Step 17. Noise budget & practical scales (what actually limits you)

Plain English: three physical noise sources set the realistic floor for lapse fluctuations \(S_\Phi(\Omega)\) in labs; they add up. Their relative size depends on frequency, with Newtonian gravity‐gradient noise usually dominating at sub-Hz.

The three contributors:

1. Vacuum TT modes: linearized metric waves projected through the Hamiltonian constraint give \(S^{\rm vac}_\Phi(\Omega)=\alpha(\Omega)\,S_h^{\rm TT}(\Omega)\) with a very small kernel \(\alpha(\Omega)\) in the lab band.
2. Stress-tensor (shot/thermal) noise: radiative flux at distance \(r\) gives \(S^{\rm shot}_\Phi(\Omega)\sim\big(G/c^4 r\big)^{\!2}\,S_L(\Omega)\), and for Poisson photons \(S_L\simeq 2\hbar\omega_\gamma L\).
3. Newtonian gravity-gradient (seismic/air-density) noise: in the quasi-Newtonian limit,

\[ S^{\rm N}_\Phi(\Omega)=(4\pi G)^2\!\int\!\frac{d^3k}{(2\pi)^3}\;\frac{S_\rho(\mathbf{k},\Omega)}{k^4}\,|W(\mathbf{k})|^2, \]

and it typically dominates below ~1 Hz in ground labs.

Hierarchy of magnitudes (order-of-magnitude guide): around 1 Hz, \(S_\Phi^{\rm N}\sim10^{-32}\,\text{s}\); around 1 kHz, \(S_\Phi^{\rm vac}\sim10^{-45}\,\text{s}\). These figures are site- and setup-dependent but capture why sub-Hz is hard.

Compare signal vs instrument: the paper’s scale table contrasts flux-induced drifts with present clock capability—lab sources are 12–22 orders below today’s stability and are therefore pipeline tests (software injections), while astrophysical bursts (e.g., a Galactic SN) approach interesting levels.

Reporting requirement (visibility channel): when turning visibilities into bounds via \(-\ln V=\frac{\omega^2}{2}\!\int S_\Phi(\Omega)|F(\Omega)|^2\,\frac{d\Omega}{2\pi}\), publish both a filter-weighted bound and, if you give it, the Markovian \(S_\Phi(0)\) bound—non-white noise is common. Also include a source-off baseline and subtract it.

Scale constant to remember: \(G/c^{4}=8.262\times10^{-45}\ \text{s}^2/(\text{kg}\,\text{m})\) sets all amplitudes and appears in both the flux→drift law and the shot-noise map.

Glossary (for this step)