We make use of deep 1.2mm-continuum observations (12.7microJy/beam RMS) of a 1 arcmin^2 region in the Hubble Ultra Deep Field to probe dust-enshrouded star formation from 330 Lyman-break galaxies spanning the redshift range z=2-10 (to ~2-3 Msol/yr at 1sigma over the entire range). Given the depth and area of ASPECS, we would expect to tentatively detect 35 galaxies extrapolating the Meurer z~0 IRX-beta relation to z>~2 (assuming T_d~35 K). However, only 6 tentative detections are found at z>~2 in ASPECS, with just three at >3sigma. Subdividing z=2-10 galaxies according to stellar mass, UV luminosity, and UV-continuum slope and stacking the results, we only find a significant detection in the most massive (>10^9.75 Msol) subsample, with an infrared excess (IRX=L_{IR}/L_{UV}) consistent with previous z~2 results. However, the infrared excess we measure from our large selection of sub-L* (<10^9.75 Msol) galaxies is 0.11(-0.42)(+0.32) and 0.14(-0.14)(+0.15) at z=2-3 and z=4-10, respectively, lying below even an SMC IRX-beta relation (95% confidence). These results demonstrate the relevance of stellar mass for predicting the IR luminosity of z>~2 galaxies. We furthermore find that the evolution of the IRX-stellar mass relationship depends on the evolution of the dust temperature. If the dust temperature increases monotonically with redshift (as (1+z)^0.32) such that T_d~44-50 K at z>=4, current results are suggestive of little evolution in this relationship to z~6. We use these results to revisit recent estimates of the z>~3 SFR density. One less obvious implication is in interpreting the high Halpha EWs seen in z~5 galaxies: our results imply that star-forming galaxies produce Lyman-continuum photons at twice the efficiency (per unit UV luminosity) as implied in conventional models. Star-forming galaxies can then reionize the Universe, even if the escape fraction is <10%.
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