Inspiral time probability distribution for two black holes captured by emitting gravitational radiation

If two initially unbound black holes of masses ${M}_{1}$ and ${M}_{2}$, total mass $M={M}_{1}+{M}_{2}$, reduced mass $\ensuremath{\mu}={M}_{1}{M}_{2}/({M}_{1}+{M}_{2})$, and initial relative velocity $v\ensuremath{\ll}c(4\ensuremath{\mu}/M)$ 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 ${b}_{c}=[340\ensuremath{\pi}{G}^{7}{M}^{6}\ensuremath{\mu}/(3{c}^{5}{v}^{9}){]}^{1/7}=[(85\ensuremath{\pi}/384)(4\ensuremath{\mu}/M){]}^{1/7}(2GM/{c}^{2})(v/c{)}^{\ensuremath{-}9/7}$ for capture. Assuming a uniform flux of impinging black holes with $b\ensuremath{\le}{b}_{c}$, 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 $[M{v}^{2}/(4\ensuremath{\mu}{c}^{2}){]}^{2/7}\ensuremath{\ll}P<1$ is $T=(2\ensuremath{\pi}GM/{v}^{3}){P}^{21/4}\ensuremath{\zeta}(3/2,1\ensuremath{-}{P}^{7/2})$, and closed-form approximate expressions for the inverse function $P(T/{T}_{0})$ with ${T}_{0}=2\ensuremath{\pi}GM/{v}^{3}$ are also given.