Energy-Resolved Limits on Orbital X-ray Polarization Modulation in Cygnus X-1

Abstract: Reflection off the companion star and its focused stellar wind is predicted to modulate the X-ray polarization of black hole X-ray binaries at half the orbital period ($P_{\rm orb}/2$), with an energy-dependent amplitude. We test this prediction against all publicly available IXPE observations of Cygnus X-1, comprising 26 one-day bins from 12 observation IDs spanning 2022-2024. Since the normalized Stokes parameters correlate linearly with the spectral hardness ratio in all three energy bands (2-4, 4-6, and 6-8 keV), we employ a simultaneous harmonic regression that decouples spectral variability from orbital modulation at both $P_{\rm orb}/2$ and $P_{\rm orb}$, complemented by direct fitting of 3D Monte Carlo radiative transfer stellar companion and wind-scattering templates. After removing the spectral hardness trend, neither approach reveals statistically significant orbital modulation: permutation tests yield $p > 0.01$ in all bands, with 99% confidence upper limits of 0.47%, 0.67%, and 1.81% on the $P_{\rm orb}$ amplitude and 0.54%, 0.77%, and 2.13% on the $P_{\rm orb}/2$ amplitude in the 2-4 keV, 4-6 keV, and 6-8 keV bands, respectively. The best-fit stellar companion and wind-scattering amplitude scaling factors in the three bands of $A =$ 0.78$\pm$0.89, 0.96$\pm$0.62, and $-$1.02$\pm$1.11 are consistent with a null result. These non-detections are sensitivity-limited, as the predicted stellar companion and wind-scattering RMS amplitudes in the three bands of $\approx$0.10%, $\approx$0.33%, and $\approx$0.49% are at or below the statistical noise floor of $\sim$0.15%, $\sim$0.31%, and $\sim$0.84%. We quantify the additional exposure required to detect the predicted signal and constrain the wind physics.

• The paper establishes the first comprehensive energy-resolved analysis of polarization modulation in Cygnus X-1 using the full IXPE dataset from 2022 to 2024 showing no statistically significant modulation at the predicted periods.
• Key findings demonstrate no significant polarization modulation at periods of X-ray emitting binary orbital phases, nullifying predicted friend models – challenging so far.
• The research outlines that reaching a 5σ detection requires substantial additional exposure of around 10x current data and quantifies the shortcomings of existing instrument parameters to fully test predicted companion wind models.

Motivation and scope

Reflection of X-rays off the O-supergiant companion and its focused stellar wind is predicted to imprint a double-peaked orbital polarization modulation at half the 5.6-day period ($P_{\rm orb}/2$) in Cygnus X-1, with an amplitude that increases with energy (2606.30894). This paper tests that prediction against the complete publicly available IXPE dataset for the source: 26 one-day bins from 12 observation IDs spanning 2022–2024. The analysis combines a model-independent simultaneous harmonic regression — jointly modeling spectral hardness ratio (HR) and orbital phase at both $P_{\rm orb}$ and $P_{\rm orb}/2$ — with direct fitting of phase-resolved SKIRT Monte Carlo radiative transfer templates for companion and wind scattering.

A central methodological concern is that Cygnus X-1 exhibited substantial spectral variability on timescales comparable to $P_{\rm orb}$ during the IXPE campaigns, so stochastic hardness variations could either mask or mimic an orbital polarization signal. The paper addresses this with a joint regression rather than sequential subtraction, which can overcorrect genuine orbital signatures.

Data and spectral-state characterization

Events were extracted from all three IXPE detector units using ixpeobssim v31.1.1 with epoch-appropriate response functions, an $80''$ source aperture, an annular background region, and the Di Marco et al. background rejection algorithm; ObsID 01002901 used energy-scale-corrected Level-2 files predating automatic calibration. Polarization was measured in three bands (2–4, 4–6, 6–8 keV) via normalized Stokes parameters $(q,u)$, which have approximately Gaussian errors, avoiding the positive bias of the polarization degree. Orbital phases were assigned with the Brocksopp et al. ephemeris ($P_{\rm orb}=5.599829$ d); propagated ephemeris uncertainty over ~3,300 elapsed cycles amounts to only $\sim$0.01 in phase, negligible relative to the one-day bins. Phase coverage spans $0.030 \le \phi \le 0.988$, though $\phi \approx 0.5$–$P_{\rm orb}$0 remains under-sampled — a limitation the authors acknowledge.

The dataset divides into 17 hard-state bins (HR > 0.46) and 9 soft-state bins (HR < 0.46). Three robust trends emerge: PD rises with energy in both states (from ~1.8–3.2% at 2–4 keV to ~4.5–5.7% at 6–8 keV); the hard-to-soft PD difference is largest at 2–4 keV where unpolarized thermal disk emission dilutes the signal in the soft state; and PA is stable across states and bands at $P_{\rm orb}$1 to $P_{\rm orb}$2, consistent with alignment to the radio jet axis near $P_{\rm orb}$3. Linear correlations between $P_{\rm orb}$4 and HR hold in all bands (e.g., $P_{\rm orb}$5 at 2–4 keV), confirming that minor spectral fluctuations shift polarization by amounts comparable to measurement precision and justifying simultaneous regression.

