We aim to place new, strengthened constraints on the luminosity function (LF) of H-alpha emitting galaxies at redshift z=2.2, and to further constrain the instantaneous star-formation rate density of the universe (rho*). We have used the new HAWK-I instrument at ESO-VLT to obtain extremely deep narrow-band (line; NB2090) and broad-band (continuum; Ks) imaging observations. The target field is in the GOODS-South, providing us with a rich multi-wavelength auxiliary data set, which we utilise for redshift confirmation and to estimate dust content. We use this new data to measure the faint-end slope (alpha) of LF(H-alpha) with unprecedented precision. The data are well fit by a Schechter function and also a single power-law, yielding alpha=(-1.72 +/- 0.20) and (-1.77 +/- 0.21), respectively. Thus we are able to confirm the steepening of alpha from low- to high-z predicted by a number of authors and observed at other wavelengths. We combine our LF data-points with those from a much shallower but wider survey at z=2.2 (Geach et al. 2008), constructing a LF spanning a factor of 50 in luminosity. Re-fitting the Schechter parameters, we obtain log L*=(43.07+/-0.22)erg s^-1 ; log phi*=(-3.45+/-0.52)Mpc^-3 ; alpha=(-1.60+/-0.15). We integrate over LF(Halpha) and apply a correction for dust attenuation to determine the instantaneous cosmic star-formation rate density at z=2 without assuming alpha or extrapolating it from lower-z. Our measurement of rho* is (0.215+/-0.090) Msun yr^-1 Mpc^-3, integrated over a range of 37 <log(LHhalpha / erg s^-1) < 47.
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