Mass-observable Relations Research Articles

Weak lensing mass bias and the alignment of centre proxies

ABSTRACT Galaxy cluster masses derived from observations of weak lensing suffer from a number of biases affecting the accuracy of mass-observable relations calibrated from such observations. In particular, the choice of the cluster centre plays a prominent role in biasing inferred masses. In the past, empirical miscentring distributions have been used to address this issue. Using hydrodynamic simulations, we aim to test the accuracy of weak lensing mass bias predictions based on such miscentring distributions by comparing the results to mass biases computed directly using intracluster medium (ICM)-based centres from the same simulation. We construct models for fitting masses to both centred and miscentred Navarro–Frenk–White profiles of reduced shear, and model the resulting distributions of mass bias with normal and lognormal distributions. We find that the standard approach of using miscentring distributions leads to an overestimation of cluster masses at levels of between 2 per cent and 6 per cent when compared to the analysis in which actual simulated ICM centres are used, even when the underlying miscentring distributions match in terms of the miscentring amplitude. While we find that neither lognormal nor normal distributions are generally reliable for accurately modelling the shapes of the mass bias distributions, both models can serve as reasonable approximations in practice.

Cluster cosmology redux: a compact representation for the halo mass function

ABSTRACT Groups and clusters of galaxies imprint coherent, arcminute-scale features across the spectrophotometric sky, especially optical-IR clusters of galaxies, spectral distortions in the cosmic microwave background, and extended sources of X-ray emission. The space–time density of the host dark matter halo population – the halo mass function (HMF) – is a common theoretical basis for modelling such observable features. We explore a compact representation – a dual-quadratic (DQ-HMF) form – that features readily interpretable parameters representing polynomial expansions of the space–time number density surface, first in terms of log-mass, then in redshift. The DQ-HMF form fits Mira-Titan N-body emulator expectations for halo masses $10^{13.7-14.5} \, h^{-1}\, {\rm M}_\odot$ over redshifts, 0.1 < z < 1.5 to within $\sim \! 5~{{\ \rm per\ cent}}$. We provide best-fitting parameters for a Planck 2018 cosmology and demonstrate model self-similarity in the Ωm−$\, \sigma _8$ plane. Convolving with a minimal mass–observable relation (MOR) yields closed-form expressions for counts, mean mass, and mass variance of cluster samples characterized by an observable property. Performing information-matrix forecasts of potential parameter constraints from existing and future surveys, we demonstrate the potential for percent-level constraints on model parameters by an LSST-like optical cluster survey of 300 000 clusters and a richness–mass variance of 0.32. Even better constraints could potentially be achieved by a survey with one-tenth the sample size but with a reduced selection property variance of 0.12. Potential benefits and extensions to the basic MOR parametrization are discussed.

Intrinsic Mass–Richness Relation of Clusters from THE THREE HUNDRED Hydrodynamic Simulations

The main systematics in cluster cosmology is the uncertainty in the mass–observable relation. In this paper, we focus on the most direct cluster observable in optical surveys, i.e., richness, and constrain the intrinsic mass–richness (MR) relation of clusters in THE THREE HUNDRED hydrodynamic simulations with two runs: GIZMO-SIMBA and GADGET-X. We find that modeling the richness at a fixed halo mass with a skewed Gaussian distribution yields a simpler and smaller scatter compared to the commonly used lognormal distribution. Additionally, we observe that baryon models have a significant impact on the scatter while exhibiting no influence on the mass dependence and a slight effect on the amplitude in the MR relation. We select member galaxies based on both stellar mass M ⋆ and absolute magnitude M . We demonstrate that the MR relations obtained from these two selections can be converted to each other by using the M⋆–M relation. Finally, we provide a seven-parameter fitting result comprehensively capturing the dependence of the MR relation on both stellar mass cutoff and redshift.

