Ensemble Xray variability of active galactic nuclei
II. Excess variance and updated structure function^{⋆}
^{1} Dipartimento di FisicaUniversità di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy
email: fausto.vagnetti@roma2.infn.it
^{2} Dipartimento di Matematica e Fisica, Università Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy
^{3} Dipartimento di Scienze Fisiche, Università di Napoli Federico II, via Cinthia 9, 80126 Napoli, Italy
Received: 5 June 2016
Accepted: 9 July 2016
Context. Most investigations of the Xray variability of active galactic nuclei (AGN) have been concentrated on the detailed analyses of individual, nearby sources. A relatively small number of studies have treated the ensemble behaviour of the more general AGN population in wider regions of the luminosityredshift plane.
Aims. We want to determine the ensemble variability properties of a rich AGN sample, called MultiEpoch XMM Serendipitous AGN Sample (MEXSAS), extracted from the fifth release of the XMMNewton Serendipitous Source Catalogue (XMMSSCDR5), with redshift between ~0.1 and ~5, and Xray luminosities in the 0.5–4.5 keV band between ~10^{42} erg/s and ~10^{47} erg/s.
Methods. We urge caution on the use of the normalised excess variance (NXS), noting that it may lead to underestimate variability if used improperly. We use the structure function (SF), updating our previous analysis for a smaller sample. We propose a correction to the NXS variability estimator, taking account of the light curve duration in the rest frame on the basis of the knowledge of the variability behaviour gained by SF studies.
Results. We find an ensemble increase of the Xray variability with the restframe time lag τ, given by SF ∝ τ^{0.12}. We confirm an inverse dependence on the Xray luminosity, approximately as SF ∝ L_{X}^{0.19}. We analyse the SF in different Xray bands, finding a dependence of the variability on the frequency as SF ∝ ν^{0.15}, corresponding to a socalled softer when brighter trend. In turn, this dependence allows us to parametrically correct the variability estimated in observerframe bands to that in the rest frame, resulting in a moderate (≲15%) shift upwards (Vcorrection).
Conclusions. Ensemble Xray variability of AGNs is best described by the structure function. An improper use of the normalised excess variance may lead to an underestimate of the intrinsic variability, so that appropriate corrections to the data or the models must be applied to prevent these effects.
Key words: catalogs / galaxies: active / quasars: general / Xrays: galaxies
Full Table 1 is only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/593/A55
© ESO, 2016
1. Introduction
Variability is a distinctive feature shared by all classes of active galactic nuclei (AGN), occurring in all the wavebands and on different timescales from a fraction of a day up to years. In the Xray band, variability is observed on timescales as short as hours, giving insight into the innermost AGN regions, but also on longer timescales, where variability is seen to increase up to at least a few years (see e.g. Markowitz & Edelson 2004; Vagnetti et al. 2011; Shemmer et al. 2014).
A large number of studies have investigated the detailed properties of the Xray variability for many individual AGN, mostly at low redshifts and luminosities (e.g. Uttley et al. 2002; Uttley & McHardy 2005; Ponti et al. 2012). In cases with sufficient sampling and high signaltonoise ratios, power spectral density (PSD) analyses have evidenced the typical rednoise character of Xray variability (Green et al. 1993; Lawrence & Papadakis 1993).
For AGN in wider intervals of redshift and luminosity, including luminous quasars, faint fluxes and sparse sampling usually prevent detailed individual variability studies, nevertheless, average properties of the Xray variability have been investigated in several ensemble analyses (e.g. Almaini et al. 2000; Manners et al. 2002; Paolillo et al. 2004; Mateos et al. 2007; Papadakis et al. 2008; Vagnetti et al. 2011).
Different methods are used to estimate the variability of these sources and one of the most popular is the normalised excess variance (NXS), which is defined as the difference between the total variance of the light curve and the mean squared error that is normalised for the average of the N flux measurements squared (e.g. Nandra et al. 1997; Turner et al. 1999); see Sect. 3. This estimator provides an easy way to quantify the AGN variability even for poorly sampled light curves. However, Allevato et al. (2013) have shown that NXS represents a biased estimator of the intrinsic light curve variance, especially when used for individual, sparsely sampled light curves, which results in overestimates or underestimates of the intrinsic variance that depend on the sampling pattern and the PSD slope below the minimum sampled frequency.
Moreover, it has been pointed out that NXS also depends on the length of the monitoring time interval from the rednoise character of the PSD, and decreasing with redshift from the effect of cosmological time dilation (e.g. Lawrence & Papadakis 1993; Papadakis et al. 2008; Vagnetti et al. 2011).
The structure function (SF) allows one to compute variability as a function of the restframe time lag, and is therefore suitable for ensemble analyses. In Vagnetti et al. (2011, Paper I), for example, we used multiepoch observations of an AGN sample extracted from the XMMNewton serendipitous source catalogue (XMMSSC) to compute the ensemble Xray SF. In the present paper, we take advantage of the recent releases of XMMSSC (Rosen et al. 2016), and of the Sloan Digital Sky Survey (SDSS) Quasar Catalogue (Pâris et al. 2016), to compute the normalised excess variance and to update the study of the structure function. Moreover, we show that the latter can be also used to correct the time dilation effect present in the estimates of the former.
The paper is organised as follows. Section 2 describes the data extracted from the archival catalogues. Section 3 computes the light curve duration effect on the NXS estimates. Section 4 updates the SF computation for the new samples. Finally, in Sect. 5, we discuss and summarise the results.
Throughout the paper, we adopt the cosmology H_{0} = 70 km s^{1} Mpc^{1}, Ω_{m} = 0.3, and Ω_{Λ} = 0.7.
2. Data
The XMMSSC catalogue was recently updated to its release 3XMMDR5 (Rosen et al. 2016), which includes 565 962 Xray detections between February 2000 and December 2013, related to 396 910 unique Xray sources^{1}.
MEXSAS sample.
A large number of sources (70 453) are observed more than once (up to 48 times) for a total of 239 505 multiepoch observations, which makes this catalogue very appropriate for variability studies. In Paper I we used the 2XMMiDR3 release (Watson et al. 2009) that contains 41 979 multiepoch sources with a total of 132 268 observations; thus, with the present release, the number of multiepoch sources and observations is almost doubled.
To extract a set of Xray observations for a sample of quasars, we used the software TOPCAT^{2} to crosscorrelate the XMMSSC catalogue with the SDSS quasar catalogues, using both data release 7 (DR7Q, Schneider et al. 2010) and data release 12 (DR12Q, Pâris et al. 2016). We took into account the quality of the observations, indicated by the parameter SUM_FLAG, selecting only detections with SUM_FLAG < 3, as suggested by the XMMSSC Team. We then searched for coordinate matches within a radius of 5 arcsec, finding 14 648 matches between the XMMSSC and the SDSS catalogues. Increasing the correlation radius to 10 arcsec produces 15 095 matches, indicating a possible incompleteness of the order of 3%. On the other side, repeating the crosscorrelation with a set of false coordinates, shifted by 1 arcmin in declination with respect to the true coordinates, we obtained 44 spurious matches, indicating a possible contamination ~0.3% within the adopted radius.
Fig. 1 Distribution of the sources in the L_{X}z plane. The blue dots represent the average values of the Xray luminosities computed on the available data of each light curve. Lines of constant Xray flux are also shown. 

