On the cosmological evolution of alpha and mu and the dynamics of dark energy

We study the cosmological evolution of the fine structure constant, $\alpha$, and the proton-to-electron mass ratio, $\mu=m_p/m_e$, in the context of a generic class of models where the gauge kinetic function is a linear function of a quintessence-type real scalar field, $\phi$, described by a Lagrangian with a standard kinetic term and a scalar field potential, $V(\phi)$. We further assume that the scalar field potential is a monotonic function of $\phi$ and that the scalar field is always rolling down the potential. We show that, for this class of models, low-redshift constrains on the evolution of $\alpha$ and $\mu$ can provide very stringent limits on the corresponding variations at high-redshift. We also demonstrate that these limits may be relaxed by considering more general models for the dynamics of $\alpha$ and $\mu$. However, in this case, the ability to reconstruct the evolution of the dark energy equation of state using varying couplings could be seriously compromised.