A Second-Order Bias Model for the Logarithmic Halo Mass Density

We present an analytic model for the local bias of dark matter halos in a ΛCDM universe. The model uses the halo mass density instead of the halo number density and is searched for various halo mass cuts, smoothing lengths, and redshift epochs. We find that, when the logarithmic density is used, the second-order polynomial can fit the numerical relation between the halo mass distribution and the underlying matter distribution extremely well. In this model, the logarithm of the dark matter density is expanded in terms of log halo mass density to the second order. The model remains excellent for all halo mass cuts (from M cut = 3 × 10 11 to 3 × 10 12 h −1 M ☉ ), smoothing scales (from R = 5 h −1 Mpc to 50 h −1 Mpc), and redshift ranges (from z = 0 to 1.0) considered in this study. The stochastic term in the relation is found to be not entirely random, but a part of the term can be determined by the magnitude of the shear tensor.