Searching for fast extragalactic X-ray transients in Chandra surveys

TL;DR

This paper develops a method to detect fast extragalactic X-ray transients in Chandra data, identifies 13 candidates, and estimates their event rate, highlighting the potential for future large-scale transient discoveries with upcoming missions.

Contribution

The paper introduces a new detection method for fast extragalactic X-ray transients and applies it systematically to archival Chandra surveys, identifying candidates and estimating their occurrence rate.

Findings

13 transient candidates identified in archival data

Event rate estimated at approximately 59 events per year per square degree

Future missions could discover thousands of such transients

Abstract

Recent works have discovered two fast ($\approx 10$ ks) extragalactic X-ray transients in the Chandra Deep Field-South (CDF-S XT1 and XT2). These findings suggest that a large population of similar extragalactic transients might exist in archival X-ray observations. We develop a method that can effectively detect such transients in a single Chandra exposure, and systematically apply it to Chandra surveys of CDF-S, CDF-N, DEEP2, UDS, COSMOS, and E-CDF-S, totaling 19~Ms of exposure. We find 13 transient candidates, including CDF-S XT1 and XT2. With the aid of available excellent multiwavelength observations, we identify the physical nature of all these candidates. Aside from CDF-S XT1 and XT2, the other 11 sources are all stellar objects, and all of them have $z$-band magnitudes brighter than 20. We estimate an event rate of $59^{+77}_{-38}\ \rm{evt\ yr^{-1} deg^{-2}}$ for CDF-S XT-like transients with 0.5-7 keV peak fluxes $\log F_{\rm peak} \gtrsim -12.6$ (erg cm$^{-2}$ s$^{-1}$). This event rate translates to $\approx 15^{+20}_{-10}$ transients existing among Chandra archival observations at Galactic latitudes $|b|>20^{\circ}$, which can be probed in future work. Future missions such as Athena and the Einstein Probe with large grasps (effective area $\times$ field of view) are needed to discover a large sample ($\sim$ thousands) of fast extragalactic X-ray transients.

Tables

Table 1. Table 1: Properties of X-ray Surveys Analyzed in this Work

Survey	Area	Total Exp.	Obs. Num.	Src. Num.	Reference
(1)	(2)	(3)	(4)	(5)	(6)
CDF-S	0.13	6.9	101	1008	Luo et al. (2017)
CDF-N	0.12	2.0	20	683	Xue et al. (2016)
DEEP2	3.28	3.7	139	2976	Goulding et al. (2012); Nandra et al. (2015)
UDS	0.33	1.2	25	868	Kocevski et al. (2018); Suh et al. in prep.
COSMOS	2.20	4.5	117	4016	Civano et al. (2016); Marchesi et al. (2016)
E-CDF-S	0.31	1.0	9	1003	Xue et al. (2016)
All	6.38	19.3	411	10554	–

Table 2. Table 2: X-ray  Properties of Transient Candidates

ID	Survey	RA	DEC	Pos. Unc.	Obs. ID	Off. Ang.	HR	$\log F_{\mathrm{peak}}$	Method
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
1	CDF-S	53.16156	$-27.85934$	0.32$″$	16454	4.3$\prime$	$-{0.13}_{-0.10}^{+0.09}$	$-11.41$	1,2
2	CDF-S	53.07648	$-27.87339$	0.31$″$	16453	4.1$\prime$	$-{0.32}_{-0.09}^{+0.08}$	$-12.18$	1,2
3	CDF-N	189.02046	$62.33728$	0.20$″$	957	6.6$\prime$	$-{0.54}_{-0.12}^{+0.08}$	$-12.59$	1
4	CDF-N	189.10587	$62.23467$	0.10$″$	3389	3.2$\prime$	$-{0.82}_{-0.06}^{+0.05}$	$-12.82$	1
5	DEEP2	215.07414	$53.10650$	0.36$″$	9875	6.7$\prime$	$-{0.72}_{-0.12}^{+0.08}$	$-12.53$	1
6	DEEP2	214.96015	$52.74344$	0.26$″$	9456	6.6$\prime$	$-{0.63}_{-0.14}^{+0.12}$	$-12.97$	1
7	DEEP2	214.61007	$52.54347$	0.20$″$	9735	4.8$\prime$	$-{0.83}_{-0.17}^{+0.04}$	$-12.75$	1,2
8	DEEP2	214.66798	$52.66658$	0.11$″$	5849	3.0$\prime$	$-{0.77}_{-0.10}^{+0.08}$	$-13.21$	2
9	DEEP2	252.12761	$34.96337$	0.53$″$	8636	7.5$\prime$	$-{0.62}_{-0.12}^{+0.09}$	$-12.46$	1
10	UDS	34.48317	$-5.09118$	0.96$″$	17305	0.7$\prime$	$-{0.53}_{-0.18}^{+0.15}$	$-13.14$	1
11	COSMOS	149.75403	$2.14188$	0.30$″$	8021	4.0$\prime$	$-{0.80}_{-0.14}^{+0.09}$	$-13.38$	1
12	COSMOS	149.82641	$2.71812$	0.30$″$	15214	5.9$\prime$	$-{0.58}_{-0.09}^{+0.07}$	$-12.59$	1
13	COSMOS	149.99794	$2.77972$	0.90$″$	15211	6.5$\prime$	$-{0.69}_{-0.14}^{+0.12}$	$-12.66$	2

Table 3. Table 3: Counterpart Properties of Transient Candidates

ID	Source	RAc	DECc	Offset	Magz	$z$	$z$ type	Gaia
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
1	CANDELS	53.16157	$-27.85936$	0.07$″$	27.9	2.14	phot	n/a
2	CANDELS	53.07659	$-27.87329$	0.50$″$	24.5	0.74	spec	n/a
3	WIRCam	189.02037	$62.33728$	0.14$″$	16.8	0.00	spec	star
4	CANDELS	189.10575	$62.23467$	0.21$″$	16.4	0.00	spec	star
5	DEEP2-1	215.07411	$53.10657$	0.26$″$	19.6	0.00	spec	n/a
6	DEEP2-1	214.95966	$52.74351$	1.10$″$	14.1	n/a	n/a	star
7	DEEP2-1	214.61031	$52.54338$	0.62$″$	17.1	0.00	spec	n/a
8	DEEP2-1	214.66805	$52.66666$	0.33$″$	16.8	n/a	n/a	star
9	DEEP2-2	252.12746	$34.96339$	0.45$″$	15.8	0.00	spec	star
10	HSC	34.48311	$-5.09118$	0.25$″$	18.1	0.00	spec	star
11	UltraVISTA	149.75412	$2.14183$	0.38$″$	16.7	0.00	spec	star
12	UltraVISTA	149.82649	$2.71803$	0.41$″$	15.8	0.00	spec	star
13	UltraVISTA	149.99794	$2.77960$	0.40$″$	16.5	0.00	spec	n/a

