tokens: ratios compared to the theoretical values expected from Doppler boost model. In three of the four candidates (J0247, J0249, and J0254), the observed values show deviations from the expected values at the ≳2σ level in at least two band pairs with p-values of 0.189, 0.013, and 0.113. In J0246, the observed values are generally consistent with those expected from Doppler boost. Figure 4. Observed frequency-dependent variability amplitude ratios compared with the expected values from the Doppler boost model. Different colours represent different combinations of band pairs. The black lines represent the one-to-one relation. The error bars denote 1σ uncertainties. See Section 5.2 for details. Fig. 5 shows the allowed parameter space for the Doppler boost model to explain the observed multiband variability amplitudes (Table 4 ). The parameters considered are the total black hole mass M BH , binary orbital inclination angle i , the fraction of the total emission from the secondary black hole f 2 , and the mass ratio q . Our other fiducial parameters are the rest frame orbital period P orb = P GLS /(1 + z ) and α ν in griz bands (Table 4 ). Figure 5. Doppler boost model parameter space estimates for the four candidate periodic quasars. The four sub-panels for each quasar represent griz bands. The dashed (shaded) contours denote f 2 = 1.0 ( f 2 = 0.8), where f 2 is defined as the fraction of the total emission from the secondary black hole. Different colours represents different mass ratios with q = 0.0, 0.05, 0.1, and 0.2 for blue, green, orange, and red, respectively. The black vertical line with the grey-shaded region shows the virial mass estimated from SDSS spectra (Fig. 3 ) and its 1σ statistical error. The orbital inclination angle i = 90° for an edge-on view and i = 0° for a face-on view. For J0246, while the observed frequency-dependent variability amplitudes are generally consistent with the expected values under the Doppler boost hypothesis (Fig. 4 ), there is almost no allowed parameter space unless the following conditions are all met: 1. The virial black mass is significantly underestimated, even after considering a 0.4-dex systematic error (e.g. Shen 2013 ), 2. A high fraction (e.g. ) of the total emission is from the secondary black hole fueled by its mini accretion disc, and 3. The binary system is close to being an edge-on view. For the other three quasars, the observed multiband variability amplitudes are generally consistent with the expected values under the Doppler boost hypothesis, although there is a stringent requirement on the allowed parameter space. Liao et al. ( 2020 ) have tested the Doppler boost hypothesis for J0252. They suggest that the Doppler boost model is disfavoured for J0252 based on both the frequency-dependent variability amplitude ratios and the multiband variability amplitudes. In summary, there is evidence against the Doppler boost hypothesis in all five periodic candidates in our parent sample based on the
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