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A schematic plot of the inflationary potential with an inflection point. Adapted from Ref.~\cite{Garcia-Bellido:2017fdg}.

Power spectrum $\mathcal{P}_{\mathcal{R}=\zeta} (k) $ for the single-field Critical Higgs Inflation model, with an inflection point at $N\approx 36$ satisfying the Planck 2018 constraints. Adapted from Ref.~\cite{Garcia-Bellido:2017mdw}.

Examples of primordial power spectra of curvature fluctuations, leading to (stellar-mass) PBH formation: power-law, broken power-law, Gaussian and log-normal models (solid lines), and particular examples of multifield and axion-gauge models (dashed lines).

Representation of a typical hybrid inflation potential, $V$, with a possible trajectory in two-field space (dotted line). CMB perturbations are created along the valley at $\psi=0$, during a first phase of inflation. Curvature perturbations suitable for production of PBHs are generated in a second flat part of the potential (red dotted line), {when} the mass-squared of the field $\psi$ changes sign.

Power spectrum of curvature perturbations for hybrid inflation parameters values $M = 0.1 \Mpl$, $\mu_1 = 3 \times 10^5 \Mpl$ and $\phi_{\rm c} = 0.125 \Mpl $ (red)$, 0.1 \Mpl $ (blue), $0.075 \Mpl $ (green), and $0.05 \Mpl $ (cyan). Those parameters correspond respectively to $\Pi^2 = 375 / 300 / 225/150$. The power spectrum is degenerate for lower values of $M,\phi$ and larger values of $\mu_1$, keeping the combination $\Pi^2$ constant. For larger values of $M, \phi_{\rm c}$ the degeneracy is broken: power spectra in orange and brown are obtained respectively for $ M = \phi_{\rm c} = \Mpl$ and $ \mu_1 = 300 \Mpl / 225 \Mpl$. Dashed lines assume $\psi_c = \psi_0$ whereas solid lines are obtained after averaging over 200 power spectra obtained from initial conditions on $\psi_c$ distributed according to a Gaussian of width $\psi_0$. The power spectra corresponding to these realizations are plotted in dashed light gray for illustration. The $\Lambda$ parameter has been fixed so that the spectrum amplitude on CMB anisotropy scales is in agreement with Planck data. The parameter $\mu_2 = 10 \Mpl$ so that the scalar spectral index on those scales is given by $n_{\rm s} = 0.96$. Figure from Ref.~\cite{Clesse:2015wea}.

The stochastic GWs produced by axion inflation at four main scales of interferometers {for which we show the approximate sensitivities}, nHz (PTAs and SKA), $\mu$Hz (Gaia and Theia), mHz {(LISA and BBO) and Hz (LVK, Einstein Telescope, and Cosmic Explorer)}. The enhanced density perturbations may produce PBHs which are a significant fraction of dark matter for $1-100 M_{\odot}$ and the totality in the $10^{-14}-10^{-11}M_{\odot}$ mass range and, remarkably, the enhanced perturbations leave inevitable GW backgrounds at the most sensitive regimes of GW detectors: for the first range of BHs this corresponds to PTA frequencies (and possibly future SKA) and LISA frequencies for the second range. The PBHs formed from fluctuations peaking at Hz scales will be so light that they are expected to be part of thermal history through Hawking radiation. The Figure is updated (with slight modifications) from Refs.~\cite{Garcia-Bellido:2017aan,Unal:2018yaa,Garcia-Bellido:2021zgu}

Illustration of the effect of a positive skewness, $\bar{\kappa}_3$, and a positive (excess) kurtosis, $\bar{\kappa}_4$, for the probability density function $P(x)$ (left) and its logarithm (right) in comparison with a Gaussian distribution ($\bar{\kappa}_{n>2}=0$). The dominant effect in the tail is given by a positive kurtosis. Plots reproduced from \cite{Ezquiaga:2018gbw}.

Illustration of the effect of a positive skewness, $\bar{\kappa}_3$, and a positive (excess) kurtosis, $\bar{\kappa}_4$, for the probability density function $P(x)$ (left) and its logarithm (right) in comparison with a Gaussian distribution ($\bar{\kappa}_{n>2}=0$). The dominant effect in the tail is given by a positive kurtosis. Plots reproduced from \cite{Ezquiaga:2018gbw}.

