Robust baryonic corrections to the halo mass function and halo bias for next-generation cluster cosmology

Feedback from star formation and accreting supermassive black holes changes the masses of galaxy groups and clusters relative to their dark-matter-only counterparts, biasing cosmological inference from cluster number counts. Castro et al. (2021) showed with the Magneticum simulations that the baryonic impact on the halo mass function and halo bias can be recast as a deterministic, median-based mass correction of the matched-halo mass ratio, reducing the forecast inconsistency of a Euclid-like cluster survey from about 8 sigma to about 1 sigma, but the correction was calibrated on one simulation suite, one cosmology, and one halo finder, and neglects the stochastic distribution of the mass mapping. The five open strands it posed remain live through mid-2026: independent suites (Simba, FLAMINGO) confirm that the mass mapping is feedback-model dependent, and FLAMINGO has built the observational-calibration machinery, but the systematic cross-suite inter-comparison defining priors on the mass-correction function, the full stochastic mass-mapping distribution with its covariance with observable proxies, multi-cosmology and halo-finder universality of the baryonic halo-mass-function and bias calibration, and the incorporation of the recently quantified baryonic assembly bias into survey bias modelling all remain open.

Background

Galaxy clusters, the most massive gravitationally bound objects in the Universe, are a standard cosmological probe: their abundance as a function of mass and redshift, predicted by the halo mass function (HMF — the comoving number density of dark matter halos per logarithmic mass interval), and their clustering, encoded in the linear halo bias (the ratio of halo to matter density contrast on large scales), are sensitive to the cosmic matter density $\Omega_{\rm m}$ and the amplitude of density fluctuations $\sigma_8$. Next-generation Sunyaev-Zel'dovich (SZ), optical, and X-ray surveys (eROSITA, Euclid, Rubin/LSST, Simons Observatory) will deliver samples of order $10^5$ clusters, so theoretical systematic errors must be held below the statistical precision.

Baseline HMF and halo-bias predictions come from dark-matter-only (DMO) $N$-body simulations. Full cosmological hydrodynamical simulations follow gas alongside dark matter, but the energetic feedback from supernovae and active galactic nuclei (AGN) that ejects gas from halos acts below the resolution limit and must be represented by approximate 'subgrid' prescriptions. AGN feedback heats and displaces intracluster gas, so halos matched between hydrodynamical and DMO runs are, at low redshift, on average a few percent lighter in the hydrodynamical case, and the mass ratio $M_{\rm Hydro}/M_{\rm DMO}$ correlates tightly with the halo gas and stellar mass fractions. Neglecting this effect biases cluster-count cosmology at the percent-to-ten-percent level in $\Omega_{\rm m}$, $S_8=\sigma_8(\Omega_{\rm m}/0.3)^{1/2}$, and the dark-energy parameter $w_0$ for stage-IV-like surveys.

Castro et al. (2021) analyzed the Magneticum hydrodynamical simulation suite (halos above $10^{13}\,M_\odot/h$ out to redshift 2) and showed that the baryonic impact on the HMF and halo bias can be recast as a mass shift: correcting DMO halo masses by a deterministic function $h_c(M,z)$ of mass and redshift (their Eqs. 36-37) reduces the index of inconsistency (IoI — a standard forecast metric that quantifies, in units of sigma, the disagreement between the cosmological constraints of two analyses) between DMO-based and Hydro-based constraints of a Euclid-like forecast from $8.1\sigma$ to about $1\sigma$. Two structural limitations remain. First, the correction is a median: the residual tension in the recovered nuisance parameters — the parameters describing the observable-mass scaling relations, their evolution, and intrinsic scatter, which cluster-count likelihoods marginalize over — is attributed precisely to the neglected stochasticity of the mass mapping. Second, the calibration is specific to one suite's subgrid physics, a single cosmology, and one halo-finding convention. Because the halo mass variation correlates with the halo assembly epoch (the AGN imprint is strongest when progenitors reach 30-50% of their final mass), baryons may also induce assembly-dependent 'secondary' bias — dependence of halo clustering on properties beyond mass. The authors therefore call for a systematic inter-comparison across independent simulation suites to define priors on the mass-correction function and for observational anchoring through measured gas and stellar mass fractions; they further acknowledge that their calibration is limited to a single cosmological model and one halo-finding convention, so that universality and cosmology dependence are not accounted for, and they leave the quantification of secondary effects on halo clustering for further investigation.

Problem Statement

The problem, as posed by the source and still live, is to make baryonic corrections to the halo mass function and halo bias robust enough that they no longer dominate the error budget of next-generation cluster-abundance cosmology. Concretely, for halos of group and cluster mass ($\gtrsim 10^{13}\,M_\odot/h$) over the redshift range $0\le z\le 2$:

1. Cross-suite calibration: carry out a systematic inter-comparison of the baryonic mass-correction mappings $M_{\rm Hydro}/M_{\rm DMO}$ across independently developed hydrodynamical simulation suites with different star-formation and feedback prescriptions, quantify the range of plausible mappings, and convert it into informative priors on the mass-correction function used in cosmological analyses.
2. Stochastic mass mapping: characterize the full distribution of $M_{\rm Hydro}/M_{\rm DMO}$ - its scatter, possible non-Gaussianity, and its covariance with observable mass proxies and nuisance parameters - and incorporate it into the cluster-count likelihood in place of the deterministic median correction; the source attributes the residual cosmological bias and nuisance-parameter tension to the median-only implementation.
3. Universality: establish (or delimit) the cosmology- and halo-finder-dependence of the baryonic HMF and halo-bias calibrations, extending them beyond a single cosmological model and halo-identification scheme.
4. Secondary bias: quantify the assembly-dependent effects of baryons on halo clustering beyond mass, and assess their impact on bias modelling for cluster cosmology.
5. Observational anchoring: use measured gas and stellar mass fractions of groups and clusters, and their evolution, to constrain the subgrid feedback models and thereby tighten the priors on the baryonic corrections.

These five strands serve one objective - survey-ready baryonic corrections - and are interlinked: the priors of (1) are set by the suite spread and tightened by (5); the stochastic model of (2) and the universality of (3) determine whether a single correction can be applied across analyses; and (4) determines whether mass-only bias modelling suffices. The five strands are independent, separately refutable sub-questions, each with its own scientific target: an inter-suite prior on the mass-correction function for (1); a stochastic $M_{\rm Hydro}/M_{\rm DMO}$ distribution, with its covariance with observable mass proxies, incorporated into the cluster-count likelihood for (2); a cosmology- and halo-finder-dependence delimitation of the calibration for (3); a quantified assembly-dependent effect of baryons on halo clustering with its impact on bias modelling for (4); and a gas- and stellar-mass-fraction constraint on the subgrid feedback priors for (5). A resolution of any one of these strands must not contradict what the other strands establish.

References

1. Tiago Castro, Stefano Borgani, Klaus Dolag, Valerio Marra, Miguel Quartin, Alexandro Saro, Emiliano Sefusatti (2021), 'On the impact of baryons on the halo mass function, bias, and cluster cosmology', Monthly Notices of the Royal Astronomical Society 500(2):2316-2335, DOI 10.1093/mnras/staa3473, https://doi.org/10.1093/mnras/staa3473 (arXiv:2009.01775)
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