Galaxy cluster optical mass proxies from probabilistic memberships

ABSTRACT Robust galaxy cluster mass estimates are fundamental for constraining cosmological parameters from counts. For this reason, it is essential to search for tracers that, independent of the cluster’s dynamical state, have a small intrinsic scatter and can be easily inferred from observations. This work uses a simulated data set to focus on photometric properties and explores different optical mass proxies including richness, optical luminosity, and total stellar mass. We have developed a probabilistic membership assignment that makes minimal assumptions about the galaxy cluster properties, limited to a characteristic radius, velocity dispersion, and spatial distribution. Applying the estimator to over 919 galaxy clusters with zphot < 0.45 within a mass range of 1012.8–1015 M⊙, we obtain robust richness estimates that deviate from the median true value (from simulations) by −0.01 ± 0.12. The scatter in the mass–observable relations is $\sigma _{log_{10}(M|\mathcal {R})}=0.181 \pm 0.009$ dex for richness, $\sigma _{log_{10}(M|L_\lambda)}=0.151 \pm 0.007$ dex for optical luminosity, and $\sigma _{log_{10}(M|M_\lambda ^{*})}=0.097 \pm 0.005$ dex for stellar mass. We also discuss membership assignment, completeness and purity, and the consequences of small centre and redshift offsets. We conclude that the application of our method for photometric surveys delivers competitive cluster mass proxies.

Self-calibrating optical galaxy cluster selection bias using cluster, galaxy, and shear cross-correlations

ABSTRACT The clustering signals of galaxy clusters are powerful tools for self-calibrating the mass–observable relation and are complementary to cluster abundance and lensing. In this work, we explore the possibility of combining three correlation functions – cluster lensing, the cluster–galaxy cross-correlation function, and the galaxy autocorrelation function – to self-calibrate optical cluster selection bias, the boosted clustering and lensing signals in a richness-selected sample mainly caused by projection effects. We develop mock catalogues of redMaGiC-like galaxies and redMaPPer-like clusters by applying halo occupation distribution models to N-body simulations and using counts-in-cylinders around massive haloes as a richness proxy. In addition to the previously known small-scale boost in projected correlation functions, we find that the projection effects also significantly boost three-dimensional correlation functions to scales of 100 $h^{-1} \, \rm Mpc$. We perform a likelihood analysis assuming survey conditions similar to the Dark Energy Survey and show that the selection bias can be self-consistently constrained at the 10 per cent level. We discuss strategies for applying this approach to real data. We expect that expanding the analysis to smaller scales and using deeper lensing data would further improve the constraints on cluster selection bias.

The outer stellar mass of massive galaxies: a simple tracer of halo mass with scatter comparable to richness and reduced projection effects

ABSTRACT Using the weak gravitational lensing data from the Hyper Suprime-Cam Subaru Strategic Program (HSC survey), we study the potential of different stellar mass estimates in tracing halo mass. We consider galaxies with log10(M⋆/M⊙) > 11.5 at 0.2 < z < 0.5 with carefully measured light profiles, and clusters from the redMaPPer and CAMIRA richness-based algorithms. We devise a method (the ‘Top-N test’) to evaluate the scatter in the halo mass–observable relation for different tracers, and to inter-compare halo mass proxies in four number density bins using stacked galaxy–galaxy lensing profiles. This test reveals three key findings. Stellar masses based on CModel photometry and aperture luminosity within R <30 kpc are poor proxies of halo mass. In contrast, the stellar mass of the outer envelope is an excellent halo mass proxy. The stellar mass within R = [50, 100] kpc, M⋆, [50, 100], has performance comparable to the state-of-the-art richness-based cluster finders at log10Mvir ≳ 14.0 and could be a better halo mass tracer at lower halo masses. Finally, using N-body simulations, we find that the lensing profiles of massive haloes selected by M⋆, [50, 100] are consistent with the expectation for a sample without projection or mis-centring effects. Richness-selected clusters, on the other hand, display an excess at R ∼ 1 Mpc in their lensing profiles, which may suggest a more significant impact from selection biases. These results suggest that M⋆-based tracers have distinct advantages in identifying massive haloes, which could open up new avenues for cluster cosmology. The codes and data used in this work can be found here:

The XXL survey

Context. We present the forward cosmological analysis of an XMM-selected sample of galaxy clusters out to a redshift of unity. We derive mass-observable relations in a self-consistent manner using the sample alone. Special care is given to the modelling of selection effects. Aims. Following our previous 2018 study based on the dn/dz quantity alone, we perform an upgraded cosmological analysis of the same XXL C1 cluster catalogue (178 objects), with a detailed account of the systematic errors. The results are combined with external constraints from baryon acoustic oscillations (BAO) and the cosmic microwave background (CMB). Methods. This study follows the ASpiX methodology: we analysed the distribution of the observed X-ray properties of the cluster population in a 3D observable space (count rate, hardness ratio, redshift) and modelled as a function of cosmology along with the scaling relations and the selection function. Compared to more traditional methods, ASpiX allows the inclusion of clusters down to a few tens of photons and is much simpler to use. Two M − T relations are considered: that from the Canada-France-Hawaii Telescope (hereafter CFHT) and another from the more recent Subaru lensing analyses. Results. We obtain an improvement by a factor of two compared to the previous analysis, which dealt with the cluster redshift distribution for the XXL sample alone, letting the normalisation of the M − T relation and the evolution of the L–T relation free. Adding constraints from the XXL cluster two-point correlation function and the BAO from various surveys decreases the uncertainties by 23% and 53%, respectively, and 62% when adding both. The central value is in excellent agreement with the Planck CMB constraints. Switching to the scaling relations from the Subaru analysis and leaving more parameters free to vary provides less stringent constraints, but those obtained are still consistent with the Planck CMB at the 1-sigma level. Our final constraints are $ \sigma_8 = 0.99^{+0.14}_{-0.23} $, $ \Omega_m = 0.296 \pm 0.034\,(S_8 = 0.98^{+0.11}_{-0.21} $) for the XXL sample alone. Combining XXL ASpiX, the XXL cluster two-point correlation function, and the BAO, leaving 11 parameters free to vary, and allowing for the cosmological dependence of the scaling relations in the fit induces a shift of the central values, which is reminiscent of that observed for the Planck S-Z cluster sample. We find $ \sigma_8 = 0.793^{+0.063}_{-0.12} $ and $ \Omega_m = 0.364 \pm 0.015\,(S_8 = 0.872^{+0.068}_{-0.12} $), which are still compatible with Planck CMB at 2.2σ. Conclusions. The results obtained by the ASpiX method are promising; further improvement is expected from the final XXL cosmological analysis involving a cluster sample that is twice as large. Such a study paves the way for the analysis of the eROSITA and future Athena surveys.

Likelihood-free Forward Modeling for Cluster Weak Lensing and Cosmology

Abstract Likelihood-free inference provides a rigorous approach to performing Bayesian analysis using forward simulations only. The main advantage of likelihood-free methods is their ability to account for complex physical processes and observational effects in forward simulations. Here we explore the potential of likelihood-free forward modeling for Bayesian cosmological inference using the redshift evolution of the cluster abundance combined with weak-lensing mass calibration. We use two complementary likelihood-free methods, namely Approximate Bayesian Computation (ABC) and Density-Estimation Likelihood-Free Inference (DELFI), to develop an analysis procedure for the inference of the cosmological parameters (Ωm, σ 8) and the mass scale of the survey sample. Adopting an eROSITA-like selection function and a 10% scatter in the observable–mass relation in a flat ΛCDM cosmology with Ωm = 0.286 and σ 8 = 0.82, we create a synthetic catalog of observable-selected Navarro–Frenk–White clusters in a survey area of 50 deg2. The stacked tangential shear profile and the number counts in redshift bins are used as summary statistics for both methods. By performing a series of forward simulations, we obtain convergent solutions for the posterior distribution from both methods. We find that ABC recovers broader posteriors than DELFI, especially for the Ωm parameter. For a weak-lensing survey with a source density of n g = 20 arcmin−2, we obtain posterior constraints on S 8 = σ 8 Ω m / 0.3 0.3 of 0.836 ± 0.032 and 0.810 ± 0.019 from ABC and DELFI, respectively. The analysis framework developed in this study will be particularly powerful for cosmological inference with ongoing cluster cosmology programs, such as the XMM–XXL survey and the eROSITA all-sky survey, in combination with wide-field weak-lensing surveys.

