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# Primordial black holes and inflation from double-well potentials

Alexandros Karam  Niko Koivunen  Eemeli Tomberg  Antonio Racioppi  and Hardi
Veermäe

###### Abstract

We investigate the formation of large peaks in the inflationary curvature
power spectrum from double-well potentials. In such scenarios, the initial CMB
spectrum is created at large field values. Subsequently, the inflaton will
cross one of the minima and will decelerate rapidly as it reaches the local
maximum at the origin, either falling back or crossing it. During this final
phase, a significant peak in the curvature power spectrum can be generated.
Our analysis reveals that this class of models produces more pronounced peaks
than the more commonly studied quasi-inflection point scenarios with less
tuning for the model parameters. Finally, we construct an explicit
theoretically motivated inflationary scenario that is consistent with the
latest CMB observations and capable of generating sufficiently large curvature
perturbations for primordial black holes.

##  1 Introduction

The inflationary paradigm [1, 2, 3, 4, 5, 6, 7] can explain several
cosmological conundra as well as provide a quantitative mechanism for
producing the observed primordial fluctuations [8, 9, 10, 11, 12, 13].
However, beyond the scales probed by the cosmic microwave background (CMB),
the primordial curvature power spectrum is essentially unconstrained due to
the lack of empirical data and may contain much stronger fluctuations than the
ones that have been observed in the CMB.

High peaks in the primordial power spectrum can lead to the production of
primordial black holes (PBHs) when the curvature fluctuations re-enter the
cosmological horizon in the early Universe [14, 15, 16]. Although the PBH
abundance is constrained by various observables (for a review see e.g. [17]),
asteroid mass PBHs remain a viable dark matter (DM) candidate [18, 19, 20].
They correspond to an enhanced power spectrum at scales around
$\mathcal{O}(10^{13})\ {\rm Mpc}^{-1}$. Peaked spectra at scales $k<10^{7}\
{\rm Mpc}^{-1}$ are associated with solar mass or heavier PBHs. These may be
related to the BH-BH merger events observed in LIGO-Virgo [21, 22, 23, 24] if
they make up $\mathcal{O}(0.1\%)$ of DM [25, 26, 27, 28, 29, 30, 31, 32]. An
even smaller fraction of heavier PBHs may have served as seeds of supermassive
BHs and cosmic structures [33, 34, 35, 36, 37, 38, 39].

Large peaks in the primordial power spectrum can be produced in a vast array
of inflationary models, with single-field inflation being the simplest (for a
recent overview of single-field scenarios see, e.g., [40, 41]). In this case,
the most widely considered scenarios introduce an inflaton potential that
contains a weak maximum, or a quasi-inflection point, which causes the
inflaton to slow down considerably in an ultra slow-roll (USR) phase as the
field rolls over that feature [42, 43, 44, 45]. Instead of freezing soon after
their Hubble exit, curvature perturbations can undergo super-Hubble growth
during such a phase, excited by the strong potential feature. As a result, the
curvature power spectrum can be greatly enhanced.

A similar—or even stronger—enhancement of the power spectrum can be achieved
in models in which the maximum lies between two global minima, also known as
double-well potentials. These models arise naturally in the context of
spontaneous symmetry breaking, with the best-known example being the Higgs
mechanism of the Standard Model (SM) [46, 47, 48, 49]. In beyond the SM
models, such as scalar extensions of the SM, Grand Unified Theories, etc.,
such potentials appear as a means to generate masses around the scale of new
physics. Coleman–Weinberg potentials arising when symmetry is broken via
quantum corrections [50] have been among the first models proposed for cosmic
inflation [5, 6].

Regarding PBH production from double-well potentials, there have been a
handful of proposals in the literature over the last three decades [51, 52,
53, 54]. In these papers, the considered potentials were of the
Coleman–Weinberg or Higgs-like nature. The disadvantage of these potentials in
their minimal form is that for large field values, they behave as
$V\propto\phi^{4}$ and therefore predict a scalar spectral index value of
$n_{s}<0.95$ and a tensor-to-scalar ratio value of $r=0.33$, both of which are
excluded by existing observations [55, 56]. To guarantee a sufficiently low
tensor-to-scalar ratio value, one should consider potentials that feature a
plateau at high field values.

The inflationary timeline of the models considered in this work consists of
the usual phases in single-field models for PBHs: the CMB spectrum is produced
during an initial slow-roll (SR) phase, which then transitions into a USR
phase causing the inflaton to slow down significantly. The latter is succeeded
by another phase of SR or constant roll (CR) inflation [40]. For double-well
potentials, however, the SR conditions are always maximally violated as the
field rolls through a global minimum during the SR to USR transition. This
leads to a strongly enhanced curvature power spectrum that is qualitatively
quite different from typical quasi-inflection point models: in quasi-
inflection point scenarios the enhanced power spectrum can be understood well
in terms of the duration of the USR phase as well as the Wands duality [57]
between the USR and the subsequent CR phases. In contrast, in scenarios where
the field rolls through the global minimum, the shape of the peak is mostly
dictated by the details of the SR to USR transition — it exhibits enhanced
oscillatory spectral features that can be interpreted in terms of a brief
period of preheating-like particle production. In this paper, we will discuss
the general features of double-well models for inflation and PBHs and
construct an explicit scenario consistent with CMB measurements and asteroid
mass PBH DM.

This paper is structured as follows. In section 2, we outline the general
theoretical considerations that go into building a phenomenologically feasible
model for both PBHs and CMB observables and propose an explicit double-well
model. The implications of our results are discussed in Section 3. We conclude
in Section 4. Natural units $\hbar=c=1$ with $M_{\rm P}=1$ and the $(-+++)$
metric signature are used throughout the paper.

