Galaxy clusters, type Ia supernovae and the fine structure constant

As is well known, measurements of the Sunyaev-Zeldovich effect can be combined with observations of the X-ray surface brightness of galaxy clusters to estimate the angular diameter distance to these structures. In this paper, we show that this technique depends on the fine structure constant, α. Therefore, if α is a time-dependent quantity, e.g., α = α0(z), where is a function of redshift, we argue that current data do not provide the real angular diameter distance, DA(z), to the cluster, but instead DAdata(z) = (z)2 DA(z). We use this result to derive constraints on a possible variation of α for a class of dilaton runaway models considering a sample of 25 measurements of DAdata(z) in redshift range 0.023 < z < 0.784 and estimates of DA(z) from current type Ia supernovae observations. We find no significant indication of variation of α with the present data.