UV slope of z$\sim$3 bright ($L>L^{*}$) Lyman-break galaxies in the
  COSMOS field
  We analyse a unique sample of 517 bright ($L>L^{*}$) LBGs at redshift
z$\sim$3 in order to characterise the distribution of their UV slopes $\beta$
and infer their dust extinction under standard assumptions. We exploited
multi-band observations over 750 arcmin$^2$ of the COSMOS field that were
acquired with three different ground-based facilities: the Large Binocular
Camera (LBC) on the Large Binocular Telescope (LBT), the Suprime-Cam on the
SUBARU telescope, and the VIRCAM on the VISTA telescope (ULTRAVISTA DR2). Our
multi-band photometric catalogue is based on a new method that is designed to
maximise the signal-to-noise ratio in the estimate of accurate galaxy colours
from images with different point spread functions (PSF). We adopted an improved
selection criterion based on deep Y-band data to isolate a sample of galaxies
at $z\sim 3 $ to minimise selection biases. We measured the UV slopes ($\beta$)
of the objects in our sample and then recovered the intrinsic probability
density function of $\beta$ values (PDF($\beta$)), taking into account the
effect of observational uncertainties through detailed simulations. The
galaxies in our sample are characterised by mildly red UV slopes with
$<\beta>\simeq -1.70$ throughout the enitre luminosity range that is probed by
our data ($-24\lesssim M_{1600}\lesssim -21$). The resulting dust-corrected
star formation rate density (SFRD) is $log(SFRD)\simeq-1.6
M_{\odot}/yr/Mpc^{3}$, corresponding to a contribution of about 25% to the
total SFRD at z$\sim$3 under standard assumptions. Ultra-bright LBGs at $z \sim
3$ match the known trends, with UV slopes being redder at decreasing redshifts,
and brighter galaxies being more highly dust extinct and more frequently
star-forming than fainter galaxies. [abridged]
