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Issue
A&A
Volume 536, December 2011
Article Number A84
Number of page(s) 17
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/201118072
Published online 13 December 2011

© ESO, 2011

1. Introduction

Active galactic nuclei (AGN) typically show flux variability in all wavebands and on different timescales from minutes to years. This behaviour has been widely used to constrain the size and location of the emission regions and to obtain information on the emission mechanisms as well as the processes that cause the variability itself.

In addition to the study of individual light curves, ensemble properties of statistical AGN samples have been investigated in the optical/UV band through the use of the structure function (SF) (e.g. Trevese et al. 1994; Cristiani et al. 1996; Vanden Berk et al. 2004), and in the X-rays through the analysis of the fractional variability (Almaini et al. 2000; Manners et al. 2002).

Optical variability has been found to increase with decreasing luminosity (e.g. Cristiani et al. 1996), and with increasing redshift (Giallongo et al. 1991). The average increase with redshift of the amplitude of variability can be explained by the fact that high-redshift sources are observed at a higher rest-frame frequency, where they are more variable (di Clemente et al. 1996). The stronger variability at higher frequency, in turn, is caused by a hardening of the spectral energy distribution (SED) in the brighter phase, as shown by ensemble analyses of multiband optical photometry of quasar (QSO) samples (Trevese et al. 2001; Trevese & Vagnetti 2002). More recently, Vanden Berk et al. (2004) applied an ensemble analysis to a large sample of  ~25   000 QSOs observed at two epochs only with the Sloan Digital Sky Survey (SDSS). The authors analysed variability as a function of intrinsic luminosity, redshift, rest-frame frequency and time lag between the observations, proposing a weak, intrinsic increase of variability with redshift, in addition to the amount previously explained by the stronger variability at higher rest-frame frequency (di Clemente et al. 1996), although additional analyses have not confirmed this increase (e.g. MacLeod et al. 2010).

In the X-ray domain, variability occurs on shorter time scales than in any other band, and is thought to come from a hot corona close to the central black hole (BH). Most investigations concern the light curves of individual nearby Seyfert 1 AGNs (e.g. Uttley et al. 2002; Uttley & McHardy 2005). It has been found that low-luminosity AGNs are generally more variable than higher luminosity ones (e.g. Barr & Mushotzky 1986; Lawrence & Papadakis 1993; Green et al. 1993; Nandra et al. 1997), and that the variability amplitude is higher on long time scales than on short time scales (e.g. Markowitz & Edelson 2004). In addition, it has been suggested that variability also increases with redshift (Paolillo et al. 2004).

Proposed variability models include a single coherent oscillator (e.g. Almaini et al. 2000), a superposition of individual flares or spots (e.g. Lehto 1989; Abramowicz et al. 1991; Czerny et al. 2004), variable absorption and/or reflection (e.g. Abrassart & Czerny 2000; Miniutti & Fabian 2004; Chevallier et al. 2006).

The relation between X-ray and optical/UV variability may be due to either i) Compton up-scattering in the hot corona of optical photons emitted by the disk (Haardt & Maraschi 1991); or to ii) a reprocessing of X-rays into thermal optical emission by means of irradiation and heating of the accretion disk (Collin-Souffrin 1991). In the first case, variations in the optical/UV flux would lead to X-ray variations, and vice versa in the latter case. Cross-correlation analyses of well-sampled X-ray and optical/UV light curves allow us to constrain models for the cause of the variability. The main results obtained so far indicate a cross-correlation between X-ray and UV/optical variations on the timescale of days, and in some cases delays between the two bands have been measured, with both X-rays lagging the UV (e.g. Marshall et al. 2008; Doroshenko et al. 2009), and vice versa (e.g. Shemmer et al. 2001; Arévalo et al. 2009).

Even more insight into the relation between X-ray and optical/UV variability is given by the analysis of the X-ray/UV ratio and its variability. Vagnetti et al. (2010) have shown that variability of αox1 increases as a function of time-lag for a sample of serendipitously selected AGNs with simultaneous X-ray and UV measurements. This contributes part of the observed dispersion in the αox-LUV anti-correlation, while another contribution is given by intrinsic differences among the average values of each AGN.

In the present paper, we present for the first time an ensemble structure function analysis of the variability of AGNs in the X-ray band. We adopt two sets of multi-epoch X-ray measurements extracted from the serendipitous source catalogues of XMM-Newton (Watson et al. 2009) and Swift (Puccetti et al. 2011).

The paper is organised as follows. Section 2 describes the data extracted from the archival catalogues. Section 3 describes the computation of the structure functions, and discusses their shapes, their dependence on black hole mass and bolometric luminosity, as well as on X-ray luminosity and redshift. In Sect. 4 we discuss and summarise the results.

Throughout the paper, we adopt the cosmology H0 = 70   km   s-1   Mpc-1, Ωm = 0.3, and ΩΛ = 0.7.

