consideration here is to identify and isolate the fraction of _Chandra_
observations that actively target nearby galaxies. While distant FXRTs can be
detected in any _Chandra_ observation (i.e., in the background of nearby
galaxy observations), nearby FXRTs can only be detected if nearby galaxies lie
within the _Chandra_ FoV. To this end, the fraction of useful212121That is,
ignoring all observations with $|b|{<}$10 deg. _Chandra_ observations that
target nearby galaxies at ${\leq}$100 Mpc is ${\approx}$20% of the total
sample or ${\approx}$36.7 Ms (based on a match with the GLADE catalog; Dálya
et al. 2018), while 80% is spent observing distant extragalactic sky,
respectively. In these fractions, we find 6 and 8 FXRTs, respectively.

Extrapolating from 8 FXRTs in 80% of the observations, we can expect
${\approx}$2 distant FXRTs should occur in the 20% fraction dedicated to
nearby objects, and thus the true number of nearby FXRTs would be
$6{-}2{\approx}$4\. However, given that we are in the Poisson regime, we need
to know whether this excess is significant. If we assume a null hypothesis,
$\mathcal{NH}$, whereby the sample consists of just one population of FXRTs,
such that there are 14 distant FXRTs detected in 100% of our data, then we
expect 2.8 sources in the 20% of the extragalactic fields that overlap with
local galaxies, and yet we observe 6. This corresponds to an excess at 90%
confidence level (i.e., detecting 6 is inconsistent with 2.8 at 90%
confidence), which is likely related with an additional nearby population of
FXRTs. So there is tentative (at 90% confidence level) evidence for two
different population of FXRTs.

More systematically, we explore the likelihood of whether each individual
local FXRT could in fact be a distant FXRT in projection with a nearby galaxy.
Adopting Poisson statistics, following a similar approach to Sect. 2.5.6, and
taking the density of distant FXRTs as ${=}$6.5${\times}$10-6 arcmin-2 (i.e.,
considering the number of secure distant FXRTs found among all nonlocal
_Chandra_ observations), we compute the probability of finding by chance a
distant FXRT within the specific angular offset ($P_{\rm dist.FXRT}$; see
Table 7 _column 4_) of each local FXRT to its associated host galaxy (see
Table 7 _column 2_). Additionally, we calculated the Binomial probability
[$P(k;N;P_{\rm dis.FXRT})$; see Table 7 _column 6_], where the number of
detections is taken to be the number of local FXRTs detected in a galaxy at
least as bright as the specific associated local FXRT host at a distance less
than or equal to the offset distance of each the specific local FXRT, and the
number of trials is taken to be the number of _Chandra_ observations of local
galaxies at least as bright as the specific associated local FXRT host (see
Table 7 _column 5_). Table 7 shows that only FXRT 1 has a significant
probability (${\approx}$0.3) to be related with the distant sample; in the
case of FXRT 11, it also shows a nonzero probability of being related with the
distant sample (${\approx}$8.0$\times$10-2), reinforcing the idea that this
source is not likely associated with nearby galaxies. Thus, there remains some
uncertainty as to which population FXRTs 1 and 11 belong. For the moment, we
have tentatively assigned FXRT 1 to the local sample (due to the similarity of
its associated host with others in that category) and FXRT 11 to the distant
sample. However, we interpret them throughout leaving both possibilities open
(see Sect. 5).

In summary, our results here reinforce the existence of two different
populations of FXRTs and rules out a possible relation of FXRTs 2–6 with the
distant sample.

Figure 13: Comparison of the stellar mass ($M_{*}$) and SFR of host galaxies
of different types of transients: nearby (_colored hexagons_) and distant
(_colored stars_) FXRT candidates, LGRBs and SGRBs (Li et al., 2016), 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), GW 170817 (Im et al., 2017), SN
2020bvc (Chang et al., 2015; Izzo et al., 2020; Ho et al., 2020), FXRT
14/CDF-S XT1 (Bauer et al., 2017) and CDF-S XT2 (Xue et al., 2019), and
supernovae events (Types Ia, Ib, Ic, and II; Tsvetkov & Bartunov, 1993;
Galbany et al., 2014). The _solid cyan line_ shows the best-fit local galaxy
main sequence relation from Peng et al. (2010). The _gray background contours_
are the galaxy distribution from the SDSS (data taken from Brinchmann et al.,
2004). The _dashed colored lines_ are the boundaries separating the star-
forming and quiescent galaxies and its evolution with redshift (at $z{=}$0.0,
0.1, 0.2, 0.3, and 0.4, from bottom to top; Moustakas et al., 2013).

FXRT | ID | RA (deg) | DEC (deg) | Offset |  $z$ or $d$(Mpc) | Log(Age/yr) | Log(M${}_{*}/M_{\odot}$) | Log(SFR/($M_{\odot}$/yr)) |  $A_{V}$ (mag) | References
---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
Parameters obtained from the literature |
1 | XRT 000519$\dagger$ | 186.54893 | 12.94622 | 12$\aas@@fstack{\prime}$2 | 16.4 | – | 11.89 | $-2.0$ | 0.081 | 1,15
2 | XRT 010908 | 167.87904 | 55.67411 | 0$\aas@@fstack{\prime}$4 | 9.3 | – | 11.11 | 0.34 | 0.046 | 2,3,13,14
3 | XRT 070530 | 201.36506 | -43.01911 | 5$\aas@@fstack{\prime}$5 | 4.04 | – | 11.0 | 0.0 | 0.315 | 4,5,13,16
4 | XRT 071203 | 211.25671 | 53.66222 | 0$\aas@@fstack{\prime}$4 | 6.9 | – | 8.87 | $-1.15$ | 0.029 | 6,7,13,17
5 | XRT 080331 | 170.06235 | 12.99154 | 1$\aas@@fstack{\prime}$3 | 8.1 | – | 10.80 | 0.51 | 0.091 | 8,9,13,14
6 | XRT 130822 | 345.50403 | 15.96478 | 1$\aas@@fstack{\prime}$2 | 29.3 | – | 10.40 | 0.14 | 0.211 | 10,11,13
14 | XRT 141001 | 53.16158 | -27.85938 | 0$\aas@@fstack{\prime\prime}$13 | 2.23${}_{-1.84}^{+0.98}$ | – | 7.99 | 0.06 | 0.021 | 12,18
Parameters derived from photometric data using BAGPIPES (Carnall et al., 2018) |
8 | XRT 041230 | 318.12653 | -63.49895 | 0$\aas@@fstack{\prime\prime}$7 | 0.61${}_{-0.17}^{+0.13}$ | 9.6${}_{-0.2}^{+0.2}$ | 10.5${}_{-0.2}^{+0.2}$ | $-0.3_{-1.5}^{+1.3}$ | 0.9${}_{-0.7}^{+0.8}$ | –
9 | XRT 080819 | 175.00511 | -31.91749 | 0$\aas@@fstack{\prime\prime}$5 | 0.7${}_{-0.10}^{+0.04}$ | 8.2${}_{-0.1}^{+0.1}$ | 10.4${}_{-0.1}^{+0.1}$ | 2.1${}_{-0.1}^{+0.1}$ | 1.6${}_{-0.1}^{+0.1}$ | –

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 1:_ FXRT candidate number. _Column 2:_ Candidate ID. _Column 3 and 4:_
Right ascension and declination of the host galaxies. _Column 5:_ Angular
offset between the transient and the host galaxy. _Column 6:_ Host galaxy
redshift or distance. _Columns 7, 8, 9:_ Logarithmic values of the age of the
stellar population, the stellar mass, and the SFR from the host galaxies.
_Column 10:_ Dust attenuation. _Column 11:_ Literature references.
References: (1)Rhode et al. (2007), (2)Wiegert et al. (2015), (3)Rhode et al.
(2007), (4)Espada et al. (2019), (5)Rejkuba et al. (2011), (6)Drozdovsky &
Karachentsev (2000), (7)Lanz et al. (2013), (8)Buta et al. (2015), (9)Beuther
et al. (2017), (10)Cappellari et al. (2011), (11)Davis et al. (2014),
(12)Bauer et al. (2017), (13)Helou et al. (1991), (14)Sorce et al. (2014),
(15)Jonker et al. (2013), (16)Crnojević et al. (2016), (17)Tully et al.
(2013), (18)Schlafly & Finkbeiner (2011).
$\dagger$ Assuming the FXRT is associated with galaxy M86. Figure 14: Color-
color diagram of FXRTs 8, 9, and 14 and X-ray sources classified as stars
according to Criterion 2 (see Sect. 2.5.2) with Pan-STARRS, DECam, and 2MASS
counterparts (_gray filled circles_). The expected parameter space of stars
with different ages ($\log(\rm{Age}){=}$7.0-10.3), metallicities (from
$\rm{[Fe/H]}{=}$-3.0–0.5), and attenuations ($A_{V}{=}$0.0–5.0) taken from the
MIST package (Dotter, 2016,Choi et al., 2016) is overplotted as an _orange_
region.

Figure 15: Best fitting SED models obtained using the BAGPIPES package
(Carnall et al., 2018) for FXRTs 8 (_left panels_) and 9 (_right panels_).
_Top panels:_ 16th to 84th percentile range for the posterior spectrum and
photometry (shaded _orange_). The used photometric data and their
uncertainties are given by the _blue markers_. _Bottom panels:_ Posterior
distribution for the five fitted parameters (SFR, age, galaxy stellar mass,
metallicity, and redshift). The 16th, 50th, and 84th percentile posterior
values are indicated by the vertical _dashed black lines_.

##  4 Host galaxy features

The host galaxy or host environment of an FXRT can provide important
information on its nature. Five nearby FXRTs (2, 3, 4, 5 and 6) and three
distant FXRTs (8, 9, and 14/CDF-S XT1) have very probable associated
optical/NIR host galaxy detections. It remains less certain whether FXRT 1 is
associated with M86.

###  4.1 Nearby extragalactic FXRT sample

The nearby events (FXRTs 1, 2, 3, 4, 5, and 6) are located in well-studied
local galaxies, although the association of FXRT 1 with M86 is less clear. We
collect information from the literature in Table 8, although we caution that
this is a heterogeneous data set, and deriving consistent global parameters is
beyond the scope of this work.

Figure 13 compares the host galaxy star-formation rate (SFR) versus stellar
mass (M∗) values of our sample to hosts of other transients (LGRBs, low-
luminosity LGRBs, SGRBs, CC-SNe, thermonuclear SNe, SN 2020bvc, and GW
170817/GRB 170817A). It is clear that the different classes of transients fall
in specific regions of the SFR-M∗ plane. For instance, thermonuclear type Ia
supernovae lie preferentially below the galaxy main sequence (_dashed cyan
line_ ; Peng et al., 2010), that is to say, they are related with older
(redder or recently quenched) stellar populations within galaxies; meanwhile,
CC-SNe (type Ib, Ic, and II; Tsvetkov & Bartunov, 1993; Galbany et al., 2014)
fall closer the galaxy main sequence, highlighting their relation with ongoing
star-formation in galaxies. LGRBs (Li et al., 2016), which are related to
massive progenitor stars, lie above the galaxy main sequence (where so-called
high-SFR starburst galaxies lie). In contrast, the location of SGRBs (Li et
al., 2016) shows a large spread in this figure (i.e., SGRBs occur in a mixed
population of early-type and star-forming galaxies). GW 170817/GRB 170817A (Im
et al., 2017) is singled out among SGRBs, due to the unusually low SFR
(${\approx}$0.001–0.01 M⊙ yr-1) of its host galaxy NGC 4993. We also single
out the off-axis LGRB candidate SN 2020bvc (Chang et al., 2015; Izzo et al.,
2020; Ho et al., 2020), the host galaxy UGC 9379 of which has a low SFR
($\approx$0.08 M⊙ yr-1) but a stellar mass M${}_{*}{\approx}$1.9$\times$1010
M⊙ similar to the Milky Way and other large spirals (Hjorth & Bloom, 2012;
Taggart & Perley, 2021; Ho et al., 2020).

Among the nearby sample, the hosts of FXRTs 2, 3, 4, 5, and 6 fall just below
(within a factor of two) of the local galaxy main sequence, implying they are
probably related to active star formation processes or young stellar
populations. In the cases of FXRTs 2, 3, 4, and 5, there are clear spatial
associations with compact Hii regions or young stellar clusters, strengthening
the link to young, presumably massive stars. FXRT 1/XRT 000519 is probably
related with M86, which has a low SFR (${\approx}$0.01 M⊙ yr-1), as expected
for elliptical galaxies.

###  4.2 Distant extragalactic FXRT sample

The optical/NIR hosts of the distant events FXRTs 8 and 9 are classified as
extended sources (galaxies) by the VHS catalog (McMahon et al., 2013), but
their properties have not been analyzed previously. We used photometric data
of their putative host galaxies to constrain the host properties through
spectral energy distribution (SED) model fitting.

We initially explored the spectral nature of FXRTs 8, 9, and 14 based on their
$i-K_{s}$ versus $g-i$ colors in Fig. 14. FXRTs 8, 9, and 14 were compared to
the counterparts of the X-ray sources classified as stars according to
Criterion 2 (_gray points_ ; see Sect. 2.5.2) and the expected parameter space
for stars (_orange region_) 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
bulk of the stellar X-ray variables form a much tighter sequence than what is
conceivably allowed by the full range of isochrones. The stellar X-ray sources
that appear as outliers are identified as PNe, YSOs (e.g., eruptive variable
stars, T Tauri stars), or emission-line stars. The FXRTs generally lie outside
of or at the edge of the stellar region, away from the tight stellar locus,
although the large error bars or limits in the NIR photometry preclude any
definitive statements here. We conclude that the SED by itself is not a clear-
cut discriminator and thus the spatially resolved nature of the counterparts
remains vital to their confirmation.

Next, we employ the code BAGPIPES (Bayesian Analysis of Galaxies for Physical
Inference and Parameter EStimation; Carnall et al., 2018), which fits stellar-
population models taking star-formation history and the transmission function
of neutral/ionized ISM into account to broadband photometry and spectra using
the MultiNest nested sampling algorithm (Feroz & Hobson, 2008; Feroz et al.,
2009), to derive constraints on the host-galaxy properties. BAGPIPES gives the
posterior distributions for the host-galaxy redshift ($z$), age, extinction by
dust ($A_{V}$), SFR, metallicity ($Z$), stellar mass ($M_{*}$), specific star
formation rate. To account for dust attenuation in the SEDs, we use the
parametrization developed by Calzetti et al. (2000), where $A_{V}$ is a free
parameter within the range 0.0 to 3.0 mag.

We assume an exponentially declining star formation history function
parametrized by the star formation timescale (free parameter). The Table 8
provides the best-fit parameters obtained with BAGPIPES for the hosts of FXRTs
8 and 9, while Fig. 15 shows the 16th to 84th percentile range for the
posterior spectrum and photometry. The posterior distribution for the fitted
parameters is shown in the bottom panels. We have confirmed the obtained
photometric redshifts of 0.61${}_{-0.17}^{+0.13}$ and 0.7${}_{-0.10}^{+0.04}$
for FXRTs 8 and 9 with our 2D spectra taken by X-SHOOTER (PIs: Quirola and
Bauer, program ID: 105.20HY.001). A detailed analysis of the spectral data
will be presented in future work.

FXRT 13 only has a single $i$-band DECam source associated with it. The non-
detections in other bands may suggest that the $i$-band DECam image
(${\approx}$6948–8645 Å) includes a dominant flux contributions from a high
equivalent width emission line. Considering the most important emission lines
of galaxies (such as H$\alpha$, H$\beta$, [OIII]$\lambda\lambda$4959, 5007 Å),
the expected redshift range of FXRT 13 is $z{\approx}$0.2–1.1.

Finally, Bauer et al. (2017) associated FXRT 14 (CDF-S XT1) with an extremely
faint, small $z_{\rm photo}{\sim}2.23$ host galaxy with a relatively flat SED
(see Tables 3 and 8).

Among the FXRTs identified as distant candidates, FXRTs 8 and 14 are located
above the galaxy main sequence, while FXRT 9 lies significantly below it. We
also show the host of CDF-S XT2, which also lies just above the galaxy main
sequence. Thus, a sizable fraction of distant FXRTs appear to be associated
with vigorous star formation; we should stress here, however, that the
statistics are poor and the uncertainties from the SED model fits remain large
(see Table 8).

