# First detection of Soft-lag in GRS 1915+105 at HFQPO using AstroSat
observations

Prajjwal Majumder1 , Broja G. Dutta1, Anuj Nandi2
1Rishi Bankim Chandra College, Naihati, West Bengal,743165, India
2Space Astronomy Group, ISITE Campus, U. R. Rao Satellite Centre, Outer Ring
Road, Marathahalli, Bangalore, 560037, India E-mail:
<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Galactic black hole GRS 1915+105 exhibits generic High-Frequency Quasi-
periodic Oscillations (HFQPOs) at $\sim$ 67 Hz only during the radio-quiet
‘softer’ variability classes. We present the time-lag properties associated
with HFQPOs in the wide energy band (3$-$60 keV) using all AstroSat
observations. For the first time, we detect soft-lag of 6$-$25 keV band w.r.t
3$-$6 keV band for all ‘softer’ variability classes ($\delta$, $\omega$,
$\kappa$ and $\gamma$). Moreover, our findings reveal that soft-lag increases
gradually with the energy of the photons. These features are entirely opposite
to the previous report of hard-lag obtained with the RXTE observations. The
energy-dependent time-lag study exhibits a maximum soft-lag of $\sim$ 3 ms and
$\sim$ 2.5 ms for the $\delta$ and $\omega$ classes respectively, whereas the
$\kappa$ and $\gamma$ classes both exhibit a maximum soft-lag of $\sim$ 2.1
ms. We find a coherent lag-energy correlation for all four variability
classes, where the amplitude of soft-lag increases with energy and becomes
maximum at $\sim$ 18 keV. We interpret this observed soft-lag as the
reflection of hard photons in the ‘cooler’ accretion disc. A generic lag-rms
correlation implies that the soft-lag increases with the rms amplitude of the
HFQPO. The wideband (0.7$-$50 keV) spectral study suggests a high value of the
optical depth ($\tau$ $\sim$ 6.90$-$12.55) of the Comptonized medium and the
magnitude of the soft-lag increases linearly with the increase in optical
depth ($\tau$). We explain the observed time-lag features at the HFQPOs in the
context of a possible accretion disc scenario.

###### keywords:

accretion, accretion discs $-$ black hole physics $-$ X-rays: binaries $-$
radiation: dynamics $-$ stars: individual: GRS 1915+105

††pubyear: 2023††pagerange: First detection of Soft-lag in GRS 1915+105 at
HFQPO using AstroSat observations–B

##  1 Introduction

X-ray binaries (XRBs) often exhibit excess power in certain frequencies called
Quasi-periodic Oscillations (QPOs) in its power density spectrum (PDS) (van
der Klis et al., 1985). In general, the QPOs in black hole XRBs (BH-XRBs)
appear in the range of 0.1$-$450 Hz. QPOs can be classified depending on the
frequency range in two general categories, Low-frequency QPO (LFQPO) and High-
frequency QPO (HFQPO) with centroid frequency < 40 Hz and 40$-$450 Hz
respectively (Remillard & McClintock, 2006). LFQPOs are observed commonly in
many BH-XRB systems, whereas HFQPOs are detected in a few sources. HFQPOs are
important as their frequency is in the range expected for its Keplerian motion
near the black hole (Stella & Vietri, 1998; Merloni et al., 1999; Rebusco,
2008; Stefanov, 2014) and it might also be generated due to the oscillations
of the compact corona (Majumder et al., 2022, and references therein). HFQPOs
were observed in a few sources such as GRS 1915+105 (Morgan et al., 1997;
Belloni & Altamirano, 2013; Belloni et al., 2019; Sreehari et al., 2020;
Majumder et al., 2022), GRO J1655-40 (Remillard et al., 1999; Strohmayer,
2001a), XTE J1550-564 (Miller et al., 2001), H 1743-322 (Homan et al., 2005;
Remillard et al., 2006), IGR J17091-3624 (Altamirano & Belloni, 2012). In some
cases, simultaneous observations of HFQPOs in $\sim$ 3:2 frequency ratio were
observed in GRO J1655-40, XTE J1550-564, GRS 1915+105 and H 1743-322. GRS
1915+105 has also shown a second pair of HFQPO which are not in a 3:2 ratio
(41 Hz and 67 Hz, Morgan et al., 1997; Strohmayer, 2001b). These HFQPOs are of
further interest because their centroid frequency does not drift very much in
response to considerable luminosity changes. This is an important difference
between the HFQPOs of black hole binary and the variable kHz QPOs of neutron
stars (Remillard & McClintock, 2006). Often the presence of prominent HFQPOs
can be observed during the soft/soft-intermediate states of the source,
dominated by the disc emission (Belloni et al., 2012).

The BH-XRBs exhibit complex non-linear time-lag features which can reveal the
dynamics of accretion geometry i.e., the variation of the size of the
Comptonizing region. The energy-dependent variation of time-lag can explain
the responsible physical mechanisms that produce HFQPO. Time-lag represents
the time difference in arrival time between soft photons and hard photons. The
delay in the arrival of hard photons as compared to soft photons is termed as
hard-lag and vice-versa (van der Klis, 1988). Time-lag was initially thought
to be generated due to the inverse Comptonization of soft seed photons by hot
electrons (Payne, 1980; Miyamoto et al., 1988) which only explains the hard-
lag. Several models have been proposed (Cui et al., 1997; Nowak et al., 1999;
Poutanen, 2001) to explain the hard and soft-lag observed in Galactic black
hole candidates. Reig et al. (2000) suggested that both hard-lag and soft-lag
can be explained by Compton up-scattering and down-scattering mechanisms. Qu
et al. (2010) reported an anticorrelation between phase-lag and LFQPO
frequency (0.5$-$10 Hz) for GRS 1915+105 where, the lag switches sign from
positive to negative at the QPO frequency $\sim$ 2 Hz. Dutta & Chakrabarti
(2016); Dutta et al. (2018) concluded that time-lag is a resultant effect of
multiple physical mechanisms which are repeated Compton scattering (produces
hard-lags), reflection of hard X-rays (soft-lag), focusing due to
gravitational bending of photon path (soft-lag) and geometry (Reverberation
mapping, produces both soft and hard-lag) of accretion dynamics. These
mechanisms are responsible for producing the lags at different Fourier
frequencies where time-lag decreases with the increase of the frequency and
vice versa. Dutta & Chakrabarti (2016) reported that the GX 339-4 (face-on)
shows hard-lag for all QPO frequencies, whereas for edge-on sources e.g. XTE
J1550-564 and GRS 1915+105, time-lag switches sign from positive to negative
at QPO frequency $\sim$ 3.2 Hz and $\sim$ 2.3 Hz respectively. Despite all of
these developments for lag study at LFQPOs (Patra et al., 2019), phase/time
lag has not been studied extensively for HFQPOs as it is rare to be observed.
GRS 1915+105 shows prominent stable HFQPO at $\sim$ 67 Hz for almost 25 years
as observed by both RXTE and AstroSat missions. Thus, it gives us a unique
opportunity to study time-lag features at HFQPOs observed with the AstroSat.

