Constraints on neutrino source populations from neutrino searches in the directions of IceCube alerts

Whenever the IceCube Neutrino Observatory detects a neutrino candidate with a high probability of astrophysical origin, it sends out a public alert. Followups of these alerts have led to major successes in neutrino astronomy, such as with the case of the blazar TXS 0506+056. Here, we show how searching for additional, lower energy, neutrinos in the directions of these neutrino alerts can aid in identifying cosmic neutrino sources. We also discuss how such an analysis provides a handle for understanding populations of astrophysical neutrino sources. After searching for neutrino emission coincident with these alert events on various timescales, we find no significant evidence of additional neutrino emission on time scales from minutes to years in the directions of these alert events. Correspondingly, assuming sources have the same luminosities and number densities that track star-formation rates (with no cosmic evolution), this study shows that a population of neutrino sources must be more numerous than $7\times 10^{-9}$ Mpc$^{-3}$ ($3\times 10^{-7}$ Mpc$^{-3}$) to be responsible for the entire diffuse astrophysical neutrino flux.