# Supermassive black hole formation from Affleck-Dine mechanism with
suppressed clustering on large scales

Kentaro Kasai  , Masahiro Kawasaki  , Kai Murai  ,
Shunsuke Neda

###### Abstract

We study a primordial black hole (PBH) formation model based on the framework
of the inhomogeneous Affleck-Dine (AD) mechanism, which can explain the seeds
of supermassive black holes (SMBHs). This model, however, predicts strong
clustering of SMBHs that is inconsistent with the observation of angular
correlation of quasars. In this paper, we propose a modified model that can
significantly reduce the PBH clustering on large scales by considering a time-
dependent Hubble-induced mass during inflation. The quasar angular correlation
is suppressed by the large Hubble-induced mass in the early stage of inflation
while the small Hubble-induced mass in the late stage leads to the AD field
fluctuations large enough for PBH formation as in the original model. As a
result, the modified scenario can successfully explain the seeds of SMBHs.

TU-1229

##  1 Introduction

Supermassive black holes (SMBHs) are considered to reside in the centers of
almost all massive galaxies [1, 2, 3]. Furtheremore, the pulsar timing array
experiments, NANOGrav [4], EPTA [5] with InPTA data, PPTA [6], and CPTA [7],
recently reported the detection of stochastic gravitational waves that may be
produced by inspiraling binaries of SMBHs. The astrophysical origin of such
SMBHs has been considered to be the accretion and merger of stellar black
holes. However, the observations by the James Webb Space Telescope [8] suggest
the existence of SMBHs with masses of $\sim
10^{6}\,\text{--}\,10^{10}M_{\odot}$ at high redshifts up to $z\sim 10$ [9,
10, 11, 12, 13, 14, 15, 16, 17, 18]. The super-Eddington accretion is required
for stellar black holes to grow into SMBHs at such high redshifts. Thus, it is
challenging to explain such SMBHs by astrophysical processes [19].

Another possibility for the origin of SMBHs is primordial black holes (PBHs),
i.e., black holes formed from large density fluctuations in the early universe
[20, 21, 22]. In contrast to stellar black holes, PBHs can be formed with an
initial mass as large as $M\gtrsim 10^{4}M_{\odot}$ and can be the seeds of
SMBHs. However, the formation of such heavy PBHs requires large density
fluctuations on the corresponding scales $k\sim
10^{4\,\text{--}\,5}\,\mathrm{Mpc}^{-1}$ and induces the $\mu$-distortion of
the CMB spectrum. In particular, if the density fluctuations follow the
Gaussian distribution, PBHs with $M\gtrsim 10^{4}M_{\odot}$ are severely
constrained from the non-observation of the $\mu$-distortion [23, 24]. On the
other hand, if PBHs are formed from highly non-Gaussian fluctuations, this
constraint can be weakened or evaded [25, 26, 27] (see also Refs. [28, 29] for
the effects of non-Gaussianity on the $\mu$-distortion and its implications
for the constraint on PBHs). The PBH formation models that produce such highly
non-Gaussian density fluctuations are proposed in Refs. [30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44].

In this paper, we consider the PBH formation from the inhomogeneous Affleck-
Dine (AD) mechanism [30, 31, 32, 33, 34, 35, 36] (see also Ref. [45]). In this
scenario, the AD field fluctuates spatially by the effect of quantum diffusion
during inflation. Due to the evolution of the potential for the AD field after
inflation, the universe is separated into large and small-field regions. The
small-field regions occupy most of the universe and have negligible field
values. On the other hand, in the large-field regions, the AD fields are large
enough for the AD mechanism to produce large baryon or lepton asymmetry. The
large-field regions are rare because they are produced from a large field tail
of the distribution. In such regions, the oscillation of the AD field
experiences spatial instabilities and fragments into non-topological solitons
called Q-balls [46, 47, 48, 49, 50, 51, 52]. Since Q-balls behave as non-
relativistic matter, the density contrast between inside and outside the
large-field regions grows during the radiation-dominated era, and the large-
field regions finally collapse into PBHs at the horizon reentry. Since the
large-field regions occupy a tiny volume fraction of the universe, their
density fluctuations give a negligible contribution to the $\mu$-distortion,
and hence avoid the stringent constraint from the CMB. This type of model,
however, inevitably induces the clustering of PBHs due to the large-scale
perturbations of the AD field. Such strong clustering of PBHs is constrained
by observations of isocurvature perturbations [35, 42] and angular
correlations of quasars [53, 54]. These constraints can be relaxed by
suppressing the large-scale fluctuations of the field related to the PBH
formation [44].

Therefore, in this paper, we propose an extension of the AD scenario for the
formation of the seeds of SMBHs. In this scenario, we consider a change of the
Hubble mass during inflation, which is realized, for example, in multi-
inflaton models. In particular, we suppose that a Hubble mass is large in the
first stage of inflation and small in the subsequent second stage, which can
suppress large-scale perturbations of the AD field while keeping the
advantages of the original model. Consequently, we find that the constraint
from clustering is significantly relaxed and that the origin of the SMBHs can
be explained in this scenario.

The rest of this paper is organized as follows. In Sec. 2, we briefly review
the PBH formation through the inhomogeneous AD mechanism. In Sec. 3, we
introduce the modified setup. Then, we discuss the observational constraints
on PBHs and their clustering in Sec. 4 and those on the L-ball scenario in
Sec. 5. Sec. 6 is devoted to the summary and discussion of our results.

##  2 Review of PBH formation from inhomogeneous AD mechanism

First, we briefly review PBH formation from the inhomogeneous AD mechanism
following Ref. [36].

###  2.1 Inhomogeneous Affleck-Dine mechanism

This PBH formation scenario is based on the AD mechanism [55, 56], in which a
baryon or lepton charge is generated by the dynamics of the AD field $\phi$
during and after inflation. The AD field is one of the flat directions in the
scalar potential in the minimal supersymmetric standard model (MSSM) and has a
baryon or lepton charge. The potential of the AD field arises from the
supersymmetry breaking and Planck-suppressed effects. In particular, we
consider the potential for the AD field given by

$\displaystyle V(\phi)=\left\\{\begin{aligned}
&(m_{\phi}^{2}+cH^{2})|\phi|^{2}+V_{\mathrm{NR}}+V_{\mathrm{A}}&\quad\text{(During~{}inflation)}\\\
&(m_{\phi}^{2}-\tilde{c}H^{2})|\phi|^{2}+V_{\mathrm{NR}}+V_{\mathrm{A}}+V_{\mathrm{T}}&\quad\text{(After~{}inflation)}\end{aligned}\right.\
,$ (2.1)

where $m_{\phi}$ is a soft SUSY breaking mass, $H$ is the Hubble parameter,
and $c$ and $\tilde{c}$ are positive dimensionless constants. In addition to
the mass term, we have the non-renormalizable term, $V_{\mathrm{NR}}$, A-term,
$V_{\mathrm{A}}$, and thermal potential, $V_{\mathrm{T}}$, given by

$\displaystyle V_{\mathrm{NR}}$
$\displaystyle=|\kappa|^{2}\frac{|\phi|^{2(n-1)}}{M_{\mathrm{Pl}}^{2(n-3)}}\
,$ (2.2) $\displaystyle V_{\mathrm{A}}$ $\displaystyle=\kappa
a_{M}\frac{m_{3/2}\phi^{n}}{nM_{\mathrm{Pl}}^{n-3}}+\mathrm{h.c.}\ ,$ (2.3)
$\displaystyle V_{\mathrm{T}}$ $\displaystyle=\left\\{\begin{aligned}
&d_{1}T^{2}|\phi|^{2}&\quad(|\phi|\lesssim T)\\\
&d_{2}T^{4}\log\left(\frac{|\phi|^{2}}{T^{2}}\right)&\quad(|\phi|\gtrsim
T)\end{aligned}\right.\ ,$ (2.4)

respectively. Here, $\kappa$, $a_{M}$, and $d_{1,2}$ are dimensionless
constants, $n\,(\geq 4)$ is an integer determined by the choice of the flat
direction, $M_{\mathrm{Pl}}\simeq 2.4\times 10^{18}$ GeV is the reduced Planck
mass, $m_{3/2}$ is the gravitino mass, and $T$ is the cosmic temperature after
inflation. Notice that the A-term, $V_{\mathrm{A}}$, breaks the $U(1)$
symmetry associated with baryon or lepton number conservation.

In the conventional AD mechanism, we assume $c<0$, and $\phi$ has a nonzero
field value during inflation. After inflation, the soft SUSY breaking mass
overcomes the negative Hubble mass term, and the AD field starts to oscillate.
At the same time, $\phi$ is kicked in the phase direction by $V_{\mathrm{A}}$,
and the baryon or lepton charge is generated.

