###### Abstract

Primordial black holes (PBHs) are a fascinating candidate for being the dark
matter, albeit one which has been heavily constrained. This review presents an
in depth look at those observational constraints, particularly at their
nuances and uncertainties. Despite their varied origins, the standard PBH
formation path is assumed to be collapse of perturbations after inflation,
which should leave signals visible in the CMB at certain scales. Other
constraints come from microlensing surveys, which severely limit PBHs as dark
matter in the solar to satellite range, but there are diminishing results in
regards to lower mass ranges. Gravitational waves signals and PBH evaporation
from Hawking radiation also make for useful probes, but the former requires
the next generation of experiments before making constraints beyond the solar
mass range, and the later is severely limited above $10^{-16}\,{\rm
M_{\odot}}$. Other dynamical and accretion constraints exist for PBH of large
masses. Care also has to be given, as all these constraints can carry
different implications coming from differences between monochromatic and
extended mass distributions, and their degree of clustering. Beyond all these
issues, a window still exists for primordial black holes to be all of the dark
matter between $10^{-16}$ and $10^{-11}\,{\rm M_{\odot}}$.

###### keywords:

Primordial Black Holes; Dark Matter; Observational Cosmology

1 1 0 Constraints on PBH as dark matter from observations: a review Marc
Oncins 1,2,3 Correspondence<EMAIL_ADDRESS>
##  1 Introduction

Primordial black holes (PBHs) are one of the current most interesting
candidates to be the dark matter (DM). Black holes (BHs) emerging from the
very early universe, without any of the restrictions on mass or abundance that
characterize BHs of stellar origin. Not only they fulfill the basic criteria
needed for the DM, they do so without the need to invoke a new set of existing
particles. Furthermore, while their specific mass is not set in stone, on
cosmological scales PBHs would behave like the standard particle cold dark
matter Carr and Kuhnel (2021).

While this might make them look ideal, the formation of PBHs is not so simple
and often requires some form of new physics too, so their specific origins and
abundance are still hotly debated. While first theorized in Zel’dovich and
Novikov (1967); Hawking (1971), their name and characteristics, including
formation from from primordial inhomogeinities during inflation and their
cosmological implications, were first formalized in Carr and Hawking (1974);
Chapline (1975); Carr (1975). Today a variety of formation paths exist,
including exotic ones. For a review of the most standard formation path and
possible alternatives I refer the reader to Escrivà (2022).

PBHs as a DM candidate have received considerable attention since the LIGO
LIGO Scientific Collaboration (2015) discovery of gravitational waves Abbott
and et al (2016) and its possible attribution to PBHs Bird et al. (2016).
Lately, they have been mostly ruled out of originating all the gravitational
waves detected by the LIGO collaboration Sasaki et al. (2016); Wang et al.
(2016); Wong et al. (2021), though some doubts still remain Jedamzik (2020).
PBHs however can form in a wide mass range, depending only on the
circumstances of the collapse that forms them. Despite this very wide range of
parameters, PBHs being all of the DM is strongly constrained in most other
mass ranges through a rather diverse set of observations Carr and Kuhnel
(2021); Sasaki et al. (2018); Carr et al. (2021); Smyth et al. (2020). While
the number of possibilities makes them a very exciting prospect from both a
theoretical and observational standpoints, it also means it is very hard to
track the extent to in which masses the PBHs can be all of the DM or more
particularly the nuances and uncertainties associated with these constraints.

In this review I attempt to make a brief summary of all current known
constraints on PBHs as DM, but also put special focus on the physical origin
of the constraints and the implications they carry for both their current form
and future prospects. A nuanced understanding of both is helpful when the
constraints appear to constantly be changing, and in correctly assessing the
possibilities of future detection. Previous review with extensive reference
lists can be found in Carr and Kuhnel (2021); Sasaki et al. (2018).

In section 2 I make a very brief summary of what is usually considered the
standard formation paths and what constraints can be derived from it. The
following section 3 will be an overview of all known current monochromatic
constraints, divided by their types. Section 4 will tackle the issue of how
non-monochromatic constraints work and their issues, and also how clustered
PBH distributions can become relevant and modify existing constraints.
Finally, I will detail the still existing window on PBHs as DM and works that
have tried to close it in 5, the possibility of directly detecting the PBHs in
section 6 and present my conclusions and future prospects in section 7.

##  2 Formation

As this review’s main purpose is to detail the existing observational
constraints on PBHs as DM, I will keep explanation on PBHs possible origin
brief.

The most common origin for PBHs is in the very early Universe, during
radiation domination, where large curvature perturbations generated during
inflation could have undergone gravitational collapse. This early epoch should
be radiation dominated, so our focus in on energy density, which we will call
it $\mu$. For the case of Planck units, which take $G=c=1$, the Schwarzschild
radius of a BH can be simplified from the standard form $r_{s}=2GM/c^{2}$ to
$r_{S}=2M$.

It is also obvious that the total mass within any spherical symmetric
perturbation in Planck units would be $M\sim\frac{4}{3}\pi\,\mu r^{3}$, as
$c=1$. Using that with the simplified Schwarzschild radius definition means
the condition for the perturbations to collapse is Carr and Hawking (1974):

$\displaystyle r_{S}=2\frac{4\pi}{3}r_{S}^{3}\mu\longrightarrow\mu
r^{2}\gtrsim 1\;,$ (1)

where we drop the constants on the right term. While this condition might
appear strange, it is in fact very close to the Jeans length for oscillations,
which in natural units would be
$\lambda_{J}\sim\left(\frac{\pi}{\mu}\right)^{1/2}$. Another possibility is
simply to remember that in radiation dominated universe the radiation pressure
is $p=\frac{\mu}{3}$. Matching it with the gravitational potential to see when
the latter wins out and the region will collapse ends giving the same result
as above.

A density contrast can then be introduced, which compares the energy density
of a point $\mu$ with the background density $\mu_{0}$. We will call this
contrast $\delta$ and use the following definition:

$\displaystyle\delta=\frac{\mu-\mu_{0}}{\mu_{0}}\;.$ (2)

This allows us to measure how overdense or underdense a certain region is, and
so the strength of the perturbation. In an homogeneous, isotropic and flat
universe, given condition 1, this will end requiring $\delta\gtrsim 1$ for the
perturbations to collapse Sasaki et al. (2018).

This condition is the threshold: if $\delta$ fulfills it by being greater than
the threshold, the perturbation will collapse and form a PBH. If it is lower
the pressure of radiation will win out, resulting in the perturbation
continuing its spread maybe even until today, though it will so be heavily
redshifted as to be extremely hard to detect.

This is all for a very idealized case though. From the start the delta will be
a function rather than a uniform value. Numerical simulations have found that
the exact threshold does not depend only on the maximum of the perturbation,
but also on the shape of the $\delta$ function Niemeyer and Jedamzik (1999);
Shibata and Sasaki (1999); Musco et al. (2005); Harada et al. (2015); Musco
(2019); Escrivà et al. (2020). Thus for different perturbations the threshold
will differ even if their maximum and average are the same, unlike our
estimation.

Those are not the only issues, as the definition of the threshold has to be
gauge independent and take into account different equations of state of the
universe, to account for all possible inflation models. A matter dominated
universe or a intermediate, dust like, universe for example would change the
physics of the collapse. All of these examples can radically alter the
threshold, which makes a generalized form to compute the collapse a necessity.

A way to improve upon our definitions is to use a new and more general type of
condition. Called the critical threshold $\delta_{c}$, it is not a direct
measure of the over-density like $\delta$ but rather of the mass excess per a
unit of volume, the so called compaction function Shibata and Sasaki (1999).
It not only is completely generic, the critical threshold also accepts a very
good analytical approximation with parameters describing shape and space-time
Escrivà et al. (2020, 2021), which simplifies simulations enormously. Previous
analytic estimates of the threshold were also done in Carr (1975); Harada et
al. (2013)

Despite this, however there is another major complication that brings our
first constraint. The current power spectrum expected for our universe should
be very close to scale invariant, and at the scales of the power spectrum
present in the CMB, the fluctuations are of the order of the microKelvin,
$\delta\sim 10^{-5}$ Planck Collaboration (2020). Therefore, remembering the
condition above the collapse of perturbations into PBHs should not occur in
any major way, not enough for them to be the DM.

There are a variety different solutions to this conundrum. First, some kind of
enhancement of the perturbations at only certain scales could have happened.
There are in fact a large number of ways to get this kind of enhancement
García-Bellido and Ruiz Morales (2017); Kawasaki and Takhistov (2018); Germani
and Musco (2019), from slightly modified inflation models to new physics
coming from string theory, but all have in common that they are outside the
standard model. The most common way is to have a modified inflation potential,
the so called ultra-slow roll potential Germani and Prokopec (2017); Atal et
al. (2019), which results in perturbations at a certain parameter dependent
scale being enhanced up to $\delta\sim 1$ for the isotropic flat case, while
remaining the usual value at other scales.

Another related possibility is that the spectrum of fluctuations is non-
gaussian. The fluctuations in the CMB are $\delta\sim 10^{-5}$ as measured on
average, which is too small for any kind of noticeable PBHs to form if the
fluctuations follow a gaussian spectrum. Gaussian fluctuations would be the
prediction of the standard model. But if fluctuations do not follow that
gaussian spectrum because we have new physics, then there could be a tail of
extreme results with $\delta>\delta_{c}$ without altering the perturbations
general properties. This could result in an important number of PBHs forming
Bullock and Primack (1997); Franciolini et al. (2018); Atal and Germani
(2019). Currently, in the CMB there has not been any sign of any type of non-
Gaussian fluctuations Planck Collaboration (2020); Cabass et al. (2022a),
which is why a lot of models do not contemplate them in a major way, but non-
Gaussian fluctuations are still not completely ruled out Cabass et al.
(2022b).

