Halo Mass Function Calculator

Halo mass function in practical modeling

The halo mass function links the statistics of the primordial density field to the abundance of collapsed dark matter halos. In surveys and simulations, it converts cosmological assumptions into expected counts of galaxy groups and clusters per comoving volume. This calculator focuses on the differential forms dn/dM and dn/dlog10M, which are widely used for comparing across decades in mass.

1) From variance to peak height

The central variable is the peak height ν = δc/σ(M,z). Larger ν corresponds to rarer peaks and more massive halos. Here δc is the collapse threshold (often near 1.686), and σ(M,z) is the linear RMS fluctuation on a mass scale. A small increase in ν can strongly suppress high-mass abundances.

2) Growth factor scaling with redshift

The redshift dependence enters through the linear growth factor D(z), with σ(M,z)=σ(M,0)D(z). For typical ΛCDM parameters, D(z) decreases with redshift, increasing ν and reducing halo counts. This is why cluster abundances are sensitive probes of late-time structure growth.

3) Press-Schechter versus Sheth-Tormen

Press-Schechter provides a classic analytic baseline with f(ν) ∝ ν exp(-ν²/2). Sheth-Tormen modifies the exponential tail and adds a correction factor motivated by ellipsoidal collapse. In practice, Sheth-Tormen often predicts more massive halos than Press-Schechter, especially near the exponential cutoff, improving agreement with N-body results in many regimes.

4) Interpreting σ8 and the mass scale M8

The parameter σ8 normalizes the fluctuation amplitude within spheres of radius 8 h⁻1 Mpc. The tool computes the corresponding mass M8 using the present-day mean matter density ρm,0 = Ωm ρcrit,0, where ρcrit,0 = 2.775×10¹¹ h² Msun/Mpc³. Raising σ8 typically boosts the abundance at fixed mass and redshift.

5) Effective slope control with n_eff

To keep the calculator fast, the mass variance is approximated as a power law: σ(M,0) ≈ σ8 (M/M8)^(-α). You may set α directly or infer it from an effective spectral index via α=(n_eff+3)/6. More negative n_eff generally yields a smaller α, flattening the mass dependence of σ.

6) Units and common output forms

The output dn/dM has units of 1/(Mpc³·Msun) in comoving coordinates. Many analyses prefer dn/dlog10M, which scales by M ln 10 and is convenient for logarithmic mass bins. When comparing to catalogs, match the mass definition used (e.g., overdensity-based masses) and apply consistent binning.

7) Sensitivity checks and sanity ranges

As a diagnostic, vary one parameter at a time. Increasing z should reduce high-mass halos. Increasing σ8 should raise abundances across the range. If results look unstable, narrow the mass range, increase grid points, or choose a more moderate α. The CSV export makes quick comparisons across runs straightforward.

8) When to use this approximation

This tool is best for exploratory work, teaching, and rapid sensitivity studies when you need an interpretable mapping from σ8, α, and D(z) to halo counts. For precision cosmology, compute σ(M) from the matter power spectrum with a top-hat filter and use calibrated fitting functions matched to your mass definition.