A comprehensive study of H$\alpha$ emitters at $z \sim$ 0.62 in the DAWN
  survey: the need for deep and wide regions
  We present new estimates of the luminosity function (LF) and star formation
rate density (SFRD) for an H$\alpha$ selected sample at $z\sim0.62$ from the
Deep And Wide Narrow-band (DAWN) survey. Our results are based on a new
H$\alpha$ sample in the extended COSMOS region (compared to Coughlin et al.
2018) with the inclusion of flanking fields, resulting in a total area coverage
of $\sim$1.5 deg$^2$. A total of 241 H$\alpha$ emitters were selected based on
robust selection criteria using spectro-photometric redshifts and broadband
color-color classification. We explore the effect of different dust correction
prescriptions by calculating the LF and SFRD using a constant dust extinction
correction, A{$_{\textrm{H}\alpha}=1$} mag, a luminosity-dependent correction,
and a stellar-mass dependent correction. The resulting H$\alpha$ LFs are well
fitted using Schechter functions with best-fit parameters: L$^*=10^{42.24}$ erg
s$^{-1}$, $\phi^*=10^{-2.85}$ Mpc$^{-3}$, $\alpha = -1.62$ for constant dust
correction, L$^*=10^{42.31}$ erg s$^{-1}$, $\phi^*=10^{-2.8}$ Mpc$^{-3}$,
$\alpha=-1.39$ for luminosity-dependent dust correction, and L$^*=10^{42.36}$
erg s$^{-1}$, $\phi^*=10^{-2.91}$ Mpc$^{-3}$, $\alpha = -1.48$, for stellar
mass-dependent dust correction. The deep and wide nature of the DAWN survey
effectively samples H$\alpha$ emitters over a wide range of luminosities,
thereby providing better constraints on both the faint and bright end of the
LF. Also, the SFRD estimates $\rho_{\textrm{SFR}}=10^{-1.39}$
M$_{\odot}$yr$^{-1}$Mpc$^{-3}$ (constant dust correction),
$\rho_{\textrm{SFR}}=10^{-1.47}$ M$_{\odot}$yr$^{-1}$Mpc$^{-3}$
(luminosity-dependent dust correction), and $\rho_{\textrm{SFR}}=10^{-1.49}$
M$_{\odot}$yr$^{-1}$Mpc$^{-3}$ (stellar mass-dependent dust correction) are in
good agreement with the evolution of SFRD across redshifts ($0 < z < 2$) seen
from previous H$\alpha$ surveys.
