# Primordial magnetic relics and their signatures

Arka Banerjee  Lalit Singh Bhandari  Ashwat Jain  and Arun M. Thalapillil

###### Abstract

Primordial black holes bearing magnetic charges may bypass the constraints
imposed by Hawking radiation, thereby enabling reasonable present-day
populations, even for masses below $10^{15}\,\mathrm{g}$—a range previously
considered improbable. They could, therefore, conceivably contribute to a
component of dark matter. We investigate novel Faraday rotation signatures
exhibited by primordial magnetic black holes while also establishing new
Parker-type bounds on their populations. For the latter, we bound the dark
matter fraction from intergalactic magnetic fields in cosmic voids
$\left(f_{\text{\tiny DM}}\lesssim 10^{-8}\right)$ and cosmic web filaments
$\left(f_{\text{\tiny DM}}\lesssim 10^{-4}\right)$, notably eclipsing previous
bounds. Exploring Faraday rotation effects, we discern a pronounced rotation
of the polarization angle and the rotation measure values for extremal
primordial magnetic black holes with masses $M^{\text{\tiny ex.}}_{\text{\tiny
BH}}\gtrsim 10^{-6}~{}\textup{M}_{\odot}$. This makes them potentially
detectable in current observations. A comparative investigation finds that the
effects are notably greater than for a neutron star, like a Magnetar, with a
similar magnetic field at the surface. Moreover, the polarization angle maps
for primordial magnetic black holes exhibit unique features, notably absent in
other astrophysical magnetic configurations. In this context, we also
introduce a simple integral measure, offering a quantitative measure for their
discrimination in many scenarios. These traits potentially suggest a robust
avenue for their observational detection and differentiation.

##  1 Introduction

The nature of dark matter is among the most intriguing problems in physics
today (see, for instance, [1, 2, 3, 4, 5, 6] and references therein). There
have been numerous investigations using a variety of different methods which
have strongly supported its existence—for example, galactic rotation curves
[1], observations of the bullet cluster [7], power spectrum of cosmic
microwave background (CMB) [8], gravitational lensing [9], and the large-scale
structure of the universe [10]. However, despite the extensive amount of
observational data, the microphysical nature of dark matter remains elusive. A
wide array of candidates for dark matter have been proposed, ranging from
weakly interacting massive particles (WIMPs) [11], axions [12], fuzzy dark
matter [13], hidden sector particles [14], to primordial black holes (PBHs)
[15, 16, 17, 18, 19, 20], to name a few.

In recent times, the last of these proposed candidates—PBHs—have been of
significant interest [15, 21, 22, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 20, 36, 37, 38, 39, 40, 41]. There have been keen
efforts to investigate various aspects of their formation [15, 21, 22, 19],
mass distribution [17, 19], signatures from gravitational waves [25, 31, 33],
gravitational lensing [27, 28, 29, 30], and other pertinent phenomena [37,
39]. In the early universe, their formation may have been a consequence of the
gravitational collapse of large overdensities [16, 21, 22] emerging during
inflation, and their population may have subsequently dwindled due to
evaporation by Hawking radiation [42]. This is especially true for the smaller
mass PBHs since the Hawking temperature (for Schwarzschild black holes) is
inversely proportional to the mass. Consequently, smaller mass Schwarzschild
PBHs with masses $M_{\text{\tiny BH}}\lesssim 10^{15}\,\text{g or
}10^{-18}\,\textup{M}_{\odot}$ [43] would have mostly evaporated by now (see,
for instance, [23] and references therein).

Interestingly, however, this terminal evaporation can be evaded if the PBHs
are magnetically charged. If so, their Hawking temperature is reduced, and as
evaporation continues, the Hawking temperature ultimately becomes zero for an
extremal magnetically charged black hole, for which the mass equals the
magnetic charge. For instance, in a scenario where magnetic monopoles existed
in the early universe, the absorption of $N$ monopoles by the PBHs could
statistically result in the accumulation of monopole charges proportional to
$\sqrt{N}$ [35, 44, 45, 38], leading to the formation of magnetically charged
PBHs or Magnetic Black Holes (MBHs). Following absorption, they persist while
emitting Hawking radiation, consequently losing mass and gradually approaching
the extremal limit, eventually leading to zero Hawking temperature. This leads
to a cessation of radiation, and hence even Schwarzschild PBHs with masses
$M_{\text{\tiny BH}}\lesssim 10^{15}\,\text{g or
}10^{-18}\,\textup{M}_{\odot}$ can potentially survive, allowing for a
relatively significant population of them to persist until the current epoch.

Although a similar evolution is theoretically possible for electrically
charged PBHs, electric charges are more readily neutralised in astrophysical
environments, making them less relevant. Additionally, rapidly spinning PBHs
may result in extremal Kerr black holes having spin $J=GM_{\text{\tiny
BH}}^{2}/c$, which also have vanishing Hawking temperatures [46, 47].
Nevertheless, angular momentum is rapidly degraded in typical astrophysical
environments, making these scenarios less potent compared to MBHs. We will,
therefore, primarily focus our study on non-spinning MBHs. Note also in
passing that non-extremal MBHs having masses $M_{\text{\tiny BH}}\gtrsim
10^{15}\,\text{g}$ are metastable anyway due to their low Hawking temperatures
compared to smaller mass MBHs. Consequently, they will avoid terminal Hawking
evaporation even without achieving the extremal limit, allowing them to
persist.

MBHs have garnered significant theoretical interest in the past [48, 49, 50,
51]. In more recent years, there has also been a growing interest in their
phenomenological aspects [38, 52, 53, 54, 55, 40, 41, 56, 57, 58, 59, 60, 61,
45, 62]. These include, for instance, investigations into gravitational and
electromagnetic radiation arising from a binary system of dyonic black holes
[56, 57, 58], as well as gravitational waves from NS and MBH mergers [59].
Studies have also attempted to discern between MBHs and their electric
counterparts by studying their quasinormal modes [60] and considerations of
the motion of charged test particles surrounding them [61]. Other studies have
looked into the restoration of electroweak symmetry [63, 64, 65] around MBHs,
leading to the formation of an electroweak corona near them [45], and its
phenomenological implications [53]. Accretion of charged particles by rotating
MBHs has been another aspect of interest [62], along with explorations on the
implications of a topologically induced black hole electric charge [55], dark
extremal PBHs [66], and solutions for MBHs in nonlinear electrodynamics—for
instance, with a focus on their implications for black hole shadows [67, 68,
69], gravitational lensing [68], and MBH quasinormal modes [70].

As MBHs carry monopole charges, they simultaneously serve as sources of
extreme gravitational fields and unique monopolar magnetic fields, potentially
giving rise to uncommon astrophysical effects. For instance, interactions with
magnetic fields in galaxies, galaxy clusters, and cosmic regions may result in
the acceleration of MBHs. This, in turn, could lead to a significant depletion
of these magnetic fields. It is then evident that the survival of these fields
until the present epoch may, in many cases, put a bound on the population of
the MBHs. Parker [71, 72] originally introduced this idea in the context of
pure magnetic monopoles, thereby establishing limits on the flux of such
monopoles based on the survival of galactic magnetic fields. These constraints
for pure magnetic monopoles have since been expanded through assessments of
the survival of galactic seed fields [73, 74], intracluster magnetic fields
[75], and primordial magnetic fields [76, 77]. Recently, [78] has investigated
the acceleration of magnetic monopoles to large velocities by intergalactic
magnetic fields. These, in turn, lead to modification of the galactic Parker
bounds on the flux of non-virialized, pure magnetic monopoles [78].
Additionally, in the mass range $M\lesssim\mathcal{O}(10^{19})\text{ GeV}$,
pure monopole fluxes have been estimated for which the inter-galactic fields
may be significantly affected [78].

For MBHs with masses $M_{\text{\tiny BH}}\gtrsim
10^{-6}\,\text{Kg}\sim\mathcal{O}(10^{20})\,\text{GeV}$ [16, 23], constraints
on their flux and dark matter fraction, specifically from intergalactic
magnetic fields (IGMFs) in cosmic voids and cosmic web filaments, are as yet
unexplored. We explore new bounds coming from these systems in this study.
These bounds for extremal MBHs also differ from those applicable to pure
monopoles due to the distinctive relationship between charge and mass (i.e.
$Q_{\text{\tiny BH}}=\sqrt{4\pi G/\mu_{0}}M_{\text{\tiny BH}})$. In addition,
IGMFs may have different generation mechanisms, for instance, galactic flux
leakage [79] or primordial magnetogenesis [80], which can result in
distinctive magnetic field characteristics and, therefore, different final
bounds. It is worth mentioning that bounds on the flux of MBHs based on the
survival of galactic magnetic fields and primordial magnetic fields during
radiation-dominated and reheating eras have already been explored [77].
Furthermore, [54, 53] have placed galactic Parker bounds, based on the field
in the Andromeda galaxy, on the fraction of dark matter in the form of
MBHs—$f_{\text{\tiny DM}}\lesssim\mathcal{O}(10^{-3})$. As mentioned, among
the aims of this work is to study and derive novel bounds on the flux and the
dark matter fraction in the form of MBHs. Specifically, we will apply
considerations of the survival of IGMFs in cosmic web filaments and cosmic
voids with different generation mechanisms to set strong bounds. For low mass
MBHs with $M_{\text{\tiny BH}}\lesssim 10^{15}\,\text{g}$, we will put Parker-
type bounds only for extremal MBHs. However, as discussed, for masses
$M_{\text{\tiny BH}}\gtrsim 10^{15}\,\text{g}$, we also present bounds
applicable to non-extremal MBHs along with extremal MBHs.

Apart from the Parker bounds on MBHs, we are also interested in exploring the
effectss of MBHs on polarized radio sources—for instance, synchrotron
emissions from pulsars or galaxies. The basic idea is that the plane of
polarization of linearly polarized electromagnetic waves rotate when they
travel through an ionic medium in the presence of a magnetic field. This
birefringence phenomenon is known as Faraday rotation or the Faraday effect.
Observations of Faraday rotation from point sources have been widely employed
to probe the structure of magnetic fields along the line of sight [81, 82, 83,
84]. Recently, few works have investigated the intrinsic Faraday effect in
pulsar radio emissions due to neutron star (NS) magnetospheres [85] and also
explored the quantum electrodynamic effects on birefringence in the
intermediate frequency regime [86], in similar contexts. In other related
works, investigations have also been conducted on the birefringence effects of
axion-like particles in NSs [87, 88]. Of particular relevance to us are also
works exploring the characteristics of the interstellar medium—through
observations of diffuse polarized emissions and their Faraday rotation
imprints [89, 90, 91, 92]. We will specifically leverage similar aspects to
scrutinize the presence of MBHs.

In this work, we will first explore Parker-type bounds on extremal MBHs. We
will estimate bounds on their flux and fraction as a dark matter component
using IGMFs in cosmic voids (Eqs. (2.46) and (2.48)) and cosmic web filaments
(Eq. (2.49)). Our findings indicate that these constraints are considerably
more stringent than those derived from galactic magnetic fields alone [76, 53,
54]. Then, we will investigate in detail the unique Faraday rotation effects
induced by extremal MBHs. Assuming a semi-realistic constant plasma density
profile and a more realistic galactic plasma density profile, we compute the
change in polarization angle and rotation measure (RM) values for a linearly
polarized wave due to an extremal MBH located within the Milky Way. We then do
a comparative analysis of the change in polarization angle and RM value due to
an MBH relative to an NS. Considering comparable magnetic field strengths at
the surfaces of an NS and the outer horizon of an MBH, we find that the values
obtained for an MBH are markedly greater in magnitude compared to those for an
NS with this matched characteristic. Finally, we also find a simple
quantitative measure to globally distinguish between the polarization angle
maps of an MBH and an NS in many scenarios. Apart from the relevant
quantitative expressions, some of the main results of the study are shown in
Figs. 9-13 and Tab. 1. Throughout, we have calculated the RM and change in
polarisation angle due to an MBH, and compared it with a matched NS. We find
that they are plausibly sufficient for current observations to detect for the
extremal MBH magnetic charges $Q^{\text{\tiny ex.}}_{\text{\tiny BH}}\gtrsim
10^{22}~{}\text{A-m}$, equivalently, $M^{\text{\tiny ex.}}_{\text{\tiny
BH}}\gtrsim 10^{-6}~{}\textup{M}_{\odot}$. The MBH effects could be
$\mathcal{O}(10^{8})$ times larger than for a similar, matched NS. We also
point out that due to the uncommon monopolar nature of the MBH’s magnetic
field, other astrophysical magnetic configurations, which are always non-
monopolar in nature, will not be able to mimic these MBH signatures very
readily.

The paper is organized as follows. In Sec. 2, we concisely revisit the Parker
bound calculations and then adapt these calculations to comprehensively
estimate limits on the MBH dark matter fraction from the IGMFs in the cosmic
voids and the cosmic web filaments. In Sec. 3, we then briefly present the
standard Faraday rotation calculation in the presence of a non-uniform
magnetic field and then employ these expressions to carefully investigate the
unique Faraday rotation effects caused by an MBH. In this context, we also
compare these effects with those produced by an NS. Our main findings and
conclusions are finally summarised in Sec. 4.

##  2 Parker-type bounds on primordial magnetic black holes

First, let us explore Parker-type bounds on MBHs due to galactic and galaxy
cluster magnetic fields, and IGMFs in cosmic voids and cosmic web filaments.
For our purposes, these magnetic fields may be characterized mainly by three
parameters—the field strength $B$, the regeneration time $t_{\text{\tiny
reg}}$ (the time scale over which the field may be regenerated), and the
coherence length $l_{\text{\tiny c}}$ (the length scale over which the field
remains relatively constant). It is useful to initially examine the
characteristics of these field parameters, as they will prove instrumental in
forthcoming subsections when we calculate the limits.

###  2.1 Characteristics of astrophysical and cosmic magnetic fields

Let us start by examining the magnetic fields in galaxies and galaxy clusters.
The galactic magnetic fields in many cases exhibit equipartition between their
mean and turbulent components, with each having field strength
$B\sim\mathcal{O}(10)\,\mu\text{G}\,$ (see, for instance, [93, 94, 95, 96] and
references therein). This equipartition is a common characteristic of fields
generated through turbulent dynamo processes [97, 98]. As anticipated, the
mean component of the field demonstrates greater coherence, with a coherence
length of approximately $\mathcal{O}(1)\,\text{kpc}$, while the turbulent
component exhibits coherence on scales roughly 10 times smaller [93, 94]. As
turbulent motions are the driving force behind the dynamo process, the latter
can be regenerated swiftly in around $\mathcal{O}(10^{-1})\,\text{Gyr}$ [93,
94], whereas the mean field undergoes regeneration only on scales of
approximately $\mathcal{O}(1)\,\text{Gyr}$ [93, 94]. We therefore broadly
characterise the conventional features of the mean and turbulent components of
galactic magnetic fields [93, 94] as

$\displaystyle B_{\text{\tiny mean}}^{\text{\tiny
gal}}\sim\mathcal{O}(10)\,\mu\text{G}\,,\quad l_{\text{\tiny c,
mean}}^{\text{\tiny gal}}\sim\mathcal{O}(1)\,\text{kpc}\,,$
$\displaystyle\quad t_{\text{\tiny reg, mean}}^{\text{\tiny
gal}}\sim\mathcal{O}(1)\,\text{Gyr}\;,$ $\displaystyle B_{\text{\tiny
turb}}^{\text{\tiny gal}}\sim\mathcal{O}(10)\,\mu\text{G}\,,\quad
l_{\text{\tiny c, turb}}^{\text{\tiny
gal}}\sim\mathcal{O}(10^{-1})\,\text{kpc}\,,$ $\displaystyle\quad
t_{\text{\tiny reg, turb}}^{\text{\tiny
gal}}\sim\mathcal{O}(10^{-1})\,\text{Gyr}\;.$ (2.1)

In galaxy clusters, the magnetic fields still maintain equipartition with
field strengths around $\mathcal{O}(1)\,\mu\text{G}$ (see [93, 99, 96] and
references therein). However, the fields exhibit greater coherence in most
cases, with the coherence lengths of the mean fields expected to be around
$l_{\text{\tiny c, mean}}^{\text{\tiny clus/iso}}\sim\mathcal{O}(10)\,$ kpc
and that of the turbulent fields around $l_{\text{\tiny c, turb}}^{\text{\tiny
clus/iso}}\sim\mathcal{O}(1)\,$ kpc [93, 99]. Nevertheless, the regeneration
time for galaxy cluster fields are expected to be comparable to those observed
in galactic fields—$t_{\text{\tiny reg, mean}}^{\text{\tiny
clus/iso}}\simeq\frac{1}{3}t_{\text{\tiny reg, turb}}^{\text{\tiny
clus/iso}}\sim\mathcal{O}(10^{-1})\,$ Gyr [93, 99].

In addition to magnetic fields within galaxies and galaxy clusters, there are
also cosmic magnetic fields. These fields [96, 100, 101, 102, 103, 104, 105,
79, 106, 107, 108, 109, 110, 111]—traditionally referred to as intergalactic
magnetic fields (IGMFs), represent magnetic fields that pervade large-scale
structures in the universe; manifesting themselves, for instance, in the voids
and filaments of the cosmic web. IGMFs have been an area of strong interest
due to their elusive nature and the experimental difficulties in observing
such low-strength fields. Recently, the Fermi-LAT/H.E.S.S collaboration [104]
have explored the strength of IGMFs through the electromagnetic cascades
induced by Blazar-emitted gamma rays. They have established a conservative
lower limit of approximately $\mathcal{O}(10^{-15})\,\text{G}$ [104] on the
strength of IGMFs for coherence lengths
$l_{\text{\tiny{c}}}\gtrsim\mathcal{O}(1)\,\text{Mpc}$. This is further
supported by earlier works [103, 109, 107, 112, 100], which bound the IGMF
strengths to $B\gtrsim\mathcal{O}(10^{-15})-\mathcal{O}(10^{-16})\,\text{G}$
for coherence length $l_{\text{\tiny{c}}}\gtrsim\mathcal{O}(1)\,\text{Mpc}$.
Moreover, based on helicity considerations of cosmic magnetic fields (drawing
from Fermi satellite gamma-ray observations), the IGMF has been estimated to
have a strength of approximately $B\gtrsim\mathcal{O}(10^{-14})\,$G [108,
106], for larger coherence lengths of $\mathcal{O}(10)\,$Mpc. Simultaneously,
MHD arguments assuming a disordered cosmic magnetic field at recombination
have established upper limits of $B\lesssim\mathcal{O}(10^{-9})\,\text{G}$
[113].