Something necessary to understand however is that even if we are using a model
that results in the enhancing of (gaussian) fluctuations to obtain PBHs, non-
gaussianities will still be relevant. This is due to the relation between the
$\delta_{c}$ and the curvature perturbation being non-lineal De Luca et al.
(2019); Kawasaki and Nakatsuka (2019), resulting in non gaussianities in the
density fluctuations despite a gaussian curvature perturbation. As the
abundance of PBHs is very sensitive to the values of $\delta$, any kind of
higher than expected value like the ones that would come from an unaccounted
tail of fluctuations, would modify the number of PBHs in a major way. This
adds another layer of subtlety in computing the abundance of PBHs from a
particular model, though workarounds exist Atal and Germani (2019). For more
studies of non-gaussianities in particular look Young and Byrnes (2013); Yoo
et al. (2019); Kitajima et al. (2021); Escrivà et al. (2022).

Another possibility is to preserve the standard inflationary model with scale
invariant power spectra as it is on the CMB. Even then, PBH formation is
possible by creating high subhorizon scale $\delta$ due to some phase
transition that may naturally take place as the universe cools down. Most
notably, PBHs could be formed in the epoch of QCD phase transition Jedamzik
(1997); Byrnes et al. (2018); García-Bellido et al. (2019) as there is
temporary drop in the pressure that counteracts gravity, allowing for higher
$\delta$ than would be expected from the amplitude of the perturbations. PBHs
formed in this epoch should have a mass of around the solar mass García-
Bellido et al. (2019), though using different phase transitions or other non-
standard physics can change the mass range.

Even after all that, that is but the first step of PBH formation. Collapse
into a BH does not happen immediately once the perturbation is over the
threshold, it is not until the perturbation crosses the horizon that it can
collapse, as before the perturbation is not fully causally connected due to
inflation. While this might look like just a timing issue it carries important
consequences. As the perturbation will collapse in the horizon in which it has
reentered, the mass of the resulting PBH will be of the order of the mass of
the horizon Niemeyer and Jedamzik (1998); Musco et al. (2005); Escrivà and
Romano (2021); Escrivà (2020). The lower the scale, the bigger the horizon and
the higher the mass it will have, and vice versa.The mass within the horizon
can be estimated withCarr et al. (2016):

$\displaystyle M_{h}\sim\rm\frac{c\,t^{3}}{G}\sim
10^{15}g\left(\frac{t}{10^{-23}s}\right)\;,$ (3)

with the time $\rm t$ ranging from the Planck time ($10^{-43}s$) to much later
in the universe’s life. This is what gives PBHs such enormous mass ranges. A
perturbation generated in first instants of inflation can create tiny BHs that
could could explain the whole of the dark matter, while perturbations
generated later can result in massive PBHs bigger than any current stellar BHs
and which could be the seeds of Supermassive black holes.

Of course, the perturbations that make PBHs do not have any reason to all come
from the same exact instant. This will naturally result in a spread of masses
for our PBHs. For the case of enhanced perturbations we do expect the
enhancement to only happen at certain scales, so while PBHs will have an
extended mass distribution, if there are enough for them to be a noticeable
part of our universe we also anticipate them to be around a certain peak. Even
then this is not a completely hard rule, as there can be extreme cases like
double inflation models Inomata et al. (2018) and also models where PBH form
in a completely different way that results in a more even spread Carr et al.
(2019). There are still even other relatively exotic methods of PBH formation,
such as inhomogeneous baryogenesis Hasegawa and Kawasaki (2019), domain walls
Deng et al. (2017); Belotsky et al. (2019), vacuum bubbles Garriga et al.
(2016); Deng and Vilenkin (2017), axion driven inflation García-Bellido et al.
(2016); Domcke et al. (2017) and through non-topological solitons known as
Q-balls Coleman (1986), each of which will have different results for the PBH
extended mass distribution.

Another expectation would be that they would form with spin 0 or close Chiba
and Yokoyama (2017), as they form in the radiation dominated epoch where the
combined fluid of radiation and matter sheds angular momentum almost
instantly. This can be a clear difference from stellar origin BHs, which will
inherit at least a fraction of the spin of their forming star111How much is a
point of contention, with very efficient angular moment transport models
positing very low spinFuller and Ma (2019) while others using medium
efficiency average notable spin BHsEkström et al. (2012). Observations of
X-ray binaries always have a very high spin, but they are subset of BHs and so
there could an intrinsic bias too., though again in more exotic formation
paths there could be PBHs with noticeable spin. A good example is the case
where PBHs would form in a matter dominated epoch, which should result in PBH
having all very high spin instead of close to 0 Harada et al. (2017). As spin
can be relevant for a subset of constraints, this is a rather important
distinction.

Something similar applies to the degree of clustering with which with PBHs
will form. In the simplest case the PBHs should follow a standard poissonian
matter distribution Afshordi et al. (2003), as the perturbations they form
from should also come from a homogeneous universe and the enhancement should
not a priori have any specific spatial distribution. Exotic models exist
however where this is not the case, in particular for alternative origins like
the previously mentioned domain wallsBelotsky et al. (2019). This adds still
another layer of complexity and will again affect constraints on their
abundance.

Still, as mentioned, if PBHs are formed through perturbation collapse not from
phase transition, then the perturbations will also be reflected on the CMB,
concretely its temperature angular power spectrum and CMB spectral
distortions. Only a number of perturbations will collapse, so the rest will
transmit and appear like the other CMB fluctuations. Unfortunately, either
through Planck Planck Collaboration (2020) or other measurements Bird et al.
(2011); Fixsen et al. (1996), the CMB can only constrain the power spectrum on
the lower scales, equivalent to the highest masses of PBHs. Currently, no non-
gaussianity nor enhanced fluctuation has been found, limiting these lower
scales, but only the lower scales. Saying it in another way, any PBH formation
below $10^{3}\,{\rm M_{\odot}}$ in this standard pathway is currently still
possible.

##  3 Monochromatic constraints

PBHs could be the DM, but the properties of these PBHs can be drastically
different between different mass ranges. $10^{3}\,{\rm M_{\odot}}$ BHs would
gravitationally affect any star or planet that is close to them, while much
more smaller $10^{-10}\,{\rm M_{\odot}}$ BHs could cross the Solar system
millions of times and we would not notice them Abramowicz et al. (2009).

Therefore is common sense to divide the types of PBH by their mass. While
extended mass distributions are expected for PBHs, for the standard inflation
perturbation collapse case the enhancement at certain scales should result in
a particular mass range corresponding to those scales, with a peak. A very
helpful approximation with setting constraints is taking the peak of that mass
distribution and making as if all the PBHs in it had that particular mass.
This simplifies the physics in most cases enormously, as we can avoid
integrals through all the masses and streamline the calculations needed to see
if it is even worth to pursue certain constraints. And while the final result
is an estimate, that estimate should be a fairly close and conservative
approximation to the real case of the extended mass distribution, as the peak
of that same mass distribution will likely have an abundance only slightly
below 1 if it contains all of the DM after all.

The end result of this simplifications are monochromatic PBH bounds, which are
shown in figure 1. Of further note is also that the graph is logarithmic and
covers more than 22 orders of magnitude in mass. In this way it is only thanks
to that approximation that covering all these possible ranges with different
types of constraints is feasible.

Figure 1: Plot with current existing robust monochromatic PBH constraints,
with each color and name corresponding to a different type. The abundance on
the left axis is which fraction of the dark matter they could be, while mass
is given in both $\,{\rm M_{\odot}}$ and grams. Evaporation constraints come
from Kim (2021); Boudaud and Cirelli (2019); Laha et al. (2020); Barnacka et
al. (2012); Carr et al. (2010); Laha (2019), microlensing constraints come
mostly from a number of surveys, including Subaru, Kepler, MACHO and EROS
Niikura et al. (2019); Alcock and et al. (2000); Griest et al. (2014);
Tisserand and et al. (2007); Blaineau and et al. (2022); Oguri et al. (2018),
gravitational waves bounds come from LIGO and related studies LIGO Scientific
Collaboration and Virgo Collaboration (2019); Nitz and Wang (2021); Chen and
Huang (2020); Kavanagh et al. (2018) and dynamical Monroy-Rodríguez and Allen
(2014); Lu and Wu (2019); Wang et al. (2020) and accretion Manshanden et al.
(2019); Lu et al. (2021) constraints from a variety of sources. More details
on how the constraints are obtained are in each of the corresponding
subsections. Figure obtained from PBH , where all references, other bounds and
plotting codes are available.

There are as varied constraints as there are mass ranges, but as seen above
there are 4 that dominate. Microlensing is the first one, perhaps the most
relevant in terms pf width of the constraints. Following from the previous
MACHO experiments Alcock and et al. (2000), a large number of PBHs would also
result in a large number of detectable microlensing events. Lack of such
events can be used therefore to set constraints. Another is BHs evaporation,
as the expected result of BHs emitting Hawking radiation Hawking (1974). The
smaller BHs will have significant high energy emissions, so their observation
or more particularly lack of thereof can be used to set abundance constraints
on PBHs. Gravitational waves (GWs) are perhaps among the most popular due to
the LIGO merger detection Abbott and et al (2016). If PBHs were all of the DM
in masses within LIGO sensitivity range, then there would be an increased
number of mergers over purely stellar origins. The lack of such mergers
results in strong bounds for the LIGO mass range. Last are the dynamical and
accretion constraints, only possible on very massive black holes. There are
multiple sources for these constraints, but they all share that they focus on
the large mass of these black holes and how they would have a large effect on
their surrounding which would significantly affect galactic dynamics.

