# A rapid optical and X-ray timing study of the neutron star X-ray binary
Swift J1858.6–0814

T. Shahbaz,1,2 J. A. Paice,3,4,5 K. M. Rajwade,5,6 A. Veledina,7,8 P. Gandhi,3
V. S. Dhillon,1,2,9T. R. Marsh,10 S. Littlefair,9 M. R. Kennedy,5,11 R. P.
Breton5 and C. J. Clark.5,12,13
1Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife,
Spain
2Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La
Laguna, Tenerife, Spain
3Department of Physics and Astronomy, University of Southampton, Highfield,
Southampton, SO17 1BJ, UK
4Inter-University Centre for Astronomy and Astrophysics, Pune, Maharashtra
411007, India
5Department of Physics and Astronomy, The University of Manchester, Oxford
Road, Manchester, M13 9PL, UK
6ASTRON, the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk
4, 7991 PD Dwingeloo, The Netherlands
7Department of Physics and Astronomy, FI-20014 University of Turku, Finland
8Nordita, KTH Royal Institute of Technology and Stockholm University,
Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
9Department of Physics and Astronomy, University of Sheffield, Sheffield, S3
7RH, UK
10Astronomy and Astrophysics Group, Department of Physics, University of
Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK
11Department of Physics and Astronomy, University College Cork, College Rd,
Cork Cork T12 K8AF, Ireland
12Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut),
Callinstraße 38, 30167 Hannover, Germany
13Leibniz Universität Hannover, 30167 Hannover, Germany E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a rapid timing analysis of optical (HiPERCAM and ULTRACAM) and
X-ray (NICER) observations of the X-ray transient Swift J1858.6–0814 during
2018 and 2019. The optical light curves show relatively slow, large amplitude
($\sim$1 mags in $g_{s}$) ‘blue’ flares (i.e. stronger at shorter wavelengths)
on time-scales of $\sim$minutes as well as fast, small amplitude ($\sim$0.1
mag in $g_{s}$) ‘red’ flares (i.e. stronger at longer wavelengths) on time-
scales of $\sim$seconds. The ‘blue’ and ‘red’ flares are consistent with X-ray
reprocessing and optically thin synchrotron emission, respectively, similar to
what is observed in other X-ray binaries. The simultaneous optical versus
soft- and hard-band X-ray light curves show time- and energy dependent
correlations. The 2019 March 4 and parts of the June data show a nearly
symmetric positive cross correlations (CCFs) at positive lags consistent with
simple X-ray disc reprocessing. The soft- and hard-band CCFs are similar and
can be reproduced if disc reprocessing dominates in the optical and one
component (disc or synchrotron Comptonization) dominates both the soft and
hard X-rays. A part of the 2019 June data shows a very different CCFs. The
observed positive correlation at negative lag in the soft-band can be
reproduced if the optical synchrotron emission is correlated with the hot flow
X-ray emission. The observed timing properties are in qualitative agreement
with the hybrid inner hot accretion flow model, where the relative role of the
different X-ray and optical components that vary during the course of the
outburst, as well as on shorter time-scales, govern the shape of the
optical/X-ray CCFs.

###### keywords:

accretion, accretion discs – X-rays: binaries – X-rays: individual: Swift
J1858.6-0814 – stars: neutron

††pubyear: 2021††pagerange: A rapid optical and X-ray timing study of the
neutron star X-ray binary Swift J1858.6–0814–B Table 1: Log of ULTRACAM,
HiPERCAM & NICER observations for Swift J1858.6–0814.

UT date | UT Start | UT End | Instrument | Filters | Cadence1 | Comments
---|---|---|---|---|---|---
2018/11/14 | 19:24:22 | 19:50:15 | HiPERCAM | $u_{s},g_{s},r_{s},i_{s},z_{s}$ | 46.6 ms |
2018/11/09 | 00:42:59 | 01:18:33 | ULTRACAM | $u^{\prime},g^{\prime},i^{\prime}$ | 0.93 (4.63) s |
2019/03/01 | 09:14:15 | 09:49:19 | ULTRACAM | $u_{s},g_{s},i_{s}$ | 1.00 (3.01) s |
2019/03/02 | 09:07:00 | 09:39:20 | NICER | 0.2–12 keV | 40 ns | ObsId 2200400101
2019/03/02 | 09:05:22 | 09:45:14 | ULTRACAM | $u_{s},g_{s},i_{s}$ | 0.28 s (0.29) s | Simultaneous with NICER
2019/03/04 | 08:54:22 | 09:48:59 | ULTRACAM | $u_{s},g_{s},i_{s}$ | 0.50 s (4.01) s | Simultaneous with NICER
2019/03/04 | 09:05:25 | 09:36:20 | NICER | 0.2–12 keV | 40 ns | ObsId 2200400103
2019/03/05 | 09:06:26 | 09:26:49 | ULTRACAM | $u_{s},g_{s},i_{s}$ | 0.58 s (1.17) s |
2019/05/09 | 08:13:38 | 10:23:17 | ULTRACAM | $u_{s},g_{s},i_{s}$ | 0.25 s (1.26) s |
2019/06/07 | 01:52:25 | 02:39:45 | HiPERCAM | $u_{s},g_{s},r_{s},i_{s},z_{s}$ | 47.9 ms | Simultaneous with NICER
2019/06/07 | 01:53:20 | 02:33:43 | NICER | 0.2–12 keV | 40 ns | ObsId 2541030101
2019/06/07 | 03:23:09 | 04:17:27 | HiPERCAM | $u_{s},g_{s},r_{s},i_{s},z_{s}$ | 47.9 ms |

  * •

1Numbers in brackets is the $u_{s}$-band cadence, if different from the other
wave-bands.

Figure 1: The long-term X-ray light curves and the radio and X-ray spectral
index light curves of Swift J1858.6–0814 during its 2018 and 2019 outburst.
The top panel shows the Swift/XRT PC mode X-ray data (black squares), where
the vertical lines mark the times of our ULTRACAM (blue) and HiPERCAM (red)
optical observations. The dashed lines show the times when the optical
observations were simultaneous with NICER. The bottom panel show the radio
(red stars; 4.5 GHz) and X-ray (black squares; 0.2–10 keV) spectral indices
($F_{\nu}\propto\nu^{\alpha}$) taken from van den Eijnden et al. (2020). The
blue squares show the spectral index of the optical flares observed with
ULTRACAM and HiPERCAM determined in this paper (see Section 4).

##  1 Introduction

The low-mass X-ray binary Swift J1858.6–0814 was discovered as an X-ray
transient on 2018 October (Krimm et al., 2018) with the Burst Alert Telescope
(BAT) aboard the Neil Gehrels Swift Observatory (Gehrels et al., 2004).
Subsequent multi-wavelengths observations detected the source at longer
wavelengths. The Ultraviolet and Optical Telescope (Swift-UVOT) on-board Swift
detected a variable UV source which was coincident with a previously detected
UKIRT Infrared Deep Sky Survey (UKIDSS) and Pan-STARRs source (Kennea & Krimm,
2018). Optical follow-up observations revealed that the source had brightened
by $\sim$2.5 magnitudes (Vasilopoulos et al., 2018). The source was also
detected in the radio by the Arcminute Microkelvin Imager Large Array having a
variable flux density of 300–600 $\mu$Jy at 15.5 GHz (Bright et al., 2018). At
X-ray wavelengths, the outburst was relatively faint, with a flux of
$\sim$10-11 erg s-1 cm-2 at 0.5–10 keV and a hard spectrum with a photon index
of $\Gamma$ = 2 (Reynolds et al., 2018).

Superimposed on the outburst were bright, short X-ray flare events (Ludlam et
al., 2018; Hare et al., 2019) where the observed flux increased by more than
an order of magnitude in a few seconds (Hare et al., 2020). Optical flares
were also identified (Vasilopoulos et al., 2018; Baglio et al., 2018; Rajwade
et al., 2018, 2019; Paice et al., 2018) with wavelength-dependent optical
variability on time-scales of minutes, and sporadic, fast ‘red’ flares on
time-scales of seconds (Paice et al., 2018). The timing characteristics were
reminiscent of those seen in the black hole X-ray binary V404 Cyg, which
showed long-term ‘blue’ flaring and short-term sporadic ‘red’ flaring during
its 2015 outburst (Kimura et al., 2016; Gandhi et al., 2016). The radio
emission from Swift J1858.6–0814 showed variability by up to a factor of
$\sim$8 on time-scales of minutes due to mass accretion rate fluctuations
consistent with a compact jet (Bright et al., 2018; van den Eijnden et al.,
2020). The X-ray spectrum showed evidence for significant intrinsic local
absorption (Reynolds et al., 2018; Hare et al., 2020) and the P-Cygni profile
observed in the optical spectrum (Muñoz-Darias et al., 2020), suggested that a
significant amount of mass was ejected from the inner accretion flow.

