Carr criterion and mass gaps in non-singular primordial black hole formation

Non-singular gravitational theories are expected to be relevant in the early universe. In this paper, we derive a set of effective Friedmann equations describing the dynamics of matter shells in the presence of a gravitational regulator $\mathrm{\ell }$. We find that such a regulator induces a primordial black hole mass gap such that below a certain mass $M_{\text{gap}}(\mathrm{\ell },R_H)$ no black holes can form. The order of magnitude of this mass gap is set by the regulator $\sim c^2\mathrm{\ell }\mathrm{/}G$, with subleading dependence on the horizon radius at time of formation $R_H$. Finally, we show that over a wide range of equation of state parameters $\omega =0\dots 1\mathrm{/}3$, the mass gap implies a Carr criterion of the form ${\delta }_H>2G{M}_{\text{gap}}\mathrm{/}{R}_H-1$. If the horizon size is of the same order of the regulator, $R_H\sim \mathrm{\ell }$, this new criterion is stronger than the traditional Carr criterion for primordial black hole formation. This connects the primordial black hole abundance directly to the presence of gravitational regulators.