Unbiased halo-mass-function calibration and uncertainty propagation for next-generation galaxy cluster number-count cosmology

Artis, Melin, Bartlett & Murray (A&A 649, A47, 2021) showed that the statistical uncertainties of halo mass function (HMF) parameter fits, propagated with simulation-derived priors into cluster number-count cosmology, significantly degrade cosmological constraints for Euclid/Roman/Rubin-class surveys, and identified five unresolved issues that together define one objective: making the HMF treatment accurate (not only precise), cosmology-dependent, inclusive of simulation systematics, and fully integrated into the survey likelihood. Later work resolved parts of this agenda — HMF emulators now cover massive-neutrino and dynamical-dark-energy cosmologies, baryon-correction models reach percent accuracy for virial masses, halo-finder effects have been propagated to cosmological constraints, and joint-inference frameworks have been built and validated on mocks — but the integrated objective remains open: inter-form HMF differences still bias forecasts for forthcoming surveys (FLAMINGO 2025), no real-survey cluster-count analysis yet adjusts HMF parameters simultaneously with cosmology, and the cosmology dependence of HMF calibration uncertainty inside a survey likelihood is untreated.

Background

Galaxy clusters are the most massive bound objects in the Universe, and their abundance as a function of mass and redshift is a sensitive probe of the matter density $\Omega_m$, the amplitude of density fluctuations $\sigma_8$, neutrino masses, and dark energy. The key theoretical ingredient is the halo mass function (HMF), $dn/d\ln M$, which counts dark matter halos per comoving volume per logarithmic mass interval; it is usually written as $\rho_{m,0}/M$ times the logarithmic derivative of $\sigma^{-1}(M,z)$ times a fitted multiplicity function $f(\sigma)$, where $\sigma(M,z)$ is the rms linear density fluctuation on the scale that collapses into a halo of mass $M$ at redshift $z$. Standard calibrations, such as Tinker et al. (2008) for spherical-overdensity halos and Despali et al. (2016) for friends-of-friends (FoF) and virial halos, assume universality: a single fitted $f(\sigma)$ with cosmology entering only through $\sigma(M,z)$. Cluster surveys select on observables (richness, X-ray luminosity, or the thermal Sunyaev-Zel'dovich Compton-$Y$ signal), so a cluster-count likelihood must also model the survey selection function and the observable-mass relation, whose parameters are calibrated on simulations or weak lensing. Artis, Melin, Bartlett & Murray (2021) quantified what happens when the fitted HMF parameters are freed within their simulation-derived covariance: while Planck-like catalogs are barely affected, for a Euclid-like survey of order $10^5$ clusters the HMF parameter uncertainties can significantly degrade constraints, and an improvement of order 30% in HMF parameter precision would be needed to keep the degradation below 5% on any single cosmological parameter and 10% on the figure of merit (FoM, the inverse area of the error ellipse in a given parameter plane, e.g. $(\Omega_m,\sigma_8)$); for a larger hypothetical 'Future' catalog the required improvement rises to about 70%. Their analysis assumes the Despali HMF form is unbiased and considers only statistical errors on its parameters; the published error bars of different HMF fits (e.g. Tinker vs Despali) do not even overlap, exposing an accuracy problem beyond precision. The HMF covariance itself was obtained in a single $\Lambda$CDM fit, so nothing is known about how the fitted parameter uncertainties change in extended cosmologies (massive neutrinos with total mass $\sum m_\nu$; a dark-energy equation of state $w(a)=w_0+w_a(1-a)$). Baryonic physics and the choice of halo finder alter the measured halo abundance, and the authors note that future priors must encapsulate these simulation systematics. Finally, they argue that a full cluster-count likelihood must adjust cosmology, the selection function, the HMF parameters (with simulation priors), the observable-mass relation (with priors), and the large-scale halo bias $b(M,z)$ simultaneously — a framework that did not exist for real survey data. Subsequent work has advanced individual facets: Mira-Titan (Bocquet et al. 2020) and later emulators (Aemulus $\nu$, e-MANTIS, CSST emulator) emulate the HMF across extended cosmological parameter spaces; Euclid Collaboration papers XXIV (2023, $\Lambda(\nu)$CDM calibration) and XXXIX (2024, baryon impact) address calibration and baryons; Payerne et al. (2023) tested likelihood accuracy; FLAMINGO (Kugel et al. 2025) quantified how residual model biases affect forthcoming surveys.

Problem Statement

Establish, and demonstrate on survey-realistic data, a treatment of the halo mass function in galaxy cluster number-count cosmology that closes the five unresolved issues identified by Artis et al. (2021) as one coupled objective — the validated end-to-end demonstration of (4), which must consume the calibration uncertainty of (2) and the priors of (3) and meet the bounds stated in (1) — comprising the following four components: (1) quantify and control the accuracy (systematic, inter-form) differences among HMF calibrations — e.g. Tinker vs Despali, baryonic impacts, halo-finder and simulation systematics — so that the adopted HMF is demonstrably unbiased, not merely precise; here 'unbiased' is measured against a stated reference truth — the fiducial simulation HMF at a declared halo mass definition — and the unbiasedness demonstration is made on mock data, where that truth is known so that the systematic offset of the adopted HMF is separable from its statistical error and is quantified against that reference truth across the inter-form, baryonic, and halo-finder comparisons; the separate error-budget bar, applied to the end-to-end demonstration of (4), is the Artis et al. (2021) target that the HMF treatment degrade no single recovered cosmological parameter by more than 5% and the figure of merit by no more than 10%, demonstrated for at least one representative next-generation survey class — the Euclid-, Rubin/LSST-, Roman-, eROSITA-, and Simons-Observatory names being a scope descriptor for next-generation cluster samples, not five separate mandatory demonstrations; (2) determine how the HMF calibration uncertainty itself depends on cosmology — whether the fitted parameter error bars and their correlations, including correlations with $\sum m_\nu$, $w_0$, $w_a$, grow or change in nonstandard cosmologies relative to the $\Lambda$CDM fit — and test the assumed universality of the fitted parameter errors across redshift and cosmology, yielding an explicit fitted-parameter covariance in at least one extended cosmology (massive neutrinos and/or $w_0$–$w_a$) and an explicit verdict on whether those fitted parameter errors are cosmology- and redshift-independent; (3) turn the simulation and halo-finder systematics quantified in (1) into the priors that a survey likelihood can actually use, yielding a concrete prior — a stated parameter covariance or prior distribution with its simulation and halo-finder provenance — that the joint likelihood of (4) actually consumes; and (4) implement and validate a joint likelihood that adjusts cosmology, the survey selection function, the HMF parameters (with cosmology-dependent simulation priors), the observable-mass relation (with priors), and the large-scale halo bias simultaneously, and apply it to real or fully realistic mock survey data — constructing or sourcing the mock catalogue, its survey selection function, and its observable-mass relation with priors is itself part of the deliverable, and the survey class, mass and redshift range, expected cluster count, and adopted observable-mass calibration must be stated and justified — showing that the recovered cosmological parameters are unbiased against the reference truth of (1) in the fully realistic mock branch, where that truth is known, and that the HMF contribution to the error budget is quantified and subdominant, that is, small enough that the degradation bounds in (1) are met (or explicitly corrected to that level).

References

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