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 Ly \alpha 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 Ly \alpha 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 ‘ Ly \alpha 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 Ly \alpha cells , we infer a limit on the cosmological redshift drift to be \dot { v } _ { obs } < 65 ~ { } { m~ { } s } ^ { -1 } ~ { } { year } ^ { -1 } ( 2 \sigma ) .