the SGWB is the so-called astrophysical duty cycle. The duty cycle gives the
average number of events present in the frequency bin and can be defined using
the event rate ${\rm d}R/{\rm d}z$ and the duration of the signal staying in
the frequency bin ${\rm d}\bar{\tau}/{\rm d} f$ as

$\frac{{\rm d}D}{{\rm d}f}=\int{\rm d}z\frac{{\rm d}R}{{\rm d}z}\frac{{\rm
d}\bar{\tau}}{{\rm d}f}.$ (5.30)

Here, ${\rm d}\bar{\tau}/{\rm d}f$ is determined by the chirp mass ${\cal
M}_{c}^{z}=(1+z){\cal M}_{c}$ as

$\frac{{\rm d}\bar{\tau}}{{\rm d}f}=\frac{5}{96\pi^{8/3}}(G{\cal
M}_{c}^{z})^{-5/3}f^{-11/3},$ (5.31)

where ${\cal M}_{c}^{z}=(1+z){\cal M}_{c}$. When the duty cycle is larger than
unity, the GW events overlap, and the background is in the so-called
continuous regime. On the other hand, if the interval between events is
comparable or larger than the typical duration of the signal, the duty cycle
becomes smaller than unity, and the statistical properties are strongly non-
Gaussian. The value of the duty cycle can differ depending on the mass
function and the redshift distribution. Thus, the non-Gaussian property could
be used to distinguish primordial and astrophysical scenarios [610, 611]
together with the spectral shape [612].

###  5.8 From close encounters

Overlapped GWs from PBH close encounters can also form a SGWB [451]. The
formulation is the same as the PBH mergers, Eq. (5.24). In the case of close
hyperbolic encounters (CHE), the energy emitted per logarithmic frequency bin
is given by [468, 424]

$\frac{{\rm d}E^{\rm CHE}_{\rm GW}}{{\rm d}\ln f_{\rm r}}=\nu\frac{{\rm
d}E_{\rm GW}}{{\rm
d}\nu}=\frac{4\pi}{45}\,\frac{G^{3}m_{1}^{2}m_{2}^{2}}{a^{2}c^{5}\nu_{0}}\,\nu^{5}F_{\rm
e}(\nu)\,,\\\ $ (5.32)

where we have defined $\nu\equiv 2\pi\nu_{0}\,f_{\rm r}$ and
$\nu_{0}^{2}\equiv a^{3}/GM$, and the semi-major axis $a$ is related to the
initial velocity as $a=GM/v_{0}^{2}$. The function $F_{\rm e}(\nu)$ describes
the dependence on eccentricity $e$ and it is given by [425]

$\displaystyle\nu^{5}F_{\rm
e}(\nu)\simeq\frac{12F(\nu)}{\pi\,y\,(1+y^{2})^{2}}\,e^{-2\nu\,\xi(y)}\,,$
$\displaystyle
F(\nu)=\nu^{2}\left(1-y^{2}-3\,\nu\,y^{3}+4\,y^{4}+9\,\nu\,y^{5}+6\,\nu^{2}y^{6}\right)\,,$
$\displaystyle\xi(y)=y-{\rm tan}^{-1}y\,,$ $\displaystyle y=\sqrt{e^{2}-1}\,.$
(5.33)

Substituting Eqs. (4.15) and (5.33) into Eq. (5.24), we can estimate the peak
frequency

$f_{\rm peak}\simeq 4.3\,{\rm
Hz}\,\left(\frac{y}{0.01}\right)^{-3}\left(\frac{M}{2000M_{\odot}}\right)^{1/2}\left(\frac{a}{1~{}{\rm
AU}}\right)^{-3/2}\,,$ (5.34)

and the peak amplitude

$\displaystyle\Omega_{\text{GW}}(f_{\rm peak})$ $\displaystyle\approx$
$\displaystyle 0.9\times 10^{-13}\,h\left(\frac{\Omega_{\rm
M}}{0.3}\right)^{-1/2}\left(\frac{\Omega_{\rm
DM}}{0.25}\right)^{2}\left(\frac{\delta_{\rm
loc}}{10^{8}}\right)\left(\frac{a}{0.1{\rm AU}}\right)^{-2}$ (5.35)
$\displaystyle\times\left(\frac{y}{0.01}\right)^{-5}\,\frac{m_{1}}{100M_{\odot}}\,\frac{m_{2}}{100M_{\odot}}\,\frac{m_{1}+m_{2}}{200M_{\odot}}\,.$

In Fig. 33, we show the SGWB spectrum originating from close hyperbolic
encounters, compared with the one from binary PBHs. We find that, for the
modest choice of the parameter values, it is difficult to be reached by LISA.
This difficulty can easily be understood if we rewrite Eq. (5.35) in terms of
the peak frequency,

$\displaystyle\Omega_{\text{GW}}^{\rm CHE}(f_{\rm peak})$
$\displaystyle\approx 3.1\times 10^{-13}\,h$
$\displaystyle\times\left(\frac{\Omega_{\rm
M}}{0.3}\right)^{-1/2}\left(\frac{\Omega_{\rm
DM}}{0.25}\right)^{2}\left(\frac{\delta_{\rm
loc}}{10^{8}}\right)\left(\frac{f_{\rm peak}}{50~{}{\rm Hz}}\right)^{4/3}$
$\displaystyle\times\left(\frac{y}{0.01}\right)^{-1}\,\frac{m_{1}}{100M_{\odot}}\,\frac{m_{2}}{100M_{\odot}}\,\left(\frac{m_{1}+m_{2}}{200M_{\odot}}\right)^{1/3}.$
(5.36)

We can see that the maximum amplitude $\Omega_{\text{GW}}^{\rm CHE}(f_{\rm
peak})$ grows with $f_{\rm peak}^{4/3}$. Thus, the peak amplitude of the
background decays significantly at low frequencies. However, we have observed
that the peak amplitude tends to get enhanced when we make the log-normal
distribution of the mass function wider. Thus, by considering a different PBH
mass function, we may be able to find a case where the SGWB from CHEs gives an
interesting contribution at the LISA frequency.

It is worth mentioning that it could be tested by third-generation ground-
based GW detectors such as Einstein Telescope and Cosmic Explorer. Although
the CHE curves are below the BBH curves, loud BBH events can be detected
individually and subtracted from the data, which may allow us to probe the
other SGWB component below the green curve.

Figure 33: Comparison of the SGWB spectrum originating from BBHs and CHEs,
both for $\beta=0$ (solid) and $\beta=1.28$ (dashed), where $\beta$ is a
parameter characterizing the redshift dependence of the merger rate as
$\tau^{\rm BBH}\propto(1+z)^{\beta}$. For the BBH curves, we take
$m_{1}=m_{2}=100-300~{}M_{\odot}$ and $v_{0}=30$km/s. The CHE curves
correspond to the same range of masses with $a_{0}=5$AU, $y_{0}=2\times
10^{-3}$ for frequencies around 10 Hz, and $a_{0}=5\cdot 10^{7}$AU,
$y_{0}=10^{-5}$ in the mHz range. For all cases, we assume log-normal
distributions for $a_{0}$, $y_{0}$ and the PBH mass, of respective widths
$\sigma_{a},\sigma_{y}=0.1$, $\sigma_{m}=0.5$, as well as $f_{\rm PBH}=1$. The
bands come from the possible range of parameters $a_{0}$ and $y_{0}$. For a
smaller fraction of PBHs, the GW spectral amplitude simply scales as
$\Omega_{\rm GW}\propto f^{2}_{\rm PBH}$. Figure and estimated LISA
sensitivity from [451].

###  5.9 Summary

In this section we have explored the different sources of SGWBs related to
PBHs:

  * •

The SGWB induced at second order by the large curvature fluctuations leading
to PBH formation. Interestingly, it peaks at PTA frequencies for stellar-mass
PBHs, which makes it a very complementary probe to test a possible primordial
origin of GW events seen in ground-based detectors. This SGWB could be
connected to the recent pulsar timing array observations, see sec. 7.6.4 for
more details. Future GW detectors like LISA, Cosmic Explorer and Einstein
Telescope will probe small PBHs, in particular in the asteroid-mass range that
remains so far unconstrained (see the section on the limits on the abundance
of PBHs).

  * •

The SGWB from the Poisson fluctuations in the initial PBH distribution, even
if they were so small that they have totally evaporated. This mechanism could
allow to probe the possible PBH formation at very high energy scales, up to
the reheating phase and the GUT energy.

  * •

The stochastic background from early binaries, in particular if PBHs have an
extended mass function. In such a case the SGWB is boosted by the merging of
binaries with extreme mass ratios.

  * •

The stochastic background from late binaries in clusters, which for stellar-
mass PBHs is smaller than the SGWB from early binaries at frequencies
corresponding to ground-based GW detectors. Because the merger rates are not
suppressed with the PBH mass, it would however be the dominant background at
low frequencies, i.e. in the LISA and PTA frequency range.

  * •

The stochastic background from close hyperbolic PBH encounters in clusters,
probably undetectable with LISA but in the range of future ground-based GW
detectors.

The SGWB is therefore expected to be an interesting complementary signal that
will help to distinguish between PBH formation models, mass functions,
clustering histories and binary origins.

##  6 Gravitational-waves and Large Scale Structure correlations

###  6.1 Introduction

Primordial black holes, both in the case where they comprise a large part of
the dark matter or if they exist only in small numbers, are part of the Large-
Scale Structure (LSS) of the Universe, and their observational relation with
the other matter-energy components can be generally different with respect to
black holes of stellar origin. By the time LISA will be online, there will be
data from several large-scale structure experiments, both preceding and
overlapping in time with LISA. These include observations from the ground and
from space, such as DESI [613, 614], the Vera Rubin Observatory [615], Euclid
[616], SPHEREx [617], Nancy Roman [618], ATLAS [619] and the SKAO [620], all
together providing galaxy catalogs over a very wide range of scales both in
width and deepness, at unprecedented accuracy. It will therefore be paramount
to take advantage of such data for PBH science. In this Section we discuss a
few possible ways to use gravitational wave observations in combination with
the LSS to constrain the existence and abundance of PBHs. LSS observations can
be cross-correlated with different types of GW maps, both from the resolved
mergers of compact objects and the stochastic backgrounds.

Cross-correlations between GWs from resolved compact object mergers and
galaxies have been first investigated in [621]. Later, it was suggested that
the cross-correlation of mergers with EM counterparts could be used to
constrain dark energy and modified gravity models [622]. After the first
detections of gravitational waves, [623] investigated the possibility to
constrain the distance-redshift relation using such correlations.

It was then shown that the cross-correlation of resolved mergers with LSS
catalogs can be used to test primordial black hole scenarios [624, 625, 626,
627] and different astrophysical models [628]. More recently, an additional
technique for constraining PBH abundances, formation channels and redshift
distributions using only GW data has been proposed and studied [629].

The GWxLSS probe became very popular and several different approaches have
been suggested and investigated (see e.g., [625, 626, 630, 631]), looking at
both correlations with resolved sources and the SGWB [632, 633, 634, 588, 635,
610, 636]. Here we present the main ideas and formalism developed so far;
specific predictions for LISA and investigations involving similar analyses
for extreme mass-ratio inspirals (EMRIs) and intermediate mass black holes
(IMBHs) are currently being developed.

###  6.2 GW-galaxy cross-correlations for resolved events

The cross-correlation of LSS catalogs with maps of gravitational waves coming
from the merger of compact objects will provide information on several
astrophysical and cosmological parameters and models, and has been
investigated in a variety of works from exploring ways to indirectly detect
GWs to proper data correlations.

The idea of cross-correlations with the LSS in order to constrain the
existence of PBHs and their abundances was first explored in [624] and then
further investigated in e.g., [625, 626, 628]. The possibility of using the
GWxLSS correlation to test the existence of PBHs comes from the fact that
primordial and astrophysical binary black holes trace the LSS in different
ways. This reflects into the fact that the bias of the hosts of the binary
system will be different when the BHs have different progenitors and formation
channels.

In [61], it was shown that primordial black holes preferentially merge in
lower biased objects and thus have a lower cross-correlation with luminous
galaxies. This happens because binary PBHs would preferentially merge in halos
with low velocity dispersion, which are low-mass halos and hence with very
little or absent star formation; these objects have a bias $b<1$. On the other
hand, mergers of compact objects that are the endpoint of stellar evolution
naturally happen for the vast majority within star forming-rich halos, which
have larger galaxy bias values, $b>1$. In the linear biasing scheme, $b$
quantifies the relation between the underlying matter distribution and
observed sources, $\delta_{\rm O}=b\delta_{\rm m}$. This reasoning is valid
for PBH binary systems that form by capture (late binary formation scenario),
as these systems form in the local Universe.

However, there is also another way to have mergers of PBH binaries, which we
have referred to as early binary mergers. In this case, within the standard
formation scenario (i.e., the collapse of primordial curvature perturbations
at horizon re-entry), PBHs should form in the peaks of the initial matter
distribution, and therefore they will trace the dark matter field. As a
consequence, such mergers should be unbiased tracers of the underlying density
field.

The overall catalog of PBH mergers will consequently be composed of a mixture
of early and late binaries, with the total bias being the weighted average of
the two.

Therefore, by measuring the amplitude of the angular cross-correlation of
galaxy maps with catalogues of compact-object mergers, which directly depends
on the bias of the mergers’ hosts, one can statistically probe the abundance
of PBH mergers. This information can also be used to discriminate between
different astrophysical models (see e.g., [628]).

In order to measure the mergers’ hosts bias, one can use the 3D angular power
spectrum:

$C_{\ell}^{\rm GW\times LSS}(z_{1},z_{2})=\frac{2}{\pi}\int{\rm
d}k\,k^{2}P(k)\,\Psi_{\ell}^{\rm GW}(k)\Psi_{\ell}^{\rm LSS}(k)\,,$ (6.1)

where $P(k)$ is the matter power spectrum and the kernels $\Psi_{\ell}^{X}(z)$
include the observational window functions and the relevant physical effects:

$\displaystyle\Psi_{\ell}^{X}(k)$ $\displaystyle=\int
N^{X}(z)b^{X}(z)D(z)W(z)\Delta_{\ell}^{X}(k,z)\,{\rm d}z\,;$ (6.2)

here $X$ will be either GW or LSS, and $W$ are observational window functions
related to the experiment specifications (i.e., they include information on
the redshift distribution and survey geometry and sensitivity). The
$\Delta_{\ell}$ terms are the (gauge-independent) observed overdensities and
include effects from intrinsic clustering, peculiar velocity and Doppler
contributions, lensing and gravitational potentials containing in this way
information information on the underlying cosmological and astrophysical
models. They are usually expressed as:

$\Delta_{\ell}(k)=\Delta^{\mathrm{den}}_{\ell}(k,z)+\Delta_{\ell}^{\mathrm{rsd}}(k,z)+\Delta_{\ell}^{\mathrm{dop}}(k,z)+\Delta^{\mathrm{len}}_{\ell}(k,z)+\Delta^{\mathrm{gr}}_{\ell}(k,z)\,,$
(6.3)

where one has that:

$\displaystyle\Delta_{\ell}^{\mathrm{den}}(k,z)$
$\displaystyle=b_{X}\delta(k,\tau_{z})j_{\ell},$ (6.4)
$\displaystyle\Delta_{\ell}^{\mathrm{rsd}}(k,z)$
$\displaystyle=\frac{k}{\mathcal{H}}j^{\prime\prime}_{\ell}V(k,\tau_{z}),$
$\displaystyle\Delta_{\ell}^{\mathrm{dop}}(k,z)$
$\displaystyle=\left[(f^{\mathrm{evo}}_{X}-3)\frac{\mathcal{H}}{k}j_{\ell}+\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}+\frac{2-5s_{X}}{r(z)\mathcal{H}}+5s_{X}-f^{\mathrm{evo}}_{X}\right)j^{\prime}_{\ell}\right]V(k,\tau_{z}),$
$\displaystyle\Delta_{\ell}^{\mathrm{len}}(k,z)$
$\displaystyle=\ell(\ell+1)\frac{2-5s_{X}}{2}\int_{0}^{r(z)}{\rm
d}r\frac{r(z)-r}{r(z)r}\left[\Phi(k,\tau_{z})+\Psi(k,\tau_{z})\right]j_{\ell}(kr),$
$\displaystyle\Delta_{\ell}^{\mathrm{gr}}(k,z)$
$\displaystyle=\left[\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}+\frac{2-5s_{X}}{r(z)\mathcal{H}}+5s_{X}-f^{\mathrm{evo}}_{X}+1\right)\Psi(k,\tau_{z})+\left(-2+5s_{X}\right)\Phi(k,\tau_{z})+\mathcal{H}^{-1}\Phi^{\prime}(k,\tau_{z})\right]j_{\ell}$
$\displaystyle+\int_{0}^{r(z)}{\rm
d}r\frac{2-5s_{X}}{r(z)}\left[\Phi(k,\tau)+\Psi(k,\tau)\right]j_{\ell}(kr)$
$\displaystyle+\int_{0}^{r(z)}{\rm
d}r\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}+\frac{2-5s_{X}}{r(z)\mathcal{H}}+5s_{X}-f^{\mathrm{evo}}_{X}\right)_{r(z)}\left[\Phi^{\prime}(k,\tau)+\Psi^{\prime}(k,\tau)\right]j_{\ell}(kr).$

For details on these terms, their derivation and physical meaning, see e.g.,
[637, 638, 639, 626].