In the PBH formation scenario, we consider $c>0$, and then AD field $\phi$
tends to settle down to the origin. However, since $\phi$ acquires quantum
fluctuations, $\phi$ has a non-zero field value whose distribution is
determined by the Hubble parameter during inflation and e-folds. In this
scenario, $\tilde{c}>0$ is also assumed, and the Hubble-induced mass term
becomes negative just after inflation. At the same time, the thermal bath is
produced by the inflaton decay, which induces the thermal potential. We
further assume that the effective mass due to the thermal potential is larger
than the Hubble-induced mass around the origin ($d_{1}T^{2}>\tilde{c}H^{2}$),
and the AD field is stabilized there. Since $V_{\mathrm{T}}$ has a logarithmic
shape for $\varphi\equiv|\phi|\gtrsim T$, it becomes subdominant for a large
field value. Then, another local minimum of $V(\phi)$ arises at

$\displaystyle\varphi\sim\varphi_{m}\equiv\left(\frac{\sqrt{\tilde{c}}HM_{\mathrm{Pl}}^{n-3}}{\lvert\kappa\rvert}\right)^{\frac{1}{n-2}}\
,$ (2.5)

due to the balance between the negative Hubble mass term and
$V_{\mathrm{NR}}$. Consequently, $V(\phi)$ comes to have multiple local minima
just after inflation.

The local maximum between those two minima is calculated as

$\displaystyle\varphi\sim\varphi_{c}\equiv\sqrt{\frac{d_{2}}{\tilde{c}}}\frac{T^{2}}{H}\
,$ (2.6)

from the Hubble-induced mass and the thermal potential. Here, we used
$\log(\varphi^{2}/T^{2})=\mathcal{O}(1)$. Note that this model requires
$\varphi_{c}\gtrsim H_{\rm{end}}$, where $H_{\rm{end}}$ denotes the Hubble
parameter at the end of inflation such that the origin, $\varphi=0$, is
stabilized after inflation. If $\varphi$ is larger than $\varphi_{c}$ at the
end of inflation, $\phi$ settles down to $\varphi=\varphi_{m}$, which we call
“vacuum B”. In vacuum B, the AD mechanism works as in the conventional case,
and the baryon or lepton charge is generated. On the other hand, if
$\varphi<\varphi_{c}$, $\phi$ settles down to $\varphi=0$, which we call
“vacuum A”, and the AD mechanism does not work. We show the schematic view of
the dynamics of the AD field in Fig. 1. Hereafter we consider that a lepton
charge is generated in the AD mechanism.111If a baryon charge is generated in
the AD mechanism, the inhomogeneous baryon density spoils the success of the
big bang nucleosynthesis as shown in Ref. [36].

After inflation, the negative Hubble-induced term decreases, and vacuum B is
destabilized. Then, the AD field starts to oscillate, and the lepton asymmetry
is generated. Assuming that $\varphi_{c}$ is larger than the typical amplitude
of the fluctuation of $\phi$ during inflation, the regions with
$\varphi>\varphi_{c}$ at the end of inflation become rare. Thus, the resultant
lepton asymmetric regions are also rare, and we call them high lepton bubbles.
The lepton-to-entropy ratio generated in the bubbles is given by

$\displaystyle\eta_{l}\equiv\frac{n_{l}}{s}\simeq\epsilon\frac{T_{\mathrm{RH}}m_{3/2}}{H_{\mathrm{osc}}^{2}}\left(\frac{\varphi_{\mathrm{osc}}}{M_{\mathrm{Pl}}}\right)^{2}\
,$ (2.7)

where $n_{l}$ is the lepton number density, $s$ is the total entropy density,
$\epsilon$ is an efficiency parameter of the asymmetry generation,
$T_{\mathrm{RH}}$ is the reheating temperature, and the subscripts “osc”
denote quantities when the AD field starts to oscillate. Hereafter we take
$\epsilon=0.1$.

Figure 1: Schematic view of the dynamics of the AD field during and after
inflation. Upper side: During inflation, the AD field coarse-grained over a
Hubble patch diffuses in the complex plane and takes different values in
different Hubble patches. Lower side: Just after inflation, the multi-vacuum
structure is realized. In the regions with $\varphi\lesssim\varphi_{c}$,
$\phi$ rolls down to vacuum A, and almost no lepton asymmetry is generated. On
the other hand, if $\varphi\gtrsim\varphi_{c}$, $\phi$ classically rolls down
to vacuum B, and those regions are identified as high lepton bubbles later.

###  2.2 PBH formation

Inside the high lepton bubbles, the AD field oscillates with a large
amplitude. Depending on the potential shape, the AD field experiences spatial
instabilities and forms L-balls [57], i.e., Q-balls with a lepton charge.
Since L-balls behave as non-relativistic matter, the density contrast between
inside and outside the bubbles grows as the universe expands during the
radiation-dominated era. The energy per unit entropy inside the bubble is
written as

$\displaystyle
Y_{Q}^{\mathrm{in}}\equiv\frac{\rho_{Q}^{\mathrm{in}}}{s}=\omega_{Q}\eta_{l}\
,$ (2.8)

where $\rho_{Q}^{\mathrm{in}}$ is the energy density of the L-balls inside the
bubble, and $\omega_{Q}$ is the energy of an L-ball per unit lepton charge.
Then, the density contrast is obtained as

$\displaystyle\delta\equiv\frac{\rho^{\mathrm{in}}-\rho^{\mathrm{out}}}{\rho^{\mathrm{out}}}\simeq\frac{\rho_{Q}^{\mathrm{in}}}{\rho_{\mathrm{rad}}}=\frac{4Y_{Q}^{\mathrm{in}}}{3T}\
,$ (2.9)

where $\rho^{\mathrm{in}}$ and $\rho^{\mathrm{out}}$ are the total energy
density inside and outside the bubbles, respectively, and
$\rho_{\mathrm{rad}}$ is the energy density of the background universe. More
details on the properties of L-balls are discussed in Sec. 5.

If the density contrast is large enough at the horizon reentry of the bubbles,
PBHs are formed. The threshold value of the density contrast evaluated on the
comoving slice is [58]

$\displaystyle\delta_{c}\simeq\frac{3(1+w)}{5+3w}\sin^{2}\left(\frac{\pi\sqrt{w}}{1+3w}\right)\
,$ (2.10)

where $w$ is the parameter of the equation of state inside the bubbles given
by

$\displaystyle
w=\frac{p^{\mathrm{in}}}{\rho^{\mathrm{in}}}\simeq\frac{p^{\mathrm{out}}}{\rho^{\mathrm{in}}}=\frac{1}{3(1+\delta)}\
.$ (2.11)

Combining $\delta>\delta_{c}$ and the above formulae, we obtain the condition
for PBH formation as

$\displaystyle\delta\gtrsim 0.40\ .$ (2.12)

From Eq, (2.9), we obtain the maximum temperature for PBH formation as

$\displaystyle T_{c}\simeq 3.4Y_{Q}^{\mathrm{in}}\ .$ (2.13)

For later convenience, we relate the temperature at the horizon reentry to the
corresponding wavenumber $k$. From the Friedmann equation,

$\displaystyle\frac{k}{a}\simeq
H=\sqrt{\frac{\pi^{2}g_{*}}{90}}\frac{T^{2}}{M_{\mathrm{Pl}}}\ ,$ (2.14)

and the entropy conservation,

$\displaystyle g_{*s}T^{3}a^{3}=g_{*s0}T_{0}^{3}\ ,$ (2.15)

we obtain the relation between $k$ and $T$ as

$\displaystyle T$
$\displaystyle=\sqrt{\frac{90}{\pi^{2}g_{*}}}\left(\frac{g_{*s}}{g_{*s0}}\right)^{1/3}\frac{M_{\mathrm{Pl}}k}{T_{0}}$
$\displaystyle\simeq
3.9\,\mathrm{MeV}\left(\frac{g_{*}}{10.75}\right)^{-1/2}\left(\frac{g_{*s}}{10.75}\right)^{1/3}\left(\frac{k}{4.5\times
10^{4}\,\mathrm{Mpc}^{-1}}\right)\ ,$ (2.16)

where $a$ is the scale factor, $g_{*}$ and $g_{*s}$ are the relativistic
degrees of freedom for the energy and entropy densities, respectively, and the
subscript “0” denotes quantities at the present time. Here, we used
$g_{*s0}=43/11$, $T_{0}=2.725$ K, and $a_{0}=1$.

The PBH mass, defined as $M$, is given by the background horizon mass at PBH
formation [59, 60]. Thus, we can also relate $k$ to $M$ as

$\displaystyle M$
$\displaystyle\simeq\frac{4\pi}{3}\frac{\rho_{\mathrm{rad}}}{H^{3}}=4\pi
M_{\mathrm{Pl}}\sqrt{\frac{\pi^{2}g_{*}}{90}}\left(\frac{g_{*s0}}{g_{*s}}\right)^{2/3}\left(\frac{T_{0}}{k}\right)^{2}$
$\displaystyle\simeq 1.0\times
10^{4}M_{\odot}\left(\frac{g_{*}}{10.75}\right)^{1/2}\left(\frac{g_{*s}}{10.75}\right)^{-2/3}\left(\frac{k}{4.5\times
10^{4}\,\mathrm{Mpc}^{-1}}\right)^{-2}\ .$ (2.17)

From Eqs. (2.13), (2.16), and (2.17), we obtain the minimum PBH mass as

$\displaystyle M\gtrsim M_{c}\simeq 1.3\times
10^{4}M_{\odot}\left(\frac{g_{*}}{10.75}\right)^{-1/2}\left(\frac{Y_{Q}^{\mathrm{in}}}{1\,\mathrm{MeV}}\right)^{-2}\
.$ (2.18)

The bubbles that reenter the horizon at $T<T_{c}$ become PBHs, while the other
bubbles remain as they are. To discuss the seed PBHs of SMBHs, we focus on
$M_{c}=10^{4}M_{\odot}$ below. From Eqs. (2.7), (2.8), and (2.18), we
determine $\varphi_{\mathrm{osc}}$ for given $M_{c},T_{\mathrm{RH}},$ and
$m_{3/2}$.

