# First evidence for chiral wobbling of triaxial nuclei

S. Frauendorf<EMAIL_ADDRESS>Department of Physics and Astronomy, University
of Notre Dame, Notre Dame, Indiana 46556, USA

Chirality (Greek "handiness") is a property of many complex molecules. Chiral
molecules exist in two forms, one being the mirror image of the other. Like
for our hands, it is impossible to make the images identical by a suitable
rotation. The two forms are called left-handed and right-handed. They have the
same binding energy, because the electromagnetic interaction, which holds the
molecule together, does not change under a reflection. Other properties that
are insensitive to the geometry are also the same. The different geometry is
the reason why the left-handed form turns the polarization plane of
transmitted light in one direction by some angle while of the right-handed
form turns it in the opposite direction by the same angle. However, the
geometrical differences between the two species may have other consequences.
The two species of the carvon molecule shown in Figure 1 taste quite
differently.

Figure 1:  The two chiral configurations of the carvon molecule. As usual in
organic chemistry, the C atoms are not shown. They are located at the bond
ends. The left-handed molecule is transformed into the right-handed by a
reflection through the mirror plane, which is perpendicular to the figure
plane and goes through the broken line.

.

In contrast to molecules, nuclei are very compact. They resemble deformed
droplets of liquid, with nearly constant interior density and a thin surface,
which reflects the short range of the interaction that keeps the nucleons
together. The shapes are simple: spherical, axial symmetric, mostly
footbal$l$-like, sometimes pear-shaped. Some nuclei take the shape of a
triaxial ellipsoid. Such simple shapes do not attain the property of
chirality, which was considered as an alien concept by nuclear physicists.
Surprisingly, the studies of the rotational excitations of triaxial nuclei
have revealed that they may attain chirality FM97 .

Deformed nuclei exhibit rotational spectra similar to molecules. These
rotational bands are regular sequences of states with quantized energy and
angular momentum differing solely in their rotational velocity. Concerning the
dynamics of rotation, molecules behave like classical bodies while nuclei
behave like a quantum liquid. In even-even nuclei, the rotational dynamics
necessitates that the triaxial shape deviates from rotational symmetry with
respect to a principal axis in order to facilitate rotation around it. The
corresponding moments of inertia increase with the asymmetry of the shape with
respect to the axis. As illustrated by the triaxial shape in Fig. 2, the
asymmetry is maximal for the medium ($m$-) axis and so the moment of inertia.
Therefore rotation about the $m$-axis generates the lowest (yrast) rotational
band. Only even values of the angular momentum $R$ appear because for a
quantum liquid of indistinguishable constituents a rotation by the angle of
$\pi$ leads to an identical state, which restricts the possible values of $R$
to even.

In triaxial nuclei, excited rotational bands appear when the angular momentum
vector $\vec{R}$ precesses around the $m$-axis. The energy of these "wobbling"
excitations increases with the quantized opening angles of the precession
cones, and the symmetry requires that $R$ is odd for the first wobbling
excitation, even for the second, etc.. Figure 2 shows the precession of
$\vec{R}$ with respect to the principal axes of triaxial shape. In the
laboratory frame, $\vec{R}$ stands still and the body executes a wobbling
motion which counteracts the precession in the body-fixed frame. The
accompanying motion of the charge quadrupole generates collectively enhanced
$E2$ transitions between the adjacent wobbling bands, which are the major
evidence for their wobbling nature. So far, the evidence for wobbling bands
built on the yrast band is restricted to a few even-even nuclei (see review
Frau24 ).

Figure 2: A rotating triaxial nucleus with an unpaired particle carrying the
angular momentum $\vec{j}_{p}$, an unpaired hole carrying $\vec{j}_{h}$ , and
collective angular momentum $\vec{R}$ . All angular momenta are displayed as
arrows. The total angular momentum $\vec{J}$, which is the axis of rotation,
points into the direction of sight and is depicted by the large dot
symbolizing the arrow tip. For purpose of illustration the high-$j$ orbitals
are drawn outside the nucleus, whereas they are located inside symmetric to
the principle planes. The cone of $\vec{R}$ illustrates the excitation of the
wobbling mode. The ellipse around the dot shows the precession of $\vec{J}$
for the chiral wobbling excitation.

A new pattern evolves when not all the angular momenta $\vec{j}$ of the
nucleonic orbitals are paired off, which appears in odd-$A$, odd-odd nuclei
and in bands built on quasiparticle excitations in even-even nuclei. Figure 2
illustrates the shape of the high-$j$ orbitals, which, due to the spin-orbit
splitting, are $d_{5/2},\leavevmode\nobreak\ f_{7/2},\leavevmode\nobreak\
g_{9/2},\leavevmode\nobreak\ h_{11/2},\leavevmode\nobreak\
i_{13/2},\leavevmode\nobreak\ j_{15/2}$ to good approximation. When an odd
proton on the doughnut-like orbital is added to the even-even "core" of the
nucleus, the angular momentum of the orbital $\vec{j}_{p}$ points along the
$s$-axis. This orientation corresponds to the energy minimum of the short-
range attractive interaction between the orbital and the core. The presence of
odd hole can be seen as the horizontal doughnut-like orbital carved out from
in the even-even core. The angular momentum $\vec{j}_{h}$ of this hole-orbital
points along the $l$-axis of the core, because this orientation corresponds to
the energy minimum of the short-range repulsive interaction between the hole-
orbital and the core. The core behaves like an even-even triaxial nucleus,
generating the $\vec{R}$ component centered with the $m$\- axis.

The presence of the high-$j$ orbitals dramatically changes the rotational
behavior. As they act like gyroscopes, the nucleus becomes a clockwork of
gyroscopes, which can rotate about an axis that is tilted away from one of the
principal axes of the shape.

When there is only one particle on a high-$j$ orbital the total angular
momentum is $\vec{J}=\vec{j}_{p}+\vec{R}$. At low spin the Coriolis force
exerted by particle aligns $\vec{R}$ with the $s$-axis, and the nucleus
rotates about the $s$-axis. Built on this yrast band there are wobbling
excitations with a precession cone of $\vec{R}$ about the $s$-axis. The total
angular momentum $\vec{J}$ precesses around the $s$-axis as well. The
corresponding wobbling motion of the whole nucleus in the laboratory system
generates the characteristic strong $E2$ transition between the adjacent
wobbling bands. This mode has been called transverse wobbling because the axis
of the precession cone is perpendicular to the $m$-axis of maximal moment of
inertia. At suffciently large spin the larger moment of inertia of the
$m$-axis overcomes the pull of the particle. The transverse mode becomes
unstable and changes to longitudinal wobbling about the $m$-axis. The
analogous mechanism works for a hole added to the core, where the $s$-axis is
replaced by the $l$-axis. Both types of transverse wobbling have been observed
in a number of odd-A nuclei and for two-quasiparticle bands in even-even
nuclei. The properties of the wobbling mode have been been recently reviewed
by Frauendorf Frau24 .

