# BASS-XL: X-ray variability properties of unobscured Active Galactic Nuclei

Alessia Tortosa 1,2, Claudio Ricci 1,3,4, Patricia Arévalo5, Michael J. Koss6,
Franz E. Bauer7,8,9,10, Benny Trakhtenbrot11, Richard Mushotzky12,13, Matthew
J. Temple1, Federica Ricci14,2, Alejandra Rojas Lilayu 1,15, Taiki Kawamuro16,
Turgay Caglar 17, Tingting Liu18, Fiona Harrison19, Kyuseok Oh20,21, Meredith
Clark Powell22, Daniel Stern23, Claudia Megan Urry24.
1Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile
2 INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio
Catone (RM), Italy.
3Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China
4George Mason University, Department of Physics & Astronomy, MS 3F3, 4400
University Drive, Fairfax, VA 22030, USA
5 Instituto de Fıısica y Astronomııa, Facultad de Ciencias, Universidad de
Valparaíso, Gran Bretana N 1111, Playa Ancha, Valparaíso, Chile.
6 Eureka Scientific, 2452 Delmer Street Suite 100, Oakland, CA 94602-3017,
USA.
7 Instituto de Astrofísica, Facultad de Física, Pontificia Universidad
Católica de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860, Santiago,
Chile, 7820436.
8 Centro de Astroingeniería, Facultad de Física, Pontificia Universidad
Católica de Chile, Campus San Joaquín, Av. Vicuña Mackenna 4860,
Santiago, Chile, 7820436.
9 Millennium Institute of Astrophysics, Nuncio Monseñor Sótero Sanz 100, Of
104, Providencia, Santiago, Chile.
10 Space Science Institute, 4750 Walnut Street, Suite 205, Boulder, Colorado
80301, USA.
11 School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978,
Israel.
12 Department of Astronomy, University of Maryland, College Park, MD 20742,
USA.
13 Joint Space-Science Institute, University of Maryland, College Park, MD
20742, USA.
14 Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, via
della Vasca Navale 84, 00146 Roma, Italy.
15 Centro de Astronomía (CITEVA), Universidad de Antofagasta, Avenida Angamos
601, Antofagasta, Chile.
16 RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama
351-0198, Japan.
17 Department of Physics, Southern Methodist University, 3215 Daniel Ave.,
Dallas, TX 75205, USA.
18 Department of Physics and Astronomy, West Virginia University, P.O. Box
6315, Morgantown, WV 26506, USA.
19 Cahill Center for Astronomy and Astrophysics, California Institute of
Technology, Pasadena, CA 91125, USA.
20 Korea Astronomy & Space Science institute, 776, Daedeokdae-ro, Yuseong-gu,
Daejeon 34055, Republic of Korea.
21 Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-
ku, Kyoto 606-8502, Japan.
22 Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, 452 Lomita Mall, Stanford, CA 94305, USA.
23 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak
Grove Drive, MS 169-224, Pasadena, CA 91109, USA.
24 Yale Center for Astronomy & Astrophysics and Department of Physics, Yale
University, P.O. Box 208120, New Haven, CT 06520-8120, USA.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We investigate the X-ray variability properties of Seyfert 1 Galaxies
belonging to the BAT AGN Spectroscopic Survey (BASS). The sample includes 151
unobscured (N${}_{\rm H}<10^{22}$ cm-2) AGNs observed with XMM-Newton for a
total exposure time of $\sim 27$ Ms, representing the deepest variability
study done so far with high signal-to-noise XMM-Newton observations, almost
doubling the number of observations analysed in previous works. We constrain
the relation between the normalised excess variance and the $2-10$ keV AGN
luminosities, black hole masses and Eddington ratios. We find a highly
significant correlation between $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ , with
a scatter of $\sim 0.85$ dex. For sources with high $L_{\rm 2-10}$ this
correlation has a lower normalization, confirming that more luminous (higher
mass) AGNs show less variability. We explored the $\sigma_{\rm NXS}^{2}$ vs
$M_{\rm BH}$ relation for the sub-sample of sources with $M_{\rm BH}$
estimated via the "reverberation mapping" technique, finding a tighter anti-
correlation, with a scatter of $\sim 0.65$ dex. We examine how the
$\sigma_{\rm NXS}^{2}$ changes with energy by studying the relation between
the variability in the hard ($3-10$ keV) and the soft ($0.2-1$ keV)/medium
($1-3$ keV) energy bands, finding that the spectral components dominating the
hard energy band are more variable than the spectral components dominating in
softer energy bands, on timescales shorter than 10 ks.

###### keywords:

Supermassive Black Hole – Active galaxies – Seyfert galaxies – X-rays

††pubyear: 2023††pagerange: BASS-XL: X-ray variability properties of
unobscured Active Galactic Nuclei–B

