11institutetext: Instituto de Astrofísica, Pontificia Universidad Católica de
Chile, Casilla 306, Santiago 22, Chile
11email<EMAIL_ADDRESS>22institutetext: Millennium Institute of Astrophysics
(MAS), Nuncio Monse$\tilde{\rm n}$or Sótero Sanz 100, Providencia, Santiago,
Chile 33institutetext: Department of Astrophysics/IMAPP, Radboud University,
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The Pennsylvania State University, University Park, PA 16802, USA
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State University, University Park, PA 16802, USA 99institutetext: Department
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University Park, PA 16802, USA 1010institutetext: Kapteyn Astronomical
Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The
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Postbus 800, 9700 AV Groningen, The Netherlands 1212institutetext: Department
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CAS Key Laboratory for Research in Galaxies and Cosmology, Department of
Astronomy, University of Science and Technology of China, Hefei 230026, China
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Brera Astronomical Observatory, via Bianchi 46, I–23807 Merate (LC), Italy
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RA, Leiden, the Netherlands 1717institutetext: School of Astronomy and Space
Science, Nanjing University 1818institutetext: Key Laboratory of Modern
Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210093, China

# Extragalactic Fast X-ray Transient Candidates Discovered by _Chandra_
(2014–2022)

J. Quirola-Vásquez 11223344 F. E. Bauer 112255 P. G. Jonker 3366 W. N. Brandt
778899 G. Yang 10101111 A. J. Levan 331212 Y. Q. Xue 13131414 D. Eappachen
6633 E. Camacho 2211 M. E. Ravasio 331515 X. C. Zheng 1616 B. Luo 17171818

(Received January 13, 2023; accepted XX XX, 2023)

###### Abstract

Context. Extragalactic fast X-ray transients (FXTs) are short flashes of X-ray
photons of unknown origin that last a few minutes to hours.

Aims.  We extend the search for extragalactic FXTs from Quirola et al. 2022
(Paper I; based on sources in the _Chandra_ Source Catalog 2.0, CSC2, using
data taken between 2000–2014) to further _Chandra_ archival data between
2014–2022.

Methods.  We extract X-ray data using a method similar to that employed by
CSC2 and apply identical search criteria as in Paper I.

Results. We report the detection of eight FXT candidates, with peak 0.3–10 keV
fluxes between 1$\times$10-13 to 1$\times$10-11 erg cm-2 s-1 and $T_{90}$
values from 0.3 to 12.1 ks. This sample of FXTs has likely redshifts between
0.7 to 1.8. Three FXT candidates exhibit light curves with a plateau
(${\approx}$1–3 ks duration) followed by a power-law decay and X-ray spectral
softening, similar to what was observed for a few previously reported FXTs in
Paper I. In light of the new, expanded source lists (eight FXTs with known
redshifts from Paper I and this work), we update the event sky rates derived
in Paper I, finding 36.9${}_{-8.3}^{+9.7}$ deg-2 yr-1 for the extragalactic
samples for a limiting flux of ${\gtrsim}$1${\times}$10-13 erg cm-2 s-1,
calculate the first FXT X-ray luminosity function, and compare the volumetric
density rate between FXTs and other transient classes.

Conclusions. Our latest _Chandra_ -detected extragalactic FXT candidates boost
the total _Chandra_ sample by $\sim$50%, and appear to have a similar
diversity of possible progenitors.

###### Key Words.:

X-ray: general – X-ray: bursts – Gamma-ray bursts

##  1 Introduction

The last decades have seen remarkable progress in understanding the time-
resolved sky. Wide-field optical and near-infrared (NIR) surveys identified
thousands of supernovae (SNe) and related sources. In the gamma-ray regime,
the progenitors of both long- and short-duration gamma-ray bursts (LGRBs and
SGRBs, respectively) have been identified, while in the radio bands decisive
inroads have been made into the nature of the fast radio bursts (FRBs).
Perhaps surprisingly, our understanding of sources with similar behavior
observed in soft X-rays with _Chandra_ , _XMM-Newton_ and _Swift_ -XRT remains
relatively poor. Phenomenologically, we define extra-galactic fast X-ray
transients (FXTs) as non-Galactic sources that manifest as non-repeating
flashes of X-ray photons in the soft X-ray regime $\sim$0.3–10 keV, with
durations from minutes to hours (e.g., Alp & Larsson, 2020; Quirola-Vásquez et
al., 2022). Unfortunately, they still lack a concise or singular physical
explanation (e.g., Soderberg et al., 2008; Jonker et al., 2013; Glennie et
al., 2015; Irwin et al., 2016; Bauer et al., 2017; Lin et al., 2018; Lin et
al., 2019; Xue et al., 2019; Yang et al., 2019; Alp & Larsson, 2020; Novara et
al., 2020; Lin et al., 2020; Ide et al., 2020; Pastor-Marazuela et al., 2020;
Lin et al., 2021, 2022; Quirola-Vásquez et al., 2022).

Critically, while of order 30 FXTs have been identified to date, both
serendipitously and through careful searches, only in one case, XRT 080109/SN
2008D (Soderberg et al., 2008; Mazzali et al., 2008; Modjaz et al., 2009), has
there been a detection of a multi-wavelength counterpart after the outburst.
This is because, in the vast majority of cases, the transients themselves have
only been identified long after the outburst via archival data mining (e.g.,
Alp & Larsson, 2020; De Luca et al., 2021; Quirola-Vásquez et al., 2022), so
that timely follow-up observations were not possible. Notably, the most
stringent limits come from deep optical VLT imaging serendipitously acquired
80 minutes after the onset of XRT 141001 ($m_{R}{>}$25.7 AB mag; Bauer et al.,
2017). Moreover, only a handful of FXTs have had clear host-galaxy
associations and even fewer have firm distance constraints (e.g., Soderberg et
al., 2008; Irwin et al., 2016; Bauer et al., 2017; Xue et al., 2019; Novara et
al., 2020; Lin et al., 2022; Eappachen et al., 2022; Quirola-Vásquez et al.,
2022; Eappachen et al., 2023). Hence, it is not trivial to discern their
energetics and distance scale and, by extension, their physical origin.

A variety of different physical mechanisms have been proposed for the origin
of FXTs, such as: $i)$ stochastic outbursts associated with X-ray binaries in
nearby galaxies [XRBs; including subclasses such as Ultra-luminous X-ray
sources (ULXs), soft gamma repeaters (SGRs), and anomalous X-ray pulsars
(AXPs)] providing possible explanations of FXTs with $L_{\rm
X,peak}{\lesssim}$1042 erg s-1 (see Colbert & Mushotzky, 1999; Kaaret et al.,
2006; Woods & Thompson, 2006; Miniutti et al., 2019; and references therein);
$ii)$ X-ray emission generated from the shock breakout (SBO; $L_{\rm
X,peak}{\sim}$1042–1045 erg s-1) of a core-collapse supernova (CC-SN) once it
crosses the surface of the exploding star (e.g., Soderberg et al., 2008; Nakar
& Sari, 2010; Waxman & Katz, 2017; Novara et al., 2020; Alp & Larsson, 2020);
$iii)$ off-axis GRBs could explain FXTs ($L_{\rm X,peak}{\lesssim}$1045 erg
s-1) where the X-ray emission is produced by a wider, mildly relativistic
cocoon jet (Lorentz factor of ${\lesssim}$100; Zhang et al., 2004), once it
breaks through the surface of a massive progenitor star (Ramirez-Ruiz et al.,
2002; Zhang et al., 2004; Nakar, 2015; Zhang, 2018; D’Elia et al., 2018);
$iv)$ Tidal disruption events (TDEs; $L_{\rm X,peak}^{\rm no-
Jet}{\lesssim}$1043 and $L_{\rm X,peak}^{\rm Jet}{\sim}$1043–1050 erg s-1
considering non- and jetted emission, respectively) involving a white dwarf
(WD) and an intermediate-mass black hole (IMBH), whereby X-rays are produced
by the tidal disruption and subsequent accretion of part of the WD in the
gravitational field of the IMBH (e.g., Jonker et al., 2013; Glennie et al.,
2015); and $v)$ Mergers of binary neutron stars, (BNS; $L_{\rm
X,peak}{\sim}$1044–1051 erg s-1 considering jetted and line-of-sight obscured
emission; e.g., Dai et al., 2018; Jonker et al., 2013; Fong et al., 2015; Sun
et al., 2017; Bauer et al., 2017; Xue et al., 2019), whereby the X-rays are
created by the accretion of fallback material onto the remnant black hole
(BH), a wider and mildly relativistic cocoon, or the spin-down magnetar
emission (Metzger & Piro, 2014; Sun et al., 2017; Sun et al., 2019; Metzger et
al., 2018).

In previous work, Quirola-Vásquez et al. (2022, hereafter Paper I) conducted a
systematic search for FXTs in the _Chandra_ Source Catalog (Data Release 2.0;
169.6 Ms over 592.4 deg2 using only observations with $|b|{>}10^{\circ}$ and
until 2014; Evans et al., 2010, 2019, 2020a), using an X-ray flare search
algorithm and incorporating various multi-wavelength constraints to rule out
Galactic contamination. Paper I reported the detection of 14 FXT candidates
(recovering five sources previously identified and classified as FXTs by
Jonker et al., 2013; Glennie et al., 2015; Bauer et al., 2017; Lin et al.,
2019) with peak fluxes ($F_{\rm peak}$) from 1$\times$10-13 to 2$\times$10-10
erg cm-2 s-1 (at energies of 0.5–7 keV) and $T_{90}$ (measured as the time
over which the source emits the central 90%, i.e. from 5% to 95% of its total
measured counts) values from 4 to 48 ks. Intriguingly, the sample was sub-
classified into two groups: six _”nearby”_ FXTs that occurred within
$d{\lesssim}$100 Mpc and eight _”distant”_ FXTs with likely redshifts
$\gtrsim$0.1. Moreover, after applying completeness corrections, the event
rates for the _nearby_ and _distant_ samples are 53.7${}_{-15.1}^{+22.6}$ and
28.2${}_{-6.9}^{+9.8}$ deg-2 yr-1, respectively. However, Paper I does not
analyze _Chandra_ observations beyond 2014, implying that several intriguing
FXTs likely remain undiscovered.

In this paper, we extend the selection of Paper I to public _Chandra_
observations between 2014–2022 using a nearly identical methodology. As in
Paper I, this work focuses only on the non-repeating FXTs, to help reduce
sample contamination. We further caution that the sparse nature of repeat
X-ray observations means that we cannot rule out that some current FXTs could
be repeating FXTs. The study of repeating FXTs is beyond the scope of this
manuscript.

The manuscript is organized as follows. We explain the methodology and
selection criteria in Sect. 2. We present the results of a search and cross-
match with other catalogs in Sect. 2.8, a spectral and timing analysis of our
final candidates in Sect. 3, and the properties of the identified potential
host galaxies in Sect. 4. In Sect. 6, we discuss possible interpretations of
some FXTs. We derive local and volumetric rates for the FXTs in Sect. 5, and
compare them to those of other X-ray transients. Finally, we present comments
and conclusions in Sect. 8. Throughout the paper, a concordance cosmology with
parameters $H_{0}{=}$70 km s-1 Mpc-1, $\Omega_{M}{=}$0.30, and
$\Omega_{\Lambda}{=}$0.70 is adopted. Magnitudes are quoted in the AB system.
Unless otherwise stated, all errors are at 1$\sigma$ confidence level.

##  2 Methodology and Sample Selection

###  2.1 Identification of X-ray sources

Paper I used as an input catalog of the X-ray sources detected by the CSC2.
This is not available for _Chandra_ observations beyond the end of 2014, so a
crucial first step is to generate a comparable source detection catalog for
the _Chandra_ observations used in this work (see Sect. 2.3), upon which we
will apply our FXT candidate selection algorithm (Sect. 2.2).

To generate robust X-ray source catalogs, we use the ciao source detection
tool wavdetect (Freeman et al., 2002). It detects possible source pixels using
a series of “Mexican Hat” wavelet functions with different pixel bin sizes to
account for the varying PSF size across the detector. The wavdetect tool
identifies all point sources above a threshold significance of $10^{-6}$
(which corresponds to one spurious source in a 1000$\times$1000 pixel map) and
a list of radii in image pixels from 1 to 32 (to avoid missing detections at
large off-axis angles). To avoid erroneous detections, we create exposure and
PSF maps, which enable refinement of the source properties. The exposure maps
are created by running the fluximage script with the 0.5–7 keV band (Fruscione
et al., 2006), while the PSF map, which provides information on the size of
the PSF at each pixel in the image, is made using the mkpsfmap task; the PSF
size corresponds to the 1$\sigma$ integrated volume of a 2D Gaussian
(Fruscione et al., 2006). The output of the ciao tool wavdetect is a catalog
with essential information about the X-ray sources such as the positions (RA
and DEC), positional uncertainty, and significance.

###  2.2 Transient-Candidate Selection Algorithm

We adopt the same algorithm as presented in Paper I, which augments somewhat
the initial version presented in Yang et al. (2019, see their sect. 2.1 for
more details). This method depends on the total ($N_{\rm tot}$) and background
($N_{\rm bkg}$) counts of the source, working on an unbinned _Chandra_ light
curve, which is advantageous because it does not depend on the light curve
shapes. The algorithm splits the light curves into different segments in two
passes: $i)$ in two halves and $ii)$ in three regions, covering the entire
_Chandra_ observation. FXT candidates are selected when: $i)$ $N_{\rm
tot}{>}$5-$\sigma$ Poisson upper limit of $N_{\rm bkg}$ (to exclude low
signal-to-noise ratio, S/N, sources); $ii)$ the counts in the different
segments ($N_{i}$) are statistically different at a ${>}$ 4$\sigma$
significance level (to select robust detections of short-duration variable
sources); and $iii)$ $N_{i}{>}5\times N_{j}$ or $N_{j}{>}5\times N_{i}$ (to
select large-amplitude number of counts-variations).

Finally, to mitigate the effect of background (especially for sources with
long exposure times and large instrumental off-axis angles), we additionally
chop each light curve into 20 ks segments (or time windows $T_{\rm window}{=}$
20 ks), and re-apply the conditions explained above. This reduces the
integrated number of background counts per PSF element and thus enables
identification of fainter sources at larger instrumental off-axis angles. To
maintain an efficient selection of transients across the gaps between these
arbitrary windows, we sequence through the entire light curve in three
iterations: a forward division in 20 ks intervals, a backward division in 20
ks intervals, and finally, a forward division with a 10 ks shift in 20 ks
intervals to cover gaps.

Based on simulations of the CDF-S XT1 and XT2 fiducial light curves (Bauer et
al., 2017; Xue et al., 2019), Paper I derived an efficiency of the method of
${\gtrsim}$ 90% for sources with $\log(F_{\rm peak})\leavevmode\nobreak\
{>}-12.6$ located at off-axis angles ${<}$ 11$\aas@@fstack{\prime}$0, with a
relatively sharp decline in efficiency for FXTs with lower fluxes; e.g.,
${\approx}$50% and ${\approx}$5% efficiencies for $\log(F_{\rm
peak})\leavevmode\nobreak\ {=}-12.8$ and $\log(F_{\rm
peak})\leavevmode\nobreak\ {=}-13.0$, respectively, at ${\approx}$
11$\aas@@fstack{\prime}$0\. This instrumental off-axis angle limit is enforced
because _Chandra_ ’s detection sensitivity (as measured by, e.g., effective
area and PSF size) drops significantly beyond this limit (Vito et al., 2016;
Yang et al., 2016). Importantly, this algorithm successfully recovered all
previously reported sources (XRT 000519, XRT 030511, XRT 110103, XRT 110919,
and XRT 141001; Jonker et al., 2013; Glennie et al., 2015; Bauer et al., 2017;
Lin et al., 2019; Quirola-Vásquez et al., 2022), and thus is flexible enough
to recognize FXTs with different light-curve shapes. We stress that this is a
key advantage compared to matched filter techniques that assume an underlying
light curve model profile.

Figure 1: Histogram (_red_ ; _left_ Y-axis) and cumulative (_black_ ; _right_
Y-axis) distributions of the exposure time of the 3899 _Chandra_ observations
used in this work. The inset provides a zoom-in to show the high-exposure time
tail of the distribution. The _dashed vertical blue line_ indicates the median
exposure time (${=}\leavevmode\nobreak\ 19.7$ ks) of the total sample.

###  2.3 Data selection

To extend the previous search for extragalactic FXTs in Paper I beyond the
_Chandra Source Catalog 2.0_ (CSC2) limit of 2014, we conducted a search
through all _Chandra_ ACIS imaging observations (science and calibration
observations) made publicly available between 2015-01-01 and 2022-04-01. This
includes 3899 individual _Chandra_ -ACIS observations, outside the Galactic
plane at $|b|{>}$10 deg, or ${\approx}$88.8 Ms, 264.4 deg2 conforming to the
following criteria. For uniformity, we consider only ACIS observations in the
energy range 0.5–7.0 keV, noting that HRC-I observations comprise only a few
percent of the overall observations and have poorer/softer response and
limited energy resolution compared with the ACIS detector. The _Chandra_
observations target a wide variety of astronomical objects, from galaxy
clusters to stellar objects. Based on the nature of the extragalactic FXTs
identified systematically in Paper I and the potential sources of
contamination, we limit our initial light-curve search to sources with
Galactic latitudes $|b|{>}\leavevmode\nobreak\ 10$ deg to reduce the
expectedly high contamination rate from flaring stars. An additional benefit
of considering objects outside the Galactic plane is that it helps to minimize
the effects of Galactic extinction in characterizing the spectral properties
of our candidates.

