11institutetext: Department of Physics and Institute of Theoretical and
Computational Physics, University of Crete, 71003 Heraklion, Greece 11email:
<EMAIL_ADDRESS>22institutetext: Institute of Astrophysics, FORTH,
GR-71110 Heraklion, Greece

# The X-ray variability of AGN: power-spectrum and variance analysis of the
Swift /BAT light curves

I. E. Papadakis 1122 V. Binas-Valavanis 1122

###### Abstract

Aims. We study the X–ray power spectrum of Active Galactic Nuclei (AGN) to
investigate whether Seyfert I and II power spectra are similar or not, whether
the AGN variability depends on black hole mass and accretion rate, and to
compare the AGN power spectra with the Galactic X–ray black hole binaries
power-spectra.

Methods. We used 14-195 keV band light curves from the 157th Swift /BAT hard
X–ray survey and we computed the mean power spectrum and excess variance of
AGN in narrow black-hole mass/luminosity bins. We fitted a power-law model to
the AGN power spectra, and we investigated whether the power spectrum
parameters and the excess variance depend on black hole mass, luminosity and
accretion rate.

Results. The Seyfert I and Seyfert II power spectra are identical, in
agreement with AGN unification models. The mean AGN X–ray power spectrum has
the same, power-law like shape with a slope of $-1$ in all AGN, irrespective
of their luminosity and BH mass. We do not detect any flattening to a slope of
zero at frequencies as low as $10^{-9}$ Hz. We detect an anti-correlation
between the PSD amplitude and the accretion rate, similar to what has been
seen in the past in the 2–10 keV band. This implies that the variability
amplitude in AGN decreases with increasing accretion rate. The universal AGN
power-spectrum is consistent with the mean, 2–9 keV band Cyg X-1 power
spectrum in its soft state. We detect a small difference in amplitude, but
this is probably due to the difference in energy.

Conclusions. The mean, low frequency AGN X–ray power spectrum is consistent
with the extension of the mean, 0.01–25 Hz Cyg X-1 power spectrum in its soft
state to lower frequencies. We cannot prove that the mean AGN PSD is analogous
of the mean Cyg X-1 PSD in its soft state, as we do not know the location of
the high frequency break in the hard X-ray AGN PSDs. However, if that is the
case, then the accretion disc in AGN probably extends to the radius of the
innermost circular stable orbit (as is probably the case with the black hole
binaries in their soft sate). The X-ray corona will then be located on top,
illuminating the disc and producing the X–ray reflection and disc
reverberation phenomena that are commonly observed in these objects.
Furthermore, the agreement between the AGN and the Cyg X–1 power spectrum
(either in the soft or the hard state) over many decades in frequency
indicates that the X–ray variability process is probably the same in all
accreting objects, irrespective of the mass of the compact object. We plan to
investigate this issue further in the near future.

###### Key Words.:

galaxies: active – galaxies: Seyfert – X-rays: galaxies

##  1 Introduction

Active Galactic Nuclei (AGN) vary at all wavebands, with the variability
amplitude increasing with increasing energy. X–rays vary significantly in flux
and/or spectral shape on time scales as short as minutes/hours in some cases.
This supports the hypothesis that X–rays are emitted in a small region located
close to the super-massive black hole (BH), which resides in the center of
these objects. X–ray spectral and timing studies are though to be important as
their study could help us understand the physical processes that operate in
the innermost region of these objects.

AGN variations are stochastic in nature, therefore their study involves the
use of statistical tools like the so-called normalized excess variance,
$\sigma^{2}_{NXV}$, and the power-spectral density function (PSD). The former
is equal to the integral of the latter, hence it provides limited information
on the nature of the variability when compared to the study of the PSD itself.
Power spectrum analysis of the X-ray light curves of radio-quiet AGN was first
introduced with the use of long, EXOSAT light curve thirty years ago (e.g.
Lawrence & Papadakis, 1993; Green et al., 1993). Detailed characterization of
the X–ray power spectra of AGN (in the 2–10 keV band) was later achieved with
the combination of RXTE and XMM-Newton light curves (e.g. Uttley et al., 2002;
Papadakis et al., 2002; Markowitz et al., 2003; McHardy et al., 2004; Uttley &
McHardy, 2005; Markowitz et al., 2007; Markowitz, 2010; González-Martín &
Vaughan, 2012). The past variability studies have established that the X–ray
PSDs in the 2–10 keV band have a featureless, power-law like shape with a
slope of $\sim-1$ over many decades in frequency, which then steepens to a
slope of $\sim-2$ (or even steeper) above a bend-frequency, $\nu_{b}$. This
characteristic frequency depends on BH mass and possibly on the accretion rate
rate as well (McHardy et al., 2006; González-Martín & Vaughan, 2012). This
power spectrum shape is similar to the shape of the X–ray power spectra of the
Galactic X–ray BH binaries (GBHs) in their soft states.

We use the Swift /BAT light curves from the 157 month BAT survey (Lien et al,
in preparation) of a large sample of AGN, in order to estimate their power
spectrum at low frequencies. BAT has been continuously monitoring the sky for
$\sim 18.5$ years, and the 157 month survey provides long, evenly sampled, and
high signal-to-noise light curves for a large number of AGN in the 14-195 keV
band. This energy band is well suited for the study of the intrinsic X–ray
variability in AGN, because it is unaffected by any potential absorption
variations (as long as the intrinsic hydrogen column density is less than a
few $\times 10^{24}$ cm-2).

Swift /BAT light curves have been used in the past to study the flux and
spectral variability of AGN (e.g. Beckmann et al., 2007; Caballero-Garcia et
al., 2012; Soldi et al., 2014). Our work is similar to the work of Shimizu &
Mushotzky (2013). These authors (SM13, hereafter) calculated the power
spectrum of 30 AGN, for the first time in hard X-rays (i.e. at energies higher
than 10 keV). They used 58 month long BAT light curves, and they measured the
PSD at frequencies $\sim 10^{-8}-10^{-6}$ Hz. They found that all power-
spectra were well fitted by a single power-law model with a slope of $\sim-1$,
and they did not find any significant correlation between the PSD parameters
and various AGN properties, including luminosity and BH mass (${\rm M}_{\rm
BH}$).

Our sample consists of the one hundred, X-ray brightest, radio quiet AGN from
the BASS DR2 survey111https://www.bass-survey.com/ (Koss et al., 2022). Our
main aim is to use the high quality, Swift /BAT light curves to compute their
long term power-spectrum and: 1) determine whether there are any differences
between the average PSD of Seyfert 1 and Seyfert 2s, 2) investigate if and how
does the PSD vary with AGN properties (like BH mass and luminosity), and 3)
compare the mean AGN PSD with the PSD of GBHs.

##  2 The sample

Koss et al. (2022) presented a catalog of AGN for the second data release of
the Swift BAT AGN Spectroscopic Survey (BASS). They have identified all AGN
among the 1210 sources in the BAT 70-month survey Baumgartner et al. (2013)
and have obtained BH mass estimates for almost all of them. Our sample
consists of the 100 brightest AGN (in the 14-195 keV band) in Table 6 of Koss
et al. (2022), with the exception of Q0241+622, because it lacks a black hole
mass estimate. In choosing the sources we excluded: a) ”beamed” AGN (i.e.
objects classified as ”BZQ”, ”BZG” and ”BZU” by Koss et al. (2022)) since
their emission may be dominated by the jet emission, and b) ”dual” AGN (listed
in Table 4 of Koss et al. (2022)), if the ratio of the predicted
$F^{obs}_{14-195{\rm keV}}$ of the fainter source over the total detected BAT
flux is larger than 1% (see §2.3 in Koss et al. (2022)).

The sources in our sample are listed in Table LABEL:table:sampledata. Source
classification, redshift, and logarithm of black hole mass, log(MBH), are
listed in columns 2, 4, and 5, and are taken from Koss et al. (2022). In the
third column we list the mean observed flux in the 14-195 keV,
$F^{obs}_{14-195{\rm keV}}$, taken from the Swift /BAT, 157-month, hard X–ray
survey web page222https://swift.gsfc.nasa.gov/results/bs157mon.

In the sixth column we list the logarithm of the intrinsic X–ray luminosity,
LX, in the 14–150 keV band. It is computed by dividing the bolometric
luminosity (Lbol) of Koss et al. (2022) by 8 (i.e. the bolometric correction
factor used by these authors). The 14-195 keV band X–ray luminosity should be
a measure of Lbol, which is one of the important physical properties of an AGN
(together with ${\rm M}_{\rm BH}$). Once the BH mass is known, the ratio
$\lambda_{\rm Edd}$=Lbol/LEdd (where LEdd is the Eddington luminosity), should
be a rough estimate of the ratio of the accretion rate over the Eddington
accretion rate. One of our objectives is to examine the dependence of the PSD
on $\lambda_{\rm Edd}$. However, the bolometric conversion factor between LX
and Lbol is uncertain. For example, Koss et al. (2022) assume
Lbol/L14-195keV=8, while Beckmann et al. (2009) assumed Lbol/L20-100keV=6. In
addition, the conversion between LX and Lbol may be more complicated than just
applying a constant bolometric correction (e.g. Lusso et al., 2012). For this
reason, we list the LX measurements in Table LABEL:table:sampledata, and we
will consider the dependence of the PSD on both LX and $\lambda_{\rm Edd}$ in
our study below. The AGN in our sample span a range of $\sim 3$ in log(MBH)
(from $\sim 10^{6}$ to $\sim 10^{9}$ M⊙) and a range of $\sim 2.5$ in log(LX)
(from $\sim 10^{42}$ to $\sim 10^{44.5}$ ergs/s). They should be
representative of the radio-quiet AGN population in the nearby Universe.

##  3 The light curves

We retrieved BAT light curves for all the sources in the sample from the
Swift-BAT 157 month survey web page. We considered the monthly, Crab-weighted
light curves in the 14-195 keV band (Lien et al, in preparation). The 157
month light curves are constructed as in the 105 month release (Oh et al.,
2018), with some updates in both the instrumental calibration and responses
(A. Lien, priv. comm.).