2. The data

2.1. XMM-Newton

The XMM-Newton Serendipitous Source Catalogue (XMMSSC) (Watson et al. 2009) is a comprehensive catalogue of serendipitous X-ray sources from the XMM-Newton observatory. The version presently available is 2XMMi-DR3, the latest incremental update of the second version of the catalogue2. It contains source detections drawn from 4953 XMM-Newton EPIC observations made between 2000 February 3 and 2008 October 08; all datasets were publicly available by 2009 October 31, but not all public observations are included in this catalogue. The total area of the catalogue fields is  ~814 deg2, but taking account of the substantial overlaps between observations, the net sky area covered independently is  ~504 deg2. The 2XMMi-DR3 catalogue contains 353 191 detections (above the processing likelihood threshold of 6), related to 262 902 unique X-ray sources, therefore a significant number of sources (41 979) have more than one record within the catalogue.

We used the TOPCAT3 software to extract the sources with repeated X-ray observations from the 2XMMi-DR3 catalogue and cross-correlated this list with the DR7 edition of the SDSS Quasar Catalogue (Schneider et al. 2010) to obtain redshifts and spectral classifications of the sources. We used a maximum distance of 1.5 arcsec, corresponding to the uncertainty in the X-ray position, resulting in 412 quasars that were observed from 2 to 25 epochs each for a total of 1376 observations. We refer to these sources as the XMM-Newton sample, and report them in Table 1, where Col. 1 corresponds to the source serial number; Col. 2 gives the source name; Col. 3 the redshift; Col. 4 the number of observation epochs for the source; Col. 5 the average log of the X-ray flux in the observed 0.5–4.5 keV band, in erg cm-2 s-1; Col. 6 the average log of the X-ray luminosity in the 0.5–4.5 keV band, in erg s-1, computed with a photon index Γ = 1.8; and Cols. 7 and 8 the log of the minimum and maximum lag between any two epochs of the light curve in the rest-frame of the source in days.

The sources are shown in the LX-z plane in Fig. 1 together with the sources of the Swift sample (Sect. 2.2). Here and throughout we adopted the same X-ray band 0.5–4.5 keV for the two samples. For XMMSSC, the flux was directly extracted from the EP9 band of the catalogue, while for the Swift sample the flux was computed from the Swift band 0.3–10 keV, adopting a photon index Γ = 1.8.

Typical monitoring times range from months to few years in the rest-frame. Some of the best sampled light curves with 10 or more epochs are shown in Fig. 2, with times in rest-frame days, counted from the initial epoch of each light curve.

thumbnail Fig. 1

Distribution of the sources in the LX-z plane. Dots: XMM-Newton sample; circles: Swift sample.

thumbnail Fig. 2

Some of the best-sampled light curves from the XMM-Newton sample. Times are counted from the initial epoch of each light curve in the rest-frame. Errors, proportional to the inverse square root of the photon count at each epoch, are displayed as 3-σ values. Errors are discussed in more detail in Sect. 3. Source numbers from Col. 1 of Table 1 are indicated.

2.2. Swift

In the context of serendipitous surveys, the Swift satellite provides a unique capability. Although this space observatory is designed to discover gamma-ray bursts (GRB) (Gehrels et al. 2004), it is possible to use individual pointed observations of each GRB to build a large sample of deep X-ray images by stacking the individual exposures. To this purpose, Puccetti et al. (2011) considered all Swift GRB observations from January 2005 to December 2008, with a total exposure time in the X-ray Telescope (XRT) longer than 10 ks. These authors also analysed the XRT 0.5 Ms observation of the Chandra Deep Field South (CDFS) sky region. This set of observations is called the Swift Serendipitous Survey in deep XRT GRB Fields (S3XGF). These 374 images make up an unbiased X-ray survey because GRBs explode at random positions in the sky, and Puccetti et al. (2011) used them to define a well-suited statistical sample of X-ray point sources. The total exposure time of the survey is 36.8 Ms, with  ~32% of the fields with more than 100 ks exposure time, and  ~28% with exposure time in the range 50–100 ks. The survey covers a total area of  ~32.55 deg2.

We used the preliminary version S3XGF catalogue, comprising GRB fields observed from January 2005 to June 2007, and cross-correlated it with the DR7 edition of the SDSS Quasar Catalogue (Schneider et al. 2010) to obtain redshifts and spectral classifications.

We found 27 confirmed quasars with sufficient sampling (at least 100 photons in the light curve) to be used in the following SF analysis. These sources, to which we will refer as the Swift sample, are reported in Table 2, where Col. 1 corresponds to the source serial number; Col. 2 to the source name; Col. 3 gives the redshift; Col. 4 the number of time bins into which we divide the light curve according to the procedure described in the following; Col. 5 the average log of the X-ray flux in the band 0.5–4.5 keV, in erg cm-2 s-1; Col. 6 the average log of the X-ray luminosity in the band 0.5–4.5 keV, in erg s-1; Col. 7 the GRB field where the source was observed.