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Selecting only sources with multiple matches (at least 2), we found 2112 matches between XMMSSCDR5 and SDSSDR7Q, corresponding to 616 unique sources, and 6105 matches between XMMSSCDR5 and SDSSDR12Q, corresponding to 2209 unique sources. For 122 sources with 370 Xray observations, which are found both in DR7Q and DR12Q, we chose the match with the latter, to use a more recent redshift determination. We chose the visual inspection redshift Z_VI among the different redshift estimates provided by DR12Q. After an additional check of the parameter SRCID, which identifies unique sources according to the XMMSSC catalogue, we finally produced a sample of 7837 observations for 2700 sources. To refer to this sample again in future papers, we will call it MultiEpoch XMM Serendipitous AGN Sample (MEXSAS). In this work, we use the EP9 band, 0.5–4.5 keV, unless otherwise stated. The main data of the MEXSAS sample are reported in Table 1, where Col. 1 indicates the source serial number N_{sou}; Col. 2 the IAU name; Col. 3 the redshift; Col. 4 the average flux in the 0.5–4.5 keV band, in erg/cm^{2}/s; Col. 5 the number of epochs N_{epo} in which the source has been observed; Col. 6 the length of the monitoring time interval in the rest frame; Col. 7 the uncorrected normalised excess variance; and Col. 8 the normalised excess variance corrected after Eq. (9) with days and b = 0.12.
In Fig. 1 we show the distribution of the sources in the luminosityredshift plane, where L_{X} indicates the luminosity in the Xray band 0.5–4.5 keV, which is computed from the corresponding flux in the EP9 band and directly extracted from the XMMSSC catalogue, by adopting a photon index Γ = 1.7.
It is to be remarked that the EP9 flux errors available in the previous release XMMSSCDR4 were wrongly estimated^{3}. This problem was not present in DR3 release and has been corrected in DR5 release, as shown in Fig. 2.
Fig. 2 Histograms of the relative EP9 flux errors in the XMMSSC releases. DR3, green; DR4, blue; DR5, red. Anomalously large errors are present for a subset of the DR4 catalogue. 

