# Cross-lingual Alzheimer’s Disease detection based on paralinguistic and pre-
trained features

###### Abstract

We present our submission to the ICASSP-SPGC-2023 ADReSS-M Challenge Task,
which aims to investigate which acoustic features can be generalized and
transferred across languages for Alzheimer’s Disease (AD) prediction. The
challenge consists of two tasks: one is to classify the speech of AD patients
and healthy individuals, and the other is to infer Mini Mental State
Examination (MMSE) score based on speech only. The difficulty is mainly
embodied in the mismatch of the dataset, in which the training set is in
English while the test set is in Greek. We extract paralinguistic features
using openSmile toolkit and acoustic features using XLSR-53. In addition, we
extract linguistic features after transcribing the speech into text. These
features are used as indicators for AD detection in our method. Our method
achieves an accuracy of 69.6% on the classification task and a root mean
squared error (RMSE) of 4.788 on the regression task. The results show that
our proposed method is expected to achieve automatic multilingual Alzheimer’s
Disease detection through spontaneous speech.

Index Terms—  Alzheimer’s dementia detection, paralinguistic features, pre-
trained features

##  1 Introduction

Alzheimer’s Disease (AD) is a neurodegenerative disease. When suffering from
AD, pathological changes occur in patients’ brain, resulting in cognitive
decline, expression degradation and other phenomena. Patients in different
countries have similar symptoms. Clinical research shows that early treatment
can delay the deterioration of AD. Therefore, the development of AD detection
approach is crucial for the treatment of this disease. Although some
researchers have carried out AD detection tasks on a single language [1],
there are relatively few studies on cross-lingual AD detection. The academia
still lacks a unified understanding of which speech features can be used for
cross-lingual AD detection. The ICASSP-SPGC-2023 ADReSS-M challenge task aims
to investigate how to extract generic acoustic features from speech for
multilingual AD detection [2].

We focus on paralinguistic features and pre-trained features during the
research. On the one hand, paralinguistic features have been proved reliable
for distinguishing AD patients from healthy individuals [3]. As a result of
the disease, the patient’s paralinguistic features have changed, and this
change should be similar despite the patient’s different language. On the
other hand, monolingual pre-trained models such as Wav2vec2 and BERT have
shown their effectiveness in monolingual AD detection [4], and there are
various multilingual pre-trained models that perform well in different
downstream tasks. In addition, previous studies have confirmed that speeches
of AD patients have linguistic features including fewer verbal real words, but
more imaginary words, pauses and hesitations [5]. We therefore believe
multilingual pre-trained models can extract effective pre-trained features
from speeches and their transcribed texts. For this reason, we propose a
cross-lingual AD detection method based on paralinguistic and pre-trained
features.

The paper is organized as follows. The second section introduces our proposed
methods. The third section shows the results. And the last section is a
summary of this paper.

##  2 methodology

In this section, we introduce our proposed AD detection methods based on
paralinguistic and pre-trained features. The general flow diagram is shown in
Fig. 1, which will be introduced in detail later.

Fig. 1: The general flow diagram of proposed methods. (FC: fully connected
layer; SVM: support vector machine.)

###  2.1 Paralinguistic features based approach

Table 1: Results of the classification task No. | Methods | Accuracy | Precision | Recall | F1
---|---|---|---|---|---
1 | IS10-Paralinguistics-compat + SVM | $0.6957$ | $0.7273$ | $0.6667$ | $0.6957$
2 | IS10-Paralinguistics + SVM | $0.6957$ | $0.6923$ | $0.75$ | $0.72$
3 | IS11-speaker-state + SVM | $0.6957$ | $0.75$ | $0.625$ | $0.6818$
4 | Whisper + RoBERTa + FC | $0.6522$ | $0.8333$ | $0.4167$ | $0.5556$
5 | XLSR-53 + FC | $0.5435$ | $0.5429$ | $0.7917$ | $0.6441$

Research shows that paralinguistic features can be used as an important AD
detection indicator for a single language [3]. We use openSmile [6] toolkit to
extract paralinguistic features, which include: (a) IS10-Paralinguistics-
compat feature set, (b) IS10-Paralinguistics feature set, and (c)
IS11-speaker-state feature set. In Table 1, Method 1,2,3 use these three sets
respectively. After feature extraction, we implement the standardization and
employ support vector machine (SVM) as the classification and regression
model.

###  2.2 Pre-trained acoustic features based approach

At present, pre-trained models are widely used in speech recognition,
affective computing and many other fields [7]. We extract the pre-trained
features using XLSR-53, a cross-lingual pre-trained model trained on speech
data of 53 languages [8]. Its excellent performance on BABEL benchmark ensures
its ability in cross-lingual feature extraction. Before input into the XLSR-53
model, the audio is cut into 6 second segments.

With the extracted features we train our downstream models, which contains two
fully connected layers (FC). The classification and regression tasks are
implemented by setting the number of neurons in the output layer to 2 and 1,
respectively. For the classification task, we use the majority vote of all
segments to predict the label of the original audio. For the regression task,
we use the average MMSE score instead.

###  2.3 Pre-trained linguistic features based approach

Whisper [9] is a speech recognition model and enables speech translation. We
use the large-level Whisper model to transcribe the speech of the training set
and the test set into English text.

RoBERTa [10] is a pre-trained procedure for iterative BERT with longer
training time and more data used for pre-training. Transcripts from the
training set are used to fine-tune the pre-trained RoBERTa model, which is
done by adding classification heads after the first embedding. The
classification and regression tasks are accomplished by adjusting the number
of neurons in the classification head.

##  3 Results

Table 1 shows the experimental result of each system on the classification
task. By comparing the performance of each model in Table 1, it can be seen
that the F1 score of Method 2 is the highest.

Method 4 is to transcribe the speech into English text by Whisper, and then
use RoBERTa for text classification. The precision of this method is high, but
the recall is relatively low, which leads to a low F1 score. This may be
resulted from the overfitting of the model due to the inconsistency between
the pictures described by the subjects in the test set and the training set.
And the performance of the whisper model can also have an impact on the
results.

The accuracy of Method 5 is not so high, much inferior to the performance of
pre-trained models on monolingual AD detection. The probable reason is that
the training set of the XLSR-53 model did not contain Greek, and better
results might have been obtained if the model was fine-tuned using Greek.

For the regression task, the best RMSE score in our method was 4.788,
indicating that our method can be effectively used for cross-lingual
Alzheimer’s Disease detection.

In general, the paralinguistic features perform better than the pre-trained
features in multilingual AD detection, probably because the low-level
descriptors (LLDs) contained in the paralinguistic feature set can reflect the
unexpected pauses in speeches regardless of languages. To the contrary, the
high-dimensional embeddings extracted by pre-trained models have a stronger
coupling with the language, which leads to a weaker generalization ability
across languages.

##  4 Conclusion

In this paper, we use paralinguistic and pre-trained features to detect
Alzheimer’s disease cross-lingually. Our proposed methods using paralinguistic
features show excellent results, while the method based on pre-trained
features does not show satisfactory results. The results suggest that
paralinguistic features can be more effective for cross-lingual Alzheimer’s
Disease detection. Our future work includes the integration of the features
mentioned above. And we will explore the use of ensemble models that focus on
different features for cross-lingual Alzheimer’s disease detection.

## References

  * [1] “NCMMSC AD2021: Alzheimer’s disease recognition evaluation 2021,” https://github.com/THUsatlab/AD2021.
  * [2] Saturnino Luz, Fasih Haider, Davida Fromm, et al.,  “Multilingual alzheimer’s dementia recognition through spontaneous speech: A signal processing grand challenge,”  arXiv:2301.05562, 2023.
  * [3] Jun Chen, Jieping Ye, Fengyi Tang, et al.,  “Automatic detection of alzheimer’s disease using spontaneous speech only,”  in Proc. INTERSPEECH, 2021, pp. 3830–3834.
  * [4] Lara Gauder, Leonardo Pepino, Luciana Ferrer, and Pablo Riera,  “Alzheimer disease recognition using speech-based embeddings from pre-trained models,”  in Proc. INTERSPEECH, 2021, pp. 3795–3799.
  * [5] Jiahong Yuan, Yuchen Bian, Xingyu Cai, et al.,  “Disfluencies and fine-tuning pre-trained language models for detection of alzheimer’s disease.,”  in Proc. INTERSPEECH, 2020, pp. 2162–2166.
  * [6] Florian Eyben, Martin Wöllmer, and Björn Schuller,  “Opensmile: the munich versatile and fast open-source audio feature extractor,”  in Proc. ACM MM, 2010, pp. 1459–1462.
  * [7] Jing Zhao and Wei-Qiang Zhang,  “Improving automatic speech recognition performance for low-resource languages with self-supervised models,”  IEEE Journal of Selected Topics in Signal Processing, vol. 16, no. 6, pp. 1227–1241, 2022.
  * [8] Alexis Conneau, Alexei Baevski, Ronan Collobert, et al.,  “Unsupervised cross-lingual representation learning for speech recognition,”  arXiv 2006.13979, 2020.
  * [9] Alec Radford, Jong Wook Kim, Tao Xu, et al.,  “Robust speech recognition via large-scale weak supervision,”  arXiv:2212.04356, 2022.
  * [10] Yinhan Liu, Myle Ott, Naman Goyal, et al.,  “RoBERTa: A robustly optimized BERT pretraining approach,”  arXiv:1907.11692, 2019.


11institutetext: Laboratoire de Physique Théorique et Hautes Energies (LPTHE),
UMR 7589 CNRS & Sorbonne Université, 4 Place Jussieu, F-75252, Paris,
France22institutetext: Institut d’Astrophysique de Paris, UMR 7095 CNRS &
Sorbonne Université, 98 bis boulevard Arago, F-75014 Paris,
France33institutetext: Department of Physics and Astronomy, The Johns Hopkins
University, 3400 N. Charles Street, Baltimore, MD 21218,
U.S.A.44institutetext: Beecroft Institute for Particle Astrophysics and
Cosmology, University of Oxford, Keble Road, Oxford OX1 3RH, U.K.

# Induced gravitational waves from the cosmic coincidence

Shyam Balaji 2, 3, 4 Joseph Silk 1 and Yi-Peng Wu<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The induced gravitational wave (GW) background from enhanced primordial scalar
perturbations is one of the most promising observational consequences of
primordial black hole (PBH) formation from inflation. We investigate the
induced GW spectrum $\Omega_{\textrm{IGW}}$ from single-field inflation in the
general ultra-slow-roll (USR) framework, restricting the peak frequency band
to be inside $10^{-3}$-$1$ Hz and saturating PBH abundance to comprise all
dark matter (DM) in the ultralight asteroid-mass window. By invoking
successful baryogenesis driven by USR inflation, we verify the viable
parameter space for the specific density ratio between baryons and PBH DM
observed today, the so-called “cosmic coincidence.” We show that the cosmic
coincidence requirement bounds the spectral index $n_{\rm UV}$ in the high
frequency limit, $\Omega_{\textrm{IGW}}(f\gg 1)\propto f^{-2n_{\rm UV}}$, into
$0<n_{\rm UV}<1$, which implies that baryogenesis triggered by USR inflation
for PBHs in the mass range of $10^{-16}$-$10^{-12}M_{\odot}$ can be tested by
upcoming Advanced LIGO and Virgo data and next generation experiments such as
LISA, Einstein Telescope, TianQin and DECIGO.

