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12Draft version February 22, 2012Preprint typeset using LATEX style emulateapj v. 8/13/10

MEASURING X-RAY VARIABILITY IN FAINT/SPARSELY-SAMPLED AGN

V. Allevato1,2,3, M. Paolillo3,4, I. Papadakis5, C. Pinto6

Draft version February 22, 2012

ABSTRACT

We discuss some practical aspects of measuring the variability amplitude of faint and distant activegalactic nuclei (AGN), characterized by sparsely sampled lightcurves and low statistic. In such casesthe excess variance, commonly used to estimate the intrinsic lightcurve variance, is affected by strongbiases and uncertainties since it represents a maximum likelihood variability estimator only for iden-tical/normal distributed measurements errors and uniform sampling. We performed realistic MonteCarlo simulations of AGN lightcurves, reproducing both the sampling pattern and measurement er-rors typical of multi-epoch deep surveys, such as the XMM-Newton observations of the Chandra DeepField South (CDFS), or assuming different sampling patterns that may characterize long surveys withsub-optimal observing conditions. We used the results to estimate our ability to measure the intrinsicsource variability as well as to constrain the observing strategy of future X-ray missions studyingdistant and/or faint AGN populations.

Subject headings: galaxies: active – sample text – user guide

1. INTRODUCTION

Active Galactic Nuclei (AGN) are characterized bylarge amplitude and rapid variability, especially in theX-ray band, which is probably originating in the innerregions of the accretion disk and the hot corona. Oneof the most common tools for examining AGN variabil-ity is the Power Spectral Density Function (PSD). Earlyattempts to measure the AGN X-ray PSDs showed thatthey have a power-law like shape with a slope of ∼ 1.5(Green et al. 1993; Lawrence & Papadakis 1993). Thisresult is indicative of a scale-invariant red-noise process,on timescales ranging from a few hours to years, with noevidence of periodicities.In recent years it has become increasingly clear that

there exists at least one characteristic timescale inthe AGN X-ray PSDs. This timescale reveals it-self in the form of “frequency breaks” (νbr) in thePSD, where the slope changes from a value of ∼ −1below the “break”, to ∼ −2 at frequencies higherthan νbr (see e.g. Uttley, McHardy & Papadakis 2002;Markowitz et al. 2003). In at least one case, namelyArk 564, a second break, where the slope changes from∼ −1 to zero, is also detected (Papadakis et al. 2002;McHardy et al. 2007). These time scales may be linkedto the characteristic disk time scales like the dynamical,thermal or viscuous timescale, and appear to correlatewith the BH mass and accretion rate (McHardy et al.2006; Koerding et al. 2007). Thus variability measure-ments represent a tool to investigate both the physics of

1 Max-Planck-Institut fur Plasmaphysik and Excellence Clus-ter Universe TUM, Boltzmannstrasse 2, D-85748 Garching, Ger-many

2 Max-Planck-Institute fur Extraterrestrische Physik, Giessen-bachstrasse 1, D-85748 Garching, Germany

3 Dept. of Physical Sciences, University Federico II, viaCinthia 6, 80126 Naples, Italy

4 Istituto Nazionale di Fisica Nucleare, Sez.di Napoli, Italy5 University of Crete Dept Physics, P.O. Box 2208, GR 710 03

Heraklion, Greece6 SRON Netherlands Institute for Space Research, Sorbon-

nelaan 2, 3584 CA Utrecht, the Netherlands

the accretion process, as well as the fundamental param-eters (MBH , m) of the active nucleus.So far, our knowledge of the X-ray variability proper-

ties of AGNs is mainly based on the study of a few nearby,X-ray bright objects, which have been monitored exten-sively with RXTE over many years, and for which therealso exist day-long, high signal-to-noise (S/N) XMM-Newton light curves. At the same time, deep multi-cycle surveys (e.g. Alexander et al. 2003; Brunner et al2008; Comastri et al. 2011; Xue et al. 2011; also seeBrandt & Hasinger 2005 and references therein), havebeen accumulating observations of intermediate and high(z > 0.5) redshift AGN, thus offering the opportunity toexplore AGN variability at high redshift as well. How-ever, due to the sparse sampling, and the low flux ofmost AGN detected in these surveys, it is not pos-sible to use PSD techniques to study the variabilityproperties of these objects. For that reason, a differ-ent statistic, namely the excess variance (Nandra et al.1997; Turner et al. 1999; Edelson et al. 2002) has beenused to parametrize the variability properties of the highredshift AGN (Almaini et al. 2000; Paolillo et al. 2004;Papadakis et al. 2008).Strictly speaking, the excess variance is a maximum

likelihood estimator of the intrinsic light curve varianceonly in the case of uniform sampling and identical andnormally distributed measurement errors (Almaini et al.2000). A detailed discussion of the statistical proper-ties of the excess variance and its performance in thecase of red noise PSDs of various slopes and ”break” fre-quencies, and of different S/N ratios, can be found inVaughan et al. (2003). These authors however consid-ered the case of continuously sampled data only, suchas those provided by long XMM observations of nearbyAGNs. Instead, in deep multi-cycle surveys, the effectsof sparse and uneven sampling must be taken taken intoaccount when investigating the statistical properties ofthe excess variance.The goal of this work is to investigate the performance

