# Systematic Collapse of the Accretion Disc Across the Supermassive Black Hole
Population

Scott Hagen,1 Chris Done,1 John D. Silverman,2,3,4,5 Junyao Li,6 Teng Liu,7
Wenke Ren,8,2 Johannes Buchner,9 Andrea Merloni,9 Tohru Nagao10,11 and Mara
Salvato9
1Centre for Extragalactic Astronomy, Department of Physics, Durham University,
South Road, Durham, DH1 3LE, UK
2Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU,
WPI), UTIAS, Tokyo Institutes for Advanced Study, University of Tokyo,
Chiba, 277-8583, Japan
3Department of Astronomy, School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo, Tokyo 113-0033, Japan
4Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of
Tokyo, Kashiwa, Chiba, 277-8583, Japan
5Center for Astrophysical Science, Department of Physics & Astronomy, John
Hopkins University, Baltimore, MD 21218, USA
6Department of Astronomy, Univerity of Illinois at Urbana-Champaign, Urbana,
IL 61801, USA
7Department of Astronomy, University of Science and Technology of China, Hefei
230026, China
8CAS Key Laboratory for Research in Galaxies and Cosmology, Department of
Astronomy, University of Science and Technology of China, Hefei 230026, China
9Max-Planck-Institute für Extraterrestrische Physik, Giessenbachstrasse 1,
85748, Garching bei München, Germany
10Research Center for Space and Cosmic Evolution, Ehime University, 2-5
Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan
11Amanogawa Galaxy Astronomy Research Center, Kagoshima University, 1-21-35
Korimoto, Kagoshima 890-0065, Japan E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The structure of the accretion flow onto supermassive black holes (SMBH) is
not well understood. Standard disc models match to zeroth order in predicting
substantial energy dissipation within optically-thick material producing a
characteristic strong blue/UV continuum. However they fail at reproducing more
detailed comparisons to the observed spectral shapes along with their observed
variability. Based on stellar mass black holes within our galaxy, accretion
discs should undergo a transition into an X-ray hot, radiatively inefficient
flow, below a (mass scaled) luminosity of $\sim 0.02\,L_{\rm{Edd}}$. While
this has been seen in limited samples of nearby low-luminosity active galactic
nuclei (AGN) and few rare changing-look AGN, it is not at all clear whether
this transition is present in the wider AGN population across cosmic time. A
key issue is the difficulty in disentangling a change in spectral state from
increased dust obscuration and/or host galaxy contamination, effectively
drowning out the AGN emission. Here we use the new eROSITA eFEDS Survey to
identify unobscured AGN from their X-ray emission, matched to excellent
optical imaging from Subaru’s Hyper Suprime-Cam; allowing the subtraction of
the host galaxy contamination. The resulting, uncontaminated, AGN spectra
reveal a smooth transition from a strongly disc dominated state in bright AGN,
to the collapse of the disc into an inefficient X-ray plasma in the low
luminosity AGN, with the transition occurring at $\sim 0.02\,L_{\rm{Edd}}$;
revealing fundamental aspects of accretion physics in AGN.

###### keywords:

accretion, accretion discs – black hole physics – galaxies: active

††pubyear: 2024††pagerange: Systematic Collapse of the Accretion Disc Across
the Supermassive Black Hole Population–A

##  1 Introduction

Active galactic nuclei (AGN) are powered by accretion onto a supermassive
black hole (SMBH), but the structure of this flow is not well understood.
Simple estimates of the accretion structure are that it should form a
geometrically thin, relatively cool, disc if it is dense enough to thermalise
(Shakura & Sunyaev, 1973), giving a disc temperature which increases with
decreasing radius from the black hole. This predicts a spectral energy
distribution (SED) consisting of a sum of black-body components, typically
peaking in the UV/EUV for bright AGN. This matches to zeroth order to the
strong blue/UV continuum seen in these sources, but more detailed comparisons
fail (Antonucci et al., 1989; Lawrence, 2018).

One issue is that AGN always show X-ray emission, with a non-thermal tail
extending to high energies (Elvis et al., 1994; Lusso & Risaliti, 2016),
indicating that some of the accretion energy must be dissipated outside of the
optically thick disc structure. The X-ray emission is highly variable on short
time-scales, indicating it originates in a compact coronal structure (e.g
Lawrence et al. 1987; Ponti et al. 2012) Compton scattering seed photons from
the disc to higher energies. Additionally, even the bright blue/UV continuum
does not match in detail to the disc models. There is a ubiquitous downturn in
the UV which occurs before the expected disc peak (Laor & Davis, 2014; Cai &
Wang, 2023), which appears to connect to an upturn in the X-ray spectra below
$\sim 1$ keV (generally referred to as the soft X-ray excess). These two
features can be joined together across the unobservable EUV region to form an
additional warm Comptonising component (Laor et al., 1997; Porquet et al.,
2004; Gierliński & Done, 2004), which cannot be explained by a either the
standard disc model nor the hot Comptonising corona.

Whatever the nature of the EUV component, this should carry most of the
accretion power. EUV radiation is strongly ionising, producing characteristic
broad emission lines where it irradiates nearby material (e.g Wills et al.
1985; Rees et al. 1989; Peterson & Wandel 1999; Kaspi et al. 2005). Current
optical surveys take advantage of this when identifying and classifying AGN,
by first selecting on the presence of a blue continuum in the photometry, and
then confirming with spectroscopic follow up of broad emission lines (e.g
Richards et al. 2002). These optically selected AGN span a wide range in mass,
and so any given luminosity bin will span a wide range in Eddington ratio.
However, controlling for black hole mass shows a sharp drop in the number of
objects with $\dot{m}\lesssim 0.01$ (e.g Trump et al. 2011; Mitchell et al.
2023), where $\dot{m}=\dot{M}/\dot{M}_{\rm{Edd}}=L/L_{\rm{Edd}}$ (i.e. with
the accretion efficiency factor included). This has generally been attributed
to selection effects such as an increase in dust obscuration or host galaxy
contamination, effectively hiding the AGN emission (e.g Fabian et al. 2008;
Vasudevan et al. 2013; Hickox & Alexander 2018).

