11institutetext: Peter Tinyakov 22institutetext: Université Libre de
Bruxelles, CP225 bvd. du Triomphe, 1050 Brussels, Belgium. 22email:
<EMAIL_ADDRESS>
# Primordial Black holes:
The asteroid mass window

Peter Tinyakov

###### Abstract

Primordial black holes (PBHs) are an attractive dark matter candidate,
particularly if they can explain the totality of it. At PBH masses below $\sim
10^{17}$ g and above $\sim 10^{23}$ g this possibility is excluded from the
variety of arguments and with different confidence. The range in between,
often referred to as the “asteroid mass window”, currently remains
unconstrained. The most promising, in our view, way to probe this mass range
is to use stars as the PBH detectors. If a star captures even a single PBH it
starts being accreted onto it and eventually gets destroyed — converted into a
sub-solar mass black hole. This process may have a variety of signatures form
a mere star disappearance to supernova-type explosions of a new kind. The
viability of this approach depends crucially on the probability of PBH capture
by stars. In this chapter we summarize the existing capture mechanisms and
discuss their implications for constraining the abundance of (or perhaps
discovering) PBHs in the asteroid mass window.

##  1 Introduction

Of the whole mass range where the primordial black holes (PBHs) may
potentially constitute the dark matter, the window of roughly
$10^{17}-10^{23}$ g is particularly difficult to assess. No actual
constraints, only proposals exist in this window at present Carr:2020gox .
Below $\sim 10^{17}$ g the PBHs evaporate fast enough to either completely
disappear by now or overproduce the diffuse gamma ray background which is
constrained by the existing observations Carr:2009jm ; Coogan:2020tuf . At
higher masses starting at around $\sim 10^{23}$ g (for comparison, the Moon
mass is $7\times 10^{26}$ g) the PBHs start being detectable through the
gravitational lensing of background stars PhysRevLett.111.181302 ;
Griest:2013aaa . The range between these two values, spanning some 6 orders of
magnitude and often referred to as the “asteroid mass window”, is currently
unconstrained. In this mass range the PBHs have sizes of order or smaller than
the atomic size and are very difficult to detect.

Below $\sim 10^{23}$ g the PBHs are too light to be detectable by the standard
star lensing. A proposal has been made to use instead gamma-ray bursts as
lensed sources Barnacka:2012bm  and look, instead of time variability, for
features in the energy spectrum where PBHs, if present, would induce
oscillations. However, it has been argued later that finite size of sources
together with the wave optics effects will cause the decoherence resulting in
no detectable spectral modulations for most of the gamma-ray bursts except,
perhaps, a small fraction of rapidly varying ones Katz:2018zrn . Most of other
proposals in the asteroid mass range turn around using stars as PBH detectors.
They are based either on the capture of PBHs by stars with the subsequent star
destruction Capela:2013yf ; Capela:2012jz ; Capela:2014ita ; Esser:2022owk ;
Esser:2023yut , or on ignition of nuclear reactions in white dwarfs by a
traversing PBH Graham:2015apa . An overview of these proposals can be found in
Ref. Montero-Camacho:2019jte . Here we focus on the PBH capture by stars.

As we will see below, even in the most favorable conditions a star can only
capture a few PBHs over its lifetime. This is too small to have any dynamical
or gravitational effects on the star. However, once captured inside the star,
a PBH starts to accrete the ambient matter. If the Bondi accretion rate is
assumed111While not completely obvious, it is plausible that Bondi regime
applies in the case of a small PBH inside the star. Nevertheless, this
question requires a quantitative analysis which will be given elsewhere., the
accretion is sufficiently rapid even in the case of ordinary stars to be able
to consume the whole star in a cosmologically short time. In the Bondi regime
Bondi:1952ni , the accretion rate by a PBH of mass $m$ in the center of a star
with the matter density $\rho_{*}$ is estimated as

$\dot{m}\simeq 4\pi{G^{2}m^{2}\rho_{*}\over c_{s}^{3}}$

where $G$ is the gravitational constant and $c_{s}$ the sound speed, and we
have suppressed a numerical factor of order 1 depending on the matter equation
of state. With this rate the accretion time is determined by the initial PBH
mass and reads

$t_{\text{acc}}={c_{s}^{3}\over 4\pi G^{2}\rho_{*}m}\simeq 5\times
10^{6}\text{yr}\left({c_{s}\over
500\text{km/s}}\right)^{3}\left({150\text{g/cm}^{3}\over\rho_{*}}\right)\left({10^{20}\text{g}\over
m}\right).$

We have substituted here the parameters typical of solar mass stars which have
lifetimes exceeding $\sim 10$ Gyr. We see that for PBH masses $m\gtrsim
10^{18}$ g the accretion times are much smaller than the age of the Universe,
and much smaller than the star lifetime. For compact stars such as white
dwarfs and neutron stars the accretion times are much shorter.

Thus, capturing an asteroid-mass BH is fatal for stars — they get destroyed in
a relatively short time. This may lead to several observable consequences like
disappearance of stars, creation of (sub)solar mass black holes and supernova-
type explosions of a new type. All three signatures may potentially be used to
constrain (or discover) the abundance of primordial black holes. However,
before discussing the signatures we need to consider how at all PBHs can be
captured inside a star.

##  2 Capture – generalities

Being a part (or whole) of the dark matter, PBHs and stars are both
concentrated in galactic halos. However, because the interactions of a PBH
with matter are very weak relative to its enormous mass, it is very difficult
to settle a PBH inside a star. There are basically two ways: either a PBH can
be captured by a star from the local DM density, or it can get inside the star
from the star birth. In either case, the process typically requires two
stages:

  * (i)

trapping, or getting on a star-crossing gravitationally bound orbit, and

  * (ii)

cooling, i.e., gradually loosing energy until settled completely inside the
star.

The second stage, cooling, is common for both mechanisms. Once the PBH is
completely inside the star the accretion phase starts.

During the cooling stage, and also during the first stage in case of the
direct capture, the key quantity is the energy loss at a single star crossing.
So we start this section by having a closer, more quantitative look at this
quantity.

Once a PBH is on a bound star-crossing orbit it starts to lose energy at each
subsequent passage through the star. This process has to be fast enough, at
least it should not take longer than the star lifetime to get completely
inside the star. Given the PBH mass, the only free parameter here is the
initial size of the PBH orbit, depending on which there may or may not be
enough time to cool. Thus, not all trapped PBHs will eventually get captured
and destroy the star. The cooling process depends only on the energy loss; it
will be the next topic in this section.

In the ideal case of an isolated star, once a PBH gets on a bound star-
crossing orbit it will cross the star again and again until it gets absorbed
inside. This is not necessarily the case for a star in a realistic environment
like a galaxy, because gravitational perturbations due to neighboring stars
may deviate the PBH and make it miss the star it is orbiting, so that the
energy loss will halt. So, as the last general point in this section we will
quantify the conditions under which the perturbers become important.