The origins of these IGMFs is still not understood fully, and there are
several proposed mechanisms. One of these posits that IGMFs are produced
primordially, arising in the early universe (see, for example, [102, 96] and
references therein). In this case, it is anticipated that the IGMFs would
possess a strength of $\mathcal{O}(10^{-15})\,G\lesssim
B\lesssim\mathcal{O}(10^{-9})\,\text{G}$ [104, 111], exhibiting coherence on
scales $\mathcal{O}(1)\,\text{Mpc}$ or greater [104, 111]. Alternatively, [79]
has suggested that seeding by galactic fields via flux leakage, may lead to
field strengths of $\mathcal{O}(10^{-12})\,\mathrm{G}\lesssim
B\lesssim\mathcal{O}(10^{-8})\,\mathrm{G}$ [79] over a time scale of
$\mathcal{O}(10)$ Gyr [79]. If the field is indeed primordial and lacks a
mechanism to amplify the strength (for example, a mean-field dynamo), the
regeneration time must be at least as long as the Hubble time
$\mathcal{O}(10)\,\text{Gyr}$ for the field to persist until the present day.

It has also been speculated that the source of IGMFs in cosmic voids may be
due to magnetization by cosmic rays from void galaxies [114]. These cosmic
rays, generated and accelerated by supernova events within galaxies residing
in voids, are energetic enough to surpass the galaxies’ virial velocities and
escape into the surrounding void. As they traverse the void, these cosmic rays
carry electric currents, contributing to the magnetization of the void region
at an approximate rate of about $10^{-16}\,$G/Gyr [114, 115]. Remarkably, the
estimate in [114] aligns with the recent observations by Fermi-LAT/H.E.S.S
collaboration [104], suggesting strongly the presence of magnetic field
strengths of the order of $10^{-15}\,$G in cosmic voids.

IGMFs in cosmic web filaments have also been inferred recently through
rotational measure maps [116, 117, 118, 119] and synchrotron radio emission
studies [120, 121] of extragalactic sources. These magnetic fields have
typical strengths of $\mathcal{O}(10^{-9})\,\text{G}$ [116, 117, 118, 119,
120, 121] with coherence lengths $\mathcal{O}(1)\,\text{Mpc}$ [116, 117, 118,
119]. This seems to be in accordance with the expected IGMF characteristics as
discussed earlier. Again, the actual origin of such magnetic field strengths
is unclear at the moment, but the amplification of a seed primordial field by
a dynamo mechanism and galactic outflows are two of the plausible
explanations. Conjecturing such origins, we therefore assume the regeneration
time to be $\mathcal{O}(10)\,\text{Gyr}$.

Based on the above discussions, for our subsequent analyses, we classify the
IGMFs depending on their location in the cosmic regions—IGMFs in cosmic web
filaments and those in cosmic voids. We note that independent of the origin of
the IGMFs in cosmic web filaments, one can conservatively assume the following
field characteristics [116, 117, 118, 119, 120, 121]

$B^{\text{\tiny fil}}\sim\mathcal{O}(10^{-9})\,\text{G}\,,\quad l_{\text{\tiny
c}}^{\text{\tiny fil}}\sim\mathcal{O}(1)\,\text{Mpc}\,,\quad t_{\text{\tiny
reg}}^{\text{\tiny fil}}\sim\mathcal{O}(10)\,\text{Gyr}\;.$ (2.2)

The origin of IGMFs in cosmic voids may be due to primordial fields, galactic
outflows, or magnetization by void galaxies. Unlike their filamentary
counterparts, the characteristics of the magnetic fields in cosmic voids are
thought to be more dependent upon their actual origins. For the case when the
magnetic fields are produced due to galactic outflows [79], we have

$\mathcal{O}(10^{-12})\,\text{G}\lesssim B^{\text{\tiny void,
outflow}}\lesssim\mathcal{O}(10^{-8})\,\text{G}\,,\quad l_{\text{\tiny
c}}^{\text{\tiny void, outflow}}\sim\mathcal{O}(1)\,\text{Mpc}\,,\quad
t_{\text{\tiny reg}}^{\text{\tiny void,
outflow}}\sim\mathcal{O}(10)\,\text{Gyr}\;.$ (2.3)

However, for the fields with a primordial origin, we have typically [104]

$B^{\text{\tiny void, prim}}\gtrsim\mathcal{O}(10^{-15})\,\text{G}\,,\quad
l_{\text{\tiny c}}^{\text{\tiny void,
prim}}\sim\mathcal{O}(1-10)\,\text{Mpc}\,,\quad t_{\text{\tiny
reg}}^{\text{\tiny void, prim}}\sim\mathcal{O}(10)\,\text{Gyr}\;.$ (2.4)

Finally, for fields generated via magnetization by void galaxies, we have
[114]

$B^{\text{\tiny void, gal}}\gtrsim\mathcal{O}(10^{-15})\,\text{G}\,,\quad
l_{\text{\tiny c}}^{\text{\tiny void,
gal}}\sim\mathcal{O}(1)\,\text{Mpc}\,,\quad t_{\text{\tiny reg}}^{\text{\tiny
void, gal}}\sim\mathcal{O}(10)\,\text{Gyr}\;.$ (2.5)

Here, we have estimated the regeneration time $t_{\text{\tiny
reg}}^{\text{\tiny void, gal}}$ using the field strength $B^{\text{\tiny void,
gal}}$ and assuming a magnetization rate $10^{-16}\,$G/Gyr [114, 115].

Understanding these magnetic field characteristics is crucial in estimating
the Parker-type bounds on the flux $F_{\text{\tiny BH}}$ and fraction
$f_{\text{\tiny DM}}$ of Dark Matter that is made up of MBHs. The original
idea [71] was that magnetic monopoles traversing galactic magnetic fields
would undergo acceleration and deplete the field’s energy. To ensure the
field’s persistence until the present era, it must be getting regenerated at a
rate at least equal to the depletion rate. Considering these two effects,
therefore, provides a constraint on the allowed monopole fluxes. We extend
this idea in the next section to scenarios which include large-scale cosmic
magnetic fields.

###  2.2 Theoretical framework for Parker-type bounds

Having discussed the characteristics of galactic, galaxy cluster, and cosmic
magnetic fields, let us now derive the theoretical expressions for the
constraints on MBH flux and MBH dark matter fraction from magnetic field
survival considerations. We will suitably adapt methodologies from [71, 72] to
the cases of interest.

Let us consider an MBH with mass $M_{\text{\tiny BH}}$ and magnetic charge
$Q_{\text{\tiny BH}}$, moving with a non-relativistic velocity $\vec{v}$
through a magnetic field of strength $\vec{B}_{\text{\tiny c}}$ and coherence
length $l_{\text{\tiny c}}$. The MBH will experience an acceleration

$\frac{d\vec{v}}{dt}=\frac{Q_{\text{\tiny BH}}\vec{B}_{\text{\tiny
c}}}{M_{\text{\tiny BH}}}\;.$ (2.6)

In the above equation, we have neglected the influence of free electric fields
due to their relative paucity in scenarios we are considering and consequently
negligible effects [122].

We may define a velocity scale $v_{\text{\tiny mag}}$ [72], termed the
magnetic velocity, which denotes the speed acquired by an MBH starting from
rest within a single coherence length $l_{\text{\tiny c}}$. Using Eq. (2.6),
we have,

$v_{\text{\tiny mag}}=\sqrt{\frac{2Q_{\text{\tiny BH}}l_{\text{\tiny
c}}B_{\text{\tiny c}}}{M_{\text{\tiny BH}}}}\;,$ (2.7)

where, $B_{\text{\tiny c}}=|\vec{B}_{\text{\tiny c}}|$. Although
$v_{\text{\tiny mag}}$ explicitly depends upon the charge $Q_{\text{\tiny
BH}}$ and mass $M_{\text{\tiny BH}}$, in the extremal case with
$Q_{\text{\tiny BH}}=\sqrt{4\pi G/\mu_{0}}M_{\text{\tiny BH}}$, these
dependences completely cancel. As we mentioned, a primordial BH that is
magnetically charged undergoes Hawking evaporation and naturally evolves to
the extremal limit, where the Hawking temperature vanishes, and the
evaporation then subsides. In this case of physical interest, therefore, the
magnetic field characteristics $B_{\text{\tiny c}}$ and $l_{\text{\tiny c}}$
completely determine $v_{\text{\tiny mag}}$.

Based on whether the initial velocity $v_{\text{\tiny in}}$ of the MBHs is
much larger or smaller than $v_{\text{\tiny mag}}$, as discussed in [72], we
may classify the MBHs into two categories – fast MBHs, for which
$v_{\text{\tiny in}}\gg v_{\text{\tiny mag}}$ and slow MBHs, for which
$v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$. For fast MBHs, since
$v_{\text{\tiny in}}\gg v_{\text{\tiny mag}}$, the change in velocity when an
MBH traverses a coherence length is small, meaning $\Delta v\sim
v_{\text{\tiny mag}}\ll v_{\text{\tiny in}}$. On the contrary, slow MBHs
($v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$) undergo acceleration to a
velocity comparable to $v_{\text{\tiny mag}}$, over a coherence length, and
the change in velocity is then significant, i.e. $\Delta v\sim v_{\text{\tiny
mag}}\gg v_{\text{\tiny in}}$. As we will demonstrate later, these two cases
will be relevant in different scenarios, resulting in distinct bounds. MBHs
may be further divided into two subcategories – clustered MBHs and unclustered
MBHs, i.e., which are virialized or infalling towards a gravitational source,
respectively. As we shall see later, MBHs belonging to these subcategories
will have different velocity distributions, potentially resulting in different
bounds.

Let us first re-derive a few relevant theoretical expressions and results. To
this end, consider non-relativistic MBHs all having the same charge
$Q_{\text{\tiny BH}}$ and mass $M_{\text{\tiny BH}}$ moving through a magnetic
field $\vec{B}_{\text{\tiny c}}$, which is assumed to be coherent over a
distance $l_{\text{\tiny c}}$. Using Eq. (2.6), we can express the rate of
change in kinetic energy of MBHs as

$\frac{\text{d}\mathcal{E}_{\text{\tiny k}}}{\text{d}t}=Q_{\text{\tiny
BH}}\vec{B}_{\text{\tiny c}}\cdot\vec{v}\;.$ (2.8)

Similarly, the second derivative of the kinetic energy can be written as,

$\frac{\text{d}^{2}\mathcal{E}_{\text{\tiny
k}}}{\text{d}t^{2}}=\frac{Q_{\text{\tiny BH}}^{2}B_{\text{\tiny
c}}^{2}}{M_{\text{\tiny BH}}}\;.$ (2.9)

Notice that the right-hand side of the above equation is constant, and thus,
all higher derivatives of the kinetic energy will vanish. Now, if $n$ number
of MBHs enter the coherent magnetic field at time $t$ and traverse
$l_{\text{\tiny c}}$ in a time $\Delta t$, we can express the average gain in
their kinetic energy as

$\Delta\mathcal{E}_{\text{\tiny k}}\equiv\left\langle\mathcal{E}_{\text{\tiny
k}}(t+\Delta t)\right\rangle-\left\langle\mathcal{E}_{\text{\tiny
k}}(t)\right\rangle=\Delta t\left\langle\frac{d\mathcal{E}_{\text{\tiny
k}}}{dt}\right\rangle+\frac{1}{2}(\Delta
t)^{2}\left\langle\frac{d^{2}\mathcal{E}_{\text{\tiny
k}}}{dt^{2}}\right\rangle\;.$ (2.10)

Notice that Eq. (2.10) is obtained by expanding $\mathcal{E}_{\text{\tiny
k}}(t+\Delta t)$, and subsequently taking an average111Here, we define averege
energy of MBHs as $\left\langle\mathcal{E}_{\text{\tiny
k}}\right\rangle\equiv\frac{1}{n}\sum_{i}^{n}\mathcal{E}_{\text{\tiny k,i}}$,
where $\mathcal{E}_{\text{\tiny k,i}}$ is the energy of $i$th MBH. over n
MBHs. Additionally, since the higher derivatives of $\mathcal{E}_{\text{\tiny
k}}$ are all zero, Eq. (2.10) is an exact equation. Substituting Eqs. (2.8)
and (2.9) in Eq. (2.10), we get

$\Delta\mathcal{E}_{\text{\tiny k}}=Q_{\text{\tiny
BH}}\langle\vec{B}_{\text{\tiny c}}\cdot\vec{v}_{\text{\tiny in}}\rangle\Delta
t+\frac{Q_{\text{\tiny BH}}^{2}B_{\text{\tiny c}}^{2}}{2M_{\text{\tiny
BH}}}\Delta t^{2}\;.$ (2.11)

Evidently, the first term depends on the velocity distribution of MBHs. As we
shall see later, this will lead to two physical scenarios—
clustered/virialized MBHs and unclustered/infalling MBHs. One expects an
isotropic velocity distribution for clustered MBHs (virialized case). In this
scenario, the first term in Eq. (2.11) will vanish, as
$\langle\vec{B}_{\text{\tiny c}}\cdot\vec{v}_{\text{\tiny in}}\rangle=0$.
However, as we shall see, for unclustered MBHs (infalling case), the
contribution from the first term will usually lead to stronger bounds.

The energy gained by MBHs comes at the expense of the background magnetic
field $\vec{B}_{\text{\tiny c}}$. As discussed by [72], and more recently in
the context of IGMFs by [76, 77], MBHs may also return energy to the magnetic
fields when MBHs with opposite magnetic charges oscillate with the magnetic
field; similar to electrostatic plasma oscillations or Langmuir oscillations.
However, for MBHs in the scenarios we consider, the time period of such
oscillations is significantly longer than the regeneration time, making them
less relevant. Assuming no other back reactions on the field due to the MBHs,
this leads to a depletion of the fields by the MBHs and places bounds on their
fluxes based on magnetic field persistence till the current era. This concept
forms the essence of Parker bounds and has been widely employed in previous
studies [71, 72, 54, 73, 123, 74].

Now, summing over all the contributions, the field loses energy at a rate

$\left|\frac{\text{d}\mathcal{E}_{\text{\tiny field}}}{\text{d}t}\right|=4\pi
l_{\text{\tiny c}}^{2}\Delta\mathcal{E}_{\text{\tiny k}}F_{\text{\tiny
BH}}\;.$ (2.12)

Here, $F_{\text{\tiny BH}}$ is the number of MBHs passing per unit area per
unit time through the magnetic field 222In some literature [72, 71], flux is
defined as number/(area $\cdot$ second $\cdot$ solid angle). Consequently, a
flux $\tilde{F}$ defined in this way is related to the flux $F$ defined above
by $\tilde{F}=F/\pi$. . The depletion time scale of the magnetic energy can be
expressed as the ratio of the total enclosed magnetic energy stored in the
field in a coherence region (taken to be a sphere of radius $l_{\text{\tiny
c}}$) to the rate of energy loss and is given by

$t_{\text{\tiny dep}}\simeq\frac{\mathcal{E}_{\text{\tiny
field}}}{\left|\frac{\text{d}\mathcal{E}_{\text{\tiny
field}}}{\text{d}t}\right|}=\frac{B_{\text{\tiny c}}^{2}l_{\text{\tiny
c}}}{6\mu_{0}\Delta\mathcal{E}_{\text{\tiny k}}F_{\text{\tiny BH}}}\;.$ (2.13)

Here, we have taken $\mathcal{E}_{\text{\tiny
field}}=\left(\frac{1}{2\mu_{0}}B_{\text{\tiny
c}}^{2}\right)\left(\frac{4}{3}\pi l_{\text{\tiny c}}^{3}\right)$. To ensure
the field’s persistence till the present day, the magnetic fields must
regenerate faster than they are being depleted. Equivalently, the regeneration
time must be smaller than the depletion time. Hence, we arrive at the
condition for the survival of magnetic fields

$t_{\text{\tiny reg}}\leq t_{\text{\tiny dep}}\simeq\frac{B_{\text{\tiny
c}}^{2}l_{\text{\tiny c}}}{6\mu_{0}\Delta\mathcal{E}_{\text{\tiny
k}}F_{\text{\tiny BH}}}\;.$ (2.14)

Here, $t_{\text{\tiny reg}}$ is the regeneration time. Eq. (2.14) in turn puts
a limit on the flux of MBHs

$F_{\text{\tiny BH}}\leq\frac{B_{\text{\tiny c}}^{2}l_{\text{\tiny
c}}}{6\mu_{0}\Delta\mathcal{E}_{\text{\tiny k}}t_{\text{\tiny reg}}}\;.$
(2.15)

In addition, using this bound on the flux of MBHs, we may now further
establish an upper limit on the fraction $f_{\text{\tiny DM}}$ of dark matter
(DM) in the form of extremal MBHs. The flux of MBHs moving with speed $v$ is
related to the mass density $(f_{\text{\tiny DM}}\rho_{\text{\tiny DM}})$ of
MBHs as

$F_{\text{\tiny BH}}=\frac{v\rho_{\text{\tiny DM}}f_{\text{\tiny
DM}}}{M_{\text{\tiny BH}}}\;,$ (2.16)

where, $\rho_{\text{\tiny DM}}$ is the total dark matter density. Using Eqs.
(2.15) and (2.17), we may then express the limit on the dark matter fraction
in the form of MBHs as

$f_{\text{\tiny DM}}\leq\frac{B_{\text{\tiny c}}^{2}l_{\text{\tiny
c}}M_{\text{\tiny BH}}}{6\mu_{0}\Delta\mathcal{E}_{\text{\tiny
k}}t_{\text{\tiny reg}}v\rho_{\text{\tiny DM}}}\;.$ (2.17)

Depending upon whether the MBHs are fast or slow, we may estimate the time
$\Delta t$ taken to cover $l_{\text{\tiny c}}$, and using Eqs. (2.11), (2.16)
and (2.17), we may place specific bounds on MBHs in various cases of interest.
As discussed earlier, MBHs may be clustered (virialized) or unclustered
(infalling towards a gravitational source), thereby having distinct velocity
distributions. Clearly, from Eq. (2.11), different velocity distributions will
lead to different gains in kinetic energy by MBHs. This may, as a consequence,
lead to different Parker-type bounds. Therefore, we will divide our analysis
into two parts – fast MBHs, which have $v_{\text{\tiny in}}\gg v_{\text{\tiny
mag}}$, and slow MBHs, with $v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$, and
then within each of these categories we will study clustered (virialized) and
unclustered (infalling) MBHs.

####  2.2.1 Fast MBHs ($v\sim v_{\text{\tiny in}}\gg v_{\text{\tiny mag}}$)

Fast MBHs undergo a minimal change in their velocities when they traverse
$l_{\text{\tiny c}}$. This is because their initial velocity, $v_{\text{\tiny
in}}$, significantly exceeds the change in velocity induced by the magnetic
field over $l_{\text{\tiny c}}$ (i.e., $\Delta v\sim v_{\text{\tiny mag}}\ll
v_{\text{\tiny in}}$). Therefore, we may write $v\sim v_{\text{\tiny in}}$ and
the time taken to cross $l_{\text{\tiny c}}$ may simply be taken as $\Delta
t\sim l_{\text{\tiny c}}/v_{\text{\tiny in}}$. Using Eq. (2.11), we may then
estimate the change in the kinetic energy of fast MBHs as

$\Delta\mathcal{E}_{\text{\tiny k}}^{\text{\tiny fast}}\simeq Q_{\text{\tiny
BH}}l_{\text{\tiny c}}\langle\vec{B}_{\text{\tiny c}}\cdot\hat{v}_{\text{\tiny
in}}\rangle+\frac{Q_{\text{\tiny BH}}^{2}B_{\text{\tiny c}}^{2}l_{\text{\tiny
c}}^{2}}{2M_{\text{\tiny BH}}v_{\text{\tiny in}}^{2}}\;,$ (2.18)

where, $\hat{v}_{\text{\tiny in}}=\vec{v}_{\text{\tiny in}}/v_{\text{\tiny
in}}$.

