Deepest View of AGN X-Ray Variability with the 7 Ms Chandra Deep Field-South Survey

Most sources in the CDF-S have a very low count rate and S/N. To enhance the S/N while retaining as many features in a light curve as possible, we decide to adopt a binning strategy such that each data point of the resulting light curve represents the binned result of an individual observation whose exposure time ranges from ≈30 to ≈150 ks. Although many sources are still too faint for reliable analysis given this binning scheme, the bright sources we focus on would have a high enough S/N for variability measurements.

In the light-curve extraction procedure, there are complexities from instruments that would influence our results, including vignetting, CCD gaps, bad pixels, and quantum efficiency degradation. Therefore, we adopt a similar solution to that of Young et al. (2012), using effective exposure maps to calibrate these instrumental effects. For each source, we calculate the 90% encircled-energy fraction radii R₉₀ in every observation based on point-spread function modeling results in Xue et al. (2011). Then we use a circular region with a radius ${R}_{\mathrm{src}}$ to estimate source counts and an annulus region with an inner radius ${R}_{\mathrm{bkg},\mathrm{in}}$ and an outer radius ${R}_{\mathrm{bkg},\mathrm{out}}$ to estimate background counts (${R}_{\mathrm{src}}$, ${R}_{\mathrm{bkg},\mathrm{in}}$ and ${R}_{\mathrm{bkg},\mathrm{out}}$ are listed in Table 1). These aperture choices are made after trying a series of aperture combinations to maximize the S/N of the light curves. We only select events with grades 0, 2, 3, 4, and 6, and exclude those that also fall into the source area of another object. Finally, after background subtraction, we obtain our long-term light curves in three energy bands: 0.5–2 (soft band), 2–7 (hard band), and 0.5–7 keV (full band; in the observed frame). We present the full-band light curve of the brightest source as an example in Figure 1.

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Table 1.  Aperture Radii Adopted in Light-curve Extraction

Net counts	Axis Angle (')	${R}_{\mathrm{src}}/{R}_{90}$	${R}_{\mathrm{bkg},\mathrm{in}}/{R}_{90}$	${R}_{\mathrm{bkg},\mathrm{out}}/{R}_{90}$
All	<2	1	1.2	7.5
0–1000	>2	1	1.5	5
1000–15,000	>2	1.3	2	5
>15,000	>2	1.7	2.5	5

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It should be noted that because we use the above simplified procedure instead of AE to extract the light curves, the amount of photon counts will be slightly different from that given by L17. For consistency, we adopt the total counts from our light-curve extractions in the following analysis.

We compute the errors of source and background counts using both the Gehrels approximation (Gehrels 1986) and the square root of counts as

$$\begin{eqnarray}&&{\rm{\Delta }}{n}_{\mathrm{Geh}}=\displaystyle \frac{1}{2}({\rm{\Delta }}{n}_{\mathrm{Geh},\mathrm{upper}}+{\rm{\Delta }}{n}_{\mathrm{Geh},\mathrm{lower}}),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{n}_{\mathrm{Geh},\mathrm{upper}}=1+\sqrt{n+0.75},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{n}_{\mathrm{Geh},\mathrm{lower}}=n-n{\left(1-\displaystyle \frac{1}{9n}-\displaystyle \frac{1}{3\sqrt{n}}\right)}^{3},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{n}_{\mathrm{sqrt}}=\sqrt{n}.\end{eqnarray} \tag{ 4 }$$

The Gerhels approximation is a better error estimation in the low-counts regime, but the square root of counts is the standard deviation of the Poisson distribution theoretically. These two approximations both have their respective advantages in the following analyses (see more details in Section 4).

As shown in Figure 1, the 102 individual observations are roughly distributed into four periods with 1 Ms, 1 Ms, 2 Ms, and 3 Ms exposures, respectively. Therefore, we divide the long light curve into four parts that correspond to the four epochs. These four short light curves provide variability information of four different timescales of a source.

We also use another binning strategy in order to search for transient events in the CDF-S . Given that a TDE usually has a decay time of a few months to years, we rebin the data in bins of about three months to make a new light curve of a source (more details are provided in Section 6).

As mentioned above, many faint sources do not have enough counts for a variability estimation. Furthermore, some sources were not covered by all the 102 observations. Inconsistent observing patterns could introduce large uncertainties in the following analysis. Therefore, we construct our initial sample based on the following criteria:

• 1  
  The source was classified as an AGN in L17, but not classified as a radio-loud AGN in Bonzini et al. (2013).
• 2  
  The source has more than 100 full-band net counts in the 7 Ms exposure.
• 3  
  The overall length of the long light curve is larger than 15.2 years (i.e., $4.8\times {10}^{8}\,{\rm{s}}$, $\sim 90 \%$ of the longest light curve).
• 4  
  The source was covered by more than 70 observations.
• 5  
  The source region is outside ${R}_{\mathrm{bkg},\mathrm{in}}$ of any other sources.

As a result, 283 of the 1008 sources meet these initial requirements. However, it should be noted that the 100-counts cut is still not enough to discard all sources that are not suitable for reliable variability analyses. We intend to include as many sources as possible while ensuring that the variability estimation of these sources does not suffer from the uncertainties arising form low count rates. Therefore, we have to determine what would happen when our measuring methods are used in the low-counts regime, in order to secure an unbiased sample (see Section 4 for details).

To quantify the variability amplitude of a light curve, we compute the normalized excess variance and its error (Vaughan et al. 2003) as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{nxv}}^{2}=\displaystyle \frac{1}{({N}_{\mathrm{obs}}-1)\langle \dot{n}{\rangle }^{2}}\displaystyle \sum _{i=1}^{{N}_{\mathrm{obs}}}{({\dot{n}}_{i}-\langle \dot{n}\rangle )}^{2}-\tfrac{1}{{N}_{\mathrm{obs}}\langle \dot{n}{\rangle }^{2}}\sum _{i=1}^{{N}_{\mathrm{obs}}}{\sigma }_{i,\mathrm{err},\mathrm{var}}^{2}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\mathrm{err}({\sigma }_{\mathrm{nxv}}^{2})=\sqrt{\displaystyle \frac{2}{{N}_{\mathrm{obs}}}{\left(\displaystyle \frac{\overline{{\sigma }_{\mathrm{err},\mathrm{var}}^{2}}}{\langle \dot{n}{\rangle }^{2}}\right)}^{2}+\displaystyle \frac{\overline{{\sigma }_{\mathrm{err},\mathrm{var}}^{2}}}{{N}_{\mathrm{obs}}}\displaystyle \frac{4{\sigma }_{\mathrm{nxv}}^{2}}{\langle \dot{n}{\rangle }^{2}}},\end{eqnarray} \tag{ 6 }$$

where ${N}_{\mathrm{obs}}$ is the number of observations, ${\dot{n}}_{i}$ and ${\sigma }_{i,\mathrm{err},\mathrm{var}}$ are the photon flux and its error of the source in the ith observation, and $\langle \dot{n}\rangle$ is the exposure-weighted average photon flux of the light curve.

It should be noted that instead of using ${\sigma }_{i,\mathrm{err},\mathrm{Geh}}$ (i.e., Equation (1)), the computation of ${\sigma }_{i,\mathrm{err},\mathrm{var}}$ is based on the square root of the observed counts (i.e., Equation (4)) and its corresponding error propagation. This choice has been proven to be a maximum-likelihood estimator for the Gaussian statistic in Almaini et al. (2000); furthermore, Allevato et al. (2013) proved that it could also be applied to the low-counts regime. We also design a test to show the different ${\sigma }_{\mathrm{nxv}}^{2}$ behaviors between adopting ${\sigma }_{i,\mathrm{err},\mathrm{var}}$ and ${\sigma }_{i,\mathrm{err},\mathrm{Geh}}$. We simulate 10,000 observed light curves of a non-variable source with a mean count rate of about 6 × 10⁻⁵ counts s⁻¹ (i.e., about 400 counts in the 7 Ms exposure) with a background level similar to an arbitrary real source. We plot the distributions of ${\sigma }_{\mathrm{nxv}}^{2}$ calculated with two types of error estimates in Figure 2. For a non-variable source, the variable amplitude is 0, so the mean measured ${\sigma }_{\mathrm{nxv}}^{2}$ should be close to 0. It is clear that using the Gehrels error (${\sigma }_{i,\mathrm{err},\mathrm{Geh}}$) yields ${\sigma }_{\mathrm{nxv}}^{2}$ values that are systematically smaller than 0. In contrast, ${\sigma }_{\mathrm{nxv}}^{2}$ values based on the square root error (${\sigma }_{i,\mathrm{err},\mathrm{var}}$) are distributed around 0, which means that this estimation is unbiased.

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In addition to the choice of ${\sigma }_{i,\mathrm{err},\mathrm{var}}$, the S/N and total counts also have nonnegligible effects on variability measurement. It has been known that faint sources are more difficult to classify than variable sources (e.g., Paolillo et al. 2004, 2017; Lanzuisi et al. 2014). Allevato et al. (2013) has proven that the uncertainty in ${\sigma }_{\mathrm{nxv}}^{2}$ measurement will become larger for sources with lower counts. Moreover, irregular sampling patterns can cause additional biases and scatters that can only be quantified through simulations.

