We investigate the properties of massive galaxies at z=1-3.5 using HST observations, ground-based near-IR imaging, and Spitzer Space Telescope observations at 3-24 micron. We identify 153 distant red galaxies (DRGs) with J-K > 2.3 mag (Vega) in the southern GOODS field. This sample is approximately complete in stellar mass for passively evolving galaxies above 10^11 solar masses and z < 3. The galaxies identified by this selection are roughly split between objects whose optical and near-IR rest-frame light is dominated by evolved stars combined with ongoing star formation, and galaxies whose light is dominated by heavily reddened starbursts. Very few of the galaxies (< 10%) have no indication of current star formation. Using SFR estimates that include the reradiated IR emission, the DRGs at z=1.5-3 with stellar masses > 10^11 solar masses have specific SFRs (SFRs per unit stellar mass) ranging from 0.2 to 10 Gyr^-1, with a mean value of ~2.4 Gyr^-1. The DRGs with stellar masses > 10^11 solar masses and 1.5 < z < 3 have integrated specific SFRs greater the global value over all galaxies. In contrast, we find that galaxies at z = 0.3-0.75 with these stellar masses have integrated specific SFRs less than the global value, and more than an order of magnitude lower than that for massive DRGs at z = 1.5-3. At z < 1, lower-mass galaxies dominate the overall cosmic mass assembly. This suggests that the bulk of star formation in massive galaxies occurs at early cosmic epochs and is largely complete by z~1.5. [Abridged]
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