Complex Energy-Dependent Behaviour of Quasi-Periodic Oscillation Observed in GRS 1915+105

The paper presents a detailed energy-resolved analysis of quasi-periodic oscillations (QPOs) in the black hole X-ray binary GRS 1915+105, using data from the LAXPC instrument onboard AstroSat from a single orbit (02360) of observation T01 030T01 9000000358, carried out in March 2016.

The authors first extract the background-subtracted light curve in the 4.0–40.0 keV energy range and divide it into three segments. The full-orbit power density spectrum (PDS) exhibits three distinct peaks, which is modeled with four Lorentzian components, yielding centroid frequencies of 4.58 ± 0.02 Hz, 5.51 ± 0.04 Hz, and 6.57 ± 0.06 Hz. However, each segmental PDS shows only a single peaked feature, with centroid frequencies at 5.64 ± 0.04 Hz (Segment 1), 6.64 ± 0.03 Hz (Segment 2), and 4.61 ± 0.02 Hz (Segment 3). This suggests that the three-peaked structure in the full orbit is likely a result of averaging PDSs with a single Lorentzian whose frequency changes.

The energy dependence of the QPO feature is investigated by generating PDSs in four energy bands: 4.0–8.0 keV, 8.0–12.0 keV, 12.0–20.0 keV, and 20.0–40.0 keV. Independent fits show that the QPO feature exhibits a systematic shift from 5.57 ± 0.04 Hz (4.0-8.0 keV) to 5.86 ± 0.08 Hz (12.0-20.0 keV) in Segment 1 and from 6.59 ± 0.04 Hz (4.0-8.0 keV) to 7.27 ± 0.12 Hz (20.0-40.0 keV) in Segment 2, whereas it remains consistent across energy bands in Segment 3 (centroid frequency ∼ 4.5 Hz). Dynamic PDS analysis does not reveal any significant time-dependent variation in the QPO centroid frequency in either segment, suggesting that the observed frequency shift with energy is not time-dependent.

To examine whether the QPO feature can be better described by two Lorentzian components (QPO and QPO shoulder), the authors perform simultaneous fits to all PDSs within each segment, with centroid frequency and width of each Lorentzian tied across energy bands while normalizations vary freely. This approach yields a smaller χ2 in both segments: for Segment 1, χ2 reduces from 471.1 to 445.5, and for Segment 2, from 714.7 to 671 for 660 degrees of freedom. The two components have centroid frequencies 5.52 ± 0.04 (QPO with quality factor Q ∼ 13.8) and 6.20 ± 0.15 (QPO shoulder with Q ∼ 5.34) in Segment 1, and 6.66 ± 0.03 (QPO with Q ∼ 11.68) and 7.94 ± 0.12 (QPO shoulder with Q ∼ 4.64) in Segment 2.

The fractional rms amplitude of the two components is computed. For Segment 1, the f rms amplitude of the QPO increases from 3.14 ± 0.18% in the 4-8 keV band to 5.18 ± 0.99% in the 20-40 keV band, while the QPO shoulder rises from 2.21 ± 0.39% to 6.41 ± 1.07% over the same range, though the difference between the two components remains statistically insignificant in individual energy bands (reaching only ∼ 0.8σ in the 20-40 keV band). In contrast, Segment 2 exhibits a clearer distinction: the f rms amplitude of the QPO increases from 2.12 ± 0.11% in 4-8 keV to 5.36 ± 0.58% in 20-40 keV, whereas the QPO shoulder rises more rapidly from 0.99 ± 0.23% to 7.86 ± 0.69%. In the highest-energy band, the f rms amplitude of the QPO shoulder exceeds that of the QPO by 2.50 ± 0.90%, corresponding to a ∼ 2.8σ difference. This quantitatively demonstrates that the QPO shoulder exhibits a significantly steeper energy dependence of the f rms amplitude than the QPO.

The phase-lag behaviour is investigated using the constant phase lag model approach, which involves simultaneous fitting of the PDS and the real and imaginary parts of the cross-spectrum. The PDS in the full energy band (4.0–40.0 keV) is fitted with a model consisting of one power law and five Lorentzian components, yielding χ2 of 278 for 315 degrees of freedom in Segment 1 and 404.4 for 386 degrees of freedom in Segment 2. The cross-spectrum is computed between different energy bands, taking the 4.0–8.0 keV band as the reference. The derived model reproduces the overall frequency dependence of the observed phase-lag spectra, including the behaviour around the QPO and QPO shoulder frequencies.