###  3.1 Microlensing

Microlensing is the name given to gravitational lensing on a very small scale.
Gravitational lensing itself is the bending of light when a body projects a
strong enough gravitational force. In typical optical parlance, the gravity of
the body acts as a lens to the light coming from behind it towards us. The
most illustrative case is that of the Einstein ring Einstein (1936), also
known as Chwolson ring Chwolson (1924). A source emits light, and in our line
of sight towards the light goes through the lens. For the sake of simplicity
we shall assume the lens is a black hole: then the light in the direct line of
sight will not go through the BH, but affected by gravity will bend around the
lens, eventually reaching Earth from multiple directions. This will results in
the viewer seeing multiple sources equidistant to where the source actually
is, with the direct line of sight blocked in the case the lens is a BH. In the
3 dimensional case, the light coming from the multiple directions result in a
ring surrounding the source, thus the name of ring.

Figure 2: Example of the geometry of a lensing event for a general case. Taken
from De Paolis et al. (2016) Figure 1, $\alpha_{s}$ is the same as our
$\alpha$

The Einstein ring is an ideal case, but most lensing events are likely to
differ. A generic example for the geometry of a lensing event in shown in
figure 2. One particular example is where the object and lens are so far away,
or small, that our resolution is unable to separate the two images. This is
the specific case of microlensing Paczynski (1986), where the lensed image
appears all on the same pixel rather than as two or more different images. As
we cannot perceive a position shift the typical measurements of angles for
each individual image cannot be done, and it can be hard to even notice if the
image has been lensed.

However, the presence of more ray beams than expected in a single pixel also
means that the image will appear brighter than normal. Going from the figure
above, if the rays have similar brightness receiving two or more will make the
pixel appear at a higher luminosity than if it only received a single one.
Because all stars and galaxies appear to move for an observer on Earth (a
combination of both their intrinsic orbits and the Earth’s own movement), the
lensing event will only happen for the short period of time in which both
source and lens are aligned. The increased luminosity can then be compared
with the previous one measured just before, and observers can notice and
record these kind of changes as lensing events.

To compute the expected magnification of light, a first approximation is to
treat gravitational lensing like common optic lenses, using what is called
geometric optics. First of all is the deflection angle $\alpha$, which would
be how much the ray is deflected from its path De Paolis et al. (2016) :

$\displaystyle\alpha=\frac{4GM}{bc^{2}}\;,$ (4)

where b is the impact parameter, the shortest distance between the point-like
lens and the ray it is deflecting, M is the mass of the lens, G the
gravitational constant and c the speed of light.

This also using what is called the small angle approximation,
$sin\alpha\approx\alpha$, which works for small $\alpha$. (4) comes from
general relativity but the result is a very simple term that reminds of
newtonian gravity, so it is also referred as Newtonian approximation. In fact,
if we assumed Newtonian gravity and that light is a particle with mass the end
result would be exactly half the value of the $\alpha$ in 4. Another
comparison can be with the Schwarzschild radius of a BH, which can be related
by $\alpha=2r_{s}/b$.

Another parameter of interest is the the Einstein angular radius $\theta_{1}$,
which is the angle at which we would see the previously mentioned Einstein
ring. When we are not in the ideal case like in the figure above, $\theta_{1}$
does differ from the angle at which we see the image $\theta$. Using
trigonometry, $\theta_{1}$ can be computed from equation (4) in the ideal
case, the definition of which will carry on for more complex geometries:

$\displaystyle\theta_{1}=\left(\frac{4GM}{c^{2}}\frac{D_{LS}}{D_{s}D_{l}}\right)^{1/2}\;,$
(5)

where like before there is the implicit assumption that
$sin\theta_{1}\sim\theta_{1}$. In standard lensing this can troubling, but in
microlensing where the angles are so small it always applies. For PBHs in
particular there is again an easy correspondence with the Schwarzschild
radius, $\theta_{1}=\sqrt{2R_{Schw}D_{LS}/(D_{s}D_{l})}$

Even in the non-ideal case where $\theta\neq\theta_{1}$ the Einstein angular
radius can be understood as the effective radius in which the the lens bends
the light. The timescale of lensing will be therefore be the time it takes the
lens to cross $2\theta_{1}$ from the Earth’s point of view. As the Einstein
radius has a dependence on mass $M^{1/2}$, the time scale will become
increasingly shorter for the lower mass PBHs, but the exact timescale has
heavy dependence on the location of the event with respect to the observer.

Looking at other important angles in figure 2, $\theta_{S}$ will be the angle
of the source with respect to the observer, and $\theta$ the angle the lensed
images would appear at. In the geometric optics approximation the angles
$\theta$ and $\theta_{S}$ can be related with the Einstein radius $\theta_{1}$
by applying the lens equation of geometric optics to our case Sasaki (1993):

$\displaystyle\theta-\theta_{S}=\frac{\theta_{1}^{2}}{\theta}\;.$ (6)

As $\theta_{S}$ is a single value, it is easy to see there would be two
solutions for $\theta$, corresponding to two images for the 2-dimensional
case. In the case of microlensing we are unable to distinguish them, but we
will be able to distinguish the increasing flux, the magnification of the
source the 2 images will cause.

The two images can be related to the flux by understanding that the flux (f
[erg/sec]) results from integrating the surface brightness of the object (S
[erg/sec/sr]) for a particular solid angle $\Omega$, the width of the beam so
to speak. This solid angle depends on the angle it reaches the observer.
Because for microlensing the angles are very small, this integral can be
approximated as a simple multiplication $f=Sd\Omega$, and compared with the
case where there would be no lensing:

$\displaystyle
A=\frac{f_{obs}}{f_{0}}=\frac{Sd\Omega(\theta)}{Sd\Omega(\theta_{S})}=\frac{d\Omega(\theta)}{d\Omega(\theta_{S})}=\frac{\theta
d\theta}{\theta_{S}d\theta_{S}}\;.$ (7)

Computing the magnification then is relatively straightforward if we remember
the lens equation again (6):

$\displaystyle
A=\frac{\left(\frac{\theta_{S}}{\theta_{1}}\right)^{2}+2}{2\left(\frac{\theta_{S}}{\theta_{1}}\right)^{2}\sqrt{\left(\frac{\theta_{S}}{\theta_{1}}\right)^{2}+4}}\pm\frac{1}{2}\;.$
(8)

Searching for this moment of greater emission from the source a lensing event
can thus be detected. Despite the previous complications, the final equation
as seen above depends only on the source angle and the Einstein angle. By
detecting the magnification of light coming from a source many details can be
easily obtained from the lens-source system.

This makes microlensing a very attractive prospect for PBH as DM constraints.
We know the mass we want to constrain and we can use the DM galaxy mass
distribution to estimate the likely places where the PBHs would be and their
velocity distribution. With this the expected magnification can be estimated
and if it is within the sensitivity of any scientific instruments. The birth
of dedicated surveys like the original MACHOAlcock and et al. (2000), but also
now Subaru Niikura et al. (2019) has allowed to map a mass range of 10 orders
of magnitude with constraints, as they have found few or none gravitational
lensing events, whereas if those PBHs were the dark matter all models
predicted many more.

But despite all these successes, microlensing constraints have started
struggling to increase constraints in new mass ranges. This comes from a lot
of the approximations used previously. First of all, we have used standard
geometric optics, but that approximation requires light rays to behave
completely classically. If any of the wave.like properties of light are
relevant during the lensing, the whole approximation breaks down. Indeed, as
the PBHs we are interested in are extreme objects, their Schwarzschild radius
is very small and can become of the order of the wavelength of the light we
are observing, triggering wave-like effects from the light that need to be
accounted for.

Fortunately, even without using geometric optics the magnification can still
be computed, though it will be more complex. As we do not expect the rays to
be perfectly symmetrical, the rays very likely traverse different distance to
reach the viewer, there will be a time delay between them. This time delay
will also result on the beams not reaching on the same phase of the wave. The
time delay is much harder to compute, but an analytic expression is still
possible Katz et al. (2018):

$\displaystyle\Delta
t=\frac{1}{c}\frac{D_{L}D_{S}}{D_{LS}}(1+z_{s})\left(\frac{\left|\theta-\theta_{s}\right|^{2}}{2}-\Psi(\theta)\right)\;,$
(9)

where all parameters are the same as previously, $z_{s}$ is the redshift of
the source and $\Psi$ is the so called lensing potential. This lensing
potential is related to the density of the lens by the Poisson equation, and
can be thought of as projection of the density through the plane of the lens.
For point like sources, like our black hole,
$\Psi(\theta)=\theta_{1}^{2}log\theta\,$ 222The lensing potential is the
result of integrating the density through the entire plane of the lens and
includes an indeterminate constant. This is normally not an issue as the time
delay is used to compare with the time delay from other rays, but it will not
help in computing absolute time delays for a path..

