Next Contents Previous


The key feature in our biasing formalism is the natural separation between non-linear and stochastic effects. The non-linearity is expressed by the conditional mean via b(delta), and the statistical scatter is measured by the conditional standard deviation, sigmab(delta), and higher moments if necessary. For analyses using local moments of second order, the biasing scheme is characterized by three parameters: bhat measuring the mean biasing, btilde / bhat measuring the effect of non-linearity, and sigmab/bhat measuring the effect of stochasticity.

Deviations from linear and deterministic biasing typically result in biased estimates of beta, which depend on the actual method of measurement. The non-linearity and the scatter lead to underestimates of order btilde2 / bhat2 and sigmab2 / bhat2 respectively in the different estimators of beta relative to betahat = f(Omega) / bhat. Based on N-body simulations and toy models, the effects of non-linear biasing are typically on the order of 20% or less, and the effects of scatter could be larger. One expects the beta parameters to be biased in the following order: betainv < betavar < betahat.

The stochasticity affects the linear redshift-distortion analysis only by limiting the useful range of scales. In this range, the basic expression reduces to the simple Kaiser formula for b(delta) = bhat = b1 (not bvar), and it does not involve the scatter at all. The distortion analysis is in principle sensitive to the non-linear properties of biasing, but they are expected to be weak, and of the same order as the effects of non-linear GI. This is good news for the prospects of measuring an unbiased beta from redshift distortions in the large redshift surveys of the near future (2dF and SDSS).

The study of stochastic and non-linear biasing should be extended to address the time evolution of biasing because many relevant measurements of galaxy clustering are now being done at high redshifts. As seen in Fig. 1, the biasing is clearly a strong function of cosmological epoch [21, 26, 46, 56, 55, 2, 45, 59, 58, 48]. In particular, if galaxy formation is limited to a given epoch and the biasing is linear, one can show [26] that the linear biasing factor b1 would eventually approach unity as a simple result of the continuity equation. Tegmark & Peebles [58] have recently generalized the analytic study of time evolution to the case of stochastic but still linear biasing and showed how bvar and r approach unity in this case. These studies should be extended to the general non-linear case using our formalism. Our current simulations [54] are aimed at this goal. The analysis of simulations could also be extended to include non-local biasing, using the biasing correlations as defined here.

The PDF (or count in cells) of galaxy density from a large-scale redshift survey, plus an estimate of sigma of the corresponding mass density, allow a measure of the mean biasing function b(delta) and the corresponding non-linearity parameter btilde / bhat. This can be done at low or high redshifts. Mapping of the biasing field in our cosmological neighborhood, and estimates of the biasing scatter, are feasible with current and future measurements of peculiar velocities and careful comparisons to the galaxy distribution. The reconstruction of the large-scale mass distribution based on weak gravitational lensing is also becoming promising for this purpose.

In summary, in order to use the measurements of beta for an accurate evaluation of Omega, one should consider the effects of non-linear and stochastic biasing and the associated complications of scale dependence, time dependence, and type dependence. The current different estimates are expected to span a range of ~ 30% in beta due to stochastic and non-linear biasing. The analysis of redshift distortions seems to be most promising; once it is limited to the appropriate range of scales, the analysis is independent of stochasticity and the non-linear effects are expected to be relatively small. The mean biasing function can be extracted from the galaxy PDF, and the scatter from theory and local cosmography.

Acknowledgements I thank O. Lahav, G. Lemson, A. Nusser, M.J. Rees, Y. Sigad, R. Somerville, M.A. Strauss, D.H. Weinberg and S.D.M. White for collaboration and stimulating discussions. This research was supported in part by the US-Israel Binational Science Foundation grant 95-00330, and by the Israel Science Foundation grant 950/95.

Next Contents Previous