# The Evolution of Rapid Optical/X-ray Timing Correlations in the Initial Hard
State of MAXI J1820+070

J. A. Paice1,2, P. Gandhi1, T. Shahbaz3,4, A. Veledina5,6, J. Malzac7,D.A.H.
Buckley8,9, P. A. Charles1, K. Rajwade10, V. S. Dhillon11,3,S. P.
Littlefair11, T. R. Marsh12, P. Uttley13, F. M. Vincentelli1, and R. Misra2.
1Department of Physics and Astronomy, University of Southampton, Highfield,
Southampton, SO17 1BJ, UK
2Inter-University Centre for Astronomy and Astrophysics, Pune, Maharashtra
411007, India
3Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife,
Spain
4Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La
Laguna, Tenerife, Spain
5Department of Physics and Astronomy, FI-20014 University of Turku, Finland
6Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
7IRAP Universite de Toulouse, CNRS, UPS, CNES, Toulouse, France
8South African Astronomical Observatory, Observatory Road, Observatory, 7925,
Cape Town, South Africa
9Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa
10Department of Physics and Astronomy, The University of Manchester, Oxford
Road, Manchester, M13 9PL, UK
11Department of Physics and Astronomy, University of Sheffield, Sheffield, S3
7RH, UK
12Astronomy and Astrophysics Group, Department of Physics, University of
Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK
13Astronomical Institute ’Anton Pannekoek’, University of Amsterdam, Science
Park 904, NL-1098XH Amsterdam, the Netherlands E-mail<EMAIL_ADDRESS>
(Submitted to MNRAS in original form XXXX AAA XX, Received XXXX AAA XX,
Accepted XXXX AAA XX)

###### Abstract

We report on a multi-epoch campaign of rapid optical/X-ray timing observations
of the superbright 2018 outburst of MAXI J1820+070, a black hole low-mass
X-ray binary system. The observations spanned 80 days in the initial hard-
state, and were taken with NTT/ULTRACAM and GTC/HiPERCAM in the optical
($u_{s}g_{s}r_{s}i_{s}z_{s}$ filters at time resolutions of 8–300 Hz) and with
ISS/NICER in X-rays. We find (i) a growing anti-correlation between the
optical and X-ray lightcurves, (ii) a steady, positive correlation at an
optical lag of $\sim$0.2 s (with a longer lag at longer wavelengths) present
in all epochs, and (iii) a curious positive correlation at negative optical
lags in the last, X-ray softest epoch, with longer wavelengths showing a
greater correlation and a more negative lag. To explain these we postulate the
possible existence of two synchrotron-emitting components; a compact jet and a
hot flow. In our model, the significance of the jet decreases over the
outburst, while the hot flow remains static (thus, relatively, increasing in
significance). We also discuss a previously discovered quasi-periodic
oscillation and note how it creates coherent optical time lags, stronger at
longer wavelengths, during at least two epochs.

###### keywords:

accretion, accretion discs – X-rays: binaries – X-rays: individual: MAXI
J1820+070 – stars: optical: variable – black holes

††pubyear: 2019††pagerange: The Evolution of Rapid Optical/X-ray Timing
Correlations in the Initial Hard State of MAXI J1820+070–A.4

##  1 Introduction

Low-Mass X-ray Binaries (LMXBs) are highly variable systems involving
accretion on to either a neutron star or black hole. Over the past few
decades, there have been many efforts to study this variability and detail its
behaviour, and there is an expanding body of literature detailing this (see
Belloni & Hasinger 1990a; Mushotzky et al. 1993; van der Klis 2000 and many
others). But why is the study of this variability important?

In short, because these systems are complex, and unresolvable with current
telescopes. LMXBs host a compact object accreting via a disc of material
transferred from a Roche-lobe-filling companion star. The environment is
complex, with an outer disc, hot inner flow/corona, and compact, relativistic
jets (to name just a few), which all emit across the electromagnetic spectrum.
And, during transient, violent outbursts that occur every few years or decades
and can last for weeks to months, the scale, presence, and behaviour of these
regions can change significantly. Their compact nature means that physically
important timescales can span $\sim$microseconds in the inner zones, to
decades at the other extreme. The goal of multiwavelength timing studies of
these sources is to understand the physical processes in these components, and
thus the system as a whole.

Over the relatively short history of multiwavelength astronomy, better
technology and new telescopes have improved the temporal resolution of such
studies, and with it, our understanding has advanced; Motch et al. (1982);
Motch et al. (1983) and Imamura et al. (1987) were some of the earliest
reports on rapid stochastic multiwavelength variability down to millisecond
scales, while Kanbach et al. (2001) was one of the works that showed
intriguing relations between the rapid optical and X-ray variability for the
first time. This interband relationship was then found to vary between
systems, each time showing complex behaviour, interpreted as a varying
dominance of the inflowing or outflowing plasma through the disc, the inner
flow and the jet (e.g. Gandhi et al. 2008; Durant et al. 2008; Gandhi et al.
2010; Casella et al. 2010; Durant et al. 2011; Gandhi et al. 2017; Pahari et
al. 2017). The true importance of these studies is in providing novel
quantitative constraints of the physical scales and interactions between the
accreting plasma components. For instance, a rapid optical/infrared lag of
$\sim$ 100 ms relative to X-rays has now been observed in several systems and
appears to be an important constraint for models of jet launching and
acceleration (Gandhi et al., 2017).

Yet, these studies rely on the source being both sufficiently bright and well-
observed at multiple wavelengths simultaneously, the former being rare and the
latter being marred by the inherent unpredictability of these outbursts. As
such, these studies have so far been few and far between, and rarely carried
out multiple times over the same outburst - though there are hints at an
evolution of processes at different stages of the outburst (see, e.g.,
Veledina et al., 2017; Vincentelli & Casella, 2019, though note that the
latter compares two different outbursts). Solutions are not yet unique, with
processes such as a jet and a hot flow invoked to explain certain signatures
on intermediate timescales (e.g. Veledina et al., 2013a; Malzac et al., 2018).
We still remain severely data-limited in terms of high-quality strictly
simultaneous multiwavelength time series in order to make progress.

In 2018, one particular X-ray binary was discovered. It became bright enough
and observed well enough that a good picture of its initial, several-week-long
hard state – including evolving inter-band correlations and Fourier
components, observed at over 100 Hz – has been made possible.

Discovered first as optical transient ASASSN-18ey on 6 March 2018 (Denisenko,
2018; Tucker et al., 2018) and then as an X-ray source on 11 March (Kawamuro
et al., 2018), MAXI J1820+070 (hereafter J1820) was detected during the rapid
outburst rise. It quickly rose to a brightness of $\sim$4 Crab (Shidatsu et
al., 2019), becoming the brightest extra-solar object in the X-ray sky by the
time it peaked on 23 March (Corral-Santana et al., 2016). By this point, its
brightness had led to observations at many sites (e.g. Baglio et al. 2018;
Bright et al. 2018; Bahramian et al. 2018; Del Santo & Segreto 2018); not only
did these observations quickly identify it as a likely Black Hole LMXB (Baglio
et al., 2018; Bright et al., 2018; Uttley et al., 2018), but they also
revealed rapid optical flaring (Littlefield 2018; Sako et al. 2018) and even a
significant optical/X-ray correlation (Paice et al., 2018). Later, a QPO would
be first identified in this source at around this peak (Mereminskiy et al.,
2018; Yu et al., 2018), and would be seen to evolve over the next few months
(Stiele & Kong, 2020). This stage of the outburst was the ‘hard state’, where
it is believed that the inner edge of the accretion disc is recessed, and a
relativistic jet is present (Done et al., 2007).

After the peak, J1820 entered a gradual decline in X-ray flux. In early July
2018, it transitioned rapidly to the soft state (Homan et al., 2018), where
the accretion disc extends to the Innermost Stable Circular Orbit (ISCO) and
the jet is quenched. During this time, a unique blackbody X-ray emission
signature was detected, which has been suggested as originating from within
the ISCO, the so-called ’plunge region’ (Fabian et al., 2020). J1820 remained
in this state until late September 2018, when it transitioned back to the hard
state (Negoro et al., 2018). It has since undergone a series of small
‘rebrightenings’ (Ulowetz et al., 2019; Tomsick & Homan, 2019; Zampieri et
al., 2019; Hambsch et al., 2019; Xu et al., 2019; Adachi et al., 2020), but as
of yet, it has not undergone a second outburst. Fig. 1 shows a timeline of the
hard state outburst at X-ray and radio wavelengths, using data from the Neil
Gehrels Swift Observatory, MAXI, and AMI-LA.

Radio parallax measurements have since constrained J1820 to a distance of
$2.96\pm 0.33$ kpc (Atri et al., 2020), and the optical parallax found using
Gaia EDR3 (Gaia Collaboration et al., 2020) gives a distance of
$2.94^{+0.87}_{-0.55}$ kpc (calculated using the recommended zero-point
correction – Lindegren 2020), which improves on the previous estimate reported
in Gaia DR2 (Gandhi et al., 2019).

J1820’s brightness led to several multiwavelength campaigns using high-time-
resolution instrumentation over the course of its outburst. In Paice et al.
(2019), we discussed the optical/X-ray correlations taken from a single night,
using HiPERCAM and NICER during the rising hard accretion state. Therein, we
noted the presence of a sub-second optical lag of order $\sim$ 100 ms between
the bands dependent upon wavelength, which we attributed to structure within
the compact jet, and presence of an anti-correlation, which we put in the
context of the hot accretion flow. Together with GX 339–4 and V404 Cyg (Gandhi
et al., 2017), these results make J1820 the third well-studied black hole XRB
to show the aforementioned sub-second lag.

The above results all highlight the importance of J1820 as a benchmark for
understanding accretion. Here, we expand on these results to trace the timing
properties through the primary hard state, including four new observations
between NICER and another optical instrument, ULTRACAM, as well as a second
correlated HiPERCAM/NICER observation later in the outburst. All observations
were taken during the initial hard state, cover time resolutions from 8 Hz to
300 Hz, and cover a span of roughly 80 days in total. We construct a picture
of the evolving optical/X-ray correlations over this period, and discuss to
what processes they may relate.