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3. The excess variance and the light curve duration effect
The normalised excess variance is defined by the equation (1)where is the mean flux computed over the available flux measures f_{i} of the same source, is the total variance of the light curve, while is the mean square photometric error associated with the measured fluxes f_{i}.
Fig. 3 Distribution of the light curve durations in the rest frame, Δt_{rest}, for the MEXSAS sample. 

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Because NXS is an estimate of the average variability within the monitoring time interval Δt_{obs} provided by the light curve, and because variability increases with the restframe time lag τ (e.g. Markowitz & Edelson 2004; Vagnetti et al. 2011), we expect that NXS also increases with the length of the monitoring time in the rest frame of the source, Δt_{rest} = Δt_{obs}/ (1 + z), whose distribution is shown in Fig. 3.
We then compute for the EP9 fluxes of all the 2700 sources of the MEXSAS sample, and report them in Fig. 4, as a function of the number of epochs N_{epo} sampled by the light curve. We notice two points: first, the large dispersion of the NXS values for poorly sampled light curves that quickly decreases for increasing N_{epo}, and, second, the presence of negative values that are also more frequent for small N_{epo}. In fact, NXS is computed with respect to the light curve average flux ⟨f⟩, which differs from the intrinsic mean μ (see, e.g. Allevato et al. 2013), and its expected deviation is larger for smaller numbers of sampled data. Moreover, the observed variance can be smaller than the error, resulting in a negative NXS that is more probable when the mean is less well estimated, so again for small N_{epo}.
Fig. 4 NXS as a function of the number of epochs in the light curve. 

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Fig. 5 NXS as a function of the light curve duration in the rest frame, Δt_{rest}. Black dots represent individual NXS values. Blue circles are ensemble averages in bins of Δt_{rest}. The solid blue line shows a linear leastsquares fit to the logarithms of the binned values with slope a = 0.196 ± 0.040. The Pearson correlation coefficient is r = 0.84, with null probability P( > r  ) = 0.008. 

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We now show in Fig. 5 the log of the excess variance as a function of the log of the restframe duration. Following Allevato et al. (2013), ensemble estimates of NXS are to be preferred to the individual values. We report ensemble averages in bins of Δt_{rest}, also including negative contributions. In fact, the removal of negative values might skew the distribution if not equally spread over the whole population. Individual values of NXS are also shown, when NXS > 0. A clear increase appears, as expected. The binned values can be fitted by a straight line with slope a = 0.196 ± 0.040, i.e. a powerlaw . The Pearson correlation coefficient is r = 0.84 with null probability P( > r  ) = 0.008.
We notice some possible sampling effects. First, different light curves are sampled with different patterns so this can introduce systematic differences, although this effect is not larger than 50% for PSD slopes that are shallower than −2 (Allevato et al. 2013). Second, when sampling long timescales, the red noise leak is smaller than for short timescales, and this might make the intrinsic slope of our fitted line steeper. However, our aim here is to show that NXS increases with the light curve duration Δt_{rest}. The precise value of the slope might be improved taking these additional effects into account.
4. Structure function
The structure function works in the time domain and is very helpful for an ensemble analysis of the variability, even for poor sampling of the individual sources, as in the present case. It is often used in the optical band (e.g. Trevese et al. 1994; Vanden Berk et al. 2004; Wilhite et al. 2008; Bauer et al. 2009; MacLeod et al. 2012) and is used less often in the Xrays, where the only ensemble analysis was performed by us in Paper I, in which we defined (2)f_{X}(t) and f_{X}(t + τ) as two measures of the flux, in a given Xray band, at two epochs differing by time lag τ in the rest frame. The term is the quadratic contribution of the photometric noise to the observed variations (see also the discussion by Kozłowski 2016). The average is computed within an appropriate bin of time lag around τ. The average of the absolute value of the variations was adopted because it is less sensitive to outliers and, in analogy with the expression introduced by di Clemente et al. (1996), in the optical. In the following, however, we also use the other standard expression first introduced by Simonetti et al. (1985)(3)The two expressions are equivalent if the variations follow a Gaussian distribution and the number of measured variations is large enough. If one or both the conditions are not fulfilled, the expression of Eq. (3) is sometimes preferred because it is directly related to other statistical quantities such as the autocorrelation function and the variance, although the differences are relatively small (see e.g. Bauer et al. 2009).
4.1. Updated ensemble SF
Fig. 6 Histogram of the restframe lag times for the MEXSAS sample. 

