Self consistent model for the evolution of eccentric massive black hole
  binaries in stellar environments: implications for gravitational wave
  observations
  We construct evolutionary tracks for massive black hole binaries (MBHBs)
embedded in a surrounding distribution of stars. The dynamics of the binary is
evolved by taking into account the erosion of the central stellar cusp bound to
the massive black holes, the scattering of unbound stars feeding the binary
loss cone, and the emission of gravitational waves (GWs). Stellar dynamics is
treated in a hybrid fashion by coupling the results of numerical 3-body
scattering experiments of bound and unbound stars to an analytical framework
for the evolution of the stellar density distribution and for the efficiency of
the binary loss cone refilling. Our main focus is on the behaviour of the
binary eccentricity, in the attempt of addressing its importance in the merger
process and its possible impact for GW detection with the planned Laser
Interferometer Space Antenna ({\it LISA}), and ongoing and forthcoming pulsar
timing array (PTA) campaigns. We produce a family of evolutionary tracks
extensively sampling the relevant parameters of the system which are the binary
mass, mass ratio and initial eccentricity, the slope of the stellar density
distribution, its normalization and the efficiency of loss cone refilling. We
find that, in general, stellar dynamics causes a dramatic increase of the MBHB
eccentricity, especially for initially already mildly eccentric and/or unequal
mass binaries. When applied to standard MBHB population models, our results
predict eccentricities in the ranges $10^{-3}-0.2$ and $0.03-0.3$ for sources
detectable by {\it LISA} and PTA respectively. Such figures may have a
significant impact on the signal modelling, on source detection, and on the
development of parameter estimation algorithms.