When in odd-odd nuclei an odd high-$j$ proton is combined with an odd high-$j$
neutron hole (or vice versa), the total angular momentum
$\vec{J}=\vec{j}_{p}+\vec{j}_{h}+\vec{R}$, which is the axis of rotation, is
tilted respect to all three principal axes of the triaxial shape. S.
Frauendorf and J. Meng FM97  first noticed that the rotating system attains
chirality. Looking in Figure 2 from the tip of the vector $\vec{J}$ onto the
shape, the $s$-, $m$\- and $l$-half-axes are ordered clockwise in the right-
handed configuration and counterclockwise in the left-handed configuration.
Although the two configurations look like mirror images, they are not related
by a reflection as in the case of molecules. A reflection does not change the
angular momenta ($\vec{R}=m\vec{r}\times\vec{v}$). One has to invert the
direction $\vec{R}$, which selects the half-axis, in order to convert the
left-handed into the right-handed configuration. (An additional rotation
brings it into the position shown in the figure.) This is achieved by time
reversal, which changes only the sign of the velocities. The nucleons run
"backwards" ($\vec{R}=m\vec{r}\times\vec{v}$).

The tilt of $\vec{J}$ with respect to the principal axes removes the symmetry
with respect to a rotation by $\pi$ around the rotational axis, which has the
consequence that all integer values of $I$ constitute a rotational band. In
addition to the ones shown, there are three more left-handed and three more
right-handed configurations, which are generated by changing the directions of
$\vec{j}_{p}$ and $\vec{j}_{h}$. They are combined into superpositions, which
give rise to two $\Delta I=1$ bands of rotational states where the states with
the same angular momentum are expected to have the same energy and emit
$\gamma$-radiation in the same way. The chiral partners have the same parity
quantum number, which indicates that the configurations do not change under
reflections through the principal planes of the triaxial shape. A more
detailed discussion of the symmetries of rotating nuclei can be found in Ref.
Frau01 .

There is some coupling between the left- and right-handed configurations which
perturbs the ideal pattern. The energies of the chiral partners are somewhat
different and so are the $\gamma$ emission pattern. The consequences of the
left-right coupling are discussed in Ref. Frau11  as part of an introduction
to nuclear chirality in the style of this comment. Chiral partner bands have
been identified in the mass regions around $A=$ 80, 110, 130, 190. The
evidence has been recently reviewed by Meng Meng24 .

The precession of $\vec{R}$ generates a precession of $\vec{J}$, which is
shown as the ellipse in Fig. 2. The corresponding lowest wobbling excitations
will appear as two $\Delta I=1$ bands that decay via strong $E2$ transition to
the chiral partner bands. The authors of Ref. Br74  reported the first
evidence for such chiral wobbling band. The Chinese-South African
collaboration studied the odd-odd nucleus 74Br utilizing a high-quality
dataset of prompt double and triple $\gamma$ coincidences following the fusion
evaporation reaction populating the nucleus. The data were acquired with Ge
Clover detectors in the AFRODITE array. Lifetimes of the excited states were
determined from the Doppler shifts of the $\gamma$ rays observed in the
coincidence spectra. Absolute reduced transition probabilities were obtained
by means of angular distribution from oriented state (ADO) ratios, linear
polarization values, and the level lifetimes.

Figure 3 shows the new level scheme constructed by means of the cutting-edge
experiment. As usual, the levels are arranged into rotational bands based on
the strong $\Delta I=2$ $E2$ and $\Delta I=1$ mixed $E2$-$M1$ transitions. B1
constitutes the $\Delta I=1$ yrast band characteristic for rotation about the
tilted axis. B2 is assigned to the $\Delta I=1$ chiral partner at somewhat
higher energy. The shift reflects the left-right coupling. The assignment is
supported by the dominant $M1$ character of the $\Delta I=1$ transitions
within and between the partner bands, which stagger with opposite phase. B3
constitutes one of the chiral wobbling bands. The assignment is established by
the enhanced $\Delta I=1$ $E2$ transition probabilities to B1.

Calculation using a model which couples a triaxial rotor with a proton
$h_{11/2}$ hole and a $h_{11/2}$ neutron particle well reproduce the observed
energies and transition probabilities. They support in a quantitive way the
discussion of chiral partner bands and chiral wobbling given above.

The discovery of the chiral wobbling band in Ref. Br74  revealed a new facet
of the exotic rotation modes of triaxial nuclei. Moreover, the work affirmed
the existence of chirality in the $A=80$ mass region.

Figure 3:  The partial level scheme for 74Br from Ref. Br74 . The levels are
arranged into the rotational bands B1 and B2, which are chiral partner bands,
and B3, which represent the chiral wobbling band. New transitions and levels
are marked as red. The arrow width represents the intensity of the
transitions. The intensities are normalized to to the 383 keV transition as
100. Reproduced with permission from Ref. Br74 .

.

##  I Bibliography

## References

  * (1) S. Frauendorf, J. Meng, Tilted rotation of triaxial nuclei, Nucl. Phys. A 617, 131, (1997).
  * (2) S. Frauendorf, in Chirality and wobbling in nuclei, ed. by C. P. Petrache (Taylor & Francis, to be published 2024), chapter 2, preprint available at http://arxiv.org/abs/2405.02747.
  * (3) S. Frauendorf and F. Dönau, Transverse wobbling: A collective mode in odd-A triaxial nuclei, Phys. Rev. C 89, 014322, (2014)
  * (4) S. Frauendorf Chirality in rotating nuclei in McGraw-Hill Yearbook of Science and Technology 2011, p. 44, preprint available at http://arxiv.org/abs/2405.05986.
  * (5) S. Frauendorf, Spontaneous symmetry breaking in rotating nuclei, Rev. Mod. Phys. 73, 463, (2001).
  * (6) J. Meng, in Chirality and wobbling in nuclei, ed. by C. P. Petrache (Taylor & Francis, to be published 2024), chapter 3, preprint available at http://arxiv.org/abs/2405.07805.
  * (7) R. J. Guo et al., Evidence for chiral wobbler in nuclei., Phys. Rev. Lett. 132, 092501. (2024). doi: 10.1103/PhysRevLett.132.092501