A few points in some light curves are associated with large error bars. Their
exposure time is less than 5 ksec (a small number when compared to the nominal
bin width of 1 month). We use the mean of the error squared to compute the
Poisson noise level in the power spectrum (see §4.1). Since the error of these
points can bias the mean squared error, we decided to remove these points and
to replace them using linear interpolation between the adjacent points in the
light curves. There are also a few missing points in some light curves, which
we also fill in using linear interpolation. We added a random error to each
interpolated point, assuming Gaussian statistics with a standard deviation
equal to the mean error of all points in the light curves (excluding the ones
with $\Delta t\leq 5$ ks). There are 39 light curves with missing points
and/or points with an exposure time less than 5 ks (the mean number of missing
points is 2).

We checked which sources show significant variations in the light curves using
traditional $\chi^{2}$ statistics. To this end, we computed the weighted mean
of each light curve333The count rate error in the Swift -BAT light curves is
inversely proportional to the square root of the exposure time in each bin
(this is easily checked when plotting errors versus the respective exposure
time), and it does not depend on the source’s flux. In this case, the weighted
rather than the straight mean is more representative of the source’s mean
flux., and we fitted the light curves with a constant line, equal to the mean.
The fit was done without considering the interpolated points. The resulting
$\chi^{2}$ values over the degrees of freedom (dof) are listed in the 7th
column in Table LABEL:table:sampledata. We accept that a light curve is
variable if the null hypothesis probability, $p_{null}$, is less than 0.01
(the null hypothesis is that the source is not variable). Letters ”V” or ”NV”
in the 8th column in Table LABEL:table:sampledata indicate whether a source is
variable or not (i.e. $p_{null}<0.01$ or $p_{null}\geq 0.01$, respectively).

Twenty six sources out of the 100 in the sample are non-variable. Nineteen of
the NV sources are Sy1 and the rest are Sy2 and Sy 1.9. The mean log(MBH) and
log(LX) are $\sim 7.8$ and $\sim 43.6$, respectively, for both the V and NV
group of sources. However, the ratio of the mean count rate over the mean
error (i.e. the signal-to-noise ratio of the light curve, S/N) is smaller than
3 in almost all (but two) of the NV sources. On the contrary, the same ratio
is larger than 3 in almost two thirds of the V sources. This result indicates
that the reason we do not detect significant variations in the NV sources is
mainly because their S/N ratio is low. As a result, the amplitude of the
variations due to the experimental noise is larger than the amplitude of the
intrinsic variations, and we cannot detect them.

##  4 Power spectrum estimation

As it is often the case, we used the periodogram as an estimate of the
intrinsic power spectrum. We computed the periodogram of each light curve in
the usual way, i.e.

$I_{N}(f_{j})=\frac{2\Delta t_{rf}}{N}|\sum^{N}_{i=1}x(t_{i})e^{-i2\pi
f_{j,rf}t_{i}}|^{2},$ (1)

where $\Delta t_{rf}$ is the rest frame light curve bin size ($\Delta
t_{rf}=1{\rm month}/(1+z);z$ is the source redshift), $N$ is the total number
of points in the light curve, and the data are normalized to the mean (i.e.
$x(t_{i})=[x(t_{i})-\overline{x}]/\overline{x}$, with $\overline{x}$ being the
light curve mean). The periodogram is computed at the usual set of
frequencies, i.e. $f_{j,rf}=j/(N\Delta t_{rf}),j=1,2,...,N/2$ ($N=157$ and
$j_{max}=78$).

###  4.1 Computing the Poisson noise error

In order to estimate the intrinsic PSD, we subtracted the constant Poisson
noise level, $C_{PN}$, from the periodogram. We computed $C_{PN}$ as follows,

$C_{PN}=\frac{2\Delta t_{rf}(\overline{\sigma^{2}_{err})}}{\overline{x}^{2}},$
(2)

where $\overline{\sigma^{2}_{err}}$ is the mean of the squared error of the
points in the light curve. This should be a good estimate of the light curve
variance due to the Poisson noise. Since the correct estimation of $C_{PN}$ is
important for the accurate determination of the PSD, we used the power
spectrum of the NV sources to test eq. (2).

The periodograms of the NV sources appear to be flat, as expected. To verify
this, we computed the logarithm of the periodogram, and we binned them into
groups of size $M=20$. The resulting PSD estimates are approximately Gaussian
distributed, with known error (Papadakis & Lawrence, 1993, PL93 hereafter). A
constant line fits well all the binned logarithmic periodograms. This confirms
that the power spectra of the NV sources are flat, indicating the lack of
intrinsic variations with amplitude larger than the Poisson noise variations
in the NV sources. In this case, the mean of the logarithm of the periodogram
estimates (plus 0.25068; see Vaughan (2005)) should be representative of the
logarithm of the actual Poisson noise level in the light curves, ${\rm
log}(C_{PN,obs})$444We computed the mean of log[$I_{N}(f_{j})$] instead of the
mean of $I_{N}(f_{j})$ because the error of the latter is known, and is equal
to $\sqrt{0.31/j_{\max}}$ (PL93)..

Figure 1 shows a plot of the logarithm of the ratio of $C_{PN,obs}$ over
$C_{PN}$ (computed using eq. 2) versus log$(C_{PN})$, for the NV sources. The
solid line shows the mean log($C_{PN,obs}/C_{PN})$, which is equal to
$0.020\pm 0.012$. The data are broadly consistent with this line, and since
the mean log($C_{PN,obs}/C_{PN})$ is also consistent with zero (within $\sim
1.6\sigma$), we conclude that $C_{PN}$, as defined by eq. (2), provides a good
estimate of the Poisson noise in the light curves.

Although the solid line in Fig. 1 appears to agree with data rather well, the
best-fit $\chi^{2}$ value of 63.2 for 25 dof indicates that the statistical
quality of the fit is rather poor. However, this is due to two sources mainly,
where log($C_{PN,obs}/C_{PN})$ is $\sim 3-4\sigma$ away from the mean. If we
do not consider these objects, the agreement between $C_{PN,obs}$ and $C_{PN}$
is quite good ($\chi^{2}=30.3/23$dof, $p_{null}=0.14$). We conclude that,
although $C_{PN}$ may not (always) provide an accurate measurement of the
Poisson noise on an individual light curve basis, it gives a good estimate of
$C_{PN,obs}$ on average. Since our work is based on the study of the average
PSD of many sources in various groups (see next Section), we adopt $C_{PN}$,
as defined by eq. (2), as an estimate of the Poisson noise in the Swift /BAT
light curves.

Figure 1: Plot of log$(C_{PN,obs}/C_{PN})$ versus $C_{PN}$ for the 26 NV
sources in the sample. The solid line indicates the mean
log$(C_{PN,obs}/C_{PN})$.

###  4.2 Computing and fitting the ensemble PSD of AGN

Although the periodogram is an unbiased estimator of the intrinsic PSD, its
statistical properties are not ideal. The probability distribution of the
periodogram estimates follows a $\chi^{2}$ distribution with 2 degrees of
freedom, their variance is large and unknown. In fact, the variance does not
even decrease with increasing data points. For that reason, it is customary to
smooth the periodogram using various “spectral windows”. However, smoothing in
the linear space is not ideal in the case of power-law like intrinsic PSDs
(PL93). One option would be to bin the logarithmic periodogram as suggested by
PL93, but that is not the best solution in our case either. Since
$j_{max}=78$, we would end up with just four points in the power spectrum,
over a limited frequency range, if we would follow the prescription of binning
the log periodogram into groups of size $M=20$. Fitting them with a straight
line would result in best-fit parameters with large uncertainties. To avoid
this, we followed a different approach to compute and fit the power spectra of
the variable sources in the sample.

As we describe in the next section, we consider AGN in relatively small boxes
in the log(LX) vs log(MBH) plane (so that they have similar BH mass and
luminosity). Suppose there are $n_{\rm AGN}$ in such a box (typically, $n_{\rm
AGN}=4-6$). We compute the periodogram of each one of them and we subtract the
Poisson noise level. Then we compute the mean of the $n_{\rm AGN}$
periodograms, say $\overline{I(\nu)}$, at frequencies lower than $0.1$
month-1, and we accept log$[\overline{I(\nu)}]$ as an estimate of the
ensemble, mean power spectrum of the AGN in each [log(LX),log(MBH)] box. In a
few cases, $\overline{I(\nu)}$ is negative at one or more frequencies. In
these cases, we group these values together with the two $\overline{I(\nu)}$
in the adjacent frequencies, and we replace all three with their mean.

At higher frequencies, the variance of the noise subtracted power spectrum
increases. In order to reduce the noise of the PSD at these frequencies, we
further smooth the mean $\overline{I(\nu)}$’s using a simple top-hat window of
size 5, i.e. we group every 5 successive $\overline{I(\nu)}$’s together, we
compute their mean and accept its logarithm as an estimate of the intrinsic
PSD at the mean of the respective frequencies.

Since $n_{\rm AGN}$ is small, the probability distribution of the resulting
PSD estimates is not Gaussian, even at high frequencies. Therefore we cannot
use traditional $\chi^{2}$ statistics to fit the observed power spectra. For
that reason, we assumed that the intrinsic PSD follows a power-law like shape,
and we fitted the power spectra, in the log-log space, with a line of the
form,

${\rm log(PSD)}(\nu)=\log(PSD_{amp})+PSD_{slope}\cdot{\rm
log}(\frac{\nu}{\nu_{0}}),$ (3)

where $PSD_{amp}$ is the PSD amplitude at $\nu_{0}=10^{-2}$ month-1. We chose
this frequency to define the PSD amplitude because it is within the range of
sampled frequencies and it is reasonably low, hence the best-fit value will
not be affected by aliasing (or other) effects (see the discussion below). We
fitted the data following the ordinary least square (OLS) [Y—X] prescription
of Isobe et al. (1990). In this way, even though we do not know the error of
the logarithmic estimates of the power spectrum, we can still compute the
best-fit line parameters, together with their error.

Figure 2: The mean power spectrum of the AGN which belong to group B4, as
indicated in Fig. 4. The power spectrum is computed as explained in §4. Solid
and dashed lines show the best-fit lines when we fit the low frequency and the
full range power-spectrum (filled and filled+open circles, respectively), with
the model defined by eq. 3.

