# The NICER “Reverberation Machine”: A systematic study of time lags in black
hole X-ray binaries

Jingyi Wang11affiliationmark:  , Erin Kara11affiliationmark:  , Matteo
Lucchini11affiliationmark:  , Adam Ingram22affiliationmark:
33affiliationmark:  , Michiel van der Klis44affiliationmark:  , Guglielmo
Mastroserio55affiliationmark:  , Javier A. García 55affiliationmark:
66affiliationmark:  , Thomas Dauser66affiliationmark:  , Riley
Connors55affiliationmark:  , Andrew C. Fabian77affiliationmark:  , James F.
Steiner88affiliationmark:  , Ron A. Remillard11affiliationmark:  , Edward M.
Cackett99affiliationmark:  , Phil Uttley44affiliationmark:  , Diego
Altamirano1010affiliationmark:  11affiliationmark:  MIT Kavli Institute for
Astrophysics and Space Research, MIT, 70 Vassar Street, Cambridge, MA 02139,
USA
22affiliationmark:  Department of Physics, Astrophysics, University of Oxford,
Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
33affiliationmark:  School of Mathematics, Statistics and Physics, Newcastle
University, Herschel Building, Newcastle upon Tyne, NE1 7RU, UK
44affiliationmark:  Astronomical Institute, Anton Pannekoek, University of
Amsterdam, Science Park 904, NL-1098 XH Amsterdam, Netherlands
55affiliationmark:  Cahill Center for Astronomy and Astrophysics, California
Institute of Technology, Pasadena, CA 91125, USA
66affiliationmark:  Remeis Observatory & ECAP, Universität Erlangen-Nürnberg,
96049 Bamberg, Germany
77affiliationmark:  Institute of Astronomy, University of Cambridge, Madingley
Road, Cambridge, CB3 0HA
88affiliationmark:  Harvard-Smithsonian Center for Astrophysics, 60 Garden
St., Cambridge, MA 02138, USA
99affiliationmark:  Department of Physics & Astronomy, Wayne State University,
666 W. Hancock St, Detroit, MI 48201, USA
1010affiliationmark:  School of Physics and Astronomy, University of
Southampton, Highfield, Southampton, SO17 1BJ

###### Abstract

We perform the first systematic search of all NICER archival observations of
black hole (and candidate) low-mass X-ray binaries for signatures of
reverberation. Reverberation lags result from the light travel time difference
between the direct coronal emission and the reflected disk component, and
therefore their properties are a useful probe of the disk-corona geometry. We
detect new signatures of reverberation lags in 8 sources, increasing the total
sample from 3 to 11, and study the evolution of reverberation lag properties
as the sources evolve in outbursts. We find that in all of the 9 sources with
more than 1 reverberation lag detection, the reverberation lags become longer
and dominate at lower Fourier frequencies during the hard-to-soft state
transition. This result shows that the evolution in reverberation lags is a
global property of the state transitions of black hole low-mass X-ray
binaries, which is valuable in constraining models of such state transitions.
The reverberation lag evolution suggests that the corona is the base of a jet
which vertically expands and/or gets ejected during state transition. We also
discover that in the hard state, the reverberation lags get shorter, just as
the QPOs move to higher frequencies, but then in the state transition, while
the QPOs continue to higher frequencies, the lags get longer. We discuss
implications for the coronal geometry and physical models of QPOs in light of
this new finding.

accretion, accretion disks — black hole physics

##  1 Introduction

Black holes are the most extreme laboratories to study accretion and ejection
physics, and to eventually test theories of gravity. In addition to the
supermassive black holes at the centers of galaxies, there are also stellar-
mass black holes, which are mostly discovered when they are X-ray bright in
systems called X-ray binaries (XRBs). These are binary systems comprised of a
compact object accreting material from a companion star (see Remillard &
McClintock, 2006 for a review and references therein). If the mass of the
companion star is lower than roughly one solar mass, it is called a low-mass
XRB (LMXB; e.g., Bradt & McClintock, 1983). Depending on the nature of the
central compact object, the source is categorized as either a black hole XRB
(BHXB) or a neutron star XRB (NSXB).

Figure 1: The hardness-intensity diagram (HID), where the arrows show the
typical temporal evolution of BHLMXBs through the 4 different states (hard
state, HIMS, SIMS, and soft state) represented by different colors. The HID is
encompassed by representative flux-energy spectrum and power spectrum of each
state, using the NICER data of the BHLMXB MAXI J1820+070. The flux-energy
spectrum (solid line) is fitted phenomenologically with the sum of a thermal
disk emission (diskbb, dashed line), the coronal emission approximated by a
power-law (dash-dotted line), and the relativistic iron line approximated by a
Gaussian line (dotted line).

Typically, LMXBs are transient sources which go through cycles of activity
called outbursts (Homan & Belloni, 2005; Remillard & McClintock, 2006;
Tetarenko et al., 2016; and see Fig. 1). During the full outburst of a given
black hole low-mass X-ray binary (BHLMXB), the system starts from the
quiescent state with a very low mass accretion rate, and increases in
luminosity through the “hard state”, when X-ray emission is dominated by a
hard X-ray component that can be approximately described by a power-law with
photon index $\Gamma$ in the range of 1.4–1.8 and is usually attributed to the
non-thermal coronal emission. After several weeks, the spectrum can change
dramatically, and as the BHXB transitions, it progresses through the “hard
intermediate state” (HIMS), to the “soft intermediate state” (SIMS), before
switching to the “soft state” dominated by the thermal emission of the
accretion disk (see Fig. 1). The fractional rms can reach 30%–40% in the hard
state, but drops to only a few percent in the soft state. After spending weeks
to months in the soft state, the luminosity steadily drops, and the source
makes a transition back to the hard state (at a lower luminosity than the
hard-to-soft transition) and then eventually decays to quiescence. There is a
remarkable coupling between accretion and ejection in BHLMXBs, as the
properties of the radio jets strongly depend on the accretion states (Fender
et al., 2004, 2009). In the hard state, a compact and mildly relativistic jet
is almost always observed (e.g. Dhawan et al. 2000); and there is a
correlation between the radio and X-ray luminosities (Hannikainen et al.,
1998; Corbel et al., 2000; Gallo et al., 2003). During the hard-to-soft
transition, the radio luminosity decreases, suggesting that the compact jet is
slowly quenched, and at the so-called “jet line” (which loosely corresponds to
the boundary between the HIMS and SIMS), a highly relativistic, discrete and
ballistic jet is launched (Bright et al., 2020). During the soft-to-hard
transition and the decay in the hard state, when the compact jet is re-formed
is not well understood (e.g., Miller-Jones et al., 2012; Kalemci et al.,
2013).

Though we have observed the ubiquitous non-thermal X-ray emission from XRBs
for several decades, our understanding of its nature is still limited (Done et
al., 2007; Nowak et al., 2011). The widely accepted idea is that the X-ray
corona, a hot plasma located close to the black hole with electron temperature
of the order of $\sim 100$ keV and optical depth $\tau\approx 1$, produces the
power-law-like X-rays via thermal Comptonisation of the disk photons. However,
how the X-ray corona is generated in the first place, and how it evolves
throughout an outburst, are not well understood. The most frequently used
coronal models include: (1) an inner hot flow (e.g., advection-dominated
accretion flow; Esin et al., 1997; Narayan & McClintock, 2008) interior to a
truncated thin accretion disk (Mahmoud et al., 2019; Zdziarski et al., 2021),
and (2) the corona being the base of a jet (Markoff et al., 2001, 2005) or an
aborted jet (Ghisellini et al., 2004), the simplest realization of which is
the lamppost geometry (Matt et al., 1991, 1992; Dauser et al., 2013).

It then becomes advantageous to study the spectral-timing evolution of XRBs to
be able to probe the inner disk-corona region. With a technique called
reflection spectroscopy, we model the time-averaged flux-energy spectrum of
the disk material that is irradiated by the coronal photons. The most
prominent reflection features include the Fe K emission complex ($6-7$ keV),
the Fe K-edge ($\sim 7-10$ keV), and the Compton hump ($\sim 20-30$ keV), all
broadened by relativistic effects (Fabian et al., 1989; Matt et al., 1993;
García et al., 2014). However, in modeling the flux-energy spectrum alone,
limitations arise from degeneracies such as the model choice for the coronal
emission (Dziełak et al., 2019), and the positive correlation between the iron
abundance and the BH spin parameter (García et al., 2015). These prevent us
from unequivocally answering important questions, such as how truncated is the
disk in the bright hard state (e.g., Wang et al., 2020b) or how quickly is the
BH spinning (Dauser et al., 2013).

X-ray timing analyses can help break some of these degeneracies and build a
more complete picture by providing insights about the nature of variability.
Fourier timing techniques allow us to separate variability on different
timescales that are believed to result from distinct physical processes
(Uttley et al., 2014). Besides its unique signature in the flux-energy
spectrum, the reflection process can also be seen in the form of a time lag
feature, which is commonly referred to as reverberation. It arises because of
the light travel time delay between the direct coronal emission and the
reflected emission. Reverberation lags have been detected in both AGN and
BHXBs at relatively high Fourier frequencies, in the forms of variations in
the soft-excess emission and/or the Fe K emission line lagging behind those in
the energy band dominated by the coronal emission (e.g., Fabian et al., 2009;
Kara et al., 2016; De Marco et al., 2017). At relatively low Fourier
frequencies (when probing longer timescale variability), there is a ubiquitous
hard lag whose amplitude is too large to result from light travel time
difference; the most popular model is mass accretion rate fluctuations
propagating through the disk, moving from the colder to the hotter regions,
causing soft photons to respond before hard photons (Lyubarskii, 1997).

In a typical lag-frequency spectrum, the low-frequency hard lag (positive by
convention) has a power-law-like shape and decreases with frequency, making it
possible to see the soft lags (negative by convention) caused by reverberation
at high frequencies. The soft lag flips into positive above some frequency
$\sim 1/(2\tau_{0})$ due to phase-wrapping, where $\tau_{0}$ is the intrinsic
time delay between the coronal and reflected emissions. At even higher
frequencies, the lag oscillates around zero, again as a result of phase-
wrapping (see e.g., Fig. 7 in Emmanoulopoulos et al., 2014, and also Fig. 3e).
The observed lag amplitude depends both on $\tau_{0}$ and the level of
“dilution”. The lag-frequency spectrum is measured between two energy bands,
each dominated by the coronal or reflected emission. However, in reality, both
energy bands contain contributions from coronal and reflected emission. This
dilutes the time delay causing the observed lag amplitude to be smaller than
the intrinsic lag (see e.g., Kara et al., 2013; Uttley et al., 2014; De Marco
et al., 2015).

In addition to the time lags, another observable that may encode information
about the inner disk-corona geometry is quasi periodic oscillations (QPOs; see
the review Ingram & Motta, 2019, and references therein). Type-C QPOs are
strong and narrow peaks in the power spectrum. They are prominent in the
bright hard state and HIMS, with centroid frequency rising from a few mHz to
$\sim 10$ Hz as the sources evolve. Type-B QPOs are characterized by a
relatively strong and narrow peak with centroid frequencies of 5–6 Hz. The
most rare Type-A QPOs are weak and broad in the power spectrum. Type-B and
Type-A QPOs appear in the SIMS (Homan & Belloni, 2005). Therefore, a common
definition of the SIMS is the presence of the Type-B and/or Type-A QPOs. One
promising interpretation of the Type-C QPOs invokes the relativistic
precession model, in which the fundamental frequency is simply assumed to be
the Lense-Thirring precession frequency at some characteristic radius (e.g.,
the disk truncation radius within which the inner hot flow is precessing;
Ingram et al., 2009). As the Type-B QPOs are observed to be closely related to
the launch of discrete jet ejections (Homan et al., 2020), they are likely to
have a jet-based origin, e.g., the jet is launched from a precessing flow and
precesses with it (Motta et al., 2015; Stevens & Uttley, 2016; de Ruiter et
al., 2019).

