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Maximum allowed amplitude of the primordial curvature power spectrum. Current (solid lines) and forecasted (dashed lines) constraints from the cosmic microwave background (CMB, $1\sigma\ \mathrm{CL}$)~\cite{akrami:planckng2018}, spectral distortions (FIRAS and PIXIE)~\cite{chluba:powerspectrumconstraints}, gravitational waves (PTA, SKA and LISA)~\cite{byrnes:steepestgrowth}, Silk damping (SD)~\cite{jeong:powerspectrumconstraints}, quasar light curves (QSO)~\cite{karami:powerspectrumconstraints} and PBHs (the thin and thick orange lines correspond to Ref.~\cite{josan:powerspectrumconstraints} and \cite{satopolito:powerspectrumconstraints}, respectively). The red shaded region is the result of this work. It shows upper limits from PBH abundance for the range of profiles with shape parameter~$\alpha\in \left[0.15,30.0\right]$, considering the most recent constraints on the maximum allowed fraction of PBHs. We report our constraint for PBHs masses ranging from~$M_\mathrm{PBH}=10^{-17}\ M_\odot$ ($k_t\simeq 10^{15}\ \mathrm{Mpc}^{-1}$) to~$M_\mathrm{PBH}=10^{3}\ M_\odot$ ($k_t\simeq 10^{5}\ \mathrm{Mpc}^{-1}$).

Shape of the reconstructed primordial curvature power spectra for steep (\textit{left panel}) and flat profiles (\textit{right panel}). In both cases we report the profile for~$M_\mathrm{PBH}=10^{-15}\ M_\odot$ (light blue line) and~$M_\mathrm{PBH}=10\ M_\odot$ (red line). To the right of the peak, the profile is reconstructed using equation~\eqref{eq:reconstruction_curvature_powerspectrum}, and to the left is computed assuming different model for the growth of the power spectrum (see text for details).

Linear and non-linear overdensity profile of equation~\eqref{eq:overdensity_cosmo_metric} on super-horizon scales, for a steep profiles with $\alpha=0.5$ (\textit{left panel}) and a flat profile $\alpha=30$ (\textit{right panel}). Solid lines indicate overdensity $(\delta_\mathrm{peak}>0)$ while dashed lines indicate underdensities $(\delta_\mathrm{peak}<0)$. The profile is showed in units of the typical scale of the perturbation $r_t=\hat{r}_m$ in the linear approximation or $r_t=\hat{r}_m e^{-\zeta_\mathrm{peak}(\hat{r}_m)}$ in the non-linear case. In both cases the non-linear corrections damps and shrink the overdensity profile significantly with respect to the linear case. We use $\mathcal{A}_\mathrm{peak} r^2_t = 5.18,\ 0.99$ for $\alpha=0.5,\ 30$, respectively.

Sketch of the overdensity random field in one spatial dimension. \textit{Upper panel:} total overdensity (\textit{solid line}) given by the sum of a peak overdensity (\textit{dashed line}) with typical scale $r_t$ and the four random fluctuations with different wavelengths illustrated in the lower panel. For completeness we report also the size of the sound horizon $r_s=r_t/\sqrt{3}$. \textit{Zoomed-in panel:} how the peak profile would be before and after smoothing on scale $s$ ($s\ll r_t$, $s\ll r_s$), i.e., removing ultra-short perturbations with frequency $k\gg k_t$. \textit{Lower panel:} the four components of the sketched overdensity with different wavelengths.

Same constraints of figure~\ref{fig:maximum_amplitude_powerspectrum}. Constraints from PBH abundance obtained using our methodology are indicated by the red shaded region and they assume the maximum abundance allowed by observations ($f_\mathrm{PBH}= f^\mathrm{max}_\mathrm{PBH}$), while the blue shaded region represents the constraints for the case where there is only one PBH in our Universe, i.e., $f_\mathrm{PBH}= f^\mathrm{single}_\mathrm{PBH}$.