###  6.3 GW-galaxy cross-correlations for the stochastic background

In addition to the signal from mergers detected by interferometers, there is
another signal coming from the superposition of many unresolved sources. These
will form a stochastic background of gravitational waves, as discussed in
Section 5. Obviously, primordial black holes, if they exist, would contribute
to this signal, and their associated GW backgrounds have been discussed in
Section 5. This will be particularly relevant for LISA, as the SGWB is
expected to be a product of LISA’s observations.

In the same fashion as for the catalogs of resolved events described above, we
can cross-correlate the stochastic background with the LSS; such correlation
will contain information on the redshift distribution, mass function and
clustering behavior of the sources. Moreover, the observed signal will be the
combination of the stochastic GW background from astrophysical objects,
denoted AGWB, the one from primordial black holes, and the one generated from
inflation; disentangling the signal and distinguishing between the different
sources will be of paramount importance to obtain information about black hole
and early Universe physics, and one of the most promising ways to do so is
thanks to the SGWB–LSS correlation.

Given that the GW energy density depends not only on astrophysical properties
but also on cosmological perturbations, it will correlate with other
cosmological probes; moreover, GWs will experience projection effects that
will need to be accounted for in order to observe the signal in the
appropriate frame [588, 640]. Some preliminary works on this have been done in
the last few years, including forecasts for the cross-correlation signals
between GW observatories and future galaxy surveys, as e.g., Euclid and SKA
(see, e.g., [581, 582, 641, 632, 634, 633, 642, 643, 644, 645, 588, 646, 610,
636]).

In a similar way to the case of resolved mergers, the most natural observable
is the angular cross power spectrum that correlates the energy density of the
AGWB with galaxy number counts:

$C^{\rm AGWB\,\times\,LSS}_{\ell}=\frac{2}{\pi}\int{\rm
d}k\,k^{2}P(k)\Psi^{\rm AGWB}_{\ell}(k)\Psi^{\rm LSS}_{\ell}(k),$ (6.5)

where most quantities are the same of Eq. (6.1), and the dependence on
astrophysical models and parameters can be included in the expression for the
AGWB energy density. This is usually written as the total GW energy density
(denoted by the superscript TOT) per logarithmic frequency $f_{\rm o}$ and
solid angle $\Omega_{\rm o}$ along the line-of-sight ${\bf\hat{n}}$ of a SGWB,
defined in e.g., [582, 588] as

$\Omega^{\rm TOT}_{\rm GW}\left(f_{\rm o},{\bf\hat{n}}\right)=\frac{f_{\rm
o}}{\rho_{{\rm c}}}\frac{{\rm d}\rho^{\rm TOT}_{\rm GW}}{{\rm d}f_{\rm o}{\rm
d}\Omega_{\rm o}}\;;$ (6.6)

this contains both a background, which is homogeneous and isotropic, and a
direction-dependent contribution; the total relative fluctuation can thus be
defined as (similarly to Eq. (5.17)):

$\Delta^{\mathrm{TOT}}_{\mathrm{AGWB}}(f_{\rm
o},\hat{\mathbf{n}})=\frac{\Omega^{\mathrm{TOT}}_{\mathrm{AGWB}}-\bar{\Omega}^{\mathrm{TOT}}_{\mathrm{AGWB}}}{\bar{\Omega}^{\mathrm{TOT}}_{\mathrm{AGWB}}}\;.$
(6.7)

Following [588], we report the expression for the contributions to the AGWB in
the Poisson gauge:

$\displaystyle\Delta\Omega_{\rm AGWB}(f_{\rm
o},\hat{\mathbf{n}},\mathbf{\theta})=\sum_{[i]}\int{\rm
d}z\,W(z)\,\mathcal{F}^{[i]}(f_{\rm o},z,\mathbf{\theta})\Bigg{\\{}b^{[i]}D$
(6.8)
$\displaystyle+\left(b^{[i]}_{\mathrm{evo}}-2-\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}\right)\hat{\mathbf{n}}\cdot\mathbf{V}-\frac{1}{\mathcal{H}}\partial_{\parallel}(\hat{\mathbf{n}}\cdot\mathbf{V})-(b^{[i]}_{\mathrm{evo}}-3)\mathcal{H}V$
$\displaystyle\quad+\left(3-b^{[i]}_{\mathrm{evo}}+\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}\right)\Psi+\frac{1}{\mathcal{H}}\Phi^{\prime}+\left(2-b^{[i]}_{\mathrm{evo}}+\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}\right)\int_{0}^{\chi(z)}{\rm
d}\tilde{\chi}\left(\Phi^{\prime}+\Psi^{\prime}\right)$
$\displaystyle\quad+\left(b^{[i]}_{\mathrm{evo}}-2-\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}\right)\left(\Psi_{\rm
o}-\mathcal{H}_{0}\int_{0}^{\tau_{0}}{\rm
d}\tau\left.\frac{\Psi(\tau)}{1+z(\tau)}\right|_{o}-\left(\hat{\mathbf{n}}\cdot\mathbf{V}\right)_{o}\right)\Bigg{\\}},$

where the density, velocity, gravity and observer terms, in the first, second,
third and fourth line, respectively contain all the cosmological information.
On the other hand, the function $\mathcal{F}^{[i]}(f_{o},z,\mathbf{\theta})$
contains all the astrophysical dependencies, i.e., the mass and spin
distribution of the binary, the emitted GW energy spectrum, the clustering
properties of GW events and the details of the GW detectors;
$b^{[i]}(z,\mathbf{\theta})$ and
$b^{[i]}_{\mathrm{evo}}(z,\mathbf{\theta}):=-d\log\left[(1+z)\mathcal{F}^{[i]}\right]/d\log(1+z)$
are the bias and the evolution bias of the $i$-th type of GW source, which
specify the clustering properties of GW sources and characterise the formation
of the sources.

Studies of the cross-correlation between the AGWB with galaxy number counts
are presented in [632, 610, 630, 636, 642, 635, 647]. This allows a
tomographic reconstruction of the redshift distribution of sources, and can be
also useful for the shot noise characterization (in the case of ground-based
detectors) [636]. The cross-correlation shows that the combination of galaxy
surveys with the AGWB can be a powerful probe for GW physics and can be a
robust observational probe for multi-messenger cosmology.

###  6.4 GW$\times$LSS forecasts

Constraints on the presence of PBHs from measurements of the GWxLSS have been
forecasted for the case of resolved GW signals, correlated with both galaxy
surveys [624, 625, 626, 648] and intensity mapping experiments [627].
Constraints coming from measurements of the stochastic background signal are
currently being worked out. In this Section we introduce the formalism for
obtaining such forecasts and report some of the results present in literature.

The correlation of two tracers ($X,Y$) can be estimated from tomographic maps
of their number counts and evaluating their N-point statistics; in the case we
consider here, this is the angular power spectrum $C_{\ell}^{XY}(z_{i},z_{j})$
of the tracer $X$ in redshift bin $z_{i}$ and the tracer $Y$ in $z_{j}$.
Assuming that the noise comes only from the standard shot noise contribution,
the total signal extracted from the maps is:

$\left\langle a_{\ell
m}^{X}\left(z_{i}\right)a_{\ell^{\prime}m^{\prime}}^{Y^{*}}\left(z_{j}\right)\right\rangle=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\tilde{C}_{\ell}^{XY}\left(z_{i},z_{j}\right)=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\left[C_{\ell}^{XY}\left(z_{i},z_{j}\right)+\delta_{XY}\delta_{ij}\mathcal{N}_{\ell}^{X}\left(z_{i}\right)\right]\,,$

where $\mathcal{N}_{\ell}$ is the noise angular power spectrum.

The observed signal will receive contributions from several physical effects,
coming from the cosmological perturbations described above, from density,
velocity, lensing and gravity effects. It is important to include all of them
to avoid mis-estimating parameters, best fit values and errors when doing
parameter inference, as shown in [640, 649].

Forecasts obtained so far have been based on the Multi_CLASS [650] and
GW_CLASS codes [640]. Multi_CLASS is the first public Boltzmann code that
allows to compute the angular power spectrum for multiple galaxy (and other
tracers) populations. The code lets the user specify specific properties of
different tracers, such as its number density redshift distribution and bias,
magnification bias and evolution bias parameters, while GW_CLASS calculates
the stochastic signal from astrophysical mergers and accounts for all the
astrophysical and projection effects contributions.

The first suggestion that this observable can be used to constrain PBHs was
presented in [624], where after introducing the concept, it is shown that the
cross-correlation between galaxies mapped by future radio surveys and GW
interferometers could distinguish between primordial and stellar origin of
merging black holes. Such possibility depends on the merger rate of primordial
black holes and the angular resolution of the GW interferometer. In that work,
to obtain the results, a minimum-variance estimator was introduced for the
effective correlation amplitude, $A_{\rm c}\equiv r\times b_{\rm GW}$, where
$r$ is the cross-correlation coefficient of Eq. (6.1). This cross-correlation
coefficient parametrises the extent to which two biased tracers of the matter
field are correlated [651].

The minimum-variance estimator for the effective correlation amplitude is
given by (see e.g. [652]):

$\widehat{A_{\rm c}}=\frac{\Sigma_{\ell}{\tilde{C}_{\ell}}F_{\ell}/{\rm
Var}[\tilde{C}_{\ell}]}{\Sigma_{\ell}F_{\ell}^{2}/{\rm
Var}[\tilde{C}_{\ell}]}\,,$ (6.9)

where $\tilde{C}_{\ell}$ is the measured power spectrum and
$F_{\ell}\equiv{d\tilde{C}_{\ell}}/{d\widehat{A_{\rm c}}}\propto b_{\rm g}$.
The variance of this estimator is then:

$\sigma^{2}_{\widehat{A_{\rm c}}}=\left[\sum_{\ell}\frac{F_{\ell}^{2}}{{\rm
Var}[\tilde{C}_{\ell}]}\right]^{-1}\,,$ (6.10)

which can be used to forecast the measurement error when neglecting that of
other parameters.

More generally, the measurement error for specific parameters in a given
experiment can be estimated using Fisher analysis. For this case, we write the
Fisher matrix as:

$F_{\alpha\beta}=\sum_{\ell}\frac{\partial
C_{\ell}}{\partial\vartheta_{\alpha}}\frac{\partial
C_{\ell}}{\partial\vartheta_{\beta}}{\sigma_{C_{\ell}}^{-2}}\,,$ (6.11)

where $\vartheta_{\alpha}=\\{A_{\rm c},b_{\rm g}\\}$; the derivatives of the
power spectra $C_{\ell}$ are evaluated at fiducial values
$\bar{\vartheta}_{\alpha}$ corresponding to the scenario at hand, and
$\sigma_{C_{\ell}}$ are errors in the power spectra.

The results in [624] were obtained by computing the $2\times 2$ Fisher matrix
for the parameters $\\{A_{\rm c},b_{\rm g}\\}$, using a prior on the galaxy
bias corresponding to the precision reached by fitting the amplitude of the
galaxy auto-correlation function $C_{\ell}^{\rm gg}$, using a value of
$\ell_{\rm max}=200$, which yields a $\sim 10\%$ precision in the measurement
of the bias $b_{\rm g}$. For more details on this analysis, see [624].

Subsequently, in [625], such analysis was updated to include improved merger
rate estimates from the LVK collaboration, for the correlation with different
possible surveys with the SKA. In Figure 34 we show forecasts for different
experiments, considering different versions of LVK and ET correlated with HI
and continuum SKA surveys. The bars show the constraints on $A_{\rm c}$ for
different values of a parameter $R$ that encapsulates uncertainties on the
merger rate, which is degenerate with observation times from the gravitational
wave interferometers.

Figure 34: Forecast errors on the cross-correlation amplitude, $A_{\rm c}$,
for different experiment combinations, varying merger rates and years of
observations. Each column corresponds to a GW detector experiment, for merger
rates from 1 to 10 Gpc-3yr-1. Horizontal lines show the expected difference in
the cross-correlation between (late binary) PBH and stellar binaries, for
different values of $f_{\rm PBH}$.

Following this, in [626] these forecasts are updated by including all
projection effects, and inserted in an analysis of a null hypothesis test by
comparing two models, one in which the BHs origin is stellar, the other in
which it is primordial. One model is assumed as a fiducial model. Then, by
computing the signal-to-noise ratio (SNR), it is checked whether the
alternative model can be differentiated from the fiducial one. The null
hypothesis is that the model is indistinguishable from the fiducial, which
happens for low values of the signal-to-noise ratio.

For instance, the distance of an alternative model from the fiducial can be
quantified using a $\Delta\chi^{2}$ statistics. The $\Delta\chi^{2}$ is given
by the logarithm of a likelihood, and the SNR can be written as:

$\left(\frac{S}{N}\right)=f_{\mathrm{sky}}\sum_{2}^{\ell_{\max}}(2\ell+1)\left(\mathbf{C}_{\ell}^{\mathrm{Alternative}}-\mathbf{C}_{\ell}^{\mathrm{Fiducial}}\right)^{T}\operatorname{Cov}_{\ell}^{-1}\left(\mathbf{C}_{\ell}^{\mathrm{Alternative}}-\mathbf{C}_{\ell}^{\mathrm{Fiducial}}\right)\,,$

where
$\mathbf{C}_{\ell}^{T}=\left(C_{\ell}^{\mathrm{gg}}\left(z_{1},z_{1}\right),\cdots,C_{\ell}^{\mathrm{gGW}}\left(z_{1},z_{1}\right),\cdots,C_{\ell}^{\mathrm{GWGW}}\left(z_{1},z_{1}\right),\cdots\right)$,
$f_{\mathrm{sky}}$ is the sky fraction covered by both GW and galaxy surveys
and $\mathrm{Cov}_{\ell}$ is the covariance matrix, computed from the angular
power spectra of the fiducial model.

