Light curves of BSM-induced neutrino echoes in the optically thin limit

High-energy neutrinos from astrophysical transients serve as a probe of neutrino physics beyond the Standard Model. In particular, nonstandard interaction of neutrinos with the cosmic neutrino background or dark matter (DM) may have imprints on not only their spectra but also the arrival and time-delay distributions. Assuming that the interaction occurs at most once during the neutrino propagation, we provide general analytic formulas for light curves of the neutrino echoes induced by BSM. The formulas can be used for constraining neutrino-neutrino scattering, neutrino-DM scattering, and other scattering processes experienced by relativistic particles.