However, an alternative scenario is a change in the accretion structure, where
the disc evaporates into a hot X-ray plasma, as often seen in galactic black
hole binaries (BHBs) (Done et al., 2007). These objects show strong thermal
disc emission, now with temperatures peaking in the X-ray ($\sim 10^{7}$ K),
but only for $\dot{m}\gtrsim 0.02$ (Maccarone, 2003). Below this there is a
dramatic spectral transition, from being dominated by soft (low energy) X-ray
thermal emission, to being dominated by harder (high energy) Comptonised
emission from significantly hotter $\sim 10^{9}$ K X-ray plasma (Maccarone,
2003; Done et al., 2007). These hard spectra are best described by an
alternative solution to the steady state accretion flow equations, one that is
hot, optically thin and geometrically thick rather than the cool, optically
thin, geometrically thin accretion discs. The best known of these alternative
solutions is the Advection Dominated Accretion Flow (ADAF: Narayan & Yi 1995).
More generally there is a series of solutions at low density where the ion
temperature is much higher than the electron temperature. Such a two
temperature flow can only be maintained below a critical density (hence mass
accretion rate/luminosity) as more frequent collisions give efficient
thermalisation between electrons and ions. The specific ADAF solution exists
up to a maximum of $\dot{m}\sim 0.01-0.02$ (Yuan & Narayan, 2014), which is
where we generally see the state-transitions.

If we scale these transitions (or differing accretion states) up to AGN we
expect to see strong UV/EUV emission above $\dot{m}\sim 0.02$ where the flow
can be characterised by a disc-like solution. However, below $\dot{m}\sim
0.01$, we expect little EUV emission, as now the flow would instead be
characterised by a hot X-ray plasma giving a much harder Comptonised spectrum
peaking in the hard X-rays rather than the EUV. In this model, the sharp drop
in broad-line AGN below $\dot{m}\sim 0.01$ (e.g. Steinhardt & Elvis 2010) is
an intrinsic feature rather than a selection effect, as the lack of a strong
EUV continuum would naturally lead to an insufficient number of ionising
photons required to induce the broad emission lines.

The lack of a UV excess (i.e a "big blue bump") has been seen in nearby low-
luminosity AGN (Ho, 1999, 2008; Nemmen et al., 2014), and in limited samples
at higher redshifts (Trump et al., 2011). More recently, a few, very rare,
changing-state (or changing-look) AGN exhibit behaviour where the
disappearance of their broad emission lines coincides with a dramatic drop in
the EUV continuum Noda & Done (2018), indicative of a transition in accretion
state. However, it has not been established whether this transition is present
across the wider AGN population (Maoz, 2007) and over cosmic time.

In this paper we use X-ray emission to identify AGN, as (unlike BHBs) they
always show a significant high energy tail (Elvis et al., 1994; Lusso &
Risaliti, 2016), and so should allow us to select objects both below and above
the $\dot{m}\sim 0.01$ transition (e.g Aird et al. 2018). We use the eROSITA
eFEDS X-ray data (Liu et al., 2022) combined with optical imaging of the eFEDS
field using Hyper Suprime-Cam (HSC) to select X-ray sources that are co-
incident with galactic nuclei (Li et al., 2024), and whose X-rays are
unabsorbed. Importantly, the excellent HSC multi-band (_grizy_) imaging,
provided by the Subaru Strategic Program (Aihara et al., 2022) allows for
detailed decomposition of the unresolved AGN emission from the host galaxy
(Ishino et al., 2020; Li et al., 2021a, 2024), allowing us to confidently
reconstruct optical emission from low luminosity AGN where the total emission
is generally dominated by the host galaxy. We then stack the data from each
source in bins of black hole mass and monochromatic $3500$ Å luminosity (used
as a proxy for $\dot{m}$), to obtain mean AGN SEDs within each bin. This
allows us to confidently model the emission both below and above the
$\dot{m}\sim 0.01$ transition, showing clearly a change in the accretion
structure from a flow dominated by a disc-like continuum to one dominated by
an X-ray plasma, as $\dot{m}$ decreases.

The paper is organised as follows. In section 2 we provide an overview of the
sample used in this paper. Then in section 3 we give an oveview of our SED
model, and apply it to the mean SEDs within each bin, showing conclusively the
transition in accretion state. Finally in section 4 we give a discussion on
our results along with our conclusions. Throughout we will assume a standard
cosmology, from the Planck 2018 results (Planck Collaboration et al., 2020).

##  2 The Sample

###  2.1 Sample Definition

We base our study on the combined HSC-eROSITA eFEDS AGN sample described in Li
et al. (2024) (HSC) and Liu et al. (2022) (eROSITA). Briefly, the eFEDS region
has 22079 AGN identified from X-ray point sources with ancillary optical-mid
IR data to allow photometric redshift estimations (Salvato et al., 2022). Our
study requires a confident decomposition of the AGN from the host galaxy. This
decomposition is most reliable in the redshift range $0.2\leq z\leq 0.8$, and
restricting the sample to this range results in 4975 X-ray AGN. These are
cross-matched to the HSC catalogue using the optical counter-part positions
from Salvato et al. (2022), giving 3796 X-ray AGN with excellent imaging data
for confident host-AGN flux decomposition, described in Li et al. (2021a);
Ding et al. (2021). As our study is focused on the nuclear emission
originating from the AGN itself we select only sources that are X-ray
unobscured ($N_{H}<10^{22}$ cm-2, using the values measured in Liu et al.
(2022)), which reduces the sample to 3509 AGN (Li et al., 2024). Additionally,
to ensure sufficient signal-to-noise in the X-ray detection we discard any
sources with less than 10 counts, $N_{\gamma}$ (as seen by eROSITA).

The HSC decomposition separates the unresolved nuclear emission, modelled by
the HSC PSF, from the extended stellar emission from the host galaxy modelled
with a smooth analytic Sersic function using the 2D image decomposition
software tool lenstronomy (Birrer & Amara, 2018; Ding et al., 2021), (see:
Ding et al. 2021; Li et al. 2021a). For additional details on the
decomposition methodology for our sample we refer the reader to Li et al.
(2024). We also refer the reader to Appendix A for details showing the effect
of the decomposition on the SED.

In order to confidently asses the AGN flux for the low-luminosity AGN we only
include sources where the AGN optical flux is _at least_ $5\,\%$ of the host
galaxy flux in all HSC bands, since below this the decomposition results
become unreliable. Additionally we also discard sources with a poor fit to the
HSC image (defined as a source with $\chi^{2}_{\nu}>5$).

In summary, our selection critera can be written as follows:

  * •

$F_{\rm{AGN,HSC}}\geq 0.05F_{\rm{Host,HSC}}$

  * •

HSC Image $\chi^{2}_{\nu}<5$

  * •

$N_{H,\rm{X-ray}}<10^{22}$ cm-2

  * •

$N_{\gamma,\rm{X-ray}}\geq 10$

These selection criteria reduce our final sample to 2759 sources.