In Fig. 35, we show the results where the fiducial model is taken to that of
stellar BH mergers (See [626]). The bars span values between 0.1 and 10 for
the factor $r$, (similarly to the case of Fig. 34, this parametrises the
uncertainties on the BBH merger rates, which is degenerate with the
observation time of the experiment).

Figure 35: Expected signal-to-noise ratio $(S/N)_{\Delta B/B}$ from the Fisher
matrix analysis, for cross-correlations between GW observations (from either
LVK or Einstein Telescope) and high-redshift LSS surveys (EMU, DESI and SKA),
assuming stellar black holes for the fiducial scenario. In each case, the left
bar corresponds to late PBH binaries and the right bar to early PBH binaries.
Figure from [626].

After updates on the BBH merger rate and the specifications for the DESI
survey [613], forecasts for the SNR for detecting a component of PBHs are
shown in [648] and a summary plot can be found in Fig. 36. Here the color code
is for the SNR, while on the x-axis is the fraction of PBH mergers from the
total of observed BBH mergers observed by the Einstein Telescope, and on the
y-axis the parameter $r$, as defined above.

Figure 36: SNR for detecting a fraction $\Gamma$ of PBH mergers out of the
total BBH mergers observed by the ET, as a function of the parameter $r$,
cross-correlated with DESI. The left panel shows the result using just DESI x
ET, while the right panel includes Planck priors. Results from [648])

The cross-correlation forecasts shown so far are for the correlation of GWs
with resolved galaxies; naturally, a similar analysis can be done for
observations of intensity mapping. A first investigation for the 21cm
Intensity Mapping (IM) observed with the SKAO is presented in [627], focusing
on the IM of the neutral hydrogen (HI) from the proposed 21cm IM survey with
the SKAO and on resolved GW events from the merger of BBHs as detected by the
Einstein Telescope (ET) [653]. A cross-correlation signal is expected, as both
HI and GWs trace the cosmic density field, and once again they do so in
different ways depending on astrophysical and cosmological models. In
particular, for constraining the PBH abundance, results were obtained for the
predicted IM from the SKAO correlated with GWs from ET.

Figure 37: SNR for different values of the GW observation time
$T_{\mathrm{obs}}^{\rm GW}$ (from 1 yr to 15 yr), $f_{\mathrm{sky}}$ (from 0.1
to 1.0) and mixed PBH scenarios $\Gamma_{\mathrm{pbh}}^{\mathrm{ALT}}$ (from
0.0 to 1.0), assuming the astrophysical model as fiducial. The fiducial model
assumed is astrophysical black hole binaries. Results on the right-side plots
are normalized to white at SNR=1.

###  6.5 Summary

Gravitational-wave observations of resolved black hole mergers and of the
stochastic background can be cross-correlated with the LSS to constrain the
existence and abundance of PBHs. This technique may provide additional
information to test and constrain the different PBH scenarios, the PBH
abundance within a broad range of masses, as well as binary formation
channels. This will be of particular interest for the next generation of LSS
surveys, such as DESI, Euclid and SKA, and gravitational-wave detectors like
LISA, Einstein Telescope or Cosmic Explorer.

##  7 Current limits

The most stringent limits on $f_{\rm PBH}$ as a function of the PBH mass for a
monochromatic mass function are represented in Fig. 38. Zooms on evaporation-
based limits (between $10^{-19}M_{\odot}$ and $10^{-15}M_{\odot}$), on
microlensing limits (between $10^{-11}M_{\odot}$ and $1M_{\odot}$) and on the
multiple limits above $1M_{\odot}$ are shown in Fig. 39. These astrophysical
and cosmological probes of PBHs can be divided in six categories, each related
to the physical process at the origin of a PBH signature in observations:

  1. 1.

Observations impacted by black hole evaporation

  2. 2.

Microlensing observations

  3. 3.

Observational signatures of the black hole dynamics

  4. 4.

Observations impacted by the black hole accretion

  5. 5.

Indirect constraints from density fluctuations

  6. 6.

Signatures of PBHs in Gravitational-wave observations

In this Section, we review these current limits on the PBH abundance and
comment on possible caveats and model dependences. We also discuss the
possible observational evidences for the existence for PBHs, in relation with
these limits.

Figure 38: We show the most stringent claimed constraints in the mass range of
phenomenological interest. They come from the Hawking evaporation producing
extra-galactic gamma-ray (EG $\gamma$) [654], $e^{\pm}$ observations by
Voyager 1 (V $e^{\pm}$) [655], positron annihilations in the Galactic Center
(GC $e^{+}$) [656] and gamma-ray observations by INTEGRAL (INT) [657] (for
other constraints in the ultra-light mass range see also [158, 658, 659, 660,
661, 662]). We plot microlensing searches by Subaru HSC [113, 116], MACHO/EROS
(E) [663, 664], Ogle (O) [117] and Icarus (I) [665]. Other constraints come
from CMB distortions. In black dashed, we show the ones assuming disk
accretion (Planck D in Ref. [403] and Ref. [666], from left to right) while in
black solid the ones assuming spherical accretion (Planck S in Ref. [403] and
both photo- and collisional ionization in Ref. [103], from left to right).
Only Ref. [403] includes the effect of the secondary dark matter halo in
catalysing accretion. Additionally, constraints coming from X-rays (Xr) [667]
and X-Ray binaries (XrB)[668] are shown. Dynamical limits coming from the
disruption of wide binaries (WB) [669], survival of star clusters in Eridanus
II (EII) [109] and Segue I (SI) [670, 671] are also shown. LVC stands for the
constraint coming from LVK measurements [408, 65, 441, 81]. Constraints from
Lyman-$\alpha$ forest observations (L$\alpha$) come from Ref. [672]. We
neglect the role of accretion which has been shown to affect constraints on
masses larger than ${\cal O}(10)M_{\odot}$ [402, 80] in a redshift dependent
manner. See Ref. [68] for a comprehensive review on constraints on the PBH
abundance. Notice that there are no stringent bounds in the asteroid mass
range [673, 674] where LISA may constrain PBHs through the search of a second
order SGWB.

Figure 39: Zooms over some claimed limits on the PBH abundance $f_{\rm PBH}$
for a monochromatic distribution of mass $m_{\rm PBH}$, in the asteroid-mass
range where constraints are dominated by various probes of PBH Hawking
evaporation (top panel), in the planetary-mass and low stellar-mass range up
to $m_{\rm PBH}\sim 10M_{\odot}$ coming from microlensing surveys (middle
panel), and in the range from stellar-mass up to the supermassive PBHs, from a
combination of accretion, dynamical, GW and indirect constraints (bottom
panel). The legend indicates the origin of each represented limit. It is worth
noticing that all those limits are subject to important uncertainties and can
be highly model dependent, moving up and down with different model and
astrophysical assumptions. The possible limitations and sources of
uncertainties are discussed in the text.

###  7.1 PBH evaporation

We review here the PBH constraints relying on their evaporation. Let us note
that the PBH mass is expressed in grams through this Section, and not in solar
mass as for all the other probes, for a better consistency with the literature
and because these constraints concern PBHs much lighter than a solar mass.

####  7.1.1 Big-bang nucleosythesis (BBN)

PBHs with masses $10^{9}-10^{13}\mathrm{g}$ can affect the abundance of light
elements produced during the Big-Bang Nucleosynthesis through Hawking
radiation and the related emitted particles, allowing to set constraints on
the abundance of low-mass PBHs [68]. In particular, for PBHs with masses
$m_{\mathrm{PBH}}\approx 10^{9}-10^{10}\mathrm{g}$, Hawking radiated mesons
and antinucleons induce extra interconversion processes of protons to
neutrons, increasing the neutron-to-proton ratio at the time of freeze-out of
the weak interaction [675], and consequently leading to an increase in the
final ${}^{4}\mathrm{He}$ abundance [676]. Regarding PBHs with masses
$m_{\mathrm{PBH}}\approx 10^{10}-10^{12}\mathrm{g}$, long-lived high energy
hadrons produced out of PBH evaporation, such as pions, kaons and nucleons,
remain long enough in the ambient medium to trigger dissociation processes of
light elements produced during BBN [677], reducing the abundance of
${}^{4}\mathrm{He}$ and increasing the one of
$\mathrm{D}$,${}^{3}\mathrm{He}$,${}^{6}\mathrm{Li}$ and ${}^{7}\mathrm{Li}$.
Finally, for PBHs with $m_{\mathrm{PBH}}\approx 10^{12}-10^{13}\mathrm{g}$,
photons produced from the particle cascade process further dissociate
${}^{4}\mathrm{He}$, increasing the abundance of light synthesized elements
[678, 679]. However, it is important to stress that the BBN constraints carry
out some uncertainties regarding the baryon-to-photon ratio, the reaction and
the decay rates of the elements produced during the BBN processes [680, 681].

####  7.1.2 Gamma-rays and neutrinos from Hawking radiation

The lifetime of a PBH due to its evaporation in standard model particles can
be approximated as [158]

$\tau\approx 400\left(m_{\rm PBH}\over 10^{10}\,{\rm g}\right)^{3}{\rm s}.$
(7.1)

For a PBH with mass $5\times 10^{14}$ g, its lifetime is close to the age of
Universe. Thus, such low-mass PBHs can be probed by looking for bursts of
gamma rays. Such searches have been performed by the Water Cherenkov
Observatories, Imaging Atmospheric Cherenkov Telescopes (IACTs) and gamma-ray
space observatories. So far, no gamma-ray signal that could be due to PBH
evaporation has been detected. The current limits are summarized in Table 3.
These are expressed in terms of the PBH burst rate densities $\dot{B}_{\rm
PBH}$, describing the number of bursts from PBH evaporation in the local
Universe in a unit of volume per unit of time.

The observations of Galactic gamma-ray emission by INTEGRAL [657] also put a
bound on the mass above which PBHs could constitute the entirety of DM,
$m_{\text{PBH}}>1.2\times 10^{17}$g, under some relatively conservative
assumptions. New analysis of the diffuse soft gamma-ray emission towards the
inner Galaxy measured by INTEGRAL/SPI over 16 years [682], implementing a
spatial template fit of SPI data, an improved instrumental background model,
and including the spatial distribution of MeV photons emitted by the
evaporating PBHs into the fitting procedure, strengthens this lower bound to
$m_{\text{PBH}}\gtrsim 4\times 10^{17}$g.

The prospects for detection of gamma-ray signals from PBH evaporation for the
future Southern Wide Field of View Gamma-ray Observatory (SWGO) have been
estimated in [683]: it will be sensitive to local PBH burst rate densities
$\dot{B}_{\rm PBH}$ of the order of $\sim$ 50 pc-3yr-1, improving by more than
one order of magnitude the limits from current experiments.

In addition to gamma-rays, high energy extra-galactic neutrinos can also be
used to put constraints on the local PBH burst rate density $\dot{B}_{\rm
PBH}$ for both non-rotating ($a^{\star}=0$) and rotating ($a^{\star}=0.99$)
black holes. The 90% C.L. limits from 10-years $\nu_{\mu}$ IceCube data [684]
are presented in Table 3.

Experiment |  $\dot{B}{\rm PBH}$ Upper Limit [pc-3yr-1] | Burst duration [s] | $T_{\rm obs}$ | Reference
---|---|---|---|---
Milagro |  $3.1\times 10^{5}$ (99% CL) | 0.001 | 1673 days | [685]
Milagro |  $1.2\times 10^{4}$ (99% CL) | 0.01 | 1673 days | [685]
Milagro |  $5.4\times 10^{4}$ (99% CL) | 0.1 | 1673 days | [685]
Milagro |  $3.6\times 10^{4}$ (99% CL) | 1 | 1673 days | [685]
Milagro |  $3.8\times 10^{4}$ (99% CL) | 10 | 1673 days | [685]
Milagro |  $6.9\times 10^{4}$ (99% CL) | 100 | 1673 days | [685]
VERITAS |  $2.22\times 10^{4}$ (99% CL) | 30 | 747 hours | [686]
H.E.S.S. |  $4.9\times 10^{4}$ (95% CL) | 1 | 2600 hours | [687]
H.E.S.S. |  $1.4\times 10^{4}$ (95% CL) | 30 | 2600 hours | [687]
H.E.S.S. |  $5.6\times 10^{4}$ (99% CL) | 10 | 2860 hours | [688]
H.E.S.S. |  $2.0\times 10^{3}$ (95% CL) | 120 | 4816 hours | [689]
Fermi-LAT |  $7.2^{+8.1}_{-2.4}\times 10^{3}$ (99% CL) | $1.26\times 10^{8}$ |  $1.26\times 10^{8}$ s | [690]
HAWC |  $3.3^{+0.3}_{-0.1}\times 10^{3}$ (99% CL) | 0.2 | 959 days | [691]
HAWC |  $3.5^{+0.4}_{-0.2}\times 10^{3}$ (99% CL) | 1 | 959 days | [691]
HAWC |  $3.4^{+0.4}_{-0.1}\times 10^{3}$ (99% CL) | 10 | 959 days | [691]
IceCube |  $1.7\times 10^{7}$(90% CL, $a^{\star}$=0) | 10 | 10 years | [684]
IceCube |  $8.2\times 10^{6}$ (90% CL, $a^{\star}$=0) | $10^{2}$ | 10 years | [684]
IceCube |  $4.1\times 10^{6}$ (90% CL, $a^{\star}$=0) | $10^{3}$ | 10 years | [684]
IceCube |  $1.5\times 10^{7}$ (90% CL, $a^{\star}$=0.99) | 10 | 10 years | [684]
IceCube |  $6.2\times 10^{6}$ (90% CL, $a^{\star}$=0.99) | $10^{2}$ | 10 years | [684]
IceCube |  $3.3\times 10^{6}$ (90% CL, $a^{\star}$=0.99) | $10^{3}$ | 10 years | [684]
Table 3: Current limits on the local PBH burst rate density from different
experiments.

####  7.1.3 Positron annihilation in the Galactic Center

PBH limits can be obtained with observations of the 511 keV line from the
Galactic Center with INTEGRAL. Positrons produced by evaporating PBHs
subsequently propagate throughout the Galaxy. As they annihilate, they
contribute to the Galactic 511 keV line. Both monochromatic and log-normal
mass distributions have been considered, e.g. giving the corresponding limit
of $f_{\rm PBH}<1$ for $m_{\rm PBH}\lesssim 2\times 10^{17}$ g for a log-
normal mass distribution with $\sigma=0.5$ and $m_{\rm PBH}<2\times 10^{17}$ g
with $\sigma=0.1$ [656]. Estimates for different background assumptions [658]
show that for the low ($\times 0.1$) power law background model, $f_{\rm PBH}$
can reach 1 up to $m_{\rm PBH}\sim 5\times 10^{17}$g. Estimates for different
mass distributions and spins [661] give the mass limit $m_{\rm PBH}\sim
3-5\times 10^{18}$ g for the log-normal mass distribution with $\sigma=1$,
$a^{*}=0.5-1$ and $f_{\rm PBH}=1$. It has been recently shown [692] that
future MeV-scale gamma-ray observatories e-ASTROGAM or AMEGO would be able not
only to detect the Hawking radiation from the Inner Galaxy, but also to
precisely measure the abundance and mass distribution of PBHs responsible for
the Galactic 511 keV line.

####  7.1.4 $e^{\pm}$ observations by Voyager 1

The measurements of interstellar low-energy (sub-GeV) $e^{\pm}$ flux are used
to constrain the PBH fraction of DM in the Galaxy [655]. These electrons and
positrons would be due to PBH evaporation. The obtained limits are based on
local measurements, independent of the cosmological model. This flux must be
shielded for Earth-bounded detectors by the magnetic field of the Sun.
However, limits can be obtained from Voyager-1 measurements since it is now
beyond the heliopause.

