Neutron stars harboring a primordial black hole: Maximum survival time

We explore in general relativity the survival time of neutron stars that host an endoparasitic, possibly primordial, black hole at their center. Corresponding to the minimum steady-state Bondi accretion rate for adiabatic flow that we found earlier for stiff nuclear equations of state (EOSs), we derive analytically the maximum survival time after which the entire star will be consumed by the black hole. We also show that this maximum survival time depends only weakly on the stiffness for polytropic EOSs with $\mathrm{\Gamma }\ge 5/3$, so that this survival time assumes a nearly universal value that depends on the initial black-hole mass alone. Establishing such a value is important for constraining the contribution of primordial black holes in the mass range ${10}^{-16}{M}_{\odot }\lesssim M\lesssim {10}^{-10}{M}_{\odot }$ to the dark-matter content of the Universe.