}}\ln{(-k_{\mathrm{T}}\eta)}+\frac{\Delta
A}{9H^{2}}\left[1+(k_{\mathrm{T}}\eta)^{3}\right]+\phi_{\mathrm{T}}&\text{for
}\eta>\eta_{\mathrm{T}}\,,\end{cases}$ (3.2)

where $\Delta A=A_{-}-A_{+}<0$ and $\eta_{\mathrm{T}}$ is the comoving
transition time. Here we have introduced the comoving scale
$k_{\mathrm{T}}=-1/\eta_{\mathrm{T}}$ that crosses the Hubble radius at the
transition time. We have assumed that inflation starts sufficiently far from
$\phi_{\mathrm{T}}$ such that the slow-roll attractor has been reached well
before the transition, and we have used the continuity of $\phi$ and
$\dot{\phi}$ at the transition to set the integration constants for
$\phi<\phi_{\mathrm{T}}$ (corresponding to $\eta>\eta_{\mathrm{T}}$).

The first and second Hubble-flow parameters (2.3) are then

$\epsilon_{1}(\eta)=\begin{cases}\frac{A_{+}^{2}}{18H^{4}}&\text{for
}\eta\leq\eta_{\mathrm{T}}\,,\\\ \frac{\left(\Delta
Ak_{\mathrm{T}}^{3}\eta^{3}+A_{-}\right)^{2}}{18H^{4}}&\text{for
}\eta>\eta_{\mathrm{T}}\,,\end{cases}$ (3.3)

and

$\epsilon_{2}(\eta)=\begin{cases}0&\text{for }\eta\leq\eta_{\mathrm{T}}\,,\\\
-\frac{6\Delta Ak_{\mathrm{T}}^{3}\eta^{3}}{\Delta
Ak_{\mathrm{T}}^{3}\eta^{3}+A_{-}}&\text{for
}\eta>\eta_{\mathrm{T}}\,.\end{cases}$ (3.4)

While $\epsilon_{1}$ is continuous, $\epsilon_{2}$ is discontinuous at the
transition, due the the discontinuity in $\mathrm{d}V/\mathrm{d}\phi$. In what
follows, the pre- and post-transition behaviours will be labelled by the
subscripts “$+$” and “$-$”, respectively.

###  3.2 Perturbations

To find solutions of the mode equation (2.4) we first need to find the
behaviour of $z=a\dot{\phi}/H$. From Eq. (3.3) we have

$z(\eta)=\begin{cases}\frac{A_{+}}{3H^{3}}\eta^{-1}&\text{for
}\eta\leq\eta_{\mathrm{T}}\,,\\\ \frac{A_{-}+\Delta
Ak_{\mathrm{T}}^{3}\eta^{3}}{3H^{3}}\eta^{-1}&\text{for
}\eta>\eta_{\mathrm{T}}\,.\end{cases}$ (3.5)

One can check that $z$ is continuous at $\eta=\eta_{\mathrm{T}}$ but that its
derivative is discontinuous.

The effective mass-squared $\mu^{2}=-z^{\prime\prime}/z$ appearing in the
Sasaki–Mukhanov equation (2.4) can also be obtained from Eq. (2.5). Neglecting
the sub-dominant contributions from $\epsilon_{1}\ll 1$, one finds

$\mu^{2}=\frac{1}{\eta^{2}}\left[-2+\frac{3\Delta
A\eta_{\mathrm{T}}}{A_{+}}\delta_{\mathrm{D}}(\eta-\eta_{\mathrm{T}})\right]\,,$
(3.6)

where $\delta_{\mathrm{D}}$ is the Dirac $\delta$ distribution. Apart from at
the transition, $\mu^{2}=-2/\eta^{2}$, but it experiences a sudden impulse at
$\eta=\eta_{\mathrm{T}}$ due to the discontinuity in the second slow-roll
parameter, $\epsilon_{2}$. Therefore the general solution for
$\mathcal{R}_{k}$ is given by Eq. (2.19) both before and after the transition,
but with different constants of integration.

#### Pre-transition

Before the transition we set initial conditions in the Bunch–Davies vacuum
(2.6) at early times, corresponding to $\alpha_{k+}=1$ and $\beta_{k+}=0$ in
Eq. (2.19), leading to

$\mathcal{R}_{k}=-\frac{iH}{2\sqrt{k^{3}\epsilon_{1+}}}(1+ik\eta)e^{-ik\eta}\,.$
(3.7)

In the absence of a transition (i.e. if $\Delta A=0$), the amplitude of the
growing and decaying modes (2.15) can be identified from Eq. (2.24). One finds

$\widetilde{C}_{k}=-\frac{3iH^{3}}{A_{+}\sqrt{2k^{3}}}\quad\text{and}\quad\widetilde{D}_{k}=\frac{A_{+}}{3H^{3}\sqrt{2k^{3}}}k^{3}\,.$
(3.8)

Here, tildes are used to denote the _apparent_ growing and decaying modes in
the pre-transition phase.

#### Post-transition

After the transition, the behaviour of the perturbations is still given by the
general solution, Eq. (2.19), but with different values for the Bogoliubov
coefficients compared to the pre-transition phase.

Both $\mathcal{R}_{k}$ and $\mathcal{R}_{k}^{\prime}$ are continuous at the
transition, since $z^{\prime}/z$ remains finite in Eq. (2.13). However,
although the Sasaki–Mukhanov variable, $v_{k}=\mathcal{R}_{k}/z$ is continuous
at the transition, its time derivative $v_{k}^{\prime}$ is discontinuous due
to the Dirac $\delta$-function in the effective mass-squared, $\mu^{2}$ in Eq.
(3.6), which appears in the mode equation Eq. (2.4). The jump in
$v_{k}^{\prime}$ is given by

$\displaystyle
v_{k-}^{\prime}(\eta_{\mathrm{T}}+h)-v_{k+}^{\prime}(\eta_{\mathrm{T}}-h)=$
$\displaystyle\int^{\eta_{\mathrm{T}}+h}_{\eta_{\mathrm{T}}-h}{\mathrm{d}}\eta\,v_{k}^{\prime\prime}(\eta)$
(3.9) $\displaystyle=$
$\displaystyle\int^{\eta_{\mathrm{T}}+h}_{\eta_{\mathrm{T}}-h}{\mathrm{d}}\eta\,\left[\frac{2}{\eta^{2}}-\frac{3\Delta
A}{A_{+}\eta_{\mathrm{T}}}\delta_{\mathrm{D}}(\eta-\eta_{\mathrm{T}})-k^{2}\right]v_{k}(\eta)$
$\displaystyle\underset{h\to 0}{\longrightarrow}-\frac{3\Delta
A}{A_{+}\eta_{\mathrm{T}}}v_{k}\left(\eta_{\mathrm{T}}\right)\,.$

This leads to the new Bogoliubov coefficients after the transition [60, 62]

$\alpha_{k-}=1+\frac{3i}{2}\frac{\Delta
Ak_{\mathrm{T}}}{A_{+}k}\left(1+\frac{k_{\mathrm{T}}^{2}}{k^{2}}\right)\,,$
(3.10) $\beta_{k-}=-\frac{3i}{2}\frac{\Delta
Ak_{\mathrm{T}}}{A_{+}k}\left(1+i\frac{k_{\mathrm{T}}}{k}\right)^{2}e^{2ik/k_{\mathrm{T}}}\,.$
(3.11)

We see that the sudden transition is non-adiabatic ($\beta_{k-}\neq 0$),
leading to particle production on sub-Hubble scales and time-dependence of the
curvature perturbation, even on super-Hubble scales.

The homogeneous growing and decaying modes can be obtained by rewriting the
homogeneous part of the general solution, Eq. (2.19), in the form (2.12),
which gives

$\displaystyle C_{k}=$
$\displaystyle-\frac{3iH^{3}}{A_{-}\sqrt{2k^{3}}}\left(\alpha_{k-}-\beta_{k-}\right),$
(3.12) $\displaystyle D_{k}=$
$\displaystyle\frac{A_{-}k^{3}}{3H^{3}\sqrt{2k^{3}}}\left[\alpha_{k-}+\beta_{k-}-3i\frac{\Delta
A}{A_{-}}\left(\frac{k_{\mathrm{T}}}{k}\right)^{3}\left(\alpha_{k-}-\beta_{k-}\right)\right].$

In the absence of a transition, $\Delta A=0$, one recovers $\beta_{k-}=0$ and
the apparent pre-transition growing and decaying modes given in Eq. (3.8). To
better understand the effect of the transition, it is helpful to expand Eq.
(3.12), using Eq. (3.10) and Eq. (3.11), in various limits.