As an example, Fig. 2 shows the average PSD of 4 AGN with log(MBH) $\sim 7$
and log(LX) $\sim 43.5$ (these are the objects that belong to the AGN group B4
in Fig. 4). The solid and dashed lines show the best-fit lines when we fit eq.
3 to the PSD at frequencies below 0.1 months-1 and at all frequencies,
respectively. The full-range best-fit slope is flatter than the low-frequency
slope, with $\Delta(PSD_{slope})\sim 0.5$. This is the PSD with one of the
largest differences between the low and the full frequency range best-fit
slopes. Differences between the low-frequency and the full range best-fit
slopes appear in other power spectra as well, but they are smaller than 0.5
(see Table 1).

A PSD flattening may be due to the smoothing process at high frequencies,
which may bias the observed power spectrum and it may appear flatter than the
intrinsic PSD. In addition, Swift /BAT does not observe continuously all AGN.
In quite a few cases, the total exposure time in some bins is less than a day,
which is much smaller than the bin size of the light curves. It is possible
then that aliasing effects may be significant, in which case they will also
flatten the observed PSDs at high frequencies. Such effects are difficult to
model (as they depend on the intrinsic PSD and the actual observing pattern
within each $\Delta t$). For that reason, we fitted the PSDs twice: once at
frequencies below $10^{-1}$ month-1 (the LF fits, hereafter), and the second
time over the full frequency range (the FR fits, hereafter).

Figure 3: The upper panel shows a plot of log(LX), versus log(MBH) for the
variable sources in the sample. Black and red circles indicate data for the
Sy1 and Sy2 galaxies, respectively. The mean BH mass and LX of the Sy1 and Sy2
sources in the dashed box are similar. Lower panel: Black and red points show
the ensemble PSD of the Sy1 and Sy2 galaxies which are in the dashed box drawn
in the upper panel. Black and red dashed lines show the best-fit line to the
PSDs.

##  5 Power spectrum analysis

###  5.1 The Sy1 and Sy2 PSDs

We first investigated whether the Sy1 and Sy2 power spectra are the same or
not. The top panel in Fig. 3 shows a plot of log(LX) versus log(MBH) for all
the variable sources in the sample. Black filled and red open circles indicate
the Sy1 and the Sy2 sources, respectively (hereafter, we refer as “Sy2” both
the Sy1.9 and Sy2 sources in the sample). On average, ${\rm M}_{\rm BH}$ in
Sy2s appears to be larger when compared to Sy1s, while their intrinsic X–ray
luminosity appears to be smaller. The investigation of the significance of
these differences is outside the scope of our work, however, since the PSD may
depend on ${\rm M}_{\rm BH}$ and/or LX we need to compare the mean power-
spectrum of Sy1 and Sy2 with the same ${\rm M}_{\rm BH}$ and LX in order to
test if they are similar or not.

There are 23 Sy1 and 23 Sy2s in the box indicated by the dashed lines in the
upper panel of Fig. 3. Their mean log(MBH) and log(LX) are consistent (within
the errors): $\overline{\rm log(M_{BH}})=7.5\pm 0.1$, and $7.8\pm 0.1$ for Sy1
and Sy2, while $\overline{\rm log(L_{X})}=43.8\pm 0.1$ and 43.6$\pm 0.1$,
respectively.

The lower panel in Fig. 3 shows the mean PSD of all the Sy1 and Sy2 AGNs
within the same box (black filled, and red open circles, respectively). We
fitted the power spectra with the model defined by eq. (3). Black solid and
red dashed lines show the best-fit lines to the full range Sy1 and Sy2 PSDs,
respectively. The LF(FR) best fit line parameters are: log$(PSD_{amp,\rm
Sy1})=0.36\pm 0.26(0.32\pm 0.28$), log($PSD_{amp,\rm Sy2})=0.43\pm
0.31(0.40\pm 0.32$), and $PSD_{slope,\rm Sy1}=-1.00\pm 0.07(-0.88\pm
0.09),PSD_{slope,\rm Sy2}=-0.95\pm 0.05(-0.87\pm 0.09$). Clearly, the Sy1 and
Sy2 PSDs are consistent, within the errors. This result suggests that the
variability properties of the Sy1 and Sy2 galaxies in the 14-195 keV band are
identical. Given this result, we combine together light curves from Sy1 and
Sy2 sources in order to investigate the dependence of the power spectrum on BH
mass and/or luminosity.

Figure 4: Plot of log(LX), versus log(MBH) for the variable sources in the
sample. AGN within the boxes indicated by the dashed lines have similar ${\rm
M}_{\rm BH}$ and LX.

###  5.2 The dependence of PSD slope on ${\rm M}_{\rm BH}$, LX, and
$\lambda_{\rm Edd}$.

It is not easy to model the periodogram of each individual AGN in our sample
and investigate if and how it may depend on ${\rm M}_{\rm BH}$ and/or LX.
Instead, we grouped the variable AGN into relatively narrow BH mass and
luminosity bins, as indicated by the dashed lines in Fig. 4. Their size is
small ($\Delta$log(${\rm M}_{\rm BH}$)$\sim\Delta$log(LX)$\sim 0.5$), so that
${\rm M}_{\rm BH}$ and LX of AGN in each box should be within a factor of
$\sim 2$ or so. Consequently, their ensemble PSD should provide a good
estimate of the intrinsic PSD of an AGN with BH mass and X–ray luminosity
equal to the mean ${\rm M}_{\rm BH}$ and LX of the objects in each box.

We computed the mean power spectrum of all AGN in each box (as explained in
§4), and we fitted them with a line, as defined by eq. 3. The best-fit slope
should be indicative of the intrinsic slope of the AGN PSD. On the other hand,
the best-fit amplitude may overestimate the intrinsic PSD amplitude, because
we have considered only the variable AGN in Fig. 4. This is appropriate when
studying the PSD slope, as information on the slope can be provided only by
sources with PSDs above the Poisson noise level. However, if there is a
distribution of PSD amplitudes at each ${\rm M}_{\rm BH}$,LX, then we should
consider all AGN in each [${\rm M}_{\rm BH}$,LX] bin in order to measure the
mean of the distribution, irrespective of whether they are variable or not (as
long as they are at the same distance). We therefore use the best-fit slope
from the model fits to the average PSDs of the variable AGN to study whether
$PSD_{lope}$ depends on ${\rm M}_{\rm BH}$ and/or LX, and we will discuss the
dependence of $PSD_{amp}$ on these parameters in the next Section.

The mean log(MBH) and log(LX) of the AGN in each box in Fig. 4, and their
error, are listed in the first column of Table 1. Second and third columns
list the LF, and the FR best-fit PSD slopes, respectively. The top panel in
Fig. 5 shows the LF, best-fit PSD slope plotted as a function of the mean
log(LX) for the AGN in groups (B1, B2, B3 B4), (C1, C2, C3) and (D1, D2, D3,
D4) (black circles, red circles and brown squares, respectively). All AGN in
the B, C and D groups have similar BH masses, with a mean of $\sim 7.1,7.7,$
and 8.1, respectively (see Table 1). Within each of the B, C and D groups the
PSD slope does not appear to depend on LX. PSD slopes also appear to be
similar between the various groups, which implies that $PSD_{slope}$ does not
depend on ${\rm M}_{\rm BH}$ either.

Table 1: First column lists the mean log(MBH), and log(LX) of the objects in the groups labeled in Fig. 4. The LF and FR best-fit $PSD_{slope}$ values are listed in the second and third columns, respectively.  $\overline{{\rm log(M_{BH})}},\overline{{\rm log(L_{X})}}$ | $PSD_{slope}$
---|---
| (LF) | (FR)
$6.33\pm 0.08,42.32\pm 0.11$(A1) | $-1.24\pm 0.36$ | $-0.79\pm 0.18$
$7.09\pm 0.13,44.10\pm 0.11$(B1) | $-1.07\pm 0.32$ | $-0.61\pm 0.15$
$7.20\pm 0.09,43.62\pm 0.05$(B2) | $-1.15\pm 0.47$ | $-0.54\pm 0.15$
$7.11\pm 0.09,43.16\pm 0.04$(B3) | $-1.05\pm 0.20$ | $-0.94\pm 0.13$
$6.96\pm 0.12,42.51\pm 0.11$(B4) | $-1.17\pm 0.20$ | $-0.63\pm 0.08$
$7.80\pm 0.04,44.09\pm 0.06$(C1) | $-0.97\pm 0.09$ | $-0.88\pm 0.11$
$7.66\pm 0.02,43.55\pm 0.05$(C2) | $-0.83\pm 0.23$ | $-0.55\pm 0.15$
$7.73\pm 0.07,43.26\pm 0.03$(C3) | $-0.94\pm 0.18$ | $-0.64\pm 0.10$
$8.13\pm 0.09,44.72\pm 0.09$(D1) | $-0.93\pm 0.11$ | $-0.44\pm 0.09$
$8.13\pm 0.08,44.15\pm 0.07$(D2) | $-0.38\pm 0.29$ | $-0.66\pm 0.26$
$8.13\pm 0.02,43.62\pm 0.04$(D3) | $-0.93\pm 0.20$ | $-0.76\pm 0.10$
$8.22\pm 0.08,43.18\pm 0.05$(D4) | $-1.18\pm 0.24$ | $-0.94\pm 0.10$
$8.85\pm 0.09,44.13\pm 0.05$(E1) | $-1.39\pm 0.15$ | $-1.08\pm 0.09$
$8.81\pm 0.10,43.63\pm 0.05$(E2) | $-0.47\pm 0.25$ | $-0.93\pm 0.21$
Figure 5: Upper panel: LF, best-fit $PSD_{slope}$ plotted as function of
log(LX) for the B, C and D groups (black circles, red circles, and brown
squares, respectively). Bottom panel: The LF and FR $PSD_{slope}$ plotted as a
function of $\overline{\log({\rm M_{BH}})}$ Black and red dashed lines
indicate the mean LF and FR slopes, respectively.