of the excess variance as a measure of the intrinsic AGN

2 Allevato et al.

variability. In particular, we consider sources similar tothose observed in multi-epoch surveys and characterizedby extreme sparsity, due to the observing strategy andorbital visibility of the targets. We measure the biasand the expected scatter of the excess variance measure-ments, and we investigate the dependence of the biason the sampling pattern and gap length, as well as onthe S/N ratio of the light curve. We believe that ourresults will be useful to researchers who wish to studythe variability properties of high redshift AGN (an areawhich is still largely unexplored), as well as to under-stand the possible limitations of the existing data, andto correct (in a statistical sense) for some effects thatthe uneven sampling introduces in the estimation of theintrinsic variability. Finally, our results could be of usein the determination of the optimal observing strategyeither for future surveys with the current X-ray satellitesor for future X-ray missions.The paper is organized as follows: in §2 we define the

variability estimator; in §3 we describe the Monte Carlosimulations of AGN lightcurves both reproducing thepattern of the XMM-Newton observations of the CDFSand further testing more favorable observing strategies.The applications to future X-ray surveys are presentedin §5 while our results are discussed in §6.

2. NORMALIZED EXCESS VARIANCE

The variability of accreting systems is usually investi-gated through the use of the PSD, which gives the lightcurve variance per Hz at each temporal frequency. AGNexhibit a power-law PSD, as S(f) ∝ f−β, where S(f)is the power at frequency f , with slopes usually in therange 1 . β . 2. A proper derivation of the PSD (orof the lightcurve variance, see below) is intrinsically dif-ficult: extrapolations from any single realization can bemisleading due to the stochastic nature of any red-noiselightcurve (see discussion in Vaughan et al. 2003). Forreal data this task is further affected by the signal-to-noise ratio of the data, the finite length of the observationand by the sampling pattern.The analysis of present day light curves of distant

AGNs is difficult since these sources are usually serendip-itously detected in deep surveys. As a result, their lightcurves are characterized by low signal-to-noise ratio aswell as sparse sampling. In such cases, instead of tryingto derive the PSD, it is easier (and often only possible)to estimate the total light curve variance using the socalled excess variance, which is defined as (Nandra et al.1997):

σ2NXS =

1

Nx2

N∑

i=1

[(xi − x)2 − σ2err,i], (1)

where xi and σerr,i are the count rate and its error ini-th bin, x is the mean count rate , and N is the numberof bins used to estimate σ2

NXS . With this normalizationwe are able to compare excess variance estimates derivedfrom different segments of a particular lightcurve or fromlightcurves of different sources. The statistic σ2

NXS is anestimate of the (squared) fraction of the total flux per binthat is variable, corrected for the experimental noise. Ac-cording to the Parseval theorem, the contribution to theintrinsic variance due to variations between the shortest

and longest time scales sampled, which σ2NXS measures,

should be roughly equal to the integral of the intrinsicPSD between the shortest and longest frequencies sam-pled.The error7 on σ2

NXS , asymptotically for large N , isgiven by the variance of the quantity (xi − x)2 − σ2

err,i,i.e.

∆σ2NXS =SD/[x2(N)1/2], (2)

SD =1

N − 1

N∑

i=1

{[(xi − x)2 − σ2err,i]− σ2

NXSx2}2.

As mentioned earlier, the performance of the excessvariance, under various intrinsic PSD models, in the caseof evenly sampled light curves has already been investi-gated by Vaughan et al. (2003); here we intend to ex-plore instead the performance in case of sparse samplingand low S/N. We also point out that if each point inthe lightcurve has equal weight, then σ2

NXS is indeeda maximum-likelihood estimator of the lightcurve vari-ance. This is not true anymore in cases where the errorsdiffer significantly from point to point, and a numeri-cal approach is needed in order to obtain the maximum-likelihood estimate of the intrinsic variance (see detailsin Almaini et al. 2000). We will explore this case as wellin the following sections.

3. MONTE CARLO SIMULATIONS OF AGNLIGHTCURVES

3.1. The algorithm and the simulated CDFS lightcurves

In order to quantify the bias and the uncertainty ofthe excess variance as an estimator of the intrinsic sourcevariance in the case of very unevenly sampled light curvesof faint sources, we performed Monte Carlo simulationsmodifying the original code of Timmer & Koenig (1995),that generates red-noise data with a power law PSD, inorder to reproduce the real data extraction process in-cluding filtering and background subtraction. We simu-lated, for each AGN, the actual lightcurve measurement:we first create an intrinsic AGN lightcurve with the abovealgorithm, following the appropriate PSD. Then we addto the AGN count rate, in each time bin, the contributionfrom the expected background, randomly adding Poissonfluctuations to both terms. A second local backgroundestimate is also generated (including again Poisson fluc-tuations), and then subtracted from the AGN, as donein real data.In order to account for the effect of red noise leak,

which transfers power from low to high frequencies, wegenerate lightcurves which are 5 times longer than thelargest timescale sampled by the data, and extract a seg-ment of the required length. We verified that extendingthe simulated lightcurves further does not significantlychanges our results, while increasing considerably theprocessing time.

7 Note there was a typographical error in Nandra et al. (1997), inthat the equation for the error on σ2

NXSshould have had the quan-

tity inside the rms summation squared, as clarified by Turner et al.(1999). Also see Edelson et al. (2002) and Vaughan et al. (2003)for alternative expressions and a discussions of the different formu-lae.

X-ray Variability of AGN 3

Fig. 1.— Simulated AGN lightcurve according to the input parameters in Table 1, reproducing a continuous sampling (black crosses) andthe sampling pattern of the XMM-Newton observations of the CDFS (red circles). Mean count rate and excess variance estimates refer tothe particular simulation extracted from a set of 5000 simulations.

