  Detecting the change of a cosmological object's redshift due to the time
evolution of the Universal expansion rate is an ambitious experiment that will
be attempted with future telescope facilities. In this paper, we describe the
ACCELERATION programme, which aims to study the properties of the most
underdense regions of the Universe. One of the highlight goals of this
programme is to prepare for the redshift drift measurement. Using the EAGLE
cosmological hydrodynamic simulations, we estimate the peculiar acceleration of
gas in galaxies and in the Lya forest. We find that star-forming 'cold neutral
gas' exhibits large peculiar acceleration due to the high local density of
baryons near star-forming regions. We conclude that absorption by cold neutral
gas is unlikely to yield a detection of the cosmological redshift drift. On the
other hand, we find that the peculiar accelerations of Lya forest absorbers are
more than an order of magnitude below the expected cosmological signal. We also
highlight that the numerous low H I column density systems display lower
peculiar acceleration. Finally, we propose a new 'Lya cell' technique that
applies a small correction to the wavelength calibration to secure a relative
measurement of the cosmic drift between two unrelated cosmological sources at
different redshifts. For suitable combinations of absorption lines, the
cosmological signal can be more than doubled, while the affect of the observer
peculiar acceleration is mitigated. Using current data of four suitable Lya
cells, we infer a limit on the cosmological redshift drift to be dv/dt_obs < 65
m/s/year (2 sigma).