#### Scales $k\ll k_{\mathrm{T}}$

When $k\ll k_{\mathrm{T}}$, i.e. for scales that exit the Hubble radius long
before the transition, by Taylor expanding Eq. (3.12), one finds

$\displaystyle C_{k}=$
$\displaystyle-\frac{3iH^{3}}{\sqrt{2k^{3}}A_{+}}\left\\{1+\frac{2}{5}\frac{\Delta
A}{A_{-}}\left(\frac{k}{k_{\mathrm{T}}}\right)^{2}\left[1+\frac{5i}{6}\frac{k}{k_{\mathrm{T}}}+\mathcal{O}\left(\frac{k^{2}}{k^{2}_{\mathrm{T}}}\right)\right]\right\\}\,,$
(3.13) $\displaystyle D_{k}=$
$\displaystyle\frac{A_{-}k^{3}}{3H^{3}\sqrt{2k^{3}}}\left[3i\frac{\Delta
A}{A_{-}}\left(1+\frac{3}{5}\frac{\Delta
A}{A_{+}}\right)\frac{k_{\mathrm{T}}}{k}+\frac{A_{+}}{A_{-}}+\mathcal{O}\left(\frac{k}{k_{\mathrm{T}}}\right)\right]\,.$

Note that the term $\Delta A/A_{-}$ may be large (this is the case if
$A_{-}\ll A_{+}$), which leads us to distinguish three sub-cases.

If $k/k_{\mathrm{T}}\ll\sqrt{A_{-}/|\Delta A|}$, then the non-adiabatic,
decaying mode $D_{k}$ is suppressed in Eq. (3.13) and $C_{k}$ reduces to the
apparent growing mode in the pre-transition phase, Eq. (3.8). These scales are
sufficiently far outside the Hubble scale at the time of the transition that
they are not affected by the sudden jump and $\mathcal{R}_{k}$ remains
constant after the transition. This corresponds to the left-hand scale-
invariant plateau shown in the left-hand panel of Fig. 3.

For $\sqrt{A_{-}/|\Delta A|}\sim k/k_{\mathrm{T}}$, the two leading terms in
Eq. (3.13) for $C_{k}$ are of the same order and can cancel, since $\Delta
A/A_{-}<0$. The higher-order terms are strongly suppressed for
$k/k_{\mathrm{T}}\ll 1$. This explains the well-known dip in the power
spectrum (2.18), as shown in Fig. 3.

If $\sqrt{A_{-}/|\Delta A|}\ll k/k_{\mathrm{T}}$, then the term proportional
to $(k/k_{\mathrm{T}})^{2}$ dominates in Eq. (3.13), which leads to the
intermediate region in Fig. 3 where the power spectrum scales as
$\mathcal{P}_{\mathcal{R}}\propto k^{4}$ [30, 76]. These scales are super-
Hubble at the time of the transition, but they have spent too little time
outside the Hubble radius to suppress gradients terms to the level that would
be necessary to make the non-adiabatic term irrelevant after the transition.
Therefore, as first observed in Ref. [59], the presence of gradient terms at
the sudden transition can leave a strong imprint on super-Hubble scales. More
precisely, the scale above which $\mathcal{R}$ is approximately constant is
not the Hubble radius but is somewhat larger, given by $\sqrt{|\Delta
A|/A_{-}}$ times the Hubble radius222The reason why the ratio $\Delta A/A_{-}$
appears can be understood from Eq. (3.3), which can be rewritten as
$\epsilon_{1-}=A_{-}^{2}/(18H^{4})[1-\Delta
A/A_{-}e^{-3(N-N_{\mathrm{T}})}]^{2}$. This shows that the ultra-slow-roll
phase after the transition lasts for of order $\ln(\Delta A/A_{-})/3$ e-folds
before it returns to slow roll..

Since this behaviour is key to understanding the main result of this paper,
let us further comment on it. As already explained, $\mathcal{R}$ and
$\mathcal{R}^{\prime}$ are continuous at the transition. Let us check whether
or not this is the case for their homogeneous counterparts, i.e. let us
compare the apparent homogeneous solution before the transition (3.8) to the
homogeneous solution after the transition (3.13)

$\displaystyle\frac{\Delta\mathcal{R}_{\textrm{h}}}{\widetilde{\mathcal{R}}_{\mathrm{h}}(\eta_{\mathrm{T}})}\equiv\frac{\mathcal{R}_{\textrm{h}}(\eta_{\mathrm{T}})-\widetilde{\mathcal{R}}_{\mathrm{h}}(\eta_{\mathrm{T}})}{\widetilde{\mathcal{R}}_{\mathrm{h}}(\eta_{\mathrm{T}})}$
$\displaystyle\simeq-\frac{3}{5}\frac{\Delta
A}{A_{+}}\left(\frac{k}{k_{\mathrm{T}}}\right)^{2}\,,$ (3.14)
$\displaystyle\frac{\Delta\mathcal{R}_{\textrm{h}}^{\prime}}{\widetilde{\mathcal{R}}_{\mathrm{h}}^{\prime}(\eta_{\mathrm{T}})}\equiv\frac{\mathcal{R}_{\mathrm{h}}^{\prime}(\eta_{\mathrm{T}})-\widetilde{\mathcal{R}}_{\mathrm{h}}^{\prime}(\eta_{\mathrm{T}})}{\widetilde{\mathcal{R}}_{\mathrm{h}}^{\prime}(\eta_{\mathrm{T}})}$
$\displaystyle\simeq 3i\frac{\Delta A}{A_{+}}\left(1+\frac{3}{5}\frac{\Delta
A}{A_{+}}\right)\frac{k_{\mathrm{T}}}{k}\,.$

Since $|\Delta A/A_{+}|<1$, the relative jump in $\mathcal{R}_{\mathrm{h}}$ at
the transition is suppressed by $(k/k_{\mathrm{T}})^{2}$, which is indeed
small for scales that are super-Hubble at the time the transition. However the
relative jump in $\mathcal{R}_{\mathrm{h}}^{\prime}$ is enhanced by
$k_{\mathrm{T}}/k$, even if the absolute jump is suppressed by
$k/k_{\mathrm{T}}$.

This explains what is observed in Fig. 2, where the apparent homogeneous
solution found before the transition, $\widetilde{\mathcal{R}}_{\mathrm{h}}$,
remains constant post-transition, while the actual homogeneous solution after
the transition, $\mathcal{R}_{\textrm{h}}$, shows rapid time evolution. Also
note that the actual homogeneous solution after the transition,
$\mathcal{R}_{\textrm{h}}$, propagated back to Hubble-exit shows an order of
magnitude difference compared to the apparent pre-transition homogeneous
solution, $\widetilde{\mathcal{R}}_{\mathrm{h}}$, at that time. But if we
track the change to the homogeneous behaviour,
$\Delta\mathcal{R}_{\textrm{h}}$ and $\Delta\mathcal{R}_{\textrm{h}}^{\prime}$
induced at the transition, and propagate that forward using the homogeneous
equation of motion, the correct post-transition behaviour on super-Hubble
scales is recovered. This suggests we can keep track of the homogeneous
behaviour, even if there is a sudden transition, by including the correct non-
adiabatic jump coming from gradient terms at the transition.

Figure 2: The evolution of the curvature perturbation $\mathcal{R}_{k}$ with
comoving time $\eta$ for a wavemode with $k=0.05\,k_{\mathrm{T}}$ in the
piece-wise linear potential (3.1) with $A_{-}/A_{+}=10^{-4}$. The sudden
transition from slow roll to ultra-slow roll occurs at
$\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$. The solid blue curve is found by
numerically solving the full Sasaki–Mukhanov equation (2.4). The dashed orange
curve corresponds to the homogeneous solution (2.12) with the apparent growing
and decaying mode amplitudes before the transition, Eq. (3.8), and the dot-
dashed purple curve corresponds to the homogeneous solution (2.12) with the
growing and decaying mode amplitudes after the transition Eq. (3.12). The
green data points are found by adding the discontinuous jump at the transition
(3.14) to the for the homogeneous solution (2.12) with the apparent growing
and decaying mode amplitudes before the transition, Eq. (3.8). The dotted
vertical line shows the Hubble-exit time, when $aH=0.05\,k_{\mathrm{T}}$.

