Cross-Band Spectral and Timing Synthesis

A broadband spectrum is not automatically a snapshot, and two variable light curves are not automatically causally connected. A defensible cross-band synthesis starts with one evolving, latent source state, passes it through each emitting region, instrument response, exposure window, and selection rule, and asks whether that model predicts data that were not used to tune it. The goal is therefore narrower than “find a correlation”: it is to distinguish a shared physical evolution from delayed zones, irregular sampling, and band-specific systematics. Instrument-specific gravitational-wave likelihood methods are outside this page’s scope.

Helpful background. Black-Hole Geometry and Mechanics supplies the gravitational length and time scales; Relativistic Matter and Compact Objects supplies radiative-transfer and plasma concepts; Black-Hole Astrophysics supplies accretion-state and jet phenomenology; Gravitational-Wave Astronomy supplies catalog and localization concepts; Relativistic Cosmology supplies redshift, distance, and source-frame transformations.

One latent evolution, many transfer functions

Let $x(t)$ denote a stochastic driver—perhaps an accretion-rate fluctuation—and let $s(t)$ denote a discrete or continuous source state. A useful forward model for datum $y_{b,i}$ in band $b$ is

$$y_{b,i}=c_b+\int dt\,W_{b,i}(t) \int_0^\infty du\, \Psi_b\!\left(u;s(t-u)\right)x(t-u)+\epsilon_{b,i}.$$

Here $W_{b,i}$ weights the arrival times sampled by exposure $i$, centered near $t_i$; for an exposure-averaged flux, normalize $\int W_{b,i}(t)dt=1$. The inner integral first constructs the physical band signal from earlier driver times $t-u$. Instrumental spectral response can be included in the band mapping, while $c_b$ contains a steady background or host contribution and $\epsilon_{b,i}$ has the measured covariance. This order matters: an observing gap removes received data, not the earlier illumination that later reaches the observer. A broad $\Psi_b$ describes an extended or reprocessing region; a displaced peak describes a delay; a changing $\Psi_b$ describes state dependence. Upper limits contribute censored likelihood factors rather than fabricated flux points.

For approximately stationary, evenly sampled segments, timing can be summarized by the cross spectrum. Define $X(f)=\int x(t)e^{+2\pi ift}dt$ and likewise $Y$, with inverse synthesis proportional to $e^{-2\pi ift}$. With $C_{xy}(f)=X^*(f)Y(f)$, the phase lag and time lag of $y$ behind $x$ are

$$\phi(f)=\arg C_{xy}(f),\qquad \tau(f)=\frac{\phi(f)}{2\pi f}.$$

For a pure delay $y(t)=x(t-\tau_0)$, $Y=e^{+2\pi if\tau_0}X$, so the displayed lag is $+\tau_0$ before phase wrapping. Phase wrapping makes $\tau$ ambiguous when $|2\pi f\tau|\gt\pi$; dilution by direct emission usually reduces the measured lag below a geometric light-travel time. For irregular sampling, interpolating both bands onto a dense grid can manufacture coherence. A state-space or Gaussian-process likelihood evaluated at the actual times is safer, provided its power spectrum is flexible enough not to impose the desired lag.

A versioned campaign and its calibration boundary

The 2017 M87 campaign is an unusually clear test case because it combines horizon-scale imaging with radio-to-TeV measurements. Use the EHT data product 2021-D02-01, released 2021-04-14, DOI 10.25739/mhh2-cw46, rather than a plot digitized from the paper. The machine-readable broadband fluxes, angular scales, observing intervals, detections, and upper limits are specified in EHT MWL Working Group 2021, §3.2, Figure 16, Table A8, and Appendix B. The release is a versioned spectral benchmark, not a uniform high-cadence lag experiment.

The observations were quasi-simultaneous. Most were within days of the 2017 April 5–11 EHT observations, some radio measurements were as much as a month later, and the photon-limited Fermi-LAT point integrates three months. Angular resolution also spans roughly eight orders of magnitude. Consequently, every fitted row needs its UTC interval, time standard, barycentric correction where applicable, flux or wavelength scale, response, background, aperture, and core/HST-1/host decomposition. Swift-XRT’s unresolved measurement is treated as an upper limit on the core in the published construction. These details, explained instrument by instrument in EHT MWL Working Group 2021, §2 and Tables A1–A7, are part of the measurement, not optional cleanup.

Selections must record cadence and seasonal gaps, quality and detection thresholds, flare or state definition, allowed simultaneity tolerance, band edges, censoring, and whether an alert triggered follow-up. In this campaign the low state and core dominance over HST-1 make a joint spectral snapshot unusually interpretable, but they do not establish that the transfer function is constant in another state.

Likelihood, nulls, and held-out prediction

Conditioned on latent parameters $\theta$, a Gaussian part of the likelihood can be written

$$\ln p(\mathbf y\mid\theta)=-\frac12\left[ (\mathbf y-\boldsymbol\mu_\theta)^{\mathsf T}\Sigma_\theta^{-1} (\mathbf y-\boldsymbol\mu_\theta)+\ln |2\pi\Sigma_\theta|\right],$$

with one-sided integrals replacing density values for censored points. Priors on correlation time, transfer width, and lag must be reported because sparse campaigns can leave the posterior prior-dominated. Report posterior predictive residuals and a model-comparison statistic, but separate statistical uncertainty from clock, flux-scale, aperture, absorption, host-subtraction, and response systematics.