To facilitate our search, we use the full-field per-observation event files
available from the _Chandra_ Data Archive
products.111https://cda.harvard.edu/chaser/ Figure 1 shows the cumulative and
histogram distributions of exposure time of the _Chandra_ observations used in
this work.

###  2.4 Generation of light curves

The event file contains the relevant stored photon event data, such as photon
arrival time, energy, position on the detector, sky coordinates, observing
conditions, and the good time interval (GTI) tables listing the start and stop
times.

To generate light curves, we take X-ray photons in the 0.5–7.0 keV range from
each event file using an aperture of 1.5$\times R_{90}$, where $R_{90}$ is the
radius encircling 90% of the X-ray counts. Based on simulations developed by
Yang et al. (2019), the aperture of 1.5$\times R_{90}$ encircles
${\gtrsim}$98% of X-ray counts and depends on the instrumental off-axis angle
(and depends on the photon energy; for more details, see Vito et al., 2016;
Hickox & Markevitch, 2006). We compute $N_{\rm bkg}$ taking into account an
annulus with inner and outer aperture radius of 1.5${\times}R_{90}$ and
1.5${\times}R_{90}$+20 pixels, respectively. In the particular case where the
background region overlaps with a nearby X-ray source, we mask the nearby
source (using a radius of 1.5${\times}R_{90}$), and do not include the masked
area to estimate $N_{\rm bkg}$. Also, we weigh $N_{\rm bkg}$ by the source-to-
background area ratio to correct the light curve of the sources.

###  2.5 Astrometry of X-ray sources

To improve upon the nominal absolute astrometric accuracy of Chandra
[0$\aas@@fstack{\prime\prime}$8 (1$\aas@@fstack{\prime\prime}$4) at 90% (99%)
uncertainty]222https://cxc.harvard.edu/cal/ASPECT/celmon/, we cross-match the
detected X-ray sources to optical sources from either the _Gaia_ Early Data
Release 3 (_Gaia_ -EDR3; Gaia Collaboration et al., 2021) or Sloan Digital Sky
Survey Data Release 16 (SDSS–DR16; Ahumada et al., 2020) catalogues, using the
wcs_match script in ciao. wcs_match compares two sets of source lists from the
same sky region and provides translation, rotation and plate-scale corrections
to improve the X-ray astrometric reference frame. We adopt a
2$\aas@@fstack{\prime\prime}$0 matching radius (i.e, $\leq$8 image pixels),
eliminating any source pairs beyond this limit. We typically achieve an
accuracy of 0$\aas@@fstack{\prime\prime}$08–1$\aas@@fstack{\prime\prime}$64
(90% quantile range). This improves our ability to discard contaminants
(stellar flares, essentially) and eventually measure projected offsets between
X-ray sources and host galaxies (in the case of the final sample of FXT
candidates). We combine in quadrature all astrometric errors into the X-ray
source positional uncertainty.

Figure 2: Comparison of 0.5–7.0 keV net-count distributions for the initial (_filled blue histograms_) and final (_filled black histograms_) FXT samples, as well as subsets covered by various purity criteria (_colour non-filled histograms_) for the sample. Net counts are provided by the same regions defined in Sect. 2.4. Criterion | Candidates
---|---
| # constrained | # total removed | # uniquely removed | # remaining
$(a)$ | $(b)$ | $(c)$ | $(d)$ | $(e)$
1) Archival X-ray data | 134∗ | 53 | 20 | 98
2) Cross-match with stars/_Gaia_ | 151 | 83 | 42 | 42
3) NED + SIMBAD + VizieR | 151 | 75 | 33 | 9
4) Archival images† | – | – | – | 9
5) Instrumental and variability effects† | – | 1 | 1 | 8
Table 1: Breakdown of FXT candidates as a function of the selection criteria
proposed in Sect. 2.7. _Column $(a)$:_ Criterion. _Column $(b)$:_ Number of
candidates constrained by this criterion. _Column $(c)$:_ Number of candidates
removed that would be cut at this stage if we disregard all previous stages.
_Column $(d)$:_ Number of candidates that are solely removed by this
criterion, and not any other. _Column $(e)$:_ Running total number of
candidates that remain after applying this criterion.
∗Candidates with additional _Chandra_ -ACIS, _XMM-Newton_ , or _Swift_ -XRT
observations.
$\dagger$Note that criteria 4 and 5 are only applied to the sources that
remain after the first three criteria are applied.

###  2.6 Initial candidate results

As a summary, we apply the FXT detection algorithm to the 0.5–7.0 keV light
curves of X-ray sources outside of the Galactic plane ($|b|>$10 deg),
resulting in 151 FXT candidates. This parent sample has total net counts and
instrumental off-axis angles spanning ${\approx}$15–33,000 (mean value of 590)
and ${\approx}$0.12–14.0 (mean value of 5.2) arcmin, respectively. As
expected, our selection method identifies FXTs with a wide range of light
curve shapes.

###  2.7 Initial Purity Criteria

As highlighted in both Yang et al. (2019) and Paper I, our search method does
not ensure the unique identification of real extragalactic FXTs. Therefore, it
is mandatory to adopt additional criteria considering archival X-ray data and
multiwavelength counterparts to differentiate real extragalactic FXTs from
Galactic variables and transients among the sample of 151 FXT candidates. We
describe and report these additional criteria in Sect. 2.7.1–2.7.5 and
summarize the number and percentage, relative to the total, of sources that
pass criteria (_column 5_), as well as ignoring all previous steps (_column
4_) in Table 1. Finally, we discuss the completeness of our search and
selection criteria in Sect. 2.7.6.

####  2.7.1 Criterion 1: Archival X-ray data

To confirm the transient nature of the FXT candidates, a non-detection in
prior and/or subsequent X-ray observations is important. In this way, we
consider different observations from _Chandra_ , based on other observations
in the CSC2 and individual observations (Evans et al., 2010); _XMM-Newton_ ,
based on individual observations of sources in the Serendipitous Source (4XMM-
DR11; Webb et al., 2020) and Slew Survey Source Catalogues (XMMSL2; Saxton et
al., 2008); and the Living _Swift_ XRT Point Source Catalogue (LSXPS) based on
observations between 2005-01-01 and 2023-02-12 (Evans et al., 2023). We impose
that the FXT candidate remain undetected (i.e., consistent with zero net
counts) at 3$\sigma$ confidence in all X-ray observations, aside from the
_Chandra_ observation in which the FXT candidate is detected. This requirement
is useful especially to exclude a large number of Galactic stellar flares, but
it also may discard FXTs associated with hosts with AGNs, as well as long-
lived or recurring X-ray transients (e.g., from SNe in strongly star-forming
galaxies). The success of this criterion is related to the number of times a
particular field is visited by X-ray facilities.

To discard candidates with prior and subsequent X-ray observations with
_Chandra_ , we used the CSC2 or in the cases of candidates with more recent
archival observations we downloaded and extracted photometry for these
sources, adopting consistent source and background regions and aperture
corrections compared to those used in Sect. 2.4. In total, 127 FXT candidates
were observed in multiple _Chandra_ observation IDs, while 24 candidates have
no additional _Chandra_ observations.

To identify additional _XMM-Newton_ and _Swift_ -XRT detections, we adopt a
search cone radius equivalent to the 3$\sigma$ combined positional errors of
the _Chandra_ detection and tentative _XMM-Newton_ or _Swift_ -XRT matches
from the 4XMM-DR11, XMMSL2 and LSXPS catalogs, respectively. We additionally
search the X-ray upper limit servers: Flux Limits from Images from _XMM-
Newton_ using DR7 data (FLIX),333https://www.ledas.ac.uk/flix/flix.html
LSXPS,444https://www.swift.ac.uk/LSXPS/ and the HIgh-energy LIght curve
GeneraTor (HILIGT) upper limit
servers.555http://xmmuls.esac.esa.int/upperlimitserver/ It is important to
mention that HILIGT provides upper limits for several X-ray observatory
archives (including _XMM-Newton_ pointed observations and slew surveys;
Röntgen Satellite (_ROSAT_) pointed observations and all-sky survey;
_Einstein_ pointed observations), while LSXPS generates _Swift_ -XRT upper
limits.666We used the 0.2–12 keV energy band, which we then converted to
0.5–7.0 keV assuming the default spectral parameters $\Gamma{=}2$ and
$N_{H}{=}3{\times}10^{20}$ cm-2.

We found that the reported detections are not always reliable (e.g.,
inconsistencies between catalogs using the same observations or failure to
confirm upon visual inspection), and hence we require detections to be
${\geq}$5$\sigma$. We found that: 72 candidates are observed in _XMM-Newton_
4XMM-DR11, with 12 candidates detected; 65 candidates are observed in _Swift_
-XRT LSXPS, with 4 candidates detected; 1 candidate is observed in _ROSAT_
pointed observations, with a clear detection; finally, all candidates are
observed in the _ROSAT_ All-Sky Survey, with 5 candidates detected. Also, zero
candidates are observed in _XMM-Newton_ XMMSL2 and the _Einstein_ pointed
observations. The upper limits from _Chandra_ and _XMM-Newton_ pointed
observations are similar to or lower than our FXT candidate peak fluxes. So,
we can conclude that similar transient episodes would have been detectable in
such observations if present.

In total, 134 candidates have multiple hard X-ray observations by _Chandra_ ,
_XMM-Newton_ , and/or _Swift_ -XRT, of which 127 candidates have been visited
more than once by _Chandra_. This implies re-observed fractions of at least
${\approx}$84% among the candidate sample (a large fraction of these 84% of
sources lie in fields intentionally observed multiple times; for instance, in
the vicinity of the Orion Nebula or M101). The high X-ray re-detection
fraction indicates that this is a very effective criterion if additional
_Chandra_ , _XMM-Newton_ or _Swift_ observations are available.

In summary, 98 candidates pass this criterion (see Table 1), albeit largely
because they lack multiple sensitive X-ray observations. We note that 20
candidates are discarded by this criterion but not by the others (see Table
1). The _left panel_ of Fig. 2 shows the net-count distribution for all the
sources that pass this criterion. To conclude, this criterion appears to be an
extremely effective means to identify persistent or repeat transients, when
data are available.

####  2.7.2 Criterion 2: Optical detections in _Gaia_

In previous works (e.g., Paper I and Yang et al., 2019), an important fraction
of FXT candidates had a Galactic origin, especially related to relatively
bright stars. To identify these, we cross-match with the _Gaia_ Early Data
Release 3 (_Gaia_ EDR3; employing the VizieR package; Gaia Collaboration et
al., 2021) catalog, which contains photometric and astrometric constraints for
sources in the magnitude range $G{=}3$–21 mag including accurate positions,
parallaxes, and proper motions throughout the Local Group (Lindegren et al.,
2018; Gaia Collaboration et al., 2018). We adopt the 3$\sigma$ positional
uncertainty (obtained by the ciao wavdetect task) associated with each
candidate as our cone search radius. In general, this radius is sufficiently
small to find a unique counterpart given the high spatial resolution and
astrometric precision of _Chandra_ (Rots & Budavári, 2011); 9 candidates show
multiple _Gaia_ sources in their cone search area, for which we adopt the
nearest _Gaia_ source.

From our initial sample of 151 FXT candidates, 107 sources have cross-matches
in _Gaia EDR3_. Nevertheless, we only discard FXT candidates matched to
“stellar” _Gaia_ EDR3 optical detections, where stellar is taken to mean those
with nonzero proper motion and/or parallax detected at $>$3$\sigma$
significance; this amounts to 83 candidates from the initial sample. These
likely stellar sources cover a wide range in magnitude $G{=}$9.2–20.1 mag
($\overline{G}{\approx}$16.4 mag) and proper motion $\mu{=}$0.7–154.5 mas yr-1
($\overline{\mu}{\approx}$22.1 mas yr-1).

The _middle panel_ of Fig. 2 shows the net-count distribution of the 68
sources that pass this criterion. Among the total sample, ${\approx}$55% are
associated with stellar flares of bright stars. Moreover, this criterion
discards 42 FXT candidates that the additional criteria do not (see Table 1).
However, because of the magnitude limit and optical window of _Gaia_ , this
criterion may not identify all persistent or recurring transient Galactic
objects, which we return to in the next subsection. As a running total, only
42 candidates successfully pass both this and the previous criterion (see
Table 1).

####  2.7.3 Criterion 3: NED, SIMBAD, and VizieR Search

To identify known Galactic and Local Group contaminating objects not detected
by _Gaia_ , we search for counterparts (or host galaxies) in large databases
using the astroquery package: the NASA/IPAC Extragalactic Database (NED; Helou
et al., 1991), the Set of Identifications, Measurements, and Bibliography for
Astronomical Data (SIMBAD; Wenger et al., 2000), and VizieR (which provides
the most complete library of published astronomical catalogs; Ochsenbein et
al., 2000).

We perform a cone search per FXT candidate, using a circular region with a
radius of 3$\sigma$ based on the X-ray positional uncertainty from the ciao
wavdetect task to find associated sources. These three databases contain many
catalogs across the electromagnetic (EM) spectrum, which permit us to rule out
candidates in our sample associated with previously classified stars, young
stellar objects (YSOs) embedded inside nebulae (where the absorption and
obscuration do not permit _Gaia_ detections), globular clusters, or high-mass
X-ray binaries (HMXBs) in either our Galaxy or the Local Group. This criterion
is important in our analysis because ${\approx}$80% (i.e., 121 FXT candidates)
of the initial sample show associated sources with the SIMBAD and NED
databases. We uniquely identify 33 objects, either as YSOs embedded in nebulae
or stars identified by other catalogs, for instance, the VISTA Hemisphere
Survey (VHS), the United Kingdom InfraRed Telescope (UKIRT) Infrared Deep Sky
Survey, the Sloan Digital Sky Survey (SDSS), or the all-sky Wide-field
Infrared Survey Explorer (WISE) CatWISE source catalog at 3.4 and 4.6 $\mu$m
(McMahon et al., 2013; Dye et al., 2018; Marocco et al., 2021). It is
important to mention that 33 FXT candidates are discarded solely by this
criterion (see Table 1).

The _right panel_ of Fig. 2 shows the net-count distribution for the 76 FXT
candidates that pass this criterion. Applying all criteria thus far, the
sample is reduced to nine candidates.

####  2.7.4 Archival Image Search

To rule out still fainter stellar counterparts, we carried out a search of
ultraviolet (UV), optical, NIR, and mid-infrared (MIR) image archives. We
perform a cone search within a radius equivalent to the 3$\sigma$ _Chandra_
positional uncertainty of the respective FXTs for the following archives: the
Hubble Legacy Archive;777https://hla.stsci.edu/hlaview.html the Pan-STARRS
archive (Flewelling et al., 2020);888http://ps1images.stsci.edu/cgi-
bin/ps1cutouts the National Science Foundation’s National Optical-Infrared
Astronomy Research (NOIR) Astro Data Lab
archive,999https://datalab.noao.edu/sia.php which includes images from the
Dark Energy Survey (DES; Dark Energy Survey Collaboration et al., 2016) and
the Legacy Survey (DR8); the Gemini Observatory
Archive;101010https://archive.gemini.edu/searchform the National Optical
Astronomy Observatory (NOAO) science
archive;111111http://archive1.dm.noao.edu/search/query/ the ESO archive
science portal;121212http://archive.eso.org/scienceportal the VISTA Science
Archive;131313http://horus.roe.ac.uk/vsa/ the Spitzer Enhanced Imaging
Products archive (Teplitz et al.,
2010);141414https://irsa.ipac.caltech.edu/data/SPITZER/Enhanced/SEIP/ the
UKIRT/Wide Field Camera (WFCAM) Science Archive;151515http://wsa.roe.ac.uk/
and the WISE archive (Wright et al., 2010).

For images obtained under good seeing (${<}$ 1$"$) and weather conditions, we
inspect visually for counterparts or host galaxies in the 3$\sigma$
uncertainty X-ray location of the FXT. We only apply this additional criteria
for the FXT candidates that remain after the previous three criteria (see
Sect. 2.7.1–2.7.3). If a source is found, we identify it as a star if it is
consistent with the spatial resolution of the imaging, we quantify its
significance and assess its extent and radial profile visually. We confirm
that none of the nine candidates is associated with stellar sources, leaving
the number of candidates unchanged.

FXT | Odds ratio | Prob. | Var. Index
---|---|---|---
(1) | (2) | (3) | (4)
15 | 10.18 | 0.99 | 9
16 | 93.73 | 1.0 | 10
17 | 9.09 | 0.99 | 8
18 | 8.67 | 0.99 | 8
19 | 167.0 | 1.0 | 10
20 | 34.47 | 1.0 | 10
21 | 6.19 | 0.99 | 8
22 | 29.56 | 1.0 | 9

Table 2: Variability properties of the extragalactic FXT candidates detected
and/or discussed in this work obtaining by the G-L method, ordered by
subsample and date. _Column 1:_ shorthand identifier (FXT #) used throughout
this work. _Column 2:_ logarithmic odds ratio (ratio of obtaining the observed
distribution versus obtaining a flat distribution) for variability signal.
_Column 3:_ variable signal probability (the probability that the flux
calculated from the source region is not constant throughout the observation).
_Column 4:_ variability index (ratio of obtaining the observed distribution
versus obtaining a flat distribution).

####  2.7.5 Instrumental and variability effects

Finally, we visually check the X-ray data to rule out false-positive
candidates that may arise from background flares, bad pixels or columns, or
cosmic-ray afterglows rather than intrinsic variability. Again, we only
undertake this last criteria for the remaining nine candidates after
Sect.2.7.4.