To evaluate the influence of these biases, we perform a test following the procedure below:

• 1  
  We select the 30 brightest AGNs (each with $\gtrsim 2400$ full-band net counts) in our initial source sample to construct a "bright sample." These 30 AGNs have very high-quality light curves and can be regarded as sources that are not influenced by noise.
• 2  
  We randomly choose an AGN in the bright sample and rescale its full-band light curve such that its average photon flux matches that of an arbitrary fainter AGN (i.e., with ≲2400 counts) in the L17 main catalog; note that in order to show the biased trend more clearly, here we also use faint AGNs with fewer than 100 total counts. Such a rescaling does not change the variability of the original light curve, so that we could simulate the "intrinsic" light curve of a faint source that has the same variability as an AGN in the bright sample.
• 3  
  To simulate the influence of low S/N, we add the Poisson-distributed background (i.e., noise) to the faint "intrinsic" light curve, and then extract the "observed" counts of each observation. Finally, we obtain a fake light curve of a faint source whose intrinsic variability is the same as that of an AGN in the bright sample.
• 4  
  We repeat steps 2 to 3 1000 times and compute ${\sigma }_{\mathrm{nxv}}^{2}$ of these 1000 simulated faint light curves.

In the top panel of Figure 3, we plot the ${\sigma }_{\mathrm{nxv}}^{2}$–counts relation of both the real (red and blue symbols) and fake sources (gray symbols). The trend of decreasing scatters of ${\sigma }_{\mathrm{nxv}}^{2}$ toward large counts appears apparent and similar for both the real and fake sources. In the bottom panel, we show the running averages and scatters of ${\sigma }_{\mathrm{nxv}}^{2}$ for the real faint sources and fake sources. The running bin sizes are 50 for the real faint sources and 100 for the fake sources. The averages and scatters of ${\sigma }_{\mathrm{nxv}}^{2}$ are largely similar between the faint and fake samples above ∼300 counts and the bright sample, while the scatters of ${\sigma }_{\mathrm{nxv}}^{2}$ in the faint and fake samples become unacceptably large below ∼300 counts, which can also be inferred from the top panel. The similarity in the overall trend of ${\sigma }_{\mathrm{nxv}}^{2}$–counts and associated scatters between the faint and fake samples suggests that the large ${\sigma }_{\mathrm{nxv}}^{2}$ scatters of very faint sources (i.e., with $\lesssim 300$ counts in this context) originate from low S/N (i.e., they are significantly influenced by noise). Figure 3 also reflects that above ∼300 counts, there is no significant difference in variability between the bright and faint samples.

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It should be noted that in the above procedure, we find there is a larger fraction of negative ${\sigma }_{\mathrm{nxv}}^{2}$ for sources below the 300-counts threshold in the fake sample than that in the real faint sample. This fact can be seen in the bottom panel, where the average ${\sigma }_{\mathrm{nxv}}^{2}$ of real data is always positive, while that of fake data can sometimes be smaller than zero. This discrepancy should be interpreted as being primarily due to the Eddington bias, i.e., in the low-counts regime, very faint sources with large variability and positive flux fluctuations (thus having positive average ${\sigma }_{\mathrm{nxv}}^{2}$) are more likely to be detected. Given that we only focus on sources above the 300-counts threshold (see Section 4.2), the Eddington bias would not affect our following analyses.

According to Figure 3 and the above arguments, it is clear that the influence of noise can be ignored while measuring variability amplitudes of sources with $\gtrsim 300$ full-band counts. Therefore, we are able to obtain an unbiased sample by applying this count threshold cut.

We perform an analysis similar to the hard-band and soft-band light curves and find that the 300-counts threshold can also be applied to the hard-band light-curve analysis, while the soft-band light-curve analysis requires only $\gtrsim 200$ counts. In most of the remaining analyses, we require our studied light curves to have more than 300 full-band counts, except in Section 5.1, where we only use sources with more than 300 hard-band counts and more than 200 soft-band counts. There are still a small number of sources with negative ${\sigma }_{\mathrm{nxv}}^{2}$(13 for the full band), but they will not affect our analysis significantly since we use the so-called "ensemble excess variance" (Allevato et al. 2013) by taking the average ${\sigma }_{\mathrm{nxv}}^{2}$ of sources that have similar physical properties.

Based on the initial sample constructed in Section 3.2, we find a total of 148 sources whose full-band light curves meet our requirement (i.e., with $\geqslant 300$ full-band counts each and satisfying the criteria of 1, 3, 4, and 5 in Section 3.2). These 148 sources make up Sample I. Similarly, the numbers of available sources are 110 and 98 for the soft and hard bands, respectively, while there are 77 sources that meet the requirements in both the soft (i.e., $\geqslant 200$ counts) and hard (i.e., $\geqslant 300$ counts) bands. These 77 sources are marked as Sample II.

Adopting the preferred redshifts in the L17 main catalog (i.e., the 51st column, "ZFINAL"; see Section 4.3 of L17 for the redshift selection criteria) and the spectral analysis results of J. Y. Li et al. (2017, in preparation), we present the redshift and X-ray luminosity distributions for the sources of Samples I and II in Figure 4. These two samples cover very similar wide ranges of redshift ($0\lt z\leqslant 6$) and X-ray luminosity (∼10^41–45 erg s⁻¹). In Sample I (Sample II), 101 (63) sources have spectroscopic-redshift measurements, 83 (50) of which are secure and 18 (13) are insecure, but agree well with at least one of the available photometric redshift estimates; and the remaining 47 (14) sources have photometric redshifts as their preferred redshifts, with the 25th, 50th, and 75th percentiles of zphot_error/(1+zphot) being 0.018, 0.026, and 0.057 (0.012, 0.019, and 0.026), respectively. Given the relatively high fractions of spectroscopic redshifts (101/148 = 68.2% for Sample I and 63/77 = 81.8% for Sample II) and small uncertainties of the photometric redshifts, using only (secure) spectroscopic redshifts should not affect our analysis significantly. Therefore, we choose to use the L17 preferred redshifts, which were selected carefully, in order to maximize our sample sizes.

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As mentioned before, we extract our light curves based on observed-frame energy bands, which means that we can discuss the variability of different rest-frame energy bands for sources with different redshifts. However, at least for short-term (i.e., $T\lesssim 100\,\mathrm{ks}$) variability, there is evidence implying that variability amplitudes in various energy bands have a good consistency (Ponti et al. 2012). Using the sources in Sample II, we compare ${\sigma }_{\mathrm{nxv}}^{2}$ measured from light curves in three different bands in Figure 5 to check whether the consistency remains for long-term variability.

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Generally, the ${\sigma }_{\mathrm{nxv}}^{2}$ in different energy bands are well correlated and the linear slope is close to 1. We divide our sources into three subsamples according to their redshifts and mark them with different colors. It appears that the correlation behavior of ${\sigma }_{\mathrm{nxv}}^{2}$ in different energy bands is largely not influenced at different redshifts, although the subsamples with higher redshifts tend to have relatively larger dispersions. We also mark the sources with ${N}_{{\rm{H}}}$ larger than ${10}^{22.5}\,{\mathrm{cm}}^{-2}$ using large filled symbols. The overall behavior of these obscured sources in Figure 5 is quite similar to that of the unobscured sources (see Section 5.2 for more details).

We note that ${\sigma }_{\mathrm{nxv}}^{2}$ in the soft band seem to be slightly larger than those in the full band and hard band (see the left and right panels of Figure 5), which is seen both in Sample II and in the obscured subsample. This difference may be explained by the superposition of a soft component that varies in flux and/or slope and a constant hard reflection component, which can result in the "softer when brighter" behavior (e.g., Sobolewska & Papadakis 2009; Gibson & Brandt 2012; Serafinelli et al. 2017). Additionally, the variability of absorption may be another possible reason, since the soft band is more easily affected by ${N}_{{\rm{H}}}$ variation than the hard band. However, given that ${\sigma }_{\mathrm{nxv}}^{2}$ in the soft band is systematically larger only up to a level of about 10%–30%, this difference does not materially affect most of our following analysis, except for the study of the evolution of variability (see Section 5.6).

Previous studies (e.g., Paolillo et al. 2004) found evidence of a possible connection between variability and obscuration such that hard obscured AGNs tend to have lower variability. Obscuration might smoothen the variability and lead to smaller ${\sigma }_{\mathrm{nxv}}^{2}$. On the other hand, Yang et al. (2016) and Liu et al. (2017) found some sources with ${N}_{{\rm{H}}}$ variations, which might increase AGN long-term variability. It is not clear how these effects would influence our following analysis, therefore we divide our sample into two parts: obscured (${N}_{{\rm{H}}}\gt {10}^{23}\,{\mathrm{cm}}^{-2}$, 49 sources) and less obscured (${N}_{{\rm{H}}}\leqslant {10}^{23}\,{\mathrm{cm}}^{-2}$, 99 sources) and plot their ${\sigma }_{\mathrm{nxv}}^{2}$ distributions in the top panel of Figure 6.

