  We investigate gravitational capture of magnetic monopoles by primordial
black holes (PBH) that evaporate before Big Bang Nucleosynthesis (BBN), a
hypothetical process which was once proposed as an alternative solution to the
monopole problem. Magnetic monopoles produced in phase transitions of a grand
or partially unified gauge theory are considered. We prove analytically that
for all extended PBH mass functions that preserve radiation domination, it is
impossible to reduce the monopole abundance via gravitational capture by PBHs
to values significantly below the one set by monopole annihilation (or below
its initial abundance if it is smaller), regardless of the nature of the
capture process (diffusive or non-diffusive). Therefore, the monopole problem
cannot be solved by PBH capture in a radiation-dominated era in the early
universe.