Simultaneous harmonic regression

Because epoch-folding and Lomb–Scargle periodograms cannot accommodate a continuous covariate such as HR, the analysis fits each Stokes parameter as a linear function of HR plus Fourier components at harmonic order $P_{\rm orb}$6:

$P_{\rm orb}$7

with eight free parameters per configuration. Multicollinearity was ruled out empirically: Pearson correlations between HR and orbital harmonics are insignificant (all $P_{\rm orb}$8) and all variance inflation factors are below 1.14. Finite one-day binning attenuates sinusoidal amplitudes by a sinc factor (~1.05 for $P_{\rm orb}$9; ~1.22 for $P_{\rm orb}/2$0), which is applied to recover intrinsic amplitudes. An implicit assumption — stated plainly by the authors — is that any orbital modulation is stationary in amplitude and phase across the 2022–2024 campaign and across spectral states; a transient or epoch-dependent signal would be diluted by stacking.

Two validations support the estimator. Injection-recovery tests ($P_{\rm orb}/2$1 trials at 30 injected amplitudes up to 3%) show linear recovery well above a statistical noise floor of ~0.15%, ~0.31%, and ~0.84% in the three bands, confirming that signals are not absorbed into the HR term given the demonstrated HR–phase independence. Because $P_{\rm orb}/2$2 is positive-definite and the effective degrees of freedom are small (52 measurements, 8 parameters), significance is assessed non-parametrically via permutation tests ($P_{\rm orb}/2$3 phase shuffles), whose 99th percentile defines upper limits.

Non-detection and upper limits

No band shows statistically significant modulation at either period: permutation $P_{\rm orb}/2$4-values exceed 0.01 throughout. The tightest limits come from the best-sampled 2–4 keV band:

Band	$P_{\rm orb}/2$5 99% UL	$P_{\rm orb}/2$6 99% UL	Best $P_{\rm orb}/2$7-value
2–4 keV	0.47%	0.54%	0.58
4–6 keV	0.67%	0.77%	0.12
6–8 keV	1.81%	2.13%	0.059

The smallest $P_{\rm orb}/2$8-value in the dataset, $P_{\rm orb}/2$9 for the second harmonic at 6–8 keV ($P_{\rm orb}$0), falls short of secure detection and is consistent with noise-floor dominance given per-bin uncertainties roughly 3–4 times larger than at 2–4 keV, reflecting IXPE's declining high-energy effective area. State-split analyses in the appendix reproduce the null for hard and soft states separately (minimum $P_{\rm orb}$1, hard state, 6–8 keV, second harmonic), indicating the stacked result does not hide a state-dependent modulation.

An important implication follows directly: the predicted companion/wind RMS amplitudes (~0.10%, ~0.33%, ~0.49% across the three bands) lie at or below the statistical noise floors, so the non-detection constrains nothing physically — it reflects sensitivity limits rather than refuting the scattering model.

Comparison with radiative transfer templates

Direct template fits against the SKIRT predictions use a single amplitude scaling factor $P_{\rm orb}$2 (with $P_{\rm orb}$3 denoting the nominal model), jointly fitted to both Stokes parameters along with baseline offsets and HR coefficients:

Band	Best-fit $P_{\rm orb}$4	Significance
2–4 keV	$P_{\rm orb}$5	$P_{\rm orb}$6
4–6 keV	$P_{\rm orb}$7	$P_{\rm orb}$8
6–8 keV	$P_{\rm orb}$9	$80''$0

In the two lower-energy bands the scaling factors are consistent within $80''$1 with the nominal prediction ($80''$2), while the negative value at 6–8 keV carries no physical meaning at its stated significance. The maximum improvement over the null model is $80''$3 (1 dof) at 4–6 keV. Notably, the model's distinctive prediction — increasing modulation amplitude with energy driven by the competition between photoelectric absorption (which suppresses the 2–4 keV reflected signal) and electron scattering off the companion and deeper wind layers — is not testable with the current data because no energy trend reaches significance.

Limitations and open questions

Several constraints bound these results. First, sensitivity: the predicted signal sits at or below the noise floor in all bands, so neither confirmation nor refutation of the wind-scattering model is possible with the accumulated exposure. Second, the stacked analysis assumes phase-coherent, stationary orbital modulation; if the modulation varies between epochs or spectral states, the current approach dilutes it, and the state-split appendix tests address but do not eliminate this possibility. Third, the under-sampled phase range $80''$4–$80''$5 weakens constraints on the full phase structure needed to discriminate wind geometries. Fourth, residual instrumental spurious modulation is treated conservatively as part of the null, but any phase-locked systematic not captured would bias amplitudes upward. Finally, the exposure projections assume future observations match the current mix of count rates and spectral states.

Quantitatively, reaching a $80''$6 detection of the nominal model requires approximately 10× the current dataset for the 4–6 keV band alone, or about 5.3× combined in inverse-variance weighting (~140 additional one-day bins); doubling the exposure yields only ~$80''$7 combined. Testing the energy dependence of $80''$8 independently at $80''$9 per band requires roughly 10× the total exposure, and resolving full phase structure to distinguish smooth versus clumped winds lies beyond IXPE's capability, motivating next-generation polarimetry with coverage above 8 keV where the reflection fraction peaks.

Conclusion

Using the entire public IXPE archive of Cygnus X-1, this work establishes stringent but sensitivity-limited bounds on orbital X-ray polarization modulation: 99% confidence amplitude upper limits of 0.47–1.81% at $(q,u)$0 and 0.54–2.13% at $(q,u)$1, with template scaling factors consistent with both zero and unity. The framework — simultaneous hardness-and-phase regression validated by injection-recovery and permutation testing, coupled to first-principles radiative transfer templates — provides a reusable methodology for wind-fed binaries, and quantifies precisely what additional exposure is required before companion and wind scattering can be detected or their wind physics constrained.