aainstitutetext: Department of Chemistry and Physics,
SUNY Old Westbury,
Old Westbury, NY, United Statesbbinstitutetext: Department of Physics,
Long Island University,
Brookville, NY, United Statesccinstitutetext: Department of Physics, Virginia
Tech, Blacksburg, VA, United States

#  Black hole spin-down via cosmic strings

Sameer Ahmed a Michael Kavic b Steven L. Liebling a Matthew Lippert a Mohammad
Mian c and John Simonetti<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Cosmic strings that are attached to rapidly spinning black holes can extract
significant amounts of rotational energy and angular momentum. Here we study
the effect on primordial black holes, which are expected to form with one or
more cosmic strings attached. Although large primordial black holes are
predicted to rapidly spin up due to accretion soon after forming, we argue
that cosmic strings will spin them down again. As a result, the spins of
primordial black holes should consequently be observed to be near zero. We
also investigate the effect on a supermassive black hole of capturing a cosmic
string and the possibility of observing the subsequent spin down by its effect
on a pulsar orbiting the black hole.

##  1 Introduction

Cosmic strings are conjectural remnants of phase transitions in the early
universe and represent an intriguing astronomical manifestation of high-energy
physics. They are one-dimensional topological defects naturally appearing in
many particle physics models, including many GUT theories (see
Vilenkin:2000jqa ; Hindmarsh:1994re ; Sakellariadou:2009ev  for reviews). In
the context of string theory, fundamental strings and D-strings can also
manifest as cosmic superstrings Polchinski:2004ia .

Efforts to detect cosmic strings have exploited several possible phenomena.
Oscillating strings produce a stochastic gravitational wave background
Vachaspati:1984gt . Cosmic strings can also develop cusps and kinks which emit
strong bursts of gravitational radiation Damour:2000wa ; Damour:2001bk ;
Damour:2004kw . The amplitude of this gravitational wave signal depends
primarily on the string’s mass density, or tension, $\mu$.111We work in units
where $\mu$ is dimensionless. Restoring the units, the tension is given by
$G\mu/c^{2}$.

Perhaps most exciting are the recent results from the North American Nanohertz
Observatory for Gravitational Waves (NANOGrav), looking for ultra-low-
frequency gravitational waves by observing millisecond pulsars. NANOGrav has
found evidence of a stochastic gravitational wave background compatible with a
network of cosmic strings in the range $10^{-12}<\mu<10^{-10}$
NANOGrav:2023hvm ; Ellis:2023tsl . Although not a firm detection, the
observations offer tantalizing evidence for the existence of cosmic strings in
the early universe.

Direct searches by the LIGO/Virgo/KAGRA (LVK) gravitational wave (GW)
observatory network for burst signals have set an upper bound of $\mu<3\times
10^{-7}$, and analysis of the stochastic GW background yields a tighter
constraint of $\mu<10^{-10}$ LIGOScientific:2021nrg . A stochastic background
would also induce characteristic noise in the timing of pulsar signals.
Results from the Parkes Pulsar Timing Array currently give the strongest
bound: $\mu<1.5\times 10^{-11}$ Lasky:2015lej ; Blanco-Pillado:2017rnf .
Microlensing of extra-galactic sources by strings could probe down to tensions
as small as $\mu=10^{-13}$ Chernoff:2019pjr . Future GW detectors, such as
LISA, could potentially be sensitive to $\mu=5.8\times 10^{-18}$ Blanco-
Pillado:2017rnf .

Here, we investigate the interaction between a cosmic string (CS) and a black
hole (BH), with a view toward observable signatures. A CS colliding with a BH
will be gravitationally captured, with the black hole ending up stuck like a
bead on a string DeVilliers:1997nk ; Snajdr:2002aa ; Dubath:2006vs . A short
CS loop might be expected to quickly fall in to the BH, but a very long CS, in
particular one extending beyond the cosmological horizon, could persist
attached to the BH for a long time.

The behavior a CS attached to a rotating black hole was studied in
Frolov:1996xw . Interestingly, the cosmic string can extract rotational energy
and angular momentum Kinoshita:2016lqd ; Igata:2018kry  from the BH by a
generalization of the Penrose Process Penrose:1971  using one-dimensional,
relativistic strings rather than particles. The well-known Blanford-Znajek
mechanism Blandford:1977ds , in which a force-free electromagnetic field
extracts rotational energy from an accreting BH, is also an example of this
stringy Penrose Process but with magnetic flux lines acting as the strings
Kinoshita:2017mio . Unlike the Blanford-Znajek mechanism, in which the
rotational energy powers highly visible jets emanating from accreting black
holes, the energy extracted by cosmic strings is not easily observable.
However, the spin down and energy loss of the BH is potentially detectable.

An important question to consider is how likely is it that a given BH will, in
fact, collide with and capture a CS. In Xing_2021 , the rate for a black hole
to capture cosmic strings was estimated. A stellar-mass BH presents a small
target, and the chance of capturing a CS is extremely low; the expected time
to capture a single cosmic string would be far longer than the age of the
universe. However, for a supermassive black hole, such as Sagittarius A* (Sgr
A*), and for very low cosmic string tension, $\mu\lesssim 10^{-18}$, there is
an order-one probability for one or more collisions during the black hole’s
lifetime.

Observing the spin down of a supermassive back hole due to cosmic strings,
although challenging, is possible, at least in principle. We propose a
technique by which the energy lost by the BH could be detected by its effect
on a pulsar orbiting the BH. However, in addition to finding an appropriate
pulsar sufficiently close to the BH, the precision of pulsar timing
measurements would need to improve significantly in order to detect the mass
loss due to phenomenologically viable cosmic strings with $\mu<10^{-11}$.

A particularly interesting alternative scenario is the interaction of a cosmic
string with a primordial black hole (PBH) Vilenkin:2018zol . PBHs may have
been created by large density fluctuations in the early universe with a very
wide range of possible masses, in principle down to the Planck scale (see
Sasaki:2018dmp ; Green:2020jor ; Villanueva-Domingo:2021spv ; Escriva:2022duf
for recent reviews). The possibility that large black holes recently observed
by LVK may have primordial origin Bird:2016dcv  has spurred renewed interest.

In the early universe, at the time of PBH formation, the cosmic string network
would be sufficiently dense that there will be, on average, one or more cosmic
strings in a typical over-dense region which collapses to form a PBH. The PBH
is then created with one or more cosmic strings already attached
Vilenkin:2018zol .

A typical PBH formed by over-densities during the radiation-dominated era is
not expected to form with significant spin Mirbabayi:2019uph ; DeLuca:2019buf
. However, there are mechanisms by which PBHs form with large spins, e.g.
Cotner:2017tir ; Harada:2017fjm . Even if the PBH is formed with low spin,
subsequent accretion of gas and dark matter can spin it up. In particular, it
was shown in DeLuca:2020bjf  that this accretion process is mostly completed
early in the history of the universe and that large PBHs are expected to have
higher accretion rates, resulting in higher spins. As a result, a PBH with
mass greater than about $30M_{\odot}$ is predicted to have near-extremal spin
when later observed.