##  1 Introduction

Supermassive Black Holes (SMBHs, $M_{\rm BH}>10^{6}M_{\odot}$) are
ubiquitously found at the center of massive galaxies. Mass accretion onto
SMBHs is the mechanism that powers Active Galactic Nuclei (AGNs, Salpeter
1964) which are very powerful sources of X-ray radiation, emitting through the
entire electromagnetic spectrum. Variability is a distinctive feature shared
by all classes of AGN, occurring over a wide range of timescales and
amplitudes across all the wavelengths (e.g., Ulrich et al., 1997; McHardy et
al., 2004). These flux variations can also be accompanied by prominent
spectroscopic changes (e.g., Ricci & Trakhtenbrot, 2022). In the X-ray band,
variability is observed on both short (e.g., $<10^{3}$ s; Uttley & McHardy,
2005; McHardy et al., 2004) and long timescales (e.g., years; McHardy, 2001;
Ishibashi & Courvoisier, 2009; Sartori et al., 2018) giving insight into the
innermost regions of the AGN. Thus, its study can help us to understand the
emission properties of AGNs (e.g., Mushotzky et al., 1993; Ulrich et al.,
1997; Uttley et al., 2014; Cackett et al., 2021; De Marco et al., 2022) and
better characterize the growing population of extremely variable AGNs
identified in the optical (e.g., Lawrence et al., 2016; Rumbaugh et al., 2018;
Trakhtenbrot et al., 2019; Shen, 2021; Zeltyn et al., 2022; Temple et al.,
2023) and X-rays (e.g., Timlin et al., 2020; Ricci et al., 2020, 2021;
Masterson et al., 2022).
One method used to study the temporal structure of the variations is the power
spectral density (PSD) analysis. If the temporal frequency is $\nu=1/t$, where
$t$ is the time, the observed power spectrum is generally modeled as a power-
law of the form: $P_{\nu}\propto\nu^{\alpha}$. For short timescales (high
frequencies) $\alpha\sim-2$, while for long timescales (low frequencies)
$\alpha\sim-1$ (Papadakis & McHardy, 1995). The PSD break timescales, $T_{B}$,
can be obtained by fitting a broken power laws to the observed PSD. This
parameter has been found to be positively correlated with the black hole mass
($M_{\rm BH}$ ; e.g., Lu & Yu, 2001; Bian & Zhao, 2002; Uttley et al., 2002;
Markowitz et al., 2003; Papadakis, 2004). However, Narrow Line Seyfert 1
(NLS1) galaxies, which typically accrete at very high Eddington ratios
($L_{\rm bol}$/$L_{\rm Edd}=\lambda_{\rm Edd}$; McHardy et al. 2004), display
a different behaviour, with their break timescales being shorter for a given
$M_{\rm BH}$ . To explain this, Uttley & McHardy (2005) suggested that the
break timescales could depend also on a second parameter, such as the
accretion rate or the black hole spin.
Accurately determining the AGNs power spectra can be difficult, since it
requires high-quality data, long exposures and sometimes monitoring campaigns,
to extend time coverage that adequately covers relevant PSD frequency ranges
that include potential breaks. Given such difficulties, it is common practice
to quantify the X-ray variability of AGNs in terms of the so-called normalised
excess variance ($\sigma_{\rm NXS}^{2}$ , Nandra et al. 1997). Although it
does not contain the same amount of information as the PSD, the normalised
excess variance can be used to confirm the PSD results in large samples of
AGN, and it also allows the discovery of new correlations between the X-ray
variability amplitude and other AGNs physical parameters. The normalised
excess variance of AGNs has been widely studied in the past decades, finding
that $\sigma_{\rm NXS}^{2}$ has a strong dependence on $M_{\rm BH}$ . Using
the data from the Advanced Satellite for Cosmology and Astrophysics (ASCA), Lu
& Yu (2001) and Bian & Zhao (2003) found an anti-correlation between the
excess variance (on a timescales of $\sim$1 day) and $M_{\rm BH}$ . Papadakis
(2004), using Rossi X-Ray Timing Explorer (RXTE) data on much longer
timescales ($\sim$300 days), also found an anti-correlation between these two
parameters. Ponti et al. (2012) investigated this relation using high quality
XMM-Newton data on timescales of 10 ks. They found that the $\sigma_{\rm
NXS}^{2}$ \- $M_{\rm BH}$ relation flattens for masses below $\sim
10^{6}M_{\odot}$, as confirmed later also by Ludlam et al. (2015) studying a
sample of low mass AGNs observed by XMM-Newton . Akylas et al. (2022), using
light curves of local Seyfert from the Nuclear Spectroscopic Telescope Array
hard X-ray mission (NuSTAR), extended the $\sigma_{\rm NXS}^{2}$ vs $M_{\rm
BH}$ relation to energy band higher than 10 keV, finding that it is possible
to accurately measure the $M_{\rm BH}$ in AGN using the above-mentioned
correlation in the $3-10$ and the $10-20$ keV bands. However, the minimum
necessary S/N is $\sim 3$ and duration of the light curves should be $\sim
80-100$ ks.
Several works suggested that the excess variance is related to other source
properties, such as the X-ray luminosity, $L_{\rm 2-10}$ , (Barr & Mushotzky,
1986; Nandra et al., 1997; Turner et al., 1999). However, studying a sample of
46 AGNs observed by ASCA, Papadakis (2004) found that once the dependence of
$\sigma_{\rm NXS}^{2}$ from $M_{\rm BH}$ is removed, the correlation between
$\sigma_{\rm NXS}^{2}$ and $L_{\rm 2-10}$ is no longer present, implying that
the correlation with $L_{\rm 2-10}$ was associated to the $\sigma_{\rm
NXS}^{2}\,-\,M_{\rm BH}$ relation. The same effect was recovered by O’Neill et
al. (2005).
Past studies of hard X-ray selected AGNs with Swift/BAT data, focusing on
longterm light curves show that in most of these AGNs a significant
variability on months to years timescales is present. In general this
variation is not related to changes of the absorption column density but to
variations of the power-law continuum (Soldi et al., 2014). Moreover, unlike
previous studies, no correlation between hard X-ray variability and different
properties of the AGNs including luminosity and black hole mass was found
(Shimizu & Mushotzky, 2013). Also Phillipson et al. (2023), studying the hard
X-ray variability properties of Swift/BAT AGNs, show that type 1 AGNs in the
14–150 keV band, are found to be less prone to harboring deterministic
variability than type 2 AGNs on timescales of $\sim 15$ years.
In this paper, we present the results from an excess variance analysis of a
sample of 151 hard X-ray selected, unobscured (N${}_{H}<10^{22}$ cm-2) AGNs
using $\sim$500 high signal-to-noise XMM-Newton observations, almost double of
the number of observations analysed in previous works (e.g., Ponti et al.
2012), for a total of $\sim$27 Ms exposure time.
The paper is organized as follows. Section 2 presents the selected sample and
the data reduction of the sources of our sample. Section 3 describes the
timing analysis of the data and the extrapolation of the $\sigma_{\rm
NXS}^{2}$ together with the analysis of the correlation between $\sigma_{\rm
NXS}^{2}$ and several physical parameters of the sources. We summarize and
conclude the results of our analysis in Section 4. Standard cosmological
parameters (H=70 km s${}^{-1}\rm Mpc^{-1}$, $\Omega_{\Lambda}$=0.73 and
$\Omega_{m}$=0.27) are adopted throughout the paper.

##  2 The sample and data reduction

Table 1: Summary of the XMM-Newton observations of the sources (OBSID) of our sample together with the Swift identification name (Swift ID) and the Swift identification number (ID). This table is available in its entirety in a machine-readable form in the online journal. A part is shown as guidance for the reader regarding its content. ID |  SwiftID | OBSID
---|---|---
6 | SWIFTJ0006.2+2012 | 0101040701
6 | SWIFTJ0006.2+2012 | 0510010701
… |  |
16 | SWIFTJ0029.2+1319 | 0783270201
34 | SWIFTJ0051.6+2928 | 0903040301
36 | SWIFTJ0051.9+1724 | 0801890301
39 | SWIFTJ0054.9+2524 | 0301450401
39 | SWIFTJ0054.9+2524 | 0841480101
43 | SWIFTJ0059.4+3150 | 0312190101
61 | SWIFTJ0113.8-1450 | 0147920101
73 | SWIFTJ0123.9-5846 | 0101040201
73 | SWIFTJ0123.9-5846 | 0721110201
… |  |
77 | SWIFTJ0127.5+1910 | 0112600601
77 | SWIFTJ0127.5+1910 | 0830551001
… |  |
106 | SWIFTJ0206.2-0019 | 0201090201
106 | SWIFTJ0206.2-0019 | 0554920301
… |  |

###  2.1 The BASS Sample

Since its launch in 2004, the Burst Alert Telescope (BAT; Barthelmy et al.,
2005) on board the Neil Gehrels Swift observatory (Gehrels et al., 2004) has
been carrying out an all-sky survey in the $14-195$ keV band. Our sample
consists of all the unobscured ($N_{H}<10^{22}\rm cm^{-2}$), radio quiet, type
1 AGNs belonging to the Swift/BAT AGN Spectroscopic Survey (BASS111www.bass-
survey.com) which have public XMM-Newton observations by December 2022.
Being unbiased by obscuration up to Comptont-thick levels (N${}_{\rm
H}>10^{24}$ cm-2, Ricci et al., 2015) and not affected by dust obscuration or
star formation, BASS provides an important census of AGNs. It gives a full
picture of the bright AGNs in the local Universe, providing the largest
available spectroscopic sample of Swift/BAT ultra-hard X-ray ($14-195$ keV)
detected AGNs (Oh et al., 2018), complementary with Swift, Chandra, and XMM-
Newton for X-ray broad-band ($0.5-200$ keV) spectral measurements (Ricci et
al., 2017). It also includes extensive multi-wavelength follow-up data, from
optical emission (Oh et al., 2022), high spatial resolution near-IR (Lamperti
et al., 2017; Koss et al., 2018), mid- and far-IR emission from WISE, IRAS,
Spitzer, Akari, and Herschel (Ichikawa et al., 2017; Shimizu et al., 2017;
Ichikawa et al., 2019) and mm/radio emission (Kawamuro et al., 2022; Koss et
al., 2021; Ricci et al., 2023), giving insight on the sample over the broadest
possible spectral range.
The first BASS data release (DR1, Koss et al. 2017) reported $M_{\rm BH}$ and
X-ray properties for all the 838 AGNs from the Swift/BAT 70-month catalogue
(Ricci et al., 2017), while the second BASS data release (DR2, Koss et al.
2022a) reports more secure and uniformly assessed $M_{\rm BH}$ for 780
unbeamed AGNs from the 70-month catalogue. The masses are estimated from broad
Balmer lines and/or "reverberation mapping" technique (RM) for type 1s and
masers, and from dynamics and/or velocity dispersions for type 2s (Koss et
al., 2022c; Mejía-Restrepo et al., 2022; Ricci et al., 2022). Moreover, in the
DR2, $\lambda_{\rm Edd}$ ($L_{\rm bol}$/$L_{\rm Edd}$) are computed using the
bolometric luminosities calculated from the intrinsic 14–150 keV luminosities
as shown in Ricci et al. 2017 with a bolometric correction of 8 (Koss et al.,
2022a). In this work, we considered $L_{\rm 2-10}$ , $\lambda_{\rm Edd}$ , and
$M_{\rm BH}$ values from BASS DR2 (Koss et al., 2022b).

