11institutetext: Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via
della Ricerca Scientifica 1, I-00133 Roma, Italy
11email<EMAIL_ADDRESS>22institutetext: INAF - Osservatorio
Astronomico di Roma, Via Frascati 33, I-00040 Monte Porzio Catone, Italy
33institutetext: INAF - Istituto di Astrofisica e Planetologia Spaziali, Via
del Fosso del Caveliere 100, I-00133 Roma, Italy 44institutetext: Department
of Astronomy, University of Maryland, College Park, MD 20742, USA
55institutetext: NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA
66institutetext: Dipartimento di Fisica e Astronomia “Augusto Righi”,
Università degli Studi di Bologna, Via Gobetti 93/2, I-40129 Bologna, Italy
77institutetext: INAF – Osservatorio di Astrofisica e Scienza dello Spazio di
Bologna, Via Gobetti 93/3, I-40129 Bologna, Italy 88institutetext:
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Via
della Vasca Navale 84, I-00146 Roma, Italy 99institutetext: INAF –
Osservatorio Astronomico di Padova, Vicolo Osservatorio 5, I-35122 Padova,
Italy 1010institutetext: INAF - Osservatorio Astronomico di Trieste, via G. B.
Tiepolo 11, I–34143 Trieste, Italy 1111institutetext: Instituto de Astrofísica
de Andalucía, IAA-CSIC, Glorieta de la Astronomía s/n, 18008 Granada, Spain
1212institutetext: Space Science Data Center, SSDC, ASI, via del Politecnico
snc, 00133 Roma, Italy 1313institutetext: Núcleo de Astronomía de la Facultad
de Ingeniería, Universidad Diego Portales, Av. Ejército Libertador 441,
Santiago 22, Chile 1414institutetext: Kavli Institute for Astronomy and
Astrophysics, Peking University, Beijing 100871, People’s Republic of China
1515institutetext: INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica
Milano, Via A. Corti 12, I-20133 Milano, Italy

# X-ray spectroscopic survey of highly-accreting AGN

M. Laurenti 1122 E. Piconcelli 22 L. Zappacosta 22 F. Tombesi 11224455 C.
Vignali 6677 S. Bianchi 88 P. Marziani 99 F. Vagnetti 1133 A. Bongiorno 22 M.
Bischetti 1010 A. del Olmo 1111 G. Lanzuisi 77 A. Luminari 2233 R. Middei
221212 M. Perri 221212 C. Ricci 13131414 G. Vietri 221515

Improving our understanding of the nuclear properties of high-Eddington ratio
($\lambda_{\mathrm{Edd}}$) active galactic nuclei (AGN) is necessary since at
this regime the radiation pressure is expected to affect the structure and
efficiency of the accretion disc-corona system. This may cause departures from
the typical nuclear properties of low-$\lambda_{\mathrm{Edd}}$ AGN, which have
been largely studied so far. We present here the X-ray spectral analysis of 14
radio-quiet, $\lambda_{\mathrm{Edd}}\gtrsim 1$ AGN at $0.4\leq z\leq 0.75$,
observed with _XMM-Newton_. Optical/UV data from simultaneous Optical Monitor
observations have been also considered. These quasars have been selected to
have relatively high values of black hole mass ($M_{\mathrm{BH}}\sim
10^{8-8.5}M_{\odot}$) and bolometric luminosity ($L_{\mathrm{bol}}\sim
10^{46}$ erg s-1), in order to complement previous studies of
high-$\lambda_{\mathrm{Edd}}$ AGN at lower $M_{\mathrm{BH}}$ and
$L_{\mathrm{bol}}$. We studied the relation between $\lambda_{\mathrm{Edd}}$
and other key X-ray spectral parameters, such as the photon index ($\Gamma$)
of the power-law continuum, the X-ray bolometric correction
$k_{\mathrm{bol,X}}$ and the optical/UV-to-X-ray spectral index
$\alpha_{\mathrm{ox}}$. Our analysis reveals that, despite the homogeneous
optical and SMBH accretion properties, the X-ray properties of these
high-$\lambda_{\mathrm{Edd}}$ AGN are quite heterogeneous. We indeed measured
values of $\Gamma$ comprised between 1.3 and 2.5, at odds with the
expectations based on previously reported $\Gamma-\lambda_{\mathrm{Edd}}$
relations, by which $\Gamma\geq 2$ would be an ubiquitous hallmark of AGN with
$\lambda_{\mathrm{Edd}}\sim 1$. Interestingly, we found that $\sim 30\%$ of
the sources are X-ray weak, with an X-ray emission about a factor of $\sim
10-80$ fainter than that of typical AGN at similar UV luminosities. The X-ray
weakness seems to be intrinsic and not due to the presence of absorption along
the line of sight to the nucleus. This result may indicate that
high-$\lambda_{\mathrm{Edd}}$ AGN commonly undergo periods of intrinsic X-ray
weakness. Furthermore, results from a follow-up monitoring with _Swift_ of one
of these X-ray weak sources suggest that these periods can last for several
years.

###### Key Words.:

galaxies: active – quasars: general – quasars: supermassive black holes

##  1 Introduction

Active galactic nuclei (AGN) are fuelled by accretion processes onto a central
supermassive black hole (SMBH). The AGN accretion activity is usually
quantified by the Eddington ratio, which is defined as the ratio between the
AGN bolometric luminosity and the Eddington luminosity, i.e.
$\lambda_{\mathrm{Edd}}=L_{\mathrm{bol}}/L_{\mathrm{Edd}}$. The Eddington
ratio is thus intimately connected with the accretion rate $\dot{M}$, since
$\lambda_{\mathrm{Edd}}\propto\dot{M}/M_{\mathrm{BH}}\propto\dot{M}/\dot{M}_{\mathrm{Edd}}$,
where $M_{\mathrm{BH}}$ is the black hole mass,
$\dot{M}_{\mathrm{Edd}}=L_{\mathrm{Edd}}/(\eta c^{2})$ and $\eta$ is the
radiative efficiency.

The bulk of the X-ray spectroscopic studies of AGN is focused on local
($z\lesssim 0.1$) sources with low-to-moderate ($\lesssim 0.3$)
$\lambda_{\mathrm{Edd}}$ (e.g. Nandra & Pounds 1994; Piconcelli et al. 2005;
Bianchi et al. 2009; Ricci et al. 2018). Because of their relative paucity in
the local Universe (e.g. Shankar et al. 2013; Shirakata et al. 2019), highly-
accreting (i.e. $\lambda_{\mathrm{Edd}}\gtrsim 0.5$) AGN have been largely
overlooked so far, with the notable exception of the Narrow Line Seyfert 1
galaxies (NLSy1s; e.g. Brandt et al. 1997; Gallo 2006; Costantini et al. 2007;
Jin et al. 2013; Fabian et al. 2013; Waddell & Gallo 2020). However NLSy1s
represent a peculiar and restricted class of low-mass
($10^{6}-10^{7}\,M_{\odot}$) AGN with the narrowest permitted emission lines
in type-1 AGN samples (FWHM(H$\beta$) ¡ 2000 km s-1; e.g. Osterbrock & Pogge
1985; Marziani et al. 2018, and references therein). In this sense, the NLSy1s
may be a biased population, therefore it would be desirable to expand the
study of the multi-wavelength properties of high-$\lambda_{\mathrm{Edd}}$ AGN
to sources with Balmer lines widths of FWHM $\geq 2000$ km s-1 and hosting
more massive SMBHs. Past and recent works support a close relation between the
prominence of the optical Fe II emission and $\lambda_{\mathrm{Edd}}$ (e.g.
Marziani et al. 2001; Shen & Ho 2014; Dou et al. 2016), with the strong Fe II
emission associated with high Eddington ratio. This result appears to be
independent of the line width.

A clear understanding of the nuclear properties of
high-$\lambda_{\mathrm{Edd}}$ AGN, regardless of their $L_{\mathrm{bol}}$ and
$M_{\rm BH}$, would be extremely important for several reasons, as detailed as
follows. The standard model of accretion discs, developed by Shakura & Sunyaev
(1973), predicts an optically thick, geometrically thin disc which is
radiatively efficient. While this picture may hold for AGN in the
low-$\lambda_{\mathrm{Edd}}$ regime, it is supposed to break down for
high-$\lambda_{\mathrm{Edd}}$ sources, as the radiation pressure gains more
importance with increasing accretion rate, and the circumnuclear region of the
disc thickens vertically. These systems are often called slim discs, as they
are both optically- and geometrically-thick (e.g. Abramowicz et al. 1988; Chen
& Wang 2004; Sadowski et al. 2011). Slim discs are also expected to have a low
radiative efficiency because of the photon trapping effect. The diffusion
timescale of photons in optically- and geometrically-thick discs may be longer
than the accretion timescale and they are advected into the central SMBH with
the infalling material. This implies that the observed disc luminosity ($\sim
L_{\mathrm{bol}}$) and, in turn, the $\lambda_{\mathrm{Edd}}$ are expected to
saturate to a limiting value of approximately $5-10\,L_{\mathrm{Edd}}$ for
steadily increasing accretion rates (e.g. Mineshige et al. 2000). However,
some critical issues on the theoretical side are still unsolved and the
broadband spectral energy distributions of high- and
low-$\lambda_{\mathrm{Edd}}$ AGN do not appear to exhibit clear differences
(e.g. Castelló-Mor et al. 2017; Liu et al. 2021).

Furthermore, the study of high-$\lambda_{\mathrm{Edd}}$ AGN is important in
terms of their cosmological implications. In fact, in the past few years there
has been an increasing effort to detect quasars (i.e. AGN with
$L_{\mathrm{bol}}$ $>$ 1046 erg s-1; QSOs hereafter) shining at $z\sim 6-7$,
when the universe was less than 1 Gyr old (e.g. Wu et al. 2015; Bañados et al.
2016; Mazzucchelli et al. 2017). The vast majority of these QSOs typically
host massive SMBHs (e.g. $M_{\mathrm{BH}}\geq 10^{9}\,M_{\odot}$) and the
process that allowed black hole seeds to grow up to billion solar masses in
such a relatively short amount of time is still debated. Several scenarios
have been proposed (see, e.g., the review in Valiante et al. 2017) and it has
been suggested that these SMBHs may evolve via gas accretion at a rate equal
or above the Eddington rate.

Table 1: XMM-Newton sample of high-$\lambda_{\mathrm{Edd}}$ QSOs and their
general properties.

Obj. ID SDSS Name RA Dec $z$ $\log{(L_{\mathrm{bol}}/\mathrm{erg\,s}^{-1})}$
$\log{(M_{\mathrm{BH}}/M_{\odot})}$ $\lambda_{\mathrm{Edd}}$
$N_{\mathrm{H,gal}}$ (1) (2) (3) (4) (5) (6) (7) (8) (9) J0300$-$08
J030000.01$-$080356.9 03 00 00.01 $-$08 03 56.90 0.562 46.5 8.4 1.1 5.93
J0809+46 J080908.13+461925.6 08 09 08.10 +46 19 25.30 0.657 46.5 8.5 0.9 3.60
J0820+23 J082024.21+233450.4 08 20 24.20 +23 34 50.20 0.47 46.0 7.9 1.0 3.88
J0940+46 J094033.75+462315.0 09 40 33.80 +46 23 15.00 0.696 46.4 8.3 1.1 1.06
J1048+31 J104817.98+312905.8 10 48 18.00 +31 29 05.80 0.452 46.2 8.1 1.0 2.13
J1103+41 J110312.93+414154.9 11 03 12.00 +41 41 55.00 0.402 46.4 8.3 1.0 0.81
J1127+11 J112756.76+115427.1 11 27 56.80 +11 54 27.00 0.51 46.3 8.1 1.1 3.17
J1127+64 J112757.41+644118.4 11 27 57.40 +64 41 18.20 0.695 46.4 8.2 1.2 0.79
J1206+41 J120633.07+412536.1 12 06 33.10 +41 25 35.80 0.554 46.3 8.3 0.9 1.60
J1207+15 J120734.62+150643.6 12 07 34.60 +15 06 43.60 0.75 46.5 8.4 0.9 2.26
J1218+10 J121850.51+101554.1 12 18 50.50 +10 15 54.00 0.543 46.2 8.0 1.2 1.72
J1245+33 J124511.25+335610.1 12 45 11.30 +33 56 10.10 0.711 46.6 8.5 1.1 1.20
J1301+59 J130112.93+590206.7 13 01 12.93 +59 02 06.70 0.476 46.7 8.5 1.2 1.40
J1336+17 J133602.01+172513.0 13 36 02.00 +17 25 13.10 0.552 46.6 8.5 0.9 1.64
• Notes: (1) Abbreviated object ID; (2) SDSS IAU name; (3) right ascension
(hours); (4) declination (degrees); (5) redshift from MS14; (6) bolometric
luminosity (typical error of $<0.2$ dex) from the Runnoe et al. (2012)
relation; (7) black hole mass (typical error of $0.3-0.4$ dex; see e.g.
Rakshit et al. 2020) from the Vestergaard & Peterson (2006) single-epoch
virial relation; (8) Eddington ratio; (9) Galactic column density (1020 cm-2)
from the full-sky H I map by HI4PI Collaboration et al. (2016).

