###### Abstract

In this study, we present an analysis of the fine-tuning required in various
inflationary models in order to explain the production of Primordial Black
Holes (PBHs). We specifically examine the degree of fine-tuning necessary in
two prominent single-field inflationary models: those with an inflection point
and those with step-like features in the potential. Our findings indicate that
models with step-like features generally require less fine-tuning compared to
those with an inflection point, making them more viable for consistent PBH
production. An interesting outcome of these models is that, in addition to
improved fine-tuning, they may also predict low-frequency signals that can be
detected by pulsar timing array (PTA) collaborations. Additionally, we extend
our analysis to multifield inflationary models to assess whether the
integration of additional fields can further alleviate the fine-tuning
demands. The study also explores the role of a spectator field and its impact
on the fine-tuning process. Our results indicate that although mechanisms
involving a spectator field can circumvent the issue of fine-tuning parameters
for PBH production, both multifield models and models with step-like features
present promising alternatives. While fine-tuning involves multiple
considerations, our primary objective is to evaluate various inflationary
models to identify the one that most naturally explains the formation of PBHs.
Hence, this study introduces a novel approach by categorizing existing PBH
mechanisms, paving the way for subsequent research to prioritize models that
minimize the need for extensive fine-tuning.

###### keywords:

primordial black holes; inflation; dark matter; scalar induced gravitational
waves

1 1 0 Academic Editor: Firstname
Lastname 27 April 2024 23 May 2024 28 May 2024 https://doi.org/ Mechanisms for
Producing Primordial Black Holes from Inflationary Models Beyond Fine-Tuning
Mechanisms for Producing Primordial Black Holes from Inflationary Models
Beyond Fine-Tuning Ioanna Stamou Ioanna Stamou Stamou, I

##  1 Introduction

In recent years, there has been an increasing focus on primordial black holes
(PBHs) due to their possible contribution to explaining the dark matter of the
Universe. This explanation has gained significant attention following the
detection of gravitational waves (GWs) emitted by binary black hole mergers,
as reported by the LIGO/VIRGO/Kagra (LVK) collaboration Abbott et al. (2016a,
b, 2017a, 2017b, 2017c). Moreover, the recent detection of GWs at nano-Hertz
frequencies by pulsar timing array (PTA) collaborations Agazie et al. (2023a,
b); Afzal et al. (2023); Antoniadis et al. (2023a, b, c, d) has sparked
additional interest in the potential link between these low-frequency GWs and
PBHs . Consequently, numerous theoretical studies have explored the potential
link between dark matter, PBHs, and GWs Özsoy and Tasinato (2023); Sasaki et
al. (2018); Carr et al. (2024); Khlopov (2010); Kuhnel and Stamou (2024);
Ellis et al. (2024); Vaskonen and Veermäe (2021); Escrivà et al. (2022). All
these studies must adhere to the observational constraints on inflation from
Cosmic Microwave Background (CMB) anisotropies as reported by the Planck
collaboration Ade et al. (2016); Akrami et al. (2020).

In previous theoretical research, it has been suggested that PBHs can form
through a significant enhancement of curvature perturbations at small scales.
One widely adopted mechanism for producing PBHs from inflation involves
studying an inflection point in the effective scalar potential Garcia-Bellido
and Ruiz Morales (2017); Hertzberg and Yamada (2018); Ballesteros and Taoso
(2018); Ballesteros et al. (2020); Cicoli et al. (2018); Stamou (2021); Spanos
and Stamou (2023); Dalianis et al. (2019); Germani and Prokopec (2017);
Ezquiaga et al. (2018); Özsoy et al. (2018); Özsoy and Lalak (2021); Karam et
al. (2023); Aldabergenov and Ketov (2023); Kannike et al. (2017); Kawasaki et
al. (2016); Choudhury and Mazumdar (2014). Such inflationary potentials are
generally characterized by a slow-roll phase at large scales exiting the
horizon, followed by an inflection point region at small scales where the
power spectrum undergoes a significant enhancement. However, a major drawback
of this mechanism is the high level of fine-tuning that the parameters require
to achieve such a configuration in the potential Stamou (2021); Hertzberg and
Yamada (2018); Cole et al. (2023). It is important to note that, within this
work, ‘fine-tuning’ refers to the necessity for highly precise parameters to
facilitate the abundant production of PBHs. A model is deemed fine-tuned if
minor variations in parameters significantly influence the enhancement of the
power spectrum and, consequently, the production of PBHs.

Alternative mechanisms within the framework of single-field inflation have
also been proposed to alleviate problematic fine-tuning Inomata et al. (2022);
Cotner and Kusenko (2017); Kefala et al. (2021); Dalianis et al. (2021); Cai
et al. (2022); Choudhury et al. (2023, 2024); Gu et al. (2023); Drees and
Erfani (2012); Domènech et al. (2024); Caravano et al. (2024); Sharma et al.
(2024); Dimastrogiovanni et al. (2024); Karam et al. (2023). Mechanisms
involving step-like features in potentials have been studied Inomata et al.
(2022); Kefala et al. (2021); Dalianis et al. (2021); Cai et al. (2022). These
mechanisms can lead to a significant enhancement of the power spectrum, which
is capable of explaining the production of PBHs. Interestingly, this mechanism
predicts additional oscillatory patterns in the power spectra that may lead to
a characteristic signal in the induced GWs Fumagalli et al. (2021).

The study of an amplification of scalar power spectrum has also been extended
to multi-field models. Multi-field inflationary models have been widely
explored for the production of PBHs Palma et al. (2020); Anguelova (2021);
Iacconi and Mulryne (2023); Zhou et al. (2020); Mavromatos et al. (2022);
Kawai and Kim (2023); Aldabergenov et al. (2022); Braglia et al. (2020); Pi
and Sasaki (2023); Gundhi et al. (2021); Geller et al. (2022); Chen et al.
(2023); Wang et al. (2024). Models with a non-canonical kinetic term in the
Lagrangian can lead to a significant enhancement in the scalar power spectrum
Aldabergenov et al. (2022); Braglia et al. (2020); Pi and Sasaki (2023);
Gundhi et al. (2021); Geller et al. (2022); Chen et al. (2023); Wang et al.
(2024); Heydari and Karami (2022); Pi et al. (2018). Moreover, multi-field
models within the framework of natural inflation, such as those involving
multiple axion fields Zhou et al. (2020); Mavromatos et al. (2022), have been
shown to result in a significant fractional abundance of PBHs. Finally, hybrid
models with a waterfall trajectory have acquired a lot of interest as they
provide a more natural explanation to PBHs Afzal and Ghoshal (2024); Spanos
and Stamou (2021); Braglia et al. (2023); Clesse and García-Bellido (2015);
Tada and Yamada (2023); Dimopoulos (2022).

