Searches after Gravitational Waves Using ARizona Observatories (SAGUARO): Observations and Analysis from Advanced LIGO/Virgo's Third Observing Run

With the conclusion of the third observing run for Advanced LIGO/Virgo (O3), we present a detailed analysis of both triggered and serendipitous observations of 17 gravitational wave (GW) events (7 triggered and 10 purely serendipitous) from the Searches After Gravitational-waves Using ARizona Observatories (SAGUARO) program. We searched a total of 4935 deg$^2$ down to a median 5$\sigma$ transient detection depth of 21.1 AB mag using the Mt Lemmon 1.5 m telescope, the discovery engine for SAGUARO. In addition to triggered events within 24~hours, our transient search encompassed a time interval following GW events of $<120$~hrs, providing observations on $\sim$ 1/2 of the events accessible to the Mt Lemmon 1.5 m telescope. We covered 2.1--86\% of the LVC total probability ($P_{\rm total}$) for individual events, with a median $P_{\rm total} \approx 8\%$ within $<120$~hours. Following improvements to our pipeline and the addition of serendipitous observations, we find a total of 7 new optical candidates across 5 GW events which we are unable to rule out after searching for additional information and comparing to kilonova models. Using both publicly available and our own late-time data, we investigated a total of 252 optical candidates for these 17 events, finding only 65\% were followed up in some capacity by the community. Of the total 252 candidates, we are able to rule out an additional 12 previously reported counterpart candidates. In light of these results, we discuss lessons learned from the SAGUARO GW counterpart search. We discuss how community coordination of observations and candidate follow-up, as well as the role of archival data, are crucial to improving the efficiency of follow-up efforts and preventing unnecessary duplication of effort with limited EM resources.