#### Scales $k\gg k_{\mathrm{T}}$

Finally we note that for $k\gg k_{\mathrm{T}}$, i.e. scales that exit the
Hubble radius some time after the transition, one has

$\displaystyle
C_{k}=-\frac{3iH^{3}}{\sqrt{2k^{3}}A_{-}}\left[1+\frac{3i}{2}\frac{\Delta
A}{A_{+}}\frac{k_{\mathrm{T}}}{k}\left(1+e^{2i\frac{k}{k_{\mathrm{T}}}}\right)+\mathcal{O}\left(\frac{k_{\mathrm{T}}^{2}}{k^{2}}\right)\right]\,.$
(3.15)

Here, $|\Delta A/A_{+}|$ is always smaller than one, hence the first term in
the square bracket dominates and one recovers the slow-roll result in the
post-transition region (i.e. Eq. (3.15) reduces to Eq. (3.8) where $A_{+}$ is
replaced with $A_{-}$). This corresponds to the right-hand scale-invariant
plateau in the left panel of Fig. 3, with damped sinusoidal modulation of the
power spectrum for $k\gtrsim k_{\mathrm{T}}$.

###  3.3 Homogeneous matching

Let us now employ the homogeneous-matching procedure described in Sec. 2.4 to
estimate the homogeneous perturbations in the Starobinsky piece-wise linear
model. Using Eq. (3.5) to evaluate the terms involving $z(\eta)$, Eq. (2.27)
gives rise to

$\displaystyle\hat{C}_{k}=\begin{cases}\mathcal{R}_{k*}+\left[1-\frac{\Delta
A}{A_{-}}\left(\frac{k}{\sigma
k_{\mathrm{T}}}\right)^{3}\right]\frac{\sigma}{3k}\mathcal{R}^{\prime}_{k*}&\text{for
}k\leq\sigma k_{\mathrm{T}}\vspace{0.3em}\\\
\mathcal{R}_{k*}+\left[1-\frac{\Delta A}{A_{-}}\left(\frac{k}{\sigma
k_{\mathrm{T}}}\right)^{-3}\right]\frac{\sigma}{3k}\mathcal{R}^{\prime}_{k*}&\text{for
}k>\sigma k_{\mathrm{T}}\end{cases}\,,$ (3.16)

where we recall that $\mathcal{R}_{k*}$ and $\mathcal{R}_{k*}^{\prime}$
correspond to the curvature perturbation and its time derivative at the time
where $k$ crosses the matching scale (i.e. when $k=\sigma aH$), as given by
the full perturbation theory established in Sec. 3.2. In Fig. 3, we show the
power spectrum of $\mathcal{R}$ and the evolution of a single mode with
$k=0.05\,k_{\mathrm{T}}$ calculated using the homogeneous matching procedure
for different values of $\sigma$.

At this stage, it is worth highlighting the calculation performed in Appendix
A, where the $\delta N$ program described in Sec. 2.4.3 has been implemented
for Starobinsky’s piece-wise linear model. In the linear version of the
calculation, one obtains Eq. (A.23), which exactly coincides with Eq. (3.16).
This illustrates the statement made above that, for linear perturbations, the
homogeneous-matching procedure and the $\delta N$ approach are equivalent.

#### Matching before the transition

For $k\leq\sigma k_{\mathrm{T}}$, the matching scale is crossed before the
transition, and inserting Eq. (3.7) into the first entry of Eq. (3.16) leads
to

$\displaystyle\hat{C}_{k}=-\frac{3iH^{3}e^{i\sigma}}{\sqrt{2k^{3}}A_{+}}\left[1-i\sigma-\frac{\sigma^{2}}{3}+\frac{\sigma^{2}}{3}\frac{\Delta
A}{A_{-}}\left(\frac{k}{\sigma k_{\mathrm{T}}}\right)^{3}\right].$ (3.17)

This has to be compared with the full homogeneous solution after the
transition given by Eq. (3.12), which reduces to Eq. (3.13) in the limit where
$k\ll k_{\mathrm{T}}$. If matching is done in the super-Hubble regime,
$\sigma\ll 1$, one observes different behaviours depending on how
$k/k_{\mathrm{T}}$ compares with $\sqrt{A_{-}/|\Delta A|}$.

If $k/k_{\mathrm{T}}\ll\sqrt{A_{-}/|\Delta A|}$, the late-time curvature
perturbation obtained from homogeneous matching, Eq. (3.17), reduces to the
full result (3.13), up to ${\cal O}(\sigma^{2})$ corrections. The usual
separate-universe approach is therefore able to reproduce the left-hand scale-
invariant plateau in Fig. 3, since the curvature perturbation remains a
constant on super-Hubble scales even during the ultra-slow-roll phase.

If $\sqrt{A_{-}/|\Delta A|}\ll k/k_{\mathrm{T}}\ll 1$, the $k^{2}$ term
becomes dominant in Eq. (3.13), and cannot be reproduced by the terms in Eq.
(3.17). The usual separate-universe approach thus fails in this case. The
reason is that, in the exact result, Eq. (3.13), the $k^{2}$ term comes from
gradient corrections to the growing mode, see Sec. 2.3.1. However, this
contribution is not accounted for in the homogeneous matching, Eqs. (3.16) and
(3.17), since it does not appear in the homogeneous solution (2.12).
Homogeneous matching only captures the leading order of the decaying mode,
which scales as $k^{3}$. This is demonstrated in Fig. 3 for both the power
spectrum and the time evolution of a particular mode.

#### Matching after the transition

For $k>\sigma k_{\mathrm{T}}$, the matching scale is crossed after the
transition and one may use Eq. (2.19), together with Eq. (3.3) for
$\epsilon_{1}(\eta)$ for $\eta\geq\eta_{\mathrm{T}}$, to evaluate
$\mathcal{R}_{k*}$ and $\mathcal{R}_{k*}^{\prime}$ in the second entry of Eq.
(3.16). This leads to

$\hat{C}_{k}=-\frac{3iH^{3}}{A_{-}\sqrt{2k^{3}}}\left[\alpha_{k-}e^{i\sigma}\left(1-i\sigma-\frac{\sigma^{2}}{3}\right)-\beta_{k-}e^{-i\sigma}\left(1+i\sigma-\frac{\sigma^{2}}{3}\right)\right],$
(3.18)

where $\alpha_{k-}$ and $\beta_{k-}$ after the transition are given in Eqs.
(3.10) and (3.11). At leading order in $\sigma$, this reduces to
$\hat{C}_{k}\simeq-3iH^{3}(\alpha_{k-}-\beta_{k-})/(A_{-}\sqrt{2k^{3}})$,
which coincides with the exact expression, Eq. (2.24). As a consequence, up to
${\cal O}(\sigma^{2})$ corrections, the homogeneous-matching procedure is able
to correctly reproduce the late-time behaviour of the curvature perturbation
if the matching is performed after the transition, as shown in Fig. 3.

Let us summarise our findings. The above results show that the homogeneous-
matching procedure fails for the scales within the range $\sqrt{A_{-}/|\Delta
A|}<k/k_{\mathrm{T}}<1$ _if_ they are matched before the transition, and
succeeds otherwise as long as $\sigma\ll 1$. Therefore, in order to match the
dangerous set of scales after the transition, one requires

$\displaystyle\sigma\ll\mathrm{min}\left(1,\sqrt{\frac{A_{-}}{|\Delta
A|}}\right)\,.$ (3.19)

As mentioned in Footnote 2, $A_{-}/\Delta A$ can be related to the number of
e-folds during the ultra-slow-roll phase, $N_{\mathrm{USR}}$, so we expect the
above condition to generalise to

$\displaystyle\sigma\ll e^{-\frac{3}{2}N_{\mathrm{USR}}}\,,$ (3.20)

in more general models. This condition is more stringent than the one usually
encountered in the absence of sudden transitions, $\sigma\ll 1$, but it
ensures the validity of the separate-universe approach after homogeneous
matching.

Figure 3: Left: the power spectrum (2.18) of the curvature perturbation,
$\mathcal{R}$, against wavenumber, $k/k_{\mathrm{T}}$ for the piece-wise
linear potential (3.1) with $A_{-}/A_{+}=10^{-4}$. Right: the evolution of
$\mathcal{R}_{k}$ for a mode with $k=0.05\,k_{\mathrm{T}}$, along with the
homogeneous solution resulting from matching at different values of
$\sigma=k/aH$. In both panels, the solid blue curve is found by numerically
solving the full mode equation Eq. (2.4) and the dashed curves are obtained
numerically using the matching procedure detailed in Sec. 2.4. The dotted
vertical line is Hubble-exit time. The ultra-slow-roll period starts at
$\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$.