First, we use the glvary tool to confirm variability using the Gregory-Loredo
(G-L) algorithm. The Gregory-Loredo variability algorithm is a commonly used
test to detect time variability in sources (Gregory & Loredo, 1992).161616The
G-L method splits the sources into multiple time bins and looks for
significant deviations between them. The tool assigns a variability index
based on the total odds ratio and the corresponding probability of a variable
signal. This adds a second criterion for variability, increasing the
probability that the light curves of our candidate FXTs show strong
variability during the observation. Applying the G-L task to our sample of
nine FXT candidates, we found that one of them (identified in the _Chandra_
ObsId 16302 at
$\alpha$=13${}^{\text{h}}$56${}^{\text{m}}$01.${}^{\text{s}}$10,
$\delta$=–32∘35′15.95′′) has a low probability to be a variable source
(${\approx}$0.1) with a variability index of 2.171717Although our algorithm is
designed to select only sources with large amplitude variations in the number
of counts (see Sect. 2.2 or sect. 2.1 in Paper I), this source does not vary.
The peculiar light curve of this source erroneously allows it pass our initial
method. The light curve is split into different time windows, then erroneously
our method selects this source since in one window the light curve contains
one of the two peaks and a quiescent phase (mimicing the light curve of a
transient source). Thus, the G-L test is necessary to rule out any such
source. These results guarantee that this source is inconsistent with flux
variability throughout the observation. The remaining eight sources show a
clear variability throughout the _Chandra_ observation according to their
variable probability (${\gtrsim}$0.99) and variability index (${\gtrsim}$8)
(see Table 2 for more details).

Finally, to reject possible strong background flaring episodes in the 0.5–7
keV band, we employ the dmextract and deflare tools to examine the evolution
of the background count rate during the observations. None of the FXT
candidates is affected by background flares. Furthermore, we confirm visually
that the counts from all sources are detected in dozens to hundreds of
individual pixels (discarding association with bad columns or hot pixels)
tracing out portions of _Chandra_ ’s Lissajous dither pattern (appearing as a
sinusoidal-like evolution of $x$ and $y$ detector coordinates as a function of
time; see Appendix Fig. 21) over their duration, reinforcing that they are
astrophysical sources. Therefore, we have a final sample of eight FXTs.

####  2.7.6 Completeness

It is important to keep in mind that real FXTs may have been ruled out
erroneously by the criteria above. To roughly estimate this, we compute the
probability that a FXT candidate overlaps with another X-ray source and/or
star by chance. Assuming Poisson statistics (i.e., $P(k,\lambda)$), the
probability of one source ($k{=}1$) being found by chance inside the 3$\sigma$
localization uncertainty region of another is given by

$P(k{=}1,\lambda){=}\frac{e^{-\lambda}\lambda^{k}}{k!},$ (1)

where $\lambda$ is the source density of X-ray sources and/or stars on the sky
multiplied by the 3$\sigma$ _Chandra_ localization uncertainty area. As a
reference, the mean density of X-ray sources detected by _Chandra_ , _XMM-
Newton_ and _Swift_ -XRT is 0.36, 1.68, and 0.07 arcmin-2, respectively, while
the mean density of optical sources detected by _Gaia_ is 2.0 arcmin-2. We use
the X-ray detections from the CSC2, 4XMM-DR11 and 2SXPS catalogs (Evans et
al., 2010; Webb et al., 2020; Evans et al., 2020b), and the _Gaia EDR3_
catalog for stars (Gaia Collaboration et al., 2021) to determine the X-ray or
optical source densities, respectively. The probability is 0.0024 and 0.0029
for X-ray and optical sources, respectively. Taking into account just the
X-ray sources discarded solely by Criteria 1 or 2, 20 and 42 X-ray sources
(see Table 1), respectively, we expect ${\ll}$1 of these to be ruled out
wrongly. If we consider only the 109 X-ray sources which were discarded by
both Criteria 1 and 2, the combined probability is 1${\times}$10-5, and thus
the expected number of erroneously dismissed sources is also ${\ll}$1.

Considering the densities of X-ray sources in individual _Chandra_ fields,
they span a minimum-maximum density range between 0.0042–2.302 arcmin-2,
yielding a probability range of $1.4{\times}10^{-6}$ to 0.0505. Thus, under
these extreme density conditions, the number of X-ray sources discarded
wrongly by Criteria 1 is ${\approx}0.1$. This value is relatively low, and
thus reinforces the idea that an erroneous rejection is unlikely even in
extreme conditions. As an extreme example, we can consider the X-ray positions
of CDF-S XT2 and source XID${}_{\text{4Ms}}$256 (${\approx}$30 photons
detected during the 4 Ms exposure, classified as a normal galaxy; Xue et al.,
2011; Luo et al., 2017), which differ by only
${\approx}2\aas@@fstack{\prime\prime}0$. Upon further investigation of the
flux and position of the X-ray variability, it was realized that
XID${}_{\text{4Ms}}$256 and CDF-S XT2 are distinct sources (Xue et al., 2019).
The X-ray source density (at ${\sim}$2$\aas@@fstack{\prime}$0 off-axis angle)
in the _Chandra Deep Field South_ at 7 Ms is ${\approx}$5.6 arcmin-2 (Luo et
al., 2017), leading to a chance alignment probability (using Eq. 1) of 0.019
between CDF-S XT2 and ID${}_{\text{4Ms}}$256\. Although this value is low, it
is non-zero, and thus care should be given to the spatial/temporal alignment
of X-ray sources, so as to not discard candidates erroneously.

It is not easy to assess the contribution by Criterion 3 to the completeness
given the highly disjoint nature of the databases. Similar to Paper I, we
assume that Criterion 3 does not disproportionately discard real FXTs. In
aggregate, we conclude that our rejection criteria do not apparently impact on
the completeness of our FXT candidate sample.

FXT | Id | ObId | Exp. (ks) | Date |  $T_{90}$ (ks) | RA (deg.) | DEC (deg.) | Off. Ang. | Pos. Unc. | HR | S/N
---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
15 | XRT 140507 | 16093 | 68.8 | 2014-05-07 | 4.8${}_{-5.5}^{+3.6}$ | 233.73496 | 23.46849 | 2$\aas@@fstack{\prime}$2 | 0$\aas@@fstack{\prime\prime}$8 | $-1.5\pm 0.21$ | 6.9
16 | XRT 150322/ | 16453 | 74.7 | 2015-03-21/22 | 10.3${}_{-6.3}^{+9.0}$ | 53.07672 | -27.87345 | 4$\aas@@fstack{\prime}$3 | 0$\aas@@fstack{\prime\prime}$32 | $-0.33\pm 0.08$ | 35.2
| CDF-S XT2$\dagger$ |  |  |  |  |  |  |  |  |  |
17 | XRT 151121 | 18715 | 24.5 | 2015-11-20/21 | 5.5${}_{-4.2}^{+1.2}$ | 40.82972 | 32.32390 | 8$\aas@@fstack{\prime}$6 | 2$\aas@@fstack{\prime\prime}$0 | $-0.17\pm 0.20$ | 6.4
18 | XRT 161125 | 19310 | 6.1 | 2016-11-25 | 2.9${}_{-2.4}^{+0.8}$ | 36.71489 | -1.08317 | 12$\aas@@fstack{\prime}$1 | 5$\aas@@fstack{\prime\prime}$0 | $-0.64\pm 0.18$ | 4.3
19 | XRT 170901$\dagger$ | 20635 | 77.0 | 2017-08-31/09-01 | 3.9${}_{-0.7}^{+1.2}$ | 356.26437 | -42.64494 | 3$\aas@@fstack{\prime}$1 | 0$\aas@@fstack{\prime\prime}$14 | $+0.00\pm 0.08$ | 43.7
20 | XRT 191127 | 21831 | 22.5 | 2019-11-26/27 | 0.3$\pm$0.2 | 207.34711 | 26.58421 | 5$\aas@@fstack{\prime}$9 | 1$\aas@@fstack{\prime\prime}$2 | $-0.34\pm 0.17$ | 8.6
21 | XRT 191223 | 23103 | 18.1 | 2019-12-23 | 3.7${}_{-0.5}^{+2.4}$ | 50.47516 | 41.24704 | 3$\aas@@fstack{\prime}$5 | 0$\aas@@fstack{\prime\prime}$44 | $-0.16\pm 0.20$ | 9.7
22 | XRT 210423$\dagger$ | 24604 | 26.4 | 2021-04-23/24 | 12.1${}_{-4.1}^{+4.0}$ | 207.23523 | 26.66230 | 7$\aas@@fstack{\prime}$5 | 0$\aas@@fstack{\prime\prime}$6 | $-0.18\pm 0.10$ | 12.4

Table 3: Properties of the extragalactic FXT candidates detected and/or
discussed in this work, ordered by date. _Column 1:_ shorthand identifier (FXT
#) used throughout this work. _Column 2:_ X-ray transient identifier (XRT
date), plus previous name when available. _Columns 3, 4, and 5:_ _Chandra_
observation ID, exposure time in units of ks, and date. _Column 6:_ $T_{90}$
duration, which measures the time over which the source emits the central 90%
(i.e., from 5% to 95%) of its total measured counts, in units of ks. _Columns
7 and 8:_ Right Ascension and Declination in J2000 equatorial coordinates.
_Column 9:_ Off-axis angle of the FRXT candidates, with respect to the
_Chandra_ aimpoint, in units of arcminutes. _Column 10:_ Estimated 2$\sigma$
X-ray positional uncertainty, in units of arcseconds (see Sect. 2.5). _Column
11:_ Hardness ratio (HR) and 1$\sigma$ uncertainty, defined as
HR=(H$-$S)/(H$+$S) where H=2–7 keV and S=0.5–2 keV energy bands, using the
Bayesian estimation of Park et al. (2006). _Column 12:_ Approximate signal-to-
noise (S/N).
$\dagger$ Previously reported as FXTs by Xue et al. (2019) for FXT 16, Lin et
al. (2019) for FXT 19, and Lin et al. (2021) in the case of FXT 22. Figure 3:
0.5–7 keV light curves for each FXT candidate: (_1st column_) full exposure,
in units of counts; (_2nd column_) zoom in, from the detection of first photon
to the end of the exposure, in units of count rate (cts s-1), with log-log
scaling and 5 counts per bin. The _gray dashed lines_ show the stop-time per
observation regarding the beginning of the transient; (_3rd column_) long-term
light curve, with each point representing individual _Chandra_ exposures (cyan
circles with 1-$\sigma$ error bars) to highlight the significance of
detections and non-detections, in units of counts; (_4th column_) long-term
light curve, with each point representing individual _Chandra_ (_cyan_), XMM-
Newton (_orange_) and _Swift-XRT_ (_green_) exposures in units of flux (erg
s-1 cm-2). For the long-term light curves, the observation including the
transient is denoted by a _large red star_ (1-$\sigma$ error bars), while
triangles denote observations with (3-$\sigma$) upper limits. All fluxes are
reported in the 0.5–7 keV band in the observer’s frame.

####  2.7.7 Summary

We identify eight FXT candidates, three of them have been previously
discovered and classified as FXTs by Xue et al. (2019), Lin et al. (2021) and
Lin et al. (2022); see Sect. 2.8 for more details. Table 3 shows important
information of the final sample: the coordinates, duration ($T_{90}$),
instrumental off-axis angle, positional uncertainty, hardness ratio (HR;
computed following Park et al., 2006), and S/N ratio (computed using the
wavdetect tool). Figure 3 shows the background-subtracted 0.5–7.0 keV light
curves of our final sample of FXT candidates: short-term, in units of counts
(_first column_) and logarithmic count rates (_second column_); long-term in
units of net-counts for _Chandra_ only (_third column_) and flux to compare
uniformly _Chandra_ , _XMM-Newton_ and _Swift_ -XRT data (_fourth column_). It
is important to mention that the first three criteria considered (X-ray
archival data, _Gaia_ detection cross-match, and NED/SIMBAD/VizieR catalogs)
contribute significantly and in complementary ways to clean the sample
(especially for discarding stellar contamination).

Finally, we label each FXT candidate by “XRT” followed by the date, where the
first two numbers correspond to the year, the second two numbers to the month,
and the last two numbers to the day (see Table 3, _second column_).
Nevertheless, similar to Paper I, to identify each source quickly throughout
this manuscript we also denominate them by “FXT”+# (ordered by date; see Table
3, _first column_) from 15 to 22, because this work is a sequel paper to Paper
I where FXTs were labeled until FXT 14. Furthermore, FXT 18 does not have
additional _Chandra_ , _XMM-Newton_ or _Swift_ -XRT observations to ensure its
transient nature, however, we keep it to be consistent with the selection
criteria of this work.

Figure 4: ATLAS differential photometry of the _cyan_ ($c$) and _orange_
($o$)-bands light curves performed at the position of the FXT candidates.
$<5\sigma$ data points are shown in hollow circles, while $>5\sigma$ data
points are shown in solid circles. The _blue vertical lines_ show the epochs
when the FXT candidates were detected by _Chandra_ , while the _dashed gray
lines_ represent the zero flux. Figure 5: ZTF differential photometry of the
$g$ (_green points_), $r$ (_red points_) and $i$-bands (_orange points_) light
curves performed at the position of the FXT candidates. $<5\sigma$ data points
are shown in hollow circles, while $>5\sigma$ data points are shown in solid
circles. The _blue vertical lines_ shows the epochs when the FXT candidates
were detected by _Chandra_ , while the _dashed gray lines_ represents the zero
flux.

Figure 6: Archival optical, near-infrared, mid-infrared and X-ray images of
extragalactic FXT candidates; the telescope/instrument +filter and FXT ID name
are shown in the upper-left corner. Each cutout is centered on the X-ray
position, while red circles denote 2$\sigma$ _Chandra_ errors in the source
localisation. _(1st, 2nd, 3rd and 4th columns)_ optical band (DECam, Pan-
STARRS and HST) images. _(5th and 6th columns)_ near-infrared $J$ or $H$ and
$K$ (2MASS, UKIRT or VISTA) images; _(7th column)_ 3.4$\mu$m (unWISE) images;
and _(8th column)_ X-ray _Chandra_ (ACIS) 0.5–7 keV images. The _coloured
arrows and circles_ show the localization of the possible host/counterparts of
the FXT candidates. HST images were aligned using the astrometry of Gaia.

###  2.8 Fainter Electromagnetic detections

We now focus on a detailed multiwavelength search (in Sects. 2.8.1 to 2.8.4
from radio to gamma rays) of each candidate for a contemporaneous
counterpart181818Hereafter, ”counterpart” refers to the emission relating to
the FXT candidate, not its host galaxy, during epochs close to the X-ray
trigger. and host galaxy using several archival datasets to understand their
origin.

####  2.8.1 Radio Emission

We search for any possible radio emission associated to our FXT candidates
using the _RADIO–Master Radio Catalog_ , which is a revised master catalog
with select parameters from a number of the HEASARC database tables. It holds
information on radio sources across a wide range of telescopes/surveys [e.g,
the Very Long Baseline Array, the Very Large Array (VLA), and the Australia
Telescope Compact Array] and frequencies (from 34 MHz to 857 GHz). Because of
the poor angular resolution of some associated radio catalogs, we perform an
initial cone search for radio sources with a radius of 60 arcseconds.
Following this initial 60 arcseconds cut, we repeat a search using limiting
radii consistent with the combined radio + X-ray 3$\sigma$ positional errors.
The current version of the master catalog does not yet contain the recent VLA
Sky Survey (VLASS)191919https://science.nrao.edu/vlass/ and Rapid ASKAP
Continuum Survey (RACS; Hale et al., 2021) catalogs, so we additionally query
these using resources from the Canadian Astronomy Data
Centre202020https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/ interface.
Unfortunately, our search returns no matches indicating that none of the final
sample of FXT candidates or host sites is unambiguously detected at radio
wavelengths.

####  2.8.2 Optical and Mid-Infrared Counterpart Emission

In order to explore possible optical and MIR contemporaneous counterparts of
our final sample, we examine forced differential photometry taken from the
Zwicky Transient Facility (ZTF; Bellm et al., 2019; Graham et al., 2019; Masci
et al., 2019), the Asteroid Terrestrial-impact Last Alert System (ATLAS; Tonry
et al., 2018; Smith et al., 2020), and a visual inspection of images obtained
during epochs around the X-ray trigger obtained from the unWISE time-domain
(Meisner et al., 2022) and the Legacy Surveys DR10 catalogs (Dey et al.,
2019).

ZTF is a wide-field (field of view of 55.0 deg2) time-domain survey, mapping
the sky every few nights with custom $g$, $r$, and $i$ filters to 5-$\sigma$
limiting magnitudes of $\approx$20.8, 20.6, 19.9 AB mag, respectively (Bellm
et al., 2019). ATLAS is a four-telescope asteroid impact early warning system,
which observes the sky several times every night in custom _cyan_ ($c$;
420–650 nm) and _orange_ ($o$; 560–820 nm) filters to 5-$\sigma$ limiting
magnitudes of $\approx$19.7 AB mag (Tonry et al., 2018; Smith et al., 2020).

Figures 4 and 5 show the differential photometry light curves taken from ZTF
($gri$) and ATLAS ($co$), respectively, for the FXT candidates. For visual
clarity, the ZTF and ATLAS photometry are binned by 50 days, with the errors
added in quadrature. The locations of all eight FXT candidates have been
observed by ATLAS (see Fig. 4), although FXT 15 and FXT 16 were not visited by
ATLAS around the time of the X-ray detection (highlighted by the vertical blue
lines). ZTF, on the other hand, has only observed six FXT candidate locations
(see Fig. 5), of which FXT 15, FXT 17 and FXT 18 fields not being observed
around the time of the _Chandra_ detection. Notably, the most recent FXTs
(FXTs 20, 21 and 22) have forced differential photometry light curves from ZTF
and ATLAS, covering a wide epoch both before and after the _Chandra_ X-ray
detections. Overall, none of the FXT candidates exhibits significant
(${>}5{\sigma}$) detections of optical variability or flares by ZTF and ATLAS
around the time of the FXT candidate X-ray trigger, nor are there any robust
detections in any previous or subsequent epochs. We derive 3$\sigma$ upper
limits from the closest observation taken by ZTF and ATLAS (for the available
filters and FXTs), as listed in Table 13.