Figure 16: 0.3–10 keV light curves of the five local CSC2 FXRTs, plus FXRTs 1
and 11, in luminosity units. The 0.3–10 keV light curves are obtained by
multiplying the 0.3–7 keV light curves by the factor derived from
extrapolating the best-fit PO model flux to the 0.5–7.0 keV spectrum to the
0.3–10 keV band and correcting it for the effects of Galactic plus intrinsic
absorption. For comparison, we overplot flaring episodes for several
individual well-known Galactic XRBs: GX339-4 (9 kpc, _green line_ ; Heida et
al., 2017), _Swift_ J1357.2-0933 (8 kpc, _magenta line_ ; Mata Sánchez et al.,
2015), _MAXI_ J1543-564 (5 kpc, _gray line_ ; Stiele et al., 2012), and _MAXI_
J1659-152 (6 kpc, _blue line_ ; Jonker et al., 2012b). The light curves of the
comparison sources are taken from the 2SXPS catalog (Evans et al., 2020b).
Figure 17: 0.3–10 keV light curves of the nine CSC2 FXRTs in luminosity units
(as in Fig. 16, 0.3–10 keV light curves were converted from 0.5–7 keV ones).
The X-ray afterglow light curves of 64 LGRBs plus 32 SGRBs (taken from
Bernardini et al., 2012; Lü et al., 2015) are shown as a 2D histogram, as are
the X-ray afterglows of GRB 170817A (off-axis SGRB, _solid dark green line_ ;
Nynka et al., 2018; D’Avanzo et al., 2018; Troja et al., 2020, 2022), SN
2020bvc (the first off-axis LGRB candidate, _solid light green line_ ; Izzo et
al., 2020), and the ultra-long duration GRB 111209A (_solid magenta line_ ,
$z{=}0.677$; Levan et al., 2014). Additionally, several individual transients
are overplotted: the low-luminosity supernova SBO XRF 080109/SN 2008D (_solid
cyan lines_ , 27 Mpc;), XRF 060218/SN 2006aj (_solid blue lines_ , 145 Mpc;),
XRF 100316D/SN 2010bh (_solid orange lines_ ,263 Mpc; Barniol Duran et al.,
2015; Starling et al., 2011; Modjaz et al., 2009; Evans et al., 2009;
Soderberg et al., 2008; Evans et al., 2007; Campana et al., 2006), the
relativistically beamed TDE _Swift_ J1644+57 (_solid black lines_ ,
$z{=}0.3543$; Bloom et al., 2011; Levan et al., 2011), the non-beamed TDE
J2150-05 (_solid pink line_ , $z{=}0.055$; Lin et al., 2018), and CDF-S XT2
(_solid indigo lines_ ; Xue et al., 2019). For FXRTs 1, 7, 10, 11, 12, and 13
(_open symbols_), we assume $z{=}$1.0, we adopt $z_{\rm photo}{=}$2.23 for
FXRT 14 from Bauer et al. (2017), and for FXRTs 8 and 9 we consider the values
from Table 8.

##  5 Possible interpretations

To understand the origin of our sample of FXRTs, we compare them with other
well-known transients. We split our discussion here into nearby (Sect. 5.1)
and distant (Sect. 5.2) samples. The former have well-established distances,
and therefore we can compare their light curves in luminosity units. As we do
not know the redshift of several distant FXRTs, we compare their X-ray light
curves in luminosity units assuming nominal distances. Given the uncertainty
in the associations for FXRTs 1 and 11, we discuss them under both the nearby
and distant extragalactic scenarios.

First, from the best-fit PL spectral model, we compute the X-ray peak flux
(corrected for Galactic and intrinsic absorption; $F_{\rm peak}$)222222Due to
the lack of a standardized method to estimate the $F_{\rm peak}$, we consider
the following. First, we find the shortest time interval during which 25% of
the counts are detected, and we compute a 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)., the
associated intrinsic X-ray peak luminosity ($L_{\rm X,peak}$), and the
Eddington mass (defined as $M_{\rm Edd}{=}7.7{\times}10^{-39}L_{\rm X,peak}$
in solar mass units). We report these values in Table 9 in the energy range
0.3–10.0 keV.

###  5.1 Nearby extragalactic FXRT sample

The nearby FXRTs 2, 3, 4, 5, and 6 have peak isotropic X-ray luminosities in
the range of $L_{\rm X,peak}{\approx}$1038–1040 erg s-1 (see Table 9). This
appears inconsistent with origins as SBOs ($L_{\rm X,peak}^{\rm
SBOs}{\approx}$1042–1047 erg s-1; Ensman & Burrows, 1992; Soderberg et al.,
2008; Modjaz et al., 2009; Waxman & Katz, 2017; Alp & Larsson, 2020), TDEs
($L_{\rm X,peak}^{\rm TDEs}{\approx}$1042–1050 erg s-1; Rees, 1988; MacLeod et
al., 2014; Maguire et al., 2020; Saxton et al., 2021, considering a jetted
emission), or on-axis GRBs ($L_{\rm X,peak}^{\rm GRBs}{\approx}$1047–1051 erg
s-1; Berger, 2014; Bauer et al., 2017, considering a jetted emission).

These lower luminosities fall into the realm of ULXs (extragalactic X-ray
emitters located off-center of their host galaxy and with luminosities in
excess of $L_{X}^{\rm ULX}{\approx}$1039 erg s-1, if the emission is
isotropic, well above the Eddington limit for neutron stars; Bachetti et al.,
2014; Kaaret et al., 2017) and Galactic XRBs (X-ray emitters where a compact
object accretes mass from a companion star with $L_{X}^{\rm
XRB}{\lesssim}$1039 erg s-1; Remillard & McClintock, 2006; van den Eijnden et
al., 2018). Most ULXs are semi-persistent X-ray emitters for years to decades
(Kaaret et al., 2017), and in extreme cases can reach high luminosities such
as NGC 5907 ULX1 (${\approx}$5$\times$1040 erg s-1; Walton et al., 2016). The
much shorter and stronger variability of our FXRTs compared to ULXs implies
that they are caused by a different phenomenon.

Another alternative could be XRBs. Figure 16 shows the X-ray light curves of
FXRTs 1, 2, 3, 4, 5, 6, and 11 one per panel, compared to several well-known
XRB flaring episodes. XRBs in the Milky Way exhibit pronounced variability
whereby the X-ray flux changes from quiescent to flare states on timescales of
weeks to months (Remillard & McClintock, 2006). Particularly, FXRTs 2 and 4
reach peak luminosities ($L_{\rm X,peak}{\approx}$ 1038–1039 erg s-1) similar
to some XRBs’ flares (e.g., the flare luminosity of GX339-4 of
${\approx}$4$\times$1038 erg s-1), which suggests that these FXRTs could be
related with the tip of longer flares. Nevertheless, they are not in agreement
with the duration (in the order of weeks) and timescale evolution (following a
slow PL decay $F_{X}{\propto}t^{-0.3}$) of XRBs flares. Thus, FXRTs 2, 3, 4
are unlikely to be related with XRBs. Meanwhile, FXRTs 1, 3, 5, 6, and 11 are
not related with XRBs because of their high luminosity ($L_{\rm
X,peak}{\gtrsim}$ 1039 erg s-1).

Next, we compare the FXRTs to SGRs and AXPs, which are both believed to be
related to young, highly magnetic neutron stars (Woods & Thompson, 2006). Soft
gamma repeaters and AXPs are very faint in quiescence but can flare by factors
of hundreds to thousands on timescales of tens of ms to seconds (Göğüş et al.,
1999; Aptekar et al., 2001) and on very rare occasions can generate giant
flares by factors of $10^{5}$ over several-minute timescales (e.g., SGR
1806-20, SGR 1900+14; Hurley et al., 1999; Terasawa et al., 2005; Woods &
Thompson, 2006). On such occasions they can reach X-ray luminosities as high
as $10^{40}$–$10^{41}$ ergs s-1, although SGR 1900+14 experienced a giant
flare with a peak luminosity of ${>}$1044 erg s-1 (at hard X-rays 40–700 keV;
Mazets et al., 1999; Feroci et al., 2001), and generate bolometric outputs up
to ${\sim}10^{46}$ erg in total (Palmer et al., 2005; Terasawa et al., 2005;
Strohmayer & Watts, 2005; Israel et al., 2005). The weaker flares generally
have a soft spectrum, while the giant flares are quite hard, and the flare
durations follow a log-normal distribution (Göğüş et al., 1999). At the
distances of our nearby FXRTs, we would presumably only see the most luminous
portions of these rare giant bursts (a few seconds at most) and not be
sensitive to the fainter bursts or quiescent emission. They should be quite
spectrally hard (e.g., SGR 1900+14 has a photon index range
$\Gamma{\approx}$1.0–2.0; Tamba et al., 2019) and given the relation to young,
highly magnetic neutron stars, seen to be originating from young star clusters
and Hii regions (Woods & Thompson, 2006). In this sense, FXRTs 2, 4, 5, and 6
share some similarities with the SGR and AXP phenomena. For instance, they
seem related to star-formation galaxies (see Fig. 13), although their spectra
remain relatively soft and their light curve lengths last thousands of
seconds. In the case of FXRT 5, we see multiple flares over ${\approx}$35 ks.
On the other hand, FXRTs 1, 3, and 11 are not associated with young star
clusters, and thus seem far less likely to be explained by an SGR/AXP origin.

A final point of comparison is with the FXRTs discovered in NGC 4636 and NGC
5128 (Cen A) by Irwin et al. (2016) and NGC 4627 by Sivakoff et al. (2005).
All exhibit rapid ($\sim$50–100 s) flares with peak luminosities of
${\sim}$1039–1041 erg s-1, but remain detectable by _Chandra_ in quiescence.
In two cases, multiple flares are observed across multiple observations, while
two transients are spatially associated with globular clusters in their host
galaxies (similar to FXRT 3, which intriguingly is also associated with NGC
5128). Overall, while the luminosities are comparable, the faster timescales,
multiple outbursts, and quiescent detections are unlike the behavior seen
among the nearby sample of FXRTs, although it could be the case that (some)
FXRTs have quiescent fluxes well below the sensitivity of _Chandra_ and XMM-
Newton and/or have not been observed frequently enough to see multiple
outbursts (e.g., FXRT 6 has not been observed again by _Chandra_ or _XMM-
Newton_ ; see Fig. 6).

In the case of FXRTs 1 and 11, their origin remains unclear. For FXRT 1,
assuming the association with M86 (${\approx}$16.4 Mpc; Jonker et al., 2013),
it is characterized by a peak luminosity of ${\approx}$6$\times$1042 erg s-1
(see Table 9). According Jonker et al. (2013), this X-ray flash (XRF) could
have been caused by the disruption of a compact WD by a 4.9$\times$104
$M_{\odot}$ BH. Nevertheless, other scenarios such as a highly off-axis GRB
(Dado & Dar, 2019) cannot be discarded because of distance uncertainties. For
FXRT 11, assuming the association with the galaxy cluster Abell 3581
(${\approx}$94.9 Mpc; Glennie et al., 2015; Johnstone et al., 2005), its peak
luminosity and Eddington mass are ${\approx}$2$\times$1044 erg s-1 and
${\approx}$1.9${\times}$106 $M_{\odot}$ (see Table 9). Glennie et al. (2015)
suggest that FXRT 11 could be consistent with the early X-ray emission
typically seen in GRB light curves; however, its similarities with FXRT 1 also
suggest that both events share the same origin.

Overall, the behavior of the nearby FXRTs appears to represent a genuinely new
phase space of transient phenomena. The wide variety of observed properties
strongly suggests that multiple physical origins may be at work.

###  5.2 Distant extragalactic FXRT sample

For the moment, we split the discussion of FXRTs into those with fairly secure
hosts and reasonable distance estimates (see Sect. 5.2.1) versus those with
less certain or no clear hosts (see Sect. 5.2.2). Here we analyze the scenario
where the FXRTs 1 and 11 are related to distant extragalactic objects (see
Sect. 5.2.2).

####  5.2.1 FXRTs with known distances

Using the photometric host redshifts calculated in Sect. 4, FXRTs 8 and 9
reach peak X-ray luminosities of $L_{\rm X,peak}{\approx}$1.5$\times$1044 and
1.3$\times$1045 erg s-1, respectively (see Table 9 and Fig. 17 for a light
curve comparison). The FXRTs have an isotropic fluence of
$E_{X}{\approx}$2.2${\times}$1047 and 3.9${\times}$1047 erg, respectively.

Such luminosities fall with the ranges predicted or detected for SBOs ($L_{\rm
X,peak}^{\rm SBOs}{\approx}\times$1042–1047 erg s-1; Soderberg et al., 2008;
Modjaz et al., 2009; Waxman & Katz, 2017; Alp & Larsson, 2020), although both
FXRTs exhibit energy released that are at least one to two orders of magnitude
higher than the energy predicted by SBO models (e.g., Waxman & Katz, 2017) or
detected from the enigmatic SBO XRT 080109/SN 2008D
($E_{X}{\approx}2\times$1046 erg; Soderberg et al., 2008). As such, we rule
out an SBO interpretation for FXRTs 8 and 9.

Considering an on-axis GRB origin, we note that no gamma-ray signals detected
near the time of discovery were associated with FXRTs 8 or 9, and neither
exhibits a characteristic PL decay phase ($F_{X}^{\rm
FXRTs8/9}{\propto}t^{-2.9/-2.8}$ associated with GRB afterglows $F_{X}^{\rm
GRBs}{\propto}t^{-1.2}$; Evans et al., 2009; Racusin et al., 2009), although
some GRBs show strong X-ray flaring in the tail of the X-ray afterglow
distribution that could mimic the observed temporal behavior (Barthelmy et
al., 2005; Campana et al., 2006; Chincarini et al., 2010; Margutti et al.,
2011). Critically, Fig. 17 demonstrates that the X-ray light curves of both
FXRTs are fainter than almost any known on-axis GRB X-ray afterglow over the
same timescale, with (prompt) initial luminosities $>$3–4 dex below the
luminosity ranges observed for GRBs ($L_{\rm X,peak}^{\rm GRBs}{\gtrsim}$1047
erg s-1). Based on their best-fit X-ray spectral slopes of $\Gamma_{\rm
FXRTs\leavevmode\nobreak\ 8/9}{\approx}$2.7/3.0, they formally lie at the edge
of the standard afterglow distribution ($\Gamma_{\rm GRBs}{=}$1.5–3.0; Berger,
2014; Wang et al., 2015; Bauer et al., 2017) overlapping at the 1$\sigma$
confidence level. In terms of their host-galaxy properties (see Fig 13 and
Table 8), FXRT 8’s host has a low-SFR ($\approx$0.5 $M_{\odot}$ yr-1) and old
stellar population ($\gtrsim$1 Gyr). It is classified as a quiescent galaxy
according to the criteria from Moustakas et al. (2013), and is thus a
potential host for an SGRB. Nevertheless, an association with low-luminosity
LGRBs (LL-LGRBs) could be discarded due to the high stellar mass of its host
galaxy.

FXRT 9’s galaxy is a massive blue starburst galaxy ($\approx$120 $M_{\odot}$
yr-1) with a young stellar population (${\approx}$0.15 Gyr). Hence, FXRT 9
might be related to an LGRB origin, although an association with SGRBs cannot
be discarded. On the other hand, an association with LL-LGRBs could be ruled
out because of the high stellar mass of its host galaxy relative to LL-LGRBs’
hosts (higher than one order of magnitude; see Fig 13). Unfortunately, the low
angular resolution of the current archival images does not permit us to
compute the offset from the host center.

