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tokens: we assume a completeness , relative to CRTS, 7 i.e. 2.4 Models of variability The luminosity of an object in a binary system will not necessarily vary on the orbital period or at all. The premise of photometric identification of MBHBs is that the binary period is somehow imprinted into variations of the observed luminosity. In the particularly convincing example of PG 1302-102 (Graham et al. 2015b ), sinusoidal variations in the light curve can be well explained by Doppler boosting from a mildly relativistic orbital velocity (D’Orazio, Haiman Schiminovich 2015b ; but see also Liu, Gezari Miller 2018 ). Additionally, purely hydrodynamic modulations to accretion rates have been observed in simulations (e.g. Farris et al. 2014 ). Here, we describe models for both types of variability mechanisms. Throughout our analysis, each variability mechanism is considered entirely independently, i.e. we do not consider systems which may be both hydrodynamically and Doppler variable. The physical scenario is depicted schematically in Fig. 5 , assuming thin-disc geometries and a mass ratio q ≳ 10 −2 (see Section 2.4.2 ). On the left, the two MBHs, each with a circum-single disc, are shown within a cavity (‘gap’) of material evacuated by the binary orbit that separates them from the circumbinary disc. The radii of the circum-single discs are determined by the Hill radius of each MBH (see equation 6 ). Despite the presence of the gap, the circumbinary disc continues to transport angular momentum outwards, requiring material to be accreted inwards. Material from the disc overflows across the gap as accretion streams on to each circum-single disc. The orbital period at the inner edge of the circumbinary disc is longer than that of the binary, causing periodic variations in the accretion rate on to each circum-single disc (Fig. 5 , upper right). The secondary MBH is farther from the centre of mass and closer to the circumbinary disc edge, which leads to it receiving a disproportionate fraction of the accreting material [as described by equation ( 1 ) and shown in Fig. 1 ]. Each circum-single disc, in addition to the circumbinary disc (although typically to a lesser extent), will produce AGN-like emission. 2.4.1 Doppler variability Any component of the binary orbital-velocity along the observer’s line of sight will lead to both a relativistic boost and a Doppler shift in the observed spectrum of each circum-single disc. The boost is calculated using the Doppler factor, D ≡ [γ(1 − || / c )] −1 , where the Lorentz factor γ ≡ (1 − 2 / c 2 ) −1/2 . The line-of-sight velocity, || ≡ sin ( i ), depends on the binary inclination i , which we define as zero for face-on systems. The observed flux is calculated as (D’Orazio et al. 2015b ) where the observed frequency ν is related to the rest-frame frequency as, . Assuming a power law of index α ν for the section of the spectrum being observed, the Doppler variation in flux from the source will be (Charisi et al. 2018 ) where