###  2.2 Mass Estimates

Only a small portion of our sample (275 sources) overlap with the SDSS DR 16
AGN catalogue, which provides mass estimates from broad-emission lines (Wu &
Shen, 2022). This is not necessarily surprising as we have intentionally used
an X-ray selection that picks AGN regardless of the presence of a blue-
continuum and/or broad emission lines such that we can test whether low-
luminosity AGN have these features or whether they look like the changing-
state AGN (Noda & Done, 2018). However, we do still require black hole mass
estimates for all our sources in order to estimate their $\dot{m}$. The HSC
images have stellar mass estimates for the host galaxy (see Li et al. 2024),
and so we use these to estimate the black hole mass from the local
$M_{*}-M_{\rm{BH}}$ relation (Ding et al., 2020):

$\log_{10}\left(\frac{M_{\rm{BH,local}}}{10^{7}}\right)=0.27+0.98\log_{10}\left(\frac{M_{*}}{10^{10}}\right)$
(1)

There is a redshift dependent offset to this relation, as shown in Li et al.
(2021b), which we add to Eqn. (1) to give the following $M_{*}-M_{\rm{BH}}$
relation:

$\log_{10}M_{\rm{BH}}=\log_{10}M_{\rm{BH,local}}(M_{*})+\Delta M_{\rm{BH}}(z)$
(2)

where

$\Delta M_{\rm{BH}}(z)=\gamma\log_{10}(1+z)\,\,\,\,\rm{where}\,\,\gamma=1\pm
0.07$

Figure 1: Black hole masses derived from the local $M_{*}$ relation (Ding et
al., 2020; Li et al., 2021b) versus masses measured from single epoch SDSS
spectra in the DR16 AGN catalogue (Wu & Shen, 2022). The solid black line
shows a 1:1 relation, while the magenta lines show a $\pm 0.5$ dex offset. The
contours highlight the density distribution of the points, showing the
$1\sigma$ and $2\sigma$ levels (inner and outer contours respectively).

As a sanity check we compare our derived black hole masses to the SDSS
measured masses, for the 275 sources where we have overlap. This is shown in
Fig. 1. It is clear from this that our mass-estimates work as an acceptable
proxy, with the majority lying within 0.5 dex of the SDSS measured values.
Additionally, it is also clear that our mass estimates are most reliable for
black hole masses in the range $\log_{10}M/M_{\odot}=8.0-8.5$, which is where
we will focus the main analysis as this is the range where we also have the
largest number of sources (see section 2.5).

###  2.3 Calculating Rest Frame Luminosities

Our sample covers a range in redshift (0.2-0.8), and so we need to convert all
the observed fluxes to rest-frame luminosities on a common wavelength/energy
grid before we can stack to create mean SEDs. Here we give a brief description
of our methodology, for both the HSC and eROSITA data.

####  2.3.1 HSC

The HSC filters range from the g-band ($\sim 4000$ Å$-5500$ Å) to the Y-band
($\sim 9000$ Å$-11000$ Å). For the redshift range of our sample this
corresponds to a rest frame wavelength range of $\sim 3300$ Å$-6000$ Å, to
ensure all of our sources fall within this range. We define common rest-frame
wavelength bins from 3350 Å to 5550 Å , each with a bin width of
$\Delta\lambda=550$ Å.

For each source we first de-redden the observed frame AGN flux, using the
extinction law of Fitzpatrick & Massa (2007), and $E(B-V)$ values from the
dust maps of Schlafly & Finkbeiner (2011). We then convert the observed
wavelength bins to rest frame wavelength; $\lambda_{em}=\lambda_{obs}/(1+z)$.
For each of the common rest-frame bins, $i$, we then calculate the fractional
overlap, $w$, with each of the HSC rest-frame bins, $j$. This is defined such
that $w_{j}=1$ implies that the original bin is fully contained within the
common rest-frame bin, $w_{j}<1$ implies the common bin only partially
overlaps with the original bin, and $w_{j}=0$ implies no overlap between the
common and original bin. These are then used to calculate the flux-density in
the new common rest-frame bin, $F_{c,i}$, using a weighted log-mean:

$\log_{10}F_{c,i}=\frac{\sum_{j}w_{j}\log_{10}F_{\rm{HSC},j}}{\sum_{j}w_{j}}$
(3)

where $F_{\rm{HSC},j}$ is the HSC flux-density in the filter $j$ (where $j$
iterates over all HSC filters). We note that this does approximate the
sensitivity of each HSC filter as a top-hat, but this should be adequate as
the transmittance of each HSC filter is mostly uniform over their respective
range (Kawanomoto et al., 2018). Any error introduced will be significantly
smaller than the dispersion in the stacked SEDs (see Fig. 5)

We stress here that the HSC flux-densities used are the _decomposed_ AGN
fluxes from Li et al. (2024), so do not contain host-galaxy contamination.
These flux-densities are then simply converted to luminosity densities through
the luminosity distance, $d_{L}$, of each source, calculated from their
redshift using our assumed cosmology (Planck Collaboration et al., 2020).

####  2.3.2 eROSITA

The eROSITA data need to be unfolded from the instrument response before we
can re-bin them onto a common grid. For this we use xspec v.12.13.0c (Arnaud
et al., 1985)

For each eROSITA spectrum we apply a simple absorbed power-law model using
tbabs*pow, where the galactic column in the absorption model tbabs is fixed at
that from HI4PI Collaboration et al. (2016), and pow gives the power-law
component, where we use the values derived by Liu et al. (2022). We stress
that unlike Liu et al. (2022), we _do not_ include an intrinsic absorption
component to model any potential absorption from within the AGN itself. This
is in part as we have already discarded sources showing significant intrinsic
absorption, but mostly because we wish to ensure that our analysis is in fact
showing an intrinsic change in the AGN SED, and not an effect induced by
spurious absorption.

We use the above model to unfold the raw data from the instrument response,
providing raw flux densities. We then remove the absorption component from the
model, and write out the unabsorbed model. The energy dependent absorption
correction factor for each eROSITA energy bin is simply the ratio of the
unabsorbed to absorbed model, which is then applied to the raw flux-densities
to give the galactic absorption corrected flux-densities.

The absorption corrected flux densities are then corrected for redshift and
re-binned onto a common energy grid. eROSITA extends from 0.2 keV to 8 keV.
Since the sensitivity of eROSITA drops off rapidly at higher energies (Predehl
et al., 2021), we choose to limit the upper observed frame energy to 5.3 keV.
For our redshift range this gives rest-frame energy limits from 0.36 keV to
9.6 keV. We use these to define a common rest-frame energy grid, with bin
edges at [0.4, 0.7, 1, 2, 4, 8] keV. These bins are intentionally wider than
the intrinsic eROSITA bins in order to simplify the rebinning process and
maximise signal-to-noise by providing higher photon counts per bin. First we
find all original energy bins (shifted to the AGN rest-frame) that lie within
the new energy bin. We then calculate the log-mean flux density of all
original bins that lie within the new bin, using the absorption corrected
flux-densities. This then provides the flux-density within the new energy bin.

