# Steepest growth re-examined: repercussions for primordial black hole
formation

Philippa S. Cole1<EMAIL_ADDRESS>Andrew D. Gow2,3<EMAIL_ADDRESS>Christian T. Byrnes3<EMAIL_ADDRESS>Subodh P. Patil4
<EMAIL_ADDRESS>
1) GRAPPA Institute,
University of Amsterdam, 1098 XH Amsterdam, The Netherlands

2) Institute of Cosmology and Gravitation, University of Portsmouth,
Portsmouth, PO1 3FX, United Kingdom

3) Department of Physics and Astronomy,
University of Sussex, Brighton BN1 9QH, United Kingdom

4) Instituut-Lorentz for Theoretical Physics,
Leiden University, 2333 CA Leiden, The Netherlands

(30/05/2022)

###### Abstract

Primordial black holes (PBHs) can be produced by a range of mechanisms in the
early universe. A particular formation channel that connects PBHs with
inflationary phenomenology invokes enhanced primordial curvature perturbations
at small scales. In this paper, we re-examine the impact of the growth of the
primordial power spectrum on PBH formation in terms of its implications for
the PBH mass function. We show that the mass function is relatively
insensitive to the steepness of the growth of the power spectrum and
subsequent decay, depending primarily on the peak amplitude and the presence
of any plateaus that last more than an e-fold. The shape of the power spectrum
can of course be constrained by other tracers, and so understanding the
physical limitations on its shape remains a pertinent question. We elaborate
on how rapidly the background can transition between different values of the
parameters of the Hubble hierarchy, which must ultimately derive from a
consistent derivative expansion for the background inflaton field. We discuss
artefacts associated with matching calculations, and highlight the robustness
of the steepest growth index previously found for single-field inflation with
conservatively smoothed transitions and limits on how much the power spectrum
can grow.

###### Contents

  1. I Introduction
  2. II PBH mass distribution: effects of growth and decay of the power spectrum
  3. III Steepest growth: matching calculation artefacts
  4. IV Conclusions
  5. A PBH mass distribution: asymmetric and plateau power spectra

##  I Introduction

Primordial black holes (PBHs) could teach us a lot about the initial
conditions of the universe. Large overdensities must have been left over at
the end of inflation for PBHs to have formed during radiation domination, when
pressure forces strongly oppose gravitational collapse, see
Sasaki:2018_Primordial ; Carr:2020_Constraints ; Carr:2020_Primordial ;
Green:2020_Primordial  for recent reviews. Assuming that formation occurs via
the direct collapse of these large amplitude perturbations shortly after they
re-enter the horizon post-inflation, and that the perturbations follow a
Gaussian distribution, the primordial power spectrum needs to be around seven
orders of magnitude larger than the amplitude observed on large scales
Gow:2021_ACPS . This raises questions of how quickly the power spectrum can
grow by this amount and how the shape of a peak in the power spectrum could be
observed.

Since the publication of Byrnes:2019_Steepest , where it was observed that the
fastest the primordial power spectrum can grow under certain assumptions in
single-field inflation is $k^{4}$, examples have been found of inflationary
potentials that enable the primordial power spectrum to grow slightly faster
in single-field inflation Carrilho:2019_Dissecting ; Ozsoy:2020_Slope ;
Ragavendra:2020sop ; Tasinato:2020vdk , and substantially faster in multi-
field inflation Palma:2020_Seeding ; Fumagalli:2020_Turning ; Braglia:2020taf
. As we elaborate upon in the next section, it turns out that the resultant
distribution or abundance of primordial black holes is largely insensitive to
changes in the slope of the power spectrum peak. In particular, that there is
essentially no dependence on the steepness of the power spectrum once it is
steeper than $k^{2}$, not even compared to the unrealistic (but frequently
studied) delta-function power spectrum. However, observational constraints on
primordial density fluctuations, for example via observations of the
stochastic gravitational wave background or CMB spectral distortions,
depending on the relevant comoving scales, may be sensitive to changes in the
shape of the power spectrum and can provide extra information about the
underlying mechanism that generated the enhanced power. Hence, the shape of an
enhancement to the power spectrum remains an important question if we are to
constrain them with other probes.