The Subaru HSC weak lensing mass-observable scaling relations of spectroscopic galaxy groups from the GAMA survey

ABSTRACT We utilize the galaxy shape catalogue from the first-year data release of the Subaru Hyper Suprime-Cam (HSC) survey to study the dark matter content of galaxy groups in the Universe using weak lensing. We use galaxy groups from the Galaxy Mass and Assembly galaxy survey in approximately 100 sq. degrees of the sky that overlap with the HSC survey as lenses. We restrict our analysis to the 1587 groups with at least five members. We divide these groups into six bins each of group luminosity and group member velocity dispersion and measure the lensing signal with a signal-to-noise ratio of 55 and 51 for these two different selections, respectively. We use a Bayesian halo model framework to infer the halo mass distribution of our groups binned in the two different observable properties and constrain the power-law scaling relation and the scatter between mean halo masses and the two-group observable properties. We obtain a 5 per cent constraint on the amplitude of the scaling relation between halo mass and group luminosity with 〈M〉 = (0.81 ± 0.04) × 1014 h−1 M⊙ for Lgrp = 1011.5 h−2 L⊙, and a power-law index of α = 1.01 ± 0.07. We constrain the amplitude of the scaling relation between halo mass and velocity dispersion to be 〈M〉 = (0.93 ± 0.05) × 1014 h−1 M⊙ for $\sigma = 500\, {\rm km\, s}^{-1}$ and a power-law index to be α = 1.52 ± 0.10. However, these scaling relations are sensitive to the exact cuts applied to the number of group members. Comparisons with similar scaling relations from the literature show that our results are consistent and have significantly reduced errors.

Cosmology with galaxy clusters: Impact of theoretical and observational systematic uncertainties

In this talk I focus on how the modelling of the mass-observable relation and the halo mass function can impact the accuracy and precision of cosmological constraints inferred from galaxy clusters. I present a new analysis of clusters detected in mm wavelengths by the Planck satellite, highlighting the need of an improved description and calibration for the mass-observable relation. I also discuss how to improve our analysis in view of future cluster surveys, with a particular focus on the impact of the halo mass function calibration and the need for a universal definition.

Galaxy cluster number counts with individual lensing mass estimates: Forecasts for Euclid

The ΛCDM model is gradually becoming challenged by observational data. Upcoming cosmological surveys will increase the number of detected galaxy clusters by several orders of magnitude. Therefore, clusters will shortly provide precise cosmological constraints and improve our understanding of structure formation in the Universe. In the following, we present a cluster likelihood based on individual weak lensing mass estimates and forecast Euclid’s performances within this framework. We use a matched filter for weak lensing mass estimation and model its characteristics with a set of simulations. We use the Flagship N-body simulation to emulate the expected cluster mass distribution of a Euclid-like sample and test our statistical framework against it. Finally, we simultaneously constrain the observable-mass relation and the cosmological parameters.

Weak lensing mass modeling bias and the impact of miscentring

ABSTRACT Parametric modeling of galaxy cluster density profiles from weak lensing observations leads to a mass bias, whose detailed understanding is critical in deriving accurate mass-observable relations for constraining cosmological models. Drawing from existing methods, we develop a robust framework for calculating this mass bias in one-parameter fits to simulations of dark matter haloes. We show that our approach has the advantage of being independent of the absolute noise level, so that only the number of haloes in a given simulation and the representativeness of the simulated haloes for real clusters limit the accuracy of the bias estimation. While we model the bias as a lognormal distribution and the haloes with a Navarro–Frenk–White profile, our method can be generalized to any bias distribution and parametric model of the radial mass distribution. We find that the lognormal assumption is not strictly valid in the presence of miscentring of haloes. We investigate the use of cluster centres derived from weak lensing in the context of mass bias, and tentatively find that such centroids can yield sensible mass estimates if the convergence peak has a signal-to-noise ratio (SNR) approximately greater than 4. In this context we also find that the standard approach to estimating the positional uncertainty of weak lensing mass peaks using bootstrapping severely underestimates the true positional uncertainty for peaks with low SNRs. Though we determine the mass and redshift dependence of the bias distribution for a few experimental setups, our focus remains providing a general approach to computing such distributions.