###  2.1 Energy loss

A PBH passing through a star slows down for two reasons. First, it accretes
the star matter. In the star frame the total PBH momentum does not change,
while the mass grows, so the velocity decreases. This is equivalent to the
drag force

$F_{\text{acc}}=-\dot{m}v$

where $\dot{m}$ is the mass accretion rate and we have assumed a non-
relativistic PBH velocity $v$. To estimate the mass accretion rate we note
that even those PBHs that have small (inbound) asymptotic velocities get
accelerated when they fall into a star, so that they cross it at a
(marginally) supersonic speed. Indeed, for a sun-like star, the escape
velocity from the star center is $v\sim\sqrt{3GM_{*}/R_{*}}\sim 750$ km/s,
while the sound speed in the Sun center is $\sim 500$ km/s. As one moves away
form the center towards the surface the sound speed decreases by an order of
magnitude, while the escape velocity changes by $\sqrt{2/3}$ at most. For the
purpose of the estimate we may therefore consider the stellar matter as a
collection of free particles. Then the accretion rate is
$\dot{m}=\rho_{*}\sigma_{\text{BH}}v$ where $\sigma_{\text{BH}}$ is the cross
section of particle capture by a black hole,

$\sigma_{\text{BH}}=16\pi{G^{2}m^{2}\over v^{2}}.$

We have therefore

$F_{\text{acc}}=-16\pi\rho_{*}G^{2}m^{2}$

where the velocity $v$ canceled out.

The second force that slows down the PBH is the dynamical friction
RevModPhys.21.383 ; 1987gady.book…..B . This force arises because matter
particles get dragged along with the passing PBH by the gravitational force;
correspondingly, the PBH is slowed down. The dynamical friction force is
written as follows,

$F_{\text{dyn}}=-4\pi\rho_{*}G^{2}m^{2}{\ln\Lambda\over v^{2}}$ (1)

where $\ln\Lambda\sim 30$ is a Coulomb logarithm. Thus, the dynamical friction
is dominant over the accretion by the factor $\ln\Lambda/v^{2}$ which is much
larger than 1, except perhaps in the case of neutron stars where $v\sim 1$ and
the dynamical friction is somewhat reduced due to matter degeneracy.

As in the case of accretion, eq. (1) assumes that the matter particles do not
interact among themselves, which is a good approximation at supersonic PBH
velocities. In our case the velocity of the PBH when passing through a star is
only marginally supersonic. In this case the dynamical friction (1) gives a
smaller force than a more accurate treatment that takes into account
properties of the medium 1999ApJ…513..252O . The accurate answer is some
factor $\sim 2$ larger than eq. (1) in the case of Sun-like stars (see Fig.4
of Ref. Oncins:2022djq ) due to a resonance-type enhancement at velocities
close to the sound speed 1999ApJ…513..252O . We do not include this numerical
factor in what follows.

With the force (1), the energy loss per single star crossing is $\Delta E\sim
R_{*}F_{\text{dyn}}$. More accurately, one has to average the energy loss over
the PBH impact parameter. If one assumes for simplicity the uniform
distribution of the PBH trajectories in the plane perpendicular to the PBH
velocity we get, in agreement with eq.(2) of Ref.Capela:2013yf

$E_{\text{loss}}={1\over\pi R_{*}^{2}}\int
ds\,dlF_{\text{dyn}}={4M_{*}G^{2}m^{2}\over R_{*}^{2}v^{2}}\ln\Lambda.$ (2)

Here we have ignored the change of $v^{2}$ along the PBH trajectory within the
star. We will see later, however, that the distribution of trajectories in
some cases is not uniform, and eq. (3) in these cases is only an order-of-
magnitude estimate.

Substituting in (2) the escape velocity from the star surface
$v^{2}_{\text{esc}}=2GM_{*}/R_{*}$ we can rewrite it as follows,

${E_{\text{loss}}\over m}={2Gm\over R_{*}}\ln\Lambda={R_{g}\over R_{*}}{m\over
M_{*}}\ln\Lambda,$ (3)

where $R_{g}=2GM_{*}$ is the gravitational radius of the star. Note that this
expression is only valid in the case when $v\simeq v_{\text{esc}}$, which
explains why the independence of $E_{\text{loss}}$ of $M_{*}$ is not a
problem. We see from eq. (3) that the more compact is the star, the larger are
the energy losses, but even in the best case of a neutron star with $R_{g}\sim
R_{*}/3$ the energy loss is very small for an asteroid mass PBH for which
$m/M_{*}\lesssim 10^{-10}$.

It follows immediately from this estimate that only very slow PBHs can be
trapped after one star crossing. To be trapped, the asymptotic kinetic energy
of a PBH must be smaller than $E_{\text{loss}}$, which translates into the
condition on the asymptotic velocity $v_{\infty}$ of PBH

$v_{\infty}<2\left({Gm\over R_{*}}\ln\Lambda\right)^{1/2}.$

Numerically, a PBH of mass $m=10^{23}$ g can only get trapped if its
asymptotic velocity relative to the star does not exceed some maximum value
which is $0.03$ km/s for the Sun and $9$ km/s for a neutron star. Even in the
case of a neutron star this velocity is smaller than the velocity dispersion
in most astrophysical environments, with notable exception of some dwarf
galaxies.

Another immediate consequence of this estimate is large initial size of the
PBH bound orbit after trapping. When a PBH from the ambient DM distribution
passes through a star it loses the amount of energy (3). If its initial
kinetic energy was smaller it gets trapped. A typical energy of a freshly
trapped PBH is therefore of the order $-E_{\text{loss}}$. Such PBHs have
radial orbits of the size

$r_{\text{max}}\sim{GM_{*}m\over E_{\text{loss}}}={M_{*}\over
2m\ln\Lambda}R_{*}.$ (4)

The trapped PBH continues to lose energy at each subsequent star crossing and
finally gets completely absorbed in the star, provided that (i) there is
enough time and (ii) the trajectory is not perturbed so that it stops passing
through the star and lose energy. We come now to these two conditions.

