# Phase-resolved Spectroscopy of Low-frequency Quasi-periodic Oscillations
from the Newly Discovered Black Hole X-ray Binary Swift J1727.8–1613

Qing-Cang Shui Key Laboratory of Particle Astrophysics, Institute of High
Energy Physics, Chinese Academy of Sciences, 100049, Beijing, China
University of Chinese Academy of Sciences, Chinese Academy of Sciences,
100049, Beijing, China  Shu Zhang  Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 100049,
Beijing, China  Jing-Qiang Peng Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 100049,
Beijing, China  University of Chinese Academy of Sciences, Chinese Academy of
Sciences, 100049, Beijing, China  Shuang-Nan Zhang Key Laboratory of Particle
Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,
100049, Beijing, China  University of Chinese Academy of Sciences, Chinese
Academy of Sciences, 100049, Beijing, China  Yu-Peng Chen Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 100049, Beijing, China  Long Ji School of Physics and Astronomy, Sun
Yat-Sen University, Zhuhai, 519082, China  Ling-Da Kong Institut für
Astronomie und Astrophysik, Kepler Center for Astro and Particle Physics,
Eberhard Karls, Universität, Sand 1, D-72076 Tübingen, Germany  Hua Feng Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 100049, Beijing, China  Zhuo-Li Yu  Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 100049, Beijing, China  Peng-Ju Wang Institut für Astronomie und
Astrophysik, Kepler Center for Astro and Particle Physics, Eberhard Karls,
Universität, Sand 1, D-72076 Tübingen, Germany  Zhi Chang Key Laboratory of
Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of
Sciences, 100049, Beijing, China  Hong-Xing Yin Shandong Key Laboratory of
Optical Astronomy and Solar-Terrestrial Environment, School of Space Science
and Physics, Institute of Space Sciences, Shandong University, Weihai,
Shandong 264209, China  Jin-Lu Qu Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, 100049,
Beijing, China  Lian Tao Key Laboratory of Particle Astrophysics, Institute of
High Energy Physics, Chinese Academy of Sciences, 100049, Beijing, China
Ming-Yu Ge Key Laboratory of Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, 100049, Beijing, China  Liang Zhang Key
Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese
Academy of Sciences, 100049, Beijing, China  Jian Li  CAS Key Laboratory for
Research in Galaxies and Cosmology, Department of Astronomy, University of
Science and Technology of China, Hefei 230026, China  School of Astronomy and
Space Science, University of Science and Technology of China, Hefei 230026,
China

###### Abstract

Low-frequency quasi-periodic oscillations (LFQPOs) are commonly observed in
X-ray light curves of black hole X-ray binaries (BHXRBs); however, their
origin remains a topic of debate. In order to thoroughly investigate
variations in spectral properties on the QPO timescale, we utilized the
Hilbert-Huang transform technique to conduct phase-resolved spectroscopy
across a broad energy band for LFQPOs in the newly discovered BHXRB Swift
J1727.8–1613. This is achieved through quasi-simultaneous observations from
Neutron star Interior Composition ExploreR (NICER), Nuclear Spectroscopic
Telescope ARray (NuSTAR), and Hard X-ray Modulation Telescope (Insight-HXMT).
Our analysis reveals that both the non-thermal and disk-blackbody components
exhibit variations on the QPO timescale, with the former dominating the QPO
variability. For the spectral parameters, we observe modulation of the disk
temperature, spectral indices, and reflection fraction with the QPO phase with
high statistical significance ($\gtrsim 5\sigma$). Notably, the variation in
the disk temperature is found to precede the variations in the non-thermal and
disk fluxes by $\sim 0.4-0.5$ QPO cycles. We suggest that these findings offer
further evidence that the type-C QPO variability is a result of geometric
effects of the accretion flow.

Accretion (14) — Black hole physics (1736) — X-ray binary stars (1811) —
Stellar mass black holes (1611)

##  1 Introduction

Black hole X-ray binaries (BHXRBs) are commonly observed to undergo outbursts,
in which the source exhibit transitions in their X-ray spectral and timing
variability properties (Remillard & McClintock, 2006; Done et al., 2007).
These outbursts can be categorized into distinct canonical states, each
characterized by unique X-ray spectral and variability properties (Belloni et
al., 2005; Homan & Belloni, 2005). In the hard state, the emitting spectrum is
dominated by a nonthermal power-law component, whereas in the soft state, a
thermal multi-temperature blackbody component becomes dominant (Remillard &
McClintock, 2006). The multi-temperature blackbody component is believed to
originate from a geometrically thin and optically thick accretion disk
(Shakura & Sunyaev, 1973; Lynden-Bell & Pringle, 1974). The nonthermal power-
law component is produced by the Comptonization of soft photons in an extended
cloud consisting of hot electrons, commonly referred to as the X-ray corona
(Sunyaev & Titarchuk, 1980; Titarchuk, 1994; Zdziarski et al., 1996), and/or
in a jet base (Markoff et al., 2005; You et al., 2021). Additionally,
reflection features are often detected in the X-ray spectrum as a reflection
component, formed by a portion of Comptonized photons irradiating the disk and
then being scattered into the line of sight. This component includes abundant
features, like broad emission lines and Compton hump, etc. (Ross & Fabian,
2005; Dauser et al., 2010; García et al., 2014). In the time domain, low
frequency quasi-periodic oscillations (LFQPOs, roughly 0.1–30 Hz, van der
Klis, 1989) have been routinely observed in most BHXRBs. These features are
characterized by a narrow peak with finite width in the power spectral density
(PSD), with the classifications of type A, B, and C based on the centroid
frequency, quality factor and root-mean-square (rms) amplitude (Wijnands et
al., 1999; Casella et al., 2005; Remillard & McClintock, 2006).

In the past few decades, various theoretical models, based on either the
geometric effect or the intrinsic variability of the accretion flow, have been
proposed to explain the physical origin of LFQPOs. Intrinsic models include
the trapped corrugation modes, the Accretion-ejection instability model (AEI),
and the Two-Component Advection Flow model, among others (Kato, 1990; Wagoner,
1999; Tagger & Pellat, 1999; Molteni et al., 1996). Geometric models, on the
other hand, are mainly related to the relativistic precession of the accretion
flow due to the frame dragging effect, and are developed into the relativistic
precession model (RPM; Stella et al., 1999), L-T precession of the hot flow
(Ingram et al., 2009) and jet precession model (Stevens & Uttley, 2016; de
Ruiter et al., 2019; Ma et al., 2021, 2023). Observational studies have shown
that the variability of type-C QPOs generally increases with photon energy
(Zhang et al., 2017; Huang et al., 2018; Kong et al., 2020; Zhang et al.,
2020a) and no prominent disc-like component exists in the rms spectra
(Sobolewska & Życki, 2006; Axelsson et al., 2014; Axelsson & Done, 2016; Shui
et al., 2023a). Additionally, Ma et al. (2021) reported the discovery of
tpye-C QPOs above 200 keV, indicating a strong relationship between type-C
QPOs and Comptonized emission. Recently, there has been some evidence
suggesting a geometric origin of LFQPOs. With large samples of observations,
the inclination dependence of amplitudes and time lags of type C and B QPOs
has been determined (see Motta et al., 2015; van den Eijnden et al., 2017).
Furthermore, reflection variability revealed with phase-resolved spectroscopy
provides further support for a geometrical origin of type-C QPOs (see Ingram &
van der Klis, 2015; Ingram et al., 2016, 2017; Nathan et al., 2022; Shui et
al., 2023a). We refer readers to Ingram & Motta (2019) for recent reviews of
observations and theories of LFQPOs.

Swift J1727.8–1613, a new bright X-ray transient, was discovered on August 24,
2023, by the Burst Alert Telescope of Swift (Swift/BAT) and initially
identified as GRB 230824A (Page et al., 2023). This source underwent a giant
outburst, and was also detected by Gas Slit Camera of the Monitor of All-sky
X-ray Image (MAXI/GSC), with a peak flux exceeding 7 Crab in the MAXI/GSC 2–20
keV energy band (Negoro et al., 2023; Nakajima et al., 2023). Subsequently,
this outburst was extensively monitored at optical and radio bands (Castro-
Tirado et al., 2023; Miller-Jones et al., 2023). Based on the observations
from Imaging X-ray Polarimeter Explorer (IXPE), a significant polarization was
detected with a polarization degree (PD) of $4.71\%\pm 0.27\%$ and
polarization angle (PA) of $2^{\circ}.6\pm 1^{\circ}.7$ in 2–8 keV energy
range (Dovciak et al., 2023). During this outburst, significant type-C QPOs
were detected by multiple X-ray instruments (Katoch et al., 2023; Draghis et
al., 2023; Mereminskiy et al., 2023; Veledina et al., 2023; Yu et al., 2024a;
Ingram et al., 2024). In this Letter, we conduct a phase-resolved analysis of
the type-C QPOs observed in Swift J1727.8–1613 quasi-simultaneously by NICER,
NuSTAR and Insight-HXMT. We provide an overview of the observations and our
data reduction in Section 2, followed by the presentation of the timing and
spectral analyses in Section 3. Finally, we discuss and summarize these
results in Section 4.