In order for our experiment to be as close as possibleto reality, we performed Monte Carlo simulation of AGNlightcurves assuming the sampling pattern and uncer-tainties of the XMM-Newton observations of the CDFS.In particular we take into account only the first 1 Msobservations, taken between 2001 and 2002, to study aworst-case scenario before discussing, in the next sec-tions, more favourable ones. As a starting point, we sim-ulated red-noise lightcurves with intrinsic count rate andvariance of one of the brightest AGN observed by XMM-Newton in the CDFS (source id 68 from Giacconi et al.2002) at z ∼ 0.54, using as input the set of parame-ters reported in Table 1. The source has a soft (andhard) flux of ∼ 5 × 10−14 erg/s/cm2, i.e. we expect10-20 of these sources per square degree, according to,e.g. Hasinger et al. (1993); Luo et al. (2008). Comparedto other bright AGNs in the field, this source has theadvantage of being fairly isolated and thus its flux andvariability can be robustly estimated. We explore PSDslopes ranging from 1 to 3; in the following we are goingto show the results for simulations with β = 1.5, but wewill discuss the results in all the other cases as well.Fig. 1 shows an example of a simulated lightcurve: the

red points highlight the sampling pattern of the XMM-Newton observations, compared to the whole underly-ing lightcurve. The first group of points corresponds tothe two observations of July 2001 with an effective ex-posure, after filtering high background periods, of ∼ 80ks and the second (with more data points) to the sixobservations of January 2002 for an additional 900 ks.The whole simulated lightcurve with continuous sam-pling (black crosses) thus spans ∼ 1.5×107sec, i.e. about6 months, out of which the actual XMM-CDFS observa-tions (red circles) sample ∼ 9.8 × 105 sec (∼ 11 days).This type of observing pattern is driven primarily by thetypical scheduling requirements of deep multi-cycle cam-paigns, and thus represents a recurring, although unde-sirable, observing scheme which has been the only avail-able to astronomers until the 2009 extended XMM ob-serving campaign of the CDFS (Comastri et al. 2011).

TABLE 1Input parameters of the simulated AGN lightcurves

Power-law PSD index (β) 1,1.5,2,2.5,3Number of simulations (N) 5000Mean count rate 0.1 cnt/sTime resolution (∆t) 10 ksIntrinsic lightcurve variance (σ2

in) 0.042Background level 0.06 cnt/s

The figure also reports the mean count rate and excessvariance measured over the whole lightcurve and overthe intervals sampled by the XMM observations, for thisspecific realization. As discussed in more detail below,when sampling the whole lightcurve the measured val-ues reproduce the input parameters, while in the case ofsparse sampling we obtain biased results.

3.2. The distribution of σ2NXS in the case of sparsely

sampled light curves

In Fig. 2 we present the excess variance distribution ofa set of 5000 simulations of sparsely sampled lightcurvessuch as the one shown in Fig.1, for the case of an in-trinsic PSD with power-law slope β = 1.5 (solid line).The dashed line in the same figure represents insteadthe distribution of the maximum likelihood variance es-timator as proposed by Almaini et al. (2000). The verti-cal dot dashed line in Fig.2 marks the intrinsic variance(σ2

intrinsic = 0.042, i.e. 20.5% r.m.s.).Although the errors on each point of the lightcurve

are not identical, the sample distribution of the vari-ance measured through the numerical estimate ofAlmaini et al. (2000) does not differ much from the dis-tribution of the excess variance, at such count rate lev-els. Both distributions in fact are highly peaked at valuessmaller than the intrinsic source variance. The medianvalue of the σ2

NXS distribution is listed in Table 2 for sim-ulations with (1) continuous sampling, (2) sparse sam-pling, (3) sparse sampling using the maximum-likelihoodestimator and (4) correcting for the true mean count rate.The lower and upper quartiles of the distribution within

4 Allevato et al.

Fig. 2.— Excess variance distribution based on a set of 5000simulated AGN lightcurves, such as the one shown in Fig.1, re-producing the sampling pattern of the XMM-Newton observationof the CDFS (solid black line), compared to the expected inputvalue (vertical red line). The dotted red line represents the samedistribution corrected for the true intrinsic mean count rate (seediscussion in the text), while the maximum likelihood approachis shown by the dashed blue line. The σ2

NXS distribution for acontinuous sampled lightcurve is not shown here since it is an ex-tremely narrow distribution peaked on the intrinsic value of thevariance.

90% are in brackets. Both the maximum likelihood andσ2NXS are thus ”biased” estimators of the intrinsic source

variance. In addition, both distributions are very broad,and highly skewed towards large positive values. Clearly,an individual measurement of neither σ2

NXS nor itsmaxi-mum likelihood equivalent, can be considered as a reliableestimate of the intrinsic source variance. We also notethat using Eq.2 the median error on σ2

NXS is equal to∼ 0.006 , i.e. the formal error tends to underestimatethe true scatter and does not account for the asymme-try of the distribution, as it does not include the effectof the sparse sampling. As shown in Table 2, in case ofcontinuous sampled AGN lightcurve, as expected the dis-tribution of σ2

NXS is quite narrow and strongly peakedto the intrinsic source variance. Very similar results arefound also assuming different index β of the power-lawPSD.We conclude that the sparse sampling does indeed

results in a biased distribution of the excess variance,and increases the ”uncertainty” on each individual value.Using a sparse sampling pattern the variance of thelightcurve is underestimated, mainly because each real-ization badly reproduces the intrinsic mean count rate;in fact the value derived from the sparsely sampled datawill always be closer to the sampled points than the truemean, thus minimizing the variance8. To demonstratethis point, we fixed the average count rate x in Eq. (1)to its intrinsic value, finding that in such case the meanoutput variance approaches on average the input value(see Table 2 and dotted red line in Fig. 2), while stillretaining the large scatter.