Probability density functions of the curvature perturbations generated by an inflection-point inflationary potential (see inset). Solid lines correspond to the full distribution functions computed by mean of the stochastic-$\delta N$ formalism, where different colours correspond to different locations in the potential $v(\phi)$ where the scale under consideration emerges from the Hubble radius and where $\Delta \phi_{\rm well} $ denotes the field range for which stochastic effects dominate over the classical dynamics. The dotted lines correspond to the standard result, which provides a good Gaussian approximation for the maximum of the distribution but that however fails to describe the exponential tails, where PBHs are nonetheless produced. This figure is adapted from Ref.~\cite{Ezquiaga:2019ftu}.

Threshold $\delta_c$ and critical collapse parameter ${\cal K}$ as a function of the shape parameter $\alpha$, assuming a universe dominate by perfect radiation (i.e. $w = 1/3$). Figure adapted from Ref.~\cite{Musco:2020jjb}

Threshold $\delta_c$ and critical collapse parameter ${\cal K}$ as a function of the shape parameter $\alpha$, assuming a universe dominate by perfect radiation (i.e. $w = 1/3$). Figure adapted from Ref.~\cite{Musco:2020jjb}

Relativistic degrees of freedom $g_{*}$ ({\it Left panel}) and equation-of-state parameter $w$ ({\it Right panel}), both as a function of temperature $T$ (in MeV). The grey vertical lines correspond to the masses of the electron, pion, proton/neutron, $W$, $Z$ bosons and top quark, respectively. The grey dashed horizontal lines indicate values of $g_{*} = 100$ and $w = 1 / 3$, respectively.

Relativistic degrees of freedom $g_{*}$ ({\it Left panel}) and equation-of-state parameter $w$ ({\it Right panel}), both as a function of temperature $T$ (in MeV). The grey vertical lines correspond to the masses of the electron, pion, proton/neutron, $W$, $Z$ bosons and top quark, respectively. The grey dashed horizontal lines indicate values of $g_{*} = 100$ and $w = 1 / 3$, respectively.

PBH density fraction at formation $\beta^{\rm form}$ (left panel) and the corresponding PBH mass function $f_{\rm PBH}$ today (right panel), neglecting the effects of PBH growth by accretion and hierarchical mergers, for two models with a power-law primordial power spectrum and including the effects of thermal history: Model 1 from~\cite{Carr:2019kxo,Clesse:2020ghq} with spectral index $n_{\rm s} = 0.97$; Model 2 from~\cite{DeLuca:2020agl,Byrnes:2018clq} with $n_{\rm s} = 1.$ and a cut-off mass of $10^{-14} M_\odot$. The transition between the large-scale and small-scale power spectrum is fixed at $k=10^3 {\rm Mpc}^{-1}$. The power spectrum amplitude is normalized such that both models produce an integrated PBH fraction $f_{\rm PBH} =1$, i.e. PBH constitute the totality of Dark Matter. A value of $\gamma = 0.8$ (ratio between the PBH mass and the Hubble horizon mass at formation) was assumed. Figure produced for~\cite{LISACosmologyWorkingGroup:2022jok}.

PBH density fraction at formation $\beta^{\rm form}$ (left panel) and the corresponding PBH mass function $f_{\rm PBH}$ today (right panel), neglecting the effects of PBH growth by accretion and hierarchical mergers, for two models with a power-law primordial power spectrum and including the effects of thermal history: Model 1 from~\cite{Carr:2019kxo,Clesse:2020ghq} with spectral index $n_{\rm s} = 0.97$; Model 2 from~\cite{DeLuca:2020agl,Byrnes:2018clq} with $n_{\rm s} = 1.$ and a cut-off mass of $10^{-14} M_\odot$. The transition between the large-scale and small-scale power spectrum is fixed at $k=10^3 {\rm Mpc}^{-1}$. The power spectrum amplitude is normalized such that both models produce an integrated PBH fraction $f_{\rm PBH} =1$, i.e. PBH constitute the totality of Dark Matter. A value of $\gamma = 0.8$ (ratio between the PBH mass and the Hubble horizon mass at formation) was assumed. Figure produced for~\cite{LISACosmologyWorkingGroup:2022jok}.

