Direct submm imaging has recently revealed the 850-micron background to be mostly composed of a population of distant ultraluminous infrared galaxies, but identifying the optical/NIR counterparts to these sources has proved difficult due to the poor submm spatial resolution. However, the proportionality of both cm and submm data to the star formation rate suggests that high resolution radio continuum maps with subarcsecond positional accuracy can be exploited to locate submm sources. In this paper we present results from a targeted SCUBA survey of micro-Jansky radio sources in the flanking fields of the Hubble Deep Field. Even with relatively shallow 850-micron SCUBA observations (>6 mJy at 3-sigma), we were successful at making submm detections of optical/NIR-faint (I>24 and K~21-22) radio sources, and our counts closely match the bright counts from submm surveys. Redshift estimates can be made from the ratio of the submm flux to the radio flux across the 100 GHz break in the spectral energy distribution. This millimetric redshift estimation places the bright submm population at z=1-3 where it forms the high redshift tail of the faint radio population. The star formation rate density (SFRD) due to ultraluminous infrared galaxies increases by more than two orders of magnitude from z~0 to z~1-3. The SFRD at high redshift inferred from our >6 mJy submm observations is comparable to that observed in the UV/optical. (Abridged)
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