Notice that for the two relevant terms, one is linear and one is quadratic in
$B_{\text{\tiny c}}$. The linear term depends upon the velocity distribution
of MBHs, and we will have two physical scenarios. MBHs may be clustered
(virialized), leading to an isotropic velocity distribution, for which the
term linear in $B_{\text{\tiny c}}$ vanishes. In this case, then, the dominant
contribution to the energy gain is from the quadratic term. On the other hand,
when the MBHs are infalling towards a gravitational source, a directional flow
is expected. In this case, both the linear and quadratic terms will contribute
to the energy gain. We will calculate the limits on MBHs in these two specific
cases in the next sections.

For the case where the MBHs pass through multiple coherence length cells,
uncorrelated field directions imply that the first term in Eq. (2.18)
vanishes. The second term will then scale as $N$, the number of coherence
length cells that the MBHs traverse. However, we will see later that for
cosmic voids and filaments, the very large coherence lengths imply that the
number of cells traversed is $N\sim\mathcal{O}(1)$. We thus cannot neglect the
contribution from the first term of Eq. (2.18) in those cases.

#### Clustered MBHs

As mentioned earlier, when MBHs are clustered or virialized, their velocity
distribution is largely isotropic, and therefore the term
$\langle\vec{B}_{\text{\tiny c}}\cdot\hat{v}_{\text{\tiny in}}\rangle=0$ over
$l_{\text{\tiny c}}$ (since $\vec{B}_{\text{\tiny c}}$ is approximately
constant). Substituting this in Eq. (2.18), we obtain the gain in the kinetic
energy of MBHs

$\Delta\mathcal{E}_{\text{\tiny k}}^{\text{\tiny
fast,clust}}\simeq\frac{Q_{\text{\tiny BH}}^{2}B_{\text{\tiny
c}}^{2}l_{\text{\tiny c}}^{2}}{2M_{\text{\tiny BH}}v_{\text{\tiny
in}}^{2}}\;.$ (2.19)

Using Eqs. (2.15) and (2.19), we then get the bounds on the flux of fast,
clustered MBHs

$F_{\text{\tiny BH}}^{\text{\tiny fast,clust}}\lesssim\frac{M_{\text{\tiny
BH}}v_{\text{\tiny in}}^{2}}{3\mu_{0}Q_{\text{\tiny BH}}^{2}l_{\text{\tiny
c}}t_{\text{\tiny reg}}}\;.$ (2.20)

Finally, combining this with Eq. (2.17), we find the expression for the DM
fraction in this case

$f_{\text{DM}}^{\text{\tiny fast,clust}}\lesssim\frac{M_{\text{\tiny
BH}}^{2}v_{\text{\tiny in}}}{{3\mu_{0}Q_{\text{\tiny BH}}^{2}l_{\text{\tiny
c}}t_{\text{\tiny reg}}\rho_{\text{\tiny DM}}}}\;.$ (2.21)

Notice here that the bounds on $F_{\text{\tiny BH}}^{\text{\tiny fast,clust}}$
and $f_{\text{DM}}^{\text{\tiny fast,clust}}$ are independent of the magnetic
field strength $\vec{B}_{\text{\tiny c}}$ and are inversely proportional to
the other magnetic field characteristics i.e. $l_{\text{\tiny c}}$ and
$t_{\text{\tiny reg}}$. Additionally, for extremal MBHs since $Q_{\text{\tiny
BH}}=\sqrt{4\pi G/\mu_{0}}M_{\text{\tiny BH}}$, the limit on $F_{\text{\tiny
BH}}^{\text{\tiny fast,clust}}$ is inversely proportional to the mass
$M_{\text{\tiny BH}}$ or magnetic charge $Q_{\text{\tiny BH}}$, while the
limit on $f_{\text{DM}}^{\text{\tiny fast,clust}}$ is independent of the MBH
mass or charge.

#### Unclustered MBHs

Relevant scenarios with unclustered MBHs may arise, for instance, when they
are infalling into a gravitational well before entering the coherence length
region. In such circumstances, one may expect a uniform velocity distribution
for the MBHs to be the apropos assumption.

Once they enter the coherence length region, these unclustered MBHs are
accelerated uniformly in the direction of $B_{\text{\tiny c}}$. However, since
the MBHs are fast, the change in velocity is small again. Therefore, we may
write $\vec{v}\sim\vec{v}_{\text{\tiny in}}$ and $\langle\vec{B}_{\text{\tiny
c}}\cdot\hat{v}\rangle\sim\langle\vec{B}_{\text{\tiny
c}}\cdot\hat{v}_{\text{\tiny in}}\rangle=B_{\text{\tiny c}}v_{\text{\tiny
in}}\cos\alpha$, where $\alpha$ is the angle between the velocity
$\vec{v}_{\text{\tiny in}}$ and magnetic field $\vec{B}_{\text{\tiny c}}$.
Using Eq. (2.18), we then get

$\Delta\mathcal{E}_{\text{\tiny k}}^{\text{\tiny fast,unclust}}\simeq
Q_{\text{\tiny BH}}B_{\text{\tiny c}}l_{\text{\tiny
c}}\cos\alpha+\frac{Q_{\text{\tiny BH}}^{2}B_{\text{\tiny
c}}^{2}l_{\text{\tiny c}}^{2}}{2M_{\text{\tiny BH}}v_{\text{\tiny
in}}^{2}}=\frac{M_{\text{\tiny BH}}v_{\text{\tiny
mag}}^{2}}{2}\left(\cos\alpha+\frac{v_{\text{\tiny mag}}^{2}}{4v_{\text{\tiny
in}}^{2}}\right)\;.$ (2.22)

When $\alpha<\cos^{-1}\left(-{\frac{v_{\text{\tiny mag}}^{2}}{4v_{\text{\tiny
in}}^{2}}}\right)\sim\pi/2$ (i.e., when MBHs have a component of velocity
along the magnetic field strength), the MBHs will gain energy, and there will
be a reduction in magnetic field strength. Otherwise, when they have a
component opposite to the field, the MBHs will slow down and return energy to
the field. To establish the Parker-type bounds, we focus our attention on
cases where the magnetic fields are being attenuated. In this case, we may use
Eqs. (2.15) and (2.22) to put bounds on the flux

$F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}\lesssim\frac{M_{\text{\tiny
BH}}v_{\text{\tiny in}}^{2}}{3\mu_{0}Q_{\text{\tiny BH}}^{2}l_{\text{\tiny
c}}t_{\text{\tiny reg}}}\frac{1}{\left(1+4\frac{v_{\text{\tiny
in}}^{2}}{v^{2}_{\text{\tiny mag}}}\cos\alpha\right)}\;.$ (2.23)

Using Eq. (2.17), we may, as before, place bounds on the dark matter fraction

$f_{\text{DM}}^{\text{\tiny fast,unclust}}\lesssim\frac{M_{\text{\tiny
BH}}^{2}v_{\text{\tiny in}}}{{3\mu_{0}Q_{\text{\tiny BH}}^{2}l_{\text{\tiny
c}}t_{\text{\tiny reg}}\rho_{\text{\tiny
DM}}}}\frac{1}{\left(1+4\frac{v_{\text{\tiny in}}^{2}}{v^{2}_{\text{\tiny
mag}}}\cos\alpha\right)}\;.$ (2.24)

Since $v_{\text{\tiny mag}}$ depends upon the magnetic field characteristics,
unlike the clustered case, the bounds on $F_{\text{\tiny BH}}^{\text{\tiny
fast,unclust}}$ and $f_{\text{DM}}^{\text{\tiny fast,unclust}}$ here are
dependent on the magnetic field strength ${B}_{\text{\tiny c}}$. However,
similar to the clustered case, for extremal MBHs with $Q_{\text{\tiny
BH}}=\sqrt{4\pi G/\mu_{0}}M_{\text{\tiny BH}}$, the limit on $F_{\text{\tiny
BH}}^{\text{\tiny fast,unclust}}$ is inversely proportional to the mass
$M_{\text{\tiny BH}}$ (or, equivalently, magnetic charge $Q_{\text{\tiny
BH}}$) and the $f_{\text{DM}}^{\text{\tiny fast,unclust}}$ limit is
independent of these. Also, in the limit when $4v^{2}_{\text{\tiny
in}}\cos\alpha\gg v^{2}_{\text{\tiny mag}}$, i.e. when the component of the
initial velocity of MBHs along the magnetic field is significantly greater
than the magnetic velocity, we may write

$F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}\lesssim\frac{{B}_{\text{\tiny
c}}}{6\mu_{0}Q_{\text{\tiny BH}}t_{\text{\tiny reg}}\cos\alpha}\;,$ (2.25)

and

$f_{\text{DM}}^{\text{\tiny fast,unclust}}\lesssim\frac{{B}_{\text{\tiny
c}}M_{\text{\tiny BH}}}{6\mu_{0}Q_{\text{\tiny BH}}t_{\text{\tiny
reg}}v_{\text{\tiny in}}\rho_{\text{\tiny DM}}\cos\alpha}\;.$ (2.26)

In this limit, the bounds on $F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}$
and $f_{\text{DM}}^{\text{\tiny fast,unclust}}$ are proportional to
${B}_{\text{\tiny c}}$, but independent of ${l}_{\text{\tiny c}}$.

We will use Eqs. (2.20), (2.21), (2.23) and (2.24) to estimate bounds on the
flux and the dark matter fraction of extremal MBHs in the scenarios where they
are fast, i.e. $v\sim v_{\text{\tiny in}}\gg v_{\text{\tiny mag}}$. In the
next subsection, we will compute the corresponding expressions for the case of
slow-moving MBHs with $v\sim v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$.

####  2.2.2 Slow MBHs ($v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$)

Let us now focus on slow-moving MBHs. Slow MBHs will experience an initial
acceleration due to the magnetic fields and move in the direction of magnetic
field lines. This makes them attain a velocity $v\simeq v_{\text{\tiny mag}}$
within the first coherence length they traverse. However, in subsequent
coherence lengths, due to their low velocities, they may undergo substantial
deflections, depending upon the magnetic field orientations, resulting in a
random trajectory. Furthermore, unlike the fast MBH case, the time taken to
cross the coherent length $l_{\text{\tiny c}}$ is no longer $l_{\text{\tiny
c}}/v_{\text{\tiny in}}$, and therefore Eq. (2.18) doesn’t hold. Irrespective
of whether MBHs are clustered or unclustered, they will be accelerated in the
direction of the magnetic field. And for initially clustered, slow MBHs, since
the virial velocity $v_{\text{\tiny vir}}\equiv v_{\text{\tiny in}}\ll
v_{\text{\tiny mag}}$, they will acquire a velocity $v\sim v_{\text{\tiny
mag}}$, in the first coherence length region and become unclustered.
Additionally, as we shall see later, in the galactic, cosmic void and cosmic
filament systems we consider, the virial velocities typically satisfy
$v_{\text{vir}}\gg v_{\text{mag}}$. Thus, the virialized MBHs are always
typically categorised as fast for these systems.

Using Eq. (2.6), we find that for $v_{\text{\tiny in}}\ll v_{\text{\tiny
mag}}$ (and thus $\Delta v\simeq v_{\text{\tiny mag}}$), the time taken to
cross $l_{\text{\tiny c}}$ is $\Delta t\simeq 2l_{\text{\tiny
c}}/v_{\text{\tiny mag}}$. Substituting it in Eq. (2.11), and taking
$v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$, we get the energy acquired by
the MBHs as

$\mathcal{E}^{\text{\tiny slow}}_{\text{\tiny
k,1}}\equiv\Delta\mathcal{E}^{\text{\tiny slow}}_{\text{\tiny
k,1}}=Q_{\text{\tiny BH}}B_{\text{\tiny c}}l_{\text{\tiny c}}\;.$ (2.27)

Notice that the contribution of the first term in Eq. (2.11) is negligible
since $v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$. Also, unlike fast MBHs,
the change in kinetic energy of slow MBHs doesn’t scale as $N$ when they pass
through $N$ multiple coherence lengths. We will now estimate this scaling and
find the average change in energy when the MBHs pass a single coherence
length. A similar treatment due to multiple coherence lengths has been
previously investigated by [77] for the change in monopole velocities.

Let us take $v_{\text{\tiny N}}$ to be the velocity of a slow MBH when it
crosses the $\text{N}^{\text{th}}$ coherence length region and
$\vec{B}_{\text{\tiny c,N}}={B}_{\text{\tiny c}}\hat{B}_{\text{\tiny c,N}}$ to
be the magnetic field there. Then, using Eq. (2.11), one may write the change
in its energy when it crosses the $\text{N}^{\text{th}}$ coherence length
region as

$\Delta\mathcal{E}_{\text{\tiny k,N}}^{\text{\tiny
slow}}\equiv\mathcal{E}_{\text{\tiny k,N}}^{\text{\tiny
slow}}-\mathcal{E}_{\text{\tiny k,N-1}}^{\text{\tiny slow}}\simeq
Q_{\text{\tiny BH}}\Delta t_{\text{\tiny N}}\vec{B}_{\text{\tiny
c,N}}\cdot\vec{v}_{\text{\tiny N}}+\frac{Q_{\text{\tiny BH}}^{2}B_{\text{\tiny
c}}^{2}}{2M_{\text{\tiny BH}}}\Delta t_{\text{\tiny N}}^{2}\quad\text{for N
$\geq 2$}\;,$ (2.28)

where $\mathcal{E}_{\text{\tiny k,N}}\equiv M_{\text{\tiny
BH}}v^{2}_{\text{\tiny N}}/2$ is the energy of the MBH when it crosses the
$\text{N}^{\text{th}}$ cell and $\Delta t_{\text{\tiny N}}$ is the time the
MBH takes to cross the $\text{N}^{\text{th}}$ cell. As mentioned earlier,
$\Delta t_{\text{\tiny 1}}\simeq 2l_{\text{\tiny c}}/v_{\text{\tiny mag}}$,
and we may assume that for $N\geq 2$, $\Delta t_{\text{\tiny N}}\sim
l_{\text{\tiny c}}/v_{\text{\tiny N-1}}$. Therefore, Eq. (2.28) becomes,

$\mathcal{E}_{\text{\tiny k,N}}^{\text{\tiny slow}}-\mathcal{E}_{\text{\tiny
k,N-1}}^{\text{\tiny slow}}\simeq\Delta\mathcal{E}^{\text{\tiny
slow}}_{\text{\tiny 1}}\hat{B}_{\text{\tiny c,N}}\cdot\hat{v}_{\text{\tiny
N-1}}+\frac{(\Delta\mathcal{E}^{\text{\tiny slow}}_{\text{\tiny
1}})^{2}}{4\mathcal{E}_{\text{\tiny k,N-1}}^{\text{\tiny slow}}}\quad\text{for
N $\geq 2$}\;.$ (2.29)

Here, we have used Eq. (2.27) for the expression for
$\Delta\mathcal{E}^{\text{\tiny slow}}_{\text{\tiny 1}}$. Eq. (2.29) is a
recursion relation that relates the energy of a slow MBH after it crosses the
$\text{N}^{\text{th}}$ cell with the energy when it crosses the
$(\text{N-1})^{\text{th}}$ cell. Since the magnetic fields in subsequent cells
are assumed to be randomly oriented, the first term in Eq. (2.29) doesn’t
contribute to the mean behaviour, and we may rewrite the above expression as

$\mathcal{E}_{\text{\tiny k,N}}^{\text{\tiny slow}}-\mathcal{E}_{\text{\tiny
k,N-1}}^{\text{\tiny slow}}\simeq\frac{(\Delta\mathcal{E}^{\text{\tiny
slow}}_{\text{\tiny 1}})^{2}}{4\mathcal{E}_{\text{\tiny k,N-1}}^{\text{\tiny
slow}}}\quad\text{for N $\geq 2$}\;,$ (2.30)

The above recurrence relation has an asymptotic solution
$\mathcal{E}_{\text{\tiny k,N}}^{\text{\tiny
slow}}\simeq\sqrt{({N+1})/{2}}~{}\Delta\mathcal{E}^{\text{\tiny
slow}}_{\text{\tiny 1}}$. Thus, the average gain in energy by an MBH while it
passes a single cell is given by

$\Delta\mathcal{E}_{\text{\tiny k}}^{\text{\tiny
slow}}\equiv\frac{\mathcal{E}_{\text{\tiny k,N}}^{\text{\tiny
slow}}}{N}\sim\sqrt{\frac{N+1}{2N}}Q_{\text{\tiny BH}}B_{\text{\tiny
c}}l_{\text{\tiny c}}\;,$ (2.31)

Using Eqs. (2.15) and (2.31), we arrive at

$F_{\text{\tiny BH}}^{\text{\tiny
slow}}\lesssim\sqrt{\left(\frac{2N}{N+1}\right)}\frac{B_{\text{\tiny
c}}}{6\mu_{0}Q_{\text{\tiny BH}}t_{\text{\tiny reg}}}\;.$ (2.32)

Notice that, unlike fast clustered MBHs, the bound on the flux of slow MBHs is
no longer independent of the magnetic field strength $B_{\text{\tiny c}}$ but
varies linearly with it. Also, $N\sim d/l_{\text{\tiny c}}$, where $d$ is the
total size of N coherence length regions. For $N=1$, when MBHs cover a single
coherence length, this limit is, therefore, independent of $l_{\text{\tiny
c}}$. For the slow MBH case, the average velocity after N cells can be
determined using Eq. (2.31) and is given by
$v\sim\left({(N+1)}/(2N)\right)^{1/4}v_{\text{\tiny mag}}$. Therefore, Eqs.
(2.17) and (2.31) give us a constraint on the dark matter fraction

$f_{\text{\tiny DM}}^{\text{\tiny
slow}}\lesssim\left(\frac{2N}{N+1}\right)^{3/4}\frac{B_{\text{\tiny
c}}M_{\text{\tiny BH}}}{6\mu_{0}Q_{\text{\tiny BH}}t_{\text{\tiny
reg}}v_{\text{\tiny mag}}\rho_{\text{\tiny
DM}}}=\left(\frac{2N}{N+1}\right)^{3/4}\frac{B_{\text{\tiny
c}}^{1/2}M_{\text{\tiny BH}}^{3/2}}{6\sqrt{2}\mu_{0}Q_{\text{\tiny
BH}}t_{\text{\tiny reg}}l_{\text{\tiny c}}^{1/2}\rho_{\text{\tiny DM}}}\;.$
(2.33)