The Hawking radiation flux of $e^{\pm}$ is computed within the fully general
diffusion-convection-reacceleration model of propagation with the parameters
adjusted to the AMS-02 data. For the log-normal mass distribution of PBHs with
a width $\sigma$ larger than 1 and a central value $\mu\lesssim 10^{17}$ g, a
PBH fraction larger than 1% of DM is excluded by the Voyager 1 data.

####  7.1.5 Ultra-faint dwarf galaxies

The ultra-faint dwarf galaxies (UFDGs), such as Leo T, also provide some
limits on light, evaporating PBHs, through the heating of the interstellar
medium gas [693]. The emitted energetic particles can heat and ionize the gas,
in particular the fast electrons can deposit a substantial fraction of their
energy and heat the interstellar gas through the Coulomb interaction. The
ultra-faint dwarf galaxy Leo T is of particular interest, not only because it
is highly dark matter dominated but also because it has a large amount of
neutral hydrogen and a relatively low metal abundance. In [693], limits are
obtained for non-spinning PBHs, with masses between $2\times 10^{15}$ g and
$10^{17}$ g as low as $f_{\rm PBH}\lesssim 10^{-3}$, i.e. more stringent than
the limits from galactic and extragalactic emission. The possible spin of PBHs
would affect the Hawking radiation, generally increasing the emission of
particles, in particular ones with large spins. In comparison, stronger limits
are obtained for spinning PBHs [662] than for non-spinning ones. In both
cases, $f_{\rm PBH}=1$ is claimed to be excluded for $m_{\rm
PBH}\sim(2-2.5)\times 10^{16}$g.

####  7.1.6 CMB limits on Dark radiation

Evaporating PBHs in the mass range $[3\times 10^{13},5\times 10^{16}]$ g
should have injected an exotic amount of electromagnetic energy in the
Universe, which has an impact on the CMB anisotropies. This effect has been
computed in [694], using the ExoCLASS code that includes electromagnetic
cascade calculations, initially developed in the context of WIMP annihilation.
For a monochromatic mass function, the obtained bounds exclude PBHs as the
main source of dark matter between $3\times 10^{13}$ and $2.5\times 10^{16}$g,
and are up to several orders of magnitude more stringent than the limits from
the galactic gamma-ray background between $3\times 10^{13}$ and $3\times
10^{14}$ g. A similar analysis was conducted in [679], leading to limits at
slightly lower masses, around $1.1\times 10^{13}$ g. See also [695] for the
connection with CMB spectral distortions induced by the energy injection from
evaporating PBHs. The non-standard energy injection problem has also been
reformulated to account for an extended PBH mass distribution in [660]. In
comparison with a monochromatic mass function, the marginalised and log-normal
ones have tighter exclusion bounds for heavier PBH masses, while at low PBH
masses these two distributions exhibit weaker bounds than the delta
distribution. In the future, experiments dedicated to the detection of the CMB
spectral distortions or of the 21cm signal from the dark ages could improve
these limits. Finally, it has been pointed out in [696] that the time-
dependence of the matter density, as the PBH density decreases as they lose
mass via Hawking radiation, can be formulated as an equivalent early dark
energy model with varying equation of state $w(z)$. For instance, a population
of ultra-light PBHs, decaying around neutrino decoupling, leads to a dark
matter-radiation coupling altering the expansion history of the Universe, and
alleviating the $H_{0}$ tension.

The rate of mass loss from particle emission by the PBHs through Hawking
radiation is then computed, and found to be negligible compared to the total
PBH mass. The electrons, positrons and photons are the particles emitted from
PBHs that interact most strongly with the primordial plasma, so actually the
matter of concern is the effect the emitted particles have on the early
Universe. After obtaining the formula for the total energy injected into the
primordial plasma by evaporating PBHs, the effects on CMB anisotropies are
then examined, where only the ionisation and excitation of hydrogen atoms are
considered (helium recombines earlier and has a smaller effect in the CMB).

####  7.1.7 Neutrinos

For PBH masses between $10^{15}$ and $10^{17}$ g, black hole evaporation via
Hawking radiation leads to sizeable fluxes of MeV neutrinos that could be
detected with the next generation of detectors like Hyper-Kamiokande [697],
JUNO [698], DUNE [699] and THEIA [700]. Current experiments like
SuperKamiokande already provide some limits on the PBH abundance [661, 701],
but they are not yet competitive with gamma-ray limits. Those future
experiments will have the ability to set complementary limits in the asteroid-
mass range. The expected sensitivity of these experiments (Hyper-Kamiokande,
JUNO, DUNE and THEIA) with various designs has been calculated in [702, 703,
701] for both monochromatic and log-normal PBH mass functions. Abundances as
low as $10^{-6}$ could be probed for PBH with zero spin and even lower if PBHs
have significant spin. DUNE and THEIA would also complement each other,
because they are respectively sensitive to neutrino and antineutrino fluxes.
Finally, neutrinos resulting from PBHs could also be observed in direct
detection experiments, through coherent elastic neutrino-nucleus scattering,
see [704].

####  7.1.8 21-cm signal

The 21cm line is an electromagnetic radiation spectral line created by an
atomic transition between the two hyperfine levels of the $1s$ ground state of
neutral hydrogen. This spectral line is of great interest and importance in
cosmology as it seems to be the only known way to probe the dark ages of the
Universe from the epoch of recombination to reionization. Observations of 21cm
are known to be very difficult to make, primarily because it is a faint signal
and is usually plagued by interferences from various sources. However, a
global 21cm signal was claimed to be detected by the Experiment to Detect the
Global Epoch of Reionization Signature (EDGES), which shows an absorption
feature with an amplitude of $T_{21}\sim 500\,{\rm mK}$ centered at redshift
$z\sim 17$, much stronger (by about a factor of 2) than expected in the
$\Lambda$CDM model [705]. Various explanations, such as interactions between
the dark matter particles and baryons, have been put forward to explain this
anomalous signal.

In general, during that epoch, any additional sources, e.g. the dark matter
annihilation or decay, will heat up the intergalactic medium (IGM) and
increase the kinetic temperature that eventually determines the strength of
the 21cm signal. Therefore, observations of the EDGES experiment can, in
principle, impose constraints on the properties of a dark matter scenario. It
has been discussed that, in the context of 21cm cosmology, PBHs can lead to
multiple distinct observable effects: (i) the heating of the IGM due to their
Hawking evaporation; (ii) the modification of the primordial power spectrum
(and hence the halo mass function) induced by Poisson noise due to PBHs being
point sources; (iii) a uniform heating and ionization of the IGM due to X-rays
produced during accretion and (iv) a local modification of the temperature and
density of the ambient medium surrounding isolated PBHs. Due to their Hawking
evaporation, PBHs can affect the evolution of the IGM, and therefore their
mass fraction can be constrained by the global 21cm signal. Using this, the
initial mass fraction of PBHs is constrained to be $\beta_{\rm PBH}\sim
2\times 10^{-30}$ for very light PBHs in the mass range $6\times 10^{13}\,{\rm
g}\lesssim m_{\rm PBH}\lesssim 3\times 10^{14}\,{\rm g}$ [706]. Considering
PBH evaporation, in [707] the following limits have been found : $f_{\rm
PBH}=10^{-6.84}(m_{\rm PBH}/10^{15}\,$g)3.75 in absence of X-ray heating and
$f_{\rm PBH}\leq 10^{-9.73}(m_{\rm PBH}/10^{15}\,$g)3.96 in presence of X-ray
heating. The mass and spin are fundamental properties of a black hole, and
they can substantially affect the BH evaporation rate. In [708], the lower
limits on masses of PBHs allowed to entirely constitute the DM have been found
to be 1.5$\times 10^{17}$ g, 1.9$\times 10^{17}$ g, 3.9$\times 10^{17}$ g and
6.7$\times 10^{17}$ g for PBH spins 0, 0.5, 0.9 and 0.9999, respectively.

####  7.1.9 Limits from DM particle production

If evaporating PBHs are at present times a source of the boosted light DM
particles, then the constraints on PBH mass and abundance can be obtained from
the XENON1T data [709]. Considering DM masses smaller than 1 MeV, for low DM-
nucleon cross-sections, the constraints on $f_{\rm PBH}$ are almost inversely
proportional to the cross-section. However, for larger cross-sections the
results might depend on the model of the DM propagation through Earth and
atmosphere. The constraints are valid for the mass range $0.5\lesssim m_{\rm
PBH}/(10^{15}g)\lesssim 8.0$: the kinetic energy of particles emitted by more
massive PBHs is too low, the less massive PBHs have evaporated nearly
completely. For the DM particles interacting with electrons similar analyses
are presented in [710, 711].

####  7.1.10 Discussion and limitations

It is important to notice that all the limits on the PBH abundance in the
asteroid mass range rely on the common assumption that PBHs evaporate through
Hawking radiation. If the process of black hole evaporation is well
established theoretically, its existence is not proven observationally.
Furthermore, it is still unclear whether quantum gravity effects would affect
or not this process (see for example [712, 713], where it is shown that the
existence of one mesoscopic extra-dimension leads to a longer life-time of the
PBHs and one can have an all-dark-matter interpretation in terms of PBHs in
the mass range $10^{14}\lesssim m_{\rm PBH}/g\lesssim 10^{21}$). The
efficiency of the evaporation through the different channels can also depend
on the underlying particle physics theory. It is therefore important to keep
in mind these possible limitations. Nevertheless, it is also worth noticing
that such very light PBHs, if they exist, are the ideal target to probe and
test the Hawking radiation hypothesis for the first time.

The existence and importance of the Hawking radiation are not the only source
of uncertainty on the abundance limits from PBH evaporation. The different
source of uncertainties have been recently reviewed and critically analyzed by
J. Auffinger in [714], especially coming from the instrument characteristics,
the prediction of the (extra)galactic photon flux, the statistical method of
signal-to-data comparison and the computation of the Hawking radiation rate.
The constraints on PBHs are found to vary by several orders of magnitude,
depending on the hypothesis. Even with an ”ideal” experiment, PBHs can only be
probed through Hawking radiation at masses below $10^{20}$ g, which is an
intrinsic limitation of the phenomena.

In order to obtain those results, study the model dependence of the limits
related to Hawking radiation and include all these uncertainties in the data
analysis, the BlackHawk [715] and Isatis [714] codes have been developed,
allowing to compute the evaporation primary and secondary spectra of stable or
long-lived particles, for any black hole distribution. We envisage to
interface the PBH numerical tool developed in the context of the present
review paper with these key tools.

###  7.2 Microlensing searches

One of the most stringent constraints on PBH as constituents of the dark
matter in the Universe comes from the microlensing amplification of light from
a distant star as a Massive Compact Halo Object (MACHO) moves across its line
of sight. Microlensing events from lenses of less than one solar mass towards
the Large Magellanic Cloud (LMC) where reported in the 1990s by several
groups, but in the subsequent decades only upper bounds were published, until
SUBARU and later OGLE reported events towards the galactic bulge and nearby
Andromeda. The constraints on the halo abundance of those objects depend very
much on the mass, velocity and spatial distribution in the halo. Most of these
constraints are reported for a standard Navarro-Frenk-White (NFW) halo with
virial velocities and a spatially uniform distribution of monochromatic (same
mass) MACHOs, which put stringent constraints in the range of masses from
$10^{-9}$ to $30\,M_{\odot}$. Above 10 $M_{\odot}$, it is essentially
impossible to derive bounds on MACHOs due to the fact that the amplification
light curve lasts more than a decade, while no microlensing survey has
monitored continuously the light from stars for so long. Below
$10^{-9}\,M_{\odot}$, the wavelength of visible light is comparable to the
Schwarzschild radius of the compact object, and we enter the regime of wave
optics, where diffraction makes the constraints essentially disappear. One
could search for PBHs with $m_{\rm PBH}<10^{-9}\,M_{\odot}$ with light of
shorter wavelength, like X-rays or gamma-rays, but sources that emit in those
bands are typically not as abundant and stable as stars.

In the following Subsections, we give a brief summary of the different
microlensing constraints coming from different surveys.

####  7.2.1 Microlensing from stars in the Magellanic clouds

Microlensing observations of stars in the Large and Small Magellanic Clouds
probe the fraction of the Galactic halo in MACHOs in a certain mass range
[716]. The optical depth of the halo towards LMC and SMC is related to $f_{\rm
PBH}$ by $\tau^{\rm SMC}_{\rm L}=1.4\tau^{\rm LMC}_{\rm L}=6.6\times
10^{-7}f_{\rm PBH}$ for the standard halo model [663]. The MACHO project
detected lenses with mass around $0.5M_{\odot}$ but concluded that their halo
contribution could be at most $10\%$ [717], while the EROS project excluded
$6\times 10^{-8}M_{\odot}<m_{\rm PBH}<15M_{\odot}$ from dominating the halo
[664, 39]. Since then further limits in the range $0.1M_{\odot}<m_{\rm
PBH}<20M_{\odot}$ have come from the OGLE experiment [41, 42].

####  7.2.2 Microlensing from stars in the galactic bulge

Recently Niikura et al. [117] have used data from a 5-year OGLE survey of the
Galactic bulge to place much stronger limits in the range
$10^{-6}M_{\odot}<m_{\rm PBH}<10^{-4}M_{\odot}$, although they also claim some
positive detections. The galactic bulge is a dense region of stars towards the
galactic center where microlensing events are likely to occur. In fact, the
OGLE collaboration has detected numerous microlensing events towards the bulge
[718], searching specifically for dark objects that amplify the light of stars
behind them. Moreover, these stars are sufficiently close to Earth that one
can use paralax measurements by GAIA to break degeneracies between mass and
distance which plague magnification microlensing events. This use of the so-
called astrometric microlensing has allowed Ref. [118] to conclusively detect
black holes in the mass gap, in the range $2-5~{}M_{\odot}$. Niikura et al.
[117] also report microlensing events towards the galactic bulge in a lower
mass range of Earth-mass primordial black holes. These are tantalizing hints
of a multimodal population of primordial black holes in the halo of our
galaxy, where non of the mass ranges are sufficient to comply 100% of all the
dark matter halo.

####  7.2.3 Microlensing from stars in Andromeda

The Andromeda galaxy (M31) is too far away to resolve individual stars with a
typical dedicated (small, 2 meter) telescope, as most microlensing surveys do.
Therefore they use a different technique called pixel lensing, which
corresponds to single pixels brightening up over the length of the survey with
a characteristic Paczinsky amplification curve due to a passing compact
object. This technique was used by the POINT-AGAPE collaboration [719], who
reported six microlensing events, which they argued could be interpreted as
self-lensing of stars and this allowed them to conclude that 20% of the Milky
Way halo in the direction of M31 could be in the form of $0.5-1.0~{}M_{\odot}$
primordial black holes.

More recently, it was reported by [113] the observation, with the Hyper
Supreme Cam (HSC) on Subaru Telescope, of a microlensing event towards
Andromeda by a compact body with mass in the range
$10^{-10}-10^{-5}~{}M_{\odot}$. Give the detailed statistics of the
observation, they claimed this could provide evidence for a non-zero
component, at the level of few percent, for the dark matter halo to be in the
form of primordial black holes.

####  7.2.4 Microlensing from quasars

Hawkins has claimed for many years that quasar microlensing data suggest the
existence of PBH dark matter [119]. He originally argued for Jupiter-mass PBHs
but later increased the mass estimate to $0.4M_{\odot}$ [720]. More recently,
the detection of 24 microlensed quasars [117] suggests that up to 25% of
galactic halos could be in PBHs with mass between $0.05$ and $0.45M_{\odot}$.
These events could also be explained by intervening stars, but in several
cases the stellar region of the lensing galaxy is not aligned with the quasar,
which suggests a population of subsolar halo objects with $f_{\rm PBH}\gtrsim
0.01$. Indeed, Hawkins has argued that the most plausible microlensers are
PBHs, either in galactic halos or distributed along the lines of sight to the
quasars [120].