##  2 Observations and Data Reduction

###  2.1 Observations

As mentioned in Section 1, modeling the reflection component in phase-resolved
analysis is crucial for diagnosing QPO models. Therefore, our study includes
NuSTAR data to provide further constraints on the reflection component. NICER,
NuSTAR and Insight-HXMT conducted quasi-simultaneous observations of the newly
discovered BHXRB Swift J1727.8–1613 quasi-simultaneously on MJD 60185. In
Figure 1 (a) and (b), the timing of these observations is shown with respect
to the source flux and hardness ratio variations as measured by MAXI/GSC.
Figure 1 (c) positions the observations on a hardness-intensity diagram (HID),
indicating that the quasi-simultaneous observations were carried out at the
beginning of the hard-to-soft state transition, where the luminosity of the
source is close to the peak value. Figure 2 displays the light curves of the
NICER, NuSTAR/FPMA and Insight-HXMT/LE observations with a 10-s time
resolution. Further details of these quasi-simultaneous observations can be
found in Table 1.

Table 1: Log of NICER, Insight-HXMT, and NuSTAR Observations of Swift J1727.8–1613 Used in This Work. Instrument | Start Time | Exposure Time
---|---|---
| (MJD) | (s)
NICER | 60185.04 | 3627
Insight-HXMT/LE | 60185.31 | 11450
Insight-HXMT/ME | 60185.31 | 8664
Insight-HXMT/HE | 60185.31 | 9875
NuSTAR/FPMA | 60185.42 | 923
NuSTAR/FPMB | 60185.42 | 1012
Figure 1: Evolution of X-ray properties of Swift J1727.8–1613 during the
2023–2024 outburst, measured by MAXI/GSC. The red vertical lines plotted in
(a) and (b) represent the date of the quasi-simultaneous observations of
NICER, NuSTAR and Insight-HXMT. (a) MAXI light curve in the 2–20 keV energy
range. (b) Evolution of the hardness ratio of the X-ray count rate in the 4–10
keV energy band to that in the 2–4 keV energy band. (c) Hardness-intensity
diagram (HID). The position of the source on the diagram during the quasi-
simultaneous observations of NICER, NuSTAR and Insight-HXMT is marked with the
red star. Figure 2: Light curves (counts per second) of Swift J1727.8–1613
with the 10-s time resolution from the quasi-simultaneous observations of
NICER (1–10 keV), NuSTAR/FPMA (3–78 keV) and Insight-HXMT/LE (1–10 keV).

###  2.2 Data Reduction

####  2.2.1 NICER

NICER, a payload on the International Space Station (ISS), is dedicated to
timing and spectroscopic analysis at soft X-rays (0.2–12 keV; Gendreau et al.,
2016). During the 2023 outburst of Swift J1727.8–1613, NICER conducted
frequent observations. In this study, we analyze an observation with ObsID
6203980105, which was carried out almost simultaneously with NuSTAR and
Insight-HXMT observations. Since 2023 May 22, NICER has experienced a light
leak issue that results in increased background noise during orbit day
observations when solar light reaches the X-ray detectors. The regular
detector resets, known as undershoots, which are characteristic of a
functioning Focal Plane Module (FPM), occur more frequently due to the
presence of solar light and subsequently lead to degraded spectral resolution
and increased noise levels at lower
energies111https://heasarc.gsfc.nasa.gov/docs/nicer/analysis_threads/undershoot-
intro/ (see also Ingram et al., 2024). The data reduction is performed by
using the official software HEASOFT V6.33/NICERDAS package with the latest
CALDB xti20221001. We first reprocess the NICER data with the standard
pipeline processing tool nicerl2. In this step, the default maximum undershoot
rate criterion of 500 undershoots per second is used for the observation.
Subsequently, the light curves and energy spectrum are extracted using
nicerl3-lc and nicerl3-spect tools, respectively. The nibackgen3C50 model
(Remillard et al., 2022) is used to estimate the background corresponding to
the observation. Here, FPM 14 and 34 are always excluded, since they often
exhibit episodes of increased detector noise. The adopted energy range in the
spectral analysis is 1–10 keV, and a systematic error of 1 per cent is
incorporated for 1–4 keV energy band to reduce the effects from calibration
uncertainties at low energies.

####  2.2.2 NuSTAR

NuSTAR, the first mission to use focusing telescopes to image the sky in the
high-energy X-ray (3–79 keV) region of the electromagnetic spectrum, was
launched at 9 am PDT, June 13, 2012 (Harrison et al., 2013). The data are
processed using the NuSTAR Data Analysis Software (NuSTARDAS v2.1.2) with the
latest calibration data base (CALDB 20230613). At first, the standard pipeline
routine nupipeline is used for data reduction. Since Swift J1727.8–1613 was
very bright, an additional status expression STATUS ==
b0000xxx00xxxx000222https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/ is
applied in this step. Then, nuproducts is used to generate light curves,
spectra, response matrix, and ancillary response files, where source and
background events are extracted from a 160 arcsec circular region and a 130
arcsec source-free circular region, respectively. Finally, we group the
spectra via the grppha task to ensure a signal-to-noise ratio of at least five
for each bin.

####  2.2.3 Insight-HXMT

Insight-HXMT is the first Chinese X-ray astronomy satellite (Zhang et al.,
2014, 2020b), launched on June 15, 2017. The science payload of Insight-HXMT
allows for observations across a broad energy band (1–250 keV) using three
telescopes: the High Energy X-ray telescope (HE, composed of NaI/CsI, 20–250
keV), the Medium Energy X-ray telescope (ME, with a Si pin detector, 5–30
keV), and the Low Energy X-ray telescope (LE, using an SCD detector, 0.7–13
keV). For additional information about the Insight-HXMT mission, please refer
to Zhang et al. (2020b), Liu et al. (2020), Cao et al. (2020) and Chen et al.
(2020). The event data are extracted by using Insight-HXMT Data Analysis
Software v2.05, along with the current calibration model
v2.06333http://hxmtweb.ihep.ac.cn/software.jhtml and the standard Insight-HXMT
Data Reduction Guide v2.05444http://hxmtweb.ihep.ac.cn/SoftDoc.jhtml, and
filtered with the following series of criteria recommended by the Insight-HXMT
team: (1) the elevation angle (ELV) is greater than $10^{\circ}$; (2) the
geometric cutoff rigidity (COR) is greater than 8 GeV; (3) the pointing
position offset is less than $0.04^{\circ}$; and (4) the good time intervals
(GTIs) are at least 300 s away from the South Atlantic Anomaly (SAA). The
backgrounds are produced from blind detectors using the LEBKGMAP, MEBKGMAP,
and HEBKGMAP tools, version 2.0.9, based on the standard Insight-HXMT
background models (Liao et al., 2020a, b; Guo et al., 2020). The observations
of Swift J1727.8–1613 were affected by temporary calibration issues in the LE
data, impacting the spectral analysis at the soft energies (see Peng et al.,
2024). As a result, this study only utilizes the LE data for timing analysis,
and does not include spectral analysis with the LE data. Following the
recommendation of the Insight-HXMT calibration group, the ME and HE spectra
are rebinned as follows: (1) ME: Channels 0–1023 are grouped with a rebin
factor of 2; (2) HE: Channels 0–255 are grouped with a rebin factor of 2.
Additionally, considering the current accuracy of the instrument calibration,
we add 0.5 per cent, and 1 per cent systematic errors to the energy spectra
for ME and HE, respectively.