One must also consider the case when the MBHs are unable to cross a single
coherence length in Hubble time $H_{\text{\tiny 0}}^{-1}$, where
$H_{\text{\tiny 0}}$ is the Hubble constant. This happens when $v_{\text{\tiny
mag}}\lesssim l_{\text{\tiny c}}H_{\text{\tiny 0}}$. Such a situation arises
in the physical scenarios where $B_{\text{\tiny c}}\lesssim l_{\text{\tiny
c}}M_{\text{\tiny BH}}H_{\text{\tiny 0}}^{2}/(2Q_{\text{\tiny BH}})$, i.e.
small magnetic field strength $B_{\text{\tiny c}}$ or large coherence length
$l_{\text{\tiny c}}$. We will see later that this situation is realized for
cosmic voids and cosmic web filaments, where the number of cells traversed is
always $N\sim\mathcal{O}(1)$. In such a case, the velocity acquired by the
MBHs (from Eq. (2.6)) in a Hubble time is given by $v_{\text{\tiny H0}}\sim
Q_{\text{\tiny BH}}B_{\text{\tiny c}}/(M_{\text{\tiny BH}}H_{\text{\tiny
0}})$. Consequently, the energy gained is

$\Delta\mathcal{E}_{\text{\tiny k,H0}}\equiv\frac{1}{2}M_{\text{\tiny
BH}}v_{\text{\tiny H0}}^{2}=\frac{Q_{\text{\tiny BH}}^{2}B_{\text{\tiny
c}}^{2}}{2M_{\text{\tiny BH}}H_{\text{\tiny 0}}^{2}}\;.$ (2.34)

Using Eqs. (2.15) and (2.34), we get the bounds on the flux of MBHs as,

$F_{\text{\tiny BH}}^{\text{\tiny slow,H0}}\lesssim\frac{l_{\text{\tiny
c}}M_{\text{\tiny BH}}H_{\text{\tiny 0}}^{2}}{3\mu_{0}Q^{2}_{\text{\tiny
BH}}t_{\text{\tiny reg}}}\;.$ (2.35)

Finally, using Eqs. (2.17) and (2.34) and taking $v\sim v_{\text{\tiny H0}}$ ,
we get the bound on the dark matter fraction as

$f_{\text{\tiny DM}}^{\text{\tiny slow,H0}}\lesssim\frac{l_{\text{\tiny
c}}M^{3}_{\text{\tiny BH}}H_{\text{\tiny 0}}^{3}}{3\mu_{0}B_{\text{\tiny
c}}Q^{3}_{\text{\tiny BH}}\rho_{\text{\tiny DM}}t_{\text{\tiny reg}}}\;.$
(2.36)

Notice that unlike previous cases, the bounds on $F_{\text{\tiny
BH}}^{\text{\tiny slow,H0}}$ and $f_{\text{DM}}^{\text{\tiny slow,H0}}$ are
directly proportional to the coherence length $l_{\text{\tiny c}}$. This
happens because the gain in energy is independent of the coherence length.
Moreover, the bound on $f_{\text{\tiny DM}}^{\text{\tiny slow,H0}}$ is
inversely proportional to the magnetic field strength $B_{\text{\tiny c}}$.

Figure 1: Summary of the classification of MBH Parker bounds in this work.
Please see the text for details. There is no clustered category for slow MBHs
as in the systems under consideration the virial velocities typically satisfy
$v_{\text{\tiny vir}}\gg v_{\text{\tiny mag}}$; always rendering them to be
fast.

Fig. 1 summarises the categories for Parker bounds on MBHs due to galactic,
extra-galactic and cosmic magnetic fields. As we have calculated, depending on
the initial velocity of the MBHs, they are divided into two categories—fast
MBHs, which have $v_{\text{\tiny in}}\gg v_{\text{\tiny mag}}$, and slow MBHs,
with $v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$. Additionally, depending
upon whether the MBHs are clustered (virialized) or unclustered (infalling),
bounds on MBHs are determined by Eqs. (2.20) and (2.21), or Eqs. (2.23) and
(2.24). However, within cosmic voids, there exists a flow of MBHs directed
toward gravitational sources, such as cosmic web nodes or filaments. In this
case, if the MBHs are slow ($v_{\text{\tiny in}}\ll v_{\text{\tiny mag}}$) and
clustered, they will be accelerated in the direction of the magnetic field,
and undergo a large change in velocity of the order of $v_{\text{\tiny mag}}$.
Due to this, they will have directional velocities, and hence, they will
eventually become unclustered. Additionally, as we will see in the next
section, the galactic, cosmic filament, and cosmic void systems under
consideration have virial velocities $v_{\text{\tiny vir}}\gg v_{\text{\tiny
mag}}$. Therefore, slow clustered MBHs are unlikely to exist in these systems.
Furthermore, since IGMFs typically have large coherence lengths and small
magnetic fields, the slow MBHs in cosmic voids and filaments may not cross
even a single coherence length in Hubble time. In that case, the constraints
will be determined by Eqs. (2.35) and (2.36). However, for the galactic case,
the bounds on slow MBHs in the galaxy will be determined by Eqs. (2.32) and
(2.33).

###  2.3 Bounds on primordial magnetic black holes

We will now apply our analyses in Sec. 2.2 to the different magnetic field
configurations discussed in Sec. 2.1. Specifically, the constraints will be
evaluated using Eqs. (2.20), (2.21), (2.23), (2.24), (2.32), (2.33), (2.35)
and (2.36). As previously stated, we will explore constraints specific to
different types of magnetic field configurations—IGMFs in cosmic filaments and
cosmic voids, and galactic magnetic fields. For the cosmic filaments, we take
the density of the dark matter as $\rho_{\text{\tiny DM}}^{\text{\tiny
fil}}\sim 10\,\rho_{\text{\tiny c}}\simeq 5.6\times
10^{-5}\,\text{GeV/cm}^{3}$ [124, 125]. However, for the cosmic voids, we take
the dark matter density to be $\rho_{\text{\tiny DM}}^{\text{\tiny
void}}\simeq 0.1\,\rho_{\text{\tiny c}}\sim 5.6\times
10^{-7}\,\text{GeV/cm}^{3}$ [124, 125], where $\rho_{\text{\tiny c}}$ is the
critical density of the universe [126]. As for the galactic magnetic fields,
we assume that the DM density in the galaxies are of the same order of
magnitude as the local DM density $\rho_{\text{\tiny DM}}^{\text{\tiny
loc}}\simeq 0.3\,\text{GeV/cm}^{3}$[127].

First, we estimate the typical value of the magnetic velocity $v_{\text{\tiny
mag}}$ using Eq. (2.7), which will determine whether the MBHs are fast or slow
in each system. In terms of magnetic field parameters, we may express this
magnetic velocity as

$v_{\text{\tiny mag}}\simeq 2.6\times 10^{-7}c\left(\frac{Q_{\text{\tiny
BH}}}{\text{A-m}}\right)^{1/2}\left(\frac{\text{Kg}}{M_{\text{\tiny
BH}}}\right)^{1/2}\left(\frac{l_{\text{\tiny
c}}}{1\,\text{Mpc}}\right)^{1/2}\left(\frac{B}{10^{-15}\,\text{G}}\right)^{1/2}\;.$
(2.37)

For an extremal MBH, using values presented in Sec. 2.1, we observe that
galactic magnetic fields possessing $B_{\text{\tiny
c}}\sim\mathcal{O}(10)\,\mu\text{G}$ and $l_{\text{\tiny
c}}\sim\mathcal{O}(1)\,\text{kpc}$ yield typically $v_{\text{\tiny mag}}\sim
10^{-4}\,\text{c}$ (where c is the speed of light). Similarly, for IGMFs in
cosmic filaments with $B_{\text{\tiny c}}\sim\mathcal{O}(10^{-9})\,\text{G}$
and $l_{\text{\tiny c}}\sim\mathcal{O}(1)\,\text{Mpc}$, we get $v_{\text{\tiny
mag}}\sim 10^{-5}c$. Now, since the virial velocities in galaxies and cosmic
filaments are of order $10^{-3}c$ [128, 129], the clustered MBHs are fast.
Additionally, for cosmic void IGMFs generated primordially or from
magnetizations due to void galaxies, we have $B_{\text{\tiny
c}}\simeq\mathcal{O}(10^{-15})\,\text{G}$ and $l_{\text{\tiny
c}}\gtrsim\mathcal{O}(1-10)\,\text{Mpc}$, leading to $v_{\text{\tiny mag}}\sim
10^{-7}c-10^{-8}c$. However, for a galactic outflow origin of IGMFs in cosmic
voids with $\mathcal{O}(10^{-12})\,\text{G}\lesssim B_{\text{\tiny
c}}\lesssim\mathcal{O}(10^{-8})\,\text{G}$ and $l_{\text{\tiny
c}}\sim\mathcal{O}(1)\,\text{Mpc}$, we have $v_{\text{\tiny mag}}\sim
10^{-4}c-10^{-6}c$. For fast unclustered MBHs, we use the typical velocities
of MBHs found in galaxies, filaments, and voids as $v_{\text{\tiny in}}\sim
10^{-3}c$ [130, 131, 132].

Notice that the magnetic velocity $v_{\text{\tiny mag}}$ for IGMFs is
typically very small. In addition, due to the typically large coherence
lengths ($\mathcal{O}(1)\,\text{Mpc}$), the time required by slow MBHs ($v\ll
v_{\text{\tiny mag}}$) to cross a single coherence length is much greater than
$10\,\text{Gyr}$ (i.e. typically much greater than the Hubble time).
Therefore, for the case of slow MBHs, the relevant expressions are Eqs. (2.35)
and (2.36).

Putting all these together, let us discern the typical constraints on fast and
slow MBHs. Under each of these cases, we will look at the various categories
and limits of interest. In the fast unclustered (infalling) case, one may
quantify the limits using Eqs. (2.23), and (2.24). One obtains, in this case

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}$
$\displaystyle\lesssim$ $\displaystyle\frac{2.5\times 10^{-23}}{1+0.6\times
10^{8}\cos\alpha\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{10^{-15}\text{G}}{B_{\text{\tiny
c}}}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{2}\;$ $\displaystyle\times\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)m^{-2}\text{s}^{-1}\;,$
$\displaystyle f_{\text{DM}}^{\text{\tiny fast,unclust}}$
$\displaystyle\lesssim$ $\displaystyle\frac{0.8\times 10^{-3}}{1+0.6\times
10^{8}\cos\alpha\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{10^{-15}\text{G}}{B_{\text{\tiny
c}}}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)}\left(\frac{\rho_{\text{\tiny c}}}{\rho_{\text{\tiny
DM}}}\right)\left(\frac{M_{\text{\tiny BH}}}{\text{kg}}\right)^{2}\;$
$\displaystyle\times\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{2}\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)\;.$

Specializing to the extremal case, this then reduces to

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}$
$\displaystyle\lesssim$ $\displaystyle\frac{3.7\times 10^{-20}}{1+2.3\times
10^{9}\cos\alpha\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{10^{-15}\text{G}}{B_{\text{\tiny
c}}}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)}\left(\frac{\text{kg}}{M_{\text{\tiny BH}}}\right)\;$
$\displaystyle\times\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)m^{-2}\text{s}^{-1}\;,$
$\displaystyle f_{\text{DM}}^{\text{\tiny fast,unclust}}$
$\displaystyle\lesssim$ $\displaystyle\frac{12.2}{1+2.3\times
10^{9}\cos\alpha\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{10^{-15}\text{G}}{B_{\text{\tiny
c}}}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)}\left(\frac{\rho_{\text{\tiny c}}}{\rho_{\text{\tiny
DM}}}\right)\;$ $\displaystyle\times\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)\;.$ (2.39)

However, if the MBHs are clustered or virialized, using Eqs. (2.20) and
(2.21), one obtains the constraints

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny fast,clust}}$
$\displaystyle\lesssim$ $\displaystyle 2.5\times
10^{-23}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{2}\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)m^{-2}\text{s}^{-1}\;,$
$\displaystyle f_{\text{DM}}^{\text{\tiny fast,clust}}$
$\displaystyle\lesssim$ $\displaystyle 0.8\times
10^{-2}\left(\frac{\rho_{\text{\tiny c}}}{\rho_{\text{\tiny
DM}}}\right)\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)^{2}\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)\;.$

For the extremal MBHs, these simplify to

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny fast,clust}}$
$\displaystyle\lesssim$ $\displaystyle 3.7\times
10^{-20}\left(\frac{\text{kg}}{M_{\text{\tiny
BH}}}\right)\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)^{2}\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)m^{-2}\text{s}^{-1}\;,$
$\displaystyle f_{\text{DM}}^{\text{\tiny fast,clust}}$
$\displaystyle\lesssim$ $\displaystyle 12.2\left(\frac{\rho_{\text{\tiny
c}}}{\rho_{\text{\tiny DM}}}\right)\left(\frac{v_{\text{\tiny
in}}}{10^{-3}c}\right)\left(\frac{\text{Mpc}}{l_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{reg}}\right)\;.$

Let us now turn our attention to slow MBHs. When MBHs are moving slowly
through galaxies, the bounds can be evaluated numerically using Eqs. (2.32),
and (2.33). We obtain, in this case

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny slow}}$
$\displaystyle\lesssim$ $\displaystyle 4.2\times
10^{-21}\left(\frac{2N}{N+1}\right)^{1/2}\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)\left(\frac{B_{\text{\tiny
c}}}{10\,\mu\text{G}}\right)\left(\frac{\text{Gyr}}{t_{\text{\tiny
reg}}}\right)\left(\frac{\text{kpc}}{l_{\text{\tiny
c}}}\right)^{1/2}m^{-2}\text{s}^{-1}\;,$ (2.42) $\displaystyle
f_{\text{DM}}^{\text{\tiny slow}}$ $\displaystyle\lesssim$ $\displaystyle
2.6\times 10^{-4}\left(\frac{2N}{N+1}\right)^{3/4}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)^{\frac{3}{2}}\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{\frac{3}{2}}\left(\frac{\rho_{\text{\tiny c}}}{\rho_{\text{\tiny
DM}}}\right)\left(\frac{B_{\text{\tiny
c}}}{10\,\mu\text{G}}\right)^{\frac{1}{2}}\left(\frac{\text{Gyr}}{t_{\text{\tiny
reg}}}\right)\left(\frac{\text{kpc}}{l_{\text{\tiny c}}}\right)^{1/2}\;.$

Here, $N\sim d_{\text{\tiny gal}}/l_{\text{\tiny c, mean}}^{\text{\tiny gal}}$
is an estimate of the number of coherence lengths that the slow MBHs may
cross. For instance, with $d_{\text{\tiny
gal}}\sim\mathcal{O}(10)\,\text{kpc}$ and $l_{\text{\tiny c,
mean}}^{\text{\tiny gal}}\sim\mathcal{O}(1)\,\text{kpc}$ one has
$N\sim\mathcal{O}(10)$. For extremal MBHs the above reduces to

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny slow}}$
$\displaystyle\lesssim$ $\displaystyle 1.6\times
10^{-19}\left(\frac{2N}{N+1}\right)^{1/2}\left(\frac{\text{kg}}{M_{\text{\tiny
BH}}}\right)\left(\frac{B_{\text{\tiny
c}}}{10\,\mu\text{G}}\right)\left(\frac{\text{Gyr}}{t_{\text{\tiny
reg}}}\right)\left(\frac{\text{kpc}}{l_{\text{\tiny
c}}}\right)^{1/2}m^{-2}\text{s}^{-1}\;,$ (2.43) $\displaystyle
f_{\text{DM}}^{\text{\tiny slow}}$ $\displaystyle\lesssim$ $\displaystyle
0.06\left(\frac{2N}{N+1}\right)^{3/4}\left(\frac{\rho_{\text{\tiny
c}}}{\rho_{\text{\tiny DM}}}\right)\left(\frac{B_{\text{\tiny
c}}}{10\,\mu\text{G}}\right)^{\frac{1}{2}}\left(\frac{\text{Gyr}}{t_{\text{\tiny
reg}}}\right)\left(\frac{\text{kpc}}{l_{\text{\tiny c}}}\right)^{1/2}\;.$

However, as discussed earlier, slow MBHs moving in IGMFs will not generally
cross even a single coherence length in Hubble time. Hence, the bounds from
IGMFs in cosmic web filaments and cosmic voids in the context of slow-moving
MBHs will be determined by Eqs. (2.35), and (2.36). From these, we have

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny slow,H0}}$
$\displaystyle\lesssim$ $\displaystyle 1.45\times
10^{-25}\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{2}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{l_{\text{\tiny
c}}}{\text{Mpc}}\right)\left(\frac{\text{Gyr}}{t_{\text{\tiny
reg}}}\right)m^{-2}\text{s}^{-1}\;,$ (2.44) $\displaystyle
f_{\text{DM}}^{\text{\tiny slow,H0}}$ $\displaystyle\lesssim$ $\displaystyle
343.33\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{3}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)^{3}\left(\frac{l_{\text{\tiny
c}}}{\text{Mpc}}\right)\left(\frac{10^{-15}\text{G}}{B_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{\text{\tiny reg}}}\right)\;.$

Again, the corresponding extremal MBH limits are

$\displaystyle F_{\text{\tiny BH}}^{\text{\tiny slow,H0}}$
$\displaystyle\lesssim$ $\displaystyle 2.2\times
10^{-22}\left(\frac{\text{kg}}{M_{\text{\tiny
BH}}}\right)\left(\frac{l_{\text{\tiny
c}}}{\text{Mpc}}\right)\left(\frac{\text{Gyr}}{t_{\text{\tiny
reg}}}\right)m^{-2}\text{s}^{-1}\;,$ (2.45) $\displaystyle
f_{\text{DM}}^{\text{\tiny slow,H0}}$ $\displaystyle\lesssim$ $\displaystyle
2.0\times 10^{7}\left(\frac{l_{\text{\tiny
c}}}{\text{Mpc}}\right)\left(\frac{10^{-15}\text{G}}{B_{\text{\tiny
c}}}\right)\left(\frac{\text{Gyr}}{t_{\text{\tiny reg}}}\right)\;.$

####  2.3.1 Parker-type bounds from cosmic voids

Parker-type bounds on MBHs due to cosmic void IGMFs are anticipated to result
in stringent constraints. This expectation is motivated by the small field
strengths and large coherence lengths in such cosmic structures. MBHs in
cosmic voids will largely flow towards gravitational sources in the cosmic web
nodes and filaments and, therefore, remain mostly unclustered. Furthermore,
the magnetic velocity in such systems is of the order of $v_{\text{\tiny
mag}}\sim 10^{-7}c-10^{-8}c$ for fields generated primordially or due to void
galaxies, and of the order of $v_{\text{\tiny mag}}\sim 10^{-4}c-10^{-6}c$ for
fields generated from galactic outflow. Depending upon the velocity of MBHs,
they may be fast ($v_{\text{\tiny in}}>v_{\text{\tiny mag}}$ ) or slow
($v_{\text{\tiny in}}<v_{\text{\tiny mag}}$ ), but unclustered. This will lead
to different bounds. Furthermore, depending on the production mechanism of the
IGMFs in cosmic voids (see Sec. 2.1), we have different field characteristics,
which yield distinct bounds.

