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Register a new userCosmic Flows on 100 Mpc/h Scales: Standardized Minimum Variance Bulk Flow, Shear and Octupole Moments
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Hume A. Feldman
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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 noisy, 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.
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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 maximum 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.
We measure the bulk flow of the local Universe using the 6dF Galaxy Survey peculiar velocity sample (6dFGSv), the largest and most homogeneous peculiar velocity sample to date. 6dFGSv is a Fundamental Plane sample of $sim10^4$ peculiar velocities covering the whole southern hemisphere for galactic latitude $|b| > 10^circ$, out to redshift ${z=0.0537}$. We apply the `Minimum Variance bulk flow weighting method, which allows us to make a robust measurement of the bulk flow on scales of $50$ and $70,h^{-1}{rm Mpc}$. We investigate and correct for potential bias due to the lognormal velocity uncertainties, and verify our method by constructing $Lambda{rm CDM}$ 6dFGSv mock catalogues incorporating the survey selection function. For a hemisphere of radius $50,h^{-1}{rm Mpc}$ we find a bulk flow amplitude of $U=248pm58,{rm km},{rm s}^{-1}$ in the direction $(l,b) = (318^circpm20^circ,40^circpm13^circ)$, and for $70,h^{-1}{rm Mpc}$ we find $U=243pm58,{rm km},{rm s}^{-1}$, in the same direction. Our measurement gives us a constraint on $sigma_8$ of $1.01^{+1.07}_{-0.58}$. Our results are in agreement with other recent measurements of the direction of the bulk flow, and our measured amplitude is consistent with a $Lambda{rm CDM}$ prediction.
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 between 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.
Using N-body simulations we study the impact of various systematic effects on the bulk flow (BF) and the Cosmic Mach Number (CMN). We consider two types of systematics: those related to survey properties and those induced by observers location in the Universe. In the former category we model sparse sampling, velocity errors, and survey incompleteness. In the latter, we consider Local Group (LG) analogue observers, placed in a specific location within the Cosmic Web, satisfying various observational criteria. We differentiate such LG observers from Copernican ones, who are at random locations. We report strong systematic effects on the measured BF and CMN induced by sparse sampling, velocity errors and radial incompleteness. For BF most of these effects exceed 10% for scales $Rleq100h^{-1}$Mpc. For CMN some of these systematics can be catastrophically large ($>50%$) also on bigger scales. Moreover, we find that the position of the observer in the Cosmic Web significantly affects the locally measured BF (CMN), with effects as large as $sim20%$ ($30%)$ at $Rleq50h^{-1}$Mpc for a LG-like observer as compared to a random one. This effect is comparable to the sample variance. To highlight the importance of these systematics, we additionally study a model of modified gravity (MG) with $sim15%$ enhanced growth rate. We found that the systematic effects can mimic the modified gravity signal. The worst-case scenario is realized for a case of a LG-like observer, when the effects induced by local structures are degenerate with the enhanced growth rate fostered by MG. Our results indicate that dedicated constrained simulations and realistic mock galaxy catalogs will be absolutely necessary to fully benefit from the statistical power of the forthcoming peculiar velocity data from surveys such as TAIPAN, WALLABY, Cosmic Flows-4 and SKA.
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 for $Gamma$ 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.
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