As with the HSC flux-densities, these are converted to luminosity densities
through the luminosity distance, again using the source redshift and Planck
2018 Cosmology results (Planck Collaboration et al., 2020).

###  2.4 Redshift Distribution

Figure 2: Redshift and monochromatic luminosity distributions for 3500 Å
(_top_) and 2 keV (_bottom_). In both cases the left panel shows the full
sample and the right the sub-sample within our fiducial
$\log_{10}M/M_{\odot}\in[8.0,8.5]$ mass-bin. The top histograms show the
redshift distribution of the samples, normalised such that their integral is
unity. It is clear that the distributions do not change significantly between
the full sample and our fiducial mass-bin. In all cases there is a slight
deficit in the number of sources below $z\sim 0.25$, however the remaining
redshift range (up to $z=0.8$) has a relatively even distribution.

The left panels of Figure 2 show the redshift distribution of our sample for
3500 Å monochromatic luminosity (_top_) and 2 keV monochromatic luminosity
(_bottom_) respectively, using the redshift measured in Salvato et al. (2022).
The right panel of each figure shows the distribution for the subset of
sources, with black hole masses within $\log_{10}M/M_{\odot}\in[8.0,8.5]$,
which we use for further analysis (see section 2.5).

The source numbers (top histogram) are relatively evenly distributed with
redshift in both 3500 Å luminosity and 2 keV luminosity in both the full
sample and the $\log_{10}M/M_{\odot}\in[8.0,8.5]$ mass bin apart from a drop
in numbers below $z\leq 0.25$ due to the small volume. Similarly, the mean
source luminosity increases with redshift as the larger volume includes the
rarer, more luminous objects.

###  2.5 Mass and Luminosity Distribution

Figure 3: Distribution of our sample across bins of black hole mass and
monochromatic $3500$ Å luminosity. The green shading gives the number of
sources per bin (as defined by the colour-bar on top). The black contour lines
show the distribution of sources that have matches with the SDSS DR16 AGN
catalogue (Wu & Shen, 2022). Magenta lines show the predicted $3500$ Å
luminosity at different Eddington ratios, calculated for a standard accretion
disc extending to the ISCO, while the blue box indicates the bins used for the
main analysis presented in this paper. It is clear that our eROSITA-HSC
combined sample pushes to significantly lower optical luminosity than the
broad-line SDSS sources, allowing us to probe AGN accretion below the $2\,\%$
of Eddington transition. Figure 4: The $3500$ Å luminosity distribution for
the sources within the $\log_{10}M/M_{\odot}\in[8.0,8.5]$ mass bin. The green
bars show the distribution for the full sample, while the blue-dashed bars
show the distribution of the sources in the SDSS DR16 AGN catalogue (Wu &
Shen, 2022). The shaded magenta region band shows the $3500$ Å monochromatic
luminosity at $2\,\%$ of Eddington for the black hole mass range
$\log_{10}M/M_{\odot}\in[8.0,8.5]$, calculated for a standard disc extending
to the ISCO. The dashed-dotted magenta line shows this for the geometric
centre of the mass-bin. It is clear that not only does our X-ray selected
sample extend to considerably lower optical luminosity than the SDSS matched
sample, but it also peaks below $2\,\%$ of Eddington, unlike the SDSS matched
sample.

Each AGN within our sample now has a rebinned, deabsorbed rest frame spectrum
with a black hole mass estimate. We start by subdividing our sample into black
hole mass bins, from $\log_{10}M/M_{\odot}=4$ to $\log_{10}M/M_{\odot}=9.5$
with bin width 0.5 dex. This allows us to control for changes in the SED due
to differences in black hole mass leading to intrinsic differences in the
temperature of the accretion flow (i.e a larger black hole gives a cooler
accretion disc; Shakura & Sunyaev 1973). We have chosen relatively broad bins
of 0.5 dex to allow for the intrinsic uncertainty in the black hole mass
estimates. Fig. 1 show that the majority of our mass estimates (from the host
galaxy stellar mass) lie within 0.5 dex of the single epoch mass estimates
from SDSS DR 16 (Wu & Shen, 2022) (which itself has an intrinsic uncertainty).
Hence we do expect there to be some leakage between mass bins, however by
choosing broad bins the overall effect of this on the stacked SEDs should be
minimal.

Within each mass-bin we further subdivide the sample into bins of
monochromatic optical luminosity, as this should originate from the disc and
as such trace the mass-accretion rate, which significantly impacts the shape
of the SED (Kubota & Done, 2018). We choose the monochromatic luminosity at
3500 Å, as the bluer the wavelength the closer to the SED peak it will
originate from, providing a better tracer of mass-accretion rate. Hence, the
sample is subdivided into 3500 Å luminosity bins from $\log_{10}\nu
L_{3500}=42$ ergs/s to $\log_{10}\nu L_{3500}=46$ ergs/s, again with bin
widths of 0.5 dex.

Overall this leads to the sample distributed over a 2 dimensional grid, shown
in Fig. 3. Here each mass-luminosity grid is shown as a green rectangle, where
the shade indicates the number of sources within each bin (with a darker green
indicating more sources, as shown by the colour-bar). The black-grey contours
show the distribution of our sources that are also included in the SDSS DR16
AGN catalogue (Wu & Shen, 2022). These are objects selected for spectroscopic
followup based on their photometric colours showing a blue continuum (Richards
et al., 2002), and these spectra show the characteristic broad emission lines.
This immediately shows that our X-ray selected sample extends down to
considerably lower optical AGN luminosity than that of the SDSS AGN catalogue.
This is more clear in Fig. 4, which shows the 3500 Å luminosity distribution
for a single mass bin, $\log_{10}M/M_{\odot}\in[8.0,8.5]$, for both the full
sample (green, solid) and the sources matched to the SDSS AGN catalogue (blue,
hatched). Here it is clear that not only does our sample extend to
considerably lower optical luminosity, it also peaks at lower luminosity,
indicating that bright broad-line AGN are not necessarily the dominant
unobscured AGN population.