Although Swift J1858.6–0814 entered the Sun constraint for most X-ray
telescopes in 2019 November, it was detected again with the Monitor of the
All-sky X-ray Imager (MAXI) in 2020 February in a previously unobserved X-ray
state, with significantly less variability and enhanced soft X-ray emission,
implying a transition to a soft state (Negoro et al., 2020; Buisson et al.,
2020b). During 2020 March several Type I X-ray bursts were detected with the
Neutron star Interior Composition Explorer (NICER) and the Nuclear
Spectroscopic Telescope Array (NuSTAR), identifying Swift J1858.6–0814 as a
neutron star binary system despite the fact that pulsations were not detected
(Buisson et al., 2020a). These bursts exhibited photospheric radius expansion
allowing a distance estimate of $\sim$12.8 kpc. Strong periodic drops in X-ray
flux were also detected, consistent with eclipses by the secondary star and
variable obscuration due to the thickness of the disc/accretion stream which
is also responsible for the strong variability (Buisson et al., 2021).

Here we report on high time-resolution HiPERCAM and ULTRACAM optical
observations of Swift J1858.6–0814 some of which are simultaneous with NICER
observations, taken in 2018 and 2019. We comment on the observed optical
flaring and on the optical/X-ray flux correlations and timing properties of
the light curves.

##  2 Observations

In Fig. 1 we show the long-term X-ray light curve of Swift J1858.6–0814 during
its 2018 and 2019 outburst and mark the optical and X-ray observations
presented in this paper.

###  2.1 NICER – X-rays

Swift J1858.6–0814 was observed with NICER in an intensive monitoring program
during its 2018 and 2019 X-ray outburst. NICER is an X-ray instrument on board
the International Space Station (ISS) where individual photons with energies
in the range 0.2–12 keV can be detected with a time resolution of 40 ns
(Gendreau et al., 2016). The data reduction was carried out using the
collection of NICER-specific tools nicerdas which is part of HEASARC
111https://heasarc.gsfc.nasa.gov. Full Level2 calibration and screening was
conducted with nicerl2, which calibrated, checked for good time intervals,
merged, and cleaned the data. The barycentric correction was carried out using
barycorr, and finally the photon events were binned to the times of the
optical light curves as described in the following sections. We produced a
light curve in the 0.2–12 keV energy band for each data segment using xselect
and then applied the background correction. In order to calculate the hardness
ratio, we extracted light curves in the 0.5–3.0 keV and 3–10 keV bands. For
these light curves, we normalised each incoming photon with respect to the
effective area of the telescope at that energy. We define the hardness ratio
of the X-rays as (hard-soft)/(hard+soft), where the hard and soft X-ray rates
are in the 3–10 keV and 0.5–3.0 keV range, respectively. The errors on the
hardness ratio were calculated by using 1-$\sigma$ Poisson errors (following
the example of Gehrels 1986) to simulate maximum and minimum values of the
individual X-ray bands, and then calculating the hardness ratio at each
extreme. We note that these errors are an approximation only and may
underestimate any outliers.

(a) The ULTRACAM light curves

(b) The HiPERCAM light curves

Figure 2: The observed ULTRACAM (top) and HiPERCAM (bottom) light curves of
Swift J1858.6–0814. The black dotted horizontal line shows the time of NICER
observations. The mean magnitude of Swift J1858.6–0814 is shown in each panel.
A MJD time offset of $T_{0}$ \+ 58000.0 ($T_{0}$ is in days) is applied and we
use the orbital ephemeris given in Buisson et al. (2021).

###  2.2 ULTRACAM/NTT – Optical

High-speed multi-colour photometry of Swift J1858.6–0814 was carried out using
ULTRACAM instrument (Dhillon et al., 2007) on the 3.5 m New Technology
Telescope (NTT) in La Silla, Chile. ULTRACAM uses dichroic beamsplitters to
simultaneously image three custom made Sloan Digital Sky Survey (SDSS)
filters, and can observe at frame-rates well above 100 Hz due to the frame-
transfer CCDs and the lack of a physical shutter (Dhillon et al., 2007). We
used ULTRACAM to observe Swift J1858.6–0814 during 2018 November, 2019 March
and 2019 May. The 2018 observations were carried out simultaneously with the
$u^{\prime}$, $g^{\prime}$, and $i^{\prime}$ SDSS filters (Doi et al., 2010),
whereas the 2019 observations were performed using the higher throughput
$u_{s}$, $g_{s}$, and $i_{s}$ Super-SDSS filters (Dhillon et al., 2021) which
use multi-layer coatings rather that coloured glass to define the filter
bandpasses, with the cut-on/off wavelengths designed to match higher
throughput the original SDSS filters. Unlike most observations of this type,
the times were not explicitly chosen to coincide with X-ray observations. Some
of the observations did overlap with the X-ray observations performed with the
NICER instrument and such simultaneity was purely serendipitous (see Section
2.1). On different nights, ULTRACAM was used in windowed mode (one window
containing the target and the other containing multiple comparison stars) with
1$\times$1 binning. Typically, compact binaries are faint in the $u_{s}$-band,
and so ULTRACAM’s on-chip co-adding feature was used, which provides a longer
exposure time in the $u_{^{\prime}}$-band so as to increase the signal-to-
noise ratio. The details of the observing setup for each night are given in
Table 1.

We used the HiPERCAM pipeline software222https://github.com/HiPERCAM/hipercam
to debias, flat-field and extract the target count rates using aperture
photometry with a seeing-dependent circular aperture tracking the centroid of
the source. The sky background was computed using the clipped mean of an
annular region around the target and relative photometry of Swift J1858.6–0814
was carried out with respect to the local standard star (PSO
J185832.982-081400.913). For the $r_{s}$-band and $g_{s}$-band, the field is
covered by the Pan-STARRS survey and so the calibrated $r_{s}$-band and
$g_{s}$-band magnitudes are listed in DR1 catalog (Magnier et al., 2020).
These were transformed to SDSS magnitudes (Finkbeiner et al., 2016) and then
used to calibrate the target light curves. Since the field is not covered by
any archival optical survey in the $u_{s}$-band, calibrating these data was
less straightforward. Flux standards were observed on various nights during
the ULTRACAM observations in 2019 March. These flux standards were used to
determine the $u_{s}$-band instrument zero-point. The local standards were
then calibrated which in turn were used to calibrate the target light curve.
For the nights when no flux standard was observed, we assume that the
$u_{s}$-band zero-point measured during the March observing runs was still
valid. The difference between the ULTRACAM Super-SDSS and SDSS filters leads
to an uncertainty in the flux calibration of $<$ 3 per cent (Wild et al.,
2022). The observed ULTRACAM light curves are shown in Fig. 2a.

###  2.3 HiPERCAM/GTC – Optical

Sub-second optical imaging was carried out in 2018 November and 2019 June
using HiPERCAM on the 10.4 m Gran Telescopio Canarias (GTC) in La Palma,
Spain. HiPERCAM uses dichroic beamsplitters to simultaneously image the custom
made Super-SDSS $u_{s}$, $g_{s}$, $r_{s}$, $i_{s}$ and $z_{s}$ filters.
Similar to ULTRACAM, HiPERCAM can observe at frame-rates well above 1000 Hz
which is achieved by the lack of a physical shutter and the 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 (7.8
ms) dead-times (Dhillon et al., 2021). The CCDs were binned by a factor of 4
and drift mode was used with four windows (336$\times$200 pixels each) for all
the observations. The instrument was orientated so that one window was
centered on Swift J1858.6–0814 and another window on a local standard star. We
used an exposure time of 43.6 ms and 44.9 which resulted in a cadence of 46.6
ms and 47.9 ms, for the 2018 and 2019 observations, respectively (see Table 1
for details). Observations were obtained on two nights, 2018 November 14 and
2019 June 7. The observations taken in 2019 were coordinated with the X-ray
instrument NICER. A log of the observations is given in Table 1. Similar to
the ULTRACAM data, we used the HiPERCAM pipeline software to debias, flat-
field and extract the photon counts for the target and local standard using
aperture photometry with a seeing dependent circular aperture. The local
standard stars used are listed in the Pan-STARRS survey DR1 catalog (Magnier
et al., 2020) and have $g^{\prime}$, $r^{\prime}$, $i^{\prime}$ and
$z^{\prime}$ magnitudes which were transformed to SDSS magnitudes (Finkbeiner
et al., 2016) and then used for the photometric calibration of Swift
J1858.6–0814. For the 2018 data the $u^{\prime}$-band calibration was
determined using the local standard star PSO J185827.968-081329.815 and the
full-frame acquisition images which was calibrated by determining the
instrument zero-point. As a check we also determined the $g_{s}$, $r_{s}$,
$i_{s}$ and $z_{s}$ magnitudes and found that they agreed with the Pan-STARRS
magnitudes at the $<$10 per cent level. Unfortunately, the local standard star
PSO J185826.795-081357.216 used in the 2019 observations was not detected in
the $u_{s}$-band images, and so it could not be flux calibrated. The
difference between the HiPERCAM Super-SDSS and SDSS filters leads to an
uncertainty in the flux calibration of $<$ 3 per cent (Brown et al., 2022).
Finally we convert from SDSS magnitudes to flux density, where we propagate
the uncertainty in the local standard. The observed HiPERCAM light curves are
shown in Fig. 2b.

