Black hole binary formation in the expanding universe: Three body problem approximation

We study black hole MACHO binary formation through three-body interactions in the early universe at $t\sim {10}^{-5}\mathrm{s}.$ The probability distribution functions of the eccentricity and the semimajor axis of binaries as well as of the coalescence time are obtained assuming that the black holes are randomly formed in space. We confirm that the previous order-of-magnitude estimate for the binary parameters is valid within $\sim 50\%$ error. We find that the coalescence rate of the black hole MACHO binaries is $\sim 5\times {10}^{-2}\times 2^{\pm 1}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}/\mathrm{y}\mathrm{r}/\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{x}\mathrm{y},$ taking into consideration several possible factors which may affect this estimate. This suggests that the event rate of coalescing binary black holes will be at least several events per year within 15 Mpc. The first LIGO-VIRGO interferometers in 2001 will be able to verify whether or not the MACHOs are black holes.