In light of the former considerations, we also study various properties of
enhancements to the power spectrum with the aim of understanding which of them
are artefacts of various calculational approximations, and which are physical.
We show that features such as oscillations in the power spectrum that arise
from unrealistic instant transitions in the evolution of the inflationary
parameters disappear if the transitions instead occur over the duration of
approximately an e-fold. For reasons that we hope to make clear shortly,
instantaneous in the context of inflationary cosmology should mean an order
one fraction of an e-fold (a distinction that for instance, the authors of
Felder:1998vq ; Felder:1999pv  were keenly aware of in their definition of
‘instantaneous’ in the context of pre-heating). The same effect can also lead
to the appearance of an over-optimistic growth in amplitude of the power
spectrum, which we show goes away when transitions are realistically smoothed.
In a follow-up paper we will study the fine-tuning associated with building
inflationary models capable of generating PBHs.

We do not include the impact of non-Gaussianity of the perturbations in this
paper, despite PBH formation taking place in the tail of the probability
distribution which implies that PBH formation is sensitive to non-Gaussianity.
Many studies of non-Gaussianity in ultra-slow-roll (USR) inflation have been
made, see e.g. Namjoo:2012aa ; Cai:2018dkf ; Ozsoy:2021pws ; Passaglia_2019 ,
as well as on PBH formation in this and related scenarios Atal:2018neu ;
Atal:2019erb ; Ragavendra:2020sop ; Davies:2021loj ; Figueroa:2021zah ;
Inomata:2021tpx  but the impact this has on PBH formation remains uncertain.
Issues include whether the (order unity) non-Gaussianity has a physical impact
or not (related to its value in Conformal Fermi Coordinates) Bravo:2017gct ;
Bravo:2017wyw ; Matarrese:2020why ; Suyama:2021adn  and the impact of quantum
diffusion during ultra-slow-roll inflation Pattison:2017mbe ; Pattison:2019hef
; Ezquiaga:2019ftu ; Pattison:2021oen ; Biagetti:2021eep ; Ballesteros:2020sre
.

In section II we study the impact of the shape of the power spectrum peak on
PBH mass distributions, and in section III we demonstrate the important impact
of smoothing the inflationary potential, going beyond the instantaneous
transitions assumed for analytic simplicity in matching calculations. We
conclude in section IV while the appendices contain additional details and
plots.

##  II PBH mass distribution: effects of growth and decay of the power
spectrum

Perhaps the strongest motivation for considering a significant growth in the
power spectrum is the production of primordial black holes, which can
contribute to early- and late-universe physics in a number of ways, most
notably as a dark matter candidate. A key observable for the study of PBHs is
their mass distribution, which can be sensitive to the details of the power
spectrum of the perturbations they form from. However, it is known that the
effects of critical collapse become significant for narrow power spectra
Niemeyer:1998_Critical-collapse ; Yokoyama:1998_Critical-collapse ;
Kuhnel:2016_Critical-collapse , and it has recently been shown that if the
power spectrum peak is sufficiently narrow, the resulting PBH mass
distribution is described entirely by the critical collapse physics and
becomes insensitive to the detailed shape of the power spectrum Gow:2020cou .
It is therefore sensible to ask the question of whether a steeper than $k^{4}$
growth is distinguishable when considering the PBH mass distribution.

We calculate the PBH mass distribution using the formalism described in
Gow:2021_ACPS . There is considerable discussion about which method to use
when calculating the PBH mass distribution, with many recent papers utilising
a peaks theory method. It was shown in Gow:2021_ACPS  that the mass
distribution shape is mostly independent of the calculation method once
normalised to a fixed $f_{\mathrm{PBH}}$. As there is no community consensus
about the correct way to apply peaks theory Yoo:2018kvb ; Yoo:2019pma ;
Germani:2019zez ; Young:2020_Peaks , we choose to use the Press–Schechter
method for simplicity. Additionally, we neglect the effect of the non-linear
relationship between $\zeta$ and $\delta$ Kawasaki:2019mbl ; Young:2019yug ;
DeLuca:2019qsy ; Kalaja:2019_From , which is again expected to have a minimal
impact on the mass distribution shape, and will not affect the conclusions
presented in this section. We use the modified Gaussian window function from
Gow:2021_ACPS . We consider the mass distribution