###  2.3 PBH mass spectrum

Next, we consider the size distribution of the high lepton bubbles and the PBH
mass spectrum. The distribution of the bubbles is determined by quantum
fluctuations of the AD field during inflation. The fluctuations of the AD
field coarse-grained over the Hubble scale are described by the probability
distribution, $P(N,\phi)$. Here, $N$ is the e-folding number during inflation
defined by

$\displaystyle N(k)\equiv\ln\left(\frac{k}{k_{0}}\right)\ ,$ (2.19)

where $k$ is the wavenumber that exits the horizon at $N$, and $k_{0}\simeq
2.24\times 10^{-4}$ Mpc-1 is the scale of the current observable universe. In
other words, we define $N$ so that the current horizon scale exits the horizon
at $N=0$ during inflation. The probability distribution follows the Fokker-
Planck equation

$\displaystyle\frac{\partial P(N,\phi)}{\partial
N}=\sum_{n=1,2}\frac{\partial}{\partial\phi_{n}}\left[\frac{\partial
V(\phi)}{\partial\phi_{n}}\frac{P(N,\phi)}{3H_{I}^{2}}+\frac{H_{I}^{2}}{8\pi^{2}}\frac{\partial
P(N,\phi)}{\partial\phi_{n}}\right],$ (2.20)

where $H_{I}$ is the Hubble parameter during inflation, which we assume to be
constant, and $\phi_{n}\,(n=1,2)$ is the canonical real scalar field defined
by

$\displaystyle\phi=\frac{\phi_{1}+i\phi_{2}}{\sqrt{2}}.$ (2.21)

We approximate the potential of the AD field by a Hubble-induced mass term,

$\displaystyle V(\phi)=cH_{I}^{2}{\lvert\phi\rvert}^{2}.$ (2.22)

To simplify the presentation, we use dimensionless fields defined by

$\displaystyle\tilde{\phi}\equiv\tilde{\phi}_{1}+i\tilde{\phi}_{2}\
,\quad\tilde{\phi}_{1}\equiv\frac{\sqrt{2}\pi\phi_{1}}{H_{I}}\
,\quad\tilde{\phi}_{2}\equiv\frac{\sqrt{2}\pi\phi_{2}}{H_{I}}\ .$ (2.23)

Then, the Fokker-Planck equation becomes

$\displaystyle\frac{\partial\tilde{P}(N,\tilde{\phi})}{\partial
N}=c^{\prime}\tilde{P}(N,\tilde{\phi})+\sum_{n=1,2}\left[\frac{c^{\prime}\tilde{\phi}_{n}}{2}\frac{\partial\tilde{P}(N,\tilde{\phi})}{\partial\tilde{\phi}_{n}}+\frac{\partial^{2}\tilde{P}(N,\tilde{\phi})}{\partial\tilde{\phi}_{n}^{2}}\right],$
(2.24)

where $c^{\prime}\equiv 2c/3$. Here, $\tilde{P}$ is defined by

$\displaystyle\tilde{P}(N,\tilde{\phi})\equiv\left(\frac{H_{I}}{\sqrt{2}\pi}\right)^{2}P(N,\phi)\
,$ (2.25)

which satisfies the normalization of

$\displaystyle\iint\mathrm{d}\tilde{\phi}_{1}\mathrm{d}\tilde{\phi}_{2}\,\tilde{P}(N,\tilde{\phi})=1\
.$ (2.26)

Setting the initial condition as
$\tilde{P}(N_{i},\tilde{\phi})=\delta^{2}(\tilde{\phi}-\tilde{\phi}_{i})$, we
obtain the solution as [35]

$\displaystyle\tilde{P}(N,\tilde{\phi};c^{\prime},N_{i},\tilde{\phi}_{i})=\frac{1}{2\pi\sigma^{2}(N-N_{i},c^{\prime})}\exp\left[-\frac{1}{2\sigma^{2}(N-N_{i},c^{\prime})}\Big{\lvert}\tilde{\phi}-e^{-\frac{c^{\prime}}{2}(N-N_{i})}\tilde{\phi}_{i}\Big{\rvert}^{2}\right]\
,$ (2.27)

where

$\displaystyle\sigma^{2}(N,c^{\prime})\equiv\frac{1-e^{-c^{\prime}N}}{2c^{\prime}}\
.$ (2.28)

In the following, we assume $N_{i}=0$ and $\tilde{\phi}_{i}=0$ for simplicity.

If $\tilde{\varphi}\equiv|\tilde{\phi}|$ is larger than the threshold value
$\tilde{\varphi}_{c}=2\pi\varphi_{c}/H_{I}$ at the end of inflation, bubbles
will be formed. Regarding that the super-horizon modes of the AD field obey
the classical motion, we define an effective threshold value during inflation
as

$\displaystyle\tilde{\varphi}_{c,\mathrm{eff}}(N)\equiv
e^{\lambda(N_{\mathrm{end}}-N)}\tilde{\varphi}_{c}\ ,$ (2.29)

where $N_{\mathrm{end}}$ is the e-folding number at the end of inflation, and

$\displaystyle\lambda\equiv\frac{3}{2}\left(1-\sqrt{1-\frac{2}{3}c^{\prime}}\right)\
.$ (2.30)

Note that $\lambda\simeq c^{\prime}/2$ for $c^{\prime}\ll 1$. Thus, we can
regard that a region finally becomes a high lepton bubble after inflation if
$\tilde{\varphi}$ coarse-grained over that region is larger than
$\tilde{\varphi}_{c,\mathrm{eff}}(N)$ at the horizon exit. From this
observation, we can estimate the volume fraction of the bubbles on larger
scales than $k^{-1}$ as

$\displaystyle
B(N_{k})=\int_{\tilde{\varphi}_{c,\mathrm{eff}}(N_{k})}^{\infty}\mathrm{d}\tilde{\varphi}\,\bar{P}(N_{k},\tilde{\varphi};c^{\prime},0,0)\
,$ (2.31)

where $N_{k}\equiv N(k)$, and $\bar{P}$ is the probability distribution for
$\tilde{\varphi}$ given by

$\displaystyle\bar{P}(N,\tilde{\varphi};c^{\prime},N_{i},\tilde{\varphi}_{i})\equiv\int_{0}^{2\pi}\mathrm{d}\theta\,\tilde{\varphi}\tilde{P}(N,\tilde{\varphi}e^{i\theta};c^{\prime},N_{i},\tilde{\varphi}_{i})\
.$ (2.32)

By differentiating the volume fraction with respect to $N_{k}$, we obtain the
size distribution of the bubbles by

$\displaystyle\beta_{N}(N_{k})\equiv\frac{\partial B(N_{k})}{\partial N_{k}}\
.$ (2.33)

From $M\propto k^{-2}\propto e^{-2N}$, we can translate $\beta_{N}$ into the
PBH formation rate per $\ln M$ as

$\displaystyle\beta(M)\equiv\frac{1}{2}\beta(N)\Theta(M-M_{c})\ .$ (2.34)

Here, $\Theta$ is the Heaviside step function. Finally, we obtain the present
energy density ratio of PBHs to the total dark matter as

$\displaystyle
f_{\mathrm{PBH}}\equiv\frac{\Omega_{\mathrm{PBH}}}{\Omega_{c}}=\int\mathrm{d}(\ln
M)\,\beta(M)\frac{T(M)}{T_{\mathrm{eq}}}\frac{\Omega_{m}}{\Omega_{c}}\ ,$
(2.35)

where $\Omega_{\mathrm{PBH}}$, $\Omega_{c}$, and $\Omega_{m}$ are the current
density parameters of the PBHs, dark matter, and non-relativistic matter,
respectively, $T(M)$ is the formation temperature of PBHs with a mass of $M$,
and $T_{\mathrm{eq}}$ is the cosmic temperature at matter-radiation equality.