A more interesting metric than the monochromatic luminosity is the Eddington
ratio. This is shown by the magenta lines in Fig. 3. We stress here that these
are calculating from converting the 3500 Å luminosity to bolometric luminosity
via a standard Shakura & Sunyaev (1973) accretion disc model, and as such the
precise values should be treated with caution. For $L/L_{\rm{Edd}}\lesssim
0.02$, where we see the changing-look transition, we expect a vastly different
accretion structure (Done et al., 2007; Noda & Done, 2018), and so converting
via a disc model is not accurate. However, it is sufficient to give an
estimate of what distribution of sources lie above or below the $2\,\%$
Eddington transition. It is clear that the majority of the SDSS AGN sources
lie above this transition, as seen previously in Mitchell et al. (2023), while
the majority of our sample lie below (for black hole masses
$\log_{10}M/M_{\odot}\gtrsim 7$ \- below this we do not trust the mass
estimates). Again this becomes more evident in Fig. 4, where we see that our
sample also peaks below the $2\,\%$ transition. This already gives strong
evidence for a change in SED at low Eddington ratios, where the optical
luminosity drops much faster than the X-ray luminosity, as predicted if there
is a transition in the physical nature of the accretion flow from the optical
being dominated by a disc to an ADAF.

We investigate this in more detail by stacking the spectra within each bin,
averaging in logarithmic luminosity on the common wavelength/energy binning.
This gives an average SED in each mass-luminosity bin. For the remainder of
the paper we focus on the single mass bin $\log_{10}M/M_{\odot}\in[8.0,8.5]$,
as this is the bin where the black hole mass estimates from the host galaxy
are most reliable, and also where we have most objects. This gives a sequence
of SEDs for changing monochromatic 3500 Å luminosity at a fixed mass, shown in
Fig. 5 (orange (HSC) and blue (eROSITA) points). It is immediatly clear that
clear that the 3500 Å monochromatic luminosity changes by over 3 orders of
magnitude while the X-ray power changes by less than a factor of a few. It is
also immideatly clear that the shape of the optical spectrum changes
dramatically, from a blue continuum like a standard disc in the higher
luminosity bins, to a fairly flat continuum in the middle bins, to a strongly
red continuum very unlike a standard disc in the lowest luminosity bins.
Again, this suggests a dramatic change in the accretion structure, which we
explore in greater detail in the next section (3), where we detail our SED
model and apply it to the stacked SEDs to give a physical interpretation of
the change in SED shape.

##  3 Modelling the SEDs

To explore the spectral evolution in detail we fit the stacked AGN SEDs within
our fiducial mass bin of $\log_{10}M/M_{\odot}\in[8.0,8.5]$. We do not use
standard Shakura & Sunyaev (1973) disc models, as these have multiple well
known issues. Firstly, there is the high energy X-ray tail which is always
present at a significant level (Elvis et al., 1994; Lusso & Risaliti, 2016).
More seriously for the disc models themselves, the UV spectrum often shows a
downturn before the predicted peak, which appears to connect across the data-
gap (due to ISM absorption, e.g Laor et al. 1997; Telfer et al. 2002) to an
upturn in the soft X-ray bandpass (the soft X-ray excess) (e.g Czerny & Elvis
1987; Laor et al. 1997; Mehdipour et al. 2011; Jin et al. 2012; Petrucci et
al. 2018; Kubota & Done 2018). The standard disc models also predict that the
time-scales for intrinsic variability is extremely slow (thousands of years),
in conflict with multiple observations (e.g Noda & Done 2018; Stern et al.
2018; Hernández Santisteban et al. 2020; Neustadt & Kochanek 2022; Yao et al.
2023; Neustadt et al. 2024). Nonetheless, the ubiquity of the strong UV
emission clearly indicates that there is optically thick material emitting a
large fraction of the gravitational potential energy from accretion, i.e there
is a disc but its properties (spectrum and variability) are different to those
predicted by standard models.

Hence we use modified disc models, which assume that the disc vertical
structure can be different than in standard models such that it does not
completely thermalise (Różańska et al., 2015; Jiang & Blaes, 2020). This could
then form a warm Comptonising plasma, which is optically thick, giving rise to
a Comptonised disc-spectrum, rather than a pure multi-colour black-body disc
spectrum; naturally explaining the UV turnover linking into the soft X-ray
excess (Mehdipour et al., 2011; Done et al., 2012; Mehdipour et al., 2015;
Kubota & Done, 2018; Petrucci et al., 2018), and generally comprises the
dominant component of the disc emission (Czerny & Elvis, 1987; Magdziarz et
al., 1998; Done et al., 2012; Petrucci et al., 2013; Petrucci et al., 2018).
However, the model does still allow the emission to revert back to a standard
black-body disc at larger radii (Done et al., 2012; Jin et al., 2012; Kubota &
Done, 2018).

In addition to a disc and disc-like structure, the model allows for the
accretion flow to evaporate into an optically thin hot X-ray plasma in the
innermost regions, $r_{\rm{isco}}<r<r_{\rm{hot}}$, giving rise to the
Comptonised high energy X-ray tail.

Hence, our accretion structure consist of three distinct regions: An inner
X-ray plasma for $r_{\rm{isco}}<r<r_{\rm{hot}}$, which then condenses into a
warm optically thick disc-like structure for $r_{\rm{hot}}<r<r_{\rm{warm}}$,
and then eventually reaching thermalisation forming a standard disc for
$r>r_{\rm{warm}}$ (Kubota & Done, 2018). These models generally give a good
fit to the observed SEDs of individual AGN with significantly higher
quality/broader bandpass data (e.g Matt et al. 2014; Matzeu et al. 2016; Done
& Jin 2016; Czerny et al. 2016; Hagino et al. 2017; Porquet et al. 2018, 2019;
Porquet et al. 2024), and comes with the additional feature that the non-
standard optically thick region gives the potential for significantly faster
intrinsic disc variability (Hagen et al., 2024), more in line with current
observations.

In this section we give a brief overview of the physical implementation of the
model in section 3.1, using the publicly available xspec model agnsed (Kubota
& Done, 2018), before applying it to the data in 3.2. Throughout we use the
standard notation for radii, where $R$ denotes the radius in physical units
and $r$ is the dimensionless radius in gravitational units, related through
$R=rR_{G}$ where $R_{G}=GM/c^{2}$.

###  3.1 agnsed \- Physical Implementation

agnsed (see Kubota & Done 2018 for full details) assumes a standard
(relativistic) disc Novikov & Thorne (1973) emissivity throughout the entire
flow, with a radial temperature profile given by $T_{NT}^{4}(R)\propto
R^{-3}f(R)$ where $f(R)$ describes the radial disc structure in the Kerr
metric (Page & Thorne, 1974).

In the outermost regions of the flow, $r>r_{\rm{warm}}$, this model assumes
the flow thermalises to a standard accretion disc. Hence, splitting the flow
up into radial annuli of width $\Delta R$, the emission from each annulus will
be a standard black-body $B_{\nu}(T_{NT}(R))$ with total luminosity $2\times
2\pi R\Delta R\sigma T_{NT}^{4}(R)$, where the extra factor $2$ comes from the
disc emitting from both sides and $\sigma$ is the Stefan-Boltzmann constant.

