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Observations of metal absorption systems in the spectra of distant quasarsallow to constrain a possible variation of the fine-structure constantthroughout the history of the Universe. Such a test poses utmost demands on thewavelength accuracy and previous studies were limited by systematics in thespectrograph wavelength calibration. A substantial advance in the field istherefore expected from the new ultra-stable high-resolution spectrographEspresso, recently installed at the VLT. In preparation of the fundamentalphysics related part of the Espresso GTO program, we present a thoroughassessment of the Espresso wavelength accuracy and identify possiblesystematics at each of the different steps involved in the wavelengthcalibration process. Most importantly, we compare the default wavelengthsolution, based on the combination of Thorium-Argon arc lamp spectra and aFabry-P\'erot interferometer, to the fully independent calibration obtainedfrom a laser frequency comb. We find wavelength-dependent discrepancies of upto 24m/s. This substantially exceeds the photon noise and highlights thepresence of different sources of systematics, which we characterize in detailas part of this study. Nevertheless, our study demonstrates the outstandingaccuracy of Espresso with respect to previously used spectrographs and we showthat constraints of a relative change of the fine-structure constant at the$10^{-6}$ level can be obtained with Espresso without being limited bywavelength calibration systematics.

Fundamental physics with Espresso: Towards an accurate wavelength calibration for a precision test of the fine-structure constant