###  2.4 PBH correlation

We can also estimate the two-point correlation of PBHs from the probability
distribution. In the following, we consider PBHs with a certain mass $M$
assuming a monochromatic mass distribution for simplicity. We then define the
PBH density fluctuation by

$\displaystyle\delta_{\mathrm{PBH}}(\bm{x})\equiv\frac{\rho_{\mathrm{PBH}}(\bm{x})}{\bar{\rho}_{\mathrm{PBH}}}-1=\frac{1}{\bar{n}_{\mathrm{PBH}}}\sum_{i}\delta^{(3)}(\bm{x}-\bm{x}_{i})-1\
,$ (2.36)

where $\rho_{\mathrm{PBH}}$ and $n_{\mathrm{PBH}}$ are the energy and number
densities of PBHs, the barred density denotes spatially averaged one, and
$\bm{x}_{i}$ denotes the position of the $i$-th PBH. From the above equation,
the two-point correlation function is written as

$\displaystyle\langle\delta_{\mathrm{PBH}}(0)\delta_{\mathrm{PBH}}(\bm{x})\rangle$
$\displaystyle=\left\langle\sum_{i,j}\frac{\delta^{(3)}(-\bm{x}_{i})\delta^{(3)}(\bm{x}-\bm{x}_{j})}{\bar{n}_{\mathrm{PBH}}^{2}}-\frac{\sum_{i}\delta^{(3)}(-\bm{x}_{i})+\sum_{j}\delta^{(3)}(\bm{x}-\bm{x}_{j})}{\bar{n}_{\mathrm{PBH}}}+1\right\rangle$
$\displaystyle=\sum_{i}\frac{\delta^{(3)}(\bm{x})}{\bar{n}_{\mathrm{PBH}}}+\sum_{i\neq
j}\frac{\delta^{(3)}(-\bm{x}_{i})\delta^{(3)}(\bm{x}-\bm{x}_{j})}{\bar{n}_{\mathrm{PBH}}^{2}}-1$
$\displaystyle\equiv\sum_{i}\frac{\delta^{(3)}(\bm{x})}{\bar{n}_{\mathrm{PBH}}}+\xi(\bm{x})\
,$ (2.37)

where the bracket denotes the average over the positions of PBHs. While the
first term describes the Poisson fluctuation, $\xi$ is the reduced PBH
correlation function describing the PBH clustering, which vanishes if PBHs are
not correlated.

To evaluate the correlation function, we consider the probability distribution
of the bubble formation at two points $x$ and $y$ separated by a comoving
distance $L(=2\pi/k)$ given by

$\displaystyle\beta_{N2}(N_{L},N_{x},N_{y})=$
$\displaystyle\frac{\partial^{2}}{\partial N_{x}\partial
N_{y}}\int_{0}^{\infty}\mathrm{d}\tilde{\varphi}_{L}\bar{P}(N_{k},\tilde{\varphi};c^{\prime},0,0)$
$\displaystyle\times\int_{\tilde{\varphi}_{c,\mathrm{eff}}(N_{x})}^{\infty}\mathrm{d}\tilde{\varphi}_{x}\,\bar{P}(N_{x},\tilde{\varphi}_{x};c^{\prime},N_{L},\tilde{\varphi}_{L})\int_{\tilde{\varphi}_{c,\mathrm{eff}}(N_{y})}^{\infty}\mathrm{d}\tilde{\varphi}_{y}\,\bar{P}(N_{y},\tilde{\varphi}_{y};c^{\prime},N_{L},\tilde{\varphi}_{L})\
,$ (2.38)

where $N_{x}$ and $N_{y}$ are the e-folding numbers corresponding to the PBHs
formed at $x$ and $y$, respectively, and $N_{L}$ is the e-folding number when
$L$ exits the horizon. Using $\beta_{N2}$, we obtain the correlation function
as

$\displaystyle\xi(L)=\frac{\beta_{N2}(N_{L},N(M),N(M))}{\beta_{N}(N(M))^{2}}-1\
.$ (2.39)

In this scenario, the PBHs are strongly clustered even on large scales, which
leads to large isocurvature perturbations and angular correlations. Thus, the
scenario is stringently constrained from the observations of CMB and angular
correlations of quasars [53, 54] depending on the mass and abundance. In
particular, the latter excludes the scenario explaining seed PBHs for the SMBH
as shown in Sec. 4.

##  3 Modified model with double inflation

In this section, we introduce a modification of the scenario described in the
previous section and discuss the suppression of the correlation function.

###  3.1 Setup

The PBH correlation comes from the fluctuation of the AD field on scales
larger than the PBH scale. Thus, we expect that, if the quantum diffusion of
the AD field is suppressed in the earlier stage of inflation compared to the
later stage, the correlation function is also suppressed. We can realize such
a situation by considering the time dependence of the Hubble mass. As an
example, we consider double inflation [61]. In double inflation, two scalar
fields work as the inflaton successively. Thus, the coefficient of the Hubble
mass is also naturally different in the first and second stages of inflation.
In the following, we assume that $c^{\prime}$ changes during inflation as

$\displaystyle c^{\prime}=\begin{cases}c^{\prime}_{1}&N<N_{\ast}\\\
c^{\prime}_{2}&N\geq N_{\ast}\end{cases}\ ,$ (3.1)

where $N_{\ast}$ denotes the e-folding number at the transition of
inflationary stages. For simplicity, we also assume that $H_{I}$ is constant
during inflation although this is not the case in the realistic situation. The
possible effects of the time evolution of the Hubble parameter will be
discussed later.

###  3.2 Probability density for the modified setup

For $N<N_{*}$, the probability distribution is the same as in the original
setup:

$\displaystyle\tilde{P}_{\mathrm{m}}(N,\tilde{\phi};c^{\prime},N_{i},\tilde{\phi}_{i})=\tilde{P}(N,\tilde{\phi};c^{\prime},N_{i},\tilde{\phi}_{i})\
,$ (3.2)

where $\tilde{P}_{\mathrm{m}}$ is the probability distribution in the modified
setup. On the other hand, $\tilde{P}_{\mathrm{m}}$ for $N\geq N_{*}$ is
obtained by convoluting the $\tilde{P}$ in the first and second stages as

$\displaystyle\tilde{P}_{\mathrm{m}}(N,\tilde{\phi};c^{\prime}_{1},c^{\prime}_{2},N_{\ast},N_{i},\tilde{\phi}_{i})=$
$\displaystyle\int\mathrm{d}\tilde{\phi}_{\ast}\,\tilde{P}(N,\tilde{\phi};c^{\prime}_{2},N_{\ast},\tilde{\phi}_{\ast})\tilde{P}(N_{\ast},\tilde{\phi}_{\ast};c^{\prime}_{1},N_{i},\tilde{\phi}_{i})$
$\displaystyle=$
$\displaystyle\int\mathrm{d}\tilde{\phi}_{\ast}\frac{1}{2\pi\sigma^{2}(N-N_{\ast})}\exp{\left[-\frac{1}{2\sigma^{2}(N-N_{\ast})}\Big{\lvert}\tilde{\phi}-e^{-\frac{c^{\prime}_{2}}{2}(N-N_{\ast})}\tilde{\phi}_{\ast}\Big{\rvert}^{2}\right]}$
$\displaystyle\times\frac{1}{2\pi\sigma^{2}(N_{\ast}-N_{i})}\exp{\left[-\frac{1}{2\sigma^{2}(N_{\ast}-N_{i})}\Big{\lvert}\tilde{\phi}_{\ast}-e^{-\frac{c^{\prime}_{1}}{2}(N_{\ast}-N_{i})}\tilde{\phi}_{i}\Big{\rvert}^{2}\right]}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi\tilde{\sigma}^{2}(N)}\exp\left[-\frac{{\Big{\lvert}\tilde{\phi}-\tilde{\phi}_{i}e^{-\frac{c^{\prime}_{1}}{2}(N_{\ast}-N_{i})-\frac{c^{\prime}_{2}}{2}(N-N_{\ast})}\Big{\rvert}}^{2}}{2\tilde{\sigma}^{2}(N)}\right],$
(3.3)

where

$\displaystyle\tilde{\sigma}^{2}(N)$
$\displaystyle\equiv\sigma^{2}(N-N_{\ast},c^{\prime}_{2})+e^{-c^{\prime}_{2}(N-N_{\ast})}\sigma^{2}(N_{\ast}-N_{i},c^{\prime}_{1})\
.$ (3.4)

In a similar way to $\bar{P}$, we obtain the probability distribution of
$\tilde{\varphi}$ for $N\geq N_{*}$ as

$\displaystyle\begin{aligned}
\bar{P}_{\mathrm{m}}(N,\tilde{\varphi};N_{\ast},N_{i},\tilde{\varphi}_{i})=&\frac{\tilde{\varphi}}{\tilde{\sigma}^{2}(N)}I_{0}\left(e^{-\frac{c^{\prime}_{1}}{2}(N_{\ast}-N_{i})-\frac{c^{\prime}_{2}}{2}(N-N_{\ast})}\frac{\tilde{\varphi}\tilde{\varphi}_{i}}{\tilde{\sigma}^{2}(N)}\right)\\\
&\times\exp{\left[-\frac{\tilde{\varphi}^{2}+e^{-c^{\prime}_{1}(N_{\ast}-N_{i})-c^{\prime}_{2}(N-N_{\ast})}\tilde{\varphi}_{i}^{2}}{2\tilde{\sigma}^{2}(N)}\right]}.\end{aligned}$
(3.5)

The condition for lepton bubble formation is that the field value at the end
of inflation exceeds the threshold $\tilde{\varphi}_{c}$. Considering the
classical dynamics of the AD field after the horizon exit, the threshold value
given at the e-folding number $N$ is obtained as

$\displaystyle\tilde{\varphi}_{c,\mathrm{eff}}(N)=\begin{cases}\tilde{\varphi}_{c}e^{\lambda_{2}(N_{\mathrm{end}}-N)}&N\geq
N_{\ast}\\\
\tilde{\varphi}_{c}e^{\lambda_{2}(N_{\mathrm{end}}-N_{\ast})+\lambda_{1}(N_{\ast}-N)}&N<N_{\ast}\end{cases}\
,$ (3.6)

where

$\displaystyle\lambda_{1,2}=\frac{3}{2}\left(1-\sqrt{1-\frac{2}{3}c^{\prime}_{1,2}}\right)\
.$ (3.7)

Note that $\lambda_{1,2}\simeq c^{\prime}_{1,2}/2$ for $c^{\prime}_{1,2}\ll
1$. Using the obtained $\bar{P}_{\mathrm{m}}$, we can evaluate the PBH
abundance and correlation function. By applying Eq. (2.31) to the modified
scenario, we obtain the volume fraction of the high lepton bubbles shown in
Fig. 2.