Equations

$$\mathrm{cntR}(\mathrm{t})\propto \{\begin{array}{ll}0,& t<0\\ t,& 0\le t<t_1\\ t^{{\alpha }_1},& t_1\le t<t_2\\ t^{{\alpha }_2},& t\ge t_2\end{array}$$

$$N_{\mathrm{net}}\approx 1.6\times 10^{14}{F}_{\mathrm{peak}}(\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}).$$

$$P_{\mathrm{eff}}=\frac{{\int }_{-\infty }^{+\infty }{P}_{\det }(t_{\mathrm{m}})d{t}_{\mathrm{m}}}{t_{\exp }}.$$

$$\mathcal{N}=\sum _i{R}_i{\Omega }_i{t}_i,$$

$$\begin{array}{r}\mathcal{N}=\sum _{i}R\Omega t_i=R\Omega \sum _i{t}_i,\end{array}$$

$$\begin{array}{rl}R& =\frac{\mathcal{N}}{\Omega {\sum }_i{t}_i}\end{array}$$

$$\mathcal{N}=R({\Omega }_{\mathrm{I}}{T}_{\mathrm{I}}+{\Omega }_{\mathrm{S},1}{T}_{\mathrm{S},1}+{\Omega }_{\mathrm{S},2}{T}_{\mathrm{S},2}+{\Omega }_{\mathrm{S},2}{T}_{\mathrm{S},3})=15_{-10}^{+20},$$

$$\frac{dR}{d\log F_{\mathrm{peak}}}\propto F_{\mathrm{peak}}^{-\gamma }$$

$$\begin{array}{rl}\mathcal{N}& \propto F_{\lim }^{-\gamma }\Omega T\\ & \propto F_{\lim }^{-\gamma +1}\frac{\Omega }{F_{\lim }}T\\ & \propto A^{\gamma -1}(\Omega A)T\\ & \propto A^{\gamma -1}GT,\end{array}$$

$$\begin{array}{rl}{P}_{\mathrm{bright}}& =\frac{{\int }_{-12.2}^{\infty }{F}_{\mathrm{peak}}^{-1}d\log F_{\mathrm{peak}}}{{\int }_{-12.6}^{\infty }{F}_{\mathrm{peak}}^{-1}d\log F_{\mathrm{peak}}}\\ & =\mathrm{0.40.}\end{array}$$

Full text

††pubyear: 2018††pagerange: Searching for fast extragalactic X-ray transients in Chandra surveys–B

1 Introduction

X-ray observations can provide uniquely insightful views of many astronomical phenomena such as accretion and mergers of compact objects (e.g. Brandt & Alexander 2015; Pooley et al. 2018). The X-ray sky is variable. Main-sequence stars (especially dwarfs) have strong flares powered by magnetic reconnection, generally lasting minutes to hours (e.g. Haisch et al., 1991; Güdel & Nazé, 2009). X-ray binaries have various variability behaviors such as pulsations, bursts, and quasi-periodic oscillations (e.g. van der Klis, 1989; Belloni & Stella, 2014). Active galactic nuclei (AGNs) typically have red-noise X-ray variability, with characteristic amplitudes being $\lesssim 0.5$ dex on timescales from $\sim$ an hour to $\sim$ 10 years (e.g. Markowitz et al., 2003a; Markowitz et al., 2003b; Yang et al., 2016; Paolillo et al., 2017; Zheng et al., 2017). However, some relatively rare AGN and related phenomena, e.g. tidal disruption events, changing-look AGNs, and narrow-line Seyfert 1s, can have larger X-ray variability amplitudes (e.g. Komossa 2015; Kara et al. 2016; Ricci et al. 2016; Gallo 2018).

Recently, a new type of X-ray variability phenomenon has been revealed in the form of two relatively faint X-ray transients found in the Chandra observations of the Chandra Deep Field-South (CDF-S XT1 and XT2; Bauer et al. 2017; Xue et al. 2019). Both transients are fast ($T_{90}\approx 10$ ks,111$T_{90}$ is defined as the time interval between the arrival times of the 5%-th photon and the 95%-th photon. observed-frame). Their origins are found to be extragalactic, with optical/near-infrared (NIR) counterparts at $z\approx 2.1$ (photometric redshift) and $z=0.74$ (spectroscopic redshift), respectively. Both transients have $\gtrsim 100$ counts detected, corresponding to enormous amounts of energy release ($\gtrsim 10^{48}$ erg, assuming isotropic emission). Due to the lack of simultaneous multiwavelength observations and the small sample size of transients, the physical origins are not well determined with some possibilities being off-axis gamma-ray bursts, tidal-disruption events, mergers of neutron stars, and shock-breakout events. In this paper, we regard CDF-S XT1 and XT2 as the same “type” of transients considering their observational similarities in flux, timespan, and extragalactic origin, although their physical causes might be different.

Given the short timescales ($T_{90}\approx 10$ ks) and large numbers of counts ($\gtrsim 100$) for CDF-S XT1 and XT2, such transients should be easy to detect in any $\gtrsim 10$ ks Chandra exposure. The two transients are both detected in a small survey area ($\approx 480$ arcmin2) and relatively short timespan (2014 October and 2015 March), indicating that a large population of X-ray transients might exist. Bauer et al. (2017) performed a preliminary transient search in the Chandra source catalog (CSC; Evans et al. 2010), which compiled Chandra observations before 2010 August 10. They did not find transients similar to the CDF-S transients. However, this CSC search is not conclusive, because the CSC is not dedicated to discovering fast transients and thus potential transients might be missed or poorly/incorrectly characterized. Also, many CSC sources have only a single short Chandra visit, making it difficult to ascertain the transient and quiescent levels. The CSC sources also generally lack deep optical/NIR observations, preventing further studies of the physical nature of potential transients.

To mitigate the above issues, in this work, we search for similar transients in Chandra archival observations of X-ray surveys. We develop a method to identify CDF-S XT-like transients in a single Chandra exposure, which is applicable to any Chandra imaging observation. In the surveys, most X-ray sources have been visited by two or more Chandra exposures, allowing us to inspect transients with multi-epoch X-ray data and study their quiescent behaviors. Deep multiwavelength data are critical in clarifying the physical origins of X-ray transients. CDF-S XT1 and XT2 have optical/NIR counterparts with $V\gtrsim 25$ mag and $H\gtrsim 24$ mag (Bauer et al. 2017; Xue et al. 2019), well beyond the detection limit of wide-field surveys such as SDSS (York et al., 2000) and UKIDSS (Lawrence et al., 2007). Glennie et al. (2015) discovered an X-ray transient in one Chandra archival observation, but were not able to clarify its physical origin due to the lack of deep multiwavelength data. Our selected X-ray surveys are accompanied by deep multiwavelength observations, allowing identifications of optical/NIR counterparts for the selected transients.

The main aim of this paper is to search for fast extragalactic X-ray transients that are similar to CDF-S XT1 and XT2 rather than general X-ray transients (although our search is effective for a fairly wide range of transients; see Appendix A). The structure of this paper is organized as follows. We detail our X-ray transient-selection algorithm and assess its efficiency with simulations in §2. We describe our X-ray data, selection of transient candidates, and optical/NIR counterparts in §3. We estimate the event rate of CDF-S XT-like transients based on our results and discuss the prospect of future missions in §4. We summarize our results in §5.

Throughout this paper, we assume a cosmology with $H_{0}=70$ km s*-1* Mpc*-1*, $\Omega_{M}=0.3$, and $\Omega_{\Lambda}=0.7$. Quoted uncertainties are at the $1\sigma$ (68%) confidence level, unless otherwise stated. Quoted optical/infrared magnitudes are AB magnitudes.