However, if the PBH formed with cosmic strings attached, then after being spun
up by accretion, the PBH would subsequently be spun down by those strings. As
a result, later observations would find high-mass PBHs with very small angular
momentum, in contrast to the prediction of DeLuca:2020bjf .

Accurate measurements of BH spins have become increasingly possible as
techniques have improved from gravitational wave observations of binary BH
mergers, and from X-ray emission spectroscopy of BH accretion disks (for a
review, see Reynolds:2020jwt ). With a sufficiently large data set of black
hole spin observations, it will become possible to distinguish, at least
statistically, those PBHs formed in the early universe from other BHs formed
later, for example, via stellar collapse. Observations of highly spinning PBHs
would then rule out, or at least highly constrain, the existence of cosmic
strings at the time of PBH formation.

The outline of the remainder of this paper is as follows: In section 2, we
review the mechanism by which cosmic strings remove energy from black holes.
Then, in section 3, we argue that cosmic strings would have effectively spun
down PBHs being observed in the present epoch. In section 4, we discuss the
possibility of observing the cosmic string-induced spin down of a supermassive
black hole via pulsar timing measurements. Finally, we discuss open questions
and future directions in section 5.

##  2 Energy extraction by cosmic strings

The Penrose Process Penrose:1971  is a mechanism in classical general
relativity by which energy can be removed from a rotating black hole. An
object that enters within the ergosphere of the black hole will necessarily
co-rotate with the black hole. The object is then split into two parts, and
one part launched in the backward (contra-rotating) direction. This piece will
still be co-rotating, albeit slower, and can have negative energy and angular
momentum as measured by an asymptotic observer. When this negative energy
piece ultimately falls through the event horizon, the energy and angular
momentum of the black hole are reduced. The remaining part of the object
receives a kick and can escape the black hole with greater energy and angular
momentum than it had originally.

The available energy is limited by the spin of the black hole. A Kerr black
hole222A Kerr BH with angular momentum $J$ and mass $M$ has spin parameter
$a=J/M$, an event horizon radius $r_{h}=M+\sqrt{M^{2}-a^{2}}$, and an angular
velocity $\Omega_{h}=\frac{a}{a^{2}+M^{2}}$. The radius of the ergosphere is
$r_{\mathrm{ergo}}=M+\sqrt{M^{2}-a^{2}\cos^{2}\theta}$ where $\theta$ is the
azimuthal angle in Boyer-Lindquist coordinates. The dimensionless Kerr
parameter is defined as $\chi=a/M$ and $\chi=1$ for an extremal Kerr BH. with
angular momentum $J$ has a mass $M$ given by

$M=\sqrt{M_{\mathrm{irr}}^{2}+\frac{J^{2}}{4M_{\mathrm{irr}}^{2}}}\ ,$ (1)

where the irreducible mass $M_{\mathrm{irr}}$ is given by the horizon area:
$M_{\mathrm{irr}}^{2}=\frac{A}{16\pi}$. The Penrose process can continue
extracting energy until $J=0$ and the black hole is completely spun down,
while the area theorem ensures that $M_{\mathrm{irr}}$ does not decrease. At
this point all of the rotational energy $E_{\mathrm{rot}}=M-M_{\mathrm{irr}}$
has been removed, leaving a non-rotating BH with $M=M_{\mathrm{irr}}$. As a
result, at most $29\%$ of the original BH energy may be extracted.

The Penrose process is actually a quite general description of classical
energy extraction from a rotating black hole, with a wide range of physical
scenarios. Superradiance (for a review, see Brito:2015oca ) is one such
example. Another such process involves an externally sourced magnetic field
threading the black hole, extracting rotational energy via the Blandford-
Znajek (BZ) process Blandford:1977ds . The equivalence of the BZ process and
energy extraction by a protruding CS is demonstrated in Kinoshita:2017mio ;
Igata:2018kry .

In particular, a long, relativistic CS with one end entering the ergosphere is
dragged around by the black hole. If the CS is rigidly rotating with an
angular velocity $\omega$ that is less than the angular velocity $\Omega_{h}$
of the BH, i.e. if $\omega<\Omega_{h}$, then there is an outward radial flow
of angular momentum and energy along the string Kinoshita:2016lqd . In the
limit of a slowly rotating black hole, $J<M^{2}$, and averaging over the angle
at which the strCSing emerges from the ergosphere, the outward angular
momentum flux $q$ is

$q=2M^{2}(\Omega_{h}-\omega)\,.$ (2)

For a CS with tension $\mu$, the energy extraction rate is $dE/dt=\mu
q\omega$. Restoring the dimensionful constants, this can be written as
Kinoshita:2016lqd :

$\frac{dE}{dt}=\left(4.5\times
10^{58}\frac{\textrm{erg}}{s}\right)\mu\left(\frac{q/a}{1/2}\right)\left(\frac{\omega/\Omega_{h}}{1/2}\right)u(a)\
,$ (3)

where the dimensionless function $u(a)=\left(1-\sqrt{1-a^{2}/M^{2}}\right)$.
When $a=0$ the BH is no longer spinning and $u(0)=0$, which shuts off the
energy extraction.

As the black hole loses rotational energy to the string, its total mass
decreases. For typical values of $a$, $q$, and $\omega$, the final three
factors of (3) are order one. The rate at which the BH is losing mass,
assuming $a>0$ and $\omega<\Omega_{h}$ is then approximately

$\frac{dM}{dt}\approx-\left(10^{34}\frac{\textrm{kg}}{s}\right)\mu\approx-\left(10^{4}\frac{M_{\odot}}{\rm
s}\right)\mu\,$ (4)

which agrees with the results of Xing_2021 .

If the CS is very long, energy will continually be drawn away from the BH and
dissipated down the length of the string. If the other end is sufficiently far
away, in particular if the CS extends beyond the cosmological horizon,
outward-moving fluctuations will never be reflected and return to the BH. As
the string extracts angular momentum from the BH, its angular velocity will
not increase significantly because that angular momentum propagates outward.
The BH will eventually spin down completely.

In the process of spinning down, the black hole can lose at most 29% of its
original mass. For a typical, rapidly rotating, but non-extemal BH, the mass
loss will be of order 10%. Taking $\Delta M\approx M/10$ and ignoring the
dependence of (3) on the BH spin, the timescale $t_{\mathrm{SD}}$ over which
the black hole spins down is given by:

$t_{\mathrm{SD}}\approx 10^{6}\,{\rm
s}\left(\frac{M}{M_{\odot}}\right)\left(\frac{10^{-11}}{\mu}\right)\ .$ (5)

The fiducial tension $\mu=10^{-11}$ is roughly the maximum value allowed by
the observational bounds, implying the last factor will be greater than one.
Because the actual mass-loss rate goes to zero as $a\to 0$, the spin-down time
given by (5) is an overestimate.