A systematic study of reverberation lags has been performed for both Seyfert
galaxies (De Marco et al., 2013; Kara et al., 2016) and BHXBs (De Marco et
al., 2015). Intrinsically, as the number of counts within the variability
timescale over which we detect reverberation lags is much lower in BHXBs than
in Seyfert galaxies, reverberation lag detection is more challenging for
BHXBs. With the XMM-Newton archival data of 10 BHXBs, soft reverberation lags
were detected in two sources in the hard state: GX 339–4 and H1743–322 (De
Marco et al., 2015).

Now with the Neutron Star Interior Composition Interior Explorer (NICER;
Gendreau et al., 2016) we have the opportunity to systematically search for
reverberation signatures in BHXBs, and to globally study their evolution.
NICER is an ideal instrument for spectral-timing analysis because of its large
effective area in the 0.3–12 keV range, superior timing capabilities (i.e.,
high resolution and very little dead-time), with negligible pileup and low
background. In addition, NICER has fast slew capabilities, which allows us to
monitor BHXB systems at high cadence, throughout their outbursts. NICER has
observed several outbursts, some from newly discovered black holes, covering
all spectral states. Therefore, with the rich NICER archive, we can perform a
systematic search for reverberation lags.

Thus far, reverberation lags have been reported in two BHXBs with NICER. In
the hard state, reverberation lags are detected in MAXI J1820+070 (Kara et
al., 2019) and in GX 339–4 (during its 2017 and 2019 outbursts; Wang et al.,
2020a). In Wang et al. (2021a), we were able to trace reverberation lags into
the HIMS for the first time, thanks to the unprecedented data quality of MAXI
J1820+070. Our main finding was that the Fourier frequency range of the
reverberation signature decreases during the hard-to-soft state transition
with an increasing lag amplitude, possibly due to a vertically expanding
corona. We successfully tested this hypothesis with the publicly-available
reverberation model reltrans (Ingram et al., 2019; Mastroserio et al., 2021),
whereby the longer time lags during the state transition are explained with an
increasing height of the lamppost corona over the accretion disk. This X-ray
coronal expansion preceded a radio flare by $\sim 5$ days, hinting that the
corona is the base of the jet, and the hard-to-soft transition is marked by
the corona expanding vertically and launching a jet knot that propagates along
the jet stream at relativistic velocities. These X-ray reverberation results
were confirmed independently by De Marco et al. (2021).

Motivated by these reverberation results on individual sources, in this paper
we perform the first systematic search for reverberation lags in BH and BH
candidate (BHC) LMXBs observed with NICER. We perform a detailed study on
their evolution throughout the hard and intermediate states, to learn whether
the behavior seen in MAXI J1820+070 is common to other sources. The lag
properties we focus on are which Fourier frequency range is dominated by
reverberation and what the reverberation lag amplitudes are. This paper is
organized as follows. Section 2 describes the observations (sample and data
reduction) and our method to efficiently search for reverberation lags.
Section 3 focuses on the search results and the evolution of reverberation
lags. We present the discussion in Section 4, and summarize the results in
Section 5.

##  2 Observations and method

###  2.1 The sample

The systems of interest in our study are the BH and BHC LMXBs. We cross-
matched the sample in Tetarenko et al. 2016 (see Tables 8 and 12 therein) and
included newly discovered systems since then (private communications with John
Tomsick). In total, since its launch in 2017 until June 2021, NICER has
observed 26 BH/BHC LMXBs (see Table 1 and A1 for the full list).

Table 1: The BH/BHC LMXB sample that is observed by NICER and has good data
groups resulting from our reverberation machine, sorted by the peak count
rate.

Name |  Peak count rate1 |  Peak rms1 |  Exposure1 | States |  $\\#$ of good |  $\\#$ of soft | Grouping
---|---|---|---|---|---|---|---
|  (103 counts/s) |  ($\%$) | (ks) |  achieved2 | groups | lag detection |  criteria3
MAXI J1820+070 | 61 | 24 | 419.8 |  H,I,S | 88 | 84 | [0.3, 0.3, 0.5, 250]
MAXI J1348–630 | 29 | 21 | 132.9 |  H,I,S | 34 | 15 | [0.3, 0.3, 0.5, 250]
MAXI J1535–571 | 17 | 14 | 73.9 |  H,I,S | 14 | 12 | [0.3, 0.3, 0.5, 250]
MAXI J1631–479 | 4.7 | 8 | 80.2 |  I,S | 12 | 9 | [0.3, 0.3, 0.5, 250]
GX 339–4 | 4.5 | 27 | 172.5 |  H,I,S | 14 | 9 | [0.3, 0.3, 0.5, 250]
MAXI J1803–298 | 4.0 | 10 | 42.6 |  H,I,S | 2 | 1 | [0.3, 1, 1, 200]
MAXI J1727–203 | 3.3 | 12 | 21.5 |  H,I,S | 3 | 2 | [1, 1, 1, 150]
EXO 1846–031 | 0.99 | 16 | 85.5 |  H,I,S | 3 | 2 | [1, 1, 1, 100]
AT2019wey | 0.78 | 18 | 382.0 |  H,I | 22 | 3 | [0.3, 0.5, 0.5, 250]
GRS 1915+105 | 0.32 | 18 | 133.3 |  H,I | 3 | 2 | [0.3, 0.3, 0.5, 250]

Notes.
The count rates are for $0.3-12$ keV, and are normalized for 52 FPMs. The rms
is calculated also in $0.3-12$ keV, with the Fourier frequency range of 1–10
Hz. 1These values are calculated among the data in the good groups. As we do
not have good groups in the soft state (for their low variability) where count
rate is usually higher than the hard state, and also with great NICER
coverage, the peak count rate and exposure could be underestimated compared to
the values taking into account all the archival data. The peak rms could also
be underestimated when there is no good group in the faint hard state (for
their low count rate). 2The hard state, intermediate state, and the soft state
are represented by H, I, and S respectively; the bold indicates where the soft
lag is detected. 3The first three values correspond to the fractions within
which the relative changes of the hardness ratio, the count rate, and the
fractional variance are in each group (by default, 0.3, 0.3, and 0.5
respectively); the last value is the minimal ${\rm SNR}_{\Delta t,ind}$ we
count as a good group. See text in Section 2.3 for more details.

###  2.2 Data reduction

The NICER data are processed with the data-analysis software NICERDAS version
v2020-04-23_V007a, and energy scale (gain) release “optmv10”. We use the
following filtering criteria: the pointing offset is less than $60\arcsec$,
the pointing direction is more than $30^{\circ}$ away from the bright Earth
limb, and more than $15^{\circ}$ away from the dark Earth limb, and the
spacecraft is outside the South Atlantic Anomaly (SAA). Data are required to
be collected at either a sun-angle $>60^{\circ}$ or else collected in shadow
(as indicated by the “sunshine” flag). We filter out commonly-noisy detectors
FPMs #14, 34, and 54. In addition, we flag any “hot detectors” in which X-ray
or undershoot (detector resets triggered by accumulated charge) rates are far
out of line with the others ($\sim 10\sigma$) and exclude those detectors for
the GTI in question. We select events that are not flagged as “overshoot”
(typically caused by charged particle passing through the detector and
depositing energy) or “undershoot” resets (EVENT_FLAGS=bxxxx00), or forced
triggers (EVENT_FLAGS=bx1x000), and require an event trigger on the slow chain
which is optimized for measuring the energy of the event (i.e., excluding
fast-chain-only events where the fast chain is optimized for more precise
timing). A “trumpet” filter on the “PI-ratio” is also applied to remove
particle events from the detector periphery (Bogdanov, 2019). The resulting
cleaned events are barycenter corrected using the FTOOL barycorr. The
background spectrum is estimated using the 3C50 background model (Remillard et
al., 2021). GTIs with overshoot rate $>2$ FPM-1 s-1 are excluded to avoid
unreliable background estimation. We use the RMF version “rmf6s” and ARF
version “consim135p”, which are both a part of the CALDB xti20200722.

###  2.3 The “Reverberation machine” methodology

To maximize the statistical quality of the lag spectrum, data need to be
grouped. As the evolution of BHLMXBs happens on typical timescales of days to
months, the data grouping needs some criteria to group only the data with
similar spectral-timing properties. Therefore, we start from a NICER pilot
pipeline (Steiner et al. in preparation) which processes public NICER data and
extracts the data products for each good time interval (GTI), including the
flux-energy spectrum (both source and background), the light curves (LCs) and
the power spectral densities (PSDs) in several different energy bands.

The GTI-based library of data products enables efficient data grouping. We
group data such that the relative changes compared to the first GTI in each
group in the hardness ratio, the count rate, and the fractional variance
($F_{\rm var}$) are within a fixed fraction (by default, 30%, 30%, and 50%
respectively, but the exact values used for each source in our analysis are
quoted in Table 1); and the remaining detector combination after filtering out
the noisy and hot detectors needs to be the same. Specifically, the hardness
ratio is defined as the count ratio of the hard band (4–12 keV) and the soft
band (2–4 keV), and the count rate is in 0.3–12 keV. The fractional variance
is calculated in 0.3–12 keV and within 1–10 Hz, the range where reverberation
lags are expected for BHLMXBs.

Figure 2: We classify the spectral states and systematically compare the
reverberation lag properties using the power-spectral “hue” in the power color
diagram. (a) The hardness-intensity diagram of MAXI J1820+070, for which the
MJD corresponding to different spectral states are known from previous
spectral-timing analysis (Buisson et al., 2019; Homan et al., 2020; Buisson et
al., 2021). (b) The power color diagram of MAXI J1820+070 in NICER’s 4.8–9.6
keV band. The hue ranges defined in RXTE’s 2–13 keV band (boundaries between
states shown in grey dashed lines) could distinguish the hard and hard
intermediate states well. (c) The power color diagram for the data groups of
the 10 sources in which we detect soft lags. We note that the data points in
the power color diagrams in (b) and (c) represent only those data groups with
soft lags detected, and there is no good data group or soft lag detection in
the soft state.

As the NICER BHLMXB archive is large, and lag measurements for each data group
would be time-consuming, we introduce a “lag significance indicator” to filter
only groups which are promising to generate good quality lag spectra, i.e.,
those that will result in small lag uncertainties. The phase lag uncertainty
is

$\footnotesize\begin{multlined}\Delta\phi(\nu_{j})=\sqrt{\left(\frac{P_{X,\rm
noise}}{P_{X,\rm signal}}+\frac{P_{Y,\rm noise}}{P_{Y,\rm
signal}}+\frac{P_{X,\rm noise}P_{Y,\rm noise}}{P_{X,\rm signal}P_{Y,\rm
signal}}\right)/2M},\end{multlined}\Delta\phi(\nu_{j})=\sqrt{\left(\frac{P_{X,\rm
noise}}{P_{X,\rm signal}}+\frac{P_{Y,\rm noise}}{P_{Y,\rm
signal}}+\frac{P_{X,\rm noise}P_{Y,\rm noise}}{P_{X,\rm signal}P_{Y,\rm
signal}}\right)/2M},$

where $X$, $Y$ are the two energy bands between which lag is measured, “noise”
denotes the Poisson noise level, and “signal” denotes the Poisson-noise-
subtracted intrinsic PSD, and $M$ is the number of LC segments. The last term
dominates over the former two terms in the reverberation-dominated frequencies
of BHLMXBs (Uttley et al., 2014). Therefore, if the intrinsic lag amplitude is
similar and $P_{X,\rm signal}/P_{X,\rm noise}\propto P_{Y,\rm signal}/P_{Y,\rm
noise}$, the signal-to-noise ratio (SNR) of a lag measurement is

$\begin{multlined}{\rm SNR}_{\Delta t}\propto\sqrt{M}\cdot P_{\rm
signal}/P_{\rm noise},\end{multlined}{\rm SNR}_{\Delta t}\propto\sqrt{M}\cdot
P_{\rm signal}/P_{\rm noise},$

where $P_{\rm signal}$ and $P_{\rm noise}$ are the intrinsic PSD and Poisson
noise in one energy band. Using the “fractional rms-squared” normalization,
the integrated intrinsic PSD over a frequency range is the fractional variance
$F_{\rm var}$ due to variability in that range and the above estimate becomes

$\begin{multlined}{\rm SNR}_{\Delta t}\propto\sqrt{M}\cdot F_{\rm
var}\cdot\left<x\right>,\end{multlined}{\rm SNR}_{\Delta
t}\propto\sqrt{M}\cdot F_{\rm var}\cdot\left<x\right>,$

where $\langle x\rangle$ is the mean count rate. We therefore define the lag
significance indicator as

$\begin{multlined}{\rm SNR}_{\Delta t,ind}\doteq\sqrt{M}\cdot F_{\rm
var}\cdot\left<x\right>,\end{multlined}{\rm SNR}_{\Delta
t,ind}\doteq\sqrt{M}\cdot F_{\rm var}\cdot\left<x\right>,$

which serves as a quantitative measurement of the expected quality of the lag
spectrum. The longer the exposure, the more variable the LC is, the brighter
the source gets, the larger the lag significance indicator ${\rm SNR}_{\Delta
t,ind}$ is, resulting in a larger SNR of the lag measurement. For instance, a
data group of a fairly bright BHLMXB (count rate $\sim$200 counts/sec) with
total exposure of 10 ks ($M=1000$ when segment length is 10 s) and a typical
fractional rms of 20% in the hard state ($F_{\rm var}=0.04$) will have ${\rm
SNR}_{\Delta t,ind}\sim 250$. In the bright soft state, the fractional rms
drops sharply to $\sim 0.5\%$, so even if the count rate is as high as 10,000
counts/sec with a very long total exposure of 100 ks, ${\rm SNR}_{\Delta
t,ind}\sim 25$.