2 Methodology

In §2.1, we detail our algorithm for transient-candidate searching, which is designed to find CDF-S XT-like events within individual Chandra exposures. Our algorithm is simple and fast, and can be easily implemented for any individual Chandra observations. We perform intensive Monte Carlo simulations and assess the sensitivity of our algorithm in §2.2.

2.1 Algorithm for Transient-Candidate Selection

Our algorithm works on an unbinned Chandra light curve, i.e. an array of photon arrival times of a source, for which the background has been estimated. Below, we denote ${N_{\rm tot}}$ (${N_{\rm bkg}}$) as the number of total (background) counts for the light curve. We require that the source lies within an off-axis angle of $8^{\prime}$, following previous Chandra studies (e.g. Vito et al. 2016; Yang et al. 2016). This is because Chandra’s performance (as measured by, e.g. effective area and PSF size) drops significantly beyond $8^{\prime}$. Additionally, we require that the light-curve length is shorter than 50 ks to avoid large numbers of accumulated background counts in long exposures. Exposures longer than 50 ks are chopped into a few parts to meet this requirement (§3). In §2.2.3, we show that our algorithm reaches a uniform sensitivity for nearly all observations shorter than 50 ks. Note that the choice of 50 ks is somewhat subjective; the flux limit and the results of our transient search (§2.2.3 and §3.3) do not change significantly if we adjust this value between $\approx 16$ ks and $\approx 100$ ks. Choosing a value below $\approx 16$ ks could chop some observations into $\lesssim 8$ ks parts, which are ineffective in our selection of XT-like transients (see §2.2). Choosing a value above $\approx 100$ ks could leave some long observations unchopped, which have relatively high accumulated background, affecting transient detection.222If we do not chop the observations, the actually detected extragalactic transients among our data will be the same, although there will be four more stellar flares detected (§3).

Our algorithm first calculates ${N_{1}}$ and ${N_{2}}$, defined as the numbers of counts at $t=(t_{\rm s},t_{\rm m})$ and $t=(t_{\rm m},t_{\rm e})$, respectively, where $t_{\rm s}$ and $t_{\rm e}$ are the times when the exposure starts and ends, respectively, and $t_{\rm m}=(t_{\rm s}+t_{\rm e})/2$, i.e. the midpoint of exposure time. Since typical Chandra observations are continuous and uninterrupted by background flares ($\approx 1\%$ of exposure time), our two-part division of the exposure is legitimate.

We select a source in an observation as a transient candidate if it satisfies all of the following criteria (Method 1):

1. (A)

${N_{\rm tot}}$ is larger than the 5${\sigma}$ Poisson upper limit of ${N_{\rm bkg}}$; 2. (B)

$N_{1}$ and $N_{2}$ are statistically different at a ${>4\sigma}$ significance level; 3. (C)

${N_{1}>5\times N_{2}}$ or ${N_{2}>5\times N_{1}}$.

Criterion A filters out faint sources that have low signal-to-noise ratios (S/N), and thus boosts the speed of the selection process. This criterion is also helpful in avoiding false detections caused by rare background flares, since flares can dominate the detected counts for faint sources. Criterion B selects sources that have significantly different count rates in the first-half and second-half exposures. Technically, we realize Criterion B with the $E$-test (Krishnamoorthy & Thomson, 2004). The $E$-test can test if two Poisson variables ($N_{1}$ and $N_{2}$ in our case) are drawn from the same distribution, and simultaneously considers the statistical fluctuations of both variables. Criteria A and B are based on statistical significance, and they select high-S/N sources with significant variability. However, these criteria are not sufficient, since they cannot rule out AGNs which have stochastic variability. To deal with this AGN issue, we also add Criterion C, which requires that the flux-variation amplitude is large. Since the characteristic AGN variability amplitudes (on timescales from $\sim$ an hour to $\sim$ 10 years) are a factor of $\lesssim 3$ (§1), we choose the amplitude threshold as a factor of 5 to cleanly rule out AGN variability. We note that the choice of amplitude threshold is empirical: a low value could not remove AGNs effectively; a high value could miss potential transients. We have tested adjusting the threshold slightly (e.g., by a factor of 3 or 4 instead of 5), and the number of extragalactic transients we found in §3 does not change.

Method 1 is not efficient in selecting transients that happen at $t\approx t_{\rm m}$, because these transients will have similar $N_{1}$ and $N_{2}$. To overcome this defect, we also select transients with the following method. We denote $N^{\prime}_{1}$ as the number of counts at $t=(t_{\rm bgn},t_{\rm q1})$ plus that at $t=(t_{\rm q3},t_{\rm end})$, where $t_{\rm q1}$ and $t_{\rm q3}$ are the first and third quartiles of the observation time, and $N^{\prime}_{2}$ as the number of counts at $t=(t_{\rm q1},t_{\rm q3})$. We also select a source as a transient candidate, if it satisfies (Method 2)

1. (A*′*)

${N_{\rm tot}}$ is larger than the 5${\sigma}$ Poisson upper limit of ${N_{\rm bkg}}$; 2. (B*′*)

$N^{\prime}_{1}$ and $N^{\prime}_{2}$ are statistically different at a ${>4\sigma}$ significance level; 3. (C*′*)

${N^{\prime}_{1}>5\times N^{\prime}_{2}}$ or ${N^{\prime}_{2}>5\times N^{\prime}_{1}}$.

In §2.2, we prove the necessity of adopting both Method 1 and Method 2 for transient selection.

2.2 Efficiency of the Selection Algorithm

In this Section, we assess the efficiency of our transient-selection algorithm (§3.2) with Monte Carlo simulations.

In §2.2.1, we detail our simulation configurations. In §2.2.2, we define a “gauge” to measure the efficiency of our algorithm. In §2.2.3, we present our simulation results, i.e. the performance of our algorithm.

2.2.1 Simulation Configurations

The simulations are based on a fiducial light-curve model. Since our main goal is to search for fast extragalactic transients analogous to CDF-S XT1 and XT2, we adopt a light-curve model similar to the best-fit models of these two transients (Bauer et al. 2017; Xue et al. 2019). The light-curve shape in the model is described by

[TABLE]

where cntR is the count rate in units of counts s*-1*. Here, we follow the convention that the transient starts at $t=0$. For $t$ between 0 and $t_{1}$, the cntR rises to the peak value. This time interval is very short ($\lesssim 100$ s for both CDF-S XT1 and XT2), and thus the exact functional form is not important. Here, we adopt a basic form of a linear rise and set $t_{1}=50$ s. For $t$ between $t_{1}$ and $t_{2}$, the light curve is roughly in a plateau with an index of $\alpha_{1}=-0.1$. This plateau only exists for XT2 (2.3 ks) but not for XT1, and we adopt $t_{2}=t_{1}+1$ ks. For $t>t_{2}$, the adopted cntR is a power-law decline with an index of $\alpha_{2}=-2$, which is between those of XT1 ($-1.5$) and XT2 ($-2.2$). We adopt a power-law spectral shape with photon index of ${\Gamma=1.6}$ for the model, which is consistent with those measured for both XT1 and XT2. We note that changing the model parameters slightly (e.g. changing $t_{1}$ to 100 s and $\Gamma$ to 2.0) does not significantly affect our simulation results. In Appendix A, we also perform simulations for some other types of transients that are significantly different from the CDF-S XTs, although these transients are not the main focus of this work; these simulations show that our algorithm can identify transients with timescales $\lesssim$ exposure time while the details of the light-curve shapes do not affect the sensitivity significantly.

