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

$F(M) = \int_{\nu_c}^{\infty} d\nu\, \rho(\nu, M)$

where $\nu_c(M) = \delta_c/\sigma(M)$ and $\rho(\nu, M)$ is the one-point PDF of $\nu$.

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

$$\rho(\nu, M) = \frac{e^{-\nu^2/2}}{\sqrt{2\pi}}\left[1 + p_1(\nu, M) + p_2(\nu, M) + \cdots\right]$$

with

$$p_1 = \frac{\kappa_3}{6} H_3(\nu), \quad p_2 = \frac{\kappa_2}{2} H_2(\nu) + \frac{\kappa_4}{24} H_4(\nu) + \frac{\kappa_3^2}{72} H_6(\nu)$$

where $\kappa_N$ are the reduced connected cumulants ($N\geq 3$) and $H_n(\nu)$ are probabilists’ Hermite polynomials.

The log-Edgeworth prescription expands the logarithm of the collapsed fraction $F(M)$: $$\ln F \approx \ln F_0 + \frac{F_1}{F_0} + \frac{F_2}{F_0} - \frac{1}{2}\left(\frac{F_1}{F_0}\right)^2$$ with $$F_i(M)=\int_{\nu_c}^{\infty} d\nu\, \frac{e^{-\nu^2/2}}{\sqrt{2\pi}} p_i(\nu, M)$$ and $p_0 = 1$ so $F_0$ is the complementary error function.

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

$$\frac{n_{\rm NG}(M)}{n_{\rm G}(M)} \approx \exp\left[\frac{F_1}{F_0} + \frac{F_2}{F_0} - \frac{1}{2}\left(\frac{F_1}{F_0}\right)^2\right] \left\{ 1 + \frac{F'_1 + F'_2}{F'_0} - \frac{F_1 F'_1}{F_0 F'_0} - \frac{F_1 + F_2}{F_0} + \frac{F_1^2}{F_0^2} \right\}$$

where primes denote derivatives with respect to mass. $n_{\rm G}(M)$ 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 $\ln F(M)$ at second order in the non-Gaussian cumulants ($\kappa_2$, $\kappa_3$, $\kappa_4$). This modification ensures that, even in the high-mass ($\nu_c \gg 1$) 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 $f_{\mathrm{NL}}$, $g_{\mathrm{NL}}$, and $\tau_{\mathrm{NL}}$ 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-$\nu$ expansions in the low-$\nu$ limit (Loverde et al., 2011).

3. Parameters, Assumptions, and Validity

The expansion is truncated at $\mathcal{O}(f_{\mathrm{NL}}^2)$, $\mathcal{O}(g_{\mathrm{NL}})$, and $\mathcal{O}(\tau_{\mathrm{NL}})$. It is validated for $|f_{\mathrm{NL}}|\lesssim 500$, $|g_{\mathrm{NL}}| \lesssim 5\times10^6$, and $\tau_{\mathrm{NL}} \sim$ few $\times f_{\mathrm{NL}}^2$. The model uses the spherical collapse threshold $\delta_c \approx 1.42$ as calibrated to simulations. Box-size dependence from infrared modes in local-type PNG enters via $\kappa_2$ and $\kappa_4$, with the relevant box length $L$ identified with the survey or simulation volume. The formalism is tested against $N$-body halo catalogs for $M\lesssim 3\times10^{15}\, h^{-1}M_\odot$ and redshift $z \leq 2$.

4. Comparison with Simulations

Extensive $N$-body validation was performed using the GADGET-2 code with $L=1600\,h^{-1}$ Mpc and $1024^3$ particles, identifying halos with a friends-of-friends (FoF) algorithm ($b=0.2$). The comparison covered: (i) local bispectrum models, including $f_{\mathrm{NL}}=\pm250,\pm500$ with both canonical $\tau_{\mathrm{NL}}=(6/5 f_{\mathrm{NL}})^2$ and enhanced $\tau_{\mathrm{NL}} = 2(6/5 f_{\mathrm{NL}})^2$, and (ii) pure trispectrum cases with $g_{\mathrm{NL}}=\pm10^6, \pm5\times10^6$. The log-Edgeworth mass function matched $n_{\rm NG}/n_{\rm G}$ to within $\sim$10% for $|f_{\mathrm{NL}}|=500$ up to $M\sim 3\times10^{15}\, h^{-1}M_\odot$, outperforming the ordinary Edgeworth truncation especially at high mass, negative $f_{\mathrm{NL}}$, and large $g_{\mathrm{NL}}$.

Distinct physical signatures include the modification of the high-mass tail with varying $\tau_{\mathrm{NL}}$ at fixed $f_{\mathrm{NL}}$, and the impact of pure $g_{\mathrm{NL}}$ 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 ($\kappa_3$, $\kappa_4$), the mass function can describe cases where the trispectrum dominates (e.g., pure $g_{\mathrm{NL}}$ or independent $\tau_{\mathrm{NL}}$) 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 $\ln F(M)$ 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 $n_{\rm G}(M)$, e.g., Sheth–Tormen.
2. Variance:

$$\sigma_G^2(M) = \int \frac{d^3k}{(2\pi)^3}\, W_M^2(k)\, \alpha^2(k)\, P_\Phi(k)$$

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 $F_0, F_1, F_2$ at $\nu_c$.
3. Derivatives: Differentiate $F_i$ with respect to $M$ to obtain $F'_i$.
4. Correction Factor:

$$R(M) \equiv \frac{n_{\rm NG}}{n_{\rm G}} = \exp\left[\frac{F_1+F_2}{F_0} - \frac{1}{2}\left(\frac{F_1}{F_0}\right)^2\right] \left[1 + \frac{F'_1+F'_2}{F'_0} - \frac{F_1F'_1}{F_0F'_0} - \frac{F_1+F_2}{F_0} + \frac{F_1^2}{F_0^2} \right]$$

1. Final Non-Gaussian Mass Function:

$n_{\rm NG}(M) = R(M)\, n_{\rm G}(M)$

This process enables direct prediction of the mass function for any set of $$\{f_{\mathrm{NL}}, g_{\mathrm{NL}}, \tau_{\mathrm{NL}}\}$$ parameters, with no empirical tuning (Loverde et al., 2011).