Chapter 6 — Discriminant: Plasma vs Temporal
Gravity is achromatic. Cold-plasma dispersion is chromatic. Fit both at once and let the data decide.
Goal of this chapter
- Write a two-component timing model that separates plasma and temporal signals.
- Give a minimal estimator for two bands. Give a weighted fit for many bands.
- List checks that guard against confounders.
The discriminant model
Model a timing residual or fractional delay as a sum of a chromatic plasma term and a flat temporal term.
\[
\epsilon_{\mathrm{dis}}(\nu)
\;=\;
A\,\nu^{-2} \;+\; B.
\]
Interpretation: \(A\) tracks dispersion measure or plasma-like effects. \(B\) tracks an achromatic signal from the lapse.
Scope. The \(\nu^{-2}\) law holds in the high-frequency limit \((\omega \gg \omega_p)\) for cold, unmagnetized plasma. It is the standard timing regime.
Two-band closed-form estimator
Suppose you observe at two simultaneous bands \(\nu_1\) and \(\nu_2\). Let the measured residuals be \(r_1\) and \(r_2\).
\[
r_1 \;=\; A\,\nu_1^{-2} + B,
\qquad
r_2 \;=\; A\,\nu_2^{-2} + B.
\]
Solve for \(A\) and \(B\).
\[
A \;=\; \frac{r_1 - r_2}{\nu_1^{-2} - \nu_2^{-2}},
\qquad
B \;=\; r_1 - A\,\nu_1^{-2}.
\]
Report \(B\) as the temporal candidate. Quote errors from your per-band uncertainties.
Many-band weighted fit
Use a linear model with basis functions \(x_1(\nu)=\nu^{-2}\) and \(x_2(\nu)=1\). Minimize weighted residuals.
\[
\min_{A,B}\;\sum_i w_i\,\bigl[r_i - (A\,\nu_i^{-2}+B)\bigr]^2,
\qquad
w_i \equiv \sigma_i^{-2}.
\]
Include a constant offset per instrument if bands are not perfectly synchronized. Keep observations simultaneous when possible.
Optional math peek: normal equations
\[
\begin{pmatrix}
\sum w_i \nu_i^{-4} & \sum w_i \nu_i^{-2} \\
\sum w_i \nu_i^{-2} & \sum w_i
\end{pmatrix}
\begin{pmatrix} A \\ B \end{pmatrix}
=
\begin{pmatrix}
\sum w_i \nu_i^{-2} r_i \\
\sum w_i r_i
\end{pmatrix}.
\]
Solve the \(2\times2\) system for \(A\) and \(B\). Errors follow from the inverse matrix.
Edge cases and extensions
- Very high frequencies. If \(\nu\gg\nu_p\), the plasma term is tiny. Expect \(A\to 0\) within errors.
- Magnetized plasma. Faraday rotation changes polarization, not the achromatic timing. A residual \(\propto \nu^{-3}\) or \(\nu^{-4}\) indicates higher-order plasma terms. Extend the model if needed.
- Scattering tails. Multi-path in the ISM can add a chromatic tail. It scales with \(\nu\) differently from \(\nu^{-2}\). Fit an extra basis if required.
Recipe for analysis
- Collect simultaneous multi-band timing or phase data. Calibrate time transfer first.
- Detrend slow instrumental drift with band-shared polynomials if needed.
- Fit \(\epsilon_{\mathrm{dis}}(\nu)=A\nu^{-2}+B\) using weights from per-band errors.
- Test \(A=0\) and \(B=0\) separately. The temporal candidate is \(B\neq 0\) with \(A\) consistent with zero.
- Repeat on time segments to check stability. Expect \(B\) to track lapse-driven drift; see Chapter 5.
Sanity checks
- Swap bands \(\nu_1\) and \(\nu_2\). The two-band \(B\) is unchanged.
- Inject a known \(\nu^{-2}\) term in simulations. The fit recovers \(A\) and leaves \(B\) near zero.
- Observe at optical and radio together. Plasma dominates radio. The achromatic part should agree across both.
Common pitfalls
- Non-simultaneous bands. Temporal drift leaks into the plasma fit. Prefer simultaneous data.
- Mixed conventions. Keep units and signs consistent across bands and instruments.
- Underestimated errors. Use realistic \(\sigma_i\) to avoid false positives in \(B\).
What you have now
A two-term model that separates chromatic plasma from achromatic temporal signals. A closed-form two-band estimator. A robust many-band fit with clear diagnostics.