To investigate these issues quantitatively, we fitted the data plotted in the
top panel in Fig. 5 with a line of the form:
log(PSD${}_{slope})=a+b$[log(LX)$-43.5$]. We took into account errors in both
coordinates, using the fitexy routine of Press et al. (1992). We expect the
$PSD_{slope}$ error to be similar in all of them, since the mean PSD is
computed in the same way and the number of objects is roughly the same in all
bins. The errors listed in Table 1 fluctuate mainly because of the small
number of points in the PSDs (specially in the low frequency part). The error
on the slope turns out small in cases where the PSD points happened to align
well around the best-fit line, while larger errors result in the case when the
scatter of the points around the best-fit line is large (see also discussion
in §IV in Isobe et al. (1990) on the best-fit errors in the case of small
number of data points). For that reason, we computed the mean error of all PSD
slope estimates, and we assigned it to each $PSD_{slope}$, assuming it is more
representative of their intrinsic error.

The best-fit results from fitting the points in the top panel of Fig. 5 are:
$b_{B}=0.06\pm 0.17,b_{C}=-0.08\pm 0.19,$ and $b_{D}=0.08\pm 0.14$. All three
best-fit $b-$values are consistent with zero, which shows that $PSD_{slope}$
does not depend on LX. Furthermore, $a_{B}=-1.08\pm 0.16,a_{C}=-0.92\pm 0.12,$
and $a_{D}=-0.98\pm 0.14$. The best-fit $a$ values for the various groups are
also consistent with each other, which implies that PSDslope does not depend
on ${\rm M}_{\rm BH}$ either.

The bottom panel in Fig. 5 shows the best fit PSD slope as a function of ${\rm
M}_{\rm BH}$ for all groups shown in Fig. 4, and listed in Table 1. Black and
red circles show the LF and FR PSD slopes, respectively. The mean LF and FR
$PSD_{slope}$ is $\overline{PSD_{slope,LF}}=-0.98\pm 0.06$ and
$\overline{PSD_{slope,FR}}=-0.74\pm 0.04$. The black and red horizontal lines
in the same panel indicate the mean LF and FR PSD slopes, respectively. When
fitting the LF and FR data with these lines we get $\chi^{2}=18.2/13$dof and
25.3/13dof, respectively. Both fits are statistically accepted
($p_{null}=0.15$ and 0.02, respectively).

As a final test, we also investigated the possibility that the PSD slope may
depend on accretion rate. We fitted the LF and the FR PSD slopes versus
log($\lambda_{\rm Edd}$) with a straight line of the form:
$PSD_{slope}[log(\lambda_{\rm
Edd})]=a_{\rm\lambda_{Edd}}+b_{\rm\lambda_{Edd}}[{\rm log(\lambda_{Edd})+1]}$.
The fits were done taking into account the error on both variables. When
considering the fit in the case of the LF slopes the improvement in the
goodness of fit is not significant ($\Delta\chi^{2}=0.2$ for 1 degree of
freedom). However, in the case of the FR slopes, a straight line improves the
fit significantly ($\Delta\chi^{2}=9.3$ for one extra degree of freedom;
$p_{null}=0.002$). This result implies that the PSD slope may depend on
$\lambda_{\rm Edd}$. The best-fit slope of the PSD slope versus
$\lambda_{Edd}$ relation is $b_{\rm\lambda_{Edd,FR}}=0.19\pm 0.06$, which
implies that the PSD slope may flatten with increasing $\lambda_{Edd}$.

###  5.3 The dependence of PSD amplitude on ${\rm M}_{\rm BH}$, LX, and
$\lambda_{\rm Edd}$.

Figure 6: Like Fig. 4, for AGN with S/N$\geq 2$ light curves.

As we discussed in the previous section, we should consider all AGN in each
[${\rm M}_{\rm BH}$,LX] bin in order to measure the intrinsic PSD amplitude,
irrespective of whether they are variable or not, as long as they are at the
same distance. Distance determines the S/N ratio of the light curve for
objects with the same luminosity. Even if the PSD amplitude is the same in two
sources, if one of them is located far away the observed count rate will be
small and its observed PSD may not be above the Poisson noise. The apparent
difference in the observed PSDs in this case will be due to differences in the
distance of the sources, and not in their PSD.

For that reason, we considered all AGN in the sample with lightcurve with
S/N$\geq 2$. These objects are shown in Fig. 6. Just like in Fig. 4, the
dashed lines in this figure indicate boxes with a small width so that all
sources in each box should have similar ${\rm M}_{\rm BH}$ and LX. AS before,
we computed the mean power spectrum of all AGN in each box in Fig. 6 as
explained in §4, and we fitted them with a line (as defined by eq. 3). The
best-fit amplitude should be indicative of the (logarithm of the) intrinsic
PSD amplitude in AGN. The mean log(MBH) and log(LX) of the AGN in each box in
Fig. 6, together with their error, are listed in the first column of Table 2.
Second and third columns list the LF, and the FR best-fit PSD amplitudes,
respectively.

Table 2: First column lists the mean log(MBH) and log(LX) of the objects in the boxes labeled in Fig. 6. The second and third columns list the LF and FR best-fit log($PSD_{amp})$ values, respectively. $\overline{\rm log(M_{BH})},\overline{\rm log(L_{X})}$ | $\log(PSD_{amp})$
---|---
| (LF) | (FR)
$6.76\pm 0.08$, $42.57\pm 0.10$(A1′) | $0.20\pm 0.20$ | $-0.13\pm 0.14$
$7.17\pm 0.13$, $44.13\pm 0.14$(B1′) | $0.00\pm 0.11$ | $0.00\pm 0.22$
$7.20\pm 0.11$, $43.57\pm 0.05$(B2′) | $-0.24\pm 0.16$ | $-0.34\pm 0.31$
$7.08\pm 0.07$, $43.19\pm 0.04$(B3′) | $0.17\pm 0.13$ | $0.07\pm 0.13$
$7.80\pm 0.04$, $44.09\pm 0.06$(C1′) | $0.25\pm 0.19$ | $0.21\pm 0.21$
$7.66\pm 0.03$, $43.56\pm 0.06$(C2′) | $0.10\pm 0.21$ | $-0.08\pm 0.15$
$7.67\pm 0.04$, $43.26\pm 0.03$(C3′) | $0.02\pm 0.11$ | $-0.06\pm 0.14$
$8.13\pm 0.09$, $44.72\pm 0.09$(D1′) | $-0.02\pm 0.12$ | $-0.22\pm 0.17$
$8.08\pm 0.06$, $44.20\pm 0.07$(D2′) | $-0.23\pm 0.46$ | $-0.41\pm 0.38$
$8.13\pm 0.03$, $43.63\pm 0.06$(D3′) | $0.21\pm 0.31$ | $0.06\pm 0.12$
$8.28\pm 0.07$, $43.22\pm 0.03$(D4′) | $0.64\pm 0.49$ | $0.29\pm 0.28$
$8.75\pm 0.07$, $44.09\pm 0.06$(E1′) | $0.65\pm 0.40$ | $0.41\pm 0.43$
$8.66\pm 0.07$, $43.68\pm 0.03$(E2′) | $0.20\pm 0.14$ | $0.34\pm 0.38$

The top panel in Fig. 7 shows the LF, best-fit log$(PSD_{amp}$) plotted as a
function of the mean log(LX) for the AGN in boxes (B1′, B2′, B3${}^{\prime})$,
(C1′, C2′, C3′) and (D1′, D2′, D3′, D4′) (black circles, red circles and brown
squares, respectively). We fitted the data with a line of the form:
log($PSD_{amp})=c+d$[log(LX)$-43.5$]. The best fit results are:
$c_{B}=-0.005\pm 0.08,c_{C}=0.09\pm 0.11,c_{D}=0.36\pm 0.22,$ and
$d_{B}=-0.14\pm 0.20,d_{C}=0.28\pm 0.34,d_{D}=-0.47\pm 0.32$. The best-fit $d$
values are all consistent with zero, indicating that the power-spectrum
amplitude does not depend on LX. The best-fit $c$ values appear to increase as
we go from group B to group D, indicating an increase of $PSD_{amp}$ with
increasing ${\rm M}_{\rm BH}$. In reality, since $c$ is equal to the PSD
amplitude of an AGN with log(LX)=43.5, such a dependence on ${\rm M}_{\rm BH}$
would imply $PSD_{amp}$ increases with decreasing accretion rate. However,
formally speaking, the best-fit c values are consistent with each other(within
1.5$\sigma$), suggesting that this apparent dependence of $log(PSd_{amp})$ on
${\rm M}_{\rm BH}$ is not significant.

The bottom panel in Fig. 7 shows the best fit log($PSD_{amp})$ amplitude as a
function of ${\rm M}_{\rm BH}$ for all the groups shown in Fig. 6 and listed
in Table 2. Black and red circles show the LF and FR best-fit results,
respectively. The black and red dashed lines indicate the LF and the FR mean
logarithmic PSD amplitudes respectively
($\overline{\log(PSD_{amp,LF})}=0.15\pm 0.06$ month-1, and
$\overline{\log(PSD_{amp,FR})}=0.01\pm 0.06$ month-1). These lines fit the
data well ($\chi^{2}=16.1/12$dof, and $\chi^{2}=14/12$dof).

Figure 7: Same as in Fig. 5, for the PSD amplitude. The black, red, and brown
points in the upper panel show the LF, best-fit log($PSD_{amp})$ values
plotted as a function of LX for AGN in the B′, C′ and D′ boxes in Fig. 6. The
bottom panel shows all the LF and the FR PSD amplitudes (listed in Table 2)
plotted as a function of ${\rm M}_{\rm BH}$ (black and red points,
respectively; the dashed lines indicate the respective mean PSD amplitudes).

We also investigated the possibility that the PSD amplitude may depend on
accretion rate. We therefore fitted the LF and the FR log($PSD_{amp}$) versus
log($\lambda_{\rm Edd}$) data with a straight line of the form:
log($PSD_{amp})[\log(\lambda_{\rm
Edd})]=a_{\rm\lambda_{Edd}}+b_{\rm\lambda_{Edd}}[{\rm log(\lambda_{Edd})+1]}$.
The best fit results are: $b_{LF,\rm\lambda_{Edd}}=-0.30\pm 0.12$, and
$b_{FR,\rm\lambda_{Edd}}=-0.26\pm 0.12$. The improvement in the goodness of
fit when we consider the straight line with a non zero slope is significant in
the case of the LF PSD amplitudes, and less so in the case of the FR
amplitudes: $\Delta\chi^{2}=7.3$ and 6.1 for one extra degree of freedom,
respectively ($p_{null}=0.007$ and 0.014). These results are indicative of an
anti-correlation between PSD amplitude and accretion rate (with the
variability amplitude decreasing with increasing $\lambda_{\rm Edd}$). This is
consistent with the dependence of the best-fit $c$ values on ${\rm M}_{\rm
BH}$ that we mentioned above.