###  3.4 Including $k^{2}$-corrections

As discussed below Eq. (2.24), the $k^{2}$-correction to the growing mode can
dominate over the leading-order decaying mode at late times. As a consequence,
the late-time non-adiabatic behaviour may be dominated by contributions coming
from $k^{2}$-corrections in the gradient expansion, hence the separate-
universe approach fails on some super-Hubble scales in the presence of a
sudden transition. One can further check the validity of this statement by
including the leading $k^{2}$-corrections in the matching procedure, as was
discussed in Sec. 2.4.2. This has been shown in Fig. 4 matching very close to
Hubble-exit, $\sigma=0.5$. This gives an excellent approximation to the exact
power spectrum, confirming the $k^{2}$-corrections are indeed needed to fully
explain the super-Hubble evolution in the presence of a sudden transition
after the matching.

By comparing the $k^{2}$-corrected reconstruction shown in Fig. 4 with the
homogeneous matching used in Fig. 3, it is clear that the post-transition
behaviour is recovered well (within $\sim 10\%$) even for $\sigma=0.5$, when
the $k^{2}$-corrections are included. This is in contrast to the homogeneous
matching case, where a $\sigma$-value corresponding to matching after the
transition ($\sigma<0.05$) is required. This clearly shows that the
$k^{2}$-corrections at the transition are required to describe the non-
adiabatic homogeneous behaviour on super-Hubble scales after the transition.
If $\sigma=0.1$ is used for the $k^{2}$-corrected Eq. (2.4.2), the late-time
behaviour is reconstructed within $\sim 1\%$.

Figure 4: Left: the power spectrum (2.18) of the curvature perturbation
$\mathcal{R}$ for the piece-wise linear potential (3.1) with
$A_{-}/A_{+}=10^{-4}$. The solid blue curve is found by numerically solving
Eq. (2.4) and the brown points are found by including the $k^{2}$ term in the
adiabatic mode at matching, as detailed in Sec. 2.4.2. Right: evolution of the
components of Eq. (2.31) plotted along with the full solution of the mode
equation for $k=0.05\,k_{\mathrm{T}}$. The $k^{2}$ curve is the $k^{2}$-term,
“Homogeneous” is the $k^{2}$-independent component, and “Total” is the sum of
the two. The dotted and dashed vertical lines correspond to Hubble-exit and
matching times, respectively. The ultra-slow-roll period starts at
$\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$.

##  4 Smooth potential with a Gaussian bump

Let us consider a model with a smooth slow-roll to ultra-slow-roll transition.
One way this can be achieved is by the addition of a Gaussian bump to a
working slow-roll potential, giving a model which is both in agreement with
CMB observations and can produce a significant abundance of PBHs [77]. One
such potential is given by

$V(\phi)=V_{0}\frac{\phi^{2}}{m^{2}+\phi^{2}}\left\\{1+K\exp\left[-\frac{1}{2}\frac{(\phi-\phi_{0})^{2}}{\Sigma^{2}}\right]\right\\}\,,$
(4.1)

with $m=0.5M_{\mathrm{Pl}}$, $K=1.876\times 10^{-3}$,
$\phi_{0}=2.005M_{\mathrm{Pl}}$ and $\Sigma=1.993\times
10^{-2}M_{\mathrm{Pl}}$ [77]. Note that all of the above precision is needed
to produce an asteroid-mass PBH abundance accounting for all of the dark
matter, due to fine-tuning [78]. This model has a period of ultra-slow roll of
$\sim 3$ e-folds, before smoothly transitioning to a phase of constant roll.
The transition from slow roll to ultra-slow roll takes $\sim 1$ e-fold. As
such, one cannot easily propagate modes through the transition analytically,
and a numerical approach is required.

Figure 5: Left: the power spectrum (2.18) of the curvature perturbation
$\mathcal{R}$ for a potential with a Gaussian bump (4.1). Right: the evolution
of $\mathcal{R}$ for a mode with $k=0.05\,k_{\mathrm{T}}$. The solid blue
curve is found by numerically solving Eq. (2.4), the dashed curves are found
using the numerical matching procedure detailed in Sec. 2.4, and the brown
data points are found by including the $k^{2}$ correction (2.31). The dotted
vertical line is Hubble-exit time. The ultra-slow-roll period
($\epsilon_{2}<-3$) starts at $\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$.

Determining the homogeneous growing and decaying modes can be done numerically
using the homogeneous-matching procedure, detailed in Eq. (2.27). This is
shown in Fig. 5. Phenomenologically we see similar behaviour to the previously
discussed piece-wise linear model. The power spectrum is most accurately
reconstructed when matching sufficiently long after Hubble-exit,
$\sigma=0.01$, although it is still not able to fully reproduce the dip
behaviour. This is because $\sigma=0.01$ is not sufficient to satisfy the
condition given in Eq. (3.20) for this model, as it has a longer duration of
ultra-slow-roll inflation compared to the piece-wise linear potential shown
above. The $k^{2}$-corrected data points, found using Eq. (2.31) to do the
matching, are accurate even when matching soon after Hubble-exit for
$\sigma=0.5$. This shows the effect of gradient terms on the homogeneous
behaviour on super-Hubble scales at the sudden transition from slow roll to
ultra-slow roll, in smooth models, not just in the idealised, instant
transition case.

The interpretation of the evolution of the mode seen in the right-hand panel
in Fig. 5 is the same as for the piece-wise linear model if we define the
transition to ultra-slow roll to happen when $\epsilon_{2}<-3$ first occurs.
If the homogeneous matching is done before the transition time, the early-time
behaviour is recovered but the behaviour after the transition is not. Equally,
matching after the transition reproduces the late-time behaviour (more
accurately for smaller $\sigma$) but does not reproduce the early-time
behaviour. This shows numerically that the sudden transition to ultra-slow
roll has resulted in the behaviour of the homogeneous growing/decaying modes
changing due to gradient terms at the transition.

In Appendix B we show another example of a model with a smooth potential that
leads to a sudden transition from slow roll to ultra-slow roll. In that case
we see that a potential with an inflection point shows the same qualitative
behaviour for modes before and after a sudden transition from slow roll to
ultra-slow roll, suggesting that this is a common feature of such models.

##  5 Implications for stochastic inflation

In the stochastic approach to studying inflationary dynamics [79, 80] quantum
field perturbations originating on small, sub-Hubble scales are incorporated
into the locally-homogeneous background classical field in a local patch after
crossing the coarse-graining scale, giving the local background field a
stochastic kick. In effect the coarse-graining introduces a matching scale,
$k=\sigma aH$, between the solution to the full mode equation on small scales
and a homogeneous solution used on larger scales. In the standard case the
stochastic kick is included soon after Hubble exit, corresponding to
$\sigma\sim 1$. But we have shown that a sudden transition leads to a change
in the state of the field on a finite range of super-Hubble scales. If there
is a discontinuity in the homogeneous field perturbation after coarse-
graining, then the standard stochastic approach breaks down.

One approach to address this issue is to coarse-grain only after the
transition or on sufficiently large super-Hubble scales such that any
discontinuity in the homogeneous solution at the transition is minimal. If the
condition given in Eq. (3.20) is used, then the correct post-transition
solution for $\delta\phi_{\textrm{h}}$ is captured for all modes. For example
in Refs. [81, 82, 83, 75, 84] $\sigma=0.01$ is chosen, in which case coarse-
graining captures the transition-induced change to $\delta\phi_{\textrm{h}}$
for modes with $k\gtrsim 0.01\,k_{\mathrm{T}}$, which is sufficient to avoid
any significant discontinuity for a sudden transition leading to $\lesssim 3$
e-folds.

However, incorporating the stochastic kick into the long-wavelength evolution
a long time after Hubble exit fails to account for nonlinear effects due to
field fluctuations close to the Hubble scale, so is likely to under-estimate
the full stochastic effects. This may be particularly important when modelling
the transition to ultra-slow roll as this is a transient regime. For example,
this could reduce the probability of large, rare fluctuations in $\delta N$
which may be associated with PBH production. We could attempt to evolve the
post-transition homogeneous solution back to the Hubble-exit time, and apply
the corresponding stochastic kick at that time in order to include the non-
adiabatic behaviour induced after the transition. However, this gives an
erroneously large non-adiabatic kick before the transition, at a time when the
super-Hubble evolution was in fact adiabatic. This can be seen by comparing
$\widetilde{\mathcal{R}}_{\textrm{h}}$ and $\mathcal{R}_{\textrm{h}}$ in Fig.
2 near Hubble exit. The enhanced non-adiabatic decaying mode of
$\mathcal{R}_{\textrm{h}}$ is induced by the transition and was not present
when the mode crossed the Hubble scale.