For a stellar mass black hole with, say, $M_{\mathrm{BH}}=30M_{\odot}$, the
approximate spin-down time (5) gives

$t_{\mathrm{SD}}\approx 1\,{\rm yr}\,\left(\frac{10^{-11}}{\mu}\right)\,.$ (6)

This implies that, unless $\mu$ is very small, a stellar mass black hole will
spin down rapidly unless it is being actively being spun up by, for example,
accretion. In particular, a typical observation of a stellar BH will be
extremely unlikely to see the spin down in progress.

In contrast, for a supermassive mass black hole,
$M_{\mathrm{BH}}=10^{7}M_{\odot}$, (5) gives

$t_{\mathrm{SD}}\approx 10^{6}\,{\rm
yr}\left(\frac{\mu}{10^{-11}}\right)^{-1}\ .$ (7)

The spin down of a supermassive BH is a much slower process, especially for
small $\mu$. Consequently, the chance of observing such a BH in the process of
spinning down will be much higher.

##  3 Primordial black holes and cosmic strings

###  3.1 Formation and spin up

Black holes may have been created in the early universe by a variety of
mechanisms. The collapse of large density fluctuations during the radiation
dominated era Zeldovich:1967lct ; Carr:1974nx  has received the most
attention, but other scenarios have been considered such as the collapse of
vacuum bubbles or domain walls Garriga:1992nm ; Khlopov:2004sc ;
Tanahashi:2014sma ; Garriga:2015fdk ; Deng:2016vzba ; Deng:2017uwc  and of
cosmic string loops or cusps Garriga:1992nm ; Hawking:1987bn ; Polnarev:1988dh
. These primordial black holes (PBHs) can, in principle, be created at any
mass within a very wide range. Although the idea of PBHs is not new, the
possibility that large black holes recently observed by LVK may have
primordial origin Bird:2016dcv  has spurred renewed interest. In addition,
recent work has argued that NANOgrav may have observed stochastic GW signals
from PBH formation in the early universe, suggesting that PBHs comprise a
significant fraction of dark matter Kohri:2020qqd ; DeLuca:2020agl ;
Vaskonen:2020lbd .

The abundance of PBHs is constrained by a variety of phenomenological bounds
Sasaki:2018dmp ; Green:2020jor ; Villanueva-Domingo:2021spv ; Escriva:2022duf
. Extremely light PBHs, with $M_{\mathrm{PBH}}<10^{-19}\,M_{\odot}$, would
have completely evaporated by now due to Hawking radiation and would no longer
be present. Somewhat more massive PBHs with
$M_{\mathrm{PBH}}<10^{-16}\,M_{\odot}$ would not have evaporated yet, but they
would be emitting sufficient Hawking radiation that their abundance must be
quite limited to have avoided observation. The abundance of PBHs in the
planetary to sub-solar-mass range,
$10^{-11}\,M_{\odot}<M_{\mathrm{PBH}}<10^{-1}\,M_{\odot}$, is modestly
constrained by the frequency of stellar microlensing events, though for solar-
mass and larger PBHs, $M_{\mathrm{PBH}}>M_{\odot}$, dynamical effects, limits
on radiation from accreted gas, and GW signals impose more stringent limits.
The sub-planetary mass range,
$10^{-16}\,M_{\odot}<M_{\mathrm{PBH}}<10^{-11}\,M_{\odot}$, however, is still
quite unconstrained.

The mass at which a PBH forms is sensitively related to the time at which it
forms. A region will collapse into a PBH of mass $M$ when its Schwarzschild
radius is the size of the Hubble radius. During the radiation-dominated era,
the average mass within a Hubble radius is
$M_{H}=M_{\odot}\left(\frac{t}{10^{-5}\,\rm s}\right)$ As a result, a PBH of
mass $M$ will form at a time Vilenkin:2018zol ; Villanueva-Domingo:2021spv

$t_{\mathrm{PBH}}\sim 10^{-5}\,{\rm s}\left(\frac{M}{M_{\odot}}\right)\,.$ (8)

Unlike black holes formed by stellar collapse, PBHs formed by collapse of
over-densities typically have very low spins. Estimates in Mirbabayi:2019uph ;
DeLuca:2019buf  predict an initial spin parameter of order $a\sim 10^{-2}$.
Larger spins have been suggested, however, as a possibility for PBHs formed
via other mechanisms Cotner:2017tir ; Harada:2017fjm .

Over the long history of the universe, both the mass and spin of a PBH can
evolve significantly. While many phenomena can affect the PBH during its
lifetime, accretion of the surrounding interstellar gas has been shown to be a
significant driver of mass and spin evolution DeLuca:2020bjf . In particular,
the accretion effects grow with BH mass and become significant for $M\gtrsim
10$.333The quantity of dark matter accreted is comparatively negligible,
although a dark matter halo around the PBH will enhance the accretion rate of
gas. At early times, $z>10$, the accretion rate for these large BHs can exceed
the Eddington rate of $1.44\times 10^{14}\,{\rm kg/s}\,(M/M_{\odot})$. For
super-Eddington accretion rates, the accreting matter will form a thin disk
Ricotti:2007au  which leads to efficient spin-up of the PBH.

This rapid accretion process is expected to decrease with the onset of
structure formation. Due to the attraction of large-scale structures, a
typical PBH will speed up relative to the ambient gas, and the resulting large
proper velocities strongly suppress accretion Ricotti:2007au ; Hutsi:2019hlw .
Although uncertainties remain regarding the dynamics of PBHs and structure
formation, significant accretion can be expected to end around $z\sim 10$.

The evolution of PBH mass and spin assuming a sharp accretion cut-off at
$z=10$, was computed in DeLuca:2020bjf  (labeled as Model 1). A small PBH with
initial mass $M\lesssim 30M_{\odot}$ and low spin $\chi\sim 0$ experiences
negligible accretion and remains small and slowly spinning. However, larger
PBHs with $M>30M_{\odot}$ rapidly accrete as $z\to 10$, spinning up to near
extremality $\chi\lesssim 1$. In both cases, after accretion ends at $z=10$,
PBHs are then assumed to continue with constant mass and spin until the
present (see Fig. 4 of DeLuca:2020bjf ).444In DeLuca:2020bjf , an alternative
scenario, Model 2, was considered in which the effect of structure formation
is limited and accretion continues until the present. In this case, only very
large PBHs, with $M>10^{3}M_{\odot}$, currently have high spins (see Fig. 5 of
DeLuca:2020bjf ).

The results of DeLuca:2020bjf  yield a prediction for the late-time spins of
PBHs. In the absence of further dynamical evolution, a large PBH
$M>30M_{\odot}$ observed any time after $z=10$ would be expected to have near-
extremal spin. A small PBH, by contrast, should be observed with nearly zero
spin.