Furthermore, mechanisms involving a spectator field have been extensively
explored as promising alternatives to mitigate the fine-tuning required for
PBH production Carr et al. (2017); Kohri et al. (2013); Stamou and Clesse
(2024, 2023); Stamou (2024); Cai et al. (2021); Gow et al. (2023); Hooper et
al. (2024). Although the spectator field does not drive inflation, it evolves
independently and significantly influences the formation of structures and
phenomena, including PBHs. Notably, it has been demonstrated that these models
can circumvent the need for fine-tuning the potential’s parameters to
facilitate PBH production Kohri et al. (2013); Stamou and Clesse (2024).

This paper investigates the extent of fine-tuning required across a diverse
range of mechanisms previously proposed in the literature. Specifically, we
examine four characteristic inflationary mechanisms for producing primordial
black holes (PBHs): a single-field mechanism with an inflection point, one
with a step-like feature, a two-field mechanism with a non-canonical kinetic
term, and a mechanism incorporating a waterfall trajectory. For each PBH
production mechanism, we provide a rationale for selecting a particular model.
This study diverges from the findings reported in Ref. Cole et al. (2023) as
it does not solely concentrate on single-field inflationary models with an
inflection point or bulk in their scalar potential. While acknowledging the
contributions of these mechanisms, our analysis broadens to include additional
inflationary models, such as those involving multiple fields, as discussed
earlier.

First, we delve into single-field inflation models, examining both the
inflection point mechanism and models that feature step-like characteristics
while evaluating the associated fine-tuning demands. We found that the amount
of fine-tuning required for models with an inflection point is significantly
large, whereas the step-like mechanism can lead to a more natural explanation
as the amount of fine-tuning is significantly reduced. Notably, this mechanism
can explain GW signals at low frequencies. We discuss this implementation in a
separated section and leave the details for a future paper.

In the continuation of our analysis, we extend our study to analyze the fine-
tuning of multifield inflationary models. Specifically, we explore two-field
model mechanisms: the first incorporates non-canonical kinetic terms in the
Lagrangian, and the second features a waterfall trajectory typical of hybrid
inflationary models. We find that the amount of fine-tuning is significantly
reduced in both multifield mechanisms by at least three orders of magnitude in
comparison with inflection point models. Lastly, we discuss the implications
of models that include a spectator field for explaining the production of
PBHs, highlighting that the need for fine-tuning to describe such structures
can be avoided. Therefore, the novelty of this work lies in classifying the
proposed PBH mechanisms, focusing on determining which mechanisms can more
naturally explain the production of PBHs. This approach can contribute to
future investigations towards the mechanisms that inherently require less
fine-tuning.

The layout of this paper is as follows: In Section 2, we describe our
methodology for computing the amount of fine-tuning. In Section 3, we present
an analysis of fine-tuning in single-field inflationary mechanisms. In Section
4, we extend this analysis to two-field models. In Section 5, we discuss the
amount of fine-tuning in models with a spectator field. In Section 6, we
explore an interesting application of step-like potentials to scalar-induced
GWs. Finally, in Section 7, we present our conclusions. The Appendices provide
the detailed equations that are needed for our analysis: Appendix A covers the
evaluation of the scalar power spectrum, Appendix B details the fractional
abundances of PBHs, and Appendix C discusses the energy densities of GWs.

##  2 Fine-Tuning

In this study, we focus on evaluating the degree of fine-tuning required by
various mechanisms to explain the production of PBHs. It is important to note
that there are several aspects of fine-tuning that need consideration.
However, our primary goal is to assess different inflationary mechanisms to
determine which can provide a more natural explanation for PBH formation.

To assess the fine-tuning of parameters, we employ the quantity
$\Delta_{p_{i}}$, defined as the maximum value of the following logarithmic
derivative Barbieri and Giudice (1988):

$\Delta_{p_{i}}^{Q}=\mathrm{Max}\left|\frac{\partial\ln(Q)}{\partial\ln(p_{i})}\right|$
(1)

Here, $Q$ represents the quantity under study, specifically the peak value of
the scalar power spectrum denoted as $P_{R}^{peak}$, and $p_{i}$ denotes the
corresponding parameters of each model being analyzed. This metric,
$\Delta_{p_{i}}^{Q}$, quantifies the sensitivity of the observable $Q$ to
variations in parameter $p_{i}$, thus serving as a measure of the required
fine-tuning. A high value of $\Delta_{p_{i}}^{Q}$ implies that minor changes
in $p_{i}$ result in significant alterations in $Q$, indicating a high degree
of fine-tuning within the model.

We need to remark here that throughout this work, we use reduced Planck mass
units, setting $\rm{M_{P}=1}$.

##  3 Single-Field Models

In this section, we discuss single-field inflationary models for the
production of PBHs. These models are widely recognized in the literature for
their role in the formation of PBHs and offer several advantages. First, they
provide a simple mechanism for producing PBHs during inflation. Second, they
can explain a significant fraction of dark matter. Third, they are easily
adaptable to various theoretical frameworks.

In the subsequent sections, we will explore two distinct mechanisms. The first
involves a mechanism characterized by an inflection point in the scalar
potential Garcia-Bellido and Ruiz Morales (2017); Hertzberg and Yamada (2018);
Ballesteros and Taoso (2018); Ballesteros et al. (2020); Cicoli et al. (2018);
Stamou (2021); Spanos and Stamou (2023); Dalianis et al. (2019); Germani and
Prokopec (2017); Ezquiaga et al. (2018); Özsoy et al. (2018); Özsoy and Lalak
(2021); Karam et al. (2023); Aldabergenov and Ketov (2023); Kannike et al.
(2017). The second examines models that incorporate step-like features Kefala
et al. (2021); Dalianis et al. (2021). We compute the amount of fine-tuning
for both of these two different mechanisms. Our results suggest that models
with step-like features can significantly reduce this required tuning of the
parameters.

###  3.1 Inflection Point

####  3.1.1 Description of the Mechanism

Significant peaks in the scalar power spectrum, which are indicative of the
production of PBHs and GWs, can be generated by a near inflection point in the
scalar potential. This inflection point is characterized by the conditions:

$\begin{split}&\frac{dV(\phi_{\text{inf}})}{d\phi_{\text{inf}}}\approx
0,\quad\frac{d^{2}V(\phi_{\text{inf}})}{d\phi_{\text{inf}}^{2}}=0,\end{split}$
(2)

where $\phi_{\text{inf}}$ denotes the inflection point. At this point, the
velocity of the inflaton field significantly decreases, leading to a local
enhancement while the inflaton field remains almost constant. During this
plateau, the power spectrum experiences significant enhancement, facilitating
the production of PBHs. In this work, we also include in this category models
with bump features in the potential, such as those in Mishra and Sahni (2020);
Özsoy and Lalak (2021); Fu and Chen (2023), since they satisfy the conditions
given in Equation (2).

