Astrophysical tests of the stability of fundamental couplings, such as thefine-structure constant $\alpha$, are a powerful probe of new physics. Recentlythese measurements, combined with local atomic clock tests and Type Iasupernova and Hubble parameter data, were used to constrain the simplest classof dynamical dark energy models where the same degree of freedom is assumed toprovide both the dark energy and (through a dimensionless coupling, $\zeta$, tothe electromagnetic sector) the $\alpha$ variation. One caveat of theseanalyses was that it was based on fiducial models where the dark energyequation of state was described by a single parameter (effectively its presentday value, $w_0$). Here we relax this assumption and study broader dark energymodel classes, including the Chevallier-Polarski-Linder and Early Dark Energyparametrizations. Even in these extended cases we find that the current dataconstrains the coupling $\zeta$ at the $10^{-6}$ level and $w_0$ to a fewpercent (marginalizing over other parameters), thus confirming the robustnessof earlier analyses. On the other hand, the additional parameters are typicallynot well constrained. We also highlight the implications of our results forconstraints on violations of the Weak Equivalence Principle and improvements tobe expected from forthcoming measurements with high-resolution ultra-stablespectrographs.
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