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We perform a K-S test to assess the similarity of the two samples, and the result indicates that their ${\sigma }_{\mathrm{nxv}}^{2}$ distributions are quite similar (${P}_{\mathrm{reject}}\approx 26 \%$). However, since obscured sources tend to have higher intrinsic luminosities (in our sample, obscured sources have a mean ${L}_{2\mbox{--}10\mathrm{keV}}$ of $\approx 1.3\,\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, while less obscured sources have a mean ${L}_{2\mbox{--}10\mathrm{keV}}$ of $\approx 5\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1};$ see the bottom panel of Figure 6), we would expect that they should have smaller ${\sigma }_{\mathrm{nxv}}^{2}$ based on the known anticorrelation between variability and luminosity. In fact, when we compare the median log ${\sigma }_{\mathrm{nxv}}^{2}$, obscured sources do have lower ${\sigma }_{\mathrm{nxv}}^{2}$ values, but not significantly so (${\rm{\Delta }}\mathrm{log}\,{\sigma }_{\mathrm{nxv}}^{2}\sim 0.2$ dex). From the results of Section 5.3 and other studies (e.g., Lanzuisi et al. 2014), we find that the ${L}_{{\rm{X}}}\mbox{--}{\sigma }_{\mathrm{nxv}}^{2}$ relation is enough to explain this difference. In order to distinguish the influences of redshift and ${L}_{{\rm{X}}}$ (see Figure 7 for the plot of ${L}_{{\rm{X}}}$ versus z), we also choose five complete subsamples (see Table 2), within which sources have similar redshifts and luminosities, and then perform the Spearman rank test to check the correlation between their ${N}_{{\rm{H}}}$ and ${\sigma }_{\mathrm{nxv}}^{2}$. The results are shown in Table 2, indicating that none of these subsamples shows an evident correlation between ${N}_{{\rm{H}}}$ and variability (i.e., all ${P}_{\mathrm{reject}}$ values are $\geqslant 8 \% ;$ but note the limited sizes of the subsamples).

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Table 2.  Spearman's Rank Test Results of ${\sigma }_{\mathrm{nxv}}^{2}\mbox{--}{N}_{{\rm{H}}}$

Subsample	Size	z	${L}_{2\mbox{--}10\mathrm{keV}}$ (${10}^{43}\mathrm{erg}\,{{\rm{s}}}^{-1}$)	ρ	${P}_{\mathrm{reject}}$
1	5	0.7–1.1	3–30	0.10	0.13
2	6	1.1–1.5	3–30	−0.05	0.12
3	10	1.5–2.1	3–30	−0.06	0.19
4	10	2.1–2.8	3–30	0.02	0.08
5	10	0.7–1.1	0.8–3	0.37	0.79

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Based on the above results of the K-S test and Spearman's rank tests, we conclude that the subsequent analysis of ${L}_{{\rm{X}}}\mbox{--}{\sigma }_{\mathrm{nxv}}^{2}$ does not suffer from the bias caused by obscuration.

It has long been known that X-ray variability amplitude is well anticorrelated with luminosity (e.g., Nandra et al. 1997; Paolillo et al. 2004, 2017; Papadakis et al. 2008; González-Martín et al. 2011; Ponti et al. 2012; Lanzuisi et al. 2014; Yang et al. 2016). This trend can be a result of the dependence of the AGN PSD on the black hole mass and accretion rate.

In Figure 8 we display the ${\sigma }_{\mathrm{nxv}}^{2}\mbox{--}{L}_{2\mbox{--}10\mathrm{keV}}$ relation of all sources in Sample I as defined in Section 4.2. A decreasing trend is revealed, but the trend may be not as apparent when we only look at one subsample with a certain range of redshifts because of the large scatter and relatively narrow ${L}_{2\mbox{--}10\mathrm{keV}}$ range. Therefore we bin our data and plot them in Figure 8. We only bin sources in the same subsample that have similar redshifts, because ${\sigma }_{\mathrm{nxv}}^{2}$ for different redshifts stands for the variability of different rest-frame timescales. Each binned data point represents an average ${\sigma }_{\mathrm{nxv}}^{2}$ of eight sources with close ${L}_{2\mbox{--}10\mathrm{keV}}$ values. The bin size is chosen to balance the luminosity range within each bin and the requirement of reliable average ${\sigma }_{\mathrm{nxv}}^{2}$ calculation. The error bars denote standard errors and luminosity ranges. The symbol sizes denote the average redshifts of the bins.

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After the binning, we see a clear anticorrelation between ${\sigma }_{\mathrm{nxv}}^{2}$ and ${L}_{2\mbox{--}10\mathrm{keV}}$ for the whole sample, which is also manifested by the Spearman rank test results based on individual sources, although the trend is not significant for either the low-redshift or high-redshift subsample (for all sources, $\rho =-0.31$, ${P}_{\mathrm{reject}}={10}^{-4};$ for $z\lt 1.5$ sources, $\rho =-0.17$, ${P}_{\mathrm{reject}}=0.2;$ and for $z\geqslant 1.5$ sources, $\rho =-0.17$, ${P}_{\mathrm{reject}}=0.1$). It should be noted that the reason we perform tests on $z\lt 1.5$ and $z\geqslant 1.5$ sources instead of the three subsamples we use in the binning is that the luminosity range of any of the three subsamples is narrow.

The decreasing trend of ${\sigma }_{\mathrm{nxv}}^{2}$ toward large ${L}_{2\mbox{--}10\mathrm{keV}}$ might be due to two reasons: time dilution (due to redshift) and PSD shape. As we know, ${\sigma }_{\mathrm{nxv}}^{2}$ is the integral of the PSD:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{nxv}}^{2}={\int }_{\tfrac{1}{{T}_{\mathrm{rest}}}}^{\tfrac{1}{2{\rm{\Delta }}{t}_{\mathrm{rest}}}}\mathrm{PSD}(\nu )d\nu ,\end{eqnarray} \tag{ 7 }$$

where ${T}_{\mathrm{rest}}$ and ${\rm{\Delta }}{t}_{\mathrm{rest}}$ are the length of light curve and the bin size²¹ in the rest frame, respectively; and the PSD is often assumed to be a single or broken power law. Although our sources have similar observational exposures and sampling patterns, their large redshift range makes a great difference to their rest-frame timescales. Therefore, for high-redshift AGNs, their integrating intervals in Equation (7) will shift to higher-frequency ranges because their light curves are shorter in the rest frame. If the AGN PSD follows a uniform power law $\mathrm{PSD}(\nu )\propto {\nu }^{-\beta }$, sources with higher redshifts are assumed to have smaller ${\sigma }_{\mathrm{nxv}}^{2}$ in our measurement if $\beta \gt 1$, because Equation (7) would become

$$\begin{eqnarray}&&{\sigma }_{\mathrm{nxv}}^{2}={(1+z)}^{-\beta +1}{\int }_{\tfrac{1}{{T}_{\mathrm{obs}}}}^{\tfrac{1}{2{\rm{\Delta }}{t}_{\mathrm{obs}}}}\mathrm{PSD}(\nu )d\nu .\end{eqnarray} \tag{ 8 }$$

From Equation (8), the influence of redshift uncertainties can also be estimated. As demonstrated in Section 4.2, the uncertainties of our adopted photometric redshifts are relatively small, the majority of which have values of zphot_error/(1+zphot) lower than a few percent. Even when $\beta =1.5$, the resulting deviation is only about 20% considering the photometric redshifts that have the largest uncertainties. Since we use average ${\sigma }_{\mathrm{nxv}}^{2}$ in the subsequent fitting, this influence will be further reduced.

Another influence comes from the PSD shape.²² As mentioned before, the AGN PSD can be well represented by a broken power law. Previous studies (e.g., McHardy et al. 2006; González-Martín & Vaughan 2012; Ponti et al. 2012) pointed out that the high-frequency break depends on black hole mass and Eddington ratio ${\lambda }_{\mathrm{Edd}}$, which could be expressed as ${\nu }_{\mathrm{hb}}\propto {M}_{\mathrm{BH}}^{-1}{\lambda }_{\mathrm{Edd}}^{\gamma }$, where the value of γ is still controversial. In addition, the normalization of PSD is found to be roughly inversely proportional to ${\nu }_{\mathrm{hb}}$ (Papadakis 2004). In Section 5.4 we introduce these results. Consequently, assuming a PSD ${(\nu )=A(\nu /{\nu }_{\mathrm{hb}})}^{-2}$ when $\nu \gt {\nu }_{\mathrm{hb}}$, we would derive ${\sigma }_{\mathrm{nxv}}^{2}\sim A{\nu }_{\mathrm{hb}}^{2}{T}_{\mathrm{rest}}$, which could also contribute to the anticorrelation between ${\sigma }_{\mathrm{nxv}}^{2}$ and ${L}_{2\mbox{--}10\mathrm{keV}}$. However, the lengths of our light curves exceed 16 years, which means $1/T\sim 2\times {10}^{-9}\ll {\nu }_{\mathrm{hb}}$. Moreover, since most of our observations lasted for ${10}^{4}\mbox{--}{10}^{5}\,{\rm{s}}$, the corresponding upper bound of integral $1/2{\rm{\Delta }}t$ in Equation (8) is close to ${\nu }_{\mathrm{hb}}$ for supermassive black holes (e.g., McHardy et al. 2006; González-Martín & Vaughan 2012). This means that our ${\sigma }_{\mathrm{nxv}}^{2}$ are more likely to be dominated by the low-frequency part of the PSD. Some studies assumed a power-law PSD with an index of 1 when $\nu \lt {\nu }_{\mathrm{hb}}$. The exact form of the low-frequency AGN X-ray PSD still needs to be explored with the help of long-term monitoring data, however.