Figure taken from \cite{Franciolini:2022tfm}. {Left panel:} the equation-of-state parameter $w=p/\rho$ (red) and squared speed of sound (blue) as functions of the cosmological horizon mass $M_H$. {Right panel:} Evolution of the equation-of-state dependent parameter $\Phi$, relating the density contrast to the curvature perturbation as functions of the cosmological horizon mass $M_H$.

Figure taken from \cite{Franciolini:2022tfm}. {Left panel:} the equation-of-state parameter $w=p/\rho$ (red) and squared speed of sound (blue) as functions of the cosmological horizon mass $M_H$. {Right panel:} Evolution of the equation-of-state dependent parameter $\Phi$, relating the density contrast to the curvature perturbation as functions of the cosmological horizon mass $M_H$.

Relative variation of the threshold compared to what is obtained assuming perfect radiation as a function of the horizon crossing time (parametrised here with $M_H$) induced by the QCD thermal effects. The color code indicates the different values of $\log_{10}(\alpha)$ as indicated by the bar on top of the frame. Figure adapted from Ref.~\cite{Musco:2023dak}.

Figure from \cite{Franciolini:2022tfm}. {\bf Left panel:} PBH mass $m_\PBH$ plotted as a function of $\delta-\delta_c$ computed at the cosmological horizon crossing (see Ref.~\cite{Musco:2023dak} for more details). The behavior for a radiation dominated medium is plotted with a black dashed line. {\bf Right panel:} The values of the power law coefficients in Eq.~\eqref{eq:masspbh22} found by fitting the results of numerical simulations shown in the left panel.

Figure from \cite{Franciolini:2022tfm}. {\bf Left panel:} PBH mass $m_\PBH$ plotted as a function of $\delta-\delta_c$ computed at the cosmological horizon crossing (see Ref.~\cite{Musco:2023dak} for more details). The behavior for a radiation dominated medium is plotted with a black dashed line. {\bf Right panel:} The values of the power law coefficients in Eq.~\eqref{eq:masspbh22} found by fitting the results of numerical simulations shown in the left panel.

Plot taken from \cite{Franciolini:2022tfm}. Mass function obtained with a few choices of the curvature power spectrum. This plot assumes $f_\PBH = 10^{-3}$, the minimum horizon mass to be $ \lesssim 10^{-2.5} M_\odot$, the largest mass $M_H^\text{max} = 10^{2.8} M_\odot$ and a variable tilt $n_s$. The black dashed line reports the lognormal mass distribution found as the best fit in the analysis of Ref.~\cite{Franciolini:2021tla}.

Figures taken from Ref.~\cite{Franciolini:2021xbq}. \textbf{Left:} Predicted primary ($\chi_1$) and secondary ($\chi_2$) spins as a function of primary mass $m_1$ and mass ratio $q$ for various values of $z_{\rm cut-off}$ (indicated by colors specified in the right panel). \textbf{Right:} Predicted distribution of $\chi_{\rm eff}$ as a function of PBH mass $m_1$ (assuming equal mass binaries) for three choices of $z_{\rm cut-off}$.

Figures taken from Ref.~\cite{Franciolini:2021xbq}. \textbf{Left:} Predicted primary ($\chi_1$) and secondary ($\chi_2$) spins as a function of primary mass $m_1$ and mass ratio $q$ for various values of $z_{\rm cut-off}$ (indicated by colors specified in the right panel). \textbf{Right:} Predicted distribution of $\chi_{\rm eff}$ as a function of PBH mass $m_1$ (assuming equal mass binaries) for three choices of $z_{\rm cut-off}$.

Figure taken from Ref.~\cite{Inman:2019wvr}. The abundance of halos $N_{\rm HL}$ containing a given number of PBHs, $N_{\rm PBH/HL}$, i.e. halo mass function. Solid lines report the results of the $N$-body simulations while dashed lines are indicates the theoretical prediction assuming Poisson statistics.

Suppression factor $f_{\rm sup}$ and its possible contributions $S_1$ and $S_2$ in different cases, in terms of the PBH abundance (from~\cite{Clesse:2020ghq}). When PBH masses are specified, it assumes a PBH mass function from a nearly-scale invariant primordial power spectrum of curvature fluctuations and including QCD-induced features.