In cosmic voids, IGMFs have field characteristics $B_{\text{\tiny
c}}=\mathcal{O}(10^{-15})$ G, $l_{\text{\tiny c}}=\mathcal{O}(1-10)$ Mpc and
$t_{\text{\tiny reg}}=\mathcal{O}(10)$ Gyr. Taking $v_{\text{\tiny in}}\sim
10^{-3}c$ [132] in Eq. (2.3), we can place very strong limits on unclustered
extremal MBHs in cosmic voids

$f_{\text{\tiny DM}}^{\text{\tiny fast,unclust}}\lesssim 10^{-8}\;.$ (2.46)

This bound is based on the requirement that there must exist no cosmic void
with a depleted magnetic field. If MBHs exist in the universe, the field
depletion is most drastic when their flux is along the cosmic void field lines
($\alpha=0$). This is the scenario we assume to place the bound in Eq. (2.46).

Cosmic void IGMFs, if they have galactic flux leakage origins and
characteristics (as quantified in Eq. (2.3)) lead to less stringent bounds
$f_{\text{\tiny DM}}^{\text{\tiny fast,unclust}}\lesssim 10^{-1}-10^{-5}$.
Note, in particular, that these bounds are independent of the mass of the
extremal MBHs and are much stronger than previous bounds in the literature
[54, 123]. No significant limits on slow MBHs can be placed from cosmic voids,
as seen trivially from Eq. (2.45). Furthermore, Eq. (2.3) limits the fraction
of dark matter in the form of non-extremal MBHs, using IGMFs produced either
primordially or via magnetizations due to void galaxies,

$f_{\text{\tiny DM}}^{\text{\tiny fast,unclust}}\lesssim
10^{-9}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny BH}}}\right)\;.$
(2.47)

Finally, if cosmic void IGMFs originate due to galactic flux leakage, we get
weaker bounds on non-extremal MBHs as $f_{\text{\tiny DM}}^{\text{\tiny
fast,unclust}}\lesssim 10^{-2}-10^{-6}\left({M_{\text{\tiny
BH}}}/{\text{kg}}\right)\cdot\left({\text{A-m}}/{Q_{\text{\tiny BH}}}\right)$.

Now, for fast unclustered extremal MBHs in the context of primordially sourced
or magnetization-induced cosmic void IGMFs, we obtain from Eq. (2.3) the
limits

$F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}\lesssim
10^{-29}\left(\frac{\text{kg}}{M_{\text{\tiny
BH}}}\right)\text{m}^{-2}\text{s}^{-1}\;.$ (2.48)

Also, from Eq. (2.3), constraints on the flux of non-extremal MBHs are
$F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}\lesssim 10^{-31}$
$\left({\text{A-m}}/{Q_{\text{\tiny BH}}}\right)$ $\text{m}^{-2}\text{s}^{-1}$
for these IGMFs.

Cosmic void IGMFs produced due to galactic flux leakage impose less stringent
constraints on the flux of fast unclustered extremal MBHs, $F_{\text{\tiny
BH}}^{\text{\tiny fast,unclust}}\lesssim
10^{-22}-10^{-26}\left({\text{kg}}/{M_{\text{\tiny BH}}}\right)$
$\text{m}^{-2}\text{s}^{-1}$, and the equivalent limits for non-extremal MBHs
are $F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}\lesssim
10^{-24}-10^{-28}$ $\left({\text{A-m}}/{Q_{\text{\tiny BH}}}\right)$
$\text{m}^{-2}\text{s}^{-1}$.

However, for slow MBHs, it is clear from Eqs. (2.44) and (2.45), that the
bounds on MBH fluxes are independent of magnetic field strength. Therefore,
IGMFs in cosmic voids, independent of their origin, limit $F_{\text{\tiny
BH}}^{\text{\tiny slow,H0}}\lesssim
10^{-21}-10^{-22}(\,{\text{kg}}/{M_{\text{\tiny
BH}}})\,\text{m}^{-2}\text{s}^{-1}$ for extremal MBHs, and $F_{\text{\tiny
BH}}^{\text{\tiny slow,H0}}\lesssim 10^{-24}-10^{-25}({M_{\text{\tiny
BH}}/{\text{kg}}})\cdot$ $(\,{\text{A-m}}/{Q_{\text{\tiny
BH}}})^{2}\,\text{m}^{-2}\text{s}^{-1}$ for non-extremal MBHs.

####  2.3.2 Parker-type bounds from cosmic filaments

Since the magnetic field strengths in cosmic web filaments are larger than in
cosmic voids, we expect relatively weaker bounds on the flux and dark matter
fraction of MBHs in this context. Again, the MBHs may be unclustered fast
($v>v_{\text{\tiny mag}}\simeq 10^{-5}$ c) or slow ($v<v_{\text{\tiny
mag}}\simeq 10^{-5}$ c), leading to different bounds. Following Eq. (2.2), we
assume typical characteristics of magnetic fields in cosmic web filaments as
$B_{\text{\tiny c}}=\mathcal{O}(10^{-9})$ G, $l_{\text{\tiny
c}}=\mathcal{O}(1)$ Mpc and $t_{\text{\tiny reg}}=\mathcal{O}(10)$ Gyr. Taking
$v_{\text{\tiny in}}\sim 10^{-3}c$ [131], we get bounds for the unclustered
(infalling) extremal MBHs from Eq. (2.3) as

$f_{\text{\tiny DM}}^{\text{\tiny fast,unclust}}\lesssim 10^{-4}\,,$ (2.49)

Again, this limit is based on the stipulation that no cosmic web filament must
exist with a depleted magnetic field. Additionally, for unclustered non-
extremal MBHs, Eq. (2.3) limits

$f_{\text{\tiny DM}}^{\text{\tiny fast,unclust}}\lesssim
10^{-5}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny BH}}}\right)\;.$
(2.50)

For virialized/clustered extremal MBHs, we get a weaker bound $f_{\text{\tiny
DM}}^{\text{\tiny fast,clust}}\lesssim\mathcal{O}(10^{-1})$ from Eq. (2.3).
Using Eq. (2.3), we get bounds on clustered non-extremal MBHs as
$f_{\text{\tiny DM}}^{\text{\tiny fast,unclust}}\lesssim
10^{-4}\left({M_{\text{\tiny BH}}}/{\text{kg}}\right)^{2}$
$\left({\text{A-m}}/{Q_{\text{\tiny BH}}}\right)^{2}$. Again, for slow
extremal MBHs we do not obtain any meaningful bounds on $f_{\text{\tiny DM}}$
from cosmic web filament IGMFs.

For fast unclustered extremal MBHs, using Eqs. (2.3) and (2.3), IGMFs in
filaments limit $F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}\lesssim
10^{-23}\,({\text{kg}}/{M_{\text{\tiny BH}}})\,\text{m}^{-2}\text{s}^{-1},$
however for non-extremal MBHs the corresponding limit is $F_{\text{\tiny
BH}}^{\text{\tiny fast,unclust}}\lesssim 10^{-25}$
$\left({\text{A-m}}/{Q_{\text{\tiny BH}}}\right)$
$\text{m}^{-2}\text{s}^{-1}$. In case when fast MBHs are virialized, we get
weaker constraints on their flux. Using Eq. (2.3), we obtain $F_{\text{\tiny
BH}}^{\text{\tiny fast,clust}}\lesssim 10^{-21}\,({\text{kg}}/{M_{\text{\tiny
BH}}})\,\text{m}^{-2}\text{s}^{-1}$ for fast clustered extremal MBHs, whereas
for the non-extremal MBHs, from Eq. (2.3), we get the limits $F_{\text{\tiny
BH}}^{\text{\tiny fast,clust}}\lesssim 10^{-24}\,({M_{\text{\tiny
BH}}}/{\text{kg}})\cdot({\text{A-m}}/{Q_{\text{\tiny
BH}}})^{2}\,\text{m}^{-2}\text{s}^{-1}$.

As previously mentioned, since the bounds on the flux of slow MBHs due to
IGMFs are independent of the magnetic field strength, we obtain similar bounds
to those found in the cosmic void case. Therefore, using Eqs. (2.44) and
(2.45), we get $F_{\text{\tiny BH}}^{\text{\tiny slow,H0}}\lesssim
10^{-22}(\,{\text{kg}}/{M_{\text{\tiny BH}}})\,\text{m}^{-2}\text{s}^{-1}$,
for extremal MBHs, and $F_{\text{\tiny BH}}^{\text{\tiny slow,H0}}\lesssim
10^{-25}({M_{\text{\tiny
BH}}/{\text{kg}}})\cdot(\,{\text{A-m}}/{Q_{\text{\tiny
BH}}})^{2}\,\text{m}^{-2}\text{s}^{-1}$ for non-extremal MBHs.

####  2.3.3 Parker-type bounds from galaxies

For galaxies, we can place bounds on extremal and non-extremal MBHs using Eqs.
(2.3)-(2.43). We find that with the presently available data, the galaxies M31
and NGC4501 furnish the strongest galactic bounds on MBHs. The field
characteristics of NGC4501 are [133, 134]

$B^{\text{\tiny NGC4501}}\simeq 10^{-6}\,\text{G}\,,\quad l_{\text{\tiny
c}}^{\text{\tiny NGC4501}}\gtrsim 100\,\text{kpc}\,,\quad t_{\text{\tiny
reg}}^{\text{\tiny NGC4501}}\simeq 2.5\,\text{Gyr}\;.$ (2.51)

From Eq. (2.3) this gives a bound for fast unclustered extremal MBHs

$f_{\text{\tiny DM}}^{\text{\tiny fast,unclust}}\lesssim 10^{-3}\,.$ (2.52)

For slow extremal MBHs, comparable bounds are obtained as $f_{\text{\tiny
DM}}^{\text{\tiny slow}}\lesssim 10^{-3}$. However, for virialized extremal
MBHs we get a weaker limit of $f_{\text{\tiny DM}}^{\text{\tiny
fast,clust}}\lesssim 10^{-2}$. Corresponding limits for non-extremal MBHs due
to NGC4501 can be derived using Eqs. (2.3), (2.3), and (2.42), which gives

$f_{\text{\tiny DM}}^{\text{\tiny
fast,unclust}}\lesssim\frac{10^{-6}}{1+0.6\cos\alpha\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)^{2}\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{2}\;.$ (2.53)

Additionally, for virialized non-extremal MBHs, we get the limits
$f_{\text{\tiny DM}}^{\text{\tiny fast,clust}}\lesssim
10^{-6}\left({M_{\text{\tiny
BH}}}/{\text{kg}}\right)^{2}\cdot\left({\text{A-m}}/{Q_{\text{\tiny
BH}}}\right)^{2}$, while for slow non-extremal MBHs, we get $f_{\text{\tiny
DM}}^{\text{\tiny slow}}\lesssim 10^{-6}\left({M_{\text{\tiny
BH}}}/{\text{kg}}\right)^{3/2}\cdot\left({\text{A-m}}/{Q_{\text{\tiny
BH}}}\right)^{3/2}$. Similar bounds on clustered MBHs have been derived
before, studying specific galaxies such as M31 [53, 54].

Moreover, NGC4501 puts strong constraints on the slow extremal MBH flux,

$F_{\text{\tiny BH}}^{\text{\tiny slow}}\lesssim
10^{-20}\left(\frac{\text{kg}}{M_{\text{\tiny
BH}}}\right)\text{m}^{-2}\text{s}^{-1}\;.$ (2.54)

Again, for fast unclustered extremal MBHs, we get similar limits
$F_{\text{\tiny BH}}^{\text{\tiny fast,unclust}}\lesssim
10^{-20}\left({\text{kg}}/{M_{\text{\tiny BH}}}\right)$
$\text{m}^{-2}\text{s}^{-1}$. However, for virialized MBHs we get weaker
limits of $F_{\text{\tiny BH}}^{\text{\tiny fast,clust}}\lesssim 10^{-19}$
$\left({\text{kg}}/{M_{\text{\tiny BH}}}\right)\text{m}^{-2}\text{s}^{-1}$.
The limits for the non-extremal MBH flux can be obtained from Eqs. (2.3),
(2.3), and (2.42). For fast, non-extremal, unclustered MBHs, NGC4501 bounds

$F_{\text{\tiny BH}}^{\text{\tiny
fast,unclust}}\lesssim\frac{10^{-22}}{1+0.6\cos\alpha\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)}\left(\frac{M_{\text{\tiny
BH}}}{\text{kg}}\right)\left(\frac{\text{A-m}}{Q_{\text{\tiny
BH}}}\right)^{2}\;.$ (2.55)

For slow non-extremal MBHs, we get $F_{\text{\tiny BH}}^{\text{\tiny
slow}}\lesssim 10^{-22}\left({\text{A-m}}/{Q_{\text{\tiny BH}}}\right)$, and
for fast virialized non-extremal MBHs, we get $F_{\text{\tiny
BH}}^{\text{\tiny fast,clust}}\lesssim 10^{-22}\left({M_{\text{\tiny
BH}}}/{\text{kg}}\right)\cdot\left({\text{A-m}}/{Q_{\text{\tiny
BH}}}\right)^{2}$.

In summary, in this section, we have predominantly explored bounds on MBHs in
cosmic web filaments and cosmic voids. These bounds are both qualitatively and
quantitatively different from previous bounds [75] based on intracluster (IC)
magnetic fields. While the effect of magnetic monopoles on IGMFs has been
recently explored in [78], these bounds are relevant for monopoles of masses
$M\lesssim\mathcal{O}(10^{19})\,\text{GeV}$. In contrast, the bounds on
extremal MBHs are applicable for mass $M_{\text{\tiny BH}}\gtrsim
10^{-6}\,\text{Kg}\sim\mathcal{O}(10^{20})\,\text{GeV}$ [16, 23]. The present
study is, therefore, complementary to these earlier studies.

Having established the constraints on dark matter fraction constituted by
MBHs, let us now investigate potential observational signatures of MBHs due to
Faraday rotation effects. In the next section, we will explore and
characterize the Faraday rotation properties of MBHs and juxtapose their
signatures with magnetic NS. As we will demonstrate subsequently, their
polarization angle maps exhibit significant differences. This characteristic
will be an observational tool to distinguish between an MBH and an NS.

##  3 Faraday rotations due to primordial magnetic black holes

In the last section, we have explored the constraints on MBHs due to galactic,
extra-galactic, and cosmic magnetic fields. These suggest that the population
of such objects may be very small. We would, therefore, like to envision
strategies to look for such rare objects in observational data. We now
consider one possible signature of MBHs that may be unique to them and
observationally relevant.

A germane question may be whether an MBH may be discerned from other
astrophysical bodies that may mimic some of its characteristics, such as a
neutron star. An intriguing effect arises when we consider the interaction of
MBHs with light. When a linearly polarized electromagnetic wave propagates
through a magnetized medium, its plane of polarization rotates along its
trajectory. This effect is commonly referred to as Faraday rotation. It occurs
because the left and right circular components of the linearly polarized waves
have distinct velocities in the medium, resulting in a rotation of the plane
of polarization along the trajectory of the wave. In realistic astrophysical
scenarios, like for an NS, the background magnetic field may be non-uniform
[135]. This may lead to a spatial dependence of the polarization angle in the
plane perpendicular to the direction of motion. The spatial variation in
polarization angle will depend on the characteristics of the magnetic field
and, by implication, its specific astrophysical source.

To begin the analyses, we will first review, in some detail, the occurrence of
Faraday rotation in a plasma threaded by a spatially non-uniform magnetic
field. Then, we will investigate two scenarios for the source of the non-
uniform magnetic field—an MBH and an NS. We will compare the Faraday effects
in these two cases in detail.

First, let us explore Faraday rotation on an electromagnetic wave passing
through an inhomogeneous plasma in an arbitrary non-uniform magnetic field. We
follow the methodology outlined in [136], with the detailed calculations
provided in Appendix A. Consider an electromagnetic wave propagating through
an inhomogeneous, weakly ionized hydrogen plasma333For a warm hydrogen plasma,
[137] has shown that away from resonances, the correction to the electron
contribution to Faraday rotation is negligible compared to the cold plasma
term. along the $z$ axis. We can take the background magnetic field as

$\vec{\mathcal{B}}_{\text{\tiny ext.}}(\vec{r})=B_{\text{\tiny
ext,x}}(\vec{r})~{}\hat{x}+B_{\text{\tiny
ext,y}}(\vec{r})~{}\hat{y}+B_{\text{\tiny ext,z}}(\vec{r})~{}\hat{z}\;,$ (3.1)

and, define $B_{\text{\tiny ext.}}(\vec{r})\equiv\sqrt{B_{\text{\tiny
ext,x}}^{2}+B_{\text{\tiny ext,y}}^{2}+B_{\text{\tiny ext,z}}^{2}}$. The
electrons in the plasma experience a Lorentz force due to the propagating
electromagnetic waves. The Lorentz force is given by,

$m\text{\tiny e}\frac{d^{2}\vec{r}_{\text{\tiny
e}}}{dt^{2}}=-e\left(\vec{E}+\frac{d\vec{r}_{\text{\tiny
e}}}{dt}\times\vec{B}\right)\;,$ (3.2)

where $\vec{r}_{\text{\tiny e}}$, $m_{\text{\tiny e}}$ and $-e$ are the
position, mass, and charge of electrons, respectively, and $\vec{E}$ and
$\vec{B}$ are the total electric and magnetic fields. Since the protons are
much heavier than electrons, their oscillations will be suppressed and,
therefore, will have a negligible contribution to the Faraday rotation.

