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

P. Valageas

TL;DR

This work develops a steepest-descent (saddle-point) framework to study dark matter halos under non-Gaussian initial conditions, focusing on local-type $f_{ m NL}$ perturbations. It derives the high-mass tail of the halo mass function exactly in the rare-event limit and provides a normalization-preserving, simple recipe to incorporate non-Gaussian corrections using the Gaussian fit, with the linear density threshold $δ_L$ dictated by spherical collapse. The bias of halos is treated in real space via a two-cell saddle point, yielding a nonlinear, displacement-aware expression that, upon Fourier transformation, reproduces the standard $k^{-2}$ scale-dependent bias for local non-Gaussianity and predicts BAO amplification for massive halos. The results are in good agreement with simulations without invoking ad-hoc parameters, and the real-space formulation offers a robust check on linear-bias approximations and a practical route to constrain primordial non-Gaussianity from large-scale structure observations.

Abstract

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 $δ_L$ 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 $δ_L$ 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 $\sim 100 h^{-1}$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.

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

TL;DR

This work develops a steepest-descent (saddle-point) framework to study dark matter halos under non-Gaussian initial conditions, focusing on local-type perturbations. It derives the high-mass tail of the halo mass function exactly in the rare-event limit and provides a normalization-preserving, simple recipe to incorporate non-Gaussian corrections using the Gaussian fit, with the linear density threshold dictated by spherical collapse. The bias of halos is treated in real space via a two-cell saddle point, yielding a nonlinear, displacement-aware expression that, upon Fourier transformation, reproduces the standard scale-dependent bias for local non-Gaussianity and predicts BAO amplification for massive halos. The results are in good agreement with simulations without invoking ad-hoc parameters, and the real-space formulation offers a robust check on linear-bias approximations and a practical route to constrain primordial non-Gaussianity from large-scale structure observations.

Abstract

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 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 provided one uses the correct linear threshold 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 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 . The low- 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.

Paper Structure

This paper contains 9 sections, 88 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Non-Gaussian initial conditions
  3. Mass function of dark matter halos
  4. Rare-event saddle point for the density distribution
  5. Mass function
  6. Bias of dark matter halos
  7. Two-cell saddle point and real-space bias
  8. Fourier-space bias
  9. Conclusion

Figures (10)

  • Figure 1: The radial profile (\ref{['delta0']})-(\ref{['delta1']}) of the linear density contrast $\delta_{Lq'}$ of the saddle point of the action ${\cal S}[\chi,\lambda]$. We show the profiles obtained with a $\Lambda$CDM cosmology for the masses $M=10^{11}$ and $10^{15}h^{-1}M_{\odot}$. A larger mass corresponds to a lower ratio $\delta_{Lq'}/\delta_{Lq}$ at large radii $q'/q>1$. We show our results for the Gaussian case (solid line), positive $f_{\rm NL}$ (dashed line) and negative $f_{\rm NL}$ (dotted line), for the local model (\ref{['tfNLd_local']}).
  • Figure 2: The ratio $n(M,f_{\rm NL})/n(M,0)$, of the mass functions obtained for $f_{\rm NL}^{\rm LSS}=\pm 200$ over the mass function obtained for Gaussian initial conditions, as a function of $M$ for several redshifts. The dot-dashed line labeled "$e^{S_3}$" is the multiplicative factor (\ref{['mult']}), the dashed line labeled "PS" is Eq.(\ref{['rationM']}) with the Press-Schechter mass function (\ref{['PS3']}), which also corresponds to the result of Matarrese et al. (2000) (but with $\delta_L\simeq 1.59$), the solid line labeled "f" is Eq.(\ref{['rationM']}) with the fitting function (\ref{['fitfsig']}). The data points are results from the numerical simulations of Grossi et al. (2009).
  • Figure 3: Same as Fig. \ref{['figfM_200_Grossi']}, but with $f_{\rm NL}=500$ (upper panel) and $f_{\rm NL}=-500$ (lower panel). The data points are the results of the numerical simulations of Dalal et al. (2008).
  • Figure 4: The halo bias $b_M(x)$, as a function of $\sigma(M)$, at fixed redshift $z=0$ and distance $x=50 h^{-1}$Mpc. The solid lines "$b$" are the nonlinear theoretical prediction of Eqs.(\ref{['xiM1M2']}), (\ref{['xs']}) and (\ref{['bias_sig']}), for $f_{\rm NL}=\pm 200$ and $f_{\rm NL}=0$ (i.e. Gaussian case, intermediate line), while the dot-dashed lines "$b_L$" are the linearized bias of Eq.(\ref{['xiMlin']}). The upper dashed line "$s=x$" shows the result obtained in the Gaussian case by setting $s=x$ in Eq.(\ref{['xiM1M2']}). The points are the fits to Gaussian numerical simulations, from Sheth, Mo & Tormen (2001) (crosses) and Pillepich (2010) (circles).
  • Figure 5: The halo bias $b_M(x)$ as a function of distance $x$, at redshifts $z=0$ (upper panel) and $z=1$ (lower panel) for several masses. We show the cases $f_{\rm NL}= 0$ (solid lines), $f_{\rm NL}=100$ (dashed lines), and $f_{\rm NL}=-100$ (dotted lines). The divergences at $x \sim 120 h^{-1}$Mpc come from the halo and matter correlations not changing sign at the same distance.
  • ...and 5 more figures