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, vertical bar Delta v vertical bar, 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 2.5 x 10(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 vertical bar Delta v vertical bar and Sigma(dust) for these BHs are positively correlated. For obscured (Sigma(dust) > 0) and for partially obscured (0 < Sigma(dust) less than or similar to 2.3 gm(-2)) quasars with vertical bar Delta v vertical bar = 45 km s(-1), the sample correlation coefficient between log(10)(vertical bar Delta v vertical bar) and Sigma(dust) is r(45) = 0.28 +/- 0.02 and r(45) = 0.13 +/- 0.02, respectively. Allowing for random +/- 100 km s(-1) errors in vertical bar Delta v vertical bar unrelated to the recoil dilutes the correlation for the partially obscured quasars to r(45) = 0.026 +/- 0.004 measured between vertical bar Delta v vertical bar and Sigma(dust). A random sample of greater than or similar to 3500 obscured quasars with vertical bar Delta v vertical bar >= 45 km s(-1) would allow rejection of the no-correlation hypothesis with 3 sigma significance 95 per cent of the time. Finally, we find that the fraction of obscured quasars, F-obs (vertical bar Delta v vertical bar), decreases with vertical bar Delta v vertical bar from F-obs (< 10 km s(-1)) greater than or similar to 0.8 to F-obs (> 10(3) km s(-1)) less than or similar to 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.