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

Abstract: We present complex energy-resolved properties of quasi-periodic oscillations (QPOs) in the black hole X-ray binary GRS 1915+105 using an observation from the LAXPC instrument onboard AstroSat. Power density spectra (PDSs) are constructed in multiple energy bands and modeled with multi-Lorentzian components to investigate the energy dependence of QPO properties. The QPO frequency shows a modest increase with energy. Dynamic PDS analysis does not reveal clear evidence for time-dependent evolution of the QPO frequency, suggesting that the observed frequency shift is not primarily driven by temporal variability. We perform simultaneous fitting of energy-resolved PDSs and find that a model in which the QPO feature is described by two Lorentzian components provides a better fit. The two components exhibit different evolution in fractional root mean square amplitude as a function of energy. We further examine the phase-lag properties by simultaneously modeling the PDS and the real and imaginary parts of the cross-spectrum and find distinct phase-lag behavior for the two components. Overall, these results indicate that the apparent energy-dependent evolution of the QPO feature may be a result of the presence of more than one variability component.

• The paper shows that the apparent increase in QPO frequency from about 5.6 to 7.3 Hz across 4–40 keV results from two superposed Lorentzian components rather than a single drifting oscillation.
• Simultaneous energy-resolved power-spectrum fits identify a high-quality-factor QPO near 5.5–6.7 Hz and a broader shoulder near 6.2–7.9 Hz, with distinct fractional-rms energy dependencies and improved fit statistics.
• The paper finds significantly different phase lags between the components in Segment 2, reaching a 3.6σ separation, supporting separate variability processes while leaving their physical origin and broader applicability unresolved.

Introduction and motivation

The black hole X-ray binary GRS 1915+105 has long been a benchmark source for studying low-frequency quasi-periodic oscillations (LFQPOs), and the energy dependence of the QPO centroid frequency has been a persistent puzzle. Early RXTE analyses showed that the QPO frequency decreases with energy below $\sim$3 Hz and increases above it (0912.4769), with analogous trends reported in XTE J1550–564 and Swift J1727.8–1613 [astro-ph/0504318-adjacent literature; (Yan et al., 2013)-adjacent work]. However, Méndez et al. demonstrated, using a simultaneous fit to the power density spectrum (PDS) and the real and imaginary parts of the cross-spectrum, that the apparent energy-dependent frequency shift in one such RXTE observation of GRS 1915+105 is not intrinsic: it can be reproduced by two closely spaced Lorentzian components — a QPO and a QPO "shoulder" — with different energy-dependent fractional rms ($frms$) amplitudes and time lags (Mendez et al., 2023). The paper under discussion applies and extends this framework to AstroSat/LAXPC data, providing independent evidence for the two-component interpretation in the same source.

Observations and data segmentation

The authors analyze a single orbit (02360, March 2016) from AstroSat observation T01_030T01_9000000358 using LAXPC, which offers 10 $\mu$s time resolution, 3–80 keV coverage, and $\sim$6000 cm$^2$ effective area at 15 keV. The full-orbit 4–40 keV PDS shows three distinct peaks at $4.58 \pm 0.02$, $5.51 \pm 0.04$, and $6.57 \pm 0.06$ Hz, fitted with four Lorentzians ($\chi^2 = 398/339$). Crucially, subdividing the orbit into three segments yields single-peaked PDSs in each segment, with centroids of $5.64 \pm 0.04$, $frms$0, and $frms$1 Hz. The three-peak structure of the orbit-averaged PDS is therefore an artifact of averaging over a drifting QPO frequency, not evidence for three simultaneously present oscillations.

Energy-dependent frequency shift

Fitting energy-resolved PDSs (4–8, 8–12, 12–20, 20–40 keV) independently with a zero-centred broad-band noise Lorentzian plus one QPO Lorentzian reveals a systematic frequency increase with energy in Segments 1 and 2: from $frms$2 Hz (4–8 keV) to $frms$3 Hz (12–20 keV) in Segment 1, and from $frms$4 Hz (4–8 keV) to $frms$5 Hz (20–40 keV) in Segment 2. No energy dependence is detected in Segment 3 ($frms$64.5 Hz). Dynamic PDS analysis with $frms$738 s segments shows no significant temporal evolution of the centroid frequency within either segment, ruling out time-dependent drift as the origin of the energy-dependent shift. The energy dependence therefore appears only near $frms$86 Hz and is absent at $frms$94.5 Hz, consistent with the earlier AstroSat report (Yadav et al., 2016) and with the broader picture that the slope of the frequency–energy relation varies systematically with QPO frequency.