From our experience working on previous datasets of GX 339–4 (Wang et al.,
2020a) and MAXI J1820+070 (Wang et al., 2021a), we find roughly that an
indicator value of $\sim 250$ leads to a good-quality lag-frequency spectrum.
This is not a strict limit, but rather a tool to help automate the pipeline.
Therefore, for each data group, we calculate ${\rm SNR}_{\Delta t,ind}$ and
filter only the “good groups” reaching ${\rm SNR}_{\Delta t,ind}>250$. In
three sources (MAXI J1803–298, MAXI J1727–203, and EXO 1846–031), there was no
good group with the default grouping criteria. By relaxing the criteria on the
relative changes in hardness ratio, fractional variance, and the count rate,
there were a few groups that had indicator values slightly less than 250. We
checked the lag-frequency spectra of these sources by hand, and determined
that significant soft lags were present despite the slightly lower indicator
value, and so we also include these in the sample (see Table 1 for the data
grouping criteria used after trials).

We classify the spectral state using the “power-color” (PC) methodology of
Heil et al. (2015b). PCs enable the shape of power spectrum to be
characterised regardless of its normalisation in the same manner that spectral
colors characterise the shape of flux-energy spectrum. We define PC1 as the
integrated variance between 0.25–2.0 Hz divided by that between 0.0039–0.031
Hz, and PC2 is defined identically for the 0.031–0.25 Hz and 2.0-16.0 Hz
bands. Fig. 2 (b) and (c) shows the power-color diagrams (PC2 vs PC1) of MAXI
J1820+070 and all the 10 sources for which we detect soft lag (see Section 3.1
for more details). We see, in line with the results of Heil et al. (2015b),
that each outburst follows a circular path on the PC diagram and crucially,
each source follows the same path. This is known as the PC “wheel”, and
provides a very efficient way to define spectral states during an outburst. In
a full outburst, the source follows the PC wheel in the clockwise direction in
the hard-to-soft state transition (i.e., from the hard state, through the HIMS
and SIMS, to the soft state), and follows the same path but in the counter-
clockwise direction in the soft-to-hard state transition (i.e., from the soft
state, through the SIMS and HIMS, to the hard state). In particular, the hue
is defined as the clockwise angle from the axis defined at a 45∘ angle to the
x and y-axes. The center of wheel is chosen to be (PC1, PC2)=(4.5, 0.45). The
hue ranges for different spectral states are: hard state with hue from –20∘ to
140∘, HIMS with hue from 140∘ to 220∘, SIMS with hue from 220∘ to 300∘, and
the soft state with hue from 300∘ to 20∘ (see Table 2 in Heil et al., 2015b).
We note that the original investigation of hue uses the RXTE data in 2–13 keV,
and in our analysis, we use NICER data in 4.8–9.6 keV to match the hard X-ray
bandpass. We checked the data for MAXI J1820+070 for which the MJD
corresponding to the spectral states are known from previous spectral-timing
analysis (Buisson et al., 2019; Homan et al., 2020; Buisson et al., 2021), and
found that the hue defined in this manner is indeed able to distinguish
between hard and hard intermediate states (see Fig. 2 a–b).

For the good groups, we calculate the PSDs, the power-spectral hue, and the
lag-frequency spectrum measured between 0.5–1 keV and 2–5 keV. These two
energy bands are chosen such that the soft band shows the soft excess, and the
hard band is dominated by the coronal emission. We take light curve segment
lengths of 10 s for the lag-frequency spectrum and 256 s for the PSD (to match
the definition of hue), both with 0.001 s bins. If none of the GTIs in the
group is longer than 256 s, the segment length for the PSD is also chosen to
be 10 s; in this case, we still count the group as a good group if it meets
the before-mentioned criteria because the segment length for the PSD does not
affect the lag spectrum, and the only difference lies in the lack of a hue
measurement. The logarithmic frequency rebinning factor is 0.4 for the lag-
frequency spectrum, and 0.1 for the PSD. The former is chosen to increase the
SNR of a lag measurement, and the latter is to to measure the QPO frequency
more precisely. These choices do not change our results. Both the source and
background flux-energy spectra for each GTI in the data group are combined.
The energy resolution of the source spectrum is oversampled by a factor of 3,
and is then binned with a minimum count of 1 per channel. The response
matrices for different data groups are combined according to the detectors
used.

To keep the reverberation machine as automated as possible, we define criteria
for determining if reverberation is present in the lag-frequency spectrum. At
its most basic, we determine that reverberation is present if a significant
soft lag (i.e., a negative lag) is seen in the lag-frequency spectrum. In
detail, reverberation is present if at least two frequency bins show a
negative lag (i.e., a soft lag) with its absolute amplitude larger than its
uncertainty ($>2\sigma$ confidence equivalent).

It is further necessary to determine the frequency range over which a soft
reverberation lag is present, and that task is performed as follows. As
sometimes low statistics can lead to some points having lags consistent with
zero, to avoid fragmenting the reverberation frequency range, we require that
the range starts from a frequency bin with a negative lag, and continues to
higher frequencies as long as the associated lag is consistent with being
negative within $1\sigma$ confidence. This means the range ends when a
positive (hard) lag appears with its amplitude larger than the uncertainty
($\nu_{1}$). In addition, in order to avoid biases due to phase wrapping, we
define a “phase wrapping frequency cut-off” ($\nu_{2}$) as the frequency when
the absolute lag amplitude becomes more than half of the time lag
corresponding to the phase limit $\pm\pi$. The reverberation lag range ends at
either $\nu_{1}$ or $\nu_{2}$, whichever frequency is lower.

The lags at the QPO frequencies can behave differently from their neighboring
frequencies dominated by broadband noise, likely due to a different mechanism
producing the QPOs (Van den Eijnden et al., 2016; Zhang et al., 2020b).
Therefore, as in our earlier work (Wang et al., 2021a), we fit the PSD with
the sum of several Lorentzians, and calculate the lag-energy spectra outside
of the central frequency $\pm$ half-width at half-maximum (HWHM) of those
fitted QPOs111We choose the fundamental QPO to be the component with the
largest rms and a quality factor $Q=\nu_{0}/{\rm FWHM}>2$. and their
(sub)harmonics. The reference band for the lag-energy spectra is the full band
0.3–10 keV.

All spectral fitting is done with XSPEC 12.11.1 (Arnaud, 1996). We use the PG-
statistics, for Poisson data with a Gaussian background. We use the wilm set
of abundances (Wilms et al., 2000), and vern photoelectric cross sections
(Verner et al., 1996), to account for the galactic absorption with the model
tbabs.

Figure 3: Reverberation machine summary plot for GX 339–4 (see Section 3.1 for
more details). Representative data groups in the bright hard state (blue) and
during the hard-to-soft state transition (red). (a) The GTI-based NICER
hardness-intensity diagram (HID) where the hardness ratio is defined as the
count rate ratio of the hard band (4–12 keV) and the soft band (2–4 keV), and
the count rate is in 0.3–12 keV and normalized for 52 FPMs. Each dot
represents a single GTI in the data group. (b) The 0.3–10 keV PSDs with the
shaded bands indicating the frequencies of the QPO and (sub)harmonics (central
frequency$\pm$HWHM). (c) The unfolded flux-energy spectra to a constant model.
(d) The data-to-model ratio when the entire 3–10 keV flux-energy spectra are
fitted with a (diskbb+powerlaw) model. The two dashed lines correspond to the
energies of Fe K$\alpha$ at 6.4 keV and H-like Fe XXVI at 6.97 keV. (e)–(f)
The lag-frequency spectra measured between 0.5–1 keV and 2–5 keV. Shaded
regions are where a soft reverberation lag is detected. The dashed lines
represent the time lags corresponding to phase lags of $\pm\pi$. (g)–(h) The
lag-energy spectra in the frequency ranges where we detect soft lags. The two
dashed lines are the same as in (d), corresponding to the energies of Fe
K$\alpha$ at 6.4 keV and H-like Fe XXVI at 6.97 keV.

##  3 Results

###  3.1 Reverberation lag detections

Following the procedures in Section 2.3, we run our autonomous reverberation
search algorithm (“reverberation machine”) over the sample. We find 10 sources
with good data groups (see Table 1), and 16 sources with no good data group
(see Table A1). Moreover, all the 10 sources with good groups have soft lag
detections (see the detection criteria in Section 2.3). For each source with a
reverberation lag detection, we provide a brief introduction to the source
properties, the outburst timeline, and the reverberation machine results
including the number of good groups, the states covered, and the reverberation
lag detections. We also present a summary figure that demonstrates what the
data look like in the luminous hard state, and in the state transition (if
available). As a case study, we show in this section the results for GX 339–4
which recently went on its first full outburst since NICER’s launch. The
results for the other 9 sources are detailed in the Appendix.

GX 339–4 is an archetypal BHLMXB which outbursts every 2–3 years (e.g.,
Belloni et al., 2005; Motta et al., 2009; see also Fig. 1 in García et al.,
2019). The mass of the BH is constrained to be between 2.3 and 9.5
$M_{\odot}$, and our distance towards the system is $d>5$ kpc (Heida et al.,
2017). It is a mildly absorbed system with the column density of $N_{\rm
H}\sim 6\times 10^{21}$ cm-2 (Zdziarski et al., 1998). The inclination via
reflection spectroscopy is estimated to lie in the range of 30–48 degrees
(e.g., García et al., 2015; Wang-Ji et al., 2018), which makes it a famous
low-inclination source. GX 339–4 is significant because it was the first to
show evidence for a reverberation feature in the hard state with XMM-Newton
(Uttley et al., 2011; De Marco et al., 2015, 2017). In the previous two hard-
only outbursts in 2017 and 2019, Wang et al. (2020a) detected, for the first
time with NICER, a soft reverberation lag with amplitude of $9\pm 3$ ms in the
Fourier frequency range 2–7 Hz, in the brightest epoch available at that time.

