  Measurements of the cosmological density parameter Omega using techniques
that exploit the gravity-induced motions of galaxies constrain, in linear
perturbation theory, the degenerate parameter combination beta = Omega^{0.6}/b,
where the linear bias parameter b is the ratio of the fluctuation amplitudes of
the galaxy and mass distributions. However, the relation between the galaxy and
mass density fields depends on the complex physics of galaxy formation, and it
can in general be non-linear, stochastic, and perhaps non-local. The
one-parameter linear bias model is almost certainly oversimplified, which leads
to the obvious question: What is the quantity beta that is actually measured by
different techniques? To address this question, we estimate beta from galaxy
distributions that are constructed by applying a variety of locally biased
galaxy formation models to cosmological N-body simulations. We compare the
values of beta estimated using three different techniques: a density-density
comparison similar to the POTENT analysis, a velocity-velocity comparison
similar to the VELMOD analysis, and an anisotropy analysis of the
redshift-space power spectrum. In most cases, we find that beta estimated using
all three methods is similar to the asymptotic value of
Omega^{0.6}/b_{sigma}(R) at large R, where b_{sigma}(R) is the ratio of rms
galaxy fluctuations to rms mass fluctuations on scale R. Thus, something close
to the conventional interpretation of beta continues to hold even for complex
bias models. Moreover, we find that beta estimates made using these three
methods should, in principle, agree with each other. It is thus unlikely that
non-linear or scale-dependent bias is responsible for the discrepancies that
exist among current measurements of beta from different techniques.
