Spectral–Timing Systematics

Spectral–timing inference uses the mean energy spectrum and the stochastic variability of that spectrum together. The gain is real: an iron-band lag, a covariance spectrum, and a time-averaged reflection feature can constrain one physical transfer model more sharply than any product alone. The danger is equally real: energy redistribution, calibration, absorption, gaps, reference-band leakage, nonstationarity, and an inadequate source model create correlated errors across all products. A small formal interval is trustworthy only if a response-aware simulation shows correct coverage. This page develops that test; it does not turn any one fit into a unique mass or spin claim.

Required background. Waves, Green Functions, and Geometric Optics supplies Fourier propagation and response functions used to connect variable emission to detector counts.

Helpful background. Black-Hole Geometry and Mechanics supplies relativistic transfer scales; Relativistic Matter and Compact Objects supplies absorption and radiative matter response; Black-Hole Astrophysics supplies accretion-state and inference context; and Computational Relativity supplies forward simulation, injection, and coverage tests.

The detector couples energy and time

Let $F(E,t;\theta)$ be a source photon flux, $A(E,t;\eta)$ an effective area, $R_i(E,t;\eta)$ the redistribution probability into detector channel $i$, and $B_{ik}$ the expected background in time bin $k$. The predicted counts are

$$\mu_{ik}=\int_{\mathcal T_k}dt\int dE\, R_i(E,t;\eta)A(E,t;\eta)F(E,t;\theta)+B_{ik}.$$

Here $\mathcal T_k$ is the live observing interval within bin $k$, $\theta$ contains source parameters, and $\eta$ contains calibration and background quantities. Replacing the time integral by the live duration $\Delta t_k$ times a value at $t_k$ assumes the response-weighted photon flux is effectively constant within the bin; finite integration otherwise filters rapid variability. Response mixing matters twice: it shapes the mean counts and mixes energy-dependent variability before the Fourier transform. Treating a channel as a delta-function energy measurement can therefore bias both spectral curvature and lag–energy structure.

For Fourier amplitudes $X_i(f)$, the cross-spectral matrix is

$$S_{ij}(f)=\left\langle X_i(f)X_j^*(f)\right\rangle .$$

It is Hermitian and positive semidefinite for every frequency. Its diagonal contains power, while off-diagonal phases and amplitudes contain lags and covariance. Any proposed model that fits separate pairwise lags but produces a non-positive matrix is internally inconsistent. Uttley and collaborators derive the estimators, noise corrections, covariance spectrum, and response interpretation in Uttley et al. 2014, §§ 2.1.3, 2.2.1–2.2.3, 2.3, and 6.1–6.2.

For the chapter’s transform pair and signed-lag convention, if $x$ is the reference band and $y$ is delayed, the relevant element is $S_{yx}=\langle YX^*\rangle$, whose phase is positive before wrapping. Reversing the indices conjugates the element and reverses the lag sign without changing the physics.

Counts are approximately Poisson conditional on the stochastic source, whereas averaged power and cross-spectral estimates have non-Gaussian sampling distributions when few segments are available. A joint likelihood must respect the data product actually fit. A Gaussian quadratic form can be useful after sufficient averaging,

$$-2\ln\mathcal L=(d-m)^\mathsf{T}\Sigma^{-1}(d-m)+\ln\det\Sigma+\text{constant},$$

but only if $\Sigma$ includes correlations created by shared reference bands, overlapping Fourier bins, background subtraction, and common calibration. Otherwise simulation from event counts is safer than assuming independent error bars.

Four inference architectures

The competing positions are not merely different software choices. They allocate physical coherence and model mismatch differently.

Architecture	Strength	Failure mode and diagnostic
One coherent physical transfer model	Enforces the same continuum, reflection, propagation, and geometry in mean spectrum, power, covariance, and lags	Can be precisely wrong; use residual structure and injection from omitted source families
Separate spectral and timing models with shared parameters	Easier to compute and exposes which product drives a parameter	Shared symbols need not represent the same effective quantity; compare posterior predictive products jointly
Phenomenological covariance and lag models	Makes few geometric assumptions and robustly summarizes correlated variability	Summary features may not identify a mechanism; test whether multiple physical models map to the same summary
Flexible calibration or source-mismatch terms	Can protect intervals against response edges, nonstationarity, and incomplete continua	Excess flexibility can absorb real signals; constrain it with calibration information and held-out frequencies or energies

Ingram explains how mean, rms, covariance, lag–frequency, and lag–energy spectra carry complementary information in Ingram 2023, §§ 2–3. A coherent analysis must also preserve causal phase conventions, exclude the channel of interest from its own reference band when required, and forward-fold every variable component rather than correcting a measured lag after the fact.

The most consequential source mismatch often occurs where a parameter is interpreted geometrically. Density, ionization gradients, disk thickness, absorption, illumination, returning radiation, and coronal extent can change a reflection spectrum and its impulse response together. Reynolds locates these degeneracies in the relativistic-reflection chain in Reynolds 2021, §§ 3.2.3–3.2.4. A fit statistic within one family cannot establish that the family spans those alternatives.

Named event release and reproducible boundary

The concrete release is the 2008 XMM-Newton EPIC-pn long observation of 1H 0707−495: ObsIDs 0511580101, 0511580201, 0511580301, and 0511580401. Miller and collaborators give the four exposure epochs, durations, count rates, and event selection in Miller et al. 2010, § 2 and Table 1. Their historical processing used SAS 8.0.0 and HEASoft 6.8, pn patterns 0–4, a 35-arcsec source aperture, tight background-flare screening, and the 0.3–1, 1–4, and 4–7.5 keV timing bands. This page performs no new event reduction, response generation, or likelihood fit.

