CRITICAL STAR FORMATION RATES FOR REIONIZATION: FULL REIONIZATION OCCURS AT REDSHIFT z ≈ 7

We assess the probable redshift (z(rei) approximate to 7) for full reionization of the intergalactic medium (IGM) using a prescription for the comoving star formation rate (SFR) density ((rho) over dot(SFR)) required to maintain photoionization against recombination. Our newly developed online reionization simulator allows users to assess the required SFR and ionization histories, using a variety of assumptions for galactic and stellar populations, IGM clumping factor and temperature, and Lyman continuum (LyC) escape fraction. The decline in high-redshift galaxy candidates and Ly alpha emitters at z = 6-8 suggests a rising neutral fraction, with reionization at z greater than or similar to 7 increasingly difficult owing to increased recombination rates and constraints from the ionizing background and LyC mean free path. The required rate is (rho) over dot(SFR) approximate to (0.018M(circle dot) yr(-1) Mpc(-3))[(1 + z)/8](3)(C-H/3)(0.2/f(esc)) T-4(-0.845) 4 scaled to fiducial values of clumping factor C-H = 3, escape fraction f(esc) = 0.2, electron temperature T-e = 10(4) K, and low-metallicity initial mass functions (IMFs) and stellar atmospheres. Our hydrodynamical + N-body simulations find a mean clumping factor C-H approximate to (2.9)[(1+ z)/6](-1.1) in the photoionized, photoheated filaments at z = 5-9. The critical SFR could be reduced by increasing the minimum stellar mass, invoking a top-heavy IMF, or systematically increasing f(esc) at high z. The cosmic microwave background optical depth, tau(e) = 0.088 +/- 0.015, could be explained by full reionization, producing tau(e) = 0.050 back to z(rei) approximate to 7, augmented by Delta tau(e) approximate to 0.01-0.04 in a partially ionized IGM at z > 7. In this scenario, the strongest 21 cm signal should occur at redshifted frequencies 124-167 MHz owing to IGM heating over an interval Delta z approximate to 3 in the range z approximate to 7.5-10.5.