The light curve files extracted from the Swift archive contain sequences of time intervals Δti between tstart,i and tstop,i, in which the telescope was observing, with ni the number of photons detected in each interval. We binned the light curves using a bin size Δtbin = 5 × 104 s, which is a good compromise to obtain an average number of photons/bin  ≳ 10 and a number of useful bins (i.e., bins with non-zero number of photons) in the light curve  ≳ 10. There is a negligible number of bins with zero photons, however. We assigned an average time tj to each bin j weighted by the number of photons detected in the intervals (or fractions of intervals) Δti overlapped with the bin: tj =  ∑ niti/ ∑ ni, where ti = (tstart,i + tstop,i)/2.

Some of the best-sampled light curves are shown in Fig. 3 with times in rest-frame days counted from the initial epoch of each light curve. Typical monitoring times range from some days to a few weeks in the rest-frame, and are therefore complementary to the time scales sampled by XMM-Newton .

The distribution of the Swift sample in the LX-z plane is shown in Fig. 1 together with the XMM-Newton sample.

thumbnail Fig. 3

Some of the best-sampled light curves from the Swift sample. Times are counted from the initial epoch of each light curve, in the rest-frame. Errors proportional to the inverse square root of the photon count at each epoch are displayed as 3-σ values. Errors are discussed in more detail in Sect. 3. Source numbers from Col. 1 of Table 2 are indicated.

3. The structure function

The structure function (SF) has the great advantage of working in the time domain, which allows for an ensemble analysis even for extremely poor sampling of individual objects, when the armonic content is completely lost. In this case, the structure function is to be preferred over power spectral density (PSD) analysis (e.g. Hughes et al. 1992; Collier & Peterson 2001; Favre et al. 2005). The SF was first introduced by Simonetti et al. (1985), and has since been used in various bands, including radio (e.g. Hughes et al. 1992), optical (e.g. Trevese et al. 1994; Kawaguchi et al. 1998; de Vries et al. 2003; Bauer et al. 2009), and X-ray (e.g. Fiore et al. 1998; Brinkmann et al. 2001; Gliozzi et al. 2001; Iyomoto & Makishima 2001; Zhang et al. 2002).

The SF provides a measure of the mean deviation for data points separated by a time lag τ, and is defined in various ways in the literature. A variant in the definition concerns the use of the average square difference (e.g. Simonetti et al. 1985; Hughes et al. 1992) or the average of the absolute values of the differences (di Clemente et al. 1996). Another variant concerns the use of magnitudes or fluxes: while in the optical the SF is usually defined in terms of magnitude differences, in the X-rays and in the radio band the SF is most often defined in terms of flux differences, although there are exceptions, e.g., Fiore et al. (1998) introduced X-ray magnitudes and their differences.

For an analogy with the optical, we used the logarithm of the flux instead of the flux itself, and defined the SF with the following formula: (1)Here, the average of the absolute value of the difference is used, as in di Clemente et al. (1996); σn is the contribution of the photometric noise to the observed variations. fX(t) and fX(t + τ) are two measures of the flux fX in a given X-ray band at two epochs differing by the lag τ. The factor π/2 normalises SF to the rms value in the case of a Gaussian distribution. The X-ray band adopted in this paper is 0.5–4.5 keV, and the lag τ is computed in the rest frame: (2)While a definition in terms of flux differences could also be used for studies of individual sources, our definition with logarithmic differences, Eq. (1), is certainly preferable for an ensemble analysis, otherwise the contribution of faint sources would be negligible compared to that of brighter ones.

We now computed and compared the structure functions for the two samples. While the XMM-Newton sample is much larger than the Swift sample (412 sources vs. 27 sources), the number of epochs is very small for most XMM-Newton sources (338/412 = 82% of the sources having less than 5 epochs), while the Swift light curves (with the adopted binning, see Sect. 2.2) are better sampled, 21/27 = 78% of the sources having 10 or more bins (or “epochs”), with a mean number  ~16. So the contributions of the two samples to the respective SFs are comparable in number, although different in the time scales sampled. The light-curve of the kth source, with Nk epochs, contributes Nk(Nk−1)/2 points to SF(τrest), for all the time lags τrest,ij = |ti − tj|/(1 + z), where ti and tj are two epochs in the observer frame.

This can be seen in Fig. 4, where the histograms of the rest-frame time lags are shown for the two samples, with bins of Δlog τ = 0.2: hundreds of points contribute the most populated bins of each sample, which are days-weeks for the Swift sample and months-years for the XMM-Newton sample. The latter contributes also non-negligibly in the days-weeks range, with several tens of points.

thumbnail Fig. 4

Histogram of the rest-frame time lags contributing to the structure functions. Continuous histogram: XMM-Newton sample; dashed histogram: Swift sample.