# New constraint on the Hawking evaporation of primordial black holes in the
radiation-dominated era

Seyed Sajad Tabasi  Department of Physics, Sharif University of Technology, P.
O. Box 11155-9161, Tehran, Iran<EMAIL_ADDRESS>Javad T. Firouzjaee
Department of Physics, K. N. Toosi University of Technology, P. O. Box
15875-4416, Tehran, Iran  School of Physics, Institute for Research in
Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran
<EMAIL_ADDRESS>
###### Abstract

In this paper, we revisit the evaporation and accretion of PBHs during cosmic
history and compare them to see if both of these processes are constantly
active for PBHs or not. Our calculations indicate that during the radiation-
dominated era, PBHs absorb ambient radiation due to accretion, and their
apparent horizon grows rapidly. This growth causes the Hawking radiation
process to practically fail and all the particles that escape as radiation
from PBHs to fall back into them. Nevertheless, our emphasis is that the
accretion efficiency factor also plays a very important role here and its
exact determination is essential. We have shown that the lower mass limit for
PBHs that have not yet evaporated should approximately be $10^{14}g$ rather
than $10^{15}g$. Finally, we study the effects of Hawking radiation quiescence
in cosmology and reject models based on the evaporation of PBHs in the
radiation-dominated era.

##  I Introduction

Six years ago the decades-long efforts of the LIGO collaboration have taken
consequence and led to the first straightforward detection LIGO2015  of a
signal which was for an inspiraling massive black hole binary. This detection
and next ones came somewhat as a surprise that most of these binaries were
made of fairly massive $30M_{\odot}$ black holes. Since the star formation
scenarios do not necessarily predict mass scale for merging black holes, the
theoretical community thought over the question of whether these black holes
could be primordial and constitute dark matter. This led people to an
interesting conjecture that the LIGO detectors have detected primordial black
holes (PBHs).

There are variety of mechanisms for PBHs formation (see e.g. 5 ; Carr:2020gox
for review). As sets of models focus on inflationary perturbations as a source
of PBHs, there are other formation mechanisms inflation-pbh , such as cosmic
string collapse cosmic-string , bubble collisions, domain wall collapse 5 ;
buble  as well as scalar field fragmentation 8 ; 20 ; 23  which can produce
plentiful populations of PBHs. The consequent PBHs can span many orders of
magnitude in mass depending on the formation time. People usually apply an
important constraint to the PBHs mass range that from hose formed with mass
above the Hawking evaporation limit of $10^{15}g$ survive until present. Since
this constraint has an important side effect on Diffuse gamma-ray background,
Cosmic rays, Neutrinos, Hadron injection, Photodissociation of deuterium, CMB
distortion, Photodissociation of deuterium, Present-day relic density pbh-
thesis , and Big Bang Nucleosynthesis pbh-from HR , revisiting their Hawking
radiation process is crucial.

After Hawking known paper hawking-75  on the black hole quantum radiation,
different methods for deriving the radiation properties were presented. One
can classify these methods in two sets; First, like the Hawking first paper,
people compare the quantum vacuum before black hole formation, $|0_{B}>$, and
after black hole formation in future infinity, $|0_{A}>$, and they read the
radiation properties from particle creation number expectation value,
$<0_{B}|N_{k}|0_{B}>=\sum_{j}\beta_{jk}$, where $\beta_{jk}$ is the Bogoliubov
coefficient which relates the incoming mode from past vacuum to the outgoing
mode of the future vacuum.

This quantum field theory approach to black hole radiation, which applies
usually to late-time stationary black holes, is not a suitable method for
calculating the Hawking temperature in the case of a fully dynamical black
hole, where one has to solve the field equations in a changing background. The
second sets of methods are alternative approaches allowing one to calculate
Hawking radiation in a Schwarzschild spacetime without using the field
equations, such as finding the related vacuum via the tunneling method
tunneling . These methods can be extended to calculating quantum fields in the
dynamical case Firouzjaee:2011hi . Let’s look at the essential condition for
Hawking radiation in these methods. The main point to have Hawking radiation
is to have geometric optic approximation or WKB approximation for and Hawking
radiation near the Horizon. For Schwartzhild black hole this condition is
satisfied near the apparent or event horizon $R=2M$. But in the dynamical
black hole can be written in some form eikonal approximation Visser:2001kq .
This condition can be minimized in terms of adiabatic condition which says as
long as one has an approximately exponential relation between the affine
parameters on the null generators of past and future null infinity, then
subject to a suitable adiabatic condition being satisfied, a Planck-
distributed flux of Hawking-like radiation will occur Barcelo:2010xk .

Since primordial black hole forms in the FLRW background, they are types of
cosmological black holes Carr:1974nx ; Firouzjaee:2010whc . The black hole
properties of a cosmological black hole like PBHs are attributed to the
apparent horizon rather than the event horizon Firouzjaee:2011dn . Defining
the quantum vacuum in the cosmological background which does not have Poincare
symmetric or any special killing horizon is not possible or in some special
cases is hard Firouzjaee:2015bqa . Hence, the tunneling method is appropriate
for calculating the Hawking radiation for PBHs Firouzjaee:2011hi ;
Firouzjaee:2014zfa . Nevertheless, there must be WKB or geometric optic
approximation as an essential condition for black hole radiation. This
condition puts limits on PBHs that can have quantum radiations
Firouzjaee:2015wps .

Since the issue of the PBHs Hawking radiation has been raised for decades and
many works have been done to study the PBHs dynamics and its mass fraction
rate based on this radiation, it’s necessary to revisit their dynamics in the
radiation-dominated era by concerning its accretion flux. We will see our
calculations indicate that PBHs accretion put a serious constraint on the PBHs
Hawking evaporation.

This paper is organized as follows: first, we review the Hawking radiation and
Bondi model accretion concepts and main equations in section II. Then, in
section III, the basic subtleties in the black hole radiation process are
discussed. Section IV is devoted to comparing the rate of decrease and
increase of PBHs mass through these two processes in the cosmological context.
Afterward, in section V, the possible effects of accretion flux on previous
PBHs models which consider the Hawking radiation were investigated. The
conclusion and discussions are given in Section VI.

##  II General Equations

Several models suggest PBHs creation in the early universe. We assume that the
formation of PBHs is separated from the expansion of the universe due to the
time scale of their creation that is too much shorter than the Hubble time.
The PBHs mass can be obtained by the following equation Carr:2019

$M_{PBH}\sim\frac{c^{3}t}{G}$ (1)

where t is the time after the Big Bang, c is the speed of light, $c\simeq
3\times 10^{8}m/s$, and G is the gravitational constant, $G\simeq 6.67\times
10^{-11}m^{3}/(kgs^{2})$.

