Nonlinear Bias of Cosmological Halo Formation in the Early Universe
  We present estimates of the nonlinear bias of cosmological halo formation,
spanning a wide range in the halo mass from $\sim 10^{5} M_\odot$ to $\sim
10^{12} M_\odot$, based upon both a suite of high-resolution cosmological
N-body simulations and theoretical predictions. The halo bias is expressed in
terms of the mean bias and stochasticity as a function of local overdensity
($\delta$), under different filtering scales, which is realized as the density
of individual cells in uniform grids. The sampled overdensities span a range
wide enough to provide the fully nonlinear bias effect on the formation of
haloes. A strong correlation between $\delta$ and halo population overdensity
$\delta_h$ is found, along with sizable stochasticity. We find that the
empirical mean halo bias matches, with good accuracy, the prediction by the
peak-background split method based on the excursion set formalism, as long as
the empirical, globally-averaged halo mass function is used. Consequently, this
bias formalism is insensitive to uncertainties caused by varying halo
identification schemes, and can be applied generically. We also find that the
probability distribution function of biased halo numbers has wider distribution
than the pure Poisson shot noise, which is attributed to the sub-cell scale
halo correlation. We explicitly calculate this correlation function and show
that both overdense and underdense regions have positive correlation, leading
to stochasticity larger than the Poisson shot noise in the range of haloes and
halo-collapse epochs we study.