To check if the forced photometry is consistent with zero flux (around the
X-ray trigger), we use the statistical test ConTEST (Stoppa et al., 2023),
developed explicitly to compare the consistency between the observations and a
constant zero flux model. We adopt a methodology identical to that discussed
in Eappachen et al. (2023, submitted). We applied this test considering two-
time windows, [-10;20] and [-10;100] days, with respect to the X-ray trigger,
because possible optical counterparts have timescales from days (e.g., the
afterglow of GRBs) to weeks/months (e.g., CC-SNe emission). We concluded that
for all the sources, the model of zero flux density detected by both periods
is consistent with the observations.

Furthermore, the DESI Legacy Imaging Surveys (DR10) combine three major public
projects plus additional archival data to provide imaging over a large portion
of the extragalactic sky visible from the Northern and Southern Hemispheres in
at least three optical bands ($g$, $r$, and $z$). The sky coverage
(${\sim}$30,000 deg2) is approximately bound by
$-90^{\circ}{<}\delta{<}+84^{\circ}$ in celestial coordinates and
$|b|{>}15^{\circ}$ in Galactic coordinates (Dey et al., 2019). Thus, the
Legacy Imaging survey observes most FXT locations (except for FXTs 17 and 21).
We explore visually each individual imaging epoch provided by the Legacy
survey in $g$-, $r-$, $i-$, and $z$-bands to identify potential optical
contemporaneous counterparts of the FXTs. However, no contemporaneous optical
counterparts are identified around the X-ray trigger time after a visual
inspection.

The Wide-field Infrared Survey Explorer (WISE; Wright et al., 2010) provides
an unprecedented time-domain view of the MIR sky at W1${=}$3.4 $\mu m$ and
W2${=}$4.6 $\mu m$ due to the NEOWISE mission extension (Mainzer et al., 2011,
2014). WISE has completed more than 19 full-sky epochs over a ${>}$12.5 year
baseline, with each location having been observed for ${\gtrsim}$12 single
exposures (Meisner et al., 2022). In order to search for a potential
counterpart inside the WISE and NEOWISE images, we use the time-domain API
tools provided by the unTimely Catalog, which considers data from 2010 to 2020
(Meisner et al., 2022). We visually inspect each single-epoch image of each
FXT field (for FXT 22, only up to ${\sim}$1.5 years before the X-ray trigger).
Unfortunately, none of the FXT candidates shows significant detections of
variability or flares around the time of the X-ray trigger.

####  2.8.3 Ultraviolet, Optical, and Infrared Host Galaxy Identification

To search for UV, optical, NIR and MIR emission associated with any possible
host galaxy in the vicinity of each FXT candidate, we perform a cone search
within a radius equivalent to the 3$\sigma$ _Chandra_ error position (see
Table 3) in the following catalogs: GALEX Data Release 5 (GR5; Bianchi et al.,
2011), Pan-STARRS Data Release 2 (Pan-STARRS–DR2; Flewelling, 2018), the DES
Data Release 2 (DES–DR2; Abbott et al., 2021), the SDSS Data Release 16
(SDSS–DR16; Ahumada et al., 2020), the NOAO Source Catalog Data Release 2
(NSC–DR2; Nidever et al., 2021), the _Hubble_ Source Catalog version 3 (HSCv3;
Whitmore et al., 2016), the UKIRT InfraRed Deep Sky Survey Data Release
11+(UKIDSS–DR11+; Warren et al., 2007), the UKIRT Hemisphere Survey Data
Release 1 (UHS–DR1; Dye et al., 2018), the Two Micron All Sky Survey (2MASS;
Skrutskie et al., 2006), the VHS band-merged multi-waveband catalogs Data
Release 5 (DR5; McMahon et al., 2013), the Spitzer Enhanced Imaging Products
Source List (Teplitz et al., 2010), and the unWISE (unWISE; Schlafly et al.,
2019) and CatWISE (McMahon et al., 2013; Dye et al., 2018; Marocco et al.,
2021) catalogs, as well as the ESO Catalogue Facility and the NED (Helou et
al., 1991), SIMBAD (Wenger et al., 2000), and VizieR (Ochsenbein et al., 2000)
databases. We supplement this by including any extended sources found during
our archival image analysis in Sect. 2.7.4. We assume that uncertainties in
the UV through MIR centroid positions contribute negligibly to the overall
error budget. Figure 6 shows image cutouts of the localization region of the
FXTs (one per row), typically from Pan-STARRS, DECam, or HST in the optical
(_1st–4th columns_ , using $g$, $r$, $i$ and $z$ or the corresponding HST
filters), VISTA, UKIRT, 2MASS or HST in the NIR (_5th and 6th columns_ , using
$J$, $H$, $K$ or the corresponding HST filters), unWISE in the MIR (_7th
column_ , in the 3.6$\mu$m filter) band, and the _Chandra_ -ACIS image (_8th
column_ , in the 0.5–7.0 keV band).

FXT 15 has no optical and NIR sources detected within the 3$\sigma$ X-ray
positional uncertainty of this source in the HST, DECam, 2MASS, or unWISE
images (see Fig. 6). Upper limits are given in Table 4.

FXT 16/CDF-S XT2 (identified previously by Zheng et al. 2017 and analyzed in
detail by Xue et al. 2019) was detected by _Chandra_ with a 2$\sigma$
positional uncertainty of 0$\aas@@fstack{\prime\prime}$32 (see Table 3). This
accurate _Chandra_ X-ray position allows us to identify the host galaxy, which
lies at an offset of $\approx 0$$\aas@@fstack{\prime\prime}$44 (i.e., a
projected distance of ${\approx}$3.3 kpc) using _HST_ images (see Fig. 6). The
galaxy has a spectroscopic redshift of $z_{\rm spec}{=}$0.738. The probability
of a random match between FXT 16 and a galaxy as bright as or brighter than
$m_{\rm F160W}{\approx}$24 AB mag within 0$\aas@@fstack{\prime\prime}$44 is
$\approx$0.01 (Xue et al., 2019).

FXT 17 does not have optical and NIR sources detected within the 3$\sigma$
X-ray error region of this source in the Pan-STARRS, 2MASS, or unWISE images
(see Fig. 6). Upper limits are given in Table 4.

For FXT 18, one faint source with $m_{r}{\approx}24.2$ AB mag (see Fig. 6,
_source #1_) appears inside the large localization region
($r{\approx}$7$\aas@@fstack{\prime\prime}$5 at 3$\sigma$) in the DECam $g$,
$r$, $i$ and $z$-band images with an off-set angle of
${\approx}$2$\aas@@fstack{\prime\prime}$6 from the X-ray center position; it
has a chance association probability of ${<}$0.095. Two other sources lie
slightly outside the X-ray uncertainty region, _sources #2_ and _#3_ , with
chance probabilities of 0.582 and 0.363, respectively (see Fig. 6); such high
probabilities suggest an association with either one of them is unlikely.

FXT 19 (reported previously by Lin et al. 2019 and analyzed in detail by Lin
et al. 2022) lies close to a faint ($m_{\rm F606W}{\approx}$24.8, $m_{\rm
F814W}{\approx}$24.9, $m_{\rm F110W}{\approx}$24.7 and $m_{\rm
F160W}{\approx}$24.3 AB mag, using aperture photometry) and extended optical
and NIR source in _HST_ imaging (see Fig. 6, _source #1_) with an angular
offset ${\approx}$0$\aas@@fstack{\prime\prime}$45\. The chance probability for
FXT 19 and _source #1_ to be randomly aligned in F160W is very low, only 0.005
(Lin et al., 2022).

FXT 20 was detected 6$\aas@@fstack{\prime}$812 (or ${\approx}$500 kpc in
projection) from the center of the galaxy cluster Abell 1795 (located at
${\approx}$285.7 Mpc) during a _Chandra_ calibration observation (ObsId
21831). FXT 20 lies close to a faint source $m_{r}{\approx}23.5$ AB mag (see
Fig. 6, _source #1_) identified in DECam $g$, $r$, and $z$-bands at an offset
angle of ${\approx}$0$\aas@@fstack{\prime\prime}$6\. The probability of a
false match is $P{<}0.005$ (adopting the formalism developed by Bloom et al.,
2002) for such offsets from similar or brighter objects.

FXT 21 has a faint optical source ($m_{r}{\approx}$25.1 AB mag) inside the
3$\sigma$ X-ray error position in Pan-STARRS images (see Fig. 6, _source #1_),
but no source is detected in 2MASS NIR or unWISE MIR images. The offset
between the FXT and the optical source position is
${\approx}$0$\aas@@fstack{\prime\prime}$5, with a false match probability of
$P{<}0.0085$ (adopting the formalism developed by Bloom et al., 2002) for such
offsets from similar or brighter objects.

Finally, FXT 22 (identified previously by Lin et al., 2021) was detected
4$\aas@@fstack{\prime}$079 (or ${\approx}$300 kpc in projection) from the
center of the galaxy cluster Abell 1795 (located at ${\approx}$285.7 Mpc)
during a _Chandra_ calibration observation (ObsId 24604). No sources are
detected within the 3$\sigma$ X-ray error region of this source in the DECam
optical, VISTA NIR, or unWISE MIR images (see Fig. 6). However, this source
falls close to an extended object, SDSS J134856.75+263946.7, with
$m_{r}{\approx}$21.4 AB mag that lies at a distance of
${\approx}$4$\aas@@fstack{\prime\prime}$5 from the position of the FXT
(${\approx}$40 kpc in projection) with a spectroscopic redshift of $z_{\rm
spec}{=}$1.5105 (Andreoni et al., 2021; Jonker et al., 2021; Eappachen et al.,
2023). The probability of a false match is $P{<}0.041$ (adopting the formalism
developed by Bloom et al., 2002) for such offsets from similar or brighter
objects.

To summarize, we conclude that four (FXTs 16, 19, 20 and 21) of the eight FXT
candidates have high probabilities of being associated with faint (FXT 20) or
moderately bright (FXTs 16, 19, and 21) extended sources within the 3$\sigma$
positional error circle. In the case of FXT 22, it may be associated with the
extended source SDSS J134856.75+263946.7 ($z_{\rm spec}{=}$1.5105);
nevertheless, a relation with a faint background source cannot be excluded (a
faint extended source is in the X-ray uncertainty region; Eappachen et al.,
2023). In the case of FXT 18, its large positional uncertainty does not allow
us to determine robustly the counterpart optical or NIR source. Finally, two
FXT candidates (FXTs 15 and 17) have no associated optical or NIR sources in
the available moderate-depth archival imaging, and remain likely extragalactic
FXTs. None of the FXT candidates analyzed in this work appear to be associated
with a nearby galaxy (${\lesssim}$100 Mpc). In Sect. 3.4, we explore a
scenario where these sources are related to Galactic stellar flares from faint
stars.

FXT | $m_{g}$ | $m_{r}$ | $m_{i}$ | $m_{z}$ | $m_{y}$ | $m_{Y}$ | $m_{J}$ | $m_{H}$ | $m_{K}$ | W1 | W2
---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
15 |  ${>}$24.0h |  ${>}$23.0h |  ${>}$22.3h |  ${>}$22.2h |  ${>}$20.0a | – |  ${>}$16.9f |  ${>}$16.6f |  ${>}$17.2f |  ${>}$20.3c |  ${>}$20.6c
16(S1) | 26.509$\pm$0.1j | 25.307$\pm$0.1j | 24.598$\pm$0.1j | 24.506$\pm$0.1j | – | 24.450$\pm$0.1j | 24.300$\pm$0.1j | 23.852$\pm$0.1j |  ${>}$20.1g | 22.407$\pm$0.1j | 21.928$\pm$0.1j
17 |  ${>}$24.1h |  ${>}$22.4h |  ${>}$21.8h |  ${>}$21.5h |  ${>}$20.0a | – |  ${>}$17.1f |  ${>}$17.1f |  ${>}$17.2f |  ${>}$19.8c |  ${>}$20.1c
18(S1) | 24.6${\pm}$0.2l | 24.2$\pm$0.1l | 24.0$\pm$0.1l | 23.6$\pm$0.2l |  $>$20.1a | – |  ${>}$19.8g |  ${>}$19.2g |  ${>}$19.2g |  ${>}$19.3c |  ${>}$20.2c
18(S2) | 24.9$\pm$0.2d | 24.6$\pm$0.1d | 24.2$\pm$0.2d | 25.2$\pm$0.9d | 24.3$\pm$1.1d | – |  ${>}$19.8g |  ${>}$19.2g |  ${>}$19.2g |  ${>}$19.3c |  ${>}$20.2c
18(S3) | 25.6${\pm}$0.7h | 23.0$\pm$0.04d | 21.8$\pm$0.2a | 21.2$\pm$0.1a | 21.1$\pm$0.1d | 20.2$\pm$0.2d | 20.1$\pm$0.1g |  ${>}$19.8b | 19.7$\pm$0.2b | 19.3$\pm$0.1c | 20.2$\pm$0.3c
19(S1) |  ${>}$25.6d | 24.79$\pm$0.04k‡ | 24.93$\pm$0.05k‡ |  ${>}$24.32d |  ${>}$24.04a | – | 24.67$\pm$0.04k‡ | 24.33$\pm$0.04k‡ |  ${>}$20.3g |  ${>}$20.1c |  ${>}$20.3c
20(S1) | 24.3${\pm}$0.2l | 23.5${\pm}$0.2l |  ${>}$21.6a | 23.2${\pm}$0.2l |  ${>}$19.8a | – |  ${>}$17.9f |  ${>}$17.7f |  ${>}$17.7f |  ${>}$20.2c |  ${>}$20.5c
21(S1) |  ${>}$23.6h | 25.09$\pm$1.49a | 22.68$\pm$0.13a | 21.73$\pm$0.13a | 21.55$\pm$0.28a | – |  ${>}$17.5f |  ${>}$17.4f |  ${>}$17.6f |  ${>}$19.8c |  ${>}$20.3c
22(S1) | 20.9$\pm$0.2h | 21.4$\pm$0.4h | 20.4$\pm$0.3h | 20.5$\pm$0.9h |  ${>}$19.7a | – |  ${>}$17.8f |  ${>}$17.5a |  ${>}$17.7a |  ${>}$20.1c |  ${>}$20.4c

Table 4: Host and/or counterpart’s photometric data or upper limits of FXT
candidates. All magnitudes are converted to the AB magnitude system using
González-Fernández et al. (2018) for VHS and 2MASS data, Hewett et al. (2006)
for UKIDSS data, and Wright et al. (2010) for unWISE data. If an optical/NIR
counterpart candidate is detected, we list its magnitude and 1-$\sigma$ error,
otherwise we provide 3$\sigma$ limits from several catalogs: aPan-STARRS-DR2
(Flewelling, 2018), bUKIDSS-DR11+ (Warren et al., 2007), cunWISE (Schlafly et
al., 2019), dDES-DR2 (Abbott et al., 2021), eNSC-DR2p (Nidever et al., 2021),
f2MASS (Skrutskie et al., 2006), gVHS-DR5 (McMahon et al., 2013), hSDSS-DR16
(Ahumada et al., 2020), i INT/CFHT (Jonker et al., 2013), j CANDELS (nearest
HST/Spitzer bands substituted: $g{=}F435W$, $r{=}F606W$, $i{=}F814W$,
$z{=}F850LP$, $Y{=}F105W$, $J{=}F125W$, $H{=}F160W$, $W1{=}ch1$, $W2{=}ch2$;
Guo et al., 2013), k (nearest HST bands substituted: $r{=}F606W$, $i{=}F814W$,
$J{=}F110W$, $H{=}F160W$), l Legacy Surveys DR9 (Dey et al., 2019).
$\dagger$ Photometric data of FXTs with counterpart(s) (S+# means the _source
number_).
${\ddagger}$ Obtained using a photometric aperture of
0$\aas@@fstack{\prime\prime}$6\.

####  2.8.4 Higher energy counterparts

To explore if hard X-ray and $\gamma$-ray observations covered the sky
locations of the FXTs, we developed a cone search in the Nuclear Spectroscopic
Telescope Array (_NuStar_ ; Harrison et al., 2013), _Swift_ -Burst Alert
Telescope (_Swift_ -BAT; Sakamoto et al., 2008), INTErnational Gamma-Ray
Astrophysics Laboratory (_INTEGRAL_ ; Rau et al., 2005), High Energy Transient
Explorer 2 (_HETE-2_ ; Hurley et al., 2011), _InterPlanetary Network_ (Ajello
et al., 2019), and _Fermi_ (von Kienlin et al., 2014; Narayana Bhat et al.,
2016) archives. We adopt a 10$\aas@@fstack{\prime}$0 search radius for the
_INTEGRAL_ , _Swift_ -BAT, _HETE-2_ and _Interplanetary Network_ Gamma-Ray
Bursts catalogs, while for the Gamma-ray Burst Monitor (_GBM_) and the Large
Area Telescope (LAT) _Fermi_ Burst catalogs we consider a cone search radius
of 4 deg (which is roughly the typical positional error at 1$\sigma$
confidence level for those detectors; Connaughton et al., 2015). Additionally,
we implement a time constraint criterion of $\pm$15 days in our search between
Gamma-ray and FXT triggers.