####  7.2.5 Microlensing of supernovae

One can also constrain the abundance of PBHs with stellar masses using
supernovae surveys. A certain number of supernova light curves are expected to
be lensed by PBHs, exhibiting a magnification of order $\mu\sim 0.1$ if PBHs
constitute all the dark matter [721]. Nevertheless, it has been debated
whether the current JLA and Union data already exclude $f_{\rm PBH}=1$,
especially in the case of broad PBH mass distributions [722]. The main
discrepancy has to do with the assumed statistics and the treatment of
outliers and other systematics, like the probability of amplification due to
discrete sources along the line of sight [723]. Interestingly, the SNe limits
are of the same order from $10M_{\odot}$ up to $10^{4}M_{\odot}$.

####  7.2.6 Femtolensing

Additional limits on the PBH abundance have been established based on the
femtolensing of gamma-ray bursts. Compared to microlensing, the terminology of
femtolensing is used to refer to the very small angular distance between
lensed images and applies to objects of very small masses. The effect is
probed by searching for interferometry patterns in the energy spectrum of
lensed objects [724]. One must nevertheless take into account that the
wavelength of the electromagnetic radiation can become comparable to the
Schwarzschild radius of the PBH, which implies a lower bound on the mass range
probed with this technique. In [725], a search for femtolensing events has
been performed on the FERMI observations of gamma-ray bursts for which the
redshifts were known. The absence of femtolensing events has been used to set
constraints on the dark matter fraction in PBHs of mass between $5\times
10^{17}$ g and $10^{20}$ g, with a limit down to $f_{\rm PBH}\lesssim 0.04$.

####  7.2.7 Discussion and limitations

The robustness of microlensing limits is of very high importance and still
controversial because this is the only observational technique to probe the
abundance of PBHs smaller than a solar mass, down to $10^{-10}M_{\odot}$.
Another debated question is whether microlensing surveys of quasars or towards
the galactic center only provide a limit on PBHs due to the lack of
observations or also hint at their existence due to a series of observations.

The contribution of PBHs to the dark matter is also an interesting question.
Deriving limits from microlensing surveys of stars in the Magellanic clouds
depends on numerous astrophysical assumptions (velocity dispersion, galactic
dark matter profile, etc…) and varying limits can be obtained with different
hypothesis [110, 107, 104, 120, 108]. Wide mass distributions can also
significantly alter the limits on their abundance, but without totally
removing them. The clustering of PBHs was the main argument invoked to totally
suppress the microlensing limits [106, 70], even though it has been shown that
this is not the case assuming initial Poisson distributed PBHs [500, 501].
Furthermore, in the case of a large initial spatial clustering distribution
beyond Poisson, it has been also recently shown that the combination of
constraints from microlensing and Lyman-$\alpha$ forest completely rules out
the parameter space for stellar-mass PBHs to be a dominant constituent of the
dark matter [503].

###  7.3 Dynamical limits

####  7.3.1 PBH capture by neutron stars or white dwarfs

The asteroid-mass range, between $10^{-16}M_{\odot}$ and $10^{-10}M_{\odot}$
($10^{17}-10^{23}$g), is the only one for which there is a general consensus
that PBHs can contribute up to the entirety of the dark matter. Nevertheless,
if this mass window remains open nowadays, this has not been always the case.
Some constraints had been proposed, coming from observations of neutron stars
and white dwarfs (WD) in globular clusters [115, 114, 726, 727] that can
capture asteroid-mass PBHs falling at the center of the star and rapidly
swallowing its material until it becomes a black hole. Such constraints
therefore arise from the combination of a dynamical and an accretion process.
But those limits have been relieved due to large uncertainties. It was
initially claimed that $f_{\rm PBH}$ cannot exceed between $10^{-2}$ and
$0.2$, depending on the DM density in globular clusters. However, this
approach is limited by our knowledge of the DM constitution of globular
clusters. Using updated prescriptions, the limits from [114, 726] have been
then relieved in [674], re-opening the asteroid-mass range to full PBH-DM
scenarios.

In [728], it was proposed that the transit of PBHs through a white dwarf
causes localized heating around the BH trajectory due to dynamical friction,
eventually triggering a runaway thermonuclear fusion and causing the WD to
explode as a supernova, even if the WD mass is below the Chandrasekhar limit.
The constraints on PBHs were derived by using two classes of observations. One
class comes from direct observations of WDs with known masses, whose existence
places bounds on PBHs that are abundant enough that they would have transited
the WD with high probability. The second class arises from the measured rate
of type Ia supernovae, that constrain PBHs with a low abundance and thus have
a low probability of transit through a WD. The shape of the observed
distribution of WD masses (with masses up to $1.25M_{\odot}$) excludes PBHs
with masses $\sim 10^{19}-10^{20}$g from being a dominant constituent of the
local DM density, whereas the type Ia supernova rate disfavors BHs with masses
between $10^{20}-10^{22}$g. However, the latter is not a robust exclusion due
to uncertainties in the methods used to determine the WD population with
masses larger than $\sim 0.85M_{\odot}$ (necessary condition for a WD to
explode due to a BH), and this mass could possibly change if the current
simulation data used to derive it is refined. Measurements of WD binaries in
GW observatories could be used in the future to further strengthen the BH
bounds.

More recently, limits on PBH in the interesting asteroid-mass range between
$10^{18}$ g and $10^{21}$ g based on the capture of PBHs by Sun-like stars in
ultra-faint dwarf galaxies leading to its destruction, have been calculated in
[729]. If one requires that no more than a fraction of stars $\xi=0.5$ is
destroyed, one obtains a limit on the PBH abundance of about $f_{\rm
PBH}\lesssim 0.3$ in the above-mentioned mass range. But the constraints
strongly depend on the possible value of $\xi$ that is still fairly unknown.
In [730], it was suggested that the recent discovery of a population of faint
supernovae (Ca-rich gap transients) may come from the explosion of white
dwarfs, possibly due to PBH captures. Their unusual spatial distribution,
predominantly at large distances from their presumed host galaxies, supports
this hypothesis and the authors have shown that their spatial distribution
matches well the distribution of dwarf spheroidal galaxies.

####  7.3.2 Ultra-faint dwarf galaxies

If PBHs above $10M_{\odot}$ constitute an important fraction of the dark
matter, their regular encounters in highly dark matter dominated ultra-faint
dwarf galaxies (UFDGs), such as in Segues I or in Eridianus II, should have
induced their dynamical heating, allowing them to reach a half-light radii
larger than 10 parsecs. This argument also applies to star clusters in UFDGs,
like for Eridanus II. Relatively stringent limits from UFDGs have been first
established by Brandt in [109], for monochromatic mass functions. These have
been extended to log-normal PBH mass functions, including the effect of a
central intermediate-mass BH, by Green in [110], or by Li et al (DES
collaboration) in [731]. All these works basically exclude $f_{\rm PBH}=1$ for
$m_{\rm PBH}>10M_{\odot}$. These limits are relatively solid because they rely
on well-known Newtonian gravitational dynamics. Nevertheless, one should point
out that these limits can be shifted if UFDGs host a central intermediate-mass
black hole.

Another way to constrain PBHs from UFDGs is to consider the radial luminosity
profile, given that stars must also be heated up by PBHs. In this case, Segue
1 provides the most stringent limit, excluding $f_{\rm PBH}=1$ for $m_{\rm
PBH}\sim 2M_{\odot}$ [670, 671]. However, these limits rely on astrophysical
assumptions about the stellar mass distribution and they could change for
wide-mass PBH models, depending on the existence and mass of a central
intermediate-mass black hole, etc. Finally, UFDGs also provide an argument in
favor of solar-mass PBHs constituting an important fraction of the dark
matter, given that there is no observation of such objects with a radius less
than about 20 parsecs, whereas they would still be above the magnitude limit
of the current instruments.

####  7.3.3 Dark Matter profile of dwarf galaxies (core-cusp problem)

The regular PBH-PBH interactions in a dense environment should prevent the
formation of cusps in the central region of dwarf galaxies, possibly
explaining the core-cusp problem. As first pointed out in [69], the
gravitational scattering cross-section for PBHs in the stellar-mass range are
of order $(0.1-1){\rm cm^{2}/g}$ and so PBHs are in essence a well motivated
self-interacting dark matter model. But contrary to particle dark matter, this
does not need exotic physics but relies on the only well-known interaction
that is not included in the standard model of particle physics, gravitation.
The core-cusp problem in the context of PBHs was then further explored in
[732]. They found that the mechanism works well for PBHs of mass between $20$
and $100M_{\odot}$, if they have $f_{\rm PBH}\sim 0.01$, but it would be less
efficient and would allow larger abundances for PBHs of the order of the solar
mass. Therefore, observations of low-mass dwarf galaxies provide at the same
time a new limit on the PBH abundance and a possible hint for their existence.
These limits imposed on the PBH abundance have been represented in Fig. 38.

####  7.3.4 Wide halo binaries

Wide stellar binaries can be destructed by the encounters with PBHs, therefore
the existence of the undisrupted ones in the galactic halo also provides
constraints on the PBH abundance [733].

Using the analysis of [734], the following constraint at 2$\sigma$ confidence
level has been found [68]: $f_{\rm PBH}<500M_{\odot}/m_{\rm PBH}$ flattens
above $M\sim 1000M_{\odot}$ with $f_{\rm PBH}\lesssim 0.5$. Using the results
of [735], where the low-mass limit was reduced from 500$M_{\odot}$ to
$21-78M_{\odot}$, $f_{\rm PBH}$ flattens at $m_{\rm PBH}\sim 100M_{\odot}$
with $f_{\rm PBH}\lesssim 0.1$ [68]. In [736], by using a sample of 4351 halo
wide binaries from the Gaia catalog, a break at $\sim$ 0.1 pc in the
separation distribution has been detected. Any break in the power law for the
separation distribution of wide halo binaries can be attributed to a
disruption of the binary as a result of encountering MACHOs (e.g. PBHs).
Therefore, this break implies $m_{\rm PBH}>10M_{\odot}$. These limits come
from the expected disruption of such wide binaries when they encounter PBHs.
They therefore strongly depend on the number density of PBHs, and they could
vary if, for instance, a majority of PBHs live in clusters.

####  7.3.5 Disruption of stellar streams

Stellar streams have been observed in the Milky Way halo and are likely formed
by the tidal stripping of progenitors. These would be perturbed by encounters
with dark matter subhalos, such as PBH clusters. Some methods have been
proposed to distinguish the origin of perturbers, in particular the PBH-DM
hypothesis form standard particle CDM [737]. Stellar streams like GD-1 could
probe the abundance of haloes down to $10^{3}M_{\odot}$, providing a test of
the existence of PBHs.

####  7.3.6 PBHs in the Solar System

The dynamics of objects in the Oort cloud suggests the possible existence of a
not yet observed planet, referred as Planet Nine. It could also be a dark
compact object several times the mass of the Earth, in orbit around the Sun in
the Oort cloud. It has been suggested that Planet Nine is a PBH captured by
the Sun [738] a long time ago. It was later proposed to search for the
existence of such a PBH in the Oort cloud by probing the flares from accretion
of small Oort cloud objects with LSST [739], which could result in improved
limits on PBH abundances, or by probing the associated Hawking radiation
[740].

Alternatively, it has been proposed to probe asteroid-mass PBHs in orbit
around the Earth by searching for their gravitational signatures in gravimeter
networks [741]. But the obtained limits on the PBH abundance with the current
gravimeter sensitivity are not yet relevant. Near-Earth PBHs could also leave
imprints in gravimeters or in GNSS orbit products. One way to detect a PBH
passing through the Earth would be to observe the induced seismic signatures
[742].

All this nevertheless remains highly speculative and it is not clear, for
instance, if a PBH captured by the Sun (or by the Earth) would remain on a
stable orbit, or if its orbit would likely pass near the Earth. However, it
must also be pointed out that discovering a PBH inside the Solar System would
have tremendous implications and could even, in principle, allow in-situ
investigations of a black hole [743].

####  7.3.7 Discussion and limitations

Again, there are a series of limitations and uncertainties that restrict the
range of applicability of the dynamical limits. In particular, let us mention
that if most PBHs are regrouped in clusters, the limits from wide halo
binaries would be strongly affected and possibly suppressed. These clusters
could even be the ultra-faint-dwarf galaxies, for which observations show a
minimum radius around a few tens of parsecs. Dynamical heating of UFDGs by
PBHs can therefore be seen both as a limit and as a clue for the existence of
such PBHs. For UFDGs, other sources of uncertainties come from the stellar
population model, the exact DM profile and the existence of a central
intermediate-mass black hole that typically makes limits less stringent. This
illustrates that the border between observational limits and evidence is
blurred and connected, and above all strongly model dependent. A similar
conclusion applies to the asteroid-mass region, for which uncertainties on the
DM content of globular clusters and the details of the tidal capture of small
PBHs by neutrons stars or white dwarfs still make this range viable for a full
PBH-DM scenario.

###  7.4 Accretion limits

####  7.4.1 Limits from accretion at present epoch

Primordial black holes which experience a phase of accretion at the present
epoch are subject to several constraints depending on their characteristic
environment.

First, a population of PBHs with masses around $\mathcal{O}(10-100)M_{\odot}$
in the Milky Way would be responsible for radio and X-ray emission if they
experience a phase of accretion from the interstellar gas. By comparing the
predicted emission with observation data in the VLA radio and Chandra X-ray
catalogs, one can set a constraint on the fraction of dark matter in the form
of PBHs to be of the order of $f_{\rm PBH}\lesssim(10^{-2}-10^{-3})$ in the
relevant mass range, depending on the assumed PBH mass function [112, 667].
This constraint is shown in Fig. 38 and denoted “Xr”.

Second, the abundance of PBHs in the stellar or intermediate mass ranges,
which interact with the interstellar medium emitting significant fluxes of
X-ray photons, may be constrained by using the observed number density of
X-ray binaries. In particular, by comparing the emitted radiation signals with
the electromagnetic data, one can constrain the PBH abundance to be smaller
than $f_{\rm PBH}\lesssim(10^{-2}-10^{-3})$ in the considered mass range
[668]. This constraint is labelled “XrB” in Fig. 38.

Finally, a cosmology-independent constraint can be set on the PBH abundance
based on the absence of gas heating in the interstellar medium due to PBH
interactions. In particular, photon emission from accreting PBHs, dynamical
friction, winds and jets emission from accretion disks may be constrained
using data from the Leo T dwarf galaxy, setting a bound on PBHs with masses in
the range $(10^{-2}-10^{6})M_{\odot}$ up to the percent level [744, 745].

####  7.4.2 Cosmic X-ray, infrared and radio backgrounds

Right after the first detection of gravitational-waves from a binary black
hole merger, it has been proposed that a PBH origin would be supported by the
spatial correlations between the source-subtracted infrared and X-ray
backgrounds [124], which have been measured at a $5\sigma$ level [125, 123]
and are unexplained by known sources. These correlations can be induced by DM
halos seeded at higher redshift than in the standard CDM scenario by the
Poisson fluctuations in the PBH distribution, if one has $f_{\rm PBH}m_{\rm
PBH}\sim\mathcal{O}(1)$. In turn, matter accretion onto PBHs leads to X-ray
emission while star formation in these halos leads to a spatially correlated
infrared radiation. The possibility of this scenario was confirmed and
explored in more details in [409, 126].