##  3 Reddening

Swift J1858.6–0814’s position in the sky allows us to estimate the line of
sight interstellar reddening. The Galactic neutral atomic hydrogen (H I)
column density to the target is $N_{\rm H}\sim 1.84\times 10^{21}$ cm-2 (HI4PI
Collaboration et al., 2016). Using the relation between the Galactic hydrogen
absorption column density and optical extinction (Foight et al., 2016) along
with the galactic extinction law (Cardelli et al., 1989) we determine a colour
excess of $E\left(B-V\right)$=0.21 mag. We can also estimate $N_{\rm H}$ from
spectral fits to the NICER data. Using the XSPEC (Arnaud, 1996) software
package, a blackbody and power-law model fit to the 2018 November data gives
$N_{\rm H}\sim 2.0-2.5\times 10^{21}$ cm-2, whereas fits to the 2019 June data
gives $N_{\rm H}\sim 1.6-1.7\times 10^{21}$ cm-2. The value for $N_{\rm H}$
determined from the NICER data is consistent with the value determined from
the H I maps and we assume a colour excess of 0.21 mag for the rest of this
paper.

Figure 3: ULTRACAM and HIPERCAM $g_{s}$-band light curves of Swift
J1858.6–0814 as a function of orbital phase using the orbital ephemeris given
in Buisson et al. (2021), where phase 0.0 is defined as superior conjunction
of the compact object. For clarity two orbital phases are plotted and the
light curves have been rebinned to a time resolution of 10 s.

##  4 Optical flares

In Figs. 2a and 2b we show the observed HiPERCAM and ULTRACAM light curves,
respectively, where wavelength dependent flaring activity is clearly seen.
Flaring is superimposed on a sinusoidal modulation, which is due to a
combination of the secondary star’s ellipsoidal modulation, X-ray heating and
other possible sources of light in the system (see Fig. 3). To determine the
properties of the flares first use the colour excess of
$E\left(B-V\right)$=0.21 mag determined in Section 3 with the interstellar
extinction law (Cardelli et al., 1989) to deredden the observed fluxes. We
identify and isolate the flare events by determining the start and end of the
same flare event in each waveband. We then subtract the interpolated flux
underneath the flare event which in effect subtracts the contribution of the
non-variable component. We assume during the actual flare event that the other
components that contribute to the observed flux do not vary. We define small
and large flares as events with $g_{s}$-band amplitudes of $\sim$0.1 mag and
$\sim$1 mag, respectively. A total of 102 large and 5 small flare events were
isolated, respectively. Fig. 13 of the Appendix shows some examples of the
isolated flare events where flares on different time-scales, amplitude and
colour are clearly seen. For the flare events we also determine the peak flare
flux in each waveband and flux ratio.

In Fig. 3 we show the observed $g_{s}$-band light curve of Swift J1858.6–0814
as a function of orbital phase, using the orbital ephemeris given in Buisson
et al. (2021) where phase 0.0 is defined as superior conjunction of the
compact object. Although our orbital phase coverage is relatively poor
($\sim$33 per cent), we observe flares at all orbital phases. Buisson et al.
(2021) find that the bright flares occur preferentially in the post-eclipse
phase of the orbit, around orbital phase $\sim$0.3, most likely due to
increased thickness at the disc-accretion stream. We do not find any evidence
for this in our optical data, but note our poor phase coverage. We find that
the mean flux and the intrinsic source fractional RMS variability defined as
$\sigma^{2}_{\rm source}$ = $\sigma^{2}_{\rm total}$ \- $\sigma^{2}_{\rm
noise}$ (Vaughan et al., 2003) are strongly linearly correlated with a
Pearson’s correlation coefficient of 0.84. The low RMS observed at phase 0.0
(2019 March 01) which has the lowest flux of our observations and very little
flaring is consistent with a system at a high binary inclination angle
(Buisson et al., 2021; Knight et al., 2022).

Figure 4: The rise, decay time-scales, duration and flux histograms of the
dereddened flare events.  Figure 5: The flare duration versus colour of the
dereddened small (red points) and large flare (blue points) events. Figure 6:
Colour–colour diagram for the ULTRACAM and HiPERCAM large and small dereddened
flare events. The dashed line shows a power-law model of the form
$F_{\nu}\propto\nu^{\alpha}$, where the black squares mark the value of
$\alpha$ ranging from -2.0 to +2.0 in units of 0.5. The solid black line is a
blackbody model where the crosses show the temperature in units of 1000 K. In
the top panel the red circles show the HiPERCAM small flares, whereas the blue
(2018 November 14) and green circles (2019 June 7) show the HiPERCAM large
flare events. In the bottom panel the red and blue circles show the ULTRACAM
small and large flares events, respectively.

###  4.1 Time-scales

We determine the rise, decay and duration of the dereddened flares which are
shown in Fig. 4. As one can see, the ‘red’ flares (more flux at longer
wavelengths) have a much shorter time-scale and amplitude compared to the
‘blue’ flares (more flux at shorter wavelengths) events. The ‘red‘ and ‘blue’
flares have median $g_{s}$-band amplitudes of $\sim$0.1 mag and $\sim$1 mag,
respectively. In Fig. 5 we show the flare duration versus colour. The flares
are separated into two regions: short-duration ‘red’ flares and long-duration
‘blue’ flares. Small amplitude ’red‘ flares are observed on 2018 November 14
(HiPERCAM) and 2019 February 2 (ULTRACAM), whereas large ‘blue’ flares are
present in all observations, except on 2019 March 2 where no flares are
observed. The different time-scales and amplitudes of the flares indicate that
they arise from different emission processes.

###  4.2 Spectral energy distribution

In an attempt to interpret the broad-band spectral properties of the flares,
we compare the observed fluxes with the prediction for different emission
mechanisms, namely synchrotron emission and blackbody. The latter has an
approximately power-law form on the Rayleigh–Jeans tail and so we characterise
the synchrotron and blackbody emission with a power-law form
$F_{\nu}\propto\nu^{\alpha}$, where $\nu$ is the frequency and $\alpha$ is the
spectral index. We compute the given emission spectrum and then calculate the
expected flux density ratios in the relevant filters using the synthetic
photometry package synphot in iraf/stsdas. For the blackbody emission, given
the intrinsic model flux we then determine the corresponding radius of the
region that produces the observed dereddened flux at a given distance.

In Fig. 6 we show the HiPERCAM and ULTRACAM individual peak flare flux ratios
and the expected results for different emission models. We show the $g_{s}$,
$r_{s}$ and $z_{s}$ fluxes common to the HiPERCAM 2018, 2019 and ULTRACAM 2019
data sets and the $u^{\prime}$, $g^{\prime}$ and $i^{\prime}$ fluxes for the
ULTRACAM 2018 data set. Fig. 14 of the Appendix shows some example fits to the
individual dereddened flare events observed on 2018 November 14 (HiPERCAM) and
2019 May 9 (ULTRACAM). The power-law indices obtained by fitting the broad-
band spectral energy distribution of the individual large and small flare
events are in the range $\alpha\sim$-1.0 to -2.0 (with a mean of $\alpha\sim$
-1.5) for the ‘red’ flares. In contrast the ‘blue’ flares can be represented
with a power-law of $\alpha\sim$ 1.0 (range of $\alpha\sim$0.6 to 1.2) or a
$\sim$14,000 $\pm$ 2000 K blackbody which with a mean $g_{s}$ peak flare flux
of $\sim$0.45 mJy (out of eclipse) corresponds to a radius of $\sim$1.0$\,\pm$
0.2 $\rm R_{\sun}$, assuming a distance of 12.8 kpc (Buisson et al., 2020a).
Although a single temperature blackbody has limited physical significance and
is likely a very poor description of a flare event, it is useful for
comparison with other works. The 2019 data was taken at orbital phase $\sim$
0.9 which is outside the start of eclipse ingress (Buisson et al., 2021) and
so we can rule out a decrease in $N_{\rm H}$ due to the absorption in the
atmosphere of the secondary star. However, Castro Segura et al. (2022) have
detected disc winds in the hard state and the associated variable obscuring
columns that contribute to $N_{\rm H}$ might explain the differences we
observe.