##  3 Analysis and Results

###  3.1 Timing Analysis

We first perform the timing analysis for these quasi-simultaneous
observations. The light curves with the 1/128-s time bin from the NICER and
Insight-HXMT observations are used to generate the PSDs, with a constant
Poisson noise level subtracted. As for the NuSTAR observation, we instead
calculate the co-spectrum between the two independent focal plane modules,
FPMA and FPMB (Bachetti et al., 2015), due to the NuSTAR dead time of
$\tau_{\rm d}\approx 2.5$ ms, which introduces instrumental features on the
Poisson noise in a PSD calculated in the standard way. The co-spectrum, which
is the real part of the cross-spectrum, does not include any Poisson noise
contribution. We then proceed to correct the co-spectrum for the suppression
of variability caused by the NuSTAR dead time using the simple formula
(Bachetti et al., 2015):

$\frac{\rm rms_{det}}{\rm rms_{in}}\approx\frac{1}{1+\tau_{\rm d}r_{\rm
in}}=\frac{r_{\rm det}}{r_{\rm in}}$ (1)

where $r_{\rm det}$ and $r_{\rm in}$ are the detected and intrinsic count
rate, respectively. For this observation, the ratio of detected to intrinsic
variability is $\rm{rms_{det}/rms_{in}=7.38\times 10^{-2}}$. As shown in
Figure 3 (a), all PSDs display QPOs with a strong fundamental and relatively
weaker second harmonic. The QPO fundamental frequency is detected at $\sim
0.88$ Hz, and the NuSTAR co-spectrum is in good agreement with the NICER and
Insight-HXMT data. As the total energy range of the NICER and Insight-HXMT
data covers the energy band of NuSTAR, and there are no dead time effects in
NICER and Insight-HXMT data, we concentrate solely on the NICER and Insight-
HXMT observations for the subsequent detailed timing analysis. We produce PSDs
in different energy bands and fit them with a sum of Lorentzian functions. The
fractional rms of QPO fundamental is calculated from the integral of the
corresponding Lorentzian function. Figure 3 (b) shows the QPO rms as a
function of photon energy from both NICER and Insight-HXMT data. It is evident
that the QPO amplitude increases with energy in the $\sim 1-150$ keV energy
range, with a hint of an additional enhancement seen above 30 keV. This
phenomenon is very different from that the QPO rms is observed to decrease
sightly with energy above $\sim 1$ keV and up to 200 keV in the LHS of MAXI
J1820+070 (Ma et al., 2021, 2023), but it is consistent with findings in other
sources, such as MAXI J1535–571 (Huang et al., 2018; Kong et al., 2020).

Figure 3: Panel (a): 1–10 keV power density spectrum for NICER (blue) and
Insight-HXMT/LE (orange), and 3–79 keV co-spectrum between NuSTAR FPMA and
FPMB (black). Panel (b): energy dependence of QPO fractional rms using NICER,
HXMT LE, ME and HE data. Panel (c): histogram of measured values of the phase
difference between QPO harmonics obtained using the HHT analysis for NICER
(blue) and HXMT/LE (orange) data sets (see the text for details). Panel (d):
reconstructed QPO waveform derived from the HHT phase-folding (scatters) and
inverse FT (dashed lines) methods for NICER (blue) and HXMT/LE (orange) data
sets.

QPO phase-resolved spectroscopy seeks to reveal spectral variations on QPO
timescales. This task is complicated by the ‘quasi-’ nature of the
oscillation, which hinders more direct approaches such as folding on a period.
To construct the QPO phase-resolved spectrum, we adopt the procedure outlined
by Shui et al. (2023a) that performing the Hilbert-Huang transform (HHT)
analysis. The HHT method, initially introduced by Huang et al. (1998) as an
adaptive data analysis technique, provides a powerful tool for studying
signals with non-stationary periodicity (Huang et al., 1998; Huang & Wu,
2008). This approach comprises two main components: mode decomposition and
Hilbert spectral analysis (HSA). The mode decomposition aims to decompose a
time series into several intrinsic mode functions (IMFs), while the HSA allows
obtaining both the frequency and phase functions for the desired IMFs, such as
the QPO component (e.g. Hu et al., 2014; Hsieh & Chou, 2020; Shui et al.,
2023a, 2024). We perform the HHT analysis on the light curves in the 1–10 keV
energy band from NICER and Insight-HXMT/LE observations, which is presented in
Appendix A. By employing the variational mode decomposition (VMD,
Dragomiretskiy & Zosso, 2014), the intrinsic variability of the fundamental
and the second harmonics of the 0.88-Hz QPO can be identified as the second
and third IMFs, respectively. Based on the Hilbert transform, we can obtain
the instantaneous phase functions of the QPO fundamental and harmonic
components. The phase difference between QPO fundamental and second harmonics,
as defined in Equation (3) of Ingram & van der Klis (2015), influences the
underlying waveform of the QPO. We employ the method proposed by Shui et al.
(2023a) to calculate the phase difference between the second and third IMFs
for each time bin. Figure 3 (c) presents the distribution of the phase
difference measured for each time bin. It is evident that, for both the NICER
and Insight-HXMT/LE data sets, the phase difference is clearly distributed
around a mean value, indicating that phases of the second and third IMFs are
corrected, and the results obtained from the NICER and Insight-HXMT/LE are
consistent. To determine the mean value of the phase difference, we use the
method described in Ingram & van der Klis (2015). This calculation yields the
mean phase difference of $\langle\psi\rangle/\pi=0.262\pm 0.002$ and
$\langle\psi\rangle/\pi=0.279\pm 0.002$ for NICER and Insight-HXMT/LE
observations, respectively. As proposed by Ingram & van der Klis (2015), the
mean value of the phase difference and the amplitudes of the QPO fundamental
and harmonic components can be used to construct the underlying waveform
through the inverse Fourier transform (FT). Additionally, the HHT phase
folding method can also give the QPO waveform. The waveforms obtained from
both the phase folding and inverse FT methods are presented in Figure 3 (d).
It is worth noting that, in comparison to the Fourier method, the HHT phase-
folding method presents both advantages and disadvantages. For example, as
mentioned by Ingram (2022) and Shui et al. (2023a), the contribution of the
harmonic component in the waveform derived from the HHT phase folding method
is suppressed. Nevertheless, it is important to note that while the current
Fourier method is limited to introducing only the QPO second harmonic, the HHT
phase-folding method has the capability to incorporate higher-order harmonics
(i.e., greater than twice the QPO frequency), albeit with some degree of
suppression. For the analysed QPO signals in our study, waveforms obtained
from the two methods are consistent, displaying a non-sinusoidal nature,
characterized by a slow rise and fast decay feature.

###  3.2 Energy Spectral Analysis

Table 2: Phase-averaged spectral fitting results for Swift J1727.8–1613 on MJD 60185. Model | Parameter | NICERa | Joint Viewb
---|---|---|---
TBABS |  $N_{\rm H}$ ($10^{21}{\rm cm^{-2}}$) | $2.08^{+0.02}_{-0.01}$ | $1.84^{+0.01}_{-0.01}$
DISKBB |  $T_{\rm in}$ (keV) | $0.41^{+0.01}_{-0.01}$ | $0.46^{+0.01}_{-0.01}$
|  $N_{\rm disk}^{\rm c}$ ($10^{4}$) | $9.78^{+2.06}_{-1.13}$ | $5.81^{+0.42}_{-0.62}$
RELXILL |  $i$ (∘) | $36.81^{+3.25}_{-2.34}$ | $37.39^{+1.79}_{-1.21}$
| $a*$ | $0.92^{+0.06}_{-0.09}$ | $0.97^{+0.02}_{-0.05}$
| $q^{\rm d}$ | 3 (f) | 3 (f)
|  $R_{\rm in}$ ($R_{\rm ISCO}$) | 1 (f) | 1 (f)
|  $R_{\rm out}$ ($R_{\rm g}$) | 1000 (f) | 1000 (f)
| $\Gamma_{1}$ | $1.49^{+0.01}_{-0.01}$ | $1.77^{+0.02}_{-0.02}$
| $R_{\rm f}$ | $0.08^{+0.02}_{-0.01}$ | $0.24^{+0.04}_{-0.03}$
|  $E_{\rm cut1}$ (keV) | 30 (f) | $56.9^{+2.6}_{-2.8}$
| $A_{\rm Fe}$ | 10 (f) | $3.55^{+0.51}_{-1.31}$
| $\log_{10}(\xi)^{\rm e}$ | $3.12^{+0.16}_{-0.24}$ | $3.24^{+0.06}_{-1.31}$
| $N_{\rm rel}^{\rm f}$ | $0.59^{+0.01}_{-0.01}$ | $0.31^{+0.01}_{-0.01}$
CUTOFFPL | $\Gamma_{2}$ | – | $0.98^{+0.02}_{-0.02}$
|  $E_{\rm cut2}$ (keV) | – | $11.5^{+0.2}_{-0.3}$
| $N_{\rm cut}^{\rm j}$ | – | $9.47^{+0.44}_{-0.43}$
CRABCORR | $\Delta\Gamma_{\rm NICER}$ | – | 0 (f)
| $N_{\rm NICER}$ | – | 1 (f)
| $\Delta\Gamma_{\rm ME}$ | – | $0.025^{+0.006}_{-0.006}$
| $N_{\rm ME}$ | – | $0.902^{+0.013}_{-0.012}$
| $\Delta\Gamma_{\rm HE}$ | – | $0.021^{+0.029}_{-0.016}$
| $N_{\rm HE}$ | – | $0.859^{+0.092}_{-0.048}$
| $\Delta\Gamma_{\rm FPMA}$ | – | $0.117^{+0.004}_{-0.004}$
| $N_{\rm FPMA}$ | – | $1.218^{+0.008}_{-0.007}$
| $\Delta\Gamma_{\rm FPMB}$ | – | $0.102^{+0.004}_{-0.004}$
| $N_{\rm FPMB}$ | – | $1.121^{+0.007}_{-0.008}$
| $\chi^{2}/{\rm d.o.f}$ | 164.13/162 | 2882.38/2742

Figure 4: The best-fit models (top rows) and data-to-model ratios (lower rows)
for different models. Panel (a) is for spectral analysis of NICER observation
in 1–10 keV energy range, while panel (b) is for joint spectral analysis of
NICER, NuSTAR and Insight-HXMT observations in 1–120 keV energy range.

