aainstitutetext: Key Laboratory of Cosmology and Astrophysics (Liaoning) &
College of Sciences, Northeastern University, Shenyang 110819,
Chinabbinstitutetext: Key Laboratory of Data Analytics and Optimization for
Smart Industry (Ministry of Education), Northeastern University, Shenyang
110819, Chinaccinstitutetext: National Frontiers Science Center for Industrial
Intelligence and Systems Optimization, Northeastern University, Shenyang
110819, China

#  Gravitational capture of magnetic monopoles by primordial black holes in
the early universe

Chen Zhang a,b,c and Xin Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It is intriguing to ask whether the existence of primordial black holes (PBHs)
in the early universe could significantly reduce the abundance of certain
stable massive particles (SMP) via gravitational capture, after which the PBHs
evaporate before BBN to avoid conflict with stringent bounds. For example,
this mechanism is relevant to an alternative solution of the monopole problem
proposed by Stojkovic and Freese, in which magnetic monopoles produced in the
early universe are captured by PBHs, thus freeing inflation from having to
occur during or after the corresponding phase transitions that produced the
monopoles. In this work, we reanalyze the solution by modelling the capture
process in the same way as the coexisting monopole annihilation. A subtle
issue which is not handled properly in the previous literature is the choice
of an effective capture cross section for diffusive capture. We model this
aspect properly and justify our treatment. A monochromatic PBH mass function
and a radiation-dominated era before PBH evaporation are assumed. We find that
for Pati-Salam monopoles corresponding to a symmetry breaking scale between
$10^{10}\,\text{GeV}$ and $10^{15}\,\text{GeV}$, the capture rate is many
orders of magnitude below what is needed to cause a significant reduction of
the monopole density. Within our assumptions, we also find that the magnetic
charge that is large enough to make an extremal magnetic black hole
cosmologically stable cannot be obtained from magnetic charge fluctuation via
monopole capture. The large magnetic charged required by cosmological
stability can nevertheless be obtained from magnetic charge fluctuation at PBH
formation, and if later the monopole abundance can be reduced significantly by
some non-inflationary mechanism, long-lived near-extremal magnetic black holes
of observational relevance might result.

##  1 Introduction

Stable massive particles (SMPs) whose existence is commonly due to exact or
approximate symmetries provide an intriguing connection between cosmology and
particle physics Burdin:2014xma . In the standard model (SM) of particle
physics, the approximate baryon number symmetry makes protons cosmologically
stable, while the need for baryogenesis in the early universe requires going
beyond the SM in a number of directions. Beyond the SM, certain global or
gauge symmetries, either discrete or continuous, could give rise to
cosmologically stable particles that may act as dark matter (DM), which are
crucial for explaining a number of phenomena from galactic to cosmological
scales.

The interest of the present work focuses on an interesting class of
SMPs—magnetic monopoles Dirac:1931kp ; tHooft:1974kcl ; Polyakov:1974ek ,
which arise as a result of a nontrivial second homotopy group $\pi_{2}(G/H)$
of the vacuum manifold of some spontaneous symmetry breaking pattern $G/H$
dictated by a (partially) unified gauge theory (see Refs. Goddard:1977da ;
Preskill:1984gd ; Vilenkin:2000jqa ; Shnir:2005vvi ; Weinberg:2012pjx ;
Mavromatos:2020gwk ; ParticleDataGroup:2022pth  for reviews). They can be
copiously produced in the corresponding symmetry breaking phase transitions
via the Kibble or Kibble-Zurek mechanism Kibble:1976sj ; Zurek:1985qw . Being
heavy non-relativistic objects, they tend to overclose the universe during the
cosmological evolution if there does not exist an effective mechanism to
reduce their number density Zeldovich:1978wj ; Preskill:1979zi . Moreover,
their relic abundance is more stringently constrained by the Parker’s bound
Parker:1970xv  coming from the effect of magnetic monopoles on galactic
magnetic fields, by the direct search experiments, and also by catalysis of
baryon number violation via the Callan-Rubakov effect Rubakov:1982fp ;
Callan:1982ah ; Callan:1982au  depending on the specific unification models
111In this work we are concerned with magnetic monopoles associated with the
visible electromagnetism. Hidden monopoles associated with some dark gauge
symmetry breaking may account for part or all of dark matter; see e.g. Refs.
Murayama:2009nj ; GomezSanchez:2011orv ; Evslin:2012fe ; Baek:2013dwa ;
Khoze:2014woa ; Kawasaki:2015lpf ; Nomura:2015xil ; Sato:2018nqy ;
Terning:2019bhg ; Daido:2019tbm ; Bai:2020ttp ; Graesser:2020hiv ;
Nakagawa:2021nme ; Graesser:2021vkr ; Fan:2021ntg ; Hiramatsu:2021kvu ;
Yang:2022quy ..

The standard approach to get rid of the overabundance of magnetic monopoles is
inflation Starobinsky:1980te ; Guth:1980zm , with solving the monopole problem
being one of its most important theoretical motivations. It requires inflation
to occur during or after the corresponding symmetry breaking phase transition
(and baryogenesis will be even later), thus establishing a connection between
the particle physics model and the cosmological history. Nevertheless, it is
both interesting and important to ask whether such a connection is inevitable.
First, there is the possibility that gauge coupling unification or even
partial unification does not occur, with the side effect that the elegant
explanantion for charge quantization is also lost. Second, in partial
unification scenarios such as the Pati-Salam model Pati:1974yy , it is
possible to have a low-scale ($\lesssim 10^{10}\,\text{GeV}$) strongly first-
order Pati-Salam breaking phase transition which could lead to suppressed
initial abundance of magnetic monopoles Huang:2020bbe . Other solutions to the
monopole problem include inverse symmetry breaking or symmetry nonrestoration
effect in finite-temperature field theory Langacker:1980kd ; Salomonson:1984rh
; Dvali:1995cj ; Bajc:1997ky , eliminating the monopoles by domain walls which
subsequently decay or get destroyed Dvali:1997sa ; Stojkovic:2005zh , entropy
production effects Izawa:1984ww , and gravitational capture by primordial
black holes (PBHs) which evaporate prior to BBN Stojkovic:2004hz . The
viability of these alternative solutions may break the connection between
inflation and the corresponding symmetry breaking phase transition, and thus
allowing for more possibilities of cosmological model building. However, there
are not many alternative solutions and most of them are effective in
restrictive portion of models or parameter space. The viability of these
solutions in a more general context, taking into account potential
uncertainties in the theoretical modelling and computation, entails further
detailed investigations.

In this work we revisit the solution to the monopole problem via gravitational
capture by PBHs Stojkovic:2004hz , proposed by Stojkovic and Freese. This idea
is of particular interest to us due to two main reasons. First, investigation
of monopole capture by PBHs might teach us lessons on whether PBHs may
significantly affect the abundance of other SMPs, which may have important
implications for physics of dark matter and baryogenesis. Second, recently
physics of magnetic black holes has received quite some attention
Maldacena:2020skw ; Bai:2020spd ; Liu:2020vsy ; Ghosh:2020tdu ; Liu:2020bag ;
Araya:2020tds ; Bai:2020ezy ; Diamond:2021scl ; Chen:2022qvg . Interesting
phenomenological bounds have been obtained Ghosh:2020tdu ; Bai:2020spd ;
Diamond:2021scl , however the formation mechanism of magnetic black holes
remains elusive. The difficulty is that one needs to feed a large number of
magnetic monopoles into PBHs, while keeping the remaining abundance of
monopoles low enough to avoid stringent constraints Maldacena:2020skw . It is
conceivable that the mechanism involved in the Stojkovic-Freese (SF) solution
might play an important role in some potential formation mechanism of magnetic
black holes.

In the SF solution to the monopole problem, two major physical processes that
affect the monopole abundance are monopole annihilation and gravitational
capture by PBHs. These two processes are in fact quite similar. Both processes
are driven by long-range forces obeying an inverse-square law, and in both
processes the movement of monopoles in the primordial plasma is that of a
Brownian motion. In the SF paper Stojkovic:2004hz , however, the gravitational
capture by PBHs is modelled somewhat differently from the monopole
annihilation. In this work, we have instead modelled the gravitational capture
by PBHs in the same manner as monopole annihilation. We find that our
modelling leads to significantly smaller capture rates compared to the SF
modelling. The difference can be traced to the fact that the SF modelling uses
an effective capture cross section that is not appropriate in the diffusive
regime. Moreover, we recognize that assuming radiation domination, the use of
an appropriately extended PBH mass function in the spirit of that used in Ref.
Stojkovic:2004hz  instead of a monochromatic one employed in our study should
improve significantly the efficiency in reducing the monopole abundance.

It is possible for PBHs to acquire some magnetic charge at formation if the
formation temperature is below the symmetry breaking phase transition
temperature. Even if the PBHs do not carry magnetic charge at formation, the
monopole capture process may leave a residual magnetic charge on the PBH
because there is fluctuation on the number of absorbed monopoles and
antimonopoles. This magnetic charge fluctuation is expected to leave a
magnetic charge of about $\sqrt{N}$ if the total number of absorbed monopoles
and antimonopoles is $N$ for each PBH. The fate of this residual magnetic
charge (including any initial magnetic charge) depends on its magnitude. For a
sufficiently large residual magnetic charge ($\gtrsim\mathcal{O}(10^{6}))$ for
a monopole mass ($\sim 10^{17}\,\text{GeV}$), the PBH evolves toward a
magnetically charged extremal Reissner-Nordström (RN) black hole that is
cosmologically stable Maldacena:2020skw . Otherwise, its evaporation should be
qualitatively similar to that of an uncharged PBH with the same mass, but at
the final stage it is unstable against decaying into multiple magnetic
monopoles which again should be taken into account when we compute the final
monopole abundance. In this work we also examine the magnitude of residual
magnetic charge. Assuming a monochromatic PBH mass function and a radiation-
dominated universe before PBH evaporation, we find that for the monopole
capture processes, it is not possible to get a sufficiently large residual
magnetic charge required by cosmological stability. We show that the large
magnetic charge required by cosmological stability might be obtained already
at PBH formation222In such a case we only require radiation domination before
PBH formation rather than PBH evaporation.. Although with the monochromatic
PBH mass function the monopole problem is not solved yet, such an
investigation may lead to insights about the formation of magnetic black holes
in the early universe.

It is interesting to note that evaporating PBHs create hot spots which could
reproduce magnetic monopoles during their cooling Das:2021wei ; He:2022wwy ,
an effect that is unknown at the time of the SF proposal. According to the
latest study He:2022wwy , the highest temperature achieved in the hot spot is
not larger than $\mathcal{O}(10^{10}\,\text{GeV})$, assuming a fine structure
constant of about $0.1$. Therefore, for simplicity, in this work we consider
the symmetry breaking phase transition scale to be
$\gtrsim\mathcal{O}(10^{10}\,\text{GeV})$.

This work is organized as follows. In Sec. 2 we briefly review the production
of magnetic monopoles in a symmetry breaking phase transition in the early
universe, and the subsequent monopole-antimonopole annihilation. In Sec. 3 we
present our modelling of the gravitational capture of monopoles by PBHs. The
essential difference compared to the SF modelling consists in the choice of
effective capture cross section in the diffusive capture regime, which we
explain in detail in Sec. 3.4. In Sec. 4 we outline the computation of
magnetic charge fluctuation. In Sec. 5 we collect the ingredients required for
comparing the different ways of modelling and analyzing the consequences of
magnetic charge fluctuations which are performed in Sec. 6. We present the
discussion and conclusions in Sec. 7.

##  2 Review of monopole production and annihilation

###  2.1 Monopole production

A magnetic monopole can be viewed as an extended field configuration
stabilized by nontrivial topology associated with the mapping from spatial
infinity to the vacuum manifold of some spontaneous symmetry breaking. The
order parameter of the symmetry breaking phase transition lives on the vacuum
manifold and it relies on local interactions to align order parameters of
nearby regions. In a cosmological setting, the range of interactions is
limited by the particle horizon, and thus different horizon patches will
choose their order parameter values independently. At the junctions of
multiple horizon patches there is some probability to form field
configurations with a nontrivial winding number, resulting in the production
of magnetic monopoles. This is the basic picture of producing topological
defects in the early universe via the Kibble mechanism Kibble:1976sj . Note
that in gauge theories it is possible to formulate the above discussion in a
gauge-invariant manner Weinberg:2012pjx .