(a) ULTRACAM/NICER 2019 March 2

(b) ULTRACAM/NICER 2019 March 4

(c) HiPERCAM/NICER 2019 June 7; part1

(d) HiPERCAM/NICER 2019 June 7; part2

(e) HiPERCAM/NICER 2019 June 7; part3

(f) HiPERCAM/NICER 2019 June 7; part4

Figure 7: The simultaneous optical and X-ray light curves of Swift
J1858.6–0814. From top to bottom, the optical light curves, the hard (3–10
keV; blue) and soft (0.5–3.0 keV; red) X-ray light curves and the X-ray
hardness ratio defined as the ratio of the rates (hard-soft)/(hard+soft). The
X-rays and hardness ratio light curves have been binned with a moving average
of 100 points for readability (except for 2019 March 4 where a 20 point moving
average was used due to the much higher count rates). A barycentered MJD time
offset of 58544.37641329, 58546.36898883 and 58641.08379497 has been applied
to the 2019 March 2, March 4 and June 7 data, respectively.

(a) ULTRACAM/NICER 2019 March 2

(b) ULTRACAM/NICER 2019 March 4

(c) HiPERCAM/NICER 2019 June 7; part1

(d) HiPERCAM/NICER 2019 June 7; part2

(e) HiPERCAM/NICER 2019 June 7; part3

(f) HiPERCAM/NICER 2019 June 7; part4

Figure 8: The ACF (left plot) and CCF (right plot) of the simultaneous optical
and X-ray light curves of the 2019 March 2 (a), 2019 March 4 (b) and 2019 June
7 (c to f) data. A positive lag implies that the optical flux lags the X-ray
flux. For the 2019 June 7 data (c to f) we show the corresponding ACFs and
CCFs of the data split into four sections, corresponding to the sections when
the data were simultaneous. In the left panel, the ACF of the X-ray data is
shown in black and the ACF of the $g_{s},r_{s},i_{s},$ and $z_{s}$ data are
shown in blue, green, orange and red, respectively. In the right panel the CCF
of the X-ray data with respect to the $g_{s},r_{s},i_{s}$ and $z_{s}$ data are
in shown blue, green, orange and red, respectively. The black dashed lines
represent the 5 and 95 percent confidence intervals.

##  5 Timing and correlation analysis

The auto-correlation function (ACF) analysis of the individual optical and
X-ray light curves and the cross-correlation function (CCF) of the
simultaneous optical and X-ray light curves can also be used to constrain the
emission processes and location, respectively. We perform such a timing
analysis on the simultaneous optical and X-ray data using the same
methods/techniques outlined in Paice et al. (2019). We use the NICER X-ray
light curves and the dereddened ULTRACAM and HiPERCAM optical light curves
determined in Sections 2.1 and 4, respectively. To create the simultaneous
light curves we first corrected the times of both datasets to the solar system
barycentre and then binned the X-ray photons directly to the optical time
bins. Since the optical light curves have a constant dead-time, the X-ray
photons observed during these times are not used. For the 2019 June 7 HiPERCAM
data, we show the four different simultaneous sections, whereas for the
ULTRACAM dataset we show the two simultaneous sections taken in 2019 March 2
and 4.

###  5.1 Optical/X-ray correlations

In Fig. 7 we show the simultaneous optical and X-ray light curves taken on
2019 March 2, 4 and June 7. For the X-ray data we also show the hardness ratio
of the X-ray count rates. The CCF shows the response of the optical light
curves to variations in the X-ray light curve as a function of time lag.
Positive time lags indicate a net correlation in which the optical flux lags
the X-ray flux. The CCF is produced by splitting and detrending the
simultaneous light curves into segments of equal length. We determine the CCF
for each segment and calculate the mean CCF and standard error in each bin. We
also compute the auto-correlation functions (ACFs) of the X-ray/optical light
curves. The Poisson noise dominating the X-ray ACFs at zero lag is corrected
by making use of the Wiener–Khinchin theorem, which states that the power
spectrum of a random process and its ACF are Fourier pairs. Therefore, we can
subtract the white noise from the X-ray power spectrum and then compute the
inverse Fourier transform to determine the ACF. In Fig. 8 we plot the
corresponding ACF and CCFs for all our simultaneous optical/X-ray light
curves. To determine the confidence levels in the CCFs we simulate 1000
similar (yet uncorrelated) optical light curves, compute the cross-correlation
function with respect to the X-ray light curve and then determine the the 5
and 95 percent boundaries in each bin of the CCF lag. We create the optical
light curves by first computing the Fourier transform of the optical light
curve, randomising the arguments and then performing the inverse Fourier
transform to create a lightcurve with an identical power spectrum. In the
following,for each simultaneous dataset we summarise the observed
characteristic of the light curves and average ACFs and CCFs.

  * •

For the 2019 March 2 data the mean X-ray count rate is 2.6 counts s-1 over the
length of the simultaneous ULTRACAM observation. Low optical and X-ray
variability is observed with no significant flaring behaviour, compared to
what is observed on other nights. In general the X-ray light curve has a
strong hard component. The optical ACF is broader than the X-ray ACF which is
what one expects if the optical flux arises from X-ray reprocessing. No
significant features are observed in the CCFs.

  * •

For the 2019 March 4 data, the mean count rate is relatively high at 7.9
counts s-1 over the length of the simultaneous ULTRACAM observation. A few
relatively strong X-ray flare events are observed which have a strong hard
component. The optical ACF is broader than the X-ray ACF, consistent with
X-ray reprocessing. One can clearly see that the optical and X-ray fluxes are
correlated, which provides a visual confirmation of the CCF observed. The CCF
of this observation shows the strongest positive correlation of any of our
epochs, with a peak at a time lag of $\sim$5 s in every band (a coefficient of
$\sim$0.3 is a significant value in fast-timing studies of X-ray binaries; see
e.g. Gandhi et al. 2010, 2017; Paice et al. 2019). A weak negative correlation
at negative lags is also seen at $\sim$-5 s. Furthermore, there appears to be
a repeated phenomenon in the light curves - the hard X-rays increase first and
then give way to softer X-rays. This is more clearly seen in the flare at time
$\sim$1100 s. In the CCFs, there appears to be a correlation between the
optical delay and wavelength, in which the $u_{s}$-band delay is shorter than
the $g_{s}$-band delay, which is shorter than the $i_{s}$-band delay. This
implies that reprocessing is dominant.

  * •

Finally, the 2019 June 7 data has a relatively low mean count rate of 0.65
counts s-1 coincident with the HiPERCAM observations. Although the X-ray
variability is much lower, several optical peaks do have slight increases in
X-ray count rates, where the increase seems to be slightly greater in the hard
X-rays. In general the X-ray light curve has a hard component but slightly
softer than other epochs and is dominated by a large flare event in part 4 at
7700 s, which has a strong soft component as noted by the change in the X-ray
hardness ratio. This is in contrast to the other short-term X-ray flare events
which seem to have a hard component. The ACF and CCF properties in sections 1
to 3 are very similar. The parts 1 to 4 data show the optical ACF and X-ray
ACF to be similar in shape. A relatively strong positive correlation in the
CCF with a peak at a time lag of $\sim$5 s is observed in every band for parts
1, 2 and 4, and at a time lag of $\sim$0 s in the part 3 data. A weak negative
correlation at negative lags is also observed between $\sim-$20 s and $-$10 s.

###  5.2 Optical/X-ray correlations of flaring Events

In order to further investigate the flaring events we determine the ACFs and
CCF for three clearly defined flare events on 2019 March 4. We compute the
optical and X-ray ACFs and well as the optical/X-ray CCF using a 100 s window
(see Fig. 9). As one can see, the CCFs of the flare events share many
characteristics, including a high CCF correlation (0.4–0.8) with lags between
0–5 s and a precognition dip. The 2019 March 4 and 2019 June 9 data are taken
at orbital phase $\sim$0.35 and $\sim$0.93, respectively. Indeed, the flare
events taken at different orbital phases have time delays consistent with
arising from reprocessing in the secondary star. One expects the longest time
delay to arise at orbital phase quadrature (phase 0.25) and the shortest at
superior conjunction of the secondary star (phase 0.0). Indeed, if one had
sufficient flare events across the binary orbit one could perform echo-mapping
in order to extract the fundamental binary parameters (O’Brien et al., 2002).

Figure 9: Same as Fig. 8 but for the clear flare events on 2019 March 4 and
2019 June 7.

(a) ULTRACAM/NICER 2019 March 2 (b) ULTRACAM/NICER 2019 March 4 (c)
HiPERCAM/NICER 2019 June 7

Figure 10: Results of the Fourier analysis of the simultaneous optical and
X-ray light curves of Swift J1858.6–0814. From top to bottom: the X-ray and
optical power spectra where the white noise has been removed, coherence
spectrum, phase lags and time lags. For the bottom two panels,a positive lag
mean that the optical lags the X-rays. We use a logarithmic rebinning of a
factor of 1.4 to display the data. In each plot, the X-ray data are shown in
black, whereas the purple, blue, green, orange and red show the $u_{s}$,
$g_{s}$, $r_{s}$, $i_{s}$\- and $z_{s}$-band data, respectively.

