Neutrino emission upper limits with maximum likelihood estimators for joint astrophysical neutrino searches with large sky localizations

Since the start of the gravitational wave observation era, no joint high energy neutrino and gravitational wave event has been found. These non-detections could be used for setting an upper bound on the neutrino emission properties for gravitational wave events individually or for a set of them. Although in the previous joint high energy neutrino and gravitational wave event searches upper limits have been found, there is a lack of consistent method for the calculation. The problem addressed in this paper is finding those limits for astrophysical events which are localized poorly in the sky where the sensitivities of the neutrino detectors change significantly and can also emit neutrinos, for example the gravitational wave detections. Here we describe methods for assigning limits for expected neutrino count, emission fluence and isotropically equivalent emission based on maximum likelihood estimators. Then we apply described methods on the three GW detections from aLIGO's first observing run (O1) and find upper limits for them.