The error budget of the Dark Flow measurement

We analyze the uncertainties and possible systematics associated with the "Dark Flow" measurements using the cumulative Sunyaev-Zeldovich effect combined with all-sky catalogs of clusters of galaxies. Filtering of all-sky cosmic microwave background (CMB) maps is required to remove the intrinsic cosmological signal down to the limit imposed by cosmic variance. Contributions to the errors come from the remaining cosmological signal, that integrates down with the number of clusters, and the instrumental noise, that scales with the number of pixels; the latter decreases with integration time and is subdominant for the Wilkinson Microwave Anisotropy Probe 5-year data. It is proven both analytically and numerically that the errors for 5-year WMAP data are $\simeq 15\sqrt{3/N_{\rm clusters}} \mu$K per dipole component. The measured bulk flow velocity is significant at up to the $>99.95%$ confidence level, depending on the cluster sample. We discuss different methods to compute error bars and demonstrate that they have biases that would overpredict the errors, as is the case in a recent reanalysis of our earlier results. If the signal is caused by systematic effects present in the data, such systematics must have a dipole pattern, which correlates with cluster X-ray luminosity and is present only at cluster positions. Only contributions from the Sunyaev-Zeldovich effect could provide such contaminants via several potential effects. We discuss such candidates apart from the bulk-motion of the cluster samples and demonstrate that their contributions to our measurements are negligible. (abridged)