##  1 Introduction

The cosmological gravitational wave (GW) background induced by scalar-type
density fluctuations at non-linear orders must exists in all viable scenarios
for the early universe 10.1143/PTP.37.831 ; Matarrese:1993zf ;
Matarrese:1997ay ; Ananda:2006af ; Baumann:2007zm ; Martineau:2007dj ;
Bartolo:2007vp ; Saito:2008jc ; 10.1143/PTP.123.867 ; PhysRevD.81.023517 ;
Bugaev2010 ; PhysRevD.83.083521 ; Suyama:2011pu ; Assadullahi:2009jc ;
Assadullahi:2009nf ; Arroja:2009sh ; Alabidi:2012ex ; Alabidi:2013lya ;
Kawasaki:2013xsa  (see Domenech:2021ztg  for a recent review). In most of the
scenarios, such as inflationary models or alternatives, the induced GWs
generated by primordial scalar perturbations at second order are already too
small and thus very challenging to observe Ananda:2006af ; Baumann:2007zm ;
Martineau:2007dj ; Bartolo:2007vp , yet exciting opportunities still exist for
future space-based telescopes that may search for largely enhanced GW spectra
that are closely related to primordial black hole (PBH) formation Saito:2008jc
; 10.1143/PTP.123.867 ; PhysRevD.81.023517 ; Belotsky:2014kca ; Bugaev2010 ;
PhysRevD.83.083521 ; Suyama:2011pu ; Suyama:2014vga ; Nakama:2015nea ;
Nakama:2016enz .

Since the first detection of GW events from a binary-black-hole merger in 2016
PhysRevLett.116.061102 , PBHs have turned into compelling dark matter (DM)
candidates Carr:2016drx  and the mass windows allowed for PBHs to comprise all
DM have been severely constrained by observations (see Carr:2020gox ;
Carr:2020xqk ; Green:2020jor  for recent reviews). Given that the largely
enhanced scalar power spectrum on small scales from inflation manifests the
mainstream scenario for PBH formation, the induced GW background from these
inflationary models is considered one of the most promising signatures of PBH
DM, flourishing many intensive investigations Kohri:2018awv ; Liu:2020oqe ;
Cai:2019cdl ; Cai:2019elf ; Yuan:2019wwo ; Pi:2020otn ; Cai:2018dig ; Garcia-
Bellido:2017aan ; Fumagalli:2020nvq ; Fumagalli:2021cel ; Unal:2018yaa ;
Ragavendra:2020vud ; Ozsoy:2020kat ; Bartolo:2018rku ; Tada:2019amh ;
Bartolo:2018evs ; Ballesteros:2020qam ; Wang:2019kaf ; Kawai:2021edk .
Currently, the viable window for PBH DM in the ultralight asteroid-mass range
seems to put the peak amplitude of the induced GW spectrum inside the joint
frequency band ($10^{-4}$–$10^{3}$ Hz) for LIGO and the next-generation
experiments Ragavendra:2020sop , such as Einstein Telescope (ET)
Maggiore:2019uih , LISA amaroseoane2017laser ; Barausse:2020rsu  and DECIGO
Yagi:2011wg ; Kawamura:2020pcg .

However, if we suppose that PBHs indeed occupy a significant fraction of the
DM density in today’s universe, they will exhibit a surprisingly similar
amount of energy density relative to that of the baryons and thus they are
inevitably confronted by the so-called “cosmic coincidence problem,” as is the
case for all other DM candidates. The cosmic coincidence problem can provide a
good motivation to consider PBH DM as a consequence of inflation, due to the
recent theoretical validation of successful baryogenesis triggered by single-
field inflationary models for PBH formation Wu:2021gtd ; Wu:2021mwy .

In this work, we investigate baryogenesis from single-field inflation with an
enhanced power spectrum for curvature perturbations on small scales driven by
the ultra-slow-roll (USR) transition of the rolling dynamics of the inflaton
field. We consider baryon asymmetry created by the Affleck-Dine (AD)
mechanism, but we relax the constant-mass assumption for the AD field used in
Refs. Wu:2021gtd ; Wu:2021mwy  so that it is possible to adopt a more general
inflationary power spectrum away from the exact USR limit. This is to say that
the inflationary spectrum can have a very sharp power-law decay on small
scales (from its peak amplitude), which is favourable for PBH formation with a
monochromatic mass spectrum. We review the general USR spectrum in Section 2.

After solving the generalised coherent motion of the AD field during
inflation, we obtain the initial conditions for computing the final baryon
asymmetry in the radiation dominated epoch, as provided in Section 3. In
Section 4, we explore the modified parameter space from successful
baryogenesis through general USR inflation and clarify its indication to the
cosmic coincidence problem (Section 4.2). We compute the spectrum of the
induced GWs associated with the input USR inflation and we discuss prospects
for the validity of the “cosmic coincidence” residing in the PBH DM paradigm
and the implications for current and future experiments. For completeness, we
address in Section 4.3.2 the corrections to the induced GW spectrum led by
non-Gaussianity of the curvature perturbations and we briefly illustrate the
GW counterpart that result from the binary PBHs mergers in this scenario.
Finally, our conclusions are provided in Section 5.

##  2 Generalised ultra-slow-roll inflation

Figure 1: The broken power-law template for the power spectrum of the
curvature perturbation $P_{\zeta}$ based on a 3-stage inflation. The rate-of-
rolling $\delta_{1}=0$ is used in the primary slow-roll phase (phase 1) and
$\delta_{2}<-3$ in the transient ultra-slow-roll (USR) phase (phase 2) is
allowed to be away from the exact ultra-slow-roll limit ($\delta_{2}=-3$). The
post-USR phase (phase 3) exhibits a continuous scaling from phase 2 yet it
must be an acceleration phase with $\delta_{3}=-\delta_{2}-3>0$ to terminate
inflation.

In this work, we consider PBH formation from single-field models of inflation
that experience a transient USR phase with largely enhanced curvature
perturbation $\zeta$ on very small scales. The pivot scale $k_{0}$ that
denotes the peak position of the power spectrum $P_{\zeta}(k)$ should be in
the range of $k_{0}\sim 10^{12}-10^{15}$ Mpc-1 so that the resulting PBHs
formed during radiation domination are inside the ultra-light asteroid-mass
window $M_{\rm PBH}/M_{\odot}\sim 10^{-16}-10^{-12}$ and they are still
allowed to account for all dark matter. The three most relevant parameters
that one should keep in mind throughout this work, are defined as

$\displaystyle\delta\equiv\frac{\ddot{\phi}}{H\dot{\phi}},\qquad\Delta N\equiv
N_{\ast}-N_{0},\qquad N_{\rm end},$ (1)

where $\delta$ parametrises the rate of rolling of inflaton $\phi$, $\Delta N$
measures the duration of the USR phase and $N_{\rm end}$ is the e-fold number
at the end of inflation. For convenience, we set the e-fold number $N_{0}=0$
at which the pivot scale $k_{0}$ crosses the horizon.

As shown in Figure 1, the generic scaling behavior of the power spectrum
$P_{\zeta}$ as a function of $k$ can be summarized by a template of the broken
power-law form Ballesteros:2020sre ; Byrnes:2018txb ; Carrilho:2019oqg ;
Cheng:2018qof ; Liu:2020oqe ; Ng:2021hll ; Ozsoy:2019lyy

$\displaystyle P_{\zeta}(k)=\left\\{\begin{array}[]{ll}A_{\rm CMB}&\;k<k_{\rm
min},\\\ A_{\rm PBH}(k/k_{0})^{4},&\;k_{\rm min}<k<k_{0},\\\ A_{\rm
PBH}(k/k_{0})^{6+2\delta_{2}},&\;k>k_{0},\end{array}\right.\qquad A_{\rm
PBH}=A_{\rm CMB}\left(\frac{k_{0}}{k_{\ast}}\right)^{6+4\delta_{2}}.$ (5)

Where $A_{\rm CMB}=\frac{H_{\ast}^{2}}{\epsilon_{\rm CMB}M_{P}^{2}}\approx
2.2\times 10^{-9}$ is measured by CMB experiments and $k_{\ast}$ is the
comoving scale that crosses the horizon at $N=N_{\ast}$ (the end of the
transient USR phase). The duration of the USR phase is $\Delta
N=\ln\frac{a_{\ast}}{a_{0}}=\ln\frac{k_{\ast}}{k_{0}}$.

The broken power-law template given by (5) includes three phases of inflation,
which are denoted as the primary slow-roll phase (phase 1), the transient USR
phase (phase 2) for enhancing the amplitude of the power spectrum and the
post-USR phase (phase 3) with $\delta>0$ that can terminate inflation. It is
remarkable that the curvature perturbation $\zeta_{k}$ in the range of $k_{\rm
min}\leq k<k_{0}$ are modes that have exited the horizon in phase 1, where
$k_{\rm min}\approx
k_{0}\left(\frac{k_{0}}{k_{\ast}}\right)^{-3/2-\delta_{2}}$. These modes
undergo superhorizon evolution after the USR transition into phase 2
Cheng:2018qof , and eventually settle in the final boundary measured at the
end of inflation as a $P_{\zeta}\sim k^{4}$ growth Byrnes:2018txb ;
Carrilho:2019oqg ; Leach:2001zf . The $P_{\zeta}\sim k^{4}$ growth is a
consequence of entropy perturbation domination Leach:2000yw , which is the
criterion for breaking the initial scaling power $P_{\zeta}\sim k^{0}$ led by
phase 1 Leach:2001zf ; Ng:2021hll .111The domination of entropy mode in
$\zeta$ is so far a sufficient condition for violating the continuity of the
momentum scaling $\Delta$ in the power spectrum as $P_{\zeta}\sim k^{2\Delta}$
is protected by the dilatation symmetry of the de Sitter background. Note that
$\Delta=3/2-|3/2+\delta|\rightarrow 0$ in the slow-roll and exact USR
inflation with $\delta=0$ or $\delta=-3$, respectively. It is interesting to
seek other mechanisms that can break the scaling power $\Delta$ in single-
field inflation. On the other hand, the entropy domination does not occur in
most of the single-field models across the transition from phase 2 to 3 so
that $P_{\zeta}(k)$ exhibits a continuous scaling in the power of $k$ till the
end of inflation. This adiabatic condition fixes the rate-of-rolling in phase
3 as $\delta_{3}=-\delta_{2}-3$ Ng:2021hll .

The previous studies Wu:2021mwy ; Wu:2021gtd  considered a transient quasi-USR
phase with $-3.2\lesssim\delta_{2}<-3$ to sustain the constant-mass
approximation for the charged scalar that is responsible for creating the
baryon asymmetry. In this work, we relax such a restriction by considering a
general value allowing for $\delta_{2}\ll-3$. There can be at least two
benefits for constructing the scenario in the region of $\delta_{2}\ll-3$.
First, the effective mass of the inflaton in phases 2 and 3 reads
$\frac{m_{\phi}^{2}}{H_{\ast}^{2}}\approx-\delta(\delta+3)$ Ng:2021hll ;
Wu:2021mwy  (assuming constant $\delta$), where $m_{\phi}$ is continuous
across the two phases as protected by the scaling condition
$\delta_{3}=-\delta_{2}-3$. Taking $\delta_{2}\ll-3$ thus gives
$m_{\phi}/H_{\ast}\gg 1$, which is expected to significantly reduce the effect
of quantum diffusion in the exact USR case with $\delta_{2}=-3$
Ezquiaga:2019ftu ; Figueroa:2020jkf ; Pattison:2021oen . Second,
$P_{\zeta}(k>k_{0})\sim k^{6+2\delta_{2}}$ with $\delta_{2}\ll-3$ decays
sharply in the large $k$ limit, leading to a narrow peak at $k=k_{0}$. As
shown in Section 4.1, such a narrow-peak spectrum can transfer into a narrow
distributed mass function for PBHs, which is more appropriate when comparing
with most of the observational constraints in which the monochromatic PBH mass
assumption is generically applied.

