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6.1. What is the Value of betaI?

VELMOD recovers the correct answer, betaI = 1, to less than 10% accuracy when applied to the mock catalogs. At betaI = 1, the velocity field in the mock Virgo region is significantly triple-valued. Thus, VELMOD, despite being close in spirit to Method II, properly treats triple-valuedness. If the strong triple-valuedness one sees at betaI = 1 were present in the real universe, VELMOD would not assign it an unduly small likelihood. Nonetheless, when VELMOD is applied to the real universe, it returns a value of betaI = 0.492 ± 0.068 (quadrupole modeled). This value is quite insensitive to two other quantities treated as free parameters in the velocity field model, the Local Group random velocity wLG and the small-scale velocity dispersion sigmav (Section 4.5). Tests with the mock catalogs demonstrated that we obtain an unbiased betaI using a 300 km s-1-smoothed IRAS reconstruction (Section 3.1). However, we found that changing to a 500 km s-1-smoothed reconstruction makes relatively little difference in betaI (Section 4.6). Finally, neglecting the quadrupole causes betaI to change by only ~ 1 sigma. Our conclusion that betaI appeq 0.5 ± 0.07 is thus robust against systematic effects internal to our method.

The VELMOD result is consistent with the relatively low estimates of betaI obtained from the Method II analyses of Hudson (1994), Roth (1994), Shaya et al. (1995), (14) DNW, and Schlegel (1995), as well as those derived from comparisons of the IRAS density field with the motion of the Local Group (Strauss et al. 1992b) and from some analyses of the redshift-space anisotropy of the IRAS density field (e.g., Hamilton 1993, 1995; Fisher et al. 1994a, 1994b; Cole, Fisher, & Weinberg 1995; Fisher & Nusser 1996). However, it is apparently inconsistent with estimates of betaI near unity, as have been found by the POTIRAS analysis (Sigad et al. 1997), measurements of the POTENT fluctuation amplitude (Kolatt & Dekel 1997; Zaroubi et al. 1997), and redshift-space distortions of spherical harmonic expansions of the density field (Fisher, Scharf, & Lahav 1994c; Fisher 1994; Heavens & Taylor 1995).

6.1.1. Why Do VELMOD and POTIRAS Yield Different Values of betaI?

We do not yet have a satisfactory explanation of why VELMOD and standard Method II analyses characteristically yield smaller values of betaI than the Method I POTIRAS approach. One possibility is that the differences stem from the Method I/Method II distinction. However, VELMOD corrects the principal drawback of Method II, the inability to deal with multivalued or flat zones in the redshift-distance relation. Thus, if the Method I/Method II distinction is at the root of the discrepancy, the reason must be more subtle than the drawbacks of standard Method II. Sigad et al. (1997) test for biases in POTIRAS using the same mock catalogs as this paper; they also find their determination of betaI to be essentially unbiased. The problem could lie with the Malmquist bias corrections that are so crucial to Method I (cf. the discussion in Willick et al. 1997). If these corrections are underestimated for any reason - e.g., the TF scatter is larger than estimated, or the density fluctuations are larger than modeled - a Method I approach will produce too strong velocity gradients and thus overestimate betaI. However, the TF scatters used by Sigad et al. (1997) are consistent with those obtained in this paper, and the large POTENT smoothing limits the effect of Malmquist bias in any case. Thus, it is unlikely that improper Malmquist bias corrections strongly affect the value of betaI obtained from POTIRAS.

An important difference between VELMOD and POTENT is the Gaussian smoothing scales employed, 300 and 1200 km s-1, respectively. These very different smoothings could result in different values of betaI if the effective bias parameters on these scales are different. In order to reconcile VELMOD and POTIRAS, we would need the effective bias parameter to decrease by a factor of 1.7 between scales of 300 and 1200 km s-1. Such a scale-dependent biasing has been suggested by the galaxy formation models of Kauffman, Nusser, & Steinmetz (1996), but Weinberg (1995) and Jenkins et al. (1996) do not find these trends. A recent analysis by Nusser & Dekel (1997) that fits the IRAS and peculiar velocity data simultaneously finds betaI = 1.0 for 1200 km s-1 smoothing, but only 0.6 for 600 km s-1 smoothing, approaching the value we have found in this paper. Such a change of betaI with smoothing scale could signal scale-dependent biasing.

Still another difference is the volume considered. We have restricted this analysis to cz leq 3000 km s-1 (Section 4), whereas the analysis of Sigad et al. (1997) extends to 6000 km s-1; only ~ 1/3 of the points used fall within 3000 km s-1. If, for whatever reason, bI differed locally from its global value, the VELMOD result could be biased low. In a future paper, we will extend the VELMOD analysis to larger distances; however, our preliminary results do not show an increase in betaI when we do so. In addition to probing a larger volume, the Sigad et al. analysis uses the full Mark III sample, ellipticals included; the possibility of systematic differences between the TF subset we have used in this paper and the full sample is difficult to rule out. Finally, it is conceivable that the requirement of precalibrating TF relations (POTENT), as opposed to calibrating them simultaneously with fitting the velocity field (VELMOD and Method II generally), accounts for part of the discrepancy. However, fixing the VELMOD TF parameters at their Mark III values has essentially no effect on the derived value of betaI (Section 4.7). This argues strongly against the notion that a major difference between VELMOD and POTIRAS is the TF relations themselves.

6.1.2. Effect of Cosmic Scatter

The sphere out to 3000 km s-1 is small; the rms value of density fluctuations within spheres of this radius is 20% for COBE-normalized cold dark matter (CDM). However, this does not propagate to a cosmic scatter error on our derived betaI for two reasons. First, the IRAS velocity field is determined within a sphere of radius 12,800 km s-1, within which the rms fluctuations are only a few percent. Thus, the predicted peculiar velocity field is subject to very little cosmic scatter. Second, this scatter manifests itself primarily as a monopole term (cf. the discussion in Section 4.4) and therefore is fully absorbed into the zero points of the TF relations (Section 3.3), having no effect on the derived value of betaI.

We have assumed that the bias relation, equation (2), is deterministic, without any scatter or variation in the effective value of bI with position. Such a model must be unrealistic at some level, and we look forward to realistic galaxy formation models that can quantify how large this form of cosmic scatter might be, and how they might affect statistics such as beta (cf. the discussion in Dekel 1997).

14 The Hudson and Shaya et al. papers actually derive betaopt, which must be multiplied by ~ 1.3 to obtain an equivalent betaI (cf. footnote 2). Back.

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