Alternatively, these FXRTs could be related to ultra-long duration GRBs.
Several ultra-long GRBs (longer than thousands of seconds) have been detected
(Thöne et al., 2011; Campana et al., 2011; Gendre et al., 2013; Virgili et
al., 2013; Stratta et al., 2013). Their nature still unclear. Gendre et al.
(2013) and Levan et al. (2014) argued that ultra-long duration GRBs form
another distinct group of GRBs; for example, Levan et al. (2014) argue that
the long duration of this population of GRBs may be explained by engine driven
explosions of stars of much larger radii than typical LGRB progenitors (which
are thought to have compact Wolf-Rayet progenitor stars). Figure 17 shows a
comparison of both FXRTs and the ultra-long GRB 111209A. At early times their
luminosities are ${\approx}$6–7 dex lower than that of GRB 111209A.
Nevertheless, we cannot discard an association with this population of GRBs
because of the uncertainty in the zero point of our FXRTs, which when changed
could match well with the temporal decay ($F_{X}{\propto}t^{-1.4/-5.3}$) and
spectral trend ($\Gamma{\approx}2.4$ at $t{>}$40 ks) of this GRB.

The possibility of an off-axis orphan GRB origin still remains plausible,
given the lack of an initial gamma-ray detection and lower luminosity. Here we
compare to the light curves of XRF 060218/SN 2006aj (Campana et al., 2006),
XRF 100316D/SN 2010bh (Starling et al., 2011), and SN 2020bvc (Izzo et al.,
2020), which have all been argued to be potential off-axis LGRBs, as well as
GRB 170817A (Nynka et al., 2018; D’Avanzo et al., 2018; Troja et al., 2020)
and CDF-S XT2 (Xue et al., 2019), and thus possible off-axis SGRBs (see Fig.
17). We note in particular that the plateau phases of FXRTs 8 and 9 are
${\approx}$ 1–3 dex lower than those of XRF 060218, XRF 100316D, and CDF-S
XT2, although the break and late-time light curves (to the extent that they
can be quantified) appear to match reasonably well. By extension, SN 2020bvc
and GRB 170817A appear to be even weaker, and join with the faint declining
tails of the XRFs at very late times. We speculate that perhaps FXRTs 8 and 9
could be weaker or higher inclination versions of off-axis SGRB and LGRBs
(e.g., Granot et al., 2002), respectively, somewhere intermediate between the
XRFs and SN 2020bvc/GRB 170817A along the possible viewable parameter space of
such events. Unfortunately, the poor count statistics (to constrain any
spectral evolution) and the lack of additional EM counterparts do not permit
us to analyze this picture in detail.

Finally, in the TDE scenario, if we interpret the peak luminosities as the
Eddington luminosity, we derive masses of ${\gtrsim}$1.2$\times$106 and
1.0$\times$107 M⊙ for FXRT 8 and FXRT 9, respectively. These masses fall in
the supermassive black hole (SMBH) range (Barack et al., 2019), and assuming
that a large fraction of the total stellar mass of the host galaxies as
derived in Sect. 4.2 is associated with a spheroid component, could be
approximately consistent with the stellar velocity dispersion ($\sigma$) of a
galaxy bulge and the mass of the SMBH ($M_{\rm BH}$) at its center ($M_{\rm
BH}-\sigma$ relation; e.g., Ferrarese & Merritt, 2000). These luminosities are
in rough agreement with the recent sample of TDEs published by Sazonov et al.
(2021).

Alternatively, these FXRTs could be related with an IMBH–WD or IMBH–MS TDEs
(which could occur in dwarf galaxies and stellar systems such as globular
clusters; Jonker et al., 2012a; Reines et al., 2013), assuming the observed
luminosities are super-Eddington or due to relativistic beaming. The FXRTs are
offset from the nuclei of their associated optical and NIR sources by only
0$\aas@@fstack{\prime\prime}$5 and 0$\aas@@fstack{\prime\prime}$7 (or
projected physical distances of 3 and 3.5 kpc), respectively, and hence remain
consistent with both on-axis and off-axis scenarios within the positional
uncertainties (see Fig. 7).

Saxton et al. (2021) review the observed and theoretical X-ray properties of
TDE candidates. Among confirmed SMBH–MS TDEs detected to date, several exhibit
peak luminosities similar to those of FXRTs 8 and 9. However, the X-ray
spectra of SMBH–MS TDEs are generally softer and none exhibit short-term X-ray
variability comparable to what see from the FXRTs, but instead show much
slower declines over timescales of months to years. For this reason, we
disfavor such an explanation, but cannot completely rule out a possible
detection bias here, given the limited sensitivity of current all-sky
instruments. One intriguing possibility for generating higher luminosities,
faster variability, and harder spectra is relativistic beaming from jetted
TDEs such as _Swift_ J1644+57 (Bloom et al., 2011; Levan et al., 2011). This
could also significantly relax the mass and/or accretion rate limits quoted
above. In the case of _Swift_ J1644+57, shown in Fig. 17, it has a peak
luminosity of ${\approx}$1048 erg s-1 and time-averaged photon index of
$\Gamma{=}$1.6–1.8 (Levan et al., 2011), although the photon index increases
and softens with decreasing flux (Bloom et al., 2011). Clearly FXRTs 8 and 9
remain $\sim$3 dex fainter, but otherwise have potentially consistent spectral
and temporal properties. As neither has multiple X-ray observations, we cannot
say anything about their long-term evolution. We can also compare the timing
and spectral properties of the off-nuclear ultrasoft hyper-luminous 3XMM
J215022.4-055108, an IMBH TDE candidate (hereafter TDE J2150-05; Lin et al.,
2018, 2020). TDE J2150-05 shows a peak luminosity of ${\approx}$1$\times$1043
erg s-1, a light curve PL decay of $F_{X}{\propto}t^{-5/3}$ during
${\gtrsim}$14 yr (see Fig.17), and ultrasoft X-ray spectra with
$kT{\lesssim}$0.25 keV, which soften with time (Lin et al., 2018, 2020). This
lies in stark contrast with FXRTs 8 and 9, which show a short and fast
timescale variability, and somewhat hotter/harder X-ray spectra. In summary,
FXRTs 8 and 9 do not conform to the ”traditional” expectations of TDEs, in
terms of slow temporal evolution or ultrasoft X-ray spectra, but
relativistically beamed emission from an IMBH-TDE scenario cannot be
discarded.

Unlike the other events, FXRT 14 has been constrained by multiwavelength
counterparts (Bauer et al., 2017). The available data are consistent with
expectations for off-axis SGRBs, although other possibilities might not be
ruled out. For instance, Peng et al. (2019) argue for an IMBH–WD TDE, Sun et
al. (2019) explain the X-ray emission considering a magnetar remnant after a
BNS merger observer at an off-axis viewing angle, while Sarin et al. (2021)
discuss an association with an off-axis afterglow of a BNS merger, without
discarding that its X-ray properties could be related to compact object such
as an asteroid hitting an isolated foreground neutron star (Colgate &
Petschek, 1981; van Buren, 1981; Campana et al., 2011). It is important to
mention that FXRT 14/CDF-S XT1 and XT2 seem to fall in the same host’s
properties parameter space as the LL-LGRBs and SGRBs at lower stellar masses
($\lesssim$109 $M_{\odot}$; see Fig. 13). This reinforces the likely
association with SGRBs.

####  5.2.2 FXRTs with unknown distances

FXRTs 7, 10, 12, and 13 do not have clear host associations as yet, and hence
have wildly uncertain distances. Based on their typical optical and NIR upper
limits (e.g., $m_{r}{\gtrsim}$23.3 and $m_{z}{\gtrsim}$22 AB mag), and
considering distances of other FXRT host galaxies such as FXRTs 8 and 9
($z_{\rm photo/spec}{\sim}0.7$), FXRT 14 ($m_{R}^{\rm FXRT\leavevmode\nobreak\
14}{=}$27.5 AB mag and $z_{\rm photo}^{\rm FXRT\leavevmode\nobreak\
14}{=}$0.39–3.21; Bauer et al., 2017), and CDF-S XT2 ($m_{\rm F606W}^{\rm
XT2}{=}$25.35 AB mag and $z_{\rm spec}^{\rm XT2}{=}0.738$; Xue et al., 2019),
we adopt a nominal redshift of $z{=}$1 for these sources. Figure 17 (_open
markers_) compares FXRTs 7, 10, 12 and 13 (at $z{=}$1.0) to several classes of
transients.

We note that FXRTs 7, 10, and 12 have light curves that exhibit plateau phases
of ${\approx}$1–3 ks, followed by PL decays ($F_{X}{\propto}t^{-2.4/-1.6}$)
that are accompanied by possible softening of the spectra for FXRTs 7 and 12
(see Table 6). Spectral softening has been seen previously in SBOs (e.g., XRF
080109/SN 2008D), GRBs afterglows, TDEs (e.g., MacLeod et al., 2014; Malyali
et al., 2019), and CDF-S XT2 (Xue et al., 2019). FXRTs 7, 10 and 12 have
photon indices (see Table 5) similar to the SBO XRF 080109/SN 2008D
($\Gamma{\approx}2.3$; Soderberg et al., 2008) and GRB afterglows
($\Gamma{\approx}$1.5–3.0; Berger, 2014; Wang et al., 2015) at a 1$\sigma$
confidence level. If these events lie at $z{\gtrsim}$0.5, we can discard the
SBO scenario, however, due to their high X-ray luminosities ($L_{\rm
X,peak}{\gtrsim}$1044 erg s-1); an SBO association would only be expected at
low redshift ($z{\lesssim}$0.5). The light curves (at $z{=}1.0$) also appear
inconsistent with on-axis GRBs. Although they share similar luminosities and
PL decays beyond ${\sim}10^{3}$ s, the early plateau phases of FXRTs are
inconsistent with the typical PL or BPL decays of on-axis GRBs and afterglows.
A subset of SGRBs exhibit plateau phases (Rowlinson et al., 2010, 2013),
although these generally have plateau luminosities ${\gtrsim}$1046 erg s-1
(although if no redshift is known the mean SGRB redshift is assumed,
$z{\sim}$0.72; energy band 0.3–10 keV), which are inconsistent with FXRTs 10
and 12 lying at $z{\lesssim}2.1$. An off-axis GRB afterglow scenario seems
unlikely. To observe luminosities similar to SN 2020bvc ($L_{\rm
X,max}{\approx}$1.8${\times}$1041 erg s-1) and GRB 170817 ($L_{\rm
X,max}{\approx}$4${\times}$1039 erg s-1), our sources must be at low
redshifts, $z{\lesssim}$0.1, which could be discarded by the non-detection of
hosts. Furthermore, Fig. 17 shows a comparison of these FXRTs with the ultra-
long GRB 111209A. Assuming $z{=}1.0$, at early times their luminosities are
orders of magnitude lower than GRB 111209A.

On the other hand, the luminosities and light curve shapes of FXRTs 7, 10, and
12 share remarkable similarities to X-ray flashes XRF 060218/SN 2006aj and XRF
100316D/SN 2010bh (which may be related to shock breakout from choked GRB
jets; Campana et al., 2006; Bromberg et al., 2012; Nakar & Sari, 2012), as
well as CDF-S XT2 (which is consistent with being powered by a millisecond
magnetar; Xue et al., 2019; Sun et al., 2019). The light curves of FXRTs 7,
10, and 12 follow the expected shape for IMBH–WD TDEs (e.g., see MacLeod et
al., 2014; Malyali et al., 2019). For instance, the photon index and flux PL
decay of these FXRTs are similar to the IMBH TDE candidate TDE J2150-05
($\Gamma{\lesssim}$4.8 and $F_{X}{\propto}t^{-5/3}$; Lin et al., 2018).
Assuming $z{=}$1, only FXRT 7 reaches a luminosity close to the beamed TDE
_Swift_ J1644+57 ($L_{\rm X,peak}{\approx}$1046–1047 erg s-1; see Fig. 17;
Bloom et al., 2011; Levan et al., 2011), but without flaring episodes. Again,
the poor count statistics (to constrain any spectral evolution) and the lack
of host or additional EM counterparts do not permit us to analyze this picture
in detail.

FXRT 13 exhibits a single PL light curve with a slow decay
($F_{X}{\propto}t^{-0.2}$). This seems to exclude a SBO nature for this FXRT.
There is a faint optical source likely associated with this FXRT, only visible
in $i$-band DECam images ($m_{i}{\approx}$24.7 AB mag), which does not
constrain its origin significantly.

Finally, assuming FXRTs 1 and 11 are actually background objects that randomly
overlap with nearby sources, we find that their light curves remain unique.
Given the uncertainties in their distances, we adopt nominal redshifts of
$z{=}$1 as above (see Fig. 17). Their X-ray luminosities of reach values
$L_{\rm X,peak}{\approx}$1047 and 5${\times}$1047 erg s-1, respectively,
ruling out an association with SBOs but falling in the range of XRFs (e.g.,
XRF 060218/SN 2006aj and XRF 100316D/SN 2010bh; Campana et al., 2006; Bromberg
et al., 2012; Nakar & Sari, 2012) and beamed TDEs (e.g., TDE J1644+57; see
Fig. 17). The duration and shapes do not appear consistent with XRFs, but do
resemble individual flares seen from TDE J1644+57.

Unfortunately, the unknown distances of these FXRTs do not permit better
constraints on their origin.

FXRT | ID |  $F_{\rm peak}$(erg cm-2 s-1) |  $L_{\rm X,peak}$(erg s-1) |  $M_{\rm Edd}$($M_{\odot}$)
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
Nearby sample
1 | XRT 000519†† | (1.9$\pm$0.1)${\times}$10-10 | (6.1${\pm}$0.3)${\times}$1042 | (4.9${\pm}$0.3)${\times}$104
2 | XRT 010908†† | (1.7$\pm$0.5)${\times}$10-13 | (1.8$\pm$0.5)${\times}$1039 | 13.9$\pm$4.1
3 | XRT 070530†† | (2.7$\pm$1.1)${\times}$10-12 | (5.3$\pm$2.1)${\times}$1039 | 41.8$\pm$17.0
4 | XRT 071203†† | (6.4$\pm$2.5)${\times}$10-14 | (3.6$\pm$1.4)${\times}$1038 | 2.9$\pm$1.1
5 | XRT 080331†† | (1.7$\pm$0.3)${\times}$10-12 | (1.3$\pm$0.2)${\times}$1040 | 105.9$\pm$18.7
6 | XRT 130822†† | (2.3$\pm$0.9)${\times}$10-13 | (2.4$\pm$0.9)${\times}$1040 | 187.5$\pm$73.4
Distant sample
7 | XRT 030511† | (2.3$\pm$0.3)${\times}$10-12 | (1.3$\pm$0.2)${\times}$1046 | (9.9${\pm}$1.3)${\times}$107
8 | XRT 041230†† | (6.9$\pm$3.4)${\times}$10-14 | (1.1$\pm$0.5)${\times}$1044 | (8.8$\pm$4.3)${\times}$105
9 | XRT 080819†† | (6.5$\pm$2.9)${\times}$10-13 | (1.5$\pm$0.7)${\times}$1045 | (1.2$\pm$0.5)${\times}$107
10 | XRT 100831† | (8.9$\pm$3.4)${\times}$10-13 | (4.8$\pm$1.8)${\times}$1045 | (3.8${\pm}$1.5)${\times}$107
11 | XRT 110103a | (2.2$\pm$0.2)${\times}$10-10 | (2.4$\pm$0.2)${\times}$1044 | (1.9$\pm$0.2)${\times}$106
12 | XRT 110919† | (5.6$\pm$1.3)${\times}$10-13 | (3.0$\pm$0.7)${\times}$1045 | (2.4${\pm}$0.6)${\times}$107
13 | XRT 140327† | (1.2$\pm$0.5)${\times}$10-13 | (6.3$\pm$2.8)${\times}$1044 | (4.9${\pm}$2.3)${\times}$106
14 | XRT 141001b | (4.3$\pm$1.1)${\times}$10-12 | (1.7$\pm$0.4)${\times}$1047 | (1.3$\pm$0.3)${\times}$109

Table 9: Energetics of the FXRT sample (fluxes are corrected for Galactic and
intrinsic absorption, and calculated over the energy range 0.3–10 keV).
_Column 1:_ FXRT candidate number. _Column 2:_ Candidate ID. _Column 3 and 4:_
X-ray peak flux and isotropic luminosity in cgs units (corrected for Galactic
and intrinsic absorption). _Column 5:_ Eddington mass (defined as $M_{\rm
Edd}{=}7.7{\times}10^{-39}L_{\rm X,peak}$) in solar mass units ($M_{\odot}$).
a Assuming an association with Abell 3581 at 94.9 Mpc (Glennie et al., 2015).
b Assuming a mean redshift of $z{=}2.23$ (Bauer et al., 2017).
† Assuming a redshift of $z{=}1$.
†† The distance or redshift is taken from Table 8.

