# Accretion Efficiency Evolution of Central Supermassive Black Holes in
Quasars

Arta Khosravi<EMAIL_ADDRESS>PDAT Laboratory, Department of Physics, K.
N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran  Alireza
Karamzadeh<EMAIL_ADDRESS>PDAT Laboratory, Department of
Physics, K. N. Toosi University of Technology, P.O. Box 15875-4416, Tehran,
Iran  Department of Physics, Shahid Beheshti University, P.O. Box 19839-69411,
Tehran, Iran  Seyed Sajad Tabasi<EMAIL_ADDRESS>Department of Physics,
Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran  PDAT
Laboratory, Department of Physics, K. N. Toosi University of Technology, P.O.
Box 15875-4416, Tehran, Iran  Javad T. Firouzjaee<EMAIL_ADDRESS>Department of Physics, K.N. Toosi University of Technology, P.O. Box
15875-4416, Tehran, Iran  PDAT Laboratory, Department of Physics, K. N. Toosi
University of Technology, P.O. Box 15875-4416, Tehran, Iran

###### Abstract

The ongoing debate regarding the most accurate accretion model for
supermassive black holes at the center of quasars has remained a contentious
issue in astrophysics. One significant challenge is the variation in
calculated accretion efficiency, with values exceeding the standard range of
$0.038<\epsilon<0.42$. This discrepancy is especially pronounced in high
redshift supermassive black holes, necessitating the development of a
comprehensive model that can address the accretion efficiency for supermassive
black holes in both the low and high redshift ranges. In this study, we have
focused on low redshift ($z<0.5$) PG quasars (79 quasars) and high redshift
($z\geq 3$) quasars with standard disks from the flux- and volume-limited
QUOTAS+QuasarNET dataset (76 quasars) to establish a model for accretion
efficiency. An interesting trend is revealed where in redshift larger than 3,
accretion efficiency increases as redshift decreases, while in redshift lower
than 0.5, accretion efficiency decreases with reducing redshift. This suggests
a peak in accretion efficiency between the low and high redshift quasars. This
peak is recognized for the flux- and volume-limited QUOTAS+QuasarNET+DL11
dataset, which is $z\sim 2.675$, and it seems to be related to the peak of the
star formation rate. ($1<z_{SFR}<3$). This result can potentially lead to a
more accurate correlation between the star formation rate in quasars and their
relationship with the mass of the central supermassive black holes with a more
comprehensive disk model in future studies.

Active Galactic Nuclei, Supermassive Black Hole, Standard Accretion Disks, and
Star Formation

iint TXFiint

##  1 Introduction

Within quasars, gas is drawn towards a central Supermassive Black Hole (SMBH),
generating electromagnetic radiation. The gas emits intense light as a result
of gravitational and frictional interactions. This characteristic renders
quasars among the most radiant entities in the cosmos, shining nearly a
thousand times more brightly than the Milky Way (Melia, 2019).

SMBHs are massive objects with masses ranging from below $10^{6}$ to above
$10^{10}$ solar mass (Gezari, 2014). They are usually located at the center of
quasars, which can be observed through various methods. One way is to measure
the amount of energy and the duration of variability in a specific region of
space, specifically at the center of quasars (Kozłowski et al., 2009). Another
way is to observe objects that orbit around an invisible mass at high speeds
(Komossa & Zensus, 2014).

High redshift quasars enable exploration of the early universe, less than 1
Gyr after the Big Bang. The early evolution of SMBHs can only be studied with
the help of the early-forming quasars. The mass of supermassive black holes in
quasars can be determined by observing the reverberation between the
fluctuations in broad emission lines and the continuum (Morganson et al.,
2012). The X-ray emission follows a power-law spectral pattern resulting from
inverse Compton scattering of photons from the accretion disk of relativistic
electrons in the hot corona and a soft X-ray excess. The optical-to-
ultraviolet continuum emission can be explained by a standard thin accretion
disk extending to the Innermost Stable Circular Orbit (ISCO) (Khangulyan et
al., 2023).

From optical to near-infrared spectroscopy, measurements of quasars imply that
SMBHs were already in place when the universe was merely 700 million years old
(Shen et al., 2019). Numerous theoretical ideas, including the super-Eddington
accretion process (Kawaguchi et al., 2004) and the use of primordial density
seeds (Kawasaki et al., 2012), have been put forth to explain the existence of
SMBHs (Regan et al., 2019).

