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3.2. Accuracy of Determination of sigmav and wLG

The mock catalogs also enable us to determine the reliability of the small-scale velocity dispersion sigmav derived from VELMOD. This quantity may be viewed as the quadrature sum of true velocity noise (sigmanv) and IRAS velocity prediction errors (sigmaIv) resulting from shot noise and imperfectly modeled nonlinearities. (For the real data, there is an additional contribution from redshift measurement errors, which are zero in the mock catalog.) We can measure both sigmanv and sigmaIv directly from the mock catalogs. To measure velocity noise, we determined sigmau, the rms value of pair velocity differences cz(ri) - cz(rj) of mock catalog TF galaxies within 3500 km s-1 outside of the mock Virgo core, for |ri - rj| leq rmax. We found sigmau to be insensitive to the precise value of rmax, provided it was ltapprox150 km s-1, implying that we are not including the gradient of the true velocity field on these scales. Taking rmax = 150 km s-1, we found sigmau = 71 km s-1, corresponding to sigmanv = sigmau / 21/2 = 50 km s-1. This value is so small because the PM code does not properly model particle-particle interactions on small scales.

We measured the IRAS prediction errors sigmaIv as follows. For each mock TF particle (again, within 3500 km s-1 and outside the mock Virgo core), we computed an IRAS-predicted redshift czI = ri + u(ri) + fri - wLG . n hati, where ri was the true distance of the object, u(ri) was the IRAS-predicted radial peculiar velocity in the Local Group frame (for betaI = 1), f was a zero-point error in the IRAS model (cf. Section 3.3), and wLG was the mock Local Group peculiar velocity, which (just as in the real data) is not known precisely and was treated also as a free parameter. We then minimized the mean squared difference between czI and the actual redshifts czi over the entire TF sample with respect to f and wLG. The rms value of (czi - czI) at the minimum was then our estimate of the quadrature sum of IRAS prediction error and true velocity noise, which we found to be 98 ± 2 km s-1 after averaging over the 20 mock catalogs. Subtracting off the small value of sigmanv found above, we obtain sigmaIv appeq 84 km s-1. This surprisingly small value is indicative of the high accuracy of the IRAS predictions for nearby galaxies not in high-density environments.

The value sigmav = 98 km s-1 is somewhat smaller than the real universe value of sigmav = 125 km s-1 (Section 4.5). Because we wanted the mock catalogs to reflect the errors in the real data, we added artificial velocity noise of 110 km s-1 to the redshift of each mock TF galaxy before applying the VELMOD algorithm, increasing sigmav to 147 km s-1. (11) The mean value of sigmav from the VELMOD runs on the 20 mock catalogs was <sigmav> = 148.7 ± 4.6 km s-1, in excellent agreement with the expected value. We conclude that VELMOD produces an unbiased estimate of the sigmav, just as it does of betaI. The rms error in the determination of sigmav from a single realization is ~ 20 km s-1.

The calculation in which we minimized (czi - czI)2 also yielded estimates of the Cartesian components of Local Group random velocity vector wLG. Their mean values over 20 mock catalogs are given in Table 1, together with the corresponding mean values returned from VELMOD over the 20 mock catalog runs. The two are in excellent agreement. These values reflect an offset between the cosmic microwave background (CMB)-to-LG transformation assigned to the simulation and the average value of wLG assigned by the mock IRAS reconstruction for betaI = 1. We conclude that VELMOD properly measures the Cartesian components of wLG to within ~ 50 km s-1 accuracy per mock catalog.

11 In retrospect, we added more noise than was necessary, but at the time we had a higher estimate of the real universe sigmav. Back

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