A possible solution is to apply two kicks for each mode with $k<k_{\rm T}$:
the first applied in the usual stochastic way at or soon after Hubble exit
before the transition, and the second, correlated kick applied at or just
after the transition to reproduce the correct non-adiabatic behaviour post-
transition. As we already saw in Fig. 2, for linear perturbations, the correct
behaviour before and after the transition can be recovered if we include the
discontinuity in the homogeneous solution at the transition. The pre-
transition kick can be found in a number of ways. The simplest is to use the
slow-roll result, $\delta\phi_{\textrm{h}}\simeq iH/\sqrt{2k^{3}}$ at Hubble
exit. Alternatively, this can be done numerically, either directly matching
$\delta\phi_{\textrm{h}}$ to $\delta\phi_{k}$, or using a Bessel function
approximation at Hubble exit as described in Appendix C. The latter allows
$\delta\phi_{\textrm{h}}$ to be found close to Hubble-scale crossing and it
explicitly removes spurious gradient terms from the homogeneous solutions. The
post-transition non-adiabatic kick would need to be found from the solution to
the full mode equations at the transition. This could again either be done
directly from a numerical solution, or with the Bessel function approximation.
Since the discontinuity in $\delta\phi_{\textrm{h}}$ occurs at the transition
for super-Hubble modes, $k<k_{\rm T}$, the second kicks for all these modes
would be applied at the same time. Thus a single non-adiabatic kick at the
transition, found by summing the contributions from all of the $k<k_{\rm T}$
modes, could incorporate the non-adiabatic change to the homogeneous behaviour
induced by the transition.

Finally we note that in the stochastic-inflation formalism it is assumed that
the quantum field fluctuations can be described by a classical noise. This is
consistent with the initial adiabatic vacuum state becoming a highly squeezed
state on super-Hubble scales [85, 86, 87]. However the discontinuity in the
homogeneous solution at the transition may have implications for the quantum-
to-classical transition [88, 89, 72, 90]. A full study of classicalisation at
such transitions is beyond the scope of this paper, and we leave further
investigation of this interesting question for future work.

##  6 Conclusion

Models of inflation incorporating a transition from slow roll to ultra-slow
roll have been intensively studied recently as a mechanism to produce
primordial black holes (PBHs). As well as using linear perturbation theory to
study the power spectrum produced in such models, non-perturbative approaches,
such as the $\delta N$ or stochastic $\delta N$ formalisms, have been used to
determine the full probability distribution function for the curvature
perturbation. These typically rely on the separate-universe approach, using
solutions to the spatially-homogeneous equations of motion to model the
evolution of inhomogeneities beyond the Hubble scale.

In this paper we have shown that the separate-universe approach breaks down on
a finite range of super-Hubble scales at a sudden transition during inflation
from slow roll to ultra-slow roll. The sudden transition leads to particle
production in the inflaton field on sub-Hubble scales and non-adiabatic
pressure perturbations on super-Hubble scales.

We emphasise that the separate-universe approach works in a piece-wise
fashion, before the transition and in the ultra-slow-roll phase after333It was
previously shown in Ref. [72] that the separate universe approach applies
beyond slow roll, in contrast to the conclusions of Ref. [91]. In general one
must allow for perturbations in the lapse function on large scales, which
vanish only in the slow roll limit where $\epsilon_{1}\to 0$., but a solution
to the spatially-homogeneous equations is unable to reproduce the
discontinuous change in the super-Hubble behaviour at the transition, which is
due to small, but finite spatial gradients still present on super-Hubble
scales. The role of spatial gradients even on super-Hubble scales in a model
of ultra-slow-roll inflation was first noted in Ref. [59].

Figure 6: The power spectrum (2.18) of the curvature perturbation
$\mathcal{R}$ for the piece-wise linear potential (3.1) with
$A_{-}/A_{+}=10^{-4}$. The solid blue curve is found by numerically solving
the full mode equation (2.4). The dashed curves use the linear $\delta N$
formalism (2.34) with
$\delta\phi_{\mathrm{h}}=\dot{\phi}\mathcal{R}_{\textrm{h}}/H$ given in a
piece-wise manner, where Eq. (3.8) is used before the transition and Eq.
(3.12) after. The $\delta N$ formalism is applied at $k=\sigma aH$. The pink
dotted curve uses the result (2.26) for a massless field at Hubble-exit in de
Sitter, $\delta\phi_{\mathrm{h}}=iH/\sqrt{2k^{3}}$ with $\sigma=1$. The ultra-
slow-roll period starts at comoving time
$\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$.

The discontinuity in the homogeneous behaviour was shown analytically in
(3.14) for an idealised test case corresponding to the piece-wise linear
potential (3.1). In this case the general solution to the full mode equation
before and after the transition is given in (2.19) allowing us to identify the
homogeneous solution (2.25) from a formal gradient expansion (2.14). We have
also demonstrated numerically that a discontinuity in the homogeneous solution
occurs at a sudden transition in models with a smooth feature in the potential
(either a Gaussian bump, or an inflection point, shown in Figs. 5 and 8
respectively) using numerical matching to estimate the homogeneous solution
after Hubble exit.

This discontinuous behaviour at the transition time,
$\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$, has implications for the $\delta N$
formalism, where one uses the homogeneous solution to follow the evolution of
the perturbed universe on super-Hubble scales. The naïve approach, using the
Bunch–Davies vacuum state for a massless field in de Sitter at Hubble-exit to
set the amplitude of fluctuations in the spatially-homogeneous field,
$\mathcal{P}^{1/2}_{\delta\phi_{h}}=H/2\pi$, fails completely to describe the
growth of the power spectrum for $k<k_{\mathrm{T}}$ or its peak at $k\sim
k_{\mathrm{T}}$, see Fig. 6. This is particularly important in models of
primordial black hole production from inflation where the PBH abundance can be
exponentially sensitive to the peak of the power spectrum [78]. We can only
reconstruct the full behaviour by matching fluctuations in the homogeneous
field to the mode functions at (or soon after) Hubble-exit if
$k^{2}$-corrections (2.31) are included at the matching, showing the essential
role of spatial gradients on super-Hubble scales at the transition. If the
corrected homogeneous solution is used post-transition, including a non-
adiabatic decaying mode, the correct power spectrum can be recovered at the
end of inflation, showing that a modified separate-universe approach may still
be used in a piece-wise manner even in the presence of a sudden transition.

In the stochastic-inflation formalism, this would correspond to coarse-
graining only on scales very much larger than the Hubble radius, such that all
the affected modes are coarse-grained after the transition. However this may
under estimate stochastic effects by failing to include all super-Hubble
fluctuations in the coarse-grained field before the transition. An alternative
method would be to coarse-grain close to the Hubble scale before the
transition, but then to include the discontinuity in the homogeneous behaviour
by summing all the affected modes into a single non-adiabatic kick to the
coarse-grained field at the transition. We leave the implementation of a
modified stochastic-inflation approach in such models for future work.

## Note Added

While this work was in preparation, Ref. [92] appeared where it was shown that
the usual $\delta N$ formalism could only be applied sufficiently long after a
sudden transition, such that $\delta\phi_{\mathrm{h}}$ is dominated by the
growing mode, which is consistent with our results.

## Acknowledgments

The authors would like to thank Laura Iacconi, Eemeli Tomberg, Diego Cruces,
David Mulryne and the participants of Cosmo’23 for insightful discussions.
This work was supported by the Science and Technology Facilities Council
(grant numbers ST/T506345/1 and ST/W001225/1). For the purpose of open access,
the authors have applied a Creative Commons Attribution (CC-BY) licence to any
Author Accepted Manuscript version arising from this work. Supporting research
data are available on reasonable request from the corresponding author, Joseph
Jackson.

##  Appendix A $\delta N$ for piece-wise linear model

In this section, we implement the linearised version of the $\delta N$
approach presented in Sec. 2.4.3, to Starobinsky’s piece-wise linear model
discussed in Sec. 3. The first step is to derive the function
$N(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$, which returns the number of
e-folds realised from the spatially-homogeneous initial field configuration
given by $\phi_{\mathrm{in}}$ and $\dot{\phi}_{\mathrm{in}}$ to the end of
inflation. This can be done using the general solution of the Klein–Gordon
equation (2.1) with $N$ as the time variable, namely

$\phi(N)=-\frac{A_{\pm}}{3H^{2}}N+B_{\pm}^{(1)}+B_{\pm}^{(2)}e^{-3N}\,.$ (A.1)

The integration constants $B_{\pm}^{(1)}$ and $B_{\pm}^{(2)}$ depend on the
initial phase-space coordinates
$(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$.

