The spin of the second-born black hole in coalescing binary black holes
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Various binary black hole formation channels have been proposed since the first gravitational event GW150914 was discovered by the Advanced Laser Interferometer Gravitational-Wave Observatory (AdLIGO). For all evolutionary channels based on the evolution of isolated binaries, the immediate progenitor of the binary black hole is a close binary system composed of a black hole and a helium star. We perform detailed binary evolution and systematically explore the parameter space of initial binary properties, including initial black hole and helium star masses, initial rotation of the helium star as well as metallicity. We argue that the spin of the first-born black hole at its birth is negligible ($\lesssim 0.1$), hence the second-born black hole's spin dominates the measured effective spin, $\chi_{\rm eff}$, from gravitational wave events of double black hole mergers. We find that tides can be important only when orbital periods are shorter than 2 days. Upon core collapse, the helium star produces a black hole (the second-born black hole in the system) with a spin that can span the entire range from zero to maximally spinning. We show that the bimodal distribution of the spin of the second-born black hole obtained in recent papers is mainly due to oversimplifying assumptions. We find an anti-correlation between the merging timescale of the two black holes, T$_{\rm merger}$, and the effective spin $\chi_{\rm eff}$. Finally, we provide new prescriptions for the tidal coefficient E$_2$ for both H-rich and the helium-rich stars. We predict that, with future improvements to AdLIGO's sensitivity, the sample of merging binary black hole systems will show an overdensity of sources with positive but small $\chi_{\rm eff}$. originating from lower mass black hole mergers born at low redshift.
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