ACTIVE GALACTIC NUCLEUS X-RAY VARIABILITY IN THE XMM-COSMOS SURVEY

COSMOS is a deep and wide extragalactic survey designed to probe the formation and evolution of galaxies as a function of cosmic time and large-scale structural environment. The survey covers a 2 deg² equatorial (10^h, +02°) field with imaging by most of the major space-based telescopes (Hubble Space Telescope (HST), Spitzer, Galaxy Evolution Explorer, XMM-Newton, Chandra), and a number of large ground based telescopes (Subaru, Very Large Array, European Southern Observatory Very Large Telescope (ESO-VLT), UKIRT, NOAO, Canada–France–Hawaii Telescope (CFHT), and others). Large dedicated ground-based spectroscopy programs in the optical with Magellan/IMACS (Trump et al. 2007), VLT/VIMOS (Lilly et al. 2009), Subaru-FMOS, and DEIMOS-Keck have been completed or are well under way.

This wealth of data has resulted in an 35-band photometric catalog of ∼10⁶ objects (Capak et al. 2007) resulting in photo-z for the galaxy population accurate to Δz/(1 + z) <1% (Ilbert et al. 2009) and to Δz/(1 + z) ∼ 1.5% for the AGN population (Salvato et al. 2009, 2011).

The XMM-Newton wide-field survey in the COSMOS field (hereafter XMM-COSMOS; Hasinger et al. 2007) is a crucial component of the multi-wavelength coverage of the COSMOS field. The ∼2 deg² area of the HST/Advanced Camera for Surveys program has been surveyed with XMM-Newton for a total of ∼1.55 Ms during the AO3, AO4, and AO6 cycles of XMM observations, providing an unprecedented large sample of point-like X-ray sources (∼1800).

The XMM-COSMOS project is described in Hasinger et al. (2007, hereafter Paper I), while the X-ray point source catalog and counts from the complete XMM-COSMOS survey are presented in Cappelluti et al. (2009, hereafter Paper II). Brusa et al. (2010, hereafter Paper III) present the optical identifications of the X-ray point sources in the XMM-COSMOS survey and the multi-wavelength properties of this large sample of X-ray selected AGNs.

The catalog used in this work includes 1797 point-like sources detected in the 0.5–2 (1545 sources), 2–8 (1078 sources), and 4.5–10 (246 sources) keV bands. The nominal limiting fluxes are ∼5 × 10⁻¹⁶, ∼3 × 10⁻¹⁵, and ∼7 × 10⁻¹⁵ erg cm⁻² s⁻¹, respectively. The adopted likelihood threshold corresponds to a probability 4.5 × 10⁻⁵ that a catalog source is a spurious background fluctuation. In this analysis, we used the source list created from the 53 XMM-COSMOS fields.

The XMM-COSMOS catalog has almost 100% redshift completeness (Paper III). 884 (out of 1797) sources have a spectroscopic redshift, 748 sources have a photometric redshift, 97 are classified as stars, and 68 remain unclassified. The sources with spectroscopic redshift are divided almost equally between BL (FWHM >2000 km s⁻¹) and non-BL AGNs, plus a small fraction of non AGN sources, i.e., passive galaxies at low redshift or stars. The sources for which only photometric redshift is available have been classified on the basis of the best SED fitting template. The agreement between SED classification and spectral classification is good (Lusso et al. 2010; Salvato et al. 2009; Brusa et al. 2010). The final classification breakdown is: 611 type 1, 941 type 2, 80 Galaxies, 97 Stars, and 68 Unclassified.

We underline that in the following analysis type 1 refers to sources that either have a spectroscopic redshift, showing at least one broad emission line, or have as best fit SED templates an unobscured quasar template with different degrees of contamination by star-forming galaxy templates (from 10% to 90% of the optical–near-IR flux). Type 2 refers to sources with optical spectra of narrow lines AGNs or passive/star-forming galaxies, or best fit SED templates of obscured AGNs, with different degrees of contamination by passive or star-forming galaxy templates. We exclude sources classified as stars and those which are unclassified from the following analysis.

