Detectability of Cosmic Dark Flow in the Type Ia Supernova Redshift-Distance Relation

We re-analyze the detectability of large scale dark flow (or local bulk flow) with respect to the CMB background based upon the redshift-distance relation for Type Ia supernovae (SN Ia). We made two independent analyses: one based upon identifying the three Cartesian velocity components; and the other based upon the cosine dependence of the deviation from Hubble flow on the sky. We apply these analyses to the Union2.1 SN Ia data and to the SDSS-II supernova survey. For both methods, results for low redshift, $z < 0.05$, are consistent with previous searches. We find a local bulk flow of $v_{\rm bf} \sim 300$ km s$^{-1}$ in the direction of $(l,b) \sim (270, 35)^{\circ}$. However, the search for a dark flow at $z>0.05$ is inconclusive. Based upon simulated data sets, we deduce that the difficulty in detecting a dark flow at high redshifts arises mostly from the observational error in the distance modulus. Thus, even if it exists, a dark flow is not detectable at large redshift with current SN Ia data sets. We estimate that a detection would require both significant sky coverage of SN Ia out to $z = 0.3$ and a reduction in the effective distance modulus error from 0.2 mag to $\lesssim 0.02$ mag. We estimate that a greatly expanded data sample of $\sim 10^4$ SN Ia might detect a dark flow as small as 300 km s$^{-1}$ out to $z = 0.3$ even with a distance modulus error of $0.2$ mag. This may be achievable in a next generation large survey like LSST.