In the warm disc-like region, $r_{\rm{hot}}<r<r_{\rm{warm}}$, the vertical
disc structure is assumed to change, such that the dissipating region moves up
into the photoshpere forming a slab geometry above an underlying passive disc
structure. Here the seed photons are tied to the underlying disc, such that
their temperature is simply $T_{NT}$ and each annulus has luminosity $4\pi
R\Delta R\sigma T_{NT}^{4}(R)$. Additionally, the seed photons are assumed to
form a black-body distribution, which then Compton scatters through the
photosphere forming a Comptonised black-body spectral shape; modelled
internally using nthcomp (Zdziarski et al., 1996; Zycki et al., 1999). It is
assumed that the disc photosphere is optically thick, and has a covering
fraction of unity such that all seed-photons are Comptonised (Petrucci et al.,
2018). Within agnsed the warm Comptonisation region is assumed to have a
single electron temperature, $kT_{e,w}$, and photon index, $\Gamma_{w}$, left
as free parameters.

Below a radius $r_{\rm{hot}}$ the warm disc-like structure evaporates into a
hot, optically thin, geometrically thick X-ray plasma; responsible for forming
the high energy X-ray tail (Narayan & Yi, 1995; Liu et al., 1999; Różańska &
Czerny, 2000). This plasma is powered by the energy dissipated in the flow
between $r_{\rm{hot}}$ and $r_{\rm{isco}}$, still assuming standard Novikov &
Thorne (1973) emissivity, giving $L_{\rm{diss}}$. In addition there is a
contribution to the total emission from the seed-photons entering the corona
from the disc, $L_{\rm{seed}}$, such that the total power emitted by the
corona is $L_{\rm{hot}}=L_{\rm{diss}}+L_{\rm{seed}}$ (see Kubota & Done 2018
for details on calculating thse). As with the warm Comptonisation in the disc-
like region, it is assumed that the X-ray plasma has a single electron
temperature $kT_{e,h}$ and photon index $\Gamma_{h}$, which are left as free
parameters.

The total SED is then simply found by adding together the integrated emission
from each component within the flow.

$\log\nu L_{3500}$ (ergs/s) | $\log\dot{m}$ |  $kT_{e,w}$ (keV) | $\Gamma_{h}$ | $\Gamma_{w}$ | $r_{h}$ |  $T_{hi}$ (K) | [$r_{w}(T_{hi})$]
---|---|---|---|---|---|---|---
$[42.0,42.5]$ | $-1.71$ | † | $1.50$ | † | $376^{\dagger\dagger}$ | $4565$ | [$376$]
$[42.5,43.0]$ | $-2.00$ | † | $1.69$ | † | $217^{\dagger\dagger}$ | $5690$ | [$217$]
$[43.0,43.5]$ | $-1.85$ | † | $1.78$ | † | $222^{\dagger\dagger}$ | $6122$ | [$222$]
$[43.5,44.0]$ | $-1.70$ | $0.41$ | $1.83$ | $2.34$ | $76.1$ | $6810$ | [$217$]
$[44.0,44.5]$ | $-1.39$ | $0.56$ | $1.71$ | $2.56$ | $32.9$ | $6810^{*}$ | [$279$]
$[44.5,45.0]$ | $-0.91$ | $0.33$ | $1.64$ | $2.81$ | $20.0$ | $6810^{*}$ | [$403$]
$[45.0,45.5]$ | $-0.39$ | $0.21$ | $1.98$ | $3.50$ | $9.69$ | $6810^{*}$ | [$610$]
$[45.5,46.0]$ | $0.23$ | $0.10$ | $2.25$ | $2.93$ | $7.50$ | $6810^{*}$ | [$990$]
Table 1: Fit values for each of the luminosity bins (left column) for the
stacked SEDs within the mass bin $\log_{10}M/M_{\odot}\in[8.0,8.5]$. We only
show parameters that were kept free during the fitting process. The $r_{w}$
values in square brackets show the radius where the disc temperature is
$T_{hi}$.
$\dagger$ The warm Compton region is not required in this bin.
$\dagger\dagger$ No warm Compton region is required so $r_{w}=r_{h}$, and
$T_{hi}=T_{bb}(r_{h})$.
$*$ In these bins the standard outer disc is subdominant and $T_{hi}$ cannot
be well constrained. Hence we freeze it to the last bin where it could make an
impact (i.e $[43.5,44.0]$).

A key feature of agnsed is the energy balance. The total available power
within the flow is always set by the mass, $M_{BH}$, and Mass-Accretion rate,
$\dot{M}$, assumed constant at all radii in the flow. The relative
contribution to the total luminosity from each of the three regions within the
flow is then set by the integrated emissivity over the radii where each
component dominates. As an example, a flow where the disc-like structure
covers the majority of the available emitting area down to $r\sim r_{isco}$
will have an SED dominated by the warm Comptonisation, whereas a flow with an
extensive X-ray plasma with $r_{hot}\gg r_{isco}$ will have an SED dominated
by the hot Comptonisation component with very little disc. In this sense,
fitting the AGN SED gives us an estimate of not only the relative contribution
to the total output power from each component, but also the relative size
scales of each component.

An important note is that although agnsed takes into account the energy
balance, it does not consider the conservation of photon number although
Compton scattering does. In some sense this is allowing for some uncertainty
in the source of seed photons. The model only considers seed photons
originating from the disc regions, which in disc dominated states will
certainly be sufficient (e.g Malzac & Belmont 2009). However, in states
dominated by the X-ray plasma there could be an issue with the spectrum being
photon-starved when only considering seed-photons from the disc, similar to
hard state BHBs (e.g Poutanen et al. 2018), and will almost certainly require
additional seed photons originating within the flow through, e.g, cyclo-
scynchrotron.

###  3.2 Fitting the data

We fit the stacked data within each bin with the agnsed model, using the
standard $\chi^{2}$ minimisation routine within xspec v.12.13.0c (Arnaud,
1996). For simplicity we treat the $1\sigma$ dispersion on the flux within
each energy/wavelength bin as the error on the flux during the fitting. For
mass-luminosity bins with too few sources to confidently calculate the
dispersion ($<10$, only relevant for the highest luminosity bin in our
fiducial mass bin), we assume a $10\,\%$ error on the flux. Due to the large
dispersion in the data, especially the X-ray, we stress that these fits are
meant to display the rough power-output and evolution of the SED, rather than
a detailed determination of model parameters.

