Fundamental physics constraints from testing the stability of the fine-structure constant with the ELTs

The increased collecting area of the ELTs will bring fainter high-z targets within the reach of high-resolution ultra-stable spectrographs, thus enabling a new generation of precision consistency tests, including tests of the stability of nature's fundamental couplings. For example, the stability of the fine-structure constant can be tested by looking at metal absorption lines produced by the intervening clouds along the line of sight of distant quasars. In this contribution, we discuss the performance that can be expected from the ELTs in testing the stability of the fine-structure constant, based on the early ESPRESSO observations, and some comparative forecasts of the impact of these measurements for representative models of fundamental physics and cosmology.