##  2 Inflation with double-well potentials

###  2.1 Theoretical background

In order to construct a double-well model that can accommodate both the CMB
observables and produce PBHs, we start from the general Jordan frame action
for single-field inflation

$\mathcal{S}=\int{{\rm
d}^{4}x\sqrt{-g}\left[\frac{1}{2}\Omega(\phi)R-\frac{1}{2}K(\phi)(\partial\phi)^{2}-U(\phi)\right]}\,,$
(2.1)

where $R$ is the Ricci scalar and $U(\phi)$ is the Jordan frame potential.
When working with inflationary perturbations it is more convenient to work in
the Einstein frame. The Weyl rescaling of the metric $g_{\mu\nu}\to
g_{\mu\nu}/\Omega(\phi)$ and the field redefinition ${\rm d}\varphi/{\rm
d}\phi=\sqrt{K/\Omega+(3/2)(\Omega_{,\phi}/\Omega)^{2}}$ transforms the Jordan
frame action (2.1) to the canonical form (see e.g. [58, 59, 60, 61])

$\mathcal{S}=\int{{\rm
d}^{4}x\sqrt{-g}\quantity[\frac{1}{2}R-\frac{1}{2}(\partial\varphi)^{2}-V]}\,,$
(2.2)

where $V\equiv U/\Omega^{2}$ is the Einstein frame potential.

As a simple realization of symmetry breaking, we will consider models in which
the Jordan frame field $\phi$ is a real scalar singlet with a $\mathbb{Z}_{2}$
symmetry under field reversal $\phi\to-\phi$. This symmetry will carry over to
the Einstein frame if the field redefinition satisfies $\varphi(\phi=0)=0$. To
obtain a double-well potential, we will further assume that the
$\mathbb{Z}_{2}$ symmetry is spontaneously broken, implying that the Einstein
frame potential possesses a maximum at the origin $\phi=0$ and the field
acquires a vacuum expectation value at $\phi=v$.

In the Einstein frame, the homogeneous FRW background evolves according to

$\ddot{\varphi}+3H\dot{\varphi}+V^{\prime}(\varphi)=0\,,\qquad
3H^{2}=\frac{1}{2}\dot{\varphi}^{2}+V(\varphi)\,,$ (2.3)

where $H\equiv\dot{a}/a$ is the Hubble parameter, $a$ is the scale factor
describing the expansion of the universe, and a dot indicates derivative with
respect to the cosmic time $t$. The background evolution is characterized by
the Hubble SR parameters

$\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\varphi}^{2}}{2H^{2}}\,,\qquad\eta_{H}\equiv-\frac{\ddot{H}}{2H\dot{H}}=-\frac{\ddot{\varphi}}{H\dot{\varphi}}\,.$
(2.4)

Inflation corresponds to $\epsilon_{H}<1$, and SR inflation to
$\epsilon_{H}\,,\,|\eta_{H}|\ll 1$.

Linear quantum fluctuations on top of the homogeneous background are described
by the Mukhanov–Sasaki variable $u$ that obeys the Fourier space equations of
motion [62, 63, 64]

$u_{k}^{\prime\prime}+\quantity(k^{2}-\frac{z^{\prime\prime}}{z})u_{k}=0\,,\qquad
z\equiv a\frac{\dot{\varphi}}{H}\,.$ (2.5)

Here $k$ is the wavenumber and a prime denotes a derivative with respect to
the conformal time $\tau$, with ${\rm d}t=a{\rm d}\tau$. The Mukhanov–Sasaki
variable gives the curvature power spectrum

$\mathcal{P}_{\mathcal{R}}=\frac{k^{3}}{2\pi^{2}}\frac{|u_{k}|^{2}}{z^{2}}\,,$
(2.6)

which freezes to a time-independent shape by the end of inflation. The frozen
power spectrum is the standard tool to describe primordial perturbations,
which, at large scales, are imprinted on the CMB. At the CMB pivot scale,
$k_{*}=0.05\mathrm{Mpc}^{-1}$, observations give [65, 56]

$A_{s}\equiv\mathcal{P}_{\mathcal{R}}(k_{*})=2.1\times 10^{-9}\,,\quad
n_{s}\equiv 1+\frac{{\rm d}\mathcal{P}_{\mathcal{R}}(k_{*})}{{\rm d}\ln
k}=0.9649\pm 0.0042\,,\quad r<0.036\,.$ (2.7)

Here $r$ is the tensor-to-scalar ratio, describing the strength of primordial
gravitational waves. The CMB observations guide model building. For an SR
model, the observables in terms of the Hubble SR parameters are given as111For
completeness, we also report the spectral parameters during CR [66]
$\displaystyle\mathcal{P}_{\mathcal{R}}$
$\displaystyle=\frac{\Gamma(3/2-\eta_{H})^{2}}{2^{1-2\eta_{H}}\pi^{3}}\frac{H^{2}}{\epsilon_{H}}\,,\qquad
n_{s}=4-\Bigl{\lvert}3+2\eta_{H}\Bigr{\rvert}\,,\qquad r=16Q\epsilon_{H}\,,$
(2.8) where $Q\equiv 2^{3-\lvert
3-2\eta_{H}\rvert}\Gamma\left(3/2\right)^{2}/\Gamma\left(\lvert
3-2\eta_{H}\rvert/2\right)^{2}$ and we neglected next-to-leading order terms
in $\epsilon_{H}$ as they are subleading when
$\epsilon_{H}\ll\eta_{H}=\mathcal{O}(1)$.

$A_{s}=\frac{H^{2}}{8\pi^{2}\epsilon_{H}}\,,\qquad
n_{s}=1+2\eta_{H}-4\epsilon_{H}\,,\qquad r=16\epsilon_{H}\,.$ (2.9)

For CMB, these quantities are to be evaluated when the pivot scale exits the
Hubble radius, $k_{*}=aH$. In typical models, the crossing happens 50–60
$e$-folds before the end of inflation.

###  2.2 Peaks in $\mathcal{P}_{\mathcal{R}}$ and primordial black holes

For the purposes of our paper, it is sufficient to adapt a simplified order-
of-magnitude approach to PBH formation. Critical collapse [67, 68, 69, 70, 71,
72, 73] of density fluctuations exceeding a critical threshold during the
radiation-dominated era in the early Universe yields a PBH whose mass is
comparable to the horizon mass

$M_{k}\approx 2.8\times 10^{18}\ {\rm g}\,\left(\frac{k}{10^{14}\ {\rm
Mpc}^{-1}}\right)^{-2}\,,$ (2.10)

while a non-negligible PBH abundance can be produced if the curvature power
spectrum contains a peak that is of the order

$\mathcal{P}_{\mathcal{R}}(k)\approx\mathcal{O}(10^{-2})\,.$ (2.11)

We will not attempt to make precise predictions of the PBH abundance and
instead aim for constructing scenarios that produce spectral features with
such a height. This is justified considering that the height of the peak is
quite sensitive to the model parameters and can therefore be tuned by a factor
of a few without changing the other predictions, for instance, the CMB power
spectrum. This freedom can be used to “correct” the height in order to obtain
any desired PBH abundance, which itself is exponentially sensitive to the
height of the power spectrum peak. Moreover, the PBH abundance estimates
contain potential theoretical uncertainties related to the shape of the
spectrum [71] and to non-Gaussianities, e.g., in the form of exponential tails
[74, 75, 76, 77, 78, 79, 80].

