Plain English: The paper is explicit: all numbers you quote are bounds, not promised detections. There are two kinds you should publish:
\[ -\ln V=\frac{\omega^{2}}{2}\!\int_{0}^{\infty}\frac{d\Omega}{2\pi}\,S_{\Phi}(\Omega)\,|F(\Omega)|^{2}. \]
Include both in results.
\[ \mathrm{Var}_{\text{tot}}(\hat\lambda)\approx \mathrm{Var}_{\text{stat}}(\hat\lambda)+\hat\lambda^{2}\!\left(\sigma_r^2+\sigma_L^2\right)+\Delta_S, \]
with the linearized sensitivity \(\delta\hat\lambda/\hat\lambda\simeq -\epsilon_r-\epsilon_L+O(\epsilon_S)\). In practice, the calibration on distance \(r\) and power \(L\) dominates.
Noise budget you must list (visibility channel): three gravitational pieces plus standard lab systematics: • Newtonian gravity-gradient (seismic/air density), typically dominant < 1 Hz; • Stress-tensor (shot/thermal) from radiative fluxes; • Vacuum TT modes mapped through the constraint kernel (tiny in lab band). Always provide a source-off baseline and subtract it. (The paper even gives typical magnitudes: around 1 Hz, \(S_{\Phi}\sim10^{-32}\,\text{s}\); around 1 kHz, \(S_{\Phi}^{\mathrm{vac}}\sim10^{-45}\,\text{s}\).)
Framing in the paper (why “bounds-first”): it explicitly states that the sensitivity numbers assume ideal filters with measured \(S_y(f)\); real measurements are limited by current tech, and no detection timelines are claimed.
| Symbol | Name | Meaning (units) | Typical value/example | Metaphor | ||
|---|---|---|---|---|---|---|
| \(S_{\Phi}(\Omega)\) | Lapse PSD | One-sided spectrum of \(\delta\Phi\) (s) | Report Markovian and filter-weighted bounds | “Noise color of time” | ||
| ( | F() | ^{2}) | Filter kernel | From your sequence \(f(t)\) (s²) | sinc² for rectangular windows | “Sieve for noise tones” |
| \(\hat\lambda\) | Flux–drift gain | Fit slope in \(\dot y=-\lambda L\) (s²·kg\(^{-1}\)·m\(^{-1}\)) | Publish \(\mathrm{Var}_{\text{stat}}\) and \(\mathrm{Var}_{\text{tot}}\) | “Gear ratio” from flux to drift | ||
| \(\sigma_r,\sigma_L\) | Calibrations | Fractional errors in \(r\), \(L\) (–) | Often dominate lab totals | “Ruler & wattmeter wiggle” | ||
| \(S_{\Phi}^{\text{Newt}}\) | Gravity-gradient | From density/seismic noise (s) | \(\sim10^{-32}\,\text{s}\) @ 1 Hz | “Breathing of the ground” |
Ready for Step 12 — Registered workflow & publishing checklist (exact items to pre-register and what every result must include).