In a steady state, we can now solve this Lorentz force equation by considering
an oscillating perturbation around the background solution. This is given by

$\displaystyle\vec{r}_{\text{\tiny e}}$ $\displaystyle=$
$\displaystyle\vec{r}^{\,\text{(0)}}_{\text{\tiny
e}}+\vec{r}^{\,\text{(1)}}_{\text{\tiny e}}e^{-i\omega t}\;,$
$\displaystyle\vec{E}(\vec{r})$ $\displaystyle=$ $\displaystyle
0+\vec{E}^{\,\text{(1)}}(\vec{r})e^{-i\omega t}\;,$ (3.3)
$\displaystyle\vec{B}(\vec{r})$ $\displaystyle=$
$\displaystyle\vec{\mathcal{B}}_{\text{\tiny
ext.}}(\vec{r})+\vec{B}^{\,\text{(1)}}(\vec{r})e^{-i\omega t}\;,$

where $\vec{r}^{\,\text{(0)}}_{\text{\tiny e}}$ and
$\vec{r}^{\,\text{(1)}}_{\text{\tiny e}}$ are the mean position and amplitude
of displacement of the electron, respectively, and $\omega$,
$\vec{E}^{\,\text{(1)}}$ and $\vec{B}^{\,\text{(1)}}$, are respectively the
frequency, the electric field and the magnetic field of the traversing wave.
Using Eqs. (3.2), and (3), we then get

$\frac{e\vec{E}^{\,\text{(1)}}}{m_{\text{\tiny
e}}}=\omega^{2}\vec{r}^{\,\text{(1)}}_{\text{\tiny
e}}+\frac{ie\omega}{m_{\text{\tiny
e}}}\left(\vec{r}^{\,\text{(1)}}_{\text{\tiny
e}}\times\vec{\mathcal{B}}_{\text{\tiny ext.}}(\vec{r})\right)\;.$ (3.4)

The above equation can be solved to obtain the displacement
$\vec{r}^{\,\text{(1)}}_{\text{\tiny e}}$ in terms of the electric field
$\vec{E}^{\,\text{(1)}}$ of the passing wave. Furthermore, these electron
oscillations will produce a time-dependent dipole moment with an amplitude
$\vec{P}\equiv-\mathcal{N}_{\text{\tiny
e}}e\vec{r}^{\,\text{(1)}}_{\text{\tiny e}}$, where $\mathcal{N}_{\text{\tiny
e}}$ is the number density of electrons. We can obtain the dielectric tensor
using the relation between the induced dipole moment and the dielectric
tensor, i.e., $\vec{P}=\epsilon_{0}({\bm{\epsilon}_{\text{\tiny
r}}}-{\bm{I}})\cdot\vec{E}^{\,\text{(1)}}$, where $\epsilon_{\text{\tiny r}}$,
$\epsilon_{0}$, and ${\bm{I}}$ are the dielectric tensor, the vacuum
permittivity and the identity matrix respectively. Following Eqs. (A.4)-(A.8),
we obtain the dielectric tensor as

${\bm{\epsilon}_{\text{\tiny
r}}}=\begin{pmatrix}1-\frac{\omega^{2}_{\text{\tiny
p}}\left(\omega^{2}-\tilde{\omega}^{2}_{\text{\tiny
x}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}&\frac{\omega^{2}_{\text{\tiny
p}}\left(\tilde{\omega}_{\text{\tiny x}}\tilde{\omega}_{\text{\tiny
y}}+i\omega\tilde{\omega}_{\text{\tiny
z}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}&\frac{\omega^{2}_{\text{\tiny
p}}\left(\tilde{\omega}_{\text{\tiny x}}\tilde{\omega}_{\text{\tiny
z}}-i\omega\tilde{\omega}_{\text{\tiny
y}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}\\\
\frac{\omega^{2}_{\text{\tiny p}}\left(\tilde{\omega}_{\text{\tiny
x}}\tilde{\omega}_{\text{\tiny y}}-i\omega\tilde{\omega}_{\text{\tiny
z}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}&1-\frac{\omega^{2}_{\text{\tiny
p}}\left(\omega^{2}-\tilde{\omega}^{2}_{\text{\tiny
y}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}&\frac{\omega^{2}_{\text{\tiny
p}}\left(\tilde{\omega}_{\text{\tiny y}}\tilde{\omega}_{\text{\tiny
z}}+i\omega\tilde{\omega}_{\text{\tiny
x}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}\\\
\frac{\omega^{2}_{\text{\tiny p}}\left(\tilde{\omega}_{\text{\tiny
x}}\tilde{\omega}_{\text{\tiny z}}+i\omega\tilde{\omega}_{\text{\tiny
y}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}&\frac{\omega^{2}_{\text{\tiny
p}}\left(\tilde{\omega}_{\text{\tiny y}}\tilde{\omega}_{\text{\tiny
z}}-i\omega\tilde{\omega}_{\text{\tiny
x}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}&1-\frac{\omega^{2}_{\text{\tiny
p}}\left(\omega^{2}-\tilde{\omega}^{2}_{\text{\tiny
z}}\right)}{\omega^{2}\left(\omega^{2}-\tilde{\omega}^{2}\right)}\\\
\end{pmatrix}\;.$ (3.5)

Here, $\tilde{\omega}_{\text{\tiny i}}\equiv eB_{\text{\tiny
ext,i}}(\vec{r})/m_{\text{\tiny e}}$ and $\tilde{\omega}\equiv eB_{\text{\tiny
ext.}}(\vec{r})/m_{\text{\tiny e}}$ are the component-wise and total electron
cyclotron frequencies. $\omega_{\text{\tiny p}}^{2}\equiv
e^{2}\mathcal{N}_{\text{\tiny e}}/(\epsilon_{0}m_{\text{\tiny e}})$ is the
electron plasma frequency.

Evidently, due to the presence of a background magnetic field, the plasma is
anisotropic in nature. In such cases, a medium has different refractive
indices along different directions. To understand this behaviour, we will
solve the Maxwell’s equations to determine the characteristic modes and the
corresponding refractive indices.

The Maxwell equations in the present context may be expressed as

$\displaystyle\nabla\cdot\vec{E}$ $\displaystyle=$
$\displaystyle\frac{e}{\epsilon_{0}}(\mathcal{N}_{\text{\tiny
p}}-\mathcal{N}_{\text{\tiny e}})\;,$ (3.6) $\displaystyle\nabla\cdot\vec{B}$
$\displaystyle=$ $\displaystyle\mu_{0}Q_{\text{\tiny
BH}}\delta^{3}(\vec{r})\;,$ (3.7)
$\displaystyle\nabla\times\vec{E}+\frac{\partial\vec{B}}{\partial t}$
$\displaystyle=$ $\displaystyle 0\;,$ (3.8)
$\displaystyle\nabla\times\vec{B}-\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}$ $\displaystyle=$
$\displaystyle\mu_{0}\vec{J}=\mu_{0}e(\mathcal{N}_{\text{\tiny
p}}\vec{v}_{\text{\tiny p}}-\mathcal{N}_{\text{\tiny e}}\vec{v}_{\text{\tiny
e}})\;.$ (3.9)

Here, $Q_{\text{\tiny BH}}$ is the monopole magnetic charge, which may be one
of the sources for the background magnetic field. Additionally,
$\mathcal{N}_{\text{\tiny p}}$ and $\mathcal{N}_{\text{\tiny e}}$ represent
the number densities of protons and electrons, while $\vec{v}_{\text{\tiny
p}}$ and $\vec{v}_{\text{\tiny e}}$ denote the velocities of protons and
electrons, respectively. Once again, we can neglect $\vec{v}_{\text{\tiny
p}}$, since protons being heavier than electrons lead to $v_{\text{\tiny
p}}\ll v_{\text{\tiny e}}$.

Eqs. (3.8) and (3.9) can be linearized using Eq. (3) and they can be
simplified to obtain (See Appendix A for details)

$\nabla\left(\nabla\cdot\vec{E}^{\,\text{(1)}}\right)-\nabla^{2}\vec{E}^{\,\text{(1)}}-\frac{\omega^{2}}{c^{2}}{\bm{\epsilon}_{\text{\tiny
r}}}\cdot\vec{E}^{\,\text{(1)}}=0\;.$ (3.10)

We can further write $\vec{E}^{\,\text{(1)}}(\vec{r})\propto
e^{i\psi_{\text{\tiny ph.}}(\vec{r})}$ for a plane electromagnetic wave. Here,
$\psi_{\text{\tiny ph.}}$ is the phase of the electric field and is related to
the wave vector as $\vec{k}=\nabla\psi_{\text{\tiny ph.}}(\vec{r})$. Putting
it in Eq. (3.10) and taking the eikonal limit, i.e. $|\nabla{k}|/k^{2}\ll 1$,
we get

$\left(\vec{k}\cdot\vec{E}^{\,\text{(1)}}\right)\vec{k}-k^{2}\vec{E}^{\,\text{(1)}}+\frac{\omega^{2}}{c^{2}}{\bm{\epsilon}_{\text{\tiny
r}}}\cdot\vec{E}^{\,\text{(1)}}\approx 0\;.$ (3.11)

Here, $k\equiv\absolutevalue{\vec{k}}$ is the magnitude of the wave vector,
and note that it is also a function of $\vec{r}$. For a wave propagating along
the $z$ direction, the wave vector takes the form
$\vec{k}=(0,0,d\psi_{\text{\tiny ph.}}(\vec{r})/dz)$ and the phase can be
written as

$\psi_{\text{\tiny ph.}}(\vec{r})=\int dz~{}k(\vec{r})=\frac{c}{\omega}\int
dz~{}n(\vec{r})\;.$ (3.12)

Here, $n(\vec{r})\equiv ck(\vec{r})/\omega$ is the refractive index of the
wave. Using Eqs. (3.5) and (3.11), we get the refractive indices as

$n_{{(\pm)}}(\vec{r})=\left(1-\frac{\omega^{2}_{\text{\tiny
p}}\left(\omega^{2}-\omega^{2}_{\text{\tiny
p}}\right)}{\omega^{2}\left(\omega^{2}-\omega^{2}_{\text{\tiny
p}}-\frac{1}{2}\left(\tilde{\omega}^{2}_{\text{\tiny
x}}+\tilde{\omega}^{2}_{\text{\tiny
y}}\right)\pm\left(\frac{1}{4}\left(\tilde{\omega}^{2}_{\text{\tiny
x}}+\tilde{\omega}^{2}_{\text{\tiny
y}}\right)^{2}+(\omega^{2}-\omega^{2}_{\text{\tiny
p}})^{2}\frac{\tilde{\omega}^{2}_{\text{\tiny
z}}}{\omega^{2}}\right)^{1/2}\right)}\right)^{1/2}\;.$ (3.13)

Here, $n_{{(\pm)}}$ are the refractive indices for the two characteristic
modes. Notice that the refractive index is symmetric about the exchange of
$\tilde{\omega}_{\text{\tiny x}}$ and $\tilde{\omega}_{\text{\tiny y}}$.
Furthermore, its dependence on $\tilde{\omega}_{\text{\tiny z}}$ is different
compared to $\tilde{\omega}_{\text{\tiny x}}$ and $\tilde{\omega}_{\text{\tiny
y}}$, this is due to our choice of the direction of propagation being along
the $z$-axis. Additionally, as we will see later, these two characteristic
modes will be further divided into branches, with our focus being solely on
the propagating branches.

Figure 2: Refractive indices (given by Eq. (3.13)) as a function of $\omega$
in a magnetic, cold, inhomogeneous plasma medium. Red and blue curves
represent $(+)$ and $(-)$ modes respectively. We have set $\omega_{\text{\tiny
p}}=\tilde{\omega}_{\text{\tiny x}}=\tilde{\omega}_{\text{\tiny
y}}=\tilde{\omega}_{\text{\tiny z}}=1~{}\text{hz}$. There are four modes
$(+)_{\text{\tiny Low(High)}}$ and $(-)_{\text{\tiny Low(High)}}$ separated by
cut-offs and resonances. The resonances are represented by vertical lines.

Let us now briefly discuss the cut-offs and the resonances of these modes. The
cut-offs are obtained by taking $n_{\pm}=0$. For a particular branch, the
refractive index is imaginary below the cut-off frequencies, resulting in
complete attenuation of the branch in the plasma at those frequencies. For the
$(+)$ mode, a cut-off exists at

$\omega_{\text{\tiny$(+)$,cut.}}\equiv\omega_{\text{\tiny p}}\;.$ (3.14)

However, for the $(-)$ mode, there exist two cut-offs at

$\omega_{\text{\tiny$(-)$,Low.cut.}}\equiv-\frac{\tilde{\omega}}{2}+\frac{1}{2}\sqrt{\tilde{\omega}^{2}+4\omega_{\text{\tiny
p}}^{2}}\;,\quad\omega_{\text{\tiny$(-)$,High.cut.}}\equiv\frac{\tilde{\omega}}{2}+\frac{1}{2}\sqrt{\tilde{\omega}^{2}+4\omega_{\text{\tiny
p}}^{2}}\;.$ (3.15)

Apart from these cut-offs, there exist resonances when $n_{\pm}\to\infty$.
They occur at characteristic frequencies

$\omega_{\text{\tiny($\pm$),res.}}=\frac{\sqrt{\tilde{\omega}^{2}+\omega_{\text{\tiny
p}}^{2}\mp\sqrt{\left(\tilde{\omega}^{2}+\omega_{\text{\tiny
p}}^{2}\right)^{2}-4\tilde{\omega}^{2}_{\text{\tiny z}}\omega^{2}_{\text{\tiny
p}}}}}{\sqrt{2}}\;.$ (3.16)

At these resonances, electrons gyrate synchronously with the wave, allowing
them to continuously extract energy and causing the waves to be absorbed by
the plasma. As shown in Fig. 2, these frequency cutoffs and resonances divide
$(\pm)$ modes into lower and higher frequency branches. Therefore, the higher
frequency branches exist for

$\omega_{\text{\tiny$(+)$,High.}}>\omega_{\text{\tiny$(+)$,cut.}}\;,\hskip
30.00005pt\omega_{\text{\tiny$(-)$,High.}}>\omega_{\text{\tiny$(-)$,High.cut.}}\;.$
(3.17)

while the lower frequency branches exist for

$0<\omega_{\text{\tiny$(+)$,Low.}}<\omega_{\text{\tiny$(+)$,res.}}\;,\hskip
30.00005pt\omega_{\text{\tiny$(-)$,Low.cut.}}<\omega_{\text{\tiny$(-)$,Low.}}<\omega_{\text{\tiny$(-)$,res.}}\;.$
(3.18)

The $(+)$ mode is non-propagating at frequencies
$\omega_{\text{\tiny$(+)$,res.}}<\omega_{\text{\tiny$(+)$}}<\omega_{\text{\tiny$(+)$,cut.}}$,
while the $(-)$ mode is non-propagating at frequencies
$\omega_{\text{\tiny$(-)$,res.}}<\omega_{\text{\tiny$(-)$}}<\omega_{\text{\tiny$(-)$,High.cut.}}$.
Furthermore, if a wave, which has frequency
$\omega<\omega_{\text{\tiny($\pm$),res.}}$, travels along the direction of
decreasing magnetic field or equivalently decreasing $\tilde{\omega}$, then
along its propagation, we see from Eq. (3.16),
$\omega_{\text{\tiny($\pm$),res.}}$ also decreases. Hence, when
$\omega_{\text{\tiny($\pm$),res.}}$ becomes equal to $\omega$, the resonance
occurs, resulting in absorption of the corresponding mode. Consequently,
throughout propagation, the wave frequency must exceed
$\omega_{\text{\tiny($\pm$),res.}}$. As mentioned before, only higher
frequency branches exist for $\omega>\omega_{\text{\tiny($\pm$),res.}}$, and
therefore solely these branches will propagate, which is pertinent to our
context.

Based on the above discussions, we will restrict ourselves to those
frequencies where higher branches, for both $(\pm)$ modes, exist. This happens
for frequencies satisfying

$\omega>\omega_{\text{\tiny$(-)$,High.
cut.}}=\frac{\tilde{\omega}}{2}+\frac{1}{2}\sqrt{\tilde{\omega}^{2}+4\omega_{\text{\tiny
p}}^{2}}\;.$ (3.19)

Notice, from Eq. (3.15) $\omega_{\text{\tiny$(-)$,High.
cut.}}>\text{max}\left({\omega_{\text{\tiny p}},\tilde{\omega}}\right)$,
consequently the relevant frequencies for propagation will have
$\omega>\text{max}\left({\omega_{\text{\tiny p}},\tilde{\omega}}\right)$.
Additionally, from Eqs. (3.5) and (3.11), one can obtain the relation between
the transverse components of the characteristic modes as

$\left(\frac{E_{\text{x}}^{\,\text{(1)}}}{E_{\text{y}}^{\,\text{(1)}}}\right)_{{(\pm)}}=\frac{i\left(\omega\left(\tilde{\omega}_{\text{\tiny
x}}^{2}-\tilde{\omega}_{\text{\tiny
y}}^{2}\right)\pm\sqrt{{4\tilde{\omega}_{\text{\tiny
z}}^{2}\left(\omega^{2}-\omega_{\text{\tiny
p}}^{2}\right)^{2}}+\omega^{2}\left(\tilde{\omega}_{\text{\tiny
x}}^{2}+\tilde{\omega}_{\text{\tiny
y}}^{2}\right)^{2}}\right)}{2\left(\left(\omega^{2}-\omega_{\text{\tiny
p}}^{2}\right)\tilde{\omega}_{\text{\tiny
z}}+i\omega\tilde{\omega}_{\text{\tiny x}}\tilde{\omega}_{\text{\tiny
y}}\right)}\;.$ (3.20)

Since $\omega>\tilde{\omega}$, and $\tilde{\omega}$ decreases away from the
magnetic field source, we may expand the above expression in powers of
$\tilde{\omega}_{\text{\tiny(x,y,z)}}/\omega$, as given by Eq. (A.23). Then,
it is revealed that the characteristic waves approximately manifest as left
and right circular polarizations, i.e.
$E_{\text{x}}^{\,\text{(1)}}/E_{\text{y}}^{\,\text{(1)}}\simeq\pm
i\,\text{sgn}(\tilde{\omega}_{\text{\tiny z}})$, when

$\absolutevalue{\frac{\omega\left(\tilde{\omega}_{\text{\tiny
x}}^{2}+\tilde{\omega}_{\text{\tiny
y}}^{2}\right)}{2(\omega^{2}-\omega^{2}_{\text{\tiny
p}})\tilde{\omega}_{\text{\tiny z}}}}\ll 1\;.$ (3.21)

Therefore, in the above limit, we infer that for $\tilde{\omega}_{\text{\tiny
z}}>0$, $(+)$ and $(-)$ modes are close to left and right circular modes,
respectively. However, for $\tilde{\omega}_{\text{\tiny z}}<0$, $(+)$ and
$(-)$ modes are close to right and left circular modes. We can therefore
write,

$n_{\text{\tiny L(R)}}=\begin{cases}n_{{+(-)}}\;;&\tilde{\omega}_{\text{\tiny
z}}>0\;,\\\ n_{{-(+)}}\;;&\tilde{\omega}_{\text{\tiny z}}<0\;.\\\ \end{cases}$
(3.22)

Here, $n_{\text{\tiny L(R)}}$ are the refractive indices for left and right
circular modes. It is evident from Eqs. (3.13) and (3.22) that the refractive
index for left and right circularly polarized waves are distinct, and for
$\tilde{\omega}_{\text{\tiny z}}>0$, $n_{\text{\tiny L}}>n_{\text{\tiny R}}$
and for $\tilde{\omega}_{\text{\tiny z}}<0$, $n_{\text{\tiny
L}}<n_{\text{\tiny R}}$. Again, this dependence specifically on the $z$
component of the magnetic field comes from our choice of the direction of
propagation.

