for fixed $M$. Near the sharp dips, the realistic sensitivity would be reduced by the effects of detector energy resolution. We emphasize that we use $\tau$ just for illustration; for our main result, we calculate spectra using Eq. (\[eq:boltzmann\]).

Line emission with $\nu$SI {#sec:Line}
--------------------------

Before considering astrophysical scenarios with broad energy spectra, it is instructive to show the effects of $\nu$SI on a mono-energetic neutrino spectrum. For rough consistency with the IceCube data, we choose 1 PeV for the line energy and define the flux in the same units as the deduced IceCube spectrum.

In Fig. \[fig:astro\_line\], we compare cases with free streaming, $\nu$SI with attenuation only, and $\nu$SI with attenuation and regeneration. The energy dependence of the $\nu$SI effects depends on the neutrino-neutrino cross section. Using the general form of Eq. (\[eq:Breit\]), we choose example resonance energies $E_{\rm res}$ well above, near, and well below the emission energy of 1 PeV. For each $E_{\rm res}$, the couplings are tuned so that $\sim e^{-1}$ of the energy spectrum is unattenuated after propagation.

For the free-streaming case, as in Eq. (\[eq:freestreaming2\]), the spectrum of neutrino energies simply reflects the distribution of source redshifts through the relation $E = 1\,{\rm\ PeV} 