We plot the adopted light-curve model in Fig. 1. The $T_{90}$ for this light-curve setting is 9.4 ks, similar to those of XT1 and XT2. This similarity is expected, because our model in Eq. 1 is based on the light-curve shapes of XT1 and XT2.

Under the fiducial-model configuration, the conversion between peak flux and total net counts is

[TABLE]

The conversion factor is calculated with pimms, assuming a typical off-axis angle of 5$\arcmin$ when accounting for vignetting (i.e. the drop of photon-collecting area toward large off-axis angle; see Appendix B for other off-axis angles).333See http://cxc.harvard.edu/toolkit/pimms.jsp for pimms; see http://cxc.harvard.edu/proposer/POG/html/chap4.html for vignetting.

Background noise is also needed for the simulations. Here, background includes both detector background and sky X-ray background for 0.5–7 keV. The background-extraction region is an annulus centered at the X-ray source (see §3.1 for details). The background level rises as a function of off-axis angle. In the simulations, we assume a background of ${5.6\times 10^{-5}}$ cnt s*-1*, which is the typical background level at an off-axis angle of 5$\arcmin$ (see Appendix B for other off-axis angles). The adopted background is also approximately the median value for all X-ray sources in our studied surveys. This background level only corresponds to ${\approx 3}$ background counts for a 50 ks light curve, which is the longest light curve analyzed (see §2.1).

2.2.2 Efficiency Gauge

For a given set of ${F_{\rm peak}}$ and ${t_{\rm exp}}$ (exposure time), we can estimate the probability of transient detection (${P_{\rm det}}$) as a function of ${t_{\rm m}}$ (observation midpoint; §2.1) with the simulation procedures described below.

Since the transient starts at $t=0$ (§2.2.1), $t_{\rm m}$ actually means the relative time between the exposure midpoint and the transient start time.

First, we simulate light curves in the time interval of $t=(-t_{\rm exp},\ t_{\rm exp})$. We divide $t=(-t_{\rm exp},\ t_{\rm exp})$ into small bins with $\Delta t=5$ s. We then calculate the expected total counts in each bin. Using these values, we generate the counts in each bin with a Poisson distribution, which gives a simulated light curve. We repeat the procedures and generate 1,000 light curves. We apply both Method 1 and Method 2 (§3.2) for these light curves and calculate the fraction of successful detections. We adopt this fraction as the detection probability ($P_{\rm det}$).

Fig. 2 displays an example of ${P_{\rm det}}$ vs. ${t_{\rm m}}$ for ${\log F_{\rm peak}=-12.7}$ (cgs) and ${t_{\rm exp}=30}$ ks. Besides showing the ${P_{\rm det}}$ when using both Method 1 and Method 2 (see §2.1), Fig. 2 also displays the ${P_{\rm det}}$ when using Method 1 and Method 2 separately. Note that ${P_{\rm det}}$ drops significantly for some ${t_{\rm m}}$ values when using Method 1 and Method 2 separately. However, such drops are greatly alleviated when using both Methods, indicating the necessity of our combined method strategy.

From Fig. 2, ${P_{\rm det}}$ (using both Methods) is not constant for different ${t_{\rm m}}$. This ${P_{\rm det}}$ variation makes it difficult to use ${P_{\rm det}}$ as a direct measure of algorithm performance as a function of ${F_{\rm peak}}$ and ${t_{\rm exp}}$. Therefore, we define an “effective” detection probability (${P_{\rm eff}}$) averaged over different ${t_{\rm m}}$ as a gauge to measure the efficiency, i.e.

[TABLE]

From this definition, ${P_{\rm eff}}$ ranges from 0 to ${\approx 1}$ for a given set of ${F_{\rm peak}}$ and ${t_{\rm exp}}$,444${P_{\rm eff}}$ might be slightly greater than unity, because a transient may be detected even when it is partially covered by the observations. with higher values indicating higher average detection efficiency.

2.2.3 Simulation Results

We calculate ${P_{\rm eff}}$ for different ${t_{\rm exp}}$ and ${F_{\rm peak}}$ and show the results in Fig. 3. As expected, ${P_{\rm eff}}$ rises toward high ${F_{\rm peak}}$ at a given ${t_{\rm exp}}$, because brighter sources have higher S/N. We choose ${\log F_{\rm peak}\approx-12.6}$ (cgs) as our detection limit, above which ${P_{\rm eff}\approx 1}$ for a wide range of ${t_{\rm exp}=8\text{--}50}$ ks. Note that this flux limit is much lower than the peak fluxes of CDF-S XT1 and XT2 (see Table 2). The estimated flux limit is mainly used to estimate the event rate in §4, we note that there are still non-zero probabilities to detect transients below this limit (see Fig. 3). Here, we remind readers that ${t_{\rm exp}=50}$ ks is the maximum exposure time accepted by our algorithm (see §2.1). We note that the simulation results above are calculated from our fiducial model which is similar to CDF-S XTs (§2.2.1; see Appendix A for some other transient models), since our main purpose is to find CDF-S XT-like transients. The simulation results are based on the instrumental response and background at a typical off-axis angle of ${5^{\prime}}$ (§2.2.1), and we present the results at other off-axis angles in Appendix B.

Below ${t_{\rm exp}=8}$ ks, ${P_{\rm eff}}$ drops significantly at a given ${F_{\rm peak}}$ (see Fig. 3). This is because, when the exposure time becomes significantly shorter than the transient timescale, the observed light curve will be similar to a normal variable source, and thus may not be selected by our algorithm. In our estimation of event rate (§4), we do not include observations that are shorter than 8 ks, although we do not discard these observations in our transient search (§3). Only a negligible fraction of observation time (${\approx 0.1\%}$; see §4) in our analyzed X-ray data is from ${<8}$ ks exposures.

3 Data and Analyses

The scope of this paper is to search for CDF-S XT-like extragalactic transients. Utilizing the methodology detailed in §2, we first perform an initial search for transient candidates in the X-ray survey data (§3.1 and §3.2). Since stellar objects can have strong X-ray flares that might be selected by our algorithm, we need to exclude stars from our selected transient candidates. We perform this task with the high-quality multiwavelength data available for the surveys (§3.3).

3.1 X-ray Data and Processing

In this work, we analyze the Chandra survey data from the CDF-S, CDF-N, DEEP2, UDS, COSMOS, and E-CDF-S regions. The survey properties are summarized in Table 1. DEEP2 includes the full field of EGS (DEEP2-1) and three other fields (DEEP2-2, DEEP2-3, and DEEP2-4) with shallower ($\approx 10$ ks) exposures. The total exposure time of these surveys is 19 Ms. All the surveys are at high Galactic latitude ($|b|\gtrsim 40^{\circ}$), matching our main interest of searching for extragalactic transients. Also, these surveys have deep multiwavelength coverage, allowing us to study the physical origins of the transients (§1 and §3.3).