##  6 Rates

We computed the event rates of FXRTs and compared them with those for other
transients to explore possible associations and interpretations. We derived
the event rates (deg-2 yr-1; Sect. 6.1), the volumetric rate for nearby and
distant samples (yr-1 Gpc-3; Sect. 6.2), the local density rate (Sect. 6.3),
and the expected number of events for current and future X-ray missions (Sect.
6.4).

###  6.1 Event-rate estimation

We found 14 FXRTs (including XRT 000519, XRT 110103 and CDF-S XT1; Jonker et
al., 2013; Glennie et al., 2015; Bauer et al., 2017) within 160.96 Ms of CSC2
data. For a set of _Chandra_ observations, the number of transients can be
written as

$\mathcal{N}=\sum_{i}\mathcal{R}_{i}\varepsilon_{i}\Omega_{i}t_{i},$ (5)

where $\mathcal{R}_{i}$ is the event rate, $\Omega_{i}$ and $t_{i}$ are the
FoV and exposure time, respectively, and $\varepsilon_{i}$ is an area
correction factor, with the subscript $i$ denoting each _Chandra_ observation.

The area correction factor, $\varepsilon_{i}$, is important for the faintest
FXRTs and captures the changes in sensitivity over the _Chandra_ detector.
$\varepsilon_{i}$ is defined as the area within which we expect successful
FXRT detections (S/N ratio$\gtrsim$3.0) normalized by the total detection
area. To determine $\varepsilon_{i}$, we simulate 1,000 fake instances of each
FXRT, randomly distributed in position (using MARX and simulate_psf scripts
taking into account the particular features per _Chandra_ observation) within
_Chandra_ ’s FoV for each individual observation. We compute the S/N ratio for
fake FXRTs in the energy range of 0.3–10 keV. Thus, $\varepsilon_{i}$ falls in
the range $\varepsilon_{i}{\in}[0.0,1.0]$. For the brightest FXRTs,
$\varepsilon_{i}{\approx}$1.0, meaning that they are detectable across the
entire detector FoV, while for fainter FXRTs, $\varepsilon_{i}{\lesssim}$1.0,
such that only a portion of the detector is sensitive to them.

We assume that $\mathcal{R}_{i}$ is constant (such that
$\mathcal{R}_{i}{=}\mathcal{R}$), because the universe is isotropic on large
scales and we are focusing on extragalactic sources (Yang et al., 2019).
$\Omega_{i}$ depends on which chips of the detector are turned on; due to the
degradation of the PSF at higher instrumental off-axis angles, we consider
only chips I0–I3 for ACIS-I and chips S1–S4 for ACIS-S, respectively.
Therefore, the expected number of events depends on $\Omega_{i}$, $t_{i}$, and
$\varepsilon_{i}$ per observation as

$\mathcal{N}=\mathcal{R}\sum_{i}\varepsilon_{i}\Omega_{i}t_{i},$ (6)

such that the event rate, $\mathcal{R}$, is

$\mathcal{R}=\frac{\mathcal{N}}{\sum_{i}\varepsilon_{i}\Omega_{i}t_{i}}.$ (7)

We derive the rate of our sample considering two cases: $(i)$ five nearby
events (seven if we include FXRT 1/XRT 000519 and FXRT 11/XRT 110103, which
have unclear associations with M86 and the galaxy cluster Abell 3581,
respectively; called _Case I_), and $(ii)$ seven distant events (nine if we
include FXRT 1/XRT 000519 and FXRT 11/XRT 110103; called _Case II_). Because
our algorithm does not have good efficiency in detecting objects in
observations with exposure times ${<}$8 ks (in fact, we do not detect any
candidates for such exposures), we do not consider such observations to derive
the rates. Another consideration when estimating the event rates for both FXRT
samples is to identify and isolate the fraction of observations that target
nearby galaxies. While distant FXRTs can be detected in any _Chandra_
observation (i.e., in the background of nearby galaxy observations), nearby
FXRTs can only be detected if nearby galaxies lie within the _Chandra_ FoV.
Thus for _Case II_ , we consider just _Chandra_ observations that target non-
nearby galaxies, while for _Case I_ , we only consider the fraction of
_Chandra_ observations that target nearby galaxies at ${<}$100 Mpc
(${\approx}$21% of the total sample; see Sect. 3.5).

Therefore, we estimate the event-rates (fully accounting for the ambiguity of
FXRTs 1 and 11 in the errors) of nearby FXRTs to be $\mathcal{R}_{\rm
Case\leavevmode\nobreak\ I}{=}$53.7${}_{-15.1}^{+22.6}$ deg-2 yr-1; while for
distant FXRTs it is $\mathcal{R}_{\rm Case\leavevmode\nobreak\
II}{=}$28.2${}_{-6.9}^{+9.8}$ deg-2 yr-1 (for $F_{\rm
X,peak}{\gtrsim}$1$\times$10-13 erg cm-2 s-1). The distant rate is consistent
with the rate of $\mathcal{R}_{\rm Yang+19}{\approx}$59${}_{-38}^{+77}$ deg-2
yr-1 at the Poisson 1$\sigma$ confidence level, as derived by Yang et al.
(2019), but is $\approx$0.9 dex higher than the rate of ${\approx}$3.4 deg-2
yr-1 derived by Glennie et al. (2015). The latter discrepancy is not
surprising, however, since Glennie et al. (2015) calculated the rate for a
much higher peak flux of $F_{\rm X,peak}{\gtrsim}$10-10 erg cm-2 s-1.

It is essential to mention again that FXRTs previously discovered as CDF-S XT2
(XRT 150321; Xue et al., 2019), XRT 170831 (Lin et al., 2019) and XRT 210423
(Lin et al., 2021) are not part of this work because of the date cut-off of
CSC2. As we showed in Sect. 2.5.6, the number of FXRTs that is removed from
our sample by our selection criteria erroneously is probably less than 1.
Therefore, the estimated event rates are robust results for FXRT candidates
brighter than $\log(F_{\rm peak}){\gtrsim}-12.6$ for _Chandra_ observations
with $T_{\rm exp}{>}$8 ks.

The event rate (event rate per dex of flux) behaves as a PL function as
$\mathcal{R}{\propto}F_{\rm lim}^{-\gamma}$, where $\gamma$ is a positive
value. In Fig. 18, we plot the observed cumulative log$\mathcal{N}$–log$S$
distribution for our entire sample, which appears to follow
$\gamma{\approx}0.5$ (_red line_). We also plot the extrapolation of the best-
fit slope, $\gamma{=}1.0$, based on the estimates of FXRTs at bright fluxes
(${\gtrsim}10^{-10}$ erg cm-2 s-1) from Arefiev et al. (2003). We caution that
Arefiev et al. (2003) do not specify an exact energy band and make no
distinction between various potential Galactic and extragalactic classes,
although it is noteworthy that the sky distribution at these bright fluxes is
also isotropic. We see that the brightest sources in our CSC2 sample are
consistent with this extrapolation, while the fainter sources fall well below
it by ${\sim}$1 dex, implying a potential break around a fluence of $3\times
10^{-8}$ erg cm-2 to our best-fit slope.

For comparison, a spatially homogenous distribution of identical (standard
candle) sources would yield a Euclidean slope of 1.5. Based on this, we adopt
$\gamma{=}1.0$ when extrapolating to brighter fluxes, and $\gamma{=}0.5$ to
fainter fluxes.

Figure 18: Observed cumulative log$\mathcal{N}$–log$S$ distribution of our
sample of FXRTs as a function of fluence (in units of Crab† $\times$ second).
Also shown are two PL models, $N({>}S){\propto}S^{-\gamma}$, with slopes
$\gamma{=}$0.5 (_red line_) and 1.0 (_blue line_). The $\gamma{=}0.5$ line
denotes the best fit to the CSC2 sample. The $\gamma{=}1$ line represents the
best fit and 1$\sigma$ error of Arefiev et al. (2003) based on bright FXRTs.
The brightest sources in our sample appear to be consistent with this bright-
end extrapolation, although our fainter sources fall $\sim$1 dex below,
implying a break. For comparison with Arefiev et al. (2003), we convert the
fluence to 2–10 keV.
$\dagger$ A Crab is a standard astrophotometric unit for measurement of the
intensity of celestial X-ray sources.

###  6.2 Volumetric rate estimate

In addition to the event rate on the sky (deg-2), we compute the volumetric
density rate $\rho(z)$, in units of yr-1 Gpc-3, to compare with other known
transient classes (GRBs, SBOs, or TDEs). Following Zhang (2018), the number of
FXRTs, $\mathcal{N}$, identified per unit (observing) time, $dt$, per unit
redshift bin, $dz$, can be written as

$\frac{d\mathcal{N}}{dtdz}=\frac{\rho(z)}{1+z}\frac{dV(z)}{dz},$ (8)

where $dV(z)/dz$ is the derivative of the volume with regards to $z$.
Integrating the previous equation by $dt$ and $dz$, we can estimate the
density rate at a particular redshift $z$ as

$\rho(z)=\frac{4\pi\mathcal{N}(1+z)}{\Omega TV_{\rm c,max}},$ (9)

where $V_{\rm c,max}$ is the maximum co-moving volume (at the maximum co-
moving distance $D_{\rm c,max}$), while $\Omega$ and $T$ are the FoV and the
exposure time used in this work (corrected by $\varepsilon_{i}$; see Sect.
6.1), respectively.

For Case I (between five and seven local FXRTs), the density rate at
${\lesssim}$100 Mpc is $\rho_{\rm Case\leavevmode\nobreak\
I}{=}$(5.9${}_{-2.6}^{+2.5}$)${\times}$10-2 yr-1 Mpc-3, at a 1$\sigma$
confidence level. Due to the small distance of these FXRTs, we can approximate
this result at $z{\approx}$0, also called the local density rate (denoting as
$\rho_{0}$), that is, $\rho_{\rm Case\leavevmode\nobreak\
I}{\approx}\rho_{0,\rm Case\leavevmode\nobreak\ I}$. This value is consistent
with previously derived rates for ULXs, taking ULX M82 as an example
(1.75$\times$10-2 yr-1 Mpc-3; Kaaret et al., 2006; Swartz et al., 2011;
Pradhan et al., 2020).

For Case II (distant FXRTs), redshift and cosmological effects become
important. Currently, we only have photometric redshifts for FXRT 8 and FXRT 9
($z_{\rm phot}{\approx}$0.7), and suspect that FXRT 13 must have a broadly
similar redshift range ($z{\approx}$0.2–1.1 and $\overline{z}{\approx}$0.7;
see Sect. 4). Thus we only compute the cosmological rate for these FXRTs.
Using Eq. 9 and assuming that FXRTs 8, 9, and 13 occurred at $z{\approx}$0.7,
the density rate of these three FXRTs is $\rho_{\rm FXRTs\leavevmode\nobreak\
8/9/13}{=}$(4.8${}_{-2.6}^{+4.7}$)${\times}$103 yr-1 Gpc-3 at a maximum
redshift of $z_{\rm max}{\approx}$2.1 (assuming a mean value of $F_{\rm
X,peak}^{\rm lim}{\approx}$1$\times$10-13 erg cm-2 s-1 as the threshold limit
detection and an isotropic luminosity $L_{\rm X,peak}^{\rm
max}{\approx}$3$\times$1045 erg s-1). In a similar way, we compute the density
rate for CDF-S XT1 at $z_{\rm max}{\approx}$3 of $\rho_{\rm
CDF-S\leavevmode\nobreak\ XT1}{=}$(4.8${}_{-4.0}^{+11.1}$)${\times}$102 yr-1
Gpc-3. In Fig. 19, _left panel_ , we compare the density rates for FXRTs
8/9/13 (_cyan star_) and CDF-S XT1 (_red square_) to other transient classes.
We note that these rate could increase by a factor of up to $\approx$2,
considering that we do not have firm redshifts for ${\sim}$50% of the FXRT
candidates.

Considering first the density rate for FXRT 8/9/13, we find that $\rho_{\rm
FXRT\leavevmode\nobreak\ 8/9/13}$ is comparable to the rate of events like
CDF-S XT2, ($\rho_{\rm XT2}(z_{\rm
max}{=}1.9){=}$(1.3${}_{-1.1}^{+2.8}$)${\times}$104 yr-1 Gpc-3; _purple
square_ , Fig. 19 _left panel_ ; Xue et al., 2019), at a similar redshift.
Excluding FXRT 13 due to its uncertain redshift and computing the density rate
only for FXRTs 8 and 9, the density rate drops by a factor of ${\sim}$1.6.
$\rho_{\rm FXRT\leavevmode\nobreak\ 8/9/13}$ remains a factor ${\gtrsim}$100
lower than the expected rate of CC-SNe (_dotted orange line_ ; Madau &
Dickinson, 2014), but is in good agreement with the density rate expected at
$z{\sim}$2.0 for LGRBs [_blue-filled region_ ; assuming $\rho_{\rm
0,LGRBs}{=}$250–500 yr-1 Gpc-3 from Wanderman & Piran (2010) and Zhang (2018),
and the normalized rate redshift evolution from Sun et al. (2015)] and TDEs
[_gray-filled region_ ; assuming $\rho_{\rm 0,TDEs}{=}$104–105 yr-1 Gpc-3 from
Sun et al. (2015) and luminosities $L{\approx}$1042–1044 erg s-1] reinforcing
a possible association with these kinds of events. For LGRBs, we adopted a jet
correction factor of ${\approx}$500 (Frail et al., 2001; Zhang, 2018);
however, other works argue for lower corrections of ${\approx}$50–100 (Piran,
2004; Guetta et al., 2005). The $\rho_{\rm FXRT\leavevmode\nobreak\ 8/9/13}$
also overlaps with the expected density rate of LL-LGRBs [_green-filled
region_ ; with $L_{\rm min}{=}$5$\times$1046 erg s-1, $\rho_{\rm 0,LL-
LGRBs}{=}$100–200 yr-1 Gpc-3 and the normalized rate redshift evolution from
Sun et al. (2015)], adopting a beaming correction of ${\approx}$1 since
observations of LL-LGRBs do not show strong evidence of collimation,
suggesting wider jet opening angles (Virgili et al., 2009; Pescalli et al.,
2015).

On the other hand, $\rho_{\rm FXRT\leavevmode\nobreak\ 8/9/13}$ is a factor of
${\approx}$1.5 higher than the estimated SGRB rate [_red-filled regions_ ;
assuming $\rho_{\rm 0,SGRBs}{=}$13–75 yr-1 Gpc-3 from Wanderman & Piran (2015)
and Zhang (2018), a merger delay Gaussian model (Virgili et al., 2011;
Wanderman & Piran, 2015), and the normalized rate redshift evolution from Sun
et al. (2015)]. However, some SGRBs are collimated (Burrows et al., 2006; De
Pasquale et al., 2010), with typical jet aperture correction factors of
${\approx}$25 (Fong et al., 2015), although other authors claim a wider range
of ${\approx}$70$\pm$40 (Berger, 2014). Given the large uncertainties, the
rates of $\rho_{\rm FXRT\leavevmode\nobreak\ 8/9/13}$ and SGRBs might remain
compatible.

We now consider the density rate for CDF-S XT1/FXRT 14 ($\rho_{\rm
CDF-S\leavevmode\nobreak\ XT1}{=}$(4.8${}_{-4.0}^{+11.1}$)${\times}$102 yr-1
Gpc-3). It remains consistent with TDEs, falls on the high side of SGRBs at
$z{\approx}$3, and is a factor of ${\approx}$2–5 lower than LGRBs, and falls
on the low side of the LL-LGRBs. Keeping in mind the uncertainty in the jet
aperture correction, it is impossible to discard the association with LGRBs.
On the other hand, the rates are ${\approx}$2 order of magnitude lower than
those of CC-SNe.

Finally, five potential distant FXRTs (FXRTs 1, 7, 10, 11, and 12) lack
redshift constraints of any kind. To constrain their contribution to the
density rate, we compute upper limits assuming that they all lie in a single
redshift bin of $\Delta z{\approx}$0.5. Figure 19, _left panel_ , shows the
resulting upper limits (_black triangles_) on the rate of these FXRTs. These
limits are consistent with the density rate computed for FXRTs 8, 9, and 13,
CDF-S XT1, and CDF-S XT2 (Xue et al., 2019), but are inconsistent with CC-SNe
beyond $z{\gtrsim}$0.5. Clearly, with firmer distance constraints on these
objects, we will be able to pin down the density rates with higher precision.