GRS 1915+105, a bright microquasar (Mirabel & Rodríguez, 1994) of mass
$12.4M_{\sun}$ is a rapidly rotating (spin > 0.98) black hole X-ray binary
(BH-XRB) source (see Zhang et al., 1997; Sreehari et al., 2020, and references
therein) at a distance of $8.6$ kpc (Reid et al., 2014) with a jet axis at an
angle $\sim$ 65∘ to the line of sight (Zdziarski, 2014). Depending on its
variability of lightcurve and color-color diagram (CCD), all the observations
of this source can be classified into 14 distinct classes (Belloni et al.,
2000; Klein-Wolt et al., 2002; Hannikainen et al., 2005). The source variation
in each of these classes could be reduced to the alternations of three basic
spectral states: A, B, and C. A quiescent state (C) with a lower count rate,
an outburst state (B) with higher count rates, and a flare state (A) in which
the flux shows rapid changes between the later two states (Belloni et al.,
2000). HFQPO was studied extensively using RXTE observations with the first
detection of HFQPO at 65$-$67 Hz (Morgan et al., 1997). Belloni & Altamirano
(2013) detected 63$-$71 Hz HFQPO in seven ($\kappa$, $\gamma$, $\mu$,
$\delta$, $\omega$, $\rho$ and $\nu$) variability classes. Cui (1999) first
attempted to study the lag properties of HFQPO using RXTE data and they found
that phase-lag increases with energy. Méndez et al. (2013) found that 35 Hz
QPO shows soft-lag, whereas the 67 Hz QPO exhibits hard-lag and the magnitude
of both lag increases with the energy using RXTE data. The HFQPO of this
source was found to be associated with state B, which corresponds to the disc
(Belloni & Altamirano, 2013, and references therein) dominated energy
spectrum. The spectral analysis during this state indicates that the inner
radius of the accretion disc can reach close towards the black hole. There was
no detection of HFQPO during state C (equivalent to the hard-intermediate
state) of the source, instead strong type-C QPOs were observed.

In the AstroSat era, a few works have been done on the HFQPOs of GRS 1915+105.
Belloni et al. (2019) observed HFQPOs in the range of 67.4$-$72.3 Hz
considering observations from July to September 2017 observations. They found
hard-lag at HFQPO with respect to 5$-$10 keV. Sreehari et al. (2020) observed
$\delta$ class observations and found that HFQPOs are only present in the
6$-$25 keV energy range and inferred that the HFQPOs in GRS 1915+105 seem to
originate due to an oscillation of Comptonized ‘compact’ corona surrounding
the central source. Further, Majumder et al. (2022) also studied all
observations of this source extensively using AstroSat and observed that the
HFQPOs are significant in 6$-$25 keV and present in the $\delta$, $\kappa$,
$\omega$, $\gamma$ classes having frequency range $68.14-72.32$ Hz. They also
concluded that the HFQPOs might be the result of the modulation of the
Comptonizing corona in the vicinity of the source. However, the cross-spectrum
analysis of those observations was not carried out for calculating the time-
lag at the HFQPOs. The study of time-lag properties can be useful for a better
understanding of generation of HFQPOs and the accretion flow around the BH-
XRBs.

In this paper, for the first time to the best of our knowledge, we are
presenting the class-wise time-lag study of the HFQPOs of GRS 1915+105
observed with AstroSat. We consider the HFQPO observations as reported by
Majumder et al. (2022). Lightcurve and color-color diagrams were produced to
identify the variability classes for those observations. We study the PDS to
obtain the HFQPO parameters for further lag study. Lag spectra of the 6$-$25
keV band with respect to the 3$-$6 keV band is produced to find the time-lag
feature at HFQPO. Further, we analyze the energy-dependent time-lag with
respect to the 3$-$6 keV band and carried out spectral analysis to correlate
the time-lag features with spectral parameters.

We organize this paper as follows. In §2, we present the reduction procedures
of AstroSat data. In §3, we discuss the method of timing and spectral analysis
and present the results. We discuss the results and attempt to explain them
with the existing models present in the literature in §4. Finally, we make our
conclusion in §5.

##  2 Observation And Data Reduction

We consider all the AstroSat observations of HFQPOs of GRS 1915+105 from April
2016 to September 2017 as reported by Majumder et al. (2022). These
observations consist of total 9 Guaranteed Time (GT) observations containing
over 70 orbits. Among these, we consider 15 orbits such that they correspond
to the observation of each consecutive day. Table 1 summarises the list of
considered observation IDs along with orbit number. The maximum exposure time
available for all observations is considered for our analysis. We obtain all
of the AstroSat data from the Indian Space Science Data Center (ISSDC)
archive111https://webapps.issdc.gov.in/astro_archive/archive/Home.jsp.

AstroSat (Agrawal, 2006) is India’s first multi-wavelength satellite which was
launched in 2015. It consists of four co-aligned instruments which are capable
of studying astrophysical objects in the ultra-violet, soft X-ray and also in
the hard X-ray regime. These four instruments are Ultra-Violet Imaging
Telescope (UVIT) (Tandon et al., 2017), Soft X-ray Telescope (SXT) (Singh et
al., 2017), Large Area X-ray Proportional Counter (LAXPC) (Yadav et al., 2016;
Antia et al., 2017) and Cadmium Zinc Telluride Imager (CZTI) (Vadawale et al.,
2016). We have used SXT and LAXPC observations for our study. The timing
resolution of 10 $\mu$s in LAXPC (Yadav et al., 2016) and the unique
combination of the energy range of both SXT and LAXPC enable us to study the
broadband (0.3$-$80 keV) spectro-temporal feature of the source.

SXT observes the X-ray sources in the energy range 0.3$-$8 keV. We have
considered the level-2 data for analysing only the spectra of SXT due to the
poor time resolution (2.3775 s) compared to LAXPC. The source image,
lightcurve, and spectra are generated by using XSELECT V2.4m in HEASOFT
V6.29c. The source has been observed by SXT both in Photon Count (PC) and Fast
Window (FW) mode. We do not carry out pile-up correction for the observations
in PC mode as the source counts are less than 40 cts/s in the central 1 arcmin
circular region of the image (Majumder et al., 2022). For all PC mode
observations, a circular region of 12 arcmins is selected for extracting the
spectra. In the case of the FW mode observations, we choose the 5 arcmins
circular region as the source region as it was offset in the CCD frame
(Sreehari et al., 2020). We use the background file, redistribution matrix
files (rmfs), and ancillary response files (arfs) provided by the SXT
instrument team at
TIFR222https://www.tifr.res.in/~astrosat_sxt/dataanalysis.html. Further, we
use sxtARFModule to apply vignetting correction on the provided arf file.

LAXPC is a proportional counter which can detect X-rays in the energy range
3$-$80 keV. It consists of three identical unit, LAXPC10, LAXPC20 and LAXPC30
having combined effective area of 6000 cm2. We use LAXPC level-1 data
available in AstroSat public archive${}^{\ref{issdc}}$. The software
LAXPCsoftware, available in AstroSat science support cell333http://astrosat-
ssc.iucaa.in is used to process the level-1 data to level-2 data. This level-2
data is used for further timing analysis. We extract data from all layers of
LAXPC10 and LAXPC20 for the temporal analysis. However, we use LaxpcSoftv3.4.3
(Antia et al., 2017) for the spectral analysis of LAXPC observations. We
generate the Good time Interval (GTI) and consider the background model and
spectral response as mentioned in Athulya et al. (2022). In order to minimize
the residue in the spectra beyond 30 keV, we extract the top layer, single
event data from LAXPC20 because of its steady gain throughout the considered
observational period. We didn’t consider LAXPC30 because its gain was observed
to increase with time due to a leakage (Antia et al., 2021).

##  3 Analysis and Results

###  3.1 Variability Classes and Color-Color Diagram (CCD)

GRS 1915+105 exhibits different variability features in its light curve and
Color-Color Diagrams (CCDs). The source displays fourteen variability classes
(Belloni et al., 2000; Klein-Wolt et al., 2002; Hannikainen et al., 2005) as
observed by the RXTE. CCDs are used to understand the hardness variation
during different spectral states and to identify the nature of the variability
class exhibited by the source. In order to produce CCDs, we generate the 1 s
binned lightcurves in the energy range 3$-$6 keV, 6$-$15 keV and 15$-$60 keV
by combining LAXPC10 and LAXPC20. We perform background correction to these
lightcurves and generate CCD to study the variation between soft color HR1 and
hard color HR2. The soft color and hard color are defined as HR1=B/A and
HR2=C/A respectively, where A, B, and C are the count rates in 3$-$6 keV,
6$-$15 keV and 15$-$60 keV respectively. Finally, we add A, B and C bands
lightcurve to produce the lightcurve in 3$-$60 keV band and perform dead-time
corrections (Sreehari et al., 2020, and references therein). However, the
power density spectra are generated from the lightcurve without any background
correction.

