A new paper studies how simple modifications of gravity that remove singularities can stop very small black holes from forming in the early universe. The authors introduce a gravitational regulator length, call it ℓ, and show that when ℓ is nonzero there is a minimum black hole mass M_gap below which no horizon forms. The order of magnitude of that mass gap is set by M_gap ~ c^2 ℓ / G, with smaller corrections that depend on the horizon radius at the time of formation, R_H.

To reach this result the researchers build effective equations for spherical matter shells in a cosmological setting. They model the regulator by a nonlocal modification of the Newtonian operator, using a form factor exp(−ℓ^2 ∇^2). That choice “smears” point sources over a length of order ℓ, turning delta-function densities into three-dimensional Gaussians. Using a spectral decomposition of the radial Laplacian, they compute the modified gravitational potentials and effective densities for a point mass, a hollow spherical shell, and a solid sphere. These ingredients let them derive a set of modified Friedmann-type equations and an effective potential for collapsing shells.

One direct physical consequence they report is an increase of the gravitational free-fall time when ℓ > 0. Numerically, the free-fall time becomes radius dependent and diverges as the object size R becomes much smaller than ℓ. In cosmological terms this leads to a primordial black hole mass gap M_gap(ℓ, R_H) so that below that mass no black-hole horizon can form. For a wide range of fluid equations of state (ω from 0 to 1/3) they show this mass gap implies a modified Carr criterion at horizon crossing of the form δ_H > 2 G M_gap / R_H − 1. If the horizon size at formation is similar to the regulator (R_H ~ ℓ), this new criterion can be stricter than the traditional Carr density-contrast condition.