3.3. The σ2NXS bias

8 Note that while the mean count rate that we measure for eachindividual realization of the sparsely sampled lightcurves is biased,and thus minimizes the variance, its distribution over the entire setof 5000 simulations peaks at the expected input value.

Fig. 3.— Bias distribution based on a set of 5000 simulatedAGN lightcurves such as shown in fig. 1, reproducing the samplingpattern of the XMM-Newton observation of the CDFS.

The numerical experiment we discussed above can beused in principle to correct the measured variances, inorder to retrieve the true intrinsic value. To this end, foreach of the 5000 simulated lightcurves we computed theratio between the intrinsic variance σ2

in and the actualexcess variance measured for the particular simulatedlightcurve σ2

sim. The sample distribution of this ratiosis plotted in Fig. 3. This distribution has a large scat-ter and it is highly asymmetric, due to the large scatterand highly skewed nature of the σ2

sim distribution itself.For about . 25% of the simulated light curves this ra-tio is ≤ 1, but for the majority of them it is > 1. Wecan then define the ”median bias” of the estimated ex-cess variance, which in essence indicates the correctionfactor that is needed to retrieve the intrinsic variance, asfollows:

b =σ2in

med(σ2sim)

(3)

where med(σ2sim) is the median of the σ2

sim distri-bution. This definition of the bias is similar to theAlmaini et al. (2000) definition, although in the lat-ter case, the authors defined the bias using the stan-dard deviation instead of the variance of the lightcurve.They used an average correction factor using lightcurvesspread over periods from 2 to 14 days, in the range 1-1.34 with the largest values for the faintest QSO withonly two widely spaced temporal bins. In the case of thesampling pattern reproducing the XMM-Newton obser-vations of the bright source n.68, we derived b = 1.8 forβ = 1.5. We also calculated the bias values for differentintrinsic power spectra, finding that the bias changes fordifferent slopes but in all the cases the intrinsic varianceis . 2 times larger than the median excess variance σ2

simthat we measure from the sparse lightcurves (see Table2).If the bias was know a priori, it could be used to rescale

the measured excess variance, and correct for the ef-fects of both the red noise leak and sampling patternin the measurement of this quantity. However, the biasof the individual realizations has a large scatter due tothe large and strongly asymmetric σ2

sim distribution, aneffect which has not been properly considered in previ-

X-ray Variability of AGN 5

TABLE 2Median σ2

NXV and bias for continuous and sparse sampling

β Continuous Sparse Max.likelihood Mean Corr. b

(1) (2) (3) (4) (5)

1 0.0418(0.0415, 0.0420) 0.030(0.004, 0.043) 0.017(0.005, 0.036) 0.040(0.015, 0.050) 1.4(1.,10.4)1.5 0.0418(0.0415, 0.0420) 0.023(0.007, 0.035) 0.015(0.008, 0.035) 0.041(0.021, 0.070) 1.8(1.2,6.0)2 0.0418(0.0415, 0.0420) 0.025(0.005, 0.042) 0.016(0.007, 0.043) 0.052(0.022, 0.078) 1.7(1.,8.36)2.5 0.0418(0.0415, 0.0420) 0.032(0.008, 0.042) 0.018(0.010, 0.052) 0.064(0.030, 0.110) 1.3(0.9,5.2)3 0.0418(0.0415, 0.0420) 0.037(0.011, 0.061) 0.020(0.012, 0.054) 0.070(0.033, 0.150) 1.1(0.7,3.8)

TABLE 3Median σ2

NXV and bias as a function of S/N ratio for β = 1.5

Mcr SN

Source Flux Median σ2

NXS b

cnt/s (cnt/bin) (erg s−1cm−2)0.1 (1000) 25 6.25 ×10−13 0.022(0.007, 0.034) 1.9(1.2, 6)0.05 (500) 22.6 3.12 ×10−13 0.022(0.007, 0.034) 1.9(1.2, 6)0.01 (100) 6.3 6.25 ×10−14 0.022(0.005, 0.036) 1.9(0.007, 10.2)0.005 (50) 3.4 3.12 ×10−14 0.021(0.002, 0.038) 2(1, 21)0.002 (20) 1.4 1.25 ×10−14 0.016(−0.54, 0.66) 2.52(0.06,∞)0.001 (10) 0.8 6.25 ×10−15 < 0 ...

TABLE 4Median σ2

NXVand bias as a function of gap length

Temporal Gap Median σ2

NXSb

(days)

5.8 0.039(0.027,0.050) 1.02(0.74,1.48)11.6 0.036(0.022,0.050) 1.12(0.80,1.82)28.9 0.030(0.016,0.050) 1.32(0.80,2.54)57.9 0.027(0.13,0.52) 1.48(0.76, 3.07)115.7 0.024(0.10,0.50) 1.62(0.80, 4.03)231.5 0.021(0.07, 0.47) 1.90(0.85, 5.71)

ous works. The bias factors shown in Table 2 are medianvalues over 5000 simulations while the individual excessvariances can differ much more from the intrinsic vari-ance. Therefore, given the large and skewed distributionshown in Fig. 2, the bias on individual lightcurve can be2-3 times higher than the one estimated using Eq. 3 andthe extreme care must be employed when inferring thevariability parameters from single observations of AGNwith such extreme sampling patterns.