Following the classification scheme in Athulya et al. (2022) and Majumder et
al. (2022), we identify that the considered AstroSat observations of HFQPOs
belong to the four variability classes $\omega$, $\delta$, $\kappa$ and
$\gamma$. The background subtracted and dead-time corrected lightcurves of
four classes are shown in Fig. 1 with CCD in the inset at the top right corner
of each panel.

Figure 1: Lightcurve and CCD of the source GRS 1915+105 during HFQPO
observations using AstroSat. The background subtracted and dead-time corrected
1 s binned LAXPC lightcurve corresponding to four variability classes namely
$\delta$, $\kappa$, $\omega$, and $\gamma$ are plotted in the 3$-$60 keV
energy range with CCD (top right inset). See text for detail.

###  3.2 HFQPOs in PDS and RMS Power

We limit our analysis to those observations where only HFQPOs are observed. We
produce the PDS of all considered observations from the four variability
classes (see §3.1 and Table 1) in order to calculate the HFQPO parameters for
further cross-spectral study. We generate 1 ms binned lightcurve in 3$-$60 keV
band to produce a PDS with Nyquist frequency of 500 Hz. We consider intervals
of 32.768 s and compute the Fast Fourier Transform (FFT) for each interval,
and we combined them using a geometric binning factor of 1.02 to get an
averaged PDS. The PDS are standardized and converted into the Leahy power
spectrum (Leahy et al., 1983). For each observation, we subtracted the Poisson
noise contribution (Zhang et al., 1995; Yadav et al., 2016) and also corrected
the dead-time effects on rms amplitude (Bachetti et al., 2015; Sreehari et
al., 2019) to produce the final PDS in rms2/Hz plane. The power spectra were
then fitted with a combination of constant and multiple Lorentzian (see Nowak,
2000). The PDS are shown for all four classes in Fig. 2 with the HFQPOs in the
inset at the upper right corner. The variability class along with the orbit
are mentioned in the lower right corner of each panel.

Figure 2: Power density spectra generated from 1 ms binned lightcurve for four
variability classes ($\delta$, $\kappa$, $\omega$, $\gamma$) in 3$-$60 keV
energy range. In each panel variability class along with orbit number is
mentioned, and the HFQPO feature (zoomed view) is also shown in the upper
right inset. See text for detail.

In order to obtain the HFQPO parameters only, we further fit the PDS in the
frequency range of 20$-$200 Hz. We fit all the PDS by using a constant and a
Lorentzian feature with centroid frequency at $\sim$ 68 Hz. This model yields
the reduced $\chi^{2}$ ($\chi^{2}_{red}=\chi^{2}/dof$) in the range 1.0$-$1.3
for all PDS which ensures the satisfactory fit. The best-fitted values of
centroid frequency, FWHM and percentage rms amplitude are produced in Table 1.
The percentage rms is calculated by finding the square root of the definite
integral of the fitted Lorentzian and multiplying it by 100 (see Athulya et
al., 2022). We calculate the error in rms by using the error propagation rule.
We use centroid frequency and FWHM for the calculation of time-lag at the
HFQPOs.

Table 1: Details of the best fitted PDS parameters from LAXPC observations of
GRS 1915+105 in 3$-$60 keV energy range for four variability classes are
presented. Results are obtained after combining the LAXPC10 and LAXPC20. All
PDS are fitted in the frequency range 20$-$200 Hz in rms2/Hz plane. Here, in
this table, model fitted parameters of HFQPO (centroid frequency and FWHM) are
given along with percentage rms amplitude. The errors of fitted parameters are
computed with 68% confidence range. Time-lags are calculated for 6$-$25 keV
with respect to 3$-$6 keV and tabulated in units of milliseconds. See text for
details.

Obs ID | MJD | Orbit | HFQPOfreq | FWHM | HFQPOrms% | Time-lag | Class
---|---|---|---|---|---|---|---
|  |  | (Hz) | (Hz) |  | (ms) |
G05_214T01_9000000428 | 57503.63 | 03117 | 70.77${}^{+0.11}_{-0.11}$ | 1.72* | 1.24$\pm$0.07% | -0.51$\pm$ 0.15 | $\omega$
| 57504.02 | 03124 | 70.78${}^{+0.24}_{-0.30}$ | 3.64${}^{+1.36}_{-1.35}$ | 1.34$\pm$0.28% | -0.40$\pm$0.07 | $\omega$
G05_189T01_9000000492 | 57551.04 | 03819 | 68.02${}^{+0.06}_{-0.05}$ | 3.00* | 1.57$\pm$0.05% | -1.06$\pm$0.17 | $\delta$
| 57552.35 | 03839 | 68.42${}^{+0.45}_{-0.23}$ | 1.46* | 0.74$\pm$0.10% | -0.49$\pm$0.16 | $\delta$
| 57553.88 | 03860 | 68.13${}^{+0.20}_{-0.24}$ | 2.93${}^{+0.70}_{-0.81}$ | 1.40$\pm$0.19% | -0.74$\pm$0.12 | $\delta$
| 57554.09 | 03864 | 68.14${}^{+0.10}_{-0.09}$ | 3.00* | 1.49$\pm$0.07% | -1.00$\pm$0.16 | $\delta$
G07_028T01_9000001370 | 57943.69 | 09633 | 70.67${}^{+0.45}_{-0.46}$ | 4.25${}^{+1.48}_{-1.42}$ | 1.77$\pm$0.35% | -0.54$\pm$0.22 | $\kappa$
G07_046T01_9000001374 | 57945.97 | 09667 | 71.14${}^{+0.11}_{-0.11}$ | 2.83${}^{+0.48}_{-0.41}$ | 2.01$\pm$0.17% | -0.80$\pm$0.21 | $\kappa$
| 57946.34 | 09670 | 70.30${}^{+0.22}_{-0.23}$ | 5.10${}^{+0.54}_{-0.48}$ | 2.50$\pm$0.15% | -0.91$\pm$0.18 | $\kappa$
G07_028T01_9000001406 | 57961.42 | 09895 | 71.16${}^{+0.08}_{-0.08}$ | 2.92${}^{+0.31}_{-0.29}$ | 2.09$\pm$0.12% | -0.79$\pm$0.13 | $\kappa$
G07_046T01_9000001408 | 57962.07 | 09902 | 70.03${}^{+0.28}_{-0.28}$ | 5.66${}^{+0.86}_{-0.74}$ | 2.58$\pm$0.21% | -0.81$\pm$0.21 | $\kappa$
G07_028T01_9000001500 | 57995.30 | 10394 | 67.85${}^{+0.17}_{-0.19}$ | 4.17${}^{+0.47}_{-0.51}$ | 2.20$\pm$0.16% | -1.50$\pm$0.15 | $\omega$
G07_046T01_9000001506 | 57996.46 | 10411 | 69.10${}^{+0.13}_{-0.14}$ | 4.92${}^{+0.31}_{-0.31}$ | 2.46$\pm$0.09% | -1.07$\pm$0.09 | $\omega$
G07_046T01_9000001534 | 58007.59 | 10575 | 71.62${}^{+0.14}_{-0.14}$ | 4.04${}^{+0.40}_{-0.35}$ | 2.07$\pm$0.11% | -0.88$\pm$0.11 | $\omega$
| 58008.08 | 10583 | 71.41${}^{+0.08}_{-0.08}$ | 2.70${}^{+0.26}_{-0.23}$ | 2.32$\pm$0.12% | -1.68$\pm$0.20 | $\gamma$

  * *

Fixed parameter.

###  3.3 Time Lag properties of HFQPO

####  3.3.1 Lag-Spectra

X-ray lags are measured using the cross Fourier spectrum of X-ray light curves
with different energy bands. The soft energy band is chosen on the basis of
the detector’s lower energy limit and also on the nature of the source. The
harder photons, energized via inverse Compton process (Sunyaev & Titarchuk,
1980), are used to cross-correlate with the soft photons to generate frequency
dependent time-lag spectra.

