Evolution mapping III: A new recipe for the halo mass function

TL;DR

This paper addresses the challenge of non-universal halo mass function (HMF) modelling arising from the growth history and local linear power spectrum shape. It introduces an Evolution Mapping-based nine-parameter function $f(\nu)$ that depends on the shape parameter $n_{\rm eff}$ and the history memory $\tilde{x}$, with a memory parameter $\eta$, calibrated against high-resolution Aletheia/AletheiaMass simulations and interpolated across overdensity $\Delta$. The authors demonstrate percent-level accuracy across a wide range of halo masses, redshifts, and cosmologies, and show that the interpolation to virial overdensity definitions remains accurate to within ~5%. Compared with existing prescriptions, the new model better captures non-universal features of the HMF and provides a flexible tool for cluster abundance analyses, illustrating the power of Evolution Mapping for non-linear observables in structure formation.

Abstract

We present a new prescription for the halo mass function (HMF) built upon the Evolution Mapping framework. This approach provides a physical motivation to parametrise the non-universality of the HMF in terms of the recent history of structure formation and the local shape of the linear matter power spectrum. Our model was calibrated against measurements from N-body simulations, with halo samples defined by ten overdensity thresholds, $Δ$, ranging from 150 to 1600 times the mean background matter density. For our reference mass definition, $Δ=200$, the calibrated fitting function achieves per cent-level accuracy across a wide range of masses, redshifts, and structure formation histories, and maintains this performance when tested on cosmologies with different linear power spectrum shapes. This high level of accuracy is maintained across other mass definitions, degrading only slightly to the 5 per cent level at the highest values of $Δ$. We also provide fitting formulae to interpolate the parameters as a function of $Δ$, which allows for accurate modelling of HMFs defined by intermediate overdensities, with accuracy still well within 5 per cent when tested on halo catalogues defined by the virial overdensity threshold. Compared to other commonly used recipes, our prescription yields competitive or superior accuracy across all redshifts and cosmologies, successfully capturing the non-universal features of the HMF where other models exhibit systematic deviations. This work provides a high-precision modelling tool for cluster abundance analyses, and demonstrates the power of the evolution mapping framework for building accurate models of observables in the non-linear regime.

Figures (14)

• Figure 1: Ratio of the halo multiplicity functions measured in the Aletheia simulations (see Section \ref{['subsec:sims']}) to the results from Model 0 at equal clustering amplitude $\sigma_{12}$. The grey bands correspond to a 1 per cent difference.
• Figure 2: Upper panel: scaling of $x$ as a function of $\sigma_{12}$ in the Aletheia cosmologies defined in Table \ref{['tab:models_aletheia']}. Bottom panel: difference between these cosmologies and that of Model 0, $x_\mathrm{M0}$, the reference $\Lambda\mathrm{CDM}$ cosmology. The vertical dashed lines mark the values of $\sigma_{12}$ characterising the simulation snapshots. The differences $x - x_\mathrm{M0}$ at each of the marked values of $\sigma_{12}$ show a remarkable similarity to the multiplicity function ratios shown in the corresponding panel of Fig. \ref{['fig:aletheia_ratio_tomodel0']}.
• Figure 3: Stability of the measurements of the multiplicity function with respect to changes in the binning scheme, in two example snapshots of the AletheiaMass simulations. Assigning to each bin a representative mass value $M(\nu)$ evaluated at the central $\nu$ of the bin, as shown in the upper panels, can bias the estimate by a few per cent. Instead, summing masses directly into the estimator, as in equation (\ref{['eq:fnu_estimator_masses']}), does not, as the lower panels show.
• Figure 4: Upper panel: multiplicity function $\hat{f}(\nu)$ measured in the AletheiaMass simulations, corresponding to an overdensity threshold of $\Delta = 200$. Lower panel: ratios of the measured values to the calibrated fitting function given in equation (\ref{['eq:fnu_ours']}). To avoid crowding the plots with the measurements from all the AletheiaMass simulations, here and in subsequent figures we show, at each snapshot and for each mass bin, a weighted average of the measurements from different boxes. The grey band in the lower panel corresponds to a 1 per cent difference.
• Figure 5: Ratios of $\hat{f}(\nu)$ measured in the Aletheia simulations for an overdensity threshold $\Delta = 200$ and the calibrated fitting function given by equation (\ref{['eq:fnu_ours']}).