Log-Edgeworth Halo Mass Function

• Log-Edgeworth halo mass function is an analytic approach that extends the Press–Schechter framework with non-Gaussian corrections based on reduced cumulants.
• It models dark matter halo abundances by expanding the logarithm of the collapsed fraction to second order, incorporating parameters fNL, gNL, and τNL.
• The formulation is validated against N-body simulations, yielding halo abundance predictions with percent-level accuracy across a wide mass range.

The log-Edgeworth halo mass function is an analytic prescription that models the abundance of dark matter halos under non-Gaussian initial conditions. Developed in the context of primordial non-Gaussianity characterized by parameters $f_{\mathrm{NL}}$, $g_{\mathrm{NL}}$, and $\tau_{\mathrm{NL}}$, it extends the Press–Schechter framework by expanding the logarithm of the collapsed fraction to second order in the reduced cumulants of the smoothed linear density field. The formalism provides a non-Gaussian correction to the halo mass function directly linked to the primordial bispectrum and trispectrum, introducing no free parameters. The log-Edgeworth approach yields physically sensible halo abundances across halo mass and redshift, demonstrating percent-level agreement with $N$-body simulations for a wide range of non-Gaussian parameters (Loverde et al., 2011).

1. Theoretical Formulation

The log-Edgeworth mass function is rooted in the Press–Schechter paradigm, which considers the probability $F(M)$ that the smoothed, linearly evolved density fluctuation $\delta_M$ on mass scale $M$ exceeds a collapse threshold $\delta_c \approx 1.42$. Defining the normalized variable $\nu = \delta_M/\sigma(M)$ with $\sigma^2(M)=\langle\delta_M^2\rangle$, the collapsed fraction is

$g_{\mathrm{NL}}$0

where $g_{\mathrm{NL}}$1 and $g_{\mathrm{NL}}$2 is the one-point PDF of $g_{\mathrm{NL}}$3.

For non-Gaussian initial conditions, the PDF is expanded using the Edgeworth series,

$g_{\mathrm{NL}}$4

with

$g_{\mathrm{NL}}$5

where $g_{\mathrm{NL}}$6 are the reduced connected cumulants ($g_{\mathrm{NL}}$7) and $g_{\mathrm{NL}}$8 are probabilists’ Hermite polynomials.

The log-Edgeworth prescription expands the logarithm of the collapsed fraction $g_{\mathrm{NL}}$9: $\tau_{\mathrm{NL}}$0 with $\tau_{\mathrm{NL}}$1 and $\tau_{\mathrm{NL}}$2 so $\tau_{\mathrm{NL}}$3 is the complementary error function.

The corresponding non-Gaussian correction factor to the mass function is

$\tau_{\mathrm{NL}}$4

where primes denote derivatives with respect to mass. $\tau_{\mathrm{NL}}$5 can be any accurate Gaussian mass function (e.g., Sheth–Tormen).

2. Derivation and Expansion Properties

The log-Edgeworth series is based on an Edgeworth expansion of the PDF for the smoothed density field, but instead of truncating the PDF, the truncation is applied to $\tau_{\mathrm{NL}}$6 at second order in the non-Gaussian cumulants ($\tau_{\mathrm{NL}}$7, $\tau_{\mathrm{NL}}$8, $\tau_{\mathrm{NL}}$9). This modification ensures that, even in the high-mass ($N$0) tail, the mass function remains positive, monotonic, and physically sensible. The cumulants arising from primordial non-Gaussianity are calculated through windowed integrals of the bispectrum and trispectrum, with analytic approximations for the local $N$1, $N$2, and $N$3 models.

This approach reproduces earlier prescriptions in relevant limits: it reduces to Press–Schechter for vanishing cumulants, matches Matarrese–Viel–Jimenez in the high-peak regime, and agrees with small-$N$4 expansions in the low-$N$5 limit (Loverde et al., 2011).