Finally, a high $\lambda_{\mathrm{Edd}}$ is invoked as a key ingredient to
launch powerful nuclear outflows (e.g. Proga 2005; Zubovas & King 2013; King &
Pounds 2015) which could be capable of controlling the growth and evolution of
the host galaxy (e.g. Hopkins & Elvis 2010; Fiore et al. 2017). These outflows
are believed to deposit large amounts of energy and momentum into the
interstellar medium (e.g. Zubovas & King 2012) and affect the host galaxy gas
reservoir available for both star formation and SMBH accretion, offering a
possible explanation for the observed $M_{\mathrm{BH}}-\sigma$ relation (e.g.
Ferrarese & Merritt 2000). Accordingly, the AGN feedback mechanism should
manifest itself in full force in high-$\lambda_{\mathrm{Edd}}$ sources, making
them the ideal laboratory for probing the real impact of nuclear activity on
the evolution of massive galaxies (e.g. Reeves et al. 2009; Nardini et al.
2015; Tombesi et al. 2015; Marziani et al. 2016; Bischetti et al. 2017, 2019;
Laurenti et al. 2021).

The study of the X-ray properties of high-$\lambda_{\mathrm{Edd}}$ AGN allows
to enlarge the dynamic range of the relations involving
$\lambda_{\mathrm{Edd}}$ and X-ray spectral parameters and luminosity, by
significantly populating the poorly-explored extreme end of the
$\lambda_{\mathrm{Edd}}$ distribution and, thus, imposing crucial constraints
on the strength of any possible correlation. A long-standing issue concerning
X-ray properties of high-$\lambda_{\mathrm{Edd}}$ AGN is the existence of a
correlation between $\Gamma$ and $\lambda_{\mathrm{Edd}}$, whereby a very
steep (i.e. $\Gamma>$ 2) continuum is a prerogative of sources with
$\lambda_{\mathrm{Edd}}\gtrsim 0.3$ (e.g. Shemmer et al. 2008; Risaliti et al.
2009; Brightman et al. 2013; Fanali et al. 2013; Huang et al. 2020; Liu et al.
2021). This correlation has been explained in terms of an enhanced UV emission
from the accretion disc due to the high accretion rate, which leads to an
increase of the radiative cooling of the X-ray corona (and a decrease of the
electron temperature) and, in turn, to a steepening of the emerging X-ray
continuum. The discovery of such a strong dependence of $\Gamma$ on
$\log\lambda_{\mathrm{Edd}}$, i.e. $\Gamma\sim
0.3\,\times\,\log{\lambda_{\mathrm{Edd}}}+2$ (e.g. Shemmer et al. 2008;
Risaliti et al. 2009; Brightman et al. 2013), has encountered an immediate
interest for a possible application in X-ray extragalactic surveys, as it
would allow to estimate the SMBH mass of AGN with reliable measurement of
$\Gamma$, including type-2 AGN, for which commonly-used “single-epoch”
$M_{\mathrm{BH}}$ estimators are not applicable. However, Trakhtenbrot et al.
(2017) questioned the existence of a strong correlation between $\Gamma$ and
$\lambda_{\mathrm{Edd}}$ reporting only a very weak correlation by analyzing a
large sample of low-$z$ ($0.01<z<0.5$) sources with broadband X-ray spectra
from the BAT AGN Spectroscopic Survey.

In addition, Lusso et al. (2010) studied the hard X-ray bolometric correction,
$k_{\mathrm{bol,X}}=L_{\mathrm{bol}}/L_{\mathrm{2-10\,keV}}$, and the
optical/UV to X-ray spectral slope $\alpha_{\mathrm{ox}}$ of AGN in the COSMOS
survey as a function of $\log\lambda_{\mathrm{Edd}}$. They claimed for the
existence of a linear relation between the logarithm of the Eddington ratio
and the two X-ray quantities. Although affected by large scatter, these
relations lend support to a more physically-motivated scenario for the
increase of $k_{\mathrm{bol,X}}$ as a function of $L_{\mathrm{bol}}$ (Marconi
et al. 2004; Martocchia et al. 2017; Duras et al. 2020) which suggests an
X-ray corona-accretion disc system with different properties in the most
luminous (i.e. highly-accreting) AGN, producing a relatively weaker X-ray
emission (as compared to the optical-UV emission from the accretion disc) than
in “standard” AGN. A more refined description of the
$k_{\mathrm{bol,X}}-\lambda_{\mathrm{Edd}}$ relation has been recently
provided by Duras et al. (2020), who found
$k_{\mathrm{bol,X}}\propto(\lambda_{\mathrm{Edd}})^{0.61}$ once a sizeable
number of highly-accreting QSOs at $z\sim 2-4$ was included. However, a full
understanding of the quantitative behaviour of these relations, along with the
$\Gamma-\lambda_{\mathrm{Edd}}$ one, and possible trends with
$M_{\mathrm{BH}}$ or ${L_{\mathrm{bol}}}$ is still hampered by the lack of a
large number of sources at $\lambda_{\mathrm{Edd}}\gtrsim 0.9$. Indeed, the
bulk of AGN that have been considered so far are accreting at low-to-moderate
$\lambda_{\mathrm{Edd}}$ regimes ($\lesssim 0.3$).

To mitigate this bias, we present the X-ray spectroscopy of a sample of QSOs
outside the local Universe ($z>0.1$) with $\lambda_{\mathrm{Edd}}\gtrsim 1$
and $M_{\mathrm{BH}}$ exceeding $\sim 10^{8}M_{\odot}$. This paper is
organised as follows. We describe the general properties of our sample in
Sect. 2, as well as the XMM-Newton observations and data reduction. Spectral
fits in the X-ray band and optical/UV photometry are detailed in Sect. 3. We
then present our results in Sect. 4, which are discussed in Sect. 5. A
$\Lambda$CDM cosmology ($H_{0}=67.7$ km s-1 Mpc-1,
$\Omega_{\mathrm{m}}=0.307$, $\Omega_{\Lambda}=0.693$) from _Planck_ 2015 data
is adopted throughout the paper (Planck Collaboration et al. 2016). All errors
are quoted at $68\%$ confidence level ($\Delta{C}=1$; Avni 1976; Cash 1979).

Table 2: Journal of the _XMM-Newton_ observations.

Obj. ID Obs. ID $t_{\mathrm{net}}$ $E|_{\mathrm{h}}$ cts—bb (cts—h) (1) (2)
(3) (4) (5) J0300–08 0802220401 33.1/44.9/45.5 1.28 74/14/34 (24/4/9) J0809+46
0843830101 11.5/16.3/16.4 1.21 323/93/108 (187/59/73) J0820+23 0843830201
9.9/16.4/16.4 1.36 507/223/244 (114/63/74) J0940+46 0843830301 13.0/19.4/19.5
1.18 1831/505/469 (224/94/84) J1048+31 0843830401 8.5/16.4/16.4 1.38
392/183/203 (172/89/118) J1103+41 0843830501 13.7/16.8/16.8 1.43 999/269/270
(161/57/63) J1127+11 0843830601 2.5/12.5/13.8 1.32 380/504/568 (69/138/145)
J1127+64 0843830701 16.5/24.7/24.7 1.18 51/20/30 (8/5/15) J1206+41 0843830801
4.1/10.9/11.7 1.29 230/114/112 (26/24/32) J1207+15 0843830901 10.8/16.4/16.4
1.14 948/331/321 (298/146/137) J1218+10 0843831001 4.1/18.7/17.8 1.30
449/340/445 (54/68/64) J1245+33 0843831101 12.1/16.1/16.1 1.17 58/15/13
(22/6/4) J1301+59 0304570101 8.5/11.2/12.1 1.36 4037/1336/1297 (696/337/339)
J1336+17 0843831201 4.8/11.8/11.8 1.29 1801/1070/1205 (574/479/546) • Notes:
(1) Abbreviated object ID; (2) _XMM-Newton_ observation ID; (3) net exposure
for the pn/MOS1/MOS2 cameras (ks); (4) lower observer-frame bound of the hard
X-ray energy range adopted to constrain the slope of the underlying primary
continuum, $E|_{\mathrm{h}}=2/(1+z)$ energy (keV); (5) net counts of the
pn/MOS1/MOS2 cameras in the broadband (hard) $E=0.3-10$ keV ($E>2$ keV rest
frame) observer-frame energy interval.

##  2 Observations and data reduction

In this work we present the _XMM-Newton_ (Jansen et al. 2001) observations of
a sample of 14 AGN drawn from a larger sample of high-$\lambda_{\mathrm{Edd}}$
radio-quiet QSOs reported in Marziani & Sulentic (2014), hereinafter MS14.
Specifically, they collected a sample of 43 radio-quiet QSOs in the range
$z\sim 0.4-0.75$ with H$\beta$ coverage from a good-quality SDSS spectrum (see
Table 1 in MS14). This ensures an accurate measure of $M_{\mathrm{BH}}$ and,
in turn, $\lambda_{\mathrm{Edd}}$, being the full width at half maximum (FWHM)
of the H$\beta$ emission line a more reliable single-epoch $M_{\mathrm{BH}}$
estimator than the FWHM of C IV (e.g. Baskin & Laor 2005; Vietri et al. 2018,
2020), which is typically used for QSOs at $z\gg 1$. The $M_{\mathrm{BH}}$ of
these MS14 QSOs are in the range $10^{8-8.5}M_{\odot}$, while their
$L_{\mathrm{5100\,\AA}}$-based $L_{\mathrm{bol}}$ are clustered around a few
$10^{46}$ erg s-1, with $\lambda_{\mathrm{Edd}}$ values spanning from $\sim
0.5$ to $\sim 2$. Such highly-accreting objects are also characterised by
strong Fe II emission with
$R_{\mathrm{Fe{\,\scaleto{\mathrm{II}}{3pt}}}}=\mathrm{EW(Fe{\,\scriptstyle
II}}\,\lambda 4570)/\mathrm{EW(H\beta)}$ $>1$, and MS14 dubbed them as extreme
Population A (xA) QSOs. For our XMM-Newton observing program, we considered
the 14 xA QSOs from the MS14 sample with $\lambda_{\mathrm{Edd}}\gtrsim 1$ and
an estimated $L_{\mathrm{bol}}\geq 10^{46}$ erg s-1.