To further probe whether there may be weak $\gamma$-ray emission below the
trigger criteria of _Fermi_ -GBM at the location of the FXTs, we investigated
the _Fermi_ -GBM daily data, the _Fermi_ position history
files212121https://fermi.gsfc.nasa.gov/ssc/data/access/gbm/, and the GBM Data
Tools (Goldstein et al.,
2022)222222https://fermi.gsfc.nasa.gov/ssc/data/analysis/gbm/gbm_data_tools/gdt-
docs/install.html. We confirmed that FXTs 15, 16, 17, 20, and 21 were in the
FoV of _Fermi_ -GBM instruments during the X-ray trigger time${\pm}50$ s,
while FXTs 18, 19, and 22 were behind the Earth around the X-ray burst trigger
time; thus, their fields were not visible. Table 12 summarizes the visibility
of the sources and the _Fermi_ -GBM instruments covering the fields around the
X-ray trigger time (at a distance of ${\lesssim}60$ degrees). In summary, we
find no hard X-ray or $\gamma$-ray counterparts associated with _NuSTAR_ ,
_INTEGRAL_ , _Swift_ -BAT, _HETE-2_ , _Interplanetary Network_ and the GBM and
LAT _Fermi_ Burst catalogs, but cannot rule out weak $\gamma$-ray emission for
FXTs 18, 19, and 22.

Figure 7: Sky positions, in Galactic coordinate projection, of FXT candidates:
the initial 151 FXT candidates of this work are represented by _blue
triangles_ (see Sect. 2.6; some symbols overlap on the sky); the _final
sample_ of eight extragalactic FXT candidates from this work are denoted by
_large red stars_ (FXTs 20 and 22 overlap on this scale); and the final sample
of 14 extragalactic FXTs analyzed in Paper I are shown as _orange circles_
(FXTs 14 and 16 overlap on this scale). The background grey scale encodes the
location and number of distinct co- or closely-located observations among the
combined 3899 and 5303 _Chandra_ observations used in this work and Paper I,
respectively.

FXT |  $T_{0}$(UTC) | Model | T${}_{\text{break}}$(ks) | $\tau_{1}$ | $\tau_{2}$ |  $F_{0}$ (erg cm-2 s-1) |  $\ln{\mathcal{L}}$(dof) | BIC
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
15 | 2014-05-07 04:26:03.74 | BPL | 3.7$\pm$0.7 | -0.4$\pm$0.1 | 2.9$\pm$1.2 | (6.8$\pm$1.9)$\times$10-14 | 1.3(2) | -2.1
|  | PL | – | 0.0$\pm$0.02 | – | (1.0$\pm$1.1)$\times$10-14 | 9.1(4) | 6.1
16 | 2015-03-22 07:02:29.30 | BPL | 2.2$\pm$0.3 | 0.09$\pm$0.1 | 2.0$\pm$0.3 | (4.1$\pm$0.8)$\times$10-13 | 26.5(26) | 9.8
|  | PL | – | 0.6$\pm$0.1 | – | (1.0$\pm$0.1)$\times$10-11 | 79.3(28) | 35.9
17 | 2015-11-21 03:26:16.08 | BPL | 0.4$\pm$2.5 | 0.4$\pm$0.1 | 0.4$\pm$1.0 | (1.8$\pm$3.9)$\times$10-13 | 6.8(4) | 6.9
|  | PL | – | 0.3$\pm$0.1 | – | (8.6$\pm$6.7)$\times$10-13 | 6.3(6) | 2.2
18 | 2016-11-25 10:12:22.22 | BPL | – | – | – | – | – | –
|  | PL | – | 0.5$\pm$0.1 | – | (4.6$\pm$3.8)$\times$10-11 | 1.6(2) | -0.8
19 | 2017-09-01 13:27:02.10 | BPL | 2.1$\pm$0.3 | 0.1$\pm$0.1 | 1.9$\pm$0.5 | (5.5$\pm$0.7)$\times$10-13 | 36.9(38) | 9.5
|  | PL | – | 0.3$\pm$0.1 | – | (3.3$\pm$1.5)$\times$10-12 | 1.7(40) | 27.7
20 | 2019-11-27 02:12:13.71 | BPL | 0.1$\pm$0.01 | -1.0$\pm$0.2 | 2.5$\pm$0.3 | (4.2$\pm$0.5)$\times$10-12 | 1.5(6) | -9.7
|  | PL | – | 0.5$\pm$0.3 | – | (3.5$\pm$4.6)$\times$10-12 | 21.9(8) | 12.5
21 | 2019-12-23 10:40:46.48 | BPL | – | – | – | – | – | –
|  | PL | – | 0.4$\pm$0.1 | – | (1.1$\pm$0.8)$\times$10-12 | 4.6(4) | 1.9
22 | 2021-04-23 22:15:36.63 | BPL | 4.4$\pm$0.4 | 0.2$\pm$0.1 | 3.8$\pm$1.2 | (3.2$\pm$0.4)$\times$10-13 | 17.9(26) | -1.7
|  | PL | – | 0.4$\pm$0.1 | – | (3.5$\pm$1.6)$\times$10-12 | 46.1(28) | 19.7

Table 5: Best-fit parameters obtained using either a broken power-law (BPL) or
a power-law (PL) model fit the X-ray light curves. _Column 2:_ start time when
the count rate is $\gtrsim$3-$\sigma$ higher than the Poisson background
level. _Column 3:_ the model used. _Column 4:_ the break time for the BPL
model. _Columns 5 and 6:_ the slope(s) for the BPL or PL model. _Column 7:_
the normalization for the BPL or PL model. _Columns 8 and 9:_ log-likelihood
($\ln{\mathcal{L}}$)/degrees-of-freedom (dof) and _Bayesian Information
Criterion_ (BIC) of the fit, respectively. Errors are quoted at the 1-$\sigma$
confidence level. Figure 8: Top panels: the observed 0.5–7.0 keV X-ray light
curves in cgs units (_blue points_), starting at $T{=}20$ seconds. We also
plot the best-fit broken power-law or simple power-law model (_red solid
lines_). The light curves contain 5 counts per bin. _Bottom panels:_ the
hardness ratio evolution (the soft and hard energy bands are 0.5–2.0 keV and
2.0–7.0 keV, respectively), following the Bayesian method of Park et al.
(2006). The _red dashed line_ denotes a hardness ratio equal to zero.
$T_{0}=0$ s is defined here as the time when the count rate is 3$\sigma$
higher than the Poisson background level.

##  3 Spatial, Temporal and X-ray Spectral properties

We analyze the spatial distribution of our final sample of FXT candidates in
Sect. 3.1. Furthermore, the time evolution and spectral properties could give
important information about the physical processes behind the FXT candidates,
and thus we explore and describe these in Sects. 3.2 and 3.3, respectively.
Finally, we explore a Galactic stellar flare origin of this sample in Sect.
3.4.

###  3.1 Spatial properties

If the FXT candidates are extragalactic and arise from large distances, then
given the isotropy of the universe on large scales, we expect them to be
randomly distributed on the sky. Figure 7 shows the locations, in Galactic
coordinates, of the final FXT candidates of Paper I and this work, the initial
FXT candidates of this work, and the _Chandra_ observations analyzed in Paper
I and this work. We investigate the randomness of the FXT candidate
distribution on the sky compared to all _Chandra_ observations considered in
this work. For this, we use the non-parametric Kolmogorov–Smirnov (K-S) test
(Kolmogorov, 1933; Massey Jr, 1951; Ishak, 2017).

We explore the randomness of the spatial distribution of our final sample of
eight FXTs. For this, we simulate 10,000 samples of 40,000 random sources
distributed over the sky, taking as a prior distribution the _Chandra_ sky
positions used in this work (which are functions of the pointings and
exposures). Out of these fake sources, we randomly select eight sources, which
we compare to the spatial distribution of the eight real FXT candidates using
a 2D K-S test (following the methods developed by Peacock, 1983 and Fasano &
Franceschini, 1987). We can reject the $\mathcal{NH}$ that these sources are
drawn from the same (random) distribution only in ${\approx}$0.2% of the
draws. Therefore, the positions of the eight FXT candidates are consistent
with being randomly distributed over the total _Chandra_ observations on the
sky.

Intriguingly, FXTs 14 and 16 lie in the same field of view (i.e., in the
_Chandra_ Deep Field South), as do FXTs 20 and 22 (i.e., in the direction of
the galaxy cluster Abell 1795). Thus, we explore the probability that two FXTs
occur in the same field, which is given by the Poisson statistic [i.e.,
$P(k,\alpha)$, using Eq. 1], where $k{=}2$ and $\alpha$ is the ratio between
the total _Chandra_ exposure time in a particular field (for the _Chandra_
Deep Field South and the cluster Abell 1795 are 6.8 and 3.1 Ms,
respectively232323Both values taken from https://cda.harvard.edu/chaser) and
the total _Chandra_ exposure time analyzed in Paper I and this work
(${\approx}$169.6 and 88.8 Ms, respectively) normalized to the total number of
FXTs identified (i.e., 22 FXTs). The chance probabilities for FXTs 14, 16 and
20, 22 are 0.115 and 0.029, respectively. We can conclude that the occurrence
of two FXTs being found in these particular fields is unusual, but not ruled
out at high significance.

###  3.2 Temporal properties

To characterize and measure the break times and light-curve slopes in the
X-ray light curves of the candidate FXTs, we consider a single power-law (PL)
model with index $\tau_{1}$, or a broken power-law (BPL) model with indices
$\tau_{1}$, $\tau_{2}$ and break time $T_{\text{break}}$ (for more detail, see
Paper I, sect. 3.2).

Both models describe well the majority of the FXT X-ray light curves in this
work. To fit the data, we use the least-square method implemented by the lmfit
Python package.242424https://github.com/lmfit/lmfit-py/ The best-fit model
parameters and statistics are given in Table 5, while the light curves (in
flux units; light curves have five counts per bin) and best-fit models are
shown in Fig. 8. We define the light-curve zeropoint ($T_{0}{=}0$ sec) as the
time when the count rate is 3$\sigma$ higher than the Poisson background level
(see Table 5). To confirm the zeropoint, we divide the light curves in bins of
$\Delta t{=}$100 and 10 seconds, and compute the chance probability that the
photons per bin come from the background ($P_{\rm bkg}$)252525We use the
Poisson probability mass function, $P_{\rm
bkg}{=}\exp{(-\mu)}\frac{\mu^{k}}{k!}$, where $k$ and $\mu$ are the number of
photons per bin and the background rate multiple by the bin time,
respectively.. We found that the bins after $T_{0}$ have a $P_{\rm
bkg}{\lesssim}$0.01, while $P_{\rm bkg}$ immediately before $T_{0}$ is higher
$P_{\rm bkg}{\gtrsim}$0.1–0.2. We use the _Bayesian Information Criterion_
(BIC)262626BIC${=}-2\ln{\mathcal{L}}+k\ln{N}$, where $\mathcal{L}$ is the
maximum value of the data likelihood, $k$ is the number of model parameters,
and $N$ is the number of data points (Ivezić et al., 2014). to understand
which of the two models describes better the data. We consider the threshold
criterion of $\Delta$BIC${=}$BICh-BIC${}_{l}{>}2$ to discriminate when
comparing two different models, where BICh is the higher model BIC, and BICl
is the lower model BIC. The larger $\Delta$BIC, the stronger the evidence
against the model with a higher BIC is (Liddle, 2007).

The parameters of the best-fitting models of the light curves are listed in
Table 5, while Figure 8 shows the best-fit broken power-law or simple power-
law models. Five sources (FXTs 15, 16, 19, 20, and 22) require a break time
(based on the BIC criterion), while three do not (FXTs 17, 18, and 21). In two
of the former (FXT 15 and 20), $\tau_{1}$ is negative, indicating a
discernible rise phase; the other three (FXTs 16, 19, and 22) are consistent
with an early plateau phase.

FXT | $z$ | $N_{\rm H,Gal}$ | $N_{H}$ | $\Gamma$ | $\log{\rm Norm}$ | Flux | C-stat(dof) | $\ln\mathcal{Z}$
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
15 | 0.0 | 0.5 | 3.8${}_{-3.5}^{+12.5}$ | 2.1${}_{-1.2}^{+2.0}$ | $-5.9_{-0.5}^{+0.9}$ | 0.3${\pm}$0.1 | 14.3(20) | $-15.4{\pm}0.01$
| 1.0 | 0.5 | 18.7${}_{-18.5}^{+58.9}$ | 2.1${}_{-1.2}^{+1.8}$ | $-5.9_{-0.5}^{+0.8}$ | 0.3${\pm}$0.7 | 14.3(20) | $-13.7{\pm}0.01$
16 | 0.0 | 0.2 | 2.4${}_{-2.3}^{+7.3}$ | 2.1${}_{-0.6}^{+0.7}$ | $-4.9_{-0.2}^{+0.3}$ | 2.9${\pm}$0.3 | 72.8(88) | $-46.8{\pm}0.02$
| 0.738 | 0.2 | 8.0${}_{-7.4}^{+13.4}$ | 2.1${}_{-0.5}^{+0.7}$ | $-5.0_{-0.2}^{+0.3}$ | 2.9${\pm}$0.3 | 73.1(88) | $-45.8{\pm}0.02$
17 | 0.0 | 0.2 | 18.1${}_{-14.0}^{+16.6}$ | 3.4${}_{-1.8}^{+1.5}$ |  $-4.2{\pm}$0.9 | 2.3${}_{-0.3}^{+0.5}$ | 24.9(34) | $-19.4{\pm}0.02$
| 1.0 | 0.2 | 66.8${}_{-55.4}^{+31.8}$ | 2.6${\pm}1.4$ | $-4.6_{-0.7}^{+0.6}$ | 2.5${}_{-0.3}^{+0.5}$ | 25.0(34) | $-18.4{\pm}0.02$
18 | 0.0 | 0.2 | 1.1${}_{-1.0}^{+3.1}$ |  ${>}$6.5 | $-4.1_{-0.3}^{+0.5}$ | 88.9${}_{-17.7}^{+17.9}$ | 10.9(11) | $-15.4{\pm}0.02$
| 0.35 | 0.2 | 1.3${}_{-1.1}^{+5.3}$ |  ${>}$6.5 |  $-4.1{\pm}$0.3 | 94.9${}_{-20.8}^{+23.1}$ | 11.1(11) | $-15.2{\pm}0.02$
19 | 0.0 | 0.5 | 5.3${}_{-4.3}^{+6.0}$ | 2.2${\pm}$0.6 | $-4.8_{-0.3}^{+0.4}$ | 3.1${\pm}$0.3 | 99.5(126) | $-59.9{\pm}0.02$
| 1.44 | 0.5 | 47.1${}_{-36.2}^{+42.6}$ | 2.2${\pm}$0.6 |  $-4.9{\pm}$0.3 | 3.1${\pm}$0.2 | 99.9(126) | $-57.9{\pm}0.02$
20 | 0.0 | 0.3 | 4.7${}_{-4.4}^{+11.9}$ | 3.0${}_{-1.3}^{+1.8}$ | $-4.7_{-0.5}^{+0.8}$ | 2.2${\pm}$0.5 | 30.6(36) | $-23.4{\pm}0.02$
| 1.0 | 0.3 | 25.3${}_{-23.6}^{+57.9}$ | 3.0${}_{-1.3}^{+1.8}$ | $-4.7_{-0.5}^{+0.8}$ | 2.2${}_{-0.3}^{+0.5}$ | 30.6(36) | $-21.6{\pm}0.01$
21 | 0.0 | 0.7 | 2.5${}_{-2.3}^{+8.3}$ | 3.1${}_{-1.1}^{+1.4}$ | $-4.8_{-0.4}^{+0.6}$ | 1.9${\pm}$0.4 | 16.6(60) | $-17.0{\pm}0.01$
| 0.85 | 0.7 | 11.6${}_{-11.4}^{+40.0}$ | 3.1${}_{-1.1}^{+1.5}$ | $-4.8_{-0.4}^{+0.6}$ | 1.9${\pm}$0.5 | 16.8(60) | $-15.6{\pm}0.01$
22 | 0.0 | 0.3 | 1.9${}_{-1.8}^{+5.9}$ | 2.3${}_{-0.6}^{+0.9}$ | $-4.5_{-0.2}^{+0.4}$ | 8.7${}_{-0.7}^{+0.9}$ | 78.1(62) | $-49.4{\pm}0.02$
| 1.5105 | 0.3 | 17.0${}_{-15.5}^{+50.1}$ | 2.2${}_{-0.5}^{+0.7}$ | $-4.5_{-0.2}^{+0.3}$ | 8.6${}_{-1.0}^{+0.9}$ | 78.1(62) | $-47.3{\pm}0.02$

Table 6: Results of the 0.5–7 keV X-ray spectral fits for the final sample of
FXT candidates. _Column 2:_ Redshift assumed ($z{=}$0, 1, or from Table 8).
_Columns 3 and 4:_ Galactic and intrinsic column density absorption
($\times$1021) in units of cm-2, respectively. The former is kept fixed during
the fit. _Column 5:_ Photon index from the power-law model. _Column 6:_
Normalization parameter (in units of photons keV-1 cm-2 s-1). _Column 7:_
Absorbed fluxes ($\times$10-14) in units of erg cm-2 s-1 (0.5–7.0 keV).
_Column 8:_ C-stat value and the number of degrees of freedom. _Column 9:_ the
log-evidence ($\ln\mathcal{Z}$) values for each model. The errors are quoted
at the 3$\sigma$ confidence level from the posterior distributions obtained by
BXA except for the flux (which is quoted at 1$\sigma$). Figure 9: _Top
panels:_ X-ray spectra (_black crosses_), in units of counts s-1 keV-1. We
also plot the best-fit absorbed power-law (_blue lines_) spectral model; see
Table 6 for the corresponding best-fitting parameters. _Bottom panels:_
residuals (defined as data-model normalized by the uncertainty) of each
spectral model.