Expected merging rates of PBH of masses $m_1$ and $m_2$, for the two mass models represented on Fig.~\ref{fig:fPBH2models} (top panels: Model 1, bottom panels: Model 2), for the two considered binary formation channel: primordial binaries (see Eq.~\ref{eq:cosmomerg}) on the left panels, and tidal capture in halos (see Eq.~\ref{eq:ratescatpure2}) on the right panels. Figure produced for~\cite{LISACosmologyWorkingGroup:2022jok}.

Expected merging rates of PBH of masses $m_1$ and $m_2$, for the two mass models represented on Fig.~\ref{fig:fPBH2models} (top panels: Model 1, bottom panels: Model 2), for the two considered binary formation channel: primordial binaries (see Eq.~\ref{eq:cosmomerg}) on the left panels, and tidal capture in halos (see Eq.~\ref{eq:ratescatpure2}) on the right panels. Figure produced for~\cite{LISACosmologyWorkingGroup:2022jok}.

The scattering of one BH of mass $m_2$ on another of mass $m_1$ induces the emission of gravitational waves which is maximal at the point of closest approach, $r_{\rm p}$.

Figure taken from Ref.~\cite{Franciolini:2021xbq}. Schematic flowchart representing how to systematically rule out or potentially assess the primordial origin of a binary merger. These criteria are based on measurements of the redshift $z$, eccentricity $e$, tidal deformability $\Lambda$, component masses $m$, and dimensionless spin $\chi$. Each arrow indicates if the condition in the box is met (green) or violated (red), while the marks indicate: \cmark) likely to be a PBH binary; \xmark) cannot to be a PBH binary; \textcolor{Black}{\textbf{?}}) may be a PBH binary.

Induced GWs spectra for both examples considered in Eqs.~\eqref{Pz-delta} and \eqref{Pz-gauss}, for the parameters choice $A_{\rm s} = 0.033$, $A_\zeta = 0.044$ and $\sigma = 0.5$. For comparison, we also show the estimated sensitivity for LISA \cite{Audley:2017drz}, {following the proposed design (4y, 2.5 Gm of length, 6 links)}. The PBH mass corresponding to the characteristic frequency is depicted on the top horizontal axis, according to Eq.~\ref{eq:f_to_mPBH}.

Nonvanishing diagrams for $ {\cal O} (f_{\rm NL}^{0})$ (left) and $ {\cal O} (f_{\rm NL}^{2})$ : center and right, called Hybrid and Walnut diagrams due to their topology and shape \cite{Unal:2018yaa}.

Nonvanishing diagrams for $ {\cal O} (f_{\rm NL}^{0})$ (left) and $ {\cal O} (f_{\rm NL}^{2})$ : center and right, called Hybrid and Walnut diagrams due to their topology and shape \cite{Unal:2018yaa}.

Nonvanishing diagrams for $ {\cal O} (f_{\rm NL}^{0})$ (left) and $ {\cal O} (f_{\rm NL}^{2})$ : center and right, called Hybrid and Walnut diagrams due to their topology and shape \cite{Unal:2018yaa}.

Diagrams of ${\cal O} (f_{\rm NL}^{4})$, called (from left to right) Reducible, Planar and non-Planar due to their topological properties \cite{Garcia-Bellido:2017aan,Unal:2018yaa}.

Diagrams of ${\cal O} (f_{\rm NL}^{4})$, called (from left to right) Reducible, Planar and non-Planar due to their topological properties \cite{Garcia-Bellido:2017aan,Unal:2018yaa}.

Diagrams of ${\cal O} (f_{\rm NL}^{4})$, called (from left to right) Reducible, Planar and non-Planar due to their topological properties \cite{Garcia-Bellido:2017aan,Unal:2018yaa}.

Second peak produced by primordial non-Gaussian component of curvature perturbations. Figure taken from \cite{Unal:2018yaa}.

Second peak produced by primordial non-Gaussian component of curvature perturbations. Figure taken from \cite{Unal:2018yaa}.

Contour plot showing the amount of GWs anisotropy in the parameter space of $f_{\rm PBH}$ and $f_{\rm NL}$ allowed by the Planck constraints for the choice of a monochromatic and log-normal small-scale power spectrum, respectively, with peak frequency around the maximum sensitivity of LISA. The dot-dashed lines identify the corresponding present GWs abundance. Figure taken from Ref.~\cite{Bartolo:2019zvb}.