The position of the peak is roughly determined by the duration of the first
phase of SR inflation during which the CMB spectrum is generated. In the
double-well models considered here, inflation always briefly stops as the
field rolls through the global minimum. We will denote the duration of the
first inflationary phase in $e$-folds by $N_{\rm I}$. To produce PBHs, the
inflaton must typically slow down during a USR-like phase and enter a second
phase of SR (or CR) inflation. The duration of the second phase of inflation
will be denoted by $N_{\rm II}$. It includes the duration of the USR phase.

We will focus on models that would produce PBHs in the asteroid mass window
$10^{17}-10^{22}\ \rm g$ [81, 18], where they can constitute all DM without
clashing with existing constraints. This window corresponds to comoving scales
in the range $k=(5\times 10^{12}-5\times 10^{14})\ {\rm Mpc}^{-1}$ which exit
the Hubble horizon $31-37$ $e$-folds after the CMB pivot scale. Assuming a
typical total of 55 $e$-folds of inflation, we thus require an additional
$18-24$ $e$-folds between the end of inflation and the creation of the
enhanced feature in the power spectrum. All in all, we are looking for models
with

$N_{\rm I}\in(31,37)\,,\qquad N_{\rm II}\approx 55-N_{\rm I}\in(18,24)\,.$
(2.12)

The second phase of inflation can, however, be much shorter in the case where
the total duration of inflation is small, for instance, due to a low scale of
inflation or modified early universe cosmologies.

###  2.3 An explicit model

Before proposing an explicit model, let us consider the general criteria it
must satisfy. We are looking for double-well potentials that generate the CMB
at large field values and PBHs around the origin.

To guarantee consistency with the CMB, potentials with a plateau at large
field values are preferred. A simple way to flatten a double-well potential
with chaotic behaviour for large field values is to assume that the inflaton
field is non-minimally coupled to gravity and/or it has a non-canonical
kinetic term of the $\alpha$-attractor type [60, 82]. In such a case, the
transformation to the Einstein frame will result in a potential that is
asymptotically flat for large positive and negative field values, while
retaining the local maximum at the origin and the two global minima on each
side of it.

In a setup like this, the field first experiences SR on one of the plateaus
for $N_{\rm I}$ $e$-folds, and then inflation halts while the field crosses
the minimum. Afterward, field evolution can proceed in three different ways:

  1. 1.

Roll-back: The field does not have enough energy to reach the origin and it
turns around. Inflation starts again shortly after the field has passed the
closest minimum; this second stage of inflation starts in USR and transitions
into its dual SR/CR phase. Inflation ends before the field returns to the
minimum.

  2. 2.

Roll-over: The inflaton has enough energy to cross the hilltop, but it slows
down so much that inflation restarts near the origin. The field crosses the
potential hill in USR inflation and transitions into a dual SR/CR phase as it
rolls down. Inflation stops again before the field reaches the second minimum.

  3. 3.

Fast roll-over: With a lot of energy, the inflaton quickly rolls over the
hilltop at $\phi=0$, not causing the universe to inflate. It starts to
oscillate in the second minimum on the opposite side and the universe reheats.

Production of PBHs is possible only in the first two cases in which inflation
restarts at the end of a USR period and lasts for $N_{\rm II}$ $e$-folds. On
the boundary between these cases, the classical field equations allow the
inflaton to stop completely on top of the hill, corresponding to the $N_{\rm
II}\to\infty$ limit. Tuning the model parameters close to this critical point
lets us tailor the number of $e$-folds spent near the origin. This period
comprises both the USR phase and its dual SR/CR phase as was recently
discussed in Ref. [40] (see also [57]).

Figure 1: _Left panel:_ Einstein frame potential corresponding to parameter
values $\lambda=1.52537\times 10^{-8}$, $\xi=0.01$, $v=0.3851983$ and
$\alpha=2v^{2}/3$. The value of $v$ is tuned to be close to the roll-
over/roll-back boundary and corresponds to a roll-over scenario. The green and
yellow bands correspond to the USR and temporarily halted inflation,
respectively. The orange star corresponds to the CMB scale. _Right panel:_
Zoomed-in potential near the origin which represents two cases: roll-over and
roll-back. The blue and red dots are the end of the second phase of inflation
for roll-over and roll-back, respectively. Note that while the potential and
blue dot corresponds to $v=0.3851983$, the red dot corresponds to
$v=0.3854377$. The shape of the potential between these two different
parameter values is indistinguishable by eye, and we present them in the same
figure for ease of illustration.

The in-between period, when inflation momentarily ceases, is expected to be
negligible in duration in most cases. However, as we will show shortly, with
double-well potentials, the peak in the curvature spectrum depends on the
features of this SR to USR transition and fluctuations can be enhanced by
effects similar to parametric resonance during preheating.