###  3.3 Spectral properties

Using X-ray spectra and response matrices generated following standard
procedures for point-like sources using ciao with the specextract script, we
analyze the spectral parameters of the FXT candidates. The source and
background regions are the same as those previously generated for the light
curves (see Sect. 2.4). To find the best-fit model, because of the low number
of counts, we consider the maximum likelihood statistics for a Poisson
distribution called Cash-statistics (C-stat; Cash, 1979).272727The Cash-
statistic, C-stat, is defined as $C{=}-2\ln{L_{\rm Poisson}}{+}{\rm const}$
Because of the Poisson nature of the X-ray spectral data, the C-stat is not
distributed like $\chi^{2}$ and the standard goodness-of-fit is inapplicable
(Buchner et al., 2014; Kaastra, 2017). Thus, similarly to Paper I, we use _the
Bayesian X-ray Astronomy package_ (BXA; Buchner et al., 2014), which joins the
Monte Carlo nested sampling algorithm MultiNest (Feroz et al., 2009) with the
fitting environment of XSPEC (Arnaud, 1996). BXA computes the integrals over
the parameter space, called the evidence ($\mathcal{Z}$), which is maximized
for the best-fit model, and assuming uniform model priors.

We consider one simple continuum model: an absorbed power-law model
(phabs*zphabs*po, hereafter the PO model), which is typically thought to be
produced by a non-thermal electron distribution. We choose this simple model
because we do not know the origin and the processes behind the emission of
FXTs. Furthermore, the low number of counts does not warrant more complex
models. The spectral absorption components phabs and zphabs represent the
Galactic and intrinsic contribution to the total absorption, respectively.
During the fitting process, the Galactic absorption ($N_{\rm H,Gal}$) was
fixed according to the values of Kalberla et al. (2005) and Kalberla & Haud
(2015), while for the intrinsic neutral hydrogen column density ($N_{\rm H}$),
we carried out fits for both $z{=}0$ (which provides a lower bound on $N_{\rm
H}$ since firm redshifts are generally not known, and is useful for comparison
with host-less FXTs) and the redshift values from Table 8 or fiducial values
of $z{=}1$ for host-less sources.

The best-fitting absorbed power-law models (and their residuals) and their
parameters are provided in Fig. 9 and Table 6, respectively; additionally,
Fig. 10 shows the histograms of the best-fit intrinsic neutral hydrogen column
densities ($N_{\rm H}$; _top panel_) and photon index ($\Gamma$; _bottom
panel_) for extragalactic FXTs candidates of this manuscript (_orange
histograms_) and from Paper I (_blue histograms_). The candidates show a range
of $N_{\rm H}{\approx}$(1.1–18.1)$\times$1021 cm-2 (assuming $z{=}0$), and a
mean value of $\overline{N}_{\rm H}{\approx}$5.0$\times$1021 cm-2, consistent
with the range for sources reported by Paper I (see Fig. 10, _top panel_). We
note that in all cases here, the best-fit $N_{\rm H}$ is higher than the
$N_{\rm H,Gal}$ estimates from Kalberla et al. (2005) and Kalberla & Haud
(2015) by a factor of $\approx$4–90. In every case, intrinsic absorption and
the Galactic component are needed, with at least ${\approx}95$% confidence,
and in some cases even ${\approx}99$% confidence level. Therefore, two
absorption components are needed in the fitting process in general.

Furthermore, excluding the soft candidate FXT 18, the best-fit power-law
photon index ranges between $\Gamma{\approx}$2.1–3.4 for the candidate FXTs,
with a mean value of $\overline{\Gamma}{=}$2.6. FXT 18 is an exceptionally
soft source ($\Gamma{\gtrsim}$6.5) compared to both this sample and the FXT
candidates presented in Paper I (see Fig. 10, _bottom panel_). Finally, FXTs
17, 18, and 21, whose light curves are best-fitted by a PL model, have some of
the softest photon indices ($\Gamma{\gtrsim}$3).

Figure 10: _Top panel:_ Intrinsic neutral hydrogen column density
distribution, evaluated at $z{=}0$ and in units of cm-2, obtained using the
_power-Law_ model for extragalactic FXT candidates from this work (_orange
histogram_) and Paper I (_blue histogram_). The _arrows_ indicate that the
$z{=}0$ intrinsic hydrogen column densities are lower bounds. _Bottom panel:_
Photon index distribution, obtained using a _power-law_ model, for FXT
candidates from this work (_orange histogram_) and Paper I (_blue histogram_).
Note that the uncertainties on these parameter values for individual sources
can be considerable (see Table 6.) Figure 11: _Top panel:_ Hardness ratio of
each FXT candidate (using the Bayesian BEHR code; Park et al., 2006) at
1$\sigma$ confidence level. _Bottom panel:_ Hardness-ratio distributions of
our final samples of FXTs (_orange_ histogram), compared to the X-ray
transients classified as “stars” by _Criterion 2_ using _Gaia_ (filled _cyan_
histogram) and the sources identified previously as _distant_ FXTs (_blue
histogram_) by Paper I.

####  3.3.1 Hardness ratio and photon index evolution

The hardness ratio (HR) can be used to distinguish between X-ray sources, and
permit us to explore their spectral evolution, especially in cases with low-
count statistics (e.g., Lin et al., 2012; Peretz & Behar, 2018). In this work,
the HR is defined as:

$HR=\frac{H-S}{H+S},$ (2)

where $H$ and $S$ are the number of X-ray photons in the 0.5–2.0 keV soft and
2.0–7.0 keV hard energy bands. For each source candidate, we calculate the HR
using the Bayesian code BEHR (Park et al., 2006), which we list in Table 3,
_column 11_ , and plot in Fig. 11 (_top panel_).

We compare the HR of the 90 objects identified as “stars” by _Criterion 2_
(see Fig. 11, _bottom panel, cyan histogram_) in Sect. 2.7.2 with the final
sample of FXTs in this work (_orange histogram_) and the sample of FXTs
reported by Paper I (_blue histogram_). Stars typically have very soft X-ray
spectra (Güdel & Nazé, 2009), confirmed by the fact that ${\approx}$90% of the
star candidates strongly skew toward soft HRs (${\lesssim}$0.0). Clearly, Fig.
11 shows that FXTs do not stand out in the HR plane; thus, HR is not a useful
discriminator on its own between stellar contamination and extragalactic FXTs.

We also analyze how the HR and power-law index of the X-ray spectrum change
with time. To this end, we compute the time-dependent HR, with the requirement
of 10 counts per time bin from the source region (to improve the statistics),
which we show in the _lower panels_ of Fig. 8. For sources that are well-fit
by a BPL model, we also split the event files at $T_{\rm break}$ and extract
and fit the spectra to compute spectral slopes ”before” and ”after” $T_{\rm
break}$ ($\Gamma_{\rm before}$ and $\Gamma_{\rm after}$, respectively; see
Table 7) using the value for the absorption derived from the fit to the full
spectrum (see Table 6). We fit both spectral intervals together assuming
fixed, constant $N_{\rm H,Gal}$ and $N_{H}$ (taken from Table 6).

The spectra of FXT 16 clearly softens after the plateau phase (Fig. 8 and
Table 7) at $>$90% confidence. Similar spectral evolution was also seen from
previous FXT candidates XRT 030511 and XRT 110919 (Paper I). FXTs 15 and 20
exhibit similar spectral softening trends, with $T_{\rm break}$ as a pivot
time, although with only marginal significance, while the rest show no obvious
evidence of such trends (Fig. 8 and Table 7). Finally, it is important to
mention that the FXTs whose light curves follow a PL model (FXTs 17, 18, and
21) show hardening trends in their HR evolution (see Fig. 8).

FXT |  $\Gamma_{\rm before}$($T{<}T_{\rm break}$) |  $\Gamma_{\rm after}$($T{\geq}T_{\rm break}$)
---|---|---
(1) | (2) | (3)
15 | 1.2${}_{-1.0}^{+1.0}$ | 2.8${}_{-1.1}^{+1.2}$
16 | 1.6${}_{-0.4}^{+0.4}$ | 2.9${}_{-0.5}^{+0.5}$
19 | 2.1${}_{-0.3}^{+0.3}$ | 2.4${}_{-0.5}^{+0.5}$
20 | 2.5${}_{-1.2}^{+2.0}$ | 3.3${}_{-1.1}^{+1.2}$
22 | 2.4${}_{-0.5}^{+0.5}$ | 2.1${}_{-0.6}^{+0.8}$
Table 7:  _Columns 2 and 3:_ spectral photon index computed before and after
the $T_{\rm break}$ for light curves that are well-fit with a broken power-
law. Errors are quoted at the 90% confidence level.

###  3.4 Galactic origin

From our sample, FXTs 16 and 19 are clearly aligned with extended objects,
proving an extragalactic origin. FXTs 18, 20, 21, and 22, based on their low
random match probabilities, could be associated with potential hosts,
supporting an extragalactic association (see Sect. 2.8 for more details). In
the next paragraphs, similar to Paper I (see its Sect. 3.4 for more details),
we explore any of the FXT candidates could still be associated with
magnetically active M- or brown-dwarf flares, which are known to produce X-ray
flares on timescales of minutes to hours, with flux enhancements up to two
orders of magnitude (not only at X-ray wavelengths; Schmitt & Liefke, 2004;
Mitra-Kraev et al., 2005; Berger, 2006; Welsh et al., 2007).

Stellar flares typically show soft thermal spectra with temperatures of the
order of $kT{\sim}$0.5–1 keV. M-dwarf stars (brown-dwarfs) have optical and
NIR absolute magnitudes in the range of $M_{z}{\sim}$8–13 AB mag
($M_{z}{\sim}$13–18 AB mag) and $M_{K_{s}}{\sim}$3–10 AB mag
($M_{J}{\sim}$15–25 AB mag), respectively (Hawley et al., 2002; Avenhaus et
al., 2012; Tinney et al., 2014).

The enhanced X-ray emission of M dwarfs shows flares on the order of
$L_{X}^{\rm M-dwarf}{\sim}$1028–1032 erg s-1 (Pallavicini et al., 1990; Pandey
& Singh, 2008; Pye et al., 2015), while brown dwarf flares cover a luminosity
range of $L_{X}^{\rm B-dwarf}{\sim}$1027–1030 erg s-1 (Berger, 2006; Robrade
et al., 2010). Empirically, the ratio between the X-ray luminosity ($L_{X}$)
and bolometric luminosity ($L_{\rm bol}$) of cool M dwarfs and L dwarfs
typically exhibits values no larger than $\log(L_{X}/L_{\rm
bol}){\lesssim}$0.0 and ${\lesssim}-3.0$, respectively (e.g., García-Alvarez
et al., 2008; De Luca et al., 2020). Adopting this limiting ratio, we rule out
a stellar flare scenario for FXT candidates. As in Paper I, we compute the
ratio $\log(L_{X}/L_{\rm bol})$ considering stellar synthetic models of dwarf
stars (taken from Phillips et al., 2020, $1000{\lesssim}T_{\rm
eff}{\lesssim}3000$ K and $2.5{\lesssim}\log{g}{\lesssim}5.5$), normalised to
the deepest photometric upper limits and/or detections (as listed in Table 4),
and compute bolometric fluxes by integrating the normalized models. We
describe the constraints for each FXT below:

For FXT 15, the $m_{g}{>}$24.0 and $m_{r}{>}$23.0 AB mag limits imply
distances to any putative M- and brown dwarfs responsible for the X-ray flares
of ${\gtrsim}$0.2–1.7 kpc and ${\gtrsim}$0.02–0.2 kpc, respectively. The
corresponding X-ray flare luminosities would be $L_{X}^{\rm
M-dwarf}{\gtrsim}$(7.0–690)$\times$1029 and $L_{X}^{\rm
B-dwarf}{\gtrsim}$(7.0–700)$\times$1027 erg s-1, respectively. These are not
enough to discard a Galactic stellar flare nature. Furthermore, the ratio
$\log(F_{X}/F_{\rm bol}){\gtrsim}-0.9$ to $-1.4$ remains consistent with the
extreme spectral type L1 stars (e.g., J0331-27 with $\log(F_{X}/F_{\rm
bol}){\sim}0.0$; De Luca et al., 2020). Thus, we cannot completely rule out an
extreme stellar flare origin for FXT 15.

In the case of FXT 17, the $m_{g}{>}24.1$ and $m_{r}{>}22.4$ AB mag limits
imply distances of ${>}$0.2–1.8 kpc and ${>}$0.02–0.2 kpc for M- and brown-
dwarfs, respectively, and corresponding X-ray flare luminosities are
$L_{X}^{\rm M-dwarf}{\gtrsim}$(1.5–149)$\times$1031 and $L_{X}^{\rm
B-dwarf}{\gtrsim}$(1.5–150)$\times$1029 erg s-1, respectively. The X-ray-to-
total flux ratio is $\log(F_{X}/F_{\rm bol}){\gtrsim}-0.1$ to $+0.4$. Based on
these, we cannot discard a Galactic stellar flare association for FXT 17.

For FXT 18, due to the large X-ray positional uncertainty, there are several
possible optical counterparts. We consider only _source #1_ here, as it is
closest, lying inside the 2$\sigma$ X-ray uncertainty position (see Fig. 6).
The $m_{g}{=}$24.6 and $m_{i}{=}$24.9 AB mag DECam detections (see Table 4)
implies distances of ${\approx}$0.3–3.1 kpc and ${\approx}$0.03–0.3 kpc for M-
and brown-dwarfs, respectively, and corresponding X-ray flare luminosities of
$L_{X}^{\rm M-dwarf}{\approx}$(5.9–600)$\times$1033 and $L_{X}^{\rm
B-dwarf}{\approx}$(5.9–590)$\times$1031 erg s-1, respectively. The ratio
$\log(F_{X}/F_{\rm bol})$ is ${\approx}$2.5 to 3.0. These allow us to rule out
robustly any Galactic stellar flare origin.

In the case of FXT 20, we consider _source #1_ for the stellar flaring
analysis (see Fig. 6). The detections $m_{g}{=}24.3$ and $m_{r}{=}23.5$ AB mag
equate to distance ranges of ${\approx}$0.2–2.5 kpc and ${\approx}$0.02–0.3
kpc for M- and brown-dwarfs, respectively, and corresponding X-ray flare
luminosities of $L_{X}^{\rm M-dwarf}{\approx}$(1.5–147)$\times$1032 and
$L_{X}^{\rm B-dwarf}{\approx}$(1.5–150)$\times$1030 erg s-1, respectively. The
ratio $\log(F_{X}/F_{\rm bol})$ is ${\approx}$0.9 to 1.4. These allow us to
discard a Galactic stellar flare origin for FXT 20.

Finally, for FXT 21, the $m_{i}{=}$22.7 AB mag PanSTARRS detection yields
distance ranges of ${\approx}$0.4–4.1 kpc and ${\approx}$0.04–0.4 kpc for M-
and brown-dwarfs, respectively, and corresponding X-ray flare luminosities of
$L_{X}^{\rm M-dwarf}{\approx}$(3.0–300)$\times$1031 and $L_{X}^{\rm
B-dwarf}{\approx}$(3.1–300)$\times$1029 erg s-1, respectively. The ratio
$\log(F_{X}/F_{\rm bol})$ is ${\approx}$0.2 to 0.7. Thus, we can rule out FXT
21 as a stellar flare.

In summary, the multi-wavelength photometry indicate that three FXTs (FXTs 18,
20, and 21) appear inconsistent with stellar flaring episodes from Galactic M
dwarfs and brown dwarfs, while deeper observations are required to completely
rule out this option for FXTs 15 and 17.

FXT | RA (deg) | DEC (deg) | Offset | $z$ | Log(M${}_{*}/M_{\odot}$) | SFR/($M_{\odot}$/yr) |  $A_{V}$ (mag) | Ref.
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Parameters obtained from the literature |
16 | 53.07658 | -27.87332 | 0$\aas@@fstack{\prime\prime}$44 | 0.738 | 9.07 | 0.81 | 0.02 | 1,2
Parameters derived from photometric data using BAGPIPES (Carnall et al., 2018) |
16 | 53.07658 | -27.87332 | 0$\aas@@fstack{\prime\prime}$44 | 0.738 | 8.91${\pm}$0.04 | 2.98${}_{-0.57}^{+0.78}$ | 1.12${\pm}0.12$ | –
18(S1) | 36.7144 | -1.0826 | 2$\aas@@fstack{\prime\prime}$63 | 0.35${}_{-0.15}^{+0.05}$ | 7.87${}_{-0.33}^{+0.27}$ | 0.17${}_{-0.09}^{+0.38}$ | 0.36${}_{-0.21}^{+0.76}$ | –
19(S1) | 356.26812 | -42.64566 | 0$\aas@@fstack{\prime\prime}$45 | 1.44${\pm}0.08$ | 8.67${\pm}0.11$ | 2.59${}_{-0.48}^{+0.66}$ | 0.16${\pm}0.10$ | –
21(S1) | 50.47531 | 41.24695 | 0$\aas@@fstack{\prime\prime}$46 | 0.85${\pm}0.14$ | 11.20${}_{-0.27}^{+0.24}$ | 1.66${}_{-1.64}^{+26.53}$ | 1.48${}_{-0.77}^{+0.37}$ | –
22(S1)a | 207.23646 | 26.66300 | 4$\aas@@fstack{\prime\prime}$6 | 1.5105 | 10.73${\pm}0.62$ | 35.23${}_{-19.65}^{+50.66}$ | 0.63${\pm}0.45$ | 3,4

Table 8: Parameters obtained from the literature and by our SED fitting to
archival photometric data using the BAGPIPES package (Carnall et al., 2018).
_Column 2 and 3:_ Right ascension and declination of the host galaxies.
_Column 5:_ Angular offset between the transient and the host galaxy. _Column
5:_ Host galaxy redshift or distance. _Columns 6 and 7:_ Logarithmic values of
the stellar mass, and the SFR from the host galaxies. _Column 8:_ Dust
attenuation. _Column 9:_ Literature references.
References: (1) Xue et al. (2019), (2) Schlafly & Finkbeiner (2011), (3)
Jonker et al. (2021), (4) Andreoni et al. (2021)
a Assuming an association with _source #1_ at $z=1.5105$.