Because of the pattern of observations, sources in XMM-COSMOS can be observed in a minimum of one to a maximum of 11 pointings. We have the possibility of following up every source for a period of up to 3.5 yr.

By using the XMM-SAS tool emldetect, we selected all sources with a detection significance det_ml > 10 (corresponding to a probability that a Poissonian fluctuation of the background is detected as a spurious source of P = 10⁻¹⁰), in each pointing. We then produced light curves of all those sources observed in more than one pointing. As a result, we produced light curves for 995 of the 1797 sources in the full band catalog. The number of detections spans from two to nine, depending on the position of the source and on its flux. Figure 1 (left panel) shows the distribution of all of the detections of sources in the total catalog, during the 3.5 yr observing campaign (between 2003 December 6 and 2007 May 18). We have a total of 3849 single detections, distributed among the observations depending on the exposure time of each pointing. In order to study variability in these sources, we limited the following analysis to sources with ⩾3 or more detections.

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To obtain an estimate of the typical rest-frame time scales sampled by the available 3.5 yr observing campaign, we computed the time intervals between the first and the last detections for each source, rescaled for the redshift of the source, for time dilation, i.e., Δt_rest = Δt/(1 + z). Figure 1 (right panel) shows the distribution of the time intervals for all of the 638 sources with ⩾3 detections. The inset shows the distribution of the number of detections for each source in the sample: 190 sources have only three detections in total and in the last bin we have ten sources with nine detections, which is the maximum number of samplings available. The plot shows that, given the distribution of the observations and the redshift distribution of the sources, the vast majority of the light curves have a total length in the range of 100–500 days (rest-frame), i.e., the low frequency limit of the light curve is typically at frequencies lower than ν_b for most of our sources. As a point of reference, we recall that the typical break frequency in the PSD of an AGN with a 10⁸ M_☉ BH is 1/ν_b ∼ 30 days. The final breakdown of sources with light curves is 340 type 1, 291 type 2, and seven galaxies.

The light curves are measured in the fixed 0.5–10 keV observed band, implying that we are measuring variability at increasingly high intrinsic energies, up to 2–40 keV at z = 3. Even if variability can, in principle, be highly energy dependent, given the different spectral components (with different origins) which observationally dominate different bands, it is known to be well correlated between the 0.3–0.7, 0.7–2, and 2–10 keV bands (e.g., P12). In soft X-rays the effect of observing variability in different bands is small: typically ∼10% from 0.5 to 10 keV for local Seyferts (Vaughan & Fabian 2004; Gallo et al. 2007). In hard X-rays (above 10 keV) less data are available. However, Caballero-Garcia et al. (2012) found that the variability amplitude above 10 keV is similar to that below 10 keV for a sample of five bright AGNs in the Swift-BAT 58 month catalog (see also Beckmann et al. 2008; Soldi et al. 2011). Soldi et al. (2013, submitted) reported for a sample of 110 radio quiet QSO from the same Swift-BAT 58 month catalog, that the variability is very well correlated even between the 14–24 and 35–100 keV bands (again with a 10% difference).

Furthermore, all of the variability estimators are strongly dependent on the available number of counts (see Section 3), hence we decided to exploit all of the X-ray counts available using the full XMM-Newton observing band for all sources, instead of restricting ourselves to a much narrower fixed rest-frame band. It is true, however, that we do not know if the variability is energy dependent in AGNs at high redshift, contrary to what is observed in the local universe. To answer this question, however, one needs a large sample of sources spanning a large redshift range, which are not so severely affected by low signal-to-noise ratio (S/N). The almost uniform 200 ks Chandra coverage on the entire 2 deg² COSMOS field, which will be available by the end of 2013, will be the perfect data set for such a study.

3.1. V versus Counts

The V parameter is strongly dependent on the number of counts available, given that the ability to constrain the variability of one source depends on the error on each measurement. Figure 2 (left panel) shows the distribution of the V parameter as a function of the total number of counts for each source. In Figure 2 (right panel) we show the fraction of variable sources (V > 1.3) in bins of 0.5–10 keV counts. The fraction decreases from ∼75% ± 13% above 1000 counts, to ∼15% ± 10% in the last bin, i.e., given sufficient statistics to detect it, variability is almost ubiquitous in X-ray selected AGNs. However, even at a very high number of counts, a small fraction of sources is found to be non-variable with high reliability.

