Accurate determination of the Lagrangian bias for the dark matter halos

We use a new method, the cross-power spectrum between the linear density field and the halo number density field, to measure the Lagrangian bias for dark matter halos. The method has several important advantages over the conventional correlation function analysis. By applying this method to a set of high-resolution simulations of 256(3) particles, we have accurately determined the Lagrangian bias, over 4 mag in halo mass, for four scale-free models with the index n = -0.5, -1.0, -1.5, and -2.0 and three typical cold dark matter models. Our result for massive halos with M greater than or equal to M* (M* is a characteristic nonlinear mass) is in very good agreement with the analytical formula of Mo & White for the Lagrangian bias, but the analytical formula significantly underestimates the Lagrangian clustering for the less massive halos, M < M*. Our simulation result, however, can be described satisfactorily, with an accuracy better than 15%, by the fitting formula of Jing far Eulerian bias under the assumption that the Lagrangian clustering and the Eulerian clustering are related with a linear mapping. It implies that it is the failure of the Press-Schechter theories for describing the formation of small halos that leads to the inaccuracy of the Mo & White formula for the Eulerian bias. The nonlinear effect in the mapping between the Lagrangian clustering and the Eulerian clustering, which was speculated as another possible cause for the inaccuracy of the Mo & White formula, must be negligible compared with the linear mapping. Our result indicates that the halo formation model adopted by the Press-Schechter theories must be improved, as independently stressed by Porciani et al.