During the last SR/CR phase, one can approximate the Hubble SR parameters with
the potential ones and apply Eq. (2.8) to estimate the power spectrum.222In
principle, the duality between the USR and SR/CR allows us to extrapolate the
spectral features computed in the SR/CR into the preceding USR phase. However,
this is not needed here as the USR has only a mild impact on the full
spectrum. For this, we look at the behaviour of the inflaton close to the
origin at $\phi=0$. Given a $\mathbb{Z}_{2}$ symmetry, the action (2.1) is
expected to depend on $\phi$ through only $\phi^{2}$ and thus the linear terms
vanish in the small-field expansion, for which we assume the form
$\Omega=1+\xi\phi^{2}+\mathcal{O}(\phi^{4})$, $K=1+\mathcal{O}(\phi^{2})$, and
$U=U_{0}-m^{2}\phi^{2}/2+\mathcal{O}(\phi^{4})$. Working in the Einstein
frame, the relevant quantity in these models is the second SR parameter during
the second inflationary phase,

$\eta_{H,\rm
II}\approx\left.\frac{V^{\prime\prime}}{V}\right|_{\phi=0}=-\frac{m^{2}}{U_{0}}-4\xi\,,$
(2.13)

which determines the slope of the spectrum. Although $\eta_{H}\gtrsim 1$ is
theoretically allowed during CR, an extended second inflationary phase $N_{\rm
II}\approx\mathcal{O}(20)$ typically requires $\eta_{H}\lesssim 1$. This
implies $\xi\lesssim 1$ and $m^{2}\lesssim U_{0}$. The last inequality is
especially constraining for a tree-level Higgs mechanism,
$U=\lambda(\phi^{2}-v^{2})^{2}/4$, for which $m^{2}/U_{0}=2/v^{2}$ indicating
the need for super-Planckian vacuum expectation values ($v\gtrsim 1$) for
reasonably small values of $\eta_{H}$.333We tested this claim explicitly by
computing the curvature power spectra for a non-minimally coupled Higgs-like
field with $U=\lambda(\phi^{2}-v^{2})^{2}/4$ and found that PBH production
together with a consistent CMB was not possible in any region of the parameter
space.

Figure 2: Example power spectra in the roll-over (left) and roll-back (right)
cases with $n_{s}=0.9649$, $\lambda=1.52537\times 10^{-8}$ and
$\alpha=2v^{2}/3$. The solid curve is the exact numerical curvature power
spectrum. The dashed and dotted curves show power spectra computed using the
SR approximation (2.9), and the analytical approximation developed for quasi-
inflection point models (3.1), respectively. The green band shows the USR
epoch. The yellow band indicates modes that temporarily re-enter the horizon
when inflation is halted and exit again when inflation re-starts. The dashed
horizontal line at $\mathcal{P}_{\mathcal{R}}=10^{-2}$ indicates efficient PBH
production. The red dashed line shows CMB observations [65] and the green
dashed line denotes the constraint from $\mu$-distortions [83].

Based on the above reasoning, we will not rely on a non-minimal coupling to
generate a plateau at large field values and assume that the symmetry breaking
is due to quantum corrections, thus avoiding a large mass around the origin.
We will consider a general class of non-minimally coupled $\alpha$-attractor
models with a Coleman–Weinberg-like potential [5, 6], characterized by

$\Omega(\phi)=1+\xi\phi^{2}\,,\quad
K(\phi)=\quantity(1-\frac{\phi^{2}}{6\alpha})^{-n}\,,\quad
U(\phi)=\frac{\lambda}{4}\left[\frac{1}{2}\phi^{4}\left(\ln\frac{\phi^{2}}{v^{2}}-\frac{1}{2}\right)+\frac{v^{4}}{4}\right]\,.$
(2.14)

Near $\phi=0$, the Coleman–Weinberg-like potential $U$ introduces a double
well with minima at $\phi=\pm v$. Near $|\phi|=\sqrt{6\alpha}$, the non-
canonical kinetic factor $K$ makes the derivative ${\rm d}\varphi/{\rm d}\phi$
diverge, so the Einstein frame potential $V(\varphi)$ develops plateaus
suitable for SR inflation. The same mechanism is used in $\alpha$-attractor
models with $n=2$ [60, 82]. However, to obtain a sufficiently high $n_{s}$ at
the CMB scales, we will choose $n=4$. This choice will be discussed below in
more detail. We require $v<\sqrt{6\alpha}$ to retain both the double-well and
the plateau. The non-minimal coupling $\Omega$ tunes the relative heights of
the hilltop and the plateaus, but, as we will show later, its contribution is
rather minor for scenarios in which PBHs can be produced. An example of the
Einstein frame potential for the model (2.14) is plotted in Fig. 1.

To study the cosmological predictions of the model in the interesting cases of
roll-over and roll-back, we solve the background field equations (2.3)
numerically, starting from parameter values $\xi$, $\alpha$, $\lambda$, and
$v$. We then solve the Mukhanov–Sasaki equation (2.5) for multiple $k$-modes
to obtain the power spectrum (see Appendix A for details). In all of our
examples, we set the CMB pivot scale on the plateau at a point that produces a
desirable $n_{s}$ according to (2.9). We then fix $\lambda$ by requiring the
correct power spectrum normalization $A_{s}$ at this scale. We tune $v$ to fix
the duration of the second inflationary phase $N_{\rm II}$.