Therefore, we take into account the bin size and the power-law indexes of different parts of the PSD to fit our ${\sigma }_{\mathrm{nxv}}^{2}\mbox{--}{L}_{2\mbox{--}10\mathrm{keV}}$ results, and try to determine how these parameters affect the observed anticorrelation trend. Furthermore, the bias caused by irregular sampling needs to be assessed with the use of light-curve simulations assuming a certain type of AGN PSD. It should be noted that similar PSD analyses could also be found in Paolillo et al. (2017), who tried to study the accretion history of SMBHs, while we aim to constrain the exact form of AGN PSD.

Previous studies (e.g., González-Martín et al. 2011; also see, e.g., Figure 1 of Zhu & Xue 2016 for an illustration) suggest that the AGN PSD can be expressed as

$$\begin{eqnarray}\mathrm{PSD}(\nu )=\left\{\begin{array}{ll}A{(\nu /{\nu }_{\mathrm{hb}})}^{-\alpha } & (\nu \gt {\nu }_{\mathrm{hb}})\\ A{(\nu /{\nu }_{\mathrm{hb}})}^{-\beta } & ({\nu }_{\mathrm{hb}}\geqslant \nu \gt {\nu }_{\mathrm{lb}}),\\ A{({\nu }_{\mathrm{lb}}/{\nu }_{\mathrm{hb}})}^{-\beta } & (\nu \leqslant {\nu }_{\mathrm{lb}})\end{array}\right.\end{eqnarray} \tag{ 9 }$$

where the high-frequency slope α is close to 2, while the low-frequency slope β is found to be about 1 in some bright sources (e.g., Uttley & McHardy 2005; Breedt et al. 2009). Papadakis (2004) found that ${C}_{1}=A{\nu }_{\mathrm{hb}}$ is roughly a constant of 0.017. Furthermore, although the ratio ${\nu }_{\mathrm{lb}}/{\nu }_{\mathrm{hb}}$ is about 0.1 in Galactic BHBs, a study on Ark 564 reported a ratio of about 10⁻⁴ (McHardy et al. 2007). As shown in Figure 8, however, there is no sign of a second PSD break, which suggests that the very low frequency part of PSD does not play an important role in the relation. Therefore we only consider the high-frequency break and PSD normalization in the following analysis.

We test four models, labeled Models 1 to 4, that link the PSD to the black hole mass and the Eddington ratio ${\lambda }_{\mathrm{Edd}}$:

• 1.  
  We use the ${\nu }_{\mathrm{hb}}$ computation given by McHardy et al. (2006),

  $$\begin{eqnarray*}&&{\nu }_{\mathrm{hb}}=0.003{\lambda }_{\mathrm{Edd}}{({M}_{\mathrm{BH}}/{10}^{6}{M}_{\odot })}^{-1},\end{eqnarray*}$$

  assuming a PSD amplitude of ${\nu }_{\mathrm{hb}}\times \mathrm{PSD}({\nu }_{\mathrm{hb}})=0.017$, as suggested by Papadakis (2004).
• 2.  
  We adopt the same PSD amplitude as in Model 1, but use the break frequency computed according to González-Martín & Vaughan (2012, also see Pan et al. 2015):

  $$\begin{eqnarray*}&&{\nu }_{\mathrm{hb}}=0.001{\lambda }_{\mathrm{Edd}}^{0.24}{({M}_{\mathrm{BH}}/{10}^{6}{M}_{\odot })}^{-1}.\end{eqnarray*}$$
• 3.  
  We use the same break frequency as in Model 1, but adopt the PSD amplitude that depends on Eddington ratio, as suggested by Ponti et al. (2012):

  $$\begin{eqnarray*}&&{\nu }_{\mathrm{hb}}\times \mathrm{PSD}({\nu }_{\mathrm{hb}})=0.003{\lambda }_{\mathrm{Edd}}^{-0.8}.\end{eqnarray*}$$
• 4.  
  We adopt the break frequency in González-Martín & Vaughan (2012) and the PSD amplitude in Ponti et al. (2012).

We then use the empirical relation between bolometric correction ${k}_{\mathrm{bol}}$ and ${\lambda }_{\mathrm{Edd}}$, which is computed by studying spectral energy distributions (Lusso et al. 2010), to calculate ${L}_{\mathrm{bol}}$ and ${M}_{\mathrm{BH}}$ from ${L}_{2\mbox{--}10\mathrm{keV}}$ for a given ${\lambda }_{\mathrm{Edd}}$. Based on these assumptions and Equation (7), we are able to connect the PSD and ${L}_{2\mbox{--}10\mathrm{keV}}$, and derive model ${L}_{2\mbox{--}10\mathrm{keV}}\mbox{--}{\sigma }_{\mathrm{nxv}}^{2}$ relations and compare with real data.

In Figure 9 we show how different parameters affect the ${L}_{2\mbox{--}10\mathrm{keV}}\mbox{--}{\sigma }_{\mathrm{nxv}}^{2}$ relation using Model 1. Generally, the observed ${L}_{2\mbox{--}10\mathrm{keV}}\mbox{--}{\sigma }_{\mathrm{nxv}}^{2}$ relation could be explained by this PSD model with proper parameters. Particularly, we may see that the relation is sensitive to ${\lambda }_{\mathrm{Edd}}$ and β. In contrast, if there were a universal PSD, sources with different redshifts would have close ${\sigma }_{\mathrm{nxv}}^{2}$ values in our observations, which is consistent with our expectation in Section 5.3 that the uncertainties of redshifts would not affect ${\sigma }_{\mathrm{nxv}}^{2}$ significantly.

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Based on these PSD models, we use the emcee code (Foreman-Mackey et al. 2013), which is based on the maximum-likelihood Markov chain Monte Carlo (MCMC) method, to fit our observed relations, and we show the results in Figures 10, 11, and Table 3. The best-fit values and their error bars are the median, 16%, and 84% percentiles of parameter distributions in the MCMC simulation.

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Table 3.  ${L}_{2\mbox{--}10\mathrm{keV}}\mbox{--}{\sigma }_{\mathrm{nxv}}^{2}$ Fitting Results

Sample	Model	Typical log ${\lambda }_{\mathrm{Edd}}$	β	${\chi }_{\nu }^{2}$	dof
All	1	$-{1.82}_{-0.11}^{+0.12}$	${1.16}_{-0.05}^{+0.05}$	1.4	17
All	2	$-{3.35}_{-0.25}^{+0.28}$	${1.16}_{-0.05}^{+0.05}$	1.4	17
All	3	${0.6}_{-0.4}^{+0.5}$	${1.31}_{-0.04}^{+0.04}$	1.6	17
All	4	${0.07}_{-0.19}^{+0.22}$	${1.30}_{-0.04}^{+0.04}$	1.5	17
$z\lt 1.5$	1	$-{1.94}_{-0.18}^{+0.21}$	${1.20}_{-0.07}^{+0.07}$	1.6	7
$z\lt 1.5$	2	$-{3.6}_{-0.4}^{+0.5}$	${1.20}_{-0.07}^{+0.07}$	1.6	7
$z\lt 1.5$	3	${0.7}_{-0.6}^{+1.0}$	${1.31}_{-0.06}^{+0.06}$	1.6	7
$z\lt 1.5$	4	${0.13}_{-0.30}^{+0.37}$	${1.31}_{-0.06}^{+0.06}$	1.6	7
$z\geqslant 1.5$	1	$-{1.77}_{-0.15}^{+0.16}$	${1.14}_{-0.10}^{+0.10}$	1.5	8
$z\geqslant 1.5$	2	$-{3.22}_{-0.34}^{+0.37}$	${1.14}_{-0.10}^{+0.10}$	1.5	8
$z\geqslant 1.5$	3	${1.2}_{-1.0}^{+1.7}$	${1.37}_{-0.09}^{+0.07}$	1.8	8
$z\geqslant 1.5$	4	${0.2}_{-0.4}^{+0.5}$	${1.34}_{-0.10}^{+0.10}$	1.8	8
All	1	$-{1.51}_{-0.11}^{+0.11}$	1 (f)a	2.1	18
All	2	$-{2.62}_{-0.26}^{+0.27}$	1 (f)	2.1	18
All	3	$-{0.85}_{-0.05}^{+0.05}$	1 (f)	3.1	18
All	4	$-{0.80}_{-0.04}^{+0.04}$	1 (f)	3.3	18
All	1	$-{1.89}_{-0.07}^{+0.06}$	1.2 (f)	1.4	18
All	2	$-{3.50}_{-0.15}^{+0.14}$	1.2 (f)	1.4	18
All	3	$-{0.22}_{-0.05}^{+0.06}$	1.2 (f)	1.8	18
All	4	$-{0.32}_{-0.04}^{+0.04}$	1.2 (f)	1.7	18
All	1	$-{2.29}_{-0.05}^{+0.04}$	1.4 (f)	2.6	18
All	2	$-{4.39}_{-0.11}^{+0.10}$	1.4 (f)	2.6	18
All	3	${2.15}_{-0.11}^{+0.12}$	1.4 (f)	1.8	18
All	4	${0.64}_{-0.05}^{+0.05}$	1.4 (f)	1.8	18

Note.