##  3 Baryogenesis triggered by the ultra–slow–roll transition

The breakdown of the constant mass approximation Wu:2021mwy  for the charged
scalar is the price to pay for entering into the $\delta_{2}\ll-3$ regime. In
this section, we investigate the coherent motion of the charged scalar with
time-varying mass induced by the transient USR transition. Given that we
consider the charged scalar, $\sigma$, as the source field for generating
baryon asymmetry via the AD mechanism, we also identify $\sigma$ as the AD
field.

As introduced in Ref. Wu:2021mwy , we consider $\sigma$ possessing a $U(1)$
baryon number given by
$n_{B}=j^{0}=i(\sigma^{\ast}\dot{\sigma}-\sigma\dot{\sigma}^{\ast})$. The
dynamics of inflaton $\phi$ enters the effective mass of $\sigma$ through
derivative couplings described in the Lagrangian

$\displaystyle\mathcal{L}_{\sigma}=|\partial\sigma|^{2}+m_{\sigma}^{2}|\sigma|^{2}+\frac{c_{1}}{\Lambda}\left|\sigma^{2}\right|\square\phi+\frac{c_{2}}{\Lambda}\partial_{\mu}\phi\left[\sigma\partial^{\mu}\sigma+\sigma^{\ast}\partial^{\mu}\sigma^{\ast}\right]+\frac{c_{3}}{\Lambda^{2}}\left(\partial\phi\right)^{2}|\sigma|^{2},$
(6)

where $c_{1}$, $c_{2}$, $c_{3}$ are $\mathcal{O}(1)$ coupling constants. The
Lagrangian (6) is C/CP invariant but the $c_{2}$ term violates baryon number.
Note that the imaginary phase in $c_{2}$ can be absorbed into the phase term
of $\sigma$. The cutoff scale $\Lambda$ should be in the range of
$H_{\ast}\ll\Lambda\leq M_{\textrm{P}}$ to justify the effective field theory
description during inflation.

To study the coherent motion of the AD field, it is more convenient to
decompose the Lagrangian into the mass eigenstates $\sigma_{\pm}$ via
$\sigma\equiv\frac{\sigma_{-}+i\sigma_{+}}{\sqrt{2}}$, where (6) becomes a
system of two real scalars as

$\displaystyle\mathcal{L}_{\pm}=$
$\displaystyle\frac{1}{2}\left(\partial\sigma_{+}\right)^{2}+\frac{1}{2}\left[m_{\sigma}^{2}+\frac{c_{1}+c_{2}}{\Lambda}\square\phi+\frac{c_{3}}{\Lambda^{2}}\left(\partial\phi\right)^{2}\right]\sigma_{+}^{2}$
$\displaystyle+\frac{1}{2}\left(\partial\sigma_{-}\right)^{2}+\frac{1}{2}\left[m_{\sigma}^{2}+\frac{c_{1}-c_{2}}{\Lambda}\square\phi+\frac{c_{3}}{\Lambda^{2}}\left(\partial\phi\right)^{2}\right]\sigma_{-}^{2}.$
(7)

The effective masses are therefore controlled by the coherent motion of
$\phi$, where

$\displaystyle m_{\pm}^{2}=m_{\sigma}^{2}+\frac{c_{1}\pm
c_{2}}{\Lambda}\square\phi+\frac{c_{3}}{\Lambda^{2}}\left(\partial\phi\right)^{2}.$
(8)

We note that the necessary CP violation for successful baryogenesis is
spontaneously realised by the emergence of a CP-violating initial VEV in a
local universe Wu:2020pej ; Hook:2015foa  (similar to the idea of spontaneous
T violation Lee:1973iz ).

###  3.1 Time evolving scalar masses

Let us now apply the inflationary background with the transient USR transition
introduced in Section 2. Based on the definitions in Eq. (1), we get the
equations of motion

$\displaystyle\square\phi=-\ddot{\phi}-3H\dot{\phi}\approx-(\delta+3)\sqrt{2\epsilon_{H}}M_{P}H_{\ast}^{2}$
(9)

and

$\displaystyle(\partial\phi)^{2}=\dot{\phi}^{2}\approx
2\epsilon_{H}M_{P}^{2}H_{\ast}^{2},$ (10)

where $\epsilon_{H}=-\frac{\dot{H}}{H^{2}}$ is the first slow-roll parameter.
Therefore, with the transition of $\delta$ in each phase of inflation, the
effective masses take different values like

$\displaystyle m_{i\pm}^{2}=m_{\sigma}^{2}+\frac{c_{1}\pm
c_{2}}{\Lambda}[-(\delta_{i}+3)]\sqrt{2\epsilon_{i}}M_{P}H_{\ast}^{2}+\frac{c_{3}}{\Lambda^{2}}2\epsilon_{i}M_{P}^{2}H_{\ast}^{2}.$
(11)

Note that $\frac{d\ln\epsilon_{H}}{dN}\approx 2\delta$ and
$\epsilon_{1}\equiv\epsilon_{H}(N<0)=\epsilon_{\rm CMB}$ is approximately a
constant. This yields $\epsilon_{2}\equiv\epsilon_{\rm CMB}e^{2\delta_{2}N}$
and $\epsilon_{3}=\epsilon_{\ast}e^{2\delta_{3}(N-N_{\ast})}$ where
$\epsilon_{\ast}\equiv\epsilon_{H}(N_{\ast})=\epsilon_{\rm
CMB}e^{2\delta_{2}N_{\ast}}$.

Figure 2:  The time evolution of $M_{3}^{2}(N)$ (left panel) and the baryon
asymmetry at the end of inflation $Y_{B}(N_{\rm end})$ (right panel) with
various choices of $\delta_{2}$ and $\delta_{3}=-\delta_{2}-3$. In both
panels, $\frac{m_{\sigma}}{H_{\ast}}=0.1$, $\frac{\Lambda}{M_{P}}=1$, $\Delta
N=N_{\ast}=2$ are used.

In order to address the time evolution of $\epsilon_{i}$ in the effective
masses (11), we use the dimensionless parametrisation

$\displaystyle
M_{i\pm}^{2}\equiv\frac{m_{i\pm}^{2}}{H_{\ast}^{2}}=M_{\sigma}^{2}+A_{i\pm}e^{\delta_{i}N}+B_{i}e^{2\delta_{i}N},$
(12)

where $M_{\sigma}=m_{\sigma}/H_{\ast}$. $A_{i\pm}$ and $B_{i}$ are
dimensionless constants, and the coherent motion of the mass eigenstates in
each phase, $\sigma_{i\pm}$ are governed by the equation of motion

$\displaystyle\frac{d^{2}\sigma_{i\pm}}{dN^{2}}+3\frac{d\sigma_{i\pm}}{dN}+M_{i\pm}^{2}(N)\sigma_{i\pm}=0.$
(13)

The dimensionless constants are solved to be

$\displaystyle A_{1,2\pm}$ $\displaystyle=\frac{(c_{1}\pm
c_{2})[-(\delta_{1,2}+3)]\sqrt{2\epsilon_{\rm CMB}}M_{P}}{\Lambda},$
$\displaystyle B_{1}=B_{2}=\frac{2c_{3}\epsilon_{\rm
CMB}M_{P}^{2}}{\Lambda^{2}}.$ (14)

Since $\delta_{1}\rightarrow 0$ in phase 1, one can easily find that
$M_{1\pm}$ are constants so that we may adopt the initial conditions from
Wu:2021mwy  as

$\displaystyle\sigma_{1\pm}(N_{0})=\sqrt{\frac{3}{8\pi^{2}}}\frac{H_{\ast}^{2}}{m_{1\pm}},\quad\dot{\sigma}_{1\pm}(N_{0})=0,$
(15)

where $c_{1}$, $c_{2}$ are chosen such that $m_{1\pm}^{2}$ are positive.

The general solution of (13) is very complicated. Fortunately, the duration of
the transient USR phase (phase 2) is constrained by the PBH abundance and in
fact $\Delta N=N_{\ast}<3$ in all physical models with an arbitrary choice of
statistical method. We have checked that $M_{2\pm}$ being approximately
constant holds in the whole parameter space of interest for $N_{\ast}\leq 3$,
and this allow us to import the constant mass results from Ref. Wu:2021mwy ,
where at the of end phase 2 we have

$\displaystyle\sigma_{2\pm}(N_{\ast})$
$\displaystyle=\sqrt{\frac{3}{8\pi^{2}}}\frac{H_{\ast}^{2}}{m_{1\pm}}\frac{1}{2\nu_{2\pm}}\left(\Delta_{2\pm}^{+}e^{-\Delta_{2\pm}^{-}N_{\ast}}-\Delta_{2\pm}^{-}e^{-\Delta_{2\pm}^{+}N_{\ast}}\right),$
(16) $\displaystyle\left.\frac{d\sigma_{2\pm}}{dN}\right|_{N=N_{\ast}}$
$\displaystyle=\sqrt{\frac{3}{8\pi^{2}}}\frac{m_{2\pm}^{2}}{m_{1\pm}}\frac{1}{2\nu_{2\pm}}\left(e^{-\Delta_{2\pm}^{+}N_{\ast}}-e^{-\Delta_{2\pm}^{-}N_{\ast}}\right),$
(17)

with definitions $\nu_{2\pm}=\sqrt{\frac{9}{4}-M_{2\pm}^{2}}$ and
$\Delta_{2\pm}^{\pm}=\frac{3}{2}\pm\nu_{2\pm}$.

Now only the solutions for phase 3 remain. In phase 3, we restrict ourselves
to the broken power-law template (5) with $\delta_{3}=-\delta_{2}-3$, where

$\displaystyle A_{3\pm}$ $\displaystyle=(c_{1}\pm
c_{2})[-(\delta_{3}+3)]\sqrt{2\epsilon_{\rm
CMB}}\frac{M_{P}}{\Lambda}e^{(\delta_{2}-\delta_{3})N_{\ast}},$ $\displaystyle
B_{3}$ $\displaystyle=2c_{3}\epsilon_{\rm
CMB}\frac{M_{P}^{2}}{\Lambda^{2}}e^{2(\delta_{2}-\delta_{3})N_{\ast}}.$

Some examples for the time evolution of $M_{3}^{2}$ with respect to the e-fold
number $N$ are given in Figure 2. One can see that the $B_{3}$ term in Eq.
(12) (illustrated by the dashed line in Figure 2) only comes to dominate the
effective mass with sufficiently long duration in phase 3. Hence, if we
consider the end of inflation with $N_{\rm end}<20$, then the $B_{3}$ term can
be neglected in most cases of interest. Under this condition the solution of
(13) reads

$\displaystyle\sigma_{3\pm}(N)=$
$\displaystyle\;C_{31}e^{-3N/2}\Gamma\left(1-2\frac{\nu_{\sigma}}{\delta_{3}}\right)J_{-\frac{2\nu_{\sigma}}{\delta_{3}}}\left(\frac{2}{\delta_{3}}\sqrt{A_{3\pm}e^{\delta_{3}N}}\right)$
$\displaystyle+C_{32}e^{-3N/2}\Gamma\left(1+2\frac{\nu_{\sigma}}{\delta_{3}}\right)J_{\frac{2\nu_{\sigma}}{\delta_{3}}}\left(\frac{2}{\delta_{3}}\sqrt{A_{3\pm}e^{\delta_{3}N}}\right),$
(18)

where $J_{n}(x)$ is Bessel function of the first kind and we have denoted
$\nu_{\sigma}=\sqrt{\frac{9}{4}-M_{\sigma}^{2}}$. For each mass eigenstate,
one can solve the coefficients $C_{31}$, $C_{32}$ by matching the solutions
with boundary conditions $\sigma_{2\pm}=\sigma_{3\pm}$,
$\frac{d\sigma_{2\pm}}{dN}=\frac{d\sigma_{3\pm}}{dN}$ at $N=N_{\ast}$. The
results are given by