The most well-known processes that can change the mass of a black hole are
evaporation, accretion, and merging. The relevant parameters of a black hole,
such as radius, temperature, etc., change with the gain and loss of mass.
Immediately after their formation, they start evaporating by the Hawking
radiation process. This process causes a black hole to lose its mass by
emitting radiation. Also, a black hole, due to its gravitational attraction,
can swallow what is around it, thereby increasing its mass. This process,
known as accretion, can also lead to binary merging, however, its efficiency
is controversial so that we will not consider it in our paper for simplifying.

###  II.1 Evaporation

It is known that the point mass black holes like those are described with
Schwarzschild and Kerr metric can have quantum radiation which is called
Hawking radiation. PBHs which is modeled with these types of metrics evaporate
through the Hawking process and lose their mass. The rate of mass change in
this way depends on the mass of PBHs. The lower the mass, the higher the
radiation emission rate and vice versa. The rate of losing mass for PBHs is
given by Carr:2010

$\frac{dM_{PBH}}{dt}=-f_{eva}4\pi R_{PBH}^{2}c\rho_{PBH},$ (2)

where $f_{eva}$ is the evaporation efficiency factor, $R_{PBH}$ is the PBH
radius, and $\rho_{PBH}$ can be obtained by

$\rho_{PBH}=\frac{\pi^{2}g_{*}(T_{PBH})k_{B}T_{PBH}^{4}}{120\hbar^{3}c^{5}},$
(3)

where g is the number of degrees of freedom,
$g_{*}(T)=\sum_{B}g_{B}+7/8\sum_{F}g_{F}$, $k_{B}$ is the Boltzmann constant,
$k_{B}\simeq 8.62\times 10^{-5}eV/K$, $\hbar$ is the reduced Planck constant,
$\hbar\simeq 6.58\times 10^{-15}eV.s$, and $T_{PBH}$ is the PBHs temperature.
The emitted particles temperature is exactly the PBH temperature that is given
by

$T_{PBH}=\frac{\hbar c^{3}}{8\pi Gk_{B}M_{PBH}}.$ (4)

We can rewrite Eq. (2) in terms of the temperature and the mass of PBHs

$\frac{dM_{PBH}}{dt}=-\frac{f_{eva}2\pi^{3}g_{*}(T_{PBH})k_{B}M_{PBH}^{2}T_{PBH}^{4}}{15\hbar
c^{6}M_{Pl}^{4}},$ (5)

where $M_{Pl}$ is the Planck mass, $M_{Pl}=1.22\times 10^{19}GeV$. If we
consider only evaporation as a dominant process that rules PBHs, we can simply
calculate their lifetime Broda:2017

$\tau_{PBH}=\frac{5120\pi G^{2}M_{PBH}^{3}}{\hbar c^{4}}.$ (6)

###  II.2 Accretion

As we stated before, we only focus on the accretion process, regardless of
merging PBHs, as a process that causes PBHs to gain mass. By assuming that all
PBHs have been formed in the radiation-dominated era, their mass increases by
falling the photons from the thermal bath into them. The spherically
symmetrical accretion rate that has been derived by Bondi Bondi  is

$\frac{dM_{PBH}}{dt}=f_{acc}4\pi R_{PBH}^{2}c\rho_{R},$ (7)

where $f_{acc}$ is the accretion efficiency factor and $\rho_{U}$ is the
radiation mass density at a high temperature which approximately is

$\rho_{R}=\frac{\pi^{2}g_{*}k_{B}^{4}T_{U}^{4}}{30\hbar^{3}c^{5}}$ (8)

where $T_{U}$ is the thermal bath temperature. We write Eq. (7) in terms of
PBHs mass and temperature, so we have

$\frac{dM_{PBH}}{dt}=\frac{f_{acc}8\pi^{3}g_{*}(T_{U})k_{B}M_{PBH}^{2}T_{U}^{4}}{15\hbar
c^{6}M_{Pl}^{4}}.$ (9)

The crucial question that pops up here is that do both accretion and
evaporation always play an important role in PBHs dynamics? In the following,
we will use two perspectives to answer this question, one is the quantum
perspective and the other is the cosmological perspective.

##  III Quantum Standpoint

The first derivation for the black hole radiation by Hawking is based on
assuming that a future eternal event horizon forms, and that the subsequent
exterior geometry is static. Developing the semi-classical physics in curved
spacetime helped us to calculate the renormalized energy-momentum tensor,
$<T_{\mu\nu}>$, and extract the spectrum of the Hawking radiation book ;
Firouzjaee:2015bqa . To know the origin of the Hawking radiation,
extrapolating the $<T_{\mu\nu}>$ properties in the spherically symmetric case
proposed that the Hawking effect is associated with the apparent horizon
rather than the event horizon Hajicek:1986hn .

Extending the Hawking radiation process by tunneling method tunneling  opened
our hand to derive the black hole quantum radiation for non-stationary cases.
We know that the quantum fluctuations around the horizon are of the order of
the Planck length Jacobson:1993 . To have real particles from virtual
particles in these quantum fluctuations, the tunneling scenario proposes a
known process around the black holes tunneling . Tunneling calculation is
based on the WKB approximation assumed for the light coming from the apparent
horizon to the observer Visser:2001kq . The required condition for black hole
radiation can be minimized in terms of the adiabatic condition which says as
long as one has an approximately exponential relation between the affine
parameters on the null generators of past and future null infinity, then
subject to a suitable adiabatic condition being satisfied, a Planck-
distributed flux of Hawking-like radiation will occur Barcelo:2010xk .

All these methods give the tunneling probability rate $\Gamma_{em}\sim
e^{-\frac{\hbar w}{k_{B}T_{H}}}$ turns to be a thermal form where $\hbar w$ is
the emitted particle energy and $T_{H}$ is the radiation temperature which is
related to the black hole’s surface gravity$T_{h}=\frac{\kappa}{2\pi}$. The
basic point is that if a black hole is in the dynamical phase for example due
to the accretion Ashtekar:2003hk , one cannot have the essential conditions
such as WKB approximation our adiabatic condition around the apparent horizon
for Hawking radiation. This leads to quenching the black hole radiation in the
dynamical phase Firouzjaee:2014zfa ; Firouzjaee:2015wps . Moreover, in any
dynamical spacetime with the in-falling matter, the apparent (dynamical)
horizon is inside the event horizon Ashtekar:2003hk , thus any virtual pair
particle created due to these effects cannot fall outside the event horizon,
become real, and reach future infinity. You can see this feature in Fig. (1)
for PBHs. This figure is the Penrose diagram for a PBH which is formed in the
radiation-dominated era. PBHs’ apparent horizon grows in the radiation-
dominated era till its accretion flux is stopped by the end of this era. After
that in the matter-dominated era, these black holes start to radiate.