We generate time-lag spectra for each observation. Time-lag between the
photons of two different energy bands can be calculated from the argument of
cross spectra. Let, the photon count in two different simultaneous energy
bands is observed to be $x_{1}(k)$ and $x_{2}(k)$ at time $t_{k}$ and their
Fourier transforms are $X_{1}(j)$ and $X_{2}(j)$ respectively at frequency
$\nu_{j}$. The Fourier transform is defined as,

$X(j)=\sum_{k=0}^{m-1}x(k)e^{2\pi i\nu_{j}t_{k}}.$ (1)

Here, $x(k)$ is the count in the $k$th bin of the lightcurve which consists of
$m$ evenly spaced time-bin of length $\Delta t$ in seconds. $X(j)$ is the
complex Fourier transform of $x(k)$ corresponding to frequency
$\nu_{j}=j/(m\Delta t)$ where, $j\in[-m/2,m/2]$ (see van der Klis, 1989, for
details). Cross spectra are defined as, $C(j)=X^{*}_{1}(j)X_{2}(j)$
corresponding to Fourier frequency $\nu_{j}$. $X^{*}_{1}(j)$ refers to the
complex conjugate of $X_{1}(j)$. The position angle of $C(j)$ in complex plane
is defined as phase-lag, $\phi(j)=arg[C(j)]$ and the corresponding time-lag is
defined as, $\delta t(j)=\phi(j)/2\pi\nu_{j}$ (see Vaughan & Nowak, 1997, for
details). The variation of this time-lag $\delta t(j)$ as a function of
Fourier frequency $\nu_{j}$ is known as time-lag spectra. Following the
methodology mentioned above, we divide each observation into two energy bands,
3$-$6 keV and 6$-$25 keV and produce the time-lag of 6$-$25 keV photons as a
function of Fourier frequency with respect to 3$-$6 keV photons. We consider
Nyquist frequency of lag spectra to be 500 Hz in order to perform time-lag
analysis near HFQPOfreq (see §3.2) which is well below the region where
binning effects are important (see Crary et al., 1998, for more details). The
Fourier frequency is binned at an interval $\sim$ 1 Hz as it offers much less
error on time-lag. We did not perform any dead-time correction on lag-spectra
since this effect was found to be negligible (Bachetti et al., 2015). The lag-
spectra of all four variability classes are shown in Fig. 3. The variability
class and the orbit are mentioned in the bottom right corner of each panel.
The vertical black dashed line shows the HFQPOfreq and the vertical black
dotted lines represent the FWHM range of the HFQPOs. The negative lag means
soft-lag where soft photons (i.e., 3$-$6 keV) are lagging behind the hard
photons (i.e., 6$-$25 keV). All the variability classes exhibit a similar
coherent soft-lag feature of different magnitudes associated with the HFQPOs.

Figure 3: Lag spectra (time-lag as a function of Fourier frequency) is plotted
for photons in the energy band 6$-$25 keV with respect to those in 3$-$6 keV
band for four variability classes. All variability classes show soft-lag at
HFQPO. The centroid frequency of HFQPO is shown as the vertical black dashed
line and the vertical black dotted lines represent the range of FWHM of the
HFQPO.

We calculate the time-lag from this lag-spectra following Reig et al. (2000)
by averaging the time-lag over the frequency range
($\nu_{QPO}-\Delta\nu_{QPO}/2$) to ($\nu_{QPO}+\Delta\nu_{QPO}/2$), where
$\nu_{QPO}$ and $\Delta\nu_{QPO}$ is the centroid frequency and FWHM of
Lorentzian respectively, obtained from the fitted PDS in §3.2. We calculate
the time-lag from each observation and presented in Table 1. Among all
classes, the $\kappa$ variability class shows the soft-lag in the range of
0.54$-$0.91 ms. The only observation of the $\gamma$ class shows the maximum
soft-lag of 1.68 ms. The $\delta$ and $\omega$ classes show a wide range of
time-lag. The $\delta$ class exhibits soft-lag in the range 0.49$-$1.06 ms,
whereas the soft-lag of the $\omega$ class lies in the range of 0.40$-$1.50
ms.

Figure 4: Energy-dependent time lag w.r.t 3$-$6 keV band at the HFQPO
frequency is plotted for the four variability class observations. Variability
classes and the orbits of all observations are mentioned in the upper right
and lower right corners of each panel respectively. See text for details.

####  3.3.2 Energy Dependent Time-lag

Time-lag is a manifestation of different non-linear mechanisms. These
mechanisms produce different energy photons at different regions of the
accretion flow. Thus, the observed energy-dependent time-lag feature can
explain the responsible mechanism for producing different energy photons.
Time-lag of any particular energy band depends on the respective reference
energy band. In order to study the time-lag variations with energy, we divide
the entire energy band 3$-$25 keV into seven different energy bands. The
chosen energy bands are: 3$-$6 keV, 6$-$9 keV, 9$-$12 keV, 12$-$15 keV,
15$-$18 keV, 18$-$21 keV and 21$-$25 keV. We generate lag-spectra for each
energy band with respect to the softest energy band 3$-$6 keV. Lag-spectra is
produced in a similar way as mentioned in the previous subsection §3.3.1. From
each lag-spectra, we calculate the average time-lag over the FWHM of the HFQPO
(see Table 1) and we plotted them with respect to their corresponding energy.

The time-lag as a function of energy for all four variability classes and for
all orbits are plotted in Fig. 4. The variability class and the MJD are
mentioned in the upper left and lower left corners respectively of each panel.
It is clear from the figure that the time-lag at the HFQPO depends strongly on
photon energy. We find for all of the observations that the magnitude of the
soft-lag increases with energy up to $\sim$ 18 keV. However, different
observations in each class show different time-lag variations with energy. In
the $\delta$ variability class, maximum soft-lag of $\sim$ 3 ms is observed on
MJD 57551.04 (orbit 3819), whereas among the $\omega$ class observations,
maximum soft-lag of $\sim$ 2.5 ms on MJD 57995.30 (orbit 10394) is observed.
Moreover, in the $\kappa$ and $\gamma$ variability classes, maximum soft-lag
of $\sim$ 2.1 ms is observed on MJD 57943.69 (orbit 9633) and MJD 58008.08
(orbit 10583). It may be noted that we do not encounter the problem of dead-
time-driven cross-talk. In this problem, a $+\pi$ or $-\pi$ phase-lag is
produced in the frequency domain where Poisson noise is dominated (see van der
Klis et al., 1987; Méndez et al., 2013).

###  3.4 Spectral Analysis and Results

In order to correlate the time-lag with spectral parameters, we perform wide
band (0.7$-$50 keV) spectral analysis from simultaneous data of SXT and LAXPC
for each variability class (see Majumder et al., 2022, for details). SXT
spectra are extracted in the energy range 0.7$-$7.0 keV and LAXPC spectra are
extracted in the energy range 3$-$50 keV. SXT spectra are grouped with 30
counts in each bin before modelling, whereas grouping is not applied for
LAXPC20 spectra. We ignore spectral analysis for some of the observations due
to the poor quality of SXT data.

Both SXT and LAXPC20 spectra are modelled using the XSPEC V12.12.0 which is
available in HEASOFT V6.29.c. A systematic error of 2% is considered for both
spectra (Antia et al., 2017; Leahy & Chen, 2019; Sreehari et al., 2019). We
considered gain fit command in SXT spectra to alter the response to fit the
edges at 1.8 keV and 2.2 keV produced due to the Si and Au (Singh et al.,
2017). During the application of gain fit, we fix the slope at 1 and allow the
offset to vary.

We fitted all the spectra with the model combination of
constant$\times$TBabs$\times$(smedge$\times$nthComp + edge$\times$powerlaw) as
discussed in Majumder et al. (2022). During the spectral fitting of all
observations, we have kept the seed photon temperature ($kT_{bb}$) fixed at
0.1 keV. All the best-fitted model parameters are shown in Table 2. The
instrumental Xenon edge model is used with power law to take care of the
instrumental absorption feature at $\sim$ 30 keV in the spectra (Sreehari et
al., 2019). In order to obtain the best fit, the model component smedge is
also required due to a broad absorption feature at $\sim$ 9 keV. The best-
fitted wide band spectra of four variability classes are shown in Fig. 5. The
optical depth ($\tau$) of the Comptonizing medium is obtained using the
relation given by Zdziarski et al. (1996). The calculated optical depth is
found to be in the range 6.90$-$12.55, and the HFQPOs are observed during the
softer spectral state, which in turn implies the presence of an optically
thick Comptonizing region close to the black hole.

