The Faint-end of the UV Luminosity Function at z~6 to z~8: Combined Constraints from the Hubble Frontier Fields Clusters and Parallels

We use deep Hubble Space Telescope imaging of the Frontier Fields to accurately measure the galaxy rest-frame ultraviolet luminosity function (UV LF) in the redshift range $z \sim 6-8$. We combine observations in three lensing clusters A2744, MACS0416, MACS0717 and their associated parallels fields to select high-redshift dropout candidates. We use the latest lensing models to estimate the flux magnification and the effective survey volume in combination with completeness simulations performed in the source plane. We report the detection of 227 galaxy candidates at $z=6-7$ and 25 candidates at $z \sim 8$. While the total survey area is about 4 arcmin$^{2}$ in each parallel field, it drops to about 0.6 to 1 arcmin$^{2}$ in the cluster core fields because of the strong lensing. We compute the UV luminosity function at $z \sim 7$ using the combined galaxy sample and perform Monte Carlo simulations to determine the best fit Schechter parameters. We are able to reliably constrain the LF down to an absolute magnitude of $M_{UV}=-15.25$, which corresponds to 0.005$L^{\star}$. More importantly, we find that the faint-end slope remains steep down to this magnitude limit with $\alpha=-2.04_{-0.17}^{+0.13}$. Our results confirm the most recent results in deep blank fields but extend the LF measurements more than two magnitudes deeper. The UV LF at $z \sim 8$ is not very well constrained below $M_{UV}=-18$ due to the small number statistics and incompleteness uncertainties. To assess the contribution of galaxies to cosmic reionization we derive the UV luminosity density at $z\sim7$ by integrating the UV LF down to an observationally constrained limit of $M_{UV} = -15$. We show that our determination of Log($\rho_{UV}$)=$26.2\pm0.13$ (erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$) can be sufficient to maintain the IGM ionized.