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 $\ell$. We find that such a regulator induces a primordial black hole mass gap such that below a certain mass $M_\text{gap}(\ell, R_H)$ no black holes can form. The order of magnitude of this mass gap is set by the regulator $\sim c^2\ell/G$, 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 $\omega = 0 \dots 1/3$, the mass gap implies a Carr criterion of the form $\delta_H > 2G M_\text{gap}/R_H - 1$. If the horizon size is of the same order of the regulator, $R_H \sim \ell$, 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.