We introduce a new estimator of the peculiar velocity of a galaxy or group of galaxies from redshift and distance estimates. This estimator results in peculiar velocity estimates which are statistically unbiased and that have errors that are Gaussian
distributed, thus meeting the assumptions of analyses that rely on individual peculiar velocities. We apply this estimator to the SFI++ and the Cosmicflows-2 catalogs of galaxy distances and, using the fact that peculiar velocity estimates of distant galaxies are error dominated, examine their error distributions, The adoption of the new estimator significantly improves the accuracy and validity of studies of the large-scale peculiar velocity field and eliminates potential systematic biases, thus helping to bring peculiar velocity analysis into the era of precision cosmology. In addition, our method of examining the distribution of velocity errors should provide a useful check of the statistics of large peculiar velocity catalogs, particularly those that are compiled out of data from multiple sources.
The low order moments, such as the bulk flow and shear, of the large scale peculiar velocity field are sensitive probes of the matter density fluctuations on very large scales. In practice, however, peculiar velocity surveys are usually sparse and no
isy, which can lead to the aliasing of small scale power into what is meant to be a probe of the largest scales. Previously, we developed an optimal ``minimum variance (MV) weighting scheme, designed to overcome this problem by minimizing the difference between the measured bulk flow (BF) and that which would be measured by an ideal survey. Here we extend this MV analysis to include the shear and octupole moments, which are designed to have almost no correlations between them so that they are virtually orthogonal. We apply this MV analysis to a compilation of all major peculiar velocity surveys, consisting of 4536 measurements. Our estimate of the BF on scales of ~ 100 Mpc/h has a magnitude of |v|= 416 +/- 78 km/s towards Galactic l = 282 degree +/- 11 degree and b = 6 degree +/- 6 degree. This result is in disagreement with LCDM with WMAP5 cosmological parameters at a high confidence level, but is in good agreement with our previous MV result without an orthogonality constraint, showing that the shear and octupole moments did not contaminate the previous BF measurement. The shear and octupole moments are consistent with WMAP5 power spectrum, although the measurement noise is larger for these moments than for the BF. The relatively low shear moments suggest that the sources responsible for the BF are at large distances.
The amplitude of cosmological density fluctuations, sigma_8, has been studied and estimated by analysing many cosmological observations. The values of the estimates vary considerably between the various probes. However, different estimators probe the
value of sigma_8 in different cosmological scales and do not take into account the nonlinear evolution of the parameter at late times. We show that estimates of the amplitude of cosmological density fluctuations derived from cosmic flows are systematically higher than those inferred at early epochs from the CMB because of nonlinear evolution at later times. We discuss the past and future evolution of linear and nonlinear perturbations, derive corrections to the value of sigma_8 and compare amplitudes after accounting for these differences.
Peculiar velocity surveys have non-uniform spatial distributions of tracers, so that the bulk flow estimated from them does not correspond to that of a simple volume such as a sphere. Thus bulk flow estimates are generally not strictly comparable bet
ween surveys, even those whose effective depths are similar. In addition, the sparseness of typical surveys can lead to aliasing of small scale power into what is meant to be a probe of the largest scales. Here we introduce a new method of calculating bulk flow moments where velocities are weighted to give an optimal estimate of the bulk flow of an idealized survey, with the variance of the difference between the estimate and the actual flow being minimized. These minimum variance estimates can be designed to estimate the bulk flow on a particular scale with minimal sensitivity to small scale power, and are comparable between surveys. We compile all major peculiar velocity surveys and apply this new method to them. We find that most surveys we studied are highly consistent with each other. Taken together the data suggest that the bulk flow within a Gaussian window of radius 50 Mpc/h is 407 km/s toward l=287 and b=8. The large-scale bulk motion is consistent with predictions from the local density field. This indicates that there are significant density fluctuations on very large scales. A flow of this amplitude on such a large scale is not expected in the WMAP5-normalized LCDM cosmology, for which the predicted one-dimensional r.m.s. velocity is ~110 km/s. The large amplitude of the observed bulk flow favors the upper values of the WMAP5 error-ellipse, but even the point at the top of the WMAP5 95% confidence ellipse predicts a bulk flow which is too low compared to that observed at >98% confidence level.
We propose to use spatial correlations of the kinetic Sunyaev-Zeldovich (KSZ) flux as an estimator of the peculiar velocity power spectrum. In contrast with conventional techniques, our new method does not require measurements of the thermal SZ signa
l or the X-ray temperature. Moreover, this method has the special advantage that the expected systematic errors are always sub-dominant to statistical errors on all scales and redshifts of interest. We show that future large sky coverage KSZ surveys may allow a peculiar velocity power spectrum estimates of an accuracy reaching ~10%.
We find the nine bulk--flow and shear moments from the SFI++ survey, as well as for subsamples of group and field galaxies. We constrain the velocity power spectrum shape parameter $Gamma$ in linear theory using these moments. A likelihood function f
or $Gamma$ was found after marginalizing over the power spectrum amplitude $sigma_8Omega_m^{0.6}$ using constraints obtained from comparisons between redshift surveys and peculiar velocity data. We have estimated the velocity noise $sigma_*$ from the data since without it our results may be biased. We also performed a statistical analysis of the difference between the field and group catalogues and found that the results from each reflect the same underlying large scale flows. We found that we can constrain the power spectrum shape parameter to be $Gamma=0.15^{+0.18}_{-0.08}$ for the groups catalogue and $Gamma=0.09^{+0.04}_{-0.04}$ for the field galaxy catalogue in fair agreement with the value from WMAP.
To study galactic motions on the largest available scales, we require bulk flow moments whose window functions have as narrow a peak as possible and having as small an amplitude as possible outside the peak. Typically the moments found using the maxi
mum likelihood estimate weights do not meet these criteria. We present a new method for calculating weights for moments that essentially allow us to design the moments window function, subject, of course, to the distribution and uncertainties of the available data.