$\displaystyle C_{31}=$
$\displaystyle\frac{\pi}{2\delta_{3}}\frac{e^{3N_{\ast}/2}}{\Gamma\left(1-\frac{2\nu_{\sigma}}{\delta_{3}}\right)}\left\\{\left[-2d\sigma_{2\ast}+(-3+2\nu_{\sigma})\sigma_{2\ast}\right]J_{\frac{2\nu_{\sigma}}{\delta_{3}}}\left(\frac{2}{\delta_{3}}\sqrt{A_{3}e^{\delta_{3}N_{\ast}}}\right)\right.$
$\displaystyle\left.-2\sigma_{2\ast}\sqrt{A_{3}e^{\delta_{3}N_{\ast}}}J_{1+\frac{2\nu_{\sigma}}{\delta_{3}}}\left(\frac{2}{\delta_{3}}\sqrt{A_{3}e^{\delta_{3}N_{\ast}}}\right)\right\\}\csc\left(\frac{2\nu_{\sigma}}{\delta_{3}}\pi\right),$
(19) $\displaystyle C_{32}=$
$\displaystyle-\frac{e^{3N_{\ast}/2}}{4\nu_{\sigma}\delta_{3}^{2}}\left[\frac{1}{\delta_{3}}\sqrt{A_{3}e^{\delta_{3}N_{\ast}}}\right]^{-\frac{2\nu_{\sigma}}{\delta_{3}}}\Gamma\left(-\frac{2\nu_{\sigma}}{\delta_{3}}\right)\times$
$\displaystyle\left\\{2\delta_{3}\nu_{\sigma}\left[2\nu_{\sigma}\sigma_{2\ast}+(2d\sigma_{2\ast}+3\sigma_{2\ast})\right]{}_{0}\tilde{F}_{1}\left[1-\frac{2\nu_{\sigma}}{\delta_{3}};-\frac{A_{3}}{\delta_{3}^{2}}e^{\delta_{3}N_{\ast}}\right]\right.$
$\displaystyle\left.+4\nu_{\sigma}A_{3}e^{\delta_{3}N_{\ast}}\sigma_{2\ast}\;{}_{0}\tilde{F}_{1}\left[2-\frac{2\nu_{\sigma}}{\delta_{3}};-\frac{A_{3}}{\delta_{3}^{2}}e^{\delta_{3}N_{\ast}}\right]\right\\},$
(20)

where ${}_{0}\tilde{F}_{1}[n;x]=\frac{{}_{0}F_{1}[n;x]}{\Gamma(n)}$ is the
regularized confluent hypergeometric function, and
$\sigma_{2\ast}\equiv\sigma_{2}(N_{\ast})$,
$d\sigma_{2\ast}\equiv\frac{d\sigma_{2}}{dN}|_{N=N_{\ast}}$ are given by (16)
and (17) respectively. It is important to note that we have suppressed the
notation for each mass eigenstate in $C_{31}$, $C_{32}$, $A_{3}$,
$\sigma_{2\ast}$ and $d\sigma_{2\ast}$.

We can estimate the (temporal) baryon asymmetry at the end of inflation by
assuming instantaneous reheating soon after $N=N_{\rm end}$. The temperature
$T_{\ast}=[\frac{30}{\pi^{2}g_{\ast}}\rho_{r\ast}]^{1/4}$ is given by the
energy density of radiation $\rho_{r\ast}\approx 3M_{P}^{2}H_{\ast}^{2}$,
where $g_{\ast}=106.75$ is the number of relativistic degrees of freedom above
$300$ GeV. The baryon asymmetry is then given by

$\displaystyle Y_{B}(N_{\rm end})=\frac{n_{B}(N_{\rm
end})}{s_{\ast}}=\frac{H_{\ast}}{s_{\ast}}\left.\left(\sigma_{3+}\frac{d\sigma_{3-}}{dN}-\sigma_{3-}\frac{d\sigma_{3+}}{dN}\right)\right|_{N=N_{\rm
end}},$ (21)

where $s_{\ast}=\frac{2\pi^{2}g_{\ast}T_{\ast}^{3}}{45}$ is the entropy
production. According to the right panel of Figure 2, one can see that in the
regime of $\delta_{2}\ll-3$, the $A_{3}$ term in Eq. (12) grows very fast in
phase 3. If $M_{3}\gg 1$ with sufficiently large $N_{\rm end}$, the AD field
starts to oscillate rapidly, which washes away the generated baryon asymmetry.

###  3.2 Final baryon asymmetry

Having acquired the initial conditions of the AD field at the end of
inflation, in this section we compute the final baryon asymmetry in the
radiation domination epoch through reheating of the universe via inflaton
decay. We consider that the decay of inflaton $\phi$, is dominated by a
perturbative channel with the decay width $\Gamma_{I}$. As in the typical
reheating scenario, inflation is terminated by a rapid oscillation of $\phi$
with an effective mass $\frac{\partial^{2}V(\phi)}{\partial\phi^{2}}\approx
m_{I}^{2}\gg H_{\ast}^{2}$ so that the inflaton density $\rho_{I}\sim a^{-3}$
decays as dust-like matter at the beginning of reheating when $H>\Gamma_{I}$.

Denoting $t_{\rm end}=0$ as the physical time at the end of inflation, we can
write down an analytic solution in the limit of $H_{\ast}(t-t_{\rm end})\gg 1$
as Wu:2021mwy ; Wu:2020pej ; Wu:2019ohx

$\displaystyle\phi(t)=\phi_{\rm max}a^{-3/2}\cos\left[m_{I}(t-t_{\rm
end})\right]e^{-\Gamma_{I}(t-t_{\rm end})/2},$ (22)

where $\phi_{\rm max}=\frac{\Lambda_{I}^{2}}{m_{I}}$ is the maximal amplitude
for $\phi$ at $t=t_{\rm end}$ and
$\Lambda_{I}^{4}=3M_{P}^{2}H_{\ast}^{2}=\rho_{I0}$ is the definition of the
energy scale of inflation. It is easy to check that $t_{r}\equiv 1/\Gamma_{I}$
is the time scale at which the density of radiation $\rho_{r}$ starts to
overcome $\rho_{I}$.

Using the background evolution $\square\phi=-\ddot{\phi}-3H\dot{\phi}\approx
m_{I}^{2}\phi$ and $(\partial\phi)^{2}=\dot{\phi}^{2}$ with $\phi(t)$ given by
(22), we can solve the equations of motion for the mass eigenstates according
to

$\displaystyle\ddot{\sigma}_{+}+3H\dot{\sigma}_{+}+\left(m_{\sigma}^{2}+\frac{c_{1}+c_{2}}{\Lambda}m_{I}^{2}\phi+\frac{c_{3}}{\Lambda^{2}}\dot{\phi}^{2}\right)\sigma_{+}$
$\displaystyle=0,$ (23)
$\displaystyle\ddot{\sigma}_{-}+3H\dot{\sigma}_{-}+\left(m_{\sigma}^{2}+\frac{c_{1}-c_{2}}{\Lambda}m_{I}^{2}\phi+\frac{c_{3}}{\Lambda^{2}}\dot{\phi}^{2}\right)\sigma_{-}$
$\displaystyle=0,$ (24)

where initial conditions at the end of inflation ($N=N_{\rm end}$) are given
by (3.1). The final baryon asymmetry at some time $t_{f}\gg t_{r}$ well inside
the radiation dominated epoch reads

$\displaystyle
Y_{B}(t_{f})=\frac{n_{B}(t_{f})}{s(t_{f})}=\frac{45}{2\pi^{2}g_{\ast}T^{3}(t_{f})}\left(\sigma_{+}\dot{\sigma}_{-}-\sigma_{-}\dot{\sigma}_{+}\right)_{t=t_{f}}.$
(25)

Since initial conditions $\sigma_{\pm 0}$ and $\dot{\sigma}_{\pm 0}$ imported
from (3.1) depend on the USR parameters $\delta_{2}$, $N_{\ast}$ and $N_{\rm
end}$, the final baryon asymmetry (25) is therefore controlled by the
inflationary scenario considered in Section 2.222Initial VEVs of the mass
eigenstates can realise successful baryogenesis ($Y_{B}\sim 10^{-10}$) in the
regime of $\sigma_{\pm 0}\ll H_{\ast}$. In contrast to the conventional
scenario for AD baryogenesis from flat directions Dine:1995kz , initial VEVs
for the AD field are usually much larger than $H_{\ast}$.

The value of $N_{\ast}$ for successful baryogenesis $Y_{B}\sim 10^{-10}$ is
our main concern as it is the key parameter that determines the fraction of
PBH density that comprises DM (see Section 4.1). To be more precise, we are
interested in finding the threshold of $N_{\ast}$ (denoted as $N_{\ast c}$)
from which the final baryon asymmetry is nearly unchanged, namely
$Y_{B}(N_{\ast c})\approx Y_{B}(N_{\ast}\gg N_{\ast c})$. Our numerical tests
for various choices of $\delta_{2}$ indicate that $N_{\ast c}$ is not
sensitive to the AD mass for $m_{\sigma}\lesssim H_{\ast}$ but it is
significantly affected by the cutoff scale $\Lambda$. In general, the smaller
the cutoff $\Lambda$, the larger the value of $N_{\ast c}$ is obtained since
the $A_{2}$ term in Eq. (12) needs a longer time to be diluted.

Figure 3:  The final baryon asymmetry in radiation domination with various
choices of $\delta_{2}$ and $\delta_{3}=-\delta_{2}-3$, where
$\frac{m_{\sigma}}{H_{\ast}}=0.1$, $\frac{\Lambda}{M_{P}}=1$ and $N_{\rm
end}=18$ is used.

In the rest of this work, we shall focus on the highest cutoff $\Lambda=M_{P}$
since this gives the lowest $N_{\ast c}$ possible among all the viable
scenarios so that the parameter space for the constant $Y_{B}$ is maximized.
As shown in Figure 3, the constant $Y_{B}$ plateau is recovered in the USR
limit when $\delta_{2}\to-3$, which are the cases investigated in Refs.
Wu:2021mwy ; Wu:2021gtd . With the generalization of $\delta_{2}$ away from
$-3$, the constant $Y_{B}$ plateau is narrowed down towards large $N_{\ast}$.
This result has important consequences for the “cosmic coincidence” as will be
addressed in the next Section.

##  4 The cosmic coincidence of dark matter and baryons

CMB observations reported that the cold dark matter (CDM) density in the
standard $\Lambda$CDM scenario today is $\Omega_{\rm CDM0}=0.265$ and the
redshift at matter-radiation equality is $z_{\rm eq}=3402$ Planck:2018vyg .
These findings indicate that $\Omega_{\rm CDMeq}=0.42$ and $\Omega_{\rm
Beq}=m_{B}n_{\rm Beq}=0.08$, where $m_{B}=0.938$ GeV is the averaged nucleon
mass and $n_{\rm Beq}=|Y_{B}|s(t_{\rm eq})$ is the baryon number density. The
specific ratio between the two species, namely

$\displaystyle\frac{\Omega_{\rm CDMeq}}{\Omega_{\rm Beq}}\approx 5,$ (26)

seems to imply a non-trivial connection of their origin, since one would
expect that the densities between CDM and baryons to differ by orders of
magnitude if their genesis is completely uncorrelated in the early universe.
Therefore the nearly $\mathcal{O}(1)$ ratio shown in Eq. (26) is sometimes
remarked referred to as the “cosmic coincidence problem” and in this section
we address a possible scenario for the cosmic coincidence in which dark matter
is composed by PBHs fostered from USR inflation.