In Figs. 5 and 6 we show the structure functions computed with Eq. (1) for the XMM-Newtonand Swift samples, respectively.

thumbnail Fig. 5

Structure function for the XMM-Newton sample in bins of Δlog τ = 0.5. The small empty circles and the continuous line connecting them show the uncorrected SF (i.e., neglecting σn in Eq. (1)). The larger filled circles and the line connecting them, show the SF corrected for the noise. The continuous line without data points indicates the average value of the noise in each bin, and the dashed, horizontal line is its weighted average, according to the number of points in each bin, adopted in Eq. (1). The dotted line is a weighted least-squares fit to the data of the bins. The small dots are the contributions from pairs of individual measurements at times differing by τ.

thumbnail Fig. 6

Structure function for the Swift sample, in bins of Δlog τ = 0.5. The small empty circles and the continuous line connecting them show the uncorrected SF (i.e., neglecting σn in Eq. (1)). The larger, filled circles and the line connecting them show the SF corrected for the noise. The continuous line without data points indicates the average value of the noise in each bin, and the dashed, horizontal line is their weighted average, according to the number of points in each bin, adopted in Eq. (1). The dotted line is a weighted least-squares fit to the data of the bins. The small dots are the contributions from pairs of individual measurements at times differing by τ.

To estimate the photometric noise σn in Eq. (1), we evaluated its contribution in each bin with the following considerations. The quadratic contribution of the noise to the SF is (3)where δfX are the flux variations caused by noise alone (excluding source variability), and we assume δfX/fX = 1/, N being the number of counted photons at a given epoch, and its reciprocal is mediated in any given bin of the SF among the Np points, which are contributed by the various light-curves; Nk is the average photon count per epoch of the kth light-curve; the factor 2 is due to the contribution of 2 independent measurements to each flux variation. The values obtained in each bin are connected and shown in Figs. 5 and 6 as thin, continuous lines, while their average values, weighted with the numbers of points in each bin, are shown as dashed lines.

The average values σn = 0.031 (XMM-Newton sample) and σn = 0.163 (Swift sample) were then inserted in Eq. (1) to compute the SF, which is shown in Figs. 5 and 6, both with and without noise subtraction. The noise so estimated is almost negligible for the XMM-Newton sample, and quite high for the Swift sample. This is mainly because of the smaller effective area of Swift, and also because of the longer exposures of the XMM-Newton observations, which are typically several tens of ks per epoch, while for Swift the light-curves are binned in intervals of 50 ks, with effective exposures within a small fraction of the bin, around 10 ks.

Although the two SFs appear different before noise subtraction, their slopes and amplitudes agree quite well after correction. We stress that noise subtraction is not parametrical, but consistently derived by the photon counts. The fits shown in Figs. 5 and 6 are least squares of the bin representative points, weighted with the number of individual points in each bin, log SF = a + blog τrest, or (4)with consistent slopes, b = 0.10 ± 0.01 for the XMM-Newton sample, and b = 0.07 ± 0.04 for the Swift sample.

3.1. Relation with the PSD

X-ray variability of individual sources is usually analysed in terms of the PSD. This has been often described by a power-law, P(f) ∝ f−α,   α ~ 1.5, (e.g. Lawrence & Papadakis 1993). However, one or two breaks in the PSD of nearby AGNs have also been detected (e.g. Markowitz et al. 2003; O’Neill et al. 2005), and the PSD has been found to have a power-law exponent α ≈ 2 for f > fHFB, α ≈ 1 for fLFB < f < fHFB, and in some cases α ≈ 0 for f < fLFB. In turn, the high-frequency break has been found to be related to the mass of the central BH (e.g. Papadakis 2004).

An SF with the form of a single power-law as in Eq. (4) is equivalent to a single power-law PSD if the frequency range extends from 0 to ∞. Then a simple relation between the exponents holds (e.g. Kawaguchi et al. 1998; Bauer et al. 2009; Emmanoulopoulos et al. 2010): (5)The slope of our SF, b ≲ 0.1, would then correspond to a PSD exponent α ≲ 1.2, slightly flatter than the reference value α ~ 1.5 (Lawrence & Papadakis 1993).

However, Eq. (5) does not straightforwardly apply when the PSD contains a break. Emmanoulopoulos et al. (2010) produced 2000 artificial light-curves with a PSD shaped as a broken power-law with a break at a given value fB, and estimate the corresponding SFs. Figures 10 and 11 of Emmanoulopoulos et al. (2010) show that SFs also display a break whose distribution peaks around τB ~ 1/fB, but the SF slopes before and after this break do not agree with the relation of Eq. (5). In particular, the SF appears flatter than Eq. (5) below the break, and steeper above the break, resulting in less bending.

To analyse the relation between the shapes of PSD and SF, we evaluated the SF numerically via fast Fourier transform (FFT) techniques according to the relation (e.g. Emmanoulopoulos et al. 2010), for a PSD shaped as a broken power-law. The result, shown in Fig. 7 for input PSD spectral indexes α1 = 1.2, α2 = 2, is an SF shaped approximately as a broken power-law, but with a slope changing gradually and with less bending, which confirms the result by Emmanoulopoulos et al. (2010).

