The Star Formation Rate Function for Redshift z~4-7 Galaxies: Evidence for a Uniform Build-Up of Star-Forming Galaxies During the First 3 Gyr of Cosmic Time

We combine recent estimates of dust extinction at z~4-7 with UV luminosity function (LF) determinations to derive star formation rate (SFR) functions at z~4, 5, 6 and 7. SFR functions provide a more physical description of galaxy build-up at high redshift and allow for direct comparisons to SFRs at lower redshifts determined by a variety of techniques. Our SFR functions are derived from well-established z~4-7 UV LFs, UV-continuum slope trends with redshift and luminosity, and IRX-beta relations. They are well-described by Schechter relations. We extend the comparison baseline for SFR functions to z~2 by considering recent determinations of the H{\alpha} and mid-IR luminosity functions. The low-end slopes of the SFR functions are flatter than for the UV LFs, \Delta\alpha\sim+0.13, and show no clear evolution with cosmic time (z~0-7). In addition, we find that the characteristic value SFR* from the Schechter fit to SFR function exhibits consistent, and substantial, linear growth as a function of redshift from ~5 M_sun/yr at z~8, 650 Myr after the Big Bang, to ~100 M_sun/yr at z~2, ~2.5 Gyr later. Recent results at z~10, close to the onset of galaxy formation, are consistent with this trend. The uniformity of this evolution is even greater than seen in the UV LF over the redshift range z~2-8, providing validation for our dust corrections. These results provide strong evidence that galaxies build up uniformly over the first 3 Gyr of cosmic time.