As we mentioned before, the concept of an inflection point in the potential
has been widely adopted in theoretical studies. The degree of fine-tuning
required for inflection point models has been extensively analyzed, notably in
Ref. Cole et al. (2023). According to this study, polynomial potentials appear
to require substantial fine-tuning both to explain the formation of PBHs and
to meet the observational constraints of inflation at CMB scales.

####  3.1.2 the Issue of Fine-Tuning

Among the various models proposing an inflection point, we focus on the one
detailed in Ref. Garcia-Bellido and Ruiz Morales (2017). This model is also
highlighted in Ref. Cole et al. (2023) as one of the models requiring minimal
fine-tuning within the inflection point framework. The potential, as described
in Ref. Garcia-Bellido and Ruiz Morales (2017), is given by:

$V(\phi)=\frac{\lambda}{12}\phi^{2}\upsilon^{2}\left(\frac{6-4a\frac{\phi}{\upsilon}+3\frac{\phi^{2}}{\upsilon^{2}}}{\left(1+b\frac{\phi^{2}}{\upsilon^{2}}\right)^{2}}\right)$
(3)

Here, $a$, $b$, $\lambda$, and $\upsilon$ are the parameters of the model. The
parameter $\lambda$ sets the scale of the power spectrum at CMB scales, while
the others primarily determine the shape of the potential. A significant
enhancement of the power spectrum, capable of facilitating the production of
PBHs, can be achieved with the following specific selection of parameters:

$a=1/\sqrt{2},\quad b={3/2},\quad\lambda=1.86\times
10^{-6},\quad\upsilon=0.19669$ (4)

and for initial condition we assume $\phi_{ic}=2.4719$ (we assume always that
the initial conditions are taken at the pivot scales
($\rm{k=0.05M_{pc}^{-1}}$) in order to make a prompt comparison with the
observable constraints of inflation). This choice of initial condition led to
the following prediction of the spectral index, $\rm{n_{s}}$, and the tensor-
to-scalar ration, $\rm{r}$ (for the evaluation of spectral index $n_{s}$ and
$r$, we assume that $n_{s}\simeq 1+2\eta_{V}-6\varepsilon_{V}$ and $r\simeq
16\varepsilon_{V}$, where
$\varepsilon_{V}=\frac{1}{2}\left({V^{\prime}(\phi)}/{V(\phi)}\right)^{2},\quad\eta_{V}={V^{\prime\prime}(\phi)}/{V(\phi)}$.):

$n_{s}=0.956,\quad r=0.0064\,.$ (5)

In Figure 1, we depict the power spectrum for a small variation of the
parameter $b$, which influences the inflection point and consequently requires
more fine-tuning. This figure clearly demonstrates that the peak value is
highly sensitive to variations in this parameter.

Figure 1: The power spectrum derived from the potential described in Equation
(3), showing the effects of small variations in the parameter b of order
$\mathcal{O}(10^{-6})$. The purple line corresponds to the parameters given in
Equation (4).

To systematically quantify the fine-tuning, we evaluate the quantity of
Equation (1). (The evalution of power spectrum is briefly described in
Appendix A.) In Table 1, we present the evaluation of Equation (1) for the
potential (3). Our results are consistent with those proposed by Cole et al.
Cole et al. (2023) in their study of fine-tuning for the peak of the scalar
power spectrum. The parameter $\rm b$ can be given by the parameter $\rm a$ by
the following expression Garcia-Bellido and Ruiz Morales (2017); Cole et al.
(2023):

$b=1-\frac{a^{2}}{3}+\frac{a^{2}}{3}\left(\frac{9}{2a^{2}}-1\right)^{2/3}\,.$
(6)

This parameter defines the inflection point. In contrast to Cole et al.
(2023), we also evaluate the fine-tuning of the parameter $b$ individually,
raising questions about the necessity of redefining this function and
uncovering hidden fine-tuning, and we found that this parameter need a lot of
fine-tuning. As discussed in Cole et al. (2023), the amount of fine-tuning
required for models with an inflection point is significant; with variations
in the studied models, it can reach an order of magnitude as high as
$\rm{10^{8}}$. The fine-tuning associated with models featuring an inflection
point has also been studied in Stamou (2021); Spanos and Stamou (2023).

Table 1: Analysis of fine-tuning for the potential given in Equation (3). Parameter | $\Delta_{p_{i}}^{P_{R}^{peak}}$
---|---
a | $\rm{7.1\times 10^{2}}$
$\upsilon$ | $\rm{4.6\times 10^{2}}$
b | $\rm{4.2\times 10^{5}}$

In our analysis, we also evaluate the degree of fine-tuning required for the
fractional abundance of PBHs, denoted as $f\rm{{}_{PBH}}$. So, we compute the
$\Delta_{p_{i}}^{f_{\rm{PBH}}}$ from Equation (1). We found that

${\Delta_{p_{i}}^{f_{\rm{PBH}}}}/{\Delta_{p_{i}}^{P_{R}^{peak}}}\approx 30$
(7)

which indicates that achieving the desired fractional abundance necessitates
fine-tuning that is 1–2 orders of magnitude more stringent than that required
for the parameters outlined in Table 1. This elevated level of fine-tuning
underscores the sensitivity of PBH formation to the specific dynamics of the
inflationary model parameters. The evaluation of the fractional abundance of
PBHs, $f_{\text{PBH}}$, is detailed in Appendix B. In this Appendix, we also
present the fractional abundances of PBHs for all the mechanisms discussed in
this work.

###  3.2 Step-Like Potential

####  3.2.1 An Alternative Single-Field Mechanism

In the pursuit of models that require minimal fine-tuning, the step-like
potential emerges as a promising approach in single-field inflationary
theories Kefala et al. (2021); Dalianis et al. (2021). This potential reads
as:

$V(\phi)=V_{0}\left(1+\frac{1}{2}\sum_{i}A_{i}\left(1+\tanh\left[c_{i}\left(\phi-\varphi_{i}\right)\right]\right)+B\phi\right)\,,$
(8)

where $V_{0}$ is fixed for the requirement of the amplitude of power spectrum
at CMB scales; $\phi$ is the inflaton field; and $A_{i}$, $B$, $c_{i}$, and
$\varphi_{i}$ are parameters that define the amplitude, steepness, and
position of the steps, respectively. These steps introduce localized features
in the potential, enabling significant enhancements in the scalar power
spectrum.

A step-like feature in the inflaton potential results in a rapid decrease in
potential energy and leads to spectral distortions Kefala et al. (2021);
Dalianis et al. (2021); Dalianis (2023). Such features, where the potential
energy abruptly transitions from one constant value to another at one or more
points, can significantly enhance the spectrum over a specific range of
scales.