With these results, let us calculate an expression for the expected change in
the polarization angle. If the light source is linearly polarized, one can
analyse the propagation of the left and right circularly polarised wave
components in the medium and then compare the $x$ and $y$ components in the
linear basis to obtain the polarization angle (See Appendix A for details).
Therefore, the change in polarization angle of the passing electromagnetic
wave is given by

$\psi_{\text{\tiny pol.}}=\frac{\omega}{2c}\int~{}dz\left(n_{\text{\tiny
L}}(\vec{r})-n_{\text{\tiny R}}(\vec{r})\right)\;.$ (3.23)

Here, the integration is over the path of the electromagnetic wave. We have
neglected the influence of gravitational lensing on the trajectory of the
electromagnetic wave. Our estimates suggest that for the lensed trajectory,
$\psi_{\text{\tiny pol.}}$ differs only by up to $\mathcal{O}(10^{-3})\,\%$
for the masses pertinent to our study. Now, since the left and right circular
waves have distinct refractive indices, and moreover, may even each change
along the path, the $\psi_{\text{\tiny pol.}}$ given by Eq. (3.23) will evolve
along the wave’s trajectory. If the initial polarization angle of the source
is $\psi_{\text{\tiny pol,0}}$, the observer will detect the electromagnetic
wave with polarization,

$\psi_{\text{\tiny obs.}}=\psi_{\text{\tiny pol,0}}+\psi_{\text{\tiny
pol.}}\;.$ (3.24)

The rotation measure (RM), which is a closely associated quantity, is defined
as

$\text{RM}(\lambda)\equiv\frac{d\psi_{\text{\tiny
pol.}}(\lambda)}{d\lambda^{2}}\;.$ (3.25)

Here, $\lambda$ is the wavelength of light. The RM is useful for determining
the properties of both the magnetic field and the plasma density along the
wave’s path.

Let us now calculate explicit expressions for these quantities of interest in
our context. As mentioned before, the frequencies of interests are
$\omega>\text{max}\left({\omega_{\text{\tiny p}},\tilde{\omega}}\right)$ and
in the limit when $\omega\gg\text{max}\left({\omega_{\text{\tiny
p}},\tilde{\omega}}\right)$, using Eqs. (3.13) and (3.22) we get

$n_{\text{\tiny L}}(\vec{r})-n_{\text{\tiny
R}}(\vec{r})\simeq\frac{\omega^{2}_{\text{\tiny
p}}}{\omega^{3}}\tilde{\omega}_{\text{\tiny z}}+\frac{\omega^{2}_{\text{\tiny
p}}\tilde{\omega}_{\text{\tiny
z}}}{\omega^{5}}\left(\tilde{\omega}^{2}_{\text{\tiny
x}}+\tilde{\omega}^{2}_{\text{\tiny y}}+\frac{\tilde{\omega}^{2}_{\text{\tiny
x}}\tilde{\omega}^{2}_{\text{\tiny y}}}{4\tilde{\omega}^{2}_{\text{\tiny
z}}}\right)+\mathcal{O}\left(\left(\frac{\omega^{2}_{\text{\tiny
p}}}{\omega^{2}}\right)^{2},\left(\frac{\tilde{\omega}}{\omega}\right)^{4}\right)\;.$
(3.26)

The change in polarization may hence be written, using Eq. (3.23), as

$\psi_{\text{\tiny pol.}}\simeq\int dz~{}\left(\frac{\omega^{2}_{\text{\tiny
p}}}{2\omega^{2}c}\tilde{\omega}_{\text{\tiny
z}}+\frac{\omega^{2}_{\text{\tiny p}}\tilde{\omega}_{\text{\tiny
z}}}{2\omega^{4}c}\left(\tilde{\omega}^{2}_{\text{\tiny
x}}+\tilde{\omega}^{2}_{\text{\tiny y}}+\frac{\tilde{\omega}^{2}_{\text{\tiny
x}}\tilde{\omega}^{2}_{\text{\tiny y}}}{4\tilde{\omega}^{2}_{\text{\tiny
z}}}\right)\right)+\mathcal{O}\left(\left(\frac{\omega^{2}_{\text{\tiny
p}}}{\omega^{2}}\right)^{2},\left(\frac{\tilde{\omega}}{\omega}\right)^{4}\right)\;.$
(3.27)

This may be further simplified in terms of the background magnetic field as

$\displaystyle\psi_{\text{\tiny pol.}}$ $\displaystyle\simeq$
$\displaystyle\frac{e^{3}\lambda^{2}}{8\pi^{2}\epsilon_{0}m^{2}_{\text{\tiny
e}}c^{3}}\int dz~{}\mathcal{N}_{\text{\tiny e}}(r)B_{\text{\tiny z}}(\vec{r})$
$\displaystyle+\frac{e^{5}\lambda^{4}}{32\pi^{4}\epsilon_{0}m^{4}_{\text{\tiny
e}}c^{5}}\int dz~{}\mathcal{N}_{\text{\tiny e}}(r)B_{\text{\tiny
z}}(\vec{r})\left(B_{\text{\tiny x}}(\vec{r})^{2}+B_{\text{\tiny
y}}(\vec{r})^{2}+\frac{B_{\text{\tiny x}}(\vec{r})^{2}B_{\text{\tiny
y}}(\vec{r})^{2}}{4B_{\text{\tiny z}}(\vec{r})^{2}}\right)\;.$

For the RM, using Eq. (3.25), we get

$\displaystyle\text{RM}(\lambda)$ $\displaystyle\simeq$
$\displaystyle\frac{e^{3}}{8\pi^{2}\epsilon_{0}m^{2}_{\text{\tiny
e}}c^{3}}\int dz~{}\mathcal{N}_{\text{\tiny e}}(r)B_{\text{\tiny z}}(\vec{r})$
$\displaystyle+\frac{e^{5}\lambda^{2}}{16\pi^{4}\epsilon_{0}m^{4}_{\text{\tiny
e}}c^{5}}\int dz~{}\mathcal{N}_{\text{\tiny e}}(r)B_{\text{\tiny
z}}(\vec{r})\left(B_{\text{\tiny x}}(\vec{r})^{2}+B_{\text{\tiny
y}}(\vec{r})^{2}+\frac{B_{\text{\tiny x}}(\vec{r})^{2}B_{\text{\tiny
y}}(\vec{r})^{2}}{4B_{\text{\tiny z}}(\vec{r})^{2}}\right)\;.$

We note that in the limit $\omega\gg\text{max}\left({\omega_{\text{\tiny
p}},\tilde{\omega}}\right)$, one obtains the dominant contribution to the
change in the polarisation angle and the RM from the component of the magnetic
field along the wave’s trajectory. However, for comprehensiveness in all the
numerical analyses, we will always use Eqs. (3.13), (3.23) and (3.25) directly
which take into account all the components. Also, we will work in the limit
where the modes are propagating and close to circular modes. These
requirements are embodied in Eqs. (3.19) and (3.21).

We will now consider two sources for the background magnetic field—an NS and
an MBH. In turn, we will compute the change of the polarization angle and the
RM assuming these two sources and compare the spatial dependencies. This
spatial dependence may be useful in observationally distinguishing between an
NS and an MBH. Apart from this, we will also establish a quantitative measure
based on the spatial dependence of the polarisation angle to further
discriminate them.

###  3.1 Faraday rotations due to a neutron star

Let us now compute the Faraday effect produced by an NS, which, for our
purposes, can approximately be modelled as a magnetic dipole. Consider the
schematic of Fig. 3—the origin is taken at the location of the NS, and an
observer O is located at a distance $d_{\text{\tiny Q}}$ from it. The NS has a
magnetic dipole moment $\vec{\mathfrak{m}}$ and mass $M_{\text{\tiny NS}}$. A
source (S) located at a distance $d_{\text{\tiny QS}}$ from the NS is emitting
plane-polarized light. We denote by $d_{\text{\tiny S}}$ the distance between
the light source (S) and the observer, and by $\xi$ the distance of closest
approach of light with the NS. In this setup, we assume that $d_{\text{\tiny
S}}\simeq d_{\text{\tiny Q}}\gg\xi$, and we can write $d_{\text{\tiny
QS}}\simeq d_{\text{\tiny S}}-d_{\text{\tiny Q}}$. Furthermore, we assume that
there is a hydrogen plasma background with electron density $N_{\text{\tiny
e}}(\vec{r})$ permeating the region under consideration.

The orientation of the magnetic moment of the NS is determined by the polar
angle $\theta_{\text{\tiny NS}}$ and the azimuthal angle $\phi_{\text{\tiny
NS}}$. We can, therefore, write

$\vec{\mathfrak{m}}=\mathfrak{m}\left(\sin\theta_{\text{\tiny
NS}}\cos\phi_{\text{\tiny NS}}\hat{x}+\sin\theta_{\text{\tiny
NS}}\sin\phi_{\text{\tiny NS}}\hat{y}+\cos\theta_{\text{\tiny
NS}}\hat{z}\right)\;,$ (3.30)

where $\mathfrak{m}$ is the magnitude of the magnetic dipole moment. The
magnetic field at an arbitrary point A with spherical coordinates
$(r,\theta,\phi)$ is then given by

$\vec{B}_{\text{\tiny
NS}}(\vec{r})=\frac{\mu_{0}}{4\pi}\left(\frac{3(\vec{\mathfrak{m}}\cdot\vec{r})\vec{r}}{r^{5}}-\frac{\vec{\mathfrak{m}}}{r^{3}}\right)\;.$
(3.31)

Figure 3: Schematic for Faraday rotation analysis in the case of a NS. An NS
with magnetic moment $\vec{\mathfrak{m}}$ is at the origin. An observer (O) is
located at a distance $d_{\text{\tiny Q}}$ from it, and a source (S) emitting
linearly polarized light is located at a distance $d_{\text{\tiny S}}$ from
the observer. The distance between the NS and the source is $d_{\text{\tiny
QS}}$. The light rays are moving along the $z$ direction, with an impact
parameter $\xi=\sqrt{x^{2}+y^{2}}$ with respect to the NS.

Notice that the magnetic field scales as

$\absolutevalue{\vec{B}_{\text{\tiny
NS}}}\sim\frac{\mu_{0}}{4\pi}\frac{\mathfrak{m}}{r^{3}}\;.$ (3.32)

Furthermore, the cyclotron frequency of electrons at a radial distance $r$ due
to the presence of an NS in the plasma is given by

$\tilde{\omega}_{\text{\tiny NS}}(\vec{r})=\absolutevalue{\frac{\mu_{0}e}{4\pi
m_{\text{\tiny
e}}}\left(\frac{3(\vec{\mathfrak{m}}\cdot\vec{r})\vec{r}}{r^{5}}-\frac{\vec{\mathfrak{m}}}{r^{3}}\right)}\;.$
(3.33)

In this case, as appropriate, we take $\\{\tilde{\omega}_{\text{\tiny
NS,x}},\tilde{\omega}_{\text{\tiny NS,y}},\tilde{\omega}_{\text{\tiny
NS,z}}\\}\equiv\\{eB_{\text{\tiny NS, x}}/m_{\text{\tiny e}},eB_{\text{\tiny
NS,y}}/m_{\text{\tiny e}},eB_{\text{\tiny NS,z}}/m_{\text{\tiny e}}\\}$, where
$B_{\text{\tiny NS, (x,y,z)}}$ are the components of the magnetic field of the
NS. Using Eqs. (3.13), (3.22) and (3.33), we may calculate the refractive
index for the left and right circularly polarized light at position $\vec{r}$

$n_{\text{\tiny L,(R)}}^{\text{\tiny
NS}}=\begin{cases}\left(1-\frac{\omega^{2}_{\text{\tiny
p}}\left(\omega^{2}-\omega^{2}_{\text{\tiny
p}}\right)}{\omega^{2}\left(\omega^{2}-\omega^{2}_{\text{\tiny
p}}-\frac{1}{2}\left(\tilde{\omega}_{\text{\tiny
NS,x}}^{2}+\tilde{\omega}_{\text{\tiny
NS,y}}^{2}\right)\pm\left(\frac{1}{4}\left(\tilde{\omega}_{\text{\tiny
NS,x}}^{2}+\tilde{\omega}_{\text{\tiny
NS,y}}^{2}\right)^{2}+(\omega^{2}-\omega^{2}_{\text{\tiny
p}})^{2}\frac{\tilde{\omega}_{\text{\tiny
NS,z}}^{2}}{\omega^{2}}\right)^{1/2}\right)}\right)^{1/2}\;,\,\tilde{\omega}_{\text{\tiny
NS,z}}>0\;,\\\ \left(1-\frac{\omega^{2}_{\text{\tiny
p}}\left(\omega^{2}-\omega^{2}_{\text{\tiny
p}}\right)}{\omega^{2}\left(\omega^{2}-\omega^{2}_{\text{\tiny
p}}-\frac{1}{2}\left(\tilde{\omega}_{\text{\tiny
NS,x}}^{2}+\tilde{\omega}_{\text{\tiny
NS,y}}^{2}\right)\mp\left(\frac{1}{4}\left(\tilde{\omega}_{\text{\tiny
NS,x}}^{2}+\tilde{\omega}_{\text{\tiny
NS,y}}^{2}\right)^{2}+(\omega^{2}-\omega^{2}_{\text{\tiny
p}})^{2}\frac{\tilde{\omega}_{\text{\tiny
NS,z}}^{2}}{\omega^{2}}\right)^{1/2}\right)}\right)^{1/2}\;,\,\tilde{\omega}_{\text{\tiny
NS,z}}<0\;.\\\ \end{cases}$ (3.34)

Here, $\omega_{\text{\tiny p}}^{2}\equiv e^{2}\mathcal{N}_{\text{\tiny
e}}(r)/(\epsilon_{0}m_{\text{\tiny e}})$ is the plasma frequency.

The spatial dependencies of these refractive indices are highly dependent upon
the orientation of the magnetic dipole. For instance, we have found that when
the magnetic moment is oriented along the direction of propagation (i.e.
$(\theta_{\text{\tiny NS}},\phi_{\text{\tiny NS}})=(0,0)$), the refractive
index of the left circular wave decreases while the refractive index of the
right circular wave increases between the NS and observer. Moreover, in this
case, the magnetic field is axially symmetric along the direction of
propagation, and consequently, the refractive index variation also has this
axial symmetry. When the magnetic moment of an NS is instead oriented
perpendicular to the direction of propagation, the refractive index of the
left circular wave first increases and then decreases, between the NS and
observer. In contrast, for this alignment, the refractive index of the right
circular wave first decreases and then increases. At infinity, the refractive
indices for both the left and right circular components match and acquire the
value $\left(1-\omega^{2}_{\text{\tiny p}}/\omega^{2}\right)^{1/2}$.

Now, for the NS case, we can calculate the change in polarization using Eq.
(3.23)

$\psi_{\text{\tiny pol.}}^{\text{\tiny
NS}}=\frac{\omega}{2c}\int_{-d_{\text{\tiny QS}}}^{d_{\text{\tiny
Q}}}~{}dz\left(n_{\text{\tiny L}}^{\text{\tiny NS}}(\vec{r})-n_{\text{\tiny
R}}^{\text{\tiny NS}}(\vec{r})\right)\;.$ (3.35)

Similarly, the RM value accrued due to an NS along the light’s path will be
given by

$\text{RM}^{\text{\tiny
NS}}(\lambda)=\frac{d}{d\lambda^{2}}\left(\psi_{\text{\tiny
pol.}}^{\text{\tiny NS}}\right)\;.$ (3.36)

For the reasons discussed earlier, we are working in a regime where Eq. (3.21)
is satisfied for the NS parameters, i.e.

$\absolutevalue{\frac{\omega\left(\tilde{\omega}_{\text{\tiny
NS,x}}^{2}+\tilde{\omega}_{\text{\tiny
NS,y}}^{2}\right)}{2(\omega^{2}-\omega^{2}_{\text{\tiny
p}})\tilde{\omega}_{\text{\tiny NS,z}}}}\ll 1\;.$ (3.37)

For a fixed $\mathfrak{m},\,\omega$ and $\omega_{\text{\tiny p}}$, this will
help us identify the regions where we can consider the characteristic modes as
left and right circularly polarized. Hence, we are implicitly assuming the
applicability of Eq. (3.35) in the regime permitted by the above equation in
conjunction with Eq. (3.19).

If we assume a constant plasma density and use Eq. (3.32), then Eq. (3.19)
gives us a radius cut-off, above which the modes are propagating. The result
is

$r^{\text{\tiny NS}}_{\text{\tiny
cut}}\sim\left(\frac{\mu_{0}e\,\mathfrak{m}}{4\pi m_{\text{\tiny
e}}}\frac{\omega}{\left(\omega^{2}-\omega_{\text{\tiny
p}}^{2}\right)}\right)^{1/3}\;.$ (3.38)

For a fixed $\mathfrak{m},\,\omega$ and $\omega_{\text{\tiny p}}$, we may
calculate this distance explicitly. We may then utilize Eqs. (3.34) and (3.35)
in a bonafide way for the evaluation of the change in polarization.

Intriguingly, we will have potentially interesting observational signatures in
the region $r\lesssim r^{\text{\tiny NS}}_{\text{\tiny cut}}$. Only the $(+)$
characteristic mode would be observed inside $r\lesssim r^{\text{\tiny
NS}}_{\text{\tiny cut}}$, since the $(-)$ modes will be optically blocked. For
the polarization angle and RM maps we will focus mainly on the larger
$r\gtrsim r^{\text{\tiny NS}}_{\text{\tiny cut}}$ regions.

Now, we will consider two semi-realistic plasma profiles to compute the
Faraday rotations by an NS. We will examine both a constant plasma density and
a more realistic galactic plasma density profile to evaluate this effect.

Figure 4: Rotation measure (RM) values due to an NS located inside the Milky
Way galaxy as a function of the magnetic moment $\mathfrak{m}$ and the
distance between the NS and the source ($d_{\text{\tiny QS}}$). The plasma
density is assumed to be constant. The plots are for when the magnetic moment
is parallel i.e., $(\theta_{\text{\tiny NS}},\phi_{\text{\tiny NS}})=(0,0)$
(top) and perpendicular i.e., $(\theta_{\text{\tiny NS}},\phi_{\text{\tiny
NS}})=(\pi/2,0)$ (bottom). The alignments are with respect to the direction of
wave propagation, i.e. $z$-axis. The surface magnetic field
($B^{*}_{\text{\tiny NS}}$) of the NS is in the range $10^{4}-10^{11}$ T.
Here, we have evaluated RM values at the impact parameter $\xi=r^{\text{\tiny
NS}}_{\text{\tiny cut}}$ (See Eq. (3.38)), and taken the distance between the
source and observer to be $d_{\text{\tiny S }}=8.5~{}\text{kpc}$. The
wavelength of light is taken to be $\lambda=1~{}\text{m}$ and the MW electron
number density is taken to be $N_{\text{\tiny e},0}^{\text{\tiny MW}}$ = 0.015
cm-3 [138]. The gray region depicts the parameter space for which $(-)$ modes
are blocked. Notice that in the parameter space regions shown, the RM values
are all too small for observation i.e. $\text{RM}^{\text{\tiny NS}}\ll 0.01$.