For definiteness, we consider a radiation-dominated universe. The energy
density $\rho$ and entropy density $s$ at temperature $T$ are given by

$\displaystyle\rho=K_{1}T^{4},\quad s=K_{2}T^{3},$ (1)

with

$\displaystyle K_{1}=\frac{\pi^{2}}{30}\mathcal{N},\quad
K_{2}=\frac{2\pi^{2}}{45}\mathcal{N},$ (2)

with $\mathcal{N}$ being the number of effective relativistic degrees of
freedom at temperature $T$. Here for simplicity we approximate $\mathcal{N}$
appearing in $\rho$ and $s$ as the same, and neglect the change of
$\mathcal{N}$ with temperature. $\mathcal{N}$ typically ranges from $100$ to
$1000$ for temperatures above the electroweak scale, depending on the specific
particle physics model. According to the Friedmann equation, the Hubble
parameter can be expressed as

$\displaystyle H=K\frac{T^{2}}{M_{\text{Pl}}},$ (3)

with

$\displaystyle K=\bigg{(}\frac{4\pi^{3}\mathcal{N}}{45}\bigg{)}^{1/2},$ (4)

and the Planck mass ($G$ is the Newton constant)

$\displaystyle M_{\text{Pl}}\equiv G^{-1/2}=1.2\times
10^{19}\,\text{GeV}=2.2\times 10^{-5}\text{g}.$ (5)

The particle horizon $d_{H}$ is given by the inverse of the Hubble parameter

$\displaystyle d_{H}=H^{-1}.$ (6)

On average each volume of $d_{H}^{3}$ should contain $p_{M}$ monopoles, with
$p_{M}$ being a number that is not much less than $1$. Therefore the monopole
number density $n_{M}$ at production should satisfy

$\displaystyle n_{M}(T_{c})\gtrsim
p_{M}d_{H}^{-3}(T_{c})=p_{M}H^{3}(T_{c})=p_{M}K^{3}\frac{T_{c}^{6}}{M_{\text{Pl}}^{3}}.$
(7)

Here $T_{c}$ denotes the temperature at which the monopoles are produced. As
an approximation in this work we identify it with the critical temperature of
the phase transition, although strictly speaking it can be somewhat lower than
the true critical temperature Weinberg:2012pjx . We now define the monopole
yield $r$ as

$\displaystyle r\equiv\frac{n_{M}}{s},$ (8)

then the monopole yield at $T=T_{c}$, denoted $r_{i}$ hereafter, satisfies

$\displaystyle r_{i}\equiv r(T_{c})=\frac{n_{M}(T_{c})}{s(T_{c})}\gtrsim
p_{M}K^{3}K_{2}^{-1}\bigg{(}\frac{T}{M_{\text{Pl}}}\bigg{)}^{3}.$ (9)

This just gives the Kibble estimate, which can be expressed as

$\displaystyle r_{i}\gtrsim
p(8\pi)^{3/2}\mathcal{N}^{1/2}\bigg{(}\frac{T_{c}}{M_{\text{Pl}}}\bigg{)}^{3},$
(10)

where

$\displaystyle p\equiv p_{M}\frac{\pi}{12\sqrt{10}}$ (11)

is a number not much less than $0.1$.

It should be emphasized that the Kibble estimate only gives a lower bound on
the initial monopole abundance, while the actual initial abundance can be much
larger. There are three scenarios that can be envisioned which we discuss
below: first-order phase transitions, second-order phase transitions, and
crossover phase transitions.

(i) First-order phase transitions

In the case of first-order phase transitions, the phase transition proceeds by
bubble nucleation. Inside a single bubble the order parameter should be
uniform, while order parameters in different bubbles should be uncorrelated.
Suppose the characteristic bubble size at bubble coalescence is $R$, then the
number density of monopoles at production is estimated to be

$\displaystyle n_{M}\simeq p_{M}R^{-3}.$ (12)

$R$ is related to the parameter $\beta$ that characterizes the inverse
duration of the phase transition (see Ref. Hindmarsh:2020hop  for definition
and discussion) and the bubble wall velocity $v_{w}$ as

$\displaystyle R=\frac{(8\pi)^{1/3}v_{w}}{\beta}.$ (13)

Introducing the dimensionless version of the $\beta$ parameter

$\displaystyle\tilde{\beta}\equiv\frac{\beta}{H(T_{p})},$ (14)

with $T_{p}$ being the percolation temperature, it is possible to express
$r_{i}$ as

$\displaystyle r_{i}\simeq
p(\tilde{\beta}v_{w}^{-1})^{3}(8\pi)^{3/2}\mathcal{N}^{1/2}\bigg{(}\frac{T_{c}}{M_{\text{Pl}}}\bigg{)}^{3}.$
(15)

Again $p$ is some number not much less than $0.1$ and for our purpose we have
made the approximation $T_{p}\approx T_{c}$. For a strongly first-order phase
transition, one gets $\tilde{\beta}v_{w}^{-1}\simeq\mathcal{O}(1)$, so that
Eq. (15) just saturates the Kibble estimate in Eq. (10). For a weakly first-
order phase transition, with typical values
$\tilde{\beta}v_{w}^{-1}\simeq\mathcal{O}(10\sim 10^{3})$, obviously the
monopole yield can be orders of magnitude larger than the Kibble estimate.

(ii) Second-order phase transitions

In the case of second-order phase transitions, monopole density is determined
via the Kibble-Zurek mechanism (see Refs. Vachaspati:2006zz ; delCampo:2013nla
; Graesser:2020hiv  for reviews). The basic picture is as follows. What
replaces $R$ in the case of first-order phase transitions should be some
correlation length. Usually, the correlation length diverges at the critical
point. However the cosmic expansion in the early universe determines a finite
quench time $\tau_{Q}$ which is just the inverse of the Hubble parameter. To
discuss the relevant physics, we introduce the reduced distance parameter
$\epsilon$, defined as

$\displaystyle\epsilon=\frac{\omega_{c}-\omega}{\omega_{c}},$ (16)

with $\omega$ being some control parameter (such as temperature), and
$\omega_{c}$ being its value at the critical point. Obviously, $\epsilon$
characterizes how close the system is to the critical point. As
$\epsilon\rightarrow 0$, the _equilibrium_ correlation length $\xi$ and the
_equilibrium_ relaxation time $\tau$ then scale as

$\displaystyle\xi(\epsilon)=\frac{\xi_{0}}{|\epsilon|^{\nu}},\quad\tau(\epsilon)=\frac{\tau_{0}}{|\epsilon|^{\mu}},$
(17)

with $\mu,\nu$ being two critical exponents. We assume a linear quench, that
is (we take $t=0$ corresponding to the critical temperature $T=T_{c}$)

$\displaystyle\epsilon(t)=\frac{t}{\tau_{Q}},\quad\text{for}\,t\in[-\tau_{Q},\tau_{Q}].$
(18)

Now the crucial thing is to note that there is a specific time $\hat{t}$,
defined in such a way that

$\displaystyle\tau(\hat{t})=\hat{t},$ (19)

which means that the equilibrium relaxation time is about the same as with the
time elapsed after crossing the critical point. $\hat{t}$ is known as the
freeze-out time, since in the Kibble-Zurek mechanism it is assumed that during
the time interval $[-t,t]$ the dynamics (i.e. the correlation length $\xi$) is
frozen while outside the $[-t,t]$ interval, the dynamics follows the usual
adiabatic behavior. Thus, the maximum correlation length reached during the
second-order phase transition is

$\displaystyle\hat{\xi}\equiv\xi(\hat{\epsilon}),\quad\hat{\epsilon}\equiv|\epsilon(\hat{t})|.$
(20)

It is then straightforward to obtain

$\displaystyle\hat{\xi}=\xi_{0}\bigg{(}\frac{\tau_{Q}}{\tau_{0}}\bigg{)}^{\frac{\nu}{1+\mu}},\quad\tau_{Q}=\frac{1}{H(T_{c})}.$
(21)

Each volume of $\hat{\xi}^{3}$ should contain about $1$ monopole, thus the
estimated initial monopole number density is

$\displaystyle
n_{M}(T_{c})\simeq\frac{1}{\xi_{0}^{3}}\bigg{(}\frac{\tau_{0}}{\tau_{Q}}\bigg{)}^{\frac{3\nu}{1+\mu}}.$
(22)

To proceed further, suppose the second-order phase transition is driven by
some scalar field dynamics with a dimensionless quartic coupling $\lambda$ and
the thermal effective mass $m_{\sigma}$ for a typical scalar resonance. Then
we may approximate

$\displaystyle\xi_{0}\simeq\tau_{0}\simeq
m_{\sigma}^{-1}\simeq\frac{1}{\sqrt{\lambda}T_{c}},$ (23)

then Eqs. (21)–(22) can be expressed as

$\displaystyle\hat{\xi}$
$\displaystyle\simeq\frac{1}{\sqrt{\lambda}T_{c}}\bigg{[}\frac{\sqrt{\lambda}T_{c}}{H(T_{c})}\bigg{]}^{\frac{\nu}{1+\mu}},$
(24) $\displaystyle n_{M}(T_{c})$
$\displaystyle\simeq\big{(}\sqrt{\lambda}T_{c}\big{)}^{3}\bigg{[}\frac{H(T_{c})}{\sqrt{\lambda}T_{c}}\bigg{]}^{\frac{3\nu}{1+\mu}}.$
(25)

Using Eqs. (1)–(4) we may obtain

$\displaystyle
r_{i}\simeq\lambda^{3/2}K_{2}^{-1}\bigg{[}\frac{KT_{c}}{\sqrt{\lambda}M_{\text{Pl}}}\bigg{]}^{\frac{3\nu}{1+\mu}}.$
(26)

We will typically consider

$\displaystyle\mu=\nu,\quad\lambda\simeq 1,$ (27)

then

$\displaystyle r_{i}\simeq
0.02\bigg{(}\frac{17T_{c}}{M_{\text{Pl}}}\bigg{)}^{\frac{3\nu}{1+\nu}},\quad\text{for}\,\,\mathcal{N}=100,~{}\lambda=1.$
(28)

Typical value of the critical exponent $\nu$ is $0.5\sim 0.8$ Murayama:2009nj
, leading to an initial monopole abundance orders of magnitude larger than the
bound set by the Kibble estimate (10).

(iii) Crossover phase transitions

One can model the dynamics of a crossover phase transition in the same spirit
as that of a second-order phase transition, with appropriate modification to
the scaling behavior of $\xi$ and $\tau$ so that their critical scaling
behavior tapers off very close to $\epsilon=0$ Zurek:1999ym . In such a case
it is possible that the correlation length saturates at an even smaller value
compared to the case of a second-order phase transition, leading to a larger
rate of defect formation. Lacking a generally accepted way to model the
crossover phase transition, here we will not attempt to estimate the initial
monopole abundance in crossover phase transitions in a more quantitative
manner.

###  2.2 Monopole annihilation

Consider a magnetic monopole with mass $m$ that carries a magnetic charge of

$\displaystyle\chi g,$ (29)

where $\chi$ is an integer (for a unit magnetic monopole, $\chi$=1), and $g$
is the unit magnetic charge in the natural Gaussian system, that is

$\displaystyle g=\frac{1}{2e},\quad e=\sqrt{\alpha}.$ (30)

Using $\alpha=\frac{1}{137}$ we obtain that

$\displaystyle g\simeq 5.9.$ (31)

When a magnetic monopole moves with velocity v the primordial plasma in the
early universe, it experiences a drag force which can be expressed as
Vilenkin:2000jqa

$\displaystyle\textbf{F}_{\text{drag}}=-CT^{2}f(v)\textbf{v},$ (32)

with $f(v)$ being a slowly varying function with $f(0)=1$ and $f(1)=3/2$, and

$\displaystyle C=\frac{2\pi}{9}\bar{C}\chi^{2}g^{2}\sum_{a}b_{a}e_{a}^{2},$
(33)

where the summation is over the spin states of light charged particles, $b=1$
for bosons and $b=\frac{1}{2}$ for fermions, and $\bar{C}$ is an angular
integral that is roughly $5\sim 10$. It is expected that
$C\chi^{-2}\sim(1-5)\mathcal{N}_{c}$, with $\mathcal{N}_{c}$ being the number
of relativistic effective charged degrees of freedom Vilenkin:2000jqa .

In the following we approximate $f(v)=1$, then the equation of motion of a
nonrelativistic magnetic monopole in the presence of the drag force is

$\displaystyle m\dot{\textbf{v}}=-CT^{2}\textbf{v}.$ (34)

Its solution $\textbf{v}=\textbf{v}_{0}e^{-t/\tau_{M}}$ is characterized by a
timescale $\tau_{M}$

$\displaystyle\tau_{M}=\frac{m}{CT^{2}}.$ (35)

$\tau_{M}$ can be regarded as the characteristic timescale the monopole needs
to forget its initial velocity, or the mean free time of the monopole. We may
then derive the monopole mean free path $\ell$ by multiplying it with the
monopole’s thermal velocity $v_{T}\sim(3T/m)^{1/2}$333Magnetic monopoles are
not in chemical equilibrium with the plasma, but due to the electromagnetic
interactions with the charged particles, they are kept in kinetic equilibrium,
leading to a thermal velocity of $\sim(3T/m)^{1/2}$.. The result is

$\displaystyle\ell\simeq\frac{1}{CT}\bigg{(}\frac{m}{T}\bigg{)}^{1/2}.$ (36)

One should note however the mean free time $\tau_{M}$ and the mean free path
$\ell$ do not correspond to the average time spent and distance traversed by
the monopole during two scatterings with the plasma particles. Instead, from
the derivation above we see that they correspond to the average time spent and
distance traversed by the monopole to have a significant change of its
velocity.

Now besides the drag force exerted by the plasma, we consider the attractive
force between a monopole and an antimonopole at a distance $\bar{R}$. When the
attractive force is balanced by the drag force, the monopole will attain a
drift velocity

$\displaystyle v_{\text{D}}\simeq\frac{\chi^{2}g^{2}}{CT^{2}\bar{R}^{2}}.$
(37)

For the typical monopole separation $d_{\text{ann}}\sim n_{M}^{-1/3}$, we may
estimate the capture time as

$\displaystyle\tau_{\text{ann}}\simeq\frac{d_{\text{ann}}}{v_{\text{D}}(d_{\text{ann}})}=\frac{CT^{2}}{\chi^{2}g^{2}n_{M}}.$
(38)

This implies that we may express the time evolution of the monopole number
density $n_{M}$ as

$\displaystyle\dot{n}_{M}=-Dn_{M}^{2}-3\frac{\dot{a}}{a}n_{M},$ (39)

with

$\displaystyle
D=\frac{1}{\tau_{\text{ann}}n_{M}}=\frac{\chi^{2}g^{2}}{CT^{2}},$ (40)

and the $-3\frac{\dot{a}}{a}n_{M}$ term obviously takes into account of the
effect of cosmic expansion. Eq. (39) can be solved by considering the
evolution of $r=n_{M}/s$ with respect to the temperature $T$. Introducing the
reduced temperature variable $z$,

$\displaystyle z(T)\equiv\frac{T}{T_{c}},$ (41)

the evolution of $r$ from $z_{1}\equiv(T_{1})$ to $z_{2}\equiv(T_{2})$ can be
expressed as

$\displaystyle
r_{2}=\bigg{[}\frac{1}{r_{1}}+\frac{\Delta}{T_{c}}\bigg{(}\frac{1}{z_{2}}-\frac{1}{z_{1}}\bigg{)}\bigg{]}^{-1},$
(42)

where $r_{1}=r(z_{1}),r_{2}=r(z_{2})$, and $\Delta$ is given by

$\displaystyle\Delta\equiv
K_{2}K^{-1}M_{\text{Pl}}DT^{2}=K_{2}K^{-1}C^{-1}\chi^{2}g^{2}M_{\text{Pl}}.$
(43)

Note that $\Delta$ is a temperature-independent quantity with mass dimension
$1$.