Figure 5: Wide band (0.7$-$50.0 keV) spectra of the source GRS 1915+105 are
plotted for four variability classes. The spectra are modelled with
constant$\times$TBabs$\times$(smedge$\times$nthComp + edge$\times$powerlaw).
The variability classes and the orbits are mentioned. Table 2: Model
parameters of the observations fitted with model
constant$\times$TBabs$\times$(smedge$\times$nthComp + edge$\times$powerlaw).
All errors are calculated with 90% confidence range.

MJD (orbit) | $kT_{e}$ | $\Gamma_{nth}$ | $norm_{nth}$ | $\Gamma_{PL}$ | $norm_{PL}$ | $\chi^{2}_{red}$ | Optical depth | Class
---|---|---|---|---|---|---|---|---
| (keV) |  |  |  |  |  | ($\tau$) |
57504.02 (03124) | 2.59${}_{-0.09}^{+0.10}$ | 2.36${}_{-0.09}^{+0.09}$ | 38.85${}_{-6.63}^{+7.85}$ | 3.26${}_{-0.02}^{+0.02}$ | 28.09${}_{-0.96}^{+0.87}$ | 1.13 | 8.60$\pm$0.31 | $\omega$
57551.04 (03819) | 2.76${}_{-0.14}^{+0.16}$ | 2.28${}_{-0.11}^{+0.10}$ | 30.52${}_{-7.15}^{+8.43}$ | 3.16${}_{-0.06}^{+0.06}$ | 30.03${}_{-4.53}^{+4.00}$ | 1.12 | 8.68$\pm$0.58 | $\delta$
57552.35 (03839) | 2.67${}_{-0.13}^{+0.16}$ | 2.77${}_{-0.12}^{+0.09}$ | 126.39${}_{-26.39}^{+28.92}$ | 3.42${}_{-0.05}^{+0.04}$ | 46.87${}_{-5.39}^{+2.55}$ | 1.07 | 6.90$\pm$0.48 | $\delta$
57553.88 (03860) | 2.83${}_{-0.08}^{+0.06}$ | 2.45${}_{-0.06}^{+0.03}$ | 48.88${}_{-4.19}^{+4.19}$ | 3.19${}_{-0.01}^{+0.01}$ | 32.59${}_{-1.51}^{+2.33}$ | 1.11 | 7.75$\pm$0.28 | $\delta$
57554.09 (03864) | 2.85${}_{-0.17}^{+0.14}$ | 2.14${}_{-0.04}^{+0.08}$ | 19.12${}_{-4.01}^{+4.79}$ | 3.04${}_{-0.24}^{+0.10}$ | 15.44* | 1.05 | 9.28$\pm$0.62 | $\delta$
57943.69 (09633) | 2.79${}_{-0.11}^{+0.15}$ | 2.55${}_{-0.07}^{+0.09}$ | 20.02${}_{-3.27}^{+4.22}$ | 3.28${}_{-0.03}^{+0.05}$ | 12.81${}_{-0.43}^{+0.58}$ | 1.18 | 7.45$\pm$0.31 | $\kappa$
57946.34 (09670) | 2.62${}_{-0.07}^{+0.14}$ | 2.33${}_{-0.08}^{+0.12}$ | 15.65${}_{-2.69}^{+2.31}$ | 3.12${}_{-0.01}^{+0.03}$ | 11.56${}_{-0.28}^{+0.42}$ | 1.03 | 8.69$\pm$0.50 | $\kappa$
57961.42 (09895) | 2.83${}_{-0.07}^{+0.11}$ | 2.27${}_{-0.05}^{+0.06}$ | 18.25${}_{-2.46}^{+3.40}$ | 3.07${}_{-0.02}^{+0.03}$ | 13.15${}_{-0.34}^{+0.44}$ | 1.09 | 8.61$\pm$0.33 | $\kappa$
57962.07 (09902) | 2.70* | 2.27${}_{-0.04}^{+0.04}$ | 18.37${}_{-1.84}^{+2.12}$ | 2.97${}_{-0.02}^{+0.02}$ | 12.29${}_{-0.49}^{+0.50}$ | 1.02 | 8.84$\pm$0.21 | $\kappa$
57995.30 (10394) | 2.60* | 2.14${}_{-0.05}^{+0.04}$ | 20.33${}_{-2.38}^{+2.24}$ | 2.93${}_{-0.01}^{+0.01}$ | 17.48${}_{-0.47}^{+0.50}$ | 1.02 | 9.78$\pm$0.31 | $\omega$
57996.46 (10411) | 2.89${}_{-0.11}^{+0.11}$ | 2.27${}_{-0.08}^{+0.06}$ | 24.97${}_{-3.94}^{+3.90}$ | 3.01${}_{-0.03}^{+0.03}$ | 17.09${}_{-0.56}^{+0.53}$ | 1.12 | 8.50$\pm$0.40 | $\omega$
58007.59 (10575) | 2.71${}_{-0.10}^{+0.11}$ | 2.19${}_{-0.07}^{+0.09}$ | 24.47${}_{-3.84}^{+5.15}$ | 3.05${}_{-0.03}^{+0.03}$ | 19.00${}_{-0.62}^{+0.56}$ | 1.05 | 9.25$\pm$0.62 | $\omega$
58008.08 (10583) | 2.40* | 1.85${}_{-0.04}^{+0.05}$ | 13.45${}_{-1.92}^{+2.08}$ | 2.90${}_{-0.01}^{+0.01}$ | 19.30${}_{-0.71}^{+0.70}$ | 0.95 | 12.55$\pm$0.48 | $\gamma$

  * *

Parameter is fixed.

##  4 Discussion

In this work, we perform a detailed energy dependent variability study for all
HFQPOs ($\sim 67$ Hz) observed by AstroSat. For the first time, we find soft-
lag for all ‘softer’ state variability classes i.e., $\delta$, $\omega$,
$\kappa$ and $\gamma$. We also study the correlation of soft-lag with the rms
power (Fig. 6) and the same with the optical depth (Fig. 7) of the
Comptonizing region/corona from where the hard photons are being generated.
The additional powerlaw could indicate the presence of an extended corona
around the central corona (Majumder et al., 2022). It is also evident from the
energy dependent power-spectral study that the photons in 6$-$25 keV are
responsible for producing HFQPOs (Majumder et al., 2022).

Figure 6: Time-lag of 6$-$25 keV band with respect to 3$-$6 keV band is
plotted as a function of rms amplitude of HFQPOs in full energy band. The
variability classes are also mentioned in the inset at the lower left corner
of the panel. See the text for details. Figure 7: Variation of time-lag is
plotted as a function of optical depth ($\tau$). The variability classes are
presented in different symbols marked in the inset at the upper right corner.
See text for details.

