We present results from a survey of galaxies in the fields of six z>3 Damped Lyman alpha systems (DLAs) using the Multi Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope (VLT). We report a high detection rate of up to ~80% of galaxies within 1000 km/s from DLAs and with impact parameters between 25 and 280 kpc. In particular, we discovered 5 high-confidence Lyman alpha emitters associated with three DLAs, plus up to 9 additional detections across five of the six fields. The majority of the detections are at relatively large impact parameters (>50 kpc) with two detections being plausible host galaxies. Among our detections, we report four galaxies associated with the most metal-poor DLA in our sample (Z/Z_sun = -2.33), which trace an overdense structure resembling a filament. By comparing our detections with predictions from the Evolution and Assembly of GaLaxies and their Environments (EAGLE) cosmological simulations and a semi-analytic model designed to reproduce the observed bias of DLAs at z>2, we conclude that our observations are consistent with a scenario in which a significant fraction of DLAs trace the neutral regions within halos with a characteristic mass of 10^11-10^12 M_sun, in agreement with the inference made from the large-scale clustering of DLAs. We finally show how larger surveys targeting ~25 absorbers have the potential of constraining the characteristic masses of halos hosting high-redshift DLAs with sufficient accuracy to discriminate between different models.
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