Since $D\propto T^{-2}$ (see Eq. (40)), the diffusive capture is more
efficient at low temperature than at high temperature. This can be traced to
the fact that at low temperature, the smaller drag force allowed a larger
drift velocity of the monopole, thus reducing the capture time. Once the
monopole and the antimonopole are captured into a Coulomb bound state, then
they cannot escape annihilation in the diffusive environment.

However, the above picture only holds when the monopole mean free path is
smaller than the capture radius. The capture radius $r_{c}^{\text{ann}}$ is
found by letting the negative Coulomb potential energy be comparable to the
monopole’s thermal kinetic energy. Therefore

$\displaystyle r_{c}^{\text{ann}}\simeq\frac{\chi^{2}g^{2}}{T}.$ (44)

Requiring $\ell<r_{c}^{\text{ann}}$ leads to

$\displaystyle T>T_{\text{ann}},\quad
T_{\text{ann}}=\frac{m}{C^{2}\chi^{4}g^{4}}.$ (45)

Thus, if the temperature drops below $T_{\text{ann}}$, diffusive capture of
monopoles will cease to be effective, and monopoles and antimonopoles can only
annihilate by radiative capture via bremsstrahlung emission, leading to a much
smaller capture rate which we neglect in this study Weinberg:2012pjx .

We introduce the reduced variables

$\displaystyle\delta=\frac{m}{T_{c}},\quad x=\frac{T_{c}}{M_{\text{Pl}}},$
(46)

then using Eq. (42) for an initial monopole yield $r_{i}$ at $T=T_{c}$, the
final monopole yield $r_{\text{ann}}$ at $T=T_{\text{ann}}$ is approximately
given by

$\displaystyle r_{\text{ann}}\simeq\min(r_{i},r_{\star}),\quad r_{\star}\equiv
K_{2}^{-1}KC^{-1}\chi^{-6}g^{-6}\delta x.$ (47)

This means that if $r_{i}<r_{\star}$, then the monopole yield cannot be
reduced further through annihilation. On the other hand, if $r_{i}>r_{\star}$,
annihilation can always reduce the monopole yield to $r_{\star}$ which is
independent of the initial yield, and the annihilation is most efficient at
temperature close to $T_{\text{ann}}$. Note also that $r_{\star}$ is very
sensitive to $\chi$ and $g$.

##  3 Capture of monopoles by primordial black holes

###  3.1 The SF modelling of monopole capture by PBHs

First consider PBHs with a universal mass $m_{\text{bh}}$ (so the Schwarzchild
radius is $R_{\text{bh}}=2m_{\text{bh}}M_{\text{Pl}}^{-2}$), uniformly
distributed in the early universe with number density $n_{\text{bh}}$. In the
SF paper Stojkovic:2004hz , the gravitational capture of monopoles by PBHs is
modelled by a new term in the evolution equation for $n_{M}$, so that Eq. (39)
is modified to be

$\displaystyle\dot{n}_{M}=-Dn_{M}^{2}-F_{\text{SF}}n_{M}-3\frac{\dot{a}}{a}n_{M},$
(48)

where

$\displaystyle F_{\text{SF}}\equiv n_{\text{bh}}\sigma_{g}v_{M},$ (49)

with $v_{M}$ being the average incident velocity of monopoles on a PBH, and
$\sigma_{g}$ being the gravitational capture cross section, given by
Frolov:2011bhp

$\displaystyle\sigma_{g}=4\pi v_{M}^{-2}R_{\text{bh}}^{2},\quad
R_{\text{bh}}=2m_{\text{bh}}M_{\text{Pl}}^{-2}.$ (50)

Then the key issue is to determine $v_{M}$. One naturally tries to link
$v_{M}$ with the monopole thermal velocity $v_{T}=(3T/m)^{1/2}$. In the SF
modelling, $v_{M}$ is taken to be a random walk velocity, such that if there
is just one random walk step, $v_{M}$ is equal to $v_{T}$. $v_{M}$ and $v_{T}$
are then related by

$\displaystyle v_{M}=\frac{v_{T}}{\sqrt{N}},$ (51)

in which $N$ is the number of random walk steps. $N$ is determined as follows.
For the gravitational capture of monopoles by PBHs, one may determine a
capture radius $r_{c}^{\text{gc}}$ in a similar manner as
$r_{c}^{\text{ann}}$. That is, the negative gravitational energy should be
comparable to the monopole’s thermal kinetic energy when the distance between
the monopole and the PBH is $r_{c}^{\text{gc}}$,

$\displaystyle r_{c}^{\text{gc}}=\frac{mm_{\text{bh}}}{M_{\text{Pl}}^{2}T},$
(52)

$\sqrt{N}$ is then taken as the ratio between $r_{c}^{\text{gc}}$ and the
monopole mean free path $\ell$,

$\displaystyle\sqrt{N}=\frac{r_{c}^{\text{gc}}}{\ell}.$ (53)

Thus, $v_{M}$ can be viewed as the random walk velocity for the monopole to
generate a root-mean-square displacement of about the capture radius
$r_{c}^{\text{gc}}$. Using $v_{T}=(3T/m)^{1/2}$, and Eqs. (51) and (53), we
obtain the incident velocity $v_{M}$ as

$\displaystyle v_{M}\simeq\frac{\sqrt{3}M_{\text{Pl}}^{2}}{Cmm_{\text{bh}}}.$
(54)

Interestingly, the dependence on $T$ disappears in the expression for $v_{M}$.
The gravitational capture cross section then becomes

$\displaystyle\sigma_{g}\simeq\frac{16}{3}\pi
C^{2}\frac{m^{2}m_{\text{bh}}^{4}}{M_{\text{Pl}}^{8}}.$ (55)

By introducing two parameters $f_{\text{SF}},\beta_{\text{SF}}$, defined
through

$\displaystyle
n_{\text{bh}}=\frac{f_{\text{SF}}\mathcal{N}T^{4}}{m_{\text{bh}}},\quad
m_{\text{bh}}=\beta_{\text{SF}}\times
0.2\mathcal{N}^{-1/2}\frac{M_{\text{Pl}}^{3}}{T^{2}},$ (56)

one may express $F_{\text{SF}}$ in Eq. (49) as

$\displaystyle F_{\text{SF}}\simeq f_{\text{SF}}\beta_{\text{SF}}^{2}Cm.$ (57)

Note that $f_{\text{SF}}$ characterizes the energy density fraction of PBHs,
while $\beta_{\text{SF}}$ characterizes the ratio between a single PBH mass
and the mass within the particle horizon. Assuming a monochromatic PBH mass
function and that the accretion does not change the PBH mass significantly, we
expect

$\displaystyle f_{\text{SF}}\propto T^{-1},\quad\beta_{\text{SF}}\propto
T^{2},\quad\text{monochromatic},$ (58)

so that $f_{\text{SF}}\beta_{\text{SF}}^{2}\propto T^{3}$. Nevertheless in the
SF paper Stojkovic:2004hz  the assumption on the PBH mass function is such
that

$\displaystyle
f_{\text{SF}}=\text{const},\quad\beta_{\text{SF}}=\text{const},\quad\text{SF
mass function}.$ (59)

In order for Eq. (59) to hold, new PBHs must keep forming while the old PBHs
must be keep evaporating in some manner to keep
$f_{\text{SF}},\beta_{\text{SF}}$ constant. As was pointed out in Ref.
Stojkovic:2004hz , $f_{\text{SF}}$ and $\beta_{\text{SF}}$ then represent some
average feature of the system.

For definiteness we only use the phrase “SF modelling” to refer to the
treatment of the gravitational capture of monopoles by PBHs in the SF paper
Stojkovic:2004hz  without any assumption on the PBH mass function. The
specific requirement in Eq. (59) will be termed as the “SF mass function”.

In retrospect, the use of Eq. (50) as the effective capture cross section in
the diffusive regime is subject to one serious criticism. Eq. (50) is derived
from solving the geodesic evolution of a massive particle in the Schwarzschild
geometry, without the frequent bombardment from other particles at all
Frolov:2011bhp . In the diffusive regime the monopole is likely to be
deflected through collisions with other particles along its journey to the PBH
event horizon, even it is already inside the capture radius. Therefore we set
out to find alternative modelling of the gravitational capture process.

###  3.2 The drift modelling of monopole capture by PBHs

As noted previously, the gravitational capture of monopoles by PBHs are quite
similar to the monopole-antimonopole annihilation. We are thus motivated to
model the two processes in a similar manner. Therefore, let us consider the
balancing between the gravitational force and the drag force exerted on a
monopole, which determines a monopole drift velocity $u_{\text{D}}$ as a
function of the monopole-PBH distance $\bar{R}$

$\displaystyle
u_{\text{D}}(\bar{R})=\frac{m_{\text{bh}}m}{M_{\text{Pl}}^{2}}\frac{1}{CT^{2}\bar{R}^{2}}.$
(60)

The typical separation between PBHs is $n_{\text{bh}}^{-1/3}$, thus we will
use a typical drift velocity for the capture process as

$\displaystyle
u_{\text{D}}(n_{\text{bh}}^{-1/3})=\frac{m_{\text{bh}}m}{M_{\text{Pl}}^{2}}\frac{n_{\text{bh}}^{2/3}}{CT^{2}}.$
(61)

Thus the typical capture time is

$\displaystyle\tau_{\text{gc}}=\frac{n_{\text{bh}}^{-1/3}}{u_{\text{D}}(n_{\text{bh}}^{-1/3})}=\frac{M_{\text{Pl}}^{2}CT^{2}}{n_{\text{bh}}m_{\text{bh}}m}.$
(62)

The typical capture frequency per monopole is

$\displaystyle
F\equiv\tau_{\text{gc}}^{-1}=\frac{n_{\text{bh}}m_{\text{bh}}m}{M_{\text{Pl}}^{2}CT^{2}}.$
(63)

The evolution of the monopole number density satisfies

$\displaystyle\dot{n}_{M}=-Dn_{M}^{2}-Fn_{M}-3\frac{\dot{a}}{a}n_{M}.$ (64)

In analogy with monopole annihilation, gravitational capture of monopoles by
PBHs should be effective only when the monopole mean free path $\ell$ is less
than the gravitational capture radius

$\displaystyle r_{c}^{\text{gc}}=\frac{m_{\text{bh}}m}{M_{\text{Pl}}^{2}T}.$
(65)

This leads to the requirement

$\displaystyle T>T_{\text{gc}},\quad
T_{\text{gc}}\equiv\frac{M_{\text{Pl}}^{4}}{C^{2}m_{\text{bh}}^{2}m}.$ (66)

Another requirement is, of course, the gravitational capture of monopoles by
PBHs can be effective only when the PBHs have not evaporated yet. Consider a
PBH with mass $m_{\text{bh}}$, its “lifetime” $\tau_{\text{bh}}$ can be
parameterized as

$\displaystyle\tau_{\text{bh}}=\varepsilon\frac{m_{\text{bh}}^{3}}{M_{\text{Pl}}^{4}}.$
(67)

For a non-rotating PBH with a negligible charge to mass ratio, we have
Hooper:2020otu ; Gehrman:2022imk

$\displaystyle\varepsilon=\frac{10240\pi}{\mathcal{G}\langle
g_{\star,H}\rangle},$ (68)

with $\mathcal{G}\simeq 3.8$ is the grey body factor, and $\langle
g_{\star,H}\rangle\simeq\mathcal{N}$ depends on the particle physics model.
Assuming radiation domination, the temperature $T_{\text{ev}}$ of the universe
at PBH evaporation can be estimated from

$\displaystyle\tau_{\text{bh}}=\frac{1}{2H(T_{\text{ev}})},$ (69)

which leads to

$\displaystyle T_{\text{ev}}=(2\varepsilon
K)^{-1/2}\bigg{(}\frac{m_{\text{bh}}}{M_{\text{Pl}}}\bigg{)}^{-3/2}M_{\text{Pl}}.$
(70)

The gravitational capture of monopoles by PBHs can thus be effective only when

$\displaystyle T_{s}<T<T_{t},\quad
T_{s}\equiv\max\\{T_{\text{ev}},T_{\text{gc}}\\},\quad
T_{t}\equiv\min\\{T_{c},T_{b}\\},$ (71)

where $T_{b}$ is the temperature of the universe when the PBHs form.