###  5.3 Fourier Analysis

In order to understand the nature of the different components contributing to
the CCF, we decomposed the observed variability into different time-scales
using Fourier techniques. We performed a Fourier analysis of the light curves
using the X-ray spectral-timing software package
stingray333https://github.com/StingraySoftware/stingray (Huppenkothen et al.,
2019). The coherence and corresponding errors were determined using the method
described in Vaughan & Nowak (1997). We computed the Fourier transform of the
light curves and then analysed them at each frequency. The power spectra
represent the amplitude of the variability at each Fourier frequency, the
coherence shows how the variability in the power of the correlated signal is
distributed over the Fourier frequencies, and the phase lags represent a
measure of the lag between the bands at each frequency as a function of phase.
Sometimes, the time lags are a more intuitive representation of the delays,
which are connected to the phase lags through $\Delta t=\Delta\phi/2\pi f$,
where $f$ is the frequency of the bin and $\phi$ is the phase lag. Positive
phase lags correspond to the delay of the optical light curve with respect to
the X-rays.

Good Time Intervals (GTIs) are used based on the individual epoch of the X-ray
observations and the average cross-spectrum is computed over independent light
curve segments with 2048 bins in length. We use 2, 1, and 6 segment(s) for the
2019 March 2, 2019 March 4 and 2019 June 7 data, respectively, where the white
noise is fitted to each power spectrum and removed prior to the calculation of
coherence. The standard root-mean-squared (RMS) normalisation is applied
(Belloni & Hasinger, 1990). In Fig. 10 we show the frequency-dependent
products binned logarithmically in frequency. The HiPERCAM data were binned by
a factor of 8, and then all data were averaged over segments of 2048 bins, or
$\sim$572, 1028, and 784 s respectively (except for the $u_{s}$ band data in
March 4, which was co-added and thus sampled differently; this was averaged
over segments of 1024 bins, or 1028 s).

For Swift J1858.6–0814 on the nights where there is significant optical and
X-ray variability (2019 March 4 and June 7), the power spectra for the optical
and X-ray light curves are very similar. However, there is consistently higher
power in the X-ray variability compared to the optical, which suggests that
the optical variability is a result of reprocessing of faster X-ray variations
at frequencies above the optical power spectrum peak.

In the 2019 March 4 and June 7 data, the coherence function shows a linear
decline with increasing frequency. The declining absolute value of the
optical/X-ray coherence means that a single component is not a good
representation of the broad-band variability. In the 2019 March 4 and June 7
data there is a plateau in the coherence between 0.01–0.1 Hz at $\sim$10 s
(most notable in the March 4 data), during which a rise in phase lags is
observed. Beyond $\sim$0.1 Hz, the data become white-noise dominated, and it
is not possible to find meaningful results. Frequency-dependent time-lags are
also observed. Below $\sim$0.01 Hz, the time lags rise towards low Fourier
frequencies and a plateau is observed between 0.01–0.1 Hz. Beyond $\sim$0.1 Hz
the time-lags are observed to decrease with frequency, a natural consequence
of the large scatter and randomly distributed phase lags. The time lag
observed at $\sim$0.1 Hz on March 4 is longer than what is observed in the
CCFs; this is likely because the lower frequency lags contribute more to the
CCF than the high frequency lags, as evidenced by the higher coherence below
0.05 Hz, and the sharp drop thereafter with time lags between 3–10 s. This
combination gives rise to a culmination of many frequencies which results in
the time-lag of $\sim$5 s observed in the CCFs.

Figure 11: The absorption-corrected spectral energy distribution of Swift
J1858.6–0814 on 2019 June 7 (black points). The NICER (0.5–3.0 keV and
3.0–10.0 keV) and optical ($g_{s},r_{s},i_{s},z_{s}$) data are simultaneous,
whereas the radio data (1.4, 4.5, and 15.5 GHz) are interpolated values taken
within $\sim$2 months (Bright et al., 2018; van den Eijnden et al., 2020;
Rhodes et al., 2022). We assume a distance of 12.8 kpc. (Buisson et al.,
2021). The blue points are the scaled X-ray luminosities according to the
$L_{\rm OIR}-L_{\rm X}$ relation of Russell et al. (2006). The radio–X-ray
spectral energy distribution can be described with a power-law of the form
$F_{\nu}\propto\nu^{\alpha}$ with an index of $\alpha=-0.84\pm 0.02$ (dashed
line).

##  6 Discussion

###  6.1 Flare spectra

Generally, in the optical/near-IR region, a negative power-law index is
expected if there is an optically-thin synchrotron spectrum from a flow/jet,
whereas a positive power-law index is expected (with spectral index $\sim$ 1)
if the optical emission is dominated by blackbody emission from regions in the
accretion disc (Hynes, 2005). The fast, ‘red’ optical flares observed in the
ULTRACAM and HiPERCAM data have a power-law index of $\alpha\sim$-1.3, steeper
than what is typically observed in XRBs, $\alpha\sim$-0.7 (Hynes et al., 2003;
Gandhi et al., 2011; Russell et al., 2013). However, it should be noted that
flares with similarly steep spectral properties have been observed before with
a power-law in the range -1.3 to -1.5 (Russell et al., 2010; Russell et al.,
2013; Shahbaz et al., 2013; Gandhi et al., 2016) and indeed the fast ‘red’
flares are reminiscent of the ‘red’ flares observed during the outburst of
V404 Cyg (Gandhi et al., 2016). Indeed, for V404 Cyg, based on the cooling
timescales of the flaring events the emission has been attributed to
synchrotron processes (Dallilar et al., 2017). For optically thin synchrotron
emission, the only parameter which changes the spectral index is the particle
energy distribution ($p$) of the emitting electrons, which is related to the
observed spectral slope in the optically thin plasma; $\alpha_{\rm
thin}=\left(1-p\right)$/$2$. If the observed quiescent power-law index of
$\alpha\sim-1.3$ is interpreted as optically thin synchrotron, then $p$ = 3.6,
which is steeper than $p\sim 2.4$ (or $\alpha\sim-0.7$), which is typical for
optically thin synchrotron in XRBs. A mixture of thermal and non-thermal
particle energies could potentially explain such a steep slope observed in
Swift J1858.6–0814. In contrast, slow, large ‘blue’ flares are observed with a
power-law index $\sim$ 1.0, consistent with blackbody emission from an
irradiated accretion disc (Hynes, 2005); the spectrum from an irradiated
accretion disc has a power-law index of 1.2 in the $g_{s}$ to $i_{s}$ bands.

We estimate the binary separation to be $\sim$5.1 $\rm R_{\sun}$($M_{\rm
1}$=2.0 $\rm M_{\sun}$, $M_{\rm 2}$=0.25 $\rm M_{\sun}$, $P_{\rm orb}$=0.883
d) assuming that the accretion disc extends to its tidal truncation radius
($R_{\rm d}$ = 0.9 $R_{\rm L1}$, where $R_{\rm L1}$ is the equivalent radius
of the Roche lobe of a sphere with the same volume), we find $R_{\rm d}<$
2.5$\pm$0.2 $\rm R_{\sun}$. The large flare events can be represented by a
$\sim$14,000$\pm$2,000 K blackbody and an equivalent blackbody radius of
$\sim$1.0$\pm$0.2 $\rm R_{\sun}$ (see Section 4.2) which is consistent with
arising from regions in the accretion disc or from an extended disc atmosphere
or wind (Buisson et al., 2021). The optical multi-wavelength spectral
properties are reminiscent of those observed in the black hole X-ray binary
V404 Cyg, where slow, ‘blue’ as well as fast, ‘red’ flares were observed
during its 2015 outburst (Kimura et al., 2016; Gandhi et al., 2016). From the
strongest observed flare on 2019 June 6 we estimate the optical
($u_{s}$–$z_{s}$) and X-ray (0.5–10 keV) unabsorbed flare power to be
$\sim$0.1% and $\sim$0.33% of the Eddington luminosity, assuming a 1.8 $\rm
M_{\sun}$neutron star and a distance of 12.8 kpc (Buisson et al., 2020a). The
optical flare in Swift J1858.6–0814 is a factor $\sim$5 less powerful compared
to the optical flares in GX 339-4 (Gandhi et al., 2010) and V404 Cyg (Gandhi
et al., 2016).