^aLabel (f) means that the parameter β is fixed in the fitting.

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With each model, we were able to find a set of best-fit parameters to fit the data well. When β is not fixed in the fitting (see the top row of Figure 10 and the top part of Table 3), we obtain a typical Eddington ratio of about 0.015 and $\beta \approx 1.2$ with Model 1. Model 2 leads to a much lower Eddington ratio of lower than 10⁻³, which appears slightly low for black hole growth, while β is consistent with Model 1. The fitting results of Models 3 and 4 seem to be unrealistic, given that all best-fit ${\lambda }_{\mathrm{Edd}}$ values are close to or larger than 1, which implies that AGNs are in the super-Eddington accreting state all the time. However, despite of the implausibly high ${\lambda }_{\mathrm{Edd}}$ values, these two models also suggest a low-frequency PSD index of $\beta \approx 1.3$, very similar to the results for Models 1 and 2.

Comparing the fitting results for $z\lt 1.5$ and $z\geqslant 1.5$ subsamples in Figure 11, we find a weak tendency that ${\lambda }_{\mathrm{Edd}}$ is larger in the high-redshift subsample for all models, but the difference is too small compared with the uncertainties. Same as ${\lambda }_{\mathrm{Edd}}$, the variation of β is not apparent either. We cannot draw a reliable conclusion about whether there is indeed an evolution with these model-fitting results alone. This problem is discussed further in Section 5.6.

We note that the results above are not consistent with those reported by Paolillo et al. (2017). The reason is probably that we did not fix the low-frequency index β and/or that our results are based on only one long timescale and thus are less sensitive to the break position and more to the PSD normalization.

So we also try to fit the data with β fixed to determine whether a larger β is necessary. The results are shown in the bottom row of Figure 10 and the bottom part of Table 3. Apparently, for low-luminosity and low-redshift sources that have the longest rest-frame light curves and highest break frequencies (i.e., which are most sensitive to the low-frequency part of the PSD), model ${\sigma }_{\mathrm{model}}^{2}$ is too small when β is fixed to 1; when β is fixed to 1.4, conversely, model ${\sigma }_{\mathrm{model}}^{2}$ is too large. The only well-fit situation is when β is fixed to 1.2, which is very close to the results inferred from the top part of Table 3. We also compare the results with those in Paolillo et al. (2017). Within the uncertainties, our results when $\beta =1$ are in agreement with their results.

It should be noted that irregular sampling and red-noise leakage (e.g., Allevato et al. 2013; Zhu & Xue 2016, and references therein) may introduce a bias to the estimation, making

$$\begin{eqnarray}&&{\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}=b{\int }_{\tfrac{1}{T}}^{\tfrac{1}{2{\rm{\Delta }}t}}\mathrm{PSD}(\nu )d\nu .\end{eqnarray} \tag{ 10 }$$

The bias factor b could only be obtained through a simulation, especially when the two intervals between observations and observation times are irregular and make the choice of ${\rm{\Delta }}t$ ambiguous. Therefore, according to the fitting results, we use the light-curve simulating code in Zhu & Xue (2016) to generate 2000 light curves assuming a PSD model whose $\beta =1.2$, $\alpha =2$, and ${\nu }_{\mathrm{hb}}=5\times {10}^{-4}\,\mathrm{Hz}$. We calculate b of these simulated light curves and find that $b\approx 1$ for all redshifts. This result indicates that our ${\sigma }_{\mathrm{nxv}}^{2}$ estimation and thus ${\sigma }_{\mathrm{nxv}}^{2}\mbox{--}{L}_{2\mbox{--}10\mathrm{keV}}$ fitting are not subject to the bias caused by irregular sampling and red-noise leakage.

Based on Equations (7) and (9), we may also perform a simple estimation of β using the ${\sigma }_{\mathrm{nxv}}^{2}\mbox{--}T$ relation. Assuming $\beta \ne 1$ and the high-frequency break ${\nu }_{\mathrm{hb}}=1/{t}_{\mathrm{hb}}$ is between ${\nu }_{1}=1/T$ and ${\nu }_{2}=1/(2{\rm{\Delta }}t)$, we have

$$\begin{eqnarray}{\sigma }_{\mathrm{nxv}}^{2} & = & {\displaystyle \int }_{\tfrac{1}{T}}^{\tfrac{1}{2{\rm{\Delta }}t}}\mathrm{PSD}(\nu )d\nu \\ & = & \displaystyle \frac{A}{\beta -1}\displaystyle \frac{{T}^{\beta -1}}{{t}_{\mathrm{hb}}^{\beta }}-C\quad (\beta \ne 1)\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&C=\displaystyle \frac{A}{(\beta -1){t}_{\mathrm{hb}}}+\displaystyle \frac{A}{(\alpha -1){t}_{\mathrm{hb}}}\left[{\left(\displaystyle \frac{2{\rm{\Delta }}t}{{t}_{\mathrm{hb}}}\right)}^{\alpha -1}-1\right].\end{eqnarray} \tag{ 12 }$$

For ${n}_{\mathrm{src}}$ sources with similar T and ${\rm{\Delta }}t$ but different A and C, we obtain the average ${\sigma }_{\mathrm{nxv}}^{2}$ as follows:

$$\begin{eqnarray}&&\langle {\sigma }_{\mathrm{nxv}}^{2}\rangle =\left\langle \displaystyle \frac{A}{(\beta -1){t}_{\mathrm{hb}}^{\beta }}\right\rangle {T}^{\beta -1}-\langle C\rangle .\end{eqnarray} \tag{ 13 }$$

If ${t}_{\mathrm{hb}}\leqslant 2{\rm{\Delta }}t$, Equation (12) becomes

$$\begin{eqnarray}&&C=\displaystyle \frac{A}{\beta -1}\displaystyle \frac{{(2{\rm{\Delta }}t)}^{\beta -1}}{{t}_{\mathrm{hb}}^{\beta }}.\end{eqnarray} \tag{ 14 }$$

Taking the redshifts into account, we write down the equation in the observed frame as

$$\begin{eqnarray}&&\langle {\sigma }_{\mathrm{nxv},\mathrm{obs}}^{2}\rangle ={(1+z)}^{-\beta +1}\left(\left\langle \displaystyle \frac{A}{(\beta -1){t}_{\mathrm{hb}}^{\beta }}\right\rangle {T}_{\mathrm{obs}}^{\beta -1}-\langle {C}_{\mathrm{obs}}\rangle \right).\end{eqnarray} \tag{ 15 }$$

Therefore, when Equation (15) is dominated by the first term, we should observe $\langle {\sigma }_{\mathrm{nxv},\mathrm{obs}}^{2}\rangle \propto {T}_{\mathrm{obs}}^{\beta -1}$ for sources with similar redshifts. If we can find a set of light curves with enough lengths, we should be able to constrain β in this way. It should be pointed out that the deduction is similar when adopting ${T}_{\mathrm{rest}}$ instead of ${T}_{\mathrm{obs}}$, if we are only concerned about constraining β using samples with small redshift ranges.

Based on the observations, we divide the light curves in the 7 Ms CDF-S into four segments, whose lengths are $\sim 4\times {10}^{6}\,{\rm{s}}$, $\sim 1\times {10}^{7}\,{\rm{s}}$, $\sim 3\times {10}^{7}\,{\rm{s}}$, and $\sim 4\times {10}^{7}\,{\rm{s}}$ in the observed frame, as shown in Figure 1. We perform tests similar to Section 4.1 to obtain light-curve samples that not biased by low counts. Furthermore, these four segments are all unevenly sampled light curves, therefore we also perform similar simulations to quantify the bias factor b as in Section 5.4. It should be noted that we do not use other types of light curves such as the combination of two or three epochs to prevent using a segment repeatedly, so that each point in the ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}{T}_{\mathrm{obs}}$ relation is based on an independent measurement.

