The spatial and velocity bias of linear density peaks and proto-haloes
  in the Lambda cold dark matter cosmology
  We use high resolution N-body simulations to investigate the Lagrangian bias
of cold dark matter haloes within the LCDM cosmology. Our analysis focuses on
"proto-haloes", which 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 proto-haloes and
estimate their auto spectral 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 and 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 velocity fields. We also
show that the bias coefficients are accurately predicted for the velocity
divergence. On the contrary, 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.