In January 2021, GX 339–4 underwent its first full outburst in the NICER era
(starting from obsID 3133010101 on 2021-01-20) and started the hard-to-soft
state transition some time between 2021-03-20 and 03-26 (MJD 59293 and 59299,
corresponding to obsID 4133010102 and 4133010103; Wang et al., 2021b).
Following the procedures described in Section 2.3, we identify 14 good groups,
and a soft reverberation lag is detected in 9 of them, in both the hard and
intermediate states (covering data until obsID 4133010105 on 2021-03-28). An
example of the reverberation machine summary figure is shown in Fig. 3 for GX
339–4 which includes (a) the hardness-intensity diagram (HID) with a
representative hard state group and an intermediate state group shown in blue
and red, respectively, (b) the full-band PSDs and any possible QPO feature,
(c) the unfolded flux-energy spectra to a constant model, (d) the data-to-
model ratio when the 3–10 keV flux-energy spectra are fitted with a
(diskbb+powerlaw) model to see phenomenologically the iron line profile,
(e)–(f) the lag-frequency spectra, and (g)–(h) the lag-energy spectra in the
frequency ranges we detect soft lags. We see that the intermediate state
exhibits a much larger lag amplitude that dominates at a lower frequency
compared with the hard state; similar to what was previously observed for MAXI
J1820+070 (Wang et al., 2021a). We note that for clarity in the summary plots,
we show only the lag-energy spectra in the full Fourier frequency band in
which we detect soft lags (i.e., not outside of the QPO frequencies), but an
example of the lag-energy spectra in the discontinuous frequency bands outside
of QPO frequencies are shown in Fig. A1. We also exclude the measured lag
within QPO frequencies when we estimate the reverberation lag amplitude for
each data group in Section 3.2.

The non-detections in the other 5 good groups of GX 339–4 are due to tentative
soft lags in the lag-frequency spectra that do not meet the $2\sigma$
confidence criteria. Examples of these low-significance lags can be found in
Wang et al. (2020a). We also note the detection of a soft lag depends on the
chosen energy bands between which we measure the lags. The analysis presented
in this systematic study adopts a uniform choice of a hard band of 2–5 keV
(with soft band 0.5–1 keV). We have also tested a hard band of 1.5–3 keV as
used in Wang et al. (2020a, 2021a), and in that case, the soft lags in the
lag-frequency spectra become more significant for the groups in the hard
state, and less significant in the intermediate state. Fundamentally, this
difference results from the different shapes of the lag-energy spectra, i.e.,
the corresponding energy of the “valley” in which the lag is negative is
approximately 1.5–3 keV in the hard state, and becomes closer to 2–5 keV
during state transition (see panels g and h in Fig. 3).

###  3.2 The evolution of reverberation lag properties throughout the
outbursts

Figure 4: A systematic look at the detected soft reverberation lags in the 10
BH/BHC LMXBs in which we detect significant soft lags. The evolution of soft
lag amplitude with the power-spectral hue (see Sections 2.3 and 3.2 for hue
and lag amplitude estimates respectively). The reverberation lag becomes
shorter in the hard state when hue$>20$ deg, and becomes longer after the
state transition (HIMS shaded in grey and SIMS in blue).  Figure 5: A
systematic look at the detected soft reverberation lags and the QPO
frequencies in the 10 BH/BHC LMXBs in which we detect significant soft lags.
(a) The upper bound of the frequency band over which we observe a soft
reverberation lag, and (b) the QPO frequency. These two quantities both
increase in the hard state (suggesting they are correlated), while they show
opposite trends after state transition.  Figure 6: (a) The soft lag amplitude
and (b) the full Fourier frequency band where we detect soft lag rebinned by
hue (i.e., averaged over continuous ranges of hue with widths equaling 15
degrees), for the 8 BH/BHC LMXBs excluding EXO 1846–031 and MAXI J1803–298
(see discussion in Section 4.3). The soft lag amplitudes for each individual
source are also plotted in (a). The colors matching the colorbar in (b)
indicate the soft lag amplitudes shown in (a). The uncertainties in (a) are
the statistical errors of soft lag amplitudes in each range of hue. We also
note that the vertical lines in (b) represent the full Fourier frequency band
where we detect soft lag, not the upper bound of that frequency band as shown
in Fig. 5a. In the hard state, the reverberation occurs at increasing
frequencies, with a decreasing lag amplitude; while after the state transition
(HIMS is shaded in grey and SIMS in blue), the reverberation happens at much
lower frequencies, with a much larger lag amplitude.

With reverberation lags detected in 10 different BH/BHC LMXBs (see Table 1),
we can perform a systematic comparison of the reverberation lag properties
from source to source, which could help us to understand the disk-corona
geometry across spectral states.

As mentioned in Section 2.3, we use the power-spectral hue as an indicator of
the state the sources are in during their outbursts (see Section 2.3). The
reverberation lag amplitude is approximated with a variance-weighted average
of the soft lag amplitude in the lag-frequency spectrum (i.e., the soft lag
amplitude is weighted based on its uncertainty $\sigma$ in each frequency bin
we detect a soft lag, by a factor of $1/\sigma^{2}$). We exclude a frequency
bin if its centroid lies within the QPO (and their harmonics or sub-harmonics)
centroid frequency $\pm$ HWHM.

The resulting soft lag amplitude of the 10 BH/BHC LMXBs as a function of the
hue is shown in Fig. 4. Our 10 sources show a remarkably similar reverberation
lag behaviour as a function of spectral state. We discuss this in more detail
later in Section 4.1. As the averaged lag amplitude depends on the level of
dilution, a useful quantity to measure the reverberation lag properties is the
upper bound of the frequency band where we detect a soft lag, which should not
be affected by dilution (while the lower bound can be affected by a change in
the low-frequency hard lags). The evolution of the upper bound of the soft lag
frequency band across values of hue is shown in Fig. 5a, together with the
fundamental QPO centroid frequency in Fig. 5b. Except for two outliers in the
hard state (EXO 1846–031 and MAXI J1803–298, see Section 4.3 for the
discussion), all sources appear to follow the same evolution of soft lag
amplitude with hue. We therefore average over all sources except for the two
outliers by binning on hue. The resulting rebinned evolution of the frequency
and amplitude of the soft lag as functions of hue for the remaining 8 sources
are shown in Fig. 6, and the soft lag amplitude as a function of QPO frequency
is shown in Fig. 7. We summarize the findings as follows:

Figure 7: The soft lag amplitude versus the QPO centroid frequency rebinned by
hue. In the hard state, the QPO frequency is $<1$ Hz, and the soft lag
amplitude decreases with QPO frequency, as expected if QPO frequency and the
frequency where reverberation is detected are correlated. However, QPO and
reverberation become anticorrelated in the intermediate state. The
uncertainties are the statistical errors of soft lag amplitudes in each range
of hue. See Section 4.2 for the implication.

  * •

In the hard state (hue $<140^{\circ}$), as hue increases, we see an increase
in soft lag amplitude at the beginning (when hue is $\lesssim 20^{\circ}$, see
Section 4.4 for the discussion), and then all the 3 sources with more than 1
reverberation lag detection in this state (GX 339–4, MAXI J1820+070, and MAXI
J1348–630) agree on a steadily decreasing trend of lag amplitude ($\sim 2$ ms
to $\lesssim 1$ ms, see Fig. 4 and Fig. 6a). Meanwhile, we see a tentative
increase of soft lag frequency with hue from tens of Hz to $\sim 100$ Hz (see
Fig. 5a and Fig. 6b). The evolution of both quantities indicates that the size
of the region causing reverberation (i.e., the inner disk-corona region)
reduces as the hard state evolves. This is consistent with what has previously
been reported for MAXI J1820+070 and interpreted as a contracting corona in
the luminous hard state (Kara et al., 2019; Buisson et al., 2019). The QPO
frequency increases from $\sim 0.1$ Hz to $\sim 1$ Hz (Fig. 5b). This can
explain why both the Type-C QPO and reverberation evolve to shorter timescales
(higher frequencies) before the transition to HIMS. The correlation between
QPO and reverberation can also be seen in Fig. 7 as long as the QPO frequency
is $<1$ Hz, as a higher upper bound of the frequency band over which
reverberation occurs generally comes with a shorter lag.

Figure 8: The NICER HIDs for the BH/BHC LMXBs in which we detect soft
reverberation lags (GRS 1915+105 is not included because of its unconventional
spectral states). (a) The original HIDs, in which the high galactic absorption
of MAXI J1535–571 (purple), MAXI J1631–479 (orange), and EXO 1846–031 (pink)
make their HIMS appear in the same hardness as the hard state in other
sources. (b) The combined HID (grey) after making linear transformations for
both axes in the 8 sources except for the benchmark MAXI J1820+070. The
transformation aims at making observations with similar hues lie in the same
region of the HID. The values of hue for the data groups with soft
reverberation lag detected are shown with colors matching the colorbar.

  * •

In the HIMS ($140^{\circ}<$ hue $<220^{\circ}$), as the hue increases, the
reverberation lag amplitude shows a sharp increase by a factor of 10, from
$\lesssim 1$ to $\sim 10$ ms (see Fig. 4 and Fig. 6a). Of the 9 sources with
more than 1 reverberation detection, all agree with this increase. In EXO
1846–031, the only hard state detection has an anomalously high lag amplitude
and correspondingly low upper bound frequency for which reverberation
dominates (see Section 4.3 for the discussion). The reverberation lag
amplitudes are mostly consistent (within uncertainties) with MAXI J1820+070,
which has the best coverage and the most detections. At the same time, the
upper bound of the frequency where reverberation is observed drops from $\sim
100$ Hz at the end of hard state to $\sim 10$ Hz (see Fig. 5a). In Fig. 6 (b),
we see it is not only the upper bound of the reverberation lag frequency that
decreases, but also the lower bound, i.e., we could safely say that the
reverberation signal evolves to lower frequencies. This result agrees with our
MAXI J1820+070 result (Wang et al., 2021a) and provides further evidence for a
longer reverberation lag in the HIMS, and is consistent with a growing disk-
corona distance, possibly due to a vertically expanding corona sitting at the
base of a jet. By comparing the timescale of the reverberation lag to that of
the Type-C QPO, we see that unlike in the hard state, during the state
transition, the Type-C QPOs continue to evolve to higher frequencies (from
$\sim 1$ Hz to $\sim 10$ Hz), but the reverberation lags, instead, get longer
(see Fig. 7 when the QPO frequency is $>1$ Hz). We discuss implications of
these trends between QPO and reverberation in Section 4.2.

  * •

In the SIMS (hue $>220^{\circ}$), out of the 10 sources in our sample, we have
only 2 good groups (in MAXI J1820+070 and MAXI J1348–630) because of the
sudden decrease in the variability level compared to the HIMS. A reverberation
lag is detected in both data groups, and the mean lag amplitude is even larger
($\sim 0.03$ s) compared to the HIMS.

Figure 9: The (left) soft lag amplitude and (right) the QPO centroid frequency
in the combined HID (grey). During the state transition, the reverberation lag
amplitude increases from $\sim 1$ ms to $\sim 10$ ms. The QPO frequency
increases in the hard state and the HIMS. Notice the values of soft lag
amplitudes and QPO frequencies are the same as those shown in Figs. 4 and 5b,
and are represented by the colors matching the colorbars. In addition to the
different colors matching the colorbars, the marker sizes also grow with the
soft lag amplitude and QPO frequency. See Fig. 8 for more details on the
combined HID.

To compare the reverberation lag amplitude and the QPO frequency in a more
intuitive way, we attempt to unify the HIDs for the 9 sources (except for GRS
1915+105, due to its nonstandard spectral evolution). Simply overplotting the
raw HIDs (Fig. 8a) can be misleading because each source is at a different
distance, with a different associated galactic absorption, etc. This means,
for example, that a hardness of 0.3 for MAXI J1820+070 corresponds to its hard
state (according to the power-spectral hue), but the same hardness for MAXI
J1535–571 corresponds to the HIMS because MAXI J1535–571 is so heavily
obscured. We attempt to scale all sources so that each state (defined by its
hue) matches the HID of our best sampled source, MAXI J1820+070. Five of the
sources (GX 339–4, MAXI J1348–630, MAXI J1727–203, AT2019wey, and MAXI
J1803–298) mostly only require a shift in the count rate axis to match the
benchmark (likely simply due to the fact that they are at different
distances), but the other three sources (MAXI J1535–571, MAXI J1631–479, and
EXO 1846–031) require both a shift in hardness ratio and count rate. Indeed,
these three sources have larger Galactic column densities compared to the rest
of the sample (in unit of $10^{22}$ cm-2, $N_{\rm H}\sim 5$ in MAXI J1535–571,
Xu et al., 2018; $N_{\rm H}\sim 3$ in MAXI J1631–479, Monageng et al., 2021;
$N_{\rm H}\sim 11$ in EXO 1846–031, Draghis et al., 2020). The larger galactic
absorption diminishes the count rate in the 2–4 keV band when calculating the
hardness ratio, and therefore shifts the hardness ratio to higher values,
which is exactly what we observe in Fig. 8 (a).

