The coalescence of a binary neutron star (BNS) pair is expected to produce gravitational waves (GW) and electromagnetic (EM) radiation, both of which may be detectable with currently available instruments. We describe a search for a theoretically predicted r-process optical transient from these mergers, dubbed the kilonova (KN), using griz broadband data from the Dark Energy Survey Supernova Program (DES-SN). Some models predict KNe to be redder, shorter-lived, and dimmer than supernovae (SNe), but at present the event rate of KNe is poorly constrained. We simulate observations of KN and SN light curves with the Monte-Carlo simulation code SNANA to optimize selection requirements, determine search efficiency, and predict SN backgrounds. We also perform an analysis using fake point sources on images to account for anomalous efficiency losses from difference-imaging on bright low-redshift galaxies. Our analysis of the first two seasons of DES-SN data results in 0 events, and is consistent with our prediction of 1.1 background events based on simulations of SN. Given our simulation prediction, there is a 33 percent chance of finding 0 events in the data. Assuming no underlying galaxy flux, our search sets 90 percent upper limits on the KN volumetric rate of $1.0times10^7$ Gpc$^{-3}$ yr$^{-1}$ for the dimmest KN model we consider (peak i-band absolute magnitude $M_i=-11.4$ mag) and $2.4times10^4$ Gpc$^{-3}$ yr$^{-1}$ for the brightest ($M_i=-16.2$ mag). Accounting for efficiency loss from host galaxy Poisson noise, these limits are 1.1 times higher; accounting for anomalous subtraction artifacts on bright galaxies, these limits are ~3 times higher. While previous KN searches were based on triggered follow-up, this analysis is the first untriggered optical KN search and informs selection requirements and strategies for future KN searches and GW follow-up observations.
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