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In Figure 2 (top) three sources are highlighted, namely XMM-Newton ID (XID) 3, 5323, and 2186. The X-ray and optical light curves for these sources are shown in the Figure 3 upper and lower panels, respectively, (optical light curves from M. Salvato et al. 2013, in preparation) Each one represents one extreme of the distribution: source 3 has the highest number of X-ray counts (and the maximum number of detections, nine) and is highly variable, both in X-ray and optical; source 5323 again has very good X-ray statistics, but is non-variable with very high confidence (only a 5% probability of being variable) and is also non-variable in optical; source 2186 has a more typical number of counts for our sample (∼200 net counts in 0.5–10 keV) and is variable, both in X-ray and optical. Sources 5323 and 2186 have the same observed light curve duration, while source 3 was detected during the entire 3.5 yr of observations. However source 5323 is a luminous quasar at high z (log(L_X) = 45.4 and z = 1.509, with log(M_BH) = 8.9), while 2186 is a low luminosity Seyfert at low redshift (log(L_X) = 42.1 and z = 0.235, with log(M_BH) = 7.3). As we will show in Section 5, the different luminosities (and M_BH) appear to have a major role in producing these different variability properties.

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In total, 232 of 638 sources (36.4% of the sample) exhibit significant flux variability on rest-frame time scales of months to years. We can compare this result with those of Young et al. (2012) on CDF-S Chandra data. They found 50% of AGNs to be variable. This difference is probably due to the different background contribution for Chandra and XMM-Newton: for a given total number of net counts, the S/N and hence the error on the flux measurement is typically lower (with larger errors) for XMM-Newton data. The Chandra background accounts for only 5%–10% of source counts even for faint sources, while the background for faint sources in the XMM-Newton-COSMOS survey can account for up to 50% of source counts in the full band.

3.2. V versus L_0.5–10 and z

We are interested in the possible correlation between X-ray variability and more intrinsic properties of the sources in our sample, such as L_0.5–10 and redshift. However, the strong dependency of V on the number of counts could produce biased correlations between these quantities. We therefore explored the distribution of L_0.5–10 and redshift as a function of the number of counts (Figure 4 left and right, top panels), for the whole sample (gray points) and for variable sources (V > 1.3, black points). The resulting fraction of variable sources in bins of L_0.5–10 and z is show in the lower panels.

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In Figure 4 (top) there is a clear upper boundary in the distribution of the luminosities (L_0.5–10 ∼5 × 10⁴⁵ erg s⁻¹) due to the angular size of the COSMOS survey and the low density of very luminous AGNs and a lower boundary in the number of counts (∼50) due to the detection threshold of each observation and the minimum number of detections of three. Furthermore, it can be seen that there is a general decreasing density of sources toward low luminosities, due to the smaller volume sampled at low z. Finally there is a zone of missing sources at low luminosities and a high number of counts due to the typical exposure time of the survey (∼40 ks vignetting corrected). However, the fraction of variable sources (i.e., the relative distribution of variable and non-variable sources) is constant, close to 40%, in the entire L_0.5–10 range we considered (10⁴¹ < L_0.5–10 <5 × 10⁴⁵ erg s⁻¹).

The distribution of counts as a function of redshift is shown in Figure 4 (bottom). Here the zone of missing sources at high redshift and a high number of counts is again due to the properties of the survey (exposure time and angular size): there are no sources more luminous than 5 × 10⁴⁵ erg s⁻¹ and the almost constant exposure time dictates a decreasing number of counts with z. In this case, however, the final effect is a decrease in the fraction of variable sources at high redshift, from 55% to 25% from low to high redshift.

We stress that the distribution of the V parameter is an indicator of our ability to detect variability in our sample and therefore Figure 4 (bottom) gives an indication that we are less able to detect and measure variability at high z (see discussion in Section 5).

