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 with cosmic background or dark matter (DM) may have imprints on not only their spectra but also arrival and time-delay distributions. Assuming that occurs at most once during propagation, we provide general analytic formulas for light curves echoes induced by BSM. The can be used constraining neutrino-neutrino scattering, neutrino-DM other scattering processes experienced relativistic particles.