The distinctive behavior of step-like features in the potential contrasts
markedly with other inflationary mechanisms, such as the inflection point
mentioned previously. While the inflection point leads to a temporary stop in
the field’s motion, leading to a spike in the power spectrum due to prolonged
inflation at specific field values, a step-like feature triggers accelerations
and decelerations in the inflaton’s velocity. For more details on the
differences between these mechanisms, see Appendix A. Unlike the inflection
point, the enhancement in the power spectrum in a step-like potential does not
rely on the minimum value of the field’s velocity. For this reason, we
anticipate that this mechanism can reduce the amount of fine-tuning required.

####  3.2.2 Studying the Fine-Tuning

The potential described in Equation (8) can be integrated into the
$\alpha$-attractor framework of supergravity, providing a reliable model that
adheres to the observable constraints of the CMB, and can lead to an
enhancement in the power spectrum Dalianis et al. (2021). However, this model
does not achieve a significant enough enhancement of the power spectrum for
the production of PBHs. For this reason, in our study focusing on assessing
fine-tuning, we modified the well-known Starobinsky potential to incorporate
step-like features. Hence, in our study, we have adopted the following form of
the potential that aligns with the observable constraints:

$V(\phi)=(1-e^{-d\phi})^{2}\left(V_{0}+\sum_{i=0}^{N}B_{i}\left(-1+\tanh[c(\phi-\varphi_{i})]\right)\right)$
(9)

where $\rm{V_{0}}$, $\rm{c}$, $\rm{d}$, and ${\varphi_{i}}$ are parameters.
The parameters $\rm{V_{0}}$ and $\rm{d}$ ensure that the potential with
respect to the CMB constraints of inflation and the others identifies the
steepness and the position of the step as previously.

In this study, we focus on the potential given in Equation (9) to explore a
substantial enhancement of the scalar power spectrum. The following parameters
are proposed to achieve such an enhancement at $\mathcal{O}(10^{-2})$ orders
of magnitude:

$\begin{split}V_{0}=&4\times 10^{-9},\quad B_{i}=0.05(V_{0}/2),\quad
d=0.76\quad c=1000\\\
\varphi_{0}=&0,\quad\varphi_{1}=5,\quad\varphi_{2}=5.2,\quad\varphi_{3}=5.4,\quad\varphi_{4}=5.6\quad\,.\end{split}$
(10)

For the initial condition, we assume $\phi_{ic}=6$. The prediction for the
spectral index, $\rm{n_{s}}$, and the tensor to scalar ratio, $\rm{r}$, are
given, respectively, as:

$n_{s}=0.971,\quad r=0.0026\,$ (11)

and they respect the observable constraints of inflation Akrami et al. (2020).

Potential (9) includes the four steps necessary to achieve the desired
enhancement level. The dependence of the power spectrum’s peak height on the
parameter $c$ is depicted in Figure 2. The selection of this potential
facilitates the formation of PBHs at $10^{-6}M_{\odot}$. In Appendix B, there
is a detailed analysis for this mass value and the corresponding fractional
abundances of PBH. Different outcomes can be achieved by varying the choice of
the parameter. A proper choice of parameters that can lead to significant
results is presented in a subsequent section.

Figure 2: The power spectrum derived from the potential described in Equation
(9) for various choices of the parameter c.

As previously described, we evaluate the parameter
$\Delta_{p_{i}}^{P_{R}^{peak}}$, defined in Equation (1), and present our
findings in Table 2. We study the parameters ${B_{i}}$, ${c}$, and ${d}$,
which are responsible for the step features. We also examine the distance of
the steps as $|\phi_{i+2}-\phi_{i+1}|$, where $\rm{i=0,1}$, etc. As we can
notice by Table 2, the parameter that needs more fine-tuning in this case is
the distances of the steps $|\varphi_{i+2}-\varphi_{i+1}|$. However, it is
noteworthy that this mechanism exhibits a substantial reduction in the amount
of fine-tuning compared to the earlier case of an inflection point in the
scalar potential, see Tables 1 and 2.

Table 2: Analysis of fine-tuning for the potential given in Equation (9). Parameter | $\Delta_{p_{i}}^{P_{R}^{peak}}$
---|---
$B_{i}/V_{0}$ | 18.2
$c$ | 3.10
$d$ | 53.5
$|\varphi_{i+2}-\varphi_{i+1}|$ | $80.6$

As we observe in this section, the distinct dynamics of single-field
mechanisms—such as the inflection point and step-like features—lead to
localized enhancements in the power spectrum. Models with step-like features
in their potentials are characterized by sharper peaks and additional
oscillatory patterns, unlike the smoother power spectra seen in other
potential forms, such as the previously discussed inflection point. Such
localized enhancements are crucial for producing PBHs within narrower mass
ranges, thus influencing their distinct characteristics (see Appendix B)

Finally, it is crucial to acknowledge that both of the single-field mechanisms
discussed have faced scrutiny regarding one-loop corrections to the evaluation
of the power spectrum, as highlighted in studies such as Refs. Kristiano and
Yokoyama (2022); Fumagalli et al. (2023); Choudhury et al. (2023a, b);
Choudhury (2024); Firouzjahi (2023); Inomata et al. (2023). Consequently,
exploring multi-field inflationary models could not only reduce the amount of
fine-tuning but also offer a viable alternative for the study of PBHs.

##  4 Multi-Field Models

Multi-field inflationary models have been extensively studied for their role
in generating PBHs. In the context of multi-field mechanisms, it is
anticipated that the degree of fine-tuning required will be lessened. This is
expected because the constraints imposed by the CMB power spectrum and the
demands for PBH production may not be fully addressed by a single-field.
However, within multi-field inflation, various mechanisms may lead to
differing degrees of fine-tuning, necessitating further investigation to
determine their efficacy. In this section, we discuss the amount of fine-
tuning required in multi-field scenarios, focusing on two distinct mechanisms:
one with a non-canonical kinetic term Aldabergenov et al. (2022); Braglia et
al. (2020); Pi and Sasaki (2023); Gundhi et al. (2021); Geller et al. (2022);
Chen et al. (2023); Wang et al. (2024) and another with a canonical kinetic
term accompanied by a waterfall trajectory Afzal and Ghoshal (2024); Spanos
and Stamou (2021); Braglia et al. (2023); Clesse and García-Bellido (2015);
Tada and Yamada (2023); Dimopoulos (2022).

###  4.1 Two Field Model with a Non-Canonical Kinetic Term

In this subsection, we introduce a two-field model incorporating a non-
canonical kinetic term. The enhancement of the power spectrum in this model
arises through a large non-canonical kinetic coupling within a two-field
inflationary framework. This mechanism is characterized by a temporary
tachyonic instability of the isocurvature perturbations during the transition
between two distinct inflationary phases Aldabergenov et al. (2022); Braglia
et al. (2020); Gundhi et al. (2021); Geller et al. (2022). Such instability
leads to a significant amplification of the curvature perturbations, which can
be imprinted in both the production of PBHs and GWs .

