ESPRESSO's Early Commissioning Results and Performance Related to Tests of Fundamental Constant Stability

ESPRESSO is a new high-resolution ultra-stable spectrograph for the VLT, which had its first light on Telescope on November 27th, 2017. The instrument is installed in the Combined Coud\'{e} Laboratory and linked to the 4 Units of Telescope through optical Coud\'{e} Trains, being the first spectrograph able to collect the light from the 4 UTs simultaneously. One of the key science goals of the instrument is to test the stability of nature's fundamental couplings with unprecedented resolution and stability. ESPRESSO will allow to eliminate current known systematics and test the claim by Webb et al 2012 of a spatial dipole in the variation of the fine-structure constant. These improved results (either null or variation detections) will put strong constraints on a range of cosmological and particle physics parameters.