Evolution of axial-perturbations in space-time of a non-rotating uncharged primordial black hole and stability criteria for it

We derive the equation governing the axial-perturbations in the space-time of a non-rotating uncharged primordial black hole (PBH), produced in early Universe, whose metric is taken as generalized McVittie metric. The generalized McVittie metric is a cosmological black hole metric, proposed by V. Faraoni and A. Jacques in 2007 [Phys. Rev. D 76, 063510 (2007)]. This describes the space-time of a Schwarzschild black hole embedded in FLRW-Universe, while allowing its mass-change. Our derivation is quite similar to the procedure of derivation of S. Chandrasekhar, for deriving the Regge-Wheeler equation for Schwarzschild metric [S. Chandrasekhar, The Mathematical Theory of Black holes ; Oxford University Press (1983)]. The equation we derive, is the equivalent counterpart of the Regge-Wheeler equation, in case of generalized McVittie metric. Using this equation, we have derived a condition for which the imaginary part of the frequency, of any mode of perturbations, would be greater than zero or in other words, the condition for which the corresponding modes grow exponentially with time. This is actually the stability criteria for the axial-perturbations in the generalized McVittie metric.