###  3.3 Drift modelling with a monochromatic PBH mass function

In this work we will focus on a monochromatic PBH mass function with a fixed
$m_{\text{bh}}$. It is then possible express $n_{\text{bh}}$ in terms of the
energy density fraction of PBHs $\beta$ at the time of PBH formation as

$\displaystyle(n_{\text{bh}}m_{\text{bh}})|_{\text{formation}}=\beta
K_{1}T_{b}^{4},$ (72)

with $T_{b}$ being the temperature of the plasma at PBH formation. Thus at any
temperature $T$ we have

$\displaystyle n_{\text{bh}}m_{\text{bh}}=\beta K_{1}T_{b}T^{3}.$ (73)

$T_{b}$ can be related to $m_{\text{bh}}$ in the following manner. The PBH
mass can be related to the Hubble parameter at formation Sasaki:2018dmp

$\displaystyle m_{\text{bh}}=\frac{\gamma}{2G}H_{\text{form}}^{-1},\quad
H_{\text{form}}^{-1}=K\frac{T_{b}^{2}}{M_{\text{Pl}}},$ (74)

Typically $\gamma\simeq 0.2$ Carr:1975qj  which is the ratio between the PBH
mass and the horizon mass at formation. $T_{b}$ can then be expressed as

$\displaystyle
T_{b}=\bigg{(}\frac{\gamma}{2K}\bigg{)}^{1/2}\bigg{(}\frac{m_{\text{bh}}}{M_{\text{Pl}}}\bigg{)}^{-1/2}M_{\text{Pl}}.$
(75)

We then obtain using Eq. (63)

$\displaystyle
F=K_{1}\bigg{(}\frac{\gamma}{2K}\bigg{)}^{1/2}C^{-1}\delta\times\beta\bigg{(}\frac{m_{\text{bh}}}{M_{\text{Pl}}}\bigg{)}^{-1/2}\bigg{(}\frac{T_{c}}{M_{\text{Pl}}}\bigg{)}T.$
(76)

Now we return to the evolution equation (64). It is straightforward to
transform it into a differential equation for $r(T)$ as follows

$\displaystyle\frac{dr}{dT}=\frac{\Delta}{T^{2}}r^{2}+\frac{\Phi}{T^{2}}r,$
(77)

where $\Delta$ is still given by Eq. (43), while $\Phi$ is defined as

$\displaystyle\Phi\equiv\frac{M_{\text{Pl}}F}{KT}.$ (78)

It is interesting to note that for a monochromatic PBH mass function, $\Phi$
is also a temperature-independent quantity of mass dimension $1$. Eq. (77) can
then be solved analytically. For $j=1,2$, suppose at temperature $T_{j}$, the
yield $r$ is $r_{j}$. Then if $\Phi=0,\Delta>0$ (i.e. only monopole
annihilation is effective), the solution to Eq. (77) is still given by Eq.
(42). When $\Phi>0,\Delta>0$, we introduce

$\displaystyle r_{\text{cr}}\equiv\frac{\Phi}{\Delta},$ (79)

and

$\displaystyle\bar{\Phi}\equiv\frac{\Phi}{T_{c}}.$ (80)

The solution is then ($z_{j}=T_{j}/T_{c}$, $j=1,2$)

$\displaystyle
r_{2}=\Bigg{\\{}\bigg{(}\frac{1}{r_{\text{cr}}}+\frac{1}{r_{1}}\bigg{)}\exp\bigg{[}\bar{\Phi}\bigg{(}\frac{1}{z_{2}}-\frac{1}{z_{1}}\bigg{)}\bigg{]}-\frac{1}{r_{\text{cr}}}\Bigg{\\}}^{-1}.$
(81)

A special case is when $\Delta=0,\Phi>0$. The solution in this case can be
obtained by simply letting $r_{\text{cr}}\rightarrow\infty$ in Eq. (81). The
resulting expression is simple:

$\displaystyle
r_{2}=r_{1}\exp\bigg{[}-\bar{\Phi}\bigg{(}\frac{1}{z_{2}}-\frac{1}{z_{1}}\bigg{)}\bigg{]}.$
(82)

###  3.4 Flux description of the drift modelling

In the previous subsections two ways of modelling the monopole capture by PBHs
are presented, however some steps involved may look ad hoc, or lead to
questions that remain to be explained. For example:

  1. 1.

Eq. (50) is derived in a non-diffusive context. Does it apply here and why
should we use a random walk velocity as discussed previously for $v_{M}$?

  2. 2.

In the drift modelling, the drift velocity is a function of the monopole-PBH
distance. Why is it appropriate to use $n_{\text{bh}}^{-1/3}$ as the typical
separation and determine the typical capture time $\tau_{\text{gc}}$? Does
this treatment still apply if $\tau_{\text{gc}}>H^{-1}$?

  3. 3.

Which way of modelling is appropriate?

Motivated by these questions, in this section we present a more intuitive
account of the SF modelling and the drift modelling, justifying our preference
for drift modelling in this work.

In the current discussion a monopole in the primordial plasma is subject to
three types of forces: gravitational attraction from nearby PBHs, magnetic
attraction/repulsion from nearby monopoles/antimonopoles, and frequent
collisions by surrounding electrically charged particles. To simplify
discussion we neglect the mutual influence between monopole annihilation and
capture by PBHs, that is, either process can be studied independently. Here we
focus on monopole capture by PBHs, so we drop the magnetic
attraction/repulsion from nearby monopoles/antimonopoles. In the remaining two
types of forces, gravitational attraction from nearby PBHs is given by the
usual inverse-square law, which for simplicity we consider only the effect of
the nearest PBH from the monopole under consideration. The effect of the
frequent collisions by surrounding electrically charged particles is
completely analogous to the force experienced by a particle that undergoes
Brownian motion, which can be modelled as the sum of a drag force
$\textbf{F}_{\text{drag}}$ that is approximately proportional to $-\textbf{v}$
(v is the monopole velocity) and a rapidly fluctuating force
$\textbf{F}_{\text{fl}}(t)$ Pathria:2022hda . $\textbf{F}_{\text{fl}}(t)$
averages to zero over long intervals of time.444By “long intervals of time” we
mean when compared to $\tau^{\star}$ which is the characteristic time between
two collisions with electrically charged particles. Note $\tau_{M}$ introduced
in Eq. (35) which we call the mean free time is conceptually different from
and numerically much larger than $\tau^{\star}$. For non-relativistic
monopoles, we may use Newtonian mechanics to describe its motion:

$\displaystyle
m\frac{d\textbf{v}}{dt}=-\frac{m_{\text{bh}}m}{M_{\text{Pl}}^{2}}\frac{1}{\bar{R}^{2}}\bar{\textbf{R}}+\textbf{F}_{\text{drag}}+\textbf{F}_{\text{fl}}(t),\quad\overline{\textbf{F}_{\text{fl}}(t)}=0,$
(83)

Here $\bar{\textbf{R}}$ is the distance vector that points from the PBH to the
monopole, and $\overline{\textbf{F}_{\text{fl}}(t)}$ denotes the average of
$\textbf{F}_{\text{fl}}(t)$ over long intervals of time. Then if we take the
ensemble average of Eq. (83), we obtain ($\langle\textbf{v}\rangle$ denotes
the ensemble average of v)555By “ensemble average” we are implicitly consider
a large number of systems similar to the one originally under consideration
and studying their motion in a _statistical_ sense. That is, the initial
positions and velocities of the monopoles and PBHs and the macroscopic
proerties of the primordial plasma are the same among ensemble members, but
the microscopic properties of the primordial plasma differ from member of
member.

$\displaystyle
m\frac{d\langle\textbf{v}\rangle}{dt}=-\frac{m_{\text{bh}}m}{M_{\text{Pl}}^{2}}\frac{1}{\bar{R}^{2}}\bar{\textbf{R}}-CT^{2}\langle\textbf{v}\rangle,$
(84)

For definiteness we have plugged in Eq. (32) for the drag force (setting
$f=1$), and used the fact that $\langle\textbf{F}_{\text{fl}}(t)\rangle=0$ by
the very nature of $\textbf{F}_{\text{fl}}(t)$. Regardless of the initial
velocity, eventually $\langle\textbf{v}\rangle$ will tend to its equilibrium
value, determined from the balancing between the gravitational attraction and
the drag force. Nevertheless, this does not mean the monopole velocity of all
ensemble members will eventually be the same. This is because we are
considering the monopole motion in a statistical sense, while the ensemble-
averaged equation Eq. (84) only constrains the first moment of the random
variable v. For example, a simpler case would be if there is no PBH, so the
gravitational attraction does not exist. Then the equilibrium value of
$\langle\textbf{v}\rangle$ will be zero. Of course this does not mean
monopoles in all ensemble members will sit still. In any case there is a non-
zero temperature $T$, thus in thermal equilibrium monopoles must exhibit a
root-mean-square velocity that satisfy $\langle\textbf{v}^{2}\rangle=3T/m$,
which is related to the second moment of v.

Therefore, while the evolution of the first moment of v is determined by the
competition between the gravitational attraction and the drag force,the
evolution of the second moment of v is further controlled by
$\textbf{F}_{\text{fl}}(t)$. If the gravitational attraction is negligible,
which is the case for very large monopole-PBH distance $\bar{R}$, then the
equilibrium value of $\langle\textbf{v}\rangle$ is zero. For such a monopole
with an initial thermal velocity of $v_{T}=(3T/m)^{1/2}$, the drag force will
make it forget this initial velocity in a characteristic time $\tau_{M}$ (see
Eq. (35)), however $\textbf{F}_{\text{fl}}(t)$ will boost the monopole
velocity to $\sim(3T/m)^{1/2}$ during a time interval of order $\sim\tau_{M}$
to maintain $\langle\textbf{v}^{2}\rangle=3T/m$ (though the direction of the
velocity may change randomly). It is in this sense that $\tau_{M}$ can be
viewed as the characteristic time for the monopole to change its velocity
significantly, and $\ell=v_{T}\tau_{M}$ can be viewed as the monopole mean
free path.

For small gravitational attraction, the equilibrium value of
$\langle\textbf{v}\rangle$ (the drift velocity) is small but nonzero. In such
a case, if we trace the motion of the monopole, then in short periods of time
(but still larger than $\tau_{M}$) it would be very difficult to tell the
influence of the drift velocity. That is, the motion of the monopole is
thermally dominated (i.e. dominated by $\textbf{F}_{\text{fl}}(t)$). For large
gravitational attraction, however, the monopole drift velocity is large, and
the motion of the monopole is gravitationally dominated, while the thermal
effect only leads to relatively small corrections. The boundary value of
$\bar{R}$ between small and large gravitational attraction regimes can be
estimated by requiring

$\displaystyle\frac{m_{\text{bh}}m}{M_{\text{Pl}}^{2}}\frac{1}{\bar{R}}=T,$
(85)

That is, the amount of gravitational potential energy is equal to the thermal
energy of the monopole. The solution of Eq. (85) is just the gravitational
capture radius $r_{c}^{\text{gc}}$ introduced in Eq. (52).

The gravitational capture of monopoles by PBHs can be divided into two
regimes, according to whether $r_{c}^{\text{gc}}>\ell$ or
$r_{c}^{\text{gc}}<\ell$. If $r_{c}^{\text{gc}}>\ell$, then for a monopole
satisfying $\bar{R}<r_{c}^{\text{gc}}$, its motion is dominated by the
drifting towards the PBH, and the probability for its thermal motion
(characterized by $\ell$) to successfully counteract the drifting and kick it
away from the PBH is small (since $\ell<r_{c}^{\text{gc}}$). Thus, once a
monopole is inside the region $\bar{R}<r_{c}^{\text{gc}}$, its fate is doomed.
This $r_{c}^{\text{gc}}>\ell$ regime is called diffusive capture. On the other
hand, if $r_{c}^{\text{gc}}<\ell$, then even if a monopole satisfies
$\bar{R}<r_{c}^{\text{gc}}$, there still can be a quite large probability that
its thermal motion may successfully counteract the drifting and kick the
monopole away from the PBH since $\ell>r_{c}^{\text{gc}}$. This
$r_{c}^{\text{gc}}<\ell$ regime is called non-diffusive capture. For a
monochromatic PBH mass function, which is the case studied in this work,
$r_{c}^{\text{gc}}\propto T^{-1}$ while $\ell\propto T^{-3/2}$, diffusive
capture is effective for $T>T_{\text{gc}}$, with $T_{\text{gc}}$ given in Eq.
(66). The SF paper Stojkovic:2004hz  however considers an extended PBH mass
function such that the ratio between the PBH mass and the horizon mass at any
given time is effectively a constant, leading to a situation in which the
diffusive capture starts at some high temperature and practically never ends.

Based on the above physical picture, we now demonstrate that the drift
modelling introduced in Sec. 3 can also be reformulated in a flux language. In
the diffusive capture regime, the reasonable candidate for the cross section
should be

$\displaystyle\sigma_{g\text{D}}\equiv\pi(r_{c}^{\text{gc}})^{2},$ (86)

since once a monopole is inside the region $\bar{R}<r_{c}^{\text{gc}}$, its
fate is doomed. The appropriate candidate for the flux velocity should be the
drift velocity at $\bar{R}=r_{c}^{\text{gc}}$, which is found to be

$\displaystyle v_{M\text{D}}\equiv
u_{\text{D}}(r_{c}^{\text{gc}})=\frac{m_{\text{bh}}m}{M_{\text{Pl}}^{2}}\frac{1}{CT^{2}(r_{c}^{\text{gc}})^{2}}=\frac{M_{\text{Pl}}^{2}}{Cmm_{\text{bh}}},$
(87)

Then the capture term coefficient $F$ in Eq. (64) can be expressed as, in the
flux description 666Here we neglect the potential inhomogeneity of $n_{M}$
during evolution, as in the parameter region considered in this work, one may
verify that $\frac{r_{c}^{\text{gc}}}{n_{\text{bh}}^{-1/3}}\ll 1$.

