A statistical method to search for recoiling supermassive black holes in active galactic nuclei

We propose an observational test for gravitationally recoiling supermassive black holes (BHs) in active galactic nuclei, based on a correlation between the velocities of BHs relative to their host galaxies, |\Delta v|, and their obscuring dust column densities, \Sigma_{dust} (both measured along the line of sight). We use toy models for the distribution of recoil velocities, BH trajectories, and the geometry of obscuring dust tori in galactic centres, to simulate 2.5x10^5 random observations of recoiling quasars. BHs with recoil velocities comparable to the escape velocity from the galactic centre remain bound to the nucleus, and do not fully settle back to the centre of the torus due to dynamical friction in a typical quasar lifetime. We find that |\Delta v| and \Sigma_ {dust} for these BHs are positively correlated. For obscured (\Sigma_{dust}>0) and for partially obscured (0<\Sigma_{dust}<~2.3 g/m^2) quasars with |\Delta v|>=45 km/s, the sample correlation coefficient between log10(|\Delta v|) and \Sigma_{dust} is r_{45} = 0.28+/-0.02 and r_{45} = 0.13+/-0.02, respectively. Allowing for random +/-100 km/s errors in |\Delta v| unrelated to the recoil dilutes the correlation for the partially obscured quasars to r_{45} = 0.026+/-0.004 measured between |\Delta v| and \Sigma_{dust}. A random sample of >~3,500 obscured quasars with |\Delta v|>=45 km/s would allow rejection of the no-correlation hypothesis with 3 sigma significance 95% of the time. Finally, we find that the fraction of obscured quasars, F_{obs}(|\Delta v|), decreases with |\Delta v| from F_{obs}(<10 km/s)>~0.8 to F_{obs}(>10^3 km/s)<~0.4. This predicted trend can be compared to the observed fraction of type II quasars, and can further test combinations of recoil, trajectory, and dust torus models.