###  3.2 Spin down by cosmic strings

The presence of cosmic strings changes the predicted distribution of PBH spins
significantly. Cosmic strings form early in the history of the universe,
generally before the formation of PBHs. In order to be relevant to PBHs,
however, they must have formed after inflation; otherwise, the string network
at the time of PBH formation would be much too sparse. Focusing on cosmic
strings created during symmetry-breaking phase transitions, the energy scale
$\eta$ of the phase transition, which is related to the string tension
$\mu=\left(\frac{m_{p}}{\eta}\right)^{2}$, determines the time of string
formation Vilenkin:2018zol :

$t_{s}\sim\left(\frac{m_{p}}{\eta}\right)^{2}t_{p}\sim\frac{t_{p}}{\mu}$ (9)

where $t_{p}$ is the Planck time. Compared with the PBH formation time (8),
cosmic string form much earlier than PBHs, as long as the CS tension $\mu$ is
not extremely small; for example, for a $30M_{\odot}$ PBHs any tension
$\mu>10^{-38}$ will yield $t_{s}\ll t_{\mathrm{PBH}}$.

After the strings are created, the network begins to evolve. Long strings
spanning across Hubble volumes stretch as the universe expands. Through self-
intersections and intersections with other strings they can reconnect and
break off smaller loops. Loops oscillate, lose energy by emitting
gravitational waves, and shrink. Numerical simulations predict a scale-
invariant string network, in which a given Hubble patch contains several long
strings and a larger number of loops ranging over all sizes Bennett:1989 ;
Allen:1990tv ; Blanco-Pillado:2011egf .

Later, after this CS network has emerged, PBHs begin to form. Overdense
regions begin to collapse when their apparent horizon comes within the
cosmological horizon (8). This roughly cosmological-horizon-sized region,
which includes $\mathcal{O}(10)$ long strings crossing through it, then
collapses, forming a new PBH. For every long CS originally in the collapsing
region, the new PBH will have a pair of long strings sticking out of the
horizon Vilenkin:2018zol .555However, alternative PBH formation scenarios,
such as collapsing bubbles, in which the black hole horizon is much smaller
than the cosmological horizon, will result in few, if any, captured strings.

The subsequent evolution of the coupled PBH-CS system was studied in detail in
Vilenkin:2018zol . A long string attached to a black hole horizon exerts a
pull on the black hole, but can not, by itself, become unattached from the
black hole. If, for example, a string connects two black holes, they can be
pulled toward each other, eventually merging into a single larger black hole
with no cosmic string. However, if the typical distance between PBHs is large,
the string pulling the PBHs together will have to compete with the
cosmological expansion.

A pair of strings emerging from a BH horizon can intersect, potentially
reconnecting to form a half loop, whose ends both enter the horizon, and a
long string which is disconnected from the PBH. The half loop will eventually
fall into the horizon and disappear, leaving the PBH with two fewer strings.
Initially, with $\mathcal{O}(10)$ strings attached to the PBH, the probability
for any two to intersect is high, but, as the number of strings falls,
intersections become increasingly unlikely. Once the number of strings drops
to 2, the chance that they then intersect can become very low Vilenkin:2018zol
.

The details are complicated and depend on the properties of the cosmic
strings, the density of the CS network, and the distance between PBHs. The
likelihood that a PBH will eventually lose all its strings, and the timescale
for this to occur, are open questions and will likely require numerical
simulations to address. However, it seems reasonable to assume at least one
pair of long strings remains attached.

As described above in Sec. 3.1, a typical PBH forms with very low spin, but
large PBHs will subsequently be spun up by accretion. However, assuming the
large PBH has one or more cosmic strings attached during the accretion period,
the angular momentum will then be extracted by the cosmic strings, spinning
the PBH back down.

The spin-down rate for a $30M_{\odot}$ mass PBH (6) is quite rapid, as long as
the cosmic string tension is not extremely small. Even for $\mu=10^{-20}$,
orders of magnitude below current observational bounds, the spin-down time is
$10^{9}$ years; by the present day, the PBH will have long since lost its
angular momentum. For larger string tensions, closer to the observational
bound $\mu\sim 10^{-11}$, the spin-down rate is faster than the accretion rate
DeLuca:2020bjf . In this case, the PBH never even gets spun up, with the
angular momentum extracted as fast as it falls in.

This allows us to establish a clear prediction and a definitive test for the
existence of cosmic strings in an unprobed range of string tensions. If there
are cosmic strings with $\mu>10^{-20}$, PBHs with mass larger than
$30M_{\odot}$ will be observed to have near-zero spin. Otherwise, according to
DeLuca:2020bjf , these high mass PBHs will instead have near-extremal spin.

###  3.3 Observations of black hole spin

Determining the spin of a BH presents a significant observational challenge.
Currently, spin measurements are possible for two types of BH systems: Spins
of accreting BHs can be measured through electromagnetic radiation, mostly
through X-ray reflection spectroscopy, and thermal continuum fitting, while
spins of merging black holes are constrained through GW observation (see
Reynolds:2020jwt  for a review).

For accreting BHs in the regime between geometrically thin and optically thick
accretion disks, X-ray reflection spectroscopy is the primary means for
determining the spin. X-ray reflection spectroscopy refers to the soft X-ray
lines emitted by the photo-ionized outer region of the optically thick
accretion disk. The applicable regime corresponds to accretion rates between
about 0.01 and 0.3 of the Eddington accretion rate and, in some circumstances,
is applicable up to about the Eddington limit. This technique is applicable to
classical Seyfert galaxies, moderately luminous quasars, and black hole X-ray
binaries in their luminous hard X-ray state.

Thermal continuum fitting utilizes the fact that the spin of an accreting BH
influences the location of the innermost stable circular orbit (ISCO) in the
accretion disk, thus determining the temperature of the blackbody radiation
from that region. For prograde accretion, the higher the spin rate of the BH,
the smaller the ISCO and the higher the temperature of the inner disk. This
technique has been applied more to stellar-mass BH X-ray binaries than to
accreting supermassive BHs.

Using these methods, many accreting supermassive BHs have been found to be
rapidly spinning. However, there is a population of more slowly spinning BHs
with masses greater than $3\times 10^{7}M_{\odot}$, consistent with what is
expected from structure formation models. Accreting stellar-mass BHs in X-ray
binary systems have also been observed to be rapidly spinning, but, in many
cases, their spins must have been rapid at birth instead of having been
produced by spin-up accretion. Spin-up from near zero spin to maximal spin
requires that a significant fraction of the BH mass be provided by accretion.
In the case of high-mass X-ray binaries, the accretion process is time-limited
due to the rapid evolution of the massive companion star. Even Eddington-
limited cases would not have sufficient time to spin up the BH by accretion.
For low-mass X-ray binaries, however, spin up by accretion is possible.