$\displaystyle
F=\sigma_{g\text{D}}v_{M\text{D}}n_{\text{bh}}=\pi\frac{n_{\text{bh}}m_{\text{bh}}m}{M_{\text{Pl}}^{2}CT^{2}},$
(88)

Apart from an $\mathcal{O}(1)$ factor, this expression coincides with Eq. (63)
found in the drift modelling derivation. Therefore the drift modelling with
the typical drift velocity $u_{\text{D}}(n_{\text{bh}}^{-1/3})$ can be
justified by the flux description with a capture cross section
$\sigma_{g\text{D}}$ and an incident monopole velocity $v_{M\text{D}}$.

In the drift modelling the typical capture time $\tau_{\text{gc}}=F^{-1}$ (see
Eq. (62)) can become larger than the Hubble time $t=\frac{1}{2H}$, raising
concerns about the self-consistency of the treatment. Nevertheless, this issue
is fixed in the flux description. In the flux description, we may define a
characteristic timescale

$\displaystyle\tau_{\text{cap}}\equiv\frac{r_{c}^{\text{gc}}}{u_{\text{D}}(r_{c}^{\text{gc}})}=\frac{Cm^{2}m_{\text{bh}}^{2}}{M_{\text{Pl}}^{4}T},$
(89)

which represents the typical time needed for a monopole to get into the PBH
once it is inside the capture radius. The analysis based on the flux
description is valid as long as $\tau_{\text{cap}}<H^{-1}$, that is

$\displaystyle\tau_{\text{cap}}H<1,$ (90)

which reads in terms of reduced variables
($x\equiv\frac{T_{c}}{M_{\text{Pl}}},y\equiv\frac{m_{\text{bh}}}{M_{\text{Pl}}},z\equiv\frac{T}{T_{c}}$)

$\displaystyle CK\delta^{2}x^{3}y^{2}z<1,$ (91)

On the other hand, the defining condition in the diffusive capture regime
$r_{c}^{\text{gc}}>\ell$ becomes

$\displaystyle C\delta^{1/2}xyz^{1/2}>1,$ (92)

The compatibility between Eq. (91) and Eq. (92) only requires

$\displaystyle K\delta x<C,$ (93)

which is always satisfied for the value of $x$ under consideration in this
work. Thus $\tau_{\text{cap}}<H^{-1}$ is always satisfied, which in turn
justifies the self-consistency of the drift modelling even if
$\tau_{\text{gc}}>H^{-1}$.

The flux description now allows a more direct comparison between the two ways
of modelling of the monopole capture by PBHs. In fact, apart from an
$\mathcal{O}(1)$ factor, the incident velocity of the drift modelling obtained
in Eq. (87) is just incident velocity $v_{M}$ used in the SF modelling, c.f.
Eq. (54). Therefore, when both ways of modelling are framed in a flux
description, we see that the same incident velocity is used. So the difference
between them essentially comes from the capture cross section. In the SF
modelling, the capture cross section is taken to be $\sigma_{g}$ (see Eq. (50)
and Eq. (55)), while in the drift modelling, the effective capture cross
section is $\sigma_{g\text{D}}$ (see Eq. (86)). Their ratio is found to be

$\displaystyle\frac{\sigma_{g}}{\sigma_{g\text{D}}}=\frac{16}{3}C^{2}\frac{m_{\text{bh}}^{2}T^{2}}{M_{\text{Pl}}^{4}}=\frac{16}{3}C^{2}x^{2}y^{2}z^{2},$
(94)

Thus for a given PBH mass, at high temperature we have
$\frac{\sigma_{g}}{\sigma_{g\text{D}}}\gg 1$, while at low temperature it is
possible to have $\frac{\sigma_{g}}{\sigma_{g\text{D}}}<1$.

From the physical picture of diffusive capture described above, it should be
clear that the appropriate capture cross section to be used is
$\sigma_{g\text{D}}$ rather than $\sigma_{g}$. The region
$\bar{R}<r_{c}^{\text{gc}}$ (corresponding to $\sigma_{g\text{D}}$) acts as an
effective extended event horizon of the PBH for the monopole, since once a
monopole gets inside, the probability for it to escape is practically small.
On the other hand, $\sigma_{g}$ is derived in a non-diffusive setting (see
Chapter 7 of Ref. Frolov:2011bhp ), by solving the geodesic trajectory of a
test massive particle in a Schwarzschild geometry and finding the critical
impact parameter that leads to gravitational capture. Such a computation does
not make sense if the motion of the test particle is also seriously affected
by collisions with other particles along the journey. In the current monopole
capture problem, this means $\sigma_{g}$ does not make sense if it exceeds
$\pi\ell^{2}$, since $\ell$ roughly defines the maximum distance of the
monopole motion that is not seriously affected by collisions with plasma
particles.777There is a similar reasoning used in determining the integration
limits in calculating the parameter $C$ that characterizes strength of the
drag force; see discussion on page 404 of Ref. Vilenkin:2000jqa . However, in
the SF modelling $\sigma_{g}>\pi(r_{c}^{\text{gc}})^{2}>\pi\ell^{2}$ in most
of the temperature range, and thus the use of $\sigma_{g}$ as the effective
capture cross section is problematic.

##  4 Magnetic charge fluctuation

There are two types of magnetic charge fluctuation regarding magnetically
charged PBHs formed in the early universe. First, even if a PBH is formed in a
magnetically neutral manner, it may capture monopoles and antimonopoles with
fluctuating numbers, resulting in some residual magnetic charge. Second, if a
PBH is formed after the symmetry breaking phase transition, it is likely that
at formation a horizon volume contains monopoles and antimonopoles with
fluctuating numbers, resulting in a net magnetic charge carried by the PBH
already at formation. We discuss these two types of magnetic charge
fluctuation in turn.

###  4.1 Magnetic charge fluctuation from monopole capture

We first estimate the total number of monopoles and antimonopoles (i.e.
regardless of the sign of the magnetic charge) captured by each PBH on
average. The drift modelling and a monochromatic PBH mass function is assumed.
To simplifying the expressions, we will use the reduced variables $z\equiv
T/T_{c}$ and

$\displaystyle y\equiv\frac{m_{\text{bh}}}{M_{\text{Pl}}}.$ (95)

The value of $r$ reduced due to gravitational capture between $T_{t}$ and
$T_{s}$ is (see Eq. (77))

$\displaystyle\kappa\equiv\Phi\int_{T_{s}}^{T_{t}}r(T)T^{-2}dT,$ (96)

here $r(T)$ is the solution of Eq. (77), keeping in mind the temperature
ranges in which monopole annihilation and/or capture by PBHs are effective. In
terms of the reduced variables $z_{s}\equiv T_{s}/T_{c},z_{t}\equiv
T_{t}/T_{c},\bar{\Phi}=\Phi/T_{c}$, $\kappa$ can be expressed as

$\displaystyle\kappa=\bar{\Phi}\int_{z_{s}}^{z_{t}}r(z)z^{-2}dz.$ (97)

The average number of monopoles (including antimonopoles) captured by each PBH
is then

$\displaystyle n_{2}=\kappa\frac{s(T_{t})}{n_{\text{bh}}(T_{t})},$ (98)

which is computed to be

$\displaystyle
n_{2}=\frac{1}{3}\bigg{(}\frac{\pi\mathcal{N}}{5}\bigg{)}^{1/2}C^{-1}\delta
y\times\int_{z_{s}}^{z_{t}}r(z)z^{-2}dz.$ (99)

The residual magnetic charge obtained by the PBH is

$\displaystyle\chi_{\text{gc}}=\chi\sqrt{n_{2}}.$ (100)

Eq. (100) is based on the assumption that for each PBH, monopoles and
antimonopoles are captured independently. This assumption breaks down when a
PBH already obtains a large residual magnetic charge of $+\chi_{\text{bh}}$,
since then it is preferable for it to capture a monopole with charge $-\chi$
instead of $+\chi$. If we require the gravitational force between it and a
monopole with charge $+\chi$ be larger than the corresponding magnetic force,
then this sets a bound on $\chi_{\text{bh}}$ Ghosh:2020tdu , which can be
expressed in terms of reduced variables as

$\displaystyle\chi_{\text{bh}}\lesssim\chi_{\text{bh}}^{\text{lim}},\quad\chi_{\text{bh}}^{\text{lim}}\equiv\delta\chi^{-1}g^{-2}xy.$
(101)

###  4.2 Magnetic charge fluctuation at PBH formation

If PBHs were formed after the production of magnetic monopoles, magnetic
charge fluctuation in a horizon volume at formation may already lead to the
formation of PBH with an initial magnetic charge.888For reviews and
discussions on PBHs and their production mechanism, see e.g. Refs.
Khlopov:2008qy ; Calmet:2015fua ; Sasaki:2018dmp ; Carr:2020gox ; Carr:2020xqk
; Villanueva-Domingo:2021spv ; Carr:2021bzv ; Escriva:2022duf ; Liu:2021svg ;
Choudhury:2013woa ; Choudhury:2023vuj ; Choudhury:2023jlt . This mechanism is
similar to the formation mechanism of primordial dark extremal black holes as
discussed in Ref. Bai:2019zcd . The only difference is that the number density
of magnetic monopoles must be obtained as a solution of the evolution equation
(77).

At $T=T_{b}<T_{c}$ the expected number of monopoles (or antimonopoles) per
horizon volume is

$\displaystyle\langle
N_{\text{col}}\rangle\simeq\frac{4\pi}{3}n_{M}(T_{b})H_{\text{form}}^{-3}.$
(102)

Using

$\displaystyle n_{M}(T_{b})=r(T_{b})s(T_{b}),\quad
H_{\text{form}}=K\frac{T_{b}^{2}}{M_{\text{Pl}}},$ (103)

$\langle N_{\text{col}}\rangle$ can be expressed as (in terms of reduced
variables)

$\displaystyle\langle
N_{\text{col}}\rangle\simeq\frac{4\pi}{3}r(z_{b})K_{2}K^{-3/2}\bigg{(}\frac{\gamma}{2}\bigg{)}^{-3/2}y^{3/2}.$
(104)

For a monochromatic PBH mass function, between $T_{c}$ and $T_{b}$ only
monopole annihilation is effective. $r(z_{b})$ is given by

$\displaystyle
r(z_{b})=\bigg{[}\frac{1}{r_{i}}+\frac{\bar{\Phi}}{r_{\text{cr}}}\bigg{(}\frac{1}{z_{b}}-1\bigg{)}\bigg{]}^{-1}.$
(105)

If the numbers of monopoles and antimonopoles follow independent Poisson
distributions, then the resulting magnetic charge distribution has a mean
value zero and a standard deviation of Bai:2019zcd

$\displaystyle\chi_{\text{col}}=\chi\sqrt{(2\langle N_{\text{col}}\rangle)},$
(106)

thus it is typical to have an initial magnetic charge of $\chi_{\text{col}}$
when the PBHs form.

We emphasize here that the above treatment of magnetic charge fluctuation at
PBH formation is oversimplified, as it neglects the correlation between
monopoles and antimonopoles, and also potential correlation between energy
density perturbation and charge asymmetry fluctuation Bai:2019zcd .

We note that magnetic charge fluctuation at PBH formation is also considered
in Ref. Araya:2020tds  in which magnetic PBHs are proposed to be responsible
for the generation of cosmic magnetic fields. The difference from our scenario
is that in Ref. Araya:2020tds  the magnetic PBHs do not evaporate
significantly prior to BBN, and thus they will neither evolve to near-extremal
magnetic black holes, nor evaporate completely. Instead, they are born as
magnetic PBHs with a tiny magnetic charge-to-mass ratio.

###  4.3 The issue of cosmological stability

The residual magnetic charge obtained by a PBH, either from its formation or
from monopole capture, may have a large impact on its fate. If a PBH carries
some magnetic charge, it is unstable against emitting magnetic monopoles via
pair creation or breaking into smaller magnetic black holes.999This is related
to the weak gravity conjecture; see e.g. Refs. Arkani-Hamed:2006emk ;
Cheung:2014vva ; Heidenreich:2019zkl ; Aharony:2021mpc ; Antipin:2021rsh . For
a near-extremal magnetic black hole with magnetic charge $\chi_{\text{bh}}$,
its lifetime can be roughly estimated as Khriplovich:2002qn ;
Khriplovich:1999gm

$\displaystyle\tau_{\text{mbh}}\sim
M_{\text{Pl}}^{-1}\exp\bigg{(}\frac{m^{2}}{M_{\text{Pl}}^{2}}\pi
g\chi_{\text{bh}}\bigg{)},$ (107)

so the requirement of cosmological stability ($\tau_{\text{bh}}\gtrsim
10^{18}\,\text{s}$) translates into

$\displaystyle\chi_{\text{bh}}\gtrsim 10^{-2}x^{-2}.$ (108)

For example, if $x=10^{-4}$, this requires $\chi_{\text{bh}}\gtrsim 10^{6}$
Maldacena:2020skw .

##  5 Preparation for the analysis

In the previous sections we described the modelling of the key physical
processes in this study, namely monopole annihilation and gravitational
capture of monopoles by PBHs. We also outline the computation of two types of
magnetic charge fluctuations. In this section we set the stage for concrete
analyses to be performed in Sec. 6.

###  5.1 Particle physics scenarios

Magnetic monopoles are generic predictions of grand unified or partially
unified gauge theories. For definiteness, in the analysis we consider
monopoles coming from a Pati-Salam gauge theory with Pati-Salam breaking scale
between $10^{10}\,\text{GeV}$ and $10^{15}\,\text{GeV}$101010For early study
of magnetic monopoles related to Pati-Salam models, see Refs. Lazarides:1980va
; Lazarides:1980cc .. The main reasons are:

  1. 1.

