  We study the dynamical evolution of quasi-circular equal mass massive black
hole binaries embedded in circumbinary discs from separations of $\sim
100R_{\rm g}$ down to the merger, following the post merger evolution. The
binary orbit evolves owing to the presence of the gaseous disc and the addition
of Post-Newtonian (PN) corrections up to the 2.5 PN order, therefore including
the dissipative gravitational wave back-reaction. We investigate two cases of a
relatively cold and warm circumbinary discs, with aspect ratios
$H/R=0.03,\,0.1$ respectively, employing 3D hyper-Lagrangian resolution
simulations with the {\sc gizmo}-MFM code. We extract spectral energy
distributions and light curves in different frequency bands (i.e. X-ray,
optical and UV) from the simulations. We find a clear two orders of magnitude
drop in the X-ray flux right before merger if the disc is warm while we
identify a significant increase in the UV flux regardless of the disc
temperature.
  The optical flux shows clear distinctive modulations on the binary orbital
period and on the cavity edge period, regardless of the disc temperature.
  We find that the presence of a cold disc can accelerate the coalescence of
the binary by up to 130 seconds over the last five days of inspiral, implying a
phase shift accumulation of about $\pi\,$radians compared to the binary
evolution in vacuum. These differences are triggered by the presence of the
gaseous disc and might have implications on the waveforms that can be in
principle detected.
  We discuss the implications that these distinctive signatures might have for
existing and upcoming time domain surveys and for multimessenger astronomy.