Due to wave nature of light, this time delay will create a phase difference
between the different rays. As the rays converge, this phase difference can
result in both destructive and positive interference. As there are multiple
rays and both the lens and source are moving with respect to each other,
rather than fully positive or fully negative there will be instead
interference fringes. This will make the magnification equation grow more
complex Katz et al. (2018):

$\displaystyle A=\left(\frac{W}{2\pi i}\int d^{2}xe^{i\omega\Delta
t(x,\theta_{S})}\right)^{2}\;.$ (10)

We can no longer avoid an integral or the use of imaginary numbers. W is a
dimensionless frequency, related to the frequency of the light $\omega$ and
the time delay $\Delta t$. For more details on the complete derivation of $A$
I refer the reader to Katz et al. (2018).

While these changes might appear minor and too cumbersome, they are relevant
because this mistake did in fact happen with Subaru, where an early
prepublication version of the paper did not take this into account as noted in
Inomata et al. (2018). Even with the added complexities, the amplification in
equation (10) can still be computed to get the corrected constraints as they
were finally present in the printed version Niikura et al. (2019) and as they
appear in figure 1. The integral makes it harder to compute, but the
magnification is still there. Note however that this also makes the
constraints much more sensitive to assumptions about the lens and sources used
to obtain said magnification. Smyth et al. (2020) claimed that the constraints
obtained by the HSC were overestimated even in the printed version, as they
had assumed all the sources from their stellar observations to be of one solar
radius, rather than account for the selection bias towards brighter (and so
larger) stars.

It is much more problematic however when finite source effects start
appearing. These come into play when the source can no longer be taken to be
point-like: if the source is big in respect to the lens, emission from
different points of the source will be lensed differently. Possibilities
include the source not being small enough or far enough, resulting in
different time delays between even close by or symmetrical rays and the
interference fringes being washed out the larger the effect is. The end result
is that the lensing effect disappears, there being no difference with the
normal luminosity. Assuming that the emission radius of the source is roughly
the same as the radius of the source itself $R_{s}$, it results in the
following condition Katz et al. (2018):

$\displaystyle\frac{R_{s}/D_{s}}{R_{E}/D_{L}}<<1\;,$ (11)

where $R_{E}$ is the Einstein radius in units of distance,
$R_{E}=\theta_{1}D_{L}$. If the condition is not fulfilled, the microlensing
effects will start disappearing. Most microlensing surveys of interest are
focused on far away objects like M31 for Subaru Niikura et al. (2019), with
both lens and source expected to be much closer to each other than Earth. In
that case $D_{s}\sim D_{L}$, leaving a much simpler condition:

$\displaystyle\frac{R_{s}}{R_{E}}<<1\;.$ (12)

Small PBHs have very small Schwarzschild radius and so Einstein radius. This
means in effect that distant sources used in lensing, stars from neighbouring
galaxies, are simply too big to reliably produce any constraints for PBHs
below $10^{-11}\,{\rm M_{\odot}}$. Therefore either close stars or other very
small sources are needed to constrain the mass range. Katz et al. (2018)
estimated that we would need a subset of Gamma Ray Burts (GRB) with very small
sizes and very small wavelengths, to reliably constrain this mass range but
found current GRB detection rates and GRB’s intrinsic variability made it very
challenging for new constraints to be created. Other proposals include using
pulsar timing arrays Dror et al. (2019) in the radio spectrum for this kind of
purpose, as pulsars are smaller than stars, but also reached the conclusion it
would take a future and very ambitious observatory to reach mass ranges which
are not currently constrained. The use of microlensing in X-ray pulsars
however is much more promising, even if current instruments cannot constrain
PBHs as DM Bai and Orlofsky (2019).

Other ideas include using Imaging Atmospheric Cherenkov telescopes to detect
the extremely fast (t<1s) microlensing events expected for lower asteroid mass
range, but prospects are also pessimistic Pfrang (2022). Another possibility
would be the use of GRB lensing parallax rather than standard microlensing. In
a lensing parallax we observe the same GRB via two spatially separated
instruments, if the GRB is lensed and the separation of the instruments is of
the order of the Einstein radius of the lensing system, then each of the
observers will see a different magnification than the other. Asteroid mass
PBHs can be constrained if we observe enough GRBs with this technique Nemiroff
and Gould (1995); Jung and Kim (2020), but the large separations needed
between the instruments (of the order of 1 Astronomical unit) means it would
require brand new spacecraft detectors.

Microlensing can still remain useful for PBHs of high mass, as recent bounds
using quasars showcase Esteban-Gutiérrez et al. (2022), but the much lower
number of PBHs means much lower expected number of events, increasing the
observation time required for adequate constraints, on top of low mass PBHs
being much more interesting for the purpose of dark matter as figure 1
illustrates.

In summary, while extremely useful, microlensing seems to have reached the
limit in what it can constraint, or at least reached very diminishing results.
While planet mass PBHs cannot be all of the dark matter, asteroid mass PBHs
are free from such constraints.

###  3.2 Black hole evaporation

PBH evaporation is a process resulting from Hawking radiation emission Hawking
(1974). Hawking radiation itself is the conclusion of applying quantum effects
to a BH’s horizon and results in the emission, mostly in terms of particles,
of energy depending on the effective temperature of the Black Hole. This
emitted radiation would have an almost thermal black body spectrum of the
corresponding effective temperature, and this temperature depends mostly on
the mass. The smaller the black hole is, the higher the effective temperature
and vice-versa.

The derivation of this effective temperature comes from the boundary
conditions of the horizon. Assuming an observer just outside the horizon, this
observer has to have an acceleration in order to not fall into the black hole.
This accelerated frame of reference will see a thermal radiation by the Unruh
effect Fulling (1973); Davies (1975); Unruh (1976), where acceleration results
in a thermal bath of particles. However, this accelerated frame of reference
is at local equilibrium with the horizon of the black hole, which means the
horizon shares that same thermal bath. This results in that observer seeing
the black hole horizon at following temperature:

$\displaystyle T_{obs}=\frac{\hbar c^{3}}{4\pi
G\sigma\sqrt{2M_{bh}r\left(1-\frac{2M_{bh}}{r}\right)}}\;,$ (13)

where r is the position of the observer, $\sigma$ the Stefan-Boltzmann
constant, $\hbar$ is the reduced Planck constant, c the speed of light and G
the gravitational constant. This comes straight from the black hole’s metric,
which is assumed to be a standard Schwarzschild black hole, and the observers
position.

This temperature the horizon observer sees can then be redshifted to infinity,
to obtain how a far off inertial observer like one from Earth would see the
temperature in this horizon:

$\displaystyle T_{bh}=\frac{\hbar c^{3}}{8\pi GM_{bh}\sigma}\sim
10^{-7}K\left(\frac{\,{\rm M_{\odot}}}{M}\right)\;.$ (14)

Of note is that this is for a non-rotating (0 spin) non-charged black hole.
While charged black holes are not expected to exist, as their charge will
simply attract opposite charges until the black holes becomes neutral again,
rotating black holes are expected in both stellar black holes and some
versions of PBHs as detailed in section 2.

From the temperature the emission can be estimated, which can only come from
the black hole’s own energy. Therefore the black hole will evaporate when it
emits all of its energy, the majority of that energy being the one stored in
the black hole’s own mass. For massive black holes this is extremely unlikely,
not only because their temperature is much lower, but because their mass means
they are constantly accreting matter unless completely isolated. For smaller
black holes however, their emission might surpass their accretion, especially
when not in dense environments like the usual void between stars.

This permits us to define the lifetime of a black hole as the time it will
take for it to evaporate through Hawking radiation. This is only an
approximation, because it does not take into account accretion. However,
because as the mass goes lower it evaporates even faster and the accretion
lowers, it is very hard for a black hole with a non-ridiculous lifetime (i.e.
smaller than the lifetime of the universe) to jump out of it purely through
accretion. A good estimate of the lifetime is LoPresto (2003):

$\displaystyle\tau(M)\sim 10^{64}\rm yr\left(\frac{M}{\,{\rm
M_{\odot}}}\right)^{3}\;.$ (15)

The lifetime of PBHs sets the strongest constraint, as they can not be the DM
if they do not exist after all. Following from above, only PBHs bigger than
$10^{15}\rm g$ would survive to today. Still, we must be remember that, while
expected, Hawking radiation is currently unconfirmed, and complete PBH
evaporation is still an open question due to the information loss paradox
Hawking (1976). For example, Dvali (2018) argued that the "Memory burden"
could slow or stop the black hole evaporation.

The model used to obtain the lifetime is also relevant. As mentioned, an
isolated PBH is commonly assumed, but as PBHs form just after inflation they
will interact with the radiation dominated, almost homogeneous early universe.
Sajad Tabasi and Firouzjaee (2021) found that accounting for those
interactions pushed the minimum mass of PBHs to $10^{14}\rm g$ instead, with
Giffin et al. (2021) reaching a similar conclusion via a very different method
of constantly accreting matter instead.

Another example of how variable the constraints obtained via PBH evaporation
are can be seen by using a different metric to compute the Hawking radiation.
Picker (2021) used the Thakurta metric to find very different results with
Hawking radiation being notably boosted, though note that the metric they
chose as an alternative is contentious as being a good description of PBHs
Hütsi et al. (2021), so their constraints are not reflected in figure 1.

Therefore only using $\tau$ as a constraint can be problematic. Fortunately,
PBHs of very small mass that survived to today would still emit a fair bit of
Hawking radiation. PBH of mass $10^{15}$ to $10^{17}$ g would have high enough
effective Temperature that we would expect them to emit quite a lot of high
energy emission that would be completely isotropic from our point of view. If
those PBHs are all of the DM, they would bump the extra-galactic background
light up by a noticeable amount at high frequencies, but there is no such bump
in current observations Ballesteros et al. (2020); Coogan et al. (2021). This
makes the robust constraints which can be seen in figure 1.