Our results so far show that the X–ray power spectrum in the 14-195 keV band
is the same in all AGN: it has a power-law shape, with the same average slope
and amplitude, irrespective of their BH mass and luminosity. This result holds
irrespective of whether we consider the LF or the FR best-fit results. The
mean PSD slope when we consider the LR and the FR results is different at the
3.9$\sigma$ level:
$\overline{PSD_{slope,LF}}-\overline{PSD_{slope,FR}}=-0.27\pm 0.07$. There is
also a difference between the mean LF and FR PSD amplitude, but it is small
and not significant: $\overline{PSD_{amp,LF}}-\overline{PSD_{amp,FR}}=0.14\pm
0.09$. We also find evidence that the PSD amplitude decreases with increasing
$\lambda_{\rm Edd}$.

###  5.4 Normalised excess variance analysis.

We performed an excess variance analysis of the Swift /BAT light curves. The
variance of a light curve is equal to the integral of the power spectrum from
a low to a high frequency (e.g. Allevato et al., 2013). It can be measured
relatively easy, and can be used to verify the power spectrum results that we
report in the previous section. However, the variance of a single light curve
is a poor measurement of the intrinsic variance of the variable process as
Allevato et al. (2013) showed. We can nevertheless use the Swift /BAT light
curves of the AGN in our sample to test the PSD analysis results as we
describe below.

It is important that we compute the variance using light curves with the same
duration in the rest frame of each source. The highest redshift in our sample
is $z=0.105$. The observed 157 month light curve of this source is 142 months
long in its rest frame. We therefore removed points from all light curves
until that their rest-frame duration is 142 months (we removed points from the
last part of the light curves), and we computed the normalized, excess
variance, $\sigma^{2}_{NXV}$, using eq. (1) in Allevato et al. (2013). We note
that, since the PSD slope is $-1$ (or even flatter), we do need to correct the
variance estimates for bias (see Section 5 in Allevato et al. (2013)). The
normalized excess variance measurements for each source are listed in the
ninth column of Table LABEL:table:sampledata.

Figure 8 shows a plot of $\sigma^{2}_{NXV}$ as a function of log(MBH) for all
AGN plotted within the boxes drawn in Fig. 6. These are the objects we used to
study the PSD amplitude, therefore we can test the PSD results we mentioned
above using the $\sigma^{2}_{NXV}$ measurements plotted in this figure. First
we investigated whether the excess variance depends on ${\rm M}_{\rm BH}$, LX,
and/or $\lambda_{\rm Edd}$, by fitting the data with the lines:
$\sigma^{2}_{NXV}$=$a_{\rm BH}+b_{\rm BH}[\log($M${}_{\rm BH})-7.5]$,
$\sigma^{2}_{NXV}$=$a_{\rm Lx}+b_{\rm Lx}[\log($L${}_{\rm X})-43.5]$, and
$\sigma^{2}_{NXV}$=$a_{\lambda_{\rm Edd}}+b_{\lambda_{\rm
Edd}}[\log($$\lambda_{\rm Edd}$$+1]$. All the best-fit $b_{\rm BH},b_{\rm
Lx}$, and $b_{\lambda_{\rm Edd}}$ values are consistent with zero, which
indicates that $\sigma^{2}_{NXV}$ does not depend om ${\rm M}_{\rm BH}$, LX,
or $\lambda_{\rm Edd}$. This is what we would expect if the PSD is the same in
all AGN, irrespective of their BH mass, luminosity and/or accretion rate.

To test even further this hypothesis, we computed the mean excess variance of
all AGN plotted in Fig. 8. We find $\overline{\sigma^{2}_{NXV}}$=0.14$\pm
0.02$ (indicated by the the horizontal solid line in the same figure). If the
excess variance is the same in all AGN, then $\overline{\sigma^{2}_{NXV}}$
should be an unbiased estimate and its error should be a reliable estimate of
the true error of the mean according to Allevato et al. (2013) (see point 8 in
their Section 8). Using eq. (4) in Allevato et al. (2013), and the mean
$PSD_{amp}$ and $PSD_{slope}$ we reported in the previous sections, we can
predict the excess variance of light curves with T${}_{\rm max}=142$ days, and
T${}_{\rm min}=2\Delta t=(2$months)/(1+$\bar{z}$), where $\bar{z}=0.023$ is
the mean redshift of all AGN in the sample. The dashed lines in Fig. 8
indicate the expected excess variance in the case when we consider the LF PSD
results (i.e. $PSD_{slope}\sim-1$ and log$(PSD_{amp})\sim 0.15$), while the
dotted lines in the same figure indicate the expected excess variance in the
case when we consider the FR PSD results ($PSD_{slope}\sim-0.75$ and
log$(PSD_{amp})\sim 0.01$), taking into account the error on both
log($PSD_{amp})$ and $PSD_{slope}$. The model excess variance in both the LF
and the FR cases are fully consistent with the observed
$\overline{\sigma^{2}_{NXV}}$.

We note that the results from the fit of a line to the FR $PSD_{slope}$ vs
$\lambda_{\rm Edd}$ data in §5.2 suggested that $PSD_{slope}$ may depend on
$\lambda_{\rm Edd}$. If indeed $PSD_{slope}\propto 0.2$[log($\lambda_{\rm
Edd}$)+1], as we found, then we would expect the AGN excess variance to
increase with $\lambda_{\rm Edd}$ and $b_{\lambda_{\rm Edd},FR}$= 0.05. We
find $b_{\lambda_{\rm Edd},FR}=-0.06\pm 0.035$, i.e. the opposite trend. The
difference between the predicted and the observed $b_{\lambda_{\rm Edd},FR}$
is at the 3$\sigma$ level. We conclude that it is rather unlikely that the
intrinsic PSD slope depends on $\lambda_{\rm Edd}$. On the other hand, in the
last paragraph of the previous section we noticed a trend of decreasing PSD
amplitude with increasing $\lambda_{\rm Edd}$. Assuming the LF best-fit
results, we would predict that $\sigma^{2}_{NXV}$ should also decrease with
increasing $\lambda_{\rm Edd}$, with $b_{\lambda_{\rm Edd},SR}=-0.06$, exactly
as observed: $b_{\lambda_{\rm Edd},obs}=-0.06\pm 0.04$. This result further
suggests that the PSD amplitude indeed decreases with increasing accretion
rate.

Figure 8: Plot of $\sigma^{2}_{NXV}$ versus log(MBH), for the AGN in the
various boxes plotted in Fig. 6. The solid black line indicates the mean
excess variance, while the dashed-dotted and the dashed lines indicate the
expected variance in the case when the PSD in all AGN has a slope of $-1$ and
$-0.75$, respectively.

##  6 Summary and Discussion

We have used 157 month Swift /BAT light curves to compute the mean X-ray PSD
of the brightest AGN in the BASS survey, in various BH mass and X–ray
luminosity bins, which span two orders of magnitude in BH mass
($6.5\lesssim$log(MBH)$\lesssim 8.5$) and X–ray luminosity
($42.5\lesssim$log(LX)$\lesssim 44.5$; see Tables 1 and 2). As we already
mentioned, the AGN in our sample should be representative of the radio-quiet
AGN population in the local Universe. We fitted the average power spectra with
a power-law model of the form: $PSD(\nu)=PSD_{amp}(\nu/\nu_{0})^{PSD_{slope}}$
(where $\nu_{0}=0.01$ month-1). Our results are summarized below:

1) The mean X–ray PSD of AGN (in the 4-195 keV band) at low frequencies ($\sim
6.5\times 10^{-3}-0.5$ month-1) is consistent with the power-law model. We do
not detect periodicities or QPO-like features, neither do we detect any
“bending” frequencies within the sampled frequency range, where the PSD slope
would steepen (at higher frequencies) or flatten (at lower frequencies).

2) The X–ray power spectrum of Seyfert 1 and Seyfert II galaxies are
consistent with each other (within errors). This result shows that the X–ray
variability mechanism is consistent with being the same in both classes of
AGN. This is in agreement with the AGN unification model.

3) The mean log($PSD_{amp}$) (which is defined as the PSD value at
$\nu_{0}=0.01$/month) is $0.15\pm 0.06$($0.01\pm 0.07$) month-1, based on the
LF(FR) PSD best-fit results. We expect a distribution of log($PSD_{amp}$)
values in AGN with a given BH mass and X–ray luminosity. Our results show that
the mean of the logarithm of the PSD amplitude distributions should be $\sim
0.015$month-1, the same in all AGN, irrespective of their ${\rm M}_{\rm BH}$
and/or LX.

4) We find significant evidence that the PSD amplitude of the individual AGN
should decrease with increasing accretion rate as follows:
$PSD_{amp}(\lambda_{\rm Edd})\propto(\lambda_{\rm Edd}/0.1)^{-0.3}$. This
result implies that the variability amplitude in AGN should decrease with
increasing accretion rate.

5) Just like with the PSD amplitude, our results indicate that the mean PSD
slope should also be the same in all AGN, irrespective of their BH mass or
luminosity. The average PSD slope is $-0.98\pm 0.06$, when fitting only the LF
PSD, and becomes $-0.74\pm 0.04$ when we fit the full range PSDs. We believe
that the slope flattening when we consider high frequencies as well may be due
(mainly) to aliasing effects. We also detect a significant dependence of the
PSD slope on $\lambda_{\rm Edd}$, when we consider the full range results. The
power spectrum slope appears to flatten with increasing $\lambda_{\rm Edd}$.
As we discuss below, this result is also due to aliasing effects, and does not
imply an intrinsic dependence of the the PSD slope on accretion rate.

6) The normalized excess variance, $\sigma^{2}_{NXV}$, does not depend on BH
mass or LX. This result is consistent with the PSD results. If the average
X–ray PSD is the same in all AGN, then its integral, i.e. $\sigma^{2}_{NXV}$,
should be independent of the AGN parameters, as we observe. The mean
$\sigma^{2}_{NXV}$ is consistent with the expected excess variance either when
we assume a PSD slope of $-1$ or $-0.75$.