##  2 Observations

Figure 1: Timeline of MAXI J1820+070’s 2018 outburst. Vertical black dashed
lines denote dates of observations (see Table 1). The source transitions to
the soft state just after MJD 58300. Swift/XRT fluxes and photon indices were
obtained from spectra produced by Swift’s ‘Build XRT Products’ tool (Evans et
al., 2009), using only data that corresponds to our epochs; errors are smaller
than the marker size for both datasets. Top: Continuous monitoring done by
Swift/BAT (filled blue circles) and ISS/MAXI (open red circles). Middle: X-ray
fluxes from Swift/XRT (purple triangles). Also plotted are Radio luminosities
from AMI-LA (orange diamonds), presented in Bright et al. (2020). Bottom:
Photon indexes (black squares). Table 1: Log of Simultaneous ULTRACAM/HiPERCAM
& NICER Observations for MAXI J1820+070 2018 Hard State Outburst.

Epoch | UT date |  Day Num.1 |  Time Start (MJD-58000)2 |  Time End (MJD-58000)2 |  Optical Instrument |  X-ray Instrument |  Optical Filters |  Correlated Time (s) |  Cadence (ms)3
---|---|---|---|---|---|---|---|---|---
1 | 2018-03-16 | 5 | 193.38035 | 193.38518 | ULTRACAM | NICER | $u_{s}g_{s}r_{s}$ | 417 | 13.8 (138)
2 | 2018-03-22 | 11 | 199.34590 | 199.41903 | ULTRACAM | NICER | $u_{s}g_{s}i_{s}$ | 1279 | 9.06 (27.2)
3 | 2018-04-12 | 32 | 220.40479 | 220.40736 | ULTRACAM | NICER | $u_{s}g_{s}i_{s}$ | 222 | 10.4 (31.1)
4 | 2018-04-17 | 37 | 225.17310 | 225.25110 | HiPERCAM | NICER | $u_{s}g_{s}r_{s}i_{s}z_{s}$ | 1648 | 2.93
5 | 2018-04-17 | 37 | 225.30454 | 225.31547 | ULTRACAM | NICER | $u_{s}g_{s}i_{s}$ | 780 | 23.0 (69.0)
6 | 2018-06-07 | 88 | 276.19520 | 276.22171 | HiPERCAM | NICER | $u_{s}g_{s}r_{s}i_{s}z_{s}$ | 1196 | 2.93

1 Number of days since 2018 March 11 (MJD 58188), as used by Stiele & Kong
(2020)
2 Start and end of simultaneous times only – observations may contain gaps.
3 Limiting cadences in optical. Numbers in brackets are $u_{s}$ band cadences,
if different from the other bands.

###  2.1 NTT/ULTRACAM – Optical

ULTRACAM is a fast-timing optical camera on the 3.58 m New Technology
Telescope (NTT) in La Silla, Chile. It was built for the purpose of fast
optical timing in multiple wavebands. To this end, it includes three channels
for simultaneous multiwavelength monitoring (with replaceable filters). It can
also observe at frame-rates well above 100 Hz; this is achieved by the lack of
a physical shutter, and frame-transfer CCDs that can rapidly shift charge into
a storage area for reading out, freeing up the original pixels for observation
and thereby achieving low dead times (Dhillon et al., 2007).

We used ULTRACAM to observe J1820’s initial outburst peak in the early
mornings of 2018 March 16, March 22, April 12, and April 17. All observations
were carried out with the $u_{s}$, $g_{s}$, and $i_{s}$ SDSS filters, except
for the first, which used the $r_{s}$ filter instead of $i_{s}$. Unlike most
observations of this type, the times were not explicitly chosen to coincide
with X-ray observations – instead, the overlaps were purely coincidental and
the serendipitous result of near-constant monitoring of J1820 by multiple
telescopes. ULTRACAM was used in two-window mode (one each for the target and
comparison star), with both window sizes of 50 x 50 pixels with a 2 x 2
binning for sensitivity and speed. See Table 1 for observational details.
J1820 was very faint in $u_{s}$, and so ULTRACAM’s on-chip co-adding feature
was used; this provides a longer exposure time in $u_{s}$ so as to increase
signal-to-noise ratio.

The data were reduced using the ULTRACAM pipeline v9.14 (Dhillon et al.,
2007). The bias was subtracted from each frame, and flat field corrections
were also applied. Aperture sizes scaled to the instantaneous seeing were
used, with radii between 0$\aas@@fstack{\prime\prime}$7 and
3$\aas@@fstack{\prime\prime}$5, with an annulus of between 12$\arcsec$ and
6$\aas@@fstack{\prime\prime}$3 to calculate the background. These apertures
had variable centre positions that tracked the centroids of the sources on
each frame, with a two-pass iteration (where an initial pass is made to track
the sources on the CCD before a second photometry pass) used for accuracy. Our
times were then adjusted to Barycentric Dynamical Time (BJD_TDB) using methods
given in Eastman et al. (2010).

Our comparison star is located at RA = 18 20 26.43, Dec = 07 10 11.7 (J2000),
and is listed in the PanSTARRS survey catalog (Magnier et al., 2020) with
$g_{s}$/$i_{s}$ magnitudes of 13.3083/12.233 respectively. The star was taken
to be constant, and was used for photometric calibration. For the $u_{s}$
observations, the comparison star was too faint to perform photometry within a
single frame. Hence, we used the measured zero-point magnitude for the $u_{s}$
band in photometric conditions for ULTRACAM (Vik Dhillon, priv. communication)
in order to calibrate our observations. We extracted the J1820 and comparison
star magnitudes using aperture photometry with a variable aperture size that
was dictated by the seeing conditions. The aperture also tracked the centroid
of the source of interest by using a bright star in the field as a reference.
For the $u_{s}$ observations, we used J1820 as the reference object itself so
as to not lose tracking within the field.

###  2.2 GTC/HiPERCAM – Optical

High-speed multi-colour photometry of J1820 was carried out using HiPERCAM
(Dhillon et al., 2018) on the 10.4 m Gran Telescopio Canarias on La Palma.
HiPERCAM uses 4 dichroic beamsplitters to image simultaneously 5 optical
channels covering the $u_{s}g_{s}r_{s}i_{s}z_{s}$-bands (respectively, central
wavelengths 3526, 4732, 6199, 7711 and 9156 Å). The CCDs were binned by a
factor of 8 and used in the highest-speed drift mode. We orientated the
instrument (PA = 58∘) and used two windows (96x72 pixels each), one centered
on J1820, and another on a comparison star, APASS–34569459 (Henden et al.,
2015). The observations discussed here were taken on 2018 April 17 from
03:26–06:11 UT, and 2018 June 07 from 04:41–05:39, coordinated with NICER. The
exposure time was 2 ms, the cadence 2.9 ms, the median seeing 2.2″. The sky
was affected by mild cirrus on both dates.

We used the HiPERCAM pipeline software111https://github.com/HiPERCAM/hipercam
to de-bias, flat-field and extract the target count rates using aperture
photometry with a seeing-dependent circular aperture tracking the centroid of
the source. Sky background was removed using the clipped mean of an annular
region around the target. The target was brighter than all stars in the field.
We thus used the raw target counts for the analyses presented herein; note
that our primary results are not affected when using photometry relative to
the comparison star.

###  2.3 ISS/NICER – X-ray

NICER (Neutron star Interior Composition ExploreR) is an X-ray instrument
aboard the International Space Station (ISS). It comprises 52 functioning
X-ray concentrator optics and silicon drift detector pairs, arranged in seven
groups of eight. Individual photons between 0.2-12 keV, and their energies,
can be detected to a time resolution of 40 ns (Gendreau et al., 2016).

J1820 was observed with an intensive monitoring program during the initial
hard state of its outburst. Data reduction of ObsIDs 1200120105, 1200120107,
1200120127, 1200120131, and 1200120172, were completed using nicerdas, a
collection of NICER-specific tools, and part of
HEASARC222https://heasarc.gsfc.nasa.gov. Full Level2 calibration and screening
was conducted with nicerl2, which calibrated, checked the time intervals,
merged, and cleaned the data. Barycentric correction was carried out using
barycorr, then the photon events (all between 0.2-12 keV) were binned to the
times of the optical light-curve.

##  3 Method

Our analysis of the optical and X-ray data involves creating simultaneous
lightcurves, Cross-Correlation Functions (CCFs) and Fourier analysis. In the
following we detail the methodology used.

###  3.1 Simultaneous Lightcurves

Simultaneous lightcurves are plotted in Fig. 2. The optical and X-ray data are
not, by default, binned simultaneously. However, while the optical data were
taken in discrete time bins by both instruments, NICER is a photon-counting
instrument and thus records the arrival time of each photon. Therefore, we
create simultaneous lightcurves by binning the photons directly to the optical
time bins, after barycentering both datasets. Since the optical lightcurves
have a constant deadtime (time between the bins in which no data were
recorded) the X-ray photons observed during this time are disregarded. For
X-rays, the square root of the counts per bin was used to determine the error
for each bin. Since the $u_{s}$ band data were sampled at a different rate to
the other optical bands, a separate X-ray lightcurve was created. This
lightcurve is not plotted in Fig. 2, but was used in creating the Cross-
Correlation functions and in the Fourier analysis for the $u_{s}$ band data in
epochs 1–3 and 5.

###  3.2 Cross-Correlation Functions

Cross-correlations are plotted in Fig. 3 & 4. Cross-correlations are
measurements of how much one lightcurve (or any time-series) varies dependent
on another as a function of lag. In these cases, we create optical vs. X-ray
cross-correlations; the figures therefore show the response of the optical
lightcurves to variations in the X-ray lightcurve, as a function of time lag.
Positive values indicate a net correlation at that lag, and negative values a
net anti-correlation, each normalised so that 1 and -1 indicate perfect
correlations and anti-correlations.

The cross-correlations were produced by splitting the simultaneous lightcurves
into segments of equal length. Each segment was then ‘pre-whitened’ by
removing a linear trend. A Cross-Correlation Function (CCF) was then run on
each segment, using the methodology of Venables & Ripley (2002, P. 390). The
mean CCF was then determined and the standard error on each bin was
calculated. To probe variations on different timescales we compute CCFs using
segment sizes of 10 s (Fig. 3) and 2 s (Fig 4).

###  3.3 Fourier Analysis

Fourier analysis is presented in Fig. 5–8. These involved computing the
Fourier transform of the lightcurves and then analysing them at each
frequency.

The power spectra represent the amplitude of the variability at each Fourier
frequency. The coherence represents the relative magnitude of the complex-
valued cross-spectrum, i.e. a measure of how the bands are correlated at that
frequency. The phase lags represent the relative phase angle of the complex-
valued cross-spectrum, i.e. a measure of the lag between the bands at each
frequency as a function of phase (measured in radians). The time lags show the
same data as the phase lags, but converted into the time domain.