###  2.2 Cooling time

Since the energy loss at a single star crossing is very small, the size of the
first bound orbit is very large, cf. eq. (4). Outside of the star such an
orbit is a Keplerian orbit with the apastron $a=r_{\text{max}}$ and periastron
$b<R_{*}\ll a$. One may therefore consider this orbit as radial. For such an
orbit the time between the first and the next star crossing is

$\Delta t=\pi{a^{3/2}\over\sqrt{R_{g}}}.$

If we assume that after the first passage the PBH total energy is
$E_{1}=-E_{\text{loss}}$, at the $n$-th step its energy is

$E_{n}=-nE_{\text{loss}}.$

Correspondingly, the size of the $n$-th orbit is

$a_{n}={r_{\text{max}}\over n}$

and the return time at the $n$-th step is

$\Delta t_{n}={\Delta t\over n^{3/2}}.$

The total cooling time converges and is saturated by a few first orbits.
Summing up $\Delta t_{n}$ one finds

$t_{\text{cool}}=\sum_{n=1}\Delta
t_{n}=\pi\zeta(3/2){r_{\text{max}}^{3/2}\over\sqrt{R_{g}}}\sim
8{r_{\text{max}}^{3/2}\over\sqrt{R_{g}}}.$ (5)

This equation is valid when the initial size of the orbit is $r_{\text{max}}$
given by (4), as determined by $E_{\text{loss}}$ of eq. (3).

In the case when the initial size of the orbit $a_{0}$ is much smaller than
$r_{\text{max}}$, one may either calculate a partial sum in eq. (5) and then
use the asymptotics of the resulting Hurwitz zeta function, or use the
continuous energy loss approximation. After the star crossing, the energy
changes by $-E_{\text{loss}}$, and correspondingly the apastron of the orbit
decreases by

$\Delta a=-{a^{2}\over GM_{*}m}E_{\text{loss}}.$

Approximating $da/dt$ by $\Delta a/\Delta t$ one obtains the differential
equation for the change of the apastron

${da\over dt}=-{2E_{\text{loss}}\over\pi m\sqrt{R_{g}}}\sqrt{a}.$

Solving this equation one finds the cooling time as a function of the initial
apastron $a_{0}$,

$t_{\text{cool}}(a_{0})={\pi\sqrt{R_{g}}m\over
E_{\text{loss}}}\sqrt{a_{0}}={\pi M_{*}R_{*}\over
m\sqrt{R_{g}}\ln\Lambda}\sqrt{a_{0}},$ (6)

where we have substituted $E_{\text{loss}}$ from eq. (3). This equation is
valid for $a\ll r_{\text{max}}$, with $r_{\text{max}}$ given by eq. (4). Note
that if one formally takes $a_{0}=r_{\text{max}}$ in eq. (6), the result is
parametrically the same as eq. (5) and only differs by a numerical
coefficient.

###  2.3 Perturbers

For a successful cooling the bound PBH trajectory must cross the star at each
approach, i.e. the periastron must be smaller than the star radius. For an
isolated star and PBH orbits that become bound as a result of the energy loss
inside the star this is automatically guaranteed. However, in a realistic
environment (in a galaxy) the gravitational perturbations from nearby stars
may make the trajectory that originally passed through the star to miss it at
the next approach. If this happens, the PBH starts orbiting the star without
energy loss and the cooling will never complete. Following Ref. Esser:2022owk
and assuming the perturber is a nearby star of the same mass, let us derive
the condition on the distance to this star which guarantees that the
perturbation is not big enough to halt the cooling.

For very extended nearly radial orbit the periastron $r_{\rm min}$ is
completely determined by the PBH angular momentum $J$ through the relation

$r_{\rm min}={J^{2}/(m^{2}R_{g})}.$

Requiring $r_{\rm min}<R_{*}$ sets an upper bound on the angular momentum

$J_{\rm max}/m=\sqrt{R_{*}R_{g}}.$ (7)

The condition that the orbit crosses the star $r_{\rm min}<R_{*}$ is thus
equivalent to requiring that $J<J_{\text{max}}$.

Consider the change of the PBH angular momentum under the influence of a small
gravitational potential $U(\vec{x})$. For definiteness we suppose that the
unperturbed PBH trajectory is $x(t)$, and the perturbation is in the plane
$(x,y)$. We are interested in the change of the angular momentum $\Delta J$
accumulated over one fall from $x=r_{\text{max}}$ to $x\sim R_{*}$. We have

$\Delta J/m=\int_{0}^{T/4}x^{2}(t){\partial^{2}U\over\partial x\partial y}dt,$

where $T$ is the period of the radial Keplerian orbit with the apastron
$r_{\rm max}$. To estimate the integral we use the unperturbed motion $x(t)$
and assume that the perturber is far away, so that the derivative
$\partial_{x}\partial_{y}U$ is approximately constant along the trajectory.
This gives

$\Delta J/m={5\pi\over 16}{r_{\rm max}^{7/2}\over
R_{g}^{1/2}}{\partial^{2}U\over\partial x\partial y}.$ (8)

We have to require that $\Delta J\lesssim J_{\text{max}}$.

If the perturber is a star with the same mass at a distance $d\gg r_{\rm max}$
which we assume to be static, this condition translates into

$r_{\rm max}<d\left({R_{*}\over\alpha d}\right)^{1/7}.$ (9)

Here $\alpha$ is a numerical coefficient of order 1 that depends on the
relative orientation of the PBH orbit and the direction to the perturber. In
view of the small power in eq. (9) the numerical value of this coefficient is
not so important. In a given environment the distance $d$ can be estimated
from the density of stars.

##  3 Direct capture

With these tools at hand, consider now the direct capture in more detail. It
is instructive to start by deriving the rate of trapping of PBHs by an
isolated star. So, assume an isolated star at rest is placed in a DM halo of
density $\rho_{\text{DM}}$ consisting of PBHs which have Maxwell distribution
in velocities with the dispersion $\sigma$,

$dn={\rho_{\text{DM}}\over m}\left({3\over
2\pi\sigma^{2}}\right)^{3/2}\exp\left(-{3v^{2}\over
2\sigma^{2}}\right)d^{3}v.$ (10)

Some of the PBHs from the halo will cross the star and lose enough energy to
become gravitationally bound on orbits that cross the star, i.e., will get
trapped. Averaging over the Maxwell distribution, the rate at which this
happens is Press:1985ug

${dN\over dt}=2\sqrt{6\pi}{\rho_{\text{DM}}\over\sigma
m}GM_{*}R_{*}\left\\{1-\exp\left(-{3E_{\text{loss}}\over
m\sigma^{2}}\right)\right\\}\simeq
6\sqrt{6\pi}{\rho_{\text{DM}}\over\sigma^{3}}{GM_{*}R_{*}\over
m^{2}}E_{\text{loss}},$

where in the last equality we have expanded the exponential assuming
$3E_{\text{loss}}/(m\sigma^{2})\ll 1$. Note that only the low-velocity end of
the distribution contributes at small $E_{\text{loss}}$, so the details of the
velocity distribution in fact do not matter.