In the 8 sources except for the benchmark MAXI J1820+070, we make linear
transformations for both axes, so that observations with similar hues lie in
the same region of the HID. The result of this transformation can be seen in
Fig. 8 (b). The final result is shown in Fig. 9, where the soft lag amplitudes
and the QPO frequencies are presented on the combined HID. One can clearly see
that the reverberation lags become much longer in the state transition, while
the QPO frequencies continue to increase, which is also shown in Fig. 7. We
note that the linear transformation of HID is only for the purpose of
presenting the reverberation lag amplitudes and QPO frequencies in a canonical
HID. The values of lag amplitudes and QPO frequencies are independent of the
linear transformation of HID.

##  4 Discussion

With our reverberation machine designed for systematic reverberation lag
searches, we have detected soft reverberation lags in 10 BH/BHC LMXBs in the
NICER archive. This work has not only increased the sample size by a factor of
5, but also expanded our horizons from the hard state to the intermediate
state. We compare the evolution of the soft lag amplitudes in different
outbursts in a systematic way, and find that the reverberation lags are short
in the hard state (lag amplitudes of $\sim 1$ ms) and increase by a factor of
$\sim 10$ during the state transition (e.g., see Fig. 4).

###  4.1 The disk/corona/jet geometry in the state transition

In the most naive picture, the reverberation lag, $\tau$, relates to the
distance between a lamppost corona and reflector, $d$, as $\tau=(d/c)[1+\cos
i]$ where $i$ is the inclination angle. Therefore, for a given intrinsic disk-
corona geometry in units of $R_{g}$, a higher BH mass (i.e., a larger $R_{\rm
g}$) leads to a longer reverberation lag. Alternatively, for a given BH mass,
the lag increases with increasing $d/R_{g}$, which is related to intrinsic
change of the disk-corona geometry in units of $R_{g}$.

Among our sample, MAXI J1820+070 has the most reliable measurements of its BH
mass ($8.48^{+0.79}_{-0.72}$ $M_{\odot}$; Torres et al., 2020) and inclination
(63$\pm$3 degrees; Bright et al., 2020). Assuming these values, a
reverberation lag amplitude of 1 ms corresponds to a light travel time of
$\sim 16$ $R_{\rm g}/c$ in the lamppost geometry. Then, a decrease of
reverberation lag amplitude from 2 ms to $1$ ms in the hard state corresponds
to a decrease in the distance between the corona and the inner disk of $32$
$R_{\rm g}$ to 16 $R_{\rm g}$. For MAXI J1820+070, we adopted the
reverberation model reltrans to fit the lag-energy spectra in several
frequency bands, and measured the coronal height in the lamppost geometry to
decrease from 40 to 27 $R_{\rm g}$ in the hard state (Wang et al., 2021a). The
reltrans model computes the reverberation lags from the light travel times of
the photons using general relativistic ray tracing, and accounts self-
consistently for the dilution. The estimate from the proper modeling is close
to the naive estimate, and the discrepancy within a factor of 2 is mostly due
to dilution.

Then, during the state transition, the increase of reverberation lag amplitude
to 10 ms leads to a disk-corona distance of $\sim$140 $R_{\rm g}$. This is of
the same order of magnitude as the fitted coronal height for MAXI J1820+070
with reltrans (starting from $\sim 100-200$ $R_{\rm g}$ and eventually to
$\gtrsim 300$ $R_{\rm g}$; Wang et al. 2021a). Moreover, since MAXI J1820+070
has the closest-in-time radio coverage to the X-ray state transition, we were
able to determine that the radio flare caused by the ballistic transient jet
lagged behind the X-ray corona expansion by $\sim 5$ days. This finding
suggests that the state transition is marked by the corona expanding and
ejecting a jet knot that propagates along the jet stream at relativistic
velocities, supporting the case for the corona being the base of the jet (see
the schematic referred to as “intermediate state” in Fig. 10).

Remarkably, we find that in all the 9 sources with more than 1 reverberation
lag detection, as previously reported for MAXI J1820+070 (Wang et al., 2021a),
the reverberation lag gets longer in the state transition. This result shows
this trend is a generic feature of state transitions of BHLMXBs, not depending
on, e.g., inclination, spin, or mass. Unfortunately, we cannot test for a
correlation between lag amplitude and BH mass, inclination, or spin, as most
of our sample consists of new transients lacking robust measurements of these
quantities. For different BHLMXBs, the dependency of the lag amplitude on the
BH mass and inclination can be seen from the naive picture at the beginning of
this section. The lag amplitude could also depend on the BH spin because it
could change the disk-corona distance in units of $R_{g}$ in two ways: (1) the
innermost stable circular orbit (ISCO) radius monotonically decreases with the
BH spin; (2) as the jet could be powered by the BH spin (see e.g., Narayan &
McClintock, 2012; Steiner et al., 2012), the geometry of the corona (as base
of a jet) could thus be related to the BH spin. In addition, the inclination
could only cause a difference in lag amplitude by a factor of 2 in the naive
picture, and the BH mass is usually the same within a factor of 2. That is to
say, the reverberation lag amplitude is expected to have a narrow dynamic
range, making it very challenging to test for its correlation with physical
quantities.

The generic trend of reverberation lag getting longer is highly suggestive of
what happens in such state transitions. The reverberation lag evolution during
the state transition cannot result merely from dilution, because the upper
bound of the Fourier frequency where reverberation is detected also evolves to
lower values (see Fig. 5a and 6b). This upper frequency bound is independent
of dilution. Therefore, a growing size of the disk-corona region that causes
reverberation is necessary. In our modeling of MAXI J1820+070 data, we find an
increase in the coronal height possibly due to a vertically expanding corona.

We also note that a new time lag model for accreting BHs could explain the
hard lags and soft (reverberation) lags with a relatively compact corona
(Uttley & Malzac 2021, in preparation; see the poster). Uttley & Malzac
suggested that the power-law pivoting delay which produces the hard power-law
continuum lags, modelled empirically in reltrans, is due to the delay expected
between seed photon variations and coronal heating as the driving mass
accretion fluctuations propagate from the disk (providing the seed photons) to
the corona (where coronal heating occurs). In this scenario, hard lags are
produced at lower frequencies and soft lags are produced at higher
frequencies, due to accretion propagation effects. The soft lags result from
the lag of the reverberation signal (which responds primarily to the power-law
emission in the harder X-rays, driven mainly by coronal heating) relative to
the seed photons from fluctuations propagating through the disk (which cause
the power-law flux in the soft band to rise before the power-law flux in the
hard band). However, these lags also depend on the coronal geometry, so that
more vertically extended coronae show larger soft lags, i.e., the inferred
geometric change is the same as for the reverberation lags being due to light-
travel times only. However, in the case where the soft lags are linked to
propagation, the vertical scales required to produce the lags are reduced.

The idea of the corona expanding in state transition was previously suggested
on different grounds (Pottschmidt et al., 2000; Nowak et al., 2002). The
authors used RXTE data of Cygnus X–1 and GX 339–4 and found longer hard lags
during state transition. The interpretation was that these prolonged lags were
produced in spatially very extended synchrotron-emitting outflows. More
recently, this result was generalized into the photon-index-time-lag
correlation in BHXBs (e.g., Grinberg et al., 2013; Reig et al., 2018). With
NICER’s soft X-ray bandpass and large effective area, we are now able to find
prolonged soft reverberation lags in state transitions, which provide more
direct evidence of the expanding corona since the mechanism that produces hard
lags is not well understood. In addition to longer lags, recent broadband
spectral modeling for several BHLMXBs finds that the radius of the corona
which is assumed to be the base of the jet increases from $\sim 10$ $R_{\rm
g}$ to $\sim 40$ $R_{\rm g}$ through the hard state and the HIMS; and the
reflection fraction occasionally becomes very low in the HIMS when approaching
the SIMS, suggesting an increase in also the vertical extent of the corona
and/or its bulk velocity (Lucchini et al. 2021; Cao et al. 2021, but see
Steiner et al. 2017).

###  4.2 The QPO/reverberation connection

We compare the timescale of the reverberation lag to that of the Type-C QPO in
Fig. 5. In the hard state, the reverberation lags evolve to shorter timescales
(higher frequencies) before the transition. Similarly, the Type-C QPOs also
evolve to higher frequencies. In fact, all characteristic frequencies in the
power spectrum increase during the outburst, not just the QPO frequencies
(Wijnands & van der Klis, 1999; Psaltis et al., 1999). However, during the
state transition, the Type-C QPOs continue to evolve to higher frequencies,
but the reverberation lags, instead, get longer (Fig. 7). In short,
reverberation and Type-C QPO are correlated in the hard state, but they
anticorrelate during the state transition. By modeling the reflection spectrum
with a dual-lamppost model (two lampposts at different coronal heights) to
mimic a vertically extended corona, Buisson et al. (2019) also found the upper
coronal height to decrease first in the bright hard state, and then increase
when approaching the state transition in MAXI J1820+070, while the
characteristic features of the temporal variability kept increasing in
frequency.

Type-C QPOs’ amplitudes show great evidence for an inclination dependence
(e.g., Motta et al., 2015; Heil et al., 2015a), which suggests that they arise
from some geometric effect, e.g., Lense-Thirring precession of an inner hot
flow or corona (Ingram et al., 2009). The evolution of the QPOs to higher
frequencies has been proposed as evidence for a smaller inner hot flow or
central corona (along with a less truncated disk) before the source reaches
the soft state. This could also explain the increasing disk blackbody
temperature, the softening coronal emission with a decreasing coronal
temperature (Done et al., 2007), and also the shorter reverberation lags in
the bright hard state (Kara et al., 2019). However, the anticorrelation of the
QPO frequency and reverberation lag in the state transition breaks this simple
picture.

If both the QPOs and reverberation lags depend mostly on the geometry of the
corona-inner-disk system as discussed before, the anticorrelation seems to
suggest that the corona is both radially and vertically extended. While the
QPO frequency is mostly coupled to the radial extent, the reverberation lag is
coupled to both the radial and vertical extents. The idea that the corona has
both radial and vertical extents is very natural, and is previously suggested
by reverberation features in AGN, which require both a radially extended
corona over the disk surface and a vertically extended one sitting at the
launching point of a jet (Wilkins et al., 2016, 2017). If the QPO is produced
by the Lense-Thirring precession of an inner hot flow and/or the jet (Ma et
al., 2021; Liska et al., 2021), which shrinks in the radial dimension before
the source reaches the soft state, the longer reverberation lags require an
additional component of the corona on the vertical axis of the BH that expands
during the state transition. In the narrow line Seyfert 1 galaxy Mrk 335, the
emissivity profile of the reflection spectrum indicates the radially extended
corona becomes gathered up in the center and is then launched away from the
BH, reminiscent of a jet, during a flaring event (Wilkins & Gallo, 2015). This
scenario is very similar to the physical picture we propose here to explain
the evolution in the QPO frequency and reverberation lag in the hard state and
the state transition of BHLMXBs. More detailed spectral-timing modelings
assuming extended coronal geometries and incorporating QPO mechanisms are
required to understand the QPO/reverberation connection.