There are hopes for even tighter constraints in the future by improving our
detectorsCoogan et al. (2021), but there are vastly diminishing results. In
particular, even if we had perfect detectors Hawking radiation could only
directly constrain PBHs below $10^{20}\rm g$ Auffinger (2022), still leaving a
good half of the asteroid mass range unconstrained. Another small issue that
becomes relevant when the constraints are less stringent is that effect of
spin now is non-negligible. As mentioned before the effective temperature, and
so the emission and its spectra, of the PBH depend on its rotation, with
highly rotating Kerr BHs emitting more. This change is expected to be less
than an order of magnitude on most constraints, and so can be mostly neglected
in mass ranges firmly ruled out by PBH evaporation. However the mass ranges
with lower constraints become more dubious, as regions within the window but
close to the constraints might be covered or on the opposite way ranges we
think are currently impossible might be where PBHs are hidden. Constraints
accounting for the spin are present in the literature Arbey et al. (2020), but
spin becomes another factor that should be taken with care.

There might be even more factors however, as one of the more curious
byproducts of this evaporation could be Planck relics. They are theorized
particles of Planck mass ($\sim 10^{-6}\rm g$) left after a black hole
evaporates MacGibbon (1987). They would be completely new physics, resulting
from quantum gravity, and their existence is contentious. Nonetheless, even if
they existed it is dubious if they could be all of the DMBarrau et al. (2021).
They would need a massive quantity of PBHs with very small mass to evaporate
in the early universe, and the increased emission and entropy would affect
primordial nucleosynthesis. Exotic models exist within loop quantum gravity
that could explain the lack of this effect via the relics forming before the
big bang, which would be a big bounce in this particular case. If we were
living in that specific universe with Planck relics as all of DM, it is likely
we would not be able to detect them in any way in the close future due to
their extremely low mass and unknown properties. Something to not is that the
scenario described is not the same as the one where the DM are particles
formed from the evaporation of PBHs Fujita et al. (2014), which is still
possible but not described in this review as the DM would then require a new
theoretical particle outside the Standard Model.

###  3.3 Gravitational waves detectors

With the recent detection of gravitational waves (GWs) from a binary merger
Abbott and et al (2016), LIGO opens a new window in astrophysics, the
possibility of using GWs to detect black holes. While a substantial number of
gravitational effects carry associated emission of GWs, the singular events
that emit most significant bursts of GW in the frequencies LIGO can detect are
binary mergers Finn and Chernoff (1993): 2 massive objects orbiting one
another that slowly inspiral until they end up colliding and fusing into a
single object.

Gravitational waves are not only emitted in the merger itself. Rather, they
are the reason the merger itself happens. As two massive objects orbit one
another they will emit gravitational waves from their movement, their moving
gravitational potential perturbing the local space-time Peters (1964). These
are very small and initially undetectable, but similar to the Hawking
radiation just the emission of GWs costs energy to the source. In the cases of
the binaries the energy stored in their orbit is slowly emitted as GWs,
resulting in the orbit becoming due to the energy loss. As the orbit decays
however the tidal forces coming from the other object of the binary further
perturb the source further, causing more emission of GWs and further decay.
The same happens in reverse for the other member of the binary. In a perfectly
symmetrical case the inspiralling binaries would meet in the middle, colliding
and merging despite being otherwise completely isolated. The GWs emission is
what turns the initially stable binary into a future merger.

The frequency of the GWs emission is in turn related to the orbits of the
binary, rising as the orbits get shorter and peaking at the merger. A good
benchmark is therefore the Innermost Stable Circular Orbit (ISCO), the last
stable orbit before the merger. For a binary with masses $M_{1}$ and $M_{2}$,
the frequency of the GWs from the ISCO is Franciolini et al. (2022):

$\displaystyle f_{ISCO}\approx 4.4\cdot 10^{3}\;\rm Hz\left(\frac{\,{\rm
M_{\odot}}}{M_{1}+M_{2}}\right)\;,$ (16)

While the GWs emission peaks during the merger, LIGO only has enough
resolution to see the waves from just before the merger Abramovici et al.
(1992); LIGO Scientific Collaboration (2015), the time scale of the GW
emission at the beginning of the inspiral too large and the time scale during
the merger too short. This works for using ISCO as an estimate though. From
equation 16, knowing LIGO’s frequency range of $\sim 10-10^{4}$ Hz gives total
mass ranges from 1 to $100\,{\rm M_{\odot}}$, in line with detections of
current mergers.

The GWs will have an amplitude, which is related to the characteristic strain
$h_{c}$ . For a typical merger of an equal mass $m_{PBH}$ binary at a distance
$d_{l}$, $h_{c}$ is approximately Franciolini et al. (2022):

$\displaystyle\left|h_{c}(f)\right|\approx 4.54\cdot
10^{-28}\left(\frac{m_{PBH}}{10^{-12}\,{\rm
M_{\odot}}}\right)^{5/6}\left(\frac{d_{l}}{\rm
kpc}\right)^{-1}\left(\frac{f}{\rm GHz}\right)^{-1/6}\;,$ (17)

For LIGO’s sensitivity band the limiting characteristic strain is $\sim
10^{-22}$, though note that the timescale of the signal is also important as
mentioned before. Besides the frequency range, the biggest limiting factor is
distance.

Figure 3: Figure showcasing the sensitivity band of current and future GWs
detectors together with projections for the GW emission for small PBHs. Taken
from Franciolini et al. (2022). Solid (dashed) lines indicate the $h_{c}$
spanned by an inspiralling binary with component masses
$10^{-16}<m_{PBH}<10^{-6}\,{\rm M_{\odot}}$ for $f_{PBH}=1\,(f_{PBH}=0.1)$.
The distance is taken as the minimum from which there will be a rate of 1
merger event per year. The signal ends at the ISCO, while the filled square,
empty square, filled and empty circles all signal the time left until the
merger, being $\Delta t=10$ Gyr, $\,\Delta t=1$ day, $\,\Delta t=1$ sec and
$\Delta t=10^{-3}$ sec, respectively. Main observatories shown are LIGO LIGO
Scientific Collaboration (2015), LISA Baker and et al. (2019) and the Einstein
Telescope Punturo and et al. (2010), but high frequency observatories from
Aggarwal and et al. (2021) are also shown. A final remark is that this assumes
there is no limitation in observation time, which is extremely optimistic for
the higher frequencies. For more details I refer to Franciolini et al. (2022).

An estimation of the sensitivity band for current and future observatories is
showcased in figure 3, from Franciolini et al. (2022). I must repeat their
warning that the figure assumes the observation time is only set by the
frequency evolution of the inspirals, when time resolution is rather important
factor (for more details see Franciolini et al. (2022)), but the figure is
very illustrative nonetheless.

Once detected the GWs can give individual information from the objects in the
system. If the merger comes from a binary of neutron stars (NS), then the GW
emission can be used for example to constrain the neutron star equation of
state The LIGO Scientific Collaboration and The Virgo Collaboration (2018). If
the merger is of a BH binary then in theory their mass and spin can be derived
from their GW emission. There is an important caveat however: while it can get
some information of the individual members of the binary, a lot of it is
degenerate with other parameters. It is also still very hard to distinguish
between a NS and a PBH in any individual merger without also an
electromagnetic counterpart. Currently for LIGO the standard for assigning
wheres the GW event comes from a neutron star or a BH is based purely on mass.
Without a visible counterpart LIGO cannot distinguish between a NS or PBH with
NS like mass Tsokaros et al. (2020).

For the case of the spin, LIGO can obtain what is called as the effective spin
of a binary BH Farr et al. (2017):

$\displaystyle\chi_{eff}\equiv\frac{\left(m_{1}\chi_{1}+m_{2}\chi_{2}\right)}{M}=\frac{c}{GM}\left(\frac{\vec{S}_{1}}{m_{1}}+\frac{\vec{S}_{2}}{m_{2}}\right)\;,$
(18)

where $\chi_{i}$ is a dimensionless spin parameter project over the z axis,
but which can be related to the angular momentum spin of the star
$\vec{S}_{i}$ as shown in the rightmost term above. $m_{i}$ is the mass of
each object and M the total mass of the merger.

As can be seen, the effective spin is degenerate with individual value of each
spin. The individual spin of one of the members of the binary can also be
negative from the two BHs rotating in opposite directions. Thus,
$\chi_{eff}\sim 0$ could mean low spins in both BHs or a very highly counter-
rotating duo. An average $\chi_{eff}=0$ in many mergers would simply point to
the spin of each member of the binary not correlating with each other.

Currently LIGO has showed the GW events it detects have a preference for a
smaller effective spin The LIGO Scientific Collaboration et al. (2021), very
close to averaging 0, stronger than expected Safarzadeh et al. (2020). While
it might seem encouraging for PBHs, which are expected to have spin close to 0
in most cases, the uncertainty is still too high and there have been
individual events with indications of at least one of the members of the
binaries having non-negligible spìn value Biscoveanu et al. (2021). For now
results on the spin are ambiguous, more robust statistics are needed.

Another issue can be the current sensitivity of LIGO. As seen in equations 16
and 17 comparing with figure 3, the observatory probes only GW events from
objects of the order of solar mass or higher and which also must be fairly
close (or belong to niche subset which amplifies the signal, like being ultra-
compact binaries LIGO Scientific Collaboration and Virgo Collaboration
(2018)). While any discovery of GW is greatly helpful, being a very novel
field, LIGO itself is perhaps of limited usefulness for PBHs as DM.