Figure 1:  Penrose diagram for a primordial black hole in a $\Lambda CDM$
expanding universe, where there is some matter and radiation accretion flux,
but it dies away to zero in the far future (matter-dominated era). Note that
we do not represent backreaction effects in this picture. Thus, the black hole
dynamical horizon tends to be event horizon in the matter-dominated era.

##  IV Cosmological Standpoint

According to Eq. (5) and Eq. (9), it is obvious that the only difference that
determines which process is dominant is the thermal bath and PBHs temperature
Masina:2020 . During the radiation-dominated era, the expansion time of the
universe in terms of the temperature of the universe is given by Manolis:2002

$T_{U}(Kelvin)\simeq\frac{1.52\times 10^{10}}{\sqrt{t(sec)}}.$ (10)

By considering that all PBHs are Schwarzschild black holes, by using Eq. (1)
and Eq. (4) we have the PBHs temperature in terms of the time after the Big
Bang

$T_{PBH}(Kelvin)\simeq\frac{3.04\times 10^{-13}}{t(sec)}.$ (11)

We should compare the two temperatures we got. In the time that the universe
temperature is larger than the PBHs temperature, the accretion process
dominates over the evaporation process. Then we can conclude that PBHs gain
mass and their radius increase respectively. On the other hand, in the time
when the PBHs temperature is larger than the thermal bath, PBHs lose their
mass and their radius gets shorter. With a simple calculation, we can find the
time of equality $(T_{PBH}=T_{U})$, $t_{eq}\sim 10^{-44}s$. This time is much
earlier than the time of the inflation. PBHs that are formed before the end of
the inflation are not important for us because of their negligible density
Pablo:2021 . We can see that from the beginning of the radiation-dominated era
until the starting of the matter-dominated era, the universe temperature is
much bigger than the PBHs temperature. Hence, the accretion process is the
process that rules the mass evolution of PBHs during the radiation-dominated
era. During the matter-dominated era, people have shown that accreting the
environment’s radiation does not have any significant effect on the mass of
PBHs Maxim:2020 .

We know that there are only three possible mass ranges for PBHs with a mass
larger than $10^{15}g$: (i) $10^{16}-10^{17}g$, (ii) $10^{20}-10^{24}g$, and
(iii) $10^{33}-10^{36}g$ for explaining the dark matter Carr:2016 . The other
ranges are eliminated by various constraints that are based on theoretical and
observational works. In Table (1), we compare the PBHs temperature of upper
and lower bound mass ranges with the universe temperature at the time of their
formation. Note that in all our calculations we assumed that $f_{acc}=0.5$.

$M_{i}(g)$ |  $10^{16}$ |  $10^{17}$ |  $10^{20}$ |  $10^{24}$ |  $10^{33}$ |  $10^{36}$
---|---|---|---|---|---|---
$t_{i}(s)$ |  $2.47\times 10^{-23}$ |  $2.47\times 10^{-22}$ |  $2.47\times 10^{-19}$ |  $2.47\times 10^{-15}$ |  $2.47\times 10^{-6}$ |  $2.47\times 10^{-3}$
$T_{PBH}(K)$ |  $1.23\times 10^{10}$ |  $1.23\times 10^{9}$ |  $1.23\times 10^{6}$ |  $1.23\times 10^{2}$ |  $1.23\times 10^{-7}$ |  $1.23\times 10^{-10}$
$T_{U}(K)$ |  $3.06\times 10^{21}$ |  $9.67\times 10^{20}$ |  $3.06\times 10^{19}$ |  $3.06\times 10^{17}$ |  $9.67\times 10^{12}$ |  $9.67\times 10^{11}$
Table 1: As shown in this table, in all mass range windows, the temperature of
the universe is much higher than the temperature of PBHs, and therefore only
the accretion process can be effective.

We can have the accretion rate in terms of the mass of a PBH and the time
after the Big Bang

$\frac{dM_{PBH}}{dt}=\frac{3f_{acc}G}{2c^{3}}\frac{M_{PBH}^{2}}{t^{2}}.$ (12)

By integrating from Eq. (12) we have

$M(t)=(M_{i}^{-1}+\frac{3f_{acc}G}{2c^{3}}(\frac{1}{t_{f}}-\frac{1}{t_{i}}))^{-1}.$
(13)

Then we substitute Eq. (1) into the Eq. (13) and we get

$M(t)=M_{i}(1+\frac{3f_{acc}}{2}(\frac{t_{i}}{t_{f}}-1))^{-1}.$ (14)

Eq. (14) simply gives us the final mass of PBHs with the initial mass $M_{i}$
at time $t$ after the Big Bang. Since we are looking at the evaluation of PBHs
during the radiation-dominated era, the value of $t_{f}$ is equal to the time
of the end of the radiation-dominated era, $t_{f}\simeq 1.48\times 10^{12}s$.
Moreover, for the accretion efficiency factor, we should determine it with a
value lower than 2/3. Now, in Table (2) we consider the remained mass range
windows and get their final mass at the end of the radiation-dominated era for
different $f_{acc}$.

$M_{i}(g)$ |  $10^{16}$ |  $10^{17}$ |  $10^{20}$ |  $10^{24}$ |  $10^{33}$ |  $10^{36}$
---|---|---|---|---|---|---
$f_{acc}=0.1$ |  $1.18\times 10^{16}$ |  $1.18\times 10^{17}$ |  $1.18\times 10^{20}$ |  $1.18\times 10^{24}$ |  $1.18\times 10^{33}$ |  $1.18\times 10^{36}$
$f_{acc}=0.2$ |  $1.43\times 10^{16}$ |  $1.43\times 10^{17}$ |  $1.43\times 10^{20}$ |  $1.43\times 10^{24}$ |  $1.43\times 10^{33}$ |  $1.43\times 10^{36}$
$f_{acc}=0.3$ |  $1.82\times 10^{16}$ |  $1.82\times 10^{17}$ |  $1.82\times 10^{20}$ |  $1.82\times 10^{24}$ |  $1.82\times 10^{33}$ |  $1.82\times 10^{36}$
$f_{acc}=0.4$ |  $2.5\times 10^{16}$ |  $2.5\times 10^{17}$ |  $2.5\times 10^{20}$ |  $2.5\times 10^{24}$ |  $2.5\times 10^{33}$ |  $2.5\times 10^{36}$
$f_{acc}=0.5$ |  $4\times 10^{16}$ |  $4\times 10^{17}$ |  $4\times 10^{20}$ |  $4\times 10^{24}$ |  $4\times 10^{33}$ |  $4\times 10^{36}$
$f_{acc}=0.6$ |  $1\times 10^{17}$ |  $1\times 10^{18}$ |  $1\times 10^{21}$ |  $1\times 10^{25}$ |  $1\times 10^{34}$ |  $1\times 10^{37}$
$f_{acc}=0.65$ |  $4\times 10^{17}$ |  $4\times 10^{18}$ |  $4\times 10^{21}$ |  $4\times 10^{25}$ |  $4\times 10^{34}$ |  $4\times 10^{37}$
Table 2: In this table, we have shown the final upper and lower bound of the
remained mass ranges windows of PBHs at the end of the radiation-dominated era
for different accretion efficiency factor