###  4.1 The RMS power in HFQPOs

In order to correlate the time-lag with the rms power of HFQPO, we study the
PDS of all observations in the 3$-$60 keV energy band and plotted only for
each class in Fig. 2. The rms power of the HFQPO lies in the range
0.74$-$2.58% and is given in Table 1 along with time-lag. The nthComp flux is
maximum in the 6$-$25 keV energy band and the HFQPOs are significant in the
6$-$25 keV energy band, indicating that Comptonized photons are responsible
for producing the HFQPOs (Majumder et al., 2022). The rms power increases
linearly with energy up to $\sim 20$ keV (Majumder et al., 2022) as well
observed by Morgan et al. (1997) at the first detection of 67 Hz HFQPO using
RXTE observations. In the RXTE era, only 49 observations of GRS 1915+105
exhibit HFQPO at $\sim 67$ Hz out of 1807 observations analyzed by Belloni &
Altamirano (2013) and they also found that the rms amplitude increases
linearly with energy. It is very interesting that none of the HFQPOs were
observed in the $\chi$ variability class where a systematic evolution of type
C QPOs was observed (Dutta et al., 2018). Therefore, the disc flux alone can’t
explain the observed rms-energy variation though the spectrum belongs to the
disc dominated spectral state. Thus, it is now obvious that the Comptonization
process plays an important role in producing the fast variability in the
‘softer’ classes of the source and HFQPO can be produced due to the
oscillation or modulation of the ‘Comptonizing region’ which may be small and
‘compact’ in size during the disc dominated spectral state (Méndez et al.,
2013; Aktar et al., 2017; Dihingia et al., 2019; Sreehari et al., 2020;
Majumder et al., 2022). On the contrary, a similar explanation for the case of
kHz QPOs in neutron-star systems (see Méndez, 2006; Sanna et al., 2010) states
that if we consider the HFQPOs are produced due to the oscillation mechanism
in the disc, the signal/soft-radiation has to be amplified and modulated in
the ‘Comptonizing region’ to contribute hard-lags at the HFQPOs.

###  4.2 Time-lag variation with Energy

The energy dependent time-lag study is important to know the exact physical
mechanism associated with the generation of HFQPOs. In this work, we performed
a detailed time-lag study of all HFQPOs observed by AstroSat choosing a
similar reference band (3$-$6 keV) to compare with RXTE observation. We
produce the lag-spectra of 6$-$25 keV photons with respect to the 3$-$6 keV
photons because HFQPO can only be seen in this band (Majumder et al., 2022).
For the first time, we find soft-lag associated with the HFQPO for all
observations during the ‘softer’ variability classes observed with AstroSat.
This implies that soft disc photons (3$-$6 keV) lag behind the harder photons
(6$-$25 keV). The hard photons are generally produced via inverse Compton
processes (Sunyaev & Titarchuk, 1980) and subsequently suffer scattering delay
(hard-lag) in respect of soft photons. The soft-lag associated with the HFQPO
can be clearly seen in the lag spectra as shown in Fig. 3 for all four
variability classes. The soft-lag for all the classes is found to be in the
range 0.40$-$1.68 ms (see Table 1). Each of the variability class shows a
similar lag-energy correlation (see Fig. 4) where soft-lag increases with the
photon energy implying that higher energy photons always arrive earlier than
the lower energy photons. The $\delta$ and $\omega$ classes show maximum soft-
lag of $\sim$ 3 ms and $\sim$ 2.5 ms respectively at $\sim$ 18 keV, whereas
both the $\kappa$ and $\gamma$ classes exhibit maximum soft-lag of $\sim$ 2.1
ms.

However, an opposite correlation was reported by Cui (1999) and Méndez et al.
(2013) using RXTE observations, i.e., hard-lag at the HFQPO during the similar
‘softer’ variability classes. They found hard-lag (w.r.t 2$-$5 keV) increases
gradually with the energy and obtained maximum hard-lag of $\sim$ 5.6 ms and
$\sim$ 3.7 ms for $\gamma$ and $\delta$ classes respectively. Thus, a gradual
increase of hard-lag with energy, a rise of nthComp flux (see Appendix B), and
an increase in Comptonized photons are observed during the RXTE observations.
In contrast, we find an increase in soft photons and increase in soft-lag with
energy during the AstroSat observations. We also find the presence of
reflection features in the NuSTAR spectra during the simultaneous observations
(see Appendix A). Thus, from the above findings, the observed increase in soft
photons during AstroSat observations may be attributed to the dominant
reflection mechanism, which may lead to the detection of the soft-lag. This
finding contradicts the situation during RXTE observations, where a dominant
inverse Comptonization mechanism from seed photons in the accretion disc led
to the detection of hard-lag. However, lack of AstroSat observation, poor
spectral resolution, and limitations in data products restrict us from probing
further to find concrete reasons behind this opposite time-lag feature.
Therefore, we need a deeper study of higher spectral resolution (e.g., NuSTAR)
and extensive analysis of all RXTE observations with HFQPOs to find concrete
reasons behind this opposite lag behaviour, which is beyond the scope of this
paper.

Belloni et al. (2019) also found hard-lag for 10$-$20 keV and 20$-$30 keV
photons w.r.t 5$-$10 keV photons associated with HFQPO of this source using
AstroSat observation. We also find similar results using the LaxpcSoftware
(FORTRAN- based) as obtained by Belloni et al. (2019) using the GHATS package,
which is IDL-based. Thus, this different lag nature (hard/soft) observed for
the same AstroSat observation could be due to the choice of different
reference bands.

A satisfactory explanation (Cui, 1999; Nowak et al., 1999; Poutanen & Fabian,
1999; Kara et al., 2013; Dutta & Chakrabarti, 2016; Chatterjee et al., 2017b,
2020; Nandi et al., 2021) of time-lag includes the major presence of
Comptonization, reflection, outflow/jet and the effects of accretion disc
geometry. However, the hard-lag can be explained by the inverse Comptonization
(Compton up-scattering) of soft seed photons from the accretion disc by the
energetic electrons in the Comptonizing region or corona (Miyamoto et al.,
1988). Some of these up-scattered photons or a sudden and small dissipation of
hard photons from the corona (Malzac & Jourdain, 2000) may revert to the
cooler accretion disc and get reprocessed again. This mechanism is called
feedback (or reflection) which could be responsible for producing soft-lag
(Lee & Miller, 1998). Bellavita et al. (2022) showed that the hard/soft-lag
depends on the feedback fraction (fraction of up-scattered photons which
return towards the disc) which in turn depends on the size of the corona and
the inclination of the source. Moreover, the feedback fraction depends mainly
on accretion geometry (Malzac & Jourdain, 2000). Méndez et al. (2013) favoured
the Compton up-scattering scenario to explain the soft-lag at 35 Hz and hard-
lag at 67 Hz in GRS 1915+105 proposed by Lee et al. (2001) where the disc and
the corona are coupled and exchange energy.

The observed soft-lag associated with HFQPO can be attributed to the
significant dominance of the reflection mechanism during the softer
variability classes when corona shrinks (see the lower panel of the diagram in
Fig. 8) and the inner edge of the Keplerian disc approaches close to the inner
most stable circular orbit (ISCO). The hard photons of higher energy take a
longer time to reduce their energy through the down-scattering and hence
suffer greater time-lag i.e., soft-lag. This explains the increase of soft-lag
with energy as evident from Fig. 4. In the present scenario, the magnitude of
the soft-lag (0.40$-$1.68 ms) can be a rough estimator of the distance between
the source of photons and the reflector. Here, using time-lag to be light
crossing time, we find a wide distance of $\sim 7-27R_{g}$ (where,
$R_{g}=GM_{BH}/c^{2}$, with $M_{BH}$ referring to the mass of the black hole),
which is assumed to produce a significant value of reflection fraction as
obtained (see Appendix A) from the spectral fitting of simultaneous NuSTAR
observation.

###  4.3 Correlation of Soft-lag and HFQPO rms

We examine the energy dependent power density spectra and lag spectra of all
observations in the 3$-$60 keV energy band to correlate the time-lag with the
rms power of HFQPO which is plotted in Fig. 2. The rms power of the HFQPO lies
in the range 0.74$-$2.58% and is given in Table 1 along with time-lag. We, for
the first time, find a different linear correlation of time-lag with rms
amplitude with slopes $-0.63$, $-0.52$ and $-0.31$ for the $\delta$, $\omega$
and $\kappa$ variability classes respectively, which are shown in Fig. 6.
However, we detect an outlier among the $\omega$ variability class which was
observed on MJD 57995.30 (Orbit 10394). The Pearson correlation coefficients
of $\delta$, $\omega$ and $\kappa$ classes (excluding the outlier) are found
to be $-0.92$, $-0.97$ and $-0.78$ respectively. The amplitude of soft-lag
increases with the increase of rms amplitude of the HFQPO. The higher rms
amplitude implies a higher number of differentially intercepted or modulated
Comptonized photons, which produces the HFQPOs. The Comptonized photons lead
with respect to soft photons that suffer a delay, i.e., soft-lag due to the
down-scattering mechanism in the cold accretion disc due to the compact corona
and higher inclination of the source. Thus, a larger magnitude of soft-lag
would imply a larger delay, which increases with the energy of the photon that
suffers more down-scattering.

