Mass function and bias of dark matter halos for non-Gaussian initial conditions

Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette, France e-mail: patrick.valageas@cea.fr

Received: 4 June 2009
Accepted: 31 January 2010

Abstract

Aims. We revisit the derivation of the mass function and the bias of dark matter halos for non-Gaussian initial conditions.

Methods. We use a steepest-descent approach to point out that exact results can be obtained for the high-mass tail of the halo mass function and the two-point correlation of massive halos. Focusing on primordial non-Gaussianity of the local type, we check that these results agree with numerical simulations.

Results. The high-mass cutoff of the halo mass function takes the same form as the one obtained from the Press-Schechter formalism, but with a linear threshold δ_L that depends on the definition of the halo (i.e. δ_L 1.59 for a nonlinear density contrast of 200). We show that a simple formula, which obeys this high-mass asymptotic and uses the fit obtained for Gaussian initial conditions, matches numerical simulations while keeping the mass function normalized to unity. Next, by deriving the real-space halo two-point correlation in the spirit of Kaiser (1984, ApJ, 284, L9) and taking a Fourier transform, we obtain good agreement with simulations for the correction to the halo bias, Δb_M(k,f_NL), due to primordial non-Gaussianity. Therefore, neither the halo mass function nor the bias require an ad-hoc parameter q (such as δ_c δ_c ), provided one uses the correct linear threshold δ_L and pays attention to halo displacements. The nonlinear real-space expression can be useful for checking that the “linearized” bias is a valid approximation. Moreover, it clearly shows how the baryon acoustic oscillation at ~100 h^-1 Mpc is amplified by the bias of massive halos and modified by primordial non-Gaussianity. On smaller scales, 30 < x < 90 h^-1 Mpc, the correction to the real-space bias roughly scales as f_NL b_M(f_NL = 0)  x². The low-k behavior of the halo bias does not imply a divergent real-space correlation, so that one does not need to introduce counterterms that depend on the survey size.