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We show in Fig. 6 the distribution of the restframe lag times τ_{rest} for the flux variations contributing to the computation of the SF for the MEXSAS sample. The histogram looks similar to that shown in Fig. 3 for the light curve durations, but it is much more populated due to all the possible combinations of pairs of observations.
We computed the SF for the MEXSAS sample, again using the EP9 fluxes. The result is shown in Fig. 7, using both Eq. (2) (red symbols and lines) and Eq. (3) (blue symbols and lines), with the SF computed in bins of log τ_{rest}. The representative points of the bins are centred weighting the individual lag values falling in each bin to take account of the nonuniform distribution of τ_{rest} shown in Fig. 6. The SF has been fitted by a powerlaw SF = kτ^{b}, through a linear leastsquares fit of the logarithms, weighted with the number of individual lag values falling in each bin.
The SF computed with the average of the square differences appears slightly flatter (b = 0.121 ± 0.004) than the SF obtained using the average of the absolute values (b = 0.143 ± 0.006), suggesting that the two expressions are not equivalent. In fact, we checked the distributions of our variations of log f_{X} for normality, applying a KolmogorovSmirnov test in each of the bins used in Fig. 7, always finding small probabilities that range from a few percent to 10^{11} depending on the bin population. Thus our distributions are not Gaussian and the expression of Eq. (3) is preferred.
Including normalisation, the SF computed with Eq. (3) is given by log SF = (0.121 ± 0.004)log τ_{rest}−(0.983 ± 0.010), so that its value at 1000 days is ≈0.24.
This updates the previous ensemble SF of Fig. 5 of Paper I, which was derived from a much smaller sample.
4.2. Correction of the NXS
We now want to use the dependence of the variability on the time lag, expressed through the SF, to estimate the expected value of the NXS in a given monitoring interval Δt_{obs}. We first rewrite Eq. (3), neglecting the photometric error (4)meaning that we refer to the intrinsic variations δlog f_{int}. Similarly, we rewrite Eq. (1), also neglecting the photometric error with the same meaning as above, as follows: (5)Here the average must be computed within the monitoring time interval Δt_{obs}. Both Eqs. (4) and (5) are expressed in terms of average square variations of log f_{int}, thus we can rewrite (6)where the factor 1/2 accounts for the two independent measures contributing to each SF flux difference, and the average must be computed within the restframe time interval Δt_{rest} = Δt_{obs}/ (1 + z).
Fig. 7 Structure function for the MEXSAS sample. Red points and continuous lines represent the SF computed for the EP9 flux variations, according to Eq. (2); blue points and continuous lines refer to the EP9 band, using Eq. (3). Red and blue shortdashed lines indicate the corresponding leastsquares fits. Blue longdashed line indicates the contribution of the photometric errors (the same for Eqs. (2) and (3)). Black dots represent the variations for the individual pairs of measurements contributing to the SF. 

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Fig. 8 Corrected NXS as a function of the light curve duration in the rest frame, Δt_{rest}, using b = 0.10 and days for individual values (black dots) and for binned averages (blue cicles). There is no correlation with Δt_{rest}. 