Although numerous models with a non-canonical kinetic term have been proposed
in the literature, we specifically focus on the one detailed in Ref. Braglia
et al. (2020). The action for this two-field toy model is expressed as:

$S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R-\frac{1}{2}(\partial\phi)^{2}-e^{-b_{1}\phi}(\partial\chi)^{2}+V(\phi,\chi)\right]\,.$
(12)

In this model, the inflaton field $\phi$ interacts with a non-canonical scalar
field $\chi$ via an exponential dilaton-like coupling Kallosh and Linde
(2022). The potential is defined by:

$V(\phi,\chi)=V_{0}\frac{\phi^{2}}{\phi_{0}^{2}+\phi^{2}}+\frac{m_{\chi}^{2}}{2}\chi^{2}$
(13)

Here, $b_{1}$ is a coupling parameter between the fields and $V_{0}$,
$\phi_{0}$, and $m_{\chi}$, which are the model parameters. We adopt the
parameter choices from Braglia et al. (2020) as follows:

$V_{0}/(m_{\chi}M_{P})^{2}=500\quad\text{and}\quad\phi_{0}=\sqrt{6}M_{P}.$
(14)

For the initial conditions, we have:

$\phi_{ic}=7,\quad\text{and}\quad\chi_{ic}=6.55$ (15)

as they lead to acceptable values for the prediction of the spectral index,
$\rm{n_{s}}$, and the tensor to scalar ratio, r Braglia et al. (2020); Akrami
et al. (2020).

In Figure 3, we depict the power spectrum for various settings of the
parameter $b_{1}$. It is evident that the enhancement of the power spectrum
varies with the choice of this parameter. However, the impact of other
parameters on the power spectrum also requires examination. Therefore, in
Table 3, we present the degree of fine-tuning by evaluating Equation (1) for
all parameters, including the potential parameters $m_{\chi}$ and $\phi_{0}$.
It should be noted that the parameter $V_{0}$ does not influence the dynamics
of the fields, and for this reason, it is not included in this table.

Figure 3: The power spectrum derived from Equations (13) and (12) for various choices of $b_{1}$ present in the kinetic term.  Table 3: Analysis of fine-tuning for the potential given in Equations (13) and (12). Parameter | $\Delta_{p_{i}}^{P_{R}^{peak}}$
---|---
$m_{\chi}$ | 1.72
$b_{1}$ | 71.7
$\phi_{0}$ | 3.02

The results presented in Table 3 indicate that fine-tuning in multi-field
inflationary models can be significantly reduced compared to single-field
inflationary models, particularly those with an inflection point. For a more
detailed analysis, it is also essential to examine how the initial conditions
of the fields may influence these results.

###  4.2 Hybrid Model

In this section, we delve into a hybrid model characterized by a waterfall
trajectory. The key aspect of this mechanism is the tachyonic instability that
occurs during the transition between two stages of inflation. This instability
results in a significant enhancement of the power spectrum at scales
corresponding to the transition. Unlike the previous mechanism where the
instability arises from strong kinetic coupling, here the instability is
triggered when the inflaton field reaches a critical point, leading the so-
called waterfall field to develop a tachyonic solution. In contrast to the
previous mechanism requiring intricate parameter setups, this hybrid model
naturally derives its instability solely from the field’s potential, enhancing
the power spectrum in a more predictable manner. This approach to PBH
production was first proposed in Clesse and García-Bellido (2015).

Before presenting a specific model for this mechanism, it is important to
outline some fundamental aspects of this framework. The hybrid model is
derived from a globally supersymmetric (SUSY) renormalizable superpotential
Copeland et al. (1994); Dvali et al. (1994):

$W=\kappa S(\bar{\Psi}_{1}\Psi_{2}-m^{2}).$ (16)

The $F$-term SUSY potential is given by:

$V_{F}^{\text{SUSY}}=\kappa^{2}\left[(\psi^{2}-m^{2})^{2}+\phi^{2}\psi^{2}\right],$
(17)

where we have set $|S|=\phi/\sqrt{2}$ and $|\Psi_{1}|=|\bar{\Psi}_{2}|=\psi$.
The field $\psi$ develops tachyonic solutions under the condition:

$\kappa^{2}(-2m^{2}+\phi^{2}+6\psi^{2})<0.$

Along the flat direction ($\psi=0$), the condition becomes:

$\phi^{2}<\phi_{c}^{2}=2m^{2}\equiv M^{2},$

where $\phi_{c}$ represents the critical value of the field $\phi$, beyond
which the field $\psi$ becomes tachyonic. The field $\psi$ is regarded as the
waterfall field and the field $\phi$ as the inflaton.

A variety of hybrid models have been explored within the literature to
elucidate the production of PBHs Afzal and Ghoshal (2024); Spanos and Stamou
(2021); Braglia et al. (2023); Clesse and García-Bellido (2015); Tada and
Yamada (2023); Dimopoulos (2022). We study the fine-tuning aspects of hybrid
models, with a particular focus on the approach outlined in Ref. Braglia et
al. (2023). We selected this model due to the unique characteristics of its
two-field potential, which not only serves as an attractor for the initial
conditions of both fields but also ensures that, regardless of these starting
conditions, the trajectory consistently leads to the potential’s plateau. Such
convergence is crucial as it initiates a slow-roll phase, an important element
in the dynamics of the model. The potential is given as follows:

$V(\phi,\psi)=M^{2}\left[\frac{(\psi^{2}-\psi_{0}^{2})^{2}}{4\psi_{0}^{2}}+\frac{m^{2}}{2}\phi^{2}+\frac{g}{2}\phi^{2}\psi^{2}+d\psi\right],$
(18)

where $M$, $m$, $g$, $d$, and $\psi_{0}$ are parameters. For the parameters we
have the following:

$M=1.47\times 10^{-5},\quad m=0.3,\quad g=0.8,\quad\psi_{0}=2.5,\quad
d=-5\times 10^{-6}$ (19)

and for initial conditions

$\phi_{ic}=3.4,\quad\psi_{ic}=0\,.$ (20)

Hence, this model aligns with the observable constraints of inflation Akrami
et al. (2020); Braglia et al. (2023).

In Figure 4, we depict the scalar power spectrum derived from the potential in
Equation (18) for various settings of the parameter $d$. This plot illustrates
that extensive fine-tuning is not necessary to achieve a significant peak in
the power spectrum. As in previous mechanisms of PBHs, we analyze the impact
of other parameters on the power spectrum. The parameters $m$ and $g$
influence the position of the power spectrum peak. We assess the amount of
fine-tuning using Equation (1) and present our results in Table 4.