Let us first consider the case where $\phi_{\mathrm{in}}<\phi_{\mathrm{T}}$.
By differentiating Eq. (A.1) with respect to time,

$\dot{\phi}=-\frac{A_{-}}{3H}-3HB_{-}^{(2)}e^{-3N}\,,$ (A.2)

and combining Eqs. (A.1) and (A.2), one finds

$N=-\frac{3H^{2}}{A_{-}}\left(\phi+\frac{\dot{\phi}}{3H}-B_{-}^{(1)}+\frac{A_{-}}{9H^{2}}\right).$
(A.3)

The constant $B_{-}^{(1)}$ can be found by requiring that
$\phi(0)=\phi_{\mathrm{in}}$ in Eq. (A.1) and
$\dot{\phi}(0)=\dot{\phi}_{\mathrm{in}}$ in Eq. (A.2), leading to

$B_{-}^{(1)}=\phi_{\mathrm{in}}+\frac{\dot{\phi}_{\mathrm{in}}}{3H}+\frac{A_{-}}{9H^{2}}\,.$
(A.4)

The number of e-folds from $(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$ to
the end of inflation at $(\phi_{\mathrm{end}},\dot{\phi}_{\mathrm{end}})$, is
then

$N_{-}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})=\frac{3H^{2}}{A_{-}}\left(\phi_{\mathrm{in}}-\phi_{\mathrm{end}}+\frac{\dot{\phi}_{\mathrm{in}}-\dot{\phi}_{\mathrm{end}}}{3H}\right)\,.$
(A.5)

For $\phi_{\mathrm{in}}>\phi_{\mathrm{T}}$, the total number of e-folds can be
decomposed into $N_{\mathrm{T}}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$,
the number of e-folds until the transition, and
$N_{-}[\phi_{\mathrm{T}},\dot{\phi}_{\mathrm{T}}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})]$,
the number of e-folds from the transition to the end of inflation. Note that
the field velocity at the transition, $\dot{\phi}_{\mathrm{T}}$, is a function
of the initial phase-space coordinate
$(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$. For
$N_{\mathrm{T}}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$, one can solve
Eq. (A.1) for $N$ with $+$ as a subscript and find

$N_{\mathrm{T}}\left(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}}\right)=\frac{3H^{2}}{A_{+}}\left(B_{+}^{(1)}-\phi_{\mathrm{T}}\right)+\frac{1}{3}W_{0}\left[\frac{9H^{2}B_{+}^{(2)}}{A_{+}}e^{-\frac{9H^{2}}{A_{+}}\left(B_{+}^{(1)}-\phi_{\mathrm{T}}\right)}\right]\,,$
(A.6)

where $W_{0}(x)$ is the $0^{\text{th}}$ branch of the Lambert function. In
this expression, $B_{+}^{(1)}$ and $B_{+}^{(2)}$ are set as before by
requiring that $\phi(0)=\phi_{\mathrm{in}}$ and
$\dot{\phi}(0)=\dot{\phi}_{\mathrm{in}}$ in Eqs. (A.1) and (A.2), leading to

$B_{+}^{(1)}=\phi_{\mathrm{in}}+\frac{\dot{\phi}_{\mathrm{in}}}{3H}+\frac{A_{+}}{9H^{2}}\quad\text{and}\quad
B_{+}^{(2)}=-\frac{\dot{\phi}_{\mathrm{in}}}{3H}-\frac{A_{+}}{9H^{2}}\,.$
(A.7)

For
$N_{-}[\phi_{\mathrm{T}},\dot{\phi}_{\mathrm{T}}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})]$,
by differentiating Eq. (A.1) with respect to time and evaluating the result at
the time $N_{\mathrm{T}}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})$ given
in Eq. (A.6), one first finds

$\dot{\phi}_{\mathrm{T}}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})=-\frac{A_{+}}{3H}\left\\{1+W_{0}\left[\frac{9H^{2}B_{+}^{(2)}}{A_{+}}e^{-\frac{9H^{2}}{A_{+}}\left(B_{+}^{(1)}-\phi_{\mathrm{T}}\right)}\right]\right\\}\,.$
(A.8)

Replacing $\phi_{\mathrm{in}}$ by $\phi_{\mathrm{T}}$ and
$\dot{\phi}_{\mathrm{in}}$ by the above expression in Eq. (A.5) then gives

$\displaystyle
N_{-}[\phi_{\mathrm{T}},\dot{\phi}_{\mathrm{T}}(\phi_{\mathrm{in}},\dot{\phi}_{\mathrm{in}})]=$
$\displaystyle\frac{3H^{2}}{A_{-}}\left(\phi_{\mathrm{T}}-\phi_{\mathrm{end}}-\frac{\dot{\phi}_{\mathrm{end}}}{3H}\right)$
(A.9)
$\displaystyle-\frac{A_{+}}{3A_{-}}\left\\{1+W_{0}\left[\frac{9H^{2}B_{+}^{(2)}}{A_{+}}e^{-\frac{9H^{2}}{A_{+}}\left(B_{+}^{(1)}-\phi_{\mathrm{T}}\right)}\right]\right\\}\,.$

Note that the dependence on $\phi_{\mathrm{in}}$ and
$\dot{\phi}_{\mathrm{in}}$ is implicit through $B_{+}^{(1)}$ and
$B_{+}^{(2)}$. One can summarise the above results as

$N(\phi,\dot{\phi})=\begin{cases}N_{\mathrm{T}}(\phi,\dot{\phi})+N_{-}[\phi_{\mathrm{T}},\dot{\phi}_{\mathrm{T}}(\phi,\dot{\phi})]&\text{if
}\phi\geq\phi_{\mathrm{T}}\,,\\\ N_{-}(\phi,\dot{\phi})&\text{if
}\phi<\phi_{\mathrm{T}}\,,\end{cases}$ (A.10)

where $N_{\mathrm{T}}(\phi,\dot{\phi})$,
$N_{-}[\phi_{\mathrm{T}},\dot{\phi}_{\mathrm{T}}(\phi,\dot{\phi})]$ and
$N_{-}(\phi,\dot{\phi})$ are given above, and where hereafter the subscripts
“$\mathrm{in}$” are dropped for convenience.

The next step is to differentiate the number of realised e-folds with respect
to the initial field value and its velocity. One obtains

$\frac{\partial
N}{\partial\phi}=\begin{cases}\frac{3H^{2}}{A_{+}}\left[1-\frac{\Delta
A}{A_{-}}\frac{W_{0}(x)}{1+W_{0}(x)}\right]&\text{if
}\phi\geq\phi_{\mathrm{T}}\,,\\\ \frac{3H^{2}}{A_{-}},&\text{if
}\phi<\phi_{\mathrm{T}}\,,\end{cases}$ (A.11)

and

$\frac{\partial
N}{\partial\dot{\phi}}=\begin{cases}\frac{H}{A_{+}}\left[1-\frac{\Delta
A}{A_{-}}\left(1+\frac{A_{+}}{9H^{2}B_{+}^{(2)}}\right)\frac{W_{0}(x)}{1+W_{0}(x)}\right]&\text{if
}\phi\geq\phi_{\mathrm{T}}\,,\\\ \frac{H}{A_{-}},&\text{if
}\phi<\phi_{\mathrm{T}}\,,\end{cases}$ (A.12)

where

$x\equiv\frac{9H^{2}B_{+}^{(2)}}{A_{+}}e^{-\frac{9H^{2}}{A_{+}}\left(B_{+}^{(1)}-\phi_{\mathrm{T}}\right)}\,.$
(A.13)

Then, for a given scale $k$, one has to evaluate the above derivatives at the
time when that scale crosses out the $\sigma$-Hubble radius, i.e. when
$k=\sigma aH$. In the de Sitter limit, this occurs at a time (denoted by a
subscript “$*$” hereafter) $\eta_{*}(k)=\sigma/k$, or equivalently,
$N_{*}(k)=N_{\mathrm{T}}+\ln[k/(\sigma k_{\mathrm{T}})]$. From Eq. (3.2), one
has

