Deep Spectroscopy with DESI for Photometric Redshift Training and Calibration

Deep spectroscopic samples can be used to improve photometric redshift (photo-$z$) estimates and reduce uncertainties on redshift distributions. Such improvements can increase the cosmological constraining power of large imaging-based experiments such as the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST) and mitigate what may be a limiting systematic effect. We present results from the ``DESI-Deep pilot'' program, which was designed to assess the capability of the Dark Energy Spectroscopic Instrument (DESI) on the 4m Mayall telescope to measure redshifts of galaxies as faint as expected lensing samples for early LSST data ($m_i \leq 24.5$). We find that DESI is remarkably efficient at this task, with redshift success rates comparable to the results of observations from 10m-class telescopes with only $\sim2\times$ longer integration time (rather than $\sim 8\times$ longer as would be expected from aperture-area scaling), while simultaneously achieving $\sim30$ times larger multiplexing. We also find that the signal-to-noise ratio of the spectra scales as expected for background-limited observations even for the longest exposure times ($\sim 7$ hours) and faintest targets in the program. These results demonstrate that DESI could provide the definitive redshift sample for the early years of LSST with a modest investment of observing time. Based upon the results of this program, we provide updated predictions for the time required to collect benchmark samples for photo-$z$ training and calibration using a variety of spectroscopic facilities. Finally, we describe a potential "DESI-Deep" survey designed to train and calibrate photo-$z$'s for imaging experiments, and provide forecasts of its impact on cosmological inference.