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Abstract Galaxy clusters are powerful laboratories for studying both cosmic structure formation and galaxy evolution. We present a comprehensive analysis of the velocity anisotropy profile,β(r), in galaxy clusters using the Uchuu-UniverseMachine mock galaxy catalog, which combines the large-volume UchuuN-body simulation with the UniverseMachine galaxy formation model. Focusing on clusters with $\log M_{200}\ge 13.9\left[h^{-1}{M}_{\odot }\right]$up to redshiftz= 1.5, we investigate the behavior ofβ(r) as a function of clustercentric radius, mass, and redshift. We find thatβ(r) exhibits a universal shape: it rises from isotropic values near the cluster core, peaks at ∼1.7R₂₀₀, declines around 3.4R₂₀₀due to orbital mixing, and increases again in the outskirts, due to the dominance of first-infalling galaxies. Our results show that more massive clusters have higher radial anisotropy and larger peakβvalues. Moreover,β(r) evolves with redshift, with high-redshift clusters displaying more radially dominated orbits and enhanced infall motions. We further derive redshift-dependent power-law scaling relations betweenM₂₀₀and key physical radii—hydrostatic (R_hs), infall ( $R_{\inf }$), and turnaround (R_ta). These findings offer a robust theoretical framework for interpreting the dynamical properties of observed galaxy clusters and provide key insights into the evolution of their dynamical state over cosmic time. 

Abstract The cluster mass–richness relation (MRR) is an observationally efficient and potentially powerful cosmological tool for constraining the matter density Ω_mand the amplitude of fluctuationsσ₈using the cluster abundance technique. We derive the MRR relation usingGalWCat19, a publicly available galaxy cluster catalog we created from the Sloan Digital Sky Survey-DR13 spectroscopic data set. In the MRR, cluster mass scales with richness as $\log M_{200}=\alpha +\beta \log N_{200}$. We find that the MRR we derive is consistent with both the IllustrisTNG and mini-Uchuu cosmological numerical simulations, with a slope ofβ≈ 1. We use the MRR we derived to estimate cluster masses from theGalWCat19catalog, which we then use to set constraints on Ω_mandσ₈. Utilizing the all-member MRR, we obtain constraints of Ω_m= ${0.31}_{-0.03}^{+0.04}$andσ₈= ${0.82}_{-0.04}^{+0.05}$, and utilizing the red member MRR only, we obtain Ω_m= ${0.31}_{-0.03}^{+0.04}$andσ₈= ${0.81}_{-0.04}^{+0.05}$. Our constraints on Ω_mandσ₈are consistent and very competitive with the Planck 2018 results. 

ABSTRACT We measure the two-point correlation function (CF) of 1357 galaxy clusters with a mass of log10M200 ≥ 13.6 h−1 M⊙ and at a redshift of z ≤ 0.125. This work differs from previous analyses in that it utilizes a spectroscopic cluster catalogue, $$\tt {SDSS-GalWCat}$$, to measure the CF and detect the baryon acoustic oscillation (BAO) signal. Unlike previous studies which use statistical techniques, we compute covariance errors directly by generating a set of 1086 galaxy cluster light-cones from the GLAM N-body simulation. Fitting the CF with a power-law model of the form ξ(s) = (s/s0)−γ, we determine the best-fitting correlation length and power-law index at three mass thresholds. We find that the correlation length increases with increasing the mass threshold while the power-law index is almost constant. For log10M200 ≥ 13.6 h−1 M⊙, we find s0 = 14.54 ± 0.87 h−1 Mpc and γ = 1.97 ± 0.11. We detect the BAO signal at s = 100 h−1 Mpc with a significance of 1.60σ. Fitting the CF with a Lambda cold dark matter model, we find $$D_\mathrm{V}(z = 0.089)\mathit{r}^{\mathrm{ fid}}_\mathrm{ d}/\mathit{r}_\mathrm{ d} = 267.62 \pm 26$$ h−1 Mpc, consistent with Planck 2015 cosmology. We present a set of 108 high-fidelity simulated galaxy cluster light-cones from the high-resolution Uchuu N-body simulation, employed for methodological validation. We find DV(z = 0.089)/rd = 2.666 ± 0.129, indicating that our method does not introduce any bias in the parameter estimation for this small sample of galaxy clusters.