Contour plot showing the amount of GWs anisotropy in the parameter space of $f_{\rm PBH}$ and $f_{\rm NL}$ allowed by the Planck constraints for the choice of a monochromatic and log-normal small-scale power spectrum, respectively, with peak frequency around the maximum sensitivity of LISA. The dot-dashed lines identify the corresponding present GWs abundance. Figure taken from Ref.~\cite{Bartolo:2019zvb}.

Frequency at which the GWs induced by a dominating gas of PBHs peak, as a function of their energy density fraction at the time they form, $\Omega_{\mathrm{PBH,f}}$ (horizontal axis), and their mass $m_{\mathrm{PBH}}$ (colour coding). The region of parameter space that is displayed corresponds to values of $m_{\mathrm{PBH}}$ and $\Omega_{\mathrm{PBH,f}}$, such that black holes form after inflation, dominate the Universe content for a transient period and Hawking evaporate before BBN. We also impose that the induced GWs do not lead to a backreaction problem before they evaporate, see~\cite{Papanikolaou:2020qtd} for more details. For comparison, the frequency detection bands of ET, LISA and SKA are shown. Figure credited to~\cite{Papanikolaou:2020qtd}.

The SGWB spectrum $\Omega_{\rm GW}h^{2}$ for early PBH binaries with a log-normal mass function (in red, with central mass $\mu = 2.5 M_\odot$ and width $\sigma = 1 $) and a broad mass distribution with scalar spectral index $n_{\rm s}$ = 0.970 (in blue), and $f_{\rm PBH}= 1$. The numerical spectrum also shows the sensitivities of the ground-based interferometers: {the LVK O3 Run, the final LVK} and the Einstein Telescope (ET). The sensitivity of future space-based interferometers is also shown (LISA, BBO/DECIGO). The Pulsar Timing Array (PTA) considered here is the Square Kilometer Array (SKA) \cite{SchmitzK}. The NANOGrav 12.5 signal is represented by the gray square {(see sec.~\ref{sec:PTAs_NG} for more details)}.

Contribution to different PBH masses to the SGWB from early PBHs binaries (with $f_{\rm PBH} = 1$). The color bar indicates the values of the quantity $\log_{10}(\Omega_{\rm GW}h^{2})$, which is represented as a function of the logarithm of the masses $m_1$ and $m_2$ of the PBHs following a mass distribution from the thermal history of the Universe (with $n_{\rm s}$ = 0.97) for LISA frequencies ($10^{-3}$ Hz). Figure from~\cite{Bagui:2021dqi}.

The SGWB spectrum $\Omega_{\rm GW}h^{2}$ for late PBH binaries in clusters with $f_{\rm PBH} = 1$ for a log-normal mass function (in red, with the central mass {$\mu=2.5\Msun$} and the width $\sigma=1$) and a broad mass distribution (in blue, with $n_{\rm s}=0.97$ and no running). Note that here we use \cite{Ajith:2009bn} for the single source GW energy spectrum instead of Eq.~\eqref{omegaGW}.

The mass contribution to the SGWB from late PBHs binaries in clusters (with $f_{\rm PBH} = 1$). The color bar indicates the values of the quantity $\log_{10}(\Omega_{\rm GW}h^{2})$, which is represented as a function of the logarithm of the masses $m_1$ and $m_2$ of the PBHs following a mass distribution from the thermal history of the Universe (with $n_{\rm s}$ = 0.97 and $v_{\rm vir}$ = 5 km/s) for LISA frequencies ($10^{-3}$ Hz).

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 $m_1=m_2=100-300~\Msun$ 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~\cite{Garcia-Bellido:2021jlq}.

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$^{-3}$yr$^{-1}$. Horizontal lines show the expected difference in the cross-correlation between (late binary) PBH and stellar binaries, for different values of $f_{\rm PBH}$.

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~\cite{Scelfo:2018sny}.

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~\cite{Bosi:2022})

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.