$\displaystyle\psi(m)$
$\displaystyle=\frac{1}{\Omega_{\mathrm{PBH}}}\frac{\mathrm{d}\Omega_{\mathrm{PBH}}}{\mathrm{d}m},$
(1)

which is normalised to unity over the PBH mass $m$, where
$\Omega_{\mathrm{PBH}}$ is the fraction of the total energy density of the
universe in PBHs today111We neglect the impact of PBH accretion and
evaporation in this paper.. This is related to another commonly used
definition for the PBH mass function, $f(m)$, by

$\displaystyle\psi(m)$
$\displaystyle=\frac{1}{f_{\mathrm{PBH}}}\frac{f(m)}{m},$ (2)

where $f(m)$ satisfies the normalisation condition

$\displaystyle\int f(m)\ \mathrm{d}\ln(m)$ $\displaystyle=f_{\mathrm{PBH}},$
(3)

with $f_{\mathrm{PBH}}=\Omega_{\mathrm{PBH}}/\Omega_{\mathrm{CDM}}$ the
fraction of dark matter in the form of PBHs.

For the shape of the primordial power spectrum, we consider a symmetric peak
with a power-law growth and decay of a constant slope, and for simplicity we
begin by assuming that the growth and decay rates of the slope are equal,

$\displaystyle\mathcal{P}_{\mathcal{R}}(k)$
$\displaystyle=A\begin{cases}\left(\frac{k}{k_{p}}\right)^{n}&k\leqslant
k_{p}\\\ \left(\frac{k}{k_{p}}\right)^{-n}&k>k_{p}\end{cases},$ (4)

with the amplitude fixed to produce a PBH abundance
$f_{\mathrm{PBH}}=10^{-3}$.

In figure 1 we show the PBH mass distributions calculated from these symmetric
power spectrum peaks, for a range of slopes. It is clear that a slope of
$k^{4}$ is already steep enough that the PBH mass distribution is extremely
close to that of a delta function in the power spectrum, and by extension any
power spectrum slope steeper than $k^{4}$.

Figure 1: PBH mass distribution from the symmetric power-law power spectrum in
equation (4). The peak positions have been normalised to the same value for
ease of comparison. For sufficiently steep peaks, the mass distribution
becomes virtually indistinguishable.

However, the assumption that the power spectrum will fall off with the same
slope as its rising edge is not normally valid in practice. For example, in
inflationary models with an inflection point, the growth is usually $k^{4}$
whilst the decrease in all known cases is less steep and related to the value
of the second slow-roll parameter after the inflection point. The slow-roll
parameters are defined as $\epsilon=-\dot{H}/H^{2}$ and
$\eta=\dot{\epsilon}/\epsilon H$, and it is the latter which tracks the
acceleration of the inflaton and determines the slope of the resulting power
spectrum. For a summary of the literature, see for instance Atal:2018neu .

In appendix A we consider two alternative cases for the shape of the power
spectrum. The first is an asymmetric peak in the power spectrum, with
different spectral indices for the rising and falling slopes. The second is a
peak followed by a plateau naïvely motivated by an $\eta$ distribution with
just one USR phase, $\eta=0\to-6\to 0$ (see figure 2 in Byrnes:2019_Steepest
), where the definition of USR is that $\eta=-6$. In the first case we can see
that varying the slope of the growing part of the spectrum between $k$ and
$k^{5}$ has only a very small impact on the mass function, irrespective of the
decay rate after the peak. The second case shows how the presence of a plateau
only becomes noticeable once the width of the plateau becomes wider than that
corresponding to one e-fold of inflation (where e-folds denoted $N$ relate to
comoving scales $k$ as $N\propto\log{k}$), and becomes significant for a
plateau width corresponding to two e-folds of inflation. However, only
accurate observations of a large number of PBHs would be able to tell the
difference between a peak with no plateau, and one with a width corresponding
to one e-fold of inflation (i.e. $k_{\rm max}/k_{\rm min}\simeq 2.7$).