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Adopting now a functional form of the SF, for example a powerlaw SF = kτ^{b} as in Paper I, we compute the average as follows: (7)and finally we obtain (8)which shows that NXS is also expected to increase with a power law of the monitoring time interval. If this is expressed in the observer frame, an obvious dependence on the redshift is also found. Using Eq. (8), it is now possible to correct for the duration effect, extrapolating the measured NXS values to a fixed restframe time interval as follows: (9)This correction can be applied to a given set of NXS values to obtain new estimates referred to a uniform duration, adopting a previously determined SF exponent from literature, for example b = 0.10 from Paper I. Choosing days, the corrected values of are shown in Fig. 8. There is no correlation with Δt_{rest}, the Pearson correlation coefficient is r = 0.12 and the probability of obtaining this by chance is P( > r  ) = 0.70. The choice of the value b = 0.12 from the updated SF of the present paper would give similar results [r = −0.16,P( > r  ) = 0.60].
On the other hand, the possible change in slope of the PSD would affect this relation for the shortest timescales; however the break is usually <100 days (for black hole masses M_{BH}< 10^{9}; GonzálezMartín & Vaughan 2012), so this effect is only relevant for the most massive BHs.
4.3. Dependence on Xray luminosity and redshift
We then update the analysis of the SF as a function of the Xray luminosity in a similar way as that performed in Paper I, dividing our sample in luminosity bins. Our present sample is much richer compared to that used in Paper I and allows us to extend our analysis to lower luminosities to between L_{X} = 10^{43} erg/s and L_{X} = 10^{45.5} erg/s. At variance with Paper I, for the present sample we find (see Fig. 9) almost uniform slopes of the SF in the different luminosity bins, while the normalisation strongly depends on L_{X}. This work differs from Paper I, where we found slopes changing with L_{X}, in that we have a much richer sample of 2700 sources compared to 412 in the fist paper. In that case, the number of unbinned SF points contributing to the shortest timelag bin was small, and therefore only a few points contributed, once they were further divided in bins of luminosity; this resulted in a large dispersion of mean SF values in bins of luminosity, thereby artificially producing a dispersion in the slopes.
Describing the SF as log SF = A + blog τ_{rest}, we show in Fig. 10 the values of the slopes b and the intercepts A for the different luminosity bins. The slopes are almost constant with an average value ⟨b⟩ = 0.115, and are compatible within 2σ with the slope b = 0.12 of the overall sample shown in Fig. 7. The intercepts are clearly anticorrelated with L_{X} (correlation coefficient r = −0.96), and a weighted leastsquares fit gives A = (6.55 ± 1.42)−(0.170 ± 0.032)log L_{X}. Assuming a fixed slope, b = 0.12, changes the estimates of the intercepts with a fit A = (7.24 ± 0.81)−(0.186 ± 0.018)log L_{X}. This corresponds to values of the structure function at 1000 days decreasing approximately from 0.35 to 0.15 for increasing L_{X} within the adopted bins.
Fig. 9 Structure function in bins of Xray luminosity. Black lines and crosses denote 10^{43} erg / s <L_{X} ≤ 10^{43.5} erg / s; blue lines and triangles denote 10^{43.5} erg / s <L_{X} ≤ 10^{44} erg / s; green lines and squares indicate 10^{44} erg / s <L_{X} ≤ 10^{44.5} erg / s; yellow lines and hexagons represent 10^{44.5} erg / s <L_{X} ≤ 10^{45} erg / s; and red lines and circles indicate 10^{45} erg / s <L_{X} ≤ 10^{45.5} erg / s. 

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Figure 11 shows the SF divided in bins of redshift. We only considered sources with 10^{44} erg / s <L_{X} ≤ 10^{45} erg / s to reduce the observational correlation between redshift and luminosity (see Fig. 1). Four bins of redshift are considered: 0 <z ≤ 1.15, 1.15 <z ≤ 1.7, 1.7 <z ≤ 2.3, and 2.3 <z ≤ 3.4. The SFs are largely overlapped with no evidence of a change in normalisation. A weak flattening of the slopes for higher redshifts might be suggested. However, at variance with Paper I, where we found a significant partial correlation coefficient of variability with redshift (compensating for the change in L_{X}), we now obtain r_{Vz,L} = 0.05, which we interpret as no evidence of a dependence on redshift.
In addition, we note that z dependence could be affected by the different restframe energy ranges probed at different redshifts. This is further discussed in Sect. 4.4.1.
Fig. 10 Structure function parameters as functions of the Xray luminosity. Upper panel: slope b; the dotted line indicates the average ⟨b⟩ = 0.115; the dashed line indicates the fixed value b = 0.12, adopting the same dependence as in the general SF of Fig. 7. Lower panel: the intercept A. The open squares represent the values derived by the SFs of Fig. 9, with free A and b. The crosses are the values derived with fixed b = 0.12. The corresponding fits are shown with dotted lines (free b, A = (6.55 ± 1.42)−(0.17 ± 0.03)log L_{X}) and dashed lines (fixed b, A = (7.24 ± 0.81)−(0.19 ± 0.02)log L_{X}). 