Likely more relevant to the PBHs being discussed here are measurements of the
spins of merging binary black holes (BBH) using the GW signal. These
observations are cleaner because details of the accretion do not come into
play. However, the effects of spin in the GW signal are subtle, and the
sensitivity of LVK is such that, for the cases observed, the results presented
in Reynolds:2020jwt  interestingly favor low spins for the merging objects.

Beyond measuring a black hole’s spin, there remains the task of identifying
whether it is primordial in origin. BBH merger events observed in the LVK
third GW Transient Catalog (GWTC-3) KAGRA:2021vkt  have widely been considered
as a mixed population of astrophysical black holes (ABHs) and PBHs
Hall:2020daa ; Hutsi:2020sol ; Chen:2024dxh . There are aspects of these
observations that suggest the existence of a sub-population of PBHs. This
includes the merger event rate of high mass BBH systems, including events that
reside in the upper mass gap that applies to ABHs LIGOScientific:2020iuh . A
recent Bayesian analysis estimated that $1/4$ of the events in the GWTC-3 are
due to PBHs Chen:2024dxh . However, because the mass spectrum of PBHs is
expected to peak at high mass ($30M_{\odot}$) Hutsi:2020sol , observations of
high-mass BBH merger events are significantly more likely to be PBH systems.
These systems are the ideal candidates for the observation described here
because of the spin-up process described above. Repeated observations of such
high-mass merger events will allow for further delineation of sub-populations
of BBH mergers and would allow for a definitive test for the existence of
cosmic strings with a string tension $\mu>10^{-20}$.

##  4 Spin-down of Supermassive Black Holes

As discussed in Sec. 3, primordial black holes form with cosmic strings pre-
attached. However, later-forming, non-primordial black holes can only become
attached to a CS if the two collide. A stellar mass BH is an extremely small
target, and the probability for a passing cosmic string to hit it is
vanishingly low. A supermassive BH, on the other hand, is large enough that
collisions with cosmic strings, although still rare, can be expected to occur
over the lifetime of the BH in certain circumstances.

The rate at which strings collide with and become captured by a supermassive
BH depends on size of the BH and the density of the CS network. Cosmic strings
tend to cluster in galactic halos Jain:2020dct  where the BHs are located,
improving the chances for a collision. The expected time for a CS collision
was estimated in Xing_2021  as

$t_{c}=3\times
10^{9}\,\rm{yr}\left(\frac{\mu}{10^{-18}}\right)^{2}\left(\frac{M}{M_{\mathrm{SgrA*}}}\right)^{-2.5}\left(\eta(p)\right)^{-1}$
(10)

where the mass of Sgr A*, $M_{\mathrm{SgrA*}}=4\times 10^{6}\ M_{\odot}$, is
taken as fiducial mass and $\eta(p)$ is an order-one function of the CS
reconnection probability $p$. In the particular case of Sgr A*, if the cosmic
strings are rather light, $\mu<10^{-18}$, we would expect at least one CS to
have been captured since the BH’s formation, order $10^{10}$ years
ago.666Although (10) was derived for supermassive black holes, if we
extrapolate the expression down to a stellar BH mass, $M_{\mathrm{BH}}\sim 10\
M_{\odot}$, the capture time becomes far longer than the age of the universe,
unless the tension is extremely small: $\mu<10^{-46}$. Consequently, the
probability that a BH formed from stellar core collapse ever captures a CS is
negligible.

If a supermassive BH were to capture a CS, there would be observational
consequences. For example, gravitational lensing could be used to detect the
acceleration of a BH caused by the net force exerted by attached cosmic
strings Ashoorioon:2022zgu . Oscillating cosmic strings, extracting rotational
energy from the BH, could emit observable gravitational waves, although the
particular signatures from such a process are still to be determinded
Xing_2021 . We propose an alternative method to observe the CS spinning down
the BH.

Supermassive BHs are expected to be rapidly spinning, and those that have been
observed all have large spins Reynolds:2020jwt . Lower mass supermassive BHs,
those with $M<3\times 10^{7}M_{\odot}$, all have near-extremal spins,
$\chi>0.9$. The heaviest supermassive BHs, on the other hand, tend to have
more modest spins.

As shown in Sec. 2, once the CS has been captured by the BH, it will begin
extracting angular momentum and, along with it, energy. This spin-down process
takes a long time (7) because there is a lot of angular momentum to extract.
In the case of Sgr A*, for $\mu<10^{14}$, the spin-down time would be
$t_{\mathrm{SD}}>10^{9}\,{\rm yr}$, implying that, however long ago the string
was captured, the spin down would be currently ongoing.

In principle, the decreasing BH spin could be measured using the methods
discussed in Sec. 3.3, for example X-ray reflection spectroscopy for an
accreting supermassive BH. However, the spin is only decreasing at a rate of
$\dot{a}/a\approx 10^{-9}\,{\rm yr^{-1}}$ and the accuracy of these techniques
is not nearly good enough to measure the spin to one part in $10^{9}$
Reynolds:2020jwt .

Although the BH’s spin is challenging to measure, its mass $M$ can be readily
determined from observations of orbiting objects. In particular, a pulsar
orbiting the BH would allow a very accurate measurement of the mass. Pulsars
are the most accurate clocks in nature and have long played an important role
in high-precision astrophysical measurements. Famously, the gradually
decreasing period of the Hulse-Taylor binary pulsar was measured to one part
in $10^{15}$, providing the first observational evidence for gravitational
waves Taylor:1982 ; Weisberg:2010zz .

Supermassive BHs are found in dense galactic centers, which are also expected
to be the home of large numbers of compact objects. In our own galaxy, around
100 pulsars are predicted to be orbiting very close to Sgr A*, with orbital
periods $P\lesssim 10\,\rm yr$ Pfahl:2003tf ; Zhang:2014kva ; Bartels:2015aea
. The Square Kilometer Array (SKA) is expected to probe the galactic center
and observe these pulsars.

Assuming such a pulsar is in fact found close to Sgr A*, timing observations
will be able to very accurately measure $P$. Ongoing observations over an
entire period will then determine the rate at which the period is changing. If
Sgr A* is losing mass due to extraction by cosmic strings, its gravitational
attraction on the pulsar will decrease, causing the pulsar to spiral outward
and the period to increase777In principle, there is a competing effect due to
gravitational wave emission of the orbiting pulsar which causes it to inspiral
and for the period to decrease. However, this leads to a change in the period
which is very small, far below the level which can be detected. For
$M_{\mathrm{BH}}=10^{6}M_{\odot}$ and $P=10\,\rm yr$, the rate of inspiral is
approximately $\dot{P}=10^{-15}$ Simonetti:2010mk . Hadjidemetriou1 ;
Hadjidemetriou2 ; Simonetti:2010mk

$\frac{\dot{P}}{P}=-2\frac{\dot{M}_{\mathrm{SgrA*}}}{M_{\mathrm{SgrA*}}}\ ,$
(11)

where the mass loss rate $\dot{M}$ is given by (4). For a pulsar with period
$P=10\,\rm yr$, the period would be increasing at a rate of $\dot{P}\sim
10^{6}\,\mu$. Only sufficiently light strings, with $\mu<10^{-18}$, would have
a significant chance of being captured by Sgr A*, implying the effect on the
period would be quite small:

$\dot{P}\lesssim 10^{-12}\ .$ (12)

Would such a small effect be observable? With current observational
technology, the orbital period of an isolated pulsar orbiting near Sgr A*
could, in principle, be measurable to an accuracy of about $\Delta P\sim
10^{-11}\ \rm yr$ Kovacs:1981 ; Larsson:1996 ; Simonetti:2020ivl . Using two
measurements spaced apart by one period, the rate of change of the period
could then be measured to an accuracy of $\dot{P}\sim\Delta P/P\sim 10^{-12}$,
which is within the order of magnitude of the size of the expected effect
(12). However, the neighborhood around Sgr A* is rather crowded, and a pulsar
in this region would likely not be sufficiently isolated to achieve this
precision. Although the mass loss is, in principle, observable via pulsar
measurements, significant observational improvements would be required to make
it viable in practice.

In addition, other factors complicate this proposed observational approach.
Sgr A* is continually accreting matter from its surroundings, and the
resulting increase to the BH mass would be both large and uneven. However, a
pulsar orbiting well outside any material surrounding the BH would measure via
Kepler’s 3rd law the mass of both the BH and the accreting matter. The
gravitational effect on the pulsar of matter in an accretion disk wouldn’t
change once it fell into the BH. In contrast, a long CS extracts the mass far
beyond the pulsar orbit and so reduces the gravitational pull on the pulsar
and increases the orbital period.

As in the PBH case discussed in Sec. 3.2, long, horizon-crossing strings are
required to effectively carry away the extracted energy and angular momentum
and spin down the BH. A small CS loop captured by the BH would not spin it
down, as the energy could just travel all the way around the loop and fall
back in. Furthermore, the loop itself would eventually fall into the BH.

##  5 Discussion and open questions

Many physical scenarios for the early universe predict cosmic strings, and,
indeed, F- or D-strings of string theory could manifest as cosmic strings.
Despite their ubiquity in such theories, scant evidence of their actual
existence has been found, but a very intriguing statistical GW background
signal was found recently by NANOGrav consistent with cosmic strings. However
this GW background turns out, alternative methods of testing for the existence
of cosmic strings are quite valuable. Here, we have considered the resulting
spin-down of a black hole due to attached cosmic strings, for various black
hole sizes, finding something of an inverse “goldilocks” problem in which the
extremes may be preferred avenues for observation.

The prospects of observing the spin-down of stellar-mass black holes (with
masses up to hundreds of solar masses) are dim for two reasons. First, the
likelihood of such a black hole encountering a cosmic string is quite small,
and, second, the spin down of the black hole would be quite fast and unlikely
to be observed.

A supermassive black hole (of order a million or more solar masses), on the
other hand, suffers from neither of these problems. Its size makes for a
significant probability of encountering a cosmic string and for a long spin-
down time. Because the spin down is very gradual, observing it requires
extremely sensitive measurements of the black hole spin. We suggest observing
a pulsar orbiting close to Sgr A* and looking for out-spiraling due to energy
being extracted from the black hole by the cosmic string. However, the signal
is likely too small even for high-precision pulsar observations, at least with
current technology.

Primordial black holes, though small, are likely to form with one or more
cosmic strings already attached. Although they typically don’t form with
significant spin, PBHs soon get spun up by accretion, and, in the absence of
cosmic strings to spin them down, large PBHs with masses over $30M_{\odot}$
are predicted to reach near-extremal spins. This allows a sharp test for the
existence of cosmic strings: Large PBHs should be observed with almost no spin
if cosmic strings exist and with very high spins if they do not.

Several factors complicate the interpretation offered above which we have
largely omitted from our analysis but which could play important roles.

We have presented a simplified description of the dynamics of the black hole-
cosmic string system. For the purposes of estimating the energy extraction
rate, the strings have been treated as rigidly rotating. Multiple long strings
attached to a BH can interact with each other, for example, by intersecting
and reconnecting, resulting in a short segment with both ends attached and
which falls into the BH and another segment no longer attached to the BH.
Other than the energy and spin extraction, other effects of the string on the
black hole have been neglected. For example, an attached string would exert a
force on the black hole, causing it to move. A more thorough discussion of
these dynamics can be found elsewhere Vilenkin:2018zol ; Xing_2021 .

We have also neglected other aspects of black hole evolution, in particular,
merger events. Typically, BH mergers result in highly spinning remnants,
affecting, for example, the distribution of masses and spins of the BH
population. As a particularly interesting example, two PBHs connected by a
cosmic string would get pulled toward each other and eventually merge. If
other strings remained attached, the resulting remnant could then be spun
down. Such a scenario merits future investigation. More discussion of PBH
evolution can be found in Hutsi:2019hlw  and DeLuca:2020bjf 888See especially,
the appendix of DeLuca:2020bjf  for discussion of PBH mergers..

Finally, differentiating PBHs from conventional stellar-sized BHs presents an
interesting statistical challenge (see Ref. DeLuca:2020bjf ), especially given
that future generations of GW detectors will greatly expand the volume range
of BH observations and that merger remnant BHs will be highly spinning.

###### Acknowledgements.

We would like to thank Edward Smakov, Michael Ramsey, and Timmy Dhakaia who
were involved in early stages of this project. The work was supported by the
National Science Foundation grants PHY-1820733, PHY-2000398, PHY-2310608
(MSL); AST-2011731 (MK); PHY-2011383, PHY-2308861 (SL); and AST-2011757 (JS).
ML would also like to thank both the APCTP and the BNL EIC Theory Institute
for warm hospitality while this work was being completed.