In contrast to grand unified theories (GUT), partially unified theories such
as the Pati-Salam extensions of the SM allows for a more flexible symmetry
breaking scale, ranging from just below the Planck scale, to as low as
$\mathcal{O}(10\,\text{TeV})$ depending on the field content Hartmann:2014fya
; DiLuzio:2020xgc ; Dolan:2020doe ; Cacciapaglia:2019dsq ;
Cacciapaglia:2020jvj .

  2. 2.

The nonobservation of tensor perturbations in the CMB requires the reheating
temperature $T_{\text{RH}}$ to satisfy $T_{\text{RH}}\lesssim
10^{16}\,\text{GeV}$, thus GUT monopoles (associated with unification scales
higher than $T_{\text{RH}}$ are likely to have already been diluted
sufficiently by inflation Lozanov:2019jxc . This is not necessarily the case
for Pati-Salam monopoles.

  3. 3.

As was mentioned in Sec. 1, we consider a symmetry breaking scale of $\gtrsim
10^{10}\,\text{GeV}$ in order to avoid complication from reproducing monopoles
from cooling of hot spots created during PBH evaporation Das:2021wei ;
He:2022wwy .

In Pati-Salam models, gauge symmetry breaking steps can be written as

$\displaystyle\quad SU(4)_{\text{PS}}\times SU(2)_{L}\times SU(2)_{R},$
$\displaystyle\rightarrow SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times
U(1)_{B-L},$ $\displaystyle\rightarrow SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y},$ $\displaystyle\rightarrow SU(3)_{c}\times U(1)_{EM}.$ (109)

Monopoles already arise in the first step of breaking as $U(1)_{B-L}$
monopoles that also carry non-Abelian magnetic charges. The $U(1)_{B-L}$
monopoles survive the next several stages of symmetry breaking and evolve into
electromagnetic magnetic monoples with also color magnetic charge. In Pati-
Salam models, the minimal magnetic monopoles carry two units of magnetic
charge, corresponding to $\chi=2$ in our notation. Their mass is

$\displaystyle m=\frac{4\pi M_{\text{PS}}}{g_{\text{PS}}},$ (110)

with $M_{\text{PS}}$ being the Pati-Salam symmetry breaking scale (which will
be identified as $T_{c}$), and $g_{\text{PS}}$ being the Pati-Salam gauge
coupling.

Thus in the early universe the Pati-Salam monopoles may correspond to $U(1)$
generators different from $U(1)_{EM}$, and also carry non-Abelian magnetic
charges that could lead to long-range forces. The monopole’s non-Abelian
magnetic charge is expected to be affected frequently by surrounding plasma
particles and the associated force is expected to be averaged to zero. This
leaves the long-range force associated with $U(1)$’s uncancelled. We will
neglect the complicated evolution of the $U(1)$ coupling due to running,
matching, and breaking and just use $g$ defined in Eq. (30) for the unit
magnetic charge, keeping in mind that $\mathcal{O}(1)$ uncertainties in the
value of $g$ is expected due to this approximation.

Though the Pati-Salam model is taken as the benchmark scenario in our study,
it is straightforward to extend the analysis to other models as one only needs
to change the relevant parameters accordingly.

###  5.2 Parameters and variables

For convenience, parameters that are relevant to the analysis are summarized
in Table 1. “Reference point value” indicates the values adopted in numerical
results, while “Floating range” indicates the range in which the parameters
are allowed to vary taking into account of uncertainties and the need to
consider alternative scenarios.111111“Floating range” is shown for
illustration purposes but are not used in numerical analysis Parameters $K$,
$K_{1}$, and $K_{2}$ frequently appear in the equations, but they are just
functions of $\mathcal{N}$, defined in Eqs. (4) and (2). Several reduced
(dimensionless) variables that appear in the analysis are

$\displaystyle r\equiv\frac{n_{M}}{s},\quad\quad z\equiv\frac{T}{T_{c}},$
(111)

and

$\displaystyle x\equiv\frac{T_{c}}{M_{\text{Pl}}},\quad
y\equiv\frac{m_{\text{bh}}}{M_{\text{Pl}}},\quad\beta,$ (112)

recalling that $\beta$ is defined in Eq. (72). Each benchmark point should be
specified by a set of $x,y,\beta$ and the initial value of $r$, which then
determines the solution of $r$ as a function of $z$ by solving Eq. (77). The
solution $r(z)$ can be employed to compute the magnetic charge fluctuation and
compare against the Parker’s bound which requires the total final yield
$r_{\text{fi}}$ to satisfy Murayama:2009nj 121212There are various versions of
the Parker’s bound based on different assumptions Turner:1982ag ;
Rephaeli:1982nv ; Adams:1993fj ; Lewis:1999zm ; Kobayashi:2022qpl , and the
bounds in fact depend on the monopole mass. These subtleties currently do not
affect the discussion in this work.

$\displaystyle r_{\text{fi}}\lesssim 10^{-26}.$ (113)

Note that $r_{\text{fi}}$ should include contributions from both the
unabsorbed magnetic monopoles, and the magnetic monopoles that are decay
products of the evaporating magnetic PBHs.

Parameter | Definition | Reference point value | Floating range
---|---|---|---
$M_{\text{Pl}}$ | Eq. (5) | $1.2\times 10^{19}\,\text{GeV}=2.2\times 10^{-5}\,\text{g}$ | NA
$\mathcal{N}$ | Eq. (2) | $100$ | $100\lesssim\mathcal{N}\lesssim 1000$
$g$ | Eq. (30) | 5.9 | $1\lesssim g\lesssim 10$
$\chi$ | Eq. (29) | 2 | $\chi=1\,\text{or}\,2$
$\delta$ | Eq. (46) | $50$ | $10\lesssim\delta\lesssim 100$
$C$ | Eq. (33) | $200$ | $100\lesssim C\lesssim 1000$
$\gamma$ | Eq. (74) | $0.2$ | $0.01\lesssim\gamma\lesssim 1$
$\varepsilon$ | Eq. (67) | 100 | $10\lesssim\varepsilon\lesssim 300$
$\nu$ | Eq. (17) | $0.5$ | $0.5\lesssim\nu\lesssim 0.8$
$p$ | Eq. (11) | $0.1$ | $0.01\lesssim p\lesssim 0.1$
Table 1:  Summary of the parameters that appear in the analysis, with their
definitions, reference point values, and floating range. In the last two rows,
$\nu$ and $p$ only affect the initial value of $r$.

###  5.3 Characteristic temperatures

Apart from the critical temperature of the symmetry breaking phase transition
$T_{c}$, in the analysis we encounter four other important characteristic
temperatures, which are

  1. 1.

$T_{b}$, the temperature of the universe when the PBHs form; see Eq. (75).

  2. 2.

$T_{\text{ann}}$, the temperature above of the universe which monopole
annihilation remains effective; see Eq. (45).

  3. 3.

$T_{\text{ev}}$, the temperature of the universe when the PBHs evaporate
completely; see Eq. (70).

  4. 4.

$T_{\text{gc}}$, the temperature of the universe when the gravitational
capture of monopoles by PBHs remains effective (assuming PBHs have not
evaporated yet); see Eq. (66).

For convenience we will often use the corresponding reduced temperatures
instead. They are defined as

$\displaystyle z_{b}\equiv\frac{T_{b}}{T_{c}},\quad
z_{\text{ann}}\equiv\frac{T_{\text{ann}}}{T_{c}},\quad
z_{\text{ev}}\equiv\frac{T_{\text{ev}}}{T_{c}},\quad
z_{\text{gc}}\equiv\frac{T_{\text{gc}}}{T_{c}}.$ (114)

Their analytic expressions in terms of the reduced variables and parameters,
and the corresponding reference point expressions are listed in Table 2.

In accordance with Eq. (71), we also introduce the corresponding reduced
temperatures

$\displaystyle
z_{s}\equiv\frac{T_{s}}{T_{c}}=\max\\{z_{\text{ev}},z_{\text{gc}}\\},\quad
z_{t}\equiv\frac{T_{t}}{T_{c}}=\min\\{1,z_{b}\\}.$ (115)

Reduced temperature | Analytic expression | Reference point expression
---|---|---
$z_{b}$ | $\big{(}\frac{\gamma}{2K}\big{)}^{1/2}x^{-1}y^{-1/2}$ | $0.078x^{-1}y^{-1/2}$
$z_{\text{ann}}$ | $\delta C^{-2}\chi^{-4}g^{-4}$ | $6.4\times 10^{-8}$
$z_{\text{ev}}$ | $(2\varepsilon K)^{-1/2}x^{-1}y^{-3/2}$ | $0.017x^{-1}y^{-3/2}$
$z_{\text{gc}}$ | $C^{-2}\delta^{-1}x^{-2}y^{-2}$ | $5\times 10^{-7}x^{-2}y^{-2}$
Table 2:  Summary of the reduced characteristic temperatures.

###  5.4 Overview of the parameter space

We now consider the requirements on the parameters $x$, $y$, and $\beta$
before solving for the $r(z)$ function. We require the Pati-Salam symmetry
breaking scale to be below the maximum reheating temperature
$T_{\text{RH}}^{\text{max}}\simeq 10^{16}\,\text{GeV}$, and not lower than
$10^{10}\,\text{GeV}$ to avoid reproducing monopoles via cooling of hot spots
created during PBH evaporation. Therefore $x$ is required to be in the range
$10^{-9}\lesssim x\lesssim 10^{-3}$.

For a fixed value of $x$, the range of $y$ is subject to the following
constraints.

  1. 1.

We require PBHs to form after inflation, that is $T_{b}\lesssim
T_{\text{RH}}^{\text{max}}\simeq 10^{16}\,\text{GeV}$. This can be translated
into the constraint

$\displaystyle
y\gtrsim\frac{\gamma}{2K}\bigg{(}\frac{M_{\text{Pl}}}{T_{\text{RH}}^{\text{max}}}\bigg{)}^{2},$
(116)

which at the reference point reads

$\displaystyle y\gtrsim 8.7\times 10^{3}.$ (117)

  2. 2.

The temperature of the universe at PBH evaporation should be higher than the
BBN temperature, that is $T_{\text{ev}}\gtrsim T_{\text{BBN}}$. This can be
translated into the constraint

$\displaystyle y\lesssim(2\varepsilon
K)^{-1/3}\bigg{(}\frac{M_{\text{Pl}}}{T_{\text{BBN}}}\bigg{)}^{2/3},$ (118)

which for a conservative value $T_{\text{BBN}}=1\,\text{MeV}$ reads at the
reference point

$\displaystyle y\lesssim 3.5\times 10^{13}.$ (119)

This corresponds to $m_{\text{bh}}\lesssim 7.7\times 10^{8}\,\text{g}$.

  3. 3.

There exists a temperature range such that gravitational capture of monopoles
by PBHs is effective. That is, $z_{s}<z_{t}$. It turns out that for the $x$
values of our interest, taking into account Eq. (117), the most stringent
bound comes from $z_{\text{gc}}<1$, leading to

$\displaystyle y>C^{-1}\delta^{-1/2}x^{-1},$ (120)

which reads at the reference point

$\displaystyle y>7.1\times 10^{-4}x^{-1}.$ (121)

We require the energy density of the PBHs be always smaller than that of
radiation. This condition only needs to be checked just prior to PBH
evaporation, which leads to the following constraint on $\beta$ for a fixed
value of $y$

$\displaystyle\beta y\lesssim(\gamma\varepsilon)^{-1/2},$ (122)

which reads at the reference point131313When the temperature drops below
$\simeq 10\,\text{MeV}$, $\mathcal{N}$ drops to about $\simeq 10$
Husdal:2016haj , one should use a more stringent constraint such as $\beta
y\lesssim 0.02$ instead. This is the case for $T_{\text{ev}}\lesssim
10\,\text{MeV}$.

$\displaystyle\beta y\lesssim 0.22.$ (123)

We summarize the constraints on $x,y,\beta$ at the reference point as follows

$\displaystyle\begin{cases}10^{-9}\lesssim x\lesssim 10^{-3},\\\
\max\big{\\{}8.7\times 10^{3},\,7.1\times 10^{-4}x^{-1}\big{\\}}\lesssim
y\lesssim 3.5\times 10^{13},\\\ \beta\lesssim 0.22y^{-1}.\end{cases}$ (124)

To ensure the self-consistency of the calculation, we also require that
monopoles do not dominate the energy density of the universe at PBH
evaporation. If for $z\simeq z_{\text{ev}}$ we have $r\simeq r_{p}$, then the
requirement implies

$\displaystyle\frac{r_{p}}{z_{\text{ev}}}\lesssim\delta^{-1},$ (125)

which will be imposed in the analysis, except when we consider magnetic charge
fluctuation at PBH formation.

###  5.5 Initial values

We consider two scenarios for the initial value of $r$, denoted $r_{i}$. The
first scenario (denoted by “Kibble”) is a strongly first-order phase
transition that makes $r_{i}$ saturate the Kibble estimate Eq. (10). The
second scenario (denoted by “KZ”) is a second-order phase transition so that
$r_{i}$ is given by the Kibble-Zurek estimate (28) at the reference point. The
initial values for representative values of $T_{c}$ are listed in Table 3 for
illustration purposes.