###  6.2 Spectral energy distribution

The radio–X-ray spectral energy distribution of the X-ray binary systems GX
339–4 (Gandhi et al., 2010), MAXI J1820+070 (Rodi et al., 2021) and GRS
1716–249 (Bassi et al., 2020), can be described by a combination of non-
thermal emission of electrons accelerated in the jet by internal shocks
(Malzac, 2013, 2014) and emission from the irradiated disc and hot corona
(Gierliński et al., 2009). In Fig. 11 we show the absorption-corrected
spectral energy distribution of Swift J1858.6–0814 observed on 2019 June 7
using $N_{\rm H}=1.84\times 10^{21}$ cm-2. The absorption-corrected NICER
soft- and hard-band fluxes were determined using the XSPEC software package
(Arnaud, 1996) with the tbabs(diskbb+bbody) model with $\Gamma=1.6$. The mean
absorption-corrected optical ($g_{s},r_{s},i_{s},z_{s}$) data were determined
using the light curves in Section 2.3. There are not many radio measurements
in 2019, so we interpolate the radio flux values at 1.4, 4.5, 15.5 GHz given
in Rhodes et al. (2022), van den Eijnden et al. (2020) and Bright et al.
(2018), respectively. From the mean optical ($u_{s}$–$z_{s}$) and X-ray
(0.5–10 keV) unabsorbed fluxes on 2019 June 7 we estimate luminosities of
$\sim 2.5\times 10^{35}$ erg s-1and $\sim 4.5\times 10^{35}$ erg s-1,
respectively, assuming a distance of 12.8 kpc (Buisson et al., 2020a). The
optical to X-ray luminosity ratio $L_{\rm opt}/L_{\rm X}$ ratio is $\sim$0.6,
which is much higher than what is typical of X-ray binaries in outburst. In
neutrons star X-ray binaries the optical and X-ray luminosity’s are described
by $L_{\rm OIR}=10^{10.8}L_{\rm X}^{0.63}$, where the optical luminosity is
dominated by X-ray reprocessing with an additional contributions from a jet
and the viscously heated accretion disc. (Russell et al., 2006). We find that
either the optical luminosity in Swift J1858.6–0814 is a factor of $\sim$140
more than what is expected or that X-ray luminosity is a factor of $\sim$2530
under-luminous. Note that the 2019 June 7 observations were taken at orbital
phase $\sim$0.93 and given the high binary inclination angle (Buisson et al.,
2021; Knight et al., 2022) the low X-ray luminosity can be explained by
optically thick material in the outer regions of the accretion disc or
secondary star blocking most of the direct X-ray emission. So what we observe
is scattered X-rays and the intrinsic X-rays is much higher. If we scale the
X-rays using the $L_{\rm OIR}-L_{\rm X}^{0.63}$ relation (Russell et al.,
2006), we find that the radio–X-ray spectral energy distribution can be
described with a power-law of the form $F_{\nu}\propto\nu^{\alpha}$ with an
index of $\alpha\sim$ 0.16. Indeed, this is similar to what is observed in the
mean spectrum of GX 339-4 (Gandhi et al., 2010) and XTE J1118+480 (Hynes et
al., 2003), where the spectral energy distribution is attributed to a mixture
of optically thin synchrotron emission from a jet and the irradiated accretion
disc/corona.

(a) ULTRACAM/NICER 2019 March 2

(b) ULTRACAM/NICER 2019 March 4

(c) HiPERCAM/NICER 2019 June 7; part 1

(d) HiPERCAM/NICER 2019 June 7; part 2

(e) HiPERCAM/NICER 2019 June 7; part 3

(f) HiPERCAM/NICER 2019 June 7; part 4

Figure 12: The HiPERCAM/NICER CCFs. The left plot shows the CCF of the optical
bands versus soft X-rays (0.5–3.0 keV) and the right plot shows the CCF of the
optical bands versus hard X-rays (3.0–10.0 keV). The black dashed lines
represent the 5 and 95 percent confidence intervals.

###  6.3 Optical/X-ray correlations

In the optical waveband, many components can potentially contribute to the
optical emission e.g. the irradiated secondary star, the cold optically-thick
accretion disc, the hot optically-thin X-ray emitting medium and hot flow/jet
(Poutanen & Veledina, 2014). Whereas in the X-rays, two separate components
are present, a soft component arising from Comptonization of disc photons and
a harder component arising from synchrotron Comptonization in the hot flow
(Veledina, 2016). Indeed, this results in optical/X-ray correlations that show
complex patterns, with both positive and negative correlations. The CCFs show
a variety of shapes: some show positive correlations with optical photons
lagging the X-rays, consistent with simple reprocessing (O’Brien et al., 2002;
Hynes et al., 2009; Paice et al., 2018; Kajava et al., 2019); some show a very
broad and nearly symmetric positive cross-correlation (Casella et al., 2010);
some show a more complex structure containing a narrow ‘precognition’ dip at
negative lags (optical photons leading X-rays) superimposed on a very broad
positive cross-correlation (Kanbach et al., 2001; Gandhi et al., 2008; Durant
et al., 2008; Durant et al., 2011; Lasso-Cabrera & Eikenberry, 2013); and some
show only a strong broad anti-correlation (Motch et al., 1983; Pahari et al.,
2017) or a narrow positive correlation superimposed on a very broad positive
cross-correlation (Hynes et al., 2019). Cyclo-synchrotron optical photons
undergoing Compton upscattering to X-rays in a hot flow can also reproduce
both the observed optical/X-ray anti-correlation and QPOs (Veledina et al.,
2011, 2013; Veledina et al., 2015). In some cases the observed features can be
explained by synchrotron emission from internal shocks within a relativistic
compact jet (Malzac, 2013; Hynes et al., 2019; Paice et al., 2019). Finally,
in some sources a fast optical delay component at $\sim$ 100 ms is observed
which is associated with the base of the optically–emitting jet close to the
compact object (Gandhi et al., 2008, 2017; Paice et al., 2019).

Although there are some strong similarities in the timing behaviour of Swift
J1858.6–0814 with well-studied XRBs, one notable difference is the lack of a
$\sim$ 100 ms positive optical time lag with respect to X-rays. This feature
has been seen in the cross-correlated timing behaviour of three sources now:
GX 339–4 (Gandhi et al., 2008), V404 Cyg (Gandhi et al., 2017) and MAXI
J1820–070 (Paice et al., 2019). Both timing and multi-wavelength spectral
properties support an origin of this feature in the inner jets of hard state
binaries, in a compact region no larger than a few thousand Schwarzschild
Radii. Malzac (2014) has shown that flicker noise Lorentz factor plasma
variations within a compact jet can naturally produce such timing lags. The
fact that Swift J1858.6–0814 does not show this feature then implies some
difference between its internal jet structure with respect to other systems.
Whether this is related to a difference in jet plasma Lorentz factors during
the state when it was observed, or perhaps even a difference in compact object
types (all three systems named above host black holes whereas Swift
J1858.6–0814 does not), remains to be investigated.

In the standard reprocessing model, X-rays arising from the inner accretion
disc photoionize and heat the surrounding regions, which later recombine and
cool producing lower energy (optical/near-IR) photons. The observed
optical/near-IR flux is thus delayed relative to the X-rays due to the light
travel time between the X-ray source and the reprocessing region. The
corresponding CCF arising from X-ray reprocessing has a characteristic orbital
phase-dependent shape, where the CCF rises from negative lags, peaks, and
subsequently falls off (Hynes et al., 1998; O’Brien et al., 2002). Depending
on the orbital phase the CCF can be very symmetric, but sometimes an extended
positive delay is observed, especially near quadrature (O’Brien et al., 2002;
Hynes et al., 2009). The shapes of the CCFs observed in Swift J1858.6–0814 are
more consistent with the shape of the CCFs in Sco X–1, Cyg X–2 (Durant et al.,
2011), rather than other XRBs such as XTE J1118+480 (Kanbach et al., 2001),
Swift J1753.5–0127 (Durant et al., 2008) and GX 339–4 (Gandhi et al., 2008)
and MAXI J1820+70 (Paice et al., 2019), where ‘pre-recognition’ dips are
observed and X-ray reprocessing is not thought to be dominant. The time delay
between the optical/near-IR and X-ray flux can be up to twice the binary
separation ($a$) and can be obtained from Kepler’s third law:
$a/c=9.77\,M^{1/3}P_{\rm d}^{2/3}$ s (where $c$ is the speed of light, $M$ is
the sum of the binary masses in solar units and $P_{\rm d}$ is the orbital
period in days). Although the binary parameters for Swift J1858.6–0814 are not
fully known, the orbital period of 21.2 hr together with estimates of the
binary masses allows one to estimate the binary separation to be $a/c\sim$ 12
s. Indeed, we observe CCFs with time delays of $\sim$5–15 s which suggests
that the delays are consistent with arising from regions in the accretion
disc.