Figure 1: Distributions of black hole masses ($M_{\rm BH}$ ; left panel), 2–10
keV luminosities ($L_{\rm 2-10}$ ; middle panel), Eddington ratios
($\lambda_{\rm Edd}$ ; right panel) of the sources analysed in this work,
compared with the parent BASS sample of unobscured AGNs (empty bars).

###  2.2 Data Reduction

The X-ray Multi-Mirror Mission (XMM-Newton , Jansen et al. 2001) high
statistics, low background and uninterrupted light-curves obtained for the
sources in this sample are crucial to compute the $\sigma_{\rm NXS}^{2}$ in
large samples of AGN.
The sample of BASS unobscured AGNs is composed by 365 sources, with a median
redshift of $z_{\rm med}=0.035$ (lower than the parent sample median
redshift). Of these, 153 had public XMM-Newton observations as of December
2022. We downloaded all the observations from the XMM-Newton Science Archive,
and extracted the EPIC-pn (Strüder et al., 2001) light curves using the
Science Analysis System (SAS) software package (v.18.0.0) (Gabriel et al.,
2004) and the calibration database CALDB 20221102. The MOS detectors (Turner
et al., 2001) and the Reflection Grating Spectrometer (RGS, den Herder et al.
2001) were not considered because their lower statistics would not
significantly improve the quality of the lightcurves.

Table 2: List of the Swift Identification number (ID), the Swift name
(SwiftID), the counterpart name, the redshift (z), the hydrogen column density
(NH [cm-2]), the mass ($M_{\rm BH}$ in solar masses), the 2–10 keV luminosity
($L_{\rm 2-10}$ [$\rm erg\,s^{-1}$]), the bolometric lumunosity ($L_{\rm
bol}$[$\rm erg\,s^{-1}$]), the Eddington ratio ($\lambda_{\rm Edd}$ ), the
normalised excess variance for the light curves binned with 100s ($\sigma_{\rm
NXS}^{2}$ 100s) and 1000s ($\sigma_{\rm NXS}^{2}$ 1000s)of time binning. This
table is available in its entirety in a machine-readable form in the online
journal, where also the errors are reported. A part is shown as guidance for
the reader regarding its content.

ID |  SwiftID | Counterpart | z | log(NH) | log($M_{\rm BH}$ ) | log($L_{\rm 2-10}$ ) | log($L_{\rm bol}$) | $\lambda_{\rm Edd}$ | log($\sigma_{\rm NXS}^{2}$ 100s) | log($\sigma_{\rm NXS}^{2}$ 1000s)
---|---|---|---|---|---|---|---|---|---|---
|  |  |  | [cm-2] | [Msun] | [erg s-1] | [erg s-1] |  |  |
6 | SWIFTJ0006.2+2012 | Mrk 335 | 0.025 | 20.48 | 7.23 | 43.23 | 44.36 | 0.068 | -2.30 | -2.30
16 | SWIFTJ0029.2+1319 | PG 0026+129 | 0.142 | 20.01 | 8.48 | 44.39 | 45.72 | 0.104 | -3.63 | -3.03
34 | SWIFTJ0051.6+2928 | UGC 524 | 0.036 | 20.02 | 7.62 | 42.99 | 44.08 | 0.032 | -3.20 | -3.20
36 | SWIFTJ0051.9+1724 | Mrk 1148 | 0.064 | 20.3 | 7.75 | 44.12 | 45.31 | 0.234 | -4.54 | -5.14
39 | SWIFTJ0054.9+2524 | PG 0052+251 | 0.155 | 20.02 | 8.46 | 44.62 | 45.89 | 0.135 | -3.36 | -3.99
43 | SWIFTJ0059.4+3150 | Mrk 352 | 0.014 | 20.01 | 7.55 | 42.72 | 44.09 | 0.016 | -3.00 | -3.00
61 | SWIFTJ0113.8-1450 | Mrk 1152 | 0.052 | 20.01 | 8.32 | 43.47 | 45.16 | 0.037 | -3.13 | -4.43
73 | SWIFTJ0123.9-5846 | Fairall 9 | 0.047 | 20.02 | 8.29 | 44.13 | 45.29 | 0.058 | -2.22 | -2.30
77 | SWIFTJ0127.5+1910 | Mrk 359 | 0.017 | 20.61 | 6.04 | 42.66 | 43.83 | 0.339 | -2.69 | -2.69
106 | SWIFTJ0206.2-0019 | Mrk 1018 | 0.042 | 20.01 | 7.81 | 43.61 | 45.08 | 0.094 | -3.61 | -4.03
… |  |  |  |  |  |  |  |  |  |

Note: The values of the $M_{\rm BH}$ are estimated from RM or broad lines
(Koss et al., 2022c; Mejía-Restrepo et al., 2022; Ricci et al., 2022).
$\lambda_{\rm Edd}$ are computed using the bolometric luminosities calculated
from the intrinsic 14–150 keV luminosities as shown in Ricci et al. 2017 with
a bolometric correction of 8 (Koss et al., 2022a). NH and $L_{\rm 2-10}$ from
Ricci et al. 2017

The XMM-Newton EPIC-pn raw data have been processed using the epchain tool of
SAS to obtain calibrated and concatenated event lists. The extraction radii
and the optimal time cuts to exclude periods of high flaring particle
background were computed via an iterative process which maximizes the signal-
to-noise ratio (SNR), as described in Piconcelli et al. (2004), filtering out
those time intervals for which the count rate of the background reach values
so high that the SNR of the source does not improve (or even worsens) when
including such time intervals in the analysis. Since the pn camera has a full-
frame time resolution of $73.3\rm ms$ per CCD, the observations generally do
not suffer significantly from pile-up, making them suitable for variability
analysis. Nonetheless, the light curves were extracted after confirming that
the data were not affected by pile-up, as indicated by the SAS task epatplot.
The resulting optimal extraction radius was $\sim 30-40$″ and the background
spectra were extracted from source-free circular regions with radii of $\sim
50-60$″ for all the observations analyzed in this work. With these regions we
extracted the EPIC-pn source and background light curves using the command
evselect and we corrected the source light curve for the background using the
command epiclccorr. We extracted the light curves using several different time
and spectral binning strategies: 100 s and 1000 s in the $0.2-10$ keV energy
band, and 100 s in the $0.2-1$ keV (soft), $1-3$ keV (medium) and $3-10$ keV
(hard) energy bands.
Following Ponti et al. (2012) we selected the observations which had cleaned
exposure times larger than 10 ks, and which had at least 10 counts in those 10
ks chunks and in each (rest-frame) energy band used in this analysis, i.e.
$0.2-1$ keV, $1-3$ keV and $3-10$ keV and for each time bin of 100 s and 1000
s. We did this selection to avoid having not enough counts in the 10 ks
independent light curve to constrain the $\sigma_{\rm NXS}^{2}$ . A total of
151 sources ($\sim 500$ observations) fulfill these criteria. The
distributions of $M_{\rm BH}$ , $L_{\rm 2-10}$ , $\lambda_{\rm Edd}$ and NH of
our sample is shown in Fig. 1.
We show in Appendix A the XMM-Newton EPIC-pn background subctracted light
curves of a sub-sample of representative sources for different $M_{\rm BH}$
values (see Fig. 8).