Many works use the Sloan Digital Sky Survey (SDSS) (Lyke et al., 2020) to
utilize the data about quasars and SMBHs from the sky. SDSS has produced a
colorful 3D map of the universe, providing data on the mass, luminosity,
redshift, and other astrophysical parameters of various objects.

The other dataset that many other works, including Beutler et al. (2017), Zhao
et al. (2019), Alam et al. (2021), and Xu et al. (2023), have used is the
Baryon Oscillation Spectroscopic Survey (BOSS) dataset. BOSS aims to explore a
large volume required for precise measurements of the Baryon Acoustic
Oscillations (BAO) scale at the percent level. Achieving this level of
precision provides essential constraints on the source of cosmic acceleration
(Schlegel et al., 2009). BOSS also has measured the redshifts of 1.5 million
luminous galaxies and 160,000 high redshift quasars (Wang et al., 2017).

Galaxies naturally form groups and clusters, and studying these formations is
crucial to comprehending the large-scale structure, the dark matter mystery,
the development of galaxies, and the underlying cosmological model. A new
chapter in cosmology has been opened by the astounding recent observations of
the Cosmic Microwave Background (CMB) from several experiments, including the
Planck Collaboration (Aghanim et al., 2020). However, because of several
degeneracies in parameter space, measurements of CMB fluctuations alone are
insufficient to limit all cosmological parameters. By merging the CMB data
with other cosmic measurements and imposing a range of priors or constraints
on parameters, these degeneracies can be eliminated, or at least reduced
(Howlett et al., 2012).

The BAO distance scale is anchored in the CMB, which means that for a given
measurement precision, the impact of BAO observations on dark energy
characteristics is most significant at low redshift (Fronenberg et al., 2023).
Furthermore, BOSS sets the stage and supplies the crucial low redshift
reference point for upcoming higher redshift baryon oscillation
investigations. The BOSS Ly$\alpha$ forest survey is set to introduce a new
approach for high-redshift BAO measurements (Rosell et al., 2022).

In this paper, we have outlined the basic equations governing the physics of
accretion and the factors influencing them in Sec. 2. We have delved into the
importance of accretion disks and their types in Sec. 3. The downsizing issue
and its implications for the data have been discussed in Sec. 4, while the
primary datasets on which this work is focused have been introduced in Sec. 5.
The process of providing flux- and volume-limited samples has been explained
in detail in Sec. 6. Subsequently, we have analyzed the relationship between
the accretion efficiency of Black Holes (BHs) and its correlation with their
mass using both high and low redshift flux- and volume-limited datasets.
Ultimately, Sec. 8 focuses on examining the finalized plots concerning
accretion efficiency regarding BH mass and redshift.

##  2 The Physics Behind Accretion

Bondi-Hoyle-Lyttleton (BHL) accretion describes the supersonic movement of a
point mass through a gas cloud (Bondi & Hoyle, 1944). The field of spherically
symmetric accretion onto a point mass is investigated by Bondi (1952). Beyond
the Bondi radius, the flow is subsonic, and the density is almost constant.
The gas there reaches supersonic speed and eventually settles into a freefall
state. The cloud is assumed to be homogeneous and free of self-gravity at
infinity. As a result of gravity, a portion of the gas may be drawn towards
the material concentrated behind the point mass, a process known as BHL
accretion (Edgar, 2004).

The BHL model has been used in various studies. According to Gregoris (2023),
the primary premise of the model is that the adiabatic speed of sound in the
accreted fluid controls the rate of mass change of the accreting item. The
mentioned work shows that the evolution of the fluid is assumed to follow
either General Relativity or the Randall-Sundrum gravity model in an
asymptotically Friedman Universe, far from the astrophysical object (Xu &
Stone, 2019).

The mass-radius relationship of the photon sphere beyond the Schwarzschild is
unaffected by this accretion process, and the event horizon conceals the
critical point of accretion in the Berthelot fluid flow (Tejeda & Aguayo-
Ortiz, 2019). The BHL model can also be used to study the accretion of
Interstellar Medium (ISM) gas onto astrophysical objects, with the
relativistic counterpart being the ability of a BH to accrete fluid (Cruz-
Osorio et al., 2023).