At first glance, it seems strange that the ratio of final mass to initial mass
per each accretion efficiency factor is the same for different initial masses.
It should be noted that the final time is fixed, and on the other hand, the
initial mass is directly related to the initial time, and therefore, by
determining the initial mass, we also set the initial time. In addition, the
increase in mass during the radiation-dominated era is perfectly consistent
with the results of Kamionkovski . It is worth mentioning that the efficiency
of disk accretion is significantly more than spherical accretionKamionkovski ;
Shapiro  that we have considered in our calculations. Ultimately, $f_{acc}$ is
the important parameter that can determine the final mass of PBHs at the end
of the radiation-dominated era in Bondi accretion. The larger the $f_{acc}$,
the greater the final mass.

We know that since $R_{PBH}=2GM_{PBH}/c^{2}$, the growth of the mass is
proportional to the growth of the Schwarzschild radius. Thus, for PBHs radius
we have

$R(t)=R_{i}(1+\frac{3f_{acc}}{2}(\frac{t_{i}}{t_{f}}-1))^{-1}.$ (15)

This means that if, for example, the mass of a PBH increases 10 times by the
end of the radiation-dominated era, its radius will increase by the same
factor.

The growth of the PBHs radius causes a huge problem for the PBHs evaporation
process. As discussed in the previous section, we write the Hawking radiation
equations for an adiabatic horizon. Many papers tried to use this phenomenon
for PBHs, but they missed something named accretion. By the accretion of PBHs,
they gain mass and their radius grows permanently during the radiation-
dominated era. Therefore, PBHs in the radiation-dominated era do not have the
adiabatic condition on the (apparent) horizon and hence we cannot write the
regular Hawking radiation formulas for them. Therefore, if the growth rate of
the (apparent) horizon is huge that at each time interval the radius of PBHs
grows much more than the order of Planck length, the produced escaping
particles fall back into PBHs and Hawking radiation turns off. This situation
remains until the end of the radiation-dominated era.

Since we have shown in Table 2 that the mass growth rate of PBHs is extremely
dependent on the accretion efficiency factor, and respectively the growth rate
of the PBHs radius, it can be concluded that the larger the $f_{acc}$, the
lower the probability of occurring tunneling and consequently evaporation.
Finally, the importance of exactly determining the $f_{acc}$ has been cleared
because it is practically what determines which process rules PBHs, accretion,
or evaporation, or both of them.

##  V Possible Effects

So far we have shown that accretion is the dominant process of PBHs during the
radiation-dominated era and the final mass of a PBH at the end of the
radiation-dominated era depends on the accretion efficiency factor. This has a
serious impact on two issues. One is the effect on the possible remaining mass
range windows of PBHs, and the second is the effect on the models that suggest
the production of particles from the evaporation of PBHs during the radiation-
dominated era. In the following, we will investigate each of these two effects
separately.

Although neglecting Hawking radiation may seem scary at first, it also raises
hopes too. It has been emphasized in previous works that PBHs with a mass of
less than $10^{15}g$ have completely evaporated and cannot be observed now.
Because of that, people almost always consider the evaporation process as an
effective process. In this work, we have shown that such an idea is wrong
because of the accretion. Given the fact that PBHs do not start to evaporate
until the beginning of the matter-dominated era, their lifetime calculations
must be considered from the beginning of this epoch. Therefore, the remained
possible PBHs mass range windows extend. By using Eq. (6), it can be shown
that the lower mass limit for PBHs that have not yet evaporated should
approximately be $10^{14}g$. In the other words, we should investigate the
possibility of the existence of PBHs in the mass range of $10^{14}g$ to
$10^{15}g$.

The other significant effect is that over the years, there have been various
suggestions that the process of evaporation of PBHs could answer the different
questions posed in cosmology. Each of them requires a specific initial mass
range for PBHs. In this paper, we have shown that evaporation of PBHs in the
radiation-dominated era is off because of the accretion. This refutes all the
suggestions that utilize Hawking radiation of PBHs during the radiation-
dominated era. Here, we mention some of the famous ones.

  * •

CMB distortions are slightly observational deviations from the black body
spectrum of the CMB. One of the important suggestions for explaining the CMB
distortions is the evaporation of PBHs. As we can see in Tashiro  and Mather ,
photons emitted from PBHs with the mass $10^{11}g$ to $10^{13}g$ in the
redshift between $z\simeq 10^{6}$ and $z\simeq 10^{3}$ can cause the CMB
distortions.

  * •

The dominance of matter over antimatter is still a huge problem in physics.
Baryogenesis is a process that causes baryonic asymmetry to provide this
dominance. Hooper  Suggests that evaporation of PBHs with the mass range of
$10^{5}g$ to $10^{9}g$ during the radiation-dominated era can elaborate GUT
baryogenesis.

  * •

Produced photons from PBHs with the mass $10^{10}g$ to $10^{13}g$ between the
end of nucleosynthesis and recombination can photodissociate deuterium. This
has been investigated in Clancy . They have described it with braneworld
cosmology in detail.

  * •

Dark matter is a main component of the universe that has not yet been properly
explained. One approach to explaining dark matter is to consider the dark
matter as special particles that we have not yet been able to detect. Works
like Gondolo  claim that PBHs can produce dark matter particles via Hawking
radiation mechanism during the radiation-dominated era and early matter-
dominated era.

  * •

The study of Kim  indicates that the diffuse gamma-ray background can relate
to PBHs with the mass $2\times 10^{13}g$ to $5\times 10^{14}g$ which have been
evaporated from $z\simeq 700$ until now. The production of the cosmic rays
Gibbon  and the diffuse neutrino backgroundBugaev  by evaporation of PBHs
circumstances are similar to the diffuse gamma-ray background.