Figure 19: Density rate as a function of redshift for several known transient
classes compared to our sample. _Left panel:_ Density rate as a function of
redshift for FXRT 8/9/13 (_cyan star_) and CDF-S XT1/FXRT 14 (_red square_),
and upper limits (_black triangle_) for FXRTs without measured redshifts. We
also show the density rate of CDF-S XT2 (_purple square_ ; Xue et al., 2019),
CC-SNe (_dotted orange line_ ; Madau & Dickinson, 2014), the redshift-
dependent intrinsic event rate densities of LGRBs (_blue-filled region_ ;
taken from Sun et al. 2015 and Wanderman & Piran 2010, normalized to the local
universe value and corrected for jet-aperture as $\rho_{\rm
0,LGRBs}{\sim}$250–500 yr-1 Gpc-3), LL-LGRBs (_green-filled region_ ; taken
from Zhang 2018, normalized to the local universe value and corrected for jet-
aperture as $\rho_{\rm 0,LGRBs}{\sim}$100–200 yr-1 Gpc-3), SGRBs considering a
merger delay Gaussian model (_red-filled region_ ; taken from Sun et al. 2015
and Wanderman & Piran 2015, normalized to the local universe value and
corrected for jet-aperture as $\rho_{\rm 0,SGRBs}{\sim}$13–75 yr-1 Gpc-3), and
TDEs (_gray-filled region_ , taken from Sun et al. 2015, normalized to the
local universe as $\rho_{\rm 0,TDEs}{\sim}$104–105 yr-1 Gpc-3 at luminosities
${\sim}$1042–1044 erg s-1). _Right panel:_ Local density rate for FXRT 8/9/13
considering they are related to GRBs (_cyan star_), TDEs (_blue star_), or
FXRT 14/CDF-S XT1 (_red square_) (see Sect. 6.3). As a comparison, we also
plot the local event rate of CDF-S XT2-like events (_purple square_ ; Xue et
al., 2019), GW 170817 (_green square_ ; Abbott et al., 2017b), and GRB 170817A
(_magenta square_ ; Zhang et al., 2018), as well other kinds of transients,
such as the new SBO candidate in _XMM-Newton data_ (_yellow square and circle_
; Xu et al., 2008; Novara et al., 2020), the TDE rate of SMBH–MS TDEs (_black
triangle_ ; Sun et al., 2015) and the IMBH–WD TDE rate (_orange triangle and
circle_ ; Malyali et al., 2019; Tanikawa et al., 2021), the CC-SN rate
(_magenta triangle_), the merger rate of BNS systems (_light green circle_ ;
Abbott et al., 2021b), and the merger rate of neutron star and BH systems
(_gray circles_ ; Abbott et al., 2021a). The local event rate of LGRBs, LL-
LGRBs, and SGRBs are plotted (_blue, green, and red horizontal shaded
regions,_ corrected for the jet aperture factor; Zhang, 2018).

###  6.3 Local density rate

Additionally, we extrapolate the density rates of FXRTs 8, 9, and 13 and CDF-S
XT1 to the local universe (i.e., $z{\approx}$0) and compare them to other
transients. The density rate of any transient evolves through redshift
following Sun et al. (2015),

$\rho(z){=}\rho_{0}f(z),$ (10)

where $f(z)$ is a function that describes the density rate evolution
(normalized to $z{=}$0) and $\rho_{0}$ is the density rate at $z{=}$0\.
Therefore, it is possible to determine the local density rate if $f(z)$ is
known.

We adopted $\rho_{\rm FXRT\leavevmode\nobreak\
8/9/13}{=}$(4.8${}_{-2.6}^{+4.7}$)${\times}$103 yr-1 Gpc-3 at $z_{\rm
max}{=}$2.1, and use the average $f(z)$ between LGRBs and SGRBs (considering
the _Gaussian merger delay model_ because of the slight overlap with our
result) taken from Sun et al. (2015); there is not much difference between the
relative evolutions of LGRBs and SGRBs. This yields a local density rate for
FXRTs 8, 9, and 13 of $\rho_{\rm 0,FXRT\leavevmode\nobreak\ 8/9/13}^{\rm
GRBs}{=}$(5.8${}_{-2.2}^{+4.0}$)${\times}$102 yr-1 Gpc-3. Meanwhile, for CDF-S
XT1, we find $\rho_{0,\rm CDF-S\leavevmode\nobreak\ XT1}^{\rm
GRBs}{=}$(5.4${}_{-4.4}^{+12.4}$)${\times}$102 yr-1 Gpc-3. The range of
$\rho_{\rm FXRT\leavevmode\nobreak\ 8/9/13}$ is also consistent with that of
TDEs (see Fig. 19, _left panel_), and based on the density rate evolution of
TDEs from Sun et al. (2015), this yields a local density rate for FXRTs 8/9/13
of $\rho_{\rm 0,FXRT\leavevmode\nobreak\ 8/9/13}^{\rm
TDEs}{=}$(5.9${}_{-3.2}^{+5.8}$)${\times}$104 yr-1 Gpc-3. This value is
${\approx}$2 order of magnitude higher than the recent extended ROentgen
Survey with an Imaging Telescope Array (eROSITA) average TDE volumetric rate
reported by Sazonov et al. (2021) of ${\approx}$210 yr-1 Gpc-3 (at
$z{=}$0.0–0.6).

The _right panel_ in Fig. 19 shows the local density rate of FXRTs 8/9/13
assuming evolution as GRBs (_cyan star_), TDEs (_blue star_) and CDF-S
XT1-like events (_red square_), as well as a comparison with CDF-S XT2-like
events (_purple square_ ; Xue et al., 2019) and other transients. The GRB
local density rates of FXRTs 8, 9, and 13 ($\rho_{\rm
0,FXRT\leavevmode\nobreak\ 8/9/13}^{\rm GRBs}$), CDF-S XT1, and CDF-S XT2-like
events remain consistent with the values observed for most flavors of GRBs
(given the large uncertainties), as well as GRB 170817A and GW 170817 (_green
and magenta squares_ , respectively; Abbott et al., 2017b), BNSs (_light green
circle_ ; Abbott et al., 2021b), and neutron star and BH mergers (NS–BH)
(_gray circle_ ; Abbott et al., 2021a). These rates are, however, 1–3 dex
below those expected for SBOs (_yellow circle and square_ ; Madau & Dickinson,
2014; Novara et al., 2020), CC-SNe (_magenta triangle_ ; Madau & Dickinson,
2014), and SMBH-MS TDEs (Sun et al., 2015). The difference with SMBH-MS TDEs
may simply be a consequence of the $f(z)$ assumption. Moreover, the TDE local
density rate of FXRTs 8, 9, and 13 ($\rho_{\rm 0,FXRT\leavevmode\nobreak\
8/9/13}^{\rm TDEs}$) remains consistent with SMBH-MS TDE rates (Sun et al.,
2015) but not with IMBH-WD TDEs (Tanikawa et al., 2021), likely due to the
different progenitor system.

###  6.4 Expected events in current and future missions

Taking the computed rates from Sect. 6.1, we examine the prospects for
detecting FXRTs in other ongoing and future X-ray missions. The expected event
rate of a new mission (called $\mathcal{R}_{\rm New}$) regarding our results
using CSC2 is

$\mathcal{R}_{\rm New}=\left(\frac{F_{\rm New,lim}}{F_{\rm
CSC2,lim}}\right)^{-\gamma}\mathcal{R}_{\rm CSC2},$ (11)

where $\mathcal{R}_{\rm New}$ and $F_{\rm New,lim}$ are the event rate and
X-ray flux limit of the new mission, respectively. Then, the expected total
number of events must be

$\mathcal{N}_{\rm New}{=}\Omega_{\rm New}T_{\rm New}\mathcal{R}_{\rm
New}=\Omega_{\rm New}T_{\rm New}\left(\frac{F_{\rm New,lim}}{F_{\rm
CSC2,lim}}\right)^{-\gamma}\mathcal{R}_{\rm CSC2},$ (12)

where $\Omega_{\rm New}$ and $T_{\rm New}$ are the FoV and the operational
time of a new mission, respectively. It is important to realize that Eq. 12
takes into account the ratio between the new mission ($F_{\rm New,lim}$) and
_Chandra_ (the limit imposed by our method $F_{\rm
CSC2,lim}{=}$1.5$\times$10-13 erg cm-2 s-1) X-ray flux limits, respectively,
which is a correction factor between both instruments. Given the low-count
statistics, we quote estimates incorporating the Poisson 1$\sigma$ errors.

Current operating observatories such as _XMM-Newton_ , _Swift_ –XRT, and
eROSITA have sufficient sensitivity and/or history in orbit to detect similar
FXRTs to those found here.

The European Photon Imaging Camera [EPIC; pn plus Metal Oxide Semi-conductor
CCD arrays] on board the _XMM-Newton_ telescope have an instantaneous
FoV${\approx}$0.25 deg2, flux sensitivity of ${\approx}$10-14 erg cm-2 s-1 in
the energy range of 0.15–12 keV, and have an archive of roughly ${\approx}$476
Ms total exposure time during ${\sim}$20 years in orbit (mean value between pn
and MOS cameras; Ehle et al., 2003). Adopting a spectral slope of
$\Gamma{=}1.7$, typical of FXRTs (e.g., CDF-S XT1), a correction factor to
account for the contribution of background flares (assuming that 30–40% of
exposure time is affected by them) and a flux cutoff of $F_{\rm
XMM,lim}{\sim}$10-13 erg cm-2 s-1 (to avoid effects from Poisson noise), we
predict up to ${\approx}$68–135 Case I and ${\approx}$37–68 Case II FXRTs,
respectively.

Similarly, _Swift_ –XRT has a FoV${\approx}$0.15 deg2, a flux sensitivity of
${\approx}$8${\times}$10-14 erg cm-2 s-1 in the energy band of 0.2–10 keV, and
has accumulated ${\approx}$ 315.4 Ms of archival data over ${\sim}$14 years
operational time (Hill et al., 2000; Burrows et al., 2003). Adopting a flux
limit of $F_{\rm XRT,lim}{\sim}$8${\times}$10-13 erg cm-2 s-1 (again, to avoid
Poisson noise effects), the expected number of FXRTs are ${\approx}$27–55 Case
I and ${\approx}$15–27 Case II events.

The above implies that there should be a substantial number of FXRTs hidden
within the _XMM-Newton_ and _Swift_ –XRT archives and catalogs. The X-ray
transient and variable sky (EXTraS) project (De Luca et al., 2021) and
systematic searches such as Alp & Larsson (2020) have reported 136 and a dozen
candidates to date, respectively, which presents a lower bound to the total
numbers estimated above. Also, in the systematic search developed by the EPIC-
pn _XMM-Newton_ Outburst Detector (EXOD) search project (Pastor-Marazuela et
al., 2020), 2536 potential XRTs have been identified, but this large number is
dominated by stellar flares, cataclysmic variables, type I X-ray bursts,
supergiant FXRTs, SBOs, AGNs, and more.

Finally, the Spectrum-Roentgen-Gamma (_SRG_)–eROSITA mission, launched in July
2019, is scanning the entire sky in the X-ray band (0.2–10 keV) with a
FoV${\approx}$0.833 deg2 during _SRG_ –eROSITA’s official 4-year survey phase.
This should provide roughly equivalent coverage in sky area per time to the
current _XMM-Newton_ archive. The _SRG_ –eROSITA all sky survey is expected to
yield flux limits of ${\approx}$10-14 and ${\approx}$10-13 erg cm-2 s-1 in the
0.5–2 and 2–10 keV energy bands, respectively. Avoiding Poisson noise effects
as above, we adopt an _SRG_ –eROSITA 0.5–2 keV flux limit for FXRTs of $F_{\rm
eROSITA,lim}{\approx}$10-13 erg cm-2 s-1. Thus, during the 4-year survey, the
expected number of FXRTs detected by _SRG_ –eROSITA (in the 0.5–2 keV band)
should be ${\approx}$50–100 and 27–50 events for Case I and Case II,
respectively.

Concerning future missions, the Advanced Telescope for High ENergy
Astrophysics (_Athena_) has been selected by European Space Agency to
characterize the hot and energetic universe, with an anticipated launch in the
mid 2030s. It is projected to have an effective area of 0.25–2.0 m2, energy
range of 0.3–12 keV, and a nominal lifetime of five years, although
consumables (such as fuel) have been rated for 10 years in the case of a
mission extension (Nandra et al., 2013; Barret et al., 2013). The Wide Field
Imager (WFI) is one of two detectors on board _Athena_ , with a spectral
resolution of $\Delta E{<}170$ eV at 7 keV, spatial resolution of $\leq$10
arcsec PSF on-axis), and FoV of 0.44 deg2 (Rau et al., 2016). To estimate the
number of extragalactic FXRTs, we conservatively assume a flux threshold 10
times higher than the nominal 60 ks (longer than the expected duration of the
FXRTs) flux limit due to Poisson fluctuations of $F_{\rm
WFI,lim}{\approx}$10-15 erg cm-2 s-1 (where the point source detection limit
is ${\approx}$10-16 erg cm-2 s-1 for the WFI deep fields). This flux limit is
a factor of 100 deeper than the _SRG_ –eROSITA sky survey flux limit. Thus,
during a ${\approx}$4 year mission, adopting $\gamma{=}$0.5 for the faint-end
slope extrapolation the expected number of FXRTs detected by _Athena_ will
${\approx}$ 130–270 and 72–130 events for _Case I_ and _Case II_ ,
respectively. This sample size of bright and fainter events can be used to
probe the multiwavelength properties with coordinated campaigns. Assuming that
the WFI observations will be spread evenly during the mission and that those
observations will also be performed during the _Athena_ ground contact,
approximately one-sixth of the events (${\approx}$9 and 16) could have
_Athena_ alerts with latencies ${<}$4 hours.

We also consider the _Einstein Probe_ (EP), which aims to monitor high-energy
transient and variable phenomena in 0.5–4.0 keV band (Yuan et al., 2015,
2017). The EP is scheduled for launch by the end of 2023, with a 3-year
operational lifetime and 5-year goal (Yuan et al., 2017). EP will carry two
scientific instruments, the Wide-field X-ray Telescope (WXT) with a large
instantaneous FoV of 3600 deg2 and a narrow-field Follow-up X-ray Telescope,
as well as a fast alert downlink system (Yuan et al., 2015). To estimate the
expected number of FXRTs, we consider just the WXT instrument, which has a
threshold sensitivity of $F_{\rm WXT}{\approx}$5$\times$10-11 erg cm-2 s-1 at
1 ks, that is, ${\approx}$500 times higher than our flux limit and
$\gamma{\approx}$1.0.

Thus, during the ${\approx}$3 year mission, the expected number of FXRTs
detected by _EP_ should be ${\approx}$69–138 and 38–69 events for Case I and
Case II, respectively.

##  7 Conclusions and future work

In this work we search for extragalactic FXRTs hidden in CSC2. We have applied
a modified version of the algorithm developed by Yang et al. (2019) to 214,701
X-ray sources identified in the CSC2 with $|b|{>}$10 deg (i.e., 5303 _Chandra_
observations, totaling $\approx$169.6 Ms and 592 deg2). Considering additional
criteria (analyzing further X-ray observations taken by _Chandra_ , _XMM-
Newton_ , _Swift_ –XRT, _Einstein,_ and _ROSAT_) and other astronomical
catalogs (_Gaia_ , NED, SIMBAD, VHS, DES, Pan-STARRS, and others), we identify
14 FXRTs that remain consistent with an extragalactic origin. We rediscover
all (five) previously reported _Chandra_ events covered by CSC2: XRT 000519
(previously identified by Jonker et al., 2013), XRT 110103 (previously
identified by Glennie et al., 2015), XRT 030511 and XRT 110919 (previously
identified by Lin et al., 2019, 2022), and XRT 141001/CDF-S XT1 (previously
identified by Bauer et al., 2017).

