  The redshifts of all cosmologically distant sources are expected to
experience a small, systematic drift as a function of time due to the evolution
of the Universe's expansion rate. A measurement of this effect would represent
a direct and entirely model-independent determination of the expansion history
of the Universe over a redshift range that is inaccessible to other methods.
Here we investigate the impact of the next generation of Extremely Large
Telescopes on the feasibility of detecting and characterising the cosmological
redshift drift. We consider the Lyman alpha forest in the redshift range 2 < z
< 5 and other absorption lines in the spectra of high redshift QSOs as the most
suitable targets for a redshift drift experiment. Assuming photon-noise limited
observations and using extensive Monte Carlo simulations we determine the
accuracy to which the redshift drift can be measured from the Ly alpha forest
as a function of signal-to-noise and redshift. Based on this relation and using
the brightness and redshift distributions of known QSOs we find that a 42-m
telescope is capable of unambiguously detecting the redshift drift over a
period of ~20 yr using 4000 h of observing time. Such an experiment would
provide independent evidence for the existence of dark energy without assuming
spatial flatness, using any other cosmological constraints or making any other
astrophysical assumption.