Why is the PBH mass function so insensitive to substantial changes in the
shape of the power spectrum peak? There are two key reasons. Firstly, the
density perturbations are due to the curvature power spectrum smoothed with a
window function, which smears out features on the scale of the PBH (see e.g.
figure 5 of Byrnes:2019_Steepest  to see the visual impact of the smoothing,
or Kalaja:2019_From ). While there has been some discussion as to the correct
choice of window function, it now seems clear that the effect on the PBH mass
distribution and abundance is sufficiently small to be indistinguishable with
current data Ando:2018_Primordial ; Young:2020_Criterion ; Gow:2021_ACPS . The
second reason is that the critical collapse phenomenon implies a minimum width
to the PBH mass distribution even if the power spectrum is a delta function
(meaning that it only has perturbations on a single scale). This ensures that
a monochromatic mass distribution is impossible, and a lognormal shape is
frequently used instead. However, for sufficiently narrow power spectra, the
shape is controlled by the critical collapse effect, leading to a non-
lognormal distribution, and an insensitivity to the power spectrum shape
Niemeyer:1998_Critical-collapse ; Yokoyama:1998_Critical-collapse ;
Gow:2020cou . The combination of these two effects means that features in the
power spectrum with a width corresponding to less than about one e-fold are
unlikely to be visible to any realistic future measurement of the PBH mass
function.

It is also important to stress that the PBH formation rate is only sensitive
to the power spectrum amplitude very close to the peak, because the collapse
rate into PBHs is exponentially sensitive to the amplitude. Hence, if the
power spectrum slope is scale dependent then the PBH mass function can (at
best) only be a function of the shape very close to the peak. Alternative
probes will be needed to constrain or detect the power spectrum on amplitudes
between the $10^{-9}$ value seen on the largest scales and the $\sim 10^{-2}$
required for PBH formation. On scales at most a few orders of magnitude
smaller than the smallest scales detectable with the CMB or LSS (relevant for
the formation of PBHs massive enough to be detected with LIGO–Virgo–KAGRA),
one can use spectral distortions of the CMB Zeldovich:1969ff ; Sunyaev:1970er
; Hu:1992dc ; Chluba:2011hw ; Khatri:2012tw  or, in a more model dependent
way, ultracompact minihaloes Bringmann:2012_UCMH ; Gosenca:2017_UCMH ;
Sten_Delos:2018_UCMH ; Sten_Delos:2018_UCMH-density ; Nakama:2019htb  or
gravitational lensing of small scale structures Karami:2018qrl ; Lee:2020wfn ;
Wagner-Carena:2022mrn . One weakness of these alternative constraints is that
the $\mu$-distortions measure an integrated value of the small scale power,
while the other two techniques can only apply for certain dark matter models,
and we note that a mixed dark matter model with a combination of WIMPs and
PBHs is incompatible over a large range of parameter space
Lacki:2010_Primordial ; Adamek:2019_WIMPs ; Carr:2020mqm ; Boudaud:2021irr .

An important method to detect the primordial perturbations on small scales is
via the primordial stochastic gravitational wave background, generated by the
large amplitude scalar perturbations sourcing second-order tensor
perturbations, see Yuan:2021qgz ; Domenech:2021ztg  for reviews. There is an
approximate 1-2-1 relation between the PBH mass and the peak of the
gravitational wave frequency. At the time of writing the stochastic
gravitational wave background constraint is only competitive with the PBH
constraint on the power spectrum amplitude on scales around one parsec, with
the corresponding frequency probed by pulsar timing arrays (PTAs), see e.g.
NANOGrav:2020bcs . Interestingly, this constraint is almost exactly equal to
the PBH constraint on the mass range in which LIGO–Virgo–KAGRA can detect
black holes, so if that experiment does detect (or has already detected) PBHs
then there should be a corresponding signal detectable by PTAs. In the future,
other experiments including SKA, LISA and the Einstein Telescope will be able
to probe the stochastic gravitational wave background with sufficient
sensitivity over such a large range of scales that a non-detection by all of
these experiments would potentially be able to rule out the existence of
almost any (non-evaporated) PBHs assuming that PBHs form from the direct
collapse of Gaussian distributed density perturbations. For the current and
forecast constraint plots see figures 5 and 6 of Gow:2021_ACPS  (see also
Wang:2019kaf ). However, if any of these experiments does detect a primordial
stochastic gravitational wave background then measurements of its frequency
dependence over a wide range of frequencies could provide a measurement of the
shape of the power spectrum peak, e.g., Pi:2020otn ; Ragavendra:2020sop ;
Dalianis:2021_Exploring ; Domenech:2021ztg .