Figure 10: (Upper) The measured soft lag amplitudes in the combined linearly-
transformed HID (same as Fig. 9a). The luminosity in Eddington units is
obtained assuming the BH mass ($8.48^{+0.79}_{-0.72}$ $M_{\odot}$; Torres et
al., 2020) and distance (2.96$\pm$0.33 kpc) of MAXI J1820+070 as it is the
benchmark whose HID is not linearly transformed. (Lower) The evolution and
geometry of the disk-corona-jet system we infer in the faint hard state,
bright hard state, and the intermediate state. In the faint hard state
($<1-2\%$ $L_{\rm Edd}$), the disk is truncated at several tens of $R_{\rm g}$
and the inner disk fills in, causing the reverberation lag to become shorter
(see Section 4.4 and Fig. 11). In the bright hard state ($>4\%$ $L_{\rm
Edd}$), the disk is no longer truncated, and the X-ray corona vertically
contracts from several tens of $R_{\rm g}$ to $\sim 10$ $R_{\rm g}$ (see
Section 4.1 and Fig. 11), while the steady jet fades. In the intermediate
state, the X-ray corona expands to $\sim 100$ $R_{\rm g}$, while the steady
jet is quenched and the transient jet is ejected (see the discussion in
Section 4.1). In the soft state, there is no good data group (due to the low
variability) or soft lag detection, but we note that there is a great amount
of evidence that the disk extends to the ISCO radius (e.g., Steiner et al.,
2010); the disk emission (with a disk temperature of $\sim 1$ keV) dominates
over the coronal emission (Done et al., 2007), and the radio jet is off
(Fender et al., 2004). During the soft-to-hard state transition and the
following decay in the hard state, the soft lag amplitudes are consistent with
those during the rise at earlier time in the outbursts. However, as the number
of detection at this stage is limited, future data and more detailed modeling
are needed to test for any hysteresis effect during the rise and decay in the
hard state, or in the upper and lower branches of the intermediate state.

###  4.3 Outliers with long reverberation lags

In the hard state, there are two outliers that show long soft lags of $\sim
10$ ms compared to the best-sampled sources (MAXI J1820+070, GX 339–4, and
MAXI J1348–630) with $\sim 1$ ms soft lags (see Fig. 4). At the same time, the
upper bound of the soft lag frequency is only $\sim 10$ Hz, which is 10 times
smaller than that of MAXI J1820+070 (see Fig. 5a). These are EXO 1846–031
(Fig. A7) and MAXI J1803–298 (Fig. A10). The reason for the much longer
reverberation lag amplitude in EXO 1846–031 is likely that it has the largest
column density among the 10 sources in our sample ($N_{\rm H}\sim 10^{23}$
cm-2). The absorption diminishes significantly the photons at soft X-rays
($<1$ keV), and therefore the lag spectral quality is not adequate to
confidently detect the reverberation lag and measure its amplitude, especially
when we search for reverberation in the soft band of 0.5–1 keV. The limit of
lag spectral quality can be seen from the large uncertainties in the lag-
frequency spectrum where soft lag is detected, and the lag-energy spectrum
below 1 keV in Fig. A7. The soft lag amplitude is measured to be $16\pm 11$
ms, also with a large uncertainty. In fact, we find the 0.5–1 keV band is
dominated by the shelf of the response matrix with such high absorption. This
means that the photons registered in these energy channels are actually the
photons redistributed downward from higher energies, most likely $\sim 2$–3
keV which is the peak area band with remaining signal. However, we note this
effect does not make our soft reverberation lag detection artificial because
an intrinsic hard lag (i.e., the hard band when measuring the lag-frequency
spectrum, 2–5 keV, lags behind the $\sim 2$–3 keV band) remains a hard lag.
Encouragingly, we observe a very promising iron lag whose energy is not
affected by the high galactic absorption in EXO 1846–031. If, instead, we use
the tentative iron lag amplitude, this source is no longer an outlier in our
sample.

The other outlier, MAXI J1803–298, is not affected by large galactic
absorption ($N_{\rm H}\sim 3\times 10^{21}$ cm-2; Homan et al. 2021), and
indeed in this source, the measured lag is highly significant. The longer
reverberation lag is more likely to be intrinsic to the system. Candidate
explanations include that MAXI J1803–298 contains a more massive black hole or
a different disk-corona geometry. Perhaps this system is unique compared to
the rest of the sample because it showed a long-lived excursion to the “steep
power-law state” (Ubach et al., 2021), which may suggest a different coronal
geometry relative to the other sources.

###  4.4 A closer look at reverberation lags in the hard state

In the hard state, we see an increase in soft lag amplitude as a function of
hue when hue is $\lesssim 20^{\circ}$ (see Figs. 4 and 6a). To explore this
further, we plot the soft lag amplitude in the hard state as a function of the
luminosity in Eddington units222MAXI J1820+070 has the most reliable
measurements of its BH mass ($8.48^{+0.79}_{-0.72}$ $M_{\odot}$; Torres et
al., 2020), distance (2.96$\pm$0.33 kpc; Atri et al., 2020), and inclination
(63$\pm$3 degrees; Bright et al., 2020). From near-infrared study, the latest
measurement of the mass function of GX 339–4 is $f(M)=1.92\pm 0.08$
$M_{\odot}$, and the mass ratio of companion star and the BH is $q=0.18\pm
0.05$; the distance to GX 339–4 has only a lower limit of $\sim 5$ kpc, and a
larger distance is favored (Heida et al., 2017). If a distance of 10 kpc and
an inclination of $\sim 40^{\circ}$ (García et al., 2015; Wang et al., 2020a)
are assumed, the mass becomes 10 $M_{\odot}$ from the mass function. As for
MAXI J1348–630, the distance is assumed to be 3.4 kpc, the mass to be 10
$M_{\odot}$ (Lamer et al., 2021), and the inclination to be $30^{\circ}$ (jet
motion constrains $i\leq 46^{\circ}$; Carotenuto et al., 2021).  for the three
sources with more than one soft lag detection in the hard state (MAXI
J1820+070, GX 339–4, and MAXI J1348–630) in Fig. 11. In the initial rise of
the hard state in MAXI J1820+070’s 2018 outburst, the source luminosity in
NICER’s obsIDs 1200120101–1200120105 increased from 1% to 5% $L_{\rm Edd}$.
During this stage, the hue decreased from $-3^{\circ}$ to $-6^{\circ}$, and
then increased to $2^{\circ}$. At the same time, the reverberation lag
amplitude decreased from 25 to 6 $R_{\rm g}/c$, and then increased to 15
$R_{\rm g}/c$. This correlation between hue and reverberation lag amplitude
results in the increase of soft lag amplitude with hue at $\lesssim
20^{\circ}$. After the initial rise, MAXI J1820+070 was in a plateau where the
luminosity slightly decreased from 20% to 18% $L_{\rm Edd}$, hue gradually
increased from $20^{\circ}$ to $120^{\circ}$, and the soft lag amplitude
decreased from 30 to 10 $R_{\rm g}/c$. After the plateau, the source went
through a decay and then a rise, before finally making the hard-to-soft state
transition (see e.g., Fig. 2a). During this stage, the hue spanned the range
of $110^{\circ}$ to $140^{\circ}$ and correlated with luminosity (in the range
of 6% to 11% $L_{\rm Edd}$), while soft lag amplitude (in the range of 7 to 10
$R_{\rm g}/c$) anticorrelated with luminosity. Therefore, after the initial
rise, the soft lag amplitude kept decreasing with hue in the hard state as
seen at hue $\gtrsim 20^{\circ}$ in Figs. 4 and 6(a). In GX 339–4 and MAXI
J1348–630, the outburst history is not as complicated, and the data groups in
the hard state are all during the initial rise.

In Fig. 11, we can see that for $L/L_{\rm Edd}\lesssim 2\%$, the reverberation
lag amplitude decreases with increasing luminosity, as expected from previous
results. However, in the $2\%\lesssim L/L_{\rm Edd}\lesssim 4\%$ range, the
lag amplitude surprisingly appears to increase with increasing luminosity,
before once again decreasing with increasing luminosity for $4\%\lesssim
L/L_{\rm Edd}\lesssim 10\%$. Among the 3 sources, only MAXI J1820+070 shows
the plateau stage at $\sim 20\%$ $L_{\rm Edd}$, where both the luminosity and
the lag amplitude steadily decrease. We also note that this finding has no
conflict with previous reverberation lag measurements in De Marco et al.
(2015, 2017), as the increase of lag amplitude we see in Fig. 11 occurs in a
luminosity range not covered in those papers.

Figure 11: The soft lag amplitude in light travel time (i.e., in units of
$R_{\rm g}/c$) as a function of luminosity in Eddington units in the hard
state. Notice the lag increase in GX 339–4 could only be seen if we use the
soft lags detected between 0.5–1 keV and 1.5–3 keV because the soft lag is not
significant enough to detect when using 2–5 keV as the hard band. The values
from previous reverberation lag measurements (De Marco et al., 2015, 2017) are
the “observed” lag amplitudes (without considering the dilution as shown in
Table 3 in De Marco et al., 2017), and the $L/L_{\rm Edd}$ values are
converted accordingly with our assumed BH mass and distance. See Section 4.4
for the assumed BH mass, distance, and inclination, and how the dip results in
the correlation of soft lag amplitude and hue in Fig. 4 and 6(a) when hue is
$<20$ degrees.

Although the increase of reverberation lag between 2% and 4% $L_{\rm Edd}$ is
subtle because of the large uncertainties of the lag amplitudes, it is
interesting that we see this behaviour in all 3 sources with more than 1 soft
lag detection in the hard state. A possible interpretation is that below 1–2%
$L_{\rm Edd}$, the disk is truncated at the order of several tens of $R_{\rm
g}$, and above that, the disk is no longer truncated (the inner edge of the
disk is only several $R_{\rm g}$), which is consistent with the spectral
analysis of GX 339–4 (García et al., 2015; Wang-Ji et al., 2018; see the
schematic referred to as “faint hard state” in Fig. 10). The levels of disk
truncation also match the soft lag amplitudes in light travel time (in units
of $R_{\rm g}/c$, see Fig. 11 below 2% $L_{\rm Edd}$). The change from a
truncated disk to a non-truncated one is very likely to be due to the
radiative efficiency of the inner disk increasing above $\sim 1\%$ $L_{\rm
Edd}$, which is theoretically predicted (Yuan & Narayan, 2014) and supported
by the change of slope in the radio/X-ray correlation (e.g., Koljonen &
Russell, 2019). Instead, above 2% $L_{\rm Edd}$, the increase and then
decrease of soft lag amplitude is more likely to be driven by changes in the
coronal geometry, which would be consistent with an evolving base of the
compact jet. Recent GRMHD simulations (Liska et al., 2019) show that, as the
inner disk is more radiatively efficient and thus thinner, the inflowing
material exerts less pressure on the jet boundary, which becomes less
collimated, resulting in the base of the jet being more extended compared to
lower luminosities (Lucchini et al., 2021; Cao et al., 2021). The decreasing
soft lag amplitude above 4% $L_{\rm Edd}$ could then be caused by a
contracting X-ray corona before the quenching of the compact radio jet (see
the schematic referred to as “bright hard state” in Fig. 10). This is similar
to the expanding X-ray corona that happens before the transient radio jet
ejection in the state transition. However, the interpretation should be taken
with some caveats. First, the evolution of soft lag amplitudes is tentative
because of the large uncertainties of the measured lags. Second, the
uncertainties in BH mass, distance, and the bolometric correction factor
preserve the trend, but shift the values of luminosities in Eddington units.
Further detailed reflection modeling of the data in the hard state (especially
measurements of the coronal height, reflection fraction, and the inner edge of
the disk), and comparison with infrared/radio data are needed.

##  5 Summary

We have constructed a NICER reverberation machine aiming to systematically
search for reverberation signatures in the BH/BHC LMXBs in the NICER archive,
and to study the reverberation lag evolution throughout the outbursts. Our
major findings are as follows:

  1. 1.

Out of 26 BH/BHC LMXBs NICER has observed, we find data groups with high
enough signal to noise to enable a soft lag detection in 10 sources, and soft
reverberation lags are detected in all 10 of these sources. Detections have
previously only been reported for MAXI J1820+070 and GX 339–4, and therefore
for the other 8 we report the first ever reverberation detections; increasing
the sample size of BHLMXBs with reverberation lag detections from 2 to 10 (3
to 11 when accounting for detections with XMM-Newton in H1743–322).

  2. 2.

We adopt the power-spectral hue to quantify what stages the sources are in
during their outbursts, and find a consistent reverberation lag evolution in
the hard state and state transition for the 10 sources.

  3. 3.

In all the 9 sources with more than 1 reverberation detection, the
reverberation lag is longer and dominates at lower Fourier frequencies during
the hard-to-soft transition. This reverberation lag evolution has been
reported only for MAXI J1820+070, and now with our systematic searches, our
result suggests this trend is a global property of state transitions of
BHLMXBs that does not depend on inclination, BH mass or spin. This
reverberation evolution suggests a growing X-ray emitting region, possibly due
to an expanding corona.

