LIKELIHOOD ANALYSIS OF LARGE-SCALE FLOWS

We apply a likelihood analysis to the data of Lauer and Postman, with P(k) parameterized by (sigma(8), Gamma), the likelihood function peaks at sigma(8) similar or equal to 0.3, Gamma less than or similar to 0.025, indicating at face value very strong large-scale power, although at a level incompatible with COBE. There is, however, a ridge of likelihood such that more conventional power spectra do not seem strongly disfavored. The likelihood calculated using as data only the components of the bulk flow solution peaks at higher sigma(8), in agreement with other analyses, but it is rather broad. The likelihood incorporating both bulk flow and shear gives a different picture. The components of the shear are all low, and this pulls the peak to lower amplitudes as a compromise. The Lauer and Postman velocity data alone are therefore consistent with models with very strong large-scale power which generates a large bulk flow, but the small shear (which also probes fairly large scales) requires that the power would have to be at very large scales, which is strongly disfavored by COBE. The velocity data also seem compatible with more conventional P(k) with 0.2 less than or similar to Gamma less than or similar to 0.5, and the likelihood is peaked around sigma(8) similar to 1, in which case the bulk flow is a moderate, but not extreme, statistical fluctuation. If we apply the same techniques to the data of Riess, Press, & Kirshner, the results are quite different. The flow is not inconsistent with the microwave dipole, and we derive only an upper limit to the amplitude of the power spectrum: sigma(8) less than or similar to 1.5 at similar to 99%.