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We use deep Hubble Space Telescope imaging of the Frontier Fields toaccurately measure the galaxy rest-frame ultraviolet luminosity function (UVLF) in the redshift range $z \sim 6-8$. We combine observations in threelensing clusters A2744, MACS0416, MACS0717 and their associated parallelsfields to select high-redshift dropout candidates. We use the latest lensingmodels to estimate the flux magnification and the effective survey volume incombination with completeness simulations performed in the source plane. Wereport 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 eachparallel field, it drops to about 0.6 to 1 arcmin$^{2}$ in the cluster corefields because of the strong lensing. We compute the UV luminosity function at$z \sim 7$ using the combined galaxy sample and perform Monte Carlo simulationsto determine the best fit Schechter parameters. We are able to reliablyconstrain the LF down to an absolute magnitude of $M_{UV}=-15.25$, whichcorresponds to 0.005$L^{\star}$. More importantly, we find that the faint-endslope remains steep down to this magnitude limit with$\alpha=-2.04_{-0.17}^{+0.13}$. Our results confirm the most recent results indeep blank fields but extend the LF measurements more than two magnitudesdeeper. 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 assessthe contribution of galaxies to cosmic reionization we derive the UV luminositydensity at $z\sim7$ by integrating the UV LF down to an observationallyconstrained limit of $M_{UV} = -15$. We show that our determination ofLog($\rho_{UV}$)=$26.2\pm0.13$ (erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$) can besufficient to maintain the IGM ionized.

Are Ultra-faint Galaxies at z=6-8 Responsible for Cosmic Reionization ? Combined Constraints from the Hubble Frontier Fields Clusters and Parallels