###  4.1 Primordial black hole dark matter

Since the inflaton decays during reheating, the enhanced curvature
perturbations are inherited by the radiation density perturbations. Therefore,
shortly after reentry into the horizon in radiation domination, PBHs are
formed at the high variance peaks of the density fluctuations. This is because
the overdense regions will cease expanding some time after they enter the
particle horizon and collapse against the pressure if they exceed the the
Jeans mass.

We shall focus on the viable window for PBHs to comprise all DM in the
asteroid-mass range $M_{\textrm{PBH}}\simeq 10^{-16}\hbox{-}10^{-12}M_{\odot}$
which indicates that the pivot scale
$k_{0}/k_{\textrm{eq}}\simeq(M_{\textrm{eq}}/M_{\textrm{PBH}})^{1/2}$ for the
initiation of a USR phase in the range $k_{0}\simeq 3\times
10^{12}\hbox{-}10^{14}\,\textrm{Mpc}^{-1}$, where $k_{\textrm{eq}}$ and
$M_{\textrm{eq}}$ denote the horizon wavenumber and the horizon mass at
matter–radiation equality respectively. The comoving horizon length is defined
$R=1/(aH)$ and the horizon mass is given by $M_{H}=\frac{4\pi\rho}{3(H)^{3}}$.
We will define $R$ as a function of $M_{H}$ below.

Given a specific primordial scalar power spectrum $\mathcal{P}_{\zeta}(k)$, we
can use standard techniques to calculate the PBH mass function $f(M)$ which
characterises the fraction of PBHs constituting DM today. Here, we shall focus
on scales of the density field that reenter the Hubble radius during the
radiation dominated epoch. At matter-radiation equality, the PBH-to-DM density
ratio is given by

$\displaystyle
f_{\textrm{PBH}}=\frac{\Omega_{\textrm{PBHeq}}}{\Omega_{\textrm{DMeq}}}=\frac{1}{\Omega_{\textrm{DMeq}}}\int
f(M_{H})d\log M_{H},$ (27)

where
$f(M_{H})=\frac{\beta(M_{H})}{\Omega_{\textrm{DM}}}\left(\frac{M_{\textrm{eq}}}{M_{H}}\right)^{1/2}$
shows the PBH mass function as a function of the horizon mass at matter-
radiation equality and $\beta(M_{H})$ is the fraction of PBH density at the
given $M_{H}$.333Strictly speaking, the PBH mass $M_{\rm PBH}$ can be much
smaller than the horizon mass $M_{H}$ at the formation epoch due to the effect
of critical collapse Niemeyer:1997mt ; Yokoyama:1998xd . Therefore the density
fraction $\beta=\beta(M_{\rm PBH},M_{H})$ in general has a distribution in
$M_{\rm PBH}$ at a given $M_{H}$. However, if we are only interested in the
value of $N_{\ast}$ for a fixed ratio for example, $f_{\rm PBH}=1$, then the
difference led by critical collapse is negligible Wu:2021gtd . Thus, for USR
inflation, we find that $M_{\textrm{PBH}}\approx M_{H}$ can be a good
approximation for resolving the PBH abundance and this relation simplifies the
density fraction as $\beta=\beta(M_{H})$.  Note that PBHs behave as dust-like
matter and thus $a_{\textrm{eq}}/a\sim(M_{\rm eq}/M_{H})^{1/2}$ describes the
relative growth of $\beta(M_{H})$ during radiation domination. The scale
dependent horizon mass $M_{H}$ and the co–moving scale $R$ can be written in
terms of $M_{H}$ as

$\displaystyle M_{H}$
$\displaystyle=M_{\textrm{eq}}\left(\frac{k_{\textrm{eq}}}{k}\right)^{2}\left(\frac{g_{\textrm{eq}}}{g_{*}}\right)^{1/3},$
$\displaystyle R(M_{H})$
$\displaystyle=\frac{1}{k_{\textrm{eq}}}\left(\frac{M_{H}}{M_{\textrm{eq}}}\right)^{1/2}\left(\frac{g_{*}}{g_{\textrm{eq}}}\right)^{1/6},$
(28)

where the horizon mass is given by $M_{\textrm{eq}}=2.94\times
10^{17}M_{\odot}$ and the number of relativistic degrees of freedom at
matter–radiation equality is given by $g_{\textrm{eq}}=3$, it also follows
from Eq. (28) that $R(M_{\textrm{eq}})\approx 1.57\times
10^{40}\,\textrm{GeV}^{-1}$. We use $k_{\textrm{eq}}=0.01\,\textrm{Mpc}^{-1}$
and $g_{*}=106.75$ since the horizon mass satisfies $M_{H}<1.5\times
10^{-7}M_{\odot}$ where the temperature of the universe exceeds $300$ GeV.

If we now consider the simplest fiducial Press-Schechter statistic Carr:1975qj
for Gaussian fluctuations to be distributed with a dispersion $\sigma_{0}$,
then the rare density peaks that exceed a critical value $\delta_{c}$, are
responsible for production of PBHs. The fraction of energy density that
collapse into BHs at the mass scale $M_{H}$ during radiation domination reads

$\displaystyle\beta(M_{H})=2\int_{\nu_{c}}d\nu\frac{e^{-\nu^{2}/2}}{\sqrt{2\pi}}=\textrm{erfc}\left(\frac{\nu_{c}}{\sqrt{2}}\right)=\textrm{erfc}\left(\frac{\delta_{c}}{\sqrt{2}\sigma_{0}}\right),$
(29)

where $\textrm{erfc}(z)$ is the complimentary error function,
$\nu=\delta_{r}/\sigma_{0}$ is the peak value of radiation fluctuations, and
we take the critical value $\delta_{c}=0.45$ from Harada:2013epa . Based on
the linear relation $P_{\delta_{r}}=(\frac{4}{9})^{2}(kR)^{4}P_{\zeta}$ in our
fiducial estimation, the variance $\sigma_{0}^{2}(R)$ for a given power
spectrum $\mathcal{P}_{\zeta}$ is computed by

$\displaystyle\sigma_{0}^{2}(R)$
$\displaystyle=\int_{0}^{\infty}\frac{dq}{q}\,\left(\frac{4}{9}\right)^{2}\left[\left(qR\right)^{4}P_{\zeta}(q)W^{2}(qR)T^{2}(qR)\right],$
(30)

where the window function $W(qR)$ and the transfer function $T(qR)$ contained
in the kernel of integral are given by

$\displaystyle W(qR)=e^{-\frac{q^{2}R^{2}}{2}},\qquad
T(qR)=\frac{9\sqrt{3}}{(qR)^{3}}\left(\sin\frac{qR}{\sqrt{3}}-\frac{qR}{\sqrt{3}}\cos\frac{qR}{\sqrt{3}}\right),$
(31)

respectively. Here we choose the volume-normalised Gaussian window function
with smoothing radius $R=1/(aH)$ led by the comoving horizon at conformal time
$\eta$. Therefore one can refer to $R$ or $M_{H}$ as the time parameter in
this calculation. Note that $T(qR)\rightarrow 1$ as $qR\rightarrow 0$ where
the $q$ mode perturbation is well outside the horizon scale.

Figure 4: The ultra-slow-roll inflation duration $\Delta N$ (solid line)
required for primordial black holes to saturate the total dark matter density
and $\Delta N_{c}$ (dashed) required for the cosmic coincidence is shown as a
function of the ultra-slow-roll rate $\delta_{2}$ (left panel). The blue and
purple shaded regions correspond to “overabundant primordial black hole dark
matter” and “no clear indication to the cosmic coincidence problem from the
USR triggered baryogenesis” respectively. The primordial black hole mass
functions for various rates are shown (right panel), we consider a broken
power-law spectrum with a pivot scale of $k_{0}=3\times
10^{12}\,\textrm{Mpc}^{-1}$ rates of $\delta_{2}=-[4,3.7,3.5,3.3,3.1]$. The
constraints coming from neutron stars (NSs) and white dwarfs (WDs) are shown
in dotted and dot-dashed respectively while the solid line with the upper
shaded region corresponds to HSC M31 microlensing limits which are obtained
from the PlotBounds bradley_j_kavanagh_2019_3538999  package.

Let us now apply the generalised USR scenario (5) to (30) to obtain the mass
function $f(M)$. Note that $P_{\zeta}$ is only controlled by $\delta$ and
$\Delta N$ and therefore $f(M)$ or $f_{\rm PBH}$ is not sensitive to $N_{\rm
end}$. In Figure 4, we show the USR duration $\Delta N=N_{\ast}$ as function
of the USR rate $\delta_{2}$ while ensuring that PBHs produced through USR
inflation in Eq. (27) saturates the DM density limit ($f_{\rm PBH}=1$). The
explicit PBH mass functions in Figure 4 are shown for a selection of
$\delta_{2}$ values. We have also shown constraints for the large mass limit
of the ultralight asteroid-mass window from neutron star disruption
Capela:2013yf  and the white dwarf explosions triggering supernovae
Graham:2015apa  as dashed and dot–dashed curves. These stellar disruption
limits are not currently considered robust since having being discredited by
Ref. Montero-Camacho:2019jte . Improving experimental limits in this mass
range presents an excellent way to test the mass spectrum of PBHs produced by
the cosmic coincidence. The more robust microlensing constraints for
sub–planetary–mass compact objects, including PBHs, coming from Subaru HSC
observations of M31 Smyth:2019whb  are shown as the red shaded region in the
right panel. This excludes PBH DM saturation above $10^{-12}M_{\odot}$ but
loses strength at lower mass. The selection of the pivot scale $k_{0}$, the
USR rates $\delta_{2}$, and durations $\Delta N$, largely evade the best
available constraints.

###  4.2 The cosmic coincidence

In the regime close to the exact USR inflation scenario
($-3.2<\delta_{2}<-3$), the required duration for PBHs to account for all DM
($\Delta N>2$, see the left panel of Figure 4) lies well within the constant
$Y_{B}$ plateau, as shown in Figure 3. Given that a $\mathcal{O}(0.2)$
variation in $\Delta N$ can cause around $10^{50}$ difference in $f_{\rm PBH}$
Wu:2021gtd , the uncertainty of $\Delta N$ (for $f_{\rm PBH}=1$) due to
various effects in the statistics of PBH abundance is at most of
$\mathcal{O}(10^{-1})$ Wu:2021mwy . This is the reason why the coincidence
ratio $\frac{\Omega_{\textrm{DMeq}}}{\Omega_{\rm Beq}}\approx 5$ is rather
guaranteed in this scenario despite the fact that we are using the simplest
statistical method shown in Eq. (29).

However, with the extension of the scenario to the regime of
$\delta_{2}<-3.2$, the constant $Y_{B}$ plateau starts to collapse towards the
large $\Delta N$ limit (see Figure 3), whereas the required $\Delta N$ for
$f_{\rm PBH}=1$ is smaller since the enhancement for $A_{\rm PBH}$ given in
Eq. (5) is more efficient. As a result, one expects that the parameter space
for PBH DM to drop out of the $Y_{B}$ plateau with a sufficiently small
$\delta_{2}$, wherein the scenario loses its indication of the cosmic
coincidence as the correct baryon asymmetry also relies on tuning of other
parameters.

