On a possible cosmological evolution of galaxy cluster $Y_{\rm X}-Y_{\rm SZE}$ scaling relation

An important result from self-similar models that describe the process of galaxy cluster formation is the simple scaling relation $Y_{\rm SZE}D_{\rm A}^{2}/C_{\rm XSZE}Y_{\rm X}= C$. In this ratio, $Y_{\rm SZE}$ is the integrated Sunyaev-Zel'dovich effect flux of a cluster, its x-ray counterpart is $Y_{\rm X}$, $C_{\rm XSZE}$ and $C$ are constants and $D_{\rm A}$ is the angular diameter distance to the cluster. In this paper, we consider the cosmic distance duality relation validity jointly with type Ia supernovae observations plus $61$ $Y_{\rm SZE}D_{\rm A}^{2}/C_{\rm XSZE}Y_{\rm X}$ measurements as reported by the Planck Collaboration to explore if this relation is constant in the redshift range considered ($z<0.5$). No one specific cosmological model is used. As basic result, although the data sets are compatible with no redshift evolution within 2$\sigma$ c.l., a Bayesian analysis indicates that other $C(z)$ functions analyzed in this work cannot be discarded. It is worth to stress that the observational determination of an universal $C(z)$ function turns the $Y_{\rm SZE}D_{\rm A}^{2}/C_{\rm XSZE}Y_{\rm X}$ ratio in an useful cosmological tool to determine cosmological parameters.