The main properties of our high-$\lambda_{\mathrm{Edd}}$ QSO sample are listed
in Tab. 1. The bolometric luminosities are calculated by adopting the relation
$\log{L_{\mathrm{bol}}}=4.891+0.912\log(\lambda\,L_{\lambda})$, with
$\lambda=5100\,$Å, from Runnoe et al. (2012). The $M_{\mathrm{BH}}$ are
estimated by using the single-epoch virial relation based on the FWHM of the
H$\beta$ and the monochromatic luminosity at $\lambda=5100\,\AA$, as described
by Vestergaard & Peterson (2006):
$\log{M_{\mathrm{BH}}(\mathrm{H}\beta)}=\log\big{[}\mathrm{FWHM}/1000\,\mathrm{km\,s}^{-1}\big{]}^{2}+\log\big{[}\lambda
L_{\lambda}/10^{44}\mathrm{erg\,s}^{-1}\big{]}^{0.5}+6.91$. The virial mass
estimates based on single-epoch spectra are typically dominated by the
systematic uncertainty of $\sim 0.3-0.4$ dex (e.g. Collin et al. 2006; Shen
2013; Rakshit et al. 2020), while the uncertainty associated with the Runnoe
et al. (2012) relation for $\log{L_{\mathrm{bol}}}$ is smaller than $0.2$ dex
for the median value of $L_{\mathrm{5100\,\AA}}\sim 7\times 10^{41}$ erg s-1
Å-1 of our sample. The errors on $\log M_{\mathrm{BH}}$ and
$\log{L_{\mathrm{bol}}}$ then propagate to uncertainties in
$\log{\lambda_{\mathrm{Edd}}}$.

_XMM-Newton_ observed 12 out of the 14 high-$\lambda_{\mathrm{Edd}}$ AGN
during the AO 18 Cycle campaign (PI: E. Piconcelli). The remaining two
sources, namely J1301$+$59 and J0300$-$08, had publicly available _XMM-Newton_
observations at the time of our investigation. Detailed information on the
_XMM-Newton_ data for each source in the sample are listed in Tab. 2. Raw data
from all sources have been retrieved from the _XMM-Newton_ Science Archive and
then processed using the _XMM-Newton_ Science Analysis System (SAS v18.0.0)
with the latest available calibration files. We take advantage of the full
potential of _XMM-Newton_ in the X-ray energy interval $E=0.3-10$ keV
(observer-frame) by collecting data by its primary instrument, i.e. the
European Photon Imaging Camera _EPIC_ , equipped with the three X-ray CCD
cameras, namely the pn (Strüder et al. 2001) and the two MOS detectors (Turner
et al. 2001). Data reduction, filtering of high background periods and
spectral extraction were performed according to the standard procedures
described in the SAS web page111https://www.cosmos.esa.int/web/xmm-newton/sas-
threads.. For each object, we chose a circular region of radius $\sim 10-20$
arcsec for the source extraction, and a nearby, larger source-free circular
region for the background. The spectra were binned to ensure at least one
count per bin and modelled within the XSPEC v12.11.1c package by minimising
the C-statistic (Cash 1979).

In addition, our study is complemented with the analysis of the simultaneous
optical/UV data taken with the Optical Monitor (OM; Mason et al. 2001). Raw OM
data are converted into science products by using the SAS task omichain. We
use the task om2pha to convert the OM photometric points into a suitable
format for XSPEC (Arnaud 1996).

##  3 Data analysis

###  3.1 X-ray spectroscopy

We adopted the following procedure to analyse the X-ray spectrum of each
source in our sample. Data from EPIC-pn, MOS1 and MOS2 were always considered
simultaneously and fitted together, taking into account an intercalibration
constant ($<10\%$) between the three instruments. We first ignored all the
data outside the $E=0.3-10$ keV observer-frame energy interval. Initially,
only data in the intrinsic hard X-ray range, i.e. with $E>2$ keV in the rest
frame of the source, corresponding to $E>2/(1+z)$ keV in the observer frame,
were considered. This hard X-ray portion of the spectrum was modelled with a
power law modified by Galactic absorption to constrain the slope of the
underlying primary continuum without any contamination from the likely soft
excess component emerging at lower energies, which may strongly affect the
measurement of the photon index $\Gamma$. From this simple spectral fit, we
found the bulk of $\Gamma$ values to be comprised between $\sim 1.7$ and
$2.3$, which is the typical range of radio-quiet AGN (e.g. Piconcelli et al.
2005; Gliozzi & Williams 2020). A flatter hard-band continuum (i.e.
$\Gamma\sim 1.2-1.3$) was found for two sources, namely J1048+31 and J1245+33,
although the latter was affected by large uncertainty. Finally, the $\Gamma$
value of J0300$-$08 resulted basically unconstrained due to the limited
statistics.

Table 3: Best-fitting spectral parameters for the broadband ($E=0.3-10$ keV)
X-ray spectra of each sample source, measured by assuming a model consisting
of a power law plus a blackbody component, modified by Galactic absorption.

Obj. ID $\Gamma$ $kT_{\mathrm{bb}}$ (C-stat/d.o.f.) (d.o.f.)
$F_{0.5-2\,\mathrm{keV}}$ $F_{2-10\,\mathrm{keV}}$ $L_{0.5-2\,\mathrm{keV}}$
$L_{2-10\,\mathrm{keV}}$ $k_{\mathrm{bol,X}}$ $R_{\mathrm{S/P}}$ (1) (2) (3)
(4) (5) (6) (7) (8) (9) (10) J0300–08 1.9 (fixed) $140\pm 30$ 1.22 (60)
$3.7_{-0.7}^{+0.3}\times 10^{-15}$ $4.3_{-0.5}^{+0.9}\times 10^{-15}$
$9.4_{-0.7}^{+0.8}\times 10^{42}$ $6_{-1}^{+1}\times 10^{42}$ $5100\pm 2500$
1.24 J0809+46a $2.10\pm 0.06$ – 1.16 (169) $2.8_{-0.2}^{+0.1}\times 10^{-14}$
$8.6_{-0.9}^{+0.7}\times 10^{-14}$ $2.4_{-0.4}^{+0.3}\times 10^{44}$
$2.3_{-0.4}^{+0.2}\times 10^{44}$ $150\pm 70$ – J0820+23 $2.26\pm 0.09$ $80\pm
20$ 1.08 (173) $8.3_{-0.6}^{+0.3}\times 10^{-14}$ $7.1_{-0.6}^{+0.5}\times
10^{-14}$ $1.13_{-0.07}^{+0.07}\times 10^{44}$ $6.9_{-0.5}^{+0.5}\times
10^{43}$ $150\pm 70$ 0.28 J0940+46 $2.3\pm 0.1$ $134\pm 4$ 0.90 (205)
$1.18_{-0.03}^{+0.03}\times 10^{-13}$ $5.3_{-0.6}^{+0.5}\times 10^{-14}$
$6.7_{-0.2}^{+0.2}\times 10^{44}$ $1.4_{-0.2}^{+0.2}\times 10^{44}$ $200\pm
90$ 2.59 J1048+31b $1.27\pm 0.08$ $120\pm 20$ 0.81 (210)
$5.1_{-0.4}^{+0.2}\times 10^{-14}$ $1.8_{-0.2}^{+0.2}\times 10^{-13}$
$4.4_{-0.3}^{+0.3}\times 10^{43}$ $1.1_{-0.1}^{+0.1}\times 10^{44}$ $140\pm
70$ 0.55 J1103+41 $1.6\pm 0.1$ $122\pm 6$ 0.84 (196) $6.9_{-0.2}^{+0.2}\times
10^{-14}$ $7.7_{-0.9}^{+0.7}\times 10^{-14}$ $6.9_{-0.3}^{+0.3}\times 10^{43}$
$4.1_{-0.5}^{+0.6}\times 10^{43}$ $600\pm 300$ 2.42 J1127+11 $2.3\pm 0.1$
$170\pm 20$ 0.91 (192) $2.24_{-0.06}^{+0.05}\times 10^{-13}$
$1.4_{-0.2}^{+0.2}\times 10^{-13}$ $3.6_{-0.3}^{+0.4}\times 10^{44}$
$1.7_{-0.3}^{+0.3}\times 10^{44}$ $100\pm 50$ 0.52 J1127+64 $1.5\pm 0.4$
$110\pm 20$ 0.77 (53) $3.8_{-1.3}^{+0.2}\times 10^{-15}$ $7_{-4}^{+2}\times
10^{-15}$ $2.4_{-0.7}^{+0.3}\times 10^{43}$ $1.2_{-0.5}^{+0.7}\times 10^{43}$
$2000_{-1300}^{+1500}$ 4.37 J1206+41 $1.4\pm 0.2$ $128\pm 7$ 0.70 (97)
$6.2_{-0.5}^{+0.2}\times 10^{-14}$ $8_{-2}^{+2}\times 10^{-14}$
$1.90_{-0.07}^{+0.09}\times 10^{44}$ $8_{-2}^{+2}\times 10^{43}$ $300\pm 100$
5.52 J1207+15 $2.0\pm 0.1$ $160^{+30}_{-50}$ 0.97 (224)
$1.04_{-0.03}^{+0.02}\times 10^{-13}$ $1.1_{-0.1}^{+0.1}\times 10^{-13}$
$3.9_{-0.4}^{+0.4}\times 10^{44}$ $3.1_{-0.4}^{+0.5}\times 10^{44}$ $100\pm
50$ 0.45 J1218+10 $2.5\pm 0.2$ $160\pm 10$ 0.98 (162)
$1.43_{-0.04}^{+0.05}\times 10^{-13}$ $5.1_{-0.8}^{+0.6}\times 10^{-14}$
$3.1_{-0.3}^{+0.3}\times 10^{44}$ $8_{-2}^{+2}\times 10^{43}$ $200\pm 100$
0.97 J1245+33 1.3 (fixed) $120\pm 20$ 0.96 (44) $5.2_{-1.0}^{+0.4}\times
10^{-15}$ $1.3_{-0.2}^{+0.2}\times 10^{-14}$ $2.6_{-0.1}^{+0.3}\times 10^{43}$
$2.1_{-0.4}^{+0.4}\times 10^{43}$ $2000\pm 1000$ 2.93 J1301+59 $2.01\pm 0.05$
$122\pm 3$ 0.98 (295) $4.65_{-0.09}^{+0.06}\times 10^{-13}$
$3.9_{-0.2}^{+0.2}\times 10^{-13}$ $7.0_{-0.2}^{+0.2}\times 10^{44}$
$3.6_{-0.2}^{+0.2}\times 10^{44}$ $140\pm 60$ 1.25 J1336+17 $1.86\pm 0.04$
$100^{+40}_{-30}$ 1.01 (336) $4.3_{-0.1}^{+0.1}\times 10^{-13}$
$6.4_{-0.2}^{+0.2}\times 10^{-13}$ $6.0_{-0.2}^{+0.2}\times 10^{44}$
$7.8_{-0.3}^{+0.3}\times 10^{44}$ $50\pm 20$ 0.10 • Notes: (1) Abbreviated
object ID; (2) photon index of the power-law continuum component; (3) rest-
frame temperature of the blackbody component (eV); (4) ratio between the
C-stat value of the spectral fit and the degrees of freedom; (5) flux in the
$0.5-2$ keV band (erg cm-2 s-1); (6) flux in the $2-10$ keV band (erg cm-2
s-1); (7) luminosity in the $0.5-2$ keV band (erg s-1); (8) luminosity in the
$2-10$ keV band (erg s-1); (9) X-ray bolometric correction; (10) value of the
$R_{\mathrm{S/P}}$ parameter. a the best-fit model of J0809$+$46 is a power
law modified by a warm absorber component (see Sect. 3.1 and Fig. 11 of
Appendix A) and it does not require a blackbody component. b the best-fit
model of J1048$+$31 also includes a Gaussian emission line component, see Fig.
11 of Appendix A.

Table 4: Optical/UV properties of the QSOs. Monochromatic luminosities,
corrected for both Galactic and local dust extinction, are reported in units
of $10^{30}$ erg s-1 Hz-1.