If we substitute here the expression (3) for $E_{\text{loss}}$ we find

${dN\over
dt}=24\sqrt{6\pi}{\rho_{\text{DM}}\over\sigma^{3}}G^{2}M_{*}\ln\Lambda.$

Remarkably, when applicable, the result does not depend neither on the PBH
mass nor on the star radius. Numerically, the trapping rate reads

${dN\over
dt}=0.5~{}\text{Gyr}^{-1}\left({7\text{km/s}\over\sigma}\right)^{3}\left({\rho_{\text{DM}}\over
100~{}\text{GeV/cm}^{3}}\right)$ (11)

for $M_{*}=M_{\odot}$. We stress again that this is the rate at which the PBHs
get trapped (settle on bound star-crossing orbits) after the first collision
with the star. While only these PBHs may repeatedly collide with the star and
eventually lose enough energy to completely sink inside, not all of them will
necessarily do it. Therefore, this is the upper bound; the actual capture rate
can only be smaller. This expression has been derived from different arguments
in Ref.Montero-Camacho:2019jte 222There is a factor 2 difference between eq.
(11) and Ref. Montero-Camacho:2019jte  which we attribute to a different
estimate of $E_{\text{loss}}$.

This bound implies that by the direct mechanism it is possible to capture a
few PBHs at most during the whole lifetime of the Universe. It applies to all
types of stars except perhaps neutron stars for which the approximations which
we have used become marginal, so a numerical factor may arise. We will
consider the case of neutron stars in more detail below.

###  3.1 Ordinary stars

Let us show that ordinary stars are very inefficient in capturing PBHs
directly. In this case the actual capture rate is much smaller than given by
eq. (11) because of further suppression resulting from cooling time and
perturbation constraints.

Consider first the cooling time (5). As explained above, it is determined by
the initial orbit size $r_{\text{max}}$ which is in turn determined by the
energy loss. Since the initial total energy of a PBH is positive, the smallest
(most negative) value the energy may have after the star crossing is
$-E_{\text{loss}}$. The larger $E_{\text{loss}}$, the smaller is
$r_{\text{max}}$ and the shorter the cooling time.

The energy loss varies depending on the impact parameter and is maximum for
the trajectory passing through the star center. But even for such a trajectory
the energy loss is ony factor $3/2$ larger than the average value (3). With
the average energy loss, the initial orbit size is

$r_{\text{max}}=3.7~{}\text{kpc}\left({10^{20}~{}\text{g}\over m}\right),$

and the corresponding cooling time is

$t_{\text{cool}}=2\cdot 10^{13}~{}\text{yr}\left({10^{20}~{}\text{g}\over
m}\right)^{3/2}.$

We see that the initial size of the orbit is unrealistically large except
maybe at the upper end of the asteroid window $m\sim 10^{23}$g. The cooling
time exceeds $10$ Gyr in the whole mass window, beign comparable to it only at
the upper end.

The constraint from perturbers cannot be satisfied either. As we will discuss
below in Sect. 5, the most promising known systems maximizing the capture rate
(11) are ultra-faint dwarf galaxies where the star-to-star distance $d$ is of
the order $d\sim 6\text{pc}$. The maximum initial orbit size has to be at
least an order of magnitude smaller: making use of eq. (4) with $\alpha=1$ we
find

$r_{\text{max}}<0.4~{}\text{pc}.$

Again, this condition cannot be satisfied unless $m>10^{24}$ g (cf. eq. (4)),
which is outside of the mass window we consider.

We conclude that direct capture does not work for ordinary Sun-like stars.
Even if a fraction of PBHs escapes gravitational perturbations by chance,
there is not enough time to lose the energy.

###  3.2 Neutron stars

In the case of neutron stars there are several differences which have to be
taken into account. First, a neutron star has radius only about factor $\sim
3$ larger than its Schwarzschild radius. This means the General Relativity
(GR) effects start becoming non-negligible. They lead, in particular, to a
marginal enhancement by a factor $1/(1-R_{g}/R_{\text{NS}})\approx 1.6$ in the
capture rate Kouvaris:2007ay .

A PBH falling onto a neutron star accelerates to a semi-relativistic velocity
$v\sim\sqrt{R_{g}/R_{\text{NS}}}\sim 0.6$. This velocity is similar or larger
than the sound speed in the neutron star core which is most often cited within
the $0.3-0.6$ range, but is not well known and depends on the neutron star
model. For semi-relativistic velocities the dynamical friction (1) loses its
$1/v^{2}$ enhancement.

Another complication is that nuclear matter is degenerate, and therefore in
the approximation of non-interacting particles (neutrons) not all scattering
states can be excited, but only those for which the momentum transfer exceeds
the Fermi momentum. Since the dynamical friction force can be viewed as a
result of the momentum transfer to scattered matter particles, less scattering
means weaker friction force.

The calculation of the dynamical friction that takes into account both effects
can be found in the Appendix of Ref. Capela:2013yf  where the contribution of
the force due to the direct accretion was also included. The resulting energy
loss, averaged over different impact parameters of PBH trajectories passing
through the star, was found to be

$E_{\text{loss}}={4G^{2}m^{2}M_{*}\over R_{*}^{2}}\left\langle{\ln\Lambda\over
v^{2}}\right\rangle$ (12)

where the $\langle...\rangle$ denotes the density-weighted average over the
star volume. Numerically one finds, using a realistic neutron star density
profile,

$\left\langle{\ln\Lambda\over v^{2}}\right\rangle\simeq 15.$

Remarkably, accounting for the degeneracy only reduces the energy loss by a
factor $\sim 6$ as compared to the naive estimate $\ln\Lambda/v^{2}\sim 83$.

Apart from the dynamical friction, one should include in the energy loss the
tidal effects, excitation of the surface waves and gravity wave emission.
Numerically these effects turn out not very important for PBH masses in the
asteroid window. With the account of all the known energy losses, the direct
capture rate by an isolated neutron star was calculated in Ref.
Genolini:2020ejw . The result can be written as follows,

${dN\over
dt}=0.26\,\text{Gyr}^{-1}\left({7\text{km/s}\over\sigma}\right)^{3}\left({\rho_{\text{DM}}\over
100~{}\text{GeV/cm}^{3}}\right){\cal C}(X),$ (13)

where we have included the GR enhancement by a factor $\approx 1.6$
Kouvaris:2007ay . Here the variable $X$ is

$X=0.018\left({m\over
10^{20}\text{g}}\right)\left({7\text{km/s}\over\sigma}\right)^{2}.$

and the function ${\cal C}(X)\approx 1$ in the range $X\in[0,20]$ and drops as
power law for larger values of $X$. Note that $X<20$ for PBH masses up to
$10^{23}$ g at $\sigma=7$ km/s. The full shape of ${\cal C}(X)$ can be found
in Ref. Genolini:2020ejw , Fig. 3.