The spectral analysis is performed using XPSEC v12.12.0 software package
(Arnaud, 1996). We first fit the phase-averaged energy spectrum from the NICER
observation. The initial spectral model is tbabs*powerlaw (Model 1), where
tbabs accounts for the galactic absorption, and powerlaw is a phenomenological
power-law model. Upon fitting with Model 1, a broad iron line around 7 keV and
a hump below 3 keV are clearly evident in the resulting data-to-model ratios
(see Figure 4). Consequently, we use Model 2: tbabs*(diskbb+relxill), to fit
the energy spectrum, where diskbb is a nonrelativistic disk-blackbody
component, and relixll is a reflection model from the relxill
family555https://github.com/thdauser/relxill/ (García et al., 2014; Dauser et
al., 2014). In the reflection model relxill, we assume the canonical power-law
emissivity profile, $\epsilon\propto r^{-3}$, for the thin disk. Since the
inner and outer radius of the disk ($R_{\rm in}$ and $R_{\rm out}$), iron
abundance ($A_{\rm Fe}$, in solar units) and $E_{\rm cut}$ cannot be well
constrained only with NICER data, we fix these parameters at the innermost
stable circular orbit, 1000 $R_{\rm g}$ ($R_{\rm g}=GM/c^{2}$), 10 and 30 keV,
respectively. Fitting the NICER energy spectrum with Model 2, we obtain a
$\chi^{2}/{\rm d.o.f}$ of 164.13/162. However, Figure 4 (a) shows that the
residuals are still strongly correlated below 4 keV. Specifically, it is
evident that the correlated residual features are a edge-like shape near
$\sim$2.2 keV and a Gaussian-like emission around 1.8 keV. These energies
correspond to specific features in NICER’s effective area versus energy, with
the 1.8 keV and 2.2 keV features attributed to silicon and gold, respectively
(see also Wang et al., 2020). Therefore, we posit that these remaining
features stem from systematic effects within NICER’s calibration. Introduction
of smaller systematic errors (e.g., 0.5%) leads to a higher reduced chi-square
value without significantly changes in the best-fitted parameter values.
Subsequently, we conduct a joint spectral fitting of NICER, NuSTAR and
Insight-HXMT data with Model 3: tbabs*crabcorr*(diskbb+relxill). The cross-
calibration among NICER, NuSTAR and Insight-HXMT is carried out by the model
crabcorr (Steiner et al., 2010; Wang et al., 2020; Zdziarski et al., 2022;
Peng et al., 2024; Yu et al., 2024b), which applies corrections to both the
slope of the power-law via the parameter $\Delta\Gamma$ and normalization,
$N$, by multiplying each model spectrum by a power-law. In this fitting, we
set $N$ and $\Delta\Gamma$ at fixed values of 1 and 0, respectively, for the
NICER spectrum. However, jointly fitting the NICER, NuSTAR and Insight-HXMT
spectra with Model 3 results in a poor goodness of fit ($\chi^{2}/{\rm
d.o.f}=4021.38/2745$) and shows significant residual features in the high-
energy band (see Figure 4 (b)). To address this, we follow Peng et al. (2024)
to add a cutoffpl component to the fitting model (Model 4):
tbabs*crabcorr*(diskbb+relxill+cutoffpl). This model modification leads to a
significant reduction in $\chi^{2}$ by 1139 with three extra free parameters,
resulting in $\chi^{2}/{\rm d.o.f}=2882.38/2742$. Furthermore, the data-to-
model ratios revealed no prominent residual features (see Figure 4 (b)). As
discussed in Peng et al. (2024), the origin of the additional cutoffpl
component remains elusive, as it can be used to account for both an excess at
$\sim$10–20 keV (as demonstrated in this analysis) and a hard tail above $\sim
70$ keV. This suggests a potential degeneracy between the extra cutoffpl
component and the incident continuum component of relxill. If this cutoffpl
component does indeed indicate an additional physical component and the
reflected emission truly originates from an individual component, different
hybrid jet/corona configurations could qualitatively explain why the cutoffpl
component is not being efficiently reflected (see Figure 7 of Peng et al.,
2024). However, it is also possible that the reflector is illuminated by both
components, but current spectroscopy is unable to distinguish between the
reflected spectra from two distinct illuminators. Additionally, the extra
cutoffpl component could potentially mimic the influence of a non-thermal
population of electrons which makes the Comptonised spectrum distinct from a
standard one. Additional details about the phase-averaged spectral fitting
with Model 4 can be found in Peng et al. (2024).After conducting the
aforementioned joint spectral analysis, we fix $E_{\rm cut}$ and $A_{\rm Fe}$
in the independent NICER spectral fitting to the best-fitting value from the
joint spectral fitting, and find no significant changes in the obtained
reduced chi-square and other free parameter values. Table 2 summarizes results
of independent spectral fitting of the NICER observation with Model 2 and
joint spectral fitting with Model 4. The uncertainties are reported at the 90%
confidence level by employing the Markov Chain Monte Carlo (MCMC) method using
the Goodman-Weare algorithm with 32 walkers and a total length of 50,000
(Goodman & Weare, 2010), and the initial 2000 elements are discarded as a
burn-in period.

Figure 5: Phase dependence of the spectral parameters and model fluxes
($F_{\rm diskbb}$, $F_{\rm relxill}$ and $F_{\rm cutoffpl}$) from the phase-
resolved spectral fitting of the NICER spectra (left panels) and joint NICER
and Insight-HXMT spectra (middle and right panels). The shown model fluxes are
in units of $10^{-8}\rm erg\ s^{-1}cm^{-2}$. Additionally, we include the
phase-folding QPO waveform in each panel as the black dashed line. The
statistical significance of each modulation is quoted in the corresponding
plot.

With the well-defined QPO phase given by the HHT analysis, we generate new
GTIs corresponding to ten distinct phase intervals for NICER and Insight-HXMT
observations, respectively. These new GTIs enable us extract spectra for each
phase bin to perform subsequent phase-resolved spectral analysis. Due to the
potential impact of NuSTAR dead time on the HHT analysis, we exclude the
NuSTAR data from the phase-resolved spectral analysis. We perform independent
spectral fitting with NICER observation with Model 2 and joint spectral
fitting with NICER and Insight-HXMT observations with Model 4, respectively.
In the phase-resolved spectral fittings, we fix the equivalent hydrogen column
($N_{\rm H}$), black hole spin parameter ($a_{*}$), disk inclination ($i$) and
ionization parameter ($\log\xi$) to the best-fit values obtained from the
phase-averaged spectral fittings. It is important to note that fixing the
ionization parameter ($\xi$) while the illuminating flux changes is not
entirely self-consistent, as $\xi$ should in principle increase when the
reflected flux increases (see e.g. García et al., 2014; Ingram et al., 2019;
Nathan et al., 2022). However, we find that the ionization parameter turns out
not sensitive to the phase-resolved spectral fitting and therefore is frozen
at its best-fitted value in the phase-averaged spectral fitting. For the joint
phase-resolved spectral fitting, we additionally fix the parameters
($\Delta\Gamma$ and $N$) from the crabcorr model to the best-fit values
obtained from the joint phase-averaged spectral fittings. Furthermore, we
employ cflux model in XSPEC to calculate unabsorbed flux contribution from
each component. Similar to the phase-averaged spectral analysis, we utilize
the MCMC method to compute the uncertainties. We find that the autocorrelation
length in the MCMC chain typically consists of 1000 elements, resulting in the
net number of independent samples in the parameter space being on the order of
$10^{4}$. To further assess the convergence of the MCMC chain, we compare the
one- and two-dimensional projections of the posterior distributions for each
parameter from the first and second halves of the chain, and find no
significant differences. In Appendix B, we provide the contour maps and
probability distributions for each free parameter in the joint phase-resolved
spectral analysis.