Figure 4: The power spectrum derived from Equation (18) for various choices of parameter $\rm{10^{6}|d|}$ .  Table 4: Analysis of fine-tuning for the potential given in Equation (18) . Parameter | $\Delta_{p_{i}}^{P_{R}^{peak}}$
---|---
$m$ | 5.2
$g$ | 0.6
$d$ | 1.9
$\psi_{0}$ | 9.5

As one can notice from Table 4, this mechanism significantly reduces the
amount of fine-tuning required, compared to both the single-field inflationary
mechanism and the two-field mechanism with a non-canonical kinetic term. An
important notice here is that this mechanism is not dependent on the initial
condition of the fields; hence, it can provide us with a more natural
explanation of PBH production.

##  5 Spectator Field

From our previous analysis, it is evident that multi-field inflationary models
can significantly reduce the fine-tuning required for the production of PBHs.
Additionally, the implementation of a spectator field as a solution to this
issue has been extensively explored in the literature Kohri et al. (2013);
Stamou and Clesse (2024, 2023); Stamou (2024), offering a promising way to
circumvent the problem.

We specifically examine the model presented in Ref. Stamou and Clesse (2024),
where the spectator field experiences quantum stochastic fluctuations. These
fluctuations lead the field to assume distinct mean values across different
Hubble patches. Considering the vast number of these patches, it is inevitable
that some will exhibit conditions allowing the spectator field to attain
values necessary for quantum fluctuations to generate significant curvature
fluctuations. The potential of the spectator field is described by:

$V(\psi)=\Lambda^{4}\left(1-\exp\left[-\frac{\psi}{M}\right]\right)$ (21)

where $\Lambda$ and $M$ are parameters.

In this work, we further evaluate the amount of fine-tuning previously
assessed for models with a spectator field. We employ the fine-tuning measure
defined in Equation (1) as we did before. This particular mechanism does not
require an enhancement in the scalar power spectrum, thereby eliminating the
need to consider peaks in our evaluation. Instead, we focus on the fractional
abundance of PBHs ($f_{\rm PBH}$) as our primary quantity $Q$. We specifically
analyze the parameter $M$ as the parameter $\Lambda$ merely sets the scale of
the spectator field and does not influence the dynamics. We found that
$\Delta_{M}^{f_{\rm{PBH}}}\sim\mathcal{O}(1)$. Therefore, in this mechanism we
can avoid the need for fine-tuning in order to explain the production of PBHs.

We need to remark here that the stochastic nature of the spectator field
during inflation plays a crucial role in suppressing fine-tuning. In this
mechanism, variations in the model’s single parameter $\rm M$, which is of the
same order as the Hubble parameter at CMB scales, can be effectively
compensated for by the specific inflationary model we study each time. A
detailed analysis can be found in Stamou and Clesse (2024).

##  6 Step-Features Potential over Inflection-Point: an Additional Advantage
in Implementation of Gws

In this section, we explore the implementation of a step-like feature
potential for the production of PBHs. We demonstrate how this mechanism can
not only lead to a significant enhancement and thus to the copious production
of PBHs but also potentially explain the signal of GWs at low frequencies.

To begin with, this approach facilitates a significant enhancement of the
power spectrum while substantially reducing the need for fine-tuning.  In this
section, we consider the potential given in Equation (9) . The reinforcement
of the power spectrum is depicted in Figure 5. For this plot, we assume the
following set of parameters:

$\begin{split}c=3000,\quad d=0.73,\quad B_{i}/(V_{0}/2)=0.049,\\\
|\phi_{i+2}-\phi_{i+1}|=0.2,\quad{\phi_{0}=0},\quad{\phi_{1}=5.08}\end{split}$
(22)

and for initial condition we have $\phi_{ic}=6$. Hence, the prediction of
spectral index and tensor-to-scalar ratio is given as follows:

$n_{s}=0.96839,\quad r=0.00354\,\,$ (23)

which are in complete accordance with the observable constraints of inflation
Akrami et al. (2020). We note here that the number of efolds we obtain for
this set of parameters is $N_{tot}-N_{CMB}\approx 50$, where $N_{CMB}$ denotes
the number of efolds at the pivot scale.

Figure 5: The power spectrum derived from the potential given in Equation (9),
with the choice of parameters given in (22).

For this enhancement, the mass of PBHs is about $10^{-1}M_{\odot}$, with the
formalism of this evaluation presented in Appendix B. It is noteworthy to
mention here that recently a few intriguing PBHs in low mass ratio binaries
have been reported in GW observations Abbott et al. (2023); Phukon et al.
(2021). Therefore, the predicted range of masses of PBHs from this step-like
mechanism can be related to the aforementioned searches, but then it needs
further investigation.

In Figure 6, we depict the energy density of GWs, denoted as $\Omega_{GW}$,
using the specific parameters discussed in this section. The calculation of
$\Omega_{GW}$ is detailed in Appendix C. The gray dots on the plot represent
the detection sensitivity of the NANOGrav collaboration. Therefore, this
mechanism has the interesting characteristic that it can be imprinted in the
GW signals detected by NANOGrav Agazie et al. (2023a, b); Afzal et al. (2023).
It is important to note that the parameter values examined in Figure 6 do not
cover the energy densities of GWs observed at larger frequencies, such as
those from LVK collaboration Abbott et al. (2016a, b, 2017a, 2017b, 2017c);
Dwyer et al. (2015) or those expected from future space-based experiments like
LISA Amaro-Seoane et al. (2017) and DECIGO Seto et al. (2001) or ground-based
observatories such as ET Maggiore et al. (2020). However, adjusting the
parameter scans may yield energy densities that can explain these larger-scale
signals. For a detailed review about the induced GWs and the frequency regions
of future and present experiments, see Ref. Domènech (2021).

Figure 6: The energy density of GWs for the potential given in Equation (9),
with the choice of parameters given in (22).

Finally, we observe that, in addition to this enhancement, depicted in Figure
5, there are also additional oscillations that could be imprinted in the
energy densities of the induced GWs shown in Figure 6. These oscillations can
potentially lead to a characteristic signal Fumagalli et al. (2021);
Mavromatos et al. (2022). Moreover, we need to mention here that in this work
we neglect the possible effects of non-Gaussianities Cai et al. (2019); Perna
et al. (2024); Liu et al. (2024); Namjoo et al. (2013). Consequently, this
mechanism needs further investigation and should be the focus of a future
study.

##  7 Conclusions

This study has provided an extensive evaluation of fine-tuning requirements
across various inflationary models for the production of PBHs. We have
demonstrated that while single-field models with inflection points require
significant fine-tuning to align with CMB observations and produce PBHs,
alternative models with step-like features in the scalar potential present a
more favourable scenario. These models significantly reduce the need for fine-
tuning and are capable of explaining the potential low-frequency GW signals
detectable by PTA collaborations.

