Step 18. Units, spectra & Fourier conventions (so nothing blows up dimensionally)

Here’s the paper’s bookkeeping so every equation is dimensionally honest and reproducible:

• Signature & basics. Metric signature \((-,+,+,+)\); \(c\) (and \(\hbar\)) are explicit unless momentarily set to 1. Areal radius \(r\) is defined by \(4\pi r^2\) area; the measured redshift is \(y\equiv\ln(\nu_\infty/\nu_r)\). In diagonal spherical gauge: \(ds^2=-e^{2\Phi}dt^2+e^{-2\Phi}dr^2+r^2d\Omega^2,\ N=e^{\Phi}\).

• Field dimensions. \(\Phi\) is dimensionless (it’s \(\ln N\)), so \(\partial_t\Phi\) carries \([{\rm s}^{-1}]\). The paper works in SI with \(c,\hbar\) explicit.

• One-sided PSD conventions. For stationary noise \(C_\Phi(t-t')\), use a one-sided spectrum with angular frequency \(\Omega\) and measure \(\int_0^\infty \frac{d\Omega}{2\pi}\): \([S_\Phi(\Omega)]=\text{s}\) and \([S_y(f)]=\text{Hz}^{-1}\). Equivalently, with ordinary frequency \(f\) (Hz) and measure \(\int_0^\infty df\), use \(S_\Phi(f)=S_\Phi(\Omega=2\pi f)/(2\pi)\) so \([S_\Phi(f)]=\text{Hz}^{-1}\).

• Visibility kernel dimensions.

  \[ -\ln V=\frac{\omega^2}{2}\!\int_0^\infty\!\frac{d\Omega}{2\pi}\,S_\Phi(\Omega)\,|F(\Omega)|^2 \]

  is dimensionless because \(|F(\Omega)|^2\) has units \(\text{s}^2\). For a rectangular interrogation of duration \(T\): \(|F(\Omega)|^2=\big(2\sin(\Omega T/2)/\Omega\big)^2\).

• Fourier definition used. \(F(\Omega)=\int f(t)\,e^{i\Omega t}\,dt\) (use the measured \(f_{\rm meas}(t)\) including finite rise/fall when turning visibilities into bounds).

• Markovian shorthand. In the low-band/white limit, \(\Gamma_\phi\simeq \frac{\omega^2}{2}S_\Phi(0)\), so over time \(T\): \(S_\Phi(0)\lesssim \dfrac{2|\ln V|}{\omega^2 T}\). Example: \(T=1\,\mathrm{s}\), \(\omega/2\pi=10^{15}\,\mathrm{Hz}\), 1% loss \(\Rightarrow S_\Phi(0)\lesssim 5\times10^{-34}\,\mathrm{s}\).

• Flux→drift units & constants. With luminosity \(L(t)\) (W) at distance \(r\) (m): \(\partial_t\Phi\simeq-\dfrac{G}{c^4}\dfrac{L(t)}{r}\) has units \(\text{s}^{-1}\). The tiny scale \(G/c^4=8.262\times10^{-45}\ \text{s}^2/(\text{kg}\,\text{m})\) sets all amplitudes used in the tables/estimates.

• “Where to find this” pointers. The paper states the conventions at the start of Section 4 and again in Appendix G (“Units and conventions”), so readers can check your numbers quickly.

That’s the bookkeeping capstone. With this, the walkthrough is complete—every preceding formula plugs into these conventions without surprises.

Glossary (for this step)