3. Parameters, Assumptions, and Validity

The expansion is truncated at $N$6, $N$7, and $N$8. It is validated for $N$9, $F(M)$0, and $F(M)$1 few $F(M)$2. The model uses the spherical collapse threshold $F(M)$3 as calibrated to simulations. Box-size dependence from infrared modes in local-type PNG enters via $F(M)$4 and $F(M)$5, with the relevant box length $F(M)$6 identified with the survey or simulation volume. The formalism is tested against $F(M)$7-body halo catalogs for $F(M)$8 and redshift $F(M)$9.

4. Comparison with Simulations

Extensive $\delta_M$0-body validation was performed using the GADGET-2 code with $\delta_M$1 Mpc and $\delta_M$2 particles, identifying halos with a friends-of-friends (FoF) algorithm ($\delta_M$3). The comparison covered: (i) local bispectrum models, including $\delta_M$4 with both canonical $\delta_M$5 and enhanced $\delta_M$6, and (ii) pure trispectrum cases with $\delta_M$7. The log-Edgeworth mass function matched $\delta_M$8 to within $\delta_M$910% for $M$0 up to $M$1, outperforming the ordinary Edgeworth truncation especially at high mass, negative $M$2, and large $M$3.

Distinct physical signatures include the modification of the high-mass tail with varying $M$4 at fixed $M$5, and the impact of pure $M$6 being confined to the very high-mass regime, leaving the low-mass abundance nearly unchanged (Loverde et al., 2011).

5. Generalization and Distinguishing Features

Unlike empirical fitting functions, the log-Edgeworth form introduces no free parameters: it is built entirely from cosmological initial statistics. By retaining both third and fourth cumulants ($M$7, $M$8), the mass function can describe cases where the trispectrum dominates (e.g., pure $M$9 or independent $\delta_c \approx 1.42$0) and yields well-behaved results in limits where simpler expansions become unreliable. For vanishing non-Gaussianity, it reduces exactly to Press–Schechter, and in appropriate limits, recovers established high-peak and low-peak expansions.

The log-Edgeworth approach remains robust across a wide dynamic range of masses. This robustness is attributed to the expansion of $\delta_c \approx 1.42$1 rather than the PDF itself, which improves physical plausibility in the high-mass halo tail where the standard Edgeworth expansion can be negative or non-monotonic (Loverde et al., 2011).

6. Implementation Workflow

The following recipe, directly reflecting the published formalism, enables practical application for any local-type PNG parameters:

1. Gaussian Mass Function: Select a reference $\delta_c \approx 1.42$2, e.g., Sheth–Tormen.
2. Variance:

$\delta_c \approx 1.42$3

1. Reduced Cumulants (using analytic fits): \begin{align*} \kappa_3(M) &\approx f_{\mathrm{NL}} \times 6.6 \times 10^{-4} \left[1 - 0.016\, \ln\frac{M}{10^{12}h^{-1}M_\odot}\right] \ \kappa_4(M) &\approx g_{\mathrm{NL}} \times 1.6 \times 10^{-7} \left[1 - 0.021\, \ln\frac{M}{10^{12}h^{-1}M_\odot}\right] \ & \qquad + \tau_{\mathrm{NL}}/(6/5)^2 \left[6.9\times10^{-7}(1-0.021\ln\frac{M}{10^{12}}) + 48\,\Delta_\Phi^2\ln\frac{L}{1600}\right] \ \kappa_2(M) &\approx \tau_{\mathrm{NL}}/(6/5)^4 f_{\mathrm{NL}}^2 \left[4.0\times10^{-8}(1-0.021\ln\frac{M}{10^{12}}) + 4\Delta_\Phi^2\ln\frac{L}{1600}\right] \end{align*}
2. Partial Collapsed Fractions: Compute $\delta_c \approx 1.42$4 at $\delta_c \approx 1.42$5.
3. Derivatives: Differentiate $\delta_c \approx 1.42$6 with respect to $\delta_c \approx 1.42$7 to obtain $\delta_c \approx 1.42$8.
4. Correction Factor:

$\delta_c \approx 1.42$9

1. Final Non-Gaussian Mass Function:

$\nu = \delta_M/\sigma(M)$0

This process enables direct prediction of the mass function for any set of $\nu = \delta_M/\sigma(M)$1 parameters, with no empirical tuning (Loverde et al., 2011).