Further exploration into multi-field models reveals that the addition of extra
fields can considerably alleviate the fine-tuning demands. Specifically,
models incorporating non-canonical kinetic terms and hybrid inflationary
models with a waterfall trajectory have shown promising results, reducing the
fine-tuning requirements by orders of magnitude compared to single-field
models.

Moreover, the implementation of a spectator field offers a novel approach to
circumvent the fine-tuning issue entirely. These models allow for the natural
stochastic variations in field values across different Hubble patches, which
can lead to the significant curvature fluctuations necessary for PBH
production without stringent constraints on the inflationary potential’s
parameters.

To sum up, we examined several inflationary models for the production of PBHs
and classified them based on the degree of fine-tuning required for explaining
their copious production. We provided a comparative analysis of single- and
multi-field inflationary mechanisms, highlighting the advantages of multi-
field approaches in reducing the need for fine-tuning. These insights help
expand the theoretical framework connecting inflation, PBHs, and GWs. The
novelty of this study lies in its contribution to guiding future research
toward models that offer a more natural explanation for the formation of PBHs,
thus paving the way for more grounded investigations in this area.

The work of I.S. from the Belgian Francqui Foundation through a Francqui
Start-up Grant, as well as the Belgian Fund for Scientific Research through an
MIS grant and an IISN grant.

No new data were created or analyzed in this study. Data sharing is not
applicable to this article.

The authors declare no conflicts of interest.

no

## Appendix A

In this Appendix A, we summarize the evaluation of the power spectrum that is
needed in our analysis. Additionally, we outline several noteworthy
characteristics of the models explored in this study.

In this framework, the field equations for ${\varphi^{i}}$, expressed in terms
of efold time (assuming ${M_{P}=1}$), are provided by the following equation:

$\ddot{\varphi}^{c}+\Gamma_{ab}^{c}\dot{\varphi}^{a}\dot{\varphi}^{b}+\left(3-\frac{1}{2}\dot{\sigma^{2}}\right)\dot{\varphi}^{c}+\left(3-\frac{1}{2}\dot{\sigma^{2}}\right)\frac{V^{c}}{V}=0$
(24)

where the dots indicate derivatives with respect to efold time, and the
indices ${i=a,b,c}$ denote derivatives relative to the fields. The Christoffel
symbols $\Gamma^{c}_{ab}$ are defined as:

$\Gamma^{c}_{ab}=\frac{1}{2}G^{cd}(G_{da,b}+G_{db,a}-G_{ab,d}),$ (25)

where ${G_{ij}}$ represents the field metric. The velocity field
$\dot{\sigma}$ is calculated as:

$\dot{\sigma}^{2}=G_{ab}\dot{{\varphi}^{a}}\dot{{\varphi}^{b}}.$ (26)

The Hubble parameter is defined by:

$H^{2}=\frac{V}{3-\frac{1}{2}\dot{\sigma^{2}}}$ (27)

and the slow-roll parameter $\varepsilon_{1}$ is determined from:

$\varepsilon_{1}=\frac{1}{2}\dot{\sigma^{2}}.$ (28)

The perturbation equations for the fields ${\delta\varphi}$ are given by:

$\begin{split}&\delta\ddot{\varphi}^{c}+(3-\varepsilon_{1})\delta\dot{\varphi}^{c}+2\Gamma^{c}_{ab}\dot{\varphi}^{a}\delta\dot{\varphi}^{b}+\left(\Gamma^{c}_{ab,d}\dot{\varphi}^{a}\dot{\varphi}^{b}+\frac{V^{c}_{d}}{H^{2}}-G^{ca}G_{ab,d}\frac{V^{b}}{H^{2}}\right)\delta\varphi^{d}\\\
&+\frac{k^{2}}{a^{2}H^{2}}\delta\varphi^{c}=4\dot{\Psi}\dot{\varphi}-2\Psi\frac{V^{c}}{H^{2}}\end{split}$
(29)

and the equation for the Bardeen potential $\Psi$ is:

$\ddot{\Psi}+(7-\varepsilon_{1})\dot{\Psi}+\left(2\frac{V}{H^{2}}+\frac{k^{2}}{a^{2}H^{2}}\right)\Psi=-\frac{V_{c}}{H^{2}}\delta\varphi^{c}.$
(30)

We assume the initial condition of a Bunch–Davies vacuum. The power spectrum
is described by:

$P_{R}=\frac{k^{3}}{(2\pi)^{2}}(R_{i}^{2})$ (31)

where:

$R_{i}=\Psi+H\sum_{i}^{N_{\text{fields}}}\frac{\dot{\varphi_{i}}\delta\varphi_{i}}{\dot{\varphi_{i}}^{2}}.$
(32)

In the case of single-field inflation $\varphi^{1}=\phi$, the equation of the
curvature perturbations takes the following form Kefala et al. (2021):

${R^{\prime\prime}}_{k}+f(N)R^{\prime}_{k}+\frac{k^{2}}{H^{2}}e^{-2N}R_{k}=0$
(33)

where

$f(N)=3+2\frac{\phi^{\prime\prime}}{\phi^{\prime}}-\frac{\phi^{\prime
2}}{2}\,.$ (34)

This function acts as a friction term.

The potential in Equation (8) can interplay between the mechanism of the
inflection point and that of the step-like potential. In Figure 7, we
illustrate the behavior of the slow-roll parameter $\varepsilon_{1}$, Equation
(28), and the $f$ from Equation (34) for two different scenarios: an
inflection point (blue curve) and a step-like potential (orange line), using
the potential specified in Equation (8). The parameters for each scenario are
provided in the figure. This plot highlights the distinct dynamics between
these two mechanisms; in the inflection point scenario, the velocity of the
field significantly decreases, whereas in the step-like potential, the field’s
velocity exhibits both increases and decreases. Moreover, the function $f$
behaves differently in the cases of the inflection point and step-like
potential. Hence, through this figure we can notice that the previously
mentioned mechanisms behave differently.

Figure 7: The different behavior of the slow-roll parameter $\varepsilon_{1}$
(left panel) and friction term $f$ (right panel) for the mechanism of an
inflection point (blue line) and the one with the step-like potential (orange
line). For blue line: $i=1$, $A_{1}=0.000605$, $c=100$, $B=-0.03$. For orange
line: $i=3$, $A_{1}=0.1$, $c=100$ $B=0.03$, $\varphi_{1}=0.4$, and $\varphi
2=0.8$. For initial condition, $\phi_{ic}=-0.3$.

