Constraining small-scale primordial magnetic fields from the abundance of primordial black holes

Abstract: The presence of magnetic fields in the early universe affects the cosmological processes, leading to the distinct signature, which allows constraining their properties and the genesis mechanisms. In this study, we revisit the method to constrain the amplitude of the magnetic fields on small scales in the radiation-dominated era from the abundance of primordial black holes. Constraints in the previous work were based on the fact that the density perturbations sourced by stronger magnetic fields become large enough to gravitationally collapse to form PBHs. However, we demonstrate that this picture is incomplete because magnetic fields also increase the threshold value of the density contrast required for PBH formation. The increase in threshold density contrast is more pronounced on smaller scales, and in extreme cases, it might even prevent PBH production despite the presence of significant magnetic field. Taking into account the relevant physical effects on the magnetized overdense region, we establish an upper-limit on the amplitude of comoving magnetic fields, approximately $0.13-0.15 {\rm \mu G}$ at a scale of $10^{17} {\rm Mpc}^{-1}$. Additionally, we compare our constraints with various small-scale probes.