## References

  * (1) A. Vilenkin and E. P. S. Shellard, _Cosmic Strings and Other Topological Defects_.  Cambridge University Press, 7, 2000.
  * (2) M. B. Hindmarsh and T. W. B. Kibble, _Cosmic strings_ , _Rept. Prog. Phys._ 58 (1995) 477–562, [hep-ph/9411342].
  * (3) M. Sakellariadou, _Cosmic Strings and Cosmic Superstrings_ , _Nucl. Phys. B Proc. Suppl._ 192-193 (2009) 68–90, [0902.0569].
  * (4) J. Polchinski, _Introduction to cosmic F- and D-strings_ , in _String theory: From gauge interactions to cosmology. Proceedings, NATO Advanced Study Institute, Cargese, France, June 7-19, 2004_ , pp. 229–253, 2004\.  hep-th/0412244.
  * (5) T. Vachaspati and A. Vilenkin, _Gravitational Radiation from Cosmic Strings_ , _Phys. Rev. D_ 31 (1985) 3052.
  * (6) T. Damour and A. Vilenkin, _Gravitational wave bursts from cosmic strings_ , _Phys. Rev. Lett._ 85 (2000) 3761–3764, [gr-qc/0004075].
  * (7) T. Damour and A. Vilenkin, _Gravitational wave bursts from cusps and kinks on cosmic strings_ , _Phys. Rev. D_ 64 (2001) 064008, [gr-qc/0104026].
  * (8) T. Damour and A. Vilenkin, _Gravitational radiation from cosmic (super)strings: Bursts, stochastic background, and observational windows_ , _Phys. Rev. D_ 71 (2005) 063510, [hep-th/0410222].
  * (9) NANOGrav collaboration, A. Afzal et al., _The NANOGrav 15 yr Data Set: Search for Signals from New Physics_ , _Astrophys. J. Lett._ 951 (2023) L11, [2306.16219].
  * (10) J. Ellis, M. Lewicki, C. Lin and V. Vaskonen, _Cosmic superstrings revisited in light of NANOGrav 15-year data_ , _Phys. Rev. D_ 108 (2023) 103511, [2306.17147].
  * (11) LIGO Scientific, Virgo, KAGRA collaboration, R. Abbott et al., _Constraints on Cosmic Strings Using Data from the Third Advanced LIGO–Virgo Observing Run_ , _Phys. Rev. Lett._ 126 (2021) 241102, [2101.12248].
  * (12) P. D. Lasky et al., _Gravitational-wave cosmology across 29 decades in frequency_ , _Phys. Rev. X_ 6 (2016) 011035, [1511.05994].
  * (13) J. J. Blanco-Pillado, K. D. Olum and X. Siemens, _New limits on cosmic strings from gravitational wave observation_ , _Phys. Lett. B_ 778 (2018) 392–396, [1709.02434].
  * (14) D. F. Chernoff, A. Goobar and J. J. Renk, _Prospects of Cosmic Superstring Detection through Microlensing of Extragalactic Point-Like Sources_ , _Mon. Not. Roy. Astron. Soc._ 491 (2020) 596–614, [1905.03796].
  * (15) J.-P. De Villiers and V. P. Frolov, _Gravitational capture of cosmic strings by a black hole_ , _Int. J. Mod. Phys. D_ 7 (1998) 957–967, [gr-qc/9711045].
  * (16) M. Snajdr and V. P. Frolov, _Capture and critical scattering of a long cosmic string by a rotating black hole_ , _Class. Quant. Grav._ 20 (2003) 1303–1320, [gr-qc/0211018].
  * (17) F. Dubath, M. Sakellariadou and C. M. Viallet, _Scattering of cosmic strings by black holes: Loop formation_ , _Int. J. Mod. Phys._ D16 (2007) 1311–1325, [gr-qc/0609089].
  * (18) V. P. Frolov, S. Hendy and J. P. De Villiers, _Rigidly rotating strings in stationary axisymmetric space-times_ , _Class. Quant. Grav._ 14 (1997) 1099–1114, [hep-th/9612199].
  * (19) S. Kinoshita, T. Igata and K. Tanabe, _Energy extraction from Kerr black holes by rigidly rotating strings_ , _Phys. Rev._ D94 (2016) 124039, [1610.08006].
  * (20) T. Igata, H. Ishihara, M. Tsuchiya and C.-M. Yoo, _Rigidly Rotating String Sticking in a Kerr Black Hole_ , _Phys. Rev._ D98 (2018) 064021, [1806.09837].
  * (21) R. Penrose and R. M. Floyd, _Extraction of Rotational Energy from a Black Hole_ , _Nature Physical Science_ 229 (Feb., 1971) 177–179.
  * (22) R. D. Blandford and R. L. Znajek, _Electromagnetic extractions of energy from Kerr black holes_ , _Mon. Not. Roy. Astron. Soc._ 179 (1977) 433–456.
  * (23) S. Kinoshita and T. Igata, _The essence of the Blandford-Znajek process_ , _PTEP_ 2018 (2018) 033E02, [1710.09152].
  * (24) H. Xing, Y. Levin, A. Gruzinov and A. Vilenkin, _Spinning black holes as cosmic string factories_ , _Physical Review D_ 103 (Apr, 2021) , [2011.00654].
  * (25) A. Vilenkin, Y. Levin and A. Gruzinov, _Cosmic strings and primordial black holes_ , _JCAP_ 11 (2018) 008, [1808.00670].
  * (26) M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, _Primordial black holes—perspectives in gravitational wave astronomy_ , _Class. Quant. Grav._ 35 (2018) 063001, [1801.05235].
  * (27) A. M. Green and B. J. Kavanagh, _Primordial Black Holes as a dark matter candidate_ , _J. Phys. G_ 48 (2021) 043001, [2007.10722].
  * (28) P. Villanueva-Domingo, O. Mena and S. Palomares-Ruiz, _A brief review on primordial black holes as dark matter_ , _Front. Astron. Space Sci._ 8 (2021) 87, [2103.12087].
  * (29) A. Escrivà, F. Kuhnel and Y. Tada, _Primordial Black Holes_ , 2211.05767.
  * (30) S. Bird, I. Cholis, J. B. Muñoz, Y. Ali-Haïmoud, M. Kamionkowski, E. D. Kovetz et al., _Did LIGO detect dark matter?_ , _Phys. Rev. Lett._ 116 (2016) 201301, [1603.00464].
  * (31) M. Mirbabayi, A. Gruzinov and J. Noreña, _Spin of Primordial Black Holes_ , _JCAP_ 03 (2020) 017, [1901.05963].
  * (32) V. De Luca, V. Desjacques, G. Franciolini, A. Malhotra and A. Riotto, _The initial spin probability distribution of primordial black holes_ , _JCAP_ 05 (2019) 018, [1903.01179].
  * (33) E. Cotner and A. Kusenko, _Primordial black holes from scalar field evolution in the early universe_ , _Phys. Rev. D_ 96 (2017) 103002, [1706.09003].
  * (34) T. Harada, C.-M. Yoo, K. Kohri and K.-I. Nakao, _Spins of primordial black holes formed in the matter-dominated phase of the Universe_ , _Phys. Rev. D_ 96 (2017) 083517, [1707.03595].
  * (35) V. De Luca, G. Franciolini, P. Pani and A. Riotto, _The evolution of primordial black holes and their final observable spins_ , _JCAP_ 04 (2020) 052, [2003.02778].