The black hole mass is fixed to the geometric centre of the mass-bin, which
for the bin considered in this paper ($\log_{10}M/M_{\odot}\in[8.0,8.5]$) is
$1.7\times 10^{8}\,M_{\odot}$. We conservatively assume spin zero, where the
general-relativistic corrections from ray-tracing (not included in the
standard version of agnsed) are small (see Hagen & Done 2023 \- certainly
smaller than the dispersion within the data). As these are unobscured AGN we
assume an on average a small inclination angle, fixing it to $\cos(i)=0.87$
throughout. Finally, we fix the X-ray coronal scale height to $h_{x}=10$, and
the outermost radius to $r_{\rm{out}}=10^{4}$.

Our initial fits (not tabulated) showed that the peak temperature of the outer
black-body disc, $T_{hi}=T_{bb}(r_{w})$ remained approximately constant ($\sim
4000-6500$ K), despite the large changes in $\dot{m}$. Hence, we slightly
modified agnsed such that this transition temperature is now an explicit free
parameter, replacing $r_{w}$, which is now an implicit parameter calculated
internally for a given $\dot{m}$ and $T_{hi}$.

The resulting best fit parameters are shown in Table 1, and the resulting SEDs
are shown in Fig. 5. As well as the (expected) increase in $\dot{m}$ with
optical luminosity, we see a correlated decrease in $r_{h}$. This shows how
the inner X-ray plasma/corona becomes less and less dominant as $\dot{m}$
increases (Lusso & Risaliti, 2016). This is also seen in the data themselves
(Fig. 5), where the X-ray emission has comparatively little change (only a
factor of a few) whereas the optical luminosity (and hence $\dot{m}$) change
by multiple orders of magnitude. Our model relies on the energy balance to
determine the size-scale of the components, so the approximate constant X-ray
power (at $L_{h}\sim 0.02L_{\rm{Edd}}$ \- Kubota & Done 2018) means that it
requires a smaller fraction of the total accretion power as $\dot{m}$
increases, and so $r_{h}$ decreases. The increasing dominance of the disc
means means there are more seed photons cooling the hot corona, and so
$\Gamma_{h}$ also increases (Kubota & Done, 2018). These relations have been
clearly seen before (Kubota & Done, 2018; Mitchell et al., 2023), but here the
host galaxy subtraction also makes the warm Compton parameters more reliable.
We see that there is a systematic increase in $\Gamma_{w}$, correlated with a
decrease in $kT_{e,w}$, which implies that the warm Comptonising disc-like
region is also systematically changing, from being consistent with all the
luminosity being emitted in the photosphere ($\Gamma_{w}\sim 2.5$, a passive
disc: Petrucci et al. 2018), to being much steeper as expected for increasing
dissipation in the warm disc itself.

Overall, our model fits show a clear and systematic evolution in the SED,
which corresponds to a change in the accretion structure itself, transitioning
from a disc(-like) dominated state at high $\dot{m}$ to one dominated by the
X-ray plasma at low $\dot{m}$ (much like the low-hard state in BHBs).

Figure 5: Stacked SEDs for the mass bin $\log_{10}M/M_{\odot}\in[8.0,8.5]$.
Each panel shows a single monochromatic, $\nu L_{\nu}$, $3500$ Å luminosity
bin, with the precise bin range shown in ergs/s by the bracketed values in the
top right corner of each panel. The number of sources, $N_{s}$, within each
bin is given in the top left corner. The binned HSC data are shown by the
orange points, while the binned eROSITA data are given by the violate points.
The orange and cyan regions show the dispersion on the HSC and eROSITA data
respectively. Each bin has been fit with agnsed, with the resulting model SED
shown by the solid black line, while the coloured dashed lines show the
individual components to the total model. Here magenta indicates the outer
standard disc, green the unstable warmly Comptonised disc, and blue the inner
X-ray corona. The dashed-dotted grey line shows the model for the middle bin
($\log_{10}\nu L_{3500}\in[44.0,44.5]$ ergs/s), provided to highlight the
relative change in the SED as we move through the luminosity bins. The bottom
right panel shows a cartoon of the change in accretion geometry with
$\dot{m}$. The colours in the cartoon corresponds to the colours in the SED
components.

##  4 Discussion and Conclusions

All the SEDs considered here are consistent with the standard outer disc
component appearing below a roughly constant temperature of $\sim 4500-6500$
K, which is also the temperature region where Hydrogen goes from being mostly
ionised (above $T_{hi}$) to mostly neutral, perhaps providing insight into the
origin of the warm Comptonising disc-like structure.

Disc temperatures around $1-5\times 10^{4}$ K predict local SEDs peaking in
the UV/EUV. Stars at these temperatures (O stars, Wolf-Rayet, and luminous
blue variables) have significant winds powered by UV line-driving (e.g Puls et
al. 2008; El Mellah et al. 2017). This can translate to AGN discs, powering
radiatively driven outflows, through e.g UV line driving (Proga et al., 2000;
Proga & Kallman, 2004). Strong winds from an accretion disc naturally lead to
significant mass-loss, which can produce a distinct change in the disc
properties themselves (Laor & Davis, 2014). Our observed $T_{hi}$ is an order
of magnitude below these characteristic UV wind temperatures, making it warm
disc region unlikely to be connected to this process.

Instead, there is a dramatic change in the disc opacity at the Hydrogen
ionisation threshold at around $6000\,K$ (Cannizzo & Reiff, 1992), leading to
a local instability (Cannizzo, 1992). In stellar mass BHB discs, which cross
this temperature at large radii where there is not much power from accretion,
the disc opacity changes so dramatically that this local instability triggers
a global limit cycle, with the disc, so it transitions between outbursts and
longer periods of quiescence (Lightman & Eardley, 1974; King & Ritter, 1998;
Lasota, 2001; Dubus et al., 2001). The lower temperature discs in AGN mean
that SMBHs cross this temperature range at much smaller radii (scaled in terms
of the gravitational radius) where there is significantly more power emitted.
Thus this instability might be expected to have an even more dramatic impact
on the disc structure. However this is not generally seen in the simulations
of this so far (Burderi et al., 1998; Siemiginowska & Elvis, 1997;
Grzȩdzielski et al., 2017), but AGN discs are also in the regime where
radiation pressure dominates, and we speculate that the non-linear outcome of
these two instabilities acting together hold the key to understanding the
origin of the warm disc region in AGN.