##  3 Analysis

Figure 2: $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ relation obtained from the
$0.2-10$ keV light curves binned with 100 s (left panels) and 1000 s (right
panels). Yellow stars with error bars correspond to the SA results for each
bin. The dashed lines are the linear regressions obtained from the fitting
over the SA results, while the shaded region represents the combined 1$\sigma$
error on the slope and normalisation. Each colored data points with error bars
represents one source of our sample. Colorbars represent the $L_{\rm 2-10}$
(top panels) and $\lambda_{\rm Edd}$ (bottom panels).

The excess variance ($\sigma_{\rm NXS}^{2}$ ) is a quantity used to describe
the variability amplitude. It is the difference between the total variance of
a light curve and the mean squared error that is normalised for the average of
the $N$ flux measurements squared (e.g. Nandra et al. 1997; Turner et al.
1999). Here $N$ is the number of good time intervals in a light curve, and
$x_{i}$ and $\sigma_{i}$ are the flux and error in each interval,
respectively. The excess variance is defined (Vaughan et al., 2003) as
follows:

$\sigma^{2}_{\rm
NXS}=\frac{S^{2}-\overline{\sigma^{2}}}{\overline{x_{i}^{2}}}$ (1)

Where $\overline{\sigma^{2}}$ is the mean square error:

$\overline{\sigma^{2}}=\frac{1}{N}\sum_{i=1}^{N}[\sigma_{i}^{2}]$ (2)

and $S^{2}$ is the sample variance:

$S^{2}=\frac{1}{N-1}\sum_{i=1}^{N}[(x_{i}-\overline{x_{i}})^{2}]$ (3)

corresponding to the integral of the PSD between two frequencies ($\nu_{1}$
and $\nu_{2}$), which yields the contribution to the expectation value of the
variance due to variations between the corresponding timescales ($1/\nu_{1}$
and $1/\nu_{2}$):

$\langle S^{2}\rangle=\int_{\nu_{1}}^{\nu_{2}}P(\nu)d\nu$ (4)

To study the correlations between $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ ,
$L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ , we calculated $\sigma_{\rm NXS}^{2}$
from the XMM-Newton light curves. The values are listed in Tab. 2.
$\sigma_{\rm NXS}^{2}$ is a good estimator of the intrinsic variance of a
source but it has some biases. It is related to the integral of the PSD
between two frequencies and thus depends on the length of the monitoring time
interval, on the red-noise character of the X-ray variability and also, due to
the effect of cosmological time dilation, on the redshift (Lawrence &
Papadakis, 1993; Green et al., 1993; Lawrence & Papadakis, 1993; Papadakis et
al., 2008; Vagnetti et al., 2011; Vagnetti, F. et al., 2016). Since our sample
of 151 type 1 AGNs is composed mainly of local AGNs ($z_{\rm med}=0.035$), the
impact of redshift is negligible. However, we need to avoid biases related to
the different exposure times of our observations and the red-noise character
of the light curves. Therefore, we computed the $\sigma_{\rm NXS}^{2}$ from 10
ks-long independent light curve sections and, for the sources with cleaned
exposure time that lasted for a multiple of 10 ks, we took the median of the
excess variances of all these independent sections in each energy band. For
the sources with more than one observation, we used the median value of the
$\sigma_{\rm NXS}^{2}$ in both the cases of the light curves with 100 s and
1000 s time bin in each energy band. We applied this procedure also for the 7
sources of our sample which are classified as ’changing-look’ (CL) AGNs (i.e.
Mrk 1018, Fairall 9, Mrk 590, NGC 3516, NGC 1566, 3C 390.3, NGC 7603, Jin et
al., 2022; Temple et al., 2023) since the $\sigma_{\rm NXS}^{2}$ computed for
these sources are consistent within the error among the different
observations.
The X-ray spectrum of AGNs in different energy bands is strongly impacted by
different components: the primary power-law component and the reflection
component are dominant in the hard energy band ($3-10$ keV, Haardt & Maraschi
1991, 1993; Haardt & Matt 1993) while soft-excess and warm-absorbers (WA) can
impact the soft ($0.2-1$ keV, Bianchi et al. 2009) and medium ($1-3$ keV,
Blustin et al. 2005; Tombesi et al. 2013) energy bands. Variations of these
different components will lead to distinct spectral variability in different
energy bands. We therefore calculated $\sigma_{\rm NXS}^{2}$ from the $0.2-1$
keV (soft), $1-3$ keV (medium) and $3-10$ keV (hard) light curves to get a
fuller picture of the AGNs X-ray variability.

Figure 3: Upper panels: $L_{\rm 2-10}$ vs $M_{\rm BH}$ relation. Lower panels:
$\lambda_{\rm Edd}$ vs $\sigma_{\rm NXS}^{2}$ relations. The relations are
obtained from the $0.2-10$ keV light curves binned with 100 s (left panels)
and 1000 s (right panels). Yellow stars with error bars correspond to the SA
results for each bin. The dashed lines are the linear regressions obtained
from the fitting over the SA results, while the shaded region represents the
combined 1$\sigma$ error on the slope and normalisation. Each colored data
points with error bars represents one source of our sample. Colorbars
represent the $M_{\rm BH}$ . Figure 4: $\sigma_{\rm NXS}^{2}$ x $M_{\rm BH}$
vs $L_{\rm 2-10}$ relation. The relations are obtained from the $0.2-10$ keV
light curves binned with 100 s (left panels) and 1000 s (right panels). Yellow
stars with error bars correspond to the SA results for each bin. The dashed
lines are the linear regressions obtained from the fitting over the SA
results. Colorbars represent the $M_{\rm BH}$ .