It is very interesting that an opposite correlation was observed by Zhang et
al. (2020) for the type-C QPOs in the range of 0.4$-$6.3 Hz where they found
that the correlation between QPO fractional rms and the average lag of the QPO
can be fitted well with a broken line which implies the amplitude of soft-lag
increases with the decrease of rms amplitude and amplitude of hard-lag also
increases as rms amplitude decreases.

###  4.4 Correlation of Soft-lag and Optical depth

A wide-band (0.7$-$50 keV) spectral study with thermal Comptonization model
nthComp along with a powerlaw component reveals a high value of the optical
depth ($\tau$) in the range of $\sim$ 6.90$-$12.55 of the Comptonized medium,
close to the black hole. The optical depth can be an important parameter to
explain the origin of soft and hard-lag because the opacity of a medium
determines the scattering probability as well as the escape probability of the
photon. A high value of optical depth ($\tau$) in the range of 6.90$-$12.55
indicates the presence of optically thick medium where the Comptonized photons
are generating.

A definite linear correlation is observed from Fig. 7 that the magnitude of
the soft-lag increases with the increase in optical depth ($\tau$). The higher
value of optical depth ($\tau$) causes a geometrically thick disc and it
enhances to occur the Compton down scattering. The higher energetic photons
take longer time to lose energy by means of the direct Compton effect. Thus,
they suffer a greater delay with respect to higher energy photons before
reaching the observer. All the variability classes exhibit a similar generic
feature. The Pearson correlation coefficient is found to be $-0.83$ which
shows that soft-lag and optical depth are strongly correlated. Thus, the
spectral study also independently validates the higher value of soft-lag for
higher energy photons which we obtained from the temporal study of this
source. It is evident from Fig. 7 that the $\omega$ and $\gamma$ classes
exhibit a maximum soft-lag for the highest value of optical depth ($\tau$).

In this present scenario of the ‘softer’ variability classes, the accretion
geometry has evolved in such a way that the reflection mechanism dominates
over the inverse Comptonization process which can be understood from the
cartoon diagram shown in Fig. 8). We expect a large fraction of the photons to
come into the Compton down scattering process to produce soft-lag as the
feedback fraction increases (Bellavita et al., 2022) due to the accretion
geometry (Malzac & Jourdain, 2000) of the soft spectral states i.e., the
smaller size of the corona.

Figure 8: Cartoon diagram of an advective flow disc (red part represents
Keplerian flow and yellow part represents Sub-Keplerian flow, see Chakrabarti
& Titarchuk, 1995) during the ‘hard’ and ‘soft’ spectral state. All possible
types of emergent photons from the possible physical processes are considered
here. The red and blue curved lines represent the ‘soft’ and ‘hard’ photons
respectively. The ‘1’ and ‘2’ lines represent ‘soft’ and ‘hard’ photons coming
from the other side of the accretion disc and have suffered gravitational
bending. Line ‘3’ goes through inverse Comptonization and reaches the observer
directly. Line ‘4’ is the reflected rays from the Keplerian disc. Line ‘5’
represents the soft photons which are generated from the disc and directly
reach the observer.

###  4.5 Soft-lag and Outflow/Radio emission

The HFQPOs are only observed from MJD 57503 to MJD 57554 and again from MJD
57943 to MJD 58808 by the AstroSat mission till now. The source is found to be
in the ‘softer state’ and exhibits four variability classes i.e., $\delta$,
$\omega$, $\kappa$ and $\gamma$ during these periods as reported by Majumder
et al. (2022). However, an active radio flux density of $\sim$ 30 mJy was
observed only during the period MJD 57600 to MJD 57700 when no HFQPO was
noticed but a tiny radio flux of $\sim$ 1.8 mJy was observed on MJD 58008 (see
Motta et al., 2021, for details) when the source exhibits soft-lag at the
HFQPO during its $\gamma$ variability class observation.

Generally, the outflow occurs in the hard-intermediate state and due to this
outflow, the amplitude of the soft-lag increases (Kara et al., 2013) as it
enhances the number of reflected photons from the Keplerian disc. Kim et al.
(2019) showed that most of the out-flowing matter, which has sub-escape
velocity, returns to the equatorial region and turns into the return flow.
This return flow down-scatters the hard photons to produce soft photons and
hence soft-lag. The radio fluxes (Vadawale et al., 2001) are associated with
the Jet/outflows or return flows (Nandi et al., 2001a, b) which contribute to
the soft-lag (see Patra et al., 2019). In order to explain the soft-lag in the
presence of outflow/Jet, we require the detection of any significant radio
flux during the HFQPO observation, which is mostly absent in the present
cases.

Fig. 8 represents a cartoon diagram of the possible accretion scenario in the
hard and soft spectral states. The upper figure represents accretion geometry
during the hard state when gravitational bending effects (see Chatterjee et
al., 2017a) and reflection mechanism (Poutanen & Fabian, 1999; Dutta &
Chakrabarti, 2016) are not dominating as the softer spectral state (lower
figure). The possible effects mentioned above are only dominating for the
edge-on (i.e., high inclination angle) sources in soft state. Chatterjee et
al. (2017a) found that gravitational bending enhances the number of hard
photons with inclination angle, whereas the multicolor blackbody part of a
spectrum becomes flat at higher inclination angles. It is a clear effect of
the focusing due to photon bending. In this scenario, the inversely
Comptonized photons (i.e., the photons that originated in the Keplerian disk
are intercepted by the corona) naturally lag behind soft photons that are
coming directly to the observer, and this mechanism dominates with the larger
size of the Comptonizing region/corona (during hard spectral state). However,
the Compton down-scatter mechanism dominates with the source’s increasing
inclination and decreasing corona size. This direct Compton effect will
eventually enhance the reflection mechanism over all other physical mechanisms
to produce soft-lag.

##  5 Conclusions

The time-lag property in the wide band energy range (3$-$60 keV) is important
to understand the accretion geometry and to identify the dominant physical
processes that produce HFQPOs. GRS 1915+105 exhibits a generic HFQPO $\sim$ 67
Hz persistently for over the last 25 years. However, a fewer observations
during the ‘softer’ variability class have been detected the HFQPO using RXTE
and AstroSat observation. Previously, only hard-lag was detected using the
RXTE observation but we discovered soft-lag associated with HFQPOs. Based on
our findings, it can be inferred that the observed soft-lag during AstroSat
observations can be attributed to the dominant reflection mechanism, whereas a
dominant inverse-Comptonization mechanism led to the detection of hard-lag in
the similar ‘softer’ variability class during RXTE observations. We summarised
here a generic feature of time-lag associated with the HFQPOs during four
variability classes i.e., $\delta$, $\omega$, $\kappa$ and $\gamma$ using all
AstroSat observations.

  * •

The first detection of soft-lag ($\sim$ 0.40$-$1.68 ms) of the 6$-$25 keV
energy band with respect to 3$-$6 energy keV associated with HFQPOs. All four
‘softer’ state variability classes exhibit a coherent lag-energy correlation.

  * •

The energy-dependent time-lag study shows the increase in soft-lag with energy
for the first time and exhibits maximum soft-lag of $\sim$ 3 ms and $\sim$ 2.5
ms for the $\delta$ and $\omega$ class, whereas the $\kappa$ and $\gamma$ both
classes show a maximum soft-lag of $\sim$ 2.1 ms.

  * •

We, for the first time, find a correlation of time-lag with rms amplitude for
each of the variability class i.e., $\delta$, $\omega$ and $\kappa$ where the
amplitude of soft-lag increases with the rms amplitude of the HFQPOs.