Figure 2: Volume fraction of lepton bubbles per $\ln k$ for different values
of $N_{\ast}$. The horizontal axis denotes the bubble size in terms of the
wave number. The lepton bubbles on the left side of the black line form PBHs,
and those on the right side remain as bubbles. Here, we fixed
$f_{\mathrm{PBH}}=3\times 10^{-9}$, $c^{\prime}_{1}=1$ and
$c^{\prime}_{2}=0.005$. The smaller bubbles have larger volume fractions. The
larger $N_{\ast}$ becomes, the more lepton bubbles remain.

##  4 Observational constraints on PBH

As mentioned in the introduction, PBH formation from the inhomogeneous AD
mechanism can avoid the constraint from the $\mu$-distortion of the CMB
spectrum. However, as for seed PBHs explaining SMBHs, heating by accretion of
PBHs changes the ionization history of the universe and affects the CMB
anisotropies, from which a stringent constraint is imposed on the PBH
abundance as $f_{\mathrm{PBH}}\lesssim 3\times 10^{-9}$ for $M\sim
10^{4}M_{\odot}$ [62, 63, 64, 65].222This constraint can be largely relaxed
depending on the choice of the accretion model and assumption on the emission
efficiency [66]. This constraint is accidentally comparable to the estimated
SMBH abundance, $f_{\mathrm{SMBH}}\sim 2.9\times 10^{-9}$ for
$M>10^{6}M_{\odot}$ [67] (see also Ref. [65]). In this paper, taking into
account the ambiguity of the growth of the PBH masses by accretion and
mergers, we consider two cases: $f_{\mathrm{PBH}}\sim 3\times 10^{-9}$ and
$3\times 10^{-10}$.

In addition, there are constraints on the PBH clustering. In Fig. 3, we show
the PBH correlation functions in this model. We fix $f_{\mathrm{PBH}}=3\times
10^{-9}$ for different values of $N_{*}$ by choosing
$\tilde{\varphi}_{c}=\mathcal{O}(10)$. Note that $\tilde{\varphi}\gtrsim 2\pi$
is required for the origin, $\varphi=0$, to be stabilized just after
inflation. Here we take $\tilde{c}\simeq d_{1}\simeq d_{2}\simeq 1$. The
original model corresponds to the case with $N_{*}=0$, and the correlation
function is more suppressed for larger $N_{*}$. In the following, we discuss
the constraints from isocurvature perturbations and the angular correlation of
quasars.

Figure 3: PBH correlation function for different values of $N_{*}$. Here, we
take $c^{\prime}_{2}=0.005$ and fix $f_{\mathrm{PBH}}$ by choosing
$\tilde{\varphi}_{c}$. The left end has apparent divergence coming from the
self-correlation at the scale of the PBH-forming bubble. The right end goes to
zero at the edge of the observable universe. In the actual numerical
calculation, it is difficult to evaluate the correlation near those endpoints.
Larger $N_{\ast}$ provides the suppression of the correlation in a wider range
of scales.

###  4.1 Isocurvature perturbation from PBH clustering

First, we discuss the constraint from isocurvature perturbations on the CMB
scales following Refs. [35, 42]. We define the power spectrum of the PBH
density perturbation by

$\displaystyle
P_{\mathrm{PBH}}=\int\mathrm{d}^{3}\bm{x}e^{-i\bm{k}\cdot\bm{x}}\langle\delta_{\mathrm{PBH}}(0)\delta_{\mathrm{PBH}}(\bm{x})\rangle=\frac{1}{\bar{n}_{\mathrm{PBH}}}+P_{\xi}(k)\
,$ (4.1)

where the first term comes from the Poisson fluctuation, and $P_{\xi}$ is
given by

$\displaystyle
P_{\xi}(k)\equiv\int\mathrm{d}^{3}\bm{x}\,\xi(x)e^{-i\bm{k}\cdot\bm{x}}=4\pi\int\mathrm{d}r\,r^{2}\xi(r)\frac{\sin
kr}{kr}\ ,$ (4.2)

where $r\equiv|\bm{x}|$. This is tested in terms of the isocurvature
perturbations of the CDM density. The Planck observation puts the constraint
[68]

$\displaystyle\beta_{\mathrm{iso}}\equiv\frac{f_{\mathrm{PBH}}^{2}\mathcal{P}_{\xi}(k_{\mathrm{CMB}})}{f_{\mathrm{PBH}}^{2}\mathcal{P}_{\xi}(k_{\mathrm{CMB}})+\mathcal{P}_{\mathcal{R}}(k_{\mathrm{CMB}})}<0.036\
,$ (4.3)

where $\mathcal{P}_{\xi}\equiv k^{3}P_{\xi}/(2\pi^{2})$, and
$\mathcal{P}_{\mathcal{R}}(k_{\mathrm{CMB}})\simeq 2\times 10^{-9}$ is the
dimensionless power spectrum of the curvature perturbations at
$k_{\mathrm{CMB}}=0.002\,\mathrm{Mpc}^{-1}$ [69]. We show
$\beta_{\mathrm{iso}}$ for different values of $N_{*}$ in Fig. 4. The
isocurvature perturbation is smaller than the observational constraint even in
the original model, and it is suppressed in the modified model.

Figure 4: Isocurvature perturbations for the different values of $N_{*}$. Here
we take $c^{\prime}_{2}=0.005$. The green-shaded region shows the excluded
region by the CMB observation. Even the original model ($N_{\ast}=0$) is not
excluded. The modified model has suppressed isocurvature perturbations, and
the suppression becomes stronger for larger $N_{\ast}$. Since the fluctuations
of the AD field on large scales are suppressed by the Hubble mass in the early
stage of inflation, the suppression of $\beta_{\mathrm{iso}}$ is more
significant for larger $c^{\prime}_{1}$. The results include numerical errors
coming from the fitting procedure.

To evaluate $\beta_{\mathrm{iso}}$, we simplify the correlation function using
a fitting function. First, we divide the correlation function into the large-
scale and small-scale regions at the comoving distance corresponding to
$N=N_{\ast}$. Then, we fit $\xi(r)$ by power-law functions of $r$ in each
region. Since we expect that $P_{\xi}(k_{\mathrm{CMB}})$ is contributed mainly
by $\xi(r)$ on large scales, we evaluate only the large-scale contribution to
avoid unphysical divergence at $r\sim 0$. In fact, a similar study for the PBH
formation from axion bubbles [44] shows that $P_{\xi}$ converges by using an
analytic expression of $\xi(r)$. Based on the physical consideration and the
result of [44], we expect that the small-scale region does not contribute to
the isocurvature perturbation.

###  4.2 Angular correlation functions of SMBH

Next, we discuss the angular correlation of the produced PBHs. We first
summarize the definition of the angular correlation function following Ref.
[53]. We define a two-dimensional PBH number density as the integral of the
comoving number density $n_{\mathrm{PBH}}(R,\theta,\phi)$ over a line of
sight,

$\displaystyle\mathcal{N}_{\mathrm{PBH}}(\theta,\phi)\equiv\int_{0}^{\infty}\mathrm{d}R\,R^{2}W(R)n_{\mathrm{PBH}}(R,\theta,\phi)\
,$ (4.4)

where $\theta$ and $\phi$ are the polar and azimuthal angles of the line of
sight, and $W(R)$ is a window function accounting for the range of $R$
relevant to the observations. The density contrast is defined for
$\mathcal{N}_{\mathrm{PBH}}$ as

$\displaystyle\Delta_{\mathrm{PBH}}(\theta,\phi)\equiv\frac{\mathcal{N}_{\mathrm{PBH}}(\theta,\phi)-\bar{\mathcal{N}}_{\mathrm{PBH}}}{\bar{\mathcal{N}}_{\mathrm{PBH}}}\
,$ (4.5)

where $\bar{\mathcal{N}}_{\mathrm{PBH}}$ is the mean value of
$\mathcal{N}_{\mathrm{PBH}}$ over $\theta$ and $\phi$. For the angular
separation $\theta$ between $(\theta_{1},\phi_{1})$ and
$(\theta_{2},\phi_{2})$, the PBH angular correlation function is defined as
[53]

$\displaystyle w_{\mathrm{PBH}}(\theta)\equiv$
$\displaystyle\braket{\Delta_{\mathrm{PBH}}(\theta_{1},\phi_{1})\Delta_{\mathrm{PBH}}(\theta_{2},\phi_{2})}$
(4.6) $\displaystyle=$
$\displaystyle\int_{R_{\mathrm{low}}}^{R_{\mathrm{high}}}\mathrm{d}R_{1}\int_{R_{\mathrm{low}}}^{R_{\mathrm{high}}}\mathrm{d}R_{2}\frac{3R_{1}^{2}}{R_{\mathrm{high}}^{3}-R_{\mathrm{low}}^{3}}\frac{3R_{2}^{2}}{R_{\mathrm{high}}^{3}-R_{\mathrm{low}}^{3}}$
$\displaystyle\times\xi\left(\sqrt{R_{1}^{2}+R_{2}^{2}-2R_{1}R_{2}\cos{\theta}}\right)\
,$ (4.7)

where $R_{\mathrm{high/low}}$ denotes the distance to the maximum/minimum
redshift of the observations, $z_{\mathrm{high/low}}$. We regard the present
universe as a flat $\Lambda$CDM one and choose $z_{\mathrm{high}}=6.49$ and
$z_{\mathrm{low}}=5.88$ to match the observations analyzed in Ref. [54].