Figure 5: Polarization angle maps ($\psi_{\text{\tiny pol.}}^{\text{\tiny
NS}}\times 10^{10}$) induced by an NS located inside the Milky Way galaxy. It
is illustrated for linearly polarized electromagnetic waves from a uniform
background source. The initial polarization is along the $\hat{x}$-axis. The
plasma density is assumed to be constant. The plots are shown when the NS
magnetic moments are parallel (left) and perpendicular (right) to the
direction of wave propagation ($z$-axis; coming out of the plane in the
figures above). Here, we have taken,
$\mathfrak{m}=10^{30}~{}\text{A-m${}^{2}$}$, or equivalently
$B^{*}_{\text{\tiny NS}}=10^{11}~{}\text{T}$, $d_{\text{\tiny
QS}}=10^{-4}~{}\text{pc}$, $d_{\text{\tiny S }}=10~{}\text{kpc}$,
$\lambda=1~{}\text{m}$ and $N_{\text{\tiny e},0}^{\text{\tiny MW}}$ = 0.015
cm-3 [138]. The black circle denotes an NS (exaggerated for visibility).

####  3.1.1 Constant density plasma

As a simplistic model, we will first consider that the plasma is uniformly
distributed inside the galaxy. This will help us explore the qualitative and
quantitative features of the Faraday effect.

We assume the NS under consideration to be situated within our Milky Way
galaxy (MW) and take the plasma density as the average Milky Way galactic
plasma density. Thus, the distribution under consideration is

$N^{\text{\tiny MW}}_{\text{\tiny e}}(r)=\begin{cases}0\;;&r<R_{\text{\tiny
NS}}\;,\\\ N_{\text{\tiny e},0}^{\text{\tiny MW}}\;;&r>R_{\text{\tiny
NS}}\;.\end{cases}$ (3.39)

Here, $R_{\text{\tiny NS}}$ is the radius of the NS and $N_{\text{\tiny
e},0}^{\text{\tiny MW}}$ is the average galactic plasma density of the Milky
Way. The plasma frequency in this scenario may be expressed as

$\omega^{2}_{\text{\tiny p}}(r)=\begin{cases}0\;;&r<R_{\text{\tiny NS}}\;,\\\
\frac{e^{2}N_{\text{\tiny e},0}^{\text{\tiny MW}}}{\epsilon_{0}m_{\text{\tiny
e}}}\;;&r>R_{\text{\tiny NS}}\;.\end{cases}$ (3.40)

A general, analytic expression for the change in polarization and RM cannot be
obtained, even in the constant plasma density case. However, when
$\omega\gg\text{max}\left({\omega_{\text{\tiny p}},\tilde{\omega}}\right)$,
using Eqs. (3.34), and (3.35), we get

$\psi_{\text{\tiny pol.}}^{\text{\tiny NS}}\simeq\int_{-d_{\text{\tiny
QS}}}^{d_{\text{\tiny Q}}}dz~{}\left(\frac{\omega^{2}_{\text{\tiny
p}}}{2\omega^{2}c}\tilde{\omega}_{\text{\tiny
NS,z}}\right)+\mathcal{O}\left(\left(\frac{\omega^{2}_{\text{\tiny
p}}}{\omega^{2}}\right)^{2},\left(\frac{\tilde{\omega}_{\text{\tiny
NS}}}{\omega}\right)^{2}\right)\;.$ (3.41)

Using the Eqs. (3.33) and (3.40), the change in polarization angle can be
written as

$\displaystyle\psi_{\text{\tiny pol.}}^{\text{\tiny NS}}$
$\displaystyle\simeq$ $\displaystyle-\frac{e^{3}\mathfrak{m}N_{\text{\tiny
e},0}^{\text{\tiny MW}}\lambda^{2}}{32\pi^{3}\epsilon_{0}m^{2}_{\text{\tiny
e}}c^{3}}\int_{-d_{\text{\tiny QS}}}^{d_{\text{\tiny
Q}}}dz~{}\left(\frac{3\left(x\sin\theta_{\text{\tiny NS}}\cos\phi_{\text{\tiny
NS}}+y\sin\theta_{\text{\tiny NS}}\sin\phi_{\text{\tiny
NS}}+z\cos\theta_{\text{\tiny
NS}}\right)z}{\left(\xi^{2}+z^{2}\right)^{5/2}}\right.$ (3.42)
$\displaystyle\hskip 30.00005pt\hskip
30.00005pt\left.-\frac{\cos\theta_{\text{\tiny
NS}}}{\left(\xi^{2}+z^{2}\right)^{\frac{3}{2}}}\right)\;.,$
$\displaystyle\simeq$ $\displaystyle-\frac{e^{3}\mathfrak{m}N_{\text{\tiny
e},0}^{\text{\tiny MW}}\lambda^{2}}{32\pi^{3}\epsilon_{0}m^{2}_{\text{\tiny
e}}c^{3}}\left(\frac{\sin\theta_{\text{\tiny NS}}(x\cos\phi_{\text{\tiny
NS}}+y\sin\phi_{\text{\tiny NS}})+d_{\text{\tiny Q}}\cos\theta_{\text{\tiny
NS}}}{\left(\xi^{2}+d_{\text{\tiny Q}}^{2}\right)^{\frac{3}{2}}}\right.$
$\displaystyle\hskip 30.00005pt\hskip
30.00005pt-\left.\frac{\sin\theta_{\text{\tiny NS}}(x\cos\phi_{\text{\tiny
NS}}+y\sin\phi_{\text{\tiny NS}})+d_{\text{\tiny QS}}\cos\theta_{\text{\tiny
NS}}}{\left(\xi^{2}+d_{\text{\tiny QS}}^{2}\right)^{\frac{3}{2}}}\right)\;.$

Using Eq. (3.25), we get the corresponding RM as

$\displaystyle\text{RM}(\lambda)$ $\displaystyle\simeq$
$\displaystyle-\frac{e^{3}\mathfrak{m}N_{\text{\tiny e},0}^{\text{\tiny
MW}}}{32\pi^{3}\epsilon_{0}m^{2}_{\text{\tiny
e}}c^{3}}\left(\frac{\sin\theta_{\text{\tiny NS}}(x\cos\phi_{\text{\tiny
NS}}+y\sin\phi_{\text{\tiny NS}})+d_{\text{\tiny Q}}\cos\theta_{\text{\tiny
NS}}}{\left(\xi^{2}+d_{\text{\tiny Q}}^{2}\right)^{\frac{3}{2}}}\right.$
(3.43) $\displaystyle\hskip 30.00005pt\hskip
30.00005pt-\left.\frac{\sin\theta_{\text{\tiny NS}}(x\cos\phi_{\text{\tiny
NS}}+y\sin\phi_{\text{\tiny NS}})+d_{\text{\tiny QS}}\cos\theta_{\text{\tiny
NS}}}{\left(\xi^{2}+d_{\text{\tiny QS}}^{2}\right)^{\frac{3}{2}}}\right)\;.$

We note that in the high-frequency limit
$\omega\gg\text{max}\left({\omega_{\text{\tiny p}},\tilde{\omega}}\right)$,
the change in polarization angle and RM exhibits a linear proportionality to
the NS magnetic moment. One also notes in passing that since the effects are
all proportional to $1/m_{e}^{2}$, the impact of ions in the plasma is
subdominant. Additionally, it’s interesting that the RM is independent of the
wavelength of light, at leading order. In the next-to-leading order term,
$\lambda^{2}$ contributes, as elucidated in equation Eq. (3).

It is also evident that the change in polarization and RM explicitly depend
upon coordinates $x$ and $y$ generically. However, when the magnetic moment of
the NS aligns with the direction of wave propagation, i.e.
$\theta_{\text{\tiny NS }}=0$, this specific dependency is eliminated. For
this alignment, quantities depend upon the impact parameter $\xi$ as expected
from the axial symmetry. Additionally, it is worth mentioning that the change
in polarization angle and RM undergoes a sign reversal when we exchange
$d_{\text{\tiny QS}}$ with $d_{\text{\tiny Q}}$. Consequently, when
$d_{\text{\tiny Q}}\simeq d_{\text{\tiny QS}}$ (i.e. when the neutrons star is
midway between source and observer), both the change in the polarization angle
and the RM will become zero.

We will employ Eqs. (3.30), (3.31), (3.34)-(3.36), and (3.40) to numerically
calculate the change in polarization and the RM due to an NS. We have assumed
the radius of the NS ($R_{\text{\tiny NS}}$) to be $10~{}\text{Km}$, and then
evaluated the RM and the change in the polarisation angles for surface
magnetic fields ($B^{*}_{\text{NS}}$) in the range
$10^{4}-10^{11}~{}\text{T}$. The relation between the dipole moment and
magnetic field is obtained using Eq. (3.31).

In Fig. 4, we have presented a density plot illustrating the magnitude of the
RM as a function of the NS magnetic dipole moment and the NS-source distance.
The magnetic moment of the NS is aligned parallel (top) and perpendicular
(bottom) to the wave propagation direction. The source-observer distance is
taken to be $d_{\text{\tiny S }}=8.5~{}\text{kpc}$ and the galactic electron
number density as $N_{\text{\tiny e},0}^{\text{\tiny MW}}$ = 0.015 cm-3 [138].
We have calculated the RM at an impact parameter $\xi=r^{\text{\tiny
NS}}_{\text{\tiny cut}}$ (using Eq. (3.38)), at which the RM value is maximum.
From the figure, we conclude that the RM values for NS are typically small and
do not generically reach observationally relevant values at present (i.e.
$\text{RM}^{\text{\tiny NS}}\ll 0.01$).

For both parallel and perpendicular orientations, it is evident that the RM
values increase as we increase the dipole moment, as should be expected.
Furthermore, increasing $d_{\text{\tiny QS}}$ leads to a drastic decrease in
the RM values because of the $1/r^{3}$ dependence of the magnetic field. Also,
notice that one gets relatively higher RM values for the parallel case since
the contribution of the $z$-component of the magnetic field is larger in this
case. It is also important to mention that inside the grey regions in Fig. 4,
the cut-off condition specified by (3.38) is not satisfied and they are
therefore excluded.

In Fig. 5, we present the polarization angle map (PA map). The uniform,
extended, linearly polarised light source is in the plane perpendicular to the
direction of wave propagation. It is located at a distance $d_{\text{\tiny
QS}}=10^{-4}~{}\text{pc}$ from the NS. The source produces linearly polarized
light, which is initially aligned along the $x$-direction. The NS is assumed
to have a magnetic dipole moment $\mathfrak{m}=10^{30}~{}\text{A-m${}^{2}$}$
or equivalently a surface magnetic field $B^{*}_{\text{\tiny
NS}}=10^{11}~{}\text{T}$. The PA map shows a patch $10^{-5}\,\text{pc}\times
10^{-5}\,\text{pc}$ as seen by the observer—for both the parallel (left) and
perpendicular (right) orientations of the NS. The actual values for the change
in polarization angle are exceedingly small to be detected in current
observations. In the case of parallel orientation, it is notable in the PA map
that for fixed impact parameters $\xi$, the electric fields are
unidirectional, rendering a uniform polarisation angle. This is because of the
axial symmetry of the NS’s magnetic field in this orientation. However, in
perpendicular orientation, it’s significant to note that the PA map has mirror
symmetry about the $x$-axis. This distinctive pattern is a consequence of the
specific alignment of the magnetic field in this configuration. Additionally,
as expected, we note that as we increase the impact parameter $\xi$, the
change in the polarization angle diminishes due to the weakening of the
magnetic field strength.

We observe that the change in polarization values within the PA map lacks
axial symmetry in the case of a general orientation of the NS. Hence, if we
choose a circular contour of any given radius on the PA map and proceed to
integrate the change in polarization values along this contour, the resulting
value for the integral will satisfy a normalized inequality given by

$\mathcal{M}^{\text{\tiny
NS}}\equiv\absolutevalue{\frac{1}{2\pi\psi_{\text{\tiny pol.}}^{\text{\tiny
NS}}(\phi_{0})}\left[\left(\oint_{C}d\phi\,\psi_{\text{\tiny
pol.}}^{\text{\tiny NS}}(\phi)\right)-2\pi\psi_{\text{\tiny
pol.}}^{\text{\tiny NS}}(\phi_{0})\right]}\geq 0\;.$ (3.44)

Here, $C$ represents any closed circular contour centred around the NS and
$\phi_{0}$ is an arbitrary azimuthal angle. The above inequality should be
approximately satisfied in many realistic scenarios for the plasma density.

As may be easily surmised, and as we shall see later, the right-hand side of
the above equation evaluates to zero for a monopolar magnetic field. This
simple observation is noteworthy, as it offers a potentially robust method to
differentiate in PA maps conventional astrophysical magnetic field
configurations—typically dipoles or higher multipoles—from MBHs, which have
unique monopolar fields.

Figure 6: Rotation measure values due to an NS located inside the Milky Way
galaxy as a function of $\mathfrak{m}$ and distance $d_{\text{\tiny QS}}$, for
a galactic plasma distribution. Again, the plots are shown when the magnetic
moment is parallel (top) and perpendicular (bottom) to the $z$-axis. As
before, the NS surface magnetic fields are in the range $10^{4}-10^{11}$ T .
We have taken $\xi=r^{\text{\tiny NS}}_{\text{\tiny cut}}$, $d_{\text{\tiny S
}}=8.5~{}\text{kpc}$, and $\lambda=1~{}\text{m}$. For the galactic plasma
density model parameters we take the values $N_{\text{\tiny e},1}^{\text{\tiny
MW}}$ = 0.025 cm-3, $N_{\text{\tiny e},2}^{\text{\tiny MW}}$ = 0.2 cm-3,
$A_{1}^{\text{\tiny MW}}$ = 20 kpc, and $A_{2}^{\text{\tiny MW}}$ = 2 kpc
[139]. The grey region again depicts the parameter space for which the $(-)$
modes are blocked. $\text{RM}^{\text{\tiny NS}}\ll 0.01$ in all the regions,
and hence observationally lackluster.

Figure 7:  NS polarization map for a galactic plasma distribution. Parallel
(left) and perpendicular (right) NS orientations are shown. The parametric
values chosen are similar to those in Fig. 5.

####  3.1.2 Galactic distribution of plasma

After exploring the general characteristics of Faraday rotation by an NS, in
the presence of a constant plasma density, we will shift to a slightly more
realistic scenario.

We will adopt the galactic plasma distribution functions from [139], and again
consider that the NS is situated within our Milky Way galaxy. We align the
direction of light propagation with the galactic plane and assume that the
source $S$ is located at the galactic centre. In this scenario, along the
direction of wave propagation, the plasma density is assumed to follow the sum
of two Gaussian profiles [139]

$\mathcal{N}^{\text{\tiny MW}}_{\text{\tiny
e}}(r)=\begin{cases}0\;;&r<R_{\text{\tiny NS}}\;,\\\ N_{\text{\tiny
e},1}^{\text{\tiny MW}}e^{-\left(\left(z-d_{\text{\tiny
QS}}\right)/A_{1}^{\text{\tiny MW}}\right)^{2}}+N_{\text{\tiny
e},2}^{\text{\tiny MW}}e^{-\left(\left(z-d_{\text{\tiny
QS}}-2A_{2}^{\text{\tiny MW}}\right)/A_{2}^{\text{\tiny
MW}}\right)^{2}}\;;&r>R_{\text{\tiny NS}}\;.\end{cases}$ (3.45)

Here, $R_{\text{\tiny NS}}$ is the radius of the NS and $N_{\text{\tiny
e},1}^{\text{\tiny MW}}$, $N_{\text{\tiny e},1}^{\text{\tiny MW}}$,
$A_{1}^{\text{\tiny MW}}$, and $A_{2}^{\text{\tiny MW}}$ are model parameters.
We will appropriate values for these parameters from [139]. As before, we will
again assume that $d_{\text{\tiny S}}\simeq d_{\text{\tiny Q}}\gg\xi$. In the
above equation, the first term represents the thick disk component of the
galactic distribution of electrons, while the second term represents the inner
thin disk component. The associated plasma frequency is

$\omega^{2}_{\text{\tiny p}}(r)=\begin{cases}0\;;&r<R_{\text{\tiny NS}}\;,\\\
\frac{e^{2}}{\epsilon_{0}m_{\text{\tiny e}}}\left(N_{\text{\tiny
e},1}^{\text{\tiny MW}}e^{-\left(\left(z-d_{\text{\tiny
QS}}\right)/A_{1}^{\text{\tiny MW}}\right)^{2}}+N_{\text{\tiny
e},2}^{\text{\tiny MW}}e^{-\left(\left(z-d_{\text{\tiny
QS}}-2A_{2}^{\text{\tiny MW}}\right)/A_{2}^{\text{\tiny
MW}}\right)^{2}}\right)\;;&r>R_{\text{\tiny NS}}\;.\end{cases}$ (3.46)

We will once again use Eqs. (3.30), (3.31), (3.34)-(3.36), and (3.46) to
numerically calculate the change in the polarization angle and the RM value in
this case.

We again assume $R_{\text{\tiny NS}}=10~{}\text{Km}$, and that the surface
field $B^{*}_{\text{NS}}$ is in the range $10^{4}-10^{11}~{}\text{T}$. The RM
density plot is presented in Fig. 6. The magnetic moment of the NS is again
assumed to be oriented parallel (top) or perpendicular (bottom) to wave
propagation. As before, we have taken $d_{\text{\tiny S }}=8.5~{}\text{kpc}$
and $\xi=r^{\text{\tiny NS}}_{\text{\tiny cut}}$. The values of the galactic
plasma density model parameters are taken to be $N_{\text{\tiny
e},1}^{\text{\tiny MW}}$ = 0.025 cm-3, $N_{\text{\tiny e},2}^{\text{\tiny
MW}}$ = 0.2 cm-3, $A_{1}^{\text{\tiny MW}}$ = 20 kpc, and $A_{2}^{\text{\tiny
MW}}$ = 2 kpc [139]. The RM values for the galactic plasma distribution case
are slightly larger than the constant density case. Additionally, it is also
found that the RM values exhibit very little variation, even when the source
is located farther from the galactic centre. The RM values are exceedingly
small, $\text{RM}^{\text{\tiny NS}}\ll 0.01$, throughout.

Fig. 7 depicts the PA map in this scenario for $B^{*}_{\text{\tiny
NS}}=10^{11}~{}\text{T}$ and $d_{\text{\tiny QS}}=10^{-4}~{}\text{pc}$. The
initial polarization direction for the wave is assumed to be along the
$x$-direction. Similar to the PA maps in Sec. 3.1.1, we have displayed
$10^{-5}\,\text{pc}\times 10^{-5}\,\text{pc}$ patches as seen by the observer;
for the parallel (left) and perpendicular (right) orientation of the magnetic
moment of an NS. Although slightly higher than the values in the constant
density case, the polarization angle values still remain exceedingly small for
current observational capabilities.