Reverberation and Polarization Constraints

Reverberation and X-ray polarization respond to different projections of the same disk–corona geometry. Complex cross-spectra measure delayed, energy-dependent variability; Stokes spectra measure the orientation and coherence of the electric field after emission, scattering, absorption, and curved-spacetime transport. Either observable can reduce geometric degeneracy, but neither becomes a spin measurement by itself. A defensible joint analysis first asks whether one inner edge, inclination, and coronal transfer model predicts both data products, including dilution, propagation, modulation response, and calibration.

Required background. Probability and Inference for Gravity supplies complex-data likelihoods, nuisance-parameter sharing, selection accounting, and posterior predictive checks.

Helpful background. Black-Hole Geometry and Mechanics supplies photon transfer and parallel transport; Relativistic Matter and Compact Objects supplies scattering atmospheres and plasma propagation; and Black-Hole Astrophysics supplies disk, corona, and source-state context.

Reverberation as a complex transfer measurement

Use the explicit transform pair

$$X(f)=\int_{-\infty}^{\infty}x(t)e^{+2\pi ift}\,dt, \qquad x(t)=\int_{-\infty}^{\infty}X(f)e^{-2\pi ift}\,df,$$

so temporal synthesis agrees with $e^{-i\omega t}$. If a driving coronal light curve $c(t)$ illuminates the disk, the count-rate variation in energy channel $E$ can be written schematically as

$$x_E(t)=D_Ec(t)+\int_0^\infty\psi_E(\tau)c(t-\tau)\,d\tau+p_E(t),$$

where $D_E$ is the direct continuum weight, $\psi_E$ is the reflected impulse response, and $p_E$ collects propagation variability not generated by simple reverberation. Fourier transformation gives

$$X_E(f)=\left[D_E+\Psi_E(f)\right]C(f)+P_E(f), \qquad \Psi_E(f)=\int_0^\infty\psi_E(\tau)e^{+2\pi if\tau}\,d\tau .$$

With cross-spectrum $G_{Er}=X_E X_r^*$, define a positive delay of channel $E$ behind reference $r$ by

$$\tau_{Er}(f)=\frac{\arg G_{Er}(f)}{2\pi f}.$$

A pure delay $x_E(t)=x_r(t-\tau_0)$ gives $X_E=e^{+2\pi if\tau_0}X_r$, hence $\tau_{Er}=+\tau_0$ before phase wrapping. Reversing either the transform sign or the cross-spectrum order reverses its phase, so comparisons must translate both definitions. The phase is defined only modulo $2\pi$, so phase wrapping can reverse an apparent lag. Direct continuum in a reflection-rich channel and reflected photons in the reference band dilute the light-travel delay. Slow inward propagation can add a lag of the opposite spectral pattern. Cross-spectrum estimation, covariance, dilution, and impulse responses are developed in Uttley et al. 2014, §§ 2.1–2.4.

The physical scale is $t_g=GM/c^3$, but a measured lag is not simply coronal height divided by $c$. The transfer function also depends on disk inner edge, inclination, coronal extent and motion, emissivity, ionization, thickness, and the detector energy response. Spin sensitivity appears mainly after $r_{\rm in}$ is tied to the ISCO, so the analysis should report the geometry and inner radius first.

Polarization as a Stokes transfer measurement

For a fixed sky basis and IAU electric-vector position-angle convention, use channel Stokes parameters $(I,Q,U)$. The linear polarization degree and position angle are

$$\Pi=\frac{\sqrt{Q^2+U^2}}{I}, \qquad \chi=\frac12\operatorname{atan2}(U,Q).$$

IXPE measures photoelectron azimuths rather than these quantities directly. Event weights, the energy-dependent modulation factor, effective area, background, detector-unit responses, and angle zero point map the event distribution to Stokes spectra. Near a null signal, $\Pi$ is positively biased; fitting $(Q,U)$ with their covariance is safer than treating a symmetric error on $\Pi$ as Gaussian. Kislat et al. give the event-level Stokes construction and uncertainty treatment in Kislat et al. 2015, §§ 2–4.

