The clustering of dark matter halos: scale-dependent bias on
  quasi-linear scales
  We investigate the spatial clustering of dark matter halos, collapsing from
$1-4 \sigma$ fluctuations, in the redshift range $0 - 5$ using N-body
simulations. The halo bias of high redshift halos ($z \geq 2$) is found to be
strongly non-linear and scale-dependent on quasi-linear scales that are larger
than their virial radii ($0.5-10$ Mpc/h). However, at lower redshifts, the
scale-dependence of non-linear bias is weaker and and is of the order of a few
percent on quasi-linear scales at $z \sim 0$. We find that the redshift
evolution of the scale dependent bias of dark matter halos can be expressed as
a function of four physical parameters: the peak height of halos, the
non-linear matter correlation function at the scale of interest, an effective
power law index of the {\it rms} linear density fluctuations and the matter
density of the universe at the given redshift. This suggests that the
scale-dependence of halo bias is not a universal function of the dark matter
power spectrum, which is commonly assumed. We provide a fitting function for
the scale dependent halo bias as a function of these four parameters. Our fit
reproduces the simulation results to an accuracy of better than 4 % over the
redshift range $0\leq z \leq 5$. We also extend our model by expressing the
non-linear bias as a function of the linear matter correlation function. It is
important to incorporate our results into the clustering models of dark matter
halos at any redshift, including those hosting early generations of stars and
galaxies before reionization.