4. WHAT AFFECTS THE OBSERVEDVARIABILITY BIAS?

4.1. Bias Dependence on the source flux

As discussed in the previous section, strongly unevenlysampled lightcurve produces a biased estimate of theintrinsic lightcurve variance. Such bias derives mainlyfrom the inability of our data to constrain the averagesource flux due to the red noise character of the AGNPSD, which implies larger power at lower frequencies.More importantly, a sparse sampling produces a wide ex-cess variance distribution, indicating that each individualmeasurement could differ significantly from the intrinsicvariance, even if an average correction is applied to ourmeasurement.Obviously we expect a dependence of the bias and its

scatter on the source flux as a result of the white noiseintroduced by Poisson fluctuations. To estimate such

effects, we simulated lightcurves assuming different av-erage count rates, corresponding to fluxes smaller thanthe one of the source 68, as is the case for the bulk ofthe AGN population detected in the CDFS. Table 3.2shows the bias dependence on the source flux, in caseof the XMM-CDFS observation pattern, fixing the PSDslope β = 1.5. The excess variance is the median of thedistribution based on a set 5000 simulations while thebias is estimated using Eq. 3 with the errors comingfrom the 90% upper and lower quartiles of the excessvariance distribution. Conversion factors from counts tofluxes were calculated assuming a power law spectrumwith αph = 1.4 and nH = 8× 1019 cm−2.The excess variance estimates and bias factor do notchange significantly with the source flux down to countrates of ∼ 0.005 cnt/s (which correspond to a S/N ra-tio per bin of 3.4 given the assumed XMM background),while the width of the excess variance distribution in-creases. At lower S/N levels the bias increases up to apoint where we are not able to detect variability any-more, since the excess variance distribution is wide andthe median value becomes negative. We verified that theresults do not depend on the specific value of β that weuse.This result suggests that a minimum S/N ratio per

bin & 1.5 − 2 is advisable for estimating the intrinsicexcess variance in case of sparse sampled ligthcurves.Moreover we verified that the same bias is observedwhen using the maximum-likelihood approach proposedby Almaini et al. (2000), for all the considered S/N ra-tios. Note that in the low count regime the Almainiapproach cannot predict a negative intrinsic variance byconstruction and thus yields a ML value of 0.

4.2. Bias Dependence on the Gap Length

Apart from such dependence on sampling pattern andsource flux, we expect that the bias will change as a func-tion of the gap length. To test this effect on the intrinsicvariance estimator, we simulated AGN lightcurves withthe total exposure time of the XMM-CDFS observations

6 Allevato et al.

(440 ks) sampled by two blocks of observations of 220 kseach, using the input parameters shown in Table 1. Thetemporal gap between the observations ranges from theextreme case of ∼ 7 months, similar to the gap in theXMM-CDFS pattern, to the more favourable case of ∼ 6days. Table 4 summarizes our results, where the excessvariance and bias estimates are as before the median ofthe distribution based on a set of 5000 lightcurves simu-lations, fixing β = 1.5, while the bias is estimated usingEq. 3 and the errors are derived from the quartiles errorsof the σ2

NXS distribution.As expected both the median bias and the width of theexcess variance distribution increase with increasing gaplength; the same trend is observed assuming differentpower-law slope values.Thus again, as discussed in §3 for the XMM sampling

pattern, because of the large uncertainties associated tothe excess variance estimate, each individual lightcurvemeasurement yields an extremely poor estimate of the in-trinsic source variability, and such uncertainties increasesas a function of the gap length, as shown by the errorsin Table 4. In such cases the only way to make a morerobust estimate is to collect repeated observations of thesame source, in order to lower the statistical uncertainties(assuming that the process producing the variability isstationary). Alternatively large samples of sources mayprovide a less biased ensemble estimate, assuming thatthe underlying PSD is similar for all sources.

4.3. Ensemble Excess Variance Estimate

A collection of several observations of the same sourceor a large sample of AGN may produce a less biased es-timate of the AGN variability under some particular as-sumptions (stationary variability process or same PSDfor all AGN). In order to verify how reliably we canconstrain the source variance through repeated/multipleobservations, we binned the 5000 simulated excess vari-ances obtained by using the XMM pattern as describedin §4.1, in groups of 5, 10, 20, 50-points. For each binwe estimated the mean excess variance and its standarddeviation. The distributions of the 5, 10, 20 and 50-points binned mean-σ2

NXV and of its standard deviationare shown in Fig. 4 for a count rate of 0.1 cnt/s andβ = 1.5.The resulting mean-σ2

NXV distributions do not peakon the intrinsic variance, as the individual realizationsare anyway biased due to the sparse sampling. However,these distributions are now more symmetric and roughlyGaussian. A Kolmogorov-Smirnov test performed on the5, 10, 20 and 50-points mean-σ2

NXV distributions indi-cates that only for the 5-points grouping we can rejectthe hypothesis of Gaussian distribution at > 95% level.Furthermore if we compare the standard of the binneddistributions in the upper panel of Fig.4 (whose valuesare shown in the inset as the errors) to the scatter of theindividual realizations in each of the n=5,10,20 and 50points bins, we find that such scatter (divided by

√n− 1)

is on average representative of the uncertainty on thebinned mean-σ2

NXV ; in fact the error in the upper panelis equal to the mean value of the distribution in the lowerpanel (due to the central limit theorem). In practice thismeans that when binning our data, we can estimate theuncertainty on each mean-σ2

NXV simply from the scatterof the individual points composing each bin.