This angular correlation function increases at small angular separations, and
then the observational constraint becomes stronger for small $\theta$. Since
$\theta=0.24^{\circ}$ is the smallest separation in the observational limit,
we require that the predicted angular correlation at $\theta=0.24^{\circ}$
should be smaller than the observational limit. We show
$w_{\mathrm{PBH}}(0.24^{\circ})$ and the observational constraint [54] in Fig.
5. We find that $0.5\leq c^{\prime}_{1}$ and $10\leq N_{\ast}$ can avoid the
observational constraint from the angular correlation of quasars.

Figure 5: Angular correlation for the different values of $N_{\ast}$. Here we
take $c^{\prime}_{2}=0.005$. The green-shaded region shows the excluded region
by the quasar observation. The suppression of the angular correlation becomes
stronger for larger $N_{\ast}$ like the isocurvature perturbations. We observe
that $N_{\ast}\geq 10$ with $c^{\prime}_{1}\geq 0.5$ can avoid the
observational constraint.

##  5 L-ball scenario

In this section, we explain the properties of L-balls and discuss the
observational constraints on the scenario following Ref. [36]. In the
following, we consider two types of L-balls: new type and delayed type. First,
we review the common properties and then discuss each type.

Although L-balls are classically stable field configurations guaranteed by
$U(1)$ symmetry, they finally decay into the standard model lepton charge via
quantum effect and thermal effect. This can affect both BBN and the effective
neutrino number density $N_{\rm{eff}}$, which is constrained by CMB. In this
section, we investigate the parameter regions consistent with these
constraints.

###  5.1 Physics with L-balls

In gauge-mediated SUSY breaking models, the scalar potential is given by [70]

$\displaystyle
V_{\mathrm{gauge}}(\phi)=\begin{cases}m_{\phi}^{2}{\lvert\phi\rvert}^{2}&({\lvert\phi\rvert}\ll
M_{S})\\\
M_{F}^{4}\left(\ln{\frac{{\lvert\phi\rvert}^{2}}{M_{S}^{2}}}\right)^{2}&({\lvert\phi\rvert}\gg
M_{S})\end{cases}\ .$ (5.1)

Here, $M_{F}$ and $M_{S}$ are the mass scales of the SUSY breaking and
messenger sectors, respectively. The gravity mediation effect, although mostly
subdominant, still exists in the gauge-mediated SUSY breaking models and
provides the scalar potential [70]

$\displaystyle
V_{\mathrm{grav}}(\phi)=m_{3/2}^{2}\left[1+K\ln{\frac{{\lvert\phi\rvert}^{2}}{M_{\mathrm{Pl}}^{2}}}\right]{\lvert\phi\rvert}^{2}\
,$ (5.2)

where $K$ is a dimensionless constant. Note that the potential (5.2) dominates
over the gauge-mediation potential (5.1) for
$\varphi\equiv|\phi|\gtrsim\varphi_{\mathrm{eq}}\equiv M_{F}^{2}/m_{3/2}$. We
assume $K<0$ for the new-type L-ball scenario (Sec. 5.2), and $K>0$ for the
delayed-type L-ball scenario (Sec. 5.3).

Putting the discussions of the L-ball formation aside, we discuss the time
evolution of L-balls here. Now, we focus on the vacuum B, which forms lepton
bubbles. After the AD field settles to vacuum B, it starts to oscillate when
the effective mass becomes comparable to the Hubble parameter. In this paper,
we assume that the oscillation starts when the gravity-mediation effects (5.2)
dominate the potential. Thus, the Hubble parameter and
$\varphi_{\mathrm{osc}}$ at the start of oscillation is given by

$\displaystyle H_{\mathrm{osc}}$ $\displaystyle\simeq m_{3/2}\ ,$ (5.3)
$\displaystyle\varphi_{\mathrm{osc}}$
$\displaystyle\simeq\left(\frac{\sqrt{\tilde{c}}m_{3/2}M_{\mathrm{Pl}}^{n-3}}{\lvert\kappa\rvert}\right)^{\frac{1}{n-2}}\
.$ (5.4)

At this time we neglect the thermal potential (2.4) since it is almost
subdominant for a parameter region that we are interested in.

After the AD-field oscillation starts, L-balls are formed as discussed later.
L-balls continuously lose and emit their charge due to the thermal effect
[71]. This process is called evaporation and its rate is written as

$\displaystyle\Gamma_{\mathrm{evap}}\equiv\frac{\mathrm{d}Q}{\mathrm{d}t}\bigg{|}_{\mathrm{evap}}\sim-4\pi\chi\left(\mu_{Q}-\mu_{\mathrm{plasma}}\right)T^{2}R_{Q}^{2}\simeq-4\pi\chi\omega_{Q}T^{2}R_{Q}^{2}\
,$ (5.5)

where $Q$ and $R_{Q}$ are the charge and radius of the L-ball, respectively,
and $\mu_{Q,\,(\mathrm{plasma})}$ is the chemical potential for lepton number
inside (outside) the L-ball. Here we use $\mu_{Q}=\omega_{Q}$ and
$\mu_{\mathrm{plasma}}\ll\mu_{Q}$, and adopt the suppression at low
temperatures given by [71]

$\displaystyle\chi(T)=\begin{cases}1&(T>m_{\phi})\\\
\left(\frac{T}{m_{\phi}}\right)^{2}&(T<m_{\phi})\end{cases}\ .$ (5.6)

For efficient L-ball evaporation, the evaporated charge should be diluted away
from the L-ball surface by diffusion [72]. The diffusion rate is given by

$\displaystyle\Gamma_{\mathrm{diff}}\equiv\frac{\mathrm{d}Q}{\mathrm{d}t}\bigg{|}_{\mathrm{diff}}\sim-4\pi
DR_{Q}\mu_{Q}T^{2}\simeq-4\pi DR_{Q}\omega_{Q}T^{2}\ ,$ (5.7)

where the diffusion constant $D\equiv A/T$ is estimated with a numerical
constant $A=4\,\text{--}\,6$. When the diffusion rate $\Gamma_{\mathrm{diff}}$
is smaller than $\Gamma_{\mathrm{evap}}$, the evaporation process is
controlled by diffusion. Thus, the rate of the evaporation process is
determined by the smaller one of $\Gamma_{\mathrm{evap}}$ and
$\Gamma_{\mathrm{diff}}$, and is given by $\Gamma_{\mathrm{diff}}$ at high
temperatures and $\Gamma_{\mathrm{evap}}$ at low temperatures. For later
convenience, we define the temperature $\tilde{T}$ for
$\Gamma_{\mathrm{diff}}(\tilde{T})=\Gamma_{\mathrm{evap}}(\tilde{T})$. Note
that $\tilde{T}<m_{\phi}$ in the typical parameter region.

The L-balls that survive evaporation finally decay into neutrinos through
quantum effects [73]. The L-ball decay rate is given by [74, 75]

$\displaystyle\left.\frac{\mathrm{d}Q}{\mathrm{d}t}\right|_{\mathrm{decay}}\simeq-
N_{\ell}\frac{\omega_{Q}^{3}}{12\pi^{2}}4\pi R_{Q}^{2}\ ,$ (5.8)

where $N_{\ell}=3$ is the number of decay channels.