## Appendix B

In Appendix B, we briefly discuss the methodology to evaluate the fractional
abundance of PBHs. First, we outline the theoretical framework and
computational approaches used to estimate the PBH abundance. Subsequently, we
present the fractional abundances calculated for the various inflationary
models discussed within this work. Finally, we discuss these results.

The fractional abundance of PBH is given by the integral:

$f_{PBH}=\int d\ln M_{PBH}\frac{\Omega_{PBH}}{\Omega_{DM}}$ (35)

where

$\frac{\Omega_{PBH}}{\Omega_{DM}}=\frac{\beta(M_{\text{PBH}}(k))}{8\times
10^{-16}}\left(\frac{\gamma}{0.2}\right)^{3/2}\left(\frac{g(T_{f})}{106.75}\right)^{-1/4}\left(\frac{M_{\text{PBH}}(k)}{10^{-18}\textsl{g}}\right)^{-1/2}.$
(36)

Here, $\gamma$ is a factor that depends on gravitation collapse, and we choose
$\gamma=0.2$ Carr (1975). $T_{f}$ denotes the temperature of PBH formation,
and $M_{\text{PBH}}$ is the mass of PBHs. $g(T_{f})$ are the effective degrees
of freedom during this formation, and counting only the SM particles we set
$g(T_{f})=106.75$.

The mass fraction of PBHs at formation, denoted by $\beta(M_{\text{PBH}})$,
can be estimated using either the Press–Schechter (PS) approach Press and
Schechter (1974) or Peak Theory Bardeen et al. (1986). In this section, we
focus on the PS approach. Hence, the mass fraction of $\beta(M_{\text{PBH}})$
is given as:

$\beta(M_{\text{PBH}})=\frac{1}{\sqrt{2\pi\sigma^{2}(M_{\text{PBH}})}}\int^{\infty}_{\delta_{c}}d\delta\,\exp\left(-\frac{\delta^{2}}{2\sigma^{2}(M_{\text{PBH}})}\right)\,,$
(37)

where the overdensity $\delta$ is above a certain threshold of collapse
$\delta_{c}$. The variance of curvature perturbation is given :

$\sigma^{2}\left(M_{\text{PBH}}(k)\right)=\frac{16}{81}\int\frac{dk^{\prime}}{k^{\prime}}\left(\frac{k^{\prime}}{k}\right)^{4}P_{\mathcal{R}}(k^{\prime})W^{2}\left(\frac{k^{\prime}}{k}\right),$
(38)

where W is a window function.

The mass as a function of the comoving wavenumber $k$ Ballesteros and Taoso
(2018):

$M_{PBH}=10^{18}\left(\frac{\gamma}{0.2}\right)\left(\frac{g(T_{f})}{106.75}\right)^{-1/6}\left(\frac{k}{7\times
10^{13}Mpc^{-1}}\right)^{-2}\textsl{g}\,.$ (39)

Figure 8: The fractional abundances of PBHs for the four different mechanisms:
inflection point (blue), step-like feauture potential (red), two field with a
non-canonical kinetic term (orange), and hybrid model (green).

In Figure 8, we illustrate the fractional abundances of PBHs for various
mechanisms: an inflection point, step-like features in the potential, a two-
field model with non-canonical kinetic terms in the Lagrangian, and a two-
field model with a waterfall trajectory, each represented by distinct colors
as detailed in the caption. We use the parameter sets presented in Figures
1–4, which lead to an enhancement of the scalar power spectrum of at least
$\rm{10^{-2}}$. We can notice that models with step-like potential can lead to
narrower mass ranges. The models shown in Figure 8 can account for a wide
range of PBH masses. However, it is important to emphasize that different
parameter choices can yield, varying fractional abundances; therefore, this
plot should not be solely relied upon for comparing these mechanisms.
Furthermore, the fractional abundances can exceed constraints, such as those
given in Refs. Capela et al. (2013); Niikura et al. (2019). As mentioned, the
proper adjustment of parameters for each model is required, but this is beyond
the scope of this paper.

For the evaluation of fractional abundance depicted in Figure 8, we adopt
critical threshold values ranging from $\delta_{c}=0.4$ to $0.6$ as referenced
in Ref. Musco et al. (2021); Escrivà et al. (2021, 2022). It is crucial to
compute the threshold value from the respective power spectrum for each
scenario, as outlined in Musco et al. (2021); Escrivà et al. (2021, 2022);
Stamou (2023). Nonetheless, by appropriately adjusting the model parameters,
one can achieve the desired fractional abundances. The choice of window
function, specifically a Gaussian window function as applied in this analysis
(see Equation (38)), significantly influences the threshold calculation.
Although the choice of window function introduces corrections in the threshold
computation, these impacts are minimized when the same window function is
consistently used across evaluations of PBH abundance Kalaja et al. (2019);
Young (2019); Tokeshi et al. (2020); Musco et al. (2021).

## Appendix C

In this Appendix, we briefly discuss how to evaluate the energy density of GWs
from the scalar power spectrum. The stochastic background of GWs can be
generated through the coupling of the scalar and tensor modes Espinosa et al.
(2018); Kohri and Terada (2018); Mollerach et al. (2004); Maggiore (2000);
Baumann et al. (2007).

The energy density of the GWs can be evaluated from the following integral:

${\Omega_{GW}(k)}=\frac{\Omega_{r}}{36}\int^{\frac{1}{\sqrt{3}}}_{0}\mathrm{d}d\int^{\infty}_{\frac{1}{\sqrt{3}}}\mathrm{d}s\left[\frac{(s^{2}-1/3)(d^{2}-1/3)}{s^{2}+d^{2}}\right]^{2}\,P_{R}(kx)\,P_{R}(ky)\,(I_{c}^{2}+I_{s}^{2})\,,$
(40)

where the radiation density $\Omega_{r}\approx 5.4\times 10^{-5}$. The
variables $x$ and $y$ are

$x=\frac{\sqrt{3}}{2}(s+d),\quad y=\frac{\sqrt{3}}{2}(s-d).$ (41)

The functions $I_{c}$ and $I_{s}$ are given by the expressions

$\displaystyle
I_{c}=-36\pi\frac{(s^{2}+d^{2}-2)^{2}}{(s^{2}-d^{2})^{3}}\Theta(s-1)$ (42)
$\displaystyle
I_{s}=-36\frac{(s^{2}+d^{2}-2)^{2}}{(s^{2}-d^{2})^{2}}\left[\frac{(s^{2}+d^{2}-2)}{(s^{2}-d^{2})}\log\left|\frac{d^{2}-1}{s^{2}-1}\right|+2\right].$
(43)

Using the fact that

$1\,{\rm Mpc}^{-1}=0.97154\times\,10^{-14}\,{\rm s}^{-1}$ and $k=2\pi\rm{f}$

we can evaluate the energy density of the power spectrum as a function of the
frequency $\rm{f}$.

References

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