Cosmology with Galaxy Cluster Weak Lensing: Statistical Limits and Experimental Design

Abstract We forecast constraints on the amplitude of matter clustering σ 8(z) achievable with the combination of cluster weak lensing and number counts, in current and next-generation weak lensing surveys. We advocate for an approach, analogous to galaxy–galaxy lensing, in which the observables in each redshift bin are the mean number counts and the mean weak lensing profile of clusters above a mass proxy threshold. The primary astrophysical nuisance parameter is the logarithmic scatter between the mass proxy and true mass near the threshold. For surveys similar to the Dark Energy Survey (DES), the Roman Space Telescope High Latitude Survey (HLS), and the Rubin Observatory Legacy Survey of Space and Time (LSST), we forecast aggregate precision on σ 8 of 0.26%, 0.24%, and 0.10%, respectively, if the mass–observable scatter is known externally to . These constraints would be degraded by about 20% for in the case of DES or HLS and for for LSST. A 1 month observing program with Roman Space Telescope targeting ∼2500 massive clusters could achieve a ∼ 0.5% constraint on σ 8(z = 0.7) on its own, or a ∼ 0.33% constraint in combination with the HLS. Realizing the constraining power of clusters Requires accurate knowledge of the mass–observable relation and stringent control of systematics. We provide analytic approximations to our numerical results that allow for easy scaling to other survey assumptions or other methods of cluster mass estimation.

Comparing different mass estimators for a large subsample of thePlanck-ESZ clusters

Context.Total mass is arguably the most fundamental property for cosmological studies with galaxy clusters. The individual cluster masses can be obtained with different methods, each with its own biases and limitations. Systematic differences in mass measurements can strongly impact the determination of the hydrostatic bias and of the mass-observable relations, key requirements of many cluster abundance studies.Aims.We investigate the present differences in the mass estimates obtained through independent X-ray, weak-lensing, and dynamical studies using a large subsample of thePlanck-ESZ clusters. We also discuss the implications for mass bias analyses.Methods.After assessing the systematic differences in the X-ray-derived masses reported by distinct groups, we examine the mass estimates obtained with independent methods and quantify the differences as the mean ratio 1-b = MHE/MWL, dyn, where HE refers to hydrostatic masses obtained from X-ray observations, WL refers to the results of weak-lensing measurements, and dyn refers to the mass estimates either from velocity dispersion or from the caustic technique. So defined, the 1-bparameter includes all possible astrophysical, observational, and methodological biases in one single value.Results.Recent X-ray masses reported by independent groups show average differences smaller than ∼10%, posing a strong limit on the systematics that can be ascribed to the differences in the X-ray analysis when studying the hydrostatic bias. The mean ratio between our X-ray masses and the weak-lensing masses in the LC2-single catalog is 1-b = 0.74 ± 0.06, which corresponds to a mass bias of 26 ± 6%, a value insufficient to reconcile thePlanckcluster abundance and cosmic microwave background results. However, the mean mass ratios inferred from the WL masses of different projects vary by a large amount, with APEX-SZ showing a bias consistent with zero (1-b = 1.02 ± 0.12), LoCuSS and CCCP/MENeaCS showing a significant difference (1-b = 0.76 ± 0.09 and 1-b = 0.77 ± 0.10, respectively), and WtG pointing to the largest deviation (1-b = 0.61 ± 0.12), which would substantially reduce the tension between thePlanckresults. Because of small differences between ourM − YXrelation and the one used by thePlanckcollaboration, our X-ray masses are on average 7% lower (4% at the same physical radius) than thePlanckmasses and can further reduce the required bias. At odds with the WL results, the dynamical mass measurements show better agreement with the X-ray hydrostatic masses, although there are significant differences when relaxed or disturbed clusters are used. However, the comparison is currently limited by the small sample sizes.Conclisions.The systematic differences between total masses obtained with recent independent X-ray analyses are smaller than those found in previous studies. This shifts the focus to WL and dynamical studies for a better convergence of the level of mass bias. However, the ratios obtained using different mass estimators suggest that there are still systematics that are not accounted for in all the techniques used to recover cluster masses. This prevents the determination of firm constraints on the level of hydrostatic mass bias in galaxy clusters.