We download all the Chandra data products of observations related to the surveys, and run the chandra_repro script in ciao 4.10.555chandra_repro cannot process observation 1431 (CDF-S), which consists of two separate exposures. For this observation, we use the data products from Luo et al. (2017), who split the observation into two continuous exposures. We perform transient searching for these two exposures independently (§3.2), but do not find transient candidates in the two exposures. The chandra_repro script performs standard cleaning and calibration processes,666http://cxc.harvard.edu/ciao/ahelp/chandra_repro.html and yields a clean event file for each observation. Based on the data products of chandra_repro, we produce the exposure map for each observation using the ciao script fluximage. The exposure maps denote the “effective” exposure times for different positions in the field of view, and instrumental factors such as bad pixels and vignetting are taken into account.

For each event file, we extract the 0.5–7 keV photons of each X-ray source presented in the X-ray catalogs (Table 1). Since the Chandra background is extremely low, any sources with $\gtrsim 10$ net counts should be detected by the X-ray surveys. This level of counts is much lower than that of our transient-selection sensitivity (see below), and thus we should not miss any transients due to their absence in the X-ray catalogs. The total events are extracted from an aperture of $1.5\times R_{90}$, where $R_{90}$ is the radius encircling 90% of the X-ray counts. We adopt $R_{90}$ as a function of off-axis angle from Table A1 of Vito et al. (2016). From simulations with the ciao script simulate_psf, we find that this aperture size (${1.5\times R_{90}}$) encircles nearly all (${\gtrsim 98\%}$) X-ray counts regardless of off-axis angle. The background events are extracted from an annulus with inner and outer radii of $1.5\times R_{90}$ and $1.5\times R_{90}+20~{}\rm{pixels}$. The background area is $9$ times larger than the source area for a typical source at an off-axis angle of $5\arcmin$. If the background region covers a nearby X-ray source, we mask the source (also with radius of $1.5\times R_{90}$), and do not include the masked area when estimating the background. We note that changing the source and background extraction regions slightly will not affect our qualitative results. We estimate the background counts in the source region (${N_{\rm bkg}}$; §2.1) by scaling the counts in the background region by a factor. Here, the scaling factor is the sum of the exposure-map values in the source area divided by that in the background area.

3.2 Selection of Transient Candidates

We apply the algorithm in §2.1 to the light curves extracted in §3.1. We note that the transient selection is only applied to sources with off-axis angle of ${<8^{\prime}}$ to avoid the low-quality X-ray data beyond ${8^{\prime}}$ (§2.1). If a light curve is longer than 50 ks (the maximum length accepted by our algorithm; §2.1), we chop it into several continuous parts with each having the same ${t_{\rm exp}}$ shorter than (or equal to) 50 ks (§2). For example, for a 80 ks exposure, we divide it into two parts each having ${t_{\rm exp}=40}$ ks. We then perform transient selection for each chopped light curve independently. After this observation-chopping process, we have 610 exposures with a median ${t_{\rm exp}}$ of 30 ks and a 20%–80% percentile range of 25–43 ks. We show the $t_{\rm exp}$ distribution of these 610 exposures in Fig. 4.

For Method 1 (2), Criterion A (A*′) selects a total of 9379 (9379) events in the 610 exposures analyzed. Among these events, Criterion B (B′) further selects 31 (24) events. Finally, Criterion C (C′) picks out 11 (5) events as the events selected by Method 1 (2). For the events filtered out by Criterion C (C′), $\approx 70\%$ of them are stellar flares, identified with the methods detailed in §3.3; the other $\approx 30\%$ have extragalactic origins. We have examined the light curves of these extragalactic sources and found all of them have significant non-zero quiescent fluxes, and thus they are likely AGNs rather than extragalactic transients. This result demonstrates the capability of Criterion C (C′*) in removing AGN variability (§2.1).

We merge the events selected by Method 1 and Method 2, leading to a sample of 13 unique transient candidates. Among these 13 candidates, 8 and 2 are uniquely selected by Method 1 and Method 2, respectively, indicating the importance of using both Methods (see §2).

We visually inspect the background light curves of these transient candidates, and do not find significant flares. We have checked the X-ray images of the transients in both sky and detector coordinates. For each source, the events are concentrated and extended in sky and detector coordinates, respectively. This indicates that the transient candidates are physical X-ray sources rather than hot pixels, because hot pixels will lead to extended (concentrated) patterns in the sky (detector) coordinates caused by Chandra dithering.

The X-ray properties of the 13 transient candidates are listed in Table 2. ID1 and ID2 are CDF-S XT1 and XT2, respectively. Their successful selection indicates that our method of transient searching is effective for selecting CDF-S XT-like transients (§2.2.3). For each transient candidate, we calculate the hardness ratio for the observation where the transient is identified. Here, hardness ratio is defined as $(H-S)/(H+S)$, where $H$ and $S$ are hard-band (2–7 keV) and soft-band (0.5–2 keV) net counts, respectively. The ${1\sigma}$ uncertainty is calculated with behr, a Bayesian code for hardness ratio estimation (Park et al., 2006). The results are listed in Table. 2. In Fig 5, we show the distribution of hardness ratios. The spectral shapes of XT1 and XT2 are harder than for other transient candidates.

In Fig. 6 (left), we show the light curves of the transient candidates during the observation when the transient happens. The light curves are derived from the X-ray events extracted in §3.2, and are binned by 5-count intervals. The data points in these light curves indicate total count rates, including contributions from the source and background. The estimated average background count rate is marked as the dashed line in each panel of Fig. 6 (left). The durations of XT1 and XT2 tend to be shorter than for other transient candidates (Fig. 6 left). The $T_{90}$ values of XT1 and XT2 are $5.0^{+4.2}_{-0.3}$ ks and $11.1^{+0.4}_{-0.6}$ ks, respectively (see Bauer et al. 2017 and Xue et al. 2019 for details). We do not derive $T_{90}$ for other sources, because $T_{90}$ cannot be derived for many transients that extend beyond the Chandra exposures (e.g. ID3 and ID9 in Fig. 6 left). Also, unlike XT1 and XT2, many of the other transient candidates have non-zero fluxes in the quiescent states, and thus their $T_{90}$ calculation requires careful subtraction of the quiescent fluxes, which is beyond the scope of this work.

We plot the long-term light curves in Fig. 6 (right), where each Chandra observation is represented by a data point. These data points indicate net count rates, which are background-subtracted. As expected, the transient observation generally has a count rate much higher than other observations. However, unlike the CDF-S XT1 and XT2 events, most of the other transient candidates have detectable signals in some of the non-transient observations. Also, CDF-S XT1 and XT2 tend to have higher hardness ratios than the rest of the selected transient candidates (Fig. 5). These differences indicate that most of the new transient candidates are physically distinct from CDF-S XT1 and XT2 (see §3.3).

3.3 Optical/NIR Counterparts

We have compiled the likelihood counterpart matching results from the survey catalogs (Table 1). All the transient candidates have optical/NIR counterparts. The counterpart properties are presented in Table 3. We also match the counterparts with the Gaia catalog (Gaia Collaboration et al., 2018) using a 1$\arcsec$ matching radius, and mark the sources with non-zero parallax and/or proper motion as “star” in Table 3.