Figure 12: Colour-colour diagrams of the potential host galaxies associated
with FXTs 16, 18, 19, 20, 21 and 22 (_coloured stars_), “distant” FXTs from
Paper I (_magenta hexagons_), and X-ray sources classified as stars in Paper I
(_gray filled circles_) and here according to Criterion 2 in Sect. 2.7.2
(_black filled circles_). The expected parameter space of stars with different
ages [$\log(\rm{Age}){=}$7.0-10.3], metallicities ($\rm{[Fe/H]}{=}-$3.0–0.5),
and attenuations ($A_{V}{=}$0.0–5.0) taken from the MIST package (Dotter,
2016; Choi et al., 2016) are overplotted as _cyan_ regions.

###  3.5 One or two populations of FXTs?

In Paper I, we found that the FXT candidates could be robustly classified into
“ _local_ ” and “ _distant_ ” populations (see their Sect. 3.5), based on the
proximity of some sources to local galaxies (distances${\lesssim}$100 Mpc). In
contrast, we find no plausible and robust association between our final sample
and local galaxies in the current work. Two FXT candidates, XRT 191127 and XRT
210423, are detected at projected distances of ${\approx}$500 and 300 kpc,
respectively, from the center of the galaxy cluster Abell 1795
(${\approx}$285.7 Mpc). However, neither is obviously associated with cluster
members and physical offsets in this range are not easily explained by any
possible physical scenario of FXTs (see Sec. 6).

The lack of local FXTs could be explained by the _Chandra_ exposure time spent
to observe local galaxies in recent years. Around 26.5% of the _Chandra_
ObsIds (amounting to ${\approx}$0.8 years of exposure and a sky area of
${\approx}$66.7 deg2) analyzed in this work covers local galaxies (based on a
match with the GLADE catalog; Dálya et al., 2018). Adopting the _local_ FXT
rate from Paper I, $\mathcal{R}_{\rm Local}{=}$53.7${}_{-15.1}^{+22.6}$ deg-2
yr-1, we thus expect ${\approx}$2.8${}_{-2.7}^{+5.6}$ _local_ FXTs in this
work, which remains consistent with our non-detection of _local_ FXTs at
3$\sigma$ confidence for Poisson statistics. On the other hand, the _distant_
FXTs rate from Paper I is $\mathcal{R}_{\rm Distant}{=}$28.2${}_{-6.9}^{+9.8}$
deg-2 yr-1, implying ${\approx}$4.5${}_{-3.7}^{+7.3}$ sources, which is
consistent with our new eight FXT candidates.

##  4 Host-galaxy features

The host galaxy or host environment can provide additional information about
the nature and origin of this FXT sample. Five FXTs (16, 19, 20, 21, and 22)
lie close to extended optical/NIR sources, which are plausible host galaxies
(see Fig. 6). The host galaxies of FXTs 19 and 22 were previously identified,
but their properties were not reported so far. For FXT 18, just one faint
source (_source #1_) falls inside the X-ray uncertainty position, but it is
not clear whether it is extended.

As a first step, in Fig. 12, we explore the nature of the hosts using
$i-K_{s}$ vs. $g-i$ (_top panel_) and $i-W1$ vs. $g-i$ (_bottom panel_)
colours, compared to the colours of X-ray sources previously classified as
stars both in Paper I (_gray points_) and this work (_black points_ ; see
Sect. 2.7.2), and the expected parameter space for stars (_cyan regions_) with
different ages [$\log(\rm{Age/yr}){=}$7.0–10.3], metallicities (from
$\rm{[Fe/H]}{=}-3.0$–0.5), and attenuations ($A_{V}{=}$0.0–5.0 mag) from
theoretical stellar isochrones (MIST; Dotter, 2016; Choi et al., 2016). The
vast majority of the X-ray flares with stellar counterparts form a tight
sequence (see Fig. 12, _cyan region_), with the outliers identified as PNe,
YSOs, eruptive variable stars, T Tauri stars, or emission-line stars. Overall,
the potential hosts of the FXT candidates appear to reside outside or at the
edge (e.g., FXT 16, 18, 19, 21 and 22) of the stellar region, although the
limits or large uncertainties (e.g., FXT 20) indicate that the current colour
estimates are not the best discriminators by themselves. Thus the spatially
resolved nature of the counterparts remains vital to their confirmation as a
candidate host galaxy.

We further constrain the host properties through spectral energy distribution
(SED) model fitting of their existing photometry using BAGPIPES (Bayesian
Analysis of Galaxies for Physical Inference and Parameter EStimation; Carnall
et al., 2018), which fits broadband photometry and spectra with stellar-
population models taking star-formation history and the transmission function
of neutral/ionized ISM into account via a MultiNest sampling algorithm (Feroz
& Hobson, 2008; Feroz et al., 2009). In Appendix C, we list the different
conditions considered for the SED fitting. Table 8 provides the best-fit
parameters obtained with BAGPIPES for the hosts of FXTs 16, 18, 19, 21 and
22282828FXT 20 only has faint $g$, $r$ and $z$-band DECam detections, which
are too few and too loosely constrained to compute a SED photometric
redshift., while Fig. 22 shows the 16th to 84th percentile range for the
posterior spectrum, photometry, and the posterior distributions for five
fitted host-galaxy parameters. Also, Figs. 13 and 14 compare the SFR and
stellar masses of the FXT hosts to those of several well-known transient
classes (such as CC- and Ia SNe, LGRBs, SGRBs and FRBs) and cumulative
functions, respectively.

Overall, in terms of stellar mass and star-formation rate, FXTs 16, 18, 19,
and 22 are located above the galaxy main sequence, while FXT 21 lies slightly
below it (see Fig. 13, _solid cyan line_). The hosts of FXTs 16, 19, and 22
have SFRs, stellar masses, and young stellar populations indicative of star-
forming galaxies (Moustakas et al., 2013). In terms of SFRs, the FXT hosts
broadly lie in the same region populated by SNe type Ib, Ic, II, SL-SNe, and
GRBs. The SFR of the host of FXT 22 compares more favorably with LGRB (30%)
and SGRB (10%) hosts over SNe (${\sim}$0%); meanwhile, its large host stellar
mass shares little overlap with the LGRB/SGRB (${\approx}$10/15%) and
partially with type-Ia/CC-SN (${\approx}$40/30%) host populations. For FXTs 16
and 19, the overlapping fractions of LGRB, SGRB, and SLSNe host galaxies with
galaxy stellar mass ${\lesssim}$109 $M_{\odot}$ are ${\approx}$20, 15, and
80%, respectively. In the particular case of FXT 18, it has a moderate SFR and
low stellar mass. This low stellar mass matches with a very small fraction of
LGRB (${\lesssim}$5%) and SGRB (${\lesssim}$2%) hosts and some SL-SNe
(${\approx}$30%) hosts.

The host of FXT 21 has a moderate SFR and high stellar mass, implying a
classification as a quiescent galaxy (Moustakas et al., 2013). Its SFR falls
in a region populated by ${\approx}$70 and 50% of LGRBs and SGRBs,
respectively, and ${\approx}$40 and 25% of CC-SNe and Ia-SNe hosts,
respectively, with SFR${\gtrsim}$2.0 $M_{\odot}$ yr-1. Meanwhile, only
${\lesssim}$10% of SNe and GRBs have similar host galaxy stellar masses
${\gtrsim}$1011 $M_{\odot}$.

Moreover, these sources fall in the same parameter space occupied by the
distant FXTs reported in Paper I (see Fig. 13, _lower panel_). Thus, we
conclude that a majority of distant FXTs appear to be associated with actively
star-forming galaxies (${\gtrsim}$108 M⊙ and ${\gtrsim}$0.5 M⊙ yr-1), while a
subset is associated with post-starburst (”green valley”) galaxies.

Another crucial parameter is the projected physical offset ($\delta R$)
between the transient’s position and the host galaxy center. Figure 14, _right
panel_ , compares the projected physical offset distribution of FXTs 16, 18,
19, 21 and 22 to several transient classes such as CC-SNe (_cyan_), Ia-SNe
(_orange_), SL-SNe (_magenta_), FRBs (_black_), LGRBs (_blue_) and SGRBs
(_red_). SGRBs have a physical offset, which is about ${\sim}$4–5 times
greater than the median offset for LGRBs (Bloom et al., 2002) and SL-SNe
(Schulze et al., 2021), and about ${\sim}$1.5 times larger than the median
offsets of CC- and Type Ia SNe (Prieto et al., 2008) and FRBs (Heintz et al.,
2020). In addition, practically no LGRBs and SL-SNe, and only ${\approx}$10%
of CC- and type Ia SNe have offsets ${\gtrsim}$10 kpc, while $\approx$40% of
SGRBs have such offsets. Moreover, ${\approx}$15% of SGRBs have offsets
${\gtrsim}$20 kpc, while essentially no SL-SNe, CC- and type Ia SNe, or LGRBs
exhibit such large offsets.

The physical offsets of FXTs 16 (${\approx}$3.3 kpc), 19 (${\approx}$3.9 kpc),
and 21 (${\approx}$3.6 kpc) overlap with the cumulative distributions of CC-
and type Ia SNe, and SGRBs at 1$\sigma$ confidence level, although only
${\approx}$10–15% of SL-SNe and LGRBs have equal or higher offset values. In
the case of FXT 18, its offset (${\approx}$13.2 kpc) resides well inside the
offset distribution of SGRBs (${\approx}$70% with $\delta R{<}$13 kpc), while
just ${\lesssim}$5% of LGRBs, and CC- and Ia-SNe have equal or higher offset
values. Nevertheless, it has a large X-ray positional uncertainty. In
contrast, FXT 22 has a physical offset of ${\approx}$40 kpc, which is just
compatible with ${\approx}$10% of SGRB hosts with equal or higher offsets.

Figure 13: Star-forming galaxy main sequence diagram, stellar mass vs. SFR,
comparing hosts of FXTs and various other transient classes (one per panel)
such as SNe type Ia, Ib, Ic, II (Tsvetkov & Bartunov, 1993; Galbany et al.,
2014; Schulze et al., 2021), super-luminous SNe (SL-SNe; Schulze et al.,
2021), LGRBs (including SN 2020bvc; Chang et al., 2015; Li et al., 2016; Izzo
et al., 2020; Ho et al., 2020), low-luminosity LGRBs (LL-LGRBs; GRB 980425,
GRB 020903, GRB 030329, GRB 031203, GRB 050826, GRB 060218, and GRB 171205A;
Christensen et al., 2008; Michałowski et al., 2014; Levesque, 2014; Krühler et
al., 2017; Wiersema et al., 2007; Wang et al., 2018; Arabsalmani et al.,
2019), SGRBs (including GW 170817/GRB 170817A; Li et al., 2016; Im et al.,
2017; Nugent et al., 2022), TDEs (French et al., 2020), and Paper I FXT
candidates (nearby and distant FXTs). _Grayscale contours_ denote the SDSS
galaxy distribution from Brinchmann et al. (2004). The _solid cyan lines_ show
the best-fit local galaxy main sequence relation from Peng et al. (2010),
while the _dashed coloured lines_ denote the upward evolution of the boundary
separating star-forming and quiescent galaxies as a function of redshift (at
$z{=}$0.0, 0.1, 0.2, 0.3, and 0.4; Moustakas et al., 2013). Figure 14:
Comparison of the host-galaxy properties of FXTs 16, 19, 21 and 22 (_colour
vertical dashed lines_) from Table 8 with the cumulative distributions of
galaxy stellar mass (_left panel_), star-formation rate (_center panel_), and
projected physical offset (_right panel_) for LGRBs (_blue lines_ ; Li et al.,
2016; Blanchard et al., 2016), SGRBs (_red lines_ ; Fong et al., 2010, 2012,
2014; Margutti et al., 2012; Sakamoto et al., 2013; Berger et al., 2013a; Fong
et al., 2022; Nugent et al., 2022), FRBs (_black lines_ ; Heintz et al.,
2020), CC-SNe and Ia-SNe (_cyan and orange lines_ ; Tsvetkov & Bartunov, 1993;
Prieto et al., 2008; Galbany et al., 2014), and SLSNe (_magenta lines_ ;
Schulze et al., 2021).

FXT | $z$ | $L_{\rm X,peak}$ | $z_{\rm max}$ | $V_{\rm max}$
---|---|---|---|---
|  | (erg s-1) |  | (Gpc3)
(1) | (2) | (3) | (4) | (5)
Paper I (Quirola-Vásquez et al., 2022)
1 | 0.1866†,a | 1.9$\times$1046 | 4.30 | 1673
7 | 1.0† | 1.3$\times$1046 | 3.59 | 1354
8 | 0.61 | 1.1$\times$1044 | 0.52 | 32
9 | 0.7 | 1.5$\times$1045 | 1.49 | 343
10 | 1.0† | 4.8$\times$1045 | 2.42 | 789
11 | 0.0216†,b | 2.4$\times$1044 | 0.72 | 70
12 | 1.0† | 3.0$\times$1045 | 2.00 | 584
13 | 1.0† | 6.5$\times$1044 | 1.08 | 177
14 | 2.23c | 1.7$\times$1047 | 11.01 | 3751
This work
15 | 1.0† | 1.0$\times$1045 | 1.30 | 261
16 | 0.738 | 2.8$\times$1045 | 1.94 | 554
17 | 1.0† | 6.3$\times$1045 | 2.73 | 946
18 | 0.35 | 1.9$\times$1047 | 11.55 | 3867
19 | 1.44 | 3.7$\times$1046 | 5.72 | 2245
20 | 1.0† | 8.1$\times$1046 | 7.98 | 2987
21 | 0.85 | 6.9$\times$1045 | 2.80 | 978
22 | 1.51 | 1.3$\times$1046 | 3.61 | 1367

Table 9: FXT properties from Paper I and this work used to compute the
volumetric density rates (Sect. 5.3) and X-ray luminosity functions (Sect.
5.2). _Column 2:_ adopted redshift for each FXT. _Column 3:_ Peak isotropic
X-ray luminosity in cgs units (corrected for Galactic and intrinsic
absorption). _Columns 4:_ maximum observable redshift. _Columns 5:_ maximum
comoving volume in Gpc3 units.
† fiducial redshifts.
a redshift taken from Eappachen et al. (2022, $z{=}$0.1866).
b redshift taken from Glennie et al. (2015, 94.9 Mpc).
c redshift taken from Bauer et al. (2017, $z{=}$2.23).

##  5 Rates

We update the FXT event rates determined in Paper I and revisit comparisons
with other transients to explore possible interpretations. Specifically, we
derive the observed event rates (deg-2 yr-1; Sect. 5.1), FXT X-ray luminosity
function (Sect. 5.2), and volumetric rates (yr-1 Gpc-3; Sect. 5.3).

###  5.1 Event-rate estimation

We compute FXT event rates following the procedure and assumptions outlined in
Paper I (their sect 6.1 and eqs. 5–7). We first estimate the rate
independently of Paper I to confirm consistency. We found eight FXTs inside
${\approx}$89 Ms of _Chandra_ data from 2014 to 2022, yielding
$\mathcal{R}_{\rm This\ work}{=}$45.6${}_{-14.3}^{+18.2}$ deg-2 yr-1 (for
sources with $F_{\rm X,peak}{\gtrsim}$1$\times$10-13 erg cm-2 s-1). This rate
is consistent with the rates derived by Yang et al. (2019, $\mathcal{R}_{\rm
Yang+19}{\approx}$59${}_{-38}^{+77}$) and Paper I ($\mathcal{R}_{\rm
Paper\leavevmode\nobreak\ I,distant}{=}$28.2${}_{-6.9}^{+9.8}$ deg-2 yr-1) at
the Poisson 1$\sigma$ confidence level and higher than the rate derived by
Glennie et al. (2015, ${\approx}$3.4 deg-2 yr-1). As already mentioned in
Paper I, this is not surprising since Glennie et al. (2015) computed the rate
for a higher peak flux of $F_{\rm X,peak}{\gtrsim}$1${\times}$10-10 erg cm-2
s-1. Then, considering all 17 distant FXTs (i.e., the nine distant FXTs from
Paper I, also including the ambiguous FXTs 1 and 11 which might be
extragalactic sources according to Eappachen et al. 2022, and the eight FXTs
from this work) detected by _Chandra_ (ACIS-I/S) instruments between 2000 and
2022, we estimate a total distant FXT event-rate of $\mathcal{R}_{\rm
Total,distant}{=}$36.9${}_{-8.3}^{+9.7}$ deg-2 yr-1. Since the number of FXTs
removed erroneously by our selection criteria is ${\ll}$1 (see Sect. 2.7.6),
the estimated event rates are robust results for FXT candidates brighter than
$F_{X}{\gtrsim}$1$\times$10-13 erg cm-2 s-1 (from the search algorithm
developed in Paper I). Finally, we found no new local FXTs among the _Chandra_
observations of nearby galaxies included in this work (i.e., 26.5% of the
ObsIds), allowing a revised estimate (considering ObsIds from Paper I and this
work) of the nearby FXT event rate of $\mathcal{R}{=}$34.3${}_{-10.8}^{+13.7}$
deg-2 yr-1, consistent with that derived in Paper I.