Impact of systematics on cosmological parameters from future galaxy cluster surveys

Galaxy clusters are a recent cosmological probe. The precision and accuracy of the cosmological parameters inferred from these objects are affected by the knowledge of cluster physics, entering the analysis through the mass-observable scaling relations, and the theoretical description of their mass and redshift distribution, modelled by the mass function. In this work we forecast the impact of different modelling of these ingredients for clusters detected by future optical and near-IR surveys. We consider the standard cosmological scenario and the case with a time-dependent equation of state for dark energy. We analyse the effect of increasing precision on the scaling relation calibration, finding improved constraints on the cosmological parameters. This higher precision exposes the impact of the mass function evaluation, which is a subdominant source of systematics for current data. We compare two different evaluations for the mass function. In both cosmological scenarios the use of different mass functions leads to biases in the parameter constraints. For the ΛCDM model, we find a 1.6σ shift in the (Ωm, σ8) parameter plane and a discrepancy of ∼7σ for the redshift evolution of the scatter of the scaling relations. For the scenario with a time-evolving dark energy equation of state, the assumption of different mass functions results in a ∼8σ tension in the w0 parameter. These results show the impact, and the necessity for a precise modelling, of the interplay between the redshift evolution of the mass function and of the scaling relations in the cosmological analysis of galaxy clusters.

Joint cosmology and mass calibration from thermal Sunyaev-Zel’dovich cluster counts and cosmic shear

We present a new method for joint cosmological parameter inference and cluster mass calibration from a combination of weak lensing measurements and the abundance of thermal Sunyaev-Zel'dovich (tSZ) selected galaxy clusters. We combine cluster counts with the spherical harmonic cosmic shear power spectrum and the cross-correlation between cluster overdensity and cosmic shear. These correlations constrain the cluster mass-observable relation. We model the observables using a halo model framework, including their full non-Gaussian covariance. Forecasting constraints on cosmological and mass calibration parameters for a combination of LSST cosmic shear and Simons Observatory tSZ cluster counts, we find competitive constraints for cluster cosmology, with a factor two improvement in the dark energy figure of merit compared to LSST cosmic shear alone. We find most of the mass calibration information will be in the large and intermediate scales of the cross-correlation between cluster overdensity and cosmic shear. Finally, we find broadly comparable constraints to traditional analyses based on calibrating masses using stacked cluster lensing measurements, with the benefit of consistently accounting for the correlations with cosmic shear.

Cosmology dependence of galaxy cluster scaling relations

ABSTRACT The abundance of galaxy clusters as a function of mass and redshift is a well known powerful cosmological probe, which relies on underlying modelling assumptions on the mass-observable relations (MOR). Some of the MOR parameters can be constrained directly from multi-wavelength observations, as the normalization at some reference cosmology, the mass-slope, the redshift evolution, and the intrinsic scatter. However, the cosmology dependence of MORs cannot be tested with multi-wavelength observations alone. We use magneticum simulations to explore the cosmology dependence of galaxy cluster scaling relations. We run fifteen hydrodynamical cosmological simulations varying Ωm, Ωb, h0, and σ8 (around a reference cosmological model). The MORs considered are gas mass, baryonic mass, gas temperature, Y and velocity dispersion as a function of virial mass. We verify that the mass and redshift slopes and the intrinsic scatter of the MORs are nearly independent of cosmology with variations significantly smaller than current observational uncertainties. We show that the gas mass and baryonic mass sensitively depends only on the baryon fraction, velocity dispersion, and gas temperature on h0, and Y on both baryon fraction and h0. We investigate the cosmological implications of our MOR parametrization on a mock catalogue created for an idealized eROSITA-like experiment. We show that our parametrization introduces a strong degeneracy between the cosmological parameters and the normalization of the MOR. Finally, the parameter constraints derived at different overdensity (Δ500c), for X-ray bolometric gas luminosity, and for different subgrid physics prescriptions are shown in the appendix.

