An Empirical Model for the Galaxy Luminosity and Star-Formation Rate Function at High Redshift

Using the most recent measurements of the ultraviolet (UV) luminosity functions (LFs) and dust estimates of early galaxies, we derive updated dust-corrected star-formation rate functions (SFRFs) at z~4-8, which we model to predict the evolution to higher redshifts, z>8. We employ abundance matching techniques to calibrate a relation between galaxy star formation rate (SFR) and host halo mass M$_{h}$ by mapping the shape of the observed SFRFs at z~4-8 to that of the halo mass function. The resulting scaling law remains roughly constant over this redshift range. We apply the average SFR-M$_{h}$ relation to reproduce the observed SFR functions at 4 <= z <= 8 and also derive the expected UV LFs at higher redshifts. At z~9 and z~10 these model LFs are in excellent agreement with current observed estimates. Our predicted number densities and UV LFs at z>10 indicate that JWST will be able to detect galaxies out to z~15 with an extensive treasury sized program. We also derive the redshift evolution of the star formation rate density and associated reionization history by galaxies. Models which integrate down to the current HUDF12/XDF detection limit (M$_{UV}$ ~ -17.7 mag) result in a SFRD that declines as (1+z)$^{-10.4\pm0.3}$ at high redshift and fail to reproduce the observed CMB electron scattering optical depth, $\tau$ ~ 0.066, to within 1$\sigma$. On the other hand, we find that the inclusion of galaxies with SFRs well below the current detection limit (M$_{UV}$ < -5.7 mag) leads to a fully reionized universe by z ~ 6.5 and an optical depth of $\tau$ ~ 0.054, consistent with the recently derived Planck value at the 1$\sigma$ level.