secondary particles, analogous to electromagnetic cascades for high energy cosmic gamma rays [@Lee:1996fp; @Murase:2012xs], $\nu_{\tau}$ regeneration in matter [@Halzen:1998be; @Dutta:2000jv; @Beacom:2001xn], and ultrahigh energy cosmic ray propagation [@Bhattacharjee:1998qc; @Berezinsky:2002nc; @Ahlers:2009rf; @Ahlers:2010fw]. As far as we know, cascade calculations have not been done for neutrino-neutrino interactions. (A similar formalism for supernova neutrinos interacting with dark matter appeared in preprint while we were finalizing this work [@Farzan:2014gza].)

We assume that there are only active neutrinos in the beam and target, that all species are comparably populated by mixing, and that this happens long before any effects due to propagation. We assume the neutrino masses are all $\simeq0.1$eV and that only one species of $\nu + \bar{\nu}$ interacts, each with half of the targets of a given species, so $n_{t}(z) = 56 (1+z)^{3} \, {\rm cm^{-3}}$ We ignore the possibility of flavor changes in scattering. We take a generic form for the total and differential cross sections to minimize model dependence. The propagation equation and our calculations could be generalized to account for changes in the assumptions, and we discuss below what happens when some of them are relaxed.

We focus on elastic scattering of neutrinos in the $s$ channel. For generality, we 