Figure 15: Observed cumulative log$\mathcal{N}$–log$S$ and
log$\mathcal{N}$–log$F_{\rm peak}$ distributions. _Left panel:_
log$\mathcal{N}$–log$S$ distribution of the sample of extragalactic FXTs
analyzed in this work (_orange line_), from Paper I (_cyan line_), and
combined (_black line_), as a function of fluence (in cgs units). Also shown
are two PL models, $N({>}Fluence){\propto}S^{-\gamma}$, with slopes
$\gamma{=}$0.52 (_red line_) and 1.0 (_blue dashed line_). The $\gamma{=}1$
line represents the best fit and 1$\sigma$ error of Arefiev et al. (2003)
based on bright FXTs (including Galactic flares). The brightest sources in our
sample appear to be consistent with this bright-end extrapolation, although
our fainter sources fall up to ${\sim}$1 dex below, implying a break. For
comparison with Arefiev et al. (2003), we convert all FXT fluences to the 2–10
keV band from their best fits. _Right panel:_ log$\mathcal{N}$–log$F_{\rm
peak}$ distribution of the total sample of extragalactic FXTs analyzed in
Paper I and this work (_black line_) and only distant FXTs (_orange line_).
The _magenta dashed line_ shows objects uniformly distributed in Euclidean
space (${\propto}F_{\rm peak}^{-3/2}$) for comparison. The _color regions_
represent the 1-$\sigma$ confidence interval.

The event rate as a function of fluence (or peak flux292929Similar to Paper I,
due to the lack of a standardized method to estimate the $F_{\rm X,peak}$,
first we find the shortest time interval during which 25% of the counts are
detected, and we compute the count rate during this shortest interval. Next,
to convert the peak-count rates to fluxes, we multiply the flux from the time-
averaged spectral fits by the ratio between the peak and the time-averaged
count rates (i.e., we assume no spectral evolution).) behaves as a power-law
function as $\mathcal{R}{\propto}F_{\rm peak}^{-\gamma}$, where $\gamma$ is a
positive value.

Figure 15, _left panel_ , shows the cumulative log$\mathcal{N}$–log$S$
distribution of the entire sample analyzed in this work (i.e., 8 FXTs; _orange
line_), FXTs identified in Paper I (i.e., 14 FXTs; _cyan line_), and finally
combining both samples (i.e., 22 FXTs; _black line_) which appears to follow
$\gamma{\approx}0.5$ (_red region/line_). We also plot the extrapolation of
the best-fit slope, $\gamma{=}1.0$, based on the estimates of FXTs at bright
fluxes (${\gtrsim}10^{-10}$ erg cm-2 s-1) from Arefiev et al. (2003).303030We
caution that Arefiev et al. (2003) does not specify an exact energy band and
makes no distinction between various potential Galactic and extragalactic
classes. It is noteworthy that the sky distribution at these bright fluxes is
also isotropic. The brightest sources in the total sample of 22 FXTs appear to
be consistent with the Arefiev et al. (2003) extrapolation at 1-$\sigma$
confidence. In contrast, the fainter sources fall well below it by ${\sim}$1
dex (following a power-law with an exponential index of $\gamma{=}$0.52),
indicating a break around a fluence of ${\sim}10^{-8}$ erg cm-2 to our best-
fit slope. A similar result was found in Paper I. Furthermore, we define a BPL
model, which fits the results obtained (see Fig. 15), for the cumulative
log$\mathcal{N}$–log$S$ distribution with a break fluence around
${\sim}1{\times}10^{-8}$ erg cm-2 and two power-law indexes of
$\gamma_{1}{=}-0.52$ and $\gamma_{2}{=}-1.0$ for the faint and bright ends,
respectively.

Finally, Fig. 15, _right panel_ , represents the cumulative
log$\mathcal{N}$–log$F_{\rm peak}$ curves considering the whole sample of 22
FXTs (_black line_) and just the 17 distant FXTs for Paper I and this work
(_orange line_). The local FXTs identified in Paper I contribute mostly to low
fluxes (compare the _orange_ and _black_ lines). The
log$\mathcal{N}$–log$F_{\rm peak}$ slope appears to be significantly shallower
at low $F_{\rm peak}$ than the Euclidean prediction (i.e.,
${\propto}S^{-3/2}$, which is expected for astrophysical objects uniformly
distributed in a Euclidean space; _dashed magenta line_). A combination of
four effects could explain this deviation: $i)$ near the sensitivity threshold
of the detector, the number of FXTs depends on the detection efficiency, which
affects the log$\mathcal{N}$–log$F_{\rm peak}$ plot; $ii)$ due to the flux
being inversely proportional to the square of the luminosity distance, which
will differ from the Euclidean distance as $z$ approaches unity (this implies
that the FXTs should be cosmological); $iii)$ the sample of FXTs likely has a
mix of origins, such that the cosmic event rate density is not constant with
redshift; and $iv)$ the sample is dominated by low-number statistical
fluctuations, particularly at the bright end, due to the _grasp_ (area
$\times$ sensitivity) of _Chandra_. New X-ray missions which are focusing on
scanning the sky, such as _eROSITA_ and _Einstein Probe_ , will increase the
number of FXTs and improve our statistics.

Figure 16: _Left panel:_ The peak X-ray luminosity of FXT candidates from
Paper I and this work (with known and fiducial redshifts; see Table 9). The
green _solid and dashed_ lines indicate peak flux limits of 10-13 (set by our
algorithm) and 10-14 erg cm-2 s-1 (approximate _Chandra_ on-axis detector
limit), respectively. _Right panel:_ FXT X-ray (0.3–10 keV) luminosity
function (XLF) of the total sample from Paper I and this work. The _red
squares_ represent the XLF using the seven FXTs with known (photometric or
spectroscopic) redshifts, while the _blue_ and _green squares_ show XLFs using
all 17 FXTs with known+fiducial redshifts, adopting $z=1.0$ and 0.5 for
unknown objects, respectively. The _solid lines_ show best-fit power-law
models (see Table 10 for values).

Case # | $\alpha$ | $\beta$ | $\rho_{0}(L_{\rm X,peak}{>}10^{44}{\rm erg\leavevmode\nobreak\ s^{-1}})$
---|---|---|---
|  | (Gpc-3 yr-1 dex-1) | (Gpc-3 yr-1)
(1) | (2) | (3) | (4)
_Case I_ (known redshifts) | $-0.26{\pm}0.13$ | $(7.90{\pm}1.49){\times}10^{2}$ | $(1.24{\pm}0.35){\times}10^{3}$
_Case II_ ($z_{\rm fiducial}{=}1$) | $-0.57{\pm}0.11$ | $(1.74{\pm}0.29){\times}10^{4}$ | $(1.32{\pm}0.11){\times}10^{4}$
_Case III_ ($z_{\rm fiducial}{=}0.5$) | $-1.13{\pm}0.27$ | $(1.14{\pm}0.45){\times}10^{5}$ | $(4.38{\pm}0.21){\times}10^{4}$

Table 10: Results of XLF fitting. _Column 1:_ _Case_ # considered. _Columns 2
and 3:_ best-fitting parameters of the XLF from Eq. 4. _Column 4:_ FXT
volumetric rate from integrating the XLF.

###  5.2 Luminosity function

Past works have constructed X-ray luminosity functions (XLFs) for GRBs (Sun et
al., 2015), SBOs (Sun et al., 2022), and TDEs (Sazonov et al., 2021). We
construct here the XLF of FXTs considering distant sources from Paper I and
this work, using the classical $V_{\rm max}$ method from Schmidt (1968). We
adopt the redshifts and peak X-ray luminosities shown in Table 9 (_columns 2
and 3_ , respectively) and Fig. 16 (_left panel_), which also plots the
limiting luminosity corresponding to a _Chandra_ detection threshold of
$F_{\rm X-peak}^{\rm lim}{\sim}$1$\times$10-13 erg cm-2 s-1 (_green solid
line_ ; representing the conservative detection limit of our search algorithm)
and 1$\times$10-14 erg cm-2 s-1 (_green dashed line_ ; representing an
approximate instrument threshold). All the FXTs except FXT 8 lie above $F_{\rm
X,peak}^{\rm lim}{\approx}$1$\times$10-13 erg cm-2 s-1.

The XLF, $\Phi(L)$, is the sum of the individual contributions by each source
($j$) in the luminosity range from $\log(L)$ to $\log(L)+d\log(L)$ (with total
$\Delta N_{L}$ sources), i.e.,

$\Phi(L)\ d\log(L)=\sum_{j=1}^{\Delta N_{L}}\frac{4\pi\
dN(L)_{j}}{[\sum_{i=1}^{N_{\rm obs}}T_{i}\Omega_{i}V_{\rm max,i}]_{j}},$ (3)

where $i$ runs over the $i$th observation, $N_{\rm obs}$ is the total number
of observations, $T_{i}$ and $\Omega_{i}$ are the exposure time and FoV per
observation, respectively, $V_{\max,i}$ is the maximum observable volume, and
$\Delta N_{L}$ is the total detectable number of sources.

$V_{\rm max}$ for a given FXT in the sample depends on its intrinsic peak
luminosity in the 0.3–10.0 keV energy band, which are both given in Table 9,
assuming a flux limit of $F_{\rm X,peak}^{\rm lim}{\approx}$10-13 erg cm-2 s-1
from our detection algorithm. To calculate the FXT XLF, we sum the derived
$V_{\rm max}^{-1}$ values of individual FXTs in five equal 0.5 dex interval
luminosity bins from $\log(L_{\rm X,peak})$ between 44.0 and 47.5. The
uncertainty within a given bin depends on the Poisson uncertainty of the
number of FXTs per bin and $V_{\rm max}$ (computed as $\sqrt{\sum(V_{\rm
max})^{-2}}$, where the summation is done over the objects within that bin).

We estimate three different cases: $I)$ considering just eight FXTs with known
redshifts, $II)$ considering 17 distant FXTs with known and fiducial
($z{=}1.0$) redshifts, and $III)$ considering 17 distant FXTs with known and
fiducial ($z{=}0.5$) redshifts. The computed FXT XLFs are shown in Fig. 16
(_right panel_). In _Case I_ (_red squares_), the largest uncertainties are
associated with the lowest luminosity bins where there is just one FXT per
luminosity bin. For _Cases II_ and _III_ (_blue_ and _green squares_ ,
respectively), the uncertainties are somewhat smaller because some luminosity
bins have more than one source.

The FXT XLFs all appear to decline with increasing X-ray luminosity and are
fit with power-law models as:

$\frac{dN}{d\log(L_{\rm X,peak})dVdt}=\beta\times\left(\frac{L_{\rm
X,peak}}{10^{44}\leavevmode\nobreak\ {\rm erg}\leavevmode\nobreak\ {\rm
s}^{-1}}\right)^{\alpha}.$ (4)

The best-fit models are plotted in Fig. 16 as _red, blue, and green lines_ for
_Cases I, II_ and _III_ , respectively, with the results summarized in Table
10. Assuming fiducial redshifts of $z{=}1.0$ ($z{=}0.5$) naturally leads to
shallower (steeper) XLF slopes.

Under the implicit assumption of no evolution, we can estimate the average FXT
volumetric rate in the $z{\approx}$0–2.2 (see Table 9) Universe by integrating
the XLF over the entire luminosity range used here ($L_{\rm X,peak}{>}10^{44}$
erg s-1). The results are tabulated in Table 10. For _Case I_ , the average
event rate density is ${\approx}1.2{\times}10^{3}$ Gpc-3 yr-1. Due to the
total galaxy volume density of ${\sim}$2$\times$107 Gpc-3 (Bell et al., 2003),
this equates to a rate of $R{\sim}6{\times}10^{-5}$ FXTs yr-1 per galaxy. On
the other hand, considering the FXTs with known+fiducial redshifts of
$z{=}1.0$ (_Case II_) and $z{=}0.5$ (_Case III_), the average FXT volumetric
rates would be ${\approx}1.3{\times}10^{4}$ and ${\approx}4.4{\times}10^{4}$
Gpc-3 yr-1, respectively, implying rates of $R{\sim}6.5{\times}10^{-4}$ and
$2.2{\times}10^{-3}$ FXTs yr-1 per galaxy, respectively. The three derived FXT
volumetric rates (which could be interpreted as a mean value for _Cases I_ or
_II/III_ , respectively) remain consistent with previous results computed in
Paper I (${\approx}5{\times}10^{3}$ Gpc-3 yr-1), Xue et al., 2019
(${\approx}1.3{\times}10^{4}$ Gpc-3 yr-1), and Bauer et al., 2017
(${\gtrsim}10^{2}$ Gpc-3 yr-1 at $z{\lesssim}1$). Notably, these values differ
by orders of magnitude from the rates computed previously for SMBH TDEs
(${\approx}$210 Gpc-3 yr-1; Donley et al., 2002; Sazonov et al., 2021) and
marginally with SBOs (${\approx}4.6{\times}$104 Gpc-3 yr-1; Sun et al., 2022)
identified in eROSITA and _XMM-Newton_ archival data, respectively. This
further helps to exclude the SBOs (considering _Case I_) and SMBH TDE
scenarios, although other transients may remain viable due to the large
beaming correction uncertainties.

Figure 17: Volumetric density rate as a function of redshift comparing FXTs
(_gray filled region_ , assuming no evolution with redshift and 1-$\sigma$
confidence; see text for details) and other transients. _Left panel:_
comparison to massive-star related sources such as CC-SNe (_dashed orange
line_ ; $k_{\rm CC-SNe}{=}0.0068$ $M_{\odot}^{-1}$ times the cosmic SFR
density from Madau & Dickinson, 2014), LGRBs (_blue-solid, dashed and dotted
lines_ show evolution normalized at $z{=}0$ to $\rho_{\rm 0,LGRBs}{=}$0.75
yr-1 Gpc-3 for $f_{b}^{-1}{=}$500, 100, and 50, respectively; Sun et al.,
2015; Wanderman & Piran, 2010), LL-LGRBs (_green-solid and dashed lines_
denote evolution normalized at $z{=}0$ to $\rho_{\rm 0,LL-LGRBs}{=}$150 yr-1
Gpc-3 for $f_{b}^{-1}{=}$14 and 1, respectively, where $f_{b}^{-1}$ is the jet
beaming correction factor; Liang et al., 2007a; Zhang, 2018). _Middle panel:_
comparison to compact-object binary systems such as SGRBs considering Gaussian
(Gau.; _red lines_), Power-law (PL; _gray lines_), and Log-Normal (LN; _purple
lines_) merger delay models (_solid, dashed and dotted lines_ denote evolution
normalized at $z{=}0$ to $\rho_{\rm 0,SGRBs}{=}$1.75 yr-1 Gpc-3 for
$f_{b}^{-1}{=}$110, 30 and 25, respectively; Sun et al., 2015; Wanderman &
Piran, 2015). _Right panel:_ comparison to SMBH-TDEs (_magenta line_ indicates
evolution normalized at $z{=}0$ to $\rho_{\rm 0,SMBH-TDEs}{=}$1.1$\times$105
yr-1 Gpc-3 for luminosities ${\gtrsim}$1044 erg s-1, assumed to be emitted
isotropically; Sun et al., 2015) and IMBH-TDEs (_cyanline_ shows evolution
normalized at $z{=}0$ to $\rho_{\rm 0,IMBH-TDEs}{=}$290 yr-1 Gpc-3 emitted
isotropically; Bloom et al., 2011; Lu & Kumar, 2018; Malyali et al., 2019;
Tanikawa et al., 2022). We also show the estimated rates from Paper I (_cyan
circle_) and Xue et al. (2019) for CDF-XT2-like objects (_purple square_).

###  5.3 Volumetric Density rate estimation

In Section 5.2, we estimated the average FXT volumetric rate in the redshift
range $z{\approx}$0–2.2 from the XLF (see Table 10), assuming zero evolution.
Now, we compare the volumetric density rate, in units of yr-1 Gpc-3, with
other known transient classes such as GRBs, SNe/SBOs, or TDEs. First, because
only eight of 17 distant FXTs have redshift estimates, we correct the FXT
volumetric rate of _Case I_ by the inverse of this fraction (i.e., multiply by
$17/8$) to account for the fact that we do not include all the sources. We
implicitly assume here that the underlying redshift distribution of the
sources without redshifts is the same as those with. Without this correction,
the luminosity functions are lower limits rather than best estimates.
Meanwhile, for _Cases II and III_ , a correction factor is not necessary
because both cases adopted fiducial redshifts ($z_{\rm fiducial}{=}1$ and
$0.5$, respectively) for all FXTs that lacked estimates. Considering this
correction, the volumetric density rate for _Cases I, II and III_ ranges
between ${\sim}1.9{\times}10^{3}{-}4.6{\times}10^{4}$ Gpc-3 yr-1 at 1$\sigma$
confidence.

The derived density rate as a function of redshift is shown in Fig. 17 (_gray
filled region_). Our result is consistent with the rates estimated in Paper I
(${\approx}$4.8$\times$103 Gpc-3 yr-1 at $z_{\rm max}{=}$2.1; _cyan circle_)
or CDF-S-XT2-like sources (_purple square_ ; Xue et al., 2019). Each panel in
Fig. 17 represents a comparison with transients related to massive stars
(LGRBs, LL-LGRBs, and CC-SNe; _left panel_), compact binary mergers (SGRBs;
_middle panel_), and tidal disruption events (SMBH and IMBH TDEs; _right
panel_).

As CC-SNe progenitors are massive, short-lived stars, the event rates should
reflect ongoing star formation at different cosmological epochs (Madau &
Dickinson, 2014, and references therein). Thus to build the cosmic density
rate shown in Fig. 17 (_left panel_), we use the star-formation history
derived by Madau & Dickinson (2014), weighted by the number of stars that
explode as SNe per unit mass $k_{\rm CC-SNe}{=}0.0068$ $M_{\odot}^{-1}$ (Madau