This analysis made use of the
Stingray333https://github.com/StingraySoftware/stingray python package
(Huppenkothen et al., 2019). Values for the intrinsic coherence, and errors on
those values, were determined using methods described in (Vaughan & Nowak,
1997, Eqn. 8), where our data fit into the category of ‘High powers, high
measured coherence’.

Good Time Intervals (GTIs) were used based on the individual epochs of X-ray
observation, and then cross-spectra were computed over independent lightcurve
segments and averaged. The segment lengths were $2^{12}$ bins for epochs 1–3
and 5, and $2^{14}$ bins for epochs 4 and 6. For observations with co-adding
in $u_{s}$, the nearest multiple of 2 was used as the bin length, so that the
lightcurve segments were of similar size compared to the other filters of the
same observation. These segment sizes were selected to balance frequency range
against statistics, making sure that all bands were averaged over at least 5
segments (aside from the $u_{s}$ bands in epoch 3 and 5, which had only 3 and
4 segments respectively).

Root-mean-squared (rms2) normalisation was applied to the power spectra
(Belloni & Hasinger, 1990b). The white noise was fitted and removed from the
power spectra before calculating the coherence (see Section 4.3.1 for
details). In Figures 5–8 the frequency-dependent products were binned
logarithmically in frequency; for the power spectra the factor was 1.1, while
for the coherence, time lags, and phase lags the factor was 1.3 (these were
chosen to balance the clarity of features with the size of the uncertainties).

Time lags were calculated by dividing the phase lags by $2\pi f$, where $f$ is
the frequency of the bin. Since the conversion is ambiguous and could be $\pm
2\pi$, we assumed that the phase lags of the frequency bins around 1 Hz were
correct, based on their relationship to the sub-second time lag seen in Figure
4. Each time lag was then arbitrarily shifted based on what would cause the
fewest discontinuities.

##  4 Results

In the Figs. 2–8, the violet plot on the left shows the timeline of the
outburst in MJD, seen by Swift/BAT (see Figure 1) – the stronger the colour,
the brighter J1820 was in hard X-rays. The epochs are marked. Each plot shows
the variation in all bands. The colour key is as follows: $u_{s}$ (blue),
$g_{s}$ (green/teal), $r_{s}$ (red), $i_{s}$ (dark red/brown), $z_{s}$
(black), and X-rays (violet).

###  4.1 Lightcurves

Figure 2: Portions of Optical and X-ray lightcurves from each epoch,
normalised to each band’s mean. Far Left: Timeline of the outburst in MJD,
seen by Swift/BAT (stronger colour = brighter in hard X-rays). Left: 60 s
overview, binned to show overall variations. Right: 4 s insets, marked by
dotted lines in the left. Arbitrary offsets have been applied to the time
axis. The $u_{s}$ band for Epoch 6 suffered from poor statistics, and was thus
not plotted.

Fig. 2 shows portions of the lightcurves from each of the epochs in all
optical bands as well as in X-rays. The lightcurves show a lot of similarities
– in the optical there are numerous sub-second flares with an increase of a
factor $\sim$1.5–2 in flux. A common property of all lightcurves is that the
variations tend to be far stronger in the red than in the blue, and is
particularly true of the sub-second flares – this is also seen in other hard-
state LMXBs (Gandhi et al., 2010, 2016). These flares become less frequent as
the epochs continue, but are still present in epoch 6. Interestingly, the
lightcurves are sometimes anti-correlated during these flares, with optical
activity rising while X-ray activity decreases - see, for example, the inset
to Epoch 4.

###  4.2 Cross-Correlation Functions

Figure 3: Optical/X-ray cross-correlation functions (i.e. a peak/trough at
positive lags means that the optical flux lags the X-ray flux). The CCFs shown
are the average CCF computed from multiple 10 s segments. The lightcurves were
binned to roughly match the lowest time resolution (epoch 5, 43 Hz) in order
to better compare CCF coefficient values (except for $u_{s}$ in epochs 1-3 and
5, due to their significantly lower time resolution). Standard errors,
averaged over the plotted range, are shown. Far left: Timeline of the outburst
– see Fig. 2 caption. Figure 4: Same as in Fig. 3, but the CCFs are created
from 2 s segments instead. This allows us to better compare the rapid
variability, and shows that the sub-second peak at positive lags is present in
a similar place in every single band and epoch.

CCFs from 10 s segments can be seen in Figure 3. Each epoch shows a sub-second
correlation peak. Epochs 1–5 also show some form of a ‘precognition dip’; i.e.
an anti-correlation at negative lags, which means that either the optical
lightcurve dips a few seconds before an X-ray flare, or that optical flares
occur before an X-ray dip.

We can also see how the CCFs evolve over time. Between epochs 1–5, the
correlation steadily decreases at positive lags, perhaps caused by an anti-
correlation component becoming more significant in the data. Additionally,
from epoch 3 onwards a new positively-correlated component appears, peaking at
negative lags, which is stronger at longer wavelengths. This feature evolves
from $\sim$-3 s in epoch 3 to -1 s in epoch 6.

To probe the more rapid variations, we also created CCFs from 2 s segments
which can be seen in Figure 4. These reveal several details. Firstly, the sub-
second correlation is strongest just before the peak of the outburst.
Secondly, the lag of the CCF peak is fairly constant from epoch 2 onwards,
even into epoch 6. Finally, the CCF peak has a ‘tail’ extending from the
initial peak out to 0.5–0.75 s and appears to shrink over time, or at least
become less significant compared to some anti-correlated component; see, in
particular, the difference between epochs 4 and 6.

The sub-second correlation was previously reported in Paice et al. (2019). In
that work, we found that the lag was greater at longer wavelengths – this can
be seen in several of these epochs, and will be shown more explicitly in
Section 4.3.4, and finally discussed in Section 5.1. Our highest-resolution
epochs, 4 and 6, also show a small spike in the CCF at 0 s lag. This is
confirmed by CCFs created from 1 s lightcurve segments and below. However, our
data do not have sufficient time resolution to study these particular
features.

A closer inspection shows that the correlated negative-lag component is
present in epoch 6, but with curious results. Firstly, the peak is now at 0 s,
not -1 s. Secondly, at negative lags the longer wavelengths have a stronger
correlation, while the shortest wavelength ($u_{s}$) shows a trend towards
anti-correlation. At positive lags, the reverse is true; it is the longer
wavelengths that now show a trend towards anti-correlation. This shows either
that this negative component affects the signals down to these rapid
timescales, or that there are two components that affect these lags – this
correlated negative component, and a new component that has appeared between
epochs 5 and 6. These possibilities will be discussed in Sections 4.4 and 4.5.

###  4.3 Fourier Analysis

To better understand the nature of different components contributing to the
CCF, we perform the decomposition of the observed variability into different
time scales using Fourier technique. Figs. 5 – 8 show various Fourier
components in each optical band plotted against X-rays.

Table 2: Fitting parameters for the power spectra. Each cell contains the following: Number of Lorentzians fitted; white noise level in fractional rms2 units $\times 10^{-5}$ (Reduced Chi2 $\chi^{2}_{\nu}$). Epoch | X-rays | $u_{s}$ | $g_{s}$ | $r_{s}$ | $i_{s}$ | $z_{s}$
---|---|---|---|---|---|---
1 | 2; 36.6 (1.47) | 2; 59.4 (0.561) | 2; 6.19 (0.847) | 2; 4.02 (1.43) |  |
2 | 2; 36.9 (1.4) | 4; 84.2 (1.13) | 4; 6.95 (2.84) |  | 4; 3.52 (1.69) |
3 | 2; 11.7 (0.992) | 2; 26.0 (0.495) | 4; 5.79 (1.08) |  | 4; 3.92 (0.894) |
4 | 5; 15.4 (0.86) | 3; 0.77 (2.07) | 6; 0.09 (3.98) | 6; 0.11 (4.98) | 7; 0.14 (7.77) | 8; 0.23 (8.93)
5 | 2; 11.5 (1.16) | 2; 32.8 (0.484) | 4; 5.8 (0.762) |  | 4; 3.25 (1.34) |
6 | 5; 29.6 (1.66) | 2; 324 (0.65) | 4; 0.38 (2.56) | 4; 0.57 (2.76) | 4; 0.66 (2.71) | 4; 10.1 (4.28)

####  4.3.1 Power Spectra

Figure 5: Power spectra of J1820. White noise has been fitted and removed from
each. Far left: Timeline of the outburst – see Fig. 2 caption.

The power spectra can be seen in Fig. 5. In addition to the five optical bands
(detailed at the beginning of Section 4) the X-ray power spectra are also
shown.444Note that these are only for the X-ray data that are strictly
simultaneous with our fast optical photometry (aside from the $u_{s}$ band in
cases of co-adding, i.e. epochs 1–3 and 5), and not the full spectra from the
NICER observations. Thus, these are not directly comparable to the power
spectra in Stiele & Kong (2020). The power is in fractional rms2 units and is
multiplied by the frequency.

As noted in Section 3.1, the separate X-ray lightcurves for the co-added
$u_{s}$ bands in epochs 1–3 and 5 are not shown. Additionally, the $u_{s}$
power spectrum in epoch 6 is not shown due to the poorer data quality.

A mix of zero-centered and non-zero-centered Lorentzians along with a constant
white noise component were fitted to each band. For these plots, that white
noise component was removed and the fitted parameters can be seen in Table 2.
The increased numbers of Lorentzians (and increased $\chi^{2}_{\nu}$ values)
for epochs 4 and 6 are due to the higher cadences, larger segment sizes, and
lower noise levels in the HiPERCAM data; these lead to far lower
uncertainties, and thus require more Lorentzians to fit numerous features in
these bands.

Regarding the evolution of the power: In all epochs, the power in the optical
bands is consistently higher at longer wavelengths, although highest overall
in X-rays. The manifestation of this can be seen in Figure 2, where one can
see activity at longer wavelengths being much stronger than that at shorter
ones.