##  III Steepest growth: matching calculation artefacts

We have seen that even large differences in power spectrum growth and decay
rates have a minimal effect on the PBH mass function. However, the simple
parametrisation of the power spectrum in equation (4) does not capture all of
the features that may affect the PBH mass distribution. Namely, rapid
oscillations in $k$-space are visible in the power spectra of many PBH-
motivated models. These are consequences of the fact that they are calculated
by stitching together the solutions for the curvature perturbation that are
each valid for a constant value of $\eta$. The solutions (and their
derivatives) are ‘matched’ at transition times between phases of constant
$\eta$, resulting in an evolution of $\eta$ that has instant transitions
between constant values. See for example figure 2, where the solid orange line
shows the $\eta$ evolution which corresponds to instant transitions between
slow-roll, followed by ultra-slow-roll $(\eta=-6)$ and then a period of
$\eta=2$ (which induces a decay in the power spectrum on small scales). Whilst
one can imagine that stitching together a large number of these solutions for
constant $\eta$ will eventually approach the limit of a smoothly evolving
function of $\eta$, for long durations of constant $\eta$, the instant
transitions are unrealistic. This is because $\eta$ and $\epsilon$ are
obtained from perturbing around a time dependent background derived from an
underlying effective action for the background inflaton, and are thus subject
to the usual requirements of deriving from a controlled derivative expansion.

Instantaneous transitions correspond to arbitrarily sharp changes in certain
parameters of the effective action (i.e. the potential and its derivatives in
the context of canonical inflation), which are unphysical. Concretely,
although one might try to engineer an arbitrarily sharp transition at the
classical level, this will not be possible at the quantum corrected level. In
addition, one must take care to distinguish between being able to engineer
arbitrarily sharp transitions in coordinate time, and in terms of e-folds —
large gradients source expansion, and transitions between different values of
the parameters of the Hubble hierarchy always end up costing an order one
fraction of an e-fold even if one were to suspend disbelief and try to arrange
for them to occur arbitrarily fast in coordinate time (this is discussed in
appendix C2 of Byrnes:2019_Steepest  in the context of how quickly inflation
can end, but the argument can be generalized). We should therefore instead
consider smoothed transitions. We model the smoothing phenomenologically by
multiplying each phase of constant $\eta$ by

$s(N,\nu)=\frac{1}{2}\left[\tanh{\left(\frac{N-N_{i}}{\nu}\right)}-\tanh{\left(\frac{N-N_{j}}{\nu}\right)}\right],$
(5)

where $N$ denotes e-folds, $N_{i}$ and $N_{j}$ are the e-fold values at the
transition times in and out of the phase of constant $\eta$, and $\nu$ is the
duration of the smoothing. The effect on the evolution of $\eta$ is
demonstrated by the dotted, dashed and solid blue lines of figure 2 which
correspond to $\nu=(0.1,0.5,1)$ e-fold transition durations respectively.
There are 2.3 e-folds of ultra-slow-roll in the case of instant transitions.

Figure 2: Trajectories of $\eta$ as a function of e-folds. Instant transitions
are shown by the orange line, and varying durations of smoothing are shown in
blue dotted (0.1 e-folds), dashed (0.5 e-folds) and solid lines (1 e-fold).