Figure 5: Relations between the $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ when the sample is divided into two bins of $L_{\rm 2-10}$ (top panels, purple lines: $L_{\rm 2-10}<L_{\rm 2-10,med}$, green lines: $L_{\rm 2-10}>L_{\rm 2-10,med}$) and $\lambda_{\rm Edd}$ (bottom panels, cyan lines: $\lambda_{\rm Edd}<\lambda_{\rm Edd,med}$, magenta lines: $\lambda_{\rm Edd}>\lambda_{\rm Edd,med}$). The best fit relation are reported in Tab. 3. Table 3: List of the best-fit relations together with their p-values. The fits are performed in the log-log space using equation Eq. 5.  Relation |  $\Delta$t(s) | Intercept (A) | Slope (B) | Pearson | 1-Pvalue
---|---|---|---|---|---
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ | 100 | $2.99\pm 1.79$ | $-0.77\pm 0.23$ | -0.94 | 0.99
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ | 1000 | $5.54\pm 2.11$ | $-1.12\pm 0.18$ | -0.96 | 0.99
$\sigma_{\rm NXS}^{2}$ vs $L_{\rm 2-10}$ | 100 | $26.29\pm 25.29$ | $-0.67\pm 0.08$ | -0.84 | 0.95
$\sigma_{\rm NXS}^{2}$ vs $L_{\rm 2-10}$ | 1000 | $29.84\pm 36.41$ | $-0.76\pm 0.06$ | -0.85 | 0.96
$\sigma_{\rm NXS}^{2}$ vs $\lambda_{\rm Edd}$ | 100 | $-3.55\pm 2.27$ | $-0.36\pm 0.61$ | -0.10 | 0.25
$\sigma_{\rm NXS}^{2}$ vs $\lambda_{\rm Edd}$ | 1000 | $-3.11\pm 3.19$ | $-0.24\pm 0.59$ | -0.12 | 0.25
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ,RM | 100 | $2.94\pm 0.29$ | $-0.75\pm 0.20$ | -0.66 | 0.75
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ,RM | 1000 | $2.91\pm 0.34$ | $-0.76\pm 0.41$ | -0.67 | 0.79
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($L_{\rm 2-10}$ $<L_{\rm 2-10,med})$ | 100 | $0.36\pm 0.51$ | $-0.43\pm 0.06$ | -0.89 | 0.97
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($L_{\rm 2-10}$ $<L_{\rm 2-10,med})$ | 1000 | $0.16\pm 0.38$ | $-0.37\pm 0.04$ | -0.99 | 0.99
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($L_{\rm 2-10}$ $>L_{\rm 2-10,med})$ | 100 | $-2.45\pm 0.48$ | $-0.14\pm 0.16$ | -0.30 | 0.38
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($L_{\rm 2-10}$ $>L_{\rm 2-10,med})$ | 1000 | $-1.71\pm 0.41$ | $-0.16\pm 0.15$ | -0.35 | 0.41
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($\lambda_{\rm Edd}$ $<\lambda_{\rm Edd,med})$ | 100 | $0.69\pm 0.76$ | $-0.48\pm 0.09$ | -0.89 | 0.97
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($\lambda_{\rm Edd}$ $<\lambda_{\rm Edd,med})$ | 1000 | $1.46\pm 1.32$ | $-0.55\pm 0.16$ | -0.89 | 0.97
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($\lambda_{\rm Edd}$ $>\lambda_{\rm Edd,med})$ | 100 | $0.16\pm 0.83$ | $-0.42\pm 0.13$ | -0.89 | 0.97
$\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ ($\lambda_{\rm Edd}$ $>\lambda_{\rm Edd,med})$ | 1000 | $2.35\pm 0.88$ | $-0.71\pm 0.11$ | -0.89 | 0.97
$\sigma^{2}_{\rm NXS}\times M_{\rm BH}$ vs $L_{\rm 2-10}$ | 100 | $14.35\pm 9.75$ | $-0.26\pm 0.27$ | -0.19 | 0.30
$\sigma^{2}_{\rm NXS}\times M_{\rm BH}$ vs $L_{\rm 2-10}$ | 1000 | $12.05\pm 7.78$ | $-0.21\pm 0.22$ | -0.20 | 0.30
$\sigma^{2}_{\rm NXS,hard}$ vs $\sigma^{2}_{\rm NXS,soft}$ | 100 | $-0.62\pm 0.26$ | $0.87\pm 0.10$ | 0.71 | 0.99
$\sigma^{2}_{\rm NXS,hard}$ vs $\sigma^{2}_{\rm NXS,med}$ | 100 | $-0.47\pm 0.22$ | $0.93\pm 0.08$ | 0.79 | 0.99

###  3.1 Correlations between the normalised excess variance and the physical
parameters

To investigate the physical parameters driving X-ray variability in our sample
of unobscured AGNs we looked for correlations between the XMM-Newton broad-
band ($0.2-10$ keV) $\sigma_{\rm NXS}^{2}$ and several key AGNs parameters
(i.e. $M_{\rm BH}$ , $L_{\rm 2-10}$ , $\lambda_{\rm Edd}$ ) by fitting a
linear model to the data in the log-log space (see Fig. 2 and Fig. 3) using
the following fitting relation:

$\log(\sigma_{\rm NXS}^{2})=\rm{A}+\rm{B}\log(x)$ (5)

where x is the value of the physical parameter. Among the 151 sources of our
sample, we found 46 objects with an intrinsic $\sigma_{\rm NXS}^{2}$ lower
than the respective error. In this case, we define the measurement as a “non-
detection”, and we consider it as an upper limit. To include the upper limits
in our analysis, we used the survival analysis method (SA; e.g., Feigelson &
Nelson 1985; Shimizu et al. 2017) using the scikit-survival (Pölsterl, 2020)
package, which applies the principles of SA to astronomical data. SA is a
statistical technique used to analyze time-to-event data and it is
particularly well-suited for analyzing data that include upper/lower limits.
Specifically, scikit-survival calculates the non-parametric Kaplan-Meier
product-limit (KMPL) estimator for a sample distribution. The KMPL estimator
is an estimate of the survival function, which is simply 1-CDF (cumulative
distribution function). Using the KMPL, we calculated, for each bin of $M_{\rm
BH}$ , $L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ , the median $\sigma_{\rm
NXS}^{2}$ , and estimated their uncertainties. Since the KMPL estimator is a
non-parametric method, it is unbiased because it does not assume any specific
distribution for the data. We divided $M_{\rm BH}$ and $\lambda_{\rm Edd}$
into 6 bins and the $L_{\rm 2-10}$ into 7 bins. These bins are not
symmetrical, since we requested each bin to have at least 15 values. We fitted
the median values obtained with the SA method using the code: linmix, a
hierarchical Bayesian model for fitting a straight line to data with errors in
both the x and y directions (Kelly, 2007). From the analysis of the
correlation between $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ , data of sources
with $M_{\rm BH}$ $>10^{9}M_{\odot}$ are excluded since, for these sources, we
found mostly upper limits on $\sigma_{\rm NXS}^{2}$ so, being the last bin
populated just by upper limits, the SA method was not reliable in computing
the median in the bin.
We report in Tab. 3 the intercepts and the slopes of the linear regression
line together with Pearson’s correlation coefficients and the correlation
probabilities for all the relations analysed in this work and for light curves
with 100 s and 1000 s bins. As shown in Tab. 3 the fitting parameters for the
100 s and 1000 s binned light curves are consistent within the errors in each
analysed relation.
Fig. 2 shows the $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ relation obtained from
the $0.2-10$ keV light curves binned with 100 s and 1000 s. We also reported
the SA results for each bin. Being the SA results the $\sigma_{\rm NXS}^{2}$
median calculated using the KMPL estimator they are representative of the
$\sigma_{\rm NXS}^{2}$ value in each $M_{\rm BH}$ bin. We report also the
linear regressions obtained from the fitting over the SA results. As expected
from previous studies (Nandra et al., 1997; Papadakis, 2004; O’Neill et al.,
2005; Ponti et al., 2012), we found, a strong anti-correlation between
$\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ (see Fig. 2). We computed the
1$\sigma$ scatter of the data around the best-fit line using the following
equation:

$\sigma_{\rm scatter}=\sqrt{\sum_{i=1}^{N}[\log(\sigma^{2}_{\rm
NXS,i})-f(M_{\rm BH,i})]^{2}/N}$ (6)

where $f(M_{\rm BH})$ is the logarithmic value of the $\sigma_{\rm NXS}^{2}$
extrapolated using the best fitting relation (see Tab. 3). We found a scatter
for this relation of $\sim 0.85$ dex in both the case of the light curves
binned with 100 s and 1000 s.
We also found a strong anti-correlation between the $\sigma_{\rm NXS}^{2}$ and
the $L_{\rm 2-10}$ (see upper panels of Fig. 3). Given the strong dependence
between the $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ , we decided to correct
the $\sigma_{\rm NXS}^{2}$ for $M_{\rm BH}$ to check if after the correction,
the relation between $\sigma_{\rm NXS}^{2}$ and $L_{\rm 2-10}$ is still
present. Previous works have shown that $\sigma_{\rm NXS}^{2}$ $\propto$
$M_{\rm BH}$ ∼(-1) (e.g., Ponti et al. 2012). In this work we found a slope
for the $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ relation of $-0.77\pm 0.23$ and
$-1.12\pm 0.18$ when considering light curves with 100 s and 1000 s bins,
respectively. Since $-1$ is still consistent within the errors with our
results, for consistency with past works, to check if the $\sigma_{\rm
NXS}^{2}$ vs $L_{\rm 2-10}$ correlation still exists when the primary
dependence is removed, we analysed the correlation between $\sigma^{2}_{\rm
NXS}\times M_{\rm BH}$ versus $L_{\rm 2-10}$ . Removing the dependence of
$\sigma_{\rm NXS}^{2}$ on $M_{\rm BH}$ , the strong correlation with $L_{\rm
2-10}$ , that was present before, is not significant anymore (see Fig.,4), as
reported from previous studies (Papadakis, 2004; O’Neill et al., 2005). In
fact we found, in both cases of $\sigma_{\rm NXS}^{2}$ obtained from the
$0.2-10$ keV light curves binned with 100 s and 1000 s, a Pearson correlation
coefficient of -0.19 and -0.20 corresponding to a $1-P_{\rm value}$ of 0.30
(see Tab. 3). Thus, the dependence between the $\sigma_{\rm NXS}^{2}$ and
$L_{\rm 2-10\,keV}$ is actually related to the dependence between $L_{\rm
2-10\,keV}$ and $M_{\rm BH}$ . From our analysis an anti-correlation between
$\sigma_{\rm NXS}^{2}$ and $\lambda_{\rm Edd}$ is present (see lower panels of
Fig. 3), but it is not significant, according to the Pearson test (see Tab.
3).
From Fig. 3 it is clear that in both relations a gradient of $M_{\rm BH}$ is
present. Thus, to check if the relation $\sigma_{\rm NXS}^{2}-M_{\rm BH}$ is
somehow affected by $L_{\rm 2-10}$ and/or $\lambda_{\rm Edd}$ , we first
computed the median values of $L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ of the
sample, which are $L_{\rm 2-10,med}=2.79\times 10^{43}$ $\rm erg\,s^{-1}$ and
$\lambda_{\rm Edd,med}=0.06$, respectively. We then used these values as
thresholds to divide the sample into two sub-samples depending on their
$L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ (see figure 5). The best-fitting values
of the correlations we found are also reported in Tab. 3. We found that the
correlation between $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ has slightly
different normalizations among the two sub-samples depending on the $L_{\rm
2-10}$ or $\lambda_{\rm Edd}$ but the same slope within the errors, which is
also in agreement with the slope of the $\sigma_{\rm NXS}^{2}$ \- $M_{\rm BH}$
relation found for the total sample. In particular, in the sub-sample
depending on $L_{\rm 2-10}$ we found that the slopes of the correlations is
similar to the one found for the total sample, while the normalization for the
sources with $L_{2-10}>L_{\rm 2-10,med}$ is lower. This is not really
surprising since sources with high luminosity (high $M_{\rm BH}$ ) show lower
variability. Also, the majority of the sources with $L_{2-10}>L_{\rm
2-10,med}$ show an upper limit of the $\sigma_{\rm NXS}^{2}$ .

###  3.2 Reverberation mapping sub-sample

Figure 6: $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ for the RM sub-sample. The
dashed black lines are the linear regressions obtained from the fitting
process. Shaded region represents the combined 1$\sigma$ error on the slope
and normalisation. The $\sigma_{\rm NXS}^{2}$ is obtained from the $0.2-10$
keV light curves binned with 100 s (left panel) and 1000 s (right panels).
Lower panels show the difference between the real $M_{\rm BH}$ and $M_{\rm
BH}$ extrapolated from the relation ($\Delta M$). For details about the
coefficients, see Tab. 3.

We checked the relation between the $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ in
the sub-sample of sources in which $M_{\rm BH}$ is obtained via RM (35
sources). Being the sub-sample smaller, we did not used the SA method. Instead
we used the method of the "censored fitting" (CF) (Guainazzi et al., 2006;
Bianchi et al., 2009), to account for upper limits. This was done by
performing a large number of least square fits, using the linmix code, on a
set of Monte-Carlo simulated data derived from the observed data points. Each
detection was substituted by a value randomly drawn from a Gaussian
distribution, whose mean is the best-fit measurement and whose standard
deviation is its statistical uncertainty. Each upper limit $U$ was substituted
by a value randomly drawn from a uniform distribution in the interval $[A,U]$,
where A was arbitrarily set to $A\ll U$. We choose $A=10^{-6}$. We found an
anticorrelation in both cases of $\sigma_{\rm NXS}^{2}$ obtained from the
$0.2-10$ keV light curves binned with 100 s and 1000 s (see Fig. 6). Using
Equation 6 we found $\sigma_{\rm scatter,100}=0.65$ and $\sigma_{\rm
scatter,1000}=0.69$, smaller than the scatters of the $\sigma_{\rm NXS}^{2}$
vs $M_{\rm BH}$ relations obtained from the total sample. This is because on
average the sources with $M_{\rm BH}$ estimated via RM are brighter and they
show a higher count rate on the same timescales. Using the $\sigma_{\rm
NXS}^{2}$ vs $M_{\rm BH}$ relations obtained from the RM sample, it is
possible to measure $M_{\rm BH}$ for a total of 87 AGNs (out of 151 AGNs in
our sample) and provide an upper/lower limit for the remaining AGN. Thus, even
if the X-ray variability is not the most accurate tool to measure $M_{\rm BH}$
, the relation obtained for the RM sub-sample gives a good $M_{\rm BH}$
estimation, with a scatter <1 dex.

###  3.3 The normalised excess variance in the soft, medium and hard energy
bands.

Figure 7: Left panel: soft ($0.2-1$ keV) vs hard ($3-10$ keV) $\sigma_{\rm
NXS}^{2}$ . Right panel: medium ($1-3$ keV) vs hard ($3-10$ keV) $\sigma_{\rm
NXS}^{2}$ . The best fit curves are plotted with solid black lines while the
shaded region represents the combined 3$\sigma$ error on the slope and
intercept. The green dashed lines represent the one-to-one relation. Black
dashed lines represent the regression lines found using the bisector method.

It is interesting to compare $\sigma_{\rm NXS}^{2}$ in various energy bands to
check if there is one or more components of the AGNs X-ray spectrum that
contributes the most in the variability. In order to verify this we calculated
$\sigma_{\rm NXS}^{2}$ in the the soft ($0.2-1$ keV), medium ($1-3$ keV) and
hard ($3-10$ keV) bands. We then looked for a correlation between the
$\sigma^{2}_{\rm NXS,hard}$ and both $\sigma^{2}_{\rm NXS,soft}$ and
$\sigma^{2}_{\rm NXS,med}$. In the left panel of Fig. 7 we show $\sigma_{\rm
NXS}^{2}$ in the soft energy band versus the same parameter calculated in the
hard band, while in the right panel of Fig. 7 we illustrate the excess
variance in the medium energy band versus that in the hard band. The best-
fitting relations, reported in Tab. 3, are obtained using the CF method (see
§3.2).
We fitted the data also with the bisector method which provides a more
balanced and symmetric estimate of the true regression line between the two
variables that in principle could be independent. The values of the slope and
intercept we found with this method are consistent within the errors with the
one found using the CF method.
The values of $\sigma_{\rm NXS}^{2}$ in the soft and medium energy bands
appear to be well correlated with $\sigma_{\rm NXS}^{2}$ in the hard energy
band, although with a slope flatter than the one-to-one relation (green dashed
line in Fig. 7). This may imply that for most of our sources, on timescales
less than 10 ks the spectral components which dominate in the hard band are
increasingly more variable than the ones dominating the soft and medium energy
bands, for higher values of the variance. In the soft energy band the dominant
component is usually the soft-excess, which can be variable. Furthermore, in
the soft-medium energy bands, the presence of absorbing material, either
neutral or ionised, can be variable or can absorb the continuum emission
responding to the continuum variations. If these variations happened on
timescales less than $\sim 10$ ks we would expect to measure a larger
variability amplitude in the soft/medium energy bands compared to the hard
band. We observe the opposite, i.e. weaker variations of the soft-excess
and/or warm absorbers than those of the primary continuum and/or reflection
component on this timescale, in agreement with previous studies (Ponti et al.,
2012; Simm et al., 2016).
For completeness we checked the relations between the $\sigma_{\rm NXS}^{2}$
in the soft, medium and hard energy bands with the physical properties of the
AGN ($M_{\rm BH}$ , $L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ ) to see whether
these relations support the results found in this work. The best fitting
results are reported in Appendix B. We found that the $\sigma_{\rm NXS}^{2}$
vs $M_{\rm BH}$ relation in the soft energy band is slightly less significant
than in the harder energy bands (see Tab. 4), but in general the best-fitting
relations in each energy band are consistent with those from the broad XMM-
Newton energy band ($0.2-10$ keV).