The power in each band evolves over the course of the outburst. At optical
wavelengths, the power above $\sim$3 Hz drops between epochs 1 and 6 by almost
an order of magnitude. This is most evident when looking at 10 Hz in the
$i_{s}$-band power spectrum. However, at the lowest frequencies it appears
more stable. This could be interpreted as a Lorentzian component peaking at
$\sim$1–2 Hz and becoming less significant as the outburst continues. However,
this does not mean that the component disappears. Furthermore, a small feature
is seen to peak at $\sim$30–40 Hz in all optical bands in epochs 4 and 6 (the
only bands that extend to this frequency with good statistics. Epochs 2 and 3
may show this too, but the uncertainties are too large to confirm this).

Meanwhile, the X-ray power spectrum behaves in the opposite manner. It remains
roughly constant between epochs at all frequencies except the lowest, where it
drops by an order of magnitude between the earliest and latest epochs.

All the power spectra show a break at around 1 Hz, and epochs 4 and 6 possibly
show higher-frequency breaks at around 40 Hz. However, Lorentzian fitting
could not sufficiently quantify these breaks, and therefore their validity and
cause will instead be left as a topic for future work.

Regarding the existence of a Quasi-Periodic Oscillation: In epochs 4 and 5, a
Quasi-Periodic Oscillation (QPO)-like feature can be seen at $\sim$0.1 Hz.
While Lorentzian fitting did not significantly improve with an additional
component at these frequencies for all bands, an X-ray QPO at these
frequencies has been previously detected; the existence and effects of such a
feature are discussed in Section 4.4.

####  4.3.2 Coherence

Figure 6: Coherence of J1820 over frequency, with the same rebinning as in
Figures 5 & 8.

Fig. 6 shows the coherence. Overall, this is generally low (<0.1) at all
frequencies, which is typical for these sources (See, e.g., Malzac et al.,
2018). During all epochs and bands, the optical is more coherent with the
X-rays at lower frequencies, and decreases with increasing frequency in every
epoch. However, the coherence at lower frequencies decreases as the outburst
continues, eventually dropping by over an order of magnitude by epoch 6.

There are numerous smaller features here, but for this work, we will just note
the peaks which occur in the later epochs – at 0.1 Hz in epochs 4 and 5, and
at 0.3 Hz in epoch 6. These will be referred to later in Section 4.4 in the
context of a QPO.

While there is no one relation for the dependence of coherence with optical
band, there are discrete sections that do show clear trends. Saliently, in the
1–5 Hz range, shorter wavelengths tend to be more coherent than longer ones
(particularly in the epochs with the best statistics, such as 4 & 6) – this
will be discussed in Section 4.3.4. There are also sections where the opposite
is true – spikes in coherence at the QPO frequency in epochs 4 and 6 are
stronger at longer wavelengths. These, again, will be noted in Section 4.4.

####  4.3.3 Phase Lags

Figure 7: Phase lags of J1820 over frequency, with the same rebinning as in
Figures 5 & 8.

The phase lags can be seen in Figure 7. Those in the range 1–10 Hz are roughly
the same across all observations, with a shift of +$\pi$ appearing at around 3
Hz; these reflect the presence and stability of the positively-correlated
peak. Above 10 Hz, there are few clear trends and it is difficult to make
definitive claims; if this regime is dominated by components with <0.1 s
delay, then we have many jumps from +$\pi$ to -$\pi$ over this period, and log
binning would average out this behaviour.

However, one difference is the behaviour of the phase lags below 0.5 Hz. In
epoch 1, the phase lags are mostly constant at +$\pi$/4, and in epoch 2, they
appear to increase towards lower frequencies. However, in epochs 3-5 (a month
after outburst peak), phase lags change to roughly $\pm\pi$ – i.e. the two
components are roughly in ‘anti-phase’, where the peak of one component
coincides with the trough of another (this is the Fourier representation of
the anti-correlation component that appears in the CCF – see Figure 3). The
transition to this anti-correlation in the phase lags occurs at around 0.2 Hz,
where there is a sudden discontinuity; analyses of epochs 4 and 5 are
inconclusive in showing whether phase lags increase from $-\pi$, or decrease
from $+\pi$ at this discontinuity. It is perhaps worth noting that negative
phase lags, sometimes approaching $\pm\pi$, are seen at lower frequencies in
multiple other sources (see Gandhi et al., 2010; Veledina et al., 2017; Malzac
et al., 2018; Vincentelli et al., 2021).

This lower-frequency behaviour then changes again much later in the outburst
during epoch 6, at which time the anti-correlation component is now bounded to
a small section at roughly 0.3 Hz, with lower frequencies being generally
above 0.

####  4.3.4 Time Lags

Figure 8: Time lags of J1820 over frequency. Inset for each plot shows a
linear version of the region around the 0.2 s lag seen in the CCFs. Open
circles denote negative optical time lags.

The time lags can be seen in Figure 8. At frequencies below $\sim$0.2 Hz in
epochs 3–5, there is confusion as to whether the time lags are positive or
negative – this depends on whether the phase lags are assumed to be positive
or negative, which is unclear from Fig. 7, as this is the point at which the
phase lags are close to $\pm\pi$.

Figure 8 also presents insets over the 1–10 Hz range, showing the similarities
over the epochs. Shorter frequencies almost uniformly have a smaller lag than
longer frequencies over this range; this is only not the case in epochs with
poorer statistics (i.e. epoch 3) or below 2.5 Hz in epoch 6. This wavelength
dependence will be discussed in Section 5.1, with epoch 6 in particular
discussed in Section 4.5.

###  4.4 Quasi-Periodic Oscillation – Its Evolution and Lags

In epochs 4–6, a feature can be seen that is similar to a QPO, with
significant effects in the coherence and the lags. In epochs 4 and 5, this
feature is at roughly 0.1 Hz, which increases to 0.3 Hz in epoch 6. Each bin
with this feature shows (i) an increase in the power spectra, (ii) higher
overall coherence (sometimes by an order of magnitude, particularly in epochs
4 and 6), (iii) greater coherence at longer wavelengths, (iv) small error bars
in the lags, and (v) negative time lags (changing from -4 s in epoch 4 to -1 s
in epoch 6). These features are best seen in epochs 4 and 6, where the
statistics are better than other epochs.

This possible QPO can also be seen in the CCFs (Figure 3). A positively
correlated component can be seen between -4 and -3 s in epochs 3–5, and at -1
s in epoch 6, as indicated by the time lags seen in Fig. 8, often stronger at
longer wavelengths. We briefly analysed the CCFs to test for the significance
of this feature – see Section A.1 in the Appendix.

As it turns out, a feature at this frequency is not a new discovery; Stiele &
Kong (2020) showed the evolution of a QPO in X-rays over time that corresponds
exactly with our feature described here. Therefore, there appears to be a
connection with this QPO and the features, including a negative lag in the
CCF, in our data. Indeed, QPOs have been associated with changes in the lags
in other LMXB sources previously (Veledina et al., 2015; Malzac et al., 2018;
Vincentelli et al., 2021).

Does this mean that the QPO shows optical variability preceding X-ray? Not
necessarily; due to the periodic nature of phases (as discussed in Section
4.3.3), phase lags between $\pi$–$2\pi$ radians would be represented as
negative lags between -$\pi$–0, and this might be occurring here.
Additionally, the negative lags seen in the CCF could just be a result of the
periodic nature of this component; epochs 4 and 6, for example, show a second
feature at positive lags (5 s and 2 s respectively). These give a time period
of 8 s and 3 s respectively between the two features; this matches the period
of the QPO in both epochs (roughly 0.125 Hz and 0.3 Hz respectively)555We also
see this behaviour in epoch 3, where we do not see clear similar QPO features.
However, the QPO is still detected by Stiele & Kong (2020) during this time at
a similar frequency. Additionally, with only 222s of correlated time, epoch 3
has the poorest statistics of any of our epochs; this may explain why we do
not see such QPO features.. See also Section 4.5, where we simulate the
Fourier components of epoch 6, and show how both the positive and negative
correlations disappear when the QPO’s Fourier components are removed. There is
no clear mechanism by which a QPO would directly cause X-ray emission to lag
optical emission in this way, while there are a number of models that would
show the opposite (e.g. see Section 5.2); we thus consider the latter case to
be the more likely one here.

###  4.5 The Oddity of Epoch 6

The QPO described in the previous section cannot, by itself, explain all the
lags in epoch 6; the phase lags that are significantly different from other
epochs extend over the frequency range 0.08–2.5 Hz, not just around the QPO
frequency. At these frequencies, shorter wavelengths have a consistently
greater lag than longer wavelengths; this is the inverse trait of the sub-
second lag seen between 1–5 Hz in other epochs (while this sub-second lag and
wavelength dependence is still seen in epoch 6, note also how this new
component supersedes it up to 2.5 Hz – Fig. 8).

The epoch 6 lightcurves show low coherence compared to other epochs ($\sim$
0.01 – the only exception here is the 0.3 Hz frequency bin coincident with the
QPO). As for the lags, over this range, $g_{s}$, $r_{s}$, $i_{s}$ and $z_{s}$
bands even have negative lags with respect to X-rays, whereas $u_{s}$ almost
always has positive lags at the same frequency. This behaviour is also evident
in the 2 s CCFs (Figure 4), where the longer-wavelength $r_{s}$, $i_{s}$ and
$z_{s}$ bands show a rising correlation at negative lags and peak at 0 s,
while the shorter-wavelength $g_{s}$ band does not, and the $u_{s}$ band shows
an anti-correlation.

The QPO, along with this different behaviour component, are both strong
features in epoch 6. To what magnitude, and in what ways, do they affect epoch
6’s cross-correlation (Figs. 3 & 4)? To find out, we simulated an
approximation of the Fourier components of the X-rays and $i_{s}$ band of
epoch 6, creating a lightcurve for each from these components, and then cross-
correlated them. We then modified the Fourier components to remove both the
Lorentzian responsible for the QPO and the negative lags; for the latter, we
instead assumed an interpolated flat distribution of 2$\pi$/5 in the phase
lags below 2 Hz. A CCF was made from these lightcurves as well, and the two
results (as well as the inputs) are shown in Figure 9.

Significantly, it can be seen how the cross-correlation is entirely different
between -2 s and 3 s lags, no longer showing the negative correlations unique
to epoch 6, nor the positive anti-correlation that is present in epochs 4 and
6. From this, we conclude that the QPO and the negative lags are the primary
cause for the oddities we see in the epoch 6 CCF. For more information,
including how each component affects the CCF individually and further evidence
of the QPO influencing positive as well as negative lags, see Appendix A.2.