The corresponding power spectra are shown in figure 3. The spectrum calculated
from matching instantaneously between constant phases of $\eta$ (solid orange
line) exhibits oscillations around the peak of the spectrum. Given that these
alter the peak amplitude, they could have a noticeable effect on the PBH mass
distribution and abundance, and they would certainly impact observations which
directly constrain the scalar perturbations. When we instead smooth the
transitions, the oscillations disappear. The dotted, dashed and solid blue
lines in figures 3 correspond to 0.1, 0.5 and 1 e-fold durations of smoothing
of the $\eta$ evolution respectively. For 0.1 e-folds of smoothing, the
oscillations survive, but for 0.5 and 1 e-folds, they disappear entirely.
Furthermore, the peak amplitude decreases too, which will have an
exponentially large effect on the abundance of PBHs produced. However, this is
in part due to the fact that when smoothing the transitions, the duration of
non-slow-roll is decreased, so one would expect less growth in the power
spectrum.

Figure 3: Analytic matching solution with instant transitions in orange,
smoothed numerical results with varying ‘duration’ of smoothing in dotted (0.1
e-folds), dashed (0.5 e-folds) and solid (1 e-fold) blue.

We therefore show for comparison the same matching calculation power spectrum
in figure 4 in orange, and then smooth the transitions by the same amount as
in figure 3, but increase the duration of the ultra-slow-roll phase such that
the integrated value of $\eta<0$ (integrated with respect to $N$) is equal in
each case. This means that there are now $2.3+(0.038,0.19,0.37)$ e-folds of
non-slow-roll for each of the smoothed trajectories. The amount of true ultra-
slow-roll where $\eta=-6$ still varies in each case. We now see that the
oscillations disappear, and the peak height is still decreased but less
significantly.

Figure 4: Analytic matching solution with instant transitions in orange,
smoothed numerical results with varying ‘duration’ of smoothing in dotted (0.1
e-folds), dashed (0.5 e-folds) and solid (1 e-fold) teal. The integrated
contribution of negative $\eta$ is kept fixed for all 4 lines.

It is important to note here that the process of smoothing the instant
transitions causes the peak of the power spectrum to decrease and shift to
larger values of $k$, and also flattens the slope in the vicinity of the peak.
These changes will have a significant effect on the PBH mass distribution,
since this is only sensitive to the region close to the peak. Figure 5 shows
the PBH mass distributions corresponding to the power spectra shown in figure
4. While the power spectrum oscillations are removed in every case by the
window function, the unsmoothed power spectrum still gives a mass distribution
with a unique double peak structure, corresponding to the two maxima. However,
it can be seen that smoothing the transitions quickly suppresses the second
peak, before removing it entirely. For 0.5 e-folds of smoothing, the power
spectrum has a plateau, producing a broad mass distribution, and for one
e-fold, the low-$k$ peak in the power spectrum no longer contributes to the
PBH mass distribution, leaving just a single narrow peak shifted to lower
masses. Therefore, we can see that any search for a bimodal mass distribution
is not motivated by ultra-slow-roll inflation once realistic transitions are
assumed.

Figure 5: PBH mass distribution corresponding to the power spectra shown in
figure 4.

Finally we also demonstrate that models which exhibit super-$k^{4}$ growth
with instant transitions between constant phases of $\eta$ suffer from similar
oscillation, slope and amplitude depletion when instant transitions are
smoothed. For example, it was shown in Carrilho:2019_Dissecting  that an extra
phase of $\eta=-1$ in between phases of slow-roll and ultra-slow-roll produces
an enhancement of the $k^{4}$ slope in the power spectrum proportional to
$k^{5}(\log{k})^{2}$. By smoothing the transitions and limiting the amplitude
increase in the primordial power spectrum to no more than seven orders of
magnitude so as to comply with both large scale CMB constraints and the
envelope of PBH non-detection constraints, we find that the super-$k^{4}$
growth is effectively lost, as shown in figure 6. This happens for exactly the
same reason as the $k^{4}$ growth depletion shown in figures 3 and 4. However,
because just one matching is required between slow-roll and ultra-slow-roll,
some $k^{4}$ growth does survive. In cases where other constant phases of
$\eta$ are required with a strong sensitivity on the duration of that phase,
for example the extra phase of $\eta=-1$ in this model, the impact of
super-$k^{4}$ power spectrum growth is overruled by the need to smooth
transitions222We note in closing that our arguments only apply to sustained
growth – that is, growth which can be achieved for multiple e-folds. There are
transients where the growth index is evidently much higher (such as in the
immediate vicinity of the dip before the sustained growth). One can imagine
positing classical inflationary backgrounds where transients of a specific
nature can be arranged to occur regularly enough so that the growth can have
periods of super $k^{4}$ growth that last more than a few e-folds, as was
shown in Ozsoy:2020_Slope ; Ragavendra:2020sop , although whether these
persist at the quantum corrected level is to be verified..

