Large-scale bias and stochasticity of haloes and dark matter

On large scales, galaxies and their haloes are usually assumed to trace the dark matter with a constant bias and dark matter is assumed to trace the linear density field. We test these assumption using several large N-body simulations with 384(3)-1024(3) particles and box sizes of 96-1152 h(-1) Mpc, which can both resolve the small galactic-size haloes and sample the large-scale fluctuations. We explore the average halo bias relation as a function of halo mass and show that existing fitting formulae overestimate the halo bias by up to 20 per cent in the regime just below the non-linear mass. We propose a new expression that fits our simulations well. We find that the halo bias is nearly constant, b similar to 0.65-0.7, for masses below one-tenth of the non-linear mass. We next explore the relation between the initial and final dark matter in individual Fourier modes and show that there are significant fluctuations in their ratio, ranging from 10 per cent rms at k similar to 0.03 h Mpc(-1) to 50 per cent rms at k similar to 0.1 h Mpc(-1). We argue that these large fluctuations are caused by perturbative effects beyond the linear theory, which are dominated by long-wavelength modes with large random fluctuations. Similar or larger fluctuations exist between haloes and dark matter and between haloes of different mass. These fluctuations must be included in attempts to determine the relative bias of two populations from their maps, which would otherwise be immune to sampling variance.