Galaxy clusters and a possible variation of the fine structure constant
  Galaxy clusters have been used as a cosmic laboratory to verify a possible
time variation of fundamental constants. Particularly, it has been shown that
the ratio $Y_{SZ}D_{A}^{2}/C_{XZS}Y_X $, which is expected to be constant with
redshift, can be used to probe a variation of the fine structure constant,
$\alpha$. In this ratio, $Y_{SZ}D_{A}^{2}$ is the integrated comptonization
parameter of a galaxy cluster obtained via Sunyaev-Zel'dovich (SZ) effect
observations multiplied by its angular diameter distance, $D_A$, $Y_X$ is the
X-ray counterpart and $C_{XSZ}$ is an arbitrary constant. Using a combination
of SZ and X-ray data, a recent analysis found $Y_{SZ}D_{A}^{2}/C_{XZS}Y_X =
{\rm{C}} \alpha(z)^{3.5}$, where ${\rm{C}}$ is a constant. In this paper,
following previous results that suggest that a variation of $\alpha$
necessarily leads to a violation of the cosmic distance duality relation,
$D_L/D_A(1+z)^2 = 1$, where $D_L$ is the luminosity distance of a given source,
we derive a new expression, $Y_{SZ}D_{A}^{2}/C_{XSZ}Y_X = {\rm{C}} \alpha^{3.5}
\eta^{-1}(z)$, where $\eta(z) = D_L/D_A(1+z)^2$. In particular, considering the
direct relation $\eta(z) \propto \alpha(z)^{1/2}$, derived from a class of
dilaton runaway models, %class of modified gravity theories, and 61
measurements of the ratio $Y_{SZ}D_{A}^{2}/C_{XSZ}Y_X$ provided by the Planck
collaboration, we discuss bounds on a possible variation of $\alpha$. We also
estimate the value of the constant $ {\rm{C}}$, which is compatible with the
unity at $2\sigma$ level, indicating that the assumption of isothermality for
the temperature profile of the galaxy clusters used in the analysis holds.