Figure 6: Power spectra including an extra phase of 6 e-folds of $\eta=-1$
evolution inbetween slow-roll and ultra-slow-roll (1.3 e-folds thereof).
Analytic matching solution with instant transitions in magenta, smoothed
numerical results with varying ‘duration’ of smoothing in dotted (0.1
e-folds), dashed (0.5 e-folds) and solid (1 e-fold) light blue.

##  IV Conclusions

We have studied the implications of the shape of a large peak in the
primordial power spectrum on the primordial black hole mass function. We have
shown that changes in the steepness of the peak, provided the steepness is
significantly greater than $k$ before the peak, or $k^{-1}$ after the peak,
have a negligible impact on the PBH mass function. As a corollary, we can also
see that growth rates faster than the “common” limit of $k^{4}$ will not have
any measurable impact on the PBH mass function, and even assuming an
unrealistic delta-function power spectrum is adequate for this purpose.
Although the conclusion that the PBH mass function has a minimum width is not
new (due to critical collapse and the impact of the smoothing when going from
the curvature perturbation to the density contrast), our detailed study builds
upon previous investigations. We have studied peaks with an equal power law
growth and decay, peaks with an unequal growth and decay rate (as motived by
single-field models of ultra-slow-roll inflation) and also peaks including a
plateau of variable width. In the case of a peak with a plateau, we have shown
that plateau widths which correspond to scales exiting over less than about
one e-fold ($N\propto\log{k}$) have a minimal impact on the PBH mass function,
but a plateau width corresponding to scales exiting over two e-folds has a
significant impact on boosting the large mass tail.

We caution that although the PBH mass function is insensitive to the shape of
the power spectrum peak, the required amplitude of the power spectrum peak
does depend on the width of the peak and also on the normalisation convention
used, as discussed in e.g. Byrnes:2019_Steepest ; Gow:2021_ACPS . Other
methods to observe the power spectrum on small scales, most notably spectral
distortions of the CMB or a stochastic gravitational wave background generated
by the non-linear mixing of scalar and tensor perturbations may be able to
probe such details in an independent manner, as discussed in section II.

In the second part of this paper we have focused on the impact of smooth vs
instant transitions between different periods of inflation. Analytic work
usually focuses on studying instantaneous transitions between periods with
different growth rates of $\eta$, because this allows an analytic solution to
be found using a matching calculation. Whilst this technique is valuable for
learning about the analytic behaviour of the power spectrum, we caution that
instant transitions are unphysical and that realistic power spectra, based on
physically realisable smooth transitions, have somewhat different behaviour.
The “steepest” growth of $k^{4}$ in most ultra-slow-roll scenarios persists
Byrnes:2019_Steepest , but the oscillations after the peak are damped into
something close to a plateau and the amplitude of the peak is also reduced.
This has a significant effect on the PBH mass distribution, thereby
demonstrating the importance of calculating PBH mass distributions from
initial conditions with smooth trajectories for the inflationary slow-roll
parameters. In at least one scenario capable of getting a steeper than $k^{4}$
growth, see Carrilho:2019_Dissecting , we have shown that this steeper growth
is erased upon smoothing the transitions to reasonable durations.