Obj. ID E$(B-V)_{\mathrm{int}}$ $L_{\mathrm{B}}$ $L_{\mathrm{U}}$
$L_{\mathrm{UVW1}}$ $L_{\mathrm{UVM2}}$ $L_{\mathrm{UVW2}}$
$L_{\mathrm{2500\,\AA}}$ $\alpha_{\mathrm{ox}}$ (1) (2) (3) (4) (5) (6) (7)
(8) (9) J0300–08 $<0.01$ – $5.68\pm 0.05$ $5.54\pm 0.04$ $4.6\pm 0.1$ $4.2\pm
0.1$ $5.78\pm 0.05$ $-2.28\pm 0.03$ J0809+46 $<0.01$ $14.6\pm 0.1$ $13.7\pm
0.1$ $14.0\pm 0.1$ – – $14.3\pm 0.1$ $-1.78\pm 0.02$ J0820+23 $<0.01$ $1.91\pm
0.06$ $1.57\pm 0.04$ $1.61\pm 0.03$ – – $1.65\pm 0.04$ $-1.60\pm 0.01$
J0940+46 0.06 $4.82\pm 0.09$ $4.11\pm 0.06$ $4.02\pm 0.07$ – – $4.66\pm 0.08$
$-1.65\pm 0.02$ J1048+31 0.20 $2.54\pm 0.08$ $1.87\pm 0.06$ $1.73\pm 0.06$ – –
$1.99\pm 0.05$ $-1.71\pm 0.02$ J1103+41 0.05 $4.89\pm 0.04$ $4.29\pm 0.03$
$3.80\pm 0.04$ – – $4.33\pm 0.03$ $-1.94\pm 0.02$ J1127+11 $<0.01$ – $2.47\pm
0.03$ $2.41\pm 0.04$ $2.33\pm 0.06$ – $2.51\pm 0.03$ $-1.51\pm 0.02$ J1127+64
$<0.01$ $4.14\pm 0.07$ $3.49\pm 0.05$ $3.39\pm 0.05$ – – $3.99\pm 0.06$
$-2.16\pm 0.04$ J1206+41 0.08 – $1.79\pm 0.04$ $1.52\pm 0.03$ – – $2.01\pm
0.05$ $-1.73\pm 0.04$ J1207+15 0.13 $5.2\pm 0.2$ $4.7\pm 0.1$ $3.8\pm 0.1$ – –
$5.1\pm 0.2$ $-1.57\pm 0.02$ J1218+10 $<0.01$ $5.38\pm 0.07$ – $3.45\pm 0.04$
– – $4.60\pm 0.05$ $-1.71\pm 0.03$ J1245+33 $<0.01$ $10.5\pm 0.1$ $10.32\pm
0.09$ $9.9\pm 0.1$ – – $10.5\pm 0.1$ $-2.25\pm 0.04$ J1301+59 $<0.01$ –
$12.51\pm 0.03$ $12.34\pm 0.04$ $12.17\pm 0.06$ – $12.58\pm 0.03$ $-1.70\pm
0.01$ J1336+17 $<0.01$ $10.1\pm 0.1$ $9.02\pm 0.08$ $8.9\pm 0.1$ – – $9.49\pm
0.09$ $-1.55\pm 0.01$ • Notes: (1) Abbreviated object ID; (2) internal
reddening; (3) $B$ rest-frame luminosity; (4) $U$ rest-frame luminosity; (5)
$UVW1$ rest-frame luminosity; (6) $UVM2$ rest-frame luminosity; (7) $UVW2$
rest-frame luminosity; (8) $2500\,\AA$ rest-frame luminosity; (9)
$\alpha_{\mathrm{ox}}$.

We then extended our spectral analysis to the whole $E=0.3-10$ keV energy
interval by including the soft X-ray portion. The extrapolation of the model
to soft energies showed for all but one sources (J0809$+$46)222A detailed
spectral analysis of J0809+46 will be presented in a forthcoming paper
(Piconcelli et al., _in prep._). The X-ray spectrum of this source shows no
soft excess: at low energies its spectrum is dominated by an ionised
absorption component with high column density (see Fig. 11). the presence of a
soft excess component in their X-ray spectra and we tried to model this
feature by considering two different spectral components, i.e., a blackbody or
an additional power law, both modified by Galactic absorption. Clearly, we are
aware that both models merely provide a phenomenological explanation of the
soft excess, whose physical origin is still debated (e.g. Sobolewska & Done
2007; Fukumura et al. 2016; Petrucci et al. 2018; Middei et al. 2020).
However, the blackbody model provided the best description of the soft excess
in terms of C-statistic for all the spectra.

From our broadband spectral analysis we obtained the best-fit values of the
photon index $\Gamma$ and the blackbody temperature $kT_{\mathrm{bb}}$ as
listed in Tab. 3. We find that our high-$\lambda_{\mathrm{Edd}}$ AGN do have
$kT_{\mathrm{bb}}$ which are consistent with those typically measured for
type-1 AGN (e.g. Piconcelli et al. 2005; Bianchi et al. 2009). For two
sources, J1245$+$33 and J0300$-$08, we fixed the photon index of the power-law
component due to the limited quality of spectral data. Specifically, for
J1245+33 we fixed $\Gamma=1.3$, which is the value derived from power-law fit
to the hard X-ray portion of the spectrum, while for J0300$-$08 we adopted the
canonical value of $\Gamma$ = 1.9 found for radio-quiet quasars in previous
works (e.g. Reeves & Turner 2000; Piconcelli et al. 2005), since the photon
index was largely unconstrained in this case. No significant evidence for Fe
K$\alpha$ emission was found in the spectra, except for J1048+31, for which we
report the presence of an emission line at $E=6.52_{-0.04}^{+0.05}$ keV with a
rest-frame equivalent width of $EW=400_{-140}^{+150}$ eV (see Fig. 11 of
Appendix A).

Figure 1: Distribution of
$R_{\mathrm{S/P}}=L^{\mathrm{0.5-2\,\mathrm{keV}}}_{\mathrm{BB}}/L^{\mathrm{0.5-2\,\mathrm{keV}}}_{\mathrm{PL}}$
for our AGN sample (in black) and the sample of PG QSOs from Piconcelli et al.
(2005) (in gold). $R_{\mathrm{S/P}}$ measures the relative strength between
the luminosity of the blackbody and power law component in the $E=0.5-2$ keV
energy band.

Also shown in Fig. 11 are the X-ray spectra, and the corresponding best fit,
of all the 14 high-$\lambda_{\mathrm{Edd}}$ AGN analysed in this work.
Moreover, in Appendix B (see also Fig. 12) we present the results of the X-ray
spectral analysis of some interesting serendipitous sources located in the
_XMM-Newton_ field of view of our targeted high-$\lambda_{\mathrm{Edd}}$ AGN.
Table 3 also lists the values of the flux and luminosity of the 14
high-$\lambda_{\mathrm{Edd}}$ QSOs in both the soft ($E=0.5-2$ keV) and the
hard X-ray ($E=2-10$ keV) band, as well as the X-ray bolometric correction
$k_{\mathrm{bol,X}}$ resulting from this broadband spectral fit. Furthermore,
we also calculated the parameter
$R_{\mathrm{S/P}}=L^{\mathrm{0.5-2\,\mathrm{keV}}}_{\mathrm{BB}}/L^{\mathrm{0.5-2\,\mathrm{keV}}}_{\mathrm{PL}}$
which is a proxy of the relative strength between the luminosity of the
blackbody and power law components (i.e. the soft excess and the primary
continuum, respectively) in the $E=0.5-2$ keV observer-frame energy band.
Figure 1 shows the distribution of $R_{\mathrm{S/P}}$ for our sample of
high-$\lambda_{\mathrm{Edd}}$ AGN compared to that derived by Piconcelli et
al. (2005) for a sample of Palomar-Green (PG) QSOs with
$-1.2\leq\log{\lambda_{\mathrm{Edd}}}\leq 0.7$. Such a sample of PG QSOs
contains 38 bright objects333We do not consider PG 1226+023, aka 3C 273, and
PG 1115+080: the former is a well-known blazar-line AGN, while the latter is a
lensed QSO at redshift $z\sim 1.7$. with an absolute magnitude $M_{B}<-23$ in
the Johnson’s $B$ band, and with redshift $z\lesssim 1.5$, which is broadly
comparable to that of our high-$\lambda_{\mathrm{Edd}}$ AGN. It is worth
noting that the distribution of $R_{\mathrm{S/P}}$ for the PG QSOs does not
extend to values larger than $\sim 3$ and is largely dominated by sources with
$R_{\mathrm{S/P}}$ $<$ 1, while for our sample the distribution appears
smoother and much broader, by reaching values of $R_{\mathrm{S/P}}\sim 6$.

###  3.2 Optical/UV photometry

Each source in the sample has at least two available measurements of
optical/UV photometry taken with the OM, which are simultaneous to the X-ray
data, with one always collected in the UVW1 (2910 Å) filter.

The other photometric points are distributed among the B (4500 Å), U (3400 Å),
UVM2 (2310 Å), and UVW2 (2120 Å) filters. The extinction-corrected optical/UV
fluxes are derived as follows. We first consider the optical spectrum of each
source from the Sloan Digital Sky Survey (SDSS; York et al. 2000) Legacy
campaign. These spectra are the same used by MS14 and were downloaded from the
SDSS SkyServer web tool444http://skyserver.sdss.org/dr16/en/home.aspx..
Galactic extinction is taken into account by considering the color excess
measurement from the reddening map of Schlafly & Finkbeiner (2011), provided
by the NASA/IPAC Infrared Science Archive (IRSA)
website555https://irsa.ipac.caltech.edu/applications/DUST/.. The spectra are
then corrected for Galactic extinction according to the Milky Way reddening
law of Fitzpatrick (1999), assuming $R_{\mathrm{V}}=3.1$.

We estimate the possible contribution of the host galaxy starlight to the
UV/optical emission by modelling the optical spectrum as a combination of AGN
and galaxy components, following the procedure described in Vagnetti et al.
(2013). We find that the host galaxy contribution is completely negligible for
$\sim 80\%$ of the sample. For the remaining sources (i.e. J1048+31, J1206+41
and J1218+10), this host galaxy contribution results only to be marginal
($\lesssim 10\%$) and can be ignored.

Furthermore, we look for indications of internal reddening by comparing the
SDSS spectra with two different templates: (i) the SDSS composite QSO spectrum
by Vanden Berk et al. (2001); and (ii) a median composite spectrum of xA QSOs
built from the full sample of highly-accreting QSOs in MS14 (see also Marziani
et al., _in prep._). For each source, the SDSS spectrum and the templates are
normalised to unit flux at rest-frame wavelengths corresponding to
$\lambda_{\mathrm{obs}}\geq 8000\,\AA$ in the observer frame. We then apply
the Small Magellanic Cloud (SMC) extinction law by Prevot et al. (1984) to
both templates as shown in Fig. 2. The choice of an SMC-like extinction law
stems from its capability of reproducing dust reddening of QSOs at all
redshifts (e.g. Richards et al. 2003; Hopkins et al. 2004; Bongiorno et al.
2007; Gallerani et al. 2010; Krawczyk et al. 2015). Although the two templates
yield similar results, we adopt the xA QSOs template since it provides a more
accurate description of the optical/UV behaviour of our sources. Even if the
majority of our QSOs do not suffer from significant internal reddening, for a
sizeable fraction of the sample ($\sim 35\%$), the estimated E$(B-V)$ is $\geq
0.05$ and two sources, namely J1048+31 and J1207+15, show an E$(B-V)$ of
$0.13$ and $0.2$, respectively (see Tab. 4). The typical uncertainty on these
reddening estimates is $\sim 0.01$. OM photometric data are corrected for the
effects of both Galactic and internal dust extinction according to the same
procedure.