Recently, it has also been investigated if the observed excess of the cosmic
X-ray and radio cosmic background by Chandra and ARCADE 2, suggesting an
undiscovered population of emitters, could be explained by a high-redshift
population of PBHs [746] distributed in DM halos and in the intergalactic
medium. It was shown that the emission should dominantly come from small DM
halos, with masses of order or smaller than $10^{6}M_{\odot}$ at high
redshifts $z\gtrsim 6$. Interestingly, it was also found that even if PBHs
account for the excess in the radio background, it cannot at the same time
explain the X-ray cosmic background. This leads to limits on the PBH
abundance, in particular in the $1-100M_{\odot}$ mass range, which are
stronger than the microlensing limits or the bounds from ultra-faint dwarf
galaxies, with $f_{\rm PBH}\lesssim 9\times 10^{-3}M_{\odot}/m_{\rm PBH}$.

####  7.4.3 21-cm signal

There are several limits based on the EDGES detection of the absorption signal
of the global redshifted 21-cm line in the range $15<z<22$. The Poisson
distribution of PBHs can lead to isocurvature perturbations and consequently
to early structure formation. The DM annihilation is enhanced in early formed
dense halos and modifies the ionization and temperature evolution of baryons.
Using the redshift dependence of the EDGES signal [747], the following
constraint on PBH abundance has been found: $f_{\rm PBH}<10^{-3.4}$ for
$f_{\rm ann}\langle\sigma v\rangle/m_{\chi}=3\times 10^{-28}$ cm3/s/GeV, where
$m_{\chi}$ denotes the DM mass.

PBHs can accrete the surrounding baryonic matter as well as DM particles from
the surrounding halo. This results in the emission of high energy photons that
are injected into the IGM, leading to changes in its thermal and ionization
history via ionization, excitation and heating. For the PBHs accreting only
baryons [748], the following approximation for the constraint on the PBHs
fraction has been found: $f_{\rm PBH}\leq C(\beta)(0.15/f_{\rm
E})(\lambda/0.01)^{-(1+\beta)}(M_{\rm PBH}/10M_{\odot})^{-(1+\beta)}$, where
$C(\beta)=0.00015+0.00051\beta+0.0091\beta^{2}$ if $\beta\leq 0.37$ and
$C(\beta)=0.019\beta^{2.5}$ if $\beta>0.37$ (here $\lambda$ is the mass
accretion parameter, $T_{k}=8\,K$ the maximal gas temperatures allowed, and
$f_{\rm E}<1$ the effective energy absorption factor, assuming a monochromatic
mass function). Including also DM particles accretion, in [749] for a
differential 21cm brightness temperature $\delta T_{21}\lesssim-100\,$mK the
limit $f_{\rm PBH}\lesssim 2.6\times 10^{-5}$ has been obtained for $m_{\rm
PBH}=10M_{\odot}$ and $f_{\rm PBH}\lesssim 2.6\times 10^{-6}$ for $m_{\rm
PBH}=10^{4}M_{\odot}$. These limits are weakened by a factor of order 3 for
$\delta T_{21}\lesssim-50\,$mK.

In [750], it has been shown that future experiments like SKA and HERA could
potentially improve the existing CMB bounds by more than an order of
magnitude. The prospects for constraining PBHs with the future 21-cm forest
observations have been discussed in [751].

In [409] the contribution of accretion on PBHs to the observed low-frequency
cosmic radio background (CRB), and thus to the EDGES signal, has been
estimated. It has been found that for radio-quiet PBHs the contribution to the
CRB constitutes at 1.4 GHz a fraction of 0.1% of the observed synchrotron
radio background, mostly accumulated at $z\gtrsim 20$. This increases the
depth of the 21cm absorption line by only about 30%. However, some fraction of
the radio-loud PBHs (e.g. 5% with 1000 times higher fluxes), as observed in
the AGN population, will easily provide the 5% excess high-redshift radio
background flux necessary to explain the EDGES observation.

####  7.4.4 Cosmic Microwave Background (CMB) anisotropies

CMB anisotropies are also impacted by accreting PBHs around the time of the
recombination and until the completion of the reionization, leading to
stringent limits on the abundance of PBHs heavier than around
$10-100M_{\odot}$. The main idea behind these constraints is the following: in
the early Universe, PBHs accrete primordial gas and then convert a fraction of
the accreted mass to radiation. This results in the injection of energy into
the primordial plasma, which affects its thermal and ionization histories,
leading to CMB distortions in its frequency spectrum, as well as signatures in
the CMB temperature and polarization angular power spectra.

When accretion comes into play, the first aspect to consider is its geometry,
i.e. spherical accretion or accretion disk formation. The former is obtained
when the characteristic angular momentum of the accreted gas at the Bondi
radius (distance from the center at which the escape velocity equals the sound
speed) is smaller than the angular momentum at the ISCO (Innermost Stable
Circular Orbit) radius. The second aspect that must be taken into account is
the local feedback. Indeed, radiation emanating from the accreting PBH may
heat and ionize the accreting gas, influencing the radiative output. Moreover,
as discussed in [103], one can assume that $1)$ in the outermost region of the
accretion flow, the ionization fraction is approximately equal to the
background value. Radiative feedback is neglected, and Compton drag and
cooling are included in the calculation. When the temperature of the gas
reaches $\sim 10^{4}$ K, the gas gets collisionally ionized. This first
limiting case is refered to as collisional ionization; 2) in the innermost
region, the radiation from the PBH photoionizes the gas, which becomes fully
ionized and adiabatically compressed. This second limiting case is refered to
as photoionization. Furthermore, the accreting PBHs are moving with respect to
the ambient gas with some relative velocity, which does not allow for a
perfectly spherically-symmetric accretion. This relative velocity is comprised
of 1) a Gaussian linear contribution on large scales $v_{\rm L}$ (derived in
[103]); 2) a small-scale contribution $v_{\rm NL}$ due to non-linear
clustering of PBHs (not examined in [103]).

For PBHs more massive than $\sim 1M_{\odot}$, strong constraints were derived
by [44], while in [103], CMB limits on the PBH abundance were re-examined in
detail, based on previous works. In particular, in order to set the most
conservative constraints on the PBH abundance, it is useful to compute the
minimum PBH luminosity that is physically plausible and the accretion rate,
taking into account the Compton drag and Compton cooling by CMB photons.
Assuming a quasi-spherical accretion flow, a steady-state flow and a Newtonian
approximation, the bounds derived show that in the collisional ionization
limit, CMB anisotropy measurements by Planck exclude PBHs with masses
$M\gtrsim 10^{2}M_{\odot}$ from being the dominant DM component. In the
photoionization limit, this threshold is lowered to $\sim 10M_{\odot}$. These
constraints are weaker than those derived in [44].

As a side note, it is shown in [103] that local Compton heating by the
radiation produced by the accreting PBH can be safely neglected, i.e. one can
neglect local thermal feedback. It is also demonstrated that it is not self-
consistent to assume that ionizations proceed exclusively through
photoionizations or collisional ionization, i.e. the level of ionization
feedback is expected to lie between the two limiting cases. Finally, it was
also derived that global heating of the plasma due to accreting PBHs does not
leave any (observable) imprint on CMB spectral distortions.

The QCD phase transition, inducing a peak in the probability to form PBHs at
the solar-mass scale, is invoked to be a good-motivated scenario where PBHs
could contribute to the BBH mergers detected by LVK (see for example [70,
73]). Nevertheless, the viability of this scenario has been revisited by [502]
in view of current CMB anisotropies constraints related to the accretion onto
PBHs, CMB spectral distortions, gravitational wave searches, and direct counts
of supermassive black holes at high redshift. For this purpose, the mass
function was computed by not only considering the QCD phase transition, but
also the following particle-antiparticle annihilation processes. Their
conclusion is that the scenario is not viable unless one introduces a mass
evolution for the PBH mass function and a cutoff in the power-spectrum close
to the QCD scale. Specifically, by defining $M_{\rm cut}$ as the mass scale
above which the PBH mass function is cut by hand, the mass function was found
for three different values of the cutoff mass, $M_{\rm
c}=10^{8},10^{4},10^{2}M_{\odot}$, each respectively corresponding to SMBH
counting, spherical accretion, and GW production. Assuming $M_{\rm
cut}=10^{8}M_{\odot}$, this leads to $f_{\rm PBH}<4\times 10^{-4}$. For
$M_{\rm cut}=10^{4}M_{\odot}$, we get $f_{\rm PBH}<3.9\times 10^{-4}$ (disk
accretion model, see below) or $f_{\rm PBH}<7.5\times 10^{-4}$ (spherical
accretion model). Further lowering $M_{\rm cut}$ below $10^{2}M_{\odot}$
relaxes the CMB anisotropy bounds to $f_{\rm PBH}\lesssim 10^{-3}$, indicating
that the PBHs from the QCD-inspired scenario have at most a tiny contribution
to the detected LVK events.

In [666], it is argued that the spherical accretion approximation is likely
not valid and that an accretion disk should form during the dark ages, between
recombination and reionization. More precisely, all plausible estimates
suggest that a disk forms soon after recombination, mainly due to the fact
that, already before recombination, stellar-mass PBHs are in a non-linear
regime (i.e. they are clustered in halos of bound objects, e.g. PBH binaries
or big clumps containing thousands of PBHs). The main criterion necessary for
disk formation considered in [666] is that the angular momentum of the
material at accretion distance is sufficient to keep the matter in Keplerian
rotation at a distance $\gg 3r_{\rm S}$, where $r_{\rm S}$ is the
Schwarzschild radius. To accumulate angular momentum, the accreted material
must have significant velocity or density differences. Based on the angular
momentum of the accreted gas, one can derive the condition required for a disk
to form [666]. For instance, the disk formation criterion is likely satisfied
if we consider an ideal, free-streaming homogeneous gas moving at a bulk
motion with velocity of order $v_{\rm L}$ without dispersion, or when PBHs
constitute an important fraction of the DM with $M\gtrsim M_{\odot}$. Another
key quantity is the radiative efficiency factor $\epsilon$, which is strongly
related to the geometry of the accretion flow and the accretion rate. It
usually takes the value $\epsilon\sim 0.1$ for moderate or low disk accretion
rate, that corresponds to the value inferred from BH observations. In the case
of spherical accretion, one gets smaller values of the order of $10^{-5}$.
Finally, to compute the CMB bound, it is crucial to evaluate the total energy
injection rate, by taking into consideration that not all radiation is equally
effective. To compute the impact on the CMB, one has to quantify what amount
of injected energy is actually deposited into the medium by using the energy
deposition functions, and the last step is to include the spectrum of the
radiation emitted through BH accretion. Let’s note that each accretion
scenario has a distinctive energy injection history, that could in principle
be distinguished in the CMB angular power spectrum [666].

The conservative constraints obtained in [666] (at 95% C.L. using data from
Planck high-l TT TE EE+lensing) are represented in Fig. 38. One notices that
the presence of disks improves the CMB constraints on PBH by at least two
orders of magnitude, so that PBHs with masses $M\gtrsim 2M_{\odot}$ (with
monochromatic mass distribution) are excluded from accounting for the totality
of the DM. The constraints derived for the monochromatic case were also
extended to a log-normal mass distribution, and become more restrictive for a
broader distribution [666].

In [403], they extended previous calculation [666] to account for the
accretion of dominant, non-PBH DM particles onto PBH. The increased
gravitational potential felt by the baryons enhances baryonic accretion and
the PBH luminosity constrained by the CMB. The accreting DM halos around each
PBH are treated via a toy model and numerical simulations, assuming that the
ionization of gas in the dark ages due to accretion onto PBHs is homogeneous.

The toy model considered in [403] is used to determine the upper limit to the
mass growth of the PBH via DM accretion. The most optimistic scenario for PBH
growth is for a cold DM with no dispersion. The time evolution of the radius
of a mass-shell around a PBH is found to be $M_{\rm
halo}\simeq(3000/1+z)M_{\rm PBH}$, which is expected to break down at very
late times. This conjecture is then verified via cosmological N-body
simulations, where in order to model the halo growth, it is required that the
DM halo is composed of many DM particles when it becomes comparable to the PBH
mass. An important result is that when the halo is compact, the accreting
baryons see a BH whose effective mass is the sum of the PBH and the DM halo
masses. If the halo is large, only a fraction of its mass contributes to the
accretion. This is the model used in [403] to compute the impact of PBH
accretion onto the CMB.

The CMB constraints of [403] are obtained by taking into account the effect of
accretion within DM halos. In the simulations previously mentioned, one gets
the necessary resolution for $100<z<1000$, encompassing the redshift range at
which the deposition of electromagnetic energy has the biggest impact on the
CMB ($300<z<600$) [752]. One important finding is that the inclusion of halos
increases the impact of the PBH on the CMB angular power spectra by up to 2
orders of magnitude. A series of runs were performed assuming spherical and
disk accretion, with the presence and absence of a DM halo around the PBH, and
assuming a monochromatic PBH mass function (for extended ones, bounds usually
become more stringent). First, supposing that PBHs are accreting at Eddington
luminosity (luminosity at which accretion is balanced by radiation pressure in
a spherical system), one gets the limit $f_{\rm PBH}<2.9\times 10^{-9}$.
Secondly, in absence of DM halos the constraints at 95% C.L. are stronger (by
a factor $\sim 4$) than bounds derived in [666]. Accounting for the DM halo
leads to improved bounds for $f_{\rm PBH}\lesssim 10^{-2}$ and $f_{\rm
PBH}\lesssim 0.2$ for the disk and spherical accretion cases, respectively.
Eventually, the bounds flatten when $M\gtrsim 10^{4}M_{\odot}$ as a
consequence of the accretion that reaches the Eddington limit. CMB constraints
exceed the ones coming from the non-observation of mergers by LVK, BBN, the
non-observation of the SGWB, etc., for masses $M\gtrsim(20-50)M_{\odot}$. The
CMB bounds remain dominant until $10^{3.5}M_{\odot}$, when they come close to
the BBN ones. Specifically, it is excluded that PBH constitute the totality of
the DM over the stellar mass range, and around $10M_{\odot}$ no more than
$f_{\rm PBH}\sim 0.1$ is allowed [403].

Finally, a possible relaxation of the limits from CMB anisotropies has
recently been suggested in [753] and [754]. The work of [753] relies on a more
realistic accretion model based on hydrodynamical simulations and conservative
assumptions for the emission efficiency. This leads to limits that are up to 2
orders of magnitude less stringent than previously estimated, between $10$ and
$10^{4}M_{\odot}$, which reopens the possibility that PBHs might explain at
the same time (at least a fraction of) the dark matter, some of the LVK binary
BH mergers, and the existence of supermassive BHs. The work of [754] takes a
step towards the development of a more realistic PBH accretion by accounting
the contribution of outflows, for various accretion geometries, ionization
models and mass distributions in absence and in presence of mechanical
feedback and non-thermal emissions due to the outflows. In general, it seems
that PBH accretion is rather complex physical process that is hard to assess,
which induces large uncertainties in all the accretion-based limits.

####  7.4.5 Discussion and limitations

Accretion limits, in particular limits from CMB temperature anisotropies, are
claimed to set the most stringent limits on the abundance of PBHs between
about $10M_{\odot}$ and $10^{4}M_{\odot}$, i.e. in the range that is
particularly relevant for GW observations with ground-based and space-based
detectors. Their strengths nevertheless depend on the details of the accretion
(e.g. disk or spherical accretion). Given the possible complexity of the
accretion process, it is possible that those limits significantly change if
different assumptions are chosen, for instance for the velocity-dependence of
the accretion process. One should also notice that the Poisson-induced
clustering of PBHs were not taken into account in the calculation of these
limits. Other sources of uncertainties come in the computation of the injected
energy and the resulting changes in the evolution of the ionized fraction at
high redshfit. Limits from the X-ray and radio emission in the galactic center
have their own uncertainties, e.g. related to the dark matter distribution.
The limits that were presented were nevertheless considered as relatively
conservative.