The above results suggest that we should expect some evidence of a break in the SF of AGNs with a typical broken power-law PSD.

thumbnail Fig. 7

SF (dashed line) and its slope (continuous line) computed from a PSD shaped as a broken power-law, with α1 = 1.2 and α2 = 2, shown in the inset. The dotted lines indicate the values of the power-law exponent b, expected from Eq. (5) for the single power-law case, b1 = 0.1, b2 = 0.5. The cut-off in the SF at long τ is caused by the finite number of Fourier frequencies used in the FFT calculation. Time lags and Fourier frequencies are in arbitrary units.

3.2. Dependence on mass and bolometric luminosity

McHardy et al. (2006) proposed that the high-frequency break is related not only to the black hole mass, MBH, but also to the accretion rate in units of its Eddington value, ṁE ≈ Lbol/LEdd, and found the following relation (6)where we abbreviate M6 = MBH/106M⊙ and L44 = Lbol/1044 erg/s.

However, while for the light curves of individual objects the relation between SF and PSD is relatively simple, for an ensemble SF the different positions of the breaks should combine in the ensemble SF, possibly smoothing the result, depending on how the variability amplitude changes with MBH and/or Lbol.

To find any break in the SF, we segregated our XMM-Newton sample according to MBH and Lbol values. Estimates of the masses and bolometric luminosities were extracted from the catalogue of quasar properties by Shen et al. (2011). We show in Fig. 8 the distribution of  ~100   000 AGNs from that catalogue in the plane MBH-Lbol, as well as the same distribution for our XMM-Newton sample, which appears quite similar, despite its smaller population (412 sources). We also show in the same figure some low-luminosity AGNs from Uttley & McHardy (2005), which will be discussed below.

thumbnail Fig. 8

Black hole masses and bolometric luminosities of AGN samples. Black dots:  ~100   000 sources from the Shen et al. (2011) catalogue. Red filled circles: XMM-Newton sample. Blue empty squares: low-luminosity AGNs from Uttley & McHardy (2005).

We then plot in Fig. 9 the structure function for XMM-Newton subsamples binned in intervals of log MBH and log Lbol, with bin width Δlog MBH = Δlog Lbol = 0.5. The SF is shown for subsamples with at least 30 SF points, in the range of masses 107.5   M⊙ < MBH < 1010   M⊙ and luminosities 1045   erg/s < Lbol < 1047.5   erg/s. The total number of SF points is reported in each box, as well as the average SF slope (weighted with the number of points in each bin of τrest), and the expected value of log τbreak, according to Eq. (6). The SF of the total XMM-Newton sample is also reported for comparison.

thumbnail Fig. 9

Structure function for XMM-Newton subsamples binned in intervals of log MBH and log Lbol, with bin width Δlog MBH = Δlog Lbol = 0.5. Values of MBH and Lbol are reported in the upper and right axes, respectively. Subsamples with less than 30 SF points are not shown. The SF of the total XMM-Newton sample is reported, for comparison, in the first box in the upper left corner. The values reported in each box are the number of SF points, the average SF slope, and the expected value of log τbreak, according to Eq. (6). Contributions from pairs of individual measurements are also shown (dots).

Our results do not support the existence of a break in the SF, expected following Eq. (6). However, we note that the analysis by McHardy et al. (2006) is based on a few AGNs with quite low luminosities and masses (see Uttley & McHardy 2005), compared to our XMM-Newton sample, and the Shen et al. (2011) catalogue, see Fig. 8.

The absence of a break in our results could be understood if Eq. (6), which appears to hold for AGNs with MBH ≲ 108   M⊙ and Lbol ≲ 1045 erg/s, would not apply for larger masses and higher luminosities.

McHardy et al. (2006) associated the break time scale to a thermal or viscous time scale related to the inner radius of the accretion disk, and identify this with the transition radius Rtr predicted by Liu et al. (1999), based on evaporation of the inner disk in low Eddington ratio AGNs, describing the transition between an external cool thin disk and an inner, hot, advection-dominated accretion flow (ADAF). This model clearly does not apply to high-luminosity QSOs (see, e.g. Narayan et al. 1998).

3.3. Dependence on X-ray luminosity and redshift

Many authors have found inverse dependences of the X-ray variability on the X-ray luminosity LX. Different variability indexes are used, so they must be briefly recalled to make comparisons.

Most authors use the normalised excess variance (e.g. Nandra et al. 1997; Vaughan et al. 2003), defined as , where S2 is the total variance of the light curve, is the mean square error, and is the mean of N total measurements; or the square root of it, which is also referred to as fractional variability amplitude, Fvar (e.g. Markowitz & Edelson 2004).

Green et al. (1993) used the normalised variability amplitude, square root of the power at a specific frequency, normalised to the mean count rate of the related light curve. Lawrence & Papadakis (1993) used the amplitude of the power spectrum at a specific frequency.

