  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.