An important implication that was recently discovered for the limits from CMB
temperature anisotropies is that the existence of a particle dark matter
component in addition to PBHs reinforce those limits on the abundance of PBHs,
in such a way that they may become incompatible with an explanation of GW
observations.

###  7.5 Indirect constraints from density fluctuations

####  7.5.1 CMB distortions

Enhanced primordial density perturbations producing PBHs will also generate
CMB distortions by dumping energy into the primordial plasma in the wave-
number range $1\lesssim k$/Mpc-1 $\lesssim 10^{4.5}$. The amount of distortion
can be roughly estimated as
$<\mu>\thinspace\thinspace\sim\,O(1)\,P_{\zeta}\,\Delta N$, where $\Delta N$
is the e-fold number duration of the enhanced perturbations, which is
equivalent to $\ln\left[\frac{k_{\rm max}}{k_{\rm min}}\right]$. Assuming
$\Delta N\sim O(1)$, one can probe the distortion level from the power
spectrum, and current bounds are $<\mu>\thinspace\thinspace\lesssim 10^{-4}$
[404, 406, 537, 217, 541, 755, 128].

The PBH production is extremely sensitive to the tail of the statistical
properties, i.e. probability distribution, of the primordial perturbations as
discussed in Sec. 3. Hence, for highly efficient PBHs producing non-Gaussian
perturbations, the distortion signal can be smaller and evade the current
bounds. However, future experiments will reach remarkable sensitivities to
probe both Gaussian and non-Gaussian primordial signals [406, 128].

If scalar induced stochastic GW background is combined with CMB distortions,
the probed PBH seed range will be $(0.1-10^{13})M_{\odot}$ [407] (equivalently
the probed wavenumber range will be $1\lesssim k$/Mpc-1 $\lesssim 10^{7}$).
These probes will also reach conclusive results about the intriguing
possibility that PBHs could be the seeds of SMBHs, and with the detection of
over a billion of solar mass BHs at redshift higher than 7, this has become an
even more pressing question. Since CMB distortions and scalar induced GWs are
global cosmic signals associated with the formation/production of PBHs, the
conclusions will be robust to modifications in: i) astrophysical evolution,
such as merger and accretion history; ii) statistical properties of the
primordial perturbations (Gaussian or non-Gaussian); iii) clustering effects.
The combined probes are shown in Figure 40, where non-detection of both
signals will robustly constrain the PBH abundance to the negligible amount of
$f_{\rm PBH}<10^{-10}$ [407].

Light PBHs, in the mass range, $10^{-15}-1M_{\odot}$, can experience
superradiance even for tiny spin values, $10^{-3}-10^{-2}$ [756], and they can
probe 17 decades of bosons (scalar, vector and spin-2) in the motivated mass
range, $10^{-12}-10^{5}\,{\rm eV}$. Bosons could be beyond Standard model
particles such as axions and dark photons, also it could be Standard Model
photon with effective mass due to plasma interactions. If spinning light PBHs
are observed, this implies non-existence of superradiance for the given BHs
and boson mass range, therefore this observation can imply either there are no
such particles or their self/external interactions prevent them from
superradiance. This results in bounds on the axion decay constant (inversely
related to the strength of self-interaction) and its energy density [756].
Furthermore, PBHs can also produce photons with an effective mass in the
plasma, which results into an energy injection in the early universe and can
lead to CMB distortion, giving rise to mass dependent bounds on PBHs [757].

Figure 40: Probing power of future PTA (SKA) and CMB distortion (PIXIE-like)
experiments on primordial fluctuations for about 7 decades in wavenumbers and
13 decades in masses. Results are shown for inflationary perturbations that
obey distinct probability distributions: Gaussian (red horizontal dashed
line), chi-sqr ($\chi^{2}$, black horizontal dashed line), cubic-Gaussian
($G^{3}$, blue horizontal dashed line), detectable if $f_{\rm PBH}>10^{-10}$.
Taken from [407].

####  7.5.2 Lyman-$\alpha$ forest

The constraints on PBHs can be obtained from the high-resolution high-redshift
Lyman-$\alpha$ forest data since the Poisson fluctuations in the PBH number
density induce a small-scale power enhancement departing from the standard CDM
prediction. This enhancement can be interpreted as an isocurvature
perturbation with the scale-invariant spectrum and isocurvature-to-adiabatic
amplitude ratio for the monochromatic distribution: $f_{\rm
iso}=\sqrt{\frac{k_{\star}^{3}m_{\rm PBH}f_{\rm PBH}}{2\pi^{2}\Omega_{\rm
CDM}\rho_{\rm cr}}\frac{1}{A_{\rm s}}}$ (here $k_{\star}=0.05$/Mpc is the
pivot scale) [672]. Using MIKE and HIRES/KECK samples of quasar spectra and a
grid of hydrodynamic simulations exploring different values of astrophysical
parameters, in [672] the marginalized upper limit $f_{\rm PBH}m_{\rm PBH}\sim
60M_{\odot}$ at 2$\sigma$ on the PBH mass have been found for a monochromatic
distribution. This has been obtained with a Gaussian prior on the reionization
redshift, preventing its posterior distribution to peak on very high values
that are disfavoured by the most recent estimates from the CMB and IGM
observations $f_{\rm PBH}m_{\rm PBH}\sim 60M_{\odot}$ at 2$\sigma$ with a
conservative flat prior.

####  7.5.3 El Gordo cluster and enhanced Halo Mass Functions

One of the most fascinating connections between PBHs and large scale structure
(LSS) arises when one considers the effect of quantum diffusion in single
field inflation with quasi-inflection points [143] as a natural way to
generate large exponential tails in the density contrast PDF a fact that
enhances the probability of collapse of small-scale fluctuations upon re-entry
leading in this way to PBH formation. These highly non-Gaussian tails in the
PDF can also be responsible at larger (galactic and cluster) scales for the
enhancement of the halo mass function at high redshift [758], possibly
explaining the increased abundance of massive galaxies seen by the James Webb
Space Telescope at redshifs $z\sim 9-20$ [759], as well as significantly more
massive clusters, like El Gordo [760] at $z\sim 1$, than expected by the
standard model $\Lambda$CDM. It remains to be shown whether such large
deviation from the $\Lambda$CDM predictions for the halo mass function are
compatible with LSS constraints. In the near future we may have deep imaging
up to redshift $z\sim 20$ with both infrared and radio surveys like SKA, to
test the enhanced exponential tails that could have given rise to PBHs in the
early universe.

####  7.5.4 High-z galaxies from JWST

The James Webb Space Telescope (JWST) imaging program via the Cosmic Evolution
Early Release Science (CEERS) survey recently reported a population of massive
galaxy candidates at redshift $z\gtrsim 8$ with stellar masses of the order of
$10^{9}M_{\odot}$ [761, 762, 763, 764, 765]. A note of caution here is in
order, as spectroscopic follow-up will be necessary to confirm current
observation only based on photometry. However, if confirmed, such early
formation of massive galaxies reported by the JWST is hardly reconcilable with
the standard $\Lambda$CDM expectations. Recently, Ref. [766] derived the
cumulative stellar mass density at $z=8$ and 10 for $M_{\star}\gtrsim 10^{10}\
M_{\odot}$ based on 14 galaxy candidates with masses in the range $\sim
10^{9}\div 10^{11}\ M_{\odot}$ at $7<z<11$ observed by the CEERS program. They
found

$\displaystyle\rho_{*}(>10^{10}M_{\odot})\simeq 1.3^{+1.1}_{-0.6}\cdot
10^{6}M_{\odot}\,{\rm Mpc}^{-3},$ (7.2)

at $z\simeq 10$. These values are larger than the $\Lambda$CDM predictions,
even allowing maximum Star Formation Efficiency (SFE) $\epsilon=1$, or
invoking extreme value statistics [767].

Recently, Refs. [768, 769] investigated the hypothesis that early structure
formation was induced by the seeding effect of PBH induced Poisson
isocurvature perturbations.171717See also Refs. [770, 771, 772] for different
potential explanations.  Their result shows that such early formation of
massive galaxies is possible if the PBH abundance is such that
$m_{\text{PBH}}f_{\text{PBH}}\gtrsim 6\times 10^{6}\ (2\times
10^{5})M_{\odot}$ for SFE $\epsilon<0.1\ (1)$. Such abundance for massive PBHs
are ruled out by observations of the CMB $\mu$-distortion, although this
conclusion relies on the assumptions that primordial density fluctuations are
sufficiently Gaussian and that PBHs formed in the standard scenario of
primordial density fluctuations not growing during the radiation-dominated
era. However, strong isocurvature perturbations of PBHs with
$m_{\text{PBH}}f_{\text{PBH}}\gtrsim 170M_{\odot}$ are ruled out by high-z
Lyman-$\alpha$ forest data [672].

####  7.5.5 Ultra-compact mini haloes

It has been claimed in [399] that a small fraction of dark matter made of PBHs
is not compatible with the rest of the dark matter being made of weakly
interacting massive particles (WIMPs) because PBHs should seed the formation
of ultra-compact mini-halos with steep density profiles. Then, the WIMPs in
the dense innermost part of these halos would annihilate, which produces a
detectable gamma-ray signal. The absence of detection allows to set limits on
the PBH fraction in presence of WIMPs, of at most one billionth of the dark
matter for PBH masses larger than $10^{-6}M_{\odot}$. An interesting corollary
is that any firm detection of PBHs implies that the remaining dark matter
cannot be constituted by WIMPs. Similar analysis, e.g. including the effect of
primordial non-Gaussianity or the mass dependence of this limit in the case of
light PBHs, have been performed in [773, 774, 775]. The s-wave and p-wave
annihilation scenarios have been compared in [776] where a stringent limit has
been obtained for the thermal relic p-wave with a WIMP mass of 100 GeV (see
also [777]). It has been also shown in [778] that WIMPs around PBHs enhance
the heating and ionization in the intergalactic medium, due to WIMP
annihilations, leading to comparable or even tighter bounds of the possible DM
fraction made of PBHs, for instance $f_{\rm
PBH}\lesssim\mathcal{O}(10^{-10}-10^{-8})$ for a WIMP mass of order
$10-10^{3}$ GeV with a conventional annihilation cross section $\langle\sigma
v\rangle=3\times 10^{-26}{\rm cm}^{3}/{\rm s}$. Some of those limits were
recently reanalyzed in [779]. WIMP annihilation in ultra-compact mini-halos
around PBHs can also lead to a potentially detectable neutrino signal [780]
and the limits can be competitive with respect to gamma-ray searches if PBHs
are heavy. Another proposed way to probe DM dresses around PBHs is by
observing the gravitational strong lensing of sources like fast radio bursts
in the PBH mass range from $10M_{\odot}$ to $10^{3}M_{\odot}$ [781].

The specific case of SU(2)L triplet fermion “winos,” has been studied in
[782]. In this case as well, after the wino kinetic decoupling, DM particles
are captured by PBHs leading to dark mini-halos, constrained by the production
of narrow line gamma rays from wino annihilation in the Galactic Center.
Another very specific scenario of parsec-size ultra-compact mini-halos around
PBHs, in the context of the QCD axion dark matter, encountering neutron stars
and leading to transient ratio emissions due to resonant axion-photon
conversion in the neutron star magnetosphere, has been explored in [783].

Finally, it has recently been argued in [784] that the large density
fluctuations required for PBH formation should also lead to ultradense DM
halos (see also [785, 404, 786, 787, 788, 773, 789, 790]) that would actually
host a large fraction of the DM (in much greater abundance than PBHs),
enabling a variety of new observational tests. This conclusion is viable as
long as clustering of the dark matter is possible on the relevant scales (e.g.
[791]). The formation of such halos can already take place before matter-
radiation equality [792, 793, 794, 795, 784], which occurred at redshift
$z\simeq 3400$ [796] Due to their early formation, ultradense DM halos could
be compact enough to produce detectable microlensing signatures.

In Ref. [797] it is investigated whether the EROS, OGLE, and HSC surveys can
probe these halos by fully accounting for finite source size and extended lens
effects. Interestingly, current data by the EROS-2 [39], OGLE-IV [117] and
Subaru-HSC [113] surveys may already constrain the amplitudes of primordial
curvature perturbations in a new region of parameter space, even though this
conclusion is sensitive to details about the internal structures of these
ultradense halos. Adopting optimistic assumptions, HSC data would constrain a
power spectrum that features an enhancement at scales $k\sim 10^{7}/$Mpc, and
an amplitude as low as ${\cal P}_{\zeta}\simeq 10^{-4}$ may be accessible.
This range of scale is particularly interesting as it connects to primordial
black hole formation in a portion of the LVK mass range and the production of
scalar induced gravitational waves in the nanohertz frequency range reachable
by pulsar timing arrays. Further dedicated numerical simulations are required
to investigate in details the central density reached by such DM halos, to
which lensing constraints are particularly sensitive.

In summary, the study of dark matter halos, seeded or induced by the existence
of PBHs, brings support for the incompatibility of the coexistence of PBHs and
WIMPs and all-or-nothing scenarios.

####  7.5.6 Discussion and limitations

The primordial fluctuations at the origin of the formation of PBHs with masses
above $60M_{\odot}$ can have left detectable signatures in the form of
spectral distortions of the CMB black-body spectrum, which have been
constrained by FIRAS, in high-redshift Lyman-$\alpha$ forest observations,
which can typically probe smaller length scales than large scale structure
surveys or CMB anisotropies, and in a GW background sourced at second order in
perturbation theory that was discussed in a dedicated section. Those indirect
probes are therefore independent of the details of the PBH formation and their
subsequent complex evolution. They also depend on the statistical properties
of primordial fluctuations, e.g. they can be suppressed in the presence of
large non-Gaussianities. These indirect probes are therefore very interesting
to discriminate between the possible PBH formation models in a way that is
independent of the complex physical processes that could have impacted the PBH
properties. This is of particular interest to constrain models with broad PBH
distributions aiming at explaining LVK observations and SMBH seeds in a
unified way. The absence of any detection of CMB distortions would bring
support to non-Gaussian models or to scenarios where PBHs are not formed by
primordial density fluctuations.

###  7.6 Gravitational waves

####  7.6.1 LVC GW mergers

The LVK collaboration has been searching for compact binary coalescences. By
the end of the O3 run, 90 binary black hole (BBH) candidates have been found
[90], providing rich information on the BH population [798]. The latest
catalog GWTC-3 indicates that the BBH merger rate is between 17.3 ${\rm
Gpc}^{-3}{\rm yr}^{-1}$ and 45 ${\rm Gpc}^{-3}{\rm yr}^{-1}$ at a fiducial
redshift $(z=0.2)$.

Soon after the first discovery of the BBH event GW150914, it has been pointed
out that the observed BBH could be of primordial origin [61, 63, 62]. More
detection of BBH events provided information on the merger rate and the chirp
mass distribution, and Ref. [69] made the first MCMC analysis to infer the PBH
mass function. Statistical tests have been performed with the GWTC-1 catalog
to test the hypothesis that all or part of observed BBHs is primordial [71].
The use of the effective spin parameter has also been discussed [77, 79]. With
more events from GWTC-2, the combination of an astrophysical and a primordial
BH population have been tested, both using phenomenological models to describe
the ABH population [82, 75] as well as state-of-the-art astrophysical
population synthesis results [83]. These analyses confirm it is difficult to
explain all the features observed in the GWTC-3 catalog of events by means of
a single binary population, while showing shortcomings of both astrophysical
and primordial scenarios, but still allows for PBH to contribute to a fraction
of the observed BBHs.