Figure 9: Two simulations of the $i_{s}$ band with X-rays from epoch 6. Top:
Input Fourier components. The red lines are a representation of the data as it
was seen in Figs. 5–8, and the blue lines are a modification that removes the
QPO and the negative lags from the $i_{s}$ band’s Fourier components between
0.02–2 Hz. Bottom: CCFs made by converting the Fourier components into
lightcurves and then cross-correlating the results. CCFs were averaged over
multiple 10s segments. Note how the behaviour completely changes between -2
and +3 s, showing the significance of epoch 6’s negative lags over this range.

##  5 Discussion

Analysis of our results has shown both features that are constant, and ones
that are varying in specific ways over the course of the outburst. To
summarise our main findings:

  1. 1.

In all epochs, J1820 shows rapid, sub-second red flares, and longer-scale
variations that are stronger at longer wavelengths (Fig. 2).

  2. 2.

Over the epochs, an anti-correlation component (stronger at longer
wavelengths) around zero lag becomes increasingly significant – until late
into the hard state decline, when it is superseded by a positive correlation
at negative lags, again stronger at longer wavelengths (Fig. 3).

  3. 3.

In all epochs, the CCF reveals a sub-second peak in the optical/X-ray
correlation function at roughly 0.2 s. The peak maintains a roughly similar
shape over the epochs, but appears to shrink in comparison to other
features(Fig. 4).

  4. 4.

Over time, optical bands become less variable (i.e. decrease in power) at
higher frequencies, but the variability/power remains roughly constant at
lower frequencies. For X-rays, this relation is inverted, showing an overall
decrease of rms2 power at low frequencies. The optical power spectra also have
consistently higher rms2 power at longer wavelengths (Fig. 5).

  5. 5.

Coherence at lower frequencies drops as the outburst continues (Fig. 6).

  6. 6.

The phase/time lags are mostly consistent between 1-10 Hz across epochs. At
lower frequencies, they change from being near +$\pi$/4 to being near $\pm\pi$
as the outburst progresses. Epoch 6, however, fits neither of these trends
(Fig. 7).

  7. 7.

All epochs have an interval between Fourier frequencies 1–5 Hz where shorter
wavelengths have shorter time lags. This behaviour is roughly consistent
(aside from in epoch 6), and neither the lag nor separation by wavelength
appear to change (Fig. 8).

In these observations lies evidence for evolving processes within the system.
We will now address several key points and theories based on these
observations.

###  5.1 Compact Jet

Jet activity has already been found in this source (e.g. Homan et al. 2018;
Bright et al. 2020), and the presence of rapid red variations and a sub-second
optical lag that we show in this paper can both result from jet activity
(Gandhi et al., 2017). Radio data (Bright et al., 2020) show the source to be
relatively bright in the radio, and the long-term lightcurve approximates that
in hard X-rays (Fig. 1). Meanwhile, Russell et al. (2018) presented evidence
that the optical emission was likely on the optically-thin tail of synchrotron
power-law emission from a jet during April 2018.

However, another interesting phenomenon ties in with this: the wavelength
dependence of the sub-second optical lag. In Paice et al. (2019) we
investigated the data shown in epoch 4, and it was first found that a
component of the optical emission lagged the X-rays by roughly 170ms. It was
also found that this lag was dependent on wavelength; shorter wavelengths
lagged less, and longer wavelengths lagged more.

In the previous paper, it was suggested that this feature is emission from a
compact jet. In this interpretation, we consider material emitting in hard
X-rays close to the compact object from a jet-emitting disk (Marcel et al.
2019; though it is also theorised that X-rays may come from the jet itself:
Markoff et al. 2005; Kylafis et al. 2008). A portion of this material is then
ejected as a jet; with a fluctuating ejection rate, this does not necessarily
lead to a uniform stream along the jet, but instead an outflow that varies in
density and/or Lorentz factor over time. We can interpret this as a series of
discrete shells of matter; since these shells vary in speed, faster shells can
thus collide with earlier, slower shells. When they do, they emit through
synchrotron radiation. This is the internal shock model (Jamil et al., 2010;
Malzac, 2013, 2014), the development of which has been motivated by research
into Gamma-Ray Bursts (GRBs) and Active Galactic Nuclei (AGN) jets (Rees &
Meszaros, 1994; Beloborodov, 2000; Tchekhovskoy, 2015). Tetarenko et al.
(2021) found that the jet in this source is highly relativistic ($\Gamma$ =
6.81); this would mean that a time delay of 170 ms corresponds to roughly
5$\times 10^{4}$ km between the X-ray and this synchrotron emitting region.

The energy of this synchrotron emission is dependent upon the variation in the
Lorentz factor of the colliding material; a larger gradient produces higher
energy dissipation. Collisions between larger gradients also occur closer to
the compact object, and thus at shorter time lags. Since the regions close to
the compact object are more compact, synchrotron emission from these regions
is more self-absorbed and peaks at shorter wavelength. Thus we see shorter
time lags for shorter wavelength. A difference of 20 ms between $z_{s}$ and
$u_{s}$ peak lags would, for a highly relativistic jet, correspond to a
spatial extent of 6$\times 10^{3}$ km.

With the new observations presented in this paper, we have found that this
behaviour is also not only present across all our epochs between 1–5 Hz in
Fourier frequency, but it also appears to be fairly consistent in that range
(with the exception of epoch 6, where a different component has the opposite
effect on wavelengths up to 2 Hz) and is independent of the shape of the X-ray
power spectrum.

However, while the behaviour stays more or less consistent, the relative
contribution of this process to the overall variability appears to decrease
over time; note the decreasing significance of the sub-second peak in Figures
3 & 4. We also note the significantly changing phase lags; Paice et al. (2019)
suggested that $\pm\pi$ phase lags at low (<$\sim$1 Hz) frequencies could be a
sign of Doppler-boosting of a jet in high-inclination systems, which was put
forward by Malzac et al. (2018). However, our analysis (see Figure 7) now
shows that $\pm\pi$ phase lags are not a constant feature of this source, and
only appear in the short timescales covered by epochs 3-5. Additionally, the
Coherence also decreases over time as the lags change, similar to what has
been seen in GX 339-4 (Vincentelli & Casella, 2019, though note that this
compares low hard-state and hard-intermediate-state observations).

Over this same range, the X-ray power spectra at these frequencies also
decrease in strength over time, with a sharp decrease between epochs 1 and 2,
where there is also a sharp decrease seen in the CCFs. Wijnands & van der Klis
(1999) notes that the Lorentzians that can describe the X-ray power spectra
move to higher frequency over an outburst, which leads to such decreases in
power at low frequencies. This is interpreted as resulting from changes in the
source geometry.

What do we know of the evolution of the geometry of the source? Kara et al.
(2019) found that the corona appears to shrink over the course of the hard
state, based on a model that assumed a disc that extends to the innermost
stable circular orbit. In our data, we see a broad anti-correlation, which is
more often attributed to a hot flow inside a truncated disc (See Section 5.2).
Zdziarski et al. (2021), meanwhile, describes a radially decreasing corona and
also features a truncated disc, inside which is a hot flow. In either
scenario, an increasingly compact corona could mean that the X-ray emission
from it would contribute less to variability at lower frequencies, and would
also correspond with a decrease in the significance of the jet component
(because both the corona and the jet are linked through fluctuations in
accretion power, which heat the X-ray emitting corona and power the jet; thus,
changes in one indicate changes in the other; Markoff et al. 2005). Overall,
the corona becoming more compact would, by itself and its effect on the jet,
explain several of features that we see.

The optical QPO could also be explained by a precessing jet. This
geometrically-based interpretation has the corona, which is connected to the
jet, precessing in such a way that it creates variability in the lightcurves.
This has been demonstrated in, e.g., Liska et al. (2018), though is still a
matter of debate (see, e.g., Ingram & Motta 2020; Marcel & Neilsen 2021). The
QPO may also contribute to the anti-correlation around zero lag in epochs 3–5;
the high coherence at the QPO frequency would mean that smooth oscillations
would be seen in the CCF, and the anti-correlation occurs between the QPO
correlation peaks of -3 s and 5 s (also worth noting is that the QPO is
stronger at longer wavelengths, a fact which is also true of the anti-
correlation). It is thus feasible that the QPO contributes to the strength of
the anti-correlation, though it need not necessarily be the sole cause of it
(for instance, an anti-correlation at negative lags is present in epochs 1 and
2, when no QPO could be seen in the optical power spectra).

###  5.2 Truncated Disc and Inner Accretion Flow

Is the disc truncated, and if so, does its inner radius evolve? Buisson et al.
(2019) noted that, using relativistic reflection models, the inner edge of the
accretion disc appears to remain steady and close to ISCO during most of the
hard state; however, Zdziarski et al. (2021) noted the inner radius of the
disc being much more truncated, and evolving over time, moving inwards overall
(though perhaps in a stochastic fashion). Likewise, De Marco et al. (2021)
reported a truncated disc moving closer to the black hole as the hard state
evolved.

If a disc’s innermost radius is recessed from the black hole, then there is
potential for a hot accretion flow to form. Both the observed fast UV/X-ray
timing (Kajava et al., 2019) and low optical polarization (Veledina et al.,
2019) properties can be explained in terms of this (optically thin,
geometrically thick) hot inner flow (Eardley et al., 1975; Ichimaru, 1977;
Narayan & Yi, 1994).

Our observations show several features that could indicate this as a
significant process. For instance, the CCFs in Figure 3 show the presence of
an anti-correlation in several of the epochs.

The anti-correlation can be expected if the hot flow broadband spectrum has a
pivoting point, e.g. if an increase of mass accretion rate leads to an
increase in X-ray luminosity, at the same time causing higher synchrotron
self-absorption within the flow (as a result of higher electron number
density), thus leading to a drop in optical emission (Veledina et al., 2011).
In this scenario, the variability amplitude is higher at energies further away
from the pivoting point, hence we expect to have stronger variability at
longer wavelengths, as observed (Fig. 2).

In order to explain the complex anti-correlations at both positive and
negative lags in epochs 4–6 in terms of the hot flow scenario, one needs to
have two sources of both X-ray and optical emission (Veledina et al., 2017);
X-rays would be produced by disc and synchrotron Comptonization, and optical
by synchrotron emission in the hot flow and irradiated disc emission. These
features may appear in the spectrum close to the state transition. The natural
expectation of such scenario is the different shape of the correlation with
soft and hard X-rays, which we indeed see (more details in Section A.3).