Fig. 4.— Upper panel : Distribution of the mean-σ2

NXV esti-mated by binning 5000 simulated excess variance (adopting theXMM sampling pattern of §4.1), in groups of 5, 10 20 and 50 points(according to the legend). The inset shows the median values ofthe binned distributions and their st.dev. The simulations are per-formed by assuming a count rate of 0.1 cnt/s and β = 1.5.Lowerpanel : Distribution of the errors on the mean-σ2

NXV estimated as

the st.dev. of the points within each bin, divided by√n− 1. The

inset reports the mean values of the distribution for the differentbinning.

According to the results described in §4.1, we expectthat the spread of the distributions of mean excess vari-ances increases with the decreasing source flux. In fact,down to count rates of ∼ 0.005 cnt/s (S/N=3.4), the10, 20 and 50-points mean-σ2

NXV distributions are stillGaussian but the errors rise such as the discrepancy be-tween the median values of the mean-σ2

NXV distributionsand the intrinsic variance. We verified that this trenddoes not depend on β. These results imply that if onebins together 10, 20 or 50 excess variances estimated for amoderately bright AGN sample, then the correspondingbinned mean σ2

NXV is roughly a Gaussian variable, andthe associated uncertainty is equal to the scatter of theindividual binned σ2

NXV , divided by√n− 1, irrespective

of β. However at low fluxes (count rates < 0.002 cnt/s,S/N<1.4), the errors become dominant and the scatteron the mean excess variance is > 100% (see Fig. 5).Similarly we expect a dependence of the average excess

variances on gap length. To test such effect we appliedthe same binning method on 5000 simulated excess vari-ances obtained by using the sampling pattern described

X-ray Variability of AGN 7

Fig. 5.— As figure 4 but for 0.001 cnt/s

in 4.2. For temporal gaps below ∼ 58 days, the mean-σ2NXV distributions obtained with the 5, 10, 20 and 50-

points binning are Gaussian, while increasing the gaplength up to ∼ 7 months, the 5-points mean excess vari-ance distribution becomes not Gaussian at > 95%. Asbefore the means of these distributions do not peat atthe input variances (e.g. they are biased) and the dis-crepancy respect to the intrinsic variance increases withthe temporal gap, as does the uncertainty on the meanvalues of the distributions.

4.4. Uniform and Progressive Sampling

In order to test more favourable scenarios, better suitedto reduce the bias in AGN variability estimates, we gen-erated two additional sets of lightcurves with the inputparameters shown in Table 1, and adopting different sam-pling patterns, which span the same maximum timescaleas the XMM observations described in §3:

1. Uniform sampling, consisting in 9 observations of50 ks each separated by constant temporal gaps of1900 ks (∼ 20 days, fig. 6, upper panel);

2. Progressive sampling, where the observations areseparated by increasing lags according to the ex-pression gap = 2n × 10 ks, with n = 1, 2, .., 8 (fig.6, lower panel);

Fig. 7 shows the normalized excess variance distribu-tion derived from 5000 simulations for the two sampling

Fig. 6.— Simulated AGN lightcurves (black crosses) with the uni-form (upper panel) and progressive (lower panel) sampling schemesmarked by red circles. The figure also reports the mean count rateand excess variance measured for the particular simulation over thewhole lightcurve and over the intervals with uniform and progres-sive sampling.

Fig. 7.— Excess variance distribution for N=5000 lightcurvessimulations of uniform (solid line) and progressive (dotted line)sampling: the errors are the 90% upper and lower quartiles of theσ2

XNVdistribution. The uniform sampling removes the bias, in

fact the mean of the distribution is in agreement with the expectedvalue of the excess variance (red line) equal to the input parameterof the simulation (σ ∼ 20%). For a progressive sampling, the biasin the intrinsic variance slightly increases.

8 Allevato et al.

schemes described above. We observe that the distribu-tions are now more symmetric, and closer to Gaussian,than was the case for the original XMM pattern (Fig.1).A regular sampling pattern also minimizes the medianbias (b = 1.01(0.80, 1.32)) in the intrinsic variance esti-mates with a median σ2

XNV consistent with the expectedvalue, although the individual measurement still have un-certainties of ∼ 25%. For the progressive sampling themedian bias is somewhat larger (b = 1.19(0.88, 1.80)).as this sampling pattern favors short time scales, whilethe dominant contribution to the total variance is due tolonger ones.Clearly, if we consider a sparsely sampled lightcurve,

the preferable observing scheme is thus a regular patternwith temporal gaps not much longer than the length ofeach observation. In this situation the observations canbe used to estimate the intrinsic source variance evenfrom single observations, although with significant uncer-tainties. The progressive sampling may be preferred if weintend to trace the whole PSD (as opposed to just thevariance), but such measurements requires higher S/Nratios and repeated measurements to average over theintrinsic scatter of any stochastic process.