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Fig. 11 Structure function in bins of redshift. Black lines and crosses represent 0 <z ≤ 1.15; blue lines and triangles indicate 1.15 <z ≤ 1.7; green lines and squares denote 1.7 <z ≤ 2.3; and red lines and circles indicate 2.3 <z ≤ 3.4. 

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4.4. Dependence on the emission band and spectral variability
Variability can of course also depend on the emission band. Results for individual Seyfert galaxies typically show a decrease of variability towards harder Xray bands (e.g. Sobolewska & Papadakis 2009), corresponding to a softer when brighter spectral variability. The same trend might also hold for quasars and high luminosity AGNs; for example Gibson & Brandt (2012) find a softer when brighter behaviour for a small sample of 16 radioquiet, nonBAL quasars extracted from the Chandra public archive. For our sample, we can investigate an ensemble behaviour indirectly, computing the structure function in different Xray bands, while a more direct analysis of the photon index variations will be presented in a future paper (Serafinelli et al., in prep.).
Fig. 12 Structure function for the XMMNewton bands EP1, EP2, EP3, and EP4 (filled circles and continuous lines). Also shown is the contribution of photometric errors, which has been subtracted from the observed variations according to Eq. (3) (dotted lines). 

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We show in Fig. 12 the structure functions for the MEXSAS sample for each of the XMMNewton spectral bands 0.2–0.5 keV (EP1), 0.5–1 keV (EP2), 1–2 keV (EP3), and 2–4.5 keV (EP4). We do not show the 4.5–12 keV band (EP5), which is strongly affected by photometric errors and is less reliable. The figure shows the structure functions computed after Eq. (3) (filled symbols and continuous lines) and the contribution of the photometric errors (dotted lines), which has been subtracted accordingly. The contribution of the errors is relatively high compared to the wider EP9 band (see Fig. 7) because of the smaller photon counts in these narrower bands, and thus these structure functions are more reliable for lags larger than ~30 days. Furthermore, we notice that there is a regular trend of decreasing variability from EP1 to EP3, while there is a more complex behaviour for band EP4 with some increase and flattening. Considering only the bands EP1, EP2, EP3, and averaging the SF in the lag interval 100 days ≤ τ_{rest} ≤ 1000 days, we find a dependence on the emission frequency given by (10)In turn, the dependence of variability on the emission frequency can be connected to the spectral variations, as was carried out by Trevese & Vagnetti (2002) for the optical band through the definition of the spectral variability parameter (11)which relates the temporal changes of the spectral index^{4} with those of the monochromatic flux. Values β> 0 correspond to a harder when brighter behaviour, typically observed in the optical band. In the Xray band, the β parameter can be rewritten as (12)in terms of changes of the photon index Γ^{5} and of the corresponding flux in the considered Xray band, f_{X}. Negative β values are expected for a softer when brighter behaviour.
Consider now a logarithmic flux variation in a given Xray band, Δlog f_{X}, which is essentially the structure function SF. When the photon index changes by ΔΓ, variations at different frequencies separated by δlog ν change as From the definition, Eq. (12), we have Thus we have δSF = β·SF δlog ν and so that from Eq. (10) we can estimate (13)This value is also in approximate agreement with a direct analysis of the photon index variations that is in progress (Serafinelli et al., in prep.).
4.4.1. Vcorrection
The dependence of variability on frequency also implies that variability in the rest frame is not the same as estimated in the observer frame. Our analysis of the variability is based on data tabulated in observerframe bands. We cannot fix the restframe band for AGNs at different redshifts. However, we can use the estimated spectral variability, Eq. (13), to simulate the shift from observer frame to rest frame, as follows.
For a source at redshift z, we are measuring variability in a restframe band shifted by δlog ν = log (1 + z), so that We can derive The average effect for a sample is a downwards shift (for β< 0), so that to correct the SF we should apply the opposite upwards shift, which we call Vcorrection: (14)We note here that the standard Kcorrection has no effect on our SFs because fluxes before and after a variation are affected by the same zdependent factor for a given source, so that the corresponding logarithmic change is not altered.
Taking β from Eq. (13) and ⟨log (1 + z)⟩ ≃ 0.38, we estimate for our sample V  corr ≃ 0.06 for the logarithm of the SF, or a ≲15% correction to the SF itself.
Fig. 13 Vcorrected SF in bins of redshift. Symbols and colours as in Fig. 11. 