The presence of a simultaneous QPO at X-ray and optical wavelengths is another
expectation of the hot flow scenario (Veledina et al., 2013b), which seems to
be confirmed by our data from epochs 4 and 6. A correlated QPO can
significantly alter the shape of the CCF (see, e.g. Veledina et al., 2017) and
can potentially explain some features of the epoch 6 CCF (see Section 4.5 and
Fig. 9 for more discussion). On the other hand, the amplitude of phase lag at
the QPO frequency, $\sim-\pi/2$, is not consistent with the expectation of the
linear theory, which suggests either 0 or $\pi$ depending on the system
orientation (Veledina et al. 2013b; though it is worth noting that the QPO
phase lag was closer to $\pm\pi$ at earlier epochs). Furthermore, the lag at
the QPO frequency can be altered by the aperiodic component – however,
quantitative conclusions on this possibility can only be drawn from dedicated
simulations, which are beyond our present scope. Alternatively, if we assume
that the true phase lag is positive (i.e., shifted by $2\pi$), the
reprocessing signal can contribute to the QPO (Veledina & Poutanen, 2015): 0.3
Hz if within the range of frequencies at which the reprocessed QPO is not
smeared out by the light travel delays.

The hot accretion flow scenario can explain most of the changing components in
the CCF from different epochs, but not the steady narrow peaks at sub-second
lags. The fast optical correlation, most probably coming from a separate
emission component, has to be added to the hot flow contribution to get the
overall CCF shape consistent with the data.

###  5.3 Epoch 6 and the Emergence of Superhumps

Towards the end of the hard state, a superhump modulation at a period of
$\sim$0.7 days was first reported in the optical light-curve of J1820 by
Patterson et al. (2018), and then later expanded upon in Patterson (2019).
This signal appeared around day 87 (MJD 58275), with post-hoc analysis
revealing that it may have appeared as early as MJD 58272. Epoch 6 took place
on MJD 58276.2, very soon after the superhump appeared. Considering the times
of maximum light noted in Patterson (2019), and assuming a period of 0.7 days,
a maximum occurred at MJD 58276.23, essentially concurrent with epoch 6.

To date, there have been very few studies into the effect of superhumps on
optical/X-ray correlations. Given that superhumps are considered to be a
property of the outer disc (see Whitehurst & King 1991), the timescales
involved will correspond to the light-travel time to the disc’s tidal radius,
which for J1820 will be $\sim$10s, and hence any correlated variations are
likely to be heavily smeared, compared to the timescales being studied here.
Actually, optical/X-ray CCFs were constructed for the black hole LMXB Swift
J1753.5–0127, and were found to be independent of the superhump period present
in that system (see Section 3 of Durant et al. 2008, and note that “orbital-
like modulation” refers to superhumps). However, there have been no studies
that examined phase lags in this scenario. Thus, this avenue of research would
be valuable in investigating whether or not they contribute to the features we
see in epoch 6, and, by extension, might be affecting the optical/X-ray
correlations and variability of LMXB systems as a whole. Further studies of
J1820’s superhump properties can be found in Thomas et al. (2021; Subm.).

###  5.4 A Combined Jet and Hot Flow Model

Let us now link our findings to the various models presented. The source shows
repeated rapid red flares, and a sub-second optical/X-ray correlation that has
a larger lag at longer wavelengths. The components dominating the correlation
at low frequencies change as the hard state evolves; the X-ray power spectra
and the optical/X-ray coherence both decrease at these frequencies, and the
phase lags move towards $\pm\pi$. The source also becomes softer over time,
and the sub-second lag in the cross-correlations becomes less significant.
Meanwhile, the X-ray power and the coherence at higher frequencies remains
static.

We do not find that the donor star is an explanation for our features; while
the star could theoretically produce a correlated component at positive lags
in our CCFs through X-ray heating and reprocessing, combining mass and orbital
period estimates from Atri et al. (2020) and Torres et al. (2019) with
Kepler’s third law gives the distance between the compact object and donor to
be $\sim$16 light-seconds, and the effect in the lags would likely vary
between epochs as we observe different phases, in disagreement with either the
smooth evolution or constant nature of the correlated components we see.
However, given the high system inclination ($\sim 75\degr$, Torres et al.
2019), the shortest delays between X-ray and (reprocessed) optical photons
from the near-side of the disc are expected to be about $\sim 0.5$ s, with
some additional smearing to longer lags due to light travel times across the
face of the disc. Hence, it is possible that X-ray reprocessing off the
accretion disc could be significant to the variability; this can be tested in
future by comparing these results to similar soft-state observations, where
the illuminating component should be more dominant.

In all, we suggest a two-component model; one correlated, and the other anti-
correlated. The correlated component we ascribe to a compact jet, which
becomes less significant over time. The anti-correlated component, meanwhile,
we ascribe to a hot flow, which remains static.

A jet as the correlated component would explain the red flares, the
optical/X-ray sub-second correlation (Gandhi et al., 2017), and the larger lag
at longer wavelengths (Malzac, 2013, 2014). X-rays coming from the inflow
would contribute more to the X-ray variability at the lowest (<0.1 Hz) and the
highest (>1 Hz) frequencies. If the corona is contracting (Evidenced either by
a change in the vertical extent, as in Kara et al. 2019, or a change in the
radial extent and a decreasing disc truncation radius, as in Zdziarski et al.
2021), the variability of hard X-rays from that corona would decrease at lower
frequencies, as would the optical/X-ray coherence over the same range – while
the jet, closely linked to the corona, would also decrease in significance,
leading to the decline of the sub-second correlation.

The latter, anti-correlated, component we ascribe to the hot flow. This
component stays mostly static, and thus, relatively, contributes more to the
overall variability as the jet declines in significance. A hot flow scenario
could feasibly also explain the QPO that we see in the data. The hot flow does
not appear to increase in significance – note that the coherence does not
increase.

####  5.4.1 Beyond the Jet and Hot Flow

Muñoz-Darias et al. (2019) and Sánchez-Sierras & Muñoz-Darias (2020) reported
the detection of optical and near-infrared winds respectively in J1820. The
effect of winds on optical/X-ray timing correlations has not yet been explored
in depth, however they would occur on similar timescales to those studied
here. V404 Cyg is a similar system to J1820 (albeit with a much longer orbital
period of 6.47 days and thus a larger physical scale; Casares et al. 1992); in
that source, the wind launching zone was found to be on the order of a few
$\times 10^{5}$ km (Muñoz-Darias et al., 2016), or about 0.5 lightseconds. For
a source inclination of 75$\degr$ (Torres et al., 2019), and using eq. (4) in
Poutanen (2002), we get minimum lags on the order of 0.01 s, so contribution
of the wind to the CCF timescales that we probe is feasible from a timing
standpoint.

However, the shallowness of the P Cygni absorption feature (1–2 percent below
the continuum level, Muñoz-Darias et al. 2019) implies that the wind is
optically thin, which would mean that there would be minimal reprocessed
emission due to the wind. Further investigation into this possibility would
require better data on the optical depth and the ionization of the wind,
combined with simulations.

##  6 Conclusions

We have presented analysis of optical and X-ray lightcurves from the black
hole Low-Mass X-ray Binary (LMXB) MAXI J1820+070 over the course of roughly 80
days. In doing so, we show an evolving Cross Correlation Function (CCF) at
longer ($\sim$10 second) timescales, a consistent sub-second correlation, and
various changes in the Fourier components, including differences between
different optical wavelengths.

This paper thus shows both the dynamic and static nature of LMXBs, even over a
single outburst. The shifting of phase lags at lower frequencies, the slowly
climbing photon index, and the increasingly significant anti-correlation shows
how the coherent components can change on a timescale of weeks. Meanwhile, the
constant nature of the correlation at sub-second lags, mid-frequency time
lags, and rapid red flares in the lightcurves show that other components are
more stable, and can be present with broadly static properties more than two
months apart. Additionally, it shows how a Quasi-Periodic Oscillation (QPO),
travelling upwards through the Fourier frequencies, can change the resultant
lags and correlation features.

We discuss our findings in terms of two synchrotron-emitting components – a
correlated jet and an anti-correlated hot flow – as major contributors to the
overall variability. If we allow for the jet to dominate at the lowest (<0.1
Hz) and the highest (>1 Hz) frequencies, and the hot flow to dominate in
between, the interaction of these components can create the features we
observe in several epochs.

A correlated component at negative lags can be seen in several epochs. Fourier
analysis showed this component to be related to the frequency of a QPO in both
the optical and X-ray lightcurves, previously reported in X-rays by Stiele &
Kong (2020). The lightcurves are consistently coherent at these frequencies,
with greater coherence (and thus correlation) at longer wavelengths. As the
QPO increases in frequency over the outburst, the lag also evolves, becoming
less negative. We note that, due to the periodic nature of the QPO, this
negative lag could easily be a Fourier artefact, and the true lag is positive,
with X-ray variability leading optical by several seconds.

Epoch 6 shows us features that are more difficult to understand. Between 0.08
to 2.5 Hz, there is some component that causes a drop in optical/X-ray phase
lags. This component is more significant at longer wavelengths, and the lags
become negative in most bands. The QPO mentioned earlier is in the middle of
these frequencies, but there is no indication as to whether it is related or
not. Further observations of LMXBs close to the intermediate state would be
highly desirable to investigate this.

The evolution of the optical/X-ray correlations over the course of an LMXB’s
outburst remains an area rich with possibility for new discoveries. This paper
highlights the fact that further, more frequent investigations of an LMXB over
its hard state (and, ideally, over the transition to the soft state) would be
invaluable in further decoding the shifting phenomena inside these sources.

## Data Availability

The NICER data underlying this article are available in the HEASARC Data
Archive (https://heasarc.gsfc.nasa.gov/docs/archive.html). The ULTRACAM and
HiPERCAM data will be shared on reasonable request to the corresponding
author. The Swift data from Fig. 1 are available from the Swift Archive
(https://www.swift.ac.uk/archive/). The AMI-LA data from Fig. 1 are available
from Bright et al. (2020) (https://www.nature.com/articles/s41550-020-1023-5).

## Acknowledgements

We acknowledge support from STFC and a UGC-UKIERI Thematic Partnership. We
would like to thank the anonymous referee for their helpful comments. We would
also like to thank Joe Bright, Piergiorgio Casella, Rob Fender, Adam Ingram,
Sera Markoff, Sara Motta, Tom Russell, Gregory Sivakoff, and Alex Tetarenko
for their helpful conversations. We thank our ULTRACAM observers Paul Chote,
Martin Dyer, and Anna Pala. We also thank Keith Gendreau, Zaven Arzoumanian,
and the rest of the NICER team for their assistance in coordinating
observations.