Indication of particular events has been also discussed. GW190521 has
attracted attention because at least the primary component mass lies in the
so-called pair-instability mass gap, where there should not be direct
formation of stellar black holes, and the possibility of a primordial origin
has been discussed in Ref. [91]. The events GW190425 and GW190814 have both
companions with masses $[1.6-2.5]M_{\odot}$ and $[2.5-2.7]M_{\odot}$, which is
unexpected if they originated from a neutron star or a stellar black hole.
Ref. [799] discussed the possibility of PBH pairing with an astrophysical BH,
and concluded that it is unlikely that such binaries form and merge within a
Hubble time (see also [92, 800]). Refs. [73, 72, 70, 74] investigated the
possibility that both the primary and secondary BHs are primordial considering
the thermal history mass function, which indeed has a bump at a few
$M_{\odot}$ and around $30-50M_{\odot}$ and explains the events very well,
together with GW190521.

Recently, Ref. [87] performed the most comprehensive Bayesian population
analysis of LVKC data, that includes both BH and NS merger events, while
testing the hypothesis that a fraction of events may come from the primordial
scenario, where the latter is derived from first principles with the inclusion
of QCD effects on the mass distribution. The results of such an analysis can
currently only set an upper bound on the PBH fraction, showing the still
limited constraining power of available data, but interestingly peaked up some
aforementioned special events with high probability of being primordial (i.e.
GW190924_021846, GW190814, GW190412, and GW190521). One peculiar property of
the PBH channel, is that if it produced mergers in the NS mass range, due to
the QCD enhancement and the critical collapse, it would necessarily produce a
mass distribution which is wider the the one expected for astrophysical NS
mergers [87]. This would inevitably predicts subsolar, as well as lower mass
gap event mergers.

Due to the much improved sensitivity of future observation runs, we notice
that O4 and O5 may have more chances to detect many PBH events. However,
unless some of these events have smoking-gun features [85], it would be hard
to distinguish them from ordinary astrophysical channels. Therefore, an
interesting prediction, reported in Table 4, is the number of subsolar and
mass-gap events detectable in O4 and O5, assuming GW190814 is primordial. Such
numbers can be considered conservative upper bounds, as well as complementary
tests of the primordial origin of the aforementioned events. In particular, in
O5 there could be as many as $\approx 8$ subsolar events per year (but the
$90\%$ confidence interval is also compatible with zero events). Then O5
should detect one to a few dozen events per year in the lower mass-gap (and up
to $\approx 50$ upper mass-gap events), which might be more difficult to
interpret in astrophysical scenarios.

LVKC observing run | $N_{\text{PBH}}^{\text{\tiny det}}$ |  $N_{\text{PBH}}^{\text{\tiny det}}$(SS) |  $N_{\text{PBH}}^{\text{\tiny det}}$(LMG) |  $N_{\text{PBH}}^{\text{\tiny det}}$(UMG)
---|---|---|---|---
O1-O3 | $[0.8,22.4]$ | $[0.0,0.6]$ | $[0.1,2.3]$ | $[0.0,6.1]$
O4 | $[1.9,43.7]$ | $[0.0,1.3]$ | $[0.3,13.0]$ | $[0.0,13.1]$
O5 | $[10.3,216.7]$ | $[0.0,8.6]$ | $[0.8,25.2]$ | $[0.0,47.3]$
Table 4:  Table adapted from Ref. [87]. Assuming GW190814 had primordial
origin, this table reports the 90% C.I. for the number of detected PBH events
within GWTC-3, and predicted events (per year) with LVKC O4 and O5
sensitivity. We also indicate forecasted detections (within the 90% C.I.) that
would fall in the subsolar ($m_{1}<M_{\odot}$, SS), lower mass gap
($m_{1}\in[2.5,5]~{}{M_{\odot}}$ and/or $m_{2}\in[2.5,5]~{}{M_{\odot}}$, LMG)
and upper mass gap ($m_{1}>50M_{\odot}$, UMG) ranges.

Future observations at third generation experiments will be able to further
constrain the abundance of PBHs in the LVKC mass range by searching for high
redshift PBH mergers, where the possible astrophysical contamination from Pop
III mergers fades out [493] (see also Refs. [801, 478, 446, 802, 490, 85,
491]). This technique would allow to constrain the PBH abundance down to
values as small as $f_{\text{PBH}}\lesssim{\cal O}(10^{-5})$ in the standard
scenario, see Fig. 41 [493].181818Such constraint becomes much more stringent
if one assumes clustered PBH initial conditions, boosting the PBH merger rate
[446]. However, further work on understanding the merger rate of initially
clustered scenarios is needed.

Figure 41: Figure taken from Ref. [493]. Projected upper limit on
$f_{\text{PBH}}$ as a function of central PBH mass scale $M_{c}$ (assuming a
narrow log-normal mass distribution) obtained from null high redshift merger
detections with one year of observations at a CE-ET network. See the main text
for a discussion of the other (non-GW constraints). In yellow, we show
forecasts for the limits that will be set by microlensing searches with the
Rubin observatory [803, 804].

####  7.6.2 LVC sub-solar searches

In addition to the standard compact binary coalescence search, the LVC has
been updating the constraint on sub-solar mass binaries, where at least one
binary component has a mass between $[0.2-1]M_{\odot}$ [805, 806, 807] (see
also [808, 809, 151, 810]). So far, no firm detection of such GWs has been
reported, but four subsolar black hole candidates with false alarm rate
smaller than $2$ yr-1 and signal-to-noise ratio SNR $>8$ were reported in
[151] after re-analysing the data from the second observing run within an
extended mass range. If such a binary system were to be found with a clear
statistical significance, it could be strong support for the existence of
PBHs, since neither black holes nor neutron stars are expected to form below
$\sim 1M_{\odot}$ through conventional stellar evolution. The recent analysis
with the data from the first half of the LVC O3 run has provided a bound on
the merger rate of subsolar binaries in the range $[220-24200]{\rm
Gpc}^{-3}{\rm yr}^{-1}$. Assuming an isotropic distribution of equal-mass
primordial black holes, the bound on the merger rate translates to the bound
on the fraction of PBHs as $f_{\rm PBH}\equiv\Omega_{\rm PBH}/\Omega_{\rm
DM}\lesssim 6\%$. Although the LVC collaboration papers have placed stringent
constraints on the PBH abundance, they can be relaxed by two orders of
magnitude if we take into account merger rate suppression due to binary
disruption by early forming clusters, matter inhomogeneities and nearby PBHs
[151].

####  7.6.3 LVC SGWB

As mentioned in Sec. 5.6, overlapped GWs from binary PBHs form a SGWB. The
most updated bound on the SGWB by the LVC O3 run is $\Omega_{\rm GW}<5.8\times
10^{-9}$ at the 95% credible level for a flat (frequency-independent) SGWB
[811]. The first constraints on PBH abundance were argued in [604] using the
O1 data. The updated constraints by O2 and O3 data can be found in [444, 75].
In addition, parameter estimation has been attempted by [612] using the O3
data by considering both astrophysical and primordial components. The analysis
provided weak constraints on the PBH merger rate and time delay parameter,
while it indicated the difficulty of separating astrophysical and primordial
BBH contributions with the current sensitivity of the detectors (see also
[606]). We may be able to obtain meaningful constraints on the PBH merger rate
with the increased sensitivity of upcoming runs and future detectors.

It is worth mentioning that the upper bound on the SGWB can provide
constraints also on the ultralight PBHs in the mass range of
$[10^{-20}-10^{-19}]M_{\odot}$ through the scalar induced GWs. Those PBHs are
strongly constrained by the Hawking radiation, but with the upgraded
sensitivity, the ground-based interferometers could provide constraints that
are several orders of magnitude stronger than existing constraints [538]. The
first constraint was demonstrated by [812] with the O2 data. See [813] for the
updated constraint by the O3 data.

####  7.6.4 PTAs and NANOGrav

As discussed in Sec. 5.1, enhanced density perturbations inevitably source
stochastic GWs, dubbed as scalar induced GWs, or SIGW. The SIGW can be used to
probe the small scale power spectrum [378, 537, 217, 539, 541, 407, 602]. The
amplitude of this signal can be estimated as $\Omega_{\rm
GW}\sim\,{\rm(symmetry\,factors)}\cdot\Omega_{\rm rad}\cdot P_{\zeta}^{2}$,
where symmetry factors include possible contractions and different diagrams.
Hence, it is seen that current PTA experiments can probe $P_{\zeta}>10^{-2.5}$
at the nano-Hertz (nHz) scale.

In 2020, the NANOGrav collaboration first reported evidence in their 12.5 year
dataset [129] for a common spectrum of a stochastic nature, representing the
first hint of the existence of a nHz SGWB (also later on confirmed in refs.
[814, 815, 816]). However, at the time, no evidence was found for a
quadrupolar spatial correlation (i.e. Hellings-Down (HD) curve [142]),
necessary to interpret the signal as a GW background consistent with General
Relativity. Interestingly, the more recent PTA data released in 2023 by the
NANOGrav [133, 134], EPTA (in combination with InPTA) [135, 136, 137], PPTA
[138, 139, 140] and CPTA [141] collaborations, gathered significant evidence
for a HD angular correlation.

Focusing on the most stringent dataset, the latest NANOGrav $15\,{\rm yr}$,
the reported observations can be fitted by a smooth power law with a scaling
$\Omega_{\rm GW}\propto f^{(1.6,2.3)}$ at $1\sigma$. It is known that
supermassive black hole binaries produce a SGWB with a scaling $\Omega_{\rm
GW}\propto f^{2/3}$ [817] if binary evolution is GW driven, which is currently
disfavoured at $2\sigma$ by the NANOGrav15 data [818, 819]. Nevertheless,
environmental and statistical effects may play a relevant role, and lead to a
steeper scaling [820, 821, 822, 823, 824, 818, 819, 825]. At the present
stage, current data does not allow to rule-out (or rule-in) a cosmological
origin for the observed signal, which may also be due to first-order phase
transitions [826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838,
839], cosmic strings and domain walls [840, 841, 842, 843, 844, 845, 846, 847,
848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861], and
inflationary first order tensor perturbations [862] (see also for other
explanations [863, 864, 865, 866, 867]).

Standard PBH formation scenarios generating a population of subsolar PBHs are
related to the emission of an induced SGWB, which may explain current PTA
detections [130, 131, 546, 868, 132, 869, 870, 871, 872, 873, 511, 874, 875,
876, 877, 514, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889,
890, 885, 891, 892, 893, 894, 895]. In Fig. 42 we show the parameter space
that would be compatible with PTA signal and how this relates to PBH
overproduction constraints (see also Sec. 3 for a discussion on the
uncertainties affecting the computation of the PBH abundance).

Figure 42: The green and purple posterior indicate the 1, 2, 3$\sigma$
contours obtained fitting to NANOGrav15 and EPTA dataset, respectively. This
plot assumed a broken power law curvature spectrum with a fixed growth index
$n=4$ and varying decay power law index after the peak located at $k_{*}$. $A$
indicates the power spectral peak amplitude and we indicate on top the
corresponding peak SGWB frequency. We also show the corresponding PBH
abundance for different models: Gaussian curvature perturbations including
non-linear effects (black), quasi-inflection-point models with $\beta=3$
(red), curvaton models with $r_{\rm dec}=0.9$ (blue) and negative
$f_{\text{\tiny NL}}$ (cyan) (see Tab. 1 and Refs. therein for more details).
The colored bands cover values of PBH abundance in the range $f_{\rm
PBH}\in(1,10^{-3})$ from top to bottom. The dashed line indicates an average
PBH mass $\langle m\rangle=M_{\odot}$. We point the reader to Sec. 3 for a
discussion on the uncertainties affecting the computation of the PBH
abundance. Figure taken from [883].

Due to the extreme sensitivity of PBHs to the statistics of the primordial
perturbations, it is also possible to reduce or enhance the abundance of PBHs
efficiently via non-Gaussian perturbations. This could result in a
larger/smaller amplitude of curvature perturbations and of the associated
SIGWs. On the one hand, assuming a fixed abundance of PBHs, negative non-
Gaussianities would be able to enhance the associated amplitude of the GW
signal, bringing it closer to the range preferred by PTA observations. On the
other hand, positive non-Gaussianities could reduce $\Omega_{\text{\tiny GW}}$
allowing this scenarios to escape the sensitivity level of current PTA
experiments.

Future nHz frequency gravitational wave detectors will reach remarkable
sensitivities, provided a potential astrophysical background can be tamed.
Therefore, with future PTA-SKA experiments we will either detect or rule out
conclusively the cosmological signals accompanying the formation of PBHs. PTA-
SKA will robustly probe PBH mass formation in the range
$[0.1-10^{4}]M_{\odot}$. This regime is extremely important for both
astrophysical sources such as stellar mass BH binaries, intermediate BHs and
SMBH seeds. On Fig. 43, from [407], we show the PTA-SKA detection capabilities
of the stochastic GW background sourced by primordial scalar perturbations
which obey different statistical distributions: Gaussian, chi-sqr ($\chi^{2}$)
and cubic-Gaussian ($G^{3}$) for the cases i) $f_{\rm PBH}\sim 1$ (i.e PBHs
constituting all the DM) and ii) $f_{\rm PBH}=10^{-10}$.

One should point out here as well the possibility of future nHz PTA-SKA GW
detectors to observe ultralight PBHs in the mass range
$10^{7}-10^{9}\mathrm{g}$ given that the peak frequency associated to the
scalar induced GWs sourced by Poisson PBH energy density fluctuations, related
to the PBH spatial distribution, is well within the PTA-SKA frequency range
[317]. This potential detectability of such small mass PBHs is very important
given the fact that these ultralight PBHs evaporate before BBN so they do not
leave another direct observational imprint of their existence.

Figure 43: PTA-SKA detection capabilities of the stochastic GW background
sourced by scalar perturbations with distinct primordial statistical
distributions: Gaussian (red), chi-sqr ($\chi^{2}$, orange) and cubic-Gaussian
($G^{3}$, blue) for i) $f_{\rm PBH}\sim 1$ (i.e PBHs constituting all the DM)
and ii) $f_{\rm PBH}=10^{-10}$, from [407].

####  7.6.5 Continuous and high-frequency GWs from planetary-mass PBHs

PBH binary systems with a planetary chirp mass should emit very weak
continuous GWs in the frequency range of ground-based interferometers, many
years before they actually merge. It has been proposed to search for
continuous waves from PBH binaries in our galaxy [896], either towards the
galactic center or in the solar system vicinity. However, the current limits
on continuous waves do not yet provide significant limits on the PBH abundance
[897], even for the models with the highest merger rates. However, future
instruments like Einstein Telescope should be able to probe $f_{\rm PBH}$
between $10^{-2}$ and $1$.

When they merge, these low-mass PBH binaries emit gravitational waves in the
MHz or GHz range. These could be probed with high-frequency GW detectors like
resonant microwave cavities, which could probe $f_{\rm PBH}$ down to about
$10^{-3}$ [898, 98]. Additionally, one recent proposal involving resonant LC
circuits may represent the best option in terms of individual merger detection
prospects in the range $(1-100)\,\text{MHz}$. It was also shown in Ref. [97]
that a detection of the stochastic gravitational wave background produced by
unresolved light PBH binaries is possible only if the theoretical sensitivity
of the proposed Gaussian beam detector is achieved. Such a detector, whose
feasibility is subject to various caveats, may be able to rule-out some
scenarios for asteroidal mass primordial black hole dark matter (see Ref. [97]
and Refs. therein for more details).

####  7.6.6 Discussion and limitations

Gravitational wave observations already provide both limits on the PBH