We show the optical/IR image cutouts in Fig. 7. From Fig. 7, the optical positions777The positional errors of the optical/NIR counterparts are not provided in the corresponding catalogs. Estimating the optical/NIR positional errors requires addressing factors such as CCD saturation and seeing (for ground-based telescopes), which are beyond the scope of this work. are within (or marginally outside, i.e. ID6 and ID7) the ${3\sigma}$ X-ray positional errors, indicating that the X-ray and optical/NIR positions are generally consistent with each other. For ID6 and ID7, in the image cutouts nearby the X-ray positions, there appear to be no other optical/NIR sources except the counterparts, and thus the counterparts are likely the same physical objects as the X-ray sources.

From Table 3, ID1 and ID2 (namely CDF-S XT1 and XT2) are likely of extragalactic origin and have already been discussed in detail (Bauer et al. 2017; Xue et al. 2019). The other transients are relatively bright ($\mathrm{mag}_{z}<20$), and all of them are reliably identified as stellar objects by optical/NIR spectroscopy and/or Gaia. Therefore, all the new transient candidates (aside from CDF-S XT1 and XT2) are stellar flares. These stellar objects have different variability properties, e.g. some have significant non-zero fluxes detected in the non-bursting observations (e.g. ID3 and ID4; see Fig. 6) while others do not (e.g. ID5 and ID9). However, since the main scope of this paper is to study extragalactic transients similar to CDF-S XT1 and XT2, we do not further classify the stellar objects into, e.g. “transient stars” vs. “variable stars”.

Since our algorithm is optimized for selecting CDF-S XT-like transients (see §2.2), the fact that only two such transients are found indicates such events are relatively rare. We further estimate the CDF-S XT-like event rate in §4. The prevalence of stars among our transient candidates is likely because stellar flares are intrinsically more common than CDF-S XT-like extragalactic transients, and it does not necessarily indicate that our algorithm is more sensitive in selecting stellar flares. There should be even more stellar flares in the survey data not identified by our algorithm, which is designed to select XT-like transients rather than stellar flares. In fact, we have tested adjusting our algorithm slightly, and the resulting stellar sample changes while the extragalactic sample remains the same. For example, if we chop the exposures to ${t_{\rm exp}<70}$ ks instead of ${t_{\rm exp}<50}$ ks (§2.1), CDF-S XT1 and XT2 will be still identified. However, this change will select 6 new stellar flares while missing 3 old stellar flares.

4 Event Rate and Future Prospects

Our transient-search algorithm is able to find CDF-S XT-like transients with ${\log F_{\rm peak}\gtrsim-12.6}$ (cgs) effectively (§2.2.3). We remind that the limiting peak flux here is estimated for a typical off-axis angle of ${5^{\prime}}$ (see §2). For an off-axis angle of ${0.5^{\prime}}$ (nearly on-axis) and ${8^{\prime}}$ (the maximum value accepted by our algorithm; §2.1), the limiting flux changes slightly (${\approx 0.1}$ dex; see Appendix B).

However, we do not find any new extragalactic transients that are similar to CDF-S XT1 and XT2, despite searching Chandra observations totaling 19 Ms exposure (§3.3). Based on this search result, we estimate the event rate of CDF-S XT-like transients in §4.1. From the estimated event rate, we discuss the prospects of future missions (Athena and Einstein Probe) in detecting CDF-S XT-like transients.

4.1 Event-Rate Estimation

Since our simulations in §2.2.3 show that the efficiency of our transient selection in short Chandra exposures (${t_{\rm exp}\lesssim 8}$ ks) is low, we do not include exposures shorter than 8 ks in when estimating the event rate below. These short exposures only add up to 0.022 Ms of observation time in total, which is negligible compared to the total observation time analyzed (19.3 Ms).

For a set of Chandra observations, the expected number of transients brighter than the flux limit (${\log F_{\rm peak}\gtrsim-12.6}$, cgs) can be written as

[TABLE]

where ${R_{i}}$ is the event rate; ${\Omega_{i}}$ and ${t_{i}}$ are the field of view (FOV) and exposure time, respectively; the subscript (${i}$) denotes different exposures. In general, ${R_{i}}$ is a function of the sky coordinate of the telescope pointing. However, considering that our focus is extragalactic transients and the Universe is largely isotropic, we assume that ${R_{i}}$ is a constant and denote it as ${R}$. ${\Omega_{i}}$ depends on the instrument used. All of our analyzed survey data are from Chandra/ACIS-I imaging observations, and thus ${\Omega_{i}}$ is a constant and we denote it as ${\Omega}={\pi\times(8^{\prime})^{2}=201}$ arcmin2. Eq. 4 can then be simplified as

[TABLE]

i.e. $\mathcal{N}$ only depends on the total exposure time of these observations. In other words, it does not matter whether our analyzed 19 Ms of data are from a single sky zone or multiple sky zones. From Eq. 6, the event rate $R$ can be calculated as

[TABLE]

Based on the fact that 2 events are detected in 19 Ms of data, we estimate ${R\approx 59^{+77}_{-38}\ \rm{evt\ yr^{-1}deg^{-2}}}$, where the uncertainties are Poisson ${1\sigma}$ errors, calculated with the astropy.stats package. We stress that the event rate estimated throughout this paper refers to that of a particular type of transients (i.e. similar to CDF-S XT1 and XT2 with ${\log F_{\rm peak}\gtrsim-12.6}$, cgs) rather than general extragalactic transients.

Given the event rate estimated above, we can estimate the number of CDF-S XT-like transients potentially existing in the Chandra archive. As of March 2019, there are 95 Ms and 94 Ms of ACIS-I and ACIS-S archival imaging observations (excluding $<8$ ks exposures) at Galactic latitudes of ${|b|>20^{\circ}}$.888Here, we do not consider the 7 Ms of observations performed by HRC, because the sensitivities and thereby flux limits of HRC and ACIS are different. We also do not include ACIS subarray-mode observations to avoid complexity in the calculation of FOV. Such observations only contribute 1% and 17% of the exposure time for ACIS-I and ACIS-S, respectively. Accounting for these observations is technically challenging, but would only affect our estimated transient number by a few percent at most.

ACIS-I and ACIS-S consist of 4 and 6 CCD chips, respectively. For ACIS-I, all the chips are front-illuminated (FI); for ACIS-S, 4 and 2 chips are FI and back-illuminated (BI), respectively. The BI chips have a slightly higher ($\approx$ 10%) flux-to-counts conversion factor than the FI chips.999http://cxc.harvard.edu/proposer/POG/html/chap6.html The former have a higher background ($\approx$ 2 times) than the latter, but still at a low level (only $\approx$ 6 counts for a 50 ks exposure). After considering these differences in conversion factor and background in our simulations (§2.2), we find the flux limits of our transient detection are similar for FI and BI chips ($\log F_{\rm lim}\approx-12.6$ for both). Therefore, the differences between the FI and BI chips should not affect our estimation of the transient number in Chandra archival observations below.

As for ACIS-I, we only account for the CCD area with off-axis angle ${<8^{\prime}}$ for ACIS-S, which covers the S2, S3, and S4 CCD chips. However, unlike the case for ACIS-I generally, ACIS-S may have some chips turned off during an observation (S3 is always on as it covers the aimpoint). When one (S3), two (S3$+$S4 or S2$+$S3), and three (S2$+$S3$+$S4) relevant chips are on, the CCD areas are $\Omega_{\rm S,1}\approx 69$ arcmin2, $\Omega_{\rm S,2}\approx 96$ arcmin2, and $\Omega_{\rm S,3}\approx 123$ arcmin2, respectively. The total exposure times for the three cases are $T_{\rm S,1}\approx 16$ Ms, $T_{\rm S,1}\approx 25$ Ms, $T_{\rm S,1}\approx 53$ Ms, respectively.