A reproducible modern analysis must pin the observation data files, SAS version and current-calibration-file release, clock correction, event flags and patterns, pile-up assessment, extraction regions, response matrix and effective area, background threshold, good-time intervals, bin width, segment length, frequency bins, reference band, excluded channels, and gap treatment. It must state whether epochs are assumed stationary or modeled hierarchically. The source likelihood or posterior must retain nuisance parameters for background and calibration rather than adding a percentage error after inference.

The minimum independent check is an end-to-end simulation. Draw a time-dependent continuum, propagation process, and impulse response; transform them into $F(E,t)$; fold every time bin through the released response; apply the observed good-time windows, background, and count statistics; reduce the synthetic events with the same selectors; and refit them under multiple covariance and mismatch models. Repeating this procedure over a declared grid tests bias and interval coverage. Injecting only the fitted model tests code consistency, not robustness to physical misspecification.

Systematic perturbations and coverage

Each nuisance source should be assigned an operation, not just a name. Perturb gain and effective area within calibration constraints; swap plausible absorption and continuum families; vary disk density, ionization profile, and emissivity; change segment length and frequency binning; split the four epochs; alter background and flare thresholds; change the reference band; and simulate both compact and extended transfer functions. Preserve the same selection procedure so failed detections and low-coherence bins enter the coverage calculation.

For a nominal $100(1-\alpha)$ per cent interval $I_j$ and injected truth $\theta_j$, empirical coverage is

$$\widehat C=\frac{1}{N}\sum_{j=1}^{N} \mathbf 1\!\left(\theta_j\in I_j\right).$$

Coverage should be stratified by count rate, lag amplitude, source state, response epoch, frequency range, and parameter regime. A global average can hide severe undercoverage near a response edge or a phase wrap. Bias, interval width, false-selection rate, and failure-to-fit rate should accompany coverage; discarding failed simulations makes the result optimistic.

Results and limitations through 2026-08-18

The mathematical need to forward-fold counts and propagate cross-spectral covariance is durable. How much physical coherence to impose, and how flexible the mismatch model should be, remains unresolved. The answer depends on instrument, source population, sampling, count rate, variability state, and the scientific parameter regime.

Position or evidence	Result	Scope and limitation
Analytic response and cross-spectral theory	Requires energy redistribution, phase convention, shared-reference covariance, and positive-semidefinite spectral matrices	Linear stationary theory does not cover all source evolution, gaps, pile-up, or dead time
Response-aware simulations	Measure bias and interval coverage for known injected spectral–timing signals	Conclusions cover only the calibration perturbations and source families actually injected
1H 0707−495 O4 event release	Supplies a named individual object with long exposures and strong energy-dependent timing structure	Four epochs of one variable active nucleus do not select a universal transfer geometry
Population comparisons	Can reveal scaling with mass, state, luminosity, and inclination	Heterogeneous responses, bands, thresholds, and publication selection can imitate population trends
Null and failed-recovery trials	Expose sensitivity limits, model failure, and false-selection rates	They must remain in the denominator; censoring low coherence or failed fits biases coverage
Competing joint-model architectures	Trade physical consistency against robustness to calibration and source mismatch	Agreement within one architecture is not independent evidence for a unique radius or spin

What could change the conclusion. The interpretation could change after an instrument calibration revision, a new response-aware simulator, replicated evidence of interval undercoverage, or a data set spanning a previously missing energy band or timescale. A model that recovers blinded injections across competing source families would also materially change the balance.

Common pitfalls

Adding systematics in quadrature at the end. Calibration and source mismatch can shift a nonlinear posterior or create multiple modes. They belong in the forward model or in a repeated perturb-and-refit analysis.

Checking only goodness of fit. A flexible model can fit every realization while its nominal intervals miss the truth too often. Coverage is a property of the full procedure, including selection and failed fits.

Solved diagnostic: a coverage failure

One hundred response-folded injections are analyzed with a procedure that reports nominal 90 per cent intervals. Only 81 intervals contain the injected parameter. Estimate the expected sampling fluctuation under correct coverage and interpret the result.

Solution

Under the null hypothesis $C=0.90$, the standard deviation of the measured coverage is

$$\sigma_{\widehat C}=\sqrt{\frac{0.90(1-0.90)}{100}}=0.030.$$

The observed difference is $0.81-0.90=-0.09$, giving the normal diagnostic

$$z=\frac{-0.09}{0.030}=-3.0.$$

Thus ordinary binomial fluctuation is an implausible sole explanation, although an exact binomial interval should be reported for a formal decision. The procedure is under-covering: its intervals are too narrow, biased, or selectively missing difficult fits. The next diagnostic is to stratify misses by injected geometry, response perturbation, count rate, frequency, and fit status. Simply inflating every interval by a fixed factor would conceal which physical or calibration assumption failed.

References

• Ingram, A. (2023). “Fourier Spectral-Timing Techniques for the Study of Accreting Black Holes.” Astronomische Nachrichten 344, e20230002. DOI; Open PDF.
• Miller, L., Turner, T. J., Reeves, J. N., and Braito, V. (2010). “X-Ray Reverberation in 1H 0707−495 Revisited.” Monthly Notices of the Royal Astronomical Society 408, 1928–1935. DOI; Open PDF.
• Reynolds, C. S. (2021). “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59, 117–154. DOI; Open PDF.
• 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; Open PDF.