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

We revisit the derivation of the mass function and the bias of dark matter halos for non-Gaussian initial conditions. 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. 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 $\delta_L$ that depends on the definition of the halo. 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) and taking a Fourier transform, we obtain good agreement with simulations for the correction to the halo bias due to primordial non-Gaussianity. Therefore, neither the halo mass function nor the bias require an ad-hoc parameter $q$ provided one uses the correct linear threshold $\delta_L$ 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 $\sim 100 h^{-1}$Mpc is amplified by the bias of massive halos and modified by primordial non-Gaussianity. On smaller scales, the correction to the real-space bias roughly scales as $\fNL \, b_M(\fNL=0) \, x^2$. 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.