As mentioned earlier, in the hybrid hot inner flow model of Veledina (2016)
two X-ray components, one arising from disc Comptonization and the other from
synchrotron Comptonization, as well as two optical components due to
synchrotron self-Compton emission from the hot inner accretion flow and disc
reprocessing are present. In the X-rays, the seed photons for Comptonization
are provided by the accretion disc (disc Comptonization) which dominates in
the hard state. However, the hot flow itself also produces synchrotron
radiation that can contribute or even dominate the seed photon flux for
Comptonization (synchrotron Comptonization). In the optical, the flux can
arise from X-ray reprocessing or from synchrotron emission in the hot inner
accretion flow. An anti-correlation and negative lags between the optical and
X-ray flux is expected because the increase in the mass accretion rate leads
to an increased X-ray flux and a higher level of synchrotron self-absorption,
leading to a drop in the optical emission (Veledina et al., 2011).
Furthermore, the optical is expected to have a stronger anti-correlation with
the hard X-rays compared to with the soft X-rays, characteristics that are
expected if the source transitions from a hard to soft state. During the
initial stages of the outburst of Swift J1858.6–0814 (in the hard state) we
observe CCFs with a positive peak at a time delay of $\sim$5–15 s and optical
ACFs which are broader than the X-ray ACFs (see Fig. blue8a,b). This implies
some underlying connection between the optical and X-ray fluxes and is
consistent with optical flux arising from X-ray reprocessing in the outer
regions of the accretion disc. For example, the 2019 March 4 CCF shows a
nearly symmetric positive correlations at positive lags which is consistent
with X-ray reprocessing, supported by the wavelength dependant optical/X-ray
delays in the CCFs, in which the longest wavelength delay has the longest
delay. On the other hand, the 2019 June 7 data taken during a softer state
cannot be described within the simple reprocessing scenario. The narrow
optical ACF (comparable with the X-ray ACF) and the negative correlation in
the optical/X-ray CCFs (see Fig. blue8c–f) are the characteristics of the
synchrotron self-Compton mechanism operating in a hot accretion flow (Veledina
et al., 2011). The presence of both synchrotron and reprocessed X-ray emission
in the optical is in line with the spectral energy distributions of the
observed fast ‘red’ flares (see Section 4.2).

The CCFs of the 2019 June 7 parts 1 and 4 data have similar shapes, with anti-
correlations at negative lags and positive correlation at positive lags. The
shape can be explained by the presence of two emission components in the
optical, with the X-rays being dominated by the synchrotron Comptonization
continuum (Veledina et al., 2017). The CCF of the 2019 June 7 part 3 data
shows a hint of positive correlation at negative lags. It looks very similar
to the CCF observed in MAXI J1820+070 (see epoch 6 in Paice et al., 2021). To
explain this shape, one requires an additional source of X-ray photons arising
from the disc Comptonization. Indeed, the hard-to-soft spectral state
transition involves the motion of the cold accretion disc towards the compact
object. As the role of the disc increases with the overall increase in the
mass accretion rate, the power dissipated in the hot accretion flow increases,
so the whole spectrum of this component increases (similar to ADAFs) resulting
in the enhancement of the synchrotron emission. The simultaneous presence of
two X-ray components, synchrotron Comptonization and disc Comptonization,
leads to the complex shape of the optical/X-ray CCF and manifests itself
through the different correlations with the soft and hard X-ray bands.

To investigate this possibility further, we separate the X-ray range into soft
(0.5–3.0 keV) and hard (3.0–10.0 keV) energy bands and show the CCFs with
respect to only one optical band ($g_{s}$) for clarity (see Fig. 12). We
systematically observe different correlations between the optical and
soft/hard-X-rays, supporting the assumption of two X-ray components. In
Appendix B we attempt to reproduce the timing and correlation properties
observed in Swift J1858.6–0814 in the context of the hot inner flow-disc
Comptonization and reprocessing model (Veledina et al., 2011; Veledina, 2018).
The low absolute value of the optical/X-ray coherence of $\sim$0.1–0.2 means
that multiple components are required to explain the broad-band variability.
We clearly observe correlations between some optical and X-ray flares which
shows that they are indeed related, some flares events have weak correlations
and so may not be related. In general, we find good qualitative agreement
between the data and the multi-component hot inner flow-disc Comptonization
and reprocessing model, and find that the relative role of the different X-ray
and optical components vary during the course of the outburst as well as on
shorter time-scales.

##  7 Conclusions

We present a rapid timing analysis of simultaneous optical (HiPERCAM and
ULTRACAM) and X-ray (NICER) observations of the X-ray transient Swift
J1858.6–0814 during 2018 and 2019. The optical light curves show rapid, small
amplitude ($\sim$0.1 mag in $g_{s}$) ‘red’ flares (i.e. stronger at longer
wavelengths) on time-scales of $\sim$seconds which have a power-law index
consistent with optically thin synchrotron emission. The optical light curves
also show relatively slow, large amplitude ($\sim$1 mag in $g_{s}$) ‘blue’
flares (i.e. stronger at shorter wavelengths) on time-scales of $\sim$minutes,
with a spectral energy distribution consistent with X-ray reprocessing in the
accretion disc.

We present a Fourier time- and energy-dependant timing analysis of the
simultaneous optical/X-ray light curves. The simultaneous optical and X-ray
data show correlated variability that has a strong hard-energy component on
2019 March 2 and 4, and a strong soft-energy X-ray component on 2019 June 7,
suggesting a spectral state change. We find that the optical ACF is broader
than the X-ray ACF during the initial outburst stages, which can be explained
by simple X-ray reprocessing. The coherence function shows a linear decline
with increasing frequency. There is also a plateau in the time lags between
0.01–0.1 Hz at $\sim$10 s. These characteristics can be attributed to thermal
reprocessing of X-ray emission in the outer regions of the accretion disc.

We find that relative roles of the different X-ray and optical components
governs the shape of the optical/X-ray CCFs and vary on shorter time-scales.
The CCFs of the simultaneous optical versus soft- and hard-band X-ray light
curves show time- and energy dependent correlations. The 2019 March 4 and 2019
June parts 1 and 4 CCFs show a nearly symmetric positive correlations at
positive lags consistent with simple X-ray disc reprocessing. The soft- and
hard-band CCFs are similar and can be reproduced if disc reprocessing
dominates in the optical and one component (disc or synchrotron
Comptonization) dominates both the soft and hard X-rays. The 2019 June 7 part
3 data obtained between parts 1 and 4, shows a very different CCFs. The
observed positive correlation at negative lags in the soft X-ray band can be
reproduced if the optical synchrotron emission is correlated with the hot flow
X-ray emission. The observed timing properties are in qualitative agreement
with the inner hot accretion flow model, where X-rays are produced by both
synchrotron and disc Comptonization and the optical emission arises from the
hot flow synchrotron and irradiated disc components.

## Acknowledgements

TS and VSD acknowledge financial support from the Spanish Ministry of Science,
Innovation and Universities (MICIU) under grant PID2020-114822GB-I00. KMR
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 694745). PG and JAP acknowledges support from Science and
Technology Facilities Council (STFC) and a UGC-UKIERI Thematic Partnership.
TRM acknowledges support from STFC, grant ST/T000406/1. M.R.K acknowledges
support from the Irish Research Council in the form of a Government of Ireland
Postdoctoral Fellowship (GOIPD/2021/670: Invisible Monsters). M.R.K., R.P.B.,
and C.J.C. acknowledge support from the ERC under the European Union’s Horizon
2020 research and innovation programme (grant agreement No.715051; Spiders).
The design and construction of HiPERCAM was funded by the European Research
Council under the European Union’s Seventh Framework Programme (FP/2007-2013)
under ERC-2013-ADG Grant Agreement no. 340040 (HiPERCAM). HiPERCAM operations
and VSD are supported by STFC grant ST/V000853/1.

Based on observations were made with the Gran Telescopio Canarias, installed
at the Spanish Observatorio del Roque de los Muchachos of the Instituto de
Astrofísica de Canarias, on the island of La Palma. Based on observations made
with ESO Telescopes at the La Silla Paranal Observatory under ESO programme
096.D-0808. We gratefully acknowledge the use of python packages: matplotlib
(Hunter, 2007) and numpy (van der Walt et al., 2011). We acknowledge to use of
Aladin (Bonnarel et al., 2000). This research has made use of data and/or
software provided by the High Energy Astrophysics Science Archive Research
Center (HEASARC), which is a service of the Astrophysics Science Division at
NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian
Astrophysical Observatory. The Pan-STARRS1 Surveys (PS1) have been made
possible through contributions of the Institute for Astronomy, the University
of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its
participating institutes, the Max Planck Institute for Astronomy, Heidelberg
and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns
Hopkins University, Durham University, the University of Edinburgh, Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las
Cumbres Observatory Global Telescope Network Incorporated, the National
Central University of Taiwan, the Space Telescope Science Institute, the
National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation under Grant No. AST-1238877, the
University of Maryland, and Eotvos Lorand University (ELTE).

Facilities: GTC (HiPERCAM), NTT (ULTRACAM), NICER

## Data availability

The ULTRACAM and HiPERCAM data can be obtained by contacting the ULTRACAM
team. The NICER data are available in the HEASARC Data Archive
(https://heasarc.gsfc.nasa.gov/docs/archive.html). The data used in this paper
will be shared on reasonable request to the corresponding author.

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##  Appendix A Optical light curves properties

In Fig. 13 we show examples of the HiPERCAM and ULTRACAM small and large flare
events. As one can see, flares on different time-scale, amplitudes and colour
are present. We also show examples of fits to the broad-band dereddened
spectral energy distribution of the observed flaring events (see Fig. 14).

Figure 13: Examples of the HiPERCAM (left) and ULTRACAM (right) small and
large flare events.