##  4 Conclusions

We analysed the variability properties of $\sim 500$ XMM-Newton observations
of a sample of 151 nearby ($z_{\rm med}=0.035$) unobscured
($N_{H}<10^{22}$cm-2) AGNs from the BASS survey, studying the correlations of
the excess variance with the physical properties of the sources and also
checking for the correlations between the excess variance computed in
different energy bands. The timescale used to compute the $\sigma_{\rm
NXS}^{2}$ is 10 ks, to avoid biases related to the differences on the exposure
times of the sources of our sample and to take into account the red-noise
character of the light curves. We analysed the relations of $\sigma_{\rm
NXS}^{2}$ with $M_{\rm BH}$ , $L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ . The
correlation between $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ is a well-know
property of AGNs (Lu & Yu, 2001; Papadakis, 2004; O’Neill et al., 2005;
Nikołajuk et al., 2006; Nikolajuk et al., 2007; Miniutti et al., 2009; Zhou et
al., 2010; Ponti et al., 2012). In agreement with this, in our sample we found
a very strong and highly significant correlation between these two quantities.
We do not find a significant correlation between $\sigma_{\rm NXS}^{2}$ and
$\lambda_{\rm Edd}$ , consistently with previous results (O’Neill et al.,
2005; Gierliński et al., 2008; Zhou et al., 2010; Ponti et al., 2012; Lanzuisi
et al., 2014). However, according to McHardy et al. (2006) the break timescale
increases proportionally as $M_{\rm BH}$ decreases ($\lambda_{\rm Edd}$
increases). Thus, if we assume a universal PDS with a single break frequency
depending on the $M_{\rm BH}$ and a equally long-timescale normalization, the
strength of the relations $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ and
$\sigma_{\rm NXS}^{2}$ vs $\lambda_{\rm Edd}$ would be the same, with higher
short-timescale variability for low $M_{\rm BH}$ (high $\lambda_{\rm Edd}$ ).
Our result could suggest that there might be no correlation between break
timescale and $\lambda_{\rm Edd}$ , as proposed by González-Martín & Vaughan
(2012) analysing a larger sample of shorter light curves. Alternatively,
following the results of McHardy et al. (2006) and Paolillo et al. (2017),
$\lambda_{\rm Edd}$ could be dependent on the break time scale but, since on
short-timescale $\sigma_{\rm NXS}^{2}$ seems mostly independent of this
parameter, the normalization of the power spectrum may be anti-correlated with
$\lambda_{\rm Edd}$ .
We found a tight anti-correlation between $\sigma_{\rm NXS}^{2}$ and $L_{\rm
2-10}$ . To remove the $M_{\rm BH}$ dependence from this correlation, we
explored the relation between the $\sigma^{2}_{\rm NXS}\times M_{\rm BH}$
versus $L_{\rm 2-10}$ , finding that, in this case, the $\sigma_{\rm NXS}^{2}$
vs $L_{\rm 2-10}$ correlation disappears, confirming that the correlation with
$L_{\rm 2-10}$ is secondary, while the primary correlation is in fact with the
mass, in agreement with what has been found by previous works (e.g. Papadakis
2004; O’Neill et al. 2005; Lanzuisi et al. 2014).
We explored the $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ relation in the sub-
sample of sources with $M_{\rm BH}$ estimated via RM, finding that the
correlation between these quantities in this sub-sample has an intrinsic
scatter of $\sim 0.65-0.69$ dex. With this relation we were able to measure
$M_{\rm BH}$ for 87 AGNs and estimate upper/lower limits for the remaining 64
AGNs of our sample. Thus, one could in principle use X-ray variability to
measure $M_{\rm BH}$ (e.g. Nikolajuk et al. 2004; Ponti et al. 2012; Akylas et
al. 2022). With the advent of future planned or proposed missions (e.g.
Athena, AXIS, etc) that will provide higher count rates, the accuracy of this
relation for mass measurement will improve significantly.
Dividing the sample into two bins of $L_{\rm 2-10}$ , the normalization of the
anti-correlation between $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ is lower for
the sources with higher luminosity. This is possibly related to the fact that
for sources with high luminosity (high mass), we detected lower variability
and mostly only an upper limit on $\sigma_{\rm NXS}^{2}$ was obtained. When
dividing the sample into two bins of $\lambda_{\rm Edd}$ the slope and the
normalisation are slightly different in the two sub-samples but still
consistent within the errors.
X-ray spectra could be dominated by different components depending on the
energy band one is analysing. Therefore we explored the relation of the
$\sigma_{\rm NXS}^{2}$ in the hard X-ray band ($3-10$ keV) with the
$\sigma_{\rm NXS}^{2}$ in the soft X-ray band ($0.2-1$ keV) and in the medium
X-ray band ($1-3$ keV), finding that $\sigma_{\rm NXS}^{2}$ calculated in
various energy bands are highly correlated, in agreement with previous studies
(Ponti et al., 2012; Simm et al., 2016). In particular we found that, in most
sources, the primary continuum and/or the reflection component are
increasingly more variable than the spectral components dominating softer
energy bands ($0.2-1$ keV and $1-3$ keV) on timescales shorter than 10 ks. In
fact, if WA components were varying, they would show more variability in the
medium energy band, while the variance in that band is lower than in the hard
energy band. Thus WA variability cannot be generally the cause of fast
(shorter than 10 ks) variations. Moreover, we found that the soft energy band
is less variable than the hard band. This implies that the soft-excess, or at
least part of it, is a less variable component (on timescales less than 10 ks)
which dilutes the $\sigma_{\rm NXS}^{2}$ by adding to the constant flux in the
denominator and not to the variable flux in the numerator, as it was found for
the Seyfert 1.5 galaxy NGC 3227 (Arévalo & Markowitz, 2014). Finally, the hard
continuum might be intrinsically more variable than the continuum in softer
bands because the break timescale of the PDS moves to shorter timescales for
higher energy X-ray photons (McHardy et al., 2004; Markowitz et al., 2007;
McHardy et al., 2007; Arévalo et al., 2008).
We examined the relation between $\sigma_{\rm NXS}^{2}$ (calculated in the
soft, medium and in the hard X-ray bands) and several important AGN physical
parameters, such as $M_{\rm BH}$ , $L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ .
Our analysis revealed that the best-fitting relations in each energy band
align with those from the broad XMM-Newton energy band ($0.2-10$ keV).
Notably, in the soft energy band, the $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$
anti-correlation appears to be slightly less significant. This lends support
to another key finding of this study, i.e. that, on timescales shorter than 10
ks, the primary continuum and/or the reflection component exhibit stronger
variability compared to the spectral components dominating softer energy
bands.
Compared with previous results (e.g., Ponti et al. 2012) we found a less steep
correlation between $\sigma_{\rm NXS}^{2}$ and $M_{\rm BH}$ . The difference
could be attributed to the larger number of black hole masses from
reverberation mapping, to the higher quality optical measurements and fitting
for the other mass measurements techniques, and also to the larger number of
observations used (a factor $\sim 2$ larger than Ponti et al., 2012), which
helped to refine the computation of $\sigma_{\rm NXS}^{2}$ . Our results are
consistent with the common picture in which, as a general rule, nearby AGNs
display similar patterns of variability once they are rescaled for $M_{\rm
BH}$ and $\lambda_{\rm Edd}$ .