## Acknowledgements

PC acknowledges support from the Institute of Physics at the University of
Amsterdam. AG acknowledges support from the Science and Technology Facilities
Council [grant numbers ST/S000550/1 and ST/W001225/1] CB acknowledges support
from the Science and Technology Facilities Council [grant number
ST/T000473/1]. We thank Sam Young for useful discussions.

Data availability Statement
No new raw data were generated or analysed in support of this research.

##  Appendix A PBH mass distribution: asymmetric and plateau power spectra

In section II, we showed that the PBH mass distribution calculated from a
symmetric peak in the primordial power spectrum becomes essentially
indistinguishable for sufficiently large slopes, such that a greater than
$k^{4}$ growth does not have a significant effect on the mass distribution.
However, it is not expected that the power spectrum will fall at the same rate
at which it grew, so it is necessary to consider the case of an asymmetric
power spectrum peak, given by

$\displaystyle\mathcal{P}_{\mathcal{R}}(k)$
$\displaystyle=A\begin{cases}\left(\frac{k}{k_{p}}\right)^{n_{1}}&k\leqslant
k_{p}\\\ \left(\frac{k}{k_{p}}\right)^{-n_{2}}&k>k_{p}\end{cases},$ (6)

with the amplitude again normalised to give $f_{\mathrm{PBH}}=10^{-3}$. For
the rising edge, we will consider the same range of indices as in section II.
For the falling slope, we choose the bounding values given in Atal:2018neu
for the selection of models they considered, $0.15\leqslant n_{2}\leqslant
2.7$. In figure A.1 we show the mass distributions for these two cases, with
the same selection of growing indices as in section II. It is clear that the
conclusion in that section holds for asymmetric power spectra.

Figure A.1: PBH mass distribution from the asymmetric power-law power spectrum
in equation 6 with $n_{2}=0.15$ (left) and 2.7 (right). The peak positions
have been normalised to the same value for each of comparison. As with the
case of a symmetric power spectrum, the mass distribution becomes virtually
indistinguishable for sufficiently fast growth.

Another possibility is that the power spectrum does not decay immediately, but
remains enhanced over some range of $k$ values. This is the case for the
example of a simple USR phase ($\eta=0\to-6\to 0$), in which the negative
$\eta$ period grows the power spectrum, after which it remains at its enhanced
value. This simple case is not possible to realise in many inflationary
models, since the end of inflation normally requires $\epsilon=1$, and so a
positive $\eta$ phase must exist to drive $\epsilon$ back to larger values,
which will also cause the power spectrum to reduce. However, it may be
possible to maintain an enhanced power spectrum for a small number of e-folds
before it decays again, or to end inflation by an alternative mechanism to the
growth of $\epsilon$, such as a waterfall field in hybrid inflation.
Therefore, we consider the case of a power spectrum that grows and decays as
$k^{4}$, but with a plateau at the top,

$\displaystyle\mathcal{P}_{\mathcal{R}}(k)$
$\displaystyle=A\begin{cases}\left(\frac{k}{k_{l}}\right)^{4}&k\leqslant
k_{l}\\\ 1&k_{l}<k\leqslant k_{u}\\\
\left(\frac{k}{k_{u}}\right)^{-4}&k>k_{u}\end{cases},$ (7)

where $k_{u}$ is defined in terms of the width of the plateau in log-space,

$\displaystyle k_{u}$ $\displaystyle=k_{l}e^{w}$ (8)

and the amplitude is fixed such that $f_{\mathrm{PBH}}=10^{-3}$.

In figure A.2 we show a range of plateau power spectra with different widths
$w$, ranging from no plateau to one that lasts for five e-folds, and the
corresponding PBH mass distributions. The power spectrum positions are chosen
such that the mass distributions peak at the same mass. We can see that for
plateau widths smaller than $\sim$ one e-fold, the mass distribution is
virtually unchanged. For broader plateaus, an enhancement to the high-mass
tail is observed.

Figure A.2: Plateau power spectra (left) and corresponding PBH mass
distributions (right) for a range of plateau widths $w$. The mass distribution
peak positions have been aligned for ease of comparison.

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