Likelihood Analysis of Large-Scale Flows

We apply a likelihood analysis to the data of \markcite{Lauer \& Postman 1994} With $P(k)$ parametrized by $(\sigma_8, \Gamma)$, the likelihood function peaks at $\sigma_8\simeq0.3$, $\Gamma\lesssim0.025$, indicating at face value very strong large-scale power, though 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 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 \& Postman velocity data alone are therefore {\em 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 {\em very}\/ large scales, which is strongly disfavored by COBE\@. The velocity data also seem compatible with more conventional $P(k)$ with $0.2\lesssim\Gamma\lesssim0.5$, and the likelihood is peaked around $\sigma_8\sim1$, in which case the bulk flow is a moderate, but not extreme, statistical fluctuation. Applying the same techniques to the data of \markcite{Riess, Press, \& Kirshner 1995}, 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: $\sigma_8\lesssim1.5$ at roughly 99\%.