Cluster cosmology with the NIKA2 SZ Large Program

The main limiting factor of cosmological analyses based on thermal Sunyaev-Zel’dovich (SZ) cluster statistics comes from the bias and systematic uncertainties that affect the estimates of the mass of galaxy clusters. High-angular resolution SZ observations at high redshift are needed to study a potential redshift or morphology dependence of both the mean pressure profile and of the mass-observable scaling relation used in SZ cosmological analyses. The NIKA2 camera is a new generation continuum instrument installed at the IRAM 30-m telescope. With a large field of view, a high angular resolution and a high-sensitivity, the NIKA2 camera has unique SZ mapping capabilities. In this paper, we present the NIKA2 SZ large program, aiming at observing a large sample of clusters at redshifts between 0.5 and 0.9, and the characterization of the first cluster oberved with NIKA2.

Tomographic measurement of the intergalactic gas pressure through galaxy–tSZ cross-correlations

ABSTRACT We cross-correlate maps of the thermal Sunyaev–Zeldovich (tSZ) Compton-y parameter published by Planck with the projected distribution of galaxies in a set of low-redshift tomographic bins. We use the nearly full-sky 2MASS Photometric Redshift and WISE × SuperCOSMOS public catalogues, covering the redshift range z ≲ 0.4. Our measurements allow us to place constraints on the redshift dependence of the mass–observable relation for tSZ cluster count analyses in terms of the so-called hydrostatic mass bias parameter $1-b_{\scriptscriptstyle \rm H}$. These results can also be interpreted as measurements of the bias-weighted average gas pressure 〈bPe〉 as a function of redshift, a quantity that can be related to the thermodynamics of gas inside haloes and used to constrain energy injection processes. We measure $1-b_{\scriptscriptstyle \rm H}$ with $\sim \!13{{\ \rm per\ cent}}$ precision in six equispaced redshift bins, and find no evidence for a redshift-dependent mass bias parameter, in agreement with previous analyses. Our mean value of $1-b_{\scriptscriptstyle \rm H}= 0.59\pm 0.03$ is also in good agreement with the one estimated by the joint analysis of Planck cluster counts and cosmic microwave background anisotropies. Our measurements of 〈bPe〉, at the level of $\sim \!10{{\ \rm per\ cent}}$ in each bin, are the most stringent constraints on the redshift dependence of this parameter to date, and agree well both with previous measurements and with theoretical expectations from shock-heating models.

Toward a Model-independent Measurement of the Halo Mass Function with Observables

Abstract In the CDM paradigm, the halo mass function is a sensitive probe of the cosmic structure. In observations, halo mass is typically estimated from its relation with other observables. The resulting halo mass function is subject to systematic bias, such as the Eddington bias, due to the scatter or uncertainty in the observable–mass relation. Exact correction for the bias is not easy, as predictions for the observables are typically model-dependent in simulations. In this paper, we point out an interesting feature in the halo mass function of the concordance ΛCDM model: the total halo mass within each evenly spaced logarithmic mass bin is approximately the same over a large mass range. We show that this property allows us to construct an almost bias-free halo mass function using only an observable (as a halo mass estimator) and stacked weak lensing measurements as long as the scatter between the true halo mass and the observable-inferred mass has a stable form in logarithmic units. The method is not sensitive to the form of the mass–observable relation. We test the idea using cosmological simulations, and show that the method performs very well for realistic observables.