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The redshifts of galaxies in the IRAS sample are affected by the same peculiar velocities that we are attempting to measure in the Mark III data set. If we measure redshifts cz in the rest frame of the Local Group, then

Equation A1 (A1)

where v(0) is the peculiar velocity of the Local Group and v(r) is the peculiar velocity at position r. Indeed, because the galaxy density field shows coherence, the galaxy density field measured in redshift space deltag(s) differs systematically from that in real space, deltag(r), as was first described in detail by Kaiser (1987; cf. SW and Strauss 1996 for reviews). The linear perturbation theory assuming gravitational instability enables us to correct for the effects of these velocities. We use here the iteration technique described by Yahil et al. (1991) and Strauss et al. (1992c), as updated by Sigad et al. (1997). The density and velocity fields are calculated within a sphere of radius 12,800 km s-1; the density fluctuation field is assumed to be zero beyond this radius. Here we very briefly reiterate the improvements described in the Sigad et al. paper and emphasize certain differences from the approach there.

In regions in which the IRAS velocity field model predicts a nonmonotonic relation between redshift and distance along a given line of sight, it becomes ambiguous as to how to assign a distance to a galaxy given its redshift (Fig. 1). Our approach is similar to that used throughout this paper: we use our assumed density and velocity fields to calculate a probability distribution of a galaxy along a given line of sight.

Along a given line of sight, we ask for the joint probability distribution of observing a galaxy along a given line of sight, with redshift cz, flux density f, and (unknown) distance r:

Equation A2 (A2)

(cf. eq. [5]). The first term is given by our velocity field model along the line of sight and thus is given by equation (9). For the iteration code, we set sigmav = 150 km s-1, independent of position, similar to the best-fit value we find when we fitted for sigmav from the velocity field data.

The second term is given by the luminosity function of galaxies:

Equation A3 (A3)

where the derivative is needed because the probability density is defined in terms of f, not L. (16) Finally, the third term in equation (A2) is given by the galaxy density distribution along the line of sight (eq. [8]).

As described in Sigad et al. (1997), the calculations of the velocity and density fields are done on a Cartesian grid. Our approach therefore is to assign each galaxy to the grid via cloud-in-cell (weighting by the selection function, of course), where (unlike Sigad et al. 1997) we distribute each galaxy along the line of sight according to the distribution function of expected distance (eq. [A2]). In order to calculate the selection function for an object, we of course need to have a definite position for it; for this purpose, we assign it the expectation value of its distance, following Sigad et al. (1997):

Equation A4 (A4)

Sigad et al. (1997) discuss the use of various filtering techniques to suppress the shot noise in the derived density and velocity fields. While they argue for the use of a power-preserving filter for the comparison of the IRAS and POTENT density fields, we have found through extensive experimentation with mock catalogs that for the VELMOD analysis, a Wiener filter gives the best comparison between the density field and the peculiar velocity data.

Finally, we found that when the iteration technique was run to values of beta gtapprox 1, the density field became unstable in the regions around triple-valued zones, oscillating between iterations. We were able to suppress these by averaging the derived density field at each iteration with that of the iteration preceding it. This has no strong effect on the derived density field for beta < 1.

16 Eq. (144) of SW mistakenly left off this last term. Back.

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