$T_{c}/\,\text{GeV}$ | $x$ |  $y$ lower bound |  $y$ upper bound |  $r_{i}$ (Kibble) |  $r_{i}$ (KZ)
---|---|---|---|---|---
$1.2\times 10^{15}$ | $10^{-4}$ | $8.7\times 10^{3}$ | $3.5\times 10^{13}$ | $1.3\times 10^{-10}$ | $3.4\times 10^{-5}$
$1.2\times 10^{14}$ | $10^{-5}$ | $8.7\times 10^{3}$ | $3.5\times 10^{13}$ | $1.3\times 10^{-13}$ | $3.4\times 10^{-6}$
$1.2\times 10^{13}$ | $10^{-6}$ | $8.7\times 10^{3}$ | $3.5\times 10^{13}$ | $1.3\times 10^{-16}$ | $3.4\times 10^{-7}$
$1.2\times 10^{12}$ | $10^{-7}$ | $8.7\times 10^{3}$ | $3.5\times 10^{13}$ | $1.3\times 10^{-19}$ | $3.4\times 10^{-8}$
$1.2\times 10^{11}$ | $10^{-8}$ | $7.1\times 10^{4}$ | $3.5\times 10^{13}$ | $1.3\times 10^{-22}$ | $3.4\times 10^{-9}$
$1.2\times 10^{10}$ | $10^{-9}$ | $7.1\times 10^{5}$ | $3.5\times 10^{13}$ | $1.3\times 10^{-25}$ | $3.4\times 10^{-10}$
Table 3:  Summary of the initial values.

It will be clear in Sec. 6 that in the scenarios we consider, a significant
reduction of the monopole yield due to capture by PBHs does not occur. This
implies that for Eq. (125), we may approximate

$\displaystyle r_{p}\simeq r(z_{\text{ann}})=\min\\{r_{i},r_{\star}\\}.$ (126)

Note that at the reference point

$\displaystyle r_{\star}\simeq 3.5\times 10^{-8}x,$ (127)

while the Kibble estimate reads (see Table 3)

$\displaystyle r_{i}^{\text{Kibble}}\simeq 1.3\times 10^{2}x^{3}.$ (128)

Thus if $r_{\star}>r_{i}$, we may deduce a bound on $x$

$\displaystyle x\lesssim 1.6\times 10^{-5},$ (129)

since the Kibble estimate sets a lower bound on $r_{i}$.

###  5.6 Evolution of $r$

The monopole annihilation is effective for $z\in[z_{\text{ann}},1]$,
gravitational capture of monopoles by PBHs is effective for
$z\in[z_{s},z_{t}]$. The evolution of $r$ as a function of $z$ can then be
obtained from solving Eq. (77) and switching to the reduced variables. The
solutions can be collected from Eqs. (42), (81), and (82). Previously, a few
derived parameters are employed to express the solutions:

$\displaystyle\bar{\Phi}\equiv\frac{\Phi}{T_{c}},\quad
r_{\text{cr}}\equiv\frac{\Phi}{\Delta}.$ (130)

We will also introduce

$\displaystyle\bar{\Delta}\equiv\frac{\Delta}{T_{c}}=\frac{\bar{\Phi}}{r_{\text{cr}}}.$
(131)

Using reduced variables, they can be cast into

$\displaystyle\bar{\Phi}$
$\displaystyle=\bigg{(}\frac{\gamma}{2}\bigg{)}^{1/2}K_{1}K^{-3/2}C^{-1}\delta\times\beta
y^{-1/2},$ (132) $\displaystyle r_{\text{cr}}$
$\displaystyle=\bigg{(}\frac{\gamma}{2}\bigg{)}^{1/2}K_{1}K_{2}^{-1}K^{-1/2}\chi^{-2}g^{-2}\delta\times\beta
xy^{-1/2},$ (133) $\displaystyle\bar{\Delta}$
$\displaystyle=K_{2}K^{-1}C^{-1}\chi^{2}g^{2}x^{-1}.$ (134)

At the reference point, these equations become

$\displaystyle\bar{\Phi}\simeq 0.038\beta y^{-1/2},\quad r_{\text{cr}}\simeq
0.021\beta xy^{-1/2},\quad\bar{\Delta}\simeq 1.84x^{-1}.$ (135)

The solution to Eq. (77) can be summarized as ($r_{1}\equiv
r(z_{1}),r_{2}\equiv r(z_{2})$)

$\displaystyle r_{2}$
$\displaystyle=\Bigg{\\{}\bigg{(}\frac{1}{r_{\text{cr}}}+\frac{1}{r_{1}}\bigg{)}\exp\bigg{[}\bar{\Phi}\bigg{(}\frac{1}{z_{2}}-\frac{1}{z_{1}}\bigg{)}\bigg{]}-\frac{1}{r_{\text{cr}}}\Bigg{\\}}^{-1},\quad
z_{1},z_{2}\in[z_{\text{ann}},1]\bigcap[z_{s},z_{t}],$ (136) $\displaystyle
r_{2}$
$\displaystyle=r_{1}\exp\bigg{[}-\bar{\Phi}\bigg{(}\frac{1}{z_{2}}-\frac{1}{z_{1}}\bigg{)}\bigg{]},\quad
z_{1},z_{2}\in[z_{s},z_{t}]\bigcap[0,z_{\text{ann}}],$ (137) $\displaystyle
r_{2}$
$\displaystyle=\bigg{[}\frac{1}{r_{1}}+\bar{\Delta}\bigg{(}\frac{1}{z_{2}}-\frac{1}{z_{1}}\bigg{)}\bigg{]}^{-1},\quad
z_{1},z_{2}\in[z_{\text{ann}},1]\bigcap\big{(}[0,z_{s}]\bigcup[z_{t},\infty]\big{)}.$
(138)

We assume when both $\Phi$ and $\Delta$ terms are turned off, $r$ remains
constant.

##  6 Analysis: modelling comparison and magnetic charge fluctuation

###  6.1 Comparison between two ways of modelling

In order to compare between the SF modelling and the drift modelling of the
monopole capture process, we note that in the SF modelling, the evolution of
$r$ obeys

$\displaystyle\frac{dr}{dT}=\frac{\Delta}{T^{2}}r^{2}+\Xi
r,\quad\Xi\equiv\frac{F_{\text{SF}}}{HT}.$ (139)

Here $F_{\text{SF}}$ is given by Eq. (57). In the following we concentrate on
the effect of gravitational capture by PBHs, and thus drop the $\Delta$ term
which characterizes monopole annihilation. By introducing

$\displaystyle w\equiv-\ln z,\quad\bar{\Xi}\equiv\Xi T_{c},$ (140)

Eq. (139) can be cast into

$\displaystyle-\frac{d\ln r}{dw}=\bar{\Xi}e^{-w},\quad\text{SF},$ (141)

while the corresponding equation for the drift modelling is

$\displaystyle-\frac{d\ln r}{dw}=\bar{\Phi}e^{w},\quad\text{Drift}.$ (142)

For a monochromatic PBH mass function, $\bar{\Xi}$ can be expressed via
reduced variables as

$\displaystyle\bar{\Xi}=2^{3/2}(\gamma K)^{1/2}C\delta\times\beta
x^{2}y^{3/2},$ (143)

which reads at the reference point

$\displaystyle\bar{\Xi}\simeq 5.2\times 10^{4}\times\beta x^{2}y^{3/2}.$ (144)

The right-hand side of Eqs. (141) and (142) may be regarded as the _fractional
efficiency_ of the monopole yield reduction due to capture by PBHs in two ways
of modelling, at a given temperature characterized by $w$. For a given
benchmark point (i.e. fixed $x$, $y$, and $\beta$), we see the fractional
efficiency shows different behavior as a function of $w$ in two ways of
modelling, assuming a monochromatic PBH mass function for both. For the SF
modelling, the fractional efficiency scales as $z=e^{-w}$, indicating an
exponential suppression at low temperature. Since in the SF modelling, $v_{M}$
and $\sigma_{g}$ are temperature-independent, this exponential suppression can
simply be traced to the decrease of $n_{\text{bh}}$ with respect to $T$; see
Eq. (49). For the drift modelling, the fractional efficiency scales as
$z^{-1}=e^{w}$, indicating an exponential enhancement at low temperature. This
can be traced to the fact that at low temperature, the drag force is reduced,
allowing for a larger drift velocity. This effect eventually overrides the
decrease of the efficiency due to the decrease of $n_{\text{bh}}$. The drift
modelling thus exhibits typical behavior of diffusive capture, as in the case
of monopole annihilation.

It is instructive to check the maximum fractional efficiency of the monopole
yield reduction that can be reached in two ways of modelling. Taking into
account the above-mentioned scaling behavior, and the fact that monopole
capture by PBHs is effective when $z\in[z_{s},z_{t}]$, the maximum fractional
efficiency that can be reached in the SF modelling is

$\displaystyle\bar{\Xi}z_{t}=\begin{cases}2^{3/2}(\gamma
K)^{1/2}C\delta\times\beta x^{2}y^{3/2},&\quad z_{b}\geq 1,\\\ 2\gamma
C\delta\times\beta xy,&\quad z_{b}<1,\end{cases}$ (145)

which reads at the reference point

$\displaystyle\bar{\Xi}z_{t}=\begin{cases}5.2\times 10^{4}\beta
x^{2}y^{3/2},&\quad z_{b}\geq 1,\\\ 4\times 10^{3}\beta xy,&\quad
z_{b}<1.\end{cases}$ (146)

The maximum fractional efficiency in the drift modelling is

$\displaystyle\bar{\Phi}z_{s}^{-1}=\begin{cases}\frac{1}{4}\big{(}\frac{\pi\mathcal{N}}{5}\big{)}^{1/2}(\gamma\varepsilon)^{1/2}C^{-1}\delta\times\beta
xy,&\quad z_{\text{ev}}\geq z_{\text{gc}},\\\
\frac{1}{4}\big{(}\frac{9\mathcal{N}}{20\pi}\big{)}^{1/4}\big{(}\frac{\gamma}{2}\big{)}^{1/2}C\delta^{2}\times\beta
x^{2}y^{3/2},&\quad z_{\text{ev}}<z_{\text{gc}},\end{cases}$ (147)

which reads at the reference point

$\displaystyle\bar{\Phi}z_{s}^{-1}=\begin{cases}2.2\beta xy,&\quad
z_{\text{ev}}\geq z_{\text{gc}},\\\ 7.7\times 10^{4}\beta x^{2}y^{3/2},&\quad
z_{\text{ev}}<z_{\text{gc}}.\end{cases}$ (148)

Using Eq. (122), it is possible to show that

$\displaystyle\bar{\Xi}z_{t}$ $\displaystyle\lesssim
2\gamma^{1/2}\varepsilon^{-1/2}C\delta x,$ (149)
$\displaystyle\bar{\Phi}z_{s}^{-1}$
$\displaystyle\lesssim\frac{1}{4}\bigg{(}\frac{\pi\mathcal{N}}{5}\bigg{)}^{1/2}C^{-1}\delta
x.$ (150)

which read at the reference point

$\displaystyle\bar{\Xi}z_{t}$ $\displaystyle\lesssim 9\times 10^{2}x,$ (151)
$\displaystyle\bar{\Phi}z_{s}^{-1}$ $\displaystyle\lesssim 0.5x.$ (152)

The maximum values are saturated when $\beta y\simeq 0.22$. We see that the
maximum fractional efficiency that can be achieved in the SF modelling is
three orders of magnitude larger than that of the drift modelling. This partly
explains why the monopole density is found to be efficiently reduced in the SF
paper Stojkovic:2004hz . However, for the $x$ values of our interest, the
maximum fractional efficiencies in both ways of modelling are not larger than
$1$, implying that a significant reduction of monopole yield cannot be
achieved in this setup, in both the SF modelling and the drift modelling. Thus
to further improve the monopole reduction efficiency, one should also consider
using an extended PBH mass function instead of a monochromatic one, as
proposed in the SF paper. The reason is that for a monochromatic PBH mass
function, since the universe is required to be radiation-domianted at the PBH
evaporation, the PBH mass density constraint is only saturated at the time of
PBH evaporation, while at earlier times the available room for PBH mass
density is not fully utilized. Since PBHs may evaporate, using an
appropriately chosen extended PBH mass function one may fully utilize the
available fraction of energy density to make PBHs and capture monopoles.

For illustrative purposes, let us consider the specific choice of parameters:

$\displaystyle x=10^{-6},\quad y=10^{9},\quad\beta=2\times 10^{-10},\quad
r_{i}=1.3\times 10^{-16}.$ (153)

This value of $r_{i}$ just saturates the Kibble estimate (see Table 3), and
the choice of $\beta$ almost saturates Eq. (122). The characteristic reduced
temperatures are computed to be

$\displaystyle z_{b}\simeq 2.45,\quad z_{\text{ann}}\simeq 6.4\times
10^{-8},\quad z_{\text{ev}}\simeq 5.5\times 10^{-10},\quad z_{\text{gc}}\simeq
5\times 10^{-13}.$ (154)

$z_{b}>1$ so the PBH forms before symmetry-breaking phase transition,
$z_{\text{ev}}>z_{\text{gc}}$ so $z_{\text{gc}}$ is irrelevant to the
analysis.

Figure 1: Comparison between the SF modelling and the drift modelling for the
fractional efficiency of the monopole yield reduction. Note the vertical axis
(the fractional efficiency) is shown on a logarithmic scale.

The comparison between the SF modelling and the drift modelling for the
fractional efficiency of the monopole yield reduction is displayed in Fig. 1
for this set of parameters. The vertical axis indicates the fractional
efficiency (FE). The horizontal axis is $w$ and a larger $w$ corresponds to a
lower temperature. The display is cut off at the PBH evaporation temperature.
It is clear that the drift modelling exhibits a typical behavior of diffusive
capture with higher fractional efficiency at the low-temperature end, while
the SF modelling exhibits a reversed behavior. With this set of parameters, in
both ways of modelling the fractional efficiency turns out to be much smaller
than the $-\frac{d\ln r}{dw}=1$ line (the green line in Fig. 1). This means in
both ways of modelling the monopole capture effect cannot keep up with the
cosmic expansion and thus cannot lead to a significant reduction of the
monopole yield. Note that this is derived from the assumption of a
monochromatic PBH mass function and radiation domination before PBH
evaporation. It is expected that the fractional efficiency can be enhanced
once these assumptions are dropped.