Figure 3: A scan over the $(\xi,v)$-space with $\alpha=2v^{2}/3$ _(upper panels)_ and $\alpha=4v^{2}/3$ _(lower panels)_. Both green-shaded regions stand for roll-back; in the lighter green region, the hilltop is higher than the plateau, preventing roll-over by a simple energy conservation argument. The blue region stands for roll-over and lighter blue for fast roll-over. PBH production can take place on the black curve between the roll-back and roll-over regions. The orange curves give the number of $e$-folds of the first phase of inflation, starting from the CMB. The three panels correspond to three values of $n_{s}$: from left to right, the central value $n_{s}=0.9649$, the $1\sigma$ value $n_{s}=0.96072$, and the $2\sigma$ value $n_{s}=0.95652$. The black dots correspond to benchmark points in Table 1. | $\xi$ |  | $v$ | $r$ | $N_{\rm I}(1\sigma)$ | $\mathcal{P}_{\mathcal{R},{\rm max}}$
---|---|---|---|---|---|---
A | $0$ | RO | $0.386{\color[rgb]{0.5,0.5,0.5}2076}$ | $0.01573$ | $37.0$ | $1.7\times 10^{-2}$
|  | RB | $0.386{\color[rgb]{0.5,0.5,0.5}5523}$ | $0.01574$ | $37.0$ | $3.5\times 10^{-2}$
B | $10^{-2}$ | RO | $0.385{\color[rgb]{0.5,0.5,0.5}1983}$ | $0.01574$ | $37.0$ | $1.5\times 10^{-2}$
|  | RB | $0.385{\color[rgb]{0.5,0.5,0.5}4377}$ | $0.01563$ | $37.0$ | $2.4\times 10^{-2}$
C | $10^{-1.5}$ | RO | $0.38{\color[rgb]{0.5,0.5,0.5}299285}$ | $0.01544$ | $37.0$ | $2.4\times 10^{-2}$
|  | RB | $0.38{\color[rgb]{0.5,0.5,0.5}30859}$ | $0.01544$ | $37.0$ | $3.0\times 10^{-2}$
D | $10^{-1}$ | RO | $0.3760{\color[rgb]{0.5,0.5,0.5}49595}$ | $0.01490$ | $36.9$ | $3.7$
|  | RB | $0.3760{\color[rgb]{0.5,0.5,0.5}524}$ | $0.01490$ | $36.9$ | $3.7$
E | $1$ | RO | $0.30842795518270935781{\color[rgb]{0.5,0.5,0.5}585}$ | $0.01091820$ | $36.5$ | $1.7\times 10^{29}$
|  | RB | $0.30842795518270935781{\color[rgb]{0.5,0.5,0.5}915}$ | $0.0109182097$ | $36.5$ | $5.9\times 10^{28}$
Table 1: Benchmark points for $\alpha=2v^{2}/3$ for roll-over (RO) and roll-
back (RB), respectively. The $\mathcal{P}_{\mathcal{R},{\rm max}}$ is the
maximum value in the peak of the power spectrum and $k_{\rm max}$ is the
location of the peak. The duration of the first phase of inflation, $N_{\rm
I}$, is determined by demanding that $n_{s}=0.96072$, which lies $1\sigma$
below the central value. The vacuum expectation value $v$ is tuned to produce
$N_{\rm II}\approx 19$. The last digits in $v$ differing in the roll-over and
roll-back cases are highlighted in gray. At all points, we obtain
$M_{k}\approx 10^{17}\ \rm g$ and $k_{\rm max}\approx 5\times 10^{14}\ {\rm
Mpc}^{-1}$.

Figure 2 shows example power spectra in the roll-over and roll-back cases.
After the low-$k$ SR plateau, the roll-over spectrum dips down, analogously to
the widely studied quasi-inflection point models [40]. Interestingly, the
roll-back spectrum shows no such dip, which is the main difference between the
two cases. Around the intermediate non-inflationary period, the power spectrum
then grows in both cases, spurred on by the violent background behaviour — the
momentary end of inflation and the subsequent USR and CR phases. A high peak
is followed by a number of oscillations and a longer plateau.

Figure 3 shows scans over the parameter space for $\alpha=2v^{2}/3$ and
$\alpha=4v^{2}/3$ and varied $\xi$ and $v$. The spectral tilt $n_{s}$ at the
CMB scales is varied between the three panels from the central value in the
first panel to values lower by $1\sigma$ and $2\sigma$, in the second and
third panels, respectively. The coloured regions correspond to the different
behaviours 1–3, with the blue shading showing models in which the inflaton can
roll over the maximum and the green shading denoting when the field rolls
back. Increasing $\xi$ decreases the height of the plateau. In the light green
region, the plateau becomes lower than the hilltop, and only the roll-back
case is possible. Decreasing $v$ lowers the local maximum at $\phi=0$ and in
the light-blue region, the field can roll over it without ever initiating a
second phase of inflation. PBHs can be produced in a fine-tuned region along
the black line that separates the roll-back and roll-over regions.

Fixing $N_{\rm II}\approx 19$ $e$-folds, singles out one roll-back and one
roll-over point for a given $\xi$. Table 1 lists such benchmark points, with
$n_{s}$ chosen to lie $1\sigma$ below the central value. We see that small
values of $\xi$ produce the desirable $\mathcal{P}_{\mathcal{R}}\sim 10^{-2}$,
with the horizon mass at the edge of the asteroid mass window. For low
$\xi\lesssim 0.1$, the background evolution, as well as the curvature power
spectrum, is nearly independent of $\xi$ and thus matches roughly the one for
minimally coupled models. On the other hand, a significant enhancement in
$\mathcal{P}_{\mathcal{R}}$ can be observed when $\xi\gtrsim 0.1$. This is
partly because, by Eq. (2.13), $\eta_{H,\rm II}\approx-4\xi$ and the spectrum
has a large slope also during the last inflationary CR period, ${\it i.e.}$,
after the peak. For a fixed $N_{\rm II}$, this can further enhance the height
of the peak and lead to unphysically large fluctuations as exemplified by case
E in Table 1.

PBHs may be produced in the asteroid mass window when $N_{\rm I}\lesssim 37$.
Models with a lower $n_{s}$ allow for a shorter initial period of inflation,
as can be seen from Fig. 3. For the central value $n_{s}=0.9649$, we see that
$N_{\rm I}>37$ for all of the parameter space and thus the produced PBHs would
be too light and evaporate. For $n_{s}=0.9607$, $1\sigma$ below the central
value, we observe that $N_{\rm I}>37$ is attainable for a sufficiently large
$\xi$. The lower bound on $\xi$ depends on $\alpha$. With $n_{s}$ lying
$2\sigma$ below the central value, $N_{\rm I}<37$ is realized in the entire
parameter space we considered. Moreover, as in quasi-inflection point models,
accounting for the running of $n_{s}$ can permit lower values of $n_{s}$, thus
relaxing the tension with the CMB observations [65, 84].

We note that low values of $n_{s}$ are a typical problem in PBH plateau
models, see e.g. [45, 85, 86, 87, 88, 89]: $\alpha$-attractor models predict
the correct $n_{s}$ when the main phase of inflation lasts for $50$ $e$-folds
and reducing this to our $30$ throws these predictions off. In equation
(2.14), we combated this by choosing the $K$-exponent $n=4$ instead of the
conventional $n=2$, making $n_{s}$ larger. With this modification, we found
workable solutions within the $2\sigma$ uncertainty bounds, if not with the
central value.

The results are qualitatively similar for both of the $\alpha$ values we
considered. Lowering $\alpha/v^{2}$ while keeping $\xi$ fixed raises the peak
of $\mathcal{P}_{\mathcal{R}}$, and below $\alpha=2v^{2}/3$, asteroid-mass
PBHs are always overproduced. On the other hand, for $\alpha=4v^{2}/3$, $\xi$
needs to be larger than in the case of $\alpha=2v^{2}/3$ to obtain the same
$n_{s}$ and $N_{\rm II}$, indicating a larger $\mathcal{P}_{\mathcal{R}}$
peak.

