  Supermassive black hole binaries (SMBHBs) produced in galaxy mergers are
thought to complete their coalescence, below separations of r_GW=10^{-3}
(M_BH/10^8 M_sun)^{3/4} pc, as their orbit decays due to the emission of
gravitational waves (GWs). It may be possible to identify such GW-driven
inspirals statistically in an electromagnetic (EM) survey for variable sources.
A GW-driven binary spends a characteristic time T_GW at each orbital separation
r_orb < r_GW that scales with the corresponding orbital time t_orb as T_GW =
(const) t_orb^{8/3}. If the coalescing binary produces variations in the EM
emission on this timescale, then it could be identified as a variable source
with a characteristic period t_var = t_orb. The incidence rate of sources with
similar inferred BH masses, showing near-periodic variability on the time-scale
t_var, would then be proportional to t_var^{8/3}. Luminosity variations
corresponding to a fraction f_Edd<0.01 of the Eddington luminosity would have
been missed in current surveys. However, if the binary inspirals are associated
with quasars, we show that a dedicated survey could detect the population of
SMBHBs with a range of periods around tens of weeks. The discovery of a
population of periodic sources whose abundance obeys N_var = (const)
t_var^{8/3} would confirm (i) that the orbital decay is indeed driven by GWs,
and (ii) that circumbinary gas is present at small orbital radii and is being
perturbed by the BHs. Deviations from the t_var^{8/3} power-law could constrain
the structure of the circumbinary gas disk and viscosity-driven orbital decay.
We discuss constraints from existing data, and quantify the sensitivity and sky
coverage that could yield a detection in future surveys.