Therefore, we can estimate the total number of CDF-S XT-like transients in these archival observations as

[TABLE]

where ${\Omega_{\rm I}}$ (${\Omega_{\rm S}}$) and ${T_{\rm I}}$ (${T_{\rm S}}$) are the FOV and total exposure time (${|b|>20^{\circ}}$) of ACIS-I (ACIS-S) in the Chandra archive. We note that, at ${|b|>20^{\circ}}$, Galactic absorption is typically low, with column density of ${N_{\rm H}\lesssim 10^{21}}$ cm*-2* (e.g. Stark et al., 1992), and such absorption only reduces the observed flux by ${\lesssim 10\%}$ (estimated with pimms). Therefore, Galactic absorption is unlikely to significantly affect the estimated number of transients above.

We will perform an extensive Chandra archival search in a separate paper (Quirola Vásquez et al. in prep.). From our results (§3.3), the stellar objects found in archival data are likely to have bright optical/NIR counterparts ($z$-band magnitudes $\lesssim 20$), and thus their stellar nature can be largely determined with current wide-field surveys, e.g. SDSS, UKIDSS, and Gaia. In contrast, the counterparts of extragalactic transients will likely be faint in the optical/NIR, and follow-up observations with large ground-based telescopes will be helpful to study their properties such as redshift and host-galaxy stellar mass. These counterparts may also be studied with future deep wide-field surveys such as LSST (Ivezić et al., 2019) and Euclid (Laureijs et al., 2011). XMM-Newton has had a similar operational time as Chandra, and it notably has a larger effective area and FOV but also higher background than Chandra. Future work could also search XMM-Newton archival data for CDF-S XT-like transients (e.g. the EXTraS project; De Luca et al. 2016).

4.2 The Perspectives for Future Missions

Future X-ray missions such as Athena and Einstein Probe should be able to discover a large number of extragalactic transients similar to CDF-S XT1 and XT2. Now, we estimate the sample sizes of transients that will be potentially detected by Athena and Einstein Probe. As a first-order approximation, we assume that the event-rate density (event rate per dex of flux) is a power-law function, i.e.

[TABLE]

Here, the power-law index (${\gamma}$) is positive, because otherwise the event rate above a given ${F_{\rm peak}}$ would be divergent. By integrating Eq. 8 from $\log F_{\rm lim}$ (limiting peak flux of the mission)101010Here, we integrate from $\log F_{\rm lim}$ rather than $-12.6$ (the limiting flux of Chandra; §2.2.3). This is because, in this Section, our goal is to estimate the number of XT-like transients detectable by future missions (i.e. sources with $F_{\rm peak}$ above $F_{\rm lim}$ of these missions). Therefore, the integration lower limit should be $F_{\rm lim}$ of the mission of interest.

to ${\infty}$ and applying Eq. 4, we can estimate the number of detected CDF-S XT-like transients as

[TABLE]

where ${A}$ and ${G}$ are the effective area and grasp (defined as ${\Omega\times A}$) of the mission. In Eq. 9, we adopt the approximation of ${F_{\rm lim}\propto A^{-1}}$. If further assuming ${\gamma=1}$ and ${T}$ is similar for different missions, we have ${N\propto G}$. Since both Athena and Einstein Probe have ${G}$ values ${\sim 200}$ times larger than that of Chandra (e.g. Nandra et al. 2013; Burrows et al. 2018; Yuan et al. 2018) which can detect ${\sim 15}$ transients (see above), we expect that Athena and Einstein Probe will each detect ${\sim 3000}$ sources if they operate for ${\approx 20}$ years. These samples will be sufficiently large for detailed sample studies. Note that the estimated sample sizes depend on the assumption that ${\gamma=1}$. If ${\gamma>1}$, Athena (Einstein Probe) will detect more (fewer) transients; if ${0<\gamma<1}$, the situation is the opposite.

Our estimation above is based on the assumption of a power-law function of event-rate density (Eq. 8) with $\gamma=1$. A natural prediction of this power-law function is that there are more faint sources than bright sources in general. One might be concerned that this prediction contradicts our results, i.e. the 19 Ms of Chandra data only contains two relatively bright sources (XT1 and XT2, both having $\log F_{\rm peak}>-12.2$; see Table 2) but no fainter sources. We now test whether this apparent inconsistency is statistically significant or not. Assuming there are two transients above the Chandra flux limit ($\log F_{\rm lim}=-12.6$) detected in the 19 Ms of data, we estimate the chance for these two both to have $\log F_{\rm peak}>-12.2$. From Eq. 8 ($\gamma=1$), the probability for one detected transient to be bright ($\log F_{\rm peak}>-12.2$) is

[TABLE]

Then, according to the binomial distribution, the probability ($p$-value) for both sources to be bright ($\log F_{\rm peak}>-12.2$) is $0.40^{2}=0.16$, only corresponding to $1.4\sigma$ significance. Therefore, the assumption of Eq. 8 ($\gamma=1$) does not contradict our results significantly. Actually, we find Eq. 8 is always consistent with our results at a $3\sigma$ level, as long as $0<\gamma<3.1$.

5 Summary

We have performed a systematic search for CDF-S XT-like extragalactic transients in 19 Ms of Chandra surveys, including CDF-S, CDF-N, DEEP2, UDS, COSMOS, and E-CDF-S. Our main results are summarized below.

We developed a method to select transients within a Chandra observation (§2). From simulations, we show that our method is efficient in discovering transients with 0.5–7 keV peak flux $\log F_{\rm peak}\gtrsim-12.6$ (erg cm*-2* s*-1*). 2. 2.

Our selection yields 13 transient candidates (§3), including CDF-S XT1 and XT2 which have been reported in previous works (Bauer et al. 2017; Xue et al. 2019). All the candidates have optical/NIR counterparts (§3.3). Except for CDF-S XT1 and XT2, all other sources are stellar objects. 3. 3.

The lack of new CDF-S XT-like transients in our search indicates that such objects are rare (§4). We estimate an event rate of ${59^{+77}_{-38}}\ \rm{evt\ yr^{-1}deg^{-2}}$, corresponding to a total of ${15^{+20}_{-10}}$ events in Chandra archival observations at $|b|>20^{\circ}$. Future X-ray missions such as Athena and the Einstein Probe  with large grasps might be able to find thousands of extragalactic transients, and sample studies will be feasible then.

Acknowledgements

We thank the referee for helpful feedback that improved this work. We thank David Burrows, Qingling Ni, John Timlin, and Fabio Vito for helpful discussions. GY, WNB, and SFZ acknowledge support from CXC grant AR8-19016X, CXC grant AR8-19011X, and NASA ADP grant 80NSSC18K0878. FEB acknowledges support from CONICYT-Chile (Basal AFB-170002, FONDO ALMA 31160033) and the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. YQX acknowledges support from the 973 Program (2015CB857004), NSFC (11890693, 11421303), and the CAS Frontier Science Key Research Program (QYZDJ-SSW-SLH006). The Guaranteed Time Observations (GTO) for the CDF-N included here were selected by the ACIS Instrument Principal Investigator, Gordon P. Garmire, currently of the Huntingdon Institute for X-ray Astronomy, LLC, which is under contract to the Smithsonian Astrophysical Observatory; Contract SV2-82024. This project uses Astropy (a Python package; see Astropy Collaboration et al. 2018).