Figure 4: The maximal value of the power spectrum
$\mathcal{P}_{\mathcal{R},\text{max}}$ as a function of $v$ for $\xi=0.01$.
The upper $x$-axis denotes the corresponding duration of the second stage of
inflation in $e$-folds. The inflaton stops at the hilltop for $v\approx
0.385325$. Roll-over (RO) and roll-back (RB) are denoted in red and blue,
respectively.

PBH-producing inflationary models typically suffer from fine-tuning (see [90]
for a recent discussion), the need to set the model parameters with great
accuracy to produce a high enough peak in the power spectrum. We illustrate
this in Fig. 4, where we display the peak value of $\mathcal{P}_{\mathcal{R}}$
as a function of the $v$ and $N_{\rm II}$. As $v$ approaches the critical
value $v\approx 0.385325$ in which the classical field can be balanced exactly
on top of the local maximum, both $N_{\rm II}$ and
$\mathcal{P}_{\mathcal{R},\rm max}$ approach infinity. However, the
sensitivity is not exponential; we find

$\mathcal{P}_{\mathcal{R},\rm max}\approx 5\times
10^{-16}|v/v_{c}-1|^{-3.9}\,$ (2.15)

when $N_{\rm II}<20$, while the power-law behaviour is slightly milder for
larger $N_{\rm II}$. All in all, to obtain $\mathcal{P}_{\mathcal{R},\rm
max}\approx\mathcal{O}(10^{-2})$ we would need a fine-tuning of the order
$v/v_{c}-1\approx 10^{-3.4}$, that is, at the level of the 3rd or 4th
significant digit. This is less than in typical inflection-point models [90].
In the same vein, Table 1 lists the tuned parameter values $v$ in example
roll-over and roll-back cases with common digits colored in gray. These
correspond to high peaks; for values in-between, the peaks are even higher.
Tuning $v$ up to 4 digits is enough to guarantee abundant PBH production.

##  3 Discussion

###  3.1 Properties of the power spectrum

To understand the shape of the power spectra of Fig. 2, we compare our results
to an analytical approximation proposed in [40], which was almost exact for
potentials built out of two downward-opening parabolas; the field starts by
rolling down the side of one parabola in SR, then crosses the local minimum
where the parabolas meet, and finally rolls over the local maximum of the
other parabola in subsequent USR and CR phases. The $z^{\prime\prime}/z$ term
in the Mukhanov–Sasaki equation (2.5) consists of two plateaus, one for the SR
phase and another for the dual USR/CR phases, with a delta function transition
in-between. The result was a peaked, oscillating power spectrum, given by
equation (4.3) of [40]:

$\mathcal{P}_{\mathcal{R}}(k)\approx\left|\theta(\mathcal{H}_{c}-k)\left[\frac{H^{2}}{2\pi\dot{\phi}}\right]_{\mathcal{H}=k}+\frac{k\,p(k/\mathcal{H}_{c},\lambda_{2})}{k_{\rm
CR}|p(k_{\rm
CR}/k_{c},\lambda_{2})|}\left[\frac{\Gamma(\lambda_{2})}{2^{3/2-\lambda_{2}}\pi^{3/2}}\frac{H^{2}}{\dot{\phi}}\right]_{k=k_{\rm
CR}}\right|^{2}\,,$ (3.1)

where the first term corresponds to SR and the second one to the USR and CR
regimes. Here

$p(\kappa,\lambda_{2})=e^{i\kappa}\left(\frac{\kappa}{2}\right)^{1-\lambda_{2}}\left(-J_{\lambda_{2}}(\kappa)+\left(i-\frac{1}{\kappa}\right)J_{\lambda_{2}+1}(\kappa)\right)\,,\quad\lambda_{2}=\frac{3}{2}\sqrt{1-\frac{4}{3}\eta_{V,\rm
II}}\,,$ (3.2)

$k_{\rm CR}$ corresponds to some scale in the CR regime, $\mathcal{H}_{c}$
corresponds to the beginning of USR, $\mathcal{H}\equiv aH$, and
$\eta_{V}\equiv V^{\prime\prime}/V$, with $\eta_{V,\rm II}$ evaluated at the
potential’s local maximum.

Figure 5: _Left panel:_ The behaviour of $(aH)^{-2}z^{\prime\prime}/z$ in the
cases of roll-over (RO) and roll-back (RB), corresponding to point $B$ in
Table 1. The yellow and green bands represent the temporarily halted inflation
and USR, respectively. The black horizontal line is at $2$, corresponding to
the value in SR. _Right panel:_ The behaviour of Hubble SR parameters
$\epsilon_{H}$ and $\eta_{H}$ for roll-over and roll-back. In the case of
roll-back, the field velocity becomes zero at the end of USR and changes sign
as the field turns around near the origin. This is reflected in the behaviour
of $\eta_{H}$: it first goes to positive infinity and then to negative
infinity as the field velocity changes sign.

The setup of this paper is close to that of [40], therefore it makes sense to
compare our power spectra to Eq. (3.1), with $\eta_{V,\rm II}=-4\xi$ by Eq.
(2.13). The comparison is made in Fig. 2. Both spectra exhibit oscillations
followed by a CR plateau, and the plateau heights match, although our spectrum
soon turns down to follow the SR approximation. Notably, the leading
oscillation peak is much stronger in our case than in the approximation,
completely overwhelming the rest of the spectrum. The difference is a
consequence of the momentary halting of inflation, absent in the simplified
model of Ref. [40]. Figure 5 shows $z^{\prime\prime}/z$ for our model, and it
still exhibits the SR and USR/CR plateaus as in [40], but the behaviour in-
between, when inflation momentarily ends, is complicated. In addition, some
modes near the peak cross the Hubble radius multiple times: they exit in the
initial phase of inflation, re-enter when inflation momentarily ceases, and
exit again after inflation continues. To resolve these modes correctly, we
must start tracking their behaviour before the first exit when they initially
leave the Bunch–Davies vacuum. The details are discussed in Appendix A. The
complicated behaviour strongly excites the Fourier modes, producing the peak.

