Exposure Required for a Relic Neutrino Detection: Target Mass, Running Time, and Background Level for Stated Significance

With a signal rate of 40 events per kilogram-year, a five-standard-deviation detection of the relic background requires an exposure of only 0.627 kilogram-years if the endpoint-window background is held to 0.1 events per kilogram-year, 0.641 kilogram-years at a background of 1 event per kilogram-year, and 0.781 kilogram-years at 10 events per kilogram-year. Three-standard-deviation evidence at the middle background needs 0.231 kilogram-years. This paper derives the exposure scaling, shows why background level matters less than intuition suggests once the signal rate is this high, and identifies the regime where the scaling inverts and background control becomes everything. The study is computed, not measured.