  * (36) C. S. Reynolds, _Observational Constraints on Black Hole Spin_ , _Ann. Rev. Astron. Astrophys._ 59 (2021) 117–154, [2011.08948].
  * (37) R. Brito, V. Cardoso and P. Pani, _Superradiance: New Frontiers in Black Hole Physics_ , _Lect. Notes Phys._ 906 (2015) pp.1–237, [1501.06570].
  * (38) Y. B. Zel’dovich and I. D. Novikov, _The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model_ , _Sov. Astron._ 10 (1967) 602.
  * (39) B. J. Carr and S. W. Hawking, _Black holes in the early Universe_ , _Mon. Not. Roy. Astron. Soc._ 168 (1974) 399–415.
  * (40) J. Garriga and A. Vilenkin, _Black holes from nucleating strings_ , _Phys. Rev. D_ 47 (1993) 3265–3274, [hep-ph/9208212].
  * (41) M. Y. Khlopov, S. G. Rubin and A. S. Sakharov, _Primordial structure of massive black hole clusters_ , _Astropart. Phys._ 23 (2005) 265, [astro-ph/0401532].
  * (42) N. Tanahashi and C.-M. Yoo, _Spherical Domain Wall Collapse in a Dust Universe_ , _Class. Quant. Grav._ 32 (2015) 155003, [1411.7479].
  * (43) J. Garriga, A. Vilenkin and J. Zhang, _Black holes and the multiverse_ , _JCAP_ 02 (2016) 064, [1512.01819].
  * (44) H. Deng, J. Garriga and A. Vilenkin, _Primordial black hole and wormhole formation by domain walls_ , _JCAP_ 04 (2017) 050, [1612.03753].
  * (45) H. Deng and A. Vilenkin, _Primordial black hole formation by vacuum bubbles_ , _JCAP_ 12 (2017) 044, [1710.02865].
  * (46) S. W. Hawking, _Black Holes From Cosmic Strings_ , _Phys. Lett. B_ 231 (1989) 237–239.
  * (47) A. Polnarev and R. Zembowicz, _Formation of Primordial Black Holes by Cosmic Strings_ , _Phys. Rev. D_ 43 (1991) 1106–1109.
  * (48) K. Kohri and T. Terada, _Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves_ , _Phys. Lett. B_ 813 (2021) 136040, [2009.11853].
  * (49) V. De Luca, G. Franciolini and A. Riotto, _NANOGrav Data Hints at Primordial Black Holes as Dark Matter_ , _Phys. Rev. Lett._ 126 (2021) 041303, [2009.08268].
  * (50) V. Vaskonen and H. Veermäe, _Did NANOGrav see a signal from primordial black hole formation?_ , _Phys. Rev. Lett._ 126 (2021) 051303, [2009.07832].
  * (51) M. Ricotti, J. P. Ostriker and K. J. Mack, _Effect of Primordial Black Holes on the Cosmic Microwave Background and Cosmological Parameter Estimates_ , _Astrophys. J._ 680 (2008) 829, [0709.0524].
  * (52) G. Hütsi, M. Raidal and H. Veermäe, _Small-scale structure of primordial black hole dark matter and its implications for accretion_ , _Phys. Rev. D_ 100 (2019) 083016, [1907.06533].
  * (53) D. P. Bennett and F. m. c. R. Bouchet, _Cosmic-string evolution_ , _Phys. Rev. Lett._ 63 (Dec, 1989) 2776–2779.
  * (54) B. Allen and E. P. S. Shellard, _Cosmic string evolution: a numerical simulation_ , _Phys. Rev. Lett._ 64 (1990) 119–122.
  * (55) J. J. Blanco-Pillado, K. D. Olum and B. Shlaer, _Large parallel cosmic string simulations: New results on loop production_ , _Phys. Rev. D_ 83 (2011) 083514, [1101.5173].
  * (56) KAGRA, VIRGO, LIGO Scientific collaboration, R. Abbott et al., _GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run_ , _Phys. Rev. X_ 13 (2023) 041039, [2111.03606].
  * (57) A. Hall, A. D. Gow and C. T. Byrnes, _Bayesian analysis of LIGO-Virgo mergers: Primordial vs. astrophysical black hole populations_ , _Phys. Rev. D_ 102 (2020) 123524, [2008.13704].
  * (58) G. Hütsi, M. Raidal, V. Vaskonen and H. Veermäe, _Two populations of LIGO-Virgo black holes_ , _JCAP_ 03 (2021) 068, [2012.02786].
  * (59) Z.-C. Chen and A. Hall, _Confronting primordial black holes with LIGO-Virgo-KAGRA and the Einstein Telescope_ , 2402.03934.
  * (60) LIGO Scientific, Virgo collaboration, R. Abbott et al., _GW190521: A Binary Black Hole Merger with a Total Mass of $150M_{\odot}$_, _Phys. Rev. Lett._ 125 (2020) 101102, [2009.01075].
  * (61) M. Jain and A. Vilenkin, _Clustering of cosmic string loops_ , _JCAP_ 09 (2020) 043, [2006.15358].
  * (62) A. Ashoorioon, M. B. Jahani Poshteh and R. B. Mann, _Distinguishing a Slowly Accelerating Black Hole by Differential Time Delays of Images_ , _Phys. Rev. Lett._ 129 (2022) 031102, [2210.10762].
  * (63) J. H. Taylor and J. M. Weisberg, _A new test of general relativity - Gravitational radiation and the binary pulsar PSR 1913+16_ , _Astrophysical J._ 253 (Feb., 1982) 908–920.
  * (64) J. M. Weisberg, D. J. Nice and J. H. Taylor, _Timing Measurements of the Relativistic Binary Pulsar PSR B1913+16_ , _Astrophys. J._ 722 (2010) 1030–1034, [1011.0718].
  * (65) E. Pfahl and A. Loeb, _Probing the spacetime around Sgr A* with radio pulsars_ , _Astrophys. J._ 615 (2004) 253–258, [astro-ph/0309744].
  * (66) F. Zhang, Y. Lu and Q. Yu, _On the Existence of Pulsars in the Vicinity of the Massive Black Hole in the Galactic Center_ , _Astrophys. J._ 784 (2014) 106, [1402.2505].
  * (67) R. Bartels, S. Krishnamurthy and C. Weniger, _Strong support for the millisecond pulsar origin of the Galactic center GeV excess_ , _Phys. Rev. Lett._ 116 (2016) 051102, [1506.05104].
  * (68) J. H. Simonetti, M. Kavic, D. Minic, U. Surani and V. Vijayan, _A Precision Test for an Extra Spatial Dimension Using Black Hole–Pulsar Binaries_ , _Astrophys. J. Lett._ 737 (2011) L28, [1010.5245].
  * (69) J. Hadjidemetriou, _Two-body problem with variable mass: A new approach_ , _Icarus_ 2 (1963) 440.
  * (70) J. Hadjidemetriou, _Analytic solutions of the two-body problem with variable mass_ , _Icarus_ 5 (1966) 34.
  * (71) G. Kovacs, _Frequency shift in Fourier analysis_ , _Astrophysics and Space Science_ 78 (Aug., 1981) 175–188.
  * (72) S. Larsson, _Parameter estimation in epoch folding analysis_ , _Astronomy and Astrophysics Suppliment Series_ 117 (May, 1996) 197–201.
  * (73) J. H. Simonetti, M. J. Kavic, D. Minic, D. Stojkovic and D.-C. Dai, _Sensitive searches for wormholes_ , _Phys. Rev. D_ 104 (2021) L081502, [2007.12184].