\textit{Upper left panel:} curvature profile~$K_\mathrm{peak}r^2_t$ at initial conditions, on super-horizon scales, for different values of the shape parameter~$\alpha$. We use $\mathcal{A}_\mathrm{peak}r_t^2=5.18,\ 2.03,\ 0.99$ for $\alpha=0.5,\ 1.0,\ 30$, respectively. While the peak amplitude~$\mathcal{A}_\mathrm{peak}$ changes when the typical scale~$r_t$ varies, the quantity $\mathcal{A}_\mathrm{peak}r_t^2$ is constant. Initial conditions are set up super-horizon, so that the typical scale of the perturbation~$r_t$ is much larger than the coming horizon at initial time, i.e.,~$a_\mathrm{ini}H_\mathrm{ini}r_t\gg 1$. \textit{Upper right panel:} Curvature profiles~$\zeta_\mathrm{peak}$, corresponding to the $K$-curvature peaks of the upper left panel, at initial time~$t_\mathrm{ini}$ for different values of the parameter~$\alpha$. Also in this case the typical scale of the perturbation is~$r_t$. \textit{Lower panel:} Corresponding overdensity profiles at initial time~$t_\mathrm{ini}$.

\textit{Left panel:} compaction function~$\mathcal{C}$ corresponding to the curvature and density perturbations of figure~\ref{fig:curvature_profiles}. \textit{Right panel:} Critical threshold as a function of the shape of the perturbation profile. Steep profiles correspond to~$\alpha\to 0$, while flat profiles correspond to~$\alpha\to\infty$. Purple dots represents values found by numerical simulation, interpolated with the orange line.

Linear-to-Non-Linear ratio between different physical scales. \textit{Left panel:} ratio of typical comoving scales~$r_t$ of the perturbation, horizon crossing times~$t_m$, masses enclosed inside the horizon~$M_\mathrm{hor}$ and overdensity zero-crossing scales~$\hat{r}_0$. \textit{Right panel:} ratio of integrated density profiles~$\delta_I(t_m,R_m)$ and integrated-density-to-peak-amplitude relation~$\mathcal{F}_\delta$.

\textit{Left panel:} dynamical behaviour of $2M/R$ against $R/R_m$ plotted at different time-slice for the critical solution of a zero mass black hole when $\delta_I=\delta_{I,c}$ obtained from equation~\eqref{eq:mass_critical_collapse} when~$\alpha=1$ (Mexican-Hat shape). The dashed line corresponds to the initial time-slice, and the peak of $2M/R$ is initially decreasing when the perturbation is still expanding, reaching afterwards nearly equilibrium state moving inward when the perturbation is collapsing. Figure taken from Ref.~\cite{musco:criticalcollapse}. \textit{Right panel:} numerical behaviour of the critical threshold~$\delta_{I,c}$ against the corresponding behaviour of the critical peak amplitude~$\delta_{\mathrm{peak},0}$ for different shapes~$(0.15 \leq \alpha \leq 30)$. Figure taken from Ref.~\cite{musco:pbhthreshold}.

\textit{Left panel:} dynamical behaviour of $2M/R$ against $R/R_m$ plotted at different time-slice for the critical solution of a zero mass black hole when $\delta_I=\delta_{I,c}$ obtained from equation~\eqref{eq:mass_critical_collapse} when~$\alpha=1$ (Mexican-Hat shape). The dashed line corresponds to the initial time-slice, and the peak of $2M/R$ is initially decreasing when the perturbation is still expanding, reaching afterwards nearly equilibrium state moving inward when the perturbation is collapsing. Figure taken from Ref.~\cite{musco:criticalcollapse}. \textit{Right panel:} numerical behaviour of the critical threshold~$\delta_{I,c}$ against the corresponding behaviour of the critical peak amplitude~$\delta_{\mathrm{peak},0}$ for different shapes~$(0.15 \leq \alpha \leq 30)$. Figure taken from Ref.~\cite{musco:pbhthreshold}.