Constraining Cosmological Parameters using the Cluster Mass-Richness Relation

TL;DR

This study derives the cluster mass-richness relation (MRR) from the spectroscopic GalWCat19 catalog and validates it against Illustris-TNG and mini-Uchuu simulations. It demonstrates a linear MRR above a richness threshold determined via a hinge analysis, with a slope near unity and modest intrinsic scatter. Using the MRR to estimate cluster masses, the authors recover cosmological constraints of ${\Omega}_m \approx 0.31$ and ${\sigma}_8 \approx 0.81$–0.82, competitive with Planck 2018, while highlighting how photometric catalogs can introduce larger systematics. The work emphasizes robustness to systematics through threshold choices and motivates future expansion to high-redshift, multi-survey cluster samples and complementary cosmological tests.

Abstract

The cluster mass-richness relation (MRR) is an observationally efficient and potentially powerful cosmological tool for constraining the mean matter density of the universe and the amplitude of fluctuations using the cluster abundance technique. We derive the MRR relation using GalWCat19, a publicly available galaxy cluster catalog we created from the Sloan Digital Sky Survey-DR13 spectroscopic dataset. The MRR shows a tail at the low-richness end. Using the Illustris-TNG and mini-Uchuu cosmological numerical simulations, we demonstrate that this tail is caused by systematical uncertainties. We show that, by means of a judicious cut, identified by the use of the Hinge function, it is possible to determine a richness threshold above which the MRR is linear i.e., where cluster mass scales with richness as logM_200 = alpha + beta logN_200. We derive the MRR and show it is consistent with both sets of simulations with a slope of beta ~ 1. We use our MRR to estimate cluster masses from the GalWCat19 catalog which we then use to set constraints on omega_m and sigma_8. Utilizing the all-member MRR, we obtain constraints of omega_m = 0.31 (+0.04-0.03) and sigma_8 = 0.82 (+0.05-0.04), and utilizing the red-member MRR, we obtain omega_m = 0.31 (+0.04-0.03) and sigma_8 = 0.81 (+0.05-0.04). Our constraints on omega_m and sigma_8 are consistent and very competitive with the Planck 2018 results.

Figures (7)

• Figure 1: Schematic diagram showing the effect on the number of clusters and their masses of varying $\Omega_\mathrm{m}$ (upper) and $\sigma_8$ (lower) independently while holding the other parameter fixed. Each circle represents a cluster with its size (large to small) and color (yellow to magenta) indicating high to low mass. A larger value of $\Omega_\mathrm{m}$ results in an approximately proportionally higher number of clusters of every mass. A larger value of $\sigma_8$ also results in a higher number of clusters of every mass but also changes the ratio of high to low mass clusters, so results in a larger number of high mass clusters relative to low mass clusters. Note that the clusters are distributed randomly in each box.
• Figure 2: Left: Distribution of clusters from the $\mathtt{GalWCat19}$ catalog in the mass-richness plane. Gray points show all 1800 clusters (with masses of $\log{M_{200}} \geq 13.5$$h^{-1} \ M_{\odot}$ and at a redshift $z\leq0.2$ (see § \ref{['sec:Rich']}). Black points show the complete fiducial ($\mathcal{S}_\mathrm{fid}$) subsample of 756 clusters to which the redshift selection function has been applied ($\log{M_{200}} \geq 13.9$$h^{-1} \ M_{\odot}$ and $0.045 \leq z \leq0.125$; see § \ref{['sec:comp']}). The mean mass of clusters in $\mathcal{S}_\mathrm{fid}$ as a function of their richness are shown as solid red circles, with error bars indicating 1$\sigma$ Poisson uncertainties. The blue solid line shows the Hinge function for $\mathcal{S}_\mathrm{fid}$ (see § \ref{['sec:threshold']}), with the dotted blue lines indicating $1\sigma$ uncertainties. The vertical blue dashed line shows the richness threshold at which the slope of the distribution changes (see § \ref{['sec:sim']} and equation \ref{['eq:Hinge']}). Right: Red circles and uncertainties are as on left. Also shown are the mean masses of clusters from two simulations (see \ref{['sec:sim']}) Black points show clusters from mini-Uchuu with a subhalo peak velocity threshold of $v_\mathrm{peak}=130$ km s$^{-1}$ and blue points show clusters from TNG with a galaxy stellar mass threshold of $M_s\geq 5\times 10^9$$h^{-1} \ M_{\odot}$. As can be seen clearly from both panels, a flattening (tail) of the MRR occurs at low richness ($\log{N}\lesssim 1.23$). The short tail in the simulations is not intrinsic. It is partially due to the threshold of simulations as well as Poisson scattering (see § \ref{['sec:appD']}).
• Figure 3: Comparison of the best-fit red MRR, $\mathrm{MRR_{red}}$, derived in this work from $\mathcal{S}\mathrm{red}_{13}$ (dark green line with shading indicating $1\sigma$ uncertainty) with results reported in the literature (see legend and Table \ref{['tab:MRR']}).
• Figure 4: Comparison of constraints obtained on $\Omega_\mathrm{m}$ and $\sigma_8$ in this work with those obtained from previous MRR analyses using different cluster catalogs and richness thresholds. Left: 68% CLs derived within $R_{200}$ for all members with $N_\mathrm{th}=17$ (yellow, $\mathcal{S}\mathrm{all}_{17}$) and red members with $N_\mathrm{th}=13$ (gray, $\mathcal{S}\mathrm{all}_{13}$). Right: uncertainties on $\Omega_\mathrm{m}$ and $\sigma_8$ estimated from the previous studies of Rozo10Costanzi19Kirby19Lesci22 (purple, brown,cyan, respectively) which use the cluster abundance technique and cluster mass estimates from the mass-richness relation ($\mathrm{CA}_\mathrm{MRR}$). Also shown are Abdullah20a (pink) which uses the cluster abundance technique and cluster mass estimates from the dynamics of member galaxies ($\mathrm{CA}_\mathrm{dyn}$) and Planck18 (blue) which uses the CMB technique (see Table \ref{['tab:Abb']} for the abbreviation).
• Figure 5: Systematical effects of cluster mass, lower and upper redshifts, and richness thresholds on our constraints on the cosmological parameters for the analysis on the fiducial samples of $\mathcal{S}\mathrm{all}_{17}$ for all members within $R_{200}$ (left) and $\mathcal{S}\mathrm{red}_{13}$ for red members within $R_{200}$ (right) (see § \ref{['sec:sys']} for details). The 68% CLs for our fiducial sample, varying mass threshold $\log M_{200}$ between $13.8$ and $14.0$ [$h^{-1} \ M_{\odot}$], fixing the upper redshift threshold to 0.125 and varying the lower redshift threshold from 0.01 to 0.07, fixing the lower redshift threshold to 0.045 and varying the upper redshift threshold from 0.11 to 0.15, varying $N_\mathrm{th}$ from 15 to 19 for $\mathcal{S}\mathrm{all}_{17}$, and varying $N_\mathrm{th}$ from 13 to 17 for $\mathcal{S}\mathrm{red}_{13}$.