Appendix A Efficiency of the Selection Algorithm for Different Transient Models

The simulations in §2.2 are based on a fiducial transient model similar to the CDF-S XTs. The employment of this fiducial model is driven by the main aim of this paper, i.e. investigating CDF-S XT-like transients in Chandra surveys. However, our algorithm might also be able to identify other types of transients as a “bonus”. In this Appendix, we perform Monte Carlo simulations for some other transient models as examples, although pursuing them is not the main focus of our paper.

The first additional transient model we test is a “time-reversed” version of our fiducial model (see Fig. 8 top for the light curve). The fiducial light curve has features of a fast rise and slow decline (Fig. 1), and thus the reverse has features of a slow rise and fast decline. The reversed model has the same flux-to-counts conversion factor and timescale as the fiducial model. We then apply the simulation process in §2.2.2 to the reversed model, and show ${P_{\rm eff}}$ as a function of ${t_{\rm exp}}$ in Fig. 8 (bottom). The simulation results are similar to those of the fiducial model, e.g. for ${\log F_{\rm peak}\gtrsim-12.6}$ (cgs, corresponding to $\approx$ 30 counts), ${P_{\rm eff}}$ is $\approx$ 1 for a wide range of ${t_{\rm exp}=8\text{--}50}$ ks. We have also tested some other light curves with different shapes but similar timescales, and found the sensitivity of our algorithm for these models is similar to the fiducial model. These results indicate that our algorithm is also capable of detecting different types of transients with timescales similar to that of the CDF-S XTs.

Another additional transient model we test is based on the ultrafast transient discovered by Glennie et al. (2015). This transient lasts only ${\approx 100}$ s with ${\log F_{\rm peak}=-9.9}$ (cgs), and has a spectral shape of ${\Gamma\approx 1.4}$. The nature of the transient remains unknown, as the optical/NIR counterpart has not been found due to the lack of deep multiwavelength data (§1). The light curve can also be approximated by the general formula in Eq. 1, with ${(t_{1},t_{2},\alpha_{1},\alpha_{2})\approx(10\ \mathrm{s},30\ \mathrm{s},0,-4)}$. This light-curve model is displayed in Fig. 9 (top). The flux-to-counts conversion factor (Eq. 2) for this model is ${3.2\times 10^{12}}$, and the ${T_{\rm 90}}$ is 47 s. Here, the conversion factor is much lower than that in Eq. 2. This is mainly because the ultrafast model has a timescale much shorter than the fiducial model, and to reach similar counts, the former must have a much higher peak flux than the latter. We show the simulation results in Fig. 9 (bottom). Unlike ${P_{\rm eff}}$ in Fig. 1, ${P_{\rm eff}}$ in Fig. 9 does not drop below ${t_{\rm exp}\approx 8}$ ks. The drop in Fig. 1 is because, when the exposure time becomes shorter than the transient timescale, the observed light curve will be similar to a normal variable source (§2.2.3). However, this is not the case in Fig. 9, since the ultrafast-transient timescale (${T_{\rm 90}=47}$ s) is even shorter than our shortest exposures (${3}$ ks). In Fig. 9, for ${\log F_{\rm peak}\lesssim-11.1}$, ${P_{\rm eff}}$ declines toward high $t_{\rm exp}$ due to high background levels for long exposures (§2.1). For ${\log F_{\rm peak}\gtrsim-11.0}$ (corresponding to $\approx$ 30 counts), ${P_{\rm eff}}$ is stable for different ${t_{\rm exp}}$, because the X-ray signal is dominated by the source rather than the background.

Glennie’s model tested above is faster than our fiducial model. Now, we test another transient model which is “slower” than the fiducial model. We extend the plateau phase of the fiducial model (§2.2.1) by setting ${t_{2}=t_{1}+5}$ ks (Eq. 1), while keeping the other parameters the same. The light curve of this slower model is displayed in Fig. 10 (top). The flux-to-counts conversion factor (Eq. 2) for this model is ${6.0\times 10^{14}}$, and the $T_{\rm 90}$ is 16.7 ks. The simulation results are displayed in Fig. 10 (bottom). For a given ${F_{\rm peak}}$, ${P_{\rm eff}}$ rises toward high ${t_{\rm exp}}$ for the aforementioned reason, i.e. our algorithm may not be able to differentiate the transient from normal variable sources when ${t_{\rm exp}\lesssim}$ transient timescale. Since most ($\approx$ 90%; §3.2) of our exposures are longer than the timescale of the slower model, our algorithm is largely capable of detecting such transients in our data.

In summary, our algorithm can detect different types of transients with timescales similar to or below that of the CDF-S XTs, as long as $\gtrsim$ 30 counts are available. For transients with longer timescales, only observations with ${t_{\rm exp}}\gtrsim$ transient timescale can have high detection probabilities. Since 80% of our exposures are longer than 25 ks (§3.2), we are potentially able to detect transients with timescales shorter than $\approx$ 25 ks in our data.

Appendix B Efficiency of Selection Algorithm at Different Off-Axis Angles

The simulations in §2.2 are performed for a typical off-axis angle of ${5^{\prime}}$. In this Appendix, we perform simulations at off-axis angles of ${0.5\arcmin}$ (nearly on-axis) and ${8\arcmin}$ (the maximum value accepted by our algorithm; §2.1). In our simulation configurations (§2.2.1), there are two parameters dependent on off-axis angle, i.e. flux-to-counts conversion factor and background noise. The conversion factors (Eq. 2) are ${\approx 1.7\times 10^{14}}$ and ${\approx 1.5\times 10^{14}}$ (cgs) at ${0.5^{\prime}}$ and ${8^{\prime}}$, respectively; the typical background count rates are ${5.9\times 10^{-6}}$ cnt s*-1* to ${2.5\times 10^{-4}}$ cnt s*-1*.

We perform our simulations under these new configurations, and display the results in Fig. 11. Similar to the results for ${5^{\prime}}$, ${P_{\rm eff}}$ drops significantly below ${t_{\rm exp}\approx 8}$ ks, because short exposures cannot differentiate between variable sources and transients (§2.2.3). Compared to that for ${5^{\prime}}$, ${P_{\rm eff}}$ for ${0.5^{\prime}}$ (${8^{\prime}}$) generally increases (decreases) for a given ${F_{\rm peak}}$ and ${t_{\rm exp}}$, as expected. As a consequence, the peak-flux limit could change if using the simulation configurations for ${0.5^{\prime}}$ (${8^{\prime}}$). In §2.2.3, we choose the peak-flux limit as the minimum flux above which ${P_{\rm eff}}$ is ${\approx 1}$ for ${t_{\rm exp}=8\text{--}50}$ ks. Applying the same criteria to Fig. 11, the peak-flux limits are ${\log F_{\rm peak}\approx-12.7}$ (${\approx-12.5}$) for ${0.5^{\prime}}$  (${8^{\prime}}$).