Primordial black holes from metric preheating: mass fraction in the excursion-set approach

Evolution of the physical scales appearing in \Eq{eq:instability:band:2}, with time parametrised by the number of \efolds~$N=\ln a$ (counted from the end of inflation). The blue line represents the Hubble radius $1/H$, the orange line the new length scale $1/\sqrt{3Hm}$ and the dotted lines the physical wavelengths of modes of interest, which may enter the instability band after inflation, during the oscillatory phase. Here the Klein-Gordon equation for the inflaton field has been solved for the quadratic potential $V(\phi) = m^2 \phi^2/2$, where $m = 10^{-6} \Mp$.

Bardeen potential $\Phi_{\bm{k}}$ rescaled by the curvature perturbation $\zeta_{\bm{k}}$ during the last \efolds~of inflation and the first \efolds~of the oscillatory phase, in the same situation as the one displayed in \Fig{fig:scale}, for a scale ${\bm{k}}$ that is sufficiently far outside the Hubble radius such that $\zeta_{\bm{k}}$ can be taken as constant. The blue line stands for the full numerical solution of \Eq{eq:zeta:Bardeen}, seen as a differential equation for $\Phi_{\bm{k}}(t)$, where $w(t)$ and $H(t)$ are extracted from \Fig{fig:scale}. The red line stands for the approximation~\eqref{eq:Phi:zeta:Appr}, $\Phi_{\bm{k}}/\zeta_{\bm{k}} = 3/5$, obtained as the late-time solution of \Eq{eq:zeta:Bardeen} when setting $w=0$ and $H=2/(3 t)$, and towards which the full numerical result asymptotes after a few oscillations. The orange line stands for \Eq{eq:zeta:Bardeen} where we neglected $\dot{\Phi}_{\bm{k}}/H$ with respect to $\Phi_{\bm{k}}$. This approximation is well justified on super-Hubble scales during inflation, since $w$ is almost constant there, but fails during the subsequent oscillatory phase where $w$ vanishes on average but otherwise undergoes large oscillations.

Example of Langevin trajectories for the density contrast evaluated on comoving slices at the end of inflation, and coarse-grained at the scale $R$, for $H_\uend = 10^{-8} \Mp$ and $H_\Gamma = 10^{-25} \Mp$. The (quasi) horizontal black dashed line shows the collapse criterion~\eqref{eq:collapse:threshold:end}. In the right panel, we isolate one realisation and the vertical dashed line denotes the first crossing ``time'' (\ie scale) of the critical threshold.

Mass fraction $\beta$ of primordial black holes for $H_\uend = 10^{-8} \Mp$ and $H_\Gamma = 10^{-25} \Mp$, as a function of the mass $M$ in grams. The vertical black bars stand for the distribution of first crossing times obtained from $10^6$ simulated realisations of the Langevin equation~\eqref{eq:Langevin:sigma2}, binned into $1000$ logarithmically spaced values of $R$. The size of the bars correspond to $5\sigma$ estimates of the statistical error by jackknife resampling. The red line corresponds to numerically solving the Volterra equation~\eqref{eq:Volterra:regular}, using the method described in \App{sec:numerical-volterra}. The blue line displays the analytical approximation developed in \Sec{sec:AnalyticalApproximation}, which provides a good fit to the full numerical. The vertical green line denotes the mass at which $\beta$ peaks, as estimated from \Eq{eq:meanMass}, and the grey shaded area stands for the $1\sigma$ deviation of $\ln(M)$ according to the distribution $\beta(M)$, centred on its mean value.

Total fraction of the universe comprised in PBHs, $\OmegaPBH$, as a function of $\rho_\mathrm{end}$, the energy density at the end of inflation, and $\rho_\Gamma$, the energy density at the end of the instability phase. On the left panel, we fix $\rho_\uend=10^{-12}\Mp^4$ and let $\rho_\Gamma$ vary. The solid red curve is the full numerical result obtained in the excursion-set approach. The dashed green line corresponds to the Press-Schechter result with the additional factor $2$, which becomes exact in the limit of a scale-invariant threshold, see \Sec{sec:ScaleInvariant:threshold}. The dashed blue line corresponds to the analytical approximation~\eqref{eq:Omegatot:anal}. On the right panel, the full parameter space is explored (where $\rho_\Gamma<\rho_\uend$ since the oscillatory phase occurs after inflation). The colour encodes the value of $\OmegaPBH$, and the transition from tiny values to values close to one is very abrupt. The dashed blue line stands for the analytical estimate~\eqref{eq:bound:Omegatot_eq_1} for the location of this transition.

Total fraction of the universe comprised in PBHs, $\OmegaPBH$, as a function of $\rho_\mathrm{end}$, the energy density at the end of inflation, and $\rho_\Gamma$, the energy density at the end of the instability phase. On the left panel, we fix $\rho_\uend=10^{-12}\Mp^4$ and let $\rho_\Gamma$ vary. The solid red curve is the full numerical result obtained in the excursion-set approach. The dashed green line corresponds to the Press-Schechter result with the additional factor $2$, which becomes exact in the limit of a scale-invariant threshold, see \Sec{sec:ScaleInvariant:threshold}. The dashed blue line corresponds to the analytical approximation~\eqref{eq:Omegatot:anal}. On the right panel, the full parameter space is explored (where $\rho_\Gamma<\rho_\uend$ since the oscillatory phase occurs after inflation). The colour encodes the value of $\OmegaPBH$, and the transition from tiny values to values close to one is very abrupt. The dashed blue line stands for the analytical estimate~\eqref{eq:bound:Omegatot_eq_1} for the location of this transition.

: Mean PBH mass

: Dispersion of the PBH masses

: Evolution of $\OmegaPBH$ at $\rho_\uend = 10^{-12} \Mp^{4}$

: $\OmegaPBH$ as a function of $\rho_\uend$ and $\rho_\Gamma$

: Average mass of PBH

: Dispersion of the PBH masses

Mass fraction $\beta$ of primordial black holes for $\rho_\uend=10^{-12}\Mp^4$ and $\rho_\Gamma=10^{-40}\Mp$. As in \Fig{fig:mass-function}, the red line corresponds to numerically solving the Volterra equation~\eqref{eq:Volterra:regular}. The olive, orange and purple lines correspond to the results of \Refa{Martin:2019nuw} and are taken from Fig.~4 of that reference. The orange line displays the ``raw'' result obtained with the estimate of \Sec{sec:simplified:PS}, which leads to the problematic $\OmegaPBH>1$. Then, ``renormalisation'' is either performed by ``premature ending'' (purple dashed line) or by ``absorption'' (olive dashed line).