To specify the lower bound of $\delta_{2}$ for preserving the predictability
of the cosmic coincidence, let us quantify the plateau threshold $\Delta
N_{c}=N_{\ast c}$ by defining the edge of the constant plateau of the final
baryon asymmetry as $Y_{B}(N_{\ast c})\equiv
Y_{B}(N_{\ast}\rightarrow\infty)/2$. For a numerical demonstration we take the
asymptotic $Y_{B}$ at $N_{\ast}=4$. The results based on this definition are
given in the left panel of Figure 4, where the intersection is around
$\delta_{2}=-3.6$. This implies that the required $\Delta N$ for $f_{\rm
PBH}=1$ is located inside the constant $Y_{B}$ plateau when $\delta_{2}>-3.6$.
Considering the uncertainties in the statistics of PBH abundance, we therefore
impose conservative bounds for the USR rate as

$\displaystyle-3.5<\delta_{2}<-3.1\qquad(\textrm{for the cosmic
coincidence}),$ (32)

where the conservative upper bound ensures that the effective mass of inflaton
during the USR phase (phase 2) is at least of order $H_{\ast}$ so that the
effect of quantum diffusion can be largely suppressed Ng:2021hll .

###  4.3 Implications for the gravitational wave background

In this section we study the relevant GWs generated in the PBH dark matter
scenario and we also investigate the implications from the cosmic coincidence
to the resulting GW spectrum for current and future observations.

####  4.3.1 Induced gravitational wave production

Here, we are interested in calculating the GW spectral density as measured in
the present universe. For a given mode in Fourier space, the frequency of GWs
today is given by

$\displaystyle f=\frac{k}{2\pi}=1.55\times
10^{-15}k\,\textrm{Mpc}\,\textrm{Hz}.$ (33)

With the scale crossing the Hubble radius at matter-radiation equality being
$k_{\textrm{eq}}=1.3\times 10^{-2}\textrm{Mpc}^{-1}$ , all modes with
frequencies $f\gtrsim 10^{-17}$ Hz have re-entered the horizon during
radiation domination, assuming the standard thermal history between the end of
inflation and the radiation-dominated era considered in Section 3.2. For the
broken power-law type spectrum adopted in Eq. (5), the peak frequency
$f_{\textrm{peak}}$ of the GW spectrum is determined by the peak scale of the
curvature perturbation spectrum $\mathcal{P_{\zeta}}$ (namely at pivot scale
$k_{0}$), which is related to the horizon mass corresponding to the pivot
scale as Cai:2019elf

$\displaystyle f_{\textrm{peak}}\sim 6.7\times
10^{-9}\left(\frac{M_{H}(k_{0})}{M_{\odot}}\right)^{-1/2}\textrm{Hz}.$ (34)

Note that the central peak in the PBH mass function depends on the choice of
$\delta_{2}$ as one can see from the right panel of Figure 4.

The induced GWs sourced by the scalar-mode perturbation (on cosmological
scales) at second order consist a stochastic background, and it is usual to
describe the induced GW spectrum by energy density per logarithmic frequency
interval normalized by the critical density. The dimensionless spectral
density associated with the induced GWs, $\Omega_{\textrm{IGW}}(k,\eta)$,
evaluated at late enough times when the modes are contained within the Hubble
radius during the radiation dominated epoch, is given by

$\displaystyle\Omega_{\textrm{IGW}}(k,\eta)$
$\displaystyle=\frac{1}{24}\left(\frac{k}{aH}\right)^{2}\overline{P_{h}(k,\eta)},$
(35)

where the conformal time is defined $\eta=(aH)^{-1}$ at horizon reentry in the
radiation dominated era and the two respective polarisation modes of GWs have
been summed over. $P_{h}$ is the power spectrum of the induced tensor-mode
perturbation sourced by linear scalar-mode perturbations at second order (A),
which can be solved via the Green’s function method Ananda:2006af ;
Baumann:2007zm  as

$\displaystyle
h_{\lambda}(\vec{k},\eta)=4\int^{\eta}d\eta_{1}G_{\vec{k}}(\eta;\eta_{1})\frac{a(\eta_{1})}{a(\eta)}S_{\lambda}(\vec{k},\eta_{1}),$
(36)

where $\lambda=+,\times$ are the two polarisations,
$G_{\vec{k}}(\eta;\eta_{1})=\frac{1}{k}\sin(k(\eta-\eta_{1}))$ is the Green’s
function in radiation domination and $S_{\lambda}$ is the source term with
detailed information provided in Appendix A. The overline in Eq. (35) denotes
average over a few wavelengths for time oscillations led by the Green’s
function and the transfer function $T(k\eta)$ given in Eq. (31). Note that the
GW density measured in today’s universe, $\Omega_{\textrm{IGW0}}h^{2}$, is
related to (35) in radiation domination like

$\displaystyle\Omega_{\textrm{IGW0}}h^{2}$
$\displaystyle=\frac{h^{2}}{3M_{P}^{2}H_{0}^{2}}\frac{d\rho_{\rm IGW}}{d\ln
k}$
$\displaystyle=\Omega_{r0}h^{2}\frac{\rho_{r}}{\rho_{r0}}\left(\frac{a}{a_{0}}\right)^{4}\Omega_{\textrm{IGW}}$
$\displaystyle\approx 1.62\times 10^{-5}\Omega_{\textrm{IGW}},$ (37)

where $\Omega_{r0}h^{2}=4.18\times 10^{-5}$ is the density fraction of
radiation today Planck:2018vyg  and the factor
$\frac{\rho_{r}(\eta)}{\rho_{r0}}\left(\frac{a(\eta)}{a_{0}}\right)^{4}=\frac{g_{\ast}}{g_{0}}\left(\frac{g_{\ast}}{g_{0}}\right)^{-4/3}\approx
0.4$ Bartolo:2018evs ; Bartolo:2018rku  is valid for $\eta\ll\eta_{\rm eq}$
where the temperature of the universe is much higher than 300 GeV. This is
correct in our case since the epoch $\eta_{k_{0}}=-1/k_{0}$ at which the pivot
scale reenters the horizon for the PBH masses of interest
($k_{0}=10^{12}-10^{14}$ Mpc-1) corresponds to the temperature $T\simeq
10^{6}-10^{7}$ GeV.

Figure 5: The present day dimensionless induced gravitational wave spectrum
$h^{2}\Omega_{\textrm{IGW0}}$ is shown as a function of frequency $f$ in Hz
assuming radiation domination for ultra–slow–roll rates of
$\delta_{2}=-[4,3.7,3.5,3.3,3.1]$ are shown (solid lines). The shaded region
between $\delta_{2}=-3.1$ and $\delta_{2}=-3.5$ corresponds to the cosmic
coincidence region. The theoretical predictions are overlaid with relevant
experimental constraints coming from IPTA (red dot-dashed), LISA (purple
dashed), LIGO (A+) (purple dot-dashed), ET (red dashed) and DECIGO (red
dotted). The experimental bounds are derived from the power-law integrated
sensitivity curves Thrane:2013oya  in terms of the gravitational wave spectral
density for some gravitational wave detectors universe at the time of
gravitational wave generation. In each case we consider a broken power-law
power spectrum with pivot scale $k_{0}=3\times 10^{12}\,\textrm{Mpc}^{-1}$.

Using (5) as an input for the primordial value of the source term, we show the
result of (35) in Figure 5 for the induced gravitational wave spectrum at
present day for the same USR rate $\delta_{2}$ and duration $\Delta N$ range
considered in Section 4.1. We emphasise once again, that the PBHs produced
during inflation ensures that the DM density of the universe is completely
explained by PBHs. For the chosen pivot scale $k_{0}=3\times 10^{12}$ Mpc-1,
we find the peak frequency to be around $f_{\textrm{peak}}\simeq 7\times
10^{-3}$ Hz with a peak amplitude around $\simeq 10^{-8}$.444For an input
inflationary spectrum parametrised by a lognormal template with the spectral
amplitude $A_{\zeta}$ and width $\Delta$, we find the ratio
$\frac{\Omega_{\textrm{IGW}}}{A_{\zeta}^{2}}<0.01$ for the entire range of the
induced GW spectrum, except for the resonance peak in the limit of
$\Delta\rightarrow 0$ where the lognormal template reproduces the delta-
function spectrum Pi:2020otn . This is the reason why the peak amplitude of
$\Omega_{\textrm{IGW}}^{\rm peak}\approx A_{\rm PBH}^{2}$ led by broken power-
law templates considered in this work is higher than induced GW spectra
provided in Refs. Domenech:2021ztg ; Bartolo:2018evs ; Bartolo:2018rku ;
Cai:2018dig  for PBHs as all DM. The peak in $\Omega_{\textrm{IGW}}$ near
$k\sim 2k_{0}/\sqrt{3}$ is due to the resonance (of the Green’s function) of
the tensor perturbation with both of the scalar perturbation sources
Domenech:2021ztg , where $1/\sqrt{3}=c_{s}$ is the sound speed of radiation
and the factor two captures the second-order effect.

The induced GW spectral shape from scalar perturbations with a broken power-
law spectrum has been well studied, which is indeed the case for our input in
Eq. (5). In what follows we summarise the model-(in)dependent results with
respect to the USR parameters.

The infrared (IR) tail: In the regime of $k\ll k_{0}$, the only contribution
from our input template is $P_{\zeta}=A_{\rm PBH}(k/k_{0})^{n_{\rm IR}}$ with
$n_{\rm IR}=4$ in the range of $k_{\rm min}<k<k_{0}$. This is the soft limit
for the induced tensor perturbation where the main contribution is from scalar
modes with $k\approx k_{0}$ that have been well inside the horizon. One can
check that the scaling dependence in the integrand of (53) only comes from the
transfer function $T(k\eta)$ given in Eq. (31), as the scalar perturbations
well inside the horizon behave like plane waves. In our case, with $n_{\rm
IR}=4$, the convolution of internal momentum of the scalar sources is
dominated at the peak scale $k_{0}$ Atal:2021jyo , where it is found that

$\displaystyle\Omega_{\textrm{IGW}}(k\ll k_{0})\sim A_{\rm
PBH}^{2}\left(\frac{k}{k_{0}}\right)^{3}\ln^{2}\left(\frac{k}{k_{0}}\right).$
(38)

The $\Omega_{\textrm{IGW}}\sim\left(\frac{k}{k_{0}}\right)^{3}\sim f^{3}$
behaviour manifests the stochastic nature of the signals as if the GW
background is a kind of random white noise Cai:2019cdl . The logarithmic
running $\sim\ln^{2}(k/k_{0})$ is due to the integration over the oscillatory
part of the transfer function (31), which is a characteristic feature of the
induced GW spectrum generated during the radiation dominated universe
Yuan:2019wwo . Note that both the $k^{3}$ scaling and the logarithmic running
are model independent features for the input spectrum $P_{\zeta}$ so that the
resulting $\Omega_{\textrm{IGW}}$ exhibits little difference among various
choices of $\delta_{2}$.