Figure 14: Examples of the broad-band dereddened spectral energy distribution
of the flaring events observed in Swift J1858.6–0814. The dotted lines show a
power-law fit to the data of the form $F_{\nu}\propto\nu^{\alpha}$. The left
panel shows the HiPERCAM data taken on 2018 November 14 and the right panel
shows the ULTRACAM data taken 2019 May 9.

##  Appendix B Modelling the timing properties in the context of the hybrid
hot inner flow model

The observed complex shape of the CCFs can be explained via an interplay
between different optical emission components, namely, reprocessing in the
accretion disc, synchrotron emission from the jet and emission from the hot
inner accretion flow (see e.g. Veledina et al., 2013; Malzac et al., 2018;
Paice et al., 2021). The systematically different correlations of the
optical/hard X-rays versus optical/soft X-rays indicate the presence of at
least two sources of X-ray photons, one dominating at soft energies and the
other dominating at hard energies. Indeed, studies of the broad-band spectral
and timing properties of many black hole X-ray binaries reveal several
emission components in the X-rays. A standard and irradiated accretion disc
dominates the soft X-rays, as demonstrated by the covariance spectra and the
soft X-ray time lags (see e.g. Uttley et al., 2011; Cassatella et al., 2012;
De Marco et al., 2015; De Marco et al., 2021); a disc Comptonization and/or
synchrotron Comptonization components stratified in a hot medium contribute to
the hard X-rays, as shown by the hard X-ray time lags (see e.g. Kotov et al.,
2001; Veledina et al., 2013; Mahmoud & Done, 2018), as well as a complex
reflection feature appearing at hard X-ray energies, studied via frequency-
resolved spectroscopy and reverberation lags (Revnivtsev et al., 1999; Fabian
& Ross, 2010; Uttley et al., 2014; Axelsson & Veledina, 2021). When the
different components contribute approximately equally, the interplay between
these components is observed in terms of complex shapes in the X-ray power
spectra and the cross-correlation function between the optical and X-ray bands
(Veledina, 2016, 2018).

To reproduce the shapes of the observed CCFs in Swift J1858.6–0814, we
investigate the scenario of the simultaneous presence of two X-ray components,
one arising from disc Comptonization and the other from synchrotron
Comptonization, as well as two optical components due to synchrotron emission
from the hot inner accretion flow and disc reprocessing. The variability of
both components are caused by accretion rate fluctuations (Lyubarskii, 1997)
and the power spectra are modelled through zero-centered Lorentzian functions.
The accretion disc is assumed to be truncated at some radius away from the
compact object. The synchrotron Comptonization continuum is assumed to be
delayed with respect to the disc Comptonization by the time it takes the
fluctuations to propagate from the radius of truncated disc to the place where
the synchrotron is effectively Comptonized. The spectral shape of the
synchrotron and disc Comptonization emission change under the changing mass
accretion rate with two major patterns: (i) an increase and decrease in flux
with a constant spectral slope and (ii) spectral pivoting. Depending on the
relative amplitudes of these variations, the number of photons in a given
X-ray band may correlate or anti-correlate with the mass accretion rate
fluctuations. This can be parameterised through the simple relation

$\displaystyle x_{\rm h}(t)$ $\displaystyle=$ $\displaystyle\varepsilon_{\rm
h}\dot{m}(t+t_{0})\ast g(t)+\dot{m}(t)$ (1) $\displaystyle x_{\rm s}(t)$
$\displaystyle=$ $\displaystyle\varepsilon_{\rm s}\dot{m}(t+t_{0})\ast
g(t)+\dot{m}(t),$ (2)

where $x_{\rm h}(t)$ and $x_{\rm s}(t)$ are the hard- and soft X-ray light
curves, respectively, $\dot{m}$ is the mass accretion rate, $t_{\rm 0}$ is the
time delay and $g(t)$ is the low-pass filter (see Veledina, 2018, for further
details). The variables $\varepsilon_{\rm h}$ and $\varepsilon_{\rm s}$
parameterise the contribution of the disc and synchrotron Comptonization
components, and their signs indicate the correlation (plus) or anti-
correlation (minus) with the accretion rate fluctuations. The optical light
curve is given by

$\displaystyle o(t)$ $\displaystyle=$
$\displaystyle-\dot{m}(t)+\varepsilon_{\rm ds}x_{s}(t)\ast r(t),$ (3)

where the first term gives the synchrotron contribution and $r(t)$ is the disc
response function whose contribution is parameterised by $\varepsilon_{\rm
ds}$ (Veledina et al., 2011). An implicit parameter of the model is the
assumed shape of the accretion rate power spectrum, which greatly affects the
exact shape (and width) of the features in the CCF. We take a single zero-
centered Lorentzian for each model. The resulting soft- and hard-band CCFs and
are shown in Figure 15.

We do not attempt to fit the 2019 March 2 and 2019 June 7 part 2 observations
because the CCFs are at the noise level. However, the 2019 March 4 and 2019
June parts 1 and 4 CCFs show a nearly symmetric positive correlations at
positive lags consistent with simple X-ray disc reprocessing. The soft- and
hard-band CCFs from 2019 March 4, June 7 parts 1 and 4 are similar in shape
indicating that one component dominates both the soft and hard X-rays. These
CCFs can be reproduced if disc reprocessing dominates in the optical
($\varepsilon_{\rm ds}=3$) and when Comptonization of either synchrotron or
disc photons dominate in the X-rays. We find that the resulting CCFs (Fig.
blue15a) can be reproduced with both $\varepsilon_{\rm h}=\varepsilon_{\rm
s}\gg 1$ and $\varepsilon_{\rm h}=\varepsilon_{\rm s}\ll 1$, hence the data do
not allow us to distinguish between these components. We assume
$\varepsilon_{\rm h}=\varepsilon_{\rm s}=0.01$.

We note that the 2019 June 7 part 3 data was observed between parts 1 and 4 in
time, but shows a very different soft- and hard- X-ray CCF. It shows a
positive correlations at negative lag in the soft-band, while in the hard
X-ray band the correlation amplitudes are not significant. To reproduce the
positive correlation at negative lags we consider an alternative scenario
where the optical synchrotron emission is correlated with the hot flow X-ray
emission, which translates to the formal description

$o(t)=\dot{m}(t)+\varepsilon_{\rm ds}x_{\rm s}(t)\ast r(t).$ (4)

This corresponds to the case when the fluctuations in the magnetic field (as
response to the mass accretion rate variations) dominates the fluctuations in
the number density and the emission in the optical band falls below the
synchrotron self-absorption frequency (see Fig. 1b of Poutanen & Vurm, 2009).
The resulting CCF is shown in Fig. blue15e and is obtained assuming the
parameters $(\varepsilon_{\rm ds},\varepsilon_{\rm s})$ to be (0,-0.8), i.e.
assuming contribution of the synchrotron alone to the optical band. The
parameter $\varepsilon_{\rm h}$ is not constrained, as the hard-band CCF does
not show any significant correlations. The parameter $t_{0}$ mildly affects
the fit; we assume $t_{\rm 0}=4$.

Overall, we see a general trend in the short-term variability. In the 2019
March and parts 1 and 4 of the June data, we find the optical flux is
dominated by a reprocessing component (with possible minor contribution from
the hot flow synchrotron emission), and the X-rays are also dominated by one
component – either synchrotron- or disc Comptonization. For the 2019 June 7
part 3 data we see an indication of a positive correlation at negative lags in
the optical/soft-band X-ray CCF (see Fig. blue12e). The lack of a significant
correlation in the optical/hard X-rays may be explained by the competing role
of two anti-correlated spectral components in the X-rays, which we attribute
to disc and synchrotron Comptonization. We find that the role of synchrotron
emission in the optical band increases. This scenario can also explain the
changing width of the ACFs. The 2019 March 4 data taken during the hard state
have optical ACFs that are broader than the X-ray ACFs, consistent with X-ray
reprocessing. The soft- and hard-band X-ray CCFs are similar implying that a
single component dominates the X-rays. In contrast, the 2019 June part 3 data
has a strong soft X-ray component with comparable optical and X-ray ACFs. This
is indicative of the interplay between two anti-correlating components. The
optical/soft- and optical/hard X-ray CCFs are very different reflecting the
comparable role of disc Comptonization and synchrotron Comptonization.

Figure 15: We show the ULTRACAM or HiPERCAM optical versus NICER X-ray CCFs.
The optical ($g_{s}$-band) versus soft (0.5–3.0 keV) and hard X-rays (3.0–10.0
keV) are shown. We only show the data where the CCFs are significantly above
the noise level. The data taken on 2019 March 4 (b), 2019 June 7 parts 1 (c),
3 (e) and 4 (f) are shown. The solid line shows the data whereas the dashed
line shows the hybrid hot inner flow model. Figure 16: We show the power
spectrum, CCF and phase lags for the 2019 March 4 (b) data (solid line) with
the corresponding hybrid hot inner flow model (dashed line).