###  6.1 The universal shape of the mean X–ray PSD of AGN

Our results suggest that, on average, the X–ray PSD (in the 14-195 keV band)
should be the same in all AGN. If that is the case, then we can compute the
mean AGN PSD if we use the periodograms of all the variable sources in our
sample. To this end, we computed the mean PSD of all variable AGN in our
sample (corrected from the Poisson noise) as explained in §4. The resulting
mean PSD is shown in Fig. 9. We have smoothed the high frequency PSD points
(above 0.1 month-1) as explained in the same section.

Figure 9: The mean PSD of the variable AGN in our sample. The solid line shows
the best-fit power-law model to the data up to $10^{-7.4}$ Hz, and the dashed
line shows the same model with aliasing effects included (see §6.1 for
details).

The mean PSD is fully consistent with a simple power-law model. The solid line
in this figure shows the best-fit model when we fit the data below $10^{-7.4}$
Hz (=0.1/month) with a power-law model defined by eq. 3. This line provides a
very good fit to the LF part of the PSD ($\chi^{2}=10.6/13$dof) with a best
fit slope of $-1.00\pm 0.06$. The PSD amplitude is larger than the mean
amplitude we report in §5.3, but that is because here we consider only the
variable AGN (in order to determine the PSD slope correctly).

The fact that a $-1$ power-law model fits so well the LF part of the mean PSD
strongly supports the hypothesis that there is no PSD flattening even at
frequencies as low as $\sim 3\times 10^{-9}$ Hz. For example, if there was a
low-frequency flattening for low mass BHs in our sample, we would expect the
lowest-frequency points in the mean PSD to show a much larger scatter, when
compared to the higher frequency points, but that is not the case. We believe
that our results clearly show that the average AGN PSD follows a power-law
shape with a slope of $-1$ down to frequencies as low as a few $10^{-9}$ Hz
(this frequency corresponds to a time scale of $\sim 12$ years).

The mean PSD flattens at frequencies higher than $\sim 4\times 10^{-8}$ Hz. We
do not believe that this is due to a systematic bias in the computation of the
Poisson noise level (see Fig. 1 and the discussion in §4.1). As we discussed
in §4.2, the high frequency flattening could be due to aliasing effects. The
dashed line in Fig. 9 shows the best-fit power-law model (the one plotted with
the solid line) when we also include the aliasing effects in the case when the
PL extends to $2\times 10{-6}$ Hz, and the light curve is evenly sampled. The
frequency of $2\times 10{-6}$ Hz indicates the high frequency break in the
2–10 keV PSD of an AGN with log(MBH)=7.5, according to the model defined by
eq. (4) in González-Martín & Vaughan (2012). The agreement between the dashed
line and the mean PSD at high frequencies is quite good. This result supports
the fact that the PSD flattening we detect at high frequencies is probably due
to aliasing effects.

Support to the hypothesis of aliasing affecting the measured PSD slope when we
consider the full frequency range is also provided by the apparent dependence
of the PSD slope on $\lambda_{\rm Edd}$, when we consider the FR results (see
§5.2). According to McHardy et al. (2006), the PSD bending frequency should
increase with increasing $\lambda_{\rm Edd}$. In this case, we would also
expect the aliasing effects to become stronger, hence resulting into a
flattening of the PSD slope with increasing $\lambda_{\rm Edd}$, as observed.

The hard X–ray PSD should steepen above a bending frequency, $\nu_{b}$, just
like in the 2–10 keV band. The fact that such a bending must exist, and is
probably located at a frequency similar to the high-frequency break in the
2-10 keV PSDs is supported by the agreement of the dashed line and the mean
PSD in Fig. 9. If the intrinsic hard X–ray PSD would extend to very high
frequencies without bending to a slope steeper than $-1$ then, irrespective of
the sampling pattern of the Swift /BAT light curves, the aliasing effects
would be much stronger, and the high frequency flattening in Fig. 9 would be
much more pronounced. If the bending frequency does not vary significantly
with energy, we would then expect $\nu_{b}\sim 470-4.7$ month-1 (i.e $\sim
1.8\times 10^{-4}-1.8\times 10^{-6}$ Hz) for AGN with log(MBH)=6.5-8.5 (which
is the majority of the AGN in our sample). These frequencies cannot be sampled
with the current Swift /BAT light curves. NuSTAR light curves would be
necessary in order to detect and investigate the bending frequency properties
at hard X-rays.

If the bending frequency does not depend significantly on energy, this would
imply that the hard X–ray PSD in AGN extends with a slope of $-1$ over $\sim
2.8-4.8$ decades for AGN with log(MBH)=8.5-6.5, respectively. This is a large
frequency range and, at least for AGN with BH masses smaller than a few 107M⊙,
this result would have interesting implications. For example, the Ark 564 PSD
shows a significant low-frequency break to slope zero at $\sim 10^{-6}$ Hz,
$\sim 3$ orders of magnitude below the high-frequency break (McHardy et al.,
2007; Papadakis et al., 2002). This is the only AGN where such a low frequency
flattening has been detected in its X–ray PSD. It is quite possible then that
such PSDs are not very common in AGN. As we argued above, if this was common,
we would expect the low frequency points in the PSD plotted in Fig. 9 to show
evidence of a low frequency flattening and to be very noisy.

###  6.2 Comparison with past results

Table 3 in SM13 lists their best-fit log($PSD_{amp}$) and $PSD_{slope}$
results when they fit an unbroken power-law model to the data. We fit the same
model to the Swift /BAT power spectra. According to the SM13 results, the PSD
at $\nu_{0}=0.01$ month-1 should be given by
$PSD_{SM13}(\nu_{0})=A_{SM13}(\nu_{0}/\nu_{SM13})^{-PSDslope,SM13}$, where
$\nu_{SM13}=10^{-6}$ Hz, and it should be consistent with the amplitude of our
PSD model. Using the SM13 best-fit results, we found that
$\overline{\log[PSD_{SM13}(\nu_{0})]}=-0.13\pm 0.10$, which is entirely
consistent with our LF, mean log$(PSD_{amp})$ measurement.

Furthermore, SM13 find a mean PSD slope of $-0.78\pm 0.29$, which becomes
$-0.92\pm 0.22$ when they restrict the average to only the 16 best-fit slopes
that were fully constrained (out of the 30 sources in their sample). The SM13
results are very similar to ours. We find a mean slope of $\sim-0.75$ when we
fit the full-range PSDs, and a steeper slope of $\sim-1$ when we consider only
the LF part of the power spectrum. As we mentioned above, we believe that the
$-1$ value is probably more representative of the intrinsic PSD slope, mainly
because aliasing effects may flatten the PSD at high frequencies. Such effects
should not be important in the SM13 power spectra because they used light
curves with a bin size of 1 day. Since the power at frequencies higher than
$\sim 1/$day should be significantly smaller than the power at low
frequencies, aliasing effects should not be important in this case. On the
other hand, as SM13 comment, their best-fit PSD slopes are well constrained
only when the power spectra are well above the Poisson noise. The fact that
the mean slope in this case is closer to our mean in the case of the LF best-
fit results reinforces our belief that $-1$ is probably closer to the
intrinsic PSD slope of the AGN X–ray PSD at low frequencies.

###  6.3 Comparison with the 2-10 keV band PSD results

As we have already mentioned in the Introduction, many works in the past have
shown that the low frequency PSD of AGN at energies below 10 keV has a slope
of $-1$. As we discussed in §6.1, this is probably the case of the low
frequency, hard X–ray PSD in AGN as well. Therefore, our results suggest that
the slope of the low-frequency PSD in the AGN does not depend on energy. This
is opposite to what is observed at high frequencies in the 2–10 keV band PSD,
where the slope above the high frequency break frequency appears to flatten
with energy.

Our results are in agreement with the results of Paolillo et al. (2023), who
find that the 2–10 keV PSD has the same shape in all AGN, over a large range
of BH mass and luminosity. Both in the 2–10 and in the 14-195 keV bands the
universal PSD slope at low frequencies is consistent with $-1$. On the other
hand, Paolillo et al. (2023) find that
$PSD_{2-10keV}(\nu_{b})\times\nu_{b}=0.008\pm 0.002$, which is smaller than
our estimate of $0.014\pm 0.003$. This could imply that the PSD amplitude is
larger at higher energies, although the difference is not statistically
significant.

Excess variance studies in the 2–10 keV band indicate that
$PSD(\nu_{b})\times\nu_{b}\propto$$\lambda_{\rm Edd}$-0.8 (Ponti et al., 2012;
Paolillo et al., 2017). If this holds in the 14-195 keV band as well, it would
imply a similar dependence of $PSD_{amp}$ on $\lambda_{\rm Edd}$. We do detect
a significant anti-correlation between PSD amplitude and $\lambda_{\rm Edd}$
in our data (see §5.3). We find that $PSD_{amp}\propto$($\lambda_{\rm
Edd}$)-0.30±0.12, which is flatter than the anti-correlation which the 2–10
keV data imply (although the 2–10 keV relation is not well defined, with no
error given on the slope). Perhaps the PSD amplitude–$\lambda_{\rm Edd}$ anti-
correlation at energies higher than 15 keV is not as steep as the one at lower
energies, although a more detailed study of this effect in the 2-10 keV band
is necessary to understand if there are differences between the two bands.

