is the rest frequency of the spectral line, $z$ is the redshift, $\Omega_\Lambda$ is the normalized dark energy density, and $\Omega_m$ is the normalized matter density. There is thus a one-to-one mapping between frequency and comoving radial distance, and as shorthand throughout this paper, we will adopt the notation $r_\nu \equiv r(\nu)$. Similarly, we will often use the symbol $\nu_r$ to denote frequency, with the subscript reminding us that the observed frequency is a function of the radial distance. Given a radial distance, transverse distances may also be computed given ${\hat{\mathbf{r}}}$ (or angle on the sky) using simple geometry.

If one’s survey occurs over a narrow radial range, the distance-frequency relation is often replaced by a linearized approximation where $$\label{eq:LinearDistanceApprox}
r - r_\textrm{ref} \approx - \alpha (\nu - \nu_\textrm{ref} ),$$ with $r_\textrm{ref}$ and $\nu_\textrm{ref}$ being a reference comoving radial distance and a reference frequency, respectively, with values constrained by Eq. , and $$\label{eq:AlphaConversion}
\alpha \equiv \frac{1}{\nu_\textrm{rest}} \frac{c}{H_0} \frac{(1+z_\textrm{ref})^2}{E(z_\textrm{ref})},$$ where $1 + z_\textrm{ref} = \nu_\textrm{rest} / \nu_\textrm{ref}$. In this paper, the symbols $\nu_r$ and $r_\nu$ will always refer to the exact nonlinear relations, and any invocations of the linearized approximations will be written out explicitly using Eq. . When using the 