Probing the time variation of fine structure constant using galaxy clusters and quintessence model

We explore a possible time variation of the fine structure constant ($\alpha \equiv e^2/\hbar c$) using the Sunyaev-Zel'dovich effect measurements of galaxy clusters along with their X-ray observations. Specifically, the ratio of the integrated Compto-ionization parameter $Y_{SZ}D_A^2$ and its X-ray counterpart $Y_X$ is used as an observable to constrain the bounds on the variation of $\alpha$. Considering the violation of cosmic distance duality relation, this ratio depends on the fine structure constant as $\sim \alpha^3$. We use the quintessence model to provide the origin of $\alpha$ time variation. In order to give a robust test on $\alpha$ variation, two galaxy cluster samples, the 61 clusters provided by the Planck collaboration and the 58 clusters detected by the South Pole Telescope, are collected for analysis. Their X-ray observations are given by the XMM-Newton survey. Our results give $\zeta=-0.203^{+0.101}_{-0.099}$ for the Planck sample and $\zeta=-0.043^{+0.165}_{-0.148}$ for the SPT sample, indicating that $\alpha$ is constant with redshift within $3\sigma$ and $1\sigma$ for the two samples, respectively.