The Building the Bridge survey for z=3 Lyα emitting galaxies II. Completion of the survey

Context. We have substantial information about the kinematics and abundances of galaxies at z approximate to 3 studied in absorption against the light of background QSOs. At the same time we have already studied 1000s of galaxies detected in emission mainly through the Lyman-break selection technique; however, we know very little about how to make the connection between the two data sets. Aims. We aim at bridging the gap between absorption-selected and emission-selected galaxies at z approximate to 3 by probing the faint end of the luminosity function of star-forming galaxies at z approximate to 3. Methods. Narrow-band surveys for Lyman-alpha (Ly alpha) emitters have proven to be an efficient probe of faint, star-forming galaxies in the high-redshift universe. We performed narrow-band imaging in three fields with intervening QSO absorbers (a damped Ly alpha absorber and two Lyman-limit systems) using the VLT. We target Ly alpha at redshifts 2.85, 3.15, and 3.20. Results. We find a consistent surface density of about 10 Ly alpha-emitters per square arcmin per unit redshift in all three fields down to our detection limit of about 3 x 10(41) erg s(-1). The luminosity function is consistent with what has been found by other surveys at similar redshifts. About 85% of the sources are fainter than the canonical limit of R = 25.5 for most Lyman-break galaxy surveys. In none of the three fields do we detect the emission counterparts of the QSO absorbers. In particular we do not detect the counterpart of the z = 2.85 damped Ly alpha absorber towards Q2138-4427. This implies that the DLA galaxy is either not a Ly alpha emitter or is fainter than our flux limit. Conclusions. Narrow-band surveys for Ly alpha emitters are excellent for probing the faint end of the luminosity function at z approximate to 3. There is a very high surface density of this class of objects; yet, we only detect galaxies with Ly alpha in emission, so the density of galaxies with similar broad band magnitudes will be substantially higher. This is consistent with a very steep slope of the faint end of the luminosity function as has been inferred by other studies. This faint population of galaxies is playing a central role in the early Universe. There is evidence that this popualtion is dominating the intergrated star-formation activity, responsible for the bulk of the ionising photons at z >= 3 and likely also responsible for the bulk of the enrichment of the intergalactic medium.