A Second-order bias model for the Logarithmic Halo Mass Density
  We present an analytic model for the local bias of dark matter halos in a
LCDM universe. The model uses the halo mass density instead of the halo number
density and is searched for various halo mass cuts, smoothing lengths, and
redshift epoches. We find that, when the logarithmic density is used, the
second-order polynomial can fit the numerical relation between the halo mass
distribution and the underlying matter distribution extremely well. In this
model the logarithm of the dark matter density is expanded in terms of log halo
mass density to the second order. The model remains excellent for all halo mass
cuts (from M_{cut}=3\times10^{11}$ to $3\times10^{12}h^{-1}M_{\odot}$),
smoothing scales (from $R=5h^{-1}$Mpc to $50h^{-1}$Mpc), and redshift ranges
(from z=0 to 1.0) considered in this study. The stochastic term in the relation
is found not entirely random, but a part of the term can be determined by the
magnitude of the shear tensor.