Now, it should be noted that as we proved before, the suggestions that are
based on Hawking evaporation of PBHs during the radiation-dominated era are
rebutted. Consequently, from the examples we just mentioned, evaporation of
PBHs cannot explain CMB distortions, Baryogenesis, photodissociation of
deuterium, and dark matter particles produced in the radiation-dominated era.

One way to save these models is to select a very small accretion efficiency
factor so that the accretion rate can be neglected. However, our work does not
affect the suggestions that occur after the end of the radiation-dominated
era. On contrary, we emphasize that our work completely agrees with the works
based on the evaporation of PBHs from the beginning of the matter-dominated
era, like the production of dark matter in the early matter-dominated era, the
diffuse gamma-ray background, production of cosmic rays, and the diffuse
neutrino background.

Another way is to look for theories for the formation of PBHs in which PBHs
are produced whose temperature at the time of formation is about or greater
than the cosmic background temperature. In this case, it can be claimed that
PBHs will have Hawking radiation. In this case, the accretion cannot turn off
the Hawking radiation and particles escape from the PBHs as Hawking radiation
and enter the cosmic environment. Anyhow, it can be said with certainty that
PBHs formed according to Eq. (1) do not have the ability to evaporate in the
radiation-dominated era.

##  VI Conclusions

In this work, we have revisited and revised the two processes that can change
the mass of PBHs, accretion and evaporation. Considering Eq. (5) and Eq. (9),
one can conclude that the only parameter that determines which process rules
PBHs is the temperature difference between PBHs and the universe. By comparing
the temperature of PBHs and the universe temperature saw that the temperature
of the universe during the radiation-dominated era is much more than the
temperature of PBHs. For instance, we have brought up the remained mass ranges
of PBHs which can explain dark matter in Table (1). Thus, the mass of PBHs
permanently increases during the radiation-dominated era and accretion is the
dominant process.

Next, the question we tried to answer was whether, with the persistent
increase in the mass of PBHs, the evaporation process occurs at all. We have
used two different approaches to answer this question. First, we have used the
Quantum standpoint and second, we have used the cosmological standpoint. In
Table (1) we have illustrated the growth of the PBHs (apparent) horizon with a
Penrose diagram in a $\Lambda$CDM expanding universe. Afterward, we have
applied the Bondi spherical accretion model to derive the final mass and final
radius of PBHs at the end of the radiation-dominated era. We have demonstrated
that the final mass totally depends on the accretion efficiency factor. We
have considered the remained mass range windows of PBHs for explaining the
dark matter in Table (2) again and have elaborated the dependency. The growth
of the (apparent) horizon by accretion turns off PBHs evaporation and the
escaping particles fall back into PBHs immediately. Thus, our calculations
indicate that PBHs accretion put a serious constraint on the PBHs Hawking
evaporation As we noted before, we have assumed Bondi spherical accretion,
nevertheless, the disk accretion is much more efficient than spherical
accretion.

In the end, we have investigated the effects of turning off the Hawking
evaporation. We illustrated that instead of considering PBHs with larger than
$10^{15}g$ initial mass as non-evaporated PBHs, we should expand this lower
bound to $10^{14}g$ for the reason that PBHs start evaporating from the
beginning of the matter-dominated era and they gain mass during the radiation-
dominated era. It is important to note that if PBHs are observed in the
future, the accretion efficiency factor can be estimated by only measuring
their mass. we questioned the models that are based on Hawking evaporation of
PBHs during the radiation-dominated era such as CMB distortions, baryogenesis,
and dark matter particles production by PBHs due to the Hawking radiation.