Finally, the value of the monochromatic UV luminosity $L_{\mathrm{2500\,\AA}}$
for each QSO is obtained by linearly interpolating (or, if necessary,
extrapolating) the OM data points at rest-frame 2500 Å. This is used jointly
with the monochromatic X-ray luminosity at 2 keV estimated from the X-ray
spectra (see Sect. 3.1) to calculate $\alpha_{\mathrm{ox}}$ as:

$\begin{split}\alpha_{\mathrm{ox}}&=\log{\Bigg{[}\frac{L_{\nu}(2\,\mathrm{keV})}{L_{\nu}(2500\,\AA)}\Bigg{]}}\,\Bigg{/}\,\log{\Bigg{[}\frac{\nu(2\,\mathrm{keV})}{\nu(2500\,\AA)}\Bigg{]}}\\\
&\simeq
0.384\,\log{\Bigg{[}\frac{L_{\nu}(2\,\mathrm{keV})}{L_{\nu}(2500\,\AA)}\Bigg{]}}\,.\end{split}$
(1)

We find that in our high-$\lambda_{\mathrm{Edd}}$ AGN sample the bulk of the
$\alpha_{\mathrm{ox}}$ values is around $-1.8$, with a minimum value of
approximately $-2.3$.

Figure 2: SDSS spectrum of J1048+31 at $z=0.452$ (in gold) compared to the
composite template of xA QSOs created by Marziani et al. (_in prep._). The
template (in black) is extinguished according to the SMC extinction law by
Prevot et al. (1984), with progressively increasing values of E$(B-V)$. The
value of E$(B-V)$ for which the reddened template (in red) matches with the
observed SDSS spectrum, corresponds to our estimate of the internal reddening
of the source. J1048+31 has the largest E$(B-V)$ among our sources, i.e.
E$(B-V)$ = 0.2  Figure 3: $\Gamma$ vs $\log{\lambda_{\mathrm{Edd}}}$ for the
high-$\lambda_{\mathrm{Edd}}$ QSOs in our sample (blue triangles). Two
sources, J0300–08 and J1245+33, are excluded since their $\Gamma$ were
unconstrained by _XMM-Newton_ data. J0940+46 and J1127+11 are described by the
same point, since they share the same values of $\Gamma$ and
$\lambda_{\mathrm{Edd}}$. Black solid, dashed and dash-dotted lines indicate
the expected relation from other studies. Gold diamonds represent the sample
of PG QSOs from Piconcelli et al. (2005). Green squares and red circles
indicate the sub- and super-Eddington (i.e. $\lambda_{\mathrm{Edd}}$ $\geq$
0.3) AGN from Liu et al. (2021), respectively.

##  4 Results

###  4.1 X-ray continuum slope

Figure 4: $k_{\mathrm{bol,X}}$ vs $\log{\lambda_{\mathrm{Edd}}}$ for the
high-$\lambda_{\mathrm{Edd}}$ AGN in our sample (blue triangles) compared to
the XMM-COSMOS type-1 AGN (light grey crosses) of Lusso et al. (2010) with
available measurement of Eddington ratio. Black solid line refers to the best-
fit relation from Duras et al. (2020). Black dashed lines describe the
$1\sigma$ spread of the above relation. Green squares and red circles indicate
the sub- and super-Eddington AGN from Liu et al. (2021), respectively.

In Sect. 2 we mentioned that our AGN have been selected to span a narrow
interval of $\lambda_{\mathrm{Edd}}$, with $0.9\leq\lambda_{\mathrm{Edd}}\leq
1.2$. This enables us to investigate the properties of these sources on a
statistical sound basis and reduce the effect of washing out possible
correlations due to considering sources in a wide range of
$\lambda_{\mathrm{Edd}}$. Nonetheless, Fig. 3 shows that the continuum slope
of the high-$\lambda_{\mathrm{Edd}}$ AGN exhibits quite a large scatter. The
values of $\Gamma$ found by our analysis indeed span from $\sim 1.3$ to $\sim
2.5$, with a non-negligible number of sources having quite flat photon indices
$\Gamma\lesssim 1.6$. Furthermore, the flat X-ray continuum does not seem to
be exclusively associated with an X-ray weak state (see Sect. 4.2). Indeed,
out of the four $\Gamma\leq 1.6$ sources in Fig. 3, only two are also X-ray
weak (namely, J1103+41 and J1127+64). Interestingly, the presence of a
consistent fraction ($\sim 30\%$) of high-$\lambda_{\mathrm{Edd}}$ AGN with
flat $\Gamma$ in our sample is at odds with the expectations based on the
popular $\Gamma-\lambda_{\mathrm{Edd}}$ relations reported in previous works
(e.g. Shemmer et al. 2008; Risaliti et al. 2009; Brightman et al. 2013), which
predict steeper $\Gamma$ of $\sim 2-2.3$ for AGN with
$\lambda_{\mathrm{Edd}}\sim 1$.

However, Trakhtenbrot et al. (2017) found that the above relation between
$\Gamma$ and $\lambda_{\mathrm{Edd}}$ could be less steep than expected. They
considered 228 hard X-ray selected AGN from the _Swift_ /BAT AGN spectroscopic
survey (BASS), with redshift $0.01<z<0.5$,
$0.001\lesssim\lambda_{\mathrm{Edd}}\lesssim 1$, and
$6\lesssim\log{(M_{\mathrm{BH}}/M_{\odot})}\lesssim 9.5$, which benefit from
high-quality and broadband X-ray spectral coverage. For their BASS sample,
Trakhtenbrot et al. (2017) found a statistically significant, albeit very weak
correlation between $\Gamma$ and $\lambda_{\mathrm{Edd}}$. To complete the
picture, in a recent paper, Liu et al. (2021) analysed a sample of 47 local,
radio-quiet AGN with an accurate $M_{\mathrm{BH}}$ measurement from
reverberation mapping, which includes both sub- and super-Eddington (i.e.
$\lambda_{\mathrm{Edd}}$ $\geq$ 0.3) sources. They found a significant
correlation between $\Gamma$ and $\lambda_{\mathrm{Edd}}$ for the full sample,
as well as for the super-Eddington subsample, with a slope of $\sim 0.3$,
consistent with that derived by, e.g, Shemmer et al. (2008) and Brightman et
al. (2013). However, it is worth noting that for their study Liu et al. (2021)
did only consider spectra during high-flux states, in case of AGN with
multiple X-ray observations, which may lead to a possible bias in estimating
the slope of the best-fit relation for high-$\lambda_{\mathrm{Edd}}$ sources,
given the trend of steeper-when-brighter typically observed in type-1 AGN
(e.g. Leighly et al. 1996; Nandra et al. 1997; Gibson & Brandt 2012; Gliozzi
et al. 2017; Serafinelli et al. 2017).

Figure 5: _Left panel_ : $\alpha_{\mathrm{ox}}$ vs $\log{L_{\mathrm{UV}}}$ for
our high-$\lambda_{\mathrm{Edd}}$ AGN (blue triangles) compared to the XMM-
COSMOS type-1 AGN (light grey crosses) of Lusso et al. (2010). Black solid
line is the best-fit relation from the same authors. Black dashed lines
describe the $1\sigma$ spread of the above relation. Blue solid line is the
best-fit relation obtained from Martocchia et al. (2017). Green squares and
red circles indicate the sub- and super-Eddington AGN from Liu et al. (2021),
respectively. Purple diamonds are the 13 WISSH QSOs from Zappacosta et al.
(2020). Red solid line marks the reference value for X-ray weakness, i.e.
$\Delta\alpha_{\mathrm{ox}}\leq-0.3$ (Pu et al. 2020). _Right panel_ :
distribution of the same sources described in the adjacent panel, divided in
two regimes of bolometric luminosity, in the
$\alpha_{\mathrm{ox}}-\log{\lambda_{\mathrm{Edd}}}$ plane. For the XMM-COSMOS
AGN sample, only those sources with available measurement of
$\lambda_{\mathrm{Edd}}$ are included. Black solid line indicates the best-fit
relation from Lusso et al. (2010). Black dashed lines describe the $1\sigma$
spread of the above relation. The shaded interval at
$\log{\lambda_{\mathrm{Edd}}}\sim 0$ is included to highlight that AGN with
very similar values of $\lambda_{\mathrm{Edd}}$ can show a widespread
distribution in $\alpha_{\mathrm{ox}}$ due to their different
$L_{\mathrm{bol}}$.

###  4.2 X-ray weakness

Another relevant aspect emerging from the X-ray analysis of our
high-$\lambda_{\mathrm{Edd}}$ AGN sample is shown in Fig. 4. Our sources (blue
triangles) are compared with the $k_{\rm bol,X}$ vs
$\log{\lambda_{\mathrm{Edd}}}$ distribution of the XMM-COSMOS sample of type-1
AGN from Lusso et al. (2010), and the
$k_{\mathrm{bol,X}}-\log{\lambda_{\mathrm{Edd}}}$ best-fit relation from Duras
et al. (2020), which holds for both absorbed (type-2) and unabsorbed (type-1)
AGN, is also reported. The bulk of our sources is on average slightly above
the range $\sim 10-100$ expected for their $\log{\lambda_{\mathrm{Edd}}}$.
Furthermore there are four high-$\lambda_{\mathrm{Edd}}$ AGN ($\sim 30\%$ of
the sample) exhibiting $k_{\mathrm{bol,X}}$ values significantly larger than
expected and spanning a range between $\sim 600-5000$, i.e. larger than the
maximum $k_{\mathrm{bol,X}}$ of $\sim 400$ predicted by the Duras et al.
(2020) relation for the most highly-accreting AGN. These
high-$k_{\mathrm{bol,X}}$ (hence X-ray weak) QSOs are J1103$+$41, J1127$+$64,
J1245$+$33 and J0300–08, with the latter showing a $k_{\mathrm{bol,X}}$ of a
few thousands. The left panel of Fig. 5 shows the distribution of the
high-$\lambda_{\mathrm{Edd}}$ AGN in terms of the $\alpha_{\mathrm{ox}}$
parameter and the monochromatic optical/UV luminosity at 2500 Å,
$\log{L_{\mathrm{UV}}}$, and provides an additional piece of evidence to
support this scenario. The same four QSOs are clearly offset from the
$\alpha_{\mathrm{ox}}$ vs $\log{L_{\mathrm{UV}}}$ relation found for large AGN
samples (e.g. Just et al. 2007; Lusso et al. 2010; Vagnetti et al. 2010;
Martocchia et al. 2017), falling in the region of the plane defined by
$\Delta\alpha_{\mathrm{ox}}\leq-0.3$, where $\Delta\alpha_{\mathrm{ox}}$
represents the difference between the observed value of $\alpha_{\mathrm{ox}}$
and the expected value from the Lusso et al. (2010) relation. In their recent
work, Pu et al. (2020) argued that those sources with
$\Delta\alpha_{\mathrm{ox}}\leq-0.3$ can be reasonably classified as X-ray
weak AGN (e.g. Brandt et al. 2000; Miniutti et al. 2009), given that such a
$\Delta\alpha_{\mathrm{ox}}$ implies a $L_{2-10\,\mathrm{keV}}$ weaker by a
factor of $f_{\rm weak}=10^{-\Delta\alpha_{\mathrm{ox}}/0.3838}\geq 6$ than
the bulk of the AGN population at a similar $\log{L_{\mathrm{UV}}}$.
Specifically, the X-ray weakness factors $f_{\rm weak}$ measured for
J1103$+$41, J1127$+$64, J1245$+$33 and J0300–08 span from $\sim 11$ to $\sim
75$, with $\Delta\alpha_{\mathrm{ox}}$ comprised between $-0.7$ and $-0.4$. It
is worth noting that the fraction of QSOs with $f_{\rm weak}\geq 6$ in our
sample is 29${}^{+22}_{-14}$% (with errors estimated according to Gehrels
1986), which is much larger (and inconsistent at the $\sim 99\%$ confidence
level) than that derived by Pu et al. (2020) for a sample of 1825 type-1 QSOs
from SDSS, i.e. $5.8\pm 0.7\%$. The
$\alpha_{\mathrm{ox}}-\log\lambda_{\mathrm{Edd}}$ distribution of our fourteen
high-$\lambda_{\mathrm{Edd}}$ AGN, the XMM-COSMOS type-1 AGN of Lusso et al.
(2010), the sub- and super-Eddington AGN of Liu et al. (2021) and the thirteen
hyper-luminous WISSH QSOs of Zappacosta et al. (2020), divided in different
regimes of bolometric luminosity, is shown in the right panel of Fig. 5. A
large scatter around the best-fit relation from Lusso et al. (2010) is clearly
visible in Fig. 5, and is probably reflecting the difficulty to derive a tight
$\alpha_{\mathrm{ox}}-\log\lambda_{\mathrm{Edd}}$ relation for large samples
of sources with different bolometric luminosities.