The work of Font et al. (1999) has been on the relativistic example of an
ideal gas accreting onto a BH (Li et al., 2023). Their study has focused on
two-dimensional cases, specifically axisymmetric accretion into a
Schwarzschild BH (Lora-Clavijo & Guzmán, 2013) and the thin-disc approximation
of accretion onto a Kerr BH, due to computational limitations (Lora-Clavijo et
al., 2015). In the Newtonian model, supersonic flow resulted in the formation
of a shock cone, which sometimes clung to the BH horizon. However, the
subsonic flow has led to a smoother flow. Font et al. (1999) has also found
that at far distances from the horizon, the accretion pattern of a Kerr BH was
similar to that of a non-spinning BH. However, as one has approached the
horizon, the shock cone, if present, has encircled the BH in a spin-dependent
manner, with this effect diminishing further away from the horizon (Blakely &
Nikiforakis, 2015).

To calculate the accretion efficiency of quasars, one needs to be familiar
enough with a few astrophysical parameters. Therefore, in this work, it is
critical to grasp the methods for computing values like bolometric luminosity,
Eddington luminosity, Eddington ratio, and accretion efficiency. The
correlation between accretion efficiency and astrophysical factors, such as
SMBH mass and redshift, is essential. Hence, employing various methodologies
to calculate accretion efficiency becomes essential, enabling the examination
of how factors like bolometric luminosity, central SMBH mass, or even redshift
individually impact the results.

Bolometric luminosity is the total power output across all electromagnetic
radiation wavelengths of an astronomical body. The way to measure the
bolometric luminosity can be obtained by (Torres, 2010)

$L_{bol}=L_{\odot}\Big{(}\frac{4.74-2.5log\big{(}\frac{L}{L_{\odot}}\big{)}}{m_{\odot}}\Big{)}^{4},$
(1)

where $m_{\odot}$ is the solar magnitude and $L_{\odot}$ is the solar
luminosity.

The Eddington luminosity, also referred to as the Eddington limit, is the
maximum luminosity that an object can reach when there is a balance between
the force of radiation acting outward and the gravitational force acting
inward, called the hydrostatic equilibrium. Since most massive stars have
luminosities far below the Eddington luminosity, their winds are driven mainly
by the less intense line absorption (Wielgus et al., 2015). The Eddington
luminosity can be measured from (Rybicki & Lightman, 1991)

$L_{edd}\simeq 1.28\times 10^{38}(\frac{M_{BH}}{M_{\odot}}),$ (2)

where $M_{BH}$ is the SMBH mass in the unit of solar mass ($M_{\odot}$ $\simeq
2.0\times 10^{30}$ Kg).

Two critical factors define the accretion mechanism, which is a crucial
component. The relationship between the bolometric luminosity and the BH
growth rate may be established by looking at the radiative accretion
efficiency, represented by $\epsilon$ (Zhang & Lu, 2017). It should be
mentioned that the Active Galactic Nuclei (AGN) bolometric luminosity is
connected to the Eddington luminosity by the Eddington ratio (Eddington, 1988)

$\lambda_{edd}=\frac{L_{bol}}{L_{edd}}.$ (3)

In addition to all of these introduced parameters, the mass of the target BH
is also very important for this work.

It is important to specify whether bolometric luminosity plays a crucial role
in determining the accretion efficiency, as demonstrated by (Bardeen et al.,
1972; Novikov & Thorne, 1973)

$\epsilon=\frac{L_{bol}c^{2}}{\dot{M}},$ (4)

where $\dot{M}$ is the accretion rate.

The Eddington accretion rate, $\dot{M}_{edd}$, is determined by comparing the
disk luminosity with the Eddington limit and calculating a mass flow rate
(Abramowicz et al., 1988; Heinzeller & Duschl, 2007)

$\dot{M}_{edd}=\frac{L_{edd}}{\epsilon c^{2}},$ (5)

where $c$ is the speed of light.

According to Piotrovich et al. (2023a), accretion efficiency is calculated for
the Narrow Line Seyfert 1 (NLS1) galaxies data (Robson, 1996; Netzer, 2015).
It has been mentioned that the main model has been used based on Du et al.
(2014), where the defined accretion efficiency is

$\epsilon(a)=0.105\bigg{(}\frac{L_{bol}}{10^{46}erg/s}\bigg{)}\bigg{(}\frac{L_{5100}}{10^{45}erg/s}\bigg{)}^{-1.5}M_{8}\mu^{1.5},$
(6)

where $L_{5100}$ is the luminosity of a BH in the wavelength of 5100 $\AA$,
$M_{8}=M_{BH}/(10^{8}M_{\odot})$, where $M_{BH}$ is the BH mass, and
$\mu=\cos{i}=0.7$, where the angle between the Line of Sight (LoS) and the
axis of the accretion disk is denoted by $i$. Assuming an average angle is the
standard practice due to the lack of information on the angles of most objects
and the lack of evidence for a preferred direction in the orientation of
galaxies (Czerny et al., 2011). Moreover, the radiative efficiency and the
observable physical properties of AGNs are linked in many models, such as
Davis & Laor (2011), Raimundo et al. (2012), Trakhtenbrot (2014), and Lawther
et al. (2017).

