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

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