Finally, Fig. 6 shows the distribution of the fourteen highly-accreting QSOs
in the plane $k_{\mathrm{bol,X}}-\alpha_{\mathrm{ox}}$, along with the
relation derived for the COSMOS type-1 AGN that exhibit $k_{\mathrm{bol,X}}$
in the range $\approx$ $10-100$. Typical local AGN in the sample of Liu et al.
(2021) are also presented for comparison. The four X-ray weak
high-$\lambda_{\mathrm{Edd}}$ AGN are also located near the
$k_{\mathrm{bol,X}}-\alpha_{\mathrm{ox}}$ relation in Lusso et al. (2010),
being these two quantities closely related with each other, both of them
indicating the relative strength between the X-ray emission produced in the
corona and the UV emission from the accretion disc. Furthermore, it appears
that the addition of a large number of X-ray weak sources might be useful to
establish the exact $k_{\mathrm{bol,X}}-\alpha_{\mathrm{ox}}$ relation in a
broad range of $\alpha_{\mathrm{ox}}$, with the latter serving as a proxy for
$\log{L_{\mathrm{bol}}}$.

Figure 6: $k_{\mathrm{bol,X}}$ vs $\alpha_{\mathrm{ox}}$ for the
high-$\lambda_{\mathrm{Edd}}$ AGN in our sample (blue triangles) compared to
the XMM-COSMOS type-1 AGN (light grey crosses) of Lusso et al. (2010). Black
solid line refers to their best-fit relation, with the black dashed lines
describing its $1\sigma$ spread. Green squares and red circles indicate the
sub- and super-Eddington AGN from Liu et al. (2021), respectively. Purple
diamonds are the hyper-luminous WISSH QSOs from Zappacosta et al. (2020).

##  5 Discussion

###  5.1 Properties of the X-ray continuum and X-ray weakness

The 14 high-$\lambda_{\mathrm{Edd}}$ AGN analysed in this paper are drawn from
the larger MS14 sample of intermediate-redshift, highly-accreting quasars,
which exhibit remarkably homogeneous optical properties (e.g. extremely strong
$\mathrm{Fe{\,\scriptstyle II}}$ emission, weak [$\mathrm{O{\,\scriptstyle
III}}$] emission, FWHM(H$\beta$) $<$ 4000 km s-1). Furthermore, our sample has
a narrow distribution in both redshift ($\sigma_{z}$ $\sim$ 0.1) centred at
$z\sim$ 0.57, and Eddington ratio ($\sigma_{\lambda_{\mathrm{Edd}}}$ $\sim$
0.1) centred at $\lambda_{\mathrm{Edd}}\sim 1.1$, while luminosities and SMBH
masses are clustered around
$\log{(L_{\mathrm{bol}}/\mathrm{erg\,s}^{-1})}\approx 46.5$ and
$\log{(M_{\mathrm{BH}}/M_{\odot})}$ $\approx$ 8.3, respectively (see Tab. 1).
The main result emerging from our analysis is that, despite the homogeneous
optical and SMBH accretion properties, the X-ray properties of these xA
quasars appear to be quite heterogeneous. This, in turn, indicates that the
structure and efficiency of the X-ray corona and innermost accretion flow in
these AGN cannot be considered a uniform and distinctive feature for reaching
a maximised radiative output per unit mass. In this sense, the scatter of the
primary continuum slope measured for our highly-accreting quasars as a
function of the logarithm of $\lambda_{\mathrm{Edd}}$ (Fig. 3) suggests
caution in the use of $\Gamma$ for providing a reliable estimate of the SMBH
mass based on the X-ray spectrum. Indeed, at these high-Eddington ratios this
may be telling us something about the structure of the disc/corona complex,
possibly indicating a (time-variable?) redistribution of disc accretion power
dissipated in the corona, such as a variation in the inner truncation radius
of the accretion disc or the geometry of the disc/corona complex (e.g. Kubota
& Done 2018). This, in turn, would affect the slope of the hard X-ray
emission. In particular, a larger number of sources with $M_{\mathrm{BH}}$ at
the high-end of the mass distribution (i.e. $\geq$ 109 $M_{\odot}$) and a
well-constrained hard X-ray photon index is needed to evaluate the presence of
possible biases due to a dependence of $\Gamma$ on $M_{\mathrm{BH}}$ (e.g.
Martocchia et al. 2017).

Another piece of evidence supporting a large variety of the X-ray properties
of our high-$\lambda_{\mathrm{Edd}}$ quasars comes from the wide distributions
of $k_{\mathrm{bol,X}}-\log{\lambda_{\mathrm{Edd}}}$ and
$\alpha_{\mathrm{ox}}-\log{L_{\mathrm{UV}}}$ reported in Fig. 4 and 5. In
particular, four sources exhibit large values of $k_{\mathrm{bol,X}}$ and
small $\Delta\alpha_{\mathrm{ox}}$, and they can be classified as X-ray weak
quasars. For these sources, we measure an X-ray emission about a factor of
$\sim 10-80$ fainter than the bulk of the AGN population at similar UV
luminosities. Bearing in mind the uncertainties due to the limited number of
sources, the fraction of QSOs with $f_{\rm weak}\geq 6$ in our sample is
$29^{+22}_{-14}\,\%$, which is much larger than that derived for the AGN
general population (e.g. Pu et al. 2020), suggesting a possible change in the
properties of the accretion disc-corona system in some objects radiating close
(or above) to the Eddington limit, possibly related to the increasing
importance of radiation pressure in shaping the efficiency of the accretion
process (e.g. Proga & Kallman 2004; Sadowski & Narayan 2016; Zappacosta et al.
2020; Nomura et al. 2020). We note that X-ray weakness in highly-accreting
quasars is often attributed to absorption. This has typically been the case
for BAL quasars (Gallagher et al. 2002; Piconcelli et al. 2005; but see also
Morabito et al. 2014 and Teng et al. 2014) and weak emission-line quasars
(WLQs; e.g. Luo et al. 2015; Ni et al. 2018), for which the presence of a
nuclear shielding gas component with a large covering factor has been proposed
to explain the weak broad emission lines and wind acceleration without over-
ionisation (e.g. Wu et al. 2011). In the case of high-$\lambda_{\mathrm{Edd}}$
AGN, this shielding gas may be identified with an optically and geometrically
thick accretion disc (Luo et al. 2015), or with a failed wind resulting from
stalling of line-driven disc winds (e.g. Proga & Kallman 2004; Proga 2005;
Nomura et al. 2020). In any case, the shielding gas should also be responsible
for the X-ray absorption along our line of sight to the nucleus. Both BAL
quasars and WLQs are indeed characterised by hard X-ray spectra, when the
broadband continuum is modelled with a single power law. Indeed, on average,
the resulting effective photon index is usually
$\langle\Gamma_{\mathrm{eff}}\rangle\sim 1.2$. Pu et al. (2020), for instance,
indicated $\Gamma_{\mathrm{eff}}=1.26$ as the threshold below which a quasar
is potentially obscured by cold absorption. However, by modelling the $0.3-10$
keV X-ray continuum with a power law, we find none of our four
high-$\lambda_{\mathrm{Edd}}$ X-ray weak sources to have flat photon indices
compatible with those typical of WLQs and other absorbed quasars. Indeed, they
show very steep $\Gamma_{\mathrm{eff}}\sim 2.2-3$, suggesting that intervening
cold obscuration is not the likely cause of their X-ray weakness, which may
therefore be an intrinsic property of these AGN. Nonetheless, we cannot rule
out the presence of a highly-ionised gas cloud with a large $N_{\mathrm{H}}$
and small covering factor along our line of sight which may scatter off a
large amount of photons, causing a decrease of the X-ray flux without altering
the spectral shape. However, this _ad hoc_ scenario is difficult to test
directly and seems quite unlikely to apply to all the four X-ray weak QSOs in
our sample. In addition, Miniutti et al. (2012) proposed a couple of
alternative scenarios for the intrinsic X-ray weakness of the $z\sim 0.4$,
high-$\lambda_{\mathrm{Edd}}$ quasar PHL 1092, which is one of the most
extreme X-ray weak quasars with no BAL and an almost constant flux in the UV.
In the former scenario, the size of the X-ray corona changes as a function of
the X-ray flux, and a very compact X-ray corona further shrinks down to the
innermost stable circular orbit around the SMBH during X-ray weak states. In
the second scenario, the corona is located within few gravitational radii from
the SMBH at all flux levels, and light-bending effects give rise to a disc
reflection-dominated X-ray spectrum in extreme low-flux states. However, this
hypothesis can be tested only by analysing additional, deeper X-ray
observations (preferably including data at $>10$ keV) to reveal typical
relativistic reflection spectral features. Similary, light-bending effects
have also been invoked by Ni et al. (2020) as a viable explanation to justify
the extreme X-ray variability of the $z=1.935$, WLQ SDSS J153913.47+395423.4.
This source was found to be in an X-ray weak state, before its flux raised by
a factor of $\sim 20$, approximately six years after its first observation
(corresponding to $\sim 2$ years in the QSO rest frame). Alternatively, the
authors also suggested that the X-ray weakness could be caused by nuclear
shielding, probably due to the inner thick disc, which may intercept the line
of sight to the central source.

High-$\lambda_{\mathrm{Edd}}$ AGN are also likely to accelerate powerful
outflows, which can provide an additional effect contributing to weaken the
X-ray emission. Laurenti et al. (2021) analysed the UFO in the NLSy1 galaxy PG
1448+273 with $\lambda_{\mathrm{Edd}}$ $\sim$ 0.75. They found that this
source underwent large variations of $\alpha_{\mathrm{ox}}$, with a maximum
offset of $\Delta\alpha_{\mathrm{ox}}=-0.7$, after the UFO was detected. As a
result, PG 1448+263 was characterised by a remarkable X-ray weakness for some
months, before the flux finally returned to the same level as prior to the UFO
detection. In this case, the authors suggested that such a powerful disc wind
could be responsible of the observed X-ray weakness, by removing a large
amount of the infalling material in the innermost part of the accretion disc
and, thus, reducing the flux of seed photons towards the corona.

In addition, Zappacosta et al. (2020) recently reported a relation between
$L_{2-10}$ and the velocity shift $v_{\mathrm{C{\,\scriptscriptstyle IV}}}$ of
the $\mathrm{C{\,\scriptstyle IV}}$ emission line profile over $\sim$ 1.5 dex
in $L_{2-10\,\mathrm{keV}}$ for a sample of thirteen very luminous quasars at
$z>$ 2, with $\lambda_{\mathrm{Edd}}\sim 0.5-3$ estimated from H$\beta$-based
$M_{\mathrm{BH}}$. These $\mathrm{C{\,\scriptstyle IV}}$ shifts are
interpreted in terms of winds produced at accretion-disc scale, and the
fastest winds appear to be associated with the lowest hard X-ray luminosities
and steepest $\alpha_{\mathrm{ox}}$ values. Interestingly, Proga (2005)
suggests that in case of highly-accreting AGN, failed winds, which are
inevitably produced along with UV line-driven winds, may affect the density of
the X-ray corona and weaken its inverse Compton X-ray emission.