5. CONSTRAINS ON THE OBSERVINGSTRATEGY OF FUTURE X-RAY SURVEYS

Several missions have been proposed over the past fewyears to study high redshift AGNs; most of these aredesigned to have larger effective area than current X-ray missions, wider Field-of-View and, depending on theplanned orbit, lower background. For instance the Inter-national X-ray Observatory (IXO, Barcons et al. 2011)and its evolution Athena9, the Wide Field X-ray Tele-scope (WFXT, Murray et al. 2010), all represent mis-sions capable of performing AGN surveys with higherspeed than Chandra or XMM. The results discussed in§4.4 allows to explore the capabilities of such future X-ray missions in the time domain. In particular we exam-ine the expectations for deep, wide-area surveys, whichwill allow to probe the highest redshift and faintest AGNpopulations at the expense of a continuous temporal cov-erage.To investigate the capabilities of such missions in mea-

suring AGN variability, we present here the performanceof a mission with 1 m2 effective area, 1 sq.deg. FOVand the low background allowed by a low earth orbit,very similar to the WFXT design (Rosati et al. 2010).This results in a large number of moderate and high red-shift AGN (see e.g. Paolillo et al. 2010). We used a totalobserving time of ∼ 400 ks and we evaluated the per-formance that can be expected assuming a uniform sam-pling scheme similar to the one presented in §4.4. Fig-ures 8 and 9 represent an example of a possible observingscheme for the survey, where observations of 50 ks eachare spread evenly over ∼ 6 months and the correspondingexcesses variance and bias distributions, respectively.In order to verify the performance of such type of mis-

sion for faint AGN populations, we explored the depen-dence of the measured excess variance on different valuesof the source mean count rate. The results are sum-marized in Table 5. The excess variance remains rela-tively small (. 20%) even at the lower count rate levels.

9 http://www.mpe.mpg.de/athena/workshop mpe 2011/index.php

Fig. 8.— Simulated AGN lightcurve, sampled in 50 ks observa-tions spread uniformly on ∼ 6 months, as expected from futurelarge effective area mission such as those described in the text.

Fig. 9.— Excess variance (upper panel) and bias (lower panel)distribution based on a set of 5000 simulated lightcurves, such asthe one shown in Figure 8; with this observing strategy we are ableto retrieve the intrinsic variance with an uncertainty of ∼ 25%.

Compared to the case discussed in §3 however, we arenow able to detect variability at flux levels10 more thanone order of magnitude lower than XMM, using approxi-

10 Conversion factors from counts to fluxes were calculated as-suming a power law spectrum with αph = 1.4 for an unabsorbedAGN at z=0.

X-ray Variability of AGN 9

TABLE 5Median σ2

NXV and bias as a function of S/N ratio for a future mission described in §5

Mcr SN

Source Flux Median σ2

NXSb

cnt/s (cnt/bin) (erg s−1cm−2)0.1 (1000) 38 4× 10−14 0.037(0.027, 0.048) 1.1(0.9, 1.5)0.01 (100) 9.3 4× 10−15 0.037(0.026, 0.047) 1.1(0.9, 1.6)0.005 (50) 7.2 2× 10−15 0.036(0.024, 0.048) 1.2(0.9, 1.7)0.002 (20) 3.9 8× 10−16 0.035(0.015, 0.055) 1.2(0.7, 2.8)0.001 (10) 2.7 4× 10−16 0.033(0.006, 0.073) 1.3(0.6, 6.9)

mately the same observing time, thus allowing variabilitystudies for hundreds of AGNs per square degree. Suchgood performances are due in part to the larger effectivearea, and in part to the low background made possibleby the considered low-earth orbital configuration.

6. DISCUSSION AND CONCLUSIONS

In this paper we discussed the performance of currentand future deep survey X-ray missions in the time do-main and their ability to measure AGN variability, usingrealistic simulations that reproduce the real data prop-erties.We show that the excess variance is a biased estima-

tor of the intrinsic lightcurve variance in sub-optimalobserving conditions, such as those characterizing the2001-2002 XMM-Newton observation of the CDFS. Thesame bias is observed when using alternative estimatorsof the intrinsic lightcurve variance, as suggested by, e.g.,Almaini et al. (2000). In fact we find that when the sam-pling pattern is very sparse, the intrinsic variance of thelightcurve is underestimated, mainly because each real-ization badly reproduces the intrinsic mean count rate.Due to the red noise nature of the AGN PSD, this bias

strongly depends on the temporal gaps between observa-tions on the longest timescales, while it is less sensitiveto the detailed distribution of the data points on shorttimescales. Furthermore, for a fixed sampling pattern,the bias does not change with the source flux as long asthe S/N ratio per bin is & 1.5; for lower values we arehardly able to detect variability at all, due to the increas-ing contribution of Poisson noise to the total variance.We then suggest as rule of thumb, to use sources with aS/N ratio per bin above 1.5-2, in estimating the intrinsicvariance for sparse sampled lightcurves. We further ver-ified that the bias depends only mildly on the power-lawPSD index, with a peak for β = 1.5, and anyway remainsbelow 2 for all slopes tested here.While in principle we can use simulations, such as

those described here, to correct the measured quantitiesand estimate the intrinsic variance, we point out thatthe uncertainties on the bias factor can be very large inthe case of irregular sampling, and the bias distributionis very asymmetrical, so that each individual lightcurveyields a very poor estimate of the intrinsic AGN prop-erties. On the other hand we showed that binning to-gether excess variances in groups of 10, 20 and 50 points,produces mean values that are approximately Gaussiandistributed and its uncertainty can simply be estimatedfrom the scatter of the individual points composing eachbin. These results are irrespective of the power law slopeβ, the temporal gap, and of the S/N, even if the thespread of the mean excess variance distributions increases

with the gap length and with decreasing S/N.Unevenly observing patterns as the ones discussed in

