We aim to place new , strengthened constraints on the luminosity function ( LF ) of H-alpha ( H \alpha ) emitting galaxies at redshift z \approx 2.2 , and to further constrain the instantaneous star-formation rate density of the universe ( \dot { \rho } _ { \star } ) .
We have used the new HAWK-I instrument at ESO-VLT to obtain extremely deep narrow-band ( line ; NB2090 ) and broad-band ( continuum ; { K _ { \mathrm { s } } } ) 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 \pm 0.20 ) and ( -1.77 \pm 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 \sim 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 _ { \star } = ( 43.07 \pm 0.22 ) erg s ^ { -1 } ; \log \phi _ { \star } = ( -3.45 \pm 0.52 ) Mpc ^ { -3 } ; \alpha = ( -1.60 \pm 0.15 ) .
We integrate over LF ( H \alpha ) and apply a correction for dust attenuation to determine the instantaneous cosmic star-formation rate density at z \sim 2 without assuming \alpha or extrapolating it from lower- z .
Our measurement of \dot { \rho } _ { \star } is ( 0.215 \pm 0.090 ) M _ { \sun } yr ^ { -1 } Mpc ^ { -3 } , integrated over a range of 37 \leq \log ( L _ { \mathrm { H } \alpha } / \mathrm { erg } ~ { } \mathrm { s } ^ { -1 } ) \leq 47 .