The spatial and velocity bias of linear density peaks and protohaloes in the Λ cold dark matter cosmology

We use high-resolution N-body simulations to investigate the Lagrangian bias of cold dark matter haloes within the Λ cold dark matter cosmology. Our analysis focuses on ‘protohaloes’ that we identify in the simulation initial conditions with the subsets of particles belonging to individual redshift-zero haloes. We then calculate the number density and velocity divergence fields of protohaloes and estimate their autospectral densities. We also measure the corresponding cross-spectral densities with the linear matter distribution. We use our results to test a Lagrangian bias model presented by Desjacques & Sheth which is based on the assumption that haloes form out of local density maxima of a specific height. Our comparison validates the predicted functional form for the scale dependence of the bias for both the density and the velocity fields. We also show that the bias coefficients are accurately predicted for the velocity divergence. In contrast, the theoretical values for the density bias parameters do not accurately match the numerical results as a function of halo mass. This is likely due to the simplistic assumptions that relate virialized haloes to density peaks of a given height in the model. We also detect appreciable stochasticity for the Lagrangian density bias, even on very large scales. These are not included in the model at leading order but correspond to higher order corrections.