Evidence for two variability components

Introducing a second Lorentzian — with centroid frequency and width tied across energy bands while normalizations vary freely — improves the simultaneous fits: $\mu$0 drops from 471.1 to 445.5 in Segment 1 and from 714.7 to 671 in Segment 2 (660 DOF). The resolved components are:

Segment	Component	Centroid (Hz)	Quality factor $\mu$1
1	QPO	$\mu$2	$\mu$313.8
1	Shoulder	$\mu$4	$\mu$55.34
2	QPO	$\mu$6	$\mu$711.68
2	Shoulder	$\mu$8	$\mu$94.64

The two components have clearly distinct $\sim$0 amplitude spectra. In Segment 2, the QPO rises from $\sim$1 (4–8 keV) to $\sim$2 (20–40 keV), while the shoulder rises more steeply from $\sim$3 to $\sim$4, exceeding the QPO by $\sim$5 ($\sim$62.8$\sim$7) in the highest band. Segment 1 shows the same qualitative trend but at only $\sim$80.8$\sim$9 significance in the 20–40 keV band — the authors are appropriately cautious that the amplitude distinction there is marginal. This mirrors the RXTE result of Méndez et al. (Mendez et al., 2023), where the shoulder saturated at $^2$05.5% versus $^2$14% for the QPO.

Phase lags from simultaneous PDS and cross-spectrum fitting

Adopting the constant phase lag model (Mendez et al., 2023), in which each mutually incoherent Lorentzian component has a frequency-independent phase $^2$2 between energy bands (so the real and imaginary cross-spectral parts scale as Lorentzian $^2$3 and Lorentzian $^2$4), the authors fit the full-band PDS (five Lorentzians plus a negative power law for deadtime) jointly with the real and imaginary cross-spectra relative to the 4–8 keV reference band. The derived phase-lag spectra reproduce the observed frequency dependence around both components.

The two components again behave differently. In Segment 1 both exhibit negative lags of increasing magnitude, but the spectra are statistically consistent (difference $^2$5 rad at 20–40 keV, $^2$60.9$^2$7). In Segment 2 the separation is significant: the QPO saturates at $^2$8 rad while the shoulder reaches $^2$9 rad, a difference of $4.58 \pm 0.02$0 rad ($4.58 \pm 0.02$13.6$4.58 \pm 0.02$2). This is the strongest quantitative result in the paper and constitutes direct evidence that the two Lorentzians are separate variability processes rather than an asymmetric extension of a single QPO.

Physical interpretation and limitations

The phenomenology is suggestive but not conclusive regarding classification. The QPO component ($4.58 \pm 0.02$313, $4.58 \pm 0.02$42–3.5%) is consistent with a Type-C QPO, while the shoulder's lower $4.58 \pm 0.02$5 ($4.58 \pm 0.02$64–5) and $4.58 \pm 0.02$7 amplitude fall in the Type-B range, qualitatively echoing the proposed coexistence of Type-B and Type-C QPOs in Swift J1727.8–1613 [10.1051/0004-6361/202555907]. The authors explicitly refrain from a Type-B identification, citing the low quality factor and the absence of additional Type-B signatures; the physical origin of the shoulder is left open.

The analysis also carries structural caveats. The constant phase lag model assumes each component has a linear, frequency-independent transfer function between bands — an assumption that may not hold if the lag arises from propagating fluctuations or phase-dependent Comptonization. The improvement in $4.58 \pm 0.02$8 from adding the second Lorentzian, while consistent across both segments, is moderate (tens of counts over 660 DOF), so the two-component decomposition rests on the convergence of three independent diagnostics — PDS shape, $4.58 \pm 0.02$9 amplitude, and phase lag — rather than on the PDS fit alone. Finally, the result is based on a single orbit, and the energy dependence appears only at $5.51 \pm 0.04$06 Hz; whether the two-component picture generalizes across the QPO frequency range where energy-dependent shifts have been reported (e.g., the sign change near $5.51 \pm 0.04$13 Hz) remains untested. The authors note that time-dependent Comptonization models reproduce rms and lag spectra but not the energy dependence of the centroid frequency, and that differential Lense–Thirring precession offers one geometric explanation; the two-component interpretation now competes with both.

Conclusion

Using AstroSat/LAXPC data from a single orbit of GRS 1915+105, this work shows that an apparent energy-dependent QPO frequency shift — increasing from $5.51 \pm 0.04$25.6 to $5.51 \pm 0.04$37.3 Hz across 4–40 keV — is not intrinsic to a single oscillation. Dynamic PDS analysis excludes temporal drift; simultaneous energy-resolved PDS fitting favors two Lorentzians with fixed centroids; and the components exhibit distinct $5.51 \pm 0.04$4 amplitude evolutions (up to $5.51 \pm 0.04$52.8$5.51 \pm 0.04$6) and phase-lag spectra (up to $5.51 \pm 0.04$73.6$5.51 \pm 0.04$8). These results independently confirm, at higher photon energies than RXTE allowed, the interpretation of Méndez et al. (Mendez et al., 2023) that energy-dependent QPO frequency shifts can be an artifact of superposed variability components, and they sharpen the open question of whether such shoulders are ubiquitous in LFQPOs and what physical mechanism produces them.