$\phi_{*}(k)=\begin{cases}-\frac{A_{+}}{3H^{2}}\ln{\Big{(}\frac{k}{\sigma
k_{\mathrm{T}}}\Big{)}}+\phi_{\mathrm{T}}&\text{for }k\leq\sigma
k_{\mathrm{T}}\,,\\\ -\frac{A_{-}}{3H^{2}}\ln{\Big{(}\frac{k}{\sigma
k_{\mathrm{T}}}\Big{)}}+\frac{\Delta
A}{9H^{2}}\Bigg{[}1-\Big{(}\frac{k}{\sigma
k_{\mathrm{T}}}\Big{)}^{-3}\Bigg{]}+\phi_{\mathrm{T}}&\text{for }k>\sigma
k_{\mathrm{T}}\,,\end{cases}$ (A.14)

and

$\dot{\phi}_{*}(k)=\begin{cases}-\frac{A_{+}}{3H}&\text{for }k\leq\sigma
k_{\mathrm{T}}\,,\\\ -\frac{A_{-}}{3H}+\frac{\Delta
A}{3H}\Big{(}\frac{k}{\sigma k_{\mathrm{T}}}\Big{)}^{-3}&\text{for }k>\sigma
k_{\mathrm{T}}\,.\end{cases}$ (A.15)

For $k\leq\sigma k_{\mathrm{T}}$, one has $\phi_{*}\geq\phi_{\mathrm{T}}$ and
inserting Eqs. (A.14) and (A.15) into Eq. (A.7) leads to
$B_{+*}^{(1)}=\phi_{\mathrm{T}}+A_{+}/(3H^{2})\ln(\sigma k_{\mathrm{T}}/k)$
and $B_{+*}^{(2)}=0$. Therefore, $x_{*}=0$ for those scales, and using that
$W_{0}(x)=x+\mathcal{O}(x^{2})$, one obtains

$\displaystyle\left.\frac{\partial
N}{\partial\phi}\right|_{*}=\begin{cases}\frac{3H^{2}}{A_{+}}&\text{for
}k\leq\sigma k_{\mathrm{T}}\\\ \frac{3H^{2}}{A_{-}}&\text{for }k>\sigma
k_{\mathrm{T}}\\\ \end{cases}$ (A.16)

and

$\displaystyle\left.\frac{\partial
N}{\partial\dot{\phi}}\right|_{*}=\begin{cases}\frac{H}{A_{+}}\left[1-\frac{\Delta
A}{A_{-}}\left(\frac{k}{\sigma k_{\mathrm{T}}}\right)^{3}\right]&\text{for
}k\leq\sigma k_{\mathrm{T}}\\\ \frac{H}{A_{-}}&\text{for }k>\sigma
k_{\mathrm{T}}\\\ \end{cases}\,.$ (A.17)

The next step is to evaluate $\delta\phi_{k}$ and $\delta\dot{\phi}_{k}$ at
the $\sigma$-Hubble crossing time, within the full linear cosmological
perturbation theory. Recalling that $v=a\delta\phi$ in the spatially-flat
gauge, and given that $\mathcal{R}=v/z$ where
$z=\mathrm{sign}(\dot{\phi})\sqrt{2a^{2}\epsilon_{1}}$, one has

$\displaystyle\delta\phi_{k}=-\sqrt{2\epsilon_{1}}\mathcal{R}_{k}\quad\text{and}\quad\delta\dot{\phi}_{k}=-\sqrt{2\epsilon_{1}}\left(\dot{\mathcal{R}}_{k}+\frac{\epsilon_{2}}{2}H\mathcal{R}_{k}\right).$
(A.18)

For $k\leq\sigma k_{\mathrm{T}}$ (i.e. $\phi_{*}\geq\phi_{\mathrm{T}}$), from
Eqs. (3.3) and (3.4) one has $\epsilon_{1*}=A_{+}^{2}/(18H^{4})$ while
$\epsilon_{2*}$ can be discarded, leading to

$\displaystyle\delta\phi_{k*}=\frac{-A_{+}}{3H^{2}}\mathcal{R}_{k*}\quad\text{and}\quad\delta\dot{\phi}_{k*}=\frac{-A_{+}}{3H^{2}}\dot{\mathcal{R}}_{k*}\,.$
(A.19)

For $k>\sigma k_{\mathrm{T}}$ (i.e. $\phi_{*}<\phi_{\mathrm{T}}$), one obtains

$\displaystyle\epsilon_{1*}=\frac{A_{-}^{2}}{18H^{4}}\left[1-\frac{\Delta
A}{A_{-}}\left(\frac{\sigma
k_{\mathrm{T}}}{k}\right)^{3}\right]^{2}\quad\text{and}\quad\epsilon_{2*}=6\frac{\frac{\Delta
A}{A_{-}}\left(\frac{\sigma k_{\mathrm{T}}}{k}\right)^{3}}{1-\frac{\Delta
A}{A_{-}}\left(\frac{\sigma k_{\mathrm{T}}}{k}\right)^{3}}\,,$ (A.20)

leading to

$\displaystyle\delta\phi_{k*}=\frac{-A_{-}}{3H^{2}}\left[1-\frac{\Delta
A}{A_{-}}\left(\frac{\sigma
k_{\mathrm{T}}}{k}\right)^{3}\right]\mathcal{R}_{k*}$ (A.21)

and

$\displaystyle\delta\dot{\phi}_{k*}=-\frac{\Delta A}{H}\left(\frac{\sigma
k_{\mathrm{T}}}{k}\right)^{3}\mathcal{R}_{k*}-\frac{A_{-}}{3H^{2}}\left[1-\frac{\Delta
A}{A_{-}}\left(\frac{\sigma
k_{\mathrm{T}}}{k}\right)^{3}\right]\dot{\mathcal{R}}_{k*}\,.$ (A.22)

Combining the above results into Eq. (2.34), one finally obtains

$\displaystyle\delta N_{k}=\begin{cases}-\mathcal{R}_{k*}-\left[1-\frac{\Delta
A}{A_{-}}\left(\frac{k}{\sigma
k_{\mathrm{T}}}\right)^{3}\right]\frac{\dot{\mathcal{R}}_{k*}}{3H}&\text{for
}k\leq\sigma k_{\mathrm{T}}\\\ -\mathcal{R}_{k*}-\left[1-\frac{\Delta
A}{A_{-}}\left(\frac{k}{\sigma
k_{\mathrm{T}}}\right)^{-3}\right]\frac{\dot{\mathcal{R}}_{k*}}{3H}&\text{for
}k>\sigma k_{\mathrm{T}}\end{cases}\,.$ (A.23)

This expression agrees with the one obtained for $\hat{C}_{k}$ (i.e. the late-
time value of $\mathcal{R}_{k}=-\zeta_{k}$) in the homogeneous-matching
procedure, see Eq. (3.16). This proves that the homogeneous-matching procedure
and the linearised $\delta N$ formalism are strictly equivalent. This should
be clear from the way these two approaches are formulated (namely, they both
rely on using linear cosmological perturbation theory below the matching scale
and the homogeneous dynamics above that scale), but here the correspondence is
shown explicitly in the case of the Starobinsky piece-wise linear potential.

##  Appendix B Example: inflection-point model

A potential with an inflection point can also lead to a phase of ultra-slow-
roll inflation and significant PBH production, with many different models
proposed, see e.g. [93, 94, 95, 96]. Here we will investigate a model
motivated by Higgs inflation with quantum corrections [97], which has been
used for simulations of stochastic inflation [82, 83]. The potential in Ref.
[97] does not have a closed analytical form, but the authors kindly provided
numerical data for $V(\phi)$ to reproduce the asteroid mass PBH case, which is
plotted in Fig. 7. There is a bump near $\phi=4$ due to an interpolation, but
this has little effect on the dynamics at the inflection point.

Figure 7: The potential for Higgs inflation modified by quantum corrections
[97]. The parameters are chosen such that sufficient PBHs of asteroid mass are
produced to provide all of the dark matter, giving the highlighted inflection
point. The dotted vertical line is where inflation ends.

Figure 8: Left: the curvature perturbation power spectrum at the end of
inflation for the inflection-point model motivated by Higgs inflation [97].
Right: the evolution of $\mathcal{R}$ for the mode with
$k=0.05k_{\mathrm{T}}$. The solid blue curve is found by numerically solving
Eq. (2.4), the dashed curves are found using the numerical matching detailed
in Sec. 2.4, and the brown data points are found by including the $k^{2}$
correction (2.31). The dotted vertical line is Hubble-exit time. The ultra-
slow-roll period ($\epsilon_{2}<-3$) starts at
$\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$.