Let us now check the conditions coming from the cooling time and the
gravitational perturbations by nearby stars. For both conditions the key
quantity is the size of the initial bound orbit $r_{\text{max}}$. Given the
energy loss by a PBH passing through the neutron star (12), the initial size
of the orbit can be found from eq. (4) which gives

$r_{\text{max}}={R_{*}^{2}\over 4Gm}\left\langle{\ln\Lambda\over
v^{2}}\right\rangle^{-1}=0.7\,\text{pc}\left({10^{20}\text{g}\over m}\right).$
(14)

In terms of $r_{\text{max}}$, the cooling time is given by eq. (5) which
contains no other parameters. This gives

$t_{\text{cool}}=5\cdot 10^{7}\,\text{yr}\left({10^{20}\text{g}\over
m}\right)^{3/2}.$

We see that the cooling is sufficiently fast as long as $m\gtrsim 3\cdot
10^{18}\,\text{g}$, i.e. almost in the whole asteroid mass range except the
lightest masses.

The condition that nearby stars do not spoil the cooling is given by eq. (9).
We may use the same estimate of the star-to-star distance as above, $d\sim 6$
pc, as found in some of the prominent dwarf galaxies. Making use of this value
and $R_{*}\simeq 10$ km for the neutron star radius we find that the condition
(9) requires

$r_{\text{max}}<0.08\,\text{pc}.$

Combining this with the expression for $r_{\text{max}}$ as a function of $m$
eq. (14), this condition translates to

$m\gtrsim 10^{21}\,\text{g}.$ (15)

Thus, the high mass end of the asteroid mass window satisfies this condition,
while the low mass end does not.

In summary, both conditions together are satisfied at the high mass end of the
asteroid window, eq. (15). For these masses the capture rate is given by eq.
(13). The rate is not vanishingly small only in environments with high DM
density and low velocity dispersion, similar to dwarf galaxies where,
numerically, the capture rate may reach values of 1 per a few Gyr.

##  4 Capture at birth

Let us now turn to the second mechanism — capture of PBHs by stars at their
birth. Stars form from dense gas clouds which at some point lose their
stability with respect to gravitational collapse. At the initial stages these
clouds are almost uniform. As the collapse progresses a dense protostellar
object develops in the center 1996ApJ…469..366B ; Bate:1998hg  which starts
accreting the surrounding gas, forming at the end a pre-main-sequence star.
This process happens at the Kelvin-Helmholtz timescale which is much longer
than the free fall time.

Star formation takes place in galaxies in the presence of DM density
$\rho_{\text{DM}}$ which we will assume, as above, to consist of PBHs. We will
also assume that the PBHs have Maxwell distribution in velocities with the
velocity dispersion $\sigma$. A typical baryonic density in a cloud of
$10^{6}-10^{7}$ GeV/cm3 is much larger than the DM density in the galactic
halos, so whatever happens to PBHs has no influence on the cloud dynamics.

The capture mechanism is the following. A gas cloud from which a star will
form is gravitationally bound and also has some amount of PBHs gravitationally
bound to it. When the cloud collapses into a pre-main-sequence star, the time-
dependent gravitational potential of the collapsing baryons drags these PBHs
along, so that their distribution becomes peaked at the star position. When
the star is finally formed, there is a probability to find a PBH on a bound
orbit either completely contained inside or passing through the star. The fist
ones are already captured; the second start cooling and will be eventually
captured if the cooling is successful.

Obviously, only the trajectories of PBHs that spend large time inside the
contracting cloud are significantly affected by the contraction, so the PBHs
that are not bound to the cloud can be neglected. The number of captured PBHs
is proportional to their total number in the corresponding part of the phase
space. This number in turn is proportional to the total DM density
$\rho_{\text{DM}}$, and to the volume of the velocity space limited from above
by the escape velocity from the cloud. The escape velocity can be estimated
from the baryonic density $\rho_{0}$ and radius $R_{0}$ of the cloud. These
parameters are known from observations Kirk:2005ng  and are summarized in
Ref.Capela:2012jz . For example, for a star of $1\,M_{\odot}$ the values are
$R_{0}=4300$ AU and $\rho_{0}=10^{6}$ GeV/cm3. For these parameters the escape
velocity is $v_{\text{esc}}=0.6$ km/s. This is much smaller than the velocity
dispersion in known dwarf galaxies. Therefore, the volume in the DM velocity
space corresponding to trajectories bound to the cloud is proportional to
$v^{3}_{\text{esc}}/\sigma^{3}$. We conclude that the number of bound
trajectories, and therefore the number of captured PBHs is proportional to the
combination ${\rho_{\text{DM}}/\sigma^{3}}$, similarly to the direct capture
case.

###  4.1 Mean captured number: ordinary stars

Consider first ordinary stars because in any case compact stars — white dwarfs
and neutron stars — are formed from main sequence stars and therefore pass
first through this stage. So, the PBHs captured by compact stars at birth are
those that they inherit from their main sequence progenitors.

In order to calculate the total number of captured PBHs one has to know the
distribution of the PBHs in position and velocity after the star is formed.
This distribution builds up from bound PBH trajectories during the star
formation. The fact that the formation happens slowly on a time scale much
longer than the free fall time implies that the motion of the bound PBHs can
be described in the adiabatic approximation. This means that the details of
the contraction process do not matter, only the initial and final state.

The maximum number of PBHs that can be captured by a star at birth can be
estimated by assuming that after the star is formed the phase space of PBHs is
filled with the maximum density all the way up to positive energies. From the
Liouville theorem, the phase space density after the star formation can
nowhere exceed the maximum phase space density before — that is, the one at
low velocities, assuming the Maxwellian DM distribution. From eq. (10), this
phase space density is

${\cal Q}={\rho_{\text{DM}}\over m}\left({3\over
2\pi\sigma^{2}}\right)^{3/2}.$ (16)

We are not, however, aware of the proof that this phase space has to be
filled. It is straightforward to argue that the Liouville theorem and the
angular momentum conservation together do not guarantee this. A counter-
example is a very fast (instantaneous) contraction which satisfies both
properties but obviously does not populate the circular orbits close to the
star.

It is instructive nevertheless to make an analytic estimate (the detailed
calculation can be found in Ref. Oncins:2022djq ). Starting from the
distribution that has constant phase space density (16), we want to calculate
the number of orbits that (i) are bound and (ii) have periastrons smaller than
the star radius $R_{*}$. In other words, we have to integrate

$dN={\cal Q}d^{3}rd^{3}v=8\pi^{2}{\cal Q}r^{2}v_{\phi}drdv_{r}dv_{\phi}$ (17)

over the part of the phase space that satisfies (i) and (ii). Here we have
used symmetries to reduce the number of variables to three: radius $r$, radial
velocity $v_{r}$ and the modulus of the tangential velocity $v_{\phi}$.

