Eccentricity evolution of massive black hole binaries from formation to
  coalescence
  Coalescing supermassive black hole binaries (BHBs) are expected to be the
loudest sources of gravitational waves (GWs) in the Universe. Detection rates
for ground or space-based detectors based on cosmological simulations and
semi-analytic models are highly uncertain. A major difficulty stems from the
necessity to model the BHB from the scale of the merger to that of inspiral. Of
particular relevance to the GW merger timescale is the binary eccentricity.
Here we present a self-consistent numerical study of the eccentricity of BHBs
formed in massive gas-free mergers from the early stages of the merger to the
hardening phase, followed by a semi-analytical model down to coalescence. We
find that the early eccentricity of the unbound black hole pair is largely
determined by the initial orbit. It systematically decreases during the
dynamical friction phase. The eccentricity at binary formation is affected by
stochasticity and noise owing to encounters with stars, but preserves a strong
correlation with the initial orbital eccentricity. Binding of the black holes
is a phase characterised by strong perturbations, and we present a quantitative
definition of the time of binary formation. During hardening the eccentricity
increases in minor mergers, unless the binary is approximately circular, but
remains largely unchanged in major mergers, in agreement with predictions from
semi-analytical models based on isotropic scattering experiments. Coalescence
times due to hardening and GW emission in gas-poor non-rotating ellipticals are
<~0.5 Gyr for the large initial eccentricities (0.5 < e < 0.9) typical of
galaxy mergers in cosmological simulations.