The ultraviolet (UV) tail: In the regime of $k\gg k_{0}$, our input template
reads $P_{\zeta}=A_{\rm PBH}\left(\frac{k}{k_{0}}\right)^{-n_{\rm UV}}$, where
the USR parameter enters the spectral index as $n_{\rm UV}=-2\delta_{2}-6$.
This regime corresponds to the generation of GWs right after horizon reentry
for the scalar modes so that there is not enough time for the Green’s function
or the transfer function to play an important role. While the results depend
on the choice of $n_{\rm UV}$ in general Domenech:2019quo ; Domenech:2020kqm ;
Atal:2021jyo , we focus on cases with $n_{\rm UV}<4$ (due to the cosmic
coincidence condition in Eq. (32) for the USR rate $\delta_{2}$), in which the
GW spectrum behaves as if it directly extracts the momentum dependence from
the input power spectrum

$\displaystyle\Omega_{\textrm{IGW}}(k\gg k_{0})\sim A_{\rm
PBH}^{2}\left(\frac{k}{k_{0}}\right)^{-2n_{\rm UV}(\delta_{2})}.$ (39)

The cosmic coincidence conservative shaded parameter region between
$-3.5\leq\delta_{2}\leq-3.1$ in Figure 5 corresponds to the condition (32)
described in Section 4.2. We remark that baryogenesis triggered by USR
inflation is still available even if the UV tail of the induced GW spectrum is
found to be outside the shaded region. It is only that both the observed
baryon asymmetry and the PBH DM rely on fine-tuning of parameters so that the
scenario discussed in Section 3 has no clear resolution of the cosmic
coincidence.

Observational implications: The predictions for the USR inflation model
considered yields compelling GW phenomenology in the observational window of
the aforementioned experiments. The relevant GW bounds include those
determined by studying the time of arrival from many pulsars in space. This
includes data from the pulsar timing array (PTA) 1979ApJ…234.1100D  which is
comprised of three constituent projects, the European Pulsar Timing Array
(EPTA) Desvignes:2016yex , the Parkes Pulsar Timing Array (PPTA) Hobbs:2013aka
, and the North American Observatory for Gravitational Waves (NANOGrav)
McLaughlin:2013ira , while the International Pulsar Timing Array (IPTA)
IPTA2016  constraint comes from a combination of all three and covers the
frequency band $10^{-9}$–$10^{-7}$ Hz. At higher frequency, we have a band
that would be observable with the advanced Laser Interferometer Gravitational
Wave Observatory (aLIGO) Thrane:2013oya  and Einstein Telescope (ET)
Maggiore:2019uih , which would be sensitive to the range $10$–$10^{3}$ Hz. In
between IPTA and aLIGO/ET we expect LISA amaroseoane2017laser ;
Barausse:2020rsu  and DECIGO Yagi:2011wg ; Kawamura:2020pcg  to be applicable.
In Figure 5, we display the recently derived future sensitivity curves for ET
and LISA using the latest experimental design specification computed in Ref.
Bavera:2021wmw . These are obtained by including the stochastic GW background,
in contrast to model selection analyses that have discriminating power only up
to the GW detector horizons. The use of the stochastic GW background to
determine sensitivity curves accounts for the redshift integration of all GW
signals in the universe and therefore presents a more appropriate projection
for comparison with model predictions.

The current bounds on the stochastic GW background from joint LIGO and Virgo
O3 data excludes $\Omega_{\textrm{IGW}}>5.8\times 10^{-9}$ in the band of
$20$-$76.6$ Hz at $2\sigma$ level KAGRA:2021kbb . Note that this is the result
for the flat GW spectrum ($\alpha=0$), from the power-law method

$\displaystyle\Omega_{\rm GW}(f)=\Omega_{\rm ref}\left(\frac{f}{f_{\rm
ref}}\right)^{\alpha},$ (40)

where $f_{\rm ref}=25$ Hz and $\alpha=2/3$ describes the stochastic background
from compact binary mergers LIGOScientific:2019vic ; KAGRA:2021kbb . The
condition for the “cosmic coincidence” in Eq. (32) for the presented scenario
translates to $-2<\alpha<-0.2$, where the posterior strength is $\Omega_{\rm
ref}<10^{-8}$ in this region by using the log-uniform prior. As shown in
Figure 5, the UV tail of the induced GW spectrum which is realised as a
consequence of PBH saturating DM (in the ultralight asteroid mass window) from
USR inflation, could be tested by the projected design sensitivity of LIGO
(A+), LISA, ET, DECIGO and TianQin Liang:2021bde . A measurement of the
spectral index $\alpha$ for the stochastic GW background in the range of
$-2<\alpha<0$ would further support the cosmic coincidence triggered by USR
baryogenesis. Non-detection of the stochastic GW background by LIGO (A+) could
exclude down to $\delta_{2}\simeq-3.2$ for $k_{0}\gtrsim 10^{14}$ Mpc-1, which
corresponds to a peak PBH mass smaller than $10^{-15}M_{\odot}$. Note that
LISA, TianQin and DECIGO would be able to test the IR tail and the peak
position of $\Omega_{\textrm{IGW0}}$ for arbitrary $k_{0}$ chosen within the
PBH mass window of interest $10^{-12}-10^{-16}M_{\odot}$.

####  4.3.2 Non-Gaussian corrections

In Section 4.3.1 above, we computed the GWs induced by a broken power-law
primordial curvature power spectrum. Previously, it has been argued that the
spectral tilt of the USR spectrum $P_{\zeta}$ in the UV regime
($k\rightarrow\infty$) is related to the magnitude of the non-Gaussianity
parameter $f_{\textrm{NL}}$ Atal:2018neu  and further argued that
$f_{\textrm{NL}}$ may be inferred directly from measurement of the UV tail of
the induced GW spectrum Atal:2021jyo . Here, $f_{\rm NL}$ is defined according
to the standard local ansatz

$\displaystyle\zeta=\zeta_{g}+\frac{3}{5}f_{\rm NL}\zeta_{g}^{2},$ (41)

where $\zeta_{g}$ is the Gaussian curvature perturbation considered in Section
2.

Similar to the generation of secondary tensor perturbations, the second-order
curvature perturbations can be generated by non-vanishing $f_{\rm NL}$ which
contributes to the total power spectrum as
$P_{\zeta}=P_{\zeta_{g}}+P_{\zeta_{g}}^{\rm NG}$, where $P_{\zeta_{g}}^{\rm
NG}\sim f_{\rm NL}^{2}A_{\textrm{PBH}}^{2}$. In the UV ($k\gg k_{0}$) and IR
($k\ll k_{0}$) regimes, one can follow the similar arguments as for the
induced GW spectrum in Section 4.3.1 to find that

$\displaystyle P_{\zeta_{g}}^{\rm NG}(k\gg
k_{0})\sim\left(\frac{k}{k_{0}}\right)^{-n_{\textrm{UV}}},\quad
P_{\zeta_{g}}^{\rm NG}(k\ll k_{0})\sim\left(\frac{k}{k_{0}}\right)^{3}.$ (42)

Note that there is no logarithmic running in the IR regime since the curvature
perturbations are always computed on superhorizon scales. More detailed
calculations for the non-Gaussian power spectrum of USR inflation can be found
in Refs. Atal:2021jyo ; Adshead:2021hnm ; Ragavendra:2020sop ;
Ragavendra:2021qdu .

For the single-field inflation scenarios considered in this work, the spectral
tilt can be related to the non-Gaussianity parameter by
$n_{\textrm{UV}}=\frac{5}{12}f_{\textrm{NL}}$ Atal:2018neu  which translates
to the USR parameter via

$\displaystyle f_{\textrm{NL}}=-\frac{5}{12}(6+2\delta_{2}).$ (43)

Even when exaggerated scenarios are considered, such as where
$f_{\textrm{NL}}\simeq 2$ in Ref. Atal:2021jyo , very little correction to the
total curvature perturbations or to the induced GW background was realised.
Our maximal value for non-Gaussianity corresponds to $f_{\textrm{NL}}=5/12$
from the steepest spectral tilt of $\delta_{2}=-3.5$ for the cosmic
coincidence, and thus we can safely conclude that non-Gaussian corrections to
$\Omega_{\textrm{IGW}}$ are subdominant in all cases of interest here as well.

####  4.3.3 Gravitational waves from binary mergers

In Section 4.3.1, we focused on the GWs induced by large primordial density
fluctuations during inflation. These large fluctuations collapse to form PBHs
if their RMS amplitude exceeds a threshold value. However, once PBHs have
formed, they may also behave as source for other GWs independent from those
produced during inflation. The other GW counterparts could be due to PBH
mergers and graviton emission by Hawking evaporation Kohri:2018awv . We note
that if PBHs are lighter than $10^{15}$ g they have already evaporated by
today and no mergers can be resolved. PBHs heavier than $10^{15}$ g have not
yet evaporated and their binaries could be merging in the nearby universe. We
avoid detailed discussion of signals from mergers in this work due to the
large astrophysical uncertainties and strong model dependence associated with
such predictions. However, there do exist some basic approximations that
provide very crude order of magnitude estimates of the frequency at which we
expect the GWs from the mergers of PBH binaries to show up at the Innermost
Stable Circular Orbit Maggiore:2007ulw

$\displaystyle f_{\textrm{GW,max}}\approx 2f_{\textrm{ISCO}}\approx
4.4\,\textrm{kHz}\left(\frac{M_{\odot}}{M}\right),$ (44)

where M is the total mass of the binary merger. If we consider a monochromatic
PBH mass function then it follows that $M=2M_{\textrm{PBH}}$. Then we may
compute the approximate frequency in today’s universe
$f_{\textrm{GW,max,0}}=\frac{f_{\textrm{GW,max}}}{1+z}$, where $z$ denotes
redshift. Hence for the PBH mergers considered in Section 4.1 of mass $\simeq
10^{-12}M_{\odot}$, we get very high frequency GWs $\gtrsim 10^{12}$ Hz that
are outside the scope of even the most futuristic experimental projections.

##  5 Conclusion

The sharp deceleration of the slow-roll dynamics on small scales into a
transient ultra-slow-roll (USR) phase is a generic mechanism to enhance the
primordial power spectrum for primordial black hole (PBH) formation in single-
field inflation. If PBHs indeed play an important role as dark matter (DM),
the cosmic coincidence problem along with the energy densities of the universe
today might hint at a correlated origin for baryons and DM. In this work, we
have explored and confirmed the viability of baryogenesis, based on the
Affleck-Dine (AD) mechanism, with modified initial conditions driven by a
generic USR transition for PBH formation. Our results include the constant-
mass AD field as a special case considered in Refs. Wu:2021gtd ; Wu:2021mwy .

In the generalised region $\delta_{2}<-3$ away from the standard USR scenario,
we find asymptotically constant behavior of the final baryon asymmetry $Y_{B}$
towards the large USR duration limit ($N_{\ast}\gg 1$) irrespective of the
rate $\delta_{2}$. We find this to be generic feature of the model and this is
directly connected to the cosmic coincidence: for PBHs to occupy a significant
fraction or saturate the DM density today (or equivalently $f_{\rm
PBH}=\Omega_{\textrm{PBHeq}}/\Omega_{\rm CDMeq}\lesssim 1$), the value of
$N_{\ast}=\Delta N$ must be precisely fixed. The allowed parameter space of
$N_{\ast}$ for PBH DM lies well within the constant plateau region for the
correct baryon asymmetry, ensuring the specific ratio $\Omega_{\rm
CDMeq}/\Omega_{\rm Beq}\approx 5$ within statistical uncertainties of the PBH
abundance Wu:2021mwy . However, the constant plateau for the correct baryon
asymmetry narrows with the decrease of the USR rate $\delta_{2}$ from $-3$,
and thus PBH as all DM can be satisfied for $\Delta N\in[1.7,2.6]$ in the
corresponding range $\delta_{2}\in[-3.5,-3.1]$. This is the condition for the
presented scenario to provide a clear indication to the cosmic coincidence
problem.