Current constraints in the mass range of LIGO exist from the merger rate. If
PBHs were the DM there would be so many PBH binaries the merger rate would be
much higher than the one we observe, even if we could not distinguish the PBHs
from the stellar origin BHs. Take note however that the modelling of the
merger rate is controversial, with perturbation of binaries by third objects
lowering the number of mergers expected. If PBHs follow the typical Poissonian
distribution the constraints should still be strong enough to exclude PBHs as
DM for the $1-100\,{\rm M_{\odot}}$ mass range Vaskonen and Veermäe (2020);
Wong et al. (2021), but results vary depending on how the PBHs clusters are
handled, with LIGO constraints not enough to rule PBHs out as all of the DM in
Jedamzik (2020, 2021).

There are still other ways in which GW detectors can be helpful however. An
option to help distinguish PBHs are more statistical arguments: for stellar
origin black holes we expect their number of mergers to peak with star
formation, as they form from stars that are very massive with very short
lives. This peak would be situated at $z\sim 1$ Abbott and et al (2016), while
in contrast PBHs would start merging at much higher redshifts and the number
of mergers would increase monotonically with redshift. Unfortunately, current
detectors are incapable of distinguishing stellar origin from non-DM PBHs this
way. Even for the most massive binaries LIGO does not have enough resolution
to resolve mergers at redshift 1Rosado et al. (2016), so trying to see where
mergers would peak in a redshift distribution is not viable.

The solution is to wait for more powerful next generation detectors Wang and
Nitz (2021), particularly LISA Baker and et al. (2019), but even with the
improved resolution there is no guarantee of much better odds looking at
figure 3, particularly if we are searching for very light PBHs in the
planetary or asteroid mass range.

An interesting alternative is not searching for individual events, but for
their added effect. Just how like multiple sources we cannot identify
individually contribute to a background in radiation, the same applies for
GWs. Mergers we do not have resolution to identify will still emit GWs, and
joined will create a non-zero background for our detectors. This incoherent
superposition of unresolved signals is commonly identified as the Stochastic
Gravitational Wave Background (SGWB).

The frequency range of the SGWB is expected to be the same as the unresolved
mergers that form it but redshifted, and the characteristic strain can be
estimated as:

$\displaystyle h_{c}(f)\approx 2\cdot
10^{-31}\left(\frac{\Omega_{GW}(f)}{10^{-7}}\right)^{-1}\left(\frac{f}{\rm
GHz}\right)^{-1}\;,$ (19)

where $\Omega_{GW}$ is the energy density of the spectra at a given $f$.

Just from the level of SGWB we would expect to see on LIGO some constraints
can already be set for solar mass ranges Raidal et al. (2017); Clesse and
García-Bellido (2017), while the prospects for future detectors are also known
Mandic et al. (2016); Chen et al. (2019).

A significant problem can also be correctly identifying the SGWB. There will
also be quite a number of mergers from stellar black holes and other massive
objects, so simply finding a signal is no smoking gun for PBHs. However, while
it is extremely hard to obtain individual values from the background (it is
the addition of a lot of unidentified mergers after all), the background
follows the same statistics as the mergers. This means if the SGWB is
dominated by signals from PBHs, which we expect if they are all of the DM, we
would see effects coming from the different redshift distribution mentioned
above, which could be used to put improved constraint on subsolar PBHs in the
futureMukherjee et al. (2022). Another possibility is to see if the anisotropy
of the SGWB mirrors that of the DM distribution which could also create strong
constraints for solar mass PBHs Wang et al. (2021).

The SGWB can have an origin different than unidentified mergers though. As it
just appears as a background in our detectors, any isotropic GW signal that
completely covers our sky could be part of it. As GWs are quite literally
aftershocks in space-time, we expect any event which significantly alters the
flat space time metric to have some GW associated to them. This does not
extend only to massive objects, but also to perturbations, like the ones
creating the PBHs. The same primordial curvature perturbations that create PBH
will at second order originate scalar-induced gravitational waves Baumann et
al. (2007). As the perturbations are expected to be isotropic, the emission
would result in another background that could give information about the
curvature perturbations, and so its PBH generation and abundance Kozaczuk et
al. (2021); Yuan and Huang (2021). Of note is that while perturbations require
physics outside the standard model to be produced in large enough quantities
to create PBHs, this background would depend only on the perturbations
themselves and thus requires no knowledge of the specific new physics to
simulate. The frequency we expect would depend only on the patch within the
horizon, being directly related to the mass within the horizon $m_{h}$
described in section 2 Saito and Yokoyama (2010):

$\displaystyle f\approx 5\rm kHz\left(\frac{M_{h}}{10^{-24}\,{\rm
M_{\odot}}}\right)^{-1/2}\;.$ (20)

Though that would be heavily redshifted today. Likewise, if the PBHs origin
was unrelated to the collapse of primordial perturbations, then the SGWB
associated to their formation may not exist. Note also that SGWBs may also be
created by other plethora of early Universe GW production mechanisms, almost
all of which would imply physics beyond the Standard Model Franciolini et al.
(2022)

While LIGO’s limitations still apply Bavera et al. (2022), this makes the
future LISA mission even more interesting. Particularly, the range of LISA
would cover the GW background associated to perturbations that would result in
$10^{-12}\,{\rm M_{\odot}}$ PBHs Bartolo et al. (2019). Considering this is
one of the few windows remaining for PBHs to be all the dark matters, as can
be seen in figure 1, this is a very exciting possibility.

###  3.4 Other constraints

While PBHs of mass in the range of planets and asteroid presents a significant
challenge to detect, this does not extend to masses higher than $10\,{\rm
M_{\odot}}$. This is natural, as the only compact objects known above that
mass are other black holes. While this makes distinguishing the primordial
nature of the BH much harder, it also means such objects cannot hide among low
luminosity objects like brown dwarves, planets or asteroids. Furthermore, the
abundance of stellar origin black holes is itself closely limited by the
number of baryons in the universe present in the primordial nucleosynthesis
Cyburt et al. (2016). As the DM is expected to be about 5 orders of magnitude
more common than baryonic matter, if such PBHs were all the DM BHs from
primordial nature would significantly outnumber their stellar origin siblings.

The typical way to detect BHs is through their dynamical effects. By studying
the motions of object within our galaxy it is possible to estimate the
gravitational forces they are subject to, and from them the masses and motions
of other close objects. This is for example how Sagittarius A*, the
supermassive black hole within our own Milky Way, was initially detected
Genzel et al. (1994). As the mass of the PBH rises, it will also rise its
gravitational potential and so their effect on nearby objects. This is the
basis for the dynamical constraints we see on figure 1, coming from how a
large number of massive PBHs would encounter and then disrupt wide binary star
systems within our Milky Way Yoo et al. (2004); Bahcall et al. (1985).

A more atypical way to obtain further constraints is from the accretion
effects such a large number of massive PBHs would bring. PBHs accrete matter
like any other BH, and like any other BH as the matter falls into the event
horizon it will emit a portion of its energy as radiation. The accretion due
to gravity for a given object commonly follows the Bondi regime, with a
squared dependence on mass $M$:

$\displaystyle\dot{M}_{\rm B}=\,$ $\displaystyle\frac{\pi
G^{2}M^{2}\rho_{s}}{c_{s}^{3}}~{},$ (21)

where $c_{s}$ is the sound speed and $\rho_{S}$ is the density of the
surrounding medium and .

For low mass PBHs, the effects of accretion into their surroundings is
completely negligible, but at higher masses, particularly above $10\,{\rm
M_{\odot}}$, accretion becomes very significant. Constraints can be made from
both the direct emission we would expect to see from such accretion in X-rays
Gaggero et al. (2017), as well as by measuring how much PBHs would heat the
surroundings with their accretion and if we see such effects in near dwarf
galaxies Lu et al. (2021).

One interesting advantage of such type of constraints is that they only depend
on the DM distribution and basic accretion physics. Formation path, spin and
even the particulars of our cosmological model are not a major factor, and
thus constraints can be made to be model independent without losing most of
the strength of their bounds Takhistov et al. (2022). The main limits on
accretion constraints come from the reliability of the DM density distribution
models for our Milky Way and other galaxies.

Currently, accretion constraints heavily limit the abundance of PBHs to be
even a very small fraction of the DM from 10 to $10^{7}\,{\rm M_{\odot}}$ Lu
et al. (2021).

##  4 Non-monochromatic constraints

Having detailed the main sources of PBH constraints, one can perhaps wonder
why monochromatic constraints have remained so popular, why extended mass
distributions of PBH are not the standard. A priori, adding multiple masses
should likewise end up adding multiple constraints in a fairly simple way, as
can be seen directly with equations (8) and (15). Those mass ranges are still
considered constrained to an extended mass distribution in our previous
figure, as the extended mass distribution should be roughly equivalent to
monochromatic mass with slightly lower abundance, but using an extended mass
distribution would be both more accurate and likely increase the strength of
the constraints, as constraints in adjacent mass ranges could contribute too.