Figure 7: Light curve of J0300$-$08 in terms of the broadband $E=0.3-10$ keV
flux. Black dashed line represents the flux level of
$F_{0.3-10\,\mathrm{keV}}=7.6\times 10^{-13}$ erg cm-2 s-1 expected from its
bolometric luminosity, once the $k_{\mathrm{bol,X}}-\log{L_{\mathrm{bol}}}$
relation from Duras et al. (2020) is assumed. Red solid line indicates the
flux reference value for X-ray weakness corresponding to
$\Delta\alpha_{\mathrm{ox}}\leq-0.3$ (Pu et al. 2020). The $3\sigma$ upper
limits from the _Swift_ observations of 2011 and 2021 are shown in green and
gold, respectively. The flux from the 2018 _XMM-Newton_ observation analysed
in this work is marked in blue.

Zappacosta et al. (2020) found that X-ray weak sources (i.e. those with
$\Delta\alpha_{\mathrm{ox}}\leq-0.3$) represent a fraction of $\sim 40\%$ of
their sample and all exhibit $v_{\mathrm{C{\,\scriptscriptstyle IV}}}\geq
5000$ km s-1. Nardini et al. (2019) have recently analysed the XMM spectra of
a sample of 30 intrinsically blue, non-BAL, $L_{\mathrm{bol}}\gtrsim 10^{47}$
erg s-1 quasars at $z\sim 3-3.3$ which are likely to shine at
$\lambda_{\mathrm{Edd}}\sim 1$ (see also Lusso et al. 2021). They reported a
fraction of $\sim 25\%$ of X-ray weak sources, according to a
$-0.4<\Delta\alpha_{\mathrm{ox}}<-0.2$. Interestingly, our finding of a
similar fraction of X-ray weak sources in our sample of $z\sim 0.6$,
$L_{\mathrm{bol}}\sim 10^{46}$ erg s-1, optically-selected quasars therefore
suggests that (i) a high frequency of intrinsically X-ray weak sources is not
a distinctive feature of the population of ultra-luminous, high-$z$ quasars
and (ii) a high-$\lambda_{\mathrm{Edd}}$ ratio may imply favourable accretion
disc conditions for enhancing the probability of catching an AGN in an X-ray
weak state.

Finally, we were recently awarded time on the _Neil Gehrels Swift_ Observatory
(Gehrels et al. 2004, hereinafter _Swift_) for monitoring the X-ray emission
of the four X-ray weak sources discovered by the XMM-Newton observations
presented here and tracking down possible rises of their X-ray emission (PI:
E. Piconcelli). The first observed source is J0300$-$08, which was targeted by
_Swift_ for 16 ks on March 2021. The source was not detected, with a 3$\sigma$
upper-limit on the $0.3-10$ keV flux of 4.1 $\times 10^{-14}$ erg cm-2 s-1,
indicating that this quasar still exhibits a weak X-ray emission. Fig. 7 shows
this value along with the fluxes derived from the 2018 _XMM-Newton_
observation and from an additional _Swift_ 6 ks archival observation performed
in 2011, for which we derive an upper-limit of 8.6 $\times 10^{-14}$ erg cm-2
s-1. According to the $k_{\mathrm{bol,X}}-\log{L_{\mathrm{bol}}}$ best-fit
relation from Duras et al. (2020), from the $\log{L_{\mathrm{bol}}}$ listed in
Tab. 1 for J0300$-$08 we would expect a flux of around
$F_{0.3-10\,\mathrm{keV}}\sim 8\times 10^{-13}$ erg cm-2 s-1. All the measured
X-ray fluxes are well below the level expected for the bulk of type-1 AGN with
comparable $\log{L_{\mathrm{bol}}}$, as shown in Fig. 7. This may indicate
that J0300$-$08 has likely been in a predominantly low-flux state during the
past decade, suggesting that the X-ray weakness regime may persist over a
timescale of several years.

###  5.2 Relations involving $\lambda_{\mathrm{Edd}}$

The results presented in Sect. 4 highlight the importance of enlarging the
number of high-$\lambda_{\mathrm{Edd}}$ quasars with dedicated X-ray
observations. This would enable new opportunities to investigate the
high-$\lambda_{\mathrm{Edd}}$ phenomenon at large and extend the dynamical
range for any possible correlation involving X-ray derived parameters and the
Eddington ratio. Specifically, Fig. 4 shows that the relation between
$k_{\mathrm{bol,X}}-\log{\lambda_{\mathrm{Edd}}}$ by Duras et al. (2020) seems
to provide a fair description of the correlation for the the bulk of the
high-$\lambda_{\mathrm{Edd}}$ AGN population, i.e. for both small- and
large-$M_{\mathrm{BH}}$ sources, except for the X-ray weak sources that
deviate considerably from the best-fit relation. Conversely, the large scatter
in the right panel of Fig. 5 indicates that only additional investigations
involving a large number of high-$\lambda_{\mathrm{Edd}}$ AGN at different
luminosity regimes may deepen our understanding of the relation between
$\alpha_{\mathrm{ox}}$ and $\log{\lambda_{\mathrm{Edd}}}$. In particular, as
shown in the same figure, sources with very similar values of
$\log{\lambda_{\mathrm{Edd}}}$ can exhibit a widespread distribution of
$\alpha_{\mathrm{ox}}$ values according to their $L_{\mathrm{bol}}$ ($\approx$
$L_{\mathrm{UV}}$ in case of type-1 AGN). This suggests that the presence of a
large scatter observed in the
$\alpha_{\mathrm{ox}}-\log{\lambda_{\mathrm{Edd}}}$ plane is intrinsic, and
the accurate description of this relation for the whole AGN population should
take into account this dependence on $L_{\mathrm{bol}}$. This may also explain
the opposite conclusions on the existence of a significant correlation between
$\alpha_{\mathrm{ox}}-\log{\lambda_{\mathrm{Edd}}}$ reported by different
studies so far (e.g. Vasudevan & Fabian 2007; Shemmer et al. 2008; Lusso et
al. 2010; Chiaraluce et al. 2018).

Figure 8: $\alpha_{\mathrm{ox}}-\lambda_{\mathrm{Edd}}-M_{\mathrm{BH}}$ plane
introduced by Liu et al. (2021). Green squares and red circles indicate their
sub- and super-Eddington AGN, respectively. Black solid and dashed lines
describe the best-fit relation and its spread as indicated by the same
authors, respectively. We extend their relation towards higher accretion rates
and black hole masses by including our high-$\lambda_{\mathrm{Edd}}$ AGN (blue
triangles).

Furthermore, Liu et al. (2021) have recently proposed the possible existence
of a more complex scenario for the relation between $\alpha_{\mathrm{ox}}$ and
$\log{\lambda_{\mathrm{Edd}}}$, which also involves another fundamental
parameter, i.e. $M_{\mathrm{BH}}$. On the one hand they found that both sub-
and super-Eddington AGN follow the same
$\alpha_{\mathrm{ox}}-\log{L_{\mathrm{UV}}}$ relation, which may indicate that
the properties of the accretion disc-corona system do not show large
differences as a function of the accretion rate for the sources in their
sample, which are all observed in a high-flux state. On the other hand, Liu et
al. (2021) found that the anti-correlation between $\alpha_{\mathrm{ox}}$ and
$\log{\lambda_{\mathrm{Edd}}}$ is more significant when the two subsamples of
sub- and super-Eddington AGN are considered separately instead of the entire
AGN sample. A similar behaviour is also observed for $\alpha_{\mathrm{ox}}$ as
a function of $\log{(M_{\mathrm{BH}}/M_{\odot})}$. Interestingly, by means of
a partial correlation analysis, Liu et al. (2021) obtained a statistically
significant non-linear relation between $\alpha_{\mathrm{ox}}$ and both
$\lambda_{\mathrm{Edd}}$ and $M_{\mathrm{BH}}$, in the form
$\alpha_{\mathrm{ox}}=\beta\,\log{\lambda_{\mathrm{Edd}}}+\gamma\,\log{(M_{\mathrm{BH}}/M_{\odot})}+\delta$,
with $\beta\simeq-0.13$, $\gamma\simeq-0.10$ and $\delta\simeq-0.69$. This
best fit is shown in Fig. 8 along with the data from the Liu et al. (2021)
sub- and super-Eddington samples. The relation appears to hold once extended
towards steeper $\alpha_{\mathrm{ox}}$ and larger $M_{\mathrm{BH}}$ by
including our high-$\lambda_{\mathrm{Edd}}$ quasars. Clearly, being based on
sources observed in a high-flux state, the Liu et al. (2021) relation cannot
describe the behaviour of the four X-ray weak sources in our sample. On the
basis of their data Liu et al. (2021) were not able to determine whether this
$\alpha_{\mathrm{ox}}-\lambda_{\mathrm{Edd}}-M_{\mathrm{BH}}$ relation
represents the fundamental plane, or a secondary effect of the
$\alpha_{\mathrm{ox}}-\log{L_{\mathrm{UV}}}$ relation. Moreover,
$\alpha_{\mathrm{ox}}$, $\lambda_{\mathrm{Edd}}$, and $M_{\mathrm{BH}}$ are
all intimately connected and cannot be considered as independent variables.
This suggests to use caution when determining relations that involve these
quantities. However, the investigation on the nature of the physical driver of
these relations will certainly be a hotly-debated topic in the future.

Finally, it is worth noting that although most of our
high-$\lambda_{\mathrm{Edd}}$ QSOs are basically unobscured in the optical,
five sources are characterised by an internal reddening E($B-V$) $\geq 0.05$,
and two of them having E($B-V$) $\geq 0.1$ (see Tab. 4). This is particularly
intriguing since Krawczyk et al. (2015) analysed both the reddening and
extinction of $\sim 35000$ type-1 QSOs from the SDSS, with $z\leq 5$ including
both BAL and non-BAL quasars. They concluded that only $9\%$ ($35\%$) of the
non-BAL (BAL) AGN has an E($B-V$) $\geq 0.05$, and $3\%$ ($11\%$) has an
E($B-V$) $\geq 0.1$. This means that, independently of their classification as
non-BAL or BAL QSOs, the five reddened, high-$\lambda_{\mathrm{Edd}}$ AGN in
our sample belong to the right-hand tail of the reddening distribution of
type-1 AGN. For what concerns the X-ray appearance of these five sources, we
do not find any trend with E($B-V$), as their main X-ray spectral properties
are widely heterogeneous. Indeed, they show a power-law continuum with photon
indices comprised between $1.3\leq\Gamma\leq 2.3$, values of
$\alpha_{\mathrm{ox}}$ between $\sim-1.6$ and $\sim-1.9$, and the
$R_{\mathrm{S/P}}$ parameter ranges from $\sim 0.5$ to $\sim 5.5$. In
addition, only one out of five reddened sources (namely J1103+41) can be
classified as X-ray weak quasar, showing a $k_{\mathrm{bol,X}}=600\pm 300$.

##  6 Summary and conclusion

In this study, we have investigated the X-ray and optical/UV properties of 14
high-$\lambda_{\mathrm{Edd}}$, radio-quiet quasars observed by _XMM-Newton_
(see Sect. 2). These sources belong to the sample of intermediate-redshift,
highly-accreting quasars presented in MS14, and are characterised by
homogeneous spectral optical and SMBH (i.e. $M_{\mathrm{BH}}$ and
$\lambda_{\mathrm{Edd}}$) properties. The main results can be summarised as
follows:

  1. 1.

