  A multi-object spectrograph on the forthcoming European Extremely Large
Telescope will be required to operate with good sky coverage. Many of the
interesting deep cosmological fields were deliberately chosen to be free of
bright foreground stars, and therefore are potentially challenging for adaptive
optics (AO) systems. Here we investigate multi-object AO performance using
sub-fields chosen at random from within the Great Observatories Origins Deep
Survey (GOODS)-S field, which is the worst case scenario for five deep fields
used extensively in studies of high-redshift galaxies. Our AO system model is
based on that of the proposed MOSAIC instrument but our findings are equally
applicable to plans for multi-object spectroscopy on any of the planned
Extremely Large Telescopes. Potential guide stars within these sub-fields are
identified and used for simulations of AO correction. We achieve ensquared
energies within 75~mas of between 25-35\% depending on the sub-field, which is
sufficient to probe sub-kpc scales in high-redshift galaxies. We also
investigate the effect of detector readout noise on AO system performance, and
consider cases where natural guide stars are used for both high-order and
tip-tilt-only AO correction. We also consider how performance scales with
ensquared energy box size. In summary, the expected AO performance is
sufficient for a MOSAIC-like instrument, even within deep fields characterised
by a lack of bright foreground stars.