Determining the growing and decaying modes can again be done numerically using
the homogeneous matching procedure detailed in Eq. (2.27), the result of which
is shown in Fig. 8. Both the power spectrum and mode evolution plots show
similar behaviour to the Gaussian bump discussed in Sec. 4. Therefore the same
conclusions drawn from the Gaussian bump model also apply here. This
demonstrates that the dynamics discussed in this paper are a general feature
of single-field models with a sudden transition from slow roll to ultra-slow
roll, followed by a slow-roll or constant-roll phase to the end of inflation.

##  Appendix C Bessel functions

Let us introduce the parameter

$\nu^{2}=-\frac{\mu^{2}}{(aH)^{2}}+\frac{1}{4}=\frac{9}{4}-\epsilon_{1}+\frac{3}{2}\epsilon_{2}-\frac{1}{2}\epsilon_{1}\epsilon_{2}+\frac{1}{4}\epsilon_{2}^{2}+\frac{1}{2}\epsilon_{2}\epsilon_{3}\,,$
(C.1)

where $\mu^{2}$ is given in Eq. (2.5). If $\nu^{2}$ is a constant, then Eq.
(2.4) has an exact solution [98]

$v_{k}=\sqrt{-\eta}[A_{k}J_{\nu}(-k\eta)+B_{k}Y_{\nu}(-k\eta)]\,,$ (C.2)

where

$J_{\nu}(x)=\bigg{(}\frac{x}{2}\bigg{)}^{\nu}\sum_{n=0}^{\infty}\bigg{(}\frac{x}{2}\bigg{)}^{2n}\frac{(-1)^{n}}{n!\Gamma(\nu+n+1)}\hskip
8.00003pt\text{and}\hskip
8.00003ptY_{\nu}(x)=\frac{J_{\nu}(x)\cos{(\nu\pi)}-J_{-\nu}(x)}{\sin{(\nu\pi)}}\,,$
(C.3)

and $\Gamma$ is the gamma function. In the standard case the integration
constants $A_{k}$ and $B_{k}$ are set by using Bunch–Davies initial conditions
(2.6). This is the more general form of Eq. (2.19), and reproduces it when
$\nu=3/2$. As both $J_{\nu}$ and $Y_{\nu}$ have power series expansions, the
growing and decaying mode interpretation of Eq. (2.14) applies. One can also
propagate Eq. (C.2) through the sharp transition of Sec. 3, using Eq. (3.9),
to find the post-transition $A_{k}$ and $B_{k}$ in an equivalent manner to
finding $\alpha_{k}$ and $\beta_{k}$. While the computation is more complex,
one again finds $A_{k}$ and $B_{k}$ are different to the pre-transition
values. This shows the conclusions of Sec. 3 are valid beyond the de Sitter
approximation.

Figure 9: Left panel: the curvature perturbation power spectrum at the end of
inflation for a potential with a Gaussian bump (4.1). The solid blue curve is
found by numerically solving the full mode equation Eq. (2.4). The orange
dashed line uses the intermediate step of finding the homogeneous behaviour
using the Bessel functions, see Eqs. (C.5) and (C.6), while the purple dotted
line uses the homogeneous matching (2.27) directly. The matching time,
$\eta_{*}$, is when $\nu^{2}$ (C.1) first becomes constant post-transition
(for modes to the left of the dotted vertical line), or when $k=0.5aH$ for
modes to the right of the dotted vertical line. Right: the percent difference
between the two matching prescriptions and the full numerical result. The
ultra-slow-roll period ($\epsilon_{2}<-3$) starts at
$\eta_{\mathrm{T}}=-1/k_{\mathrm{T}}$.

The solution given in Eq. (C.2) is valid during any period of constant
$\nu^{2}$, as is the case during both slow roll and ultra-slow roll. This is
true even if following a period when $\nu^{2}$ is varying, in a piece-wise
manner. Then, $A_{k}$ and $B_{k}$ can be found by matching onto the full
solution for $v_{k}$ at a time $\eta_{*}$, using

$\begin{split}A_{k}&=\frac{1}{2(-\eta_{*})^{3/2}}\frac{2\eta_{*}Y_{\nu}(-k\eta_{*})v_{k*}^{\prime}-Y_{\nu}(-k\eta_{*})v_{k}-2\eta_{*}[Y_{\nu}(-k\eta_{*})]^{\prime}v_{k*}}{J_{\nu}[Y_{\nu}(-k\eta_{*})]^{\prime}-Y_{\nu}(-k\eta_{*})[J_{\nu}(-k\eta_{*})]^{\prime}}\,,\\\
B_{k}&=-\frac{1}{2(-\eta_{*})^{3/2}}\frac{2\eta_{*}J_{\nu}(-k\eta_{*})v_{k*}^{\prime}-J_{\nu}(-k\eta_{*})v_{k}-2\eta_{*}[J_{\nu}(-k\eta_{*})]^{\prime}v_{k*}}{J_{\nu}[Y_{\nu}(-k\eta_{*})]^{\prime}-Y_{\nu}(-k\eta_{*})[J_{\nu}(-k\eta_{*})]^{\prime}}\,.\end{split}$
(C.4)

This allows the homogeneous matching of Sec. 2.4 to be done more accurately
with the following steps. Assuming no further sudden transitions occur, then
the homogeneous growing and decaying mode coefficients of Eq. (2.12) will be
constant until the end of inflation. This is true even if there is a slow
variation in $\nu^{2}$ later. The leading order expansion of Eq. (C.2) can
then be used, along with $\mathcal{R}=v_{k}/z$ and the homogeneous matching
(2.27), to give

$\hat{C}_{k}=\frac{\sqrt{\eta_{*}}}{z_{*}}\left(a_{k}\eta_{*}^{-\nu}+b_{k}\eta_{*}^{\nu}\right)+\\\
z_{*}\sqrt{\eta_{*}}\left\\{a_{k}\left[\left(\frac{1}{2}-\nu\right)\eta_{*}^{-1}-\frac{z^{\prime}}{z}\right]\eta_{*}^{-\nu}+b_{k}\left[\left(\frac{1}{2}+\nu\right)\eta_{*}^{-1}-\frac{z^{\prime}}{z}\right]\eta_{*}^{\nu}\right\\}\int_{\eta_{*}}^{0}\frac{\mathrm{d}\tilde{\eta}}{z^{2}(\tilde{\eta})}\,,$
(C.5)
$\hat{D}_{k}=-z_{*}\sqrt{\eta_{*}}\left\\{a_{k}\left[\left(\frac{1}{2}-\nu\right)\eta_{*}^{-1}-\frac{z^{\prime}}{z}\right]\eta_{*}^{-\nu}+b_{k}\left[\left(\frac{1}{2}+\nu\right)\eta_{*}^{-1}-\frac{z^{\prime}}{z}\right]\eta_{*}^{\nu}\right\\}\,,$
(C.6)

where

$\begin{split}a_{k}&=-\bigg{[}B_{k}\frac{\Gamma(\nu)}{\pi}\bigg{]}\bigg{(}\frac{-k}{2}\bigg{)}^{-\nu},\\\
b_{k}&=\bigg{[}\frac{A_{k}}{\Gamma(\nu+1)}-B_{k}\frac{\Gamma(-\nu)\cos{(\nu\pi)}}{\pi}\bigg{]}\bigg{(}\frac{-k}{2}\bigg{)}^{\nu}\,.\end{split}$
(C.7)

With this approach, any contamination from gradient terms can be effectively
removed. This improves the accuracy of the method, and allows the post-
transition homogeneous behaviour to be found closer to Hubble exit
(corresponding to a matching scale, $k=\sigma aH$, with larger $\sigma$).

The method detailed above has been applied numerically in Fig. 9 for the
Gaussian bump model, see Sec. 4. The power spectrum at the end of inflation
has been reconstructed to test the accuracy of this method against applying
Eq. (2.27) directly. For modes which would have left the matching scale
($k=0.5aH$) before $\nu^{2}$ becomes constant, the matching has been done at
$\eta_{*}=\eta_{\mathrm{T}}/10$, when $\nu^{2}$ becomes constant post-
transition. The later modes are simply matched when $k=0.5aH$ (i.e.
$\sigma=0.5$). The dotted line shows when the matching time changes. Using
Eqs. (C.5) and (C.6), which effectively removes any gradient terms, produces
excellent agreement with the full numerical result. The error is less than
$0.01\%$ throughout, while directly applying Eq. (2.27) gives a percent level
error. Both have vanishing errors for modes which left the Hubble scale long
before the transition.

Periods of constant $\nu^{2}$ after the slow-roll to ultra-slow-roll
transition are present in many models. This is the case for all potentials
investigated in this paper, as can be seen, using Eq. (C.1), in Fig. 1.
Therefore it is expected that the approach will be applicable in a wide
variety of models.

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