Candidates have peak 0.5–7 keV fluxes between ${\approx}$1.0$\times$10-13 and
2$\times$10-10 erg cm-2 s-1 and $T_{90}$ values from ${\approx}$4 to 40 ks.
None of the FXRTs are detected in gamma rays near the time of the detection of
the transient X-ray light. Based on multiwavelength constraints, we rule out a
Galactic origin (e.g., as Galactic M or brown-dwarf stellar flares) in all but
two cases (for these, existing data cannot yet rule out extreme stellar X-ray
flares). The origin of the extragalactic FXRT sample appears to be diverse:
five events are robustly associated with local galaxies (${\lesssim}$100 Mpc;
called the local sample); seven are likely distant events (${\gtrsim}$100 Mpc;
called the distant sample); and two events, XRT 000519 and XRT 110103, have
nearby associations that remain somewhat ambiguous. Among the distant FXRTs,
we identify hosts for four FXRTs, which span a wide range of magnitudes
($m_{i}{\approx}$20.6–27.0 AB mag), while we can only place upper limits on
five FXRTs.

We have studied the spectral and timing properties of the FXRTs. The X-ray
spectra can be well fitted by PLs with a median slope of $\Gamma{=}$2.5 and an
overall range $\Gamma{\approx}$1.7–4.0. Furthermore, we observe potential
spectral softening for six FXRTs with time (for XRT 000519 and XRT 110103, the
softening is highly significant and occurs during the main flare; Glennie et
al., 2015). In the case of timing properties, five FXRTs show plateaus in
their X-ray light curves, similar to CDF-S XT2 (Xue et al., 2019), with
durations of ${\sim}$2–10 ks followed by PL decays with slopes ranging from
${\sim}$1.2 to 2.6. For three FXRTs we see, simultaneously with the plateau
and decay, possible spectral softening (at 90% confidence), similar to CDF-S
XT2 (Xue et al., 2019).

The five local FXRTs have projected physical offsets between ${\approx}$0.7
and 9.4 kpc, with four being co-spatial with apparent star-forming regions or
young star clusters. Adopting their host distances, these local events have
peak isotropic X-ray luminosities of $L_{\rm X,peak}{\approx}$1038–1040 erg
s-1, well below expectations for GRBs, TDEs, XRFs, and supernova SBOs. Such
luminosities are comparable to those of ULXs and Galactic XRBs, although the
durations and time variability properties of the local FXRTs are quite
distinct. As such, we speculate that several may represent a new type of X-ray
phenomenon related to massive stars.

Among the distant FXRT sample, two are associated with relatively bright
optical and NIR extended sources, allowing us to derive galaxy properties
using photometric archival data. The other two host associations are very
faint extended sources; one is detected only in a single band, and hence lacks
physical constraints, while the other is fortuitously observed by the _HST_
but has only weak constraints on its properties. Both bright hosts have
similar redshifts ($z_{\rm phot}{\approx}$0.5–0.7) and stellar masses
($M_{*}{\approx}$3$\times$1010 $M_{\odot}$), but starkly different SFRs
(SFR${\approx}$0.5 vs. $\approx$125 $M_{\odot}$ yr-1), and the faint _HST_
host has an uncertain redshift ($z_{\rm phot}{\approx}$0.4–3.2) and associated
host properties (Bauer et al., 2017). Adopting $\bar{z}{=}$0.7 for all four
events, the peak luminosities, energetics, and spectro-temporal properties
robustly rule out an SBO origin but potentially remain consistent with origins
as on-axis GRBs, and even off-axis GRBs in the tail of the X-ray afterglow, or
TDEs involving an IMBH and a WD.

For the three FXRTs that lack optical and NIR host detections, interpretations
are broader. An association with SBOs remains possible at low redshifts
($z{\lesssim}$0.5), as long as potential hosts are low-mass, low-SFR dwarf
galaxies. An on-axis GRB scenario remains possible for $z{\gtrsim}$1.0 and
naturally explains the non-detection of faint host galaxies by existing
optical and NIR facilities. An off-axis GRB afterglow scenario is also viable,
except perhaps for very low redshifts ($z{\lesssim}$0.1), where the lack of
any association with a host becomes problematic. Finally, a TDE scenario
remains possible across a broad redshift range, although the lack of a
detectable host requires strong beaming, for instance, similar to _Swift_
J1644+57.

Finally, we compute the event rates of local (_Case I_) and distant (_Case
II_) FXRTs of $\mathcal{R}_{\rm Case\leavevmode\nobreak\
I}{=}$53.7${}_{-15.1}^{+22.6}$ and $\mathcal{R}_{\rm Case\leavevmode\nobreak\
II}{=}$28.2${}_{-6.9}^{+9.8}$ deg-2 yr-1, respectively. Additionally, for
three distant FXRTs (assuming $\bar{z}{=}$0.7), we derive a volumetric rate
(in units of yr-1 Gpc-3) of $\rho_{\rm FXRT\leavevmode\nobreak\
8/9/13}{=}$(4.8${}_{-2.6}^{+4.7}$)${\times}$103 yr-1 Gpc-3 at $z_{\rm
max}{=}$2.1. This value is in good agreement with the value derived by Xue et
al. (2019) at a similar redshift ($z_{\rm max}{=}$1.9), as well as with other
transient classes such as LGRBs, SGRBs, and TDEs. Nevertheless, this rate is
${\approx}$2 order of magnitude lower than that of CC-SNe.

Our investigation of 14 _Chandra_ -detected extragalactic FXRT candidates
breaks new ground in terms of characterizing their diverse properties and
nature, although the lack of firm distances and host properties for the
distant subset clearly leaves much to speculation. The _Chandra_ sample
provides the most accurate positions among existing X-ray missions, which is
critical for pinpointing potential host galaxies and potential physical
offsets. Given the low numbers of distant FXRTs (both found here and predicted
in other archives) and the diverse range of host redshifts and properties, it
will be critical to identify and follow up their associated host galaxies with
dedicated spectroscopy and/or deep multiwavelength imaging in order to place
extragalactic FXRTs in a proper physical and cosmological context. The
contemporaneous multiwavelength nature of FXRTs remains completely unknown.
Given the short duration of these events, progress here will crucially hinge
upon the ability of current and future X-ray observatories to carry out
efficient strategies for (onboard) detection and alert generation to trigger
follow-up campaigns while the FXRTs are still active in X-rays and,
presumably, at other wavelengths. The launch of narrow- and wide-field
observatories such as _Athena_ and _EP_ should provide a watershed moment for
expanding samples.

As future work, we plan to characterize this new sample of FXRTs using recent
optical and NIR observations to catch their host galaxies and thus constraint
their energetics. Also, we plan to extend our search to _Chandra_ data not
considered in the CSC2 to identify new FXRTs and thus better understand their
elusive nature.

###### Acknowledgements.

We acknowledge support from: ANID grants Programa de Capital Humano Avanzado
folio #21180886 (J.Q–V), CATA-Basal AFB-170002 (J.Q–V, F.E.B.), FONDECYT
Regular 1190818 (J.Q–V, F.E.B.), 1200495 (J.Q–V, F.E.B.) and Millennium
Science Initiative ICN12_009 (J.Q–V, F.E.B.); NSF grant AST-2106990 and
_Chandra_ X-ray Center grant GO0-21080X (W.N.B.); the National Natural Science
Foundation of China grant 11991053 (B.L.); support from NSFC grants 12025303
and 11890693 (Y.Q.X.); support from the George P. and Cynthia Woods Mitchell
Institute for Fundamental Physics and Astronomy at Texas A&M University, from
the National Science Foundation through grants AST-1614668 and AST-2009442,
and from the NASA/ESA/CSA James Webb Space Telescope through the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-03127 (G.Y.). The scientific results reported in this article are based
on observations made by the _Chandra_ X-ray Observatory. This research has
made use of software provided by the _Chandra_ X-ray Center (CXC). This
research uses services or data provided by the Astro Data Lab at NSF’s
National Optical-Infrared Astronomy Research Laboratory. NOIRLab is operated
by the Association of Universities for Research in Astronomy (AURA), Inc.
under a cooperative agreement with the National Science Foundation.