The ultralight asteroid-mass window $10^{-16}\lesssim
M_{\textrm{PBH}}/M_{\odot}\lesssim 10^{-11}$ for PBH DM corresponds to the
choice of pivot scales $k_{0}\in[10^{12},3\times 10^{14}]\,\textrm{Mpc}^{-1}$
for the USR template. As the most promising observational consequences of the
general USR inflation, we have computed the induced gravitational wave (GW)
background for $f_{\rm PBH}=1$ and found that it has a peak frequency
$10^{-3}<f_{\textrm{peak}}<1$ Hz with a maximum spectral amplitude around
$h^{2}\Omega_{\textrm{IGW0}}(f)\simeq 1\times 10^{-8}$. The frequency scaling
of the $\Omega_{\textrm{IGW}}\sim f^{-2n_{\rm UV}}$ spectrum in the limit of
$k\gg k_{0}$ (namely the UV tail) is controlled by the choice of the USR rate
$\delta_{2}$ (as $n_{\rm UV}=-2\delta_{2}-6$), where the current LIGO and
Virgo constraints on the stochastic GW background KAGRA:2021kbb  ($\gtrsim
10^{-8}$ in the vicinity of 25 Hz) is about one order of magnitude higher than
the highest UV tail for $\delta_{2}=-3.1$. For PBHs comprising all DM from USR
inflation in the mass window of interest, the IR tail of
$\Omega_{\textrm{IGW0}}$ must be measured by LISA, TianQin or DECIGO. The UV
tail of the induced GWs would be tested by future experiments such as LISA,
Advanced LIGO and Virgo (beyond O3), the Einstein Telescope (ET) and DECIGO,
regardless of whether it is compatible with the cosmic coincidence scenario
considered or not.

Further more, we find that non-Gaussianity would have subdominant effects on
the total curvature power spectrum and GW background since
$f_{\textrm{NL}}=-\frac{5}{12}(6+2\delta_{2})$ is not large enough to source
notieceable corrections Atal:2021jyo . We also conclude that gravitational
waves resulting from binary mergers would be at frequencies
$f_{\textrm{peak}}\geq 10^{12}$Hz which would be far too high for observation
even in the most optimistic future experimental scenarios. Finally, we remark
that even though the USR transition provides a striking solution to the cosmic
coincidence, the fine-tuning of inflationary parameters to properly realise
PBH DM inevitably remains.

## Acknowledgements

We would like to thank Guillem Domènech, Jacopo Fumagalli, Gabriele
Franciolini and Yoann Genolini for helpful discussions and feedback. The
project has received funding from the European Union’s Horizon 2020 research
and innovation programme under grant agreement No 101002846 (ERC CoG
“CosmoChart”) as well as support from the Initiative Physique des Infinis
(IPI), a research training program of the Idex SUPER at Sorbonne Université.

##  Appendix A The power spectrum of induced tensor perturbations

We provide the complete equations for the computation of the tensor power
spectrum $P_{h}$ used in Eq. (35). Let us begin with the definition of linear
perturbations in the conformal Newtonian gauge for the metric of the form

$\displaystyle
ds^{2}=-a^{2}(\eta)(1+2\Psi)d\eta^{2}+a^{2}(\eta)\left[(1-2\Phi)\delta_{ij}+\frac{1}{2}h_{ij}\right]dx^{i}dx^{j},$
(45)

where $\Psi$ is the Newtonian potential and $\Phi$ is the curvature potential.
We define

$\displaystyle h_{ij}(\vec{x},\eta)=\int
d^{3}k(2\pi)^{-3/2}[e_{ij}^{+}(\vec{k})h_{+}(\vec{k},\eta)+e_{ij}^{\times}(\vec{k})h_{\times}(\vec{k},\eta)]e^{i\vec{k}\cdot\vec{x}},$
(46)

which is the linear tensor perturbation including the two polarisation modes.
The transverse-traceless polarisation tensors are

$\displaystyle e_{ij}^{+}(\vec{k})$
$\displaystyle=\frac{[e_{i}^{1}(\vec{k})e_{j}^{1}(\vec{k})-e_{i}^{2}(\vec{k})e_{j}^{2}(\vec{k})]}{\sqrt{2}},$
$\displaystyle
e_{ij}^{\times}(\vec{k})=\frac{[e^{1}_{i}(\vec{k})e^{2}_{j}(\vec{k})+e^{2}_{i}(\vec{k})e^{1}_{j}(\vec{k})]}{\sqrt{2}},$
(47)

which are expressed in terms of orthonormal basis vectors $\textbf{e}^{1}$ and
$\textbf{e}^{2}$ orthogonal to $\vec{k}$.

Keeping the tensor perturbation at linear order and the linear scalar
perturbations up to second order, one can obtain the equation of motion for
each polarisation $h_{\lambda}$ from the Einstein equation as

$\displaystyle
h_{\lambda}^{\prime\prime}(\vec{k},\eta)+2\mathcal{H}h_{\lambda}^{\prime}(\vec{k},\eta)+k^{2}h_{\lambda}(\vec{k},\eta)=4S_{\lambda}(\vec{k},\eta),$
(48)

where second-order perturbations are projected away in the transverse-
traceless decomposition Baumann:2007zm  and we have neglected the anisotropic
stress in the energy momentum tensor so $\Psi=\Phi$, and thus

$\displaystyle S_{\lambda}(\vec{k},\eta)=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3/2}}e_{ij}^{\lambda}(\vec{k})q^{i}q^{j}\psi_{\vec{p}}\psi_{\vec{p}}f(p,q,\eta),$
(49) $\displaystyle f(p,q,\eta)=$ $\displaystyle
2T(p\eta)T(q\eta)+\frac{4}{3(1+w)}\left[\frac{T^{\prime}(p\eta)}{\mathcal{H}}+T(p\eta)\right]\left[\frac{T^{\prime}(q\eta)}{\mathcal{H}}+T(q\eta)\right],$
(50)

where $\mathcal{H}=aH=\frac{2}{(1+3w)\eta}$ and $w$ is the equation of state
of the universe. $\vec{k}=\vec{p}+\vec{q}$ with $p\equiv|\vec{p}|$,
$q\equiv|\vec{q}|$ are the two internal momenta. The time evolution of the
scalar potential is described by $\Psi_{\vec{k}}(\eta)=T(k\eta)\psi_{\vec{k}}$
with respect to the primordial value $\psi_{\vec{k}}$, where the transfer
function in the radiation dominated universe is given by (31). The primordial
Newtonian potential $\psi_{\vec{k}}$ well outside the horizon is related to
the (gauge-invariant) curvature perturbation $\zeta$ as
$\psi_{\vec{k}}=\frac{3+3w}{5+3w}\zeta$, or namely

$\displaystyle\left\langle\psi_{\vec{k}}\psi_{\vec{K}}\right\rangle=\delta^{(3)}\left(\vec{k}+\vec{K}\right)\frac{2\pi^{2}}{k^{3}}\left(\frac{3+3w}{5+3w}\right)^{2}P_{\zeta}(k).$
(51)

This is where parameters of the inflationary scenario given by (1) enters the
power spectrum of the tensor perturbation.

Solving the equation of motion (48) by virtue of the Green’s function method
(36), we can compute the total power spectrum of the tensor perturbation as

$\displaystyle\delta^{(3)}(\vec{k}+\vec{K})P_{h}(k,\eta)=$
$\displaystyle\frac{k^{3}}{2\pi^{2}}\sum_{\lambda}^{+,\times}\left\langle
h_{\lambda}(\vec{k},\eta)h_{\lambda}(\vec{K},\eta)\right\rangle,$
$\displaystyle=$
$\displaystyle\frac{k^{3}}{2\pi^{2}}\int^{\eta}d\eta_{1}G_{\vec{k}}(\eta;\eta_{1})\frac{a(\eta_{1})}{a(\eta)}\int^{\eta}d\eta_{2}G_{\vec{K}}(\eta;\eta_{2})\frac{a(\eta_{2})}{a(\eta)}$
$\displaystyle\qquad\times\sum_{\lambda}^{+,\times}\left\langle
S_{\lambda}(\vec{k},\eta_{1})S_{\lambda}(\vec{K},\eta_{2})\right\rangle.$ (52)

Here, it is convenient to use the dimensionless variables $u\equiv p/k$,
$v\equiv q/k$ and $z\equiv k\eta$ to rewrite the tensor spectrum as

$\displaystyle P_{h}(k,z)$
$\displaystyle=4\int_{0}^{\infty}dv\int_{|1-v|}^{1+v}du\left[\frac{v}{u}-\frac{(1-u^{2}+v^{2})}{4uv}\right]^{2}I^{2}(u,v,z)P_{\zeta}(ku)P_{\zeta}(kv),$
(53) $\displaystyle I(u,v,z)$
$\displaystyle=\frac{9(1+w)^{2}}{(5+3w)^{2}}\int_{0}^{z}dz_{1}\frac{a(z_{1})}{a(z)}kG_{\vec{k}}(z,z_{1})f(u,v,z),$
(54)

where our definition of $I(u,v,z)$ coincide with that defined in Ref.
Kohri:2018awv . Note that the projection of momentum under polarisation
tensors can be found in the Appendix B of Ref. Atal:2021jyo , where

$\displaystyle(e_{ij}^{+}q_{i}q_{j})^{2}+(e_{ij}^{\times}q_{i}q_{j})^{2}=k^{4}v^{4}\left[1-\frac{(1-u^{2}+v^{2})^{2}}{(2v)^{2}}\right]^{2}.$
(55)

For numerical evaluation, we adopt new variables $t=u+v-1$, $s=u-v$ introduced
in Ref. Kohri:2018awv , where $u=\frac{t+s+1}{2}$, $v=\frac{t-s+1}{2}$ and the
tensor spectrum now reads

$\displaystyle P_{h}(k,z)=$ $\displaystyle
2\int_{0}^{\infty}dt\int_{-1}^{1}ds\left[\frac{t(2+t)(s^{2}-1)}{(1-s+t)(1+s+t)}\right]^{2}$
(56) $\displaystyle\quad\times
P_{\zeta}\left(\frac{k(t+s+1)}{2}\right)P_{\zeta}\left(\frac{k(t-s+1)}{2}\right)I_{\rm
RD}^{2}(s,t,z).$

In the late-time limit of the radiation dominated universe i.e. where
$\eta\rightarrow\infty$ and $z\gg 1$, we have the oscillation averaged result
from Ref. Kohri:2018awv  as

$\displaystyle\overline{I^{2}_{\textrm{RD}}(s,t,k\eta\to\infty)}=\frac{288(-5+s^{2}+t(2+t))^{2}}{z^{2}(1-s+t)^{6}(1+s+t)^{6}}\times$
(57)
$\displaystyle\qquad\Bigg{\\{}\frac{\pi^{2}}{4}\left(-5+s^{2}+t(2+t)\right)^{2}\Theta\left(t-(\sqrt{3}-1)\right)+$
$\displaystyle\qquad\left[-(t-s+1)(t+s+1)+\frac{1}{2}(-5+s^{2}+t(2+t))\ln\left|\frac{(-2+t(2+t))}{3-s^{2}}\right|\right]^{2}\Bigg{\\}},$

where $\Theta$ is the usual Heaviside theta function. Hence the averaged
analytical transfer function during radiation domination (57) above can be
substituted into Eq. (56). The resulting integral will yield the oscillation
averaged power spectrum $\overline{P_{h}(k,\eta)}$ which can be substituted
into Eq. (35) and the dimensionless gravitational wave background to be
compared with experimental limits or signals can be determined. We have
verified the calculation by direct comparison with the scale invariant power
spectrum normalised to unity ($A_{\zeta}=1$), which yields a dimensionless
gravitational wave spectrum of $\Omega_{\textrm{IGW}}/A_{\zeta}^{2}=0.822$ as
expected in Ref. Kohri:2018awv .

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