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We also note that the previously discussed dependence of the SF on redshift can be affected. The z dependent Vcorrections for the four redshift bins adopted in Fig. 11 are Vcorr = 0.040, 0.058, 0.074, 0.085, in order of increasing redshift. The effect, shown in Fig. 13, is relatively small and suggests a weak increase with z for the variability at short time lags, ≲few days. However, this is not strong evidence because of both the relatively poor sampling and high error contribution for the SF at these time lags; see Figs. 6 and 7.
5. Discussion
The normalised excess variance is popularly used as a variability estimator. In most cases the method is applied correctly, using monitoring time intervals of fixed duration, for AGN samples at low redshift (e.g. as in Ponti et al. 2012). But this estimator depends on the length of the time interval in the rest frame and is therefore affected also by the cosmological time dilation (e.g. Gaskell 1981). The method is sometimes used improperly, choosing nonuniform time intervals, and/or including high redshift sources (e.g. Lanzuisi et al. 2014), thereby underestimating their variability. A few other examples of this include the work by La Franca et al. (2014), which applies NXS to the same data as Ponti et al. (2012) to derive a luminosity distance estimator, but envisages an extension of the study to higher redshift sources, where NXS would underestimate variability. The work by Cartier et al. (2015) applies NXS to the QuestLa Silla variability survey, including high redshift AGNs, whose variability is therefore underestimated. However, their main implication is a trend indicating that high redshift and more variable AGNs tend to have redder colours and this trend would be reinforced taking the NXS underestimate into account.
To demonstrate the duration effect for the NXS estimates, we used a sample of AGNs with multiepoch Xray observations (MEXSAS) extracted from the fifth release of the XMMNewton Serendipitous Source Catalogue (XMMSSCDR5). We have also shown that the effect can be corrected on the basis of the knowledge of the behaviour of variability that is gained from structure function studies; our correcting formula, Eq. (9), can be successfully applied to further NXSbased studies.
We have updated the analysis of the ensemble structure function, finding that Xray variability is well described by a powerlaw function of the restframe time lag, increasing as τ^{0.12} and extending up to ~2000 days. We have also shown that Xray variability is inversely correlated with Xray luminosity, approximately as . This anticorrelation has been reported, usually at short timescales, by many authors (e.g. Barr & Mushotzky 1986; Lawrence & Papadakis 1993 for lowz AGNs; Manners et al. 2002; Papadakis et al. 2008 for higher z) with variability approximately proportional to . At longer timescales, the analysis by Markowitz & Edelson (2004), for local AGNs, indicates . One simple interpretation of the anticorrelation is the superposition of several independently flaring subunits (e.g. Green et al. 1993; Nandra et al. 1997; Almaini et al. 2000).
We also find a dependence of variability on the emission frequency approximately as ν^{0.15}. In turn, this dependency is related to the change of the photon index, indicating a softer when brighter spectral variability behaviour, which extends a trend previously found for Seyfert galaxies (Sobolewska & Papadakis 2009) to AGNs with higher redshifts and luminosities. Because of this dependence, variability in the rest frame differs from that estimated in the observerframe bands; however the effect can be corrected and we propose a simple correction term called Vcorrection, resulting in a moderate shift upwards (≲15%) for the structure function. The same correction, applied in different bins of redshift, can affect the resulting zdependence of variability, suggesting a weak increase with z for the variability at short time lags.
We finally remark that the corrections proposed by Allevato et al. (2013) on the NXS should also be taken into account in the case of sparse sampling and for a comparison with physical models.
http://xmmsscwww.star.le.ac.uk/Catalogue/xcat_public_3XMMDR4.html#watchouts. This watchout was indeed published after the XMMSSC Team had been alerted of the problem by our group.
Acknowledgments
We acknowledge funding from PRIN/MIUR2010 award 2010NHBSBE. We thank Stefano Bianchi, Szymon Kozłowski, Fabio La Franca, Francesco Tombesi, and Dario Trevese for useful discussions. This research has made use of data obtained from the 3XMM XMMNewton serendipitous source catalogue compiled by the 10 institutes of the XMMNewton Survey Science Centre selected by ESA. Funding for SDSSIII has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSSIII web site is http://www.sdss3.org/. SDSSIII is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSSIII Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico StateUniversity, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
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All Tables
All Figures
Fig. 1 Distribution of the sources in the L_{X}z plane. The blue dots represent the average values of the Xray luminosities computed on the available data of each light curve. Lines of constant Xray flux are also shown. 