## References

  * Abbott et al. (2017a) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a, ApJ, 848, L12
  * Abbott et al. (2017b) Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017b, ApJ, 848, L13
  * Abbott et al. (2021a) Abbott, R., Abbott, T. D., Abraham, S., et al. 2021a, ApJ, 915, L5
  * Abbott et al. (2021b) Abbott, R., Abbott, T. D., Abraham, S., et al. 2021b, Physical Review X, 11, 021053
  * Abbott et al. (2021c) Abbott, T. M. C., Adamów, M., Aguena, M., et al. 2021c, ApJS, 255, 20
  * Ahumada et al. (2020) Ahumada, R., Prieto, C. A., Almeida, A., et al. 2020, ApJS, 249, 3
  * Ajello et al. (2019) Ajello, M., Arimoto, M., Axelsson, M., et al. 2019, ApJ, 878, 52
  * Alp & Larsson (2020) Alp, D. & Larsson, J. 2020, ApJ, 896, 39
  * Aptekar et al. (2001) Aptekar, R., Frederiks, D., Golenetskii, S., et al. 2001, ApJS, 137, 227
  * Arabsalmani et al. (2019) Arabsalmani, M., Roychowdhury, S., Starkenburg, T. K., et al. 2019, MNRAS, 485, 5411
  * Arefiev et al. (2003) Arefiev, V. A., Priedhorsky, W. C., & Borozdin, K. N. 2003, ApJ, 586, 1238
  * Arnaud (1996) Arnaud, K. A. 1996, Astronomical Society of the Pacific Conference Series, Vol. 101, XSPEC: The First Ten Years, ed. G. H. Jacoby & J. Barnes, 17
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33
  * Avenhaus et al. (2012) Avenhaus, H., Schmid, H. M., & Meyer, M. R. 2012, A&A, 548, A105
  * Bachetti et al. (2014) Bachetti, M., Harrison, F. A., Walton, D. J., et al. 2014, Nature, 514, 202
  * Balberg & Loeb (2011) Balberg, S. & Loeb, A. 2011, MNRAS, 414, 1715
  * Barack et al. (2019) Barack, L., Cardoso, V., Nissanke, S., et al. 2019, Classical and Quantum Gravity, 36, 143001
  * Barniol Duran et al. (2015) Barniol Duran, R., Nakar, E., Piran, T., & Sari, R. 2015, MNRAS, 448, 417
  * Barret et al. (2013) Barret, D., Nandra, K., Barcons, X., et al. 2013, in SF2A-2013: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. L. Cambresy, F. Martins, E. Nuss, & A. Palacios, 447–453
  * Barthelmy et al. (2005) Barthelmy, S. D., Chincarini, G., Burrows, D. N., et al. 2005, Nature, 438, 994
  * Bauer et al. (2017) Bauer, F. E., Treister, E., Schawinski, K., et al. 2017, MNRAS, 467, 4841
  * Berger (2006) Berger, E. 2006, ApJ, 648, 629
  * Berger (2014) Berger, E. 2014, ARA&A, 52, 43
  * Bernardini et al. (2012) Bernardini, M. G., Margutti, R., Mao, J., Zaninoni, E., & Chincarini, G. 2012, A&A, 539, A3
  * Beuther et al. (2017) Beuther, H., Meidt, S., Schinnerer, E., Paladino, R., & Leroy, A. 2017, A&A, 597, A85
  * Bianchi et al. (2011) Bianchi, L., Herald, J., Efremova, B., et al. 2011, Ap&SS, 335, 161
  * Bloom et al. (2011) Bloom, J. S., Giannios, D., Metzger, B. D., et al. 2011, Science, 333, 203
  * Bloom et al. (2002) Bloom, J. S., Kulkarni, S. R., & Djorgovski, S. G. 2002, AJ, 123, 1111
  * Brinchmann et al. (2004) Brinchmann, J., Charlot, S., White, S. D. M., et al. 2004, MNRAS, 351, 1151
  * Bromberg et al. (2012) Bromberg, O., Nakar, E., Piran, T., & Sari, R. 2012, ApJ, 749, 110
  * Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, A&A, 564, A125
  * Burrows et al. (2006) Burrows, D. N., Grupe, D., Capalbi, M., et al. 2006, ApJ, 653, 468
  * Burrows et al. (2003) Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2003, 662, 488
  * Buta et al. (2015) Buta, R. J., Sheth, K., Athanassoula, E., et al. 2015, ApJS, 217, 32
  * Calzetti et al. (2000) Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682
  * Campana et al. (2011) Campana, S., Lodato, G., D’Avanzo, P., et al. 2011, Nature, 480, 69
  * Campana et al. (2006) Campana, S., Mangano, V., Blustin, A. J., et al. 2006, Nature, 442, 1008
  * Cappellari et al. (2011) Cappellari, M., Emsellem, E., Krajnović, D., et al. 2011, MNRAS, 413, 813
  * Carnall et al. (2018) Carnall, A. C., McLure, R. J., Dunlop, J. S., & Davé, R. 2018, MNRAS, 480, 4379
  * Cash (1979) Cash, W. 1979, ApJ, 228, 939
  * Chang et al. (2015) Chang, Y.-Y., van der Wel, A., da Cunha, E., & Rix, H.-W. 2015, ApJS, 219, 8
  * Chincarini et al. (2010) Chincarini, G., Mao, J., Margutti, R., et al. 2010, MNRAS, 406, 2113
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102
  * Christensen et al. (2008) Christensen, L., Vreeswijk, P. M., Sollerman, J., et al. 2008, A&A, 490, 45
  * Colbert & Mushotzky (1999) Colbert, E. J. M. & Mushotzky, R. F. 1999, ApJ, 519, 89
  * Colgate & Petschek (1981) Colgate, S. A. & Petschek, A. G. 1981, ApJ, 248, 771
  * Connaughton et al. (2015) Connaughton, V., Briggs, M. S., Goldstein, A., et al. 2015, ApJS, 216, 32
  * Crnojević et al. (2016) Crnojević, D., Sand, D. J., Spekkens, K., et al. 2016, ApJ, 823, 19
  * Dado & Dar (2019) Dado, S. & Dar, A. 2019, ApJ, 884, L44
  * Dai et al. (2018) Dai, L., McKinney, J. C., Roth, N., Ramirez-Ruiz, E., & Miller, M. C. 2018, ApJ, 859, L20
  * Dálya et al. (2018) Dálya, G., Galgóczi, G., Dobos, L., et al. 2018, MNRAS, 479, 2374
  * Dark Energy Survey Collaboration et al. (2016) Dark Energy Survey Collaboration, Abbott, T., Abdalla, F. B., et al. 2016, MNRAS, 460, 1270
  * D’Avanzo et al. (2018) D’Avanzo, P., Campana, S., Salafia, O. S., et al. 2018, A&A, 613, L1
  * Davis et al. (2014) Davis, T. A., Young, L. M., Crocker, A. F., et al. 2014, MNRAS, 444, 3427
  * De Luca et al. (2021) De Luca, A., Salvaterra, R., Belfiore, A., et al. 2021, A&A, 650, A167
  * De Luca et al. (2020) De Luca, A., Stelzer, B., Burgasser, A. J., et al. 2020, A&A, 634, L13
  * De Pasquale et al. (2010) De Pasquale, M., Schady, P., Kuin, N. P. M., et al. 2010, ApJ, 709, L146
  * Dotter (2016) Dotter, A. 2016, ApJS, 222, 8
  * Drozdovsky & Karachentsev (2000) Drozdovsky, I. O. & Karachentsev, I. D. 2000, A&AS, 142, 425
  * Dye et al. (2018) Dye, S., Lawrence, A., Read, M. A., et al. 2018, MNRAS, 473, 5113
  * Eappachen et al. (2022) Eappachen, D., Jonker, P. G., Fraser, M., et al. 2022, arXiv e-prints, arXiv:2204.10012
  * Ehle et al. (2003) Ehle, M., Breitfellner, M., Dahlem, M., et al. 2003, Issue, 2, 2003
  * Ensman & Burrows (1992) Ensman, L. & Burrows, A. 1992, ApJ, 393, 742
  * Espada et al. (2019) Espada, D., Verley, S., Miura, R. E., et al. 2019, ApJ, 887, 88
  * Evans et al. (2019) Evans, I. N., Allen, C., Anderson, C. S., et al. 2019, in AAS/High Energy Astrophysics Division, Vol. 17, AAS/High Energy Astrophysics Division, 114.01
  * Evans et al. (2010) Evans, I. N., Primini, F. A., Glotfelty, K. J., et al. 2010, ApJS, 189, 37
  * Evans et al. (2020a) Evans, I. N., Primini, F. A., Miller, J. B., et al. 2020a, in American Astronomical Society Meeting Abstracts, Vol. 235, American Astronomical Society Meeting Abstracts #235, 154.05
  * Evans et al. (2009) Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2009, MNRAS, 397, 1177
  * Evans et al. (2007) Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379
  * Evans et al. (2014) Evans, P. A., Osborne, J. P., Beardmore, A. P., et al. 2014, ApJS, 210, 8
  * Evans et al. (2020b) Evans, P. A., Page, K. L., Osborne, J. P., et al. 2020b, ApJS, 247, 54
  * Fasano & Franceschini (1987) Fasano, G. & Franceschini, A. 1987, MNRAS, 225, 155
  * Feroci et al. (2001) Feroci, M., Hurley, K., Duncan, R. C., & Thompson, C. 2001, ApJ, 549, 1021
  * Feroz & Hobson (2008) Feroz, F. & Hobson, M. P. 2008, MNRAS, 384, 449
  * Feroz et al. (2009) Feroz, F., Hobson, M. P., & Bridges, M. 2009, MNRAS, 398, 1601
  * Ferrarese & Merritt (2000) Ferrarese, L. & Merritt, D. 2000, ApJ, 539, L9
  * Flewelling (2018) Flewelling, H. 2018, in American Astronomical Society Meeting Abstracts, Vol. 231, American Astronomical Society Meeting Abstracts #231, 436.01
  * Flewelling et al. (2020) Flewelling, H. A., Magnier, E. A., Chambers, K. C., et al. 2020, ApJS, 251, 7
  * Fong et al. (2015) Fong, W., Berger, E., Margutti, R., & Zauderer, B. A. 2015, ApJ, 815, 102
  * Frail et al. (2001) Frail, D. A., Kulkarni, S. R., Sari, R., et al. 2001, ApJ, 562, L55
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1
  * Galbany et al. (2014) Galbany, L., Stanishev, V., Mourão, A. M., et al. 2014, A&A, 572, A38
  * García-Alvarez et al. (2008) García-Alvarez, D., Drake, J. J., Kashyap, V. L., Lin, L., & Ball, B. 2008, ApJ, 679, 1509
  * Gendre et al. (2013) Gendre, B., Stratta, G., Atteia, J. L., et al. 2013, ApJ, 766, 30
  * Glennie et al. (2015) Glennie, A., Jonker, P. G., Fender, R. P., Nagayama, T., & Pretorius, M. L. 2015, MNRAS, 450, 3765
  * Göğüş et al. (1999) Göğüş, E., Woods, P. M., Kouveliotou, C., et al. 1999, ApJ, 526, L93
  * González-Fernández et al. (2018) González-Fernández, C., Hodgkin, S. T., Irwin, M. J., et al. 2018, MNRAS, 474, 5459
  * Granot et al. (2018a) Granot, J., De Colle, F., & Ramirez-Ruiz, E. 2018a, MNRAS, 481, 2711
  * Granot et al. (2018b) Granot, J., Gill, R., Guetta, D., & De Colle, F. 2018b, MNRAS, 481, 1597
  * Granot et al. (2002) Granot, J., Panaitescu, A., Kumar, P., & Woosley, S. E. 2002, ApJ, 570, L61
  * Güdel & Nazé (2009) Güdel, M. & Nazé, Y. 2009, A&A Rev., 17, 309
  * Guetta et al. (2005) Guetta, D., Piran, T., & Waxman, E. 2005, ApJ, 619, 412
  * Guo et al. (2013) Guo, Y., Ferguson, H. C., Giavalisco, M., et al. 2013, ApJS, 207, 24
  * Hajela et al. (2022) Hajela, A., Margutti, R., Bright, J. S., et al. 2022, ApJ, 927, L17
  * Hawley et al. (2002) Hawley, S. L., Covey, K. R., Knapp, G. R., et al. 2002, AJ, 123, 3409
  * Heida et al. (2017) Heida, M., Jonker, P. G., Torres, M. A. P., & Chiavassa, A. 2017, ApJ, 846, 132
  * Helou et al. (1991) Helou, G., Madore, B. F., Schmitz, M., et al. 1991, Astrophysics and Space Science Library, Vol. 171, The NASA/IPAC extragalactic database., ed. M. A. Albrecht & D. Egret, 89–106
  * Hewett et al. (2006) Hewett, P. C., Warren, S. J., Leggett, S. K., & Hodgkin, S. T. 2006, MNRAS, 367, 454
  * Hickox & Markevitch (2006) Hickox, R. C. & Markevitch, M. 2006, ApJ, 645, 95
  * Hill et al. (2000) Hill, J. E., Zugger, M. E., Shoemaker, J., et al. 2000, in X-Ray and Gamma-Ray Instrumentation for Astronomy XI, Vol. 4140, International Society for Optics and Photonics, 87–98
  * Hjorth & Bloom (2012) Hjorth, J. & Bloom, J. S. 2012, The Gamma-Ray Burst - Supernova Connection, 169–190
  * Ho et al. (2020) Ho, A. Y. Q., Kulkarni, S. R., Perley, D. A., et al. 2020, ApJ, 902, 86
  * Hurley et al. (2011) Hurley, K., Atteia, J. L., Barraud, C., et al. 2011, ApJS, 197, 34
  * Hurley et al. (1999) Hurley, K., Cline, T., Mazets, E., et al. 1999, Nature, 397, 41
  * Ide et al. (2020) Ide, S., Hayashida, K., Noda, H., et al. 2020, PASJ, 72, 40
  * Im et al. (2017) Im, M., Yoon, Y., Lee, S.-K. J., et al. 2017, ApJ, 849, L16
  * Irwin et al. (2016) Irwin, J. A., Maksym, W. P., Sivakoff, G. R., et al. 2016, Nature, 538, 356
  * Ishak (2017) Ishak, B. 2017, Contemporary Physics, 58, 99
  * Israel et al. (2005) Israel, G. L., Belloni, T., Stella, L., et al. 2005, ApJ, 628, L53
  * Ivezić et al. (2014) Ivezić, Ž., Connolly, A. J., VanderPlas, J. T., & Gray, A. 2014, Statistics, data mining, and machine learning in astronomy (Princeton University Press)
  * Izzo et al. (2020) Izzo, L., Auchettl, K., Hjorth, J., et al. 2020, A&A, 639, L11
  * Johnstone et al. (2005) Johnstone, R. M., Fabian, A. C., Morris, R. G., & Taylor, G. B. 2005, MNRAS, 356, 237
  * Jonker et al. (2013) Jonker, P. G., Glennie, A., Heida, M., et al. 2013, ApJ, 779, 14
  * Jonker et al. (2012a) Jonker, P. G., Heida, M., Torres, M. A. P., et al. 2012a, ApJ, 758, 28
  * Jonker et al. (2012b) Jonker, P. G., Miller-Jones, J. C. A., Homan, J., et al. 2012b, MNRAS, 423, 3308
  * Kaaret et al. (2017) Kaaret, P., Feng, H., & Roberts, T. P. 2017, ARA&A, 55, 303
  * Kaaret et al. (2006) Kaaret, P., Simet, M. G., & Lang, C. C. 2006, ApJ, 646, 174
  * Kaastra (2017) Kaastra, J. S. 2017, A&A, 605, A51
  * Kalberla et al. (2005) Kalberla, P. M. W., Burton, W. B., Hartmann, D., et al. 2005, A&A, 440, 775
  * Kalberla & Haud (2015) Kalberla, P. M. W. & Haud, U. 2015, A&A, 578, A78
  * Kolmogorov (1933) Kolmogorov, A. 1933, Inst. Ital. Attuari, Giorn., 4, 83
  * Kraft et al. (1991) Kraft, R. P., Burrows, D. N., & Nousek, J. A. 1991, ApJ, 374, 344
  * Krishnamoorthy & Thomson (2004) Krishnamoorthy, K. & Thomson, J. 2004, Journal of Statistical Planning and Inference, 119, 23
  * Krühler et al. (2017) Krühler, T., Kuncarayakti, H., Schady, P., et al. 2017, A&A, 602, A85
  * Lamb et al. (2021) Lamb, G. P., Fernández, J. J., Hayes, F., et al. 2021, Universe, 7, 329
  * Lanz et al. (2013) Lanz, L., Zezas, A., Brassington, N., et al. 2013, ApJ, 768, 90
  * Levan et al. (2011) Levan, A. J., Tanvir, N. R., Cenko, S. B., et al. 2011, Science, 333, 199
  * Levan et al. (2014) Levan, A. J., Tanvir, N. R., Starling, R. L. C., et al. 2014, ApJ, 781, 13
  * Levesque (2014) Levesque, E. M. 2014, PASP, 126, 1
  * Li et al. (2018) Li, L., Wu, X.-F., Lei, W.-H., et al. 2018, ApJS, 236, 26
  * Li et al. (2016) Li, Y., Zhang, B., & Lü, H.-J. 2016, ApJS, 227, 7
  * Lin et al. (2019) Lin, D., Irwin, J., & Berger, E. 2019, The Astronomer’s Telegram, 13171, 1
  * Lin et al. (2021) Lin, D., Irwin, J. A., & Berger, E. 2021, The Astronomer’s Telegram, 14599, 1
  * Lin et al. (2022) Lin, D., Irwin, J. A., Berger, E., & Nguyen, R. 2022, ApJ, 927, 211
  * Lin et al. (2018) Lin, D., Strader, J., Carrasco, E. R., et al. 2018, Nature Astronomy, 2, 656
  * Lin et al. (2020) Lin, D., Strader, J., Romanowsky, A. J., et al. 2020, ApJ, 892, L25
  * Lin et al. (2012) Lin, D., Webb, N. A., & Barret, D. 2012, ApJ, 756, 27
  * Lindegren et al. (2018) Lindegren, L., Hernández, J., Bombrun, A., et al. 2018, A&A, 616, A2
  * Liu (2011) Liu, J. 2011, ApJS, 192, 10
  * Lopes de Oliveira et al. (2007) Lopes de Oliveira, R., Motch, C., Smith, M. A., Negueruela, I., & Torrejón, J. M. 2007, A&A, 474, 983
  * Lü et al. (2019) Lü, H.-J., Yuan, Y., Lan, L., et al. 2019, arXiv e-prints, arXiv:1904.06664
  * Lü et al. (2015) Lü, H.-J., Zhang, B., Lei, W.-H., Li, Y., & Lasky, P. D. 2015, ApJ, 805, 89
  * Luo et al. (2014) Luo, B., Brandt, N., & Bauer, F. 2014, The Astronomer’s Telegram, 6541, 1
  * Luo et al. (2017) Luo, B., Brandt, W. N., Xue, Y. Q., et al. 2017, ApJS, 228, 2
  * Lyons et al. (2010) Lyons, N., O’Brien, P. T., Zhang, B., et al. 2010, MNRAS, 402, 705
  * MacLeod et al. (2014) MacLeod, M., Goldstein, J., Ramirez-Ruiz, E., Guillochon, J., & Samsing, J. 2014, ApJ, 794, 9
  * Madau & Dickinson (2014) Madau, P. & Dickinson, M. 2014, ARA&A, 52, 415
  * Maguire et al. (2020) Maguire, K., Eracleous, M., Jonker, P. G., MacLeod, M., & Rosswog, S. 2020, Space Sci. Rev., 216, 39
  * Malyali et al. (2019) Malyali, A., Rau, A., & Nandra, K. 2019, MNRAS, 489, 5413
  * Margutti et al. (2011) Margutti, R., Bernardini, G., Barniol Duran, R., et al. 2011, MNRAS, 410, 1064
  * Margutti & Chornock (2021) Margutti, R. & Chornock, R. 2021, ARA&A, 59
  * Marocco et al. (2021) Marocco, F., Eisenhardt, P. R. M., Fowler, J. W., et al. 2021, ApJS, 253, 8
  * Massey Jr (1951) Massey Jr, F. J. 1951, Journal of the American statistical Association, 46, 68
  * Mata Sánchez et al. (2015) Mata Sánchez, D., Mu$\tilde{n}$oz-Darias, T., Casares, J., Corral-Santana, J. M., & Shahbaz, T. 2015, MNRAS, 454, 2199
  * Matzner & McKee (1999) Matzner, C. D. & McKee, C. F. 1999, ApJ, 510, 379
  * Mazets et al. (1999) Mazets, E. P., Cline, T. L., Aptekar’, R. L., et al. 1999, Astronomy Letters, 25, 635
  * Mazzali et al. (2008) Mazzali, P. A., Valenti, S., Della Valle, M., et al. 2008, Science, 321, 1185
  * McMahon et al. (2013) McMahon, R. G., Banerji, M., Gonzalez, E., et al. 2013, The Messenger, 154, 35
  * Michałowski et al. (2014) Michałowski, M. J., Hunt, L. K., Palazzi, E., et al. 2014, A&A, 562, A70
  * Mineo et al. (2012) Mineo, S., Gilfanov, M., & Sunyaev, R. 2012, MNRAS, 419, 2095
  * Miniutti et al. (2019) Miniutti, G., Saxton, R. D., Giustini, M., et al. 2019, Nature, 573, 381
  * Mitra-Kraev et al. (2005) Mitra-Kraev, U., Harra, L. K., Güdel, M., et al. 2005, A&A, 431, 679
  * Modjaz et al. (2009) Modjaz, M., Li, W., Butler, N., et al. 2009, ApJ, 702, 226
  * Moustakas et al. (2013) Moustakas, J., Coil, A. L., Aird, J., et al. 2013, ApJ, 767, 50
  * Nakar (2020) Nakar, E. 2020, Phys. Rep, 886, 1
  * Nakar & Sari (2010) Nakar, E. & Sari, R. 2010, ApJ, 725, 904
  * Nakar & Sari (2012) Nakar, E. & Sari, R. 2012, ApJ, 747, 88
  * Nandra et al. (2013) Nandra, K., Barret, D., Barcons, X., et al. 2013, arXiv e-prints, arXiv:1306.2307
  * Narayana Bhat et al. (2016) Narayana Bhat, P., Meegan, C. A., von Kienlin, A., et al. 2016, ApJS, 223, 28
  * Nidever et al. (2021) Nidever, D. L., Dey, A., Fasbender, K., et al. 2021, AJ, 161, 192
  * Novara et al. (2020) Novara, G., Esposito, P., Tiengo, A., et al. 2020, ApJ, 898, 37
  * Nynka et al. (2018) Nynka, M., Ruan, J. J., Haggard, D., & Evans, P. A. 2018, ApJ, 862, L19
  * Ochsenbein et al. (2000) Ochsenbein, F., Bauer, P., & Marcout, J. 2000, A&AS, 143, 23
  * Pallavicini et al. (1990) Pallavicini, R., Tagliaferri, G., & Stella, L. 1990, A&A, 228, 403
  * Palmer et al. (2005) Palmer, D. M., Barthelmy, S., Gehrels, N., et al. 2005, Nature, 434, 1107
  * Pandey & Singh (2008) Pandey, J. C. & Singh, K. P. 2008, MNRAS, 387, 1627
  * Park et al. (2006) Park, T., Kashyap, V. L., Siemiginowska, A., et al. 2006, ApJ, 652, 610
  * Pastor-Marazuela et al. (2020) Pastor-Marazuela, I., Webb, N. A., Wojtowicz, D. T., & van Leeuwen, J. 2020, A&A, 640, A124
  * Peacock (1983) Peacock, J. A. 1983, MNRAS, 202, 615
  * Peng et al. (2010) Peng, Y.-j., Lilly, S. J., Kovač, K., et al. 2010, ApJ, 721, 193
  * Peng et al. (2019) Peng, Z.-K., Yang, Y.-S., Shen, R.-F., et al. 2019, ApJ, 884, L34
  * Peretz & Behar (2018) Peretz, U. & Behar, E. 2018, MNRAS, 481, 3563
  * Pescalli et al. (2015) Pescalli, A., Ghirlanda, G., Salafia, O. S., et al. 2015, MNRAS, 447, 1911
  * Phillips et al. (2020) Phillips, M. W., Tremblin, P., Baraffe, I., et al. 2020, A&A, 637, A38
  * Piran (2004) Piran, T. 2004, Reviews of Modern Physics, 76, 1143
  * Pradhan et al. (2020) Pradhan, P., Falcone, A. D., Kennea, J. A., & Burrows, D. N. 2020, Journal of Astronomical Telescopes, Instruments, and Systems, 6, 038002
  * Pye et al. (2015) Pye, J. P., Rosen, S., Fyfe, D., & Schröder, A. C. 2015, A&A, 581, A28
  * Racusin et al. (2009) Racusin, J. L., Liang, E. W., Burrows, D. N., et al. 2009, ApJ, 698, 43
  * Rau et al. (2005) Rau, A., Kienlin, A. V., Hurley, K., & Lichti, G. G. 2005, A&A, 438, 1175
  * Rau et al. (2016) Rau, A., Nandra, K., Aird, J., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99052B
  * Rees (1988) Rees, M. J. 1988, Nature, 333, 523
  * Reines et al. (2013) Reines, A. E., Greene, J. E., & Geha, M. 2013, ApJ, 775, 116
  * Rejkuba et al. (2011) Rejkuba, M., Harris, W. E., Greggio, L., & Harris, G. L. H. 2011, A&A, 526, A123
  * Remillard & McClintock (2006) Remillard, R. A. & McClintock, J. E. 2006, ARA&A, 44, 49
  * Rhode et al. (2007) Rhode, K. L., Zepf, S. E., Kundu, A., & Larner, A. N. 2007, AJ, 134, 1403
  * Robrade et al. (2010) Robrade, J., Poppenhaeger, K., & Schmitt, J. H. M. M. 2010, A&A, 513, A12
  * Rosen et al. (2016) Rosen, S. R., Webb, N. A., Watson, M. G., et al. 2016, A&A, 590, A1
  * Rots & Budavári (2011) Rots, A. H. & Budavári, T. 2011, ApJS, 192, 8
  * Rowlinson et al. (2013) Rowlinson, A., O’Brien, P. T., Metzger, B. D., Tanvir, N. R., & Levan, A. J. 2013, MNRAS, 430, 1061
  * Rowlinson et al. (2010) Rowlinson, A., O’Brien, P. T., Tanvir, N. R., et al. 2010, MNRAS, 409, 531
  * Sakamoto et al. (2008) Sakamoto, T., Barthelmy, S. D., Barbier, L., et al. 2008, ApJS, 175, 179
  * Sapir et al. (2013) Sapir, N., Katz, B., & Waxman, E. 2013, ApJ, 774, 79
  * Sarin et al. (2021) Sarin, N., Ashton, G., Lasky, P. D., et al. 2021, arXiv e-prints, arXiv:2105.10108
  * Saxton et al. (2021) Saxton, R., Komossa, S., Auchettl, K., & Jonker, P. G. 2021, Space Sci. Rev., 217, 18
  * Saxton et al. (2008) Saxton, R. D., Read, A. M., Esquej, P., et al. 2008, A&A, 480, 611
  * Sazonov et al. (2021) Sazonov, S., Gilfanov, M., Medvedev, P., et al. 2021, MNRAS, 508, 3820
  * Sazonov & Khabibullin (2017) Sazonov, S. & Khabibullin, I. 2017, MNRAS, 466, 1019
  * Schlafly & Finkbeiner (2011) Schlafly, E. F. & Finkbeiner, D. P. 2011, ApJ, 737, 103
  * Schlafly et al. (2019) Schlafly, E. F., Meisner, A. M., & Green, G. M. 2019, ApJS, 240, 30
  * Schmitt & Liefke (2004) Schmitt, J. H. M. M. & Liefke, C. 2004, A&A, 417, 651
  * Sivakoff et al. (2005) Sivakoff, G. R., Sarazin, C. L., & Jordán, A. 2005, ApJ, 624, L17
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163
  * Soderberg et al. (2008) Soderberg, A. M., Berger, E., Page, K. L., et al. 2008, Nature, 454, 246
  * Sorce et al. (2014) Sorce, J. G., Tully, R. B., Courtois, H. M., et al. 2014, MNRAS, 444, 527
  * Starling et al. (2011) Starling, R. L. C., Wiersema, K., Levan, A. J., et al. 2011, MNRAS, 411, 2792
  * Stiele et al. (2012) Stiele, H., Muñoz-Darias, T., Motta, S., & Belloni, T. M. 2012, MNRAS, 422, 679
  * Stratta et al. (2013) Stratta, G., Gendre, B., Atteia, J. L., et al. 2013, ApJ, 779, 66
  * Strohmayer & Watts (2005) Strohmayer, T. E. & Watts, A. L. 2005, ApJ, 632, L111
  * Sun et al. (2019) Sun, H., Li, Y., Zhang, B.-B., et al. 2019, ApJ, 886, 129
  * Sun et al. (2015) Sun, H., Zhang, B., & Li, Z. 2015, ApJ, 812, 33
  * Swartz et al. (2011) Swartz, D. A., Soria, R., Tennant, A. F., & Yukita, M. 2011, ApJ, 741, 49
  * Taggart & Perley (2021) Taggart, K. & Perley, D. A. 2021, MNRAS, 503, 3931
  * Tamba et al. (2019) Tamba, T., Bamba, A., Odaka, H., & Enoto, T. 2019, PASJ, 71, 90
  * Tananbaum et al. (2014) Tananbaum, H., Weisskopf, M., Tucker, W., Wilkes, B., & Edmonds, P. 2014, Reports on Progress in Physics, 77, 066902
  * Tanikawa et al. (2021) Tanikawa, A., Giersz, M., & Arca Sedda, M. 2021, arXiv e-prints, arXiv:2103.14185
  * Teplitz et al. (2010) Teplitz, H. I., Capak, P., Brooke, T., et al. 2010, Astronomical Society of the Pacific Conference Series, Vol. 434, The Spitzer Source List, ed. Y. Mizumoto, K. I. Morita, & M. Ohishi, 437
  * Terasawa et al. (2005) Terasawa, T., Tanaka, Y. T., Takei, Y., et al. 2005, Nature, 434, 1110
  * Thöne et al. (2011) Thöne, C. C., de Ugarte Postigo, A., Fryer, C. L., et al. 2011, Nature, 480, 72
  * Tinney et al. (2014) Tinney, C. G., Faherty, J. K., Kirkpatrick, J. D., et al. 2014, ApJ, 796, 39
  * Traulsen et al. (2019) Traulsen, I., Schwope, A. D., Lamer, G., et al. 2019, A&A, 624, A77
  * Troja et al. (2019) Troja, E., Castro-Tirado, A. J., Becerra González, J., et al. 2019, MNRAS, 489, 2104
  * Troja et al. (2007) Troja, E., Cusumano, G., O’Brien, P. T., et al. 2007, ApJ, 665, 599
  * Troja et al. (2022) Troja, E., O’Connor, B., Ryan, G., et al. 2022, MNRAS, 510, 1902
  * Troja et al. (2020) Troja, E., van Eerten, H., Zhang, B., et al. 2020, MNRAS, 498, 5643
  * Tsvetkov & Bartunov (1993) Tsvetkov, D. Y. & Bartunov, O. S. 1993, Bulletin d’Information du Centre de Donnees Stellaires, 42, 17
  * Tully et al. (2013) Tully, R. B., Courtois, H. M., Dolphin, A. E., et al. 2013, AJ, 146, 86
  * van Buren (1981) van Buren, D. 1981, ApJ, 249, 297
  * van den Eijnden et al. (2018) van den Eijnden, J., Degenaar, N., Russell, T. D., et al. 2018, Nature, 562, 233
  * Virgili et al. (2009) Virgili, F. J., Liang, E.-W., & Zhang, B. 2009, MNRAS, 392, 91
  * Virgili et al. (2013) Virgili, F. J., Mundell, C. G., Pal’shin, V., et al. 2013, ApJ, 778, 54
  * Virgili et al. (2011) Virgili, F. J., Zhang, B., O’Brien, P., & Troja, E. 2011, ApJ, 727, 109
  * Vito et al. (2016) Vito, F., Gilli, R., Vignali, C., et al. 2016, MNRAS, 463, 348
  * von Kienlin et al. (2014) von Kienlin, A., Meegan, C. A., Paciesas, W. S., et al. 2014, ApJS, 211, 13
  * Walton et al. (2016) Walton, D. J., Fürst, F., Bachetti, M., et al. 2016, ApJ, 827, L13
  * Wanderman & Piran (2010) Wanderman, D. & Piran, T. 2010, MNRAS, 406, 1944
  * Wanderman & Piran (2015) Wanderman, D. & Piran, T. 2015, MNRAS, 448, 3026
  * Wang et al. (2018) Wang, J., Zhu, Z. P., Xu, D., et al. 2018, ApJ, 867, 147
  * Wang et al. (2016) Wang, S., Liu, J., Qiu, Y., et al. 2016, ApJS, 224, 40
  * Wang et al. (2015) Wang, X.-G., Zhang, B., Liang, E.-W., et al. 2015, ApJS, 219, 9
  * Warren et al. (2007) Warren, S. J., Cross, N. J. G., Dye, S., et al. 2007, arXiv e-prints, astro
  * Waxman & Katz (2017) Waxman, E. & Katz, B. 2017, Shock Breakout Theory, ed. A. W. Alsabti & P. Murdin, 967
  * Webb et al. (2020) Webb, N. A., Coriat, M., Traulsen, I., et al. 2020, A&A, 641, A136
  * Welsh et al. (2007) Welsh, B. Y., Wheatley, J. M., Seibert, M., et al. 2007, ApJS, 173, 673
  * Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9
  * Whitmore et al. (2016) Whitmore, B. C., Allam, S. S., Budavári, T., et al. 2016, AJ, 151, 134
  * Wiegert et al. (2015) Wiegert, T., Irwin, J., Miskolczi, A., et al. 2015, AJ, 150, 81
  * Wiersema et al. (2007) Wiersema, K., Savaglio, S., Vreeswijk, P. M., et al. 2007, A&A, 464, 529
  * Wilkes & Tucker (2019) Wilkes, B. & Tucker, W., eds. 2019, The Chandra X-ray Observatory, 2514-3433 (IOP Publishing)
  * Woods & Thompson (2006) Woods, P. M. & Thompson, C. 2006, Soft gamma repeaters and anomalous X-ray pulsars: magnetar candidates, Vol. 39, 547–586
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, ApJ, 140, 1868
  * Xiao et al. (2019) Xiao, D., Zhang, B.-B., & Dai, Z.-G. 2019, ApJ, 879, L7
  * Xu et al. (2008) Xu, D., Watson, D., Fynbo, J., et al. 2008, in 37th COSPAR Scientific Assembly, Vol. 37, 3512
  * Xue et al. (2016) Xue, Y. Q., Luo, B., Brandt, W. N., et al. 2016, ApJS, 224, 15
  * Xue et al. (2019) Xue, Y. Q., Zheng, X. C., Li, Y., et al. 2019, Nature, 568, 198
  * Yang et al. (2016) Yang, G., Brandt, W. N., Luo, B., et al. 2016, ApJ, 831, 145
  * Yang et al. (2019) Yang, G., Brandt, W. N., Zhu, S. F., et al. 2019, MNRAS, 1535
  * Yi et al. (2014) Yi, S. X., Dai, Z. G., Wu, X. F., & Wang, F. Y. 2014, arXiv e-prints, arXiv:1401.1601
  * Yi et al. (2016) Yi, S.-X., Xi, S.-Q., Yu, H., et al. 2016, ApJS, 224, 20
  * Yuan et al. (2015) Yuan, W., Zhang, C., Feng, H., et al. 2015, PoS, SWIFT 10, 006
  * Yuan et al. (2017) Yuan, W., Zhang, C., Ling, Z., et al. 2017, in The X-ray Universe 2017, ed. J.-U. Ness & S. Migliari, 240
  * Zhang (2013) Zhang, B. 2013, ApJ, 763, L22
  * Zhang (2018) Zhang, B. 2018, The Physics of Gamma-Ray Bursts (Cambridge University Press)
  * Zhang et al. (2018) Zhang, B. B., Zhang, B., Sun, H., et al. 2018, Nature Communications, 9, 447
  * Zheng et al. (2017) Zheng, X. C., Xue, Y. Q., Brandt, W. N., et al. 2017, ApJ, 849, 127