Note that in [40], the oscillations in the spectrum were a consequence of the
artificial sharp features in the approximation; in our case, the end of
inflation naturally produces sharp features even with a smooth potential. The
features arise from the high mass scale that dominates the time evolution when
the field crosses the bottom of the potential. The spectral oscillations are
analogous to excitations in preheating, where the field oscillates around its
minimum after inflation and the oscillations induce particle production in
periodic resonance bands, see e.g. [91, 92, 93].

Even though (3.1) fails to produce the strong peak, it qualitatively captures
another feature of the spectrum: the presence or absence of a dip at the
SR–USR boundary. Looking at (3.1), the field velocities $\dot{\phi}$ in the
two terms have the same sign in roll-over, but different signs in roll-back,
since there the field changes direction before entering the CR regime. The
roll-over case then experiences destructive interference from the two terms,
producing a dip, while such behaviour is absent in roll-back, and thus no dip
is formed. A dip could be detectable in future $\mu$-distortion experiments
[94] or 21 cm observations [95]. In light of our results, such observations
could distinguish roll-over and roll-back-type models from each other.
However, loop corrections have been shown to significantly reduce the depth of
the dip [96].

Recently, the suggestion that one-loop corrections may rule out single-field
inflationary scenarios for PBHs [97] has sparked an active debate in the
literature [98, 99, 100, 101, 97, 102, 103, 104, 105, 96]. The most recent
study [96] showed that although appreciable one-loop corrections can be
generated, they are not in conflict with the CMB in models producing a
significant PBH abundance. The general claim is that, assuming single-field
inflation, an enhanced peak in the power spectrum at small scales resulting
from an extended period of USR will induce large corrections at the CMB
scales. The duration of USR is related to the heights of power spectra during
the first and second inflationary SR/CR periods. In the double-well scenarios,
the duration of the USR period can be significantly shorter than in the quasi-
inflection point ones, suggesting that the one-loop corrections may be
diminished when compared to the quasi-inflection point scenarios. However, a
dedicated study would be required to confirm this claim.

In addition to the case of a single roll-over or roll-back, our scenario can
be expanded to multiple roll-overs and roll-backs in a landscape of potential
minima. One can envisage, for example, the field rolling over both of the
minima of a double-well potential relatively unhindered, and entering a roll-
back phase on the subsequent slope on the other side of the potential, or even
continuing from there and undergoing a roll-over phase on its way back, when
crossing the potential maximum a second time. A case like this was considered
in [52]. In principle, it is possible to realize multiple USR–CR periods this
way, producing multiple peaks in the power spectrum. Indeed, multiple roll-
back-like phases happen in usual reheating when the field oscillates around a
minimum, switching direction periodically. Usually, no discernable amount of
inflation takes place between these; stretching the produced perturbations to
super-Hubble scales takes additional tuning.

Finally, the strong power spectrum peak will induce a stochastic GW background
(SGWB) at the second order of perturbation theory so that all the scenarios
mentioned above can be tested by future GW experiments [106, 107, 108, 109,
110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 40]. LISA [120] and AEDGE
[121, 122] may probe the SGWB generated in asteroid mass PBH scenarios, while
pulsar timing arrays (PTAs) can test the curvature fluctuations associated
with solar mass PBHs. In fact, the stochastic common-spectrum process recently
detected by pulsar timing experiments [123, 124, 125, 126] may be interpreted
as such a primordial SGWB induced by large scalar perturbations [127, 128,
129, 130, 131].

###  3.2 Comparison to previous studies

Early studies of PBH production from double-well potentials include [51, 52,
53]. They obtained a similar shape for the power spectrum, with a strong peak
followed by rapid oscillations and a plateau. However, the models of [51, 52,
53] do not reproduce the CMB observations (2.7). We have presented a model
where this is possible and concentrated on PBHs in the asteroid mass window.
We have also discussed the general properties of such models based on recent
developments in inflationary model building and studied the power spectrum of
the roll-back case, absent from [51, 52, 53].

A model similar to ours with roll-back was constructed in [132] by stitching
together two different potentials on two sides of a single minimum. Strong,
PBH-producing perturbations were produced after the inflaton had crossed the
minimum and turned back. In [133], a more physical model was constructed with
a smooth potential and a non-canonical kinetic term featuring a double pole.
The potential in terms of the canonical field has two plateaus at different
heights with a minimum in-between. Again, strong perturbations were generated,
but the model did not provide the correct CMB predictions. Both [132] and
[133] obtain power spectra with no dip before the peak, consistent with our
results. They also obtain high-frequency oscillations in the tail of the peak.
However, in our case, see Fig. 2, the leading oscillation is much more
pronounced than in [132], [133], and the subsequent oscillations die down more
quickly and do not exhibit similar deep dips.

The recent paper [54] discusses a setup similar to ours with a double-well
potential. The authors go beyond our linear analysis by using stochastic
inflation, a computational method that presents corrections to the Gaussian
PBH abundance. However, the linear perturbation spectrum is still needed as an
input for the stochastic calculation. The authors of [54] use a simple
expression for the spectrum, capturing the (approximately) constant slope
after the leading peak in our Fig. 2, but not the peak itself or the cutoff at
lower $k$. The model of [54] also restricts their analysis to very light PBHs,
and they do not try to fit the CMB. We leave the stochastic analysis of our
model for future work.

##  4 Conclusions

We examined the formation of large peaks in the curvature power spectrum from
inflation driven by double-well potentials with a $\mathbb{Z}_{2}$ symmetry.
After the initial inflationary phase, the rolling inflaton crosses one of the
minima and slows down rapidly around the local maximum, creating a peak in the
curvature power spectrum. Then the inflaton either has enough energy to cross
the hilltop, or it does not have enough energy to reach the origin and it
turns around. In both cases, an ultra-slow-roll phase ensues followed by a
slow-roll/constant-roll phase dual to it. The main difference between rolling
over or turning around is that the roll-back spectrum displays no dip before
the peak, which may be detected by future $\mu$-distortion experiments or 21
cm observations.

We constructed an explicit double-well scenario that is consistent with the
latest CMB observations and capable of generating sufficiently large curvature
perturbations for primordial black holes. The peaks in the curvature power
spectra are more prominent than those in the commonly studied quasi-inflection
point scenarios and require less tuning of the model parameters. The enhanced
peak arises from the temporary cessation of inflation as the field traverses
the minimum of the potential. Our results suggest that double-well potentials
offer a promising avenue for further investigation and could provide important
insights into primordial black hole phenomenology.

