Gravitational wave driven mergers and coalescence time of supermassive black holes

Aims. The evolution of supermassive black holes (SMBHs) initially embedded in the centres of merging galaxies realised with a stellar mass function (SMF) is studied from the onset of galaxy mergers until coalescence. Coalescence times of SMBH binaries are of great importance for black hole evolution and gravitational wave detection studies. Methods. We performed direct N-body simulations using the highly efficient and massively parallel phi-GRAPE+GPU code capable of running on high-performance computer clusters supported by graphic processing units (GPUs). Post-Newtonian terms up to order 3.5 are used to drive the SMBH binary evolution in the relativistic regime. We performed a large set of simulations with three different slopes of the central stellar cusp and different random seeds. The impact of a SMF on the hardening rate and the coalescence time is investigated. Results. We find that SMBH binaries coalesce well within one billion years when our models are scaled to galaxies with a steep cusp at low redshift. Here higher central densities provide a larger supply of stars to efficiently extract energy from the SMBH binary orbit and shrink it to the phase where gravitational wave (GW) emission becomes dominant, leading to the coalescence of the SMBHs. Mergers of models with shallow cusps that are representative of giant elliptical galaxies having central cores result in less efficient extraction of the binary's orbital energy, due to the lower stellar densities in the centre. However, high values of eccentricity witnessed for SMBH binaries in such galaxy mergers ensure that the GW emission dominated phase sets in earlier at larger values of the semi-major axis. This helps to compensate for the less efficient energy extraction during the phase dominated by stellar encounters resulting in mergers of SMBHs in about 1 Gyr after the formation of the binary. Additionally, we witness mass segregation in the merger remnant resulting in enhanced SMBH binary hardening rates. We show that at least the final phase of the merger in cuspy low-mass galaxies would be observable with the GW detector eLISA.