The shape of the Keplerian orbits is completely characterized by conserved
energy $E$ and angular momentum $L$,

$\displaystyle\epsilon$ $\displaystyle\equiv$ $\displaystyle{E\over m}={1\over
2}(v_{r}^{2}+v_{\phi}^{2})-{GM_{*}\over r},$ (18) $\displaystyle l$
$\displaystyle\equiv$ $\displaystyle{L\over m}=rv_{\phi}.$ (19)

At fixed $r$ one may trade the velocities $v_{r}$ and $v_{\phi}$ for
$\epsilon$ and $l$. From eqs. (18)–(19) the Jackobian of this transformation
is

$dv_{r}dv_{\phi}={1\over rv_{r}}d\epsilon dl.$

One has therefore from eq. (17)

$dN=8\pi^{2}{\cal Q}{l\over v_{r}}drd\epsilon dl.$

We may express the radial velocity through $\epsilon$ and $l$ as follows,

$v_{r}=\pm\sqrt{{2GM_{*}\over r}+2\epsilon-{l^{2}\over r^{2}}}$

where both signs are physical. We find therefore

$dN=8\pi^{2}{\cal Q}{1\over\sqrt{{2GM_{*}\over r}+2\epsilon-{l^{2}\over
r^{2}}}}drd\epsilon d(l^{2}).$ (20)

The extra factor 2 in this equation is introduced to account for two signs of
$v_{r}$ at given $E$ and $L$.

One more change of variables is needed to make the integration trivial. Since
the condition (ii) is imposed on the periastron, it is useful to change the
variables from $\epsilon$ and $l^{2}$ to the periastron $b$ and apastron $a$.
For any finite $a$ the condition (i) — negative total energy – is then
automatically satisfied. We will see at the end that most of the contributions
come from very elongated orbits for which $a\gg b$. In this case one can
approximate $a$ and $b$ as follows,

$\displaystyle a$ $\displaystyle=$ $\displaystyle-{GM_{*}\over\epsilon}+{\cal
O}(b/a),$ (21) $\displaystyle b$ $\displaystyle=$ $\displaystyle{l^{2}\over
2GM_{*}}+{\cal O}(b/a).$ (22)

Recall that the energy is negative, $\epsilon<0$. In the same approximation,
one can neglect the last term under the square root in eq. (20). Eqs. (20),
(21) and (22) then give

$dN=4\pi^{2}{\cal Q}{R_{g}^{3/2}r\over a^{3/2}\sqrt{ar-r^{2}}}dr\,da\,db,$

where as before $R_{g}=2GM_{*}$ is the Schwarzschild radius of the star.

We will need at the end the number of PBHs on star-crossing orbits with the
apastron smaller than a given $a_{\text{max}}\gg R_{*}$. This makes the
integral convergent. We therefore have to integrate the above expression over
$r$ from $b$ to $a$, then over $a$ from $b$ to $a_{\text{max}}$ and finally
over $b$ from 0 to $R_{*}$,

$N=4\pi^{2}{\cal
Q}R_{g}^{3/2}\int_{0}^{R_{*}}db\int_{b}^{a_{\text{max}}}{da\over
a^{3/2}}\int_{b}^{a}{r\,dr\over\sqrt{ar-r^{2}}}.$

One may notice that the dominant contribution to this integral comes from the
region $r,a\gg R_{*}$. We may therefore change the lower limit of the
integration over $r$ to 0 instead of $b$. The integral over $r$ then gives

$\int_{0}^{a}{r\over\sqrt{ar-r^{2}}}dr={a\pi\over 2},$

so that

$N=2\pi^{3}{\cal
Q}R_{g}^{3/2}\int_{0}^{R_{*}}db\int_{b}^{a_{\text{max}}}{da\over\sqrt{a}}$
(23)

The integral over $a$ now becomes explicitly convergent at the lower limit and
we can again replace $b$ by $0$. Then the integration gives

$N=4\pi^{3}{\cal
Q}R_{g}^{3/2}R_{*}\sqrt{a_{\text{max}}}={\rho_{\text{DM}}\over
m\sigma^{3}}(3\pi R_{g})^{3/2}R_{*}\sqrt{2a_{\text{max}}}.$ (24)

This expression agrees with the one obtained in Ref.Oncins:2022djq  for the
same quantity. It also agrees with the number of PBHs that settle on the star
crossing orbits calculated numerically in Ref.Esser:2022owk . This implicitly
confirms our starting assumption that when the star is formed, the whole phase
space with $E<0$ is filled with the maximum phase-space density.

The captured number grows infinitely as $a_{\text{max}}\to\infty$, i.e. when
larger and larger orbits are included. At some point, however,
$a_{\text{max}}$ becomes so big that either the cooling time becomes longer
than the age of the star, or the perturbations from nearby stars can no more
be neglected. We will see below that, in realistic conditions of dwarf
galaxies, the cooling requirement is more constraining at masses below $\sim
10^{21}$ g. In order to estimate the number of captured PBHs from eq. (24) in
this mass range it remains, therefore, to substitute in eq. (24) the maximum
size of the orbit that still has time to cool and settle inside the star. The
orbits that PBHs acquire after the contraction are much smaller than the
critical value (4), so we should use eq. (6) for the cooling time. Comparing
eqs. (24) and (6) we see that their dependence on $a$ and $m$ is the same, so
that we can rewrite the captured number $N$ as follows,

$N=3\sqrt{6\pi}R_{g}^{2}{\ln\Lambda\over
M_{*}}\,{\rho_{\text{DM}}\over\sigma^{3}}\,t_{\text{cool}}.$ (25)

From this expression we conclude that when the capture is limited by the
cooling time, the number of captured PBHs does not depend on the PBH mass $m$.
Note, however, that eq. (6) is approximate, so the estimate (25) is only valid
when the former is correct.

If we substitute in eq. (25) the cooling time of order of the star lifetime,
$t_{\text{cool}}\sim 10$ Gyr, we find that numerically

$N=2.3\,\left({\rho_{\text{DM}}\over
100\,\text{GeV/cm}^{3}}\right)\left(7\,\text{km/s}\over\sigma\right)^{3}\left({t_{\text{cool}}\over
10\,\text{Gyr}}\right).$ (26)

This is very close to the result of the numerical simulations of
Ref.Esser:2022owk  as will be discussed below. It should be stressed, however,
that several approximations have been made to obtain eq. (25), so the
numerical agreement of eq. (25) with simulations is a mere coincidence. Among
these approximations we have used the average energy loss and assumed, when
calculating the average in eq. (2), that the trajectories are distributed
uniformly over the impact parameter plane inside the star. We see now from eq.
(24) that this is not the case: if it were, the measure in eq. (23) would have
been $b\,db$, not $db$. We have also assumed that $b\ll a$. This approximation
may break for small PBH masses leading to small energy loss, cf. eq. (2),
which would imply that only relatively compact trajectories have time to cool
down.