Unfortunately, while the cases for microlensing (in the simplest form) and PBH
evaporation work very well, this is because their constraints come from
effects that have a direct relation with mass. Both equations (8) and (15) are
analytic expressions that contain the mass (through the Einstein angle of
equation (5) in the microlensing case). In other constraints the effects can
be more indirect, a good example being gravitational waves: we would not only
see the PBHs of a certain mass, but their abundance and spread would directly
affect how many mergers we expect to see if they are the dark matter, and the
difference in masses between binaries has a pretty direct effect on the
frequency and shape of the GW burst their merger would create. Another good
example can be simulations of dynamic effects the PBHs would have. While
technically possible to account for an extended mass distribution, as multiple
bodies simulations their relation is highly non-linear, so the time and
computational cost can increase exponentially. Even in cases different to
those two, if no direct analytical relation exist between the constraint and
mass it is much harder to account for the extended mass distribution, as it
would require numerically integrating over different masses all the process
rather than just one equation.

Bellomo et al. (2018) studied the issue in a thorough way, and reached a
similar conclusion. To convert a monochromatic constraint into one for an
extended mass function you would need to compute a function g:

$\displaystyle g(M_{eq},p_{j})=\int dM\frac{d\Phi_{EMD}}{dM}g(M,p_{j})\;.$
(22)

$g$ is a function that encloses the details of the underlying physics of the
particular constraint and depends on the PBH mass M and a set of astrophysical
parameters $p_{j}$. $\Phi_{EMD}$ is the shape of the extended mass
distribution function. $M_{eq}$ is the main result, the equivalent mass to
move from a monochromatic constraint to an extended distribution one, and it
is defined as the individual mass in the monochromatic constraint that the
extended function is effectively equivalent to. If an extended distribution
has a $M_{eq}=10^{-6}\,{\rm M_{\odot}}$, it would constrain the abundance of
PBHs of the extended distribution to the same degree as $M=10^{-6}\,{\rm
M_{\odot}}$ is in the monochromatic constraints.

$g$ must be determined individually for each different constraint, and should
enclose both the physics of the constraints and the way the observations were
taken (for more details I refer to Bellomo et al. (2018)). Again, for some
constraints $g$ will be an analytic expression we know and there will not be
any major issues resolving the integral above and finding $M_{eq}$. But for
other constraints g is much, much harder to determine

Something else to note is that while this allows us to find constraints for
extended mass distributions through only the monochromatic constraint, this
does not allow us to combine 2 different set of constraints. It might be
tempting to consider that 2 nearby constraints can together constrain the same
extended mass distribution in higher degree than each alone. However, the
physics between the 2 different sets of constraints differ, or in other words,
the $g$ above is different. The equivalent mass of the extended distribution
will have to be computed individually, and the more stringent constraint
taken. For purely monochromatic constraints this much clearer, as there is no
addition of different types of constraints, the bigger one is the one we
apply, simple as that. But the same applies to our extended extended
distributions, as we are converting our constraints from monochromatic ones.

Besides the one described above, other methods existCarr et al. (2017) to
translate monochromatic constraints into extended ones, but have similar
characteristics and difficulties.

The type of extended function expected is also open to debate. A lognormal
mass function is the one commonly believed to be a good approximation for the
standard PBH formation path of fluctuation collapseGreen (2016):

$\displaystyle\psi(M)=\frac{f_{PBH}}{\sqrt{2\pi}\sigma
M}exp\left(-\frac{log^{2}(M/M_{c})}{2\sigma^{2}}\right)\;,$ (23)

where $M_{c}$ is the mass of the peak, the one commonly associated with the
monochromatic constraint, and $\sigma$ is the width of the distribution. The
value of these parameters depends directly of the model used. However, as
keeps repeating, particular models can result in particular mass functions,
including power functions and even very sharp mass distributions that are very
close to a purely monochromatic one.

Of note is that the extended mass distribution can, and in many cases will,
change with time. Mergers between PBHs and accretion can bring major changes
by significantly raising the abundance of higher mass PBHs De Luca et al.
(2020), while on the opposite end very small PBHs will slowly evaporate by a
different rate depending on their own mass, changing the shape of the
distribution Mosbech and Picker (2022). While this may bring some slight
changes between the formation mass distribution and the current one it does
not affect most constraints, including microlensing, gravitational waves and
evaporation, as they are based on observations of the nearby universe and so
constrain the mass distribution at current redshift 0. Constraints based on
high redshift, like those coming from the CMB, however will have to be shifted
to what currently are higher masses.

Overall, extended mass distributions seem to create more stringent constraints
than monochromatic ones, but it is not possible to generalize for all cases
and graphs have to be individualized for each type of distribution. In
comparison, the monochromatic constraints figure is generic and much easier to
produce, and a sufficiently good estimate to merit its continued usage.

While different, something similar happens with clustered distributions. To
begin with, the curvature fluctuations that originate the PBHs in our standard
formation path are Poissonian Ballesteros et al. (2018); Desjacques and Riotto
(2018); Ali-Haïmoud (2018), and so it is expected for a large number of small
clusters of PBHs to form. However, we also expect most of those small clusters
to have evaporated by now Afshordi et al. (2003), and to be too diffuse to
affect the microlensing constraints for example Petač et al. (2022).

Other formation models can result in cases where the PBHs form in a much more
clustered distribution rather than the typical spread Belotsky et al. (2019),
and even in the typical fluctuation collapse the presence of non-gaussianities
can bring forth non-negligible clustering Suyama and Yokoyama (2019); Young
and Byrnes (2020). Like before, the change in signals and constraints must be
treated individually for each mass range in question. Care has to be given
though, because while clustering can decrease the constraints for some mass
ranges, on some cases it can also introduce new signals that should not be
overlooked. A good example are binary mergers: clustering significantly
decreases the constraints of LIGO, as most mergers would have occurred in the
early universe due to PBHs being more concentrated, but those previous mergers
also create a very significant stochastic GW background Atal et al. (2020)
that would be within the current LIGO/VIRGO sensitivity Abbott and et al.
(2021).

Another example can be with microlensing. While very strongly clustered PBHs
are expected to form a single much larger lens rather one per PBH, which
should a priori reduce the number of lenses and so of events we expect to see,
the number of events expected can actually raise if we have one of the PBH
clusters close enough by chance Gorton and Green (2022).

Naively we would expect PBH evaporation and accretion constraints to not be
strongly affected by clustering, but understanding of both the physical
reasoning of the constraints and also the particularities of the clustered
distribution are needed before applying monochromatic non-clustered
constraints.

##  5 Existing windows

Among all the discussed constraints for smaller PBHs that could be all the
dark matter, there still remains a sizable window between $10^{-16}$ and
$10^{-11}\,{\rm M_{\odot}}$. This window is extremely hard to probe with usual
methods, as they are PBHs too small to be detected either through
gravitational waves emission or microlensing events by current instruments,
but also are too big to be affected by evaporation via Hawking radiation.
Their mass, roughly corresponding to the one of an asteroid, also makes it
exceedingly hard to detect them through conventional methods in big enough
numbers to say anything about their abundance, even if somehow a few could be
found amid countless opaque asteroids and similar objects.

There have been a number of attempts at constraining the mass range still,
most trying to use the effect such PBHs would have onto more standard objects
like stars and predict what observational consequences it could leave.

A novel approach based on these ideas was proposed in Roncadelli et al.
(2009), where they expose that asteroid mass PBHs could randomly traverse
through a star and then fall to the core of said star via energy loss coming
from dynamical friction. If the DM contains PBHs, these would be present in
the cool molecular gas clouds in which star formation happens and, in cases
where the relative velocities are low enough, the PBHs would undergo adiabatic
contraction as the baryonic mass in the form of gas cools down during the
formation of a protostar, resulting in a number of PBHs bound to the newly
born star. A portion of the PBHs orbiting close or within the star would
eventually lose energy via dynamical friction and fall to the core of the
star, thus being captured. Despite the small mass of the PBHs, once settled in
the stellar core they would start accreting, growing up to a total mass that
might be a large fraction of the total mass of the star.

This raises the possibility of black hole with a typical mass of a star but
below the Chandrasekhar mass existing, which cannot be explained in ordinary
stellar evolution but is much easier to detect. If any such black hole could
be detected today, it would therefore point to a possible origin in a PBH and
thanks to forming via such indirect methods it would avoid abundance
constraints in the solar mass range. Similarly, the lack of such objects could
be used to place constraints on the asteroid mass range. However, Abramowicz
et al. (2009) questioned that this process could occur in the Milky Way,
finding the rate of this PBH capture to be negligible because capture by
dynamical friction is very slow and the current DM density too low.

The possibility of PBH capture in the present Universe was reexamined by
Capela et al. (2013), considering star formation in globular clusters formed
in dense dark matter halos made of PBHs in the asteroid mass range. If the
star become a neutron star while it had a PBH within itself, the much higher
densities of the object would result in an almost instant accretion by the
PBH. With high enough DM densities, the process is common enough as to imply
no neutron star would survive, making constraints based on the detection of
such neutron stars possible. While energy loss via dynamical friction was
still very slow, high DM density meant rare very eccentric orbits or orbits
wholly within the star could be populated by PBHs, and only one PBH was
required to be captured for the star to be accreted. The impact of eccentric
orbits to the capture rate was considered in a follow up study Capela et al.
(2014), which found an enhanced capture rate that made the constraints tighter
around the same mass range. Both works however considered globular clusters
with a very high DM and baryonic density, whereas in fact globular clusters
may form from gas cloud fragmentation without involving any DM, so their
constraints are not readily applicable Montero-Camacho et al. (2019) and so do
not appear in figure 1.