  * •

Spectral study suggests a high value of the optical depth ($\tau$ $\sim$
6.90$-$12.55) of the Comptonized medium close to the black hole. A linear
correlation is observed for all variability classes where the magnitude of the
soft-lag increases with the increase in optical depth ($\tau$).

  * •

No significant radio-emission was observed when HFQPOs are detected. However,
a tiny radio flux of $\sim$ 1.8 mJy was observed during the $\gamma$
variability class.

## Acknowledgments

Authors thank the anonymous reviewer for valuable suggestions and comments
that helped to improve the quality of this manuscript. BGD, PM and AN
acknowledge the support from ISRO sponsored project
(DS-2B-1313(2)/6/2020-Sec.2). PM, BGD thanks the Department of Physics, Rishi
Bankim Chandra College for providing the facilities to support this work. BGD
acknowledges ‘TARE’ scheme (Ref. No. TAR/2020/000141) under SERB, DST, Govt.
of India and also acknowledges Inter-University Centre for Astronomy and
Astrophysics (IUCAA) for the Visiting Associate-ship Programme. AN thanks GH,
SAG; DD, PDMSA, and Director, URSC for encouragement and continuous support to
carry out this research. This work uses the data of AstroSat mission of ISRO
which is archived at the Indian Space Science Data Centre (ISSDC). We have
used the data of Soft X-ray Telescope (SXT) which is calibrated and verified
by AstroSat-SXT team. We thank the SXT-POC team at TIFR for providing the
necessary software tool to analyse SXT data. This work has also used LAXPC
data which is verified by LAXPC-POC at TIFR. We thank AstroSat Science Support
Cell for providing the software LAXPCsoftware for the analysis of LAXPC data.

## Data Availability

Observational data used for this publication are available at the Astrobrowse
(AstroSat archive) website
https://webapps.issdc.gov.in/astro_archive/archive of the Indian Space Science
Data Centre (ISSDC).

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##  Appendix A Reflection feature in the softer variability state with NuSTAR
spectrum

We have considered a simultaneous NuSTAR observation (Obs ID 30202033002) with
one of the analyzed observations of AstroSat (Orbit 3860, MJD 57553, $\delta$
class) to find the reflection features in the spectrum. We have tried to get
the Compton hump at high energy in the spectrum of LAXPC20 using Tbabs
$\times$(diskbb$+$powerlaw) model, but unable to detect any hump-like feature
in the energy range 20 – 30 keV due to the poor spectral resolution of LAXPC.
We have extracted the cleaned events using nupipeline and obtained both FPMA
and FPMB spectra using nuproducts. We fitted the NuSTAR spectra considering
the model TBabs$\times$gabs(diskbb$+$powerlaw) (referred to as M1 hereafter).
A sharp absorption feature is observed in the residual at $\sim$ 7 keV, which
is taken care of using the Gaussian absorption model gabs. The best-fit yields
a $\chi^{2}_{red}$ of 1.48. We observed a Compton hump in the ratio plot in
the energy range 20$-$30 keV (see panel (a) in Fig. 9). This signature implies
that the reflection feature is present in the spectra. We consider another
model TBabs$\times$gabs(nthComp$+$powerlaw) (M2 hereafter), to incorporate the
reflection feature in the spectrum. In this case, the best-fit yields a
$\chi^{2}_{red}$ of 1.83. The Compton hump in 20 – 30 keV is also observed in
this model in the ratio plot (see panel (b) in Fig. 9).

Further, we use a self-consistent reflection model relxillCp (Dauser et al.,
2022) which can take care of both Comptonised emission from corona and the
reprocessed emission from the disk. The reflection model relxillCp is
considered to study the reflection feature in the spectrum quantitatively.
This model assumes the primary source spectrum model to be nthComp. We fit the
wide-band energy spectra considering the model
TBabs$\times$gabs(diskbb$+$relxillCp) (M3 hereafter), and we obtained a
satisfactory fit (see panel (c) in Fig. 9), which yielded a $\chi^{2}_{red}$
to 1.09. The inclination of the source is frozen to the standard value of 65∘.
The inner disk radius is assumed to be three times $R_{ISCO}$ of the black
hole according to the norm of diskbb obtained from the fitting in model M1.
The spectral fit suggests that the accretion disk is highly ionized. The other
parameters are considered according to Bhuvana et al. (2023). The model
parameter, reflection fraction, is found to be 0.48.

The reflection fraction is defined as the ratio of the intensity of the
primary source (nthComp flux due to the spherical corona) irradiating the disk
and the intensity directly going to infinity (Dauser et al., 2016). The non-
zero value of the reflection fraction shows the presence of reflection
features in the spectrum. Thus, the existence of the reflection mechanism can
be concluded during the soft-state of the accretion disk dynamics.

Figure 9: FPMA (black) and FPMB (red) energy spectra observed on MJD 57553 and
the ratio of the model M1, M2 and M3 shown in panel (a), (b) and (c)
respectively. A reflection signature is visible in (a) and (b) at the energy
range 20 – 30 keV. The model combination and the $\chi^{2}_{red}$ are
mentioned in the upper left corner of each panel.

##  Appendix B Evolution of energy spectra of RXTE and AstroSat observations

We considered five observations for spectral analysis, two of which are from
RXTE (Cui, 1999; Méndez et al., 2013) where hard-lag (w.r.t 2$-$5 keV) was
observed at the HFQPO. The rest of the observations are from AstroSat where
soft-lag is detected (w.r.t 3$-$6 keV). We considered the same spectral model
prescription, constant$\times$TBabs$\times$(nthComp + powerlaw) as mentioned
in §3.4, and the fitted parameters are shown in Table 3. The model parameter
$kT_{bb}$ was kept constant at 0.1 keV for all AstroSat observation, as
Sreehari et al. (2020) found the $kT_{bb}$ to be in the range of 0.1$-$0.3
keV. However, we could not constrain the $kT_{bb}$ parameter in the case of
RXTE observations. Further, we calculated the flux contribution in the energy
range of 3 – 50 keV due to nthComp and powerlaw separately.

Table 3: Model parameters of the RXTE and AstroSat observations fitted with
model constant$\times$TBabs$\times$(smedge$\times$nthComp + powerlaw)

.  Mission Obs ID/Orbit MJD $kT_{e}$ $\Gamma_{nth}$ $kT_{bb}$ $\Gamma_{PL}$
$\chi^{2}_{red}$ $Flux_{nth}$ $Flux_{PL}$ $Flux_{nth}\%$ Class (keV) (keV)
($\times 10^{-8}$erg$cm^{-2}s^{-1}$) ($\times 10^{-8}$erg$cm^{-2}s^{-1}$) RXTE
10408-01-06-00 50208.58 3.54$\pm$0.21 3.10$\pm$0.10 1.33$\pm$0.01 3.00* 0.98
3.96 0.02 99.49 $\gamma$ RXTE 80701-01-28-01 52933.69 2.95$\pm$0.10
2.28$\pm$0.07 1.12$\pm$0.06 2.63$\pm$0.04 0.90 4.08 0.25 94.22 $\delta$
AstroSat 3860 57553.88 2.83$\pm$0.06 2.45±0.03 0.1* 3.19$\pm$0.01 1.11 2.31
0.96 70.64 $\delta$ AstroSat 9895 57961.42 2.83$\pm$0.11 2.27$\pm$0.06 0.1*
3.07$\pm$0.03 1.09 1.20 0.48 71.42 $\kappa$ AstroSat 10583 58008.08 2.40*
1.85$\pm$0.04 0.1* 2.90$\pm$0.01 0.95 2.24 0.96 70.00 $\gamma$

  * *

Parameter is fixed.

We find that the flux contribution of nthComp in the AstroSat observation has
decreased nearly 30% from the RXTE observation. The reduction of the nthComp
flux that we observed during the AstroSat observation implies a smaller number
of Comptonised photons to the observer. Here, we expect a fraction of the
Comptonised photon to get down-scattered by the Keplerian disk before coming
to the observer and subsequently suffer a delay (soft lag) with respect to the
Comptonised photons that come directly.