##  Appendix A Spatial location and duration of X-ray events

To estimate the duration of the final sample of FXRTs, we computed the
$T_{90}$ duration parameter. $T_{90}$ measures the time over which the event
emits from 5% to 95% of its total measured counts (in the 0.5–7.0 keV band in
our case). Figure 20 shows the $T_{90}$ duration (_orange region_) for each
event, as well as their light curves (with a bin time of 1 ks) in unit of
counts.

Figure 20:  Light curves for each FXRT candidate in units of counts and the
region covering the $T_{90}$ (which measures the time over which the event
emits from 5% to 95% of its total measured counts; _orange region_) The light
curves have a bin width of 1 ks.

Furthermore, Fig. 21 confirms that the final sample of FXRT candidates are
real celestial sources in the sky rather than detector artifacts. Due to
_Chandra_ ’s Lissajous dither pattern, executed during observation, the X-ray
photons of the FXRTs are distributed over dozens to hundreds of individual
pixels on the detector. The _first column_ of the figure shows the light
curves, color-coded by the phase in the light curve evolution. The _second
column_ shows the spatial location in $x$ and $y$ chip detector coordinates,
also color-coded by time, tracing out a sinusoidal-like evolution in $x$ and
$y$ coordinates over time. The _third and fourth columns_ show the $x$ and $y$
position changes (in _blue_ and _purple_ , respectively, over time, with the
light curve superimposed in dark gray.

Figure 21: Lissajous dither pattern in detector coordinates. _First column:_
FXRT 0.5–7.0 keV light curves in count units, color-coded as a function of
time. _Second column:_ _Chandra_ 0.5–7.0 keV images in detector coordinates,
with the same color-coding as a function of time, demonstrating the temporal
movement of the source on the detector in response to the Lissajous dither
pattern. A flaring pixel would appear as a point on these plots. _Third and
fourth columns:_ x (_blue_) and y (_purple_) detector coordinates,
respectively, of the detected X-ray photons from the FXRTs as a function of
time, with the candidate light curves superimposed as solid dark gray lines.
Figure 21: (continued)

##  Appendix B Color-magnitude diagram of stellar matches

Figure 22: Color-magnitude diagrams, considering only Pan-STARRS and DECam
counterparts (_gray background points_) of X-ray sources classified as stars
according to Criterion 2 (see Sect. 2.5.2). As a comparison, we overplot
isochrones with different ages (from $\log(\rm{Age}){=}7.0-10.0$) taken from
the MIST package (Dotter 2016; Choi et al. 2016). Each panel represents
different metallicities (from $\mathrm{[Fe/H]}{=}-3.0$ to $+0.5$), while
_solid_ and _dashed lines_ are isochrones with attenuations of $A_{V}{=}$0.0
and 5.0, respectively.

To further demonstrate the stellar-like nature of the star candidates (beyond
identification by _Gaia_), we show an example $M_{\rm g}$ versus $g-i$ color-
magnitude diagram (see Fig. 22) considering all Pan-STARRS and DECam
counterparts of X-ray sources classified as stars according to Criterion 2
(see Sect. 2.5.2). Isochrones with different ages (from
$\log(\rm{Age}){=}7.0-10.0$) taken from the MIST package (Dotter 2016; Choi et
al. 2016) are overplotted, with each panel representing different
metallicities (from $\mathrm{[Fe/H]}{=}-3.0$ to $+0.5$). _Solid_ and _dashed
lines_ denote isochrones with attenuations of $A_{V}{=}$0.0 and 5.0,
respectively. The vast majority of the stars fall on these tracks. According
to SIMBAD, the outliers are identified as PNe, YSOs, or emission-line stars.
We additionally stress that the Pan-STARRS and DECam colors are not
necessarily taken in a purely simultaneous manner; in the case of Pan-STARRS,
they are averaged over the duration of the survey, while for DECam they come
from only a few disjoint epochs.