We use the four complete subsamples indicated in Figure 7 to plot the ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}{T}_{\mathrm{obs}}$ relation in Figure 12. In the binning process, we exclude outliers with ${\sigma }_{\mathrm{nxv}}^{2}$ beyond the 3σ range of other sources in each bin because we find the ${\sigma }_{\mathrm{nxv}}^{2}$ measurements of these outliers (one or two at most in each bin) usually suffer from one or two points in the light curves with abnormally high values that are due to large errors or bursts. This effect is negligible for the 102-point light curves, but it severely influences short light curves. As a result, it is not surprising for us to find an increasing trend in most of the subsamples except for the highest-redshift one (i.e., Sample D). Note that we only have four data points and the highest-redshift subsample is the smallest one (each point is binned by 10 to 15 sources).

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We use a power-law model to fit the ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}{T}_{\mathrm{obs}}$ relation,

$$\begin{eqnarray}&&\mathrm{log}\,{\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}=a\,\mathrm{log}\,{T}_{\mathrm{obs}}+\mathrm{Const}.\end{eqnarray} \tag{ 16 }$$

The fitting results are listed in Table 4. Theoretically, slope a in Equation (16) and the low-frequency slope of PSD β are connected in the form of $a\sim \beta -1$ if ${T}_{\mathrm{obs}}$ is long enough, but obviously, our light curves are not ideal. For a light curve with a length of 10⁷ s, a bin size ${\rm{\Delta }}{t}_{\mathrm{obs}}$ of 80 ks, and originating from a PSD with a break frequency of $\sim {10}^{5}$, the first term of Equation (13) is only about twice larger than the second term. Furthermore, the difference in source properties can also introduce bias.

Table 4.  ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}{T}_{\mathrm{obs}}$ Fitting Results

Sample	z	${L}_{2\mbox{--}10\mathrm{keV}}$ (${10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$)	a	Const
A	0.5–1	$\gt 2.8$	0.53 ± 0.19	−5.1 ± 1.5
B	1–1.5	$\gt 7.6$	0.34 ± 0.09	−3.7 ± 0.6
C	1.5–2.5	$\gt 27$	0.38 ± 0.15	−4.2 ± 1.1
D	2.5–3.5	$\gt 60$	0.41 ± 1.0	−5.0 ± 7.1

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To determine the exact dependence between a and β, we perform a simple simulation. We use a broken power-law PSD model and randomly select 100 sets of redshifts and PSD shape parameters (normalization and high-frequency break). Through Equation (10), we obtain the expected ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}$ in four timescales. Then we fit the expected ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}{T}_{\mathrm{obs}}$ relation, and find the value of a we will obtain when we use different β. We present the result in Figure 13.

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We find the corresponding β values in Figure 13 for the two low-redshift subsamples (i.e., Samples A and B) are likely to be ≈1.2–1.4, consistent with that derived from the ${L}_{{\rm{X}}}\mbox{--}{\sigma }_{\mathrm{nxv}}^{2}$ relation. We stress that this β–a relation only assumes a broken power-law PSD and does not depend on a specific model in Section 5.4. The consistency between the results obtained in these two different ways proves the reliability of our β estimation. For the two high-redshift subsamples (i.e., Samples C and D), the upward trends in Figure 12 are not significant. The reason is probably that their intrinsic variability is weak and the rest-frame lengths of light curves are short, which leads to a weak trend. Moreover, the small number of sources can also be a problem. In this case, we decide to draw our conclusion based on the low-redshift results.

We have known that by PSD model fitting it is not enough to tell whether AGN variability changes in different cosmic eras. A direct way to explore this question is to compare ${\sigma }_{\mathrm{nxv}}^{2}$ of sources from different redshift ranges. However, from Equation (10) and Section 5.3, we also know that the measured ${\sigma }_{\mathrm{nxv}}^{2}$ suffers from the differences of luminosity ranges, rest-frame timescales, and sampling patterns of different redshift samples.

In order to reduce the influence of luminosity differences, we select a complete luminosity-limited subsample. This subsample only contains AGNs with $3\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ $\lt \,{L}_{2\mbox{--}10\mathrm{keV}}\leqslant 3\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and $0.7\lt z\leqslant 2.8$. We aim to compare the variable amplitudes (i.e., ${\sigma }_{\mathrm{nxv}}^{2}$ of light curves with same rest-frame lengths) of different redshift subsamples.

Since ${t}_{\mathrm{rest}}={t}_{\mathrm{obs}}/(1+z)$, we choose four light-curve segments corresponding to four representative redshifts z = 0.9, 1.3, 1.8, and 2.4, which are noted with horizontal lines in the left panel of Figure 14, so that the rest-frame lengths of light curves ${t}_{\mathrm{rest}}$ are consistent (${t}_{\mathrm{rest}}\simeq 8\times {10}^{7}\,{\rm{s}}$), making ${\sigma }_{\mathrm{nxv}}^{2}$ from different redshift subsamples straightforwardly comparable. We also choose proper redshift bins (also noted in the left panel of Figure 14) in the source selection to make the variation of ${t}_{\mathrm{rest}}$ within each bin smaller than 10%. These light-curve segments are the best choices available to make use of the longest light curves possible and ensure the consistency of the rest-frame timescales of all sources. The bias from the irregular sampling pattern is also determined through the simulation described in Section 5.4. After all these adjustments and bias corrections, we choose the sources with photon flux $\gt 4\times {10}^{-7}\,\mathrm{counts}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$, which corresponds to a threshold of $\gt 300$ counts for reliable ${\sigma }_{\mathrm{nxv}}^{2}$ measurement (see Section 4.2), and then obtain the non-biased ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}z$ relation. The result is plotted in the right panel of Figure 14.

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Because of the above strict source-selection criteria, the available source numbers in the four redshift bins are only 5, 6, 10, and 10, respectively. According to the requirement suggested by Allevato et al. (2013), it is difficult to draw any reliable conclusion with these small bin sizes. Therefore, we cannot reach a definitive conclusion about whether there is an evolution of the variability, and only list some intriguing indications from the results below.

First, the ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}z$ relation displays an overall decreasing trend. If it is a real trend, it could be due to the changing of PSD shape rather than accretion rate (see Figure 9), because both our PSD fitting results based on Model 1 or 2 and other studies (e.g., McLure & Dunlop 2004; Paolillo et al. 2004, 2017; Papadakis et al. 2008) infer lower or constant Eddington ratios toward lower redshifts, which appears to be in contrast to what this observed evolution shows. Alternatively, the likely energy-band dependence mentioned in Section 5.1 can be another potential possibility.

Second, there appears a peak at $z\simeq 1.3$ on top of the overall decreasing trend. If this is a real feature, it is unlikely to be connected with large-scale structures (LSSs) in the E-CDF-S (e.g., Gilli et al. 2003; Treister et al. 2009; Silverman et al. 2010; Dehghan & Johnston-Hollitt 2014; Xue 2017) since LSSs do not exist only around this redshift. Paolillo et al. (2017) also ruled out this possibility in a relevant analysis. Interestingly, we note that Ueda et al. (2014) found a peak of X-ray emissivity for AGNs with $\mathrm{log}\,{L}_{2\mbox{--}10\mathrm{keV}}=43\mbox{--}44$ (see Figure 20 in that work) that is close to the peak here. We then repeat our procedure to plot the ${\sigma }_{\mathrm{nxv},\mathrm{corr}}^{2}\mbox{--}z$ relation for sources with ${L}_{2\mbox{--}10\mathrm{keV}}=8\times {10}^{42}\mbox{--}3\times {10}^{43}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and $z=0.7\mbox{--}2.1$ in the right panel of Figure 14. The trend becomes monotonically decreasing, which also seems to be in line with the peak-shifting behavior of AGN X-ray emissivity shown in Ueda et al. (2014).