Figure 5 illustrates the phase dependence of the spectral parameters from the
phase-resolved spectral fittings of NICER spectra and joint NICER and Insight-
HXMT spectra. In these plots, we additionally include the QPO waveform in each
panel as the black dashed line. We employ an F-test for each of these
parameters to compare a fit with a constant model against the best-fitting
model, in order to evaluate its variation with QPO phase and determine its
statistical significance (see also Ingram & van der Klis, 2015). In the top-
right corner of each panel in Figure 5, we provide the statistical
significance with which each spectral parameter varies with QPO phase. It is
evident that both phase-resolved spectral analyses reveal that $T_{\rm in}$
exhibits modulations with the QPO phase ($>5\sigma$). Furthermore, both
results show substantial phase differences between the $T_{\rm in}$ and flux
modulations. Since the joint spectral analysis is performed within a much
boarder energy band (1–120 keV), we focus on the results from it. We observe
strong modulations ($>5\sigma$) in the photon indices of relxill
($\Gamma_{1}$) and cutoffpl ($\Gamma_{2}$), but these two parameters exhibit
opposite variation trends. The former lags the count rate modulation by $\sim
0.5$ QPO cycles, while the later shows a same modulation phase as the count
rate. Furthermore, phase modulations ($\sim 5\sigma$) are also observed in
reflection fraction ($R_{\rm f}$) and cut-off energy of cutoffpl ($E_{\rm
cut2}$). Regarding the model fluxes, we find that the fluxes of diskbb and
cutoffpl exhibit significant modulations ($>7\sigma$), with the later showing
the largest absolute variability amplitude of $\sim 7\times 10^{-8}\ {\rm erg\
cm^{-2}s^{-1}}$. Interestingly, a significant phase difference ($\sim 0.3-0.4$
QPO cycles) between modulations of $T_{\rm in}$ and the diskbb flux ($F_{\rm
disk}$) is observed. This indicates that $T_{\rm in}$ and $F_{\rm disk}$ do
not satisfy a positive correlation over the QPO cycle.

##  4 Discussion and Conclusion

In this study, we have performed broad-band phase-resolved spectroscopy of
type-C QPOs in the newly discovered BHXRB Swift J1727.8–1613, utilizing quasi-
simultaneous NICER and Insight-HXMT observations. Additionally, to improve the
constraints on the phase-averaged spectrum, we have included NuSTAR data in
the phase-averaged spectral analysis. The results show that over the QPO
period, the energy spectrum undergoes changes not only in normalization, but
also in spectral shape. Specifically, parameters such as the temperature of
the inner radius of the thin disk ($T_{\rm in}$), spectral indices
($\Gamma_{1}$ and $\Gamma_{2}$) and reflection fraction ($R_{\rm f}$) exhibit
significant modulations. Notably, while the non-thermal component dominates
the QPO variability, the disk-blackbody component also displays changes on the
QPO timescale. Furthermore, a significant phase difference ($\sim 0.3-0.4$ QPO
cycles) between modulations of $T_{\rm in}$ and the flux of disk-blackbody
component is observed.

Figures 5 demonstrates that the non-thermal component dominates the QPO
variability. These results are consistent with the previous findings that the
power-law component dominates the type-C QPO emission in BHXRBs (e.g.
Sobolewska & Życki, 2006; Axelsson et al., 2014; Axelsson & Done, 2016). Given
that the nonthermal emission arises from the Compton upscattering of disk
photons in a hot corona/jet base region (Sunyaev & Titarchuk, 1980; Titarchuk,
1994; Zdziarski et al., 1999; Markoff et al., 2005), our findings support
models such as the corona’s L-T precession model (Ingram et al., 2009),
precession of the jet base (Ma et al., 2021) and the vkompth model (Karpouzas
et al., 2020; Bellavita et al., 2022). Phase-resolved analysis of the joint
NICER and Insight-HXMT spectra reveals a significant modulation ($\sim
5\sigma$) in the reflection fraction, which represents the ratio of
Comptonized emission intensity illuminating the disk to the intensity directly
reaching the observer (Dauser et al., 2016). The observed variation in $R_{\rm
f}$ suggests changes in accretion geometry over the cycle, providing strong
evidence for the geometric origin of the QPO. Additionally, as the non-
spherical corona/jet base precesses, the optical depth of the corona with
respect to the observer ($\tau$) could vary over the precession period,
leading to modulations in the photon index of the spectrum (Shui et al.,
2023a). Figure 5 also shows strong modulations in both $\Gamma_{1}$ and
$\Gamma_{2}$, with the two parameters exhibiting opposite variation trends.
Peng et al. (2024) proposed that the presence of an additional hard component
indicates a joint jet-corona scenario in Swift J1727.8–1613, where the jet and
corona could be somehow coupled and precess around the BH spin axis together
(see e.g. Liska et al., 2018; Shui et al., 2024). The flux variation from the
corona is attributed to the geometric wobble of the corona, altering the
projection area with respect to the observer and modulating the observed X-ray
flux (Ingram et al., 2009; You et al., 2018; Shui et al., 2023b). In the case
of jet precession, the flux variation is due to the Doppler boosting effects
(Ma et al., 2021). In a hybrid corona/jet configuration to account for the
nonthermal emission, a jet relevant to $\Gamma_{1}$ is set on top of a
radially extended corona that corresponds to $\Gamma_{2}$ (see Peng et al.,
2024). At the trough phases of the flux modulation, the corona precesses to an
edge-on case and the harder jet base can be blocked by the surrounding corona,
which can result in a softer spectral shape of the jet emission (i.e. larger
$\Gamma_{1}$). On the other hand, a smaller $\Gamma_{1}$ is expected at the
peak phases of the flux modulation when the corona is viewed face-on and the
harder jet base becomes visible again. As suggested in Shui et al. (2023a),
modulations in $\Gamma_{2}$ may be relevant to the changes of the optical
depth of the corona: a harder spectrum shall be observed in an edge-on corona
with a larger optical depth. Accordingly, although the fluxes from the corona
and jet regions could vary synchronously, the modulations of the photon
indices of the two spectral components can have different trends.

Figure 6: The disk flux vs. disk temperature $T_{\rm in}$ over the QPO cycle
from the joint phase-resolved spectral fitting with Model 4. The dashed red
line represents the relation of $F_{\rm disk}\propto T^{4}_{\rm in}$, which is
predicted by the standard Shakura-Sunyaev disk model with a stable inner
radius.

In terms of the variations in the disk component, both $T_{\rm in}$ and the
disk flux show changes over the QPO cycle. We observe that $T_{\rm in}$
precedes the varying total flux by $\sim 0.5$ QPO cycles. And more
importantly, we find a significant phase difference ($\sim 0.3-0.5$ QPO
cycles) between modulations of $T_{\rm in}$ and the flux of disk-blackbody
component. In Figure 6, we plot the disk flux versus $T_{\rm in}$ over the QPO
cycle from the joint phase-resolved spectral fitting. This plot indicates
that, over the QPO cycle, the disk emission completely deviates from the
behavior of $F_{\rm disk}\tilde{\propto}T^{4}_{\rm in}$, which is predicted by
the standard Shakura-Sunyaev disk with a stable inner radius. The inner radius
of the disk, $R_{\rm in}$, typically changes on a viscosity timescale, $t_{\rm
visc}$, which is given by