As pointed out by Lawrence & Papadakis (1993), and Fvar depend on the length of the monitored time interval. Moreover, we notice that they depend on redshift, because the time interval must be properly measured in the rest-frame of the source, as stressed by Giallongo et al. (1991) for the optical variability, and by Papadakis et al. (2008) for the X-ray case. So the comparison between different results must be taken with some caution. With these limitations in mind, and calling Ivar a generic variability index (or its square root where appropriate), most of the previous results on the variability dependence on luminosity can be expressed in power-law form, . Values for the exponent k are usually about  ~0.3, for time scales of days, and for samples including Seyfert galaxies and/or low-z QSOs (Green et al. 1993; Lawrence & Papadakis 1993; Nandra et al. 1997; Markowitz & Edelson 2004). Similar values are found also for higher redshift QSOs, e.g. by Manners et al. (2002), up to z = 2 (k = 0.27, still for time scales of days), and by Papadakis et al. (2008), up to z ~ 3.4 (k = 0.33, for time scales of tens of days). Stronger dependences are instead found by Paolillo et al. (2004) (k ~ 0.65, in the redshift range 0.5 < z < 1.3) and by Almaini et al. (2000) (k = 0.75, for z < 0.5). For longer time scales (years), a few analyses have been performed, e.g., Markowitz & Edelson (2004) found a weaker dependence, k ~ 0.13.

With the analysis of the rest-frame structure function, we can properly compare variability amplitudes at various time lags, and provide an unbiased characterisation of the dependence of variability on luminosity and redshift. In Fig. 10 we show the SFs for four luminosity bins between LX = 1043.5   erg/s and LX = 1045.5   erg/s: a clear and strong dependence on LX appears. A change in the slope of the SF is also present (between  ~0.04 and 0.14), implying that a different dependence on LX is expected for different time lags. To see this, we re-plot in Fig. 11 the SF data vs LX for two different bins of time lag, centred on 1 day and 100 days, respectively. The least-squares fits, weighted with the number of measurements in each bin, correspond to power-law exponent k = 0.42 ± 0.03 for the shorter time scale, a slightly stronger dependence, compared to the results by most previous authors. For the longer time scale (100 days), our result is k = 0.21 ± 0.07, which approximately agrees with the trend found by Markowitz & Edelson (2004).

A simple interpretation of the decrease of variability with luminosity (L) is the superposition of N randomly flaring subunits. This was already considered in early studies of optical variability (e.g. Pica & Smith 1983; Aretxaga et al. 1997), and, in its simplest version of independent and identical flares, would predict a variability amplitude  ∝ N−1/2 ∝ L−1/2. In the X-ray domain, several authors have also considered the same argument (Green et al. 1993; Nandra et al. 1997; Almaini et al. 2000; Manners et al. 2002). The observed shallower slope can be understood invoking a correlation among flares (e.g. Green et al. 1993), or a dependence of the amplitude of the flares on the luminosity of the source (Almaini et al. 2000). We stress that a simple scaling of the flare amplitude with the luminosity of the source cannot account for the change in the slope of the SF with luminosity, shown in Fig. 10, unless some correlation among the flares is also introduced.

Instead of multiple flaring subunits, models based on the variability of a single region have also been considered, e.g., Almaini et al. (2000) explained the dependence of variability on luminosity, invoking a relation between the luminosity and the size of the varying region, which produces a shift of the PSD in the frequency direction, with unchanged slope, under the assumption of self-similar scaling of the variable region. However, a PSD with slope independent on luminosity is inconsistent with our results on the SF (see Fig. 10), implying that a deviation from self-similarity should be considered.

thumbnail Fig. 10

Structure function in bins of X-ray luminosity, represented as points connected by continuous lines. Straight lines with different dash styles: least-squares fits weighted according to the number of points in each bin of time lag. 1043.5   erg/s < LX < 1044   erg/s: circles, dotted lines; 1044   erg/s < LX < 1044.5   erg/s: squares, short-dashed lines; 1044.5   erg/s < LX < 1045   erg/s: triangles, long-dashed lines; 1045   erg/s < LX < 1045.5   erg/s: crosses, dot-dashed lines.

thumbnail Fig. 11

Dependence of the SF on LX. Filled circles: τrest = 1 d; open circles: τrest = 100 d. Lines: weighted least-squares fits, according to the number of points in each bin of LX.

The stronger dependences on LX found by Almaini et al. (2000) and Paolillo et al. (2004) are accompanied by the suggestion of a possible increase of the variability with redshift. Almaini et al. (2000) find an opposite dependence on luminosity (k = −0.3) for sources at z > 0.5, which could be caused by an increase with z. Paolillo et al. (2004) measure a higher variability for sources at z > 1.3 than for their low-z counterparts of similar luminosity. Manners et al. (2002) also reported tentative evidence of a stronger variability for sources at z > 2. Finally, Papadakis et al. (2008) compared the variability of high-redshift AGNs in the Lockman Hole region with that of nearby AGNs by Markowitz & Edelson (2004), finding evidence of an increase with redshift.