A further refinement of this method of PBH capture is present in Oncins et al.
(2022), which concluded that the higher DM density and lower velocity
dispersion in DM haloes at high redshift ($z\sim 20$) would allow PBHs to be
captured by common low mass main sequence stars. While no bounds based on
neutron star survival can be made, as most captures happened at very high
redshift, a substantial number of sub-Chandrasekhar mass BHs would exist as
results of such captures if PBHs in the asteroid mass range are all of the DM.

Pani and Loeb (2014) presented a similar case of capture of PBHs but by
neutron stars directly. Rather than dynamical friction, the energy loss of the
PBH was via tidal deformations of the neutron stars, which was supposedly more
efficient and thus would end up with neutron stars capturing the PBHs much
more easily, severely constraining the mass range as DM candidates. Capela et
al. (2014); Defillon et al. (2014) however cast doubts on it, arguing both
tidal deformations and dynamical friction were roughly equivalent, and models
agreed with previous dynamical friction calculations.

Overall, despite the interesting approaches, asteroid mass range PBHs remain
the last big windows where PBHs could provide all the dark matter. There
exists more exotic PBH formation models that avoid most of the other
constraints, but if using standard inflation perturbation collapse with as few
extra elements as possible they remain perhaps the last possibility to explain
the DM. In the future LISA could constrain this simple case through the SGWBs
the PBH formation would produce Bartolo et al. (2019), but today it still
remains one of the most interesting windows in the PBH mass range.

##  6 Possible detections

Most constraints on PBh abundance are based on assuming a population of PBH
with a certain abundance and mass range, studying a possible observable
consequence their existence would bring and then using the lack of said
detections to set constraints. The inverse however is possible, that is, to
explain an observed but unexplained event through a population of PBHs.

The most famous example of this is the one previously cited, where a
population of PBHs could explain the LIGO GWs merger events observed Bird et
al. (2016). While currently the mergers can all be explained through only
stellar origin BHs, models suggest that a combined population of PBHs and
stellar origin BHs mergers is the one that fits best the rate observed by LIGO
Hütsi et al. (2021); De Luca et al. (2021), though with an abundance far below
the one needed for PBHs to be the dark matter and with uncertainties in
astrophysical and primordial formation models not allowing for an unequivocal
answer Franciolini et al. (2022). Exotic variations also exist where PBHs
outside the LIGO mass range are not all of the DM and end up creating the
mergers seen by LIGO Eroshenko (2021); Kusenko et al. (2020).

Another good example are microlensing events. Famously, while the MACHO
experiments constrained compact objects to not be all of the DM they also
found a higher number of microlensing events than expected by objects in the
disk Alcock and et al. (2000). Microlensing surveys like HSC today also find a
number of microlensing events caused by unknown compact objects Niikura et al.
(2019) that could be easily be explained by PBHs, though it has to be stressed
than in a much lower abundance than the one needed for them to be all of the
DM. Finally, Hawkins (2022) argued that PBHs as all of the DM could explain
the microlensing effects seen on the quasars, but their results contradict
other microlensing studies of quasars Esteban-Gutiérrez et al. (2022).

Another event that has commonly been explained as a possible result of PBHs is
the excess of 511 keV photons from the center of the Milky Way relative to
expectations found by telescopes SPI/INTEGRAL/COSI Churazov et al. (2005);
Kierans et al. (2020). Very small PBHs could be the source of the increased
radiation via their Hawking emission Keith and Hooper (2021), though it can
also be explained by other conventional means or particle DM Mukhopadhyay et
al. (2021), and even if the origin were PBHs they would only make up a small
fraction of the DM.

Perhaps the most famous and current case are the latest finds by NANOGrav
Arzoumanian and et al. (2020). Within a pulsar-timing data set comprising 12.5
years there was evidence of a stochastic common-spectrum process. The signal
was not found to have a quadrupole correlation that would confirm the origin
of such process as being from a SGWB, but it is still a popular and reasonable
explanation for the process.

In particular, a SGWB with parameters that could result in the process
observed in NANOGrav could in turn have the origin in PBH formation De Luca et
al. (2021); Domènech and Pi (2020). As mentioned in section 3.3, the formation
of PBHs by perturbation collapse generates its own SGWB. For the NANOgrav
case, the formation of a small abundance of subsolar mass PBHs would create a
SGWB capable of generating the stochastic process present in the data Kohri
and Terada (2021).

It is important however to stress that the abundance of PBHs that would be
generated is far below the one needed to explain all of the DM if only using
subsolar mass PBHs, at least as long as subsolar mass PBHs are the only ones
produced. There exists the possibility of a very broad PBH mass range
distribution, where a small fraction of subsolar mass PBHs would be generated
as the end point of a distribution with a peak around the asteroid mass range
De Luca et al. (2021); Sugiyama et al. (2021), which could explain all of the
DM. This extremely broad distribution means there would be a non-negligible
abundance of PBH in the microlensing range too, which the HSC would probe in
the future Sugiyama et al. (2021).

Overall, direct detection of PBHs remain a tantalizing if challenging
prospect. While several events exists which could be explained by their
presence, all of them would require them to be only a small fraction of the
DM, and several, like the galaxy center excess, have even been used to create
constraints.

##  7 Conclusions and future prospects

As of today, only PBHs within a small range of masses between
$10^{-16}-10^{-11}\,{\rm M_{\odot}}$ could make up all of the DM, assuming
they form with the smallest deviation possible to the standard model. But
while relatively small this range of masses remains challenging to constrain,
comprising of PBHs too small to distinguish from diffuse matter and too
massive to emit significant amounts of Hawking radiation.

In the years following this review it is expected for plenty of experiments
and new instruments to help further constraint the abundance of PBHs Green and
Kavanagh (2021); Bird et al. (2022). While helpful, most of these tighter
constraints are in already constrained mass ranges, so they offer gradual
improvement rather than resolving the remaining mass window.

There are two possible exceptions. The first are the so called 3rd generation
GW instruments. On the ground the Cosmic Explorer LIGO Scientific
Collaboration and Harms (2017); Reitze and et al. (2019) and Einstein
Telescope Punturo and et al. (2010); Maggiore and et al. (2020) could, if
approved and build, be able to observe binaries with total mass of $\sim
10\,{\rm M_{\odot}}$ up to redshifts $z\sim 100$. While not on the most
interesting mass ranges, the possibility of detecting mergers at extremely
large redshifts, before the first stars, remain one of the few ways in which
PBHs existence could be undeniably proved or harsher constrains be made Ng et
al. (2022); De Luca et al. (2021). Furthermore, if PBHs are all of the DM in
the asteroid mass range, they would be captured and accrete a good portion of
the lower mass longer lived first stars Oncins et al. (2022), so even the
asteroid mass range could be probed indirectly by looking for sub-
Chandrasekhar mass mergers.

On space, LISA Amaro-Seoane and et al. (2017) frequency range offers the
possibility of detecting the SGWB emitted during the formation of asteroid
mass PBH Bartolo et al. (2019), as mentioned in section 3.3. While the
emission is only expected in the case of PBH formation through the standard
perturbation collapse, it remains one of the few ways to probe the case where
PBH are all of the DM. LISA should also be able to confirm if the stochastic
process detected by NANOGrav is related to PBH formation Vaskonen and Veermäe
(2021). While only tangentially related, next generation pulsar timing arrays
like ngVLA NANOGrav Collaboration (2018) could also help with the detection of
SGWBs, particularly if the NANOGrav signal persists.

Another exciting prospect are X-ray pulsars microlensing, which avoid finite
body effects due to pulsar’s small size. Current X-ray telescope AstroSat
Singh and et al (2014) could already create some constraints around the
$10^{-14}\,{\rm M_{\odot}}$ range with 300 days of observations Bai and
Orlofsky (2019). Future missions Athena Barcons et al. (2015) and Lynx The
Lynx Team (2018) could improve upon those. eXTP Zhang and et al (2016) could
widen the constraints to reach masses $10^{-15}-10^{-13}\,{\rm M_{\odot}}$ Bai
and Orlofsky (2019), which would result in PBHs needing very narrow and fine
tuned mass distributions to make up all of the DM.

Important amounts of clustering could relax the microlensing constraints,
though by how much is still debated Gorton and Green (2022). Nonetheless,
relaxing microlensing constraints would widen the window where PBHs can make
up all of the DM considerably and change future prospects. Still, in analogy
to the case in Atal et al. (2020), significant clustering would also result in
much increased mergers at earlier redshifts that would leave a very relevant
SGWBs. Future GW detectors like the already mentioned Einstein Telescope could
make up some of the lost microlensing constraints in this case.

Overall, asteroid mass PBHs are today still a rather interesting candidate to
be all of the DM. While much more constrained that when they first awoke
interest after the LIGO detection, a window still remains, one which is very
challenging to close with current instruments. Next generation experiments
should be able to probe the window with both microlensing and GWs, either
closing it or perhaps finally answering the decades long question that is the
DM.

This work was supported in part by Spanish grant CEX-2019-000918-M funded by
MCIN/AEI/10.13039/501100011033.

###### Acknowledgements.

I would like to thank discussions on the PBH topic with Albert Escrivà, Ivan
Esteban, Cristiano Germani, Jordi Miralda, Jordi Salvadó and Javier G. Subils.
PBHs: Primordial Black Holes
DM: Dark matter
BH: Black Hole
GW: Gravitational Wave
CMB: Cosmic Microwave Background
LIGO: Laser Interferometer Gravitational-Wave Observatory
SGWB: Stochastic Gravitational Wave Background
NANOGrav: North American Nanohertz Observatory for Gravitational Waves
ISCO: Innermost Stable Circular Orbit
no References yes

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