Implications of the search for optical counterparts during the second part of the Advanced LIGO's and Advanced Virgo's third observing run: lessons learned for future follow-up observations

Joint multi-messenger observations with gravitational waves and electromagnetic data offer new insights into the astrophysical studies of compact objects. The third Advanced LIGO and Advanced Virgo observing run began on April 1, 2019; during the eleven months of observation, there have been 14 compact binary systems candidates for which at least one component is potentially a neutron star. Although intensive follow-up campaigns involving tens of ground and space-based observatories searched for counterparts, no electromagnetic counterpart has been detected. Following on a previous study of the first six months of the campaign, we present in this paper the next five months of the campaign from October 2019 to March 2020. We highlight two neutron star - black hole candidates (S191205ah, S200105ae), two binary neutron star candidates (S191213g and S200213t) and a binary merger with a possible neutron star and a "MassGap" component, S200115j. Assuming that the gravitational-wave candidates are of astrophysical origin and their location was covered by optical telescopes, we derive possible constraints on the matter ejected during the events based on the non-detection of counterparts. We find that the follow-up observations during the second half of the third observing run did not meet the necessary sensitivity to constrain the source properties of the potential gravitational-wave candidate. Consequently, we suggest that different strategies have to be used to allow a better usage of the available telescope time. We examine different choices for follow-up surveys to optimize sky localization coverage vs.\ observational depth to understand the likelihood of counterpart detection.