Owing to the strong correlation of sources in the LX-z plane (Fig. 1), we limited our analysis of the z-dependence to the sources in the luminosity interval 1044   erg/s < LX < 1045   erg/s, and divided the sample into four equally populated redshift bins, 0 < z ≤ 1, 1 < z ≤ 1.4, 1.4 < z ≤ 1.8, 1.8 < z ≲ 4.5. The result, displayed in Fig. 12, suggests the presence of a weak trend with redshift at intermediate time scales, while at short and long timescales the behaviour appears unclear and possibly non-monotonic.

To investigate this dependence in more detail, we computed partial correlation coefficients of variability with redshift, considering all individual variations that contribute to the SF. While the ordinary correlation coefficient indicates no correlation, rVz = −0.06, with probability P(>r) = 0.001, the first-order partial correlation coefficient, which takes account of the dependence on LX, is (7)with probability P(>r) = 10-12, suggesting the presence of a weak, intrinsic correlation. We also calculated the second-order partial correlation coefficient (Kendall & Stuart 1977), which compensates for both the dependences on LX and on the time lag τ, and still strengthens the correlation: (8)The probability is in this case P(>r) = 6 × 10-13.

thumbnail Fig. 12

Structure function in bins of redshift for sources in the luminosity interval 1044   erg/s < LX < 1045 erg/s, represented as points connected by continuous lines. Straight lines with different dash styles: least-squares fits, weighted according to the number of points in each bin of time lag. 0 < z ≤ 1: circles, dotted lines; 1 < z ≤ 1.4: squares, short-dashed lines; 1.4 < z ≤ 1.8: triangles, long-dashed lines; 1.8 < z ≤ 4.5: crosses, dot-dashed lines.

4. Discussion

The analysis of X-ray variability of AGNs has previously been performed mainly for individual nearby Seyferts or for small samples of them, and only a few works extend the study to large samples in wide ranges of luminosity and redshift (e.g. Almaini et al. 2000; Manners et al. 2002; Paolillo et al. 2004). Our study presents the first ensemble analysis based on the structure function. This is to be preferred for statistic studies compared with a PSD analysis, because SF operates in the time domain, is less dependent on irregular sampling, and allows for an analysis even with very few epochs. The SF is also preferable compared with the analysis of fractional variability and excess variance, because these parameters are biased by the duration of the monitoring time interval in the rest-frame, and thus on cosmological time dilation.

Our variability analysis, based on two different serendipitously selected samples extracted from the catalogues of XMM-Newton (Watson et al. 2009) and Swift (Puccetti et al. 2011), gives statistically consistent results in the two cases, with the SF described by a power law of the time lag, with exponent b = 0.10 ± 0.01 (XMM-Newton ) or b = 0.07 ± 0.04 (Swift). This would correspond to a PSD with power law exponent α ≈ 1.2 for the case of a single-power-law PSD, which is within the range of exponents found for nearby Seyferts (Lawrence & Papadakis 1993).

While the PSD of local low-luminosity AGNs often shows one or two breaks, we do not find evidence of breaks in the SF, even dividing the analysis in bins of MBH and Lbol. However, while a break at a time lag roughly proportional to the black hole mass is expected for local AGNs, our results do not support this expectation for more luminous AGNs and QSOs. This suggests that the relation found by McHardy et al. (2006), reported in Eq. (6), cannot be extrapolated to high bolometric luminosities and large black hole masses, possibly because the transition between an external cool thin disk and an inner ADAF (Liu et al. 1999) does not apply in the high-Eddington ratio regime.

We confirm a strong anti-correlation of the variability with X-ray luminosity, as and as for time lags  ~1 day and  ~100 days, respectively. This approximately agrees with most previous authors (Green et al. 1993; Lawrence & Papadakis 1993; Nandra et al. 1997; Markowitz & Edelson 2004; Papadakis et al. 2008).

The behaviour of the slope and amplitude of the SF as a function of the luminosity implies that (i) for a model of multiple flaring subunits, they cannot be uncorrelated; (ii) for a model with a single varying region self-similar scaling with luminosity cannot hold.

We find evidence in support of a weak, intrinsic, increase of the average X-ray variability with redshift. The dependence, however, appears tangled with that on the time lag. This suggests that different processes could dominate the variability at short and long time scales, and that their relative importance changes with the redshift.

1

αox ≡ log (L2   keV/L2500   Å)/log (ν2   keV/ν2500   Å).

Acknowledgments

We are grateful to Paolo Giommi, Maurizio Paolillo, Matteo Perri, and Simonetta Puccetti for useful discussions. S.T. acknowledges financial support through Grant ASI I/088/06/0. Part of this work is based on archival data, software or on-line services provided by the ASI Science Data Center (ASDC). This research made use of the XMM-Newton Serendipitous Source Catalogue, which is a collaborative project involving the whole Science Survey Center Consortium. Funding for the SDSS and SDSS-II was provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS was managed by the Astrophysical Research Consortium for the Participating Institutions.

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Online material

Table 1

XMM-Newton sample.

Table 2

Swift sample.