JAP is part supported by a University of Southampton Central VC Scholarship,
and thanks D Ashton for spectral timing help, as well as A Stevens and D
Huppenkothen for help with the Stingray software. TS thanks the Spanish
Ministry of Economy and Competitiveness (MINECO; grant AYA2017-83216). KR
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 694745). AV acknowledges the Academy of Finland grant 309308 and
the International Space Science Institute (ISSI) in Bern, Switzerland for
support. This work was supported by the Programme National des Hautes Energies
of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES. HiPERCAM and VSD
were funded by the European Research Council (FP/2007–2013) under ERC-2013-ADG
grant agreement no. 340040. ULTRACAM and VSD are funded by the STFC. FMV
acknowledges support from STFC under grant ST/R000638/1.

HiPERCAM observations were made with the GTC telescope (Spanish Observatorio
del Roque de los Muchachos, Instituto de Astrofísica de Canarias), under
Director’s Discretionary Time. SMARTNet helped to coordinate observations. We
have made use of software and web tools from the High Energy Astrophysics
Science Archive Research Center (HEASARC), and made use of data and the ’Build
XRT Products’ tool supplied by the UK Swift Science Data Centre at the
University of Leicester.

## References

  * Adachi et al. (2020) Adachi R., et al., 2020, The Astronomer’s Telegram, 13502, 1
  * Atri et al. (2020) Atri P., et al., 2020, MNRAS, 493, L81
  * Baglio et al. (2018) Baglio M. C., Russell D. M., Lewis F., 2018, The Astronomer’s Telegram, 11418
  * Bahramian et al. (2018) Bahramian A., Strader J., Dage K., 2018, The Astronomer’s Telegram, 11424, 1
  * Belloni & Hasinger (1990a) Belloni T., Hasinger G., 1990a, A&A, 227, L33
  * Belloni & Hasinger (1990b) Belloni T., Hasinger G., 1990b, A&A, 230, 103
  * Beloborodov (2000) Beloborodov A. M., 2000, ApJ, 539, L25
  * Bright et al. (2018) Bright J., Fender R., Motta S., 2018, The Astronomer’s Telegram, 11420
  * Bright et al. (2020) Bright J. S., et al., 2020, Nature Astronomy, 4, 697
  * Buisson et al. (2019) Buisson D. J. K., et al., 2019, MNRAS, 490, 1350
  * Casares et al. (1992) Casares J., Charles P. A., Naylor T., 1992, Nature, 355, 614
  * Casella et al. (2010) Casella P., et al., 2010, Monthly Notices of the Royal Astronomical Society: Letters, 404, L21
  * Corral-Santana et al. (2016) Corral-Santana J. M., et al., 2016, A&A, 587, A61
  * De Marco et al. (2021) De Marco B., Zdziarski A. A., Ponti G., Migliori G., Belloni T. M., Segovia Otero A., Dziełak M., Lai E. V., 2021, arXiv e-prints, p. arXiv:2102.07811
  * Del Santo & Segreto (2018) Del Santo M., Segreto A., 2018, The Astronomer’s Telegram, 11427, 1
  * Denisenko (2018) Denisenko D., 2018, The Astronomer’s Telegram, 11400
  * Dhillon et al. (2007) Dhillon V. S., et al., 2007, Monthly Notices of the Royal Astronomical Society, 378, 825
  * Dhillon et al. (2018) Dhillon V., et al., 2018, in Ground-based and Airborne Instrumentation for Astronomy VII. p. 107020L (arXiv:1807.00557), doi:10.1117/12.2312041
  * Done et al. (2007) Done C., Gierliński M., Kubota A., 2007, A&ARv, 15, 1
  * Durant et al. (2008) Durant M., et al., 2008, The Astrophysical Journal Letters, 682, L45
  * Durant et al. (2011) Durant M., et al., 2011, MNRAS, 410, 2329
  * Eardley et al. (1975) Eardley D. M., Lightman A. P., Shapiro S. L., 1975, ApJ, 199, L153
  * Eastman et al. (2010) Eastman J., Siverd R., Gaudi B. S., 2010, Publications of the Astronomical Society of the Pacific, 122, 935
  * Evans et al. (2009) Evans P. A., et al., 2009, MNRAS, 397, 1177
  * Fabian et al. (2020) Fabian A. C., et al., 2020, MNRAS, 493, 5389
  * Gaia Collaboration et al. (2020) Gaia Collaboration Brown A. G. A., Vallenari A., Prusti T., de Bruijne J. H. J., Babusiaux C., Biermann M., 2020, arXiv e-prints, p. arXiv:2012.01533
  * Gandhi et al. (2008) Gandhi P., et al., 2008, MNRAS, 390, L29
  * Gandhi et al. (2010) Gandhi P., et al., 2010, Monthly Notices of the Royal Astronomical Society, 407, 2166
  * Gandhi et al. (2016) Gandhi P., et al., 2016, Monthly Notices of the Royal Astronomical Society, 459, 554
  * Gandhi et al. (2017) Gandhi P., et al., 2017, Nature Astronomy, 1, 859
  * Gandhi et al. (2019) Gandhi P., Rao A., Johnson M. A. C., Paice J. A., Maccarone T. J., 2019, MNRAS, 485, 2642
  * Gendreau et al. (2016) Gendreau K. C., et al., 2016, in den Herder J.-W. A., Takahashi T., Bautz M., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray. p. 99051H, doi:10.1117/12.2231304
  * Hambsch et al. (2019) Hambsch J., Ulowetz J., Vanmunster T., Cejudo D., Patterson J., 2019, The Astronomer’s Telegram, 13014, 1
  * Henden et al. (2015) Henden A. A., Levine S., Terrell D., Welch D. L., 2015, in American Astronomical Society Meeting Abstracts #225. p. 336.16
  * Homan et al. (2018) Homan J., et al., 2018, The Astronomer’s Telegram, 11576
  * Huppenkothen et al. (2019) Huppenkothen D., et al., 2019, ApJ, 881, 39
  * Ichimaru (1977) Ichimaru S., 1977, ApJ, 214, 840
  * Imamura et al. (1987) Imamura J. N., Steiman-Cameron T. Y., Middleditch J., 1987, ApJ, 314, L11
  * Ingram & Motta (2020) Ingram A., Motta S., 2020, arXiv e-prints, p. arXiv:2001.08758
  * Jamil et al. (2010) Jamil O., Fender R. P., Kaiser C. R., 2010, MNRAS, 401, 394
  * Kajava et al. (2019) Kajava J. J. E., et al., 2019, arXiv e-prints, p. arXiv:1906.06519
  * Kanbach et al. (2001) Kanbach G., et al., 2001, Nature, 414, 180
  * Kara et al. (2019) Kara E., et al., 2019, Nature, 565, 198
  * Kawamuro et al. (2018) Kawamuro T., et al., 2018, The Astronomer’s Telegram, 11399
  * Kylafis et al. (2008) Kylafis N. D., Papadakis I. E., Reig P., Giannios D., Pooley G. G., 2008, A&A, 489, 481
  * Lindegren (2020) Lindegren L., 2020, A&A, 633, A1
  * Liska et al. (2018) Liska M., Hesp C., Tchekhovskoy A., Ingram A., van der Klis M., Markoff S., 2018, MNRAS, 474, L81
  * Littlefield (2018) Littlefield C., 2018, The Astronomer’s Telegram, 11421, 1
  * Magnier et al. (2020) Magnier E. A., et al., 2020, ApJS, 251, 6
  * Malzac (2013) Malzac J., 2013, MNRAS, 429, L20
  * Malzac (2014) Malzac J., 2014, MNRAS, 443, 299
  * Malzac et al. (2018) Malzac J., et al., 2018, MNRAS, 480, 2054
  * Marcel & Neilsen (2021) Marcel G., Neilsen J., 2021, ApJ, 906, 106
  * Marcel et al. (2019) Marcel G., et al., 2019, A&A, 626, A115
  * Markoff et al. (2005) Markoff S., Nowak M. A., Wilms J., 2005, ApJ, 635, 1203
  * Mereminskiy et al. (2018) Mereminskiy I. A., Grebenev S. A., Molkov S. V., Zaznobin I. A., Khorunzhev G. A., Burenin R. A., Eselevich M. V., 2018, The Astronomer’s Telegram, 11488, 1
  * Motch et al. (1982) Motch C., Ilovaisky S. A., Chevalier C., 1982, A&A, 109, L1
  * Motch et al. (1983) Motch C., Ricketts M. J., Page C. G., Ilovaisky S. A., Chevalier C., 1983, Astronomy and Astrophysics, 119, 171
  * Muñoz-Darias et al. (2016) Muñoz-Darias T., et al., 2016, Nature, 534, 75
  * Muñoz-Darias et al. (2019) Muñoz-Darias T., et al., 2019, ApJ, 879, L4
  * Mushotzky et al. (1993) Mushotzky R. F., et al., 1993, ARA&A, 31, 717
  * Narayan & Yi (1994) Narayan R., Yi I., 1994, ApJ, 428, L13
  * Negoro et al. (2018) Negoro H., et al., 2018, The Astronomer’s Telegram, 12057, 1
  * Pahari et al. (2017) Pahari M., et al., 2017, MNRAS, 469, 193
  * Paice et al. (2018) Paice J. A., et al., 2018, The Astronomer’s Telegram, 11432, 1
  * Paice et al. (2019) Paice J. A., et al., 2019, MNRAS, 490, L62
  * Patterson (2019) Patterson J., 2019, in Proc. 38th Annual Conf. Society for Astronomical Sciences, ed. R. K. Buchheim et al. (Rancho Cucamonga, CA: Society for Astronomical Sciences), 61, http://www.socastrosci.org/Publications.html
  * Patterson et al. (2018) Patterson J., et al., 2018, The Astronomer’s Telegram, 11756
  * Poutanen (2002) Poutanen J., 2002, Monthly Notices of the Royal Astronomical Society, 332, 257
  * Rees & Meszaros (1994) Rees M. J., Meszaros P., 1994, ApJ, 430, L93
  * Russell et al. (2018) Russell D. M., et al., 2018, The Astronomer’s Telegram, 11533
  * Sako et al. (2018) Sako S., Ohsawa R., Ichiki M., Maehara H., Morii M., Tanaka M., 2018, The Astronomer’s Telegram, 11426, 1
  * Sánchez-Sierras & Muñoz-Darias (2020) Sánchez-Sierras J., Muñoz-Darias T., 2020, A&A, 640, L3
  * Shidatsu et al. (2019) Shidatsu M., et al., 2019, ApJ, 874, 183
  * Stiele & Kong (2020) Stiele H., Kong A. K. H., 2020, ApJ, 889, 142
  * Tchekhovskoy (2015) Tchekhovskoy A., 2015, Launching of Active Galactic Nuclei Jets. Springer, p. 45, doi:10.1007/978-3-319-10356-3_3
  * Tetarenko et al. (2021) Tetarenko A. J., et al., 2021, MNRAS,
  * Timmer & Koenig (1995) Timmer J., Koenig M., 1995, A&A, 300, 707
  * Tomsick & Homan (2019) Tomsick J. A., Homan J., 2019, The Astronomer’s Telegram, 12732, 1
  * Torres et al. (2019) Torres M. A. P., Casares J., Jiménez-Ibarra F., Muñoz-Darias T., Armas Padilla M., Jonker P. G., Heida M., 2019, ApJ, 882, L21
  * Tucker et al. (2018) Tucker M. A., et al., 2018, ApJ, 867, L9
  * Ulowetz et al. (2019) Ulowetz J., Myers G., Patterson J., 2019, The Astronomer’s Telegram, 12567, 1
  * Uttley et al. (2018) Uttley P., et al., 2018, The Astronomer’s Telegram, 11423
  * Vaughan & Nowak (1997) Vaughan B. A., Nowak M. A., 1997, ApJ, 474, L43
  * Veledina (2016) Veledina A., 2016, ApJ, 832, 181
  * Veledina (2018) Veledina A., 2018, MNRAS, 481, 4236
  * Veledina & Poutanen (2015) Veledina A., Poutanen J., 2015, Monthly Notices of the Royal Astronomical Society, 448, 939
  * Veledina et al. (2011) Veledina A., Poutanen J., Vurm I., 2011, The Astrophysical Journal Letters, 737, L17
  * Veledina et al. (2013a) Veledina A., Poutanen J., Vurm I., 2013a, MNRAS, 430, 3196
  * Veledina et al. (2013b) Veledina A., Poutanen J., Ingram A., 2013b, ApJ, 778, 165
  * Veledina et al. (2015) Veledina A., Revnivtsev M. G., Durant M., Gandhi P., Poutanen J., 2015, MNRAS, 454, 2855
  * Veledina et al. (2017) Veledina A., Gandhi P., Hynes R., Kajava J. J. E., Tsygankov S. S., Revnivtsev M. G., Durant M., Poutanen J., 2017, MNRAS, 470, 48
  * Veledina et al. (2019) Veledina et al., 2019, A&A, 623, A75
  * Venables & Ripley (2002) Venables W. N., Ripley B. D., 2002, Modern Applied Statistics with S, Fourth Edition. Springer, doi:10.1007/978-0-387-21706-2
  * Vincentelli & Casella (2019) Vincentelli F. M., Casella P., 2019, Astronomische Nachrichten, 340, 319
  * Vincentelli et al. (2021) Vincentelli F. M., et al., 2021, MNRAS, 503, 614
  * Whitehurst & King (1991) Whitehurst R., King A., 1991, MNRAS, 249, 25
  * Wijnands & van der Klis (1999) Wijnands R., van der Klis M., 1999, ApJ, 514, 939
  * Xu et al. (2019) Xu Y., Harrison F., Tomsick J., 2019, The Astronomer’s Telegram, 13025, 1
  * Yu et al. (2018) Yu W., Zhang J., Yan Z., Wang X., Bai J., 2018, The Astronomer’s Telegram, 11510, 1
  * Zampieri et al. (2019) Zampieri L., Munari U., Ochner P., Manzini F., 2019, The Astronomer’s Telegram, 12747, 1
  * Zdziarski et al. (2021) Zdziarski A. A., Dzielak M. A., De Marco B., Szanecki M., Niedzwiecki A., 2021, arXiv e-prints, p. arXiv:2101.04482
  * van der Klis (2000) van der Klis M., 2000, ARA&A, 38, 717