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 $\gamma$) \cite{Arbey:2019vqx}, $e^\pm$ observations by Voyager 1 (V $e^\pm$) \cite{Boudaud:2018hqb}, positron annihilations in the Galactic Center (GC $e^+$) \cite{DeRocco:2019fjq} and gamma-ray observations by INTEGRAL (INT) \cite{Laha:2020ivk} (for other constraints in the ultra-light mass range see also \cite{Carr:2009jm, Ballesteros:2019exr,Laha:2019ssq, Poulter:2019ooo,Dasgupta:2019cae,Laha:2020vhg}). We plot microlensing searches by Subaru HSC \cite{Niikura:2017zjd, Smyth:2019whb}, MACHO/EROS (E) \cite{Alcock:2000kd, Allsman:2000kg}, Ogle (O) \cite{Niikura:2019kqi} and Icarus (I) \cite{Oguri:2017ock}. Other constraints come from CMB distortions. In black dashed, we show the ones assuming disk accretion (Planck D in Ref.~\cite{Serpico:2020ehh} and Ref.~\cite{Poulin:2017bwe}, from left to right) while in black solid the ones assuming spherical accretion (Planck S in Ref.~\cite{Serpico:2020ehh} and both photo- and collisional ionization in Ref.~\cite{Ali-Haimoud:2016mbv}, from left to right). Only Ref.~\cite{Serpico:2020ehh} includes the effect of the secondary dark matter halo in catalysing accretion. Additionally, constraints coming from X-rays (Xr) \cite{Manshanden:2018tze} and X-Ray binaries (XrB)\cite{Inoue:2017csr} are shown. Dynamical limits coming from the disruption of wide binaries (WB) \cite{2009MNRAS.396L..11Q}, survival of star clusters in Eridanus II (EII) \cite{Brandt:2016aco} and Segue I (SI) \cite{Koushiappas:2017chw,Stegmann:2019wyz} are also shown. LVC stands for the constraint coming from LVK measurements \cite{Ali-Haimoud:2017rtz,Raidal:2018bbj,Vaskonen:2019jpv,Wong:2020yig}. Constraints from Lyman-$\alpha$ forest observations (L$\alpha$) come from Ref.~\cite{Murgia:2019duy}. We neglect the role of accretion which has been shown to affect constraints on masses larger than ${\cal O}(10) M_\odot$ \cite{DeLuca:2020fpg,DeLuca:2020qqa} in a redshift dependent manner. See Ref.~\cite{Carr:2020gox} for a comprehensive review on constraints on the PBH abundance. Notice that there are no stringent bounds in the asteroid mass range \cite{Katz:2018zrn, Montero-Camacho:2019jte} where LISA may constrain PBHs through the search of a second order SGWB.

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 $m_{\rm PBH }\sim 10 M_\odot$ 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.

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 $m_{\rm PBH }\sim 10 M_\odot$ 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.

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 $m_{\rm PBH }\sim 10 M_\odot$ 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.

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 \cite{Unal:2020mts}.

Figure taken from Ref.~\cite{Ng:2022agi}. Projected upper limit on $f_\PBH$ 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~\citep{LSSTDarkMatterGroup:2019mwo,2022arXiv220308967B}.

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) $f_{\rm PBH}\sim1$ (i.e PBHs constituting all the DM) and ii) $f_{\rm PBH}=10^{-10}$, from \cite{Unal:2020mts}.

Redshift range of LISA for equal-mass BBH coalescences as a function of the total system mass and comparison with the range of other detectors and pulsar timing arrays. The color scale represents the expected SNR. Figure taken from Ref.~\cite{Burke-Spolaor:2018bvk}.

Illustration of the concept of multi-band GW astronomy, combining (e)LISA and aLIGO (LVK). Each blue line corresponds to a trajectory in the strain-frequency plane for black hole mergers. The horizontal time scale corresponds to the probed time before the merger. Figure taken from Ref.~\cite{Sesana:2016ljz}.

Distance for which a circular binary of total mass $m$ and coalescence time $\tau$ leads to quasi-continuous gravitational waves detectable with a SNR $=8$ in different detectors. The expectations for LISA are shown in the right panel. Figure from Ref.~\cite{Pujolas:2021yaw}.

Left: Expected mass of dark matter clumps or compact objects like PBHs $m_{\rm cl}$ in the inner solar system that could be detected at a distance $d$ by LISA and other types of gravitational-wave detectors, due to the transient acceleration of test masses (which for LISA corresponds to {the mirrors in the three space probes}). Right: Corresponding rate of expected observable events as a function of the object mass, if they comprise all the dark matter and assuming galactic velocities. Figure from~\cite{Baum:2022duc}.