Fisher-forecast Faraday detection layer (sky-marginalized SNR)

[christianhbye]

Summary

Adds a Fisher-matrix forecast that estimates LuSEE's sky-marginalized detectability of the Faraday-rotation amplitude — the realistic version of the optimistic SNR_far from the RM-synthesis layer. The data are linear in the unknown intrinsic (Q,U) sky given known beam+RM+rotation operators, so the Fisher Jacobian is built analytically and the unknown large-scale sky is marginalized away.

New modules (src/lusee_faraday/):

• forward.py — pol_response, a reusable linear forward operator wrapping compute_vis_fast → complex pQ + i·pU per (time, channel); alpha scales the RM (Faraday amplitude).
• skybasis.py — spin2_basis / n_modes, low-ℓ spin-2 (Q,U) harmonic basis = the marginalized intrinsic-sky nuisance.
• fisher.py — run_forecast assembles F = JᵀN⁻¹J over (α, an effective Faraday-dispersion variance τ, sky modes) and returns the sky-marginalized σ(α) and SNR vs a fixed-sky bound. α via finite difference, τ analytic, sky modes via the linear operator.

Driver notebooks/fisher_forecast.py reuses results/faraday_fullband.npz (channel table + pI_FR as T_sys), reconstructs sim inputs at nside=64 (a resolution knob), and runs run_forecast once, scaling SNR by √dt across integration times (exact: radiometer noise gives F ∝ dt).

Key result

Marginalizing the 24 low-ℓ spin-2 sky modes + τ reduces the Faraday-amplitude SNR by < 0.5% vs the fixed-sky bound — the smooth sky nuisance does not span the λ²-winding Faraday signature, i.e. the known-RM matched filter is not degeneracy-limited by these modes.

Caveat (stated in code + docs): the absolute SNR is an optimistic upper bound — the nuisance basis is restricted (low-ℓ, fixed spectral index per mode) and nside=64 smooths small-scale RM. Per-mode spectral freedom + higher ℓ are deferred follow-ups that would lower it.

Notes

• Stacks on the unmerged faraday-rmsynth (PR Faraday RM-synthesis: Step 0 (rmsynth module + calibration) #1) — base is faraday-rmsynth.
• Figure results/fisher_forecast.png is gitignored (repo convention); the driver regenerates it.
• Implementation plan: docs/superpowers/plans/2026-06-09-faraday-fisher-forecast.md.

Test Plan

• uv run pytest -q — 140 passed (new: test_forward.py, test_skybasis.py, test_fisher.py)
• Driver runs end-to-end → finite σ(α), marginalized ≤ fixed-sky in every integration-time case
• (follow-up) higher-ℓ / spectral-freedom nuisance; nside=128 precision run

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