$\displaystyle t_{\rm vsic}$
$\displaystyle\sim\frac{1}{\alpha\Omega}\left(\frac{R}{H}\right)^{2}$
$\displaystyle\approx 50{\rm
s}\left(\frac{M}{10M_{\odot}}\right)\left(\frac{R}{R_{\rm
g}}\right)^{3/2}\left(\frac{\alpha}{0.01}\right)^{-1}\left(\frac{H/R}{0.01}\right)^{-2},$
(2)

where $\alpha$ is the viscosity parameter, $\Omega$ is the Kepler angular
velocity, $R$ is the radius in the disk, $H$ is the vertical scale height of
the disk and $M$ is the mass of the black hole (see e.g. Frank et al., 2002).
Considering that the parameter combination including a black hole mass of
$10M_{\odot}$, $\alpha=0.01$, $H/R=0.01$ and $R=(10-20)R_{\rm g}$, the
viscosity timescale is expected to be $\sim 10^{2}-10^{3}$s, which is much
longer than the QPO period ($\sim 1$ s). This suggests that the inner radius
of the disk is unlikely to undergo significant changes over the QPO cycle.
Consequently, we propose that the observed significant phase difference
between $T_{\rm in}$ and $F_{\rm disk}$, which causes the $F_{\rm
disk}$-$T_{\rm in}$ relation to deviate from the expected behavior of $F_{\rm
disk}\tilde{\propto}T^{4}_{\rm in}$, could be indicative of variations in the
accretion geometry over the QPO cycle. In the context of corona/jet precession
model, the thin disk is held stationary by viscosity (Bardeen & Petterson,
1975). However, the precessing corona and/or jet could obscure different disk
azimuths at different precession phases, leading to the variable inner disk
emission (You et al., 2018; Nathan et al., 2022). Additionally, a precessing
corona/jet base could illuminate different azimuthal regions of the disk,
thereby heating the disk atmosphere and causing variations in the disk
temperature (Stevens & Uttley, 2016; Nathan et al., 2022). Considering quasi-
periodic heating of the approaching and receding sides of the disc by a
precessing Comptonizing region, one might anticipate a typical phase
difference of roughly $0.25$ QPO cycles between the variation in apparent
$T_{\rm in}$ (and also the thermal emission) and Comptonized emission reaching
the observer directly (Stevens & Uttley, 2016). This is because the
approaching side of the disk is consistently illuminated by the Comptonizing
region when the normal of the Comptonizing region aligns perpendicularly to
the observer’s line of sight (see Figure 12 of Stevens & Uttley, 2016).
However, we would also anticipate that the phase difference should be
influenced by the GR effects (e.g. light bending effects), leading to slight
deviations from the expected value. Moreover, the situation becomes more
complicated when considering the reflection component, where the phase of
modulations in apparent temperature and disk flux with respect to the
reflected flux would be determined by the azimuthal angle of the observer. An
alternative possibility is the scenario proposed by Ma et al. (2023), in which
the jet and inner disk ring precess together. According to this model, the
inner region of the jet and the innermost part of the accretion disk are
somehow coupled, leading to a situation where the precession of the disk
induces a corresponding precession of the jet. In this case, the peak phases
of the variation in the thermal and nonthermal emissions are expected to
roughly coincide. As the geometrical alignment between the jet and the inner
disk remains constant, the illumination pattern also remains stable throughout
the precession cycle. Consequently, the effective temperature ($T_{\rm eff}$)
of the accretion disk is expected to remain constant. It is important to note
that the fitted peak temperature ($T_{\rm in}$) may be affected by electron
scattering and general relativistic (GR) effects, and can be expressed as:

$T_{\rm in}=f_{\rm GR}\left(i,a_{*}\right)f_{\rm col}T_{\rm eff},$ (3)

where $f_{\rm GR}$ represents the fractional change of the fitted color
temperature due to the GR effects, $i$ is the inclination angle, $a_{*}$ is
the black hole spin, and $f_{\rm col}$ denotes the hardening factor (Zhang et
al., 1997). Numerical simulations suggest a canonical value of $f_{\rm
col}\sim 1.6-1.8$ (Shimura & Takahara, 1995; Davis et al., 2005; Shafee et
al., 2006), whereas $f_{\rm GR}$ strongly depends on the inclination. As
demonstrated in Cunningham (1975) and Zhang et al. (1997), the GR effects
cause the spectrum to be redshifted at small inclinations, but blueshifted at
large inclinations. This indicates that the apparent temperature could display
a positive correlation with the inclination. However, for the accretion disk
flux over the precession period, it is primarily influenced by the projected
area, and is therefore expected to be negatively correlated with the
inclination. Consequently, a large phase difference (roughly 0.5 QPO cycles)
between the $T_{\rm in}$ variation and the $F_{\rm disk}$ variation could be
potentially observed. Furthermore, $f_{\rm col}$ may also vary throughout the
precession cycle, due to changes in the optical depth of the electron
scattering with the inclination. This effect could also result in a higher
color temperature when the disk precesses to an edge-on orientation.
Considering the potential presence of a hybrid corona/jet configuration in
this source, our findings may suggest a scenario in which the corona, inner
disk, and jet are coupled and precess together. It is worth noting that no
significant modulations in the polarization degree and polarization angle over
the QPO cycle were observed through the phase-resolved polarimetric analysis
(Zhao et al., 2024). If the QPO is produced by the precession of the inner hot
flow, findings from the QPO phase-resolved polarimetry suggest high
inclinations of the source ($\gtrsim 60^{\circ}$, see Ingram et al., 2015). To
conduct a more precise phase-resolved analysis for a better comparison with
theoretical predictions, obtaining simultaneous polarization, high-statistics
timing, and broadband spectroscopy data is crucial. This can be accomplished
with the upcoming mission, the enhanced X-ray Timing and Polarimetry
Observatory (Zhang et al., 2019).

In summary, we have performed a detailed phase-resolved analysis of type-C
QPOs from the newly discovered BHXRB Swift J1727.8–1613. Our findings indicate
that both the source flux and spectral shape exhibit variations over the QPO
cycle. In particular, the variability in reflection fraction and disk
emission, along with a significant phase difference of the variation in the
disk temperature compared to the variation in the disk flux, align with the
concept that the QPO variability is not primarily intrinsic to the accretion
flow but geometric.

We are grateful to the anonymous referees for constructive comments that
helped us improve this paper. This research has made use of data obtained from
the High Energy Astrophysics Science Archive Research Center (HEASARC),
provided by NASA’s Goddard Space Flight Center, and the Insight-HXMT mission,
a project funded by China National Space Administration (CNSA) and the Chinese
Academy of Sciences (CAS). This work is supported by the National Key R&D
Program of China (2021YFA0718500) and the National Natural Science Foundation
of China under grants, 12333007, 12173103, 12027803, U2038101 and U1938103.
This work is partially supported by International Partnership Program of
Chinese Academy of Sciences (Grant No.113111KYSB20190020). L. D. Kong is
grateful for the financial support provided by the Sino-German (CSC-DAAD)
Postdoc Scholarship Program (57251553). P. J. Wang is grateful for the
financial support provided by the Sino-German (CSC-DAAD) Postdoc Scholarship
Program (57678375).

##  Appendix A Hilbert-Huang Transform Method

To obtain the QPO waveform and extract the QPO phase-resolved spectra, we
perform the HHT analysis on the NICER and Insight-HXMT data. Figure 7 presents
a representative HHT analysis of Insight-HXMT data. For a comprehensive review
of the method, see Shui et al. (2023a, 2024).

Figure 7: Hilbert–Huang transform analysis of a representative example of a 25
s long light curve with a 0.05 s time resolution of LE in the energy range of
1–10 keV. (a) Representative example of a 25 s long light curve and
corresponding five IMFs obtained from VMD algorithm. (b) Power density spectra
of the original LE light curve, as well as the second and third IMFs. (c) The
corresponding phase function, instantaneous frequency, and amplitude of the
QPO (the second) IMF, obtained from the Hilbert transform. In panel (b), we
additionally include the fitting model and best-fit Lorentzian functions of
the QPO fundamental and second harmonics in the plot.

##  Appendix B MCMC Parameter Probability Distributions

This appendix contains corner plots of spectral parameters from an example
MCMC analysis. We use the Goodman–Weare algorithm with 32 walkers and a total
length of 50,000 to perform the MCMC analysis, and the initial 2000 elements
are discarded as the burn-in period during which the chain reaches its
stationary state. In Figure 8, we compare the one- and two-dimensional
projections of the posterior distributions for each parameter from the first
and second halves of the chain to test the convergence. The contour maps and
probability distributions are plotted using the corner package (Foreman-
Mackey, 2016).

Figure 8: One- and two-dimensional projections of the posterior probability
distributions, and the 0.16, 0.5, and 0.84 quantile contours derived from the
MCMC analysis for each free spectral parameter, from the joint spectral
fitting of NICER and Insight-HXMT data. To test the convergence, we compare
the one- and two-dimensional projections of the posterior distributions from
the first (blue) and second (orange) halves of the chain. This illustration
corresponds to the spectral fitting of the QPO peak phase.

