Reconstructing the evolution of dark energy with variations of fundamental parameters

Under the assumption that the variations of parameters of nature and the current acceleration of the universe are related and governed by the evolution of a single scalar field, we show how information can be obtained on the nature of dark energy from observational detection of (or constraints on) cosmological variations of the fine structure constant and the proton-to-electron mass ratio. We also comment on the current observational status, and on the prospects for improvements with future spectrographs such as ESPRESSO and CODEX.