###  6.4 Comparison with the GBH PSD

Both AGN and GBHs are power by accretion of matter on BHs, a major difference
being the mass of the BH at their center. It has long been suggested that the
X–ray variability mechanism is the same in both systems. However, the PSD of
GBHs depends on the “state” of the system. There are two main states, the hard
and the soft, in GBHs. The PSD in the soft state is usually well described by
a single bending power-law, with a slope equal/steeper than $-2$ above
$\nu_{b}$, and a slope of $-1$ extending to very low frequencies. In the hard
state, the PSD is more complex, and is usually modeled as a mixture of zero-
centred Lorentzian components. It is true that similar power spectra from the
hardest part of the hard state show similar shape to the soft state PSDs (e.g.
Heil et al., 2015), however, in general, black hole X-ray binary hard states
show power spectra with slopes $\sim-1$ over at least 1-2 decades in
frequency, which bend to a constant, i.e. to a slope of zero, at lower
frequencies (e.g. Remillard & McClintock, 2006). Based on the shape of the AGN
PSDs (e.g. McHardy et al. (2004); Markowitz et al. (2003); Uttley & McHardy
(2005); see also Paolillo et al. (2023)), as well as the scaling of $-1$ to
$-2$ bending frequencies between AGN and GBHs (McHardy et al., 2006), it has
been argued that almost all of the X–ray bright AGN should be the analogue of
the GBHs in their soft state. SM13 reached the same conclusion because they
did not find any evidence for a low frequency break to a slope of zero in the
AGN Swift /BAT PSDs they studied.

Figure 10: The grey lines show the Cyg X–1 PSDs in its canonical soft state
according to Axelsson et al. (2006). The solid red line indicates their mean,
while the red dashed and dotted lines show its extension, and 1$\sigma$
scatter, respectively, to lower frequencies. The solid black and brown lines
indicate the mean AGN PSD when we consider our LF and FR best-fit results,
respectively (dashed lines indicate the 1$\sigma$ scatter due to the
$PSD_{amp}$ error).

We compared our results with results from the PSD analysis of Cyg X-1 light
curves when the source is in the soft state. Such a comparison between the
timing properties of AGN and Cyg X-1 in its soft state has been done many
times in the past (e.g. Uttley et al., 2002; McHardy et al., 2006). The solid,
grey lines in Fig. 10 show the Cyg X-1 PSDs (in $PSD(\nu)\times\nu$
representation), when the source is in its soft state. The data are taken from
Axelsson et al. (2006), who analyzed all the the archival Cyg X–1 data from
the Rossi X-ray Timing Explorer (RXTE) from 1996 to 2003 in the 2–9 keV band,
and in the frequency range of 0.01–25 Hz. They used various models to fit the
PSDs, and one of them was a cut-off power-law (their eq. 2). The grey lines
show the best-fit models, when the PSDs were fitted only by the cut-off power
law. According to the authors this model fits well the PSD only when the
source is in its canonical soft state. Figure 10 shows that the amplitude of
the Cyg X-1 PSD in the soft state varies by a factor $\sim 15$ or so. The PSD
amplitude probably varies among AGN as well, as we have discussed above, but
what is interesting is to compare the mean AGN PSD with the mean Cyg X-1 PSD
in its soft state.

The solid red line shows the mean Cyg X-1 PSD. The dashed line shows its
extension to lower frequencies, while the dotted lines indicate the 1$\sigma$
spread of the PSD normalization around its mean. The solid black line shows
the mean best-fit PSD of the AGN our sample. The frequency range extends to
$10^{-6}$ Hz, as SM13 showed the the AGN PSDs extend as a power-law up to that
frequency. The dashed lines indicates the $1\sigma$ uncertainty of the best-
fit to the mean $PSD_{amp}$. Clearly, the mean AGN PSD agrees very well with
the Cyg X-1 PSD in its soft state. The difference in BH mass between AGN in
our sample and Cyg X-1 is $\sim 10^{6}-10^{8}$, while the difference in
frequency between the Cyg X-1 and AGN PSD is over 4–5 orders of magnitude.
Despite these differences the PSD slopes are identical, and the mean PSD
amplitude differs only by a factor of 3, maximum. We suspect that this
difference is probably due to the difference in energy bands (2–9 KeV for Cyg
X-1 as opposed to 14-195 keV for AGN). We plan to investigate this issue
further in the near future.

We note, that the agreement between the mean AGN PSD and the Cyg X-1 PSD in
its soft state is quite plausible but, strictly speaking, suggestive at this
point. For example, hard state low-frequency breaks can be seen from 0.01 Hz
and higher, which would be 10-9 Hz or higher for a 108M⊙ mass black hole,
assuming linear mass-scaling. As we discussed in §6.1, if that were the case
we would have detected it in the mean AGN PSD. However, we cannot be certain
about the location of the high frequency breaks in these objects (at the hard
X-ray PSD). If the mean AGN PSD were to be similar to the GBHs PSD in the hard
state, then the $-1$ to $-2$ breaks in AGN at energies higher than 15 keV must
be located at frequencies much lower than the ones we have detected at
energies below 10 keV.

If, on the other hand, the mean AGN PSD is like the mean Cyg X-1 in its soft
state, then it is possible that the physical properties are similar in both
systems. It is generally believed that the accretion disc extends to its
innermost radius, and the X-ray corona is less luminous than the disc in GBHs
when in their soft state. This could be then the case in AGN as well. The disc
should extend to its innermost radius and its emission would dominate the
output in these objects as well (as is indeed the case with the “big blue
bump” that is observed in these objects). Given the inner disc radius, the
corona cannot be located within the disc, and it has to be located on top of
it (or above the central BH), illuminating the disc and causing all the X–ray
reflection features (i.e. soft excess, broad iron line and Compton hump) as
well as the X–ray reverberation phenomena (like e.g. driving the UV/optical
variability and producing the delays between X–rays and UV/optical
variations).

###### Acknowledgements.

We would like to thank the referee for their comments which helped us improve
the original version of the paper.