If one does not want to rely on assumptions, numerical methods must be used to
determine $N$. Such simulations have been performed in Refs.Capela:2012jz ;
Capela:2014ita ; Esser:2022owk . The calculation is greatly simplified by the
fact that the total dark matter mass on bound orbits is by many orders of
magnitude smaller than the star mass, so one can simulate PBH trajectories one
by one in the same gravitational potential of the contracting cloud, each time
with different initial conditions. Further simplification is due to the
adiabatic character of the contraction. Because of that one may chose a
convenient parameterization of the time-dependent cloud shape — all choices
should give the same result as long as the contraction is slow.

The general algorithm is as follows (see Ref. Esser:2022owk  for details). One
generates random initial conditions for PBH trajectories covering uniformly
the relevant region of the phase space. For numerical efficiency one may
sample only the region of negative starting energy and small angular momentum
containing (almost) all the trajectories that, by the end of the contraction,
may become gravitationally bound and passing through the star. Each trajectory
is then evolved in the gravitational field of the contracting cloud. The
trajectories that do pass through the star at the end of the contraction are
selected.

Among these trajectories, the ones with the apastron too large to survive the
deflections by nearby stars (see Sect. 2.3) are then discarded. For each of
the remaining trajectories the cooling time has to be checked. It is not
sufficient at this stage to rely on the average energy loss (3). In
Ref.Esser:2022owk  the energy losses were calculated with the account of the
actual star density profile (the Solar profile was used) and the parameters of
individual trajectories. The trajectories having the cooling time longer than
the star lifetime are again discarded. The remaining ones are considered
successful.

The phase space region sampled by the initial conditions contains some total
number and total mass of PBH. Knowing the fraction of the successful (that is,
leading to capture) trajectories from simulations, one may immediately
determine the mean captured number $\bar{N}$ and the mean captured mass
$\bar{M}=m\bar{N}$ of PBHs. For a $1~{}M_{\odot}$ star the result is shown in
Fig. 1, which is the same as Fig. 1 of Ref.Esser:2022owk , except $\bar{N}$ is
shown instead of $\bar{M}$.

Figure 1: The mean number $\bar{N}$ of PBHs captured by a star of
$1~{}M_{\odot}$ at birth as a function of the PBH mass. The calculation
assumes an isolated star at rest in an environment with the PBH mean mass
density of $100$ GeV/cm3 and velocity dispersion $7$ km/s. The plot is based
on the same simulation as Fig. 1 of Ref.Esser:2022owk .

On the horizontal part of the plot in the range roughly $10^{19}-10^{21}$ g,
the mean captured number is limited by the cooling time. There it is in
agreement with eq. (25) which predicts no dependence on the PBH mass. The drop
at around $10^{21}$ g occurs where the requirement of not being perturbed by
the nearby stars becomes more important than that of the cooling time. This
part of the curve depends on the assumed ambient star density $n_{*}$; the
value $n_{*}=0.009$ pc-3 as inferred from measured parameters of the
Triangulum II dwarf galaxy was used on this plot.

At PBH masses smaller than $\sim 10^{19}$ g the mean captured number starts to
decrease. This can be traced back to breaking of the approximations used to
derive eq. (25), in particular of the assumption that the trajectories with
$a\gg b$ give the dominant contribution.

###  4.2 Mean captured number: compact stars

Compact stars form as a result of evolution of main sequence stars. Massive
stars with masses in the range of $8M_{\odot}\lesssim M_{*}\lesssim
15M_{\odot}$ during their main sequence phase typically end their evolution as
neutron stars, the lighter ones become white dwarfs, while the heavier ones
collapse directly into black holes. The lifetimes of massive stars decrease as
the mass grows, reaching about $\sim 10$ Myr at the heavy mass end. If a main
sequence star at its birth has trapped one or more PBHs on bound star crossing
orbits, and if they get captured inside the progenitor star and sink to the
center before it evolves into a compact remnant, they may be inherited by the
latter.

This two-step process has been considered in Refs.Capela:2012jz ;
Capela:2014ita  where the amount of captured PBHs was estimated for neutron
stars and white dwarfs in the assumption that at the first stage the PBHs do
not destroy the progenitor. This assumption is not crucial for deriving
constraints based on the neutron star and white dwarf disappearance because
the destruction of a progenitor star has the same effect. Progenitors of
various masses have been considered. The results were presented in the form of
expected constraints on the fraction of PBHs in the total amount of dark
matter that one could impose provided a population of neutron stars and/or
white dwarfs is found in dwarf galaxies. Unfortunately, this is not presently
the case. Note, however, that perturbations by nearby stars were not taken
into account in the analysis of Refs.Capela:2012jz ; Capela:2014ita .

##  5 Implications for constraints on PBH abundance

As already mentioned, the destruction of stars — main sequence or compact — by
PBHs may potentially have several different signatures that may be used to
constrain (or detect) the PBH abundance. Of these, a mere star disappearance
due to their conversion into sub-solar mass black holes appears the most
robust one. We will now consider this signature in more detail; other
signatures will be briefly discussed in the concluding section. We will focus
on main sequence stars because they are much easier to observe, while the
deficit of compact stars due to their transmutation into black holes is much
harder to establish experimentally. In fact, as of today, no neutron stars or
white dwarfs have been detected in dwarf galaxies.

In the previous sections we have seen that PBHs of asteroid masses are very
difficult to capture inside stars. All the mechanisms that we have considered
require cosmologically long time and may lead to sizable probability of
capture only in exceptional environments. Both the probability of the direct
capture, eq. (13), and the capture at birth, eq. (26), are proportional to
$\rho_{\text{DM}}/\sigma^{3}$. Following Ref.Esser:2022owk  one may classify
the existing environments according to their “merit factor” $\eta$ defined as

$\eta={\rho_{\text{DM}}\over
100~{}\text{GeV/cm}^{3}}\left({7~{}\text{km/s}\over\sqrt{2}\sigma}\right)^{3},$

where $\sqrt{2}$ accounts for the fact that when both PBHs and stars have the
same velocity dispersion $\sigma$, the relative velocity has the dispersion
$\sqrt{2}\sigma$. In the environments with $\eta=1$ eqs.(13) and (26) apply
with $\rho_{\text{DM}}$\- and $\sigma$-dependent factors equal to 1.

Of all the astrophysical environments with experimentally confirmed DM density
only some ultra-faint dwarf galaxies (UFDs) reach the values of $\eta$ close
to 1. Among the most promising UFDs are the Triangulum II ($\eta\geq 0.95$),
Tucana III ($\eta\geq 0.51$), Draco II ($\eta\geq 0.39$), Segue 1
($\eta=0.39$) and Grus I ($\eta=0.37$). These numbers are calculated from
actually measured parameters of these galaxies Esser:2022owk ; the
inequalities are used where only the upper bounds on the velocity dispersion
exist. For comparison, the Milky Way in the solar neighborhood has $\eta\sim
2\times 10^{-8}$.

