Forecasts analysis on varying-$\alpha$ theories from gravitational wave standard sirens

Motivated by future gravitational waves observations, we perform forecasts analysis to constrain a possible time variation of the fine structure constant ($\alpha$) within the context of the so-called runaway dilaton model. For this purpose, some gravitational-wave standard sirens mock data within the perspective of Einstein Telescope and LISA mission were considered jointly with current strong gravitational lensing systems observations. We find that future standard sirens observations can also play an important role in the search for possible variations of $\alpha$ within the methodology presented in this work.