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.
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