Examining the local Universe isotropy with galaxy cluster velocity dispersion scaling relations

In standard cosmology, the late Universe is assumed to be statistically homogeneous and isotropic. However, a recent study based on galaxy clusters by Migkas et al. (2021, arXiv:2103.13904) found an apparent spatial variation of approximately $9\%$ in the Hubble constant, $H_0$, across the sky. The authors utilised galaxy cluster scaling relations between various cosmology-dependent cluster properties and a cosmology-independent property, i.e., the temperature of the intracluster gas $(T)$. A position-dependent systematic bias of $T$ measurements can, in principle, result in an overestimation of apparent $H_0$ variations. In this study, we search for directional $T$ measurement biases by examining the scaling relation between the member galaxy velocity dispersion and the gas temperature $(σ_\mathrm{v}-T)$. Additionally, we search for apparent $H_0$ angular variations independently of $T$ by analysing the relations between the X-ray luminosity and Sunyaev-Zeldovich signal with the velocity dispersion, $L_\mathrm{X}-σ_\mathrm{v}$ and $Y_\mathrm{SZ}-σ_\mathrm{v}$. We utilise Monte Carlo simulations of isotropic cluster samples to quantify the statistical significance of any observed anisotropies. We find no significant directional $T$ measurement biases, and the probability that a directional $T$ bias causes the previously observed $H_0$ anisotropy is only $0.002\%$. On the other hand, from the joint analysis of the $L_\mathrm{X}-σ_\mathrm{v}$ and $Y_\mathrm{SZ}-σ_\mathrm{v}$ relations, the maximum variation of $H_0$ is found in the direction of $(295^\circ\pm71^\circ, -30^\circ\pm71^\circ)$ with a statistical significance of $3.64σ$, fully consistent with arXiv:2103.13904. Our findings strongly corroborate the previously detected spatial anisotropy of galaxy cluster scaling relations using a new independent cluster property, $σ_\mathrm{v}$.