Inspiral Time Probability Distribution for Two Black Holes Captured by Emitting Gravitational Radiation

Abstract

If two initially unbound black holes of masses M_1 and M_2, total mass M = M_1 + M_2, reduced mass mu = M_1 M_2/(M_1+M_2), and initial relative velocity v << c(4 mu/M) in otherwise empty space are captured into a bound orbit by emitting gravitational radiation, the inspiral time to coalescence increases monotonically to infinity as the impact parameter b approaches from below the critical impact parameter b_c = [340 pi G^7 M^6 mu/(3 c^5 v^9)]^{1/7} = [(85 pi/384)(4 mu/M)]^{1/7}(2GM/c^2)(v/c)^{-9/7} for capture. Assuming a uniform flux of impinging black holes with b < b_c, the cumulative probability for impact parameters smaller than some value $b$, conditional upon the impact parameter being smaller than $b_c$, is $P = (b/b_c)^2$. Then it is shown that the inspiral time for [Mv^2/(4 mu c^2)]^{2/7} << P < 1 is T = (2 pi GM/v^3) P^{21/4} zeta(3/2,1-P^{7/2}), and closed-form approximate expressions for the inverse function P(T/T_0) with T_0 = 2 pi GM/v^3 are also given.