Non-linear bias of cosmological halo formation in the early universe

We present estimates of the non-linear bias of cosmological halo formation, spanning a wide range in the halo mass from ∼105 to ∼1012 M⊙, 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 (δ), 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 non-linear bias effect on the formation of haloes. A strong correlation between δ and halo population overdensity δh 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