## References

  * (1) LIGO Scientific and Virgo Collaborations, B. P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016), 1602.03837
  * (2) S W Hawking (1975) Comm Math Physics 43: 199-220.
  * (3) M. Yu. Khlopov, Res. Astron. Astrophys. 10, 495 (2010), arXiv:0801.0116 [astro-ph].
  * (4) B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, [arXiv:2002.12778 [astro-ph.CO]].
  * (5) B. J. Carr and J. E. Lidsey, Phys. Rev. D 48, 543-553 (1993); M. Drees and E. Erfani, JCAP 04, 005 (2011); E. D. Stewart, Phys. Rev. D 56, 2019-2023 (1997); S. Clesse and J. García-Bellido, Phys. Rev. D 92, no.2, 023524 (2015); J. Garcia-Bellido and E. Ruiz Morales, Phys. Dark Univ. 18, 47-54 (2017); C. Germani and I. Musco, Phys. Rev. Lett. 122, no.14, 141302 (2019); A. Ashoorioon, A. Rostami and J. T. Firouzjaee, JHEP 07, 087 (2021).
  * (6) E. Cotner and A. Kusenko, Phys. Rev. Lett. 119, 031103 (2017), arXiv:1612.02529 [astro-ph.CO].
  * (7) E. Cotner, A. Kusenko, and V. Takhistov, Phys. Rev. D98, 083513 (2018), arXiv:1801.03321 [astro-ph.CO].
  * (8) E. Cotner, A. Kusenko, M. Sasaki, and V. Takhistov, JCAP 1910, 077 (2019), arXiv:1907.10613 [astroph.CO].
  * (9) J. Garriga and A. Vilenkin, Phys. Rev. D47, 3265 (1993), arXiv:hep-ph/9208212 [hep-ph]; S. W. Hawking, I. G. Moss, and J. M. Stewart, Phys. Rev. D26, 2681 (1982); H. Deng, J. Garriga, and A. Vilenkin, JCAP 1704, 050 (2017), arXiv:1612.03753 [gr-qc]; A. Ashoorioon, A. Rostami and J. T. Firouzjaee, Phys. Rev. D 103, 123512 (2021).
  * (10) S. W. Hawking, Phys. Lett. B 231, 237-239 (1989); A. Polnarev and R. Zembowicz, Phys. Rev. D 43, 1106-1109 (1991); S. A. Sanidas, R. A. Battye and B. W. Stappers, Phys. Rev. D 85, 122003 (2012); C. Ringeval and T. Suyama, JCAP 12, 027 (2017); P. A. R. Ade et al. [Planck], Astron. Astrophys. 571, A25 (2014);
  * (11) Josan, Amandeep Singh. Constraints on the power spectrum of primordial perturbations from small-scale structure. Diss. University of Nottingham, 2010.
  * (12) I. Masina, Eur. Phys. J. Plus 135, no.7, 552 (2020); I. Baldes, Q. Decant, D. C. Hooper and L. Lopez-Honorez, JCAP 08, 045 (2020); C. Keith, D. Hooper, N. Blinov and S. D. McDermott, Phys. Rev. D 102, no.10, 103512 (2020).
  * (13) K. Srinivasan and T. Padmanabhan, Phys. Rev. D 60, 024007 (1999); M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85, 5042 (2000); S. Shankaranarayanan, T. Padmanabhan and K. Srinivasan, Class. Quant. Grav. 19, 2671 (2002).
  * (14) A. Ashtekar and B. Krishnan, Phys. Rev. D 68, 104030 (2003).
  * (15) J. T. Firouzjaee and R. Mansouri, Gen. Rel. Grav. 42, 2431-2452 (2010) doi:10.1007/s10714-010-0991-7
  * (16) B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc. 168, 399-415 (1974)
  * (17) Birrell, Nicholas David, Nicholas David Birrell, and P. C. W. Davies. ”Quantum fields in curved space.” (1984); Fabbri, Alessandro, and Jos Navarro-Salas. Modeling black hole evaporation. World Scientific, 2005.
  * (18) J. T. Firouzjaee, Int. J. Mod. Phys. D 21, 1250039 (2012) doi:10.1142/S0218271812500393
  * (19) J. T. Firouzjaee and R. Mansouri, EPL 97, no.2, 29002 (2012) doi:10.1209/0295-5075/97/29002 [arXiv:1104.0530 [gr-qc]].
  * (20) J. T. Firouzjaee and G. F. R. Ellis, Gen. Rel. Grav. 47, no.2, 6 (2015) doi:10.1007/s10714-014-1848-2
  * (21) P. Hajicek, Phys. Rev. D 36, 1065 (1987).
  * (22) J. T. Firouzjaee and G. F. R. Ellis, Phys. Rev. D 91, no.10, 103002 (2015) doi:10.1103/PhysRevD.91.103002
  * (23) A. Ashtekar and B. Krishnan, Phys. Rev. D 68, 104030 (2003).
  * (24) J. T. Firouzjaee and G. F. R. Ellis, Eur. Phys. J. C 76, no.11, 620 (2016) doi:10.1140/epjc/s10052-016-4479-9
  * (25) M. Visser, Int. J. Mod. Phys. D 12, 649-661 (2003) doi:10.1142/S0218271803003190
  * (26) C. Barcelo, S. Liberati, S. Sonego and M. Visser, JHEP 02, 003 (2011) doi:10.1007/JHEP02(2011)003
  * (27) B. Carr, Astrophys. Space Sci. Proc. 56, 29-39 (2019) doi:10.1007/978-3-030-31593-1_4 [arXiv:1901.07803 [astro-ph.CO]].
  * (28) B. J. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Phys. Rev. D 81, 104019 (2010) doi:10.1103/PhysRevD.81.104019 [arXiv:0912.5297 [astro-ph.CO]].
  * (29) B. Broda, Eur. Phys. J. C 77, no.11, 756 (2017) doi:10.1140/epjc/s10052-017-5336-1 [arXiv:1706.04557 [gr-qc]].
  * (30) H. Bondi, Mon. Not. Roy. Astron. Soc. 112, 195 (1952) doi:10.1093/mnras/112.2.195
  * (31) I. Masina, Eur. Phys. J. Plus 135, no.7, 552 (2020) doi:10.1140/epjp/s13360-020-00564-9 [arXiv:2004.04740 [hep-ph]].
  * (32) M. Plionis, Lect. Notes Phys. 592, 147-207 (2002) [arXiv:astro-ph/0205166 [astro-ph]].
  * (33) B. Carr, F. Kuhnel and M. Sandstad, Phys. Rev. D 94, no.8, 083504 (2016) doi:10.1103/PhysRevD.94.083504 [arXiv:1607.06077 [astro-ph.CO]].
  * (34) P. Villanueva-Domingo, O. Mena and S. Palomares-Ruiz, Front. Astron. Space Sci. 8, 87 (2021) doi:10.3389/fspas.2021.681084 [arXiv:2103.12087 [astro-ph.CO]].
  * (35) M. Khlopov, B. Paik and S. Ray, Axioms 9, no.2, 71 (2020) doi:10.3390/axioms9020071
  * (36) Y. Ali-Haïmoud and M. Kamionkowski, Phys. Rev. D 95, no.4, 043534 (2017) doi:10.1103/PhysRevD.95.043534 [arXiv:1612.05644 [astro-ph.CO]].
  * (37) S. L. Shapiro and S. A. Teukolsky,
  * (38) S. W. Hawking, Nature 248, 30-31 (1974) doi:10.1038/248030a0
  * (39) T. Jacobson, Phys. Rev. D 48, 728-741 (1993) doi:10.1103/PhysRevD.48.728 [arXiv:hep-th/9303103 [hep-th]].
  * (40) H. Tashiro and N. Sugiyama, Phys. Rev. D 78, 023004 (2008) doi:10.1103/PhysRevD.78.023004 [arXiv:0801.3172 [astro-ph]].
  * (41) J. C. Mather, E. S. Cheng, D. A. Cottingham, R. E. Eplee, D. J. Fixsen, T. Hewagama, R. B. Isaacman, K. A. Jesnsen, S. S. Meyer and P. D. Noerdlinger, et al. Astrophys. J. 420, 439-444 (1994) doi:10.1086/173574
  * (42) D. Clancy, R. Guedens and A. R. Liddle, Phys. Rev. D 68, 023507 (2003) doi:10.1103/PhysRevD.68.023507 [arXiv:astro-ph/0301568 [astro-ph]].
  * (43) H. I. Kim, C. H. Lee and J. H. MacGibbon, Phys. Rev. D 59, 063004 (1999) doi:10.1103/PhysRevD.59.063004 [arXiv:astro-ph/9901030 [astro-ph]].
  * (44) D. Hooper and G. Krnjaic, Phys. Rev. D 103, no.4, 043504 (2021) doi:10.1103/PhysRevD.103.043504 [arXiv:2010.01134 [hep-ph]].
  * (45) P. Gondolo, P. Sandick and B. Shams Es Haghi, Phys. Rev. D 102, no.9, 095018 (2020) doi:10.1103/PhysRevD.102.095018 [arXiv:2009.02424 [hep-ph]].
  * (46) J. H. MacGibbon and B. J. Carr, Astrophys. J. 371, 447-469 (1991) doi:10.1086/169909
  * (47) E. V. Bugaev and K. V. Konishchev, Phys. Rev. D 65, 123005 (2002) doi:10.1103/PhysRevD.65.123005 [arXiv:astro-ph/0005295 [astro-ph]].