##  Appendix A Appendix

###  A.1 CCF Significance Test

Firstly, we analysed the significance of our CCFs. We simulated lightcurves
based on our optical data, uncorrelated with X-rays, and then ran CCFs on
those. To do this, we Fourier transformed the lightcurves, randomised the
phases (i.e. the arguments of the resulting complex numbers), and then inverse
Fourier transformed the result (using methodology laid out in Timmer & Koenig
1995). This simulated lightcurve therefore had the same power spectrum as the
source lightcurve, but was randomised in time and would thus be uncorrelated
with respect to X-rays.

This was done 1000 times. Each time, the simulated, uncorrelated lightcurve
was cross-correlated with the X-rays, and the resultant CCF was recorded. At
the end, for each lag bin, the 5-95% intervals of all simulations was found.
We used this as a way of measuring the significance of features in the
original CCF; any features that lie outside of these intervals are considered
to be significant. The negative lag feature was found to be outside these
intervals for epochs 3–6, and thus we consider it to be a significant feature
rather than a spurious result (Fig. 10).

Figure 10: The $g_{s}$ band vs X-ray CCF from epoch 4, averaged over 162
segments 10 s in size, is shown in green – similar to that seen in Figure 3,
but without binning. A representative error bar is plotted. The faded grey
area is 1000 overlapping simulated correlation functions, and the black dotted
lines are the 5% and 95% intervals of all the simulations. The feature at -3
s, noted in Section 4.4, is above this significance line. The -1 s feature in
epoch 6 was similarly found to be significant.

###  A.2 Simulated Fourier Components

As part of our analysis, we wanted to investigate what correlations we would
see if we modified certain variability features of the source. We did this by
using a custom code that simulated Fourier features, and inputted features
akin to those seen in Figs. 5–8. The code created lightcurves out of these
features, and then carrying out cross-correlation analysis on those resultant
lightcurves. The majority of the analysis on the simulated lightcurves was
carried out by the Stingray666https://github.com/StingraySoftware/stingray
python package (Huppenkothen et al., 2019).

We were thus able to modify the lightcurves by changing the Fourier features.
Figure 9 shows two variants; a simulation meant to reproduce the Epoch 6
features (red) and one that removes all negative-lag trends and the QPO
between 0.02–2 Hz. Figure 11 shows two more variants intended to clarify the
contributions of these components – namely, a version that just removes the
QPO (green) and a version that just removes the negative-lag trends (gold).

The synthetic CCFs here reveal which features are due to which Fourier
components. The +0.5 s anti-correlation and the +2 s correlation, for example,
are primarily due to the QPO; while the -1 s correlation is mainly due to the
negative lags. In both, a sub-second lag is still present, showing that it is
independent of the variability below 2 Hz.

Figure 11: A further two simulations of the $i_{s}$ band with X-rays from
epoch 6. Top: Input Fourier components. The green lines are a modification of
the behaviour of Epoch 6 with the QPO removed, and the gold lines are a
modification that removes the negative lags. Bottom: CCFs made by converting
the Fourier components into lightcurves and then cross-correlating the
results. CCFs were averaged over multiple 10s segments. Note how each changes
affects the CCF in different waves; specifically, the presence of an anti-
correlation at -2 s and a correlation at +3 s.

###  A.3 Additional Epoch 6 CCFs

Figure 12: Two additional CCFs from epoch 6, made from 10 s segments. The CCFs
are optical bands vs. soft X-rays (0.1–1.0 keV, Top) and vs. hard X-rays
(3.0–12.0 keV, Bottom).

We studied the dependence of the CCF shape on the X-ray energy band for Epoch
6. We considered two ranges: $0.1$–$1.0$ keV and $3.0$–$12.0$ keV. The
resulting CCFs are shown in Fig. 12. The correlations look significantly
different in soft and hard X-rays; while the optical/soft-X-ray CCF is
dominated by the peak at small negative lags, the shape of the optical/hard
X-ray CCF more resembles a sinusoid. Interestingly, the narrow peak at
positive lags, which we attribute to the jet, is more prominent in the soft
CCF, though is present in both at the same lag.

Previous studies of the dependence of CCF shape on the X-ray energy band has
been performed for Swift J1753.5–0127 (Durant et al., 2008, 2011), GX 339–4,
and the neutron star binaries Sco X-1 and Cyg X-2 (Durant et al., 2011). These
show shape variations, yet such acute difference of CCF shape seen in this
paper has never been reported before.

The difference may appear due to presence of two separate components in the
X-ray band, e.g. softer component coming from comptonisation of disc photons
and harder component coming from hot flow synchrotron Comptonization
(Veledina, 2016). Recent spectral studies indeed suggest the presence of two
Comptonization continua (Zdziarski et al., 2021). The variability of both of
these is caused by the propagating fluctuations, however, their response may
alter between harder-when-brighter behaviour for the synchrotron
Comptonization to softer-when-brighter for disc Comptonization, resulting in a
complex variability pattern (Veledina, 2018). Correlation with the optical
components (disc, jet, and hot flow) is naturally expected to be different for
these X-ray components, leading to difference between X-ray energy bands, as
the fraction of synchrotron- to disc-Comptonization depends on the energy. The
weighted average of the sharply different soft and hard CCFs may then lead to
the complex CCF shape seen in Fig. 3.

###  A.4 Possibility of a Flared Disc

Figure 13: Phase lags from Epoch 4, in the $u_{s}$ band only, rebinned to best
show these variations. The red box indicates a region where there might be a
sinusoidal variation.

The phase lags for epoch 4 appear to show peaks in the range 0.1–5 Hz (see
Figure 13). Such features have previously been proposed to originate in a
highly flared disc (Poutanen 2002 – see Fig. 6 within). In this model, these
peaks can result from features in the power density spectrum, and reflection
from the outer disc.

To investigate, we rebinned the data until it best showed this feature. Then,
two models were fit to the phase lags over this range, the first being a line,
and the second being a line with a sinusoid added on top, both in linear
space. However, the second model was not found to fit the phase lags
significantly better than the first. There was ambiguity in the results;
relative to the variations in the phase lags themselves, large errors are
present which allowed for both models to be viable.

Hence, while we do not consider the peaks in epoch 4 to be evidence for a
flared disc, we note that this could be a topic for further investigation in
future observations of J1820.