Limits on primordial magnetic fields from primordial black hole
  abundance
  Primordial magnetic field (PMF) is one of the feasible candidates to explain
observed large-scale magnetic fields, for example, intergalactic magnetic
fields. We present a new mechanism that brings us information about PMFs on
small scales based on the abundance of primordial black holes (PBHs). The
anisotropic stress of the PMFs can act as a source of the super-horizon
curvature perturbation in the early universe. If the amplitude of PMFs is
sufficiently large, the resultant density perturbation also has a large
amplitude, and thereby, the PBH abundance is enhanced. Since the anisotropic
stress of the PMFs is consist of the square of the magnetic fields, the
statistics of the density perturbation follows the non-Gaussian distribution.
Assuming Gaussian distributions and delta-function type power spectrum for
PMFs, based on a Monte-Carlo method, we obtain an approximate probability
density function of the density perturbation, and it is an important piece to
relate the amplitude of PMFs with the abundance of PBHs. Finally, we place the
strongest constraint on the amplitude of PMFs as a few hundred nano-Gauss on
$10^{2}\;{\rm Mpc}^{-1} \leq k\leq 10^{18}\;{\rm Mpc}^{-1}$ where the typical
cosmological observations never reach.