For the specific set of parameters in Eq. (153), the critical value of $z$
that divides the $\frac{\sigma_{g}}{\sigma_{g\text{D}}}>1$ and
$\frac{\sigma_{g}}{\sigma_{g\text{D}}}<1$ regimes is computed to be $z\simeq
2.2\times 10^{-6}$. The corresponding value of $w$ is $w=-\ln z\simeq 13$,
which just corresponds to the intersection between the SF line and the Drift
line in Fig. 1.141414In Fig. 1 the SF line and the Drift line intersects at
$w\simeq 14$ instead of $w\simeq 13$. This slight discrepancy can be traced to
the factor of $\pi$ in Eq. (88). From the above analysis it is clear that the
overall higher capture efficiency achieved in the SF modelling (for a
monochromatic PBH mass function) can be traced to the fact that for most of
the temperature range where the diffusive capture is effective,
$\frac{\sigma_{g}}{\sigma_{g\text{D}}}\gg 1$. Though at low temperature
$\frac{\sigma_{g}}{\sigma_{g\text{D}}}<1$ is achieved, this trend is cut off
at $T_{\text{gc}}$ where the maximum FE in the drift modelling is not as large
as that of the SF modelling.

We also note that with this set of parameters

$\displaystyle r_{\star}\simeq 3.5\times 10^{-14}>r_{i}.$ (155)

Therefore monopole annihilation cannot reduce the monopole yield below
$r_{i}$, either. Since $r_{i}$ exceeds the Parker’s bound by many orders of
magnitude, without other mechanisms the monopole problem would remain
unresolved. Nevertheless, the analysis demonstrates the differences between
two ways of modelling, from which we expect that if the same extended PBH mass
function is used, in the drift modelling it would be harder to solve the
monopole problem using PBHs compared to the SF work, and thus to what extent
the monopole problem can be eliminated via capture by PBH should be
reexamined.

###  6.2 Analysis of magnetic charge fluctuations

####  6.2.1 Monopole capture

We now consider magnetic charge fluctuation from monopole capture.
Approximation to the integral $\int_{z_{s}}^{z_{t}}r(z)z^{-2}dz$ that appears
in Eq. (99) can be made by noting that the main contribution to the integral
comes from the region where $z$ is close to $z_{s}$. In that region one may
approximate $r(z)\simeq r_{p}$ as constant, then

$\displaystyle\int_{z_{s}}^{z_{t}}r(z)z^{-2}dz\simeq r_{p}z_{s}^{-1}\lesssim
r_{p}z_{\text{ev}}^{-1}\lesssim\delta^{-1},$ (156)

where we have used Eq. (125) that comes from requiring the monopole energy
density be smaller than that of radiation at PBH evaporation. Then we arrive
at an inequality for $n_{2}$

$\displaystyle
n_{2}\lesssim\frac{1}{3}\sqrt{\frac{\pi\mathcal{N}}{5}}C^{-1}y,$ (157)

and the corresponding inequality for $\chi_{\text{gc}}$ at the reference point

$\displaystyle\chi_{\text{gc}}\lesssim 0.23y^{1/2}.$ (158)

Since $y\lesssim 3.5\times 10^{13}$ (PBHs evaporate before BBN), we find an
upper bound on $\chi_{\text{gc}}$

$\displaystyle\chi_{\text{gc}}\lesssim 1.4\times 10^{6}.$ (159)

At first sight this suggests the possibility for cosmologically long-lived
magnetic black holes if $x\gtrsim 10^{-4}$. However a closer examination
suggests that this is not possible. Basically we need to consider two cases:

  1. 1.

If $r_{i}<r_{\star}$, then $r_{p}=r_{i}$ with

$\displaystyle
n_{2}\simeq\frac{1}{3}\sqrt{\frac{\pi\mathcal{N}}{5}}C^{-1}\delta y\times
r_{i}z_{\text{ev}}^{-1}.$ (160)

By adjusting the parameters, it is possible to saturate
$r_{i}z_{\text{ev}}^{-1}\simeq\delta^{-1}$ and thus achieve
$\chi_{\text{gc}}\simeq 10^{6}$. However, from Eq. (129) we have $x\lesssim
1.6\times 10^{-5}$, indicating that $\chi_{\text{gc}}\simeq 10^{6}$ is not
large enough for the magnetic black hole to be cosmologically stable.

  2. 2.

If $r_{i}>r_{\star}$, the problem is that the constraint
$r_{p}z_{\text{ev}}^{-1}\lesssim\delta^{-1}$ becomes

$\displaystyle x^{2}y^{3/2}\lesssim 10^{4},$ (161)

at the reference point, which for $x=10^{-4}$ requires $y\lesssim 10^{8}$,
which is far from the value needed to saturate Eq. (159). Decreasing $x$
further can only worsen the result.

Thus we conclude that it is not possible to obtain a large enough magnetic
charge to make light PBHs cosmologically stable from monopole capture (at
least for parameter choices not very far from our reference points). On one
hand, this is related to the drift modelling in which the fractional
efficiency is significantly reduced compared to the SF modelling, especially
at high temperature. On the other hand, one of the main constraints is due to
the requirement that the universe be radiation-dominated up to PBH
evaporation.

####  6.2.2 Magnetic charge from PBH formation

To analyze the magnetic charge fluctuation at PBH formation, we consider Eqs.
(104) and (105), and approximate $r(z_{b})$ as

$\displaystyle r(z_{b})\simeq\min\\{r_{i},\bar{\Delta}^{-1}z_{b}\\}.$ (162)

Therefore we may consider two cases, that are $r_{i}<\bar{\Delta}^{-1}z_{b}$
and $r_{i}>\bar{\Delta}^{-1}z_{b}$, separately. In both cases we impose the
requirement $\chi_{\text{col}}\gtrsim 10^{-2}x^{-2}$ at the reference point,
and also $y\lesssim 3.5\times 10^{13}$ and $x^{2}y\gtrsim 6\times 10^{-3}$
which comes from $z_{b}<1$. It turns out that in both cases these
considerations lead to

$\displaystyle x\gtrsim 1.8\times 10^{-5},$ (163)

and it is possible to have a large $\chi_{\text{col}}$ to obtain a
cosmologically stable extremal magnetic black hole. For example, a benchmark
choice can be

$\displaystyle x=10^{-4},\quad y=3.5\times 10^{13},\quad\beta=5.7\times
10^{-16},\quad r_{i}=6.7\times 10^{-9},$ (164)

and it turns out

$\displaystyle\chi_{\text{col}}\simeq 2.2\times 10^{7},$ (165)

which is sufficiently large for $x=10^{-4}$. Since here we are interested in
the magnetic charge obtained at PBH formation, we do not require the monopole
energy density to be constrained at the time of PBH evaporation in this
analysis.

##  7 Discussion and conclusions

###  7.1 Discussion: Non-diffusive monopole capture?

In the case of monopole annihilation, it is possible to have non-diffusive
capture when $\ell>r_{c}^{\text{ann}}$, though its contribution to the
reduction of monopole abundance is found to be much smaller than that of
diffusive capture. When it comes to monopole capture by PBHs, is it possible
to have non-diffusive capture as well and is its contribution negligible? The
initial expectation is that once the temperature drops below $T_{\text{gc}}$,
then $r_{c}^{\text{gc}}<\ell$, then the effective capture cross section in the
non-diffusive regime should be given by
$\sigma_{g\text{ND}}\equiv\min\\{\sigma_{g},\pi\ell^{2}\\}$. However, there
are other constraints that have to be taken into account. First, we should
require $z_{\text{ev}}<z_{\text{gc}}$, that is, the PBH evaporation should
occur after the monopole capture enters the non-diffusive regime. Otherwise
the PBH evaporation occurs first and the discussion of non-diffusive capture
makes no sense. Using expressions in Table 2 we deduce that
$z_{\text{ev}}<z_{\text{gc}}$ is equivalent to

$\displaystyle(2\varepsilon K)^{-1/2}C^{2}\delta xy^{1/2}<1,$ (166)

which at the reference point reads

$\displaystyle 3.5\times 10^{4}xy^{1/2}<1,$ (167)

On the other hand, the monopole is an extended object with the characteristic
length scale of the classical field configuration being
$r_{\text{cl}}\sim(\alpha m)^{-1}$. If this length scale is larger than the
Schwarzschild radius $R_{\text{bh}}$ of the PBH, then the monopole cannot be
viewed as a point particle to be captured. Requiring
$R_{\text{bh}}>r_{\text{cl}}$ leads to

$\displaystyle\frac{2\alpha m_{\text{bh}}m}{M_{\text{Pl}}^{2}}>1,$ (168)

which is approximately equivalent to

$\displaystyle xy>1,$ (169)

Now let us try to use Eq. (169) in Eq. (167). The left-hand side of Eq. (167)
can be written as

$\displaystyle 3.5\times 10^{4}xy^{1/2}=3.5\times
10^{4}x^{1/2}\times(xy)^{1/2}>3.5\times 10^{4}x^{1/2},$ (170)

where in the second step we used Eq. (169). However, in this work we restrict
ourselves to $x\gtrsim 10^{-9}$ (see Eq. (124)), thus Eq. (170) leads to

$\displaystyle 3.5\times 10^{4}xy^{1/2}>3.5\times 10^{4}x^{1/2}>1,$ (171)

which contradicts the constraint Eq. (167). Therefore, within the parameter
range discussed in this work, to avoid the complication due to the extended
field profile of the monopole, we do not consider the non-diffusive capture
regime.

Let us comment that besides the size of the classical field profile
$r_{\text{cl}}$, there are two other length scales one might want to compare
with $R_{\text{bh}}$. One is the Compton wavelength of the monopole, which is
$m^{-1}$. However, $m^{-1}>R_{\text{bh}}$ does not imply the monopole cannot
be captured by PBH. A massless photon by definition has infinite Compton
wavelength but can nevertheless be captured by a black hole. The proper
wavelength to be used should be the de Broglie wavelength for a massive
particle (and the optical wavelength for a massless photon) Hawking:1971ei .
If we use the thermal de Broglie wavelength
$\lambda_{\text{TdB}}\sim(mT)^{-1/2}$ for the monopole, gravitational capture
by PBH requires $R_{\text{bh}}>\lambda_{\text{TdB}}$, which at the reference
point reads $xyz^{1/2}\gtrsim 0.1$, which can be easily violated by our choice
of parameters, such as the specific choice in Eq. (153). Nevertheless, there
is an additional twist. As the monopole gets closer to the PBH, its drift
velocity increases dramatically and the corresponding de Broglie wavelength
becomes much smaller than $\lambda_{\text{TdB}}$, which can be confirmed by
computing $u_{\text{D}}(\lambda_{\text{TdB}})$. It turns out that no matter
$u_{\text{D}}(\lambda_{\text{TdB}})$ is relativistic or not, the corresponding
de Broglie wavelenth of the monopole at $\bar{R}=\lambda_{\text{TdB}}$ is
smaller than $R_{\text{bh}}$ and we thus expect the monopole can be captured
like a point particle by the PBH.

###  7.2 Conclusions

Both magnetic monopoles and black holes are objects with fascinating
theoretical properties, and it is interesting to ask whether they met before
in cosmological history, leading to mechanisms that solve the monopole problem
and produce magnetic black holes. In this regard we revisited the black holes
solution to the monopole problem proposed by Stojkovic and Freese
Stojkovic:2004hz . We propose to model monopole capture by PBHs in the same
manner as the modelling of monopole annihilation, which exhibits a typical
behavior of diffusive capture. Our drift modelling is compared to the SF one
in the case of a monochromatic PBH mass function, and we find a monopole
capture efficiency significantly less than that of SF modelling. We present an
intuitive comparison between the two ways of modelling based on the flux
description and justify our preference for the drift modelling with an
appropriate choice of capture cross section in the diffusive regime. Our
result calls for a reanalysis of this black hole solution to the monopole
problem by using an appropriately extended PBH mass function with the drift
modelling, which we left for future study.

We have also investigated two types of magnetic charge fluctuation: from PBH
formation, or from monopole capture. We find that if the magnetic charge is
acquired at PBH formation, it is possible to make it sufficiently large such
that the resulting extremal magnetic black hole is cosmologically stable.
However, if the magnetic charge is acquired from monopole capture alone, it is
not possible to have a sufficiently large residual magnetic charge to make a
cosmologically stable extremal magnetic black hole, due to the assumption that
the universe is radiation-dominated before PBH evaporation.

The analysis done in this work can be extended or refined in a number of
directions. Besides using an extended PBH mass function as mentioned above,
there are a number of issues that are not clear at the moment. For example, in
the current modelling the motion of PBHs are neglected, the correlations
between monopoles and antimonopoles are ignored, etc. Also the study is
restricted to the case of a radiation-dominated universe and relatively high
symmetry breaking scales ($\gtrsim\mathcal{O}(10^{10}\,\text{GeV})$). These
aspects are worth further explorations and being checked against numerical
simulations which are needed to determine the prospects of producing long-
lived magnetic black holes and resolving the accompanying monopole problem.

###### Acknowledgements.

Chen Zhang would like to thank Yi-Lei Tang for helpful discussion. The authors
are grateful to the anonymous referee whose report motivates an extended
discussion about the difference between the two ways of modelling and related
issues which is presented in sections 3.4 and 7.1. This work was supported by
the National Natural Science Foundation of China (Grants Nos. 11975072 and
11835009) and the National SKA Program of China (Grants Nos. 2022SKA0110200
and 2022SKA0110203).

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