Finally, we note that double-well scenarios do not necessarily rely on a
notable period of ultra-slow-roll to generate a significant primordial black
hole abundance. Thus we expect them to provide an example of single-field
inflation for PBHs for which one-loop corrections are suppressed when compared
to quasi-inflection point scenarios.

## Acknowledgments

We thank Antonio Iovino for useful discussions. This work was supported by the
Estonian Research Council grants PSG86, PRG1055, PRG1677, PSG761 and by the EU
through the European Regional Development Fund CoE program TK133 “The Dark
Side of the Universe”.

##  Appendix A Solving primordial perturbations

The evolution of scalar curvature perturbation $\mathcal{R}_{c}$ obeys the
Mukhanov–Sasaki equation. It is traditionally cast into a convenient form
using the Mukhanov field $u\equiv z\mathcal{R}_{c}$, where

$z\equiv\frac{a\dot{\phi}}{H}\,.$ (A.1)

The Fourier modes of the Mukhanov variable evolve according to the
Mukhanov–Sasaki equation [62, 63, 64]:

$u_{k}^{\prime\prime}+\left(k^{2}-\frac{z^{\prime\prime}}{z}\right)u_{k}=0\,,$
(A.2)

where the prime denotes differentiation with respect to conformal time,
$d\tau=dt/a$.

In the sub-Hubble regime, $aH\ll k$, the curvature of space-time is negligible
and the space-time is essentially Minkowski. A given mode starts its evolution
deep inside the horizon. This leads to Bunch–Davies initial conditions

$u^{\prime}_{k}=-iku_{k}\,,\quad|u_{k}|=\frac{1}{\sqrt{2k}}$ (A.3)

at $\tau\to-\infty$. The power spectrum $\mathcal{P}_{\mathcal{R}}$ is given
by

$\mathcal{P}_{\mathcal{R}}=\frac{k^{3}}{2\pi^{2}}\frac{|u_{k}|^{2}}{z^{2}}\,,$
(A.4)

evaluated at late times.

We solve the Mukhanov–Sasaki equation numerically. For the numerical
implementation, it is convenient to introduce a new variable, closely related
to the Mukhanov variable $u$ [85],

$g_{k}\equiv\frac{u_{k}e^{ik\tau}}{z}\,.$ (A.5)

The advantage of this variable is that the multiplication with $e^{ik\tau}$
cancels the rapid early oscillations in $u_{k}$. The $g_{k}$ is also not
subject to as large changes as $u_{k}$ in its order of magnitude due to the
division by $z$. The Bunch–Davies vacuum condition in terms of $g_{k}$ is

$\dot{g}_{k}=-\frac{\dot{z}}{z}g_{k}\,,\quad|g_{k}|=\frac{1}{\sqrt{2k}|z|}\,,$
(A.6)

and for mode $k$ we start our numerical evolution at time $t$, defined by
$a(t)H(t)=k/100$. In our case with inflation stopping momentarily, we need to
take special care in applying the initial condition, since some scales re-
enter the Hubble radius during the brief period with $\epsilon_{H}>1$ and exit
again when inflation continues. To make sure we capture the full non-trivial
time evolution of a mode, we start tracking each of them before their first
Hubble exit.

The Mukhanov–Sasaki equation in terms of $g_{k}$ is

$\ddot{g}_{k}+\left(H+2\frac{\dot{z}}{z}-\frac{2ik}{a}\right)\dot{g}_{k}-\frac{2ik}{a}\frac{\dot{z}}{z}g_{k}=0\,,$
(A.7)

where the dot denotes differentiation with respect to the cosmic time $t$. The
conformal time grows exponentially with cosmic time and hence the dynamical
range of cosmic time is smaller than that of conformal time, therefore the use
of cosmic time allows for an easier numerical treatment.

As is well known, in SR inflation the curvature perturbations—and thus
$g_{k}$—freeze to a constant value after they exit the Hubble radius. When the
SR conditions are broken, such as during our non-inflating period and the
subsequent phase when the inflaton climbs up towards the potential maximum,
this is not necessarily the case [134, 85]—indeed, the power spectrum
enhancement of Fig. 2 is a consequence of the super-Hubble growth of the
corresponding modes. The spectrum only freezes in the final SR phase after the
roll-over or roll-back has happened. To make sure we capture the final value
of the spectrum, we track the modes all the way to the end of inflation.

We use the variable $g_{k}$ and equations (A.6), (A.7) for the roll-over case
where the inflaton field rolls over the local maximum at the origin. However,
$g_{k}$ is not suitable for the roll-back case where the field stops near the
local maximum at the origin and starts to roll back, since $z$ becomes zero at
the turning point and $g_{k}$ is not defined. This is a gauge issue: the
comoving gauge is momentarily not defined, and neither is $\mathcal{R}$.
However, the Mukhanov variable $u_{k}$ turns out to behave well even in this
limit (in particular, the ratio $z^{\prime\prime}/z$ in (A.2) behaves
regularly); it is related to the field perturbation in the spatially flat
gauge. In the spirit of (A.5), we introduce the new variable

$h_{k}=\frac{u_{k}e^{ik\tau}}{a}\,,$ (A.8)

which stays well-defined throughout the roll-back case. The Bunch–Davies
vacuum initial condition in this case is

$\dot{h}_{k}=-\frac{\dot{a}}{a}h_{k}\,,\quad|h_{k}|=\frac{1}{\sqrt{2k}|a|}\,,$
(A.9)

and the Mukhanov–Sasaki equation is

$\ddot{h}_{k}+H\left(3-\frac{2ik}{aH}\right)\dot{h}_{k}+H^{2}\left(2-\epsilon_{H}-2i\left(\frac{k}{aH}\right)-\frac{H\dot{z}+\ddot{z}}{H^{2}z}\right)h_{k}=0\,.$
(A.10)

The variable $h_{k}$ is suitable for both roll-over and roll-back, but is
slightly slower to solve numerically. In the numerical analysis we use the
variable $g_{k}$ for roll-over; in the case of roll-back we use the variable
$g_{k}$ from the beginning until the re-start of inflation and then switch to
$h_{k}$, before the field turns around.

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