For later convenience, we clarify the transformation from the cosmic time $t$
to the temperature $T$ here:

$\displaystyle\frac{\mathrm{d}t}{\mathrm{d}T}=\begin{cases}-\frac{288}{\pi^{2}g_{\ast}}M_{\mathrm{Pl}}^{2}H(T_{\mathrm{RH}})T^{-5}&\left(T\gtrsim
T_{\mathrm{RH}}\right)\\\
-\frac{3}{\pi}\sqrt{\frac{10}{g_{\ast}}}M_{\mathrm{Pl}}T^{-3}&\left(T\lesssim
T_{\mathrm{RH}}\right)\end{cases}.$ (5.9)

###  5.2 New-type L-ball

When $V_{\mathrm{grav}}(\phi)$ dominates
($\varphi\gtrsim\varphi_{\mathrm{eq}}$) and $K<0$, “new-type” L-balls are
formed. The assumption $\varphi_{\mathrm{osc}}\gtrsim\varphi_{\mathrm{eq}}$ is
satisfied in most cases of our scenario that explains the PBHs with
$10^{4}M_{\odot}$. The properties of a new-type L-ball are given as follows:

$\displaystyle Q_{N}$
$\displaystyle\simeq\beta_{N}\left(\frac{\varphi_{\mathrm{osc}}}{m_{3/2}}\right)^{2},$
(5.10) $\displaystyle M_{Q}$ $\displaystyle\simeq m_{3/2}Q_{N},$ (5.11)
$\displaystyle R_{Q}$ $\displaystyle\simeq\lvert
K\rvert^{-\frac{1}{2}}m_{3/2}^{-1},$ (5.12) $\displaystyle\omega_{Q}$
$\displaystyle\simeq m_{3/2},$ (5.13)

where $Q_{N}$ is the charge of the new-type L-ball, and $M_{Q}$ and $R_{Q}$ is
the mass and radius of the L-ball. Here the numerical constant $\beta_{N}$ is
estimated by $0.02$ [76].

By substituting new-type L-ball properties to the evaporation and diffusion
rate, we can evaluate the lepton charge that evaporates from the L-balls. A
part of the lepton charge is changed to the baryon charge through the
sphaleron process. We are concerned with the baryon charge produced in that
way because it can affect the big bang nucleosynthesis (BBN). Taking into
account that the sphaleron process is only effective above the temperature
$T_{\mathrm{ew}}$ at the electroweak phase transition, we finally obtain the
total baryon charge evaporating and decaying from an L-ball [36],

$\displaystyle\begin{split}\Delta
Q_{b}^{\mathrm{evap}}\simeq&\frac{48M_{\mathrm{Pl}}}{23}{\lvert
K\rvert}^{-1}\Bigg{[}\frac{\sqrt{10}}{m_{\phi}^{2}m_{3/2}}\frac{\tilde{T}^{2}}{\sqrt{g_{\ast}(\tilde{T})}}+2\sqrt{10}A{\lvert
K\rvert}^{\frac{1}{2}}\left(\frac{1}{\sqrt{g_{\ast}(\tilde{T})}\tilde{T}}-\frac{1}{\sqrt{g_{\ast}(T_{\mathrm{RH}})}T_{\mathrm{RH}}}\right)\\\
&+\frac{64AM_{\mathrm{Pl}}H(T_{\mathrm{RH}})}{\pi}{\lvert
K\rvert}^{\frac{1}{2}}\frac{1}{g_{\ast}(T_{\mathrm{RH}})T_{\mathrm{RH}}^{3}}\Bigg{]}\Theta(T_{\mathrm{RH}}-\tilde{T})\\\
&+\frac{48M_{\mathrm{Pl}}}{23}{\lvert
K\rvert}^{-1}\Bigg{[}\frac{\sqrt{10}}{m_{\phi}^{2}m_{3/2}}\frac{T_{\mathrm{RH}}^{2}}{\sqrt{g_{\ast}(T_{\mathrm{RH}})}}+\frac{192M_{\mathrm{Pl}}H(T_{\mathrm{RH}})}{\pi
m_{\phi}^{2}m_{3/2}}\left(\frac{\log{\tilde{T}}}{g_{\ast}(\tilde{T})}-\frac{\log{T_{\mathrm{RH}}}}{g_{\ast}(T_{\mathrm{RH}})}\right)\\\
&+\frac{64AM_{\mathrm{Pl}}H(T_{\mathrm{RH}})}{\pi}{\lvert
K\rvert}^{\frac{1}{2}}\frac{1}{g_{\ast}(\tilde{T})\tilde{T}^{3}}\Bigg{]}\Theta(\tilde{T}-T_{\mathrm{RH}})\
,\end{split}$ (5.14)

and

$\displaystyle\begin{split}\Delta
Q_{b}^{\mathrm{decay}}\simeq\frac{576M_{\mathrm{Pl}}^{2}m_{3/2}H(T_{\mathrm{RH}})}{23\pi^{3}g_{\ast}(T_{\mathrm{RH}})T_{\mathrm{RH}}^{4}}{\lvert
K\rvert}^{-1}+\frac{12\sqrt{10}M_{\mathrm{Pl}}m_{3/2}}{23\pi^{2}\sqrt{g_{\ast}(T_{\mathrm{ew}})}T_{\mathrm{ew}}^{2}}{\lvert
K\rvert}^{-1}\ .\end{split}$ (5.15)

Here we have used $T_{\mathrm{ew}}\ll\tilde{T}$ and $T_{\mathrm{RH}}\ll
T_{\mathrm{osc}}$ in the parameter region we are interested in. The
temperature when the L-balls decay is estimated by using Eq. (5.15) as

$\displaystyle
T_{\mathrm{decay}}\simeq\left(\frac{3}{2\pi^{2}}\sqrt{\frac{10}{g_{\ast}}}{\lvert
K\rvert}^{-1}m_{3/2}M_{\mathrm{Pl}}Q_{N}^{-1}\right)^{\frac{1}{2}}\ .$ (5.16)

###  5.3 Delayed-type L-ball

In the case of $K>0$, the “delayed-type” L-balls are formed. The property of a
single L-ball is given as follows.

$\displaystyle Q_{G}$
$\displaystyle\simeq\beta_{G}\frac{\varphi_{\mathrm{eq}}^{4}}{M_{F}^{4}}=\beta_{G}\left(\frac{M_{F}}{m_{3/2}}\right)^{4},$
(5.17) $\displaystyle M_{Q}$ $\displaystyle\simeq\frac{4\sqrt{2}\pi}{3}\zeta
M_{F}Q_{G}^{\frac{3}{4}},$ (5.18) $\displaystyle R_{Q}$
$\displaystyle\simeq\frac{1}{\sqrt{2}\zeta}M_{F}^{-1}Q_{G}^{\frac{1}{4}},$
(5.19) $\displaystyle\omega_{Q}$ $\displaystyle\simeq\sqrt{2}\pi\zeta
M_{F}Q_{G}^{-\frac{1}{4}},$ (5.20)

where $Q_{G}$ is the charge of the delayed-type L-ball. Here the numerical
constant $\beta_{G}\simeq 6\times 10^{-4}$ and $\zeta\simeq 3.6$. Similarly to
the new-type L-balls, we can evaluate the total baryon charge from an L-ball
as [36]

$\displaystyle\begin{split}\Delta
Q_{b}^{\mathrm{evap}}\simeq&\frac{48M_{\mathrm{Pl}}}{23}\Bigg{[}\frac{\sqrt{5}\pi
Q_{G}^{\frac{1}{4}}}{\zeta
M_{F}m_{\phi}^{2}}\frac{\tilde{T}^{2}}{\sqrt{g_{\ast}(\tilde{T})}}+2\sqrt{10}A\pi\left(\frac{1}{\sqrt{g_{\ast}(\tilde{T})}\tilde{T}}-\frac{1}{\sqrt{g_{\ast}(T_{\mathrm{RH}})}T_{\mathrm{RH}}}\right)\\\
&+64AM_{\mathrm{Pl}}H(T_{\mathrm{RH}})\frac{1}{g_{\ast}(T_{\mathrm{RH}})T_{\mathrm{RH}}^{3}}\Bigg{]}\Theta(T_{\mathrm{RH}}-\tilde{T})\\\
&+\frac{48M_{\mathrm{Pl}}}{23}\Bigg{[}\frac{\sqrt{5}\pi
Q_{G}^{\frac{1}{4}}}{\zeta
M_{F}m_{\phi}^{2}}\frac{T_{\mathrm{RH}}^{2}}{\sqrt{g_{\ast}(T_{\mathrm{RH}})}}+\frac{96\sqrt{2}M_{\mathrm{Pl}}H(T_{\mathrm{RH}})Q_{\mathrm{G}}^{\frac{1}{4}}}{\zeta
M_{F}m_{\phi}^{2}}\left(\frac{\log{\tilde{T}}}{g_{\ast}(\tilde{T})}-\frac{\log{T_{\mathrm{RH}}}}{g_{\ast}(T_{\mathrm{RH}})}\right)\\\
&+64AM_{\mathrm{Pl}}H(T_{\mathrm{RH}})\frac{1}{g_{\ast}(\tilde{T})\tilde{T}^{3}}\Bigg{]}\Theta(\tilde{T}-T_{\mathrm{RH}})\
,\end{split}$ (5.21)

and

$\displaystyle\begin{split}\Delta
Q_{b}^{\mathrm{decay}}\simeq&\frac{8}{23}\Bigg{[}Q_{G}-\bigg{\\{}Q_{G}^{\frac{5}{4}}-\frac{15\sqrt{5}\pi\zeta
M_{F}M_{\mathrm{Pl}}}{4\sqrt{g_{\ast}(T_{\mathrm{ew}})}T_{\mathrm{ew}}^{2}}-\frac{90\sqrt{2}\zeta
M_{F}M_{\mathrm{Pl}}^{2}H(T_{\mathrm{RH}})}{g_{\ast}(T_{\mathrm{RH}})T_{\mathrm{RH}}^{4}}\bigg{\\}}^{\frac{4}{5}}\Bigg{]}\
.\end{split}$ (5.22)