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## Appendix A

Table 3:  AGN in the sample and their properties. Source | DR2 |  $F^{obs}_{14-195{\rm keV}}$(b)(b)footnotemark: $(b)$ |  $z$(a)(a)footnotemark: $(a)$ | log(${\rm M}_{\rm BH}$)(a)(a)footnotemark: $(a)$ | log(LX)(c)(c)footnotemark: $(c)$ |  $\chi^{2}$/dof(d)(d)footnotemark: $(d)$ | V/NV(e)(e)footnotemark: $(e)$ |  $\sigma^{2}_{NXV}$(f)(f)footnotemark: $(f)$
---|---|---|---|---|---|---|---|---
| Type(a)(a)footnotemark: $(a)$ |  |  |  | (erg/s) |  |  |
CenA | Sy2 | 1346.34 | 0.0019 | 7.77 | 42.33 | 80640.65/156 | V | 0.165
NGC4151 | Sy1 | 618.88 | 0.0032 | 7.56 | 43.30 | 25562.30/156 | V | 0.092
NGC2110 | Sy2 | 328.93 | 0.0075 | 8.78 | 43.60 | 7188.11/156 | V | 0.221
NGC4945 | Sy2 | 282.11 | 0.0023 | 6.15 | 42.33 | 2129.02/156 | V | 0.112
NGC4388 | Sy2 | 278.91 | 0.0083 | 6.94 | 43.05 | 5008.13/155 | V | 0.237
CircinusGalaxy | Sy2 | 273.17 | 0.0015 | 6.23 | 42.06 | 525.07/156 | V | 0.010
IC4329A | Sy1 | 263.25 | 0.0160 | 7.65 | 44.16 | 1371.12/156 | V | 0.078
NGC5506 | Sy1.9 | 239.40 | 0.0060 | 7.24 | 43.24 | 597.65/153 | V | 0.029
4C+50.55 | Sy1.9 | 210.38 | 0.0154 | 7.97 | 44.01 | 1405.12/156 | V | 0.054
MCG-5-23-16 | Sy1.9 | 209.62 | 0.0084 | 7.65 | 43.44 | 827.79/156 | V | 0.035
NGC4507 | Sy2 | 184.54 | 0.0117 | 7.81 | 43.90 | 993.72/156 | V | 0.103
NGC3783 | Sy1 | 173.84 | 0.0090 | 7.37 | 43.45 | 736.34/156 | V | 0.060
NGC7172 | Sy2 | 160.02 | 0.0085 | 8.15 | 43.18 | 875.63/152 | V | 0.074
Mrk3 | Sy1.9 | 150.12 | 0.0138 | 8.96 | 44.02 | 1343.69/156 | V | 0.115
CygnusA | Sy2 | 145.23 | 0.0565 | 9.43 | 44.97 | 947.41/156 | V | 0.080
Mrk348 | Sy1.9 | 144.81 | 0.0147 | 6.8 | 43.83 | 1357.28/156 | V | 0.122
UGC3374 | Sy1 | 141.21 | 0.0202 | 6.61 | 44.03 | 1260.05/153 | V | 0.212
IRAS05078+1626 | Sy1 | 119.80 | 0.0174 | 6.88 | 43.74 | 994.33/151 | V | 0.286
1RXSJ173728.0-290759 | Sy1.9 | 119.04 | 0.0218 | 7.84 | 44.05 | 719.70/145 | V | 0.112
3C111 | Sy1 | 118.67 | 0.0497 | 8.45 | 44.79 | 692.25/152 | V | 0.144
NGC3227 | Sy1 | 112.47 | 0.0033 | 6.77 | 42.77 | 648.91/153 | V | 0.071
NGC3516 | Sy1 | 112.42 | 0.0087 | 7.39 | 43.26 | 1561.91/156 | V | 0.224
4U1344-60 | Sy1.9 | 111.79 | 0.0128 | 9.09 | 43.52 | 211.47/156 | V | -0.002
Mrk926 | Sy1 | 110.22 | 0.0477 | 7.98 | 44.72 | 605.22/155 | V | 0.139
ESO103-35 | Sy1.9 | 106.86 | 0.0135 | 7.37 | 43.64 | 553.05/156 | V | 0.093
NGC5252 | Sy2 | 103.15 | 0.0230 | 9.0 | 44.05 | 1352.73/155 | V | 0.426
3C390.3 | Sy1 | 102.87 | 0.0556 | 8.64 | 44.82 | 772.56/156 | V | 0.086
Mrk509 | Sy1 | 100.14 | 0.0347 | 8.05 | 44.37 | 526.51/153 | V | 0.150
MR2251-178 | Sy1 | 99.53 | 0.0645 | 8.19 | 44.95 | 370.15/153 | V | 0.103
NGC6300 | Sy2 | 96.37 | 0.0031 | 6.77 | 42.29 | 701.76/156 | V | 0.117
3C120 | Sy1 | 95.38 | 0.0331 | 7.74 | 44.32 | 697.27/155 | V | 0.179
ESO506-27 | Sy2 | 90.68 | 0.0242 | 8.84 | 44.17 | 1038.69/154 | V | 0.672
NGC4593 | Sy1 | 88.30 | 0.0083 | 6.88 | 43.10 | 233.36/153 | V | 0.004
NGC5548 | Sy1 | 86.47 | 0.0167 | 7.72 | 43.62 | 1129.77/156 | V | 0.195
NGC5728 | Sy1.9 | 84.74 | 0.0103 | 8.25 | 43.24 | 611.12/154 | V | 0.341
NGC1275 | Sy1.9 | 82.64 | 0.0168 | 8.6 | 43.65 | 649.27/156 | V | 0.089
3C382 | Sy1 | 82.33 | 0.0579 | 8.01 | 44.72 | 473.91/156 | V | 0.097
NGC7582 | Sy2 | 82.28 | 0.0052 | 7.74 | 43.24 | 523.49/155 | V | 0.148
NGC3081 | Sy2 | 81.89 | 0.0081 | 7.67 | 43.18 | 526.76/156 | V | 0.243
NGC3281 | Sy2 | 81.21 | 0.0111 | 8.22 | 43.46 | 479.64/156 | V | 0.108
NGC788 | Sy2 | 77.51 | 0.0137 | 8.18 | 43.61 | 585.38/156 | V | 0.276
NGC1142 | Sy2 | 74.31 | 0.0287 | 8.96 | 44.31 | 730.35/156 | V | 0.451
Ark120 | Sy1 | 74.29 | 0.0326 | 8.07 | 44.16 | 176.15/156 | NV | -0.022
NGC526A | Sy1.9 | 73.91 | 0.0189 | 8.06 | 43.63 | 699.07/156 | V | 0.140
NGC7469 | Sy1 | 70.63 | 0.0160 | 6.96 | 43.51 | 182.87/155 | NV | -0.033
IC5063 | Sy2 | 67.76 | 0.0113 | 8.24 | 43.29 | 545.68/156 | V | 0.353
ESO297-18 | Sy2 | 67.37 | 0.0252 | 8.5 | 44.10 | 219.67/156 | V | 0.025
NGC6814 | Sy1 | 64.47 | 0.0058 | 7.04 | 42.62 | 777.90/150 | V | 0.439
NGC1365 | Sy1 | 63.52 | 0.0051 | 6.6 | 42.39 | 322.18/156 | V | 0.057
IRAS05589+2828 | Sy1 | 62.28 | 0.0331 | 8.0 | 44.14 | 836.57/147 | V | 0.578
Mrk1040 | Sy1 | 62.22 | 0.0163 | 7.41 | 43.51 | 193.46/154 | NV | 0.019
Mrk110 | Sy1 | 60.95 | 0.0352 | 7.29 | 44.16 | 214.51/156 | V | 0.024
MCG-6-30-15 | Sy1 | 59.53 | 0.0071 | 6.6 | 42.78 | 174.65/155 | NV | -0.010
ESO141-55 | Sy1 | 58.77 | 0.0371 | 7.99 | 44.18 | 174.08/156 | NV | -0.028
LEDA178130 | Sy2 | 58.27 | 0.0352 | 7.88 | 44.17 | 311.22/156 | V | 0.128
NGC7314 | Sy1.9 | 57.42 | 0.0046 | 6.3 | 42.16 | 211.06/153 | V | 0.035
Mrk1210 | Sy1.9 | 57.40 | 0.0136 | 6.86 | 43.35 | 198.27/155 | NV | 0.055
Mrk6 | Sy1 | 56.70 | 0.0190 | 8.1 | 43.62 | 586.66/156 | V | 0.274
LEDA138501 | Sy1 | 55.63 | 0.0497 | 8.01 | 44.42 | 234.89/156 | V | 0.051
NGC4992 | Sy2 | 54.42 | 0.0252 | 8.59 | 43.93 | 195.39/155 | NV | -0.012
LEDA168563 | Sy1 | 54.10 | 0.0283 | 7.88 | 43.98 | 273.67/156 | V | 0.077
NGC7603 | Sy1 | 52.96 | 0.0287 | 8.59 | 43.92 | 129.26/150 | NV | -0.119
NGC235A | Sy1.9 | 52.14 | 0.0221 | 8.49 | 43.71 | 180.41/156 | NV | -0.001
NGC6860 | Sy1.9 | 51.52 | 0.0147 | 7.71 | 43.33 | 338.31/156 | V | 0.167
4C+74.26 | Sy1 | 51.18 | 0.1050 | 9.83 | 45.10 | 183.93/156 | NV | 0.025
2MASSJ07594181-3843560 | Sy1 | 51.15 | 0.0400 | 8.82 | 44.21 | 180.72/156 | NV | 0.125
LEDA38038 | Sy2 | 49.46 | 0.0277 | 8.11 | 43.86 | 184.09/156 | NV | 0.015
LEDA86269 | Sy2 | 49.38 | 0.0105 | 7.98 | 43.02 | 484.86/152 | V | 0.496
NGC612 | Sy2 | 48.97 | 0.0299 | 8.28 | 44.26 | 322.18/156 | V | 0.210
ESO362-18 | Sy1 | 48.89 | 0.0125 | 7.11 | 43.17 | 216.20/156 | V | 0.046
NGC973 | Sy2 | 48.41 | 0.0157 | 8.48 | 43.18 | 1045.03/156 | V | 0.540
Fairall9 | Sy1 | 48.39 | 0.0459 | 8.3 | 44.34 | 172.59/156 | NV | 0.007
HE1143-1810 | Sy1 | 48.28 | 0.0326 | 7.39 | 44.05 | 215.05/156 | V | 0.074
MCG-1-24-12 | Sy2 | 46.38 | 0.0196 | 7.66 | 43.50 | 213.07/156 | V | 0.157
LEDA166252 | Sy1 | 44.90 | 0.0249 | 8.79 | 43.76 | 302.68/156 | V | 0.204
1RXSJ165605.6-520345 | Sy1 | 43.23 | 0.0538 | 8.33 | 44.44 | 177.28/156 | NV | 0.142
Mrk1498 | Sy1 | 42.97 | 0.0558 | 7.19 | 44.48 | 493.22/156 | V | 0.321
Mrk79 | Sy1 | 42.72 | 0.0221 | 7.61 | 43.64 | 457.84/156 | V | 0.265
NGC4051 | Sy1 | 42.49 | 0.0020 | 6.13 | 41.69 | 169.25/156 | NV | 0.002
2MASXiJ1802473-145454 | Sy1 | 42.36 | 0.0343 | 7.98 | 43.97 | 157.24/152 | NV | 0.017
1RXSJ174155.3-121157 | Sy1 | 41.54 | 0.0376 | 8.03 | 44.13 | 168.96/154 | NV | 0.042
ESO548-81 | Sy1 | 41.21 | 0.0144 | 7.96 | 43.26 | 337.36/156 | V | 0.253
3C445 | Sy1 | 39.82 | 0.0561 | 7.89 | 44.56 | 150.83/153 | NV | -0.016
ESO490-26 | Sy1 | 39.49 | 0.0252 | 7.15 | 43.67 | 224.91/155 | V | 0.087
IGRJ21277+5656 | Sy1 | 39.43 | 0.0149 | 7.15 | 43.21 | 172.84/156 | NV | -0.009
ESO511-30 | Sy1 | 39.27 | 0.0229 | 7.23 | 43.62 | 202.05/155 | V | -0.014
Fairall272 | Sy2 | 39.15 | 0.0222 | 8.11 | 43.73 | 223.64/155 | V | 0.170
IC4709 | Sy2 | 39.08 | 0.0169 | 7.81 | 43.40 | 171.50/156 | NV | -0.096
NGC7213 | Sy1 | 39.04 | 0.0048 | 7.13 | 42.32 | 164.71/155 | NV | 0.105
NGC4235 | Sy1 | 38.62 | 0.0079 | 7.28 | 42.36 | 443.00/153 | V | 0.474
NGC1068 | Sy1.9 | 37.90 | 0.0035 | 6.93 | 42.65 | 182.20/156 | NV | 0.040
PictorA | Sy1 | 37.32 | 0.0350 | 6.77 | 43.98 | 264.56/156 | V | 0.165
Mrk704 | Sy1 | 36.84 | 0.0295 | 8.24 | 43.76 | 169.00/151 | NV | 0.046
NGC3079 | Sy2 | 36.74 | 0.0035 | 6.38 | 42.70 | 203.91/156 | V | 0.030
Fairall51 | Sy1 | 36.51 | 0.0139 | 7.11 | 43.18 | 181.93/156 | NV | 0.040
NGC7319 | Sy2 | 36.45 | 0.0226 | 8.1 | 43.71 | 209.13/156 | V | 0.313
NGC1194 | Sy2 | 36.22 | 0.0136 | 7.83 | 43.62 | 154.44/156 | NV | -0.053
CTS109 | Sy1 | 35.80 | 0.0300 | 7.84 | 43.89 | 228.57/153 | V | 0.050
Z121-75 | Sy1 | 35.66 | 0.0331 | 7.27 | 43.94 | 177.16/152 | NV | -0.011
LEDA170194 | Sy2 | 35.33 | 0.0363 | 8.35 | 44.01 | 221.19/153 | V | 0.358

  * •

Notes. $a$$a$footnotetext: Taken from Koss et al. (2022). They classified the
unbeamed AGN in three categories, Sy1, Sy1.9 and Sy2, based on the presence
of: 1) broad H$\beta$ lines, 2) narrow H$\beta$ but broad H$\alpha$, and 3)
only narrow optical lines in their optical spectra, respectively.
$b$$b$footnotetext: Taken from the Swift-BAT 157 month survey (in units of
$10^{-12}$ ergs/s/cm2). $c$$c$footnotetext: Logarithm of the intrinsic X–ray
luminosity in the 14-195 keV band, computed as explained in §2.
$d$$d$footnotetext: The $\chi^{2}$, and the number of degrees of freedom (dof)
when we fit a constant line to the light curves. $e$$e$footnotetext:
Classification of the variable (V) and non-variable (NV) sources, depending on
$p_{null}$ (see §2 for details). $f$$f$footnotetext: The normalized excess
variance, $\sigma^{2}_{NXV}$, computed as explained in §5.4.