3.3. V versus Optical Type

Figure 5 (left panel) shows the distribution of the variability parameter, V, for the 638 sources with an available light curve in the 0.5–10 keV band, divided by class. The dashed line represents the probability threshold we choose to define a source as variable. Figure 5 (right panel) shows the fraction of variable sources for the different classes. Type 1 AGNs have the highest fraction of variable sources (∼40%), type 2 have a lower fraction (∼32%), while galaxies have only a very large upper limit (only one variable source out of seven in this class). The inset of Figure 5 (bottom) shows, however, that the higher fraction of variable type 1 (with respect to type 2 sources), could be due to the typical higher number of counts available for these sources. Therefore, as observed also in the CDF-S (Young et al. 2012) variability is common in AGN samples, independent of the optical classification.

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Figure 6 shows the distribution of the $\sigma ^{2}_{{\rm rms}}$ as a function of total net counts for each source. Sources with a detection on $\sigma ^{2}_{{\rm rms}}$ are labeled in red. Sources with an UL are in gray. Black circles mark sources considered variable on the basis of their V parameter (V > 1.3). The distribution of $\sigma ^{2}_{{\rm rms}}$ is in the log scale and ULs, defined as $\sigma ^{2}_{{\rm rms}}$ + $\sigma ^{2}_{{\rm rms}}$ _Err, for all of the sources with $\sigma ^{2}_{{\rm rms}}$-$\sigma ^{2}_{{\rm rms}}$ _Err < 0 are reported. We note that all of the ULs are positive (i.e., $\sigma ^{2}_{{\rm rms}}$ + $\sigma ^{2}_{{\rm rms}}$ _Err > 0).

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There is clearly an anti-correlation between $\sigma ^{2}_{{\rm rms}}$ and the number of counts. We emphasize that this anti-correlation is produced by the increasing minimum UL measurable for a given number of counts,¹⁴ and hence the increasing minimum level at which we can have a detection, and by the fact that we have very few sources with high number of counts, hence the region with high $\sigma ^{2}_{{\rm rms}}$ and high number of counts is less populated. The dashed line corresponds to the cut of 700 total net counts which we will use in Section 5.1 to evaluate the effect of the selection in V. We emphasize that above this line there is instead no correlation between $\sigma ^{2}_{{\rm rms}}$ and the number of counts, unlike what was found for the total sample.

If we assume that a detection in $\sigma ^{2}_{{\rm rms}}$ can be considered as an indication of intrinsic variability, it is interesting to compare the distribution of $\sigma ^{2}_{{\rm rms}}$ detections with that of sources showing V > 1.3. We recall, however, that the two methods express different concepts: the V parameter indicates at which confidence a source can be considered variable, while the excess variance measures the amplitude of the variability. Nonetheless, the two methods give very similar results: There are only 21 sources detected in $\sigma ^{2}_{{\rm rms}}$ which are non-variable according to the V parameter and 36 sources variable in V but not detected in $\sigma ^{2}_{{\rm rms}}$. The agreement is also visually represented in Figure 6, where there is a high correspondence of red dots (detections in $\sigma ^{2}_{{\rm rms}}$) and black circles (V > 1.3 sources).

Figure 7 (top) shows the distribution of σ_rms as a function of the 0.5–10 keV band luminosity for the sample of 232 sources with V > 1.3. The X-ray luminosities are corrected for absorption, either from spectral analysis (Mainieri et al. 2007) or hardness ratio (Brusa et al. 2010), and computed in the 0.5–10 keV rest-frame band. Blue squares are type 1 sources, red triangles are type 2, green crosses are galaxies. The dashed line represents the linear regression performed with the ASURV software Rev 1.2 (Isobe & Feigelson 1992), which implements the methods presented in Isobe et al. (1986). We adopted the results from the Buckley–James method, given that in all cases the censorship is present in only one variable. The results from the estimation maximization algorithm method are always consistent with those obtained with the Buckley–James method, while results from the Schmitt's binned linear regression are not suitable due to the limited number of data points. We excluded the two sources at L_0.5–10 <10⁴² erg s⁻¹ from the fit, which may have significant contamination from the host galaxy. An anti-correlation between excess variance and X-ray luminosity is observed with high significance: the Spearman's rank correlation coefficient (ρ_S = −0.375) gives a probability of P_S < 10⁻⁴. The resulting correlation is:

$$\begin{equation} {\rm log}\left(\sigma ^2_{{\rm rms}}\right)=(-0.91\pm 0.02)+(-0.23\pm 0.03){\rm log}(L_{2\hbox{--}10\ {\rm keV},44}) \end{equation} \tag{ 5 }$$

with L_0.5–10 in units of 10⁴⁴ erg s⁻¹.

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A steeper anti-correlation for AGNs has been observed in previous works both for high frequencies (e.g., Barr & Mushotzky 1986; Lawrence & Papadakis 1993; Nandra et al. 1997; Hawkins 2000) and on longer time scales (Paolillo et al. 2004; Papadakis et al. 2008; Young et al. 2012). In particular, Paolillo et al. (2004) found a slope of −1.31 ± 0.23 in the CDF-S, limiting the analysis to sources in the redshift range 0.5 < z < 1.5. Papadakis et al. (2008) found a flatter slope of −0.66 ± 0.12 in the Lockman Hole while Young et al. (2012) found a slope of −0.76 ± 0.06 in the 4 Ms CDF-S data set (for sources with L_0.5–10 >10⁴² erg s⁻¹), but not including sources with ULs. Again the latter sample is limited to z ⩽ 1.

Figure 7 (bottom) shows the distribution of $\sigma ^{2}_{{\rm rms}}$ as a function of z for the sample. No significant correlation is observed (ρ_S = −0.031 and P_S = 0.6798). This is somewhat unexpected, because at high redshift we are sampling the high luminosity part of the population and therefore we would expect to observe a decrease in the typical $\sigma ^{2}_{{\rm rms}}$ with z. This is probably balanced by the fact that at high z we are selecting a smaller fraction of variable sources (see Figure 4, right), thus selecting only the most variable ones.

The large redshift and L_0.5–10 range encompassed by the XMM-COSMOS survey allow us to study this issue more in detail. Figure 8 (top) shows the distribution of $\sigma ^{2}_{{\rm rms}}$ versus L_0.5–10 for sources divided into three redshift bins (z < 0.7, 0.7 < z < 1.5, and 1.5 < z < 3.5). The dashed lines are linear regressions computed with ASURV for each subsample. Sources and lines are color coded so that blue represent the lower redshift sample, green the intermediate, and red the high redshift sample. Dividing the population in this way, the anti-correlation between $\sigma ^{2}_{{\rm rms}}$ and L_0.5–10 is stronger: each correlation has P_S < 10⁻⁴ and the slopes go from −0.40 ± 0.07 to −0.65 ± 0.10 from low to high redshift. Furthermore, it appears that for a given luminosity range and at all luminosities, the typical excess variance becomes higher at higher redshifts. The intercept coefficients are significantly different, going from −1.21 ± 0.07 to −0.48 ± 0.07 from low to high redshift. An hint of a similar effect was observed in Paolillo et al. (2004), but the smaller sample size and L_0.5–10-z coverage did not allow for a robust investigation in that work.

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As explained in Section 4, due to the broad distribution of the bias introduced by measuring the excess variance from sparsely sampled light curves, the $\sigma ^{2}_{{\rm rms}}$ measured for a specific source can be far from the intrinsic one. To overcome this problem, we computed the average excess variance for subsamples of sources showing similar physical properties (such as L_0.5–10, z, M_BH, and Eddington ratio). Figure 8 (bottom) shows the distribution of average $\sigma ^{2}_{{\rm rms}}$ versus L_0.5–10 for sources in the same redshift bins as the left panel and further binned in luminosity bins (one bin per decade). The positions of the squares mark the average of $\sigma ^{2}_{{\rm rms}}$ and L_0.5–10 for the bin. The average is computed taking into account all of the values of $\sigma ^{2}_{{\rm rms}}$ at face value for both detections and ULs (e.g., the computed value regardless of the errors, that for ULs can be negative). Error bars represent the standard error on the mean for both quantities in each subsample. The linear regressions of the left panel are shown for reference. We stress that the linear regressions computed with ASURV for the distribution of points in Figure 8 (top) are in perfect agreement with the distribution of the binned points.