## References

  * Arnaud (1996) Arnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes, 17
  * Axelsson & Done (2016) Axelsson, M., & Done, C. 2016, MNRAS, 458, 1778, doi: 10.1093/mnras/stw464
  * Axelsson et al. (2014) Axelsson, M., Done, C., & Hjalmarsdotter, L. 2014, MNRAS, 438, 657, doi: 10.1093/mnras/stt2236
  * Bachetti et al. (2015) Bachetti, M., Harrison, F. A., Cook, R., et al. 2015, ApJ, 800, 109, doi: 10.1088/0004-637X/800/2/109
  * Bardeen & Petterson (1975) Bardeen, J. M., & Petterson, J. A. 1975, ApJ, 195, L65, doi: 10.1086/181711
  * Bellavita et al. (2022) Bellavita, C., García, F., Méndez, M., & Karpouzas, K. 2022, MNRAS, 515, 2099, doi: 10.1093/mnras/stac1922
  * Belloni et al. (2005) Belloni, T., Homan, J., Casella, P., et al. 2005, A&A, 440, 207, doi: 10.1051/0004-6361:20042457
  * Cao et al. (2020) Cao, X., Jiang, W., Meng, B., et al. 2020, Science China Physics, Mechanics, and Astronomy, 63, 249504, doi: 10.1007/s11433-019-1506-1
  * Casella et al. (2005) Casella, P., Belloni, T., & Stella, L. 2005, ApJ, 629, 403, doi: 10.1086/431174
  * Castro-Tirado et al. (2023) Castro-Tirado, A. J., Sanchez-Ramirez, R., Caballero-Garcia, M. D., et al. 2023, The Astronomer’s Telegram, 16208, 1
  * Chen et al. (2020) Chen, Y., Cui, W., Li, W., et al. 2020, Science China Physics, Mechanics, and Astronomy, 63, 249505, doi: 10.1007/s11433-019-1469-5
  * Cunningham (1975) Cunningham, C. T. 1975, ApJ, 202, 788, doi: 10.1086/154033
  * Dauser et al. (2014) Dauser, T., Garcia, J., Parker, M. L., Fabian, A. C., & Wilms, J. 2014, MNRAS, 444, L100, doi: 10.1093/mnrasl/slu125
  * Dauser et al. (2016) Dauser, T., García, J., Walton, D. J., et al. 2016, A&A, 590, A76, doi: 10.1051/0004-6361/201628135
  * Dauser et al. (2010) Dauser, T., Wilms, J., Reynolds, C. S., & Brenneman, L. W. 2010, MNRAS, 409, 1534, doi: 10.1111/j.1365-2966.2010.17393.x
  * Davis et al. (2005) Davis, S. W., Blaes, O. M., Hubeny, I., & Turner, N. J. 2005, ApJ, 621, 372, doi: 10.1086/427278
  * de Ruiter et al. (2019) de Ruiter, I., van den Eijnden, J., Ingram, A., & Uttley, P. 2019, MNRAS, 485, 3834, doi: 10.1093/mnras/stz665
  * Done et al. (2007) Done, C., Gierliński, M., & Kubota, A. 2007, A&A Rev., 15, 1, doi: 10.1007/s00159-007-0006-1
  * Dovciak et al. (2023) Dovciak, M., Ratheesh, A., Tennant, A., & Ma, G. 2023, The Astronomer’s Telegram, 16242, 1
  * Draghis et al. (2023) Draghis, P. A., Miller, J. M., Homan, J., et al. 2023, The Astronomer’s Telegram, 16219, 1
  * Dragomiretskiy & Zosso (2014) Dragomiretskiy, K., & Zosso, D. 2014, IEEE Transactions on Signal Processing, 62, 531, doi: 10.1109/TSP.2013.2288675
  * Foreman-Mackey (2016) Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24, doi: 10.21105/joss.00024
  * Frank et al. (2002) Frank, J., King, A., & Raine, D. J. 2002, Accretion Power in Astrophysics: Third Edition
  * García et al. (2014) García, J., Dauser, T., Lohfink, A., et al. 2014, ApJ, 782, 76, doi: 10.1088/0004-637X/782/2/76
  * Gendreau et al. (2016) Gendreau, K. C., Arzoumanian, Z., Adkins, P. W., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051H, doi: 10.1117/12.2231304
  * Goodman & Weare (2010) Goodman, J., & Weare, J. 2010, Communications in Applied Mathematics and Computational Science, 5, 65, doi: 10.2140/camcos.2010.5.65
  * Guo et al. (2020) Guo, C.-C., Liao, J.-Y., Zhang, S., et al. 2020, Journal of High Energy Astrophysics, 27, 44, doi: 10.1016/j.jheap.2020.02.008
  * Harrison et al. (2013) Harrison, F. A., Craig, W. W., Christensen, F. E., et al. 2013, ApJ, 770, 103, doi: 10.1088/0004-637X/770/2/103
  * Homan & Belloni (2005) Homan, J., & Belloni, T. 2005, Ap&SS, 300, 107, doi: 10.1007/s10509-005-1197-4
  * Hsieh & Chou (2020) Hsieh, H.-E., & Chou, Y. 2020, ApJ, 900, 116, doi: 10.3847/1538-4357/abacbd
  * Hu et al. (2014) Hu, C.-P., Chou, Y., Yang, T.-C., & Su, Y.-H. 2014, ApJ, 788, 31, doi: 10.1088/0004-637X/788/1/31
  * Huang & Wu (2008) Huang, N. E., & Wu, Z. 2008, Reviews of Geophysics, 46, RG2006, doi: 10.1029/2007RG000228
  * Huang et al. (1998) Huang, N. E., Shen, Z., Long, S. R., et al. 1998, Proceedings of the Royal Society of London Series A, 454, 903, doi: 10.1098/rspa.1998.0193
  * Huang et al. (2018) Huang, Y., Qu, J. L., Zhang, S. N., et al. 2018, ApJ, 866, 122, doi: 10.3847/1538-4357/aade4c
  * Ingram (2022) Ingram, A. 2022, in Handbook of X-ray and Gamma-ray Astrophysics, 113, doi: 10.1007/978-981-16-4544-0_139-1
  * Ingram et al. (2009) Ingram, A., Done, C., & Fragile, P. C. 2009, MNRAS, 397, L101, doi: 10.1111/j.1745-3933.2009.00693.x
  * Ingram et al. (2015) Ingram, A., Maccarone, T. J., Poutanen, J., & Krawczynski, H. 2015, ApJ, 807, 53, doi: 10.1088/0004-637X/807/1/53
  * Ingram et al. (2019) Ingram, A., Mastroserio, G., Dauser, T., et al. 2019, MNRAS, 488, 324, doi: 10.1093/mnras/stz1720
  * Ingram & van der Klis (2015) Ingram, A., & van der Klis, M. 2015, MNRAS, 446, 3516, doi: 10.1093/mnras/stu2373
  * Ingram et al. (2017) Ingram, A., van der Klis, M., Middleton, M., Altamirano, D., & Uttley, P. 2017, MNRAS, 464, 2979, doi: 10.1093/mnras/stw2581
  * Ingram et al. (2016) Ingram, A., van der Klis, M., Middleton, M., et al. 2016, MNRAS, 461, 1967, doi: 10.1093/mnras/stw1245
  * Ingram et al. (2024) Ingram, A., Bollemeijer, N., Veledina, A., et al. 2024, ApJ, 968, 76, doi: 10.3847/1538-4357/ad3faf
  * Ingram & Motta (2019) Ingram, A. R., & Motta, S. E. 2019, New A Rev., 85, 101524, doi: 10.1016/j.newar.2020.101524
  * Karpouzas et al. (2020) Karpouzas, K., Méndez, M., Ribeiro, E. M., et al. 2020, MNRAS, 492, 1399, doi: 10.1093/mnras/stz3502
  * Kato (1990) Kato, S. 1990, PASJ, 42, 99
  * Katoch et al. (2023) Katoch, T., Antia, H. M., Nandi, A., & Shah, P. 2023, The Astronomer’s Telegram, 16235, 1
  * Kong et al. (2020) Kong, L. D., Zhang, S., Chen, Y. P., et al. 2020, Journal of High Energy Astrophysics, 25, 29, doi: 10.1016/j.jheap.2020.01.003
  * Liao et al. (2020a) Liao, J.-Y., Zhang, S., Lu, X.-F., et al. 2020a, Journal of High Energy Astrophysics, 27, 14, doi: 10.1016/j.jheap.2020.04.002
  * Liao et al. (2020b) Liao, J.-Y., Zhang, S., Chen, Y., et al. 2020b, Journal of High Energy Astrophysics, 27, 24, doi: 10.1016/j.jheap.2020.02.010
