with the C$\nu$B in the presence of strong $\nu$SI. Besides attenuation, this properly takes into account regeneration as well as multiple scattering of the parent and daughter neutrinos, i.e., a cascade.

In Sec. \[sec:nusi\], we consider existing $\nu$SI constraints. In Sec. \[sec:nucnub\], we discuss the effects of $\nu$SI on astrophysical neutrino spectra. We conclude in Sec. \[sec:conclusion\]. Throughout, we use cosmological parameters for which the matter density fraction is $\Omega_{M} = 0.3$, the cosmological constant density fraction is $\Omega_{\Lambda} = 0.7$, and the Hubble function is $H(z) = H_{0} \sqrt{\Omega_{\Lambda}+\Omega_{M}(1+z)^{3}}$, where $H_{0} = 70 \; {\rm km\;s^{-1}\; Mpc^{-1}}$.

Secret Neutrino Interactions {#sec:nusi}
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We first review existing constraints on $\nu$SI for a phenomenological scalar interaction term, ${\mathcal L}_{int}\sim g \phi \bar{\nu}\nu$. Fig. \[fig:constraint\] shows the parameter space of neutrino coupling $g$ to a new mediator $\phi$ with mass $M$. While these parameters vary depending on the specific model, type of coupling, number of new mediators, etc., this figure gives a broad comparison of different constraints. The accuracy is adequate, especially considering the many orders of magnitude on the axes. We do not consider the vector mediator case, since the laboratory constraints are much stronger than for the scalar case [@Laha:2013xua; 