A redshift-independent theoretical halo mass function validated with the Uchuu simulations

TL;DR

The paper introduces GPS+, a redshift-independent generalised Press–Schechter framework incorporating triaxial collapse to predict the halo mass function (HMF) across an unprecedented range of masses ($6.5<\log(M/M_\odot)<16$) and redshifts ($0<z<20$). It expresses the HMF through $dn/dM = (\rho_m/M) dF/dM$, where $F(M,z)$ depends on the linear variance $\sigma(M,z)$ and includes mass-dependent corrections via $b(M)$ and $c(M)$, with no explicit $z$-dependence. Calibration against the Uchuu N-body simulations yields $A\approx1.089$, $B\approx0.652$, and $D\approx1$, achieving typical deviations of $10$–$20\%$ across the full domain and outperforming the Sheth–Tormen model at high redshift, especially for massive halos. The work also shows that the choice of halo mass definition matters, with $M_{200m}$ producing a more universal HMF than the evolving virial overdensity definition. These results provide a robust, physically grounded tool for interpreting large-scale structure surveys and early-Universe observations, with public code and data releases.

Abstract

We present a new theoretical framework for the halo mass function (HMF) that accurately predicts the abundance of dark matter haloes across an exceptionally wide range in mass and redshift. Building on a generalised Press & Schechter model and triaxial collapse (GPS+), we predict the HMF in terms of the variance of the linear density field, with only a weak explicit dependence on halo mass and no explicit dependence on redshift. The GPS+ model naturally provides the correct normalization and high-mass behaviour without requiring empirical fitting. We calibrate and validate the GPS+ model using the Uchuu N-body simulation suite, which combines large cosmological volume and high mass resolution under Planck cosmology. Using six simulations with up to 300 realizations, we obtain precision HMF measurements spanning halo masses in the range 6.5 < log($M_{\rm 200m}$/[h$^{-1}$ $M_{\odot}$]) <16 over 0 < z < 20, with reduced cosmic variance. Across this full domain, the GPS+ model reproduces the simulated HMF with deviations typically below 10-20%. Comparison with the Sheth-Tormen (ST) model shows similar performance at z < 2, but markedly improved agreement at higher redshifts, where ST can deviate by 70-80% while our model remains within ~20%. Finally, we assess the impact of the halo mass definition: adopting the evolving virial overdensity of Bryan & Norman (1998) worsens agreement at low redshift and high masses, whereas M200m yields a more universal, nearly redshift-independent HMF.

Figures (4)

• Figure 1: The markers show the measurements of the multiplicity function, $f(M,z)$ (see Equation \ref{['multfunc']}), obtained from the suite of Uchuu simulations at different redshifts. The solid lines correspond to the predictions from the multiplicity function from the GPS+ model at each redshift. Each colour represents a different redshift, ranging from $z=0$ to $z=20$ from right to left (see Table \ref{['redshifts']} for the full list). The $y$-axis shows the normalized differential halo abundance, including the derivative term $d\ln M/d\ln \sigma$. For the measurements, we only include mass bins containing more than 200 halos.
• Figure 2: The top panel shows the multiplicity function $f(\sigma, z)$ as a function of $\sigma$ at different redshifts, as predicted by the GPS+ model presented in this work. The bottom panel shows the ratio of the model at each redshift to $z=0$ prediction.
• Figure 3: Ratios between the combined simulated HMF and the predictions from GPS+ model (black line), as well as from the ST model (magenta line) computed using the Colossus tool 2018ApJS..239...35D, at different redshifts. Only mass bins containing more than 100 halos are shown.
• Figure 4: Comparison of the ratio between the simulated HMF and the GPS+ model at different redshifts, for two halo mass definitions in the data: the mass enclosed within overdensity 200$\rho_{\rm b}$, M$_{\rm 200m}$, and the mass within the virial radius, M$_{\rm vir}$.