Chapter 8 — Observables & Forecasts
Turn equations into signals. Estimate magnitudes. Pick instruments and strategies that can see them.
Goal of this chapter
- List the main observables: clock links, pulsar/FRB timing, lensed-quasar delays.
- Give scaling laws for achromatic drift from source power.
- Provide practical targets and tactics.
Core scaling laws
Start from the drift law and the flux relation. Use path-length element \(\mathrm{d}\ell\) and radius \(R\).
\[
\frac{\mathrm{d}}{\mathrm{d}t_{\mathrm{obs}}}\Delta T
\simeq -\frac{2}{c} \int \partial_t\Phi\,\mathrm{d}\ell,
\qquad
\partial_t\Phi(t,R) = -\frac{G}{c^4}\,\frac{P(t)}{R}.
\]
Combine them to estimate the drift from a segment of length \(L\) near \(R\) with power \(P\).
\[
\dot{\Delta T}\big|_{\text{seg}}
\;\simeq\;
\frac{2G}{c^5}\,\frac{P}{R}\,L.
\]
Express as fractional frequency using \(\dot{\nu}/\nu \approx -\dot{\Delta T}\).
\[
\frac{\dot{\nu}}{\nu}
\;\approx\;
-\,\frac{2G}{c^5}\,\frac{P}{R}\,L.
\]
A convenient per-year form follows by multiplying by one year \(T_{\mathrm{yr}}\).
\[
\Delta(\Delta T)\;[\mathrm{s/yr}]\;\simeq\;
\left(2.19\times10^{-14}\right)\,
\lambda_{\mathrm{Edd}}\,
\frac{M}{M_\odot}\,
\frac{L}{R}.
\]
Here \(\lambda_{\mathrm{Edd}}\) scales power to the Eddington luminosity and \(M/M_\odot\) is source mass in solar units. The factor matches the paper's benchmarks.
Observable channels
1) Redshift drift with optical clock links
Link optical clocks over a long baseline. Track phase continuously. The achromatic signal is the common-mode drift.
\[
\left|\frac{\dot{\nu}}{\nu}\right|
\;\approx\;
\left|\dot{\Delta T}\right|.
\]
- Targets. Nearby massive engines with changing power. Long lines of sight skimming potential gradients.
- Specs. Stability at \(10^{-18}\) to \(10^{-19}\) over hours to days. Phase-stable time transfer.
- Checks. Closed-loop comparisons. Swap endpoints. Expect the same achromatic drift.
2) Pulsar and FRB timing (multi-band)
Fit a chromatic plus achromatic model to simultaneous bands.
\[
\epsilon_{\mathrm{dis}}(\nu) = A\,\nu^{-2} + B.
\]
- Targets. Bright, stable pulsars. Repeaters with high S/N bursts.
- Specs. Synchronized bands. Accurate DM modeling. Weight by per-band uncertainties.
- Checks. \(A\to 0\) at high frequencies. \(B\) constant across bands.
3) Lensed-quasar time-delay monitoring
Monitor inter-image delays over long baselines. Evolving potentials along the paths produce achromatic drifts.
- Targets. Well-modeled strong lenses with seasonal sampling.
- Specs. Sub-day cadence. Stable photometry. Long time spans.
- Checks. Cross-image comparisons. Plasma variability is chromatic; temporal drift is flat.
Back-of-the-envelope forecasts
Massive source near Eddington
Assume \(\lambda_{\mathrm{Edd}}\approx 1\), \(M=10^8 M_\odot\), and \(L/R\approx 1\).
\[
\Delta(\Delta T)\;\sim\;
2.19\times10^{-14}\times 10^8
\;\approx\; 2.2\times10^{-6}\ \mathrm{s/yr}.
\]
That is a few microseconds per year. It sits within reach of precise, long-baseline phase tracking.
Galactic compact object
Let \(M=10 M_\odot\), \(\lambda_{\mathrm{Edd}}=0.1\), and \(L/R=0.3\).
\[
\Delta(\Delta T)\;\sim\;
2.19\times10^{-14}\times 10 \times 0.1 \times 0.3
\;\approx\; 6.6\times10^{-16}\ \mathrm{s/yr}.
\]
This is challenging. It motivates stacking, longer integrations, and multi-epoch comparisons.
Targets and tactics
- Clock networks. Build continental-baseline links. Tie optical combs to fiber and satellite transfer. Compare against stable references.
- Pulsar timing. Use wideband receivers. Fit \(A\) and \(B\) together. Repeat on time slices.
- Lensed quasars. Maintain cadence and calibration. Compare bands to reject chromatic contamination.
Assumptions and limits
- Quasi-static geometry along the path: \(\bigl|\partial_t\Phi/\Phi\bigr|\,T_{\mathrm{path}} \ll 1\).
- Thin or slowly varying lenses: \(\epsilon_{\mathrm{lens}} \equiv L_{\mathrm{lens}}/(c\,\tau_{\mathrm{var}}) \ll 1\).
- Achromatic tests assume the plasma term follows \(\nu^{-2}\) in the observed bands.
What you have now
Concrete observables for three channels. A scaling law from power to drift. Benchmarks that guide instrument requirements.