of the sources. Throughout this work, we solve the propagation equation numerically in redshift variables, conservatively taking the initial condition $\tilde{n}(z_{\rm max} = 4,E) = 0$.

In the free-streaming case, $J(E)$ has a convenient closed form in the redshift variable [@Ahlers:2009rf; @Berezinsky:2002nc], given by $$\label{eq:freestreaming2}
J(E) =  \int_{0}^{z_{\rm max}} \frac{c dz}{4 \pi H(z)} {\cal L}(z,E(1+z)) \, ,$$ following from the simple relationship between emitted and observed energy as a function of redshift. For the source term, we assume a universal emission spectrum with a factorized form, $${{\cal L}(z,E)} = {\cal W}(z) {\cal L}_{0}(E)\,,$$ where ${\cal L}_0(E)$ is the differential number luminosity for each source and ${\cal W}(z)$ the redshift evolution of the source density, assumed to follow the star formation rate (SFR) [@Hopkins:2006bw; @Yuksel:2008cu]. The term $H(z)$ increasingly suppresses the importance of flux contributions from higher redshifts. The SFR evolution, which rises by an order of magnitude between $z = 0$ and $z = 1$ and then begins to fall, overcomes this effect until $z = 1$, so $z = 1$ is the typical redshift of the most relevant cosmic sources.

Neutrino propagation: $\nu$SI case
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When neutrinos interact with the C$\nu$B, the effects can be calculated by adding terms 