  * Liska et al. (2018) Liska, M., Hesp, C., Tchekhovskoy, A., et al. 2018, MNRAS, 474, L81, doi: 10.1093/mnrasl/slx174
  * Liu et al. (2020) Liu, C., Zhang, Y., Li, X., et al. 2020, Science China Physics, Mechanics, and Astronomy, 63, 249503, doi: 10.1007/s11433-019-1486-x
  * Lynden-Bell & Pringle (1974) Lynden-Bell, D., & Pringle, J. E. 1974, MNRAS, 168, 603, doi: 10.1093/mnras/168.3.603
  * Ma et al. (2021) Ma, X., Tao, L., Zhang, S.-N., et al. 2021, Nature Astronomy, 5, 94, doi: 10.1038/s41550-020-1192-2
  * Ma et al. (2023) Ma, X., Zhang, L., Tao, L., et al. 2023, ApJ, 948, 116, doi: 10.3847/1538-4357/acc4c3
  * Markoff et al. (2005) Markoff, S., Nowak, M. A., & Wilms, J. 2005, ApJ, 635, 1203, doi: 10.1086/497628
  * Mereminskiy et al. (2023) Mereminskiy, I., Lutovinov, A., Molkov, S., et al. 2023, arXiv e-prints, arXiv:2310.06697, doi: 10.48550/arXiv.2310.06697
  * Miller-Jones et al. (2023) Miller-Jones, J. C. A., Sivakoff, G. R., Bahramian, A., & Russell, T. D. 2023, The Astronomer’s Telegram, 16211, 1
  * Molteni et al. (1996) Molteni, D., Sponholz, H., & Chakrabarti, S. K. 1996, ApJ, 457, 805, doi: 10.1086/176775
  * Motta et al. (2015) Motta, S. E., Casella, P., Henze, M., et al. 2015, MNRAS, 447, 2059, doi: 10.1093/mnras/stu2579
  * Nakajima et al. (2023) Nakajima, M., Negoro, H., Serino, M., et al. 2023, The Astronomer’s Telegram, 16206, 1
  * Nathan et al. (2022) Nathan, E., Ingram, A., Homan, J., et al. 2022, MNRAS, 511, 255, doi: 10.1093/mnras/stab3803
  * Negoro et al. (2023) Negoro, H., Serino, M., Nakajima, M., et al. 2023, The Astronomer’s Telegram, 16205, 1
  * Page et al. (2023) Page, K. L., Dichiara, S., Gropp, J. D., et al. 2023, GRB Coordinates Network, 34537, 1
  * Peng et al. (2024) Peng, J.-Q., Zhang, S., Shui, Q.-C., et al. 2024, ApJ, 960, L17, doi: 10.3847/2041-8213/ad17ca
  * Remillard & McClintock (2006) Remillard, R. A., & McClintock, J. E. 2006, ARA&A, 44, 49, doi: 10.1146/annurev.astro.44.051905.092532
  * Remillard et al. (2022) Remillard, R. A., Loewenstein, M., Steiner, J. F., et al. 2022, AJ, 163, 130, doi: 10.3847/1538-3881/ac4ae6
  * Ross & Fabian (2005) Ross, R. R., & Fabian, A. C. 2005, MNRAS, 358, 211, doi: 10.1111/j.1365-2966.2005.08797.x
  * Shafee et al. (2006) Shafee, R., McClintock, J. E., Narayan, R., et al. 2006, ApJ, 636, L113, doi: 10.1086/498938
  * Shakura & Sunyaev (1973) Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
  * Shimura & Takahara (1995) Shimura, T., & Takahara, F. 1995, ApJ, 445, 780, doi: 10.1086/175740
  * Shui et al. (2023a) Shui, Q. C., Zhang, S., Zhang, S. N., et al. 2023a, ApJ, 957, 84, doi: 10.3847/1538-4357/acfc42
  * Shui et al. (2023b) Shui, Q. C., Zhang, S., Chen, Y. P., et al. 2023b, ApJ, 943, 165, doi: 10.3847/1538-4357/aca7b8
  * Shui et al. (2024) Shui, Q. C., Zhang, S., Zhang, S. N., et al. 2024, ApJ, 965, L7, doi: 10.3847/2041-8213/ad374d
  * Sobolewska & Życki (2006) Sobolewska, M. A., & Życki, P. T. 2006, MNRAS, 370, 405, doi: 10.1111/j.1365-2966.2006.10489.x
  * Steiner et al. (2010) Steiner, J. F., McClintock, J. E., Remillard, R. A., et al. 2010, ApJ, 718, L117, doi: 10.1088/2041-8205/718/2/L117
  * Stella et al. (1999) Stella, L., Vietri, M., & Morsink, S. M. 1999, ApJ, 524, L63, doi: 10.1086/312291
  * Stevens & Uttley (2016) Stevens, A. L., & Uttley, P. 2016, MNRAS, 460, 2796, doi: 10.1093/mnras/stw1093
  * Sunyaev & Titarchuk (1980) Sunyaev, R. A., & Titarchuk, L. G. 1980, A&A, 86, 121
  * Tagger & Pellat (1999) Tagger, M., & Pellat, R. 1999, A&A, 349, 1003
  * Titarchuk (1994) Titarchuk, L. 1994, ApJ, 434, 570, doi: 10.1086/174760
  * van den Eijnden et al. (2017) van den Eijnden, J., Ingram, A., Uttley, P., et al. 2017, MNRAS, 464, 2643, doi: 10.1093/mnras/stw2634
  * van der Klis (1989) van der Klis, M. 1989, ARA&A, 27, 517, doi: 10.1146/annurev.aa.27.090189.002505
  * Veledina et al. (2023) Veledina, A., Muleri, F., Dovčiak, M., et al. 2023, ApJ, 958, L16, doi: 10.3847/2041-8213/ad0781
  * Wagoner (1999) Wagoner, R. V. 1999, Phys. Rep., 311, 259, doi: 10.1016/S0370-1573(98)00104-5
  * Wang et al. (2020) Wang, J., Kara, E., Steiner, J. F., et al. 2020, ApJ, 899, 44, doi: 10.3847/1538-4357/ab9ec3
  * Wijnands et al. (1999) Wijnands, R., Homan, J., & van der Klis, M. 1999, ApJ, 526, L33, doi: 10.1086/312365
  * You et al. (2018) You, B., Bursa, M., & Życki, P. T. 2018, ApJ, 858, 82, doi: 10.3847/1538-4357/aabd33
  * You et al. (2021) You, B., Tuo, Y., Li, C., et al. 2021, Nature Communications, 12, 1025, doi: 10.1038/s41467-021-21169-5
  * Yu et al. (2024a) Yu, W., Bu, Q.-C., Zhang, S.-N., et al. 2024a, MNRAS, 529, 4624, doi: 10.1093/mnras/stae835
  * Yu et al. (2024b) Yu, Z. L., Zhang, S., Chen, Y. P., et al. 2024b, MNRAS, 527, 8029, doi: 10.1093/mnras/stad3696
  * Zdziarski et al. (1996) Zdziarski, A. A., Johnson, W. N., & Magdziarz, P. 1996, MNRAS, 283, 193, doi: 10.1093/mnras/283.1.193
  * Zdziarski et al. (1999) Zdziarski, A. A., Lubiński, P., & Smith, D. A. 1999, MNRAS, 303, L11, doi: 10.1046/j.1365-8711.1999.02343.x
  * Zdziarski et al. (2022) Zdziarski, A. A., You, B., Szanecki, M., Li, X.-B., & Ge, M. 2022, ApJ, 928, 11, doi: 10.3847/1538-4357/ac54a7
  * Zhang et al. (2017) Zhang, L., Wang, Y., Méndez, M., et al. 2017, ApJ, 845, 143, doi: 10.3847/1538-4357/aa8138
  * Zhang et al. (2020a) Zhang, L., Méndez, M., Altamirano, D., et al. 2020a, MNRAS, 494, 1375, doi: 10.1093/mnras/staa797
  * Zhang et al. (2014) Zhang, S., Lu, F. J., Zhang, S. N., & Li, T. P. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9144, Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, ed. T. Takahashi, J.-W. A. den Herder, & M. Bautz, 914421, doi: 10.1117/12.2054144
  * Zhang et al. (2019) Zhang, S., Santangelo, A., Feroci, M., et al. 2019, Science China Physics, Mechanics, and Astronomy, 62, 29502, doi: 10.1007/s11433-018-9309-2
  * Zhang et al. (1997) Zhang, S. N., Cui, W., & Chen, W. 1997, ApJ, 482, L155, doi: 10.1086/310705
  * Zhang et al. (2020b) Zhang, S.-N., Li, T., Lu, F., et al. 2020b, Science China Physics, Mechanics, and Astronomy, 63, 249502, doi: 10.1007/s11433-019-1432-6
  * Zhao et al. (2024) Zhao, Q.-C., Tao, L., Li, H.-C., et al. 2024, ApJ, 961, L42, doi: 10.3847/2041-8213/ad1e6c