Regardless of the origin of the warm disc-like region, it is fundamentally
clear that it depends strongly on $\dot{m}$. The clear smooth transition in
the SEDs from a disc dominated state to one dominated by the X-ray, is most
easily explained by a change in the accretion structure itself, as shown
previously for some rare changing-look (changing-state) AGN (Noda & Done,
2018). This change cannot be due to obscuration, as the X-rays are clearly
unobscured (by selection). The inner accretion flow in AGN is relatively
compact, and so we would expect any obscuring medium present in the UV to
intercept the X-rays too. Of course, there could be some low column-density
dusty/gas on large scales that would contribute to the reddening of the
optical/UV continuum, but would not be seen in our X-ray data (e.g Czerny et
al. 1995). However, we find it unlikely that this could cause sufficient
reddening to give the extremely red continua seen in the lowest three
luminosity bins, while remaining X-ray unobscured. In addition the transition
from a strong soft X-ray excess in the highest three luminosity bins, to a
single non-thermal X-ray component in the lowest three bins, strongly suggests
a collapse of the structure responsible for the soft excess as $\dot{m}$
reduces rather than increased obscuration.

Along with the collapse of the disc, and thus the EUV continuum, we expect to
see a reduction/loss of the broad emission lines, due to a reduction/loss of
the ionising photons. This is entirely analogous with the changing-look
phenomenon (Noda & Done, 2018; Ruan et al., 2019), except now it is a generic
feature of the AGN population rather than limited to a few rare objects. In
some sense this has been previously hinted at, with studies showing a sharp
drop in the number of broad line AGN below $0.02\,L_{\rm{Edd}}$ (Trump et al.,
2011; Mitchell et al., 2023), which is both where the changing-look transition
is known to occur and where our sample shows a loss of the EUV continuum. This
can easily be tested by matching our HSC-eROSITA sample to one of the
spectroscopic surveys (e.g SDSS York et al. 2000 or DESI DESI Collaboration et
al. 2016). This is beyond the scope of our current paper, where we limit
ourselves to the SED, but will be addressed in future work.

The more obvious limitation within our work is the data-gap between the
optical/UV and the X-ray. Our modelling _predicts_ substantial power emitted
in the EUV at high $\dot{m}$ and the loss of this power at low $\dot{m}$.
However, this is predictive, and entirely based on what the model requires to
bridge the gap between the optical/UV and X-ray, while maintaining energy
conservation. It is entirely plausible that alternative accretion flow models
could predict entirely different behaviour in the EUV gap, which would fit our
data just as well. In this sense additional data in the near-UV (NUV) and far-
UV (FUV) could work to strengthen (or falsify) our conclusions. This will be
the topic of a future paper, using archival GALEX data (Kang et al. (in
prep)).

Nonetheless, there is a clear drop in the blue optical continuum from the warm
disc at $\dot{m}\lesssim 0.02$. This is clearly a generic feature in the AGN
population, and not limited to local low-luminosity AGN (LINERs) (Ho, 1999)
and a few rare individual changing-state AGN (Noda & Done, 2018). There is a
real, and quite abrupt change in the accretion structure, driven by the
changing Eddington ratio, which leads to the complete loss of the ionising EUV
emission, and hence the characteristic broad emission lines. This has a
significant impact on how we identify AGN through cosmic time. In an era where
the James Webb Space Telescope is breaking new ground with the discoveries of
quasars at cosmic dawn, any inference for the overall population requires we
understand the nature of the accretion flow itself. This study reveals a
systematic change in the accretion structure; the collapse of the accretion
disc; between high and low luminosity AGN, impacting what we infer about the
general AGN population, structure and growth through cosmic time.

## Acknowledgements

SH acknowledges support from the Japan Society for the Promotion of Science
(JSPS) through the short-term fellowship PE23722 and from the Science and
Technologies Facilities Council (STFC) through the studentship grant
ST/V506643/1. CD acknowledges support from STFC through grant ST/T000244/1 and
Kavli IPMU, University of Tokyo. Kavli IPMU was established by World Premier
International Research Center Initiative (WPI), MEXT, Japan. JS is supported
by JSPS KAKENHI (23K22533) and the World Premier International Research Center
Initiative (WPI), MEXT, Japan. This work was supported by JSPS Core-to-Core
Program (grant number: JPJSCCA20210003).

This work made use of the following python modules: astropy (Astropy
Collaboration et al., 2013, 2018, 2022) and numpy (Harris et al., 2020).
Additionally, the plots were made using a combination of matplotlib (Hunter,
2007) and seaborn (Waskom, 2021).

## Data Availability

The optical HSC-fluxes used in this paper are from Li et al. (2024), and the
corresponding catalogue can be found at:
https://member.ipmu.jp/john.silverman/HSC/.

The X-ray spectra are part of the eROSITA eFEDS early data-release AGN
catalogue, as described in Liu et al. (2022). This is available through:
https://erosita.mpe.mpg.de/edr/eROSITAObservations/Catalogues/.

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##  Appendix A Importance of the HSC host-AGN decomposition

Figure 6: Zoom-in on the optical/UV region of the SEDs from Fig. 5, for select
luminosity bins (as indicated in the top right corners). Here we show the
stacked data for the non-decomposed HSC fluxes (i.e host + AGN) in grey, with
the decomposed stacks (AGN only) in orange, highlighting the necessity of the
HSC decomposition. In all, apart from the highest (likely super-Eddington),
luminosity bins the inferred power and optical SED shape would be incorrect
with no decomposition.

The results of our study rely on confidently extracting the AGN emission from
the host galaxy emission, especially for the low luminosity bins. This is
clear in Fig. 6, where we show the optical part of the stacked SEDs for both
the de-composed AGN fluxes and the non-decomposed total fluxes for four of the
luminosity bins ($\nu
L_{3500}\in[42.5,43.0],\,[43.5,44.0],\,[44.5,45.0],\,\mathrm{and}\,[45.5,46.0]$
ergs/s). In all panels, except the highest luminosity bin, we clearly see that
the non-decomposed SED gives the wrong optical luminosity; from an error of a
factor few to an order of magnitude. Hence an attempt at this study with no
host galaxy decomposition would give the wrong estimate of the system
energetics, over predicting the contribution from a disc (especially at low
luminosity), and hence arriving at the wrong conclusion on the balance between
the X-ray and optical emitting parts of the accretion flow.

As well as systematically overestimating the normalisation, we see in Fig. 6
that the non-decomposed SEDs can give the wrong spectral shape (or slope).
This is especially clear in the bin $\nu L_{3500}\in[43.5,44.0]$ ergs/s, where
the non-decomposed SED is clearly decreasing in power with frequency (energy),
while the isolated AGN SED is more or less flat.

The AGN and host galaxy fluxes are decomposed using a 2D optical image
decomposition method, as described in Li et al. (2024). For details on how the
method works we refer the reader to Li et al. (2021a); Ding et al. (2021).