A relativistic transfer model predicts source-frame Stokes intensity along each ray, parallel-transports the polarization basis, and integrates over the image. Disk scattering, Comptonization, returning radiation, absorption, and Faraday rotation or conversion can change $\Pi(E)$ and $\chi(E)$. Similar polarization can arise from an extended slab corona, a mildly outflowing flow, or a compact source plus returning radiation. The calibration basis and the model basis must be connected by an explicit rotation; silently changing the sign of $U$ mirrors the inferred angle.

One joint likelihood and one observing window

Let $\vartheta_{\rm geo}=(M,i,r_{\rm in},a_*,\eta_{\rm corona},\eta_{\rm disk})$ be shared geometry, while $\eta_t$ and $\eta_p$ hold timing and polarimeter calibration. A joint analysis has the structure

$$p(\vartheta_{\rm geo},\eta_t,\eta_p\mid d_t,d_p) \propto \mathcal L_t\!\left[d_t\mid G_{Er}(f;\vartheta_{\rm geo},\eta_t)\right] \mathcal L_p\!\left[d_p\mid I,Q,U(\vartheta_{\rm geo},\eta_p)\right] p(\vartheta_{\rm geo},\eta_t,\eta_p).$$

Complex cross-spectral estimates require their real–imaginary covariance or the underlying segment likelihood. Stokes spectra require the full $(I,Q,U)$ response and covariance. Sharing only a point estimate of inclination discards the main benefit; multiplying separately marginalized posteriors double counts priors and can hide incompatible coronal geometries.

The 2022 May Cygnus X-1 campaign provides a named public exercise: IXPE ObsID 01002901, NICER ObsIDs 5100320101–5100320107, and NuSTAR ObsIDs 30702017002, 30702017004, and 30702017006. The identifiers and broadband analysis are documented in Krawczynski et al. 2022, pp. 650–654 and Supplementary Materials. Select the intersection of good-time intervals for joint timing and polarization; using the full exposure of each mission as if simultaneous would mix source states.

At the 2026-08-18 cutoff, pin HEASoft 6.36, NICERDAS 15, NICER CALDB xti20240206, IXPE GPD CALDB 20260610, and IXPE XRT CALDB 20260609. Record each Level-2 event-file version and observation readme because the archive can replace reprocessed files. Also record clock and barycentric correction, gain, dead time, energy and Fourier bins, reference band, modulation response, angle zero point, leakage, source and background regions, and the common good-time expression.

Results and limitations through 2026-08-18

The evidence does not establish a unique joint reverberation–polarization spin estimator. The unresolved question is whether current signal-to-noise and calibration domains distinguish corona and disk geometries strongly enough that the remaining inner-edge sensitivity can be mapped to spin.

Evidence or result	Geometry test	Scope or limitation
Fe K and soft-excess reverberation patterns occur in accreting black holes	Predict lag–energy and lag–frequency spectra with one impulse response	Dilution, propagation lags, phase wrapping, and frequency selection can imitate or shift light-travel delays
Cygnus X-1 IXPE Stokes spectra favor polarization aligned with the radio jet and constrain coronal shape	Predict energy-dependent $Q$ and $U$ together with the broadband spectrum	The inference depends on scattering plasma, inclination, absorption, and modulation calibration
Simultaneous IXPE, NICER, and NuSTAR windows can test one disk–corona state	Jointly predict complex cross-spectra, Stokes spectra, and count spectra	The published campaign was not a completed joint reverberation spin measurement; reduced simultaneity lowers precision
Returning radiation and relativistic basis rotation can create spin-sensitive polarization structure	Seek an energy-dependent angle or degree change tied to the same inner edge as the lag	Disk atmosphere, corona, absorption, and Faraday effects can produce similar structure
Compact, extended, and moving coronae make different lag and polarization predictions	Leave one observable out and predict it from the others	Current model grids and source populations are incomplete; a null discriminant does not select a universal geometry

The competing positions include a compact lamppost-like corona, an extended disk-plane corona, a vertically or radially moving Comptonizing flow, propagation-dominated lags, disk truncation, and returning-radiation polarization. The Cygnus X-1 polarization result disfavors some simple geometries in its hard state, yet it does not prove a unique coronal structure or an ISCO edge. Reverberation observations across binaries and active nuclei similarly constrain scales while retaining transfer-function ambiguity; see Uttley et al. 2014, §§ 3–5.

