  Gravitational Waves (GW) have now been detected from stellar-mass black hole
binaries, and the first observations of GW from Massive Black Hole (MBH)
Binaries are expected within the next decade. Pulsar Timing Arrays (PTA), which
can measure the years long periods of GW from MBHB, have excluded many standard
predictions for the amplitude of a stochastic GW Background (GWB). We use
coevolved populations of MBH and galaxies from hydrodynamic, cosmological
simulations ('Illustris') to calculate a predicted GWB. The most advanced
predictions so far have included binary hardening mechanisms from individual
environmental processes. We present the first calculation including all of the
environmental mechanisms expected to be involved: dynamical friction, stellar
'loss-cone' scattering, and viscous drag from a circumbinary disk. We find that
MBH binary lifetimes are generally multiple gigayears, and only a fraction
coalesce by redshift zero. For a variety of parameters, we find all GWB
amplitudes to be below the most stringent PTA upper limit of
$A_{\textrm{yr}^{-1}} \approx 10^{-15}$. Our fairly conservative fiducial model
predicts an amplitude of $A_{\textrm{yr}^{-1}} \approx 0.4\times
10^{-15}$---less than a factor of three below the current limit. At lower
frequencies, we find $A_{0.1\,\textrm{yr}^{-1}} \approx 1.5\times 10^{-15}$
with spectral indices between $-0.4$ and $-0.6$---significantly flatter than
the canonical value of $-2/3$ due to purely GW-driven evolution. Typical MBHB
driving the GWB signal come from redshifts around $0.3$, with total masses of a
few times $10^9\,M_\odot$, and in host galaxies with very large stellar masses.
Even without GWB detections, our results can be connected to observations of
dual AGN to constrain binary evolution.