We note that our decision to consider only the subsample of significantly variable sources (V > 1.3) avoids the difficulty of dealing with sparse light curves of faint sources for which the Poissonian noise is far greater than the intrinsic variability. However, we note that the 〈$\sigma ^{2}_{{\rm rms},V>1.3}$〉 averaged over the subset of the sources with V > 1.3 is not the same as the 〈$\sigma ^{2}_{{\rm rms}}$〉 averaged over the whole set of sources and 〈$\sigma ^{2}_{{\rm rms},V>1.3}$〉 will always be ⩾〈$\sigma ^{2}_{{\rm rms}}$〉. Therefore the selection V > 1.3 introduces a further bias in the averaged excess variances. The bias (in the estimation of the mean excess variance) is stronger the larger the number of faint sources left out for each luminosity and z bin.

From the light curves, we measure the variance (S²) that has a contribution due to the intrinsic variability ($\sigma ^{2}_{{\rm rms}}$) and the Poissonian noise ($\sigma ^2_{{\rm Poiss}}$), $\sigma ^{2}_{{\rm rms}}$ $= S^2 + \sigma ^2_{{\rm Poiss}}$ (Edelson et al. 2002). If, as in the majority of faint AGNs in XMM-COSMOS, $S^2 \sim \sigma ^2_{{\rm Poiss}} \gg$ $\sigma ^{2}_{{\rm rms}}$, the uncertainty in the determination of the true value of $\sigma ^2_{{\rm Poiss}}$ will lead to either an over- or underestimation of the excess variance, thus sometimes producing negative excess variances. The average of the excess variance estimates (considering both positive and negative values) is unbiased and it does tend to the real excess variance. On the other hand if, as done here, we average only the variable sources (V > 1.3) thus preferentially discarding the negative excess variance values, the obtained average excess variance will be overestimated, especially for those bins where a large fraction of faint sources are excluded. This is why we attribute the increasing observed AGN variability at high redshift (see Figure 8) to this effect. In fact, as shown in Section 3.3, the fraction of variable sources selected with a fixed values of V decreases with redshift. This means that an increasingly large number of sources are left out from the bin when moving to high redshift.

To test this and to check how much of the observed effect is due to selection, we computed the same distribution of Figure 8 (bottom) in two cases. First (Figure 9, left), we included all the 638 sources with a light curve in the average distribution without any pre-selection. The linear regressions obtained in different redshift bins from Figure 8 are shown for reference. The average $\sigma ^{2}_{{\rm rms}}$ for each bin is, as expected, lower than before, because we are including a large number of sources with $\sigma ^{2}_{{\rm rms}}$ close to 0 (positive or negative) in the average and, as expected, the difference increases when moving to high redshift, because more sources were excluded at high redshift by the V > 1.3 selection (see Figure 4, right). Using the whole sample, the evolution in redshift is no longer significant.

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Finally, we showed in Section 4 (and Figure 6) that the scatter in $\sigma ^{2}_{{\rm rms}}$ increases when moving toward low counts due to light curves with poor statistics. We want to test the effect of this in the distribution of $\sigma ^{2}_{{\rm rms}}$ and luminosity. Therefore, in Figure 9 (bottom) we selected sources on the basis of the number of counts available, i.e., sources with good statistics (>700 total net counts, 144 sources), for which we consider the estimation of variability to be more reliable. No selection in V is used in this plot. Also, in this case all points are consistent with the local relation, i.e., that measured for z < 0.7:

$$\begin{equation} {\rm log}(\sigma ^2_{{\rm rms}})=(-1.21\pm 0.06)+(-0.40\pm 0.07){\rm log}(L_{2\hbox{--}10\ {\rm keV},44}) \end{equation} \tag{ 6 }$$

even if the number of sources available for the bin is small and we are missing the low luminosity–high redshift bins. We underline that, even if the subsample used in Figure 9 (bottom) is much smaller than the one in Figure 9 (top), (144 sources instead of 638), the smaller scatter produce similar error bars.