Our high-$\lambda_{\mathrm{Edd}}$ quasars exhibit significant differences in
their X-ray properties. Specifically, we find a large scatter in the
distribution of the continuum slope, with a non-negligible fraction of these
AGN showing $\Gamma\leq 1.6$ (see Fig. 3). Bearing in mind the limited number
of sources analysed here, this result suggests that the X-ray properties of
high-$\lambda_{\mathrm{Edd}}$ AGN can be more heterogeneous than previously
reported. In this sense, the observed offsets from the $\Gamma$ vs
$\lambda_{\mathrm{Edd}}$ relations found in the past (e.g. Shemmer et al.
2008; Brightman et al. 2013; Huang et al. 2020) can be a manifestation of
distinctive physical properties of the inner accretion disc–X-ray corona in
these AGN.

  2. 2.

We also find that $\sim 30\%$ of the high-$\lambda_{\mathrm{Edd}}$ sources
considered in our study show an X-ray weakness factor $f_{\rm weak}>10$
corresponding to $\Delta\alpha_{\mathrm{ox}}$ comprised between $-0.7$ and
$-0.4$ (see Figs. 5 and 6). They can be therefore classified as X-ray weak
quasars. The steep broad-band X-ray spectral shape (i.e. $\Gamma_{\rm eff}\geq
2.2$; see Sect. 5.1) seems to rule out absorption as the cause of their
reduced X-ray emission, lending support to an intrinsic X-ray weakness.
Interestingly, this fraction is similar to that reported by Nardini et al.
(2019) and Zappacosta et al. (2020) who studied samples of very luminous
($L_{\mathrm{bol}}\gtrsim 10^{47}$ erg s-1) quasars at $z\sim 2-3$, which are
likely to shine at $\lambda_{\mathrm{Edd}}\gtrsim 1$. This indicates that a
high $\lambda_{\mathrm{Edd}}$ might be a key parameter for triggering a weak
X-ray corona state. A systematic X-ray/UV study of large samples of
$\lambda_{\mathrm{Edd}}\geq 1$ AGN would therefore be useful to shed light on
the distinctive nuclear properties of sources undergoing an intrinsically
X-ray weak phase. Furthermore, a recent _Swift_ monitoring of J0300$-$08, one
of the four X-ray weak QSOs in our sample, reveals that the source was still
in a low-flux state in March 2021. Combining this measurement with the 2018
_XMM-Newton_ and an archival 2011 _Swift_ observation, it appears that
J0300$-$08 has likely been going through a period of intrinsic X-ray weakness
for nearly a decade (see Fig. 7).

  3. 3.

The analysis of the optical spectra of the fourteen
high-$\lambda_{\mathrm{Edd}}$ quasars reveals that five sources have a
reddening E($B-V$) $\geq 0.05$. Such a fraction of reddened sources is higher
than that typically measured for the non-BAL quasar population, and we do not
report any apparent link with specific X-ray spectral and
$\Delta\alpha_{\mathrm{ox}}$ properties. Spectroscopy in the UV band would be
very useful to detect the possible presence of BAL features in these reddened,
highly-accreting quasars at intermediate redshifts and shed light on their
nature.

We note that, given the relatively bright $0.5-2$ keV fluxes of the quasars in
the present sample and the expected soft X-ray flux limits of the eROSITA
surveys (eRASS:1, i.e. first full-sky data release: $\approx 4.5\times
10^{-14}$ erg cm-2 s-1; eRASS:8, final full-sky data release: $\approx 1\times
10^{-14}$ erg cm-2 s-1; Merloni et al. 2012), we can expect that $\approx
70-80$% of our highly-accreting quasars will be detected by eROSITA. However,
the paucity of X-ray photons provided by eROSITA in survey mode will strongly
limit any detailed X-ray spectral analysis, thus calling for more sensitive
pointed X-ray observations.

Finally, the results presented in this paper inspire the following lines of
future investigation: (i) a monitoring of the X-ray flux of the
high-$\lambda_{\mathrm{Edd}}$ quasars analysed here, both X-ray “normal” and
“weak” ones, over multiple timescales (i.e. months to years). This would allow
us to get useful insights on the frequency and duration of the transitions
between these two states; (ii) deep X-ray spectroscopy of the X-ray weak
quasars is also desirable to provide unambiguous constraints on the spectral
shape and the possible presence of some absorption along our line of sight to
the nucleus.

###### Acknowledgements.

We thank the anonymous referee for her/his useful comments. This work is based
on observations obtained with _XMM-Newton_ , an ESA science mission with
instruments and contributions directly funded by ESA Member States and NASA.
Part of this work is based on archival data, software and online services
provided by the Space Science Data Center - ASI. This work has been partially
supported by the ASI-INAF program I/004/11/4. ML acknowledges financial
support from the Ph.D. programme in Astronomy, Astrophysics and Space Science
supported by MIUR (Ministero dell’Istruzione, dell’Università e della
Ricerca). EP, LZ, FT, SB, AL and GV acknowledge financial support under
ASI/INAF contract 2017-14-H.0. EP, SB, MB and GV acknowledge support from PRIN
MIUR project ”Black Hole winds and the Baryon Life Cycle of Galaxies: the
stone-guest at the galaxy evolution supper”, contract #2017PH3WAT. AdO
acknowledges financial support from the State Agency for Research of the
Spanish MCIU through the project PID2019-106027GB-C41 and the “Center of
Excellence Severo Ochoa” award to the Instituto de Astrofísica de Andalucía
(SEV-2017-0709). RM acknowledges the financial support of INAF (Istituto
Nazionale di Astrofisica), Osservatorio Astronomico di Roma, ASI (Agenzia
Spaziale Italiana) under contract to INAF: ASI 2014-049-R.0 dedicated to SSDC.
CR acknowledges support from the Fondecyt Iniciacion grant 11190831.

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##  Appendix A X-ray spectra of high-$\lambda_{\mathrm{Edd}}$ AGN

Fig. 11 shows the results of the broadband spectral fits to the EPIC spectrum
(pn, MOS1 and MOS2) of the 14 high-$\lambda_{\mathrm{Edd}}$ quasars listed in
Tab. 1. The best fit to the bulk of the spectra is provided by a model
consisting of a power law and a blackbody component, both modified for
Galactic absorption (tbabs $\cdot$ (zpowerlw + zbbody) in the XSPEC notation).

The only exceptions are J0809+46 and J1048+31. The former shows no excess
emission in the soft X-rays where the spectrum, instead, is dominated by an
ionised absorption component with high column density (Piconcelli et al., _in
prep._), while the latter requires an additional component (i.e. a Gaussian
line) to account for the Fe K$\alpha$ emission line. Further details on the
X-ray spectral analysis can be found in Sect. 3.1.

Figure 9: (Continued)

Figure 10: (Continued)

Figure 11: _XMM-Newton_ spectra of the 14 high-$\lambda_{\mathrm{Edd}}$
quasars analysed in this paper. For each source, the top panel shows the X-ray
spectrum (data from pn, MOS1 and MOS2 are indicated as black, red and blue
points, respectively) and the corresponding best-fit model (green solid line).
The bottom panel shows the ratios between the data points and the best-fit
model, except for J0809+46, whose spectrum is dominated by an absorption
feature in the soft X-rays. To highlight the presence of this component, for
this source the extension of the best-fitting hard X-ray continuum power-law
model to the soft X-ray portion is shown. Furthermore, for the quasar
J1048+31, the inset figure displays the confidence contour plot of
normalisation against rest-frame energy of the Gaussian line component which
accounts for Fe K$\alpha$ emission.

##  Appendix B Serendipitous sources

In this section we briefly summarise the results of the X-ray spectral
analysis of a small group of serendipitous X-ray sources with interesting
X-ray properties and AGN optical classification available from the literature,
located in the _XMM-Newton_ field of view of our targeted
high-$\lambda_{\mathrm{Edd}}$ AGN. Figure 12 shows their X-ray spectra along
with the corresponding best fit model.

SDSS J120753.12+150221.2. This source is an X-ray bright Seyfert 1 galaxy at
redshift $z=0.08558$ (Oh et al. 2015). Its X-ray spectrum is characterised by
a soft excess with $R_{\mathrm{S/P}}=0.16$. The broadband ($E=0.3-10$ keV
observer frame) continuum is well described by a power law ($\Gamma=2.06\pm
0.06$) modified by Galactic absorption, plus a blackbody component
($kT=120_{-20}^{+10}$ eV). In the $2-10$ keV energy interval, we measure a
flux $F_{\mathrm{2-10}}\sim 10^{-12}$ erg cm-2 s-1 and a luminosity
$L_{\mathrm{2-10}}\sim 1.9\times 10^{43}$ erg s-1. The net pn/MOS1/MOS2 photon
counts in the broadband $E=0.3-10$ keV observer-frame energy interval are
6098/2060/1963.

2MASX J12071559+1512158. This source is classified as a Seyfert 1 galaxy (Toba
et al. 2014) at redshift $z=0.10701$. Its soft X-ray spectrum is characterised
by absorption below 1 keV rest frame. A good fit of the broadband continuum
consists of a power law component with $\Gamma=1.8\pm 0.2$. The absorption
feature in the soft X-rays is modelled with a photoeletric absorber (zwabs
component in XSPEC) with a column density of $N_{\mathrm{H}}=(0.7\pm
0.2)\times 10^{22}$ cm-2. The source has a flux $F_{\mathrm{2-10}}\sim
5.7\times 10^{-13}$ erg cm-2 s-1 and a luminosity $L_{\mathrm{2-10}}\sim
1.6\times 10^{43}$ erg s-1. The net pn/MOS1/MOS2 photon counts in the
broadband $E=0.3-10$ keV observer-frame energy interval are 859/397/362.

SDSS J094012.29+461850.3. This QSO is located at redshift $z=0.7112$ (Pâris et
al. 2014). To fit the X-ray broadband continuum, we first considered a simple
model consisting of a power law modified by Galactic absorption. However, such
a model resulted in a poor fit of the underlying continuum. We suspected the
discrepancy could arise from the deficit below 2 keV rest frame. So we
included a partial covering, cold absorption component in the model. The
covering factor and column density measured with XMM-Newton are
$C_{\mathrm{f}}=0.7_{-0.2}^{+0.1}$ and $N_{\mathrm{H}}=1.7_{-1.1}^{+1.5}\times
10^{22}$ cm-2, respectively. The primary continuum is described by a power law
with photon index $\Gamma=1.4\pm 0.3$. We estimate a flux
$F_{\mathrm{2-10}}\sim 7.1\times 10^{-14}$ erg cm-2 s-1 and a luminosity
$L_{\mathrm{2-10}}\sim 1.2\times 10^{44}$ erg s-1. The net pn/MOS1/MOS2 photon
counts in the broadband $E=0.3-10$ keV observer-frame energy interval are
104/42/64.

SDSS J120558.16+412825.3. This source is a Seyfert 1 galaxy at redshift
$z=0.22648$ (Toba et al. 2014). The _XMM-Newton_ data do not require the
presence of a soft excess and a model consisting of a power law, modified by
Galactic absorption, with photon index $\Gamma=1.8\pm 0.2$ provides a best fit
to the X-ray spectrum. We estimate a flux $F_{\mathrm{2-10}}\sim 1.2\times
10^{-13}$ erg cm-2 s-1, which implies a luminosity $L_{\mathrm{2-10}}\sim
1.8\times 10^{43}$ erg s-1. The net pn/MOS1/MOS2 photon counts in the
broadband $E=0.3-10$ keV observer-frame energy interval are 121/96/53.

Figure 12: XMM-Newton spectra of the serendipitous sources described in
Appendix B, with their corresponding best fit. Each top panel shows the X-ray
spectrum and the best-fit line (in green). Each spectrum is rebinned for
plotting purposes only. The bottom panels describe the ratio between the data
points and the best-fit model. Data from EPIC-pn, MOS1 and MOS2 are shown in
black, red and blue, respectively.