For Segment 1, both variability components exhibit negative phase lags whose magnitudes generally increase with energy, but the phase-lag spectra are statistically consistent with each other. In the 20-40 keV band, the QPO and QPO shoulder exhibit phase lags of −1.07 ± 0.14 rad and −0.72 ± 0.35 rad, respectively, differing by only 0.35 ± 0.38 rad (∼ 0.9σ). In contrast, Segment 2 reveals a clear separation between the phase-lag spectra of the two components. While both exhibit increasingly negative phase lags with energy, the QPO saturates at −1.19 ± 0.10 rad, whereas the QPO shoulder reaches −1.86 ± 0.16 rad in the 20-40 keV band. The difference of 0.66 ± 0.18 rad corresponds to a significance of ∼ 3.6σ, providing strong evidence that the two variability components possess intrinsically different phase-lag spectra.

The authors conclude that the apparent energy-dependent evolution of the QPO feature may be a result of the presence of more than one variability component. The simultaneous fit of the energy-resolved PDSs, together with the corresponding f rms amplitude and phase-lag spectra, supports that the observed QPO feature is composed of two distinct components, namely the QPO and the QPO shoulder. These two components exhibit different f rms amplitude and phase lag behaviours, indicating that they are likely associated with separate variability processes. The apparent shift in the centroid frequency of the QPO feature with energy arises from changes in the relative contributions of these two components in different energy bands.

The phenomenological properties of the two Lorentzian components suggest that the QPO (with quality factor ∼ 13.1-13.6 and f rms amplitude ∼ 2.25-3.5%) is consistent with a Type-C QPO, while the QPO shoulder (with quality factor ∼ 4.2-4.5 and f rms amplitude ∼ 1.6-3.2%) has properties within the range commonly associated with Type-B QPOs. However, the relatively low quality factor of the shoulder and the lack of additional observational signatures characteristic of Type-B QPOs prevent the authors from drawing such an identification. They interpret the QPO shoulder as a distinct variability component whose physical origin remains uncertain, while noting that the possible coexistence of multiple QPOs is an intriguing possibility deserving further investigation.

Improvement 1: Energy-Resolved Decomposition of Superimposed Oscillatory Signals

The AI system can be enhanced to automatically detect and decompose multiple overlapping quasi-periodic oscillation (QPO) components in power density spectra (PDS) across different energy bands. Instead of fitting a single Lorentzian per segment, the system can implement a simultaneous multi-Lorentzian fitting routine (with tied centroid frequencies and widths across energy bands, free normalizations) to identify hidden components (e.g., QPO and QPO shoulder) that are otherwise masked by frequency shifts in averaged spectra. This enables the AI to distinguish intrinsic variability processes from apparent frequency drifts caused by relative amplitude changes.

Improvement 2: Frequency-Shift-Aware Time-Series Segmentation

The AI can be trained to segment long observational light curves into quasi-stationary intervals (e.g., using change-point detection or Bayesian blocks) before computing PDS. This prevents the artificial broadening or splitting of a single QPO peak into multiple peaks due to frequency evolution over time. The system can then flag whether a multi-peaked structure in the full-orbit PDS is an artifact of averaging, as demonstrated in the paper (three peaks in full orbit vs. single peaks in each segment).

Improvement 3: Energy-Dependent rms Amplitude and Phase-Lag Modeling

The AI can incorporate energy-resolved fractional rms amplitude and phase-lag spectra as joint constraints when fitting variability components. By simultaneously modeling the PDS, real/imaginary parts of the cross-spectrum, and rms amplitude trends (e.g., QPO shoulder rising faster with energy than QPO), the system can statistically separate components that have distinct energy dependencies. This improves classification of QPO types (e.g., Type-C vs. Type-B) and identifies intrinsically different variability processes even when their centroid frequencies are close.

Improvement 4: Automated Significance Testing for Component Distinction

The AI can implement rigorous significance tests (e.g., Δχ2 with appropriate degrees of freedom, or Bayesian evidence) to decide whether adding a second Lorentzian (shoulder) is justified. It can also compute the significance of differences in rms amplitude and phase lag between components (e.g., 2.8σ for rms, 3.6σ for phase lag) to avoid over-interpreting noise. This prevents false positives in identifying distinct variability components.

Improved AI System Capabilities:

• Multi-Component QPO Decomposition: Automatically separates overlapping QPO features in black hole X-ray binaries, providing robust centroid frequencies, quality factors, and energy-dependent amplitudes for each component.

• Artifact Detection: Identifies when apparent frequency shifts with energy are due to relative component contributions rather than true time-dependent changes, using dynamic PDS and segment-wise fitting.

• Physical Classification: Distinguishes between Type-C and Type-B QPOs based on quality factor, rms amplitude, and phase-lag behavior, even when components are blended.

• Data-Driven Hypothesis Testing: Quantitatively tests whether multiple variability processes coexist, with statistical confidence levels, aiding in the interpretation of accretion disk and corona dynamics.

• Generalization to Other Sources: Applies the same energy-resolved decomposition and significance framework to other X-ray binaries or accreting systems with similar QPO phenomena, improving robustness of variability studies.