The interpretation could change after a truly simultaneous high-throughput campaign yields both resolved reverberation and Stokes spectra; after a timing, modulation, angle, leakage, or background calibration revision; or after a joint transfer model fails a leave-one-observable-out prediction or reveals cross-method geometric inconsistency.

Adversarial checks and reproduction boundary

Fit geometry before spin. Allow $r_{\rm in}$ to vary independently of $a_*$. Establish whether lag and Stokes data prefer the same inclination, inner edge, and corona before applying $r_{\rm in}=r_{\rm ISCO}$.

Inject dilution and propagation. Generate event data with a known extended transfer function plus propagation noise, pass them through the exact timing window and instrument responses, and verify coverage. A compact-corona fit to such data is a documented counterexample if it returns a precise but wrong height or spin.

Leave out one observable. Fit count spectra and timing, then predict $(Q,U)$; separately fit count and Stokes spectra, then predict the complex lag. Agreement is stronger than a joint fit in which each data class repairs a different model error.

A truthful reproduction publishes file checksums, common good-time intervals, barycentric and clock products, channel and frequency selections, cross-spectrum code and convention, Stokes extraction and response commands, background regions, response and CALDB identifiers, transfer-model version, priors, posterior samples, and simulated recovery tests. The exercise does not claim a production joint spin result: exact inference also needs validated polarized reverberation transfer, adequate simultaneous counts, and independent calibration review.

Exercises

A reflection-rich channel has linear variability weights of $70\%$ direct continuum and $30\%$ delayed reflection, measured against a purely direct-continuum reference. In the low-frequency limit its measured reverberation lag is $0.40\ {\rm ms}$. Estimate the undiluted delay. The same energy bin has normalized Stokes values $q=Q/I=0.030$ and $u=U/I=0.040$; find $\Pi$ and $\chi$ relative to the declared sky basis. Do these numbers determine spin?

Solution

For $x(t)=0.7c(t)+0.3c(t-\tau_0)$ and small $2\pi f\tau_0$, the phase gives $\tau_{\rm obs}\simeq0.3\tau_0/(0.7+0.3)$. Thus

$$\tau_0\simeq\frac{0.40\ {\rm ms}}{0.30}=1.33\ {\rm ms}.$$

The polarization quantities are

$$\Pi=\sqrt{q^2+u^2}=0.050, \qquad \chi=\frac12\operatorname{atan2}(0.040,0.030)\simeq26.6^\circ.$$

They do not determine spin. The delay still depends on mass, corona and disk geometry, propagation, and reference-band dilution; the Stokes pair still depends on scattering geometry, inclination, absorption, and calibration. Spin becomes inferable only if a validated common transfer model constrains an inner edge and demonstrates that it is the ISCO.

References

• HEASARC. n.d. “IXPE Data Archive and Calibration Releases.” Official archive.
• Kislat, Fabian, Brian Clark, Martin Beilicke, and Henric Krawczynski. 2015. “Analyzing the Data from X-Ray Polarimeters with Stokes Parameters.” Astroparticle Physics 68:45–51. DOI.
• Krawczynski, Henric, et al. 2022. “Polarized X-Rays Constrain the Disk–Jet Geometry in the Black Hole X-Ray Binary Cygnus X-1.” Science 378:650–654. DOI.
• Uttley, Phil, et al. 2014. “X-Ray Reverberation around Accreting Black Holes.” Astronomy and Astrophysics Review 22:72. DOI.