massive mediator cases considered in KT87. For the former, the cross section is independent of $M$; for the latter, it is degenerate in $g/M$. Between these two limits, the cross section is peaked at the resonance energy defined by $s = 2 E_{\rm res} m_{\nu} = M^{2}$, where the cross section is regulated by its decay width and is $\sigma = 4\pi/M^{2}$.

Figure \[fig:constraint\] shows all three of these behaviors in the optical depth for neutrino scattering. We define this as $$\label{eq:opticaldepth}
\tau(E| g,M) = c \int_{0}^{1}dz \frac{n_{t}(0)}{H(z)} (1+z)^{2} \sigma(E|g,M)\,,$$ where $z = 1$ is a typical redshift for cosmic sources. For simplicity, we ignore the redshift dependence of $\sigma(E)$ inside the integral, which would slightly broaden the range of $M$ for which a resonance could occur. The factor $(1+z)^{2}$ comes from the target density evolution factor $(1+z)^{3}$ and a factor of $(1+z)^{-1}$ from $|dt/dz|$. Taking redshift into account only increases $\tau$ by about a factor of 3; it would matter more if high-redshift sources were dominant. We show contours of $\tau = 1$ for $E=$ 0.01, 0.1 and 1 PeV in Fig. \[fig:constraint\]. Above the contours ($\tau>1$), the effect of scattering increases exponentially with $\tau$, which increases as $g^{4}$ 