These tests indicate that the selection based on the V parameter introduces a bias at high redshift on the average excess variance. Once we average all of the excess variance values (with no selection in V), the evidence for higher AGN variability at high redshift is no longer significant for the XMM-COSMOS AGNs.

We collected all of the BH mass information available to date in the XMM-COSMOS survey. For type 1 we have 89 virial masses available from Mg ii from Merloni et al. (2010); 128 from Mg ii, 31 from Hβ, and 37 from C iv from Trump et al. (2009); and 69 from Hα and 183 from Mg ii from Matsuoka et al. (2013). We also have a recompilation of these type 1 BH masses using the same spectra, but with a self consistent re-analysis, following Trakhtenbrot & Netzer (2012). The different BH mass values for sources with multiple measurements are in general in good agreement, within 0.3 dex. For type 2 we have 481 mass estimates through scaling relations and SED fitting from Lusso et al. (2011). In total we have at least one mass estimate for 814 sources.

As we already showed in Section 5, the less biased selection to study the dependency of $\sigma ^{2}_{{\rm rms}}$ on other physical properties in our sample requires selecting all sources (with no pre-selection in V). Furthermore, we chose to rely on the subsample of sources with sufficiently good statistics (>700 counts) for which the estimation of variability is more reliable. Using the same selection, we built a sample of 111 sources with M_BH estimates. We stress that this is the first time that the correlation between X-ray variability on long time scales and BH masses can be performed in such large sample of AGNs spanning a wide range of redshift.

The distribution of excess variance as a function of the BH mass is shown in Figure 10 (top). Single sources are shown in gray (squares for type 1 and triangles for type 2). The black error bar shows the average (representative of a single measurement) error on $\sigma ^{2}_{{\rm rms}}$. The position of the filled blue squares mark the average of $\sigma ^{2}_{{\rm rms}}$ and M_BH in five bins of BH mass, where sources with ULs are also taken into account with their nominal value. Error bars represent the standard error on the mean for both quantities in each subsample.

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When only sources with good statistics are considered, the correlation between $\sigma ^{2}_{{\rm rms}}$ and M_BH is strong (ρ_S = −0.315 and P_S = 0.0007), with a slope of −0.42 ± 0.11. We note that the correlation between $\sigma ^{2}_{{\rm rms}}$ and the M_BH found in previous works sampling higher frequencies (minutes–hours) is of the order of −1 (e.g., P12). This is generally interpreted with the fact that the $\sigma ^{2}_{{\rm rms}}$, measuring the integral of the PSD in that specific frequency rangem is affected by the position of ν_b which scales linearly with the BH mass.

The lower frequencies sampled in the XMM-COSMOS survey are in the months–years regime. On the other hand, the highest frequency sampled by a sparsely sampled light curve is not well defined. However, in most light curves the minimum distance between two observations is of the order of hours–days. Therefore we are integrating the PSD both above and below ν_b. The part of the PSD integral above ν_b would introduce a linear correlation of the excess variance with M_BH, but this is weakened by the part of the integral below ν_b which would induce no correlation (see for example Soldi et al. 2013).

We also collected the AGN bolometric luminosities from Lusso et al. (2010) for type 1 (estimated from direct integration of the rest-frame SED) and Lusso et al. (2011) for type 2 (from SED-fitting by assuming a fixed covering factor of 0.67). Thanks to this, we were able to compute Eddington ratios for 74 sources with >700 counts.

Figure 10 (bottom) shows the distribution of $\sigma ^{2}_{{\rm rms}}$ as a function of Eddington ratio expressed as L_Bol/L_Edd. As in Figure 10 (top), gray points show single sources, blue filled squares represent the average of $\sigma ^{2}_{{\rm rms}}$ and L_Bol/L_Edd in five bins of Eddington ratio. There is no correlation between $\sigma ^{2}_{{\rm rms}}$ and the Eddington ratio (ρ_S = −0.122 and P_S = 0.2784, with a slope fully consistent with 0, −0.09 ± 0.14).

