increasing their numbers. We obtain the resultant spectra by numerically solving the propagation equation, Eq. (\[eq:boltzmann\]). We show the effects of attenuation alone as a comparison for the full calculation that includes regeneration. For regeneration, the total energy carried in neutrinos is conserved. We checked our numerical results by comparing the total energy in the spectrum to that of the free-streaming case, finding agreement at the percent level. Energy conservation corresponds to area conservation in a plot of $E^{2} dN/dE$ with log energy bins. In Fig. \[fig:astro\_line\], the area conservation is apparent, with the exception of the bottom panel, where we cut off the figure before showing the whole regenerated spectrum.

For the top panel, where $E_{\rm res} = 5\,{\rm\ PeV}$, the cross section increases with energy, which produces a flattish spectrum of regenerated neutrinos. For the middle panel, where $E_{\rm res} = 0.5\,{\rm\ PeV}$, the resonance energy is at the peak of the unattenuated spectrum, which causes significant absorption there and a pileup of regenerated events at slightly lower energy. Importantly, the absorption dip is broadened by the redshift effects. Neutrinos emitted with the same energy at different redshifts reach the resonance energy at different redshifts, smearing out the 