  4. 4.

The frequency range where we see reverberation correlates with the Type-C QPO
frequency in the hard state, but they become anticorrelated during the state
transition. This result hints at a corona that is both radially and vertically
extended, and while the QPO frequency is mostly coupled to the radial extent,
the reverberation lag is coupled to both extents.

JW, GM, EK and JAG acknowledge support from NASA ADAP grant 80NSSC17K0515. JAG
thanks support from the Alexander von Humboldt Foundation. AI and DA
acknowledge support from the Royal Society. RAR acknowledges support from
NICER grant 80NSSC19K1287.

##  Appendix A Reverberation lag detections in individual sources

There are 26 BH/BHC LMXBs observed by NICER. Among those, the reverberation
machine finds good data groups and detects soft reverberation lags in 10
sources (see Table 1). The reverberation machine summary plots include the
spectral-timing results for a group in the bright hard state and a group in
the intermediate state (if available). We note that an example of the lag-
energy spectra of GX 339–4 in the discontinuous frequency bands outside of QPO
frequencies is shown in Fig. A1. In the 10 reverberation machine summary plots
(Figs. 3, A2–A10), we show only the lag-energy spectra in the full Fourier
frequency band in which we detect soft lags for clarity. The results for GX
339–4 are detailed in Section 3.1, and those for the other 9 sources are
presented below. The reverberation machine finds no good data group in the
other 16 sources, and the peak count rate, peak rms, total exposure, the
states achieved, and the reasons for no good groups are shown in Table A1.

Figure A1: (Right) The lag-energy spectra of GX 339–4 in the HIMS (data
corresponding to the red group in Fig. 3) in three frequency bands chosen to
exclude the QPO frequency bands. (Lower left) The lag-frequency spectrum with
the three bands (0.1–3.0 Hz, 3.5–5.8 Hz, and 8.3–21.8 Hz) shaded in colors
corresponding to those of the lag-energy spectra. (Upper left) PSD with the
QPO bands (central frequency $\pm$ HWHM) shaded in grey.

###  A.1 MAXI J1820+070

MAXI J1820+070 is one of the brightest BHLMXBs detected in history as it
reached 5.8 Crab at the peak luminosity in the 2018 outburst. The brightness
and great exhibition of multi-wavelength activity across spectral states make
it an extensively studied new BHLMXB (e.g., Homan et al., 2020; Tetarenko et
al., 2021). The BH mass is measured to be $8.48^{+0.79}_{-0.72}$ $M_{\odot}$
(Torres et al., 2020), and the distance to be 2.96$\pm$0.33 kpc (Atri et al.,
2020). The inclination constrained from the jet motion is 63$\pm$3 degrees
(Bright et al., 2020).

Thanks to the high brightness and near-daily cadence coverage of this source,
we find 88 good groups (making use of data with obsID 1200120101 in the
initial rise of hard state to obsID 2200120309 in the quiescence), and detect
soft reverberation lags in 84 of them. The mean lag significance indicator is
3115, which is the highest among all 10 BHLMXBs with detected reverberation
lags. The non-detection groups lie in the lower branch of HIMS (i.e., soft-to-
hard transition) approaching the hard state for the significance of promising
soft lag being lower than $2\sigma$.

Figure A2: Reverberation machine summary plot for MAXI J1820+070 (see Section
A.1 for more details).

We note that with NICER, we have performed detailed time lag analysis for MAXI
J1820+070 with a conventional method of manually grouping the data (Kara et
al., 2019; Wang et al., 2021a). With the new automatic grouping, we verify in
Fig. A2 that the major result of Wang et al. (2021a), i.e., that the
reverberation lag frequency decreases during transition with growing lag
amplitude, is well reproduced.

Figure A3: Reverberation machine summary plot for MAXI J1348–630 (see Section
A.2 for more details).

###  A.2 MAXI J1348–630

MAXI J1348–630 is a bright X-ray transient reaching 4.5 Crab at its peak
luminosity in its Jan 2019 outburst (Tominaga et al., 2020), after which 4
hard-only mini-outbursts were observed (Zhang et al., 2020a). With H I
absorption, the distance of MAXI J1348–630 is measured to be
$2.2^{+0.5}_{-0.6}$ kpc (Chauhan et al., 2021), and the jet motion constrains
the ejecta inclination to our line of sight to be $i\leq 46^{\circ}$
(Carotenuto et al., 2021). Moreover, MAXI J1348–630 was believed to host a
relatively massive black hole (Tominaga et al., 2020), but with the more
recent distance measurement, the evidence has become weaker because the lower
limit of $M_{\rm BH}$ is only 3.7 $M_{\odot}$ assuming $a_{*}=0$ and
$i=0^{\circ}$ (Chauhan et al., 2021).

We find 34 good groups and detect soft reverberation lags in 15 groups. Among
them, 14 groups are during the major outburst covering both the hard and
intermediate states, and the remaining 1 group is in the first hard-only mini-
outburst after the major outburst. The non-detections all lie in the hard
state because of the intrinsic shape difference of the lag-energy spectrum
compared to the HIMS. The evolution in the soft reverberation lag frequency
and amplitude is consistent with what has been detected in MAXI J1820+070 and
GX 339–4 (see Fig. A3). We note that NICER had a visibility gap for this
source between MJD 58513.2 (obsID 1200530105) when it was in the bright hard
state and MJD 58521.9 (obsID 1200530106) when the source was already in the
HIMS, during which time a relativistic transient jet was estimated to be
ejected on MJD $58518.9\pm 2.4$ (Carotenuto et al., 2021). Therefore, we
cannot conclude when the state transition starts relative to the transient jet
launching, nor can we precisely determine when the X-ray reverberation lag
becomes longer during the state transition.

###  A.3 MAXI J1535–571

MAXI J1535–571 is a LMXB that was first detected by MAXI on 2017 September 2
(Negoro et al., 2017). By fitting the soft state spectrum with a relativistic
thin disk component kerrbb, the black hole mass is estimated to be
$10.39^{+0.61}_{-0.62}$ $M\odot$ with a distance of $5.4^{+1.8}_{-1.1}$ kpc
(Sridhar et al., 2019). Reflection spectroscopy suggests an inner disk
inclination in the range of 55–77 degrees (Xu et al., 2018; Miller et al.,
2018); but the jet inclination inferred from the motion of the superluminal
knot is $\leq$45 degrees (Russell et al., 2019). Notice that MAXI J1535–571
has a relatively high interstellar absorption with $N_{\rm H}=(3-8)\times
10^{22}$ cm-2 (Kennea et al., 2017; Gendreau et al., 2017; Xu et al., 2018),
which makes the soft lag more challenging to measure than in less obscured
sources. In the radio, a transient jet knot was estimated to be ejected on MJD
$58010^{+2.65}_{-2.5}$ (Russell et al., 2019), and the compact jet was rapidly
quenching on MJD 58013.6 (Russell et al., 2020).

Figure A4:  Reverberation machine summary plot for MAXI J1535–571 (see Section
A.3 for more details). The PSD of the only group in the hard state starts from
0.1 Hz because none of the GTIs in this group is longer than 256 s, and the
segment length is changed to 10 s in this case.

NICER started observing this transient from 2017 September 9 (MJD 58005.3),
corresponding to the observation (obsID 1050360103) that composes the first
good group we find with our reverberation machine (see Fig. A4). In this data
group, no GTI is longer than 256 s to measure the power down to 0.0039 Hz. As
one ingredient in measuring the hue is the power in 0.0039–0.031 Hz, we cannot
use the hue to determine the spectral state for this data group. Instead, we
confirm its hard state nature from spectral-timing features. The QPO centroid
frequency is relatively low at 0.9 Hz. The photon index is $\sim 1.8$, and the
disk is cool at $T_{\rm in}\sim 0.18$ keV when the flux-energy spectrum is
fitted in 0.5–10 keV range with a basic reflection model of
(tbabs*diskbb+reltransDCp) following Wang et al. (2021a). The other 13 good
groups are all in the HIMS covering MJD 58008.5–58038.4 (obsID 1050360104 to
1130360115)333It was suggested that the source entered the SIMS between MJD
58014.2 and 58015.4 (Tao et al., 2018). Meanwhile, all the good groups we find
in the intermediate state are determined to be in the HIMS based on the hue.
There is no controversy because by comparing with the observations classified
to be SIMS with the NICER data, our good groups do not cover any of those data
segments (see Fig. 1 in Stevens et al., 2018). This is owing to the criteria
of the relative change in fractional variance when grouping the data, and the
requirement on the lag significance indicator that is most sensitive to the
rms., and the 2 non-detections happen along with the two lowest lag
significance indicators. A representative group in the HIMS is also shown in
Fig. A4, and we observe again that the frequency over which soft reverberation
lag dominates decreases during state transition. Because of the gap between
MJD 58005.3 and 58008.5, and the large uncertainty in the transient jet
ejection time (MJD $58010^{+2.65}_{-2.5}$), we cannot determine the time
between the X-ray corona expansion and the launch of the transient jet.

###  A.4 MAXI J1631–479

MAXI J1631–479 was discovered as a bright hard X-ray source on 2018 December
21 by MAXI/GSC (Kobayashi et al., 2018), and was in the hard state until
December 26 (MJD 58478; Fiocchi et al., 2020). MAXI J1631–479 is likely to be
a low-inclination source based upon several arguments including from
reflection spectroscopy with NuSTAR ($i=29\pm 1^{\circ}$; Xu et al., 2020),
the shape of the HID (Monageng et al., 2021), and the phase lag of its Type-C
QPO (van den Eijnden et al., 2019). Both the black hole mass and distance are
currently unknown. A radio ﬂare was observed to peak on MJD 58560 to 58582
(though notably, the HIMS-to-soft-state transition happened earlier, between
MJD 58545 and 58560). The delay of the radio flare hints at the possibility of
multiple unresolved ﬂares and/or jet–ISM interactions (Monageng et al., 2021).

Figure A5: Reverberation machine summary plot for MAXI J1631–479 (see Section
A.4 for more details). All the good data groups are in the intermediate state.

NICER started monitoring this source on MJD 58498 (obsID 1200500101) when it
was already in the soft state, and caught a transition back to the HIMS on MJD
58506 (obsID 1200500110; van den Eijnden et al., 2019). The source stayed in
the HIMS until MJD 58545 (obsID 2200500102), after which NICER had a
visibility gap and the source was already back to the soft state when the
monitoring resumed on MJD 58560 (obsID 2200500108; Rout et al., 2021). As no
NICER data is available in the hard state, all the good groups we find are in
the HIMS, and the results for one representative group is shown in Fig. A5.
Soft reverberation lags are detected in 9 out of 12 good groups. These lags
are always found at low Fourier frequencies (i.e., even at 0.1 Hz, no
prominent hard lag is observed), consistent with results from previous sources
in the HIMS.

###  A.5 MAXI J1727–203

MAXI J1727–203 was discovered by MAXI/GSC on 2018 June 5 (Yoneyama et al.,
2018). NICER observed this source since the day it was discovered, and caught
the initial HIMS followed by the soft state, the soft-to-hard transition, and
the final hard state (see Fig. 1 in Alabarta et al., 2020). Spectral-timing
analysis with NICER data has been reported in Alabarta et al. (2020), and the
nature of the compact object in this system is by far the most uncertain.

Figure A6: Reverberation machine summary plot for MAXI J1727–203 (see Section
A.5 for more details).

For the first time, we show here the lag analysis for this source. There are
only 3 good groups found with our reverberation machine: one in the initial
HIMS (MJD 58275.0–58276.5, obsID 1200220102 to 1200220103), one in the HIMS
during the transition back to the hard state (MJD 58298.6–58301.4, obsID
1200220120–1200220123), and one in the hard state at the end of the outburst
(MJD 58305.6–58327.8, obsID 1200220127–1200220140). The group in the hard
state has only prominent low-frequency hard lags, and has no strong evidence
for a high frequency soft lag, probably for the same reason as discussed in
Section 3.1 (see the blue data group in Fig. A6). The soft reverberation lag
is detected in both HIMS groups (e.g., Fig. A6). Although the soft lag $<1$
keV is not as significant as in previous sources, we note that the iron lag is
very significant in the group shown. This confirms that reverberation is
present in this source. Interestingly, the relativistic iron line in the flux-
energy spectrum is not as prominent as the iron lag in the lag-energy
spectrum. This type of discrepancy was reported previously for several AGN,
and was thought to suggest the inadequacy of the lamppost corona assumption
(Zoghbi et al., 2020; Mastroserio et al., 2020). The detection of its
reverberation lag (especially the iron lag) hints at a black hole nature for
MAXI J1727–203.