Open with DEXTER  
In the text 
Fig. 2 Histograms of the relative EP9 flux errors in the XMMSSC releases. DR3, green; DR4, blue; DR5, red. Anomalously large errors are present for a subset of the DR4 catalogue. 

Open with DEXTER  
In the text 
Fig. 3 Distribution of the light curve durations in the rest frame, Δt_{rest}, for the MEXSAS sample. 

Open with DEXTER  
In the text 
Fig. 4 NXS as a function of the number of epochs in the light curve. 

Open with DEXTER  
In the text 
Fig. 5 NXS as a function of the light curve duration in the rest frame, Δt_{rest}. Black dots represent individual NXS values. Blue circles are ensemble averages in bins of Δt_{rest}. The solid blue line shows a linear leastsquares fit to the logarithms of the binned values with slope a = 0.196 ± 0.040. The Pearson correlation coefficient is r = 0.84, with null probability P( > r  ) = 0.008. 

Open with DEXTER  
In the text 
Fig. 6 Histogram of the restframe lag times for the MEXSAS sample. 

Open with DEXTER  
In the text 
Fig. 7 Structure function for the MEXSAS sample. Red points and continuous lines represent the SF computed for the EP9 flux variations, according to Eq. (2); blue points and continuous lines refer to the EP9 band, using Eq. (3). Red and blue shortdashed lines indicate the corresponding leastsquares fits. Blue longdashed line indicates the contribution of the photometric errors (the same for Eqs. (2) and (3)). Black dots represent the variations for the individual pairs of measurements contributing to the SF. 

Open with DEXTER  
In the text 
Fig. 8 Corrected NXS as a function of the light curve duration in the rest frame, Δt_{rest}, using b = 0.10 and days for individual values (black dots) and for binned averages (blue cicles). There is no correlation with Δt_{rest}. 

Open with DEXTER  
In the text 
Fig. 9 Structure function in bins of Xray luminosity. Black lines and crosses denote 10^{43} erg / s <L_{X} ≤ 10^{43.5} erg / s; blue lines and triangles denote 10^{43.5} erg / s <L_{X} ≤ 10^{44} erg / s; green lines and squares indicate 10^{44} erg / s <L_{X} ≤ 10^{44.5} erg / s; yellow lines and hexagons represent 10^{44.5} erg / s <L_{X} ≤ 10^{45} erg / s; and red lines and circles indicate 10^{45} erg / s <L_{X} ≤ 10^{45.5} erg / s. 

Open with DEXTER  
In the text 
Fig. 10 Structure function parameters as functions of the Xray luminosity. Upper panel: slope b; the dotted line indicates the average ⟨b⟩ = 0.115; the dashed line indicates the fixed value b = 0.12, adopting the same dependence as in the general SF of Fig. 7. Lower panel: the intercept A. The open squares represent the values derived by the SFs of Fig. 9, with free A and b. The crosses are the values derived with fixed b = 0.12. The corresponding fits are shown with dotted lines (free b, A = (6.55 ± 1.42)−(0.17 ± 0.03)log L_{X}) and dashed lines (fixed b, A = (7.24 ± 0.81)−(0.19 ± 0.02)log L_{X}). 

Open with DEXTER  
In the text 
Fig. 11 Structure function in bins of redshift. Black lines and crosses represent 0 <z ≤ 1.15; blue lines and triangles indicate 1.15 <z ≤ 1.7; green lines and squares denote 1.7 <z ≤ 2.3; and red lines and circles indicate 2.3 <z ≤ 3.4. 

Open with DEXTER  
In the text 
Fig. 12 Structure function for the XMMNewton bands EP1, EP2, EP3, and EP4 (filled circles and continuous lines). Also shown is the contribution of photometric errors, which has been subtracted from the observed variations according to Eq. (3) (dotted lines). 

Open with DEXTER  
In the text 
Fig. 13 Vcorrected SF in bins of redshift. Symbols and colours as in Fig. 11. 

Open with DEXTER  
In the text 