We use the 7 Ms CDF-S data to search for likely transient events, especially TDEs, and then constrain their occurrence rate. For this purpose, it is not appropriate to only consider the sources in L17. Since the L17 source detection is based on average fluxes over the 7 Ms timespan, it is possible that some sources lying below the nominal detection limits (thus not included in L17) may become detectable when an outburst occurs. Most TDEs were found in non-active galaxies that are usually not very bright in X-rays. Therefore we also take into account the galaxy sample described in Xue et al. (2010). This sample contains 100,318 galaxies in the E-CDF-S field (Xue et al. 2016), and the vast majority of them have redshift and stellar-mass estimates (thus, masses of potential central black holes can be roughly estimated based on the galaxy-SMBH mass scaling). Not all of these galaxies are adopted because some of them are too faint to be detected even if the central black hole is accreting at the Eddington limit level, and the central black hole masses in some galaxies do not satisfy the requirement for a TDE (Frank & Rees 1976; Rees 1988; Luo et al. 2008b). As a result, the sources in the final galaxy sample considered should meet all the following criteria:

• a.  
  The stellar mass is between $2\times {10}^{7}\,{M}_{\odot }$ and $1.5\times {10}^{11}\,{M}_{\odot }$. This stellar-mass range roughly corresponds to a central black hole mass range from $1\times {10}^{5}\,{M}_{\odot }$ to $3\times {10}^{8}\,{M}_{\odot }$ (Luo et al. 2008b) adopting a scaling factor of 200–500 between stellar mass and black hole mass (Kormendy & Ho 2013).
• b.  
  The galaxy should have an expected full-band flux of $\gtrsim 1.5\times {10}^{-7}\,\mathrm{counts}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ such that it would become detectable in an outburst if its central black hole is accreting at the Eddington limit. This flux limit is derived based on the typical background fluctuation level ${\sigma }_{\mathrm{bkg}}$ of about ${10}^{-8}\mbox{--}{10}^{-7}\,\mathrm{counts}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ in the 7 Ms CDF-S, assuming a ${\rm{\Gamma }}=1.8$ power law and adopting the ${k}_{\mathrm{bol}}\mbox{--}{L}_{\mathrm{Edd}}$ relation from Lusso et al. (2010).
• c.  
  The galaxy is covered by all 102 CDF-S observations, which ensures that each bin has enough exposure time.
• d.  
  The galaxy is located outside of the ${R}_{\mathrm{bkg},\mathrm{in}}$ (see Table 1) of any sources in the 7 Ms CDF-S main catalog.

There are a total of 19,599 galaxies (without L17 detection) in the final galaxy sample, which is supplemented by the 764 L17 main-catalog X-ray sources that are covered by all 102 CDF-S observations and not in crowded X-ray source regions (e.g., pairs or triplets). The redshifts and stellar masses of the non-X-ray galaxies in the final galaxy sample are shown in Figure 15.

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Previous studies (e.g., Auchettl et al. 2017, and references therein) have shown that TDEs usually last for a few months to a few years. Therefore, we adopt three-month bins in our analysis to increase the S/N and avoid smoothing burst-like features. We display some binned full-band light curves with blue dots in Figure 16.

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The next step is to search for transient/burst events using the 3-month-bin light curves. For each light curve, we first identify the highest flux ${f}_{\max }$ and its error ${\sigma }_{\mathrm{err},{\rm{m}}}$, and then compute the average flux ${\bar{f}}_{\mathrm{normal}}$ and standard error ${\sigma }_{\mathrm{normal}}$ of the remaining data points. If there is an outburst, the flux change ${\rm{\Delta }}f={f}_{\max }-{\bar{f}}_{\mathrm{normal}}$ should be significantly larger than the normal variability and statistical error of the source. Therefore, we select sources with ${\rm{\Delta }}f/{\sigma }_{\mathrm{normal}}\gt 3$. We also note that short exposure times of some data points (especially the first and the last) in the light curves would cause some misidentifications in this process. We therefore also require the candidates to have ${\rm{\Delta }}f/{\sigma }_{\mathrm{err},{\rm{m}}}\gt 3$.

In addition to the above ${\rm{\Delta }}f$ criteria, the variable factor is taken into account as well. It is usually defined as ${f}_{\max }/{f}_{\min }$ or ${f}_{\max }/{\bar{f}}_{\mathrm{normal}}$ (e.g., Donley et al. 2002; Luo et al. 2008b). Since the existence of negative fluxes is inevitable after background subtraction, we adopt ${f}_{\max }/{\bar{f}}_{\mathrm{normal}}$ in our analysis. If a source satisfies at least one of the following three situations, we regard this source as a candidate transient:

• (1)  
  ${\bar{f}}_{\mathrm{normal}}\gt 0$, ${f}_{\max }/{\bar{f}}_{\mathrm{normal}}\gt 20$.
• (2)  
  ${\bar{f}}_{\mathrm{normal}}\leqslant 0$, ${f}_{\max }/({\bar{f}}_{\mathrm{normal}}+{\sigma }_{\mathrm{normal}})\gt 20$.
• (3)  
  ${\bar{f}}_{\mathrm{normal}}+{\sigma }_{\mathrm{normal}}\leqslant 0$, ${f}_{\max }\gt 0$.

We stress that the second and third situations are possible because in the calculation of ${\bar{f}}_{\mathrm{normal}}$ the highest data point (and probably the highest few points; see the next paragraph) is not used. In this case, the signal of a galaxy may be smoothed by background fluctuation, leading to a negative ${\bar{f}}_{\mathrm{normal}}$ measurement.

It is possible that the transient event lasts for a very long time and we would miss it, since we will obtain elevated ${\bar{f}}_{\mathrm{normal}}$ and ${\sigma }_{\mathrm{normal}}$ by including the data points adjacent to the peak. To include such events, for sources that do not meet ${\rm{\Delta }}f$ and the variable factor criteria, we recalculate ${\bar{f}}_{\mathrm{normal}}$ and ${\sigma }_{\mathrm{normal}}$ by excluding the highest $n(=2\mbox{--}6)$ data points and then check ${\rm{\Delta }}f$ and the variable factor iteratively using the original ${f}_{\max }$ and new ${\bar{f}}_{\mathrm{normal}}$ and ${\sigma }_{\mathrm{normal}}$. If ${\rm{\Delta }}f$ and the variable factor are large enough after we exclude $n(\leqslant 6)$ data points, the source will also be considered as a candidate hosting an outburst. In contrast, a source will not be considered as a candidate with an outburst during the observations if it cannot pass the test even after the six highest data points are excluded. This step may also introduce spurious fluctuations. Therefore we perform a final visual inspection to see if the highest points are close to each other, which would be the situation for real long-duration outbursts.

Finally, we find a total of six candidate transients in our galaxy sample. Basic information and light curves of these candidates are shown in Table 5 and Figure 16. All these candidates are detected in L17 and satisfy our first criterion. From their light curves, these six sources could be roughly divided into two types. One type is a long outburst. The outbursts of XID = 297, XID = 403, XID = 541, and XID = 935 last for at least several months and they are covered by several observations. When we inspect their 102-data-point light curves, some of the candidate outbursts become less evident. Particularly, XID = 403 has been reported in L17 and will be studied in depth in W. Wang et al. (2017, in preparation). The other type is a short outburst, including XID = 330 and XID = 725. Their outbursts occurred in a single observation and become extremely evident in the 102-data-point light curves. When we examine these two candidates, we find that their count rates rose to 10⁻² to 10⁻¹ $\mathrm{counts}\,{{\rm{s}}}^{-1}$ within just a few hundred seconds and then slowly returned to the normal level after a few thousand seconds. One of these two sources, XID = 725, has also been reported in L17. More details about this source, including its likely origin, can be found in Bauer et al. (2017). We will discuss all these six candidates (particularly XID = 330) further in a future work (X. C. Zheng et al. 2017, in preparation). Inspired by the discovery of XID = 330 and XID = 725, we also perform a similar test for the 102-data-point light curves of both nomal galaxies and X-ray sources, but find no additional fast burst candidates.

Table 5.  Sources with Candidate Transient Events

XIDa	R.A.	Decl.	zb	Peak Timec	${f}_{\max }$ (${{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$)d	${\bar{f}}_{\mathrm{normal}}$ (${{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$)	${f}_{\max }/{\bar{f}}_{\mathrm{normal}}$	Type
297	53.069719	−27.777204	${1.24}_{-1.17}^{+0.08}$	2000/12	$1.39\times {10}^{-7}$	$1.14\times {10}^{-9}$	122.4	Long
330	53.076485	−27.873395	0.74	2015/03	$3.35\times {10}^{-6}$	$2.82\times {10}^{-8}$	118.8	Short
403	53.094719	−27.694609	${1.51}_{-0.01}^{+0.03}$	2015/03	$5.39\times {10}^{-7}$	$2.06\times {10}^{-8}$	26.2	Long
541	53.122333	−27.734364	−1.0	2015/06	$3.33\times {10}^{-7}$	$1.09\times {10}^{-8}$	30.4	Long
725	53.161561	−27.859342	${2.14}_{-0.56}^{+0.37}$	2014/10	$3.31\times {10}^{-7}$	$2.29\times {10}^{-9}$	144.2	Short
935	53.248664	−27.841828	0.25	1999/11	$3.43\times {10}^{-6}$	$6.23\times {10}^{-8}$	55.0	Long

Notes.

^aAll sources are included in the L17 7 Ms CDF-S main catalog, with their XIDs shown here. ^bRedshifts with upper and lower errors are photometric redshifts; the value of −1.0 indicates no reliable redshift measurement available; and the remaining are spectroscopic redshifts. ^cThis is the time when a source reached its highest flux level. The time values are directly read from the three-month-bin light curves, therefore they are not very accurate. However, accurate outburst times for the short outbursts XID = 330 and XID = 725 could be determined (see the text for details). ^dThe maximum fluxes are also derived from the three-month-bin light curves, i.e., they are the mean values over three months. For the short outbursts XID = 330 and XID = 725, their maximum fluxes calculated from the 102-data-point light curves are much higher than the values quoted here.