###  A.6 EXO 1846–031

EXO 1846–031 was first discovered in 1985 (Parmar & White, 1985). After 25
years in quiescence state, it underwent an outburst in 2019, and was detected
first by MAXI (Negoro et al., 2019). Draghis et al. (2020) performed
reflection spectroscopy analysis with NuSTAR data, and found that EXO 1846–031
has a very high spin ($a_{*}\sim 0.997$), a high inclination ($i\sim
73^{\circ}$), and a high column density of $N_{\rm H}\sim(10-12)\times
10^{22}$ cm-2, which makes it difficult to dynamically measure the black hole
mass in the optical. From the model kerrbb, the authors estimate a black hole
mass of $9\pm 5$ $M_{\odot}$. Both the extremely high spin and its position in
the radio/X-ray plane (Miller-Jones et al., 2019) hint at a black hole nature
for this system.

Figure A7: Reverberation machine summary plot for EXO 1846–031 (see Section
A.6 for more details).

After its renewed activity, NICER quickly followed up on 2019 July 31 (MJD
58695, obsID 2200760101) and confirmed the source was in the hard state (Bult
et al., 2019). With our reverberation machine, we have found 3 good groups, 1
in the initial hard state, and 2 in the intermediate state. A soft
reverberation lag is detected in the first two groups (see Fig. A7). We note
that although the soft X-ray data quality is largely affected by the large
column density towards this source (see also Section 4.3), an iron lag is
present in the lag-energy spectrum in the hard state data group. On the other
hand, the shape of the lag-energy spectrum for HIMS data group is certainly
not a typical hard lag (which would monotonically increase with energy without
any feature). Therefore, we confirm that reverberation is present in these two
data groups. By comparing results from these two groups, we see the frequency
over which soft reverberation lag is detected decreases during the hard-to-
soft transition, and the iron line seems to get blueshifted, exactly as seen
previously in MAXI J1820+070 (Wang et al., 2021a). The nature of the
blueshifting is not well understood, and physically-driven models that are
preliminarily promising include the emission from the plunging region (Fabian
et al., 2020) and returning radiation (Connors et al., 2020). Other
possibilities include extra Comptonization of the reﬂection spectrum in the
corona (Steiner et al., 2017), a change in the inclination (i.e., warps) in
the reflecting surface of the inner disk, a change in the disk thickness as
the mass accretion rate rises, or a change in the irradiation profile due to
changing coronal geometry.

###  A.7 GRS 1915+105

Figure A8: Reverberation machine summary plot for GRS 1915+105 (see Section
A.7 for more details). All the good data groups are in the HIMS determined
from the power-spectral hue.

GRS 1915+105 is the first source of detected superluminal jets in our galaxy,
and therefore is the first “microquasar” (Mirabel & Rodriguez, 1994). It
displays at least a dozen of variability states that are believed to result
from disk-jet coupling (e.g., Belloni et al., 2000). The BH mass is measured
to be $12\pm 2$ $M_{\odot}$, with an orbital period of 33.5 days (Reid et al.,
2014). GRS 1915+105 is a peculiar BHLMXB as it remained in a bright outburst
for 26 years since its discovery in 1992 (Castro-Tirado et al., 1992). Since
its launch, NICER frequently observed GRS 1915+105 in 2017 (until 2017-11-06,
obsID 1103010134). This dataset was analysed in Neilsen et al. (2018) and the
well-known disk wind evidence including Fe XXV and Fe XXVI absorption was
detected. From 2018 April to 2020 March, GRS 1915+105 surprisingly declined in
count rate, and eventually settled in a very faint and hard state (Negoro et
al., 2018; Homan et al., 2019). Very recently, in 2021 July, the source was
detected to be rebrightening (Neilsen et al., 2021).

Our reverberation machine results in 3 good data groups all during the
declining stage (making use of obsID 1103010147 to 2596010901, corresponding
to 2018-05-27 to 2019-05-01). A soft reverberation lag is detected in the
first two groups. The results of the first group is shown in Fig. A8. The lag
spectral quality is limited, and the soft lag amplitude is measured to be
$8\pm 5$ ms, with a relatively large uncertainty. However, we note that this
measurement is within uncertainties with MAXI J1820+070 (see Fig. 4). We find
a broad emission line at 6.4 keV and a prominent narrow absorption feature at
6.97 keV which corresponds to Fe XXVI line from highly ionized wind,
consistent with previous NICER result in the more luminous state (Neilsen et
al., 2018). We also note that the hue for these groups suggests that GRS
1915+105 was in the HIMS in the decay stage.

###  A.8 AT2019wey

AT2019wey was initially discovered as an optical transient in December 2019
(Tonry et al., 2019), and later was classified as a galactic XRB for its
strong X-ray emission and hydrogen lines detected at redshift $z=0$ (Yao et
al., 2020c). The source is expected to be a LMXB because the mass of the
companion star was constrained to be $<0.8$ $M_{\odot}$ (Yao et al., 2020b).
The multi-wavelength (radio to X-ray) spectral-timing features are consistent
with a BHLMXB nature for AT2019wey (Yao et al., 2020b; Mereminskiy et al.,
2021). A resolved radio source was detected and could be interpreted as the
steady compact jet common in the BHLMXB hard state (Yadlapalli et al., 2021).
Currently, we lack a mass estimate for this source, and the inclination is
constrained to be $i\leq 30^{\circ}$ from reflection spectroscopy (Yao et al.,
2020a).

During the NICER coverage, AT2019wey was in the hard state at the beginning
from MJD 59070 to MJD 59082 (obsID 3201710105–3201710117), and then
transitioned into the HIMS as reported by Yao et al. (2020a), which is in
great agreement with spectral states classified by the hue in our
reverberation machine. Though we have found in total 22 good groups (6 in the
hard state and 16 in the HIMS), the fraction of good groups with soft
reverberation lag detected is by far the lowest: only 3 out of 22, all in the
HIMS. In addition, the lag-energy spectra found in the soft lag frequencies
this way are not representative of a soft reverberation lag (i.e., there is no
strong evidence for reverberation in either the soft X-ray band or the iron
line energies, see the upper panel in Fig. A9). If we use 1.5–3 keV instead of
2–5 keV as the hard band in calculating the lag-frequency spectrum, the lag-
energy spectra show quite promising reverberation lag features (see the lower
panel in Fig. A9). The detection of reverberation lag may suggest that this
source is a BHLMXB.

Figure A9: Reverberation machine summary plot for AT2019wey. The soft lag is
detected in 3 out of 22 good data groups, but the reverberation nature is
difficult to confirm because of a lack of the iron line feature in the lag-
energy spectrum (upper). If we use 1.5–3 keV instead of 2–5 keV as the hard
band in calculating the lag-frequency spectrum, the lag-energy spectra show
more promising reverberation lag features (lower, see Section A.8 for more
details).

The reason for a very low reverberation lag detection fraction and a better-
suited hard band of 1.5–3 keV instead of 2–5 keV is again intrinsically the
shape of the lag-energy spectrum. This would mean that even in the HIMS where
we observe a prominent reverberation lag in the other sources when we match
the hard band of lag-frequency measurement to the energy of the “valley” in
the lag-energy spectrum (2–5 keV), AT2019wey still has a lag-energy spectrum
shape similar to the other sources in the hard state (valley at 1.5–3 keV).
This would imply a lower disk temperature, a lower ionization, or a lower disk
density. We note that this system has a very short orbital period ($<8.6$
hours), and a low mass accretion rate that could result in the “hard-only”
outburst implied in Yao et al. (2020a).

###  A.9 MAXI J1803–298

MAXI J1803–298 was first discovered by MAXI/GSC in May 2021 and quickly
followed up by NICER (Serino et al., 2021; Gendreau et al., 2021), making it
the most newly discovered XRB in our sample. It was initially detected in the
hard state between MJD 59336 and 59340 (obsID 4202130101–4202130104), and was
already in the intermediate state on MJD 59352.7 (obsID 4202130105) when NICER
came out of a visibility gap (Homan et al., 2021; Ubach et al., 2021). With
the promising nearly-periodic X-ray absorption dips seen by NICER and NuSTAR,
and the observed disk wind, it is likely to be a high-inclination source
($i>60^{\circ}$), and the orbital period is between 7 and 8 hours (Homan et
al., 2021; Xu & Harrison, 2021; Miller & Reynolds, 2021).

There are 2 good groups in our reverberation machine: one in the hard state
(obsID 4202130103–4202130104), and the other in the HIMS (obsID 4202130107,
very close to the SIMS judging by the value of hue), both are shown in Fig.
A10. The reverberation lag is detected only in the hard state group, while
prominent low-frequency hard lags are observed in the intermediate state
group. Notice that the intermediate state group (MJD 59354.5-59363.4) has a
type-B QPO detected at 6.7 Hz with a sub-harmonic at 3.3 Hz, which is
consistent with what was reported in Ubach et al. (2021). It was surprising at
first to see that only in MAXI J1803–298, the low-frequency hard lag remains
evident in the intermediate state, where for previous sources, we see the
reverberation lag dominates over the hard lags at the low frequencies. One
possibility is that the intermediate state here is different from the
canonical HIMS, and rather, it is in the steep power-law (SPL) state
(Remillard & McClintock, 2006), as suggested by Ubach et al. (2021) based on a
relatively high scattering fraction. In Fig. A10, besides the blue-shifted
iron line in the intermediate state, we also find evidence for absorption
feature in the Fe band at 6.97 keV, suggesting the system is observed closer
to edge-on as the disk wind is expected to be confined to the equatorial
plane, in agreement with what was seen with Swift (Miller & Reynolds, 2021).

Figure A10: Reverberation machine summary plot for MAXI J1803–298 (see Section
A.9 for more details).  Table A1: The BH/BHC LMXB samples observed by NICER
without any good reverberation data from our reverberation machine, sorted by
the peak count rate.

Name | Peak count rate (counts/s) | Peak rms (%) | Exposure (ks) | States achieved | Reasons for no good groups
---|---|---|---|---|---
4U 1543–475 | 95864 | 0.8 | 109.9 | S | Low rms
MAXI J0637–430 | 6452 | 36 | 155.6 | H,I,S | Low count rate/rms
4U 1957+11 | 1958 | 3 | 73.8 | S | Low rms
Swift J1728.9–3613 | 1683 | 13 | 135.4 | I,S | Low rms/short exposure
4U 1630–472 | 833 | 12 | 152.0 | H,I,S | Low rms
Swift J1842.5–1124 | 693 | 15 | 9.0 | H,I/S? | Low count rate/rms
MAXI J1810–222 | 602 | 24 | 108.7 | H,I,S | Low count rate/rms
Swift J1658.2–4242 | 342 | 19 | 62.1 | H,I,S | Low count rate/rms
XTE J1908+094 | 210 | 28 | 14.0 | H,S | Low count rate/rms
MAXI J1813–095 | 169 | 19 | 5.9 | H | Short exposure
H1743–322 | 140 | 24 | 26.6 | H | Low count rate/rms
GRS 1758–258 | 72 | 13 | 2.9 | H | Low count rate/rms
XTE J1859+226 | 23 | 49 | 37.3 | H | Low count rate
IGR J17379–3747 | 20 | 28 | 136.6 | H | Low count rate
Swift J1357.2–0933 | 7 | 28 | 148.5 | H | Low count rate
GRS 1739–278 | 5 | 22 | 5.6 | H | Low count rate

Notes.
The count rates are for $0.3-12$ keV, and are normalized for 52 FPMs. The rms
is calculated also in $0.3-12$ keV, with the Fourier frequency range of 1–10
Hz.

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