## Acknowledgments

This work was funded by ANID programs FONDECYT Postdoctorado - 3190213 (AT),
3220516 (MT), 3210157(ARL); FONDECYT Regular - 1230345 (CR) and1200495 (FEB);
Millennium Science Initiative Program - ICN12_009 (FEB); CATA-BASAL -
ACE210002 (FEB) and FB210003 (CR, FEB). TL acknowledges support from the
NANOGrav NSF Physics Frontiers Center No. 2020265. BT acknowledges support
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program (grant agreement number 950533) and from
the Israel Science Foundation (grant number 1849/19). TK is supported by JSPS
KAKENHI grant No. 23K13153 and acknowledges support by the Special
Postdoctoral Researchers Program at RIKEN. KO acknowledges support from the
National Research Foundation of Korea (NRF-2020R1C1C1005462) and the Korea
Astronomy and Space Science Institute under the R&D program(Project No.
2023-1-868-03) supervised by the Ministry of Science and ICT. This work is
based on observations obtained with the ESA science mission XMM-Newton , with
instruments and contributions directly funded by ESA Member States and the USA
(NASA). The authors thank the anonymous referee for constructive comments that
have helped in improving the quality of the paper .

## Data Availability

All the data utilized in this paper are publicly available in the XMM-Newton
data archive at https://nxsa.esac.esa.int/nxsa-web/#search. More details of
the observations are listed in Tab. 1.

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##  Appendix A Light Curves

Figure 8: XMM-Newton EPIC-pn light curves (background subtracted) in the
$0.2-10$ keV with a time binning of 100 s are shown for some representative
sources of the sample. From the top to the bottom and from the left to the
right we reported the light curves for: NGC 4593, Mrk 359, NGC 4748, NGC 4593,
Fairall 51, Mrk 817, Mrk 1018, Ark 120, 3C 111, PG 1545+210, PKS 2135$-$147,
4C74+26. The red solid and dashed lines indicate the average and the standard
error of the mean, respectively.

We show in Fig. A the XMM-Newton EPIC-pn light curves (background subtracted)
in the $0.2-10$ keV energy band and with a time binning of 100 s for some
representative sources of the sample:

  * •

NGC 4593, $\log(M_{\rm BH}/M_{\odot})=4.45$;

  * •

Mrk 359, $\log(M_{\rm BH}/M_{\odot})=6.05$;

  * •

NGC 4748, $\log(M_{\rm BH}/M_{\odot})=6.40$;

  * •

NGC 4593, $\log(M_{\rm BH}/M_{\odot})=6.88$;

  * •

Fairall 51, $\log(M_{\rm BH}/M_{\odot})=7.12$;

  * •

Mrk 817, $\log(M_{\rm BH}/M_{\odot})=7.59$;

  * •

Mrk 1018, $\log(M_{\rm BH}/M_{\odot})=7.81$;

  * •

Ark 120, $\log(M_{\rm BH}/M_{\odot})=8.07$;

  * •

3C111 $\log(M_{\rm BH}/M_{\odot})=8.45$;

  * •

PG 1545+210, $\log(M_{\rm BH}/M_{\odot})=8.84$;

  * •

PKS 2135-147, $\log(M_{\rm BH}/M_{\odot})=9.10$;

  * •

4C74p26, $\log(M_{\rm BH}/M_{\odot})=9.83$.

##  Appendix B Relations between the excess variance in the soft, medium and
hard energy band and the AGN physical quantities

We checked the relations between $\sigma_{\rm NXS}^{2}$ in the soft ($0.2-1$
keV), medium ($1-3$ keV) and hard ($3-10$ keV) energy bands with $M_{\rm BH}$
, $L_{\rm 2-10}$ and $\lambda_{\rm Edd}$ to see whether these relations
support the results found in this work or not. For the analysis we applied the
same method described in §3. The best fitting results are shown in Tab. 4. For
both $\sigma_{\rm NXS}^{2}$ vs $M_{\rm BH}$ and $\sigma_{\rm NXS}^{2}$ vs
$L_{\rm 2-10}$ , the relations we found in the different energy bands are
consistent with the ones in the total XMM-Newton energy band (0.2-10 keV). It
is worthwhile to underline that, in the case of the $\sigma_{\rm NXS}^{2}$ vs
$M_{\rm BH}$ relation, the anti-correlation is slightly less significant in
the soft energy band. This can be related to the other result of this paper
according to which the primary continuum and/or of the reflection component
(the spectral component dominating the hard energy band) are increasingly more
variable than the spectral components dominating other energy bands on
timescales shorter than 10 ks. In the case of the $\sigma_{\rm NXS}^{2}$ vs
$\lambda_{\rm Edd}$ we found an anti-correlation which is still not
statistically significant that became less stronger in the medium and hard
energy bands.

Table 4: List of the best-fit relations together with their p-values for the $\sigma_{\rm NXS}^{2}$ in the soft, medium and hard energy bands. The fits are performed in the log-log space using equation Eq. 5.  Relation | Intercept (A) | Slope (B) | Pearson | 1-Pvalue
---|---|---|---|---
$\sigma_{\rm NXS}^{2}$ soft vs $M_{\rm BH}$ | $1.23\pm 1.08$ | $-0.68\pm 0.16$ | -0.61 | 0.79
$\sigma_{\rm NXS}^{2}$ med vs $M_{\rm BH}$ | $2.58\pm 1.45$ | $-0.73\pm 0.25$ | -0.87 | 0.98
$\sigma_{\rm NXS}^{2}$ hard vs $M_{\rm BH}$ | $2.67\pm 1.32$ | $-0.79\pm 0.28$ | -0.89 | 0.98
$\sigma_{\rm NXS}^{2}$ soft vs $L_{\rm 2-10}$ | $16.65\pm 5.06$ | $-0.44\pm 0.11$ | -0.85 | 0.98
$\sigma_{\rm NXS}^{2}$ med vs $L_{\rm 2-10}$ | $12.06\pm 6.00$ | $-0.34\pm 0.14$ | -0.71 | 0.92
$\sigma_{\rm NXS}^{2}$ hard vs $L_{\rm 2-10}$ | $18.06\pm 5.64$ | $-0.47\pm 0.13$ | -0.83 | 0.99
$\sigma_{\rm NXS}^{2}$ soft vs $\lambda_{\rm Edd}$ | $-3.31\pm 0.25$ | $-0.67\pm 0.20$ | -0.34 | 0.57
$\sigma_{\rm NXS}^{2}$ med vs $\lambda_{\rm Edd}$ | $-2.93\pm 0.24$ | $-0.29\pm 0.19$ | -0.29 | 0.33
$\sigma_{\rm NXS}^{2}$ hard vs $\lambda_{\rm Edd}$ | $-2.77\pm 0.23$ | $-0.44\pm 0.19$ | -0.15 | 0.29