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We note that all the above six candidate transients are classified as AGNs in L17, which may not be appropriate. The reason is that the L17 source detection and classification are based on the entire 7 Ms CDF-S data (i.e., stacking all individual observations), which means that photons from a transient event could dominate the overall spectrum of the source, thus likely affecting the source classification.

With the results of the search for candidate transient events, we can make a rough estimation of the TDE rate ${\dot{N}}_{\mathrm{TDE}}$ in our sample. Based on the algorithm outlined in Luo et al. (2008b), we first compute the total rest-frame time ${T}_{\mathrm{total}}$ we inspect:

$$\begin{eqnarray}&&{T}_{\mathrm{total}}=\displaystyle \sum _{i}^{{N}_{\mathrm{src}}}{T}_{i,\mathrm{eff}}/(1+{z}_{i}).\end{eqnarray} \tag{ 17 }$$

${T}_{i,\mathrm{eff}}$ is the effective exposure time of the ith source in the observed frame. ${N}_{\mathrm{src}}$ and z_i are the number of our sources and their redshifts. For sources without any redshift estimates (only 14 sources), we assigned the median redshift of our sample ${z}_{\mathrm{med}}=1.27$. Two situations should be considered separately: long and short outbursts, with the former likely being TDEs.

For long outbursts, considering that they could last for several months, we should still be able to detect such outbursts if they occur a few months ahead of each of the four epochs (see Figure 16). If we only consider outbursts that can be detected in three months (rest-frame time), ${T}_{i,\mathrm{eff}}$ can be estimated by

$$\begin{eqnarray}{T}_{i,\mathrm{eff}} & = & 3\,\mathrm{months}\times (1+{z}_{i})\times 4\\ & & +({t}_{3}-{t}_{1})+({t}_{7}-{t}_{5})+({t}_{13}-{t}_{8}).\end{eqnarray} \tag{ 18 }$$

Here t_i is the time of the ith data point in the three-month-bin light curve (see Figure 16).

Assuming that any outburst occurring during the observations would be detected, we could obtain the event rate using

$$\begin{eqnarray}&&\langle {\dot{N}}_{\mathrm{event}}\rangle =\displaystyle \frac{{N}_{\mathrm{event}}}{{T}_{\mathrm{total}}}{\mathrm{galaxy}}^{-1}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 19 }$$

According to Gehrels (1986), we can derive the 90% confidence-level upper limit and lower limit of the amount of transient events ${N}_{\mathrm{event}}$. For long outbursts, if all candidates were associated with TDEs, we would have $\langle {\dot{N}}_{\mathrm{TDE}}\rangle \,={8.6}_{-4.9}^{+8.5}\times {10}^{-5}\,{\mathrm{galaxy}}^{-1}\,{\mathrm{yr}}^{-1}$. Our $\langle {\dot{N}}_{\mathrm{TDE}}\rangle$ estimation is consistent with other studies. Previous observational results (e.g., Donley et al. 2002; Luo et al. 2008b; van Velzen & Farrar 2014) found the TDE rates in their studied samples to be ${10}^{-6}\mbox{--}{10}^{-4}\,{\mathrm{galaxy}}^{-1}\,{\mathrm{yr}}^{-1}$, while theoretical studies (e.g., Wang & Merritt 2004; Stone & Metzger 2016) indicated ${\dot{N}}_{\mathrm{TDE}}={10}^{-5}\mbox{--}{10}^{-3}\,{\mathrm{galaxy}}^{-1}\,{\mathrm{yr}}^{-1}$. As mentioned before, our $\langle {\dot{N}}_{\mathrm{TDE}}\rangle$ calculation is approximate. Uncertainties may be introduced through the sample selection, detection efficiency, and some other issues. We briefly introduce the influences of these issues below.

The first issue is from sample selection. Our sample is flux limited and has a stellar-mass range of $2\times {10}^{7}\,{M}_{\odot }\mbox{--}1.5\,\times {10}^{11}\,{M}_{\odot }$ (see Figure 15). First, previous studies (e.g., Wang & Merritt 2004; Stone & Metzger 2016) pointed out that ${\dot{N}}_{\mathrm{TDE}}$ is anticorrelated with the black hole mass, which indicates that our estimated average TDE rate is likely to be slightly underestimated, given that massive galaxies make up a larger fraction in our sample than in a complete sample. Second, our sample volume might be overestimated (thus the TDE rate might be underestimated) with the adopted broad stellar-mass range, which originates from the galaxy-SMBH mass scaling relation that has large scatter and uncertainties (e.g., we use a scaling factor of 500 to estimate the upper limit of stellar mass and 200 to estimate the lower limit). Third, a relevant point is that if a scaling factor of around 1000 (suggested by, e.g., Häring & Rix 2004 and Sun et al. 2015) were adopted, additional very massive galaxies would be included in our sample, but this increase in our sample volume would be less than 0.1% given the scarcity of such galaxies. Fourth, we choose a scaling factor of 200 to estimate the Eddington luminosity of the central black hole, which would overestimate the black hole mass and maximum flux and thus include some sources that cannot be detected even in the outburst state, resulting in the overestimation of our sample volume. Finally, we choose a uniform flux limit in source selection, while the X-ray flux limit varies significantly across the CDF-S field of view (e.g., Xue et al. 2011; Luo et al. 2017). Therefore it is possible that we would have been able to detect some outbursts from galaxies that are not included in our galaxy sample (because of the flux-limit cut), especially for galaxies near the central field of view where the flux limit is much smaller than the adopted value. This means an underestimation of our sample volume, thus leading to an overestimate of the TDE rate. However, given that there is an anticorrelation between ${\dot{N}}_{\mathrm{TDE}}$ and black hole mass (see the first point above) and that these "missed" galaxies tend to be less massive than the sources in our sample, the inclusion of these "missed" galaxies in our sample should boost the estimated TDE rate. In fact, the peak flux of the outburst candidate XID = 297, with a small off-axis angle of $\approx 3^{\prime}$, is below the flux limit we set for source selection (see Table 5). The above various factors bring some uncertainties to the estimate of the TDE rate, most of which, if treated properly, tend to increase the estimated TDE rate.

Detection efficiency is another important issue that would influence the estimate of TDE rate. In our search, we assume that all outbursts occurring during the exposures could be found, regardless of their properties (e.g., ${N}_{{\rm{H}}}$, Γ, and off-axis angle), but this is obviously too ideal. A more realistic calculation should be

$$\begin{eqnarray}&&{N}_{\mathrm{TDE}}=\displaystyle \sum _{i}^{n}{\epsilon }_{i}\displaystyle \frac{{T}_{i,\mathrm{eff}}}{(1+{z}_{i})}\langle {\dot{N}}_{\mathrm{TDE}}\rangle .\end{eqnarray} \tag{ 20 }$$

In this expression, the detection efficiency ${\epsilon }_{i}$ should be less than 1. Therefore, the real average TDE rate $\langle {\dot{N}}_{\mathrm{TDE}}\rangle$ is again underestimated.

The light-curve profile of an outburst also plays an important role. We estimate $\langle {\dot{N}}_{\mathrm{TDE}}\rangle$ assuming that outbursts are only detectable in three months. From the results, we also find that three out of the four long outbursts (except XID = 297) are likely to be recognized in more than one data point in the three-month-bin light curves, however. The variety of the outburst profiles makes it difficult to estimate ${T}_{i,\mathrm{eff}}$. For longer outbursts, real ${T}_{\mathrm{total}}$ should be longer, leading to a smaller $\langle {\dot{N}}_{\mathrm{TDE}}\rangle$ than we obtain. For example, with our calculation, if we assume all outbursts could be detected in one year, $\langle {\dot{N}}_{\mathrm{TDE}}\rangle$ becomes ${3.4}_{-1.9}^{+3.4}\times {10}^{-5}\,{\mathrm{galaxy}}^{-1}\,{\mathrm{yr}}^{-1}$. To solve this problem, it is necessary to know the intrinsic distribution of X-ray outburst durations, which is not feasible at present.

Last but not least, further studies (e.g., X. C. Zheng et al. 2017, in preparation) are needed to confirm the nature of these outbursts. Since there are only four long outbursts, any mistake in classification will change the result significantly.

For short outbursts such as XID = 330 and XID = 725, ${T}_{i,\mathrm{eff}}$ should be the true exposure time. Because we only use sources that are fully covered by all 102 observations, ${T}_{i,\mathrm{eff}}$ becomes a constant of $\approx 7.0\times {10}^{6}\,{\rm{s}}\approx 0.225\,\mathrm{yr}$. Similarly, we can obtain the frequency of this type of events $\langle \dot{N}\rangle ={1.0}_{-0.7}^{+1.1}\times {10}^{-3}\,{\mathrm{galaxy}}^{-1}\,{\mathrm{yr}}^{-1}$. Since we still do not completely understand their nature (see Bauer et al. 2017, for detailed discussions of XID = 725), we cannot assess how the sample selection would influence the result. Because of their short durations and high variable factors, however, detection efficiency and light curve profile probalby do not have important effects on estimating the event rate of short outbursts.