Here, we approximate that the L-ball charge is constant during the evaporation
process. Also, here, we have used $T_{\mathrm{ew}}\ll\tilde{T}$ and
$T_{\mathrm{RH}}\ll T_{\mathrm{osc}}$ in the parameter region we are
interested in. The temperature when the L-balls decay is estimated by using
Eq. (5.22) as

$\displaystyle
T_{\mathrm{decay}}\simeq\left(\frac{15\pi}{4\sqrt{2}}\sqrt{\frac{10}{g_{\ast}}}\zeta
M_{\mathrm{Pl}}M_{F}Q_{G}^{-\frac{5}{4}}\right)^{\frac{1}{2}}\ .$ (5.23)

###  5.4 Observational constraints on L-ball scenario

For our L-ball scenario to successfully produce the PBHs explaining the SMBH
seeds, we consider the following constraints. To obtain the constraints, we
take $n=6$ in $V_{\rm{NR}}$ [Eq. (2.2)]. For the observational constraints, we
set parameters related to the potential of the AD field, $\tilde{c}=\lvert
a_{M}\rvert=1,~{}A=5,~{}\lvert K\rvert=0.1$. Also we set the energy scales as
$M_{F}=10^{7}\,\mathrm{GeV},~{}m_{\phi}=M_{g}=10^{4}\,\mathrm{GeV},~{}T_{\mathrm{ew}}=160\,\mathrm{GeV}$
[77].

  * •

Baryon asymmetry

In the present model, the spatial distribution of lepton bubbles is highly
inhomogeneous. Since baryon charge is produced from leptons emitted from
L-balls as we calculated in Sec. 5.2 and 5.3, the produced inhomogeneous
baryon charge should not affect the homogeneous baryon density not to spoil
the success of the BBN. Thus, we obtain the constraint as

$\displaystyle\braket{\Delta\eta_{b}}=\eta_{l}\frac{\Delta
Q_{b}^{\mathrm{evap}}+\Delta
Q_{b}^{\mathrm{decay}}(T_{\mathrm{ew}})}{Q_{N,G}}\beta_{\mathrm{HLB}}\lesssim
10^{-2}\eta_{b}^{\mathrm{obs}}\ ,$ (5.24)

where $\Delta\eta_{b}$ denotes the inhomogeneous part of the baryon asymmetry,
and $\eta_{b}^{\mathrm{obs}}\simeq 8.7\times 10^{-11}$ denotes the observed
value of its homogeneous part [69]. $\beta_{\mathrm{HLB}}$ is the volume
fraction of the high lepton bubbles.

  * •

Effective number of neutrino generations

The effective number of neutrino species is calculated as

$\displaystyle
N_{\mathrm{eff}}=\frac{8}{7}\left(\frac{11}{4}\right)^{\frac{4}{3}}\left[\frac{\rho_{\mathrm{rad}}}{\rho_{\gamma}}-1\right]\
,$ (5.25)

where $\rho_{\gamma}$ is the energy density of the total photons. Neutrinos
produced from the L-ball decay could give a significant contribution to
$N_{\mathrm{eff}}$. Using the Planck 2018 result [69], we can obtain the
constraint on the contribution from the L-balls as

$\displaystyle\Delta
N_{\mathrm{eff}}=\frac{8}{7}\left(\frac{11}{4}\right)^{\frac{4}{3}}\frac{Y_{Q}^{\mathrm{in}}s(T_{\mathrm{decay}})}{\rho_{\gamma}(T_{\mathrm{decay}})}\beta_{\mathrm{HLB}}<0.24\
.$ (5.26)

  * •

Gravitino overproduction

For the high reheating temperature, gravitinos are over-produced and exceed
the present dark matter abundance. The gravitino density parameter
$\Omega_{3/2}$ is calculated as [78, 79]

$\displaystyle\Omega_{3/2}h^{2}\simeq
0.71\left(\frac{m_{3/2}}{0.5\,\mathrm{GeV}}\right)^{-1}\left(\frac{M_{g}}{10^{4}\,\mathrm{GeV}}\right)^{2}\left(\frac{T_{\mathrm{RH}}}{10^{5}\,\mathrm{GeV}}\right)\
,$ (5.27)

where $M_{g}$ is the gluino mass. We obtain the constraint requiring that
$\Omega_{3/2}h^{2}$ should not exceed the observed dark matter density
$\Omega_{\mathrm{DM}}h^{2}\simeq 0.12$ [69].

  * •

PBH formation by L-ball

Since the present L-ball scenario assumes the PBH formation occurs due to the
density perturbation of L-balls, PBH formation must be ahead of the L-ball
decay

$\displaystyle T_{\mathrm{decay}}<T_{c}\ .$ (5.28)

  * •

Dominance of the gravity-mediation potential

We assume that the potential due to the gravity-mediation dominates over the
gauge-mediation potential when the AD field starts to oscillate. Thus we
require

$\displaystyle\varphi\gtrsim\varphi_{\mathrm{eq}}=\frac{M_{F}^{2}}{m_{3/2}}\gg
M_{S}\ .$ (5.29)

We focus on $c^{\prime}_{1}\geq 0.5$ and $N_{\ast}\geq 10$ considering the
constraints on the PBH correlation discussed in Sec. 4. We show the parameter
region for the L-ball scenario for $c^{\prime}_{1}=0.5$ and $1.0$ in Figs. 6
and 7, respectively. Here, we fix $f_{\mathrm{PBH}}=3\times 10^{-9}$ and show
the constraints listed above. For the constraints depending on
$f_{\mathrm{PBH}}$, we show those for $f_{\mathrm{PBH}}=3\times 10^{-10}$ with
the dashed lines. All these cases are stringently constrained but we have
allowed parameter regions.

(a) new type, $N_{\ast}=10$

(b) new type, $N_{\ast}=15$

(c) delayed type, $N_{\ast}=10$

(d) delayed type, $N_{\ast}=15$

Figure 6: Observational constraints on the L-ball scenario with
$c^{\prime}_{1}=0.5$. Here we take $c^{\prime}_{2}=0.005$. The solid line
shows the constraints for $f_{\mathrm{PBH}}=3\times 10^{-9}$, and the dashed
line shows that for $f_{\mathrm{PBH}}=3\times 10^{-10}$ only if they are
different.  (a) new type, $N_{\ast}=10$ (b) new type, $N_{\ast}=15$
---
(c) delayed type, $N_{\ast}=10$ (d) delayed type, $N_{\ast}=15$
Figure 7: Same as Fig. 6, but for $c^{\prime}_{1}=1.0$.

##  6 Summary and discussion

In this paper, we have studied a modified scenario to form PBHs with masses of
$M\gtrsim 10^{4}M_{\odot}$ via the inhomogeneous AD leptogenesis motivated by
the observations of SMBHs. In this type of model, the PBH formation depends on
quantum fluctuations of the AD field during inflation, and the formed PBHs
inevitably have strong spatial correlations, which are stringently constrained
on large scales by observations of CMB and the angular correlation of quasars.
By considering a time-dependent Hubble mass during inflation, which can be
realized, for example, in multi-inflaton models, we can suppress the spatial
correlation of PBHs on large scales, and the constraints from isocurvature
perturbations and angular correlation of quasars are relaxed.

This scenario accompanies the formation of L-balls. The L-balls source the
density contrast for PBH formation and protect the lepton asymmetry against
the sphaleron processes. Consequently, we can successfully explain the seeds
of the SMBHs without conflicting with the observational constraints.

While the PBH correlation is suppressed on large scales, it can have some
implications on small scales. For example, the clustered PBHs will result in
the clustering of SMBHs on smaller scales than observed today. It may affect
the radiation of gravitational waves from SMBH binaries.

In addition, small bubbles do not collapse into PBHs and result in rare
regions with large lepton asymmetry after the L-ball decay. The energy of the
L-balls is emitted as relativistic neutrinos and does not contribute to the
isocurvature perturbations. On the other hand, if the L-balls decay before the
BBN at $T\sim\mathcal{O}(1)$ MeV, the abundance of light elements can be
altered in such regions. In particular, large lepton asymmetry significantly
affects the abundance of 4He. Thus, this scenario may predict enhanced or
suppressed abundance of 4He for galaxies in the rare regions, which is left
for future study.

Finally, we comment on the assumption in our analysis. For simplicity, we have
assumed a constant Hubble parameter during inflation. However, the Hubble
parameter typically decreases during inflation. In particular, the Hubble
parameter can significantly decrease at the transition of the stages in multi-
inflaton models. If we take into account the time evolution of the Hubble
parameter, the correlations on large scales are enhanced because the field
fluctuations in the earlier period become relatively larger. This enhancement,
however, can be compensated by $c_{1,2}$. Thus, we expect our conclusion to be
unchanged by the time-dependent Hubble parameter.

## Acknowledgments

This work was supported by JSPS KAKENHI Grant Nos. 20H05851 (M.K.), 21K03567
(M.K.), 23KJ0088 (K.M.), and 24K17039 (K.M.). This work was supported by JST
SPRING, Grant Number JPMJSP2108 (K.K. and S.N.).

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