A simple toy model for the electromagnetic variability of lump-dominated circumbinary disks around binary black holes

Context. The electromagnetic detection of circumbinary disks around pre-merger binary black holes (BBHs) relies on theoretical predictions. These are generally obtained through expensive numerical simulations, but simple or fast toy models are lacking to unleash the potential of these theoretical advances for observational purposes. Aims. We present a simple toy model for computing the electromagnetic variability of circumbinary disks around circular-orbit BBHs at relativistic separations. We focus on the effect of disk nonaxisymmetries. Methods. We assumed that the disk is threaded by spiral arms and hosts a hotspot linked to an overdense structure (the lump) that is preferably reported in binaries of a close to equal mass. We built a simple temperature distribution and estimated its thermal emission, perceived by a distant observer, via a ray-tracing code in a BBH approximate metric. We propose a toy model that reproduces the main light-curve features and show that it is consistent with 2D general relativistic hydrodynamical simulations under the assumption of compressional heating and expansional cooling, except for purely dynamical effects such as the binary-lump beat. Results. The light curve exhibits a main modulation at the lump period (i.e., a few times the orbital period) due to the relativistic Doppler effect, and a shorter modulation at the orbit-like period due to spiral arms or the beat. These are more prominent in the optical/UV band for a total binary mass M = 104 − 10 M⊙, where the disk energy spectrum peaks. For M = 109 M⊙, a lump modulation with an amplitude of 4% is detectable with the Vera Rubin Observatory after six months of observations up to z = 0.5. Conclusions. We proposed a new simple toy model that can be used, for instance, to test the compatibility of the periodicity of BBH candidate sources with a circumbinary disk origin.