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

Beginning in 2016, the IceCube Neutrino Observatory has sent out alerts in real time containing the information of high-energy ($E \gtrsim 100$~TeV) neutrino candidate events with moderate-to-high ($\gtrsim 30$\%) probability of astrophysical origin. In this work, we use a recent catalog of such alert events, which, in addition to events announced in real-time, includes events that were identified retroactively, and covers the time period of 2011-2020. We also search for additional, lower-energy, neutrinos from the arrival directions of these IceCube alerts. We show how performing such an analysis can constrain the contribution of rare populations of cosmic neutrino sources to the diffuse astrophysical neutrino flux. After searching for neutrino emission coincident with these alert events on various timescales, we find no significant evidence of either minute-scale or day-scale transient neutrino emission or of steady neutrino emission in the direction of these alert events. This study also shows how numerous a population of neutrino sources has to be to account for the complete astrophysical neutrino flux. Assuming sources have the same luminosity, an $E^{-2.5}$ neutrino spectrum and number densities that follow star-formation rates, the population of sources has to be more numerous than $7\times 10^{-9}~\textrm{Mpc}^{-3}$. This number changes to $3\times 10^{-7}~\textrm{Mpc}^{-3}$ if number densities instead have no cosmic evolution.