§3 and §4, are often due to the scheduling requirements ofdeep multi-cycle campaigns; in order to show the benefitsderiving from a proper observing strategy, we tested tworegular observing schemes, which allow us to span thesame maximum timescale as the XMM-Newton observa-tion of the CDFS; we find that such schemes significantlyreduce the bias in the excess variance estimates and pro-duce more symmetrical distribution, with uncertaintiesthat range from ∼ 100% down to ∼ 20% for the brightestsources. Uniform sampling patterns are those producingthe best results, although different schemes sampling alarger range of timescales may be desirable to derive afull PSD.Finally we showed that for future X-ray mission, a

properly designed observing strategy may allow to mea-sure variability for hundreds of sources per square de-gree. Such dataset would largely overlap with the spec-troscopic sample (e.g. Gilli et al. 2011), thus resultingthousand of AGNs with both temporal and spectroscopicinformations. Since the individual variance estimateswill still be affected by significant uncertainties, a largedataset will be essential in order to constrain the averagetiming properties of high redshift AGNs (provided thatthe AGN population shares the same intrinsic proper-ties).Several dedicated timing missions have also been pro-

posed in the X-ray regime such as Lobster or LOFT(Feroci et al. 2010). In such cases the continuous mon-itoring ensures a sampling pattern very close to a con-tinuous lightcurve yielding unbiased variability estimateswith small uncertainties, thanks to the possibility to av-erage out the scatter intrinsic to any stochastic process.This type of analysis however will be possible only for thebrightest (and mostly nearby) sources due to the smallangular resolution of such missions.We want to stress that the simulations presented here

do not include additional systematics, such as for in-stance vignetting and PSF variation across the FOV. Thereaders are then encouraged to explore their specific sci-ence cases using simulations that closely reproduce theirspecific sampling pattern, S/N ratio, background con-tamination etc. Furthermore, the observing strategy offuture missions will likely be decided based on additionalscientific requirements, such as the need to discover andtrace transients with variable decay timescales, or to fol-low up observations made by observatories at other wave-lengths (e.g. LSST), which may require to adopt strate-gies that are sub-optimal for AGN studies with respectto those discussed here.

10 Allevato et al.

ACKNOWLEDGEMENTS

VA acknowledge support by the German DeutscheForschungsgemeinschaft, DFG Leibniz Prize (FKZ HA

1850/28-1). MP acknowledges support from the ItalianPRIN 2009.

REFERENCES

Alexander, D. M., et al. 2003, AJ, 126, 539O. Almaini, A. Lawrence, T. Sharks, MNRAS, 2000, 315, 325Barcons, X., Barret, D., Bautz, M., et al. 2011, arXiv:1102.2845Brandt, W. N., & Hasinger, G. 2005, ARAA, 43, 827Brunner, H., Cappelluti, N., Hasinger, G., Barcons, X., Fabian,

A. C., Mainieri, V., & Szokoly, G. 2008, A&A, 479, 283Comastri, A., et al. 2011, A&A, 526, L9Edelson, R., Turner, T. J., Pounds, K., Vaughan, S., Markowitz,

A., Marshall, H., Dobbie, P., & Warwick, R. 2002, ApJ, 568,610

Feroci, M., et al. 2010, SPIE Conference Proceedings, 7732, 57R. Giacconi, et al., ApJS, 2002, 139, 369Gilli, R., et al. 2011, Memorie della Societa Astronomica Italiana

Supplementi, 17, 85A. R. Green, I. McHardy, & H. J. Lehto, MNRAS, 1993, 265,

664Hasinger, G., et al. 1993, A&A, 26, 275, Koerding, E. G., et al. 2007, MNRAS, 380, 301A. Lawrence, & I. E. Papadakis, Apj, 1993, 414, 85Luo, B., et al. 2008, ApJS, 179, 19A. Markowitz, R. Edelson and S. Vaughan, ApJ, 2003, 598, 935I. M. McHardy, I. E. Papadakis, P. Uttley, MNRAS, 2004,359, 348I. M. McHardy, K. F. Gunn, P. Uttley, & M. R. Goad, MNRAS,

2005, 359, 1469I. M. McHardy, Koerding, E., Knigge, C., Uttley, P., Fender, R.

P., 2006, Nature, 444, 730I. M. McHardy et al. 2007, MNRAS, 382, 985S. S. Murray et al. 2010 in SPIE Conference Proceedings, Vol.

7732, 58

K. Nandra, I. M. George, R.F. Mushotzsky et al., Apj, 1997,476, 70

Paolillo, M., Schreier, E. J., Giacconi, R., Koekemoer, A. M.,Grogin, N. A., 2004, ApJ, 611, 93

Paolillo, M., et al. 2011, Memorie della Societa AstronomicaItaliana Supplementi, 17, 97

I.E. Papadakis, Brinkmann, W., Negoro, H., Gliozzi, M., 2002,A&A, 382, 1

I.E. Papadakis, E. Chatzopoulos, D. Athanasiadis et al., A&A,2008, 487, 475

Turner, T. J., George, I. M., Nandra, K., & Turcan, D. 1999,ApJ, 524, 667

P. Uttley, I. McHardy, & I. E. Papadakis, MNRAS, 2002, 332,231

Rosati, P., et al. 2011, Memorie della Societa AstronomicaItaliana Supplementi, 17, 8

J. Timmer, M. Koenig, A&A, 1995, 300, 707P. Uttley, I. McHardy, & M. Ian, MNRAS, 2005, 363, 286Uttley, P., McHardy, I. M., Papadakis, I. E., 2002, MNRAS, 332,

231Vaughan, S., Edelson, R., Warwick, R. S. and Uttley, P. 2003

MNRAS 345, 1271S. Vaughan, P. Uttley, MNRAS, 2005, 362, 235Xue, Y. Q., et al. 2011, ApJS 195, 10