Three checks are indispensable. First, inject known lags and state changes into simulated latent light curves, then sample them through the actual windows and censoring. Second, generate red-noise surrogates independently in each band while preserving sampling and marginal distributions; rerun the complete search, including every tested lag and state cut. Third, hold out one band and predict its fluxes and upper-limit probability. A model that merely interpolates the training bands is not a physical synthesis.

Results and limitations through 2026-08-18

The following matrix separates durable products from model-dependent links. Results published after 2026-08-18 are outside this assessment.

Evidence or result	Physical interpretation	Scope and limitations	Independent check
M87 product 2021-D02-01 and Table A8	A low-state, quasi-simultaneous radio-to-TeV SED constrains emitting zones	Cadence, aperture, and angular scale differ strongly; it is not a uniform lag series	Rebuild the SED from released tables and responses; predict an omitted band
M87 single-zone fits in §3.3	The EHT-mm zone cannot supply the simultaneous gamma-ray flux in the tested isotropic leptonic models	This rejects those formulations, not every compact-zone or hadronic model	Vary zone geometry and require the same VLBI size and optical-depth constraints
TXS 0506+056 IceCube-170922A association	Direction, time, and gamma-ray activity jointly motivate a connected-source hypothesis	A single alert and trial accounting do not identify the radiating zone	Compare the alert likelihood with the independent 2014–2015 archival excess
GWTC-5.0 public release, 2026-05-26	Versioned event times and source posteriors can anchor cross-messenger windows	The 390-event cumulative thresholded catalog is not a catalog of electromagnetic counterparts	Use released calibration channels and predeclared survey coverage, including null follow-up

The durable statements are the calibrated products, observing intervals, apertures, detection thresholds, and model-conditional exclusions. The number and geometry of emitting zones, whether an apparent delay is stable between states, and whether one latent driver explains all bands remain unresolved.

Competing physical explanations

Four positions can fit a visually coherent campaign. A single accretion–ejection state may drive all bands through different transfer functions. Several zones may instead respond with genuine propagation or light-travel delays. Red noise sampled through unrelated windows may produce a false peak after enough lag or state trials. Finally, absorption, host subtraction, aperture mixing, or clock and flux calibration may dominate one band. The M87 SED favors a structured jet over the tested single-zone formulations—see EHT MWL Working Group 2021, §3.3.3—but does not uniquely select longitudinal propagation over a spine–sheath geometry or a separate high-energy zone.

Formulation and parameter regime axis. Compare stationary versus state-changing drivers, one-zone versus stratified transfer functions, and strictly simultaneous high-cadence series versus sparse quasi-simultaneous SEDs. Conclusions may not be carried from one end of any axis to the other without a new fit and injection test.

Reproducing the result and testing its robustness

A reproducible analysis should archive the product identifier and checksums; one row per exposure with UTC and barycentric times, response and aperture identifiers, and calibration versions; the exact state and simultaneity selections; upper-limit likelihoods; priors; latent-process code and seed; injection-recovery and surrogate ensembles; and held-out predictions. A rerun must recover injected lags without excess false positives under the observed cadence before any physical distance is inferred from $c\tau$.

What could change the conclusion. The joint interpretation could change when a genuinely simultaneous campaign adds comparable cadence across the decisive bands, a clock or flux calibration changes, a new latent-process method alters recovered delays, or injection tests fail under the published sampling and censoring.

Formulation-choice diagnostic

An X-ray telescope obtains 2 ks exposures every day for 30 days. A millimeter array observes on days 2, 9, 16, and 23. A cross-correlation of linearly interpolated daily curves peaks at a four-day X-ray lead. May the analyst report a four-light-day separation?

Solution

No. The proposed lag lies below the seven-day millimeter spacing and was obtained after interpolation imposed unmeasured structure. Fit a latent red-noise process at the actual exposure times, convolve it with each window, include calibration covariance and any upper limits, and search the same lag range in independent red-noise surrogates. Inject several known lags—including zero and four days—through the four-epoch millimeter cadence. Only if the four-day lag is recoverable with controlled coverage and false-positive rate, survives prior and state choices, and predicts held-out measurements may it be interpreted as a delay. Even then, $c\tau$ is a response-weighted scale, not automatically the separation between two point zones.

References

• Event Horizon Telescope Multi-Wavelength Science Working Group et al. (2021), “Broadband Multi-wavelength Properties of M87 during the 2017 Event Horizon Telescope Campaign,” The Astrophysical Journal Letters 911, L11, doi:10.3847/2041-8213/abef71; data product 2021-D02-01, doi:10.25739/mhh2-cw46.
• IceCube Collaboration et al. (2018), “Multimessenger Observations of a Flaring Blazar Coincident with High-Energy Neutrino IceCube-170922A,” Science 361, eaat1378, doi:10.1126/science.aat1378.
• IceCube Collaboration et al. (2018), “Neutrino Emission from the Direction of the Blazar TXS 0506+056 Prior to the IceCube-170922A Alert,” Science 361, 147–151, doi:10.1126/science.aat2890.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration (2026), “GWTC-5.0: Observations from the Second Part of the Fourth LIGO–Virgo–KAGRA Observing Run and Updates to the Gravitational-Wave Transient Catalog,” arXiv:2605.27225, arXiv; GWTC-5.0 data release, version 6, doi:10.7935/bk00-6a89.
• Uttley, P., Cackett, E. M., Fabian, A. C., Kara, E., and Wilkins, D. R. (2014), “X-ray Reverberation around Accreting Black Holes,” Astronomy and Astrophysics Review 22, 72, doi:10.1007/s00159-014-0072-0.