Variation in the fine-structure constant, distance-duality relation and the next generation of high-resolution spectrograph

The possibility of variation of the fundamental constants of nature has been a long-standing question, with important consequences for fundamental physics and cosmology. In particular, it has been shown that variations in the fine-structure constant, $\alpha$, are directly related to violation of the distance duality relation (DDR), which holds true as long as photons travel on unique null geodesics and their number is conserved. In this paper we use the currently available measurements of ${\Delta \alpha}/{\alpha}$ to impose the most stringent constraints on departures of the DDR to date, here quantified by the parameter $\eta$. We also perform a forecast analysis to discuss the ability of the new generation of high-resolution spectrograph, like ESPRESSO/VLT and E-ELT-HIRES, to constrain the DDR parameter $\eta$. From the current data we obtain constraints on $\eta$ of the order of $10^{-7}$ whereas the forecasted constraints are two orders of magnitude lower. Considering the expected level of uncertainties of the upcoming measurements, we also estimate the necessary number of data points to confirm the hypotheses behind the DDR.