The PBH capture is a random uncorrelated process and therefore the probability
to capture a given number of PBHs follows the Poisson distribution with the
expectation equal to the mean captured number $\bar{N}$. For a solar mass star
this quantity was calculated numerically in Ref. Esser:2022owk ; it is shown
in Fig. 1 for the Trinagulum II galaxy which has $\eta\simeq 1$. In the mass
range $\sim 10^{19}-10^{21}$ g the mean captured number is $\bar{N}\simeq 2.2$
assuming PBHs constitute all of the DM. This corresponds to the survival
probability (the probability to capture zero PBHs) $p_{0}=\exp(-\bar{N})\simeq
0.11$, meaning that the stars actually observed in this galaxy would
constitute only 11% of their original number as formed about 10 Gyr ago, while
89% would have been destroyed by PBHs.

If PBHs constitute only a fraction $f=\Omega_{\text{PBH}}/\Omega_{\text{DM}}$
of the DM, the mean captured number would be $f\bar{N}$, and the survival
probability would be higher for smaller $f$. Requiring that no more than a
certain fraction of stars has been destroyed translates into constraints on
$f$ Esser:2022owk . The result is shown in Fig. 2 for two assumed values of
the maximum destroyed fraction of stars equal to 50% and 20%.

Figure 2: Upper limits on the PBH abundance $\Omega_{\rm PBH}/\Omega_{\rm DM}$
as a function of the PBH mass for the Triangulum II dwarf galaxy assuming that
no more than 50% (blue) or 20% (orange) of stars were destroyed by PBHs.
Figure adopted from Ref.Esser:2022owk .

It is not straightforward to infer the maximum allowed fraction of destroyed
stars from present-day observations as this would involve modeling of a dwarf
galaxy history. While such models exist, they have large uncertainties. A way
around has been proposed in Ref.Esser:2023yut ; it makes use of the fact that
the destruction probability grows with the mass of the star. This is
illustrated in Fig. 3 which shows the dependence of the mean captured number
of PBHs on the star mass for the case $\eta=f=1$. This dependence is close to
linear, and therefore the survival probability decreases exponentially with
the star mass. Thus, the presence of PBHs, apart form reducing the number of
stars, also strongly impacts the stellar mass function depleting its high-mass
end.

Figure 3: The mean captured number $\bar{N}$ as a function of the star mass
$M$ assuming $\eta=1$ and $f=1$. The dots show the numerical result, the blue
line represents the fit by the function $a(M/M_{\odot})^{b}+c$ with $a=3.76$,
$b=0.685$ and $c=-0.883$. Figure adapted from Ref.Esser:2023yut .

Known UFD galaxies (see Ref. Simon  for a review) have from hundreds to
thousands of individual stars resolved with masses typically going down to
$0.5M_{\odot}$, and down to $0.2M_{\odot}$ in some cases. Moreover, it has
been shown jwst  that the James Webb Space Telescope can resolve stars down to
$0.09M_{\odot}$ in the dwarf galaxy Draco II at a distance of $\sim 20$ kpc.
From these data one can, in principle, reconstruct the stellar mass function
with a good accuracy and in a sufficiently wide mass range. Importantly, with
the already existing technology, these data may be extended to low masses
$\lesssim 0.2M_{\odot}$ where the effect of PBHs should be negligible.

Modeling of the stellar mass distribution at birth, or initial mass function,
is a standard tool in the galactic astrophysics Kroupa ; Chabrier . The UFDs
are of particular interest in this context because, according to current
understanding, their stars have all been formed in a single burst some 12.8
Gyr ago. The long-lived ones — those with masses below $\sim 0.8M_{\odot}$ —
should therefore have preserved their initial mass distribution until present.

The destruction of stars by PBHs modifies this picture. Importantly, the
effect of PBHs switches on and off very sharply (exponentially) as the merit
factor goes through the value $\eta\approx 1$. Thus, comparing several UFDs
with $\eta$ scattered around 1 should give a good handle on the parameters of
the unmodified stellar mass function, as well as on its possible modifications
due to the star destruction by PBHs.

Preliminary estimates of the sensitivity of this approach have been made in
Ref.Esser:2023yut  using simulated data. A star population was generated with
the Kroupa mass function with the standard parameters and the number of stars
corresponding to a typical UFD, thus mimicking closely the observational data.
These mock data then have been fitted to a model based on a generic mass
function model (broken power law or log-normal) with a few free parameters
varying in the ranges inferred from observations. The model also included the
destruction of stars by PBHs controlled by one more free parameter, the
fraction of PBHs $f$ varying between 0 and 1. The Bayesian method was used to
infer the likely values of all the parameters, including $f$. The results show
that the method is sensitive enough to exclude large values of $f$ down to
$0.8$ at more than $3\sigma$ level. It appears therefore quite feasible to
exclude 100% of the DM being composed of PBHs with already existing
observations.

##  6 Summary and concluding remarks

Primordial black holes are an appealing candidate for the dark matter.
However, large parts of the potentially available mass range are constrained
by various arguments, and only the so-called asteroid mass window from $\sim
10^{17}$g to $\sim 10^{23}$g remains fully allowed until now. We have argued
that capture of PBHs by main sequence stars in dwarf galaxies may be a
promising way to probe this mass range. Moreover, it is conceivable that this
can be done with the existing technologies, if not with already existing data.

The approach that we have outlined consists, essentially, in comparing the
stellar mass functions of dwarf galaxies with low and high values of the merit
factor $\eta$. For that good quality data on stellar populations are required.
Such data already exist for some of the galaxies with low $\eta$ where stars
have been resolved down to $0.2M_{\odot}$. For high $\eta$ galaxies, notably
for the ones listed in Sect. 5, the current observations go down to
$0.5M_{\odot}$. This may or may not be sufficient to firmly exclude the
possibility of all of the DM consisting of PBHs,
$\Omega_{\text{PBH}}/\Omega_{\text{DM}}=1$. Better observations of these
galaxies extending to smaller star masses will be very important for this
purpose.

We have focused on the most straightforward signature of the star destruction
by PBHs, the mere star disappearance. But there are likely other signatures.
If a sizable fraction of stars in a dwarf galaxy is destroyed by PBHs, this
would create a population of sub-solar mass black holes. They are not directly
visible, but may perhaps be detected in gravitational waves. It is likely that
the star destruction itself is accompanied by a supernova-type energy release.
If this is a short event that might have happened any time in the last 10 Gyr,
we have to be very lucky to observe and identify one today, unless it has
features that make these explosions unambiguously distinguishable from other
supernova events. Alternatively, last stages of the star accretion onto a
black hole may be relatively long and luminous, in which case we have a good
chance to detect and maybe identify such an event as being due to a PHB
capture.

###### Acknowledgements.

The author is indebted to N. Esser for the help with making Fig. 1 and for
comments on the manuscript. This work is supported in part by the Institut
Interuniversitaire des Sciences Nucléaires (IISN) Grant No. 4.4503.15.

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