  We study spectral and time variability of accreting massive black hole
binaries (MBHBs) at milli-pc separations surrounded by a geometrically thin
circumbinary disc. We present the first computed spectral energy distribution
(SED) and light curves (LCs) from 3D hyper-Lagrangian resolution hydrodynamic
simulations of these systems. We model binaries with mass of $10^6$ M$_\odot$,
eccentricities e=0, 0.9 and mass ratio q=0.1,1. The circumbinary disc has
initial aspect ratio of 0.1, features an adiabatic equation of state, and
evolves under the effect of viscous heating, black body cooling and self
gravity. To compute the SED, we consider black body emission from each disc
element and we add an X-ray corona with luminosity proportional to that of the
mini-discs around each black hole. We find significant variability of the SED,
especially at high energies, which translates into LCs displaying modulations
of a factor of ~ 2 in optical and of ~ 10 in UV and X-rays. We focus on flux
variability in the optical band which will be probed by the Vera Rubin
Observatory (VRO). Modulations on the orbital period and half of the orbital
period are evident in all systems. In equal mass binaries, we find another
longer timescale modulation, linked to an over-density forming at the inner
edge of the disc. Considering the VRO properties, we find that equal mass,
circular binaries are unlikely to be identified, due to the lack of prominent
peaks in their Fourier spectra. Conversely, unequal mass and/or eccentric
binaries can be singled out up to z ~ 0.5 (for systems with $L_{\rm
bol}\approx10^{42}$ erg s$^{-1}$) and z ~ 2 (for systems with $L_{\rm
bol}\approx10^{44}$ erg s$^{-1}$). Identifying electromagnetic signatures of
MBHBs at separations $\sim 10^{-4}-10^{-2}$ pc is crucial for understanding the
physics of the future Laser Interferometer Space Antenna sources and the origin
of the GW background.