All Tables

Table 1

XMM-Newton sample.

In the text
Table 2

Swift sample.

In the text

All Figures

thumbnail Fig. 1

Distribution of the sources in the LX-z plane. Dots: XMM-Newton sample; circles: Swift sample.
In the text
thumbnail Fig. 2

Some of the best-sampled light curves from the XMM-Newton sample. Times are counted from the initial epoch of each light curve in the rest-frame. Errors, proportional to the inverse square root of the photon count at each epoch, are displayed as 3-σ values. Errors are discussed in more detail in Sect. 3. Source numbers from Col. 1 of Table 1 are indicated.
In the text
thumbnail Fig. 3

Some of the best-sampled light curves from the Swift sample. Times are counted from the initial epoch of each light curve, in the rest-frame. Errors proportional to the inverse square root of the photon count at each epoch are displayed as 3-σ values. Errors are discussed in more detail in Sect. 3. Source numbers from Col. 1 of Table 2 are indicated.
In the text
thumbnail Fig. 4

Histogram of the rest-frame time lags contributing to the structure functions. Continuous histogram: XMM-Newton sample; dashed histogram: Swift sample.
In the text
thumbnail Fig. 5

Structure function for the XMM-Newton sample in bins of Δlog τ = 0.5. The small empty circles and the continuous line connecting them show the uncorrected SF (i.e., neglecting σn in Eq. (1)). The larger filled circles and the line connecting them, show the SF corrected for the noise. The continuous line without data points indicates the average value of the noise in each bin, and the dashed, horizontal line is its weighted average, according to the number of points in each bin, adopted in Eq. (1). The dotted line is a weighted least-squares fit to the data of the bins. The small dots are the contributions from pairs of individual measurements at times differing by τ.
In the text
thumbnail Fig. 6

Structure function for the Swift sample, in bins of Δlog τ = 0.5. The small empty circles and the continuous line connecting them show the uncorrected SF (i.e., neglecting σn in Eq. (1)). The larger, filled circles and the line connecting them show the SF corrected for the noise. The continuous line without data points indicates the average value of the noise in each bin, and the dashed, horizontal line is their weighted average, according to the number of points in each bin, adopted in Eq. (1). The dotted line is a weighted least-squares fit to the data of the bins. The small dots are the contributions from pairs of individual measurements at times differing by τ.
In the text
thumbnail Fig. 7

SF (dashed line) and its slope (continuous line) computed from a PSD shaped as a broken power-law, with α1 = 1.2 and α2 = 2, shown in the inset. The dotted lines indicate the values of the power-law exponent b, expected from Eq. (5) for the single power-law case, b1 = 0.1, b2 = 0.5. The cut-off in the SF at long τ is caused by the finite number of Fourier frequencies used in the FFT calculation. Time lags and Fourier frequencies are in arbitrary units.
In the text
thumbnail Fig. 8

Black hole masses and bolometric luminosities of AGN samples. Black dots:  ~100   000 sources from the Shen et al. (2011) catalogue. Red filled circles: XMM-Newton sample. Blue empty squares: low-luminosity AGNs from Uttley & McHardy (2005).
In the text
thumbnail Fig. 9

Structure function for XMM-Newton subsamples binned in intervals of log MBH and log Lbol, with bin width Δlog MBH = Δlog Lbol = 0.5. Values of MBH and Lbol are reported in the upper and right axes, respectively. Subsamples with less than 30 SF points are not shown. The SF of the total XMM-Newton sample is reported, for comparison, in the first box in the upper left corner. The values reported in each box are the number of SF points, the average SF slope, and the expected value of log τbreak, according to Eq. (6). Contributions from pairs of individual measurements are also shown (dots).
In the text
thumbnail Fig. 10

Structure function in bins of X-ray luminosity, represented as points connected by continuous lines. Straight lines with different dash styles: least-squares fits weighted according to the number of points in each bin of time lag. 1043.5   erg/s < LX < 1044   erg/s: circles, dotted lines; 1044   erg/s < LX < 1044.5   erg/s: squares, short-dashed lines; 1044.5   erg/s < LX < 1045   erg/s: triangles, long-dashed lines; 1045   erg/s < LX < 1045.5   erg/s: crosses, dot-dashed lines.
In the text
thumbnail Fig. 11

Dependence of the SF on LX. Filled circles: τrest = 1 d; open circles: τrest = 100 d. Lines: weighted least-squares fits, according to the number of points in each bin of LX.
In the text
thumbnail Fig. 12

Structure function in bins of redshift for sources in the luminosity interval 1044   erg/s < LX < 1045 erg/s, represented as points connected by continuous lines. Straight lines with different dash styles: least-squares fits, weighted according to the number of points in each bin of time lag. 0 < z ≤ 1: circles, dotted lines; 1 < z ≤ 1.4: squares, short-dashed lines; 1.4 < z ≤ 1.8: triangles, long-dashed lines; 1.8 < z ≤ 4.5: crosses, dot-dashed lines.
In the text

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