It is also interesting to note that the slope of the $\sigma ^{2}_{{\rm rms}}$ versus M_BH in Figure 10 (top) is the same as that of the global slope between $\sigma ^{2}_{{\rm rms}}$ and L_0.5–10 in Figure 9 (bottom). This suggests that the latter is a by-product of the former one, as observed at lower frequencies (P12). This is shown if Figure 11 where the distribution of $\sigma ^{2}_{{\rm rms}}$ versus L_0.5–10 is shown for sources with >700 counts after normalizing $\sigma ^{2}_{{\rm rms}}$ for the M_BH. The linear regression between the $\sigma ^{2}_{{\rm rms}}$ normalized for the BH mass is fully consistent with 0 (slope of 0.13 ± 0.12).

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The residual scatter of $\sigma ^{2}_{{\rm rms}}$ after accounting for the M_BH/L_0.5–10 dependency is still of ∼2 orders of magnitude for individual values. This would imply that the normalization PSD below ν_b is not the same for all of the sources as is usually assumed. However, as we pointed out in Section 5.1, it has been shown in Allevato et al. (2013) that the bias due to sparse sampling can be broadly distributed between 0.1 and 10, meaning that we cannot use the observed scatter of individual $\sigma ^{2}_{{\rm rms}}$ values as a direct probe for the intrinsic scatter of the PSD normalization.

The existence of the correlation between $\sigma ^{2}_{{\rm rms}}$ and Eddington ratio is debated in the literature (O'Neill et al. 2005; McHardy et al. 2006; Körding et al. 2007; P12). At low frequencies, Young et al. (2012) found a significant anti-correlation between $\sigma ^{2}_{{\rm rms}}$ and accretion rate which they interpreted as an artifact of the $\sigma ^{2}_{{\rm rms}}$ –L_0.5–10 anti-correlation. In our dataset the correlation is globally flat. A hint of bi-modality is present in Figure 10 (bottom), with average $\sigma ^{2}_{{\rm rms}}$ increasing both for very high and very low accretion rates. However the quality of the data and the limited sample included here do not allow further investigation of this issue.

The optical variability, expressed in Δmag, was computed in Salvato et al. (2009; see details in Section 3 of that paper for the definition of Δmag) in order to correct for its effect when computing photometric redshifts. The COSMOS optical photometry has been acquired in five epochs distributed over ∼6 yr. Within each epoch (apart from 2005), several individual filter observations were distributed over less than three months, covering the whole optical range. This allowed AGN time variability to be studied and corrected over time scales of years, while shorter variability cannot easily be addressed. A source with ΔMag > 0.25 was considered variable.

The optical variability is show in Figure 12 (top) as a function of the X-ray $\sigma ^{2}_{{\rm rms}}$ for sources with V > 1.3. The two quantities are clearly not correlated (ρ_S = −0.113 and P_S = 0.0611). The optical photometry is not simultaneous to the X-ray observations but span a larger time scale. Also the cadence is different. These two factors (in addition to the variability being produced at various distances from the BH, depending on the wavelength) can explain why X-ray and optical variability are not clearly correlated.

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Figure 13 reports two examples of sources that are highly variable in one band and not in the other. Source 2016 (top) is a source classified as broad line from the optical spectrum and it is non-variable in X-ray and highly variable in optical. Source 5192 (bottom), classified as type 2 from the SED fitting, is instead variable in X-ray while showing a perfectly flat optical light curve.

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Finally, optical variability tends to be highly type-dependent (Figure 12, right): the fraction of type 1 that is optically variable (ΔMag > 0.25 Mag) is considerably higher (∼55%) than that of type 2 (∼25%). We underline, however, that variability is one of the criteria that is used in the process to identify sources with photometric redshift as AGN-dominated or galaxy-dominated (see flow chart in Figure 6 of Salvato et al. (2011) for details on the template class assignation), thus this result is partly an induced effect.