In Raimundo et al. (2012), a trend of $\epsilon\propto M_{BH}^{0.5}$ using a
simulated sample corresponds to the distribution of the Palomar-Green (PG)
quasars in the Davis & Laor (2011) dataset (DL11) along an area in the
accretion efficiency versus the SMBH mass. The DL11 dataset is set in the low
redshift quasars and uses the PG quasar sample of Schmidt & Green (1983). For
the DL11 sample, the distribution of the optical luminosity and SMBH mass with
respect to redshift is investigated

$\epsilon=0.063\Bigg{(}\frac{L_{bol}}{10^{46}erg/s}\bigg{)}^{0.99}\bigg{(}\frac{L_{opt}}{10^{45}erg/s}\bigg{)}^{-1.5}M_{8}^{0.89},$
(7)

where $L_{opt}$ is the optical luminosity in erg $s^{-1}$.

It is worth mentioning that the accretion efficiency can approximately be
calculated with BH mass based on the data of PG quasars (Davis & Laor, 2011)

$\epsilon=1.08\times 10^{-21}M_{BH}^{0.52}.$ (8)

All in all, different types of accretion disks directly impact the
calculations of accretion efficiency. Therefore, it is necessary to consider
the selected accretion disk models. Thus, in the next Section, the various
types of accretion disks and their differences will be discussed in detail.

##  3 Types of Accretion Disks

BHs are believed to be the sources of ultraviolet and soft X-ray emissions
detected from AGNs due to the accretion disks surrounding them (Plotkin et
al., 2016). The ”big blue bumps” in the ultraviolet spectrum provide
observational evidence for this (Shields, 1978; Shang et al., 2005). However,
this model has faced challenges when confronted with data from multiple
wavelengths (Liu et al., 2008).

Empirical and theoretical analyses have shed light on the current state of
this issue. As Inoue (2010) has explained, the absence of observable Lyman
discontinuity is a significant obstacle for theoretical models, which has been
examined by several works including Davidson (1976), Shukla et al. (2016), and
Sellwood & Masters (2022).

The two primary factors influencing the accretion process are the angular
momentum of the material falling into the BH and the accretion rate. High
angular momentum flow results in an accretion disk, and the accretion rate is
expressed as a dimensionless value, which is the ratio of the accretion rate
to the Eddington accretion rate (Chakrabarti, 1996; Bu & Yang, 2019).

The accretion efficiency of a thin disk is determined by the BH spin, and some
energy is trapped in the disk, reducing the efficiency of the accretion flow
(Bu & Zhang, 2023). The value of the angular momentum at the outer disk, its
connection to the local Keplerian angular momentum, as well as the Eddington
ratio of the flow are used to classify commonly employed accretion disk models
(He et al., 2022).

Within the framework of three accretion disk models, the relationship of
accretion efficiency is explored: Radiation Inefficient Accretion Flow (RIAF),
slim disk, and standard disk (Czerny et al., 2011).

###  3.1 RIAF Accretion Disk

At lower specific accretion rates ($L_{int}/L_{Edd}\lesssim 10^{-2}$, where
$L_{int}$ represents intrinsic accretion luminosity; or $\dot{m}\lesssim
10^{-2}$), some AGNs are revealed without a broad-line region, while others
lack broad lines and exhibit narrow-line features (Trump et al., 2011). In the
inner radius of the accretion disk, a growing RIAF is responsible for the
disappearance of the broad emission lines. The presence of an RIAF simplifies
the production of a radio outflow, causing AGNs with this accretion model
(narrow-line and line-less) to exhibit radio-to-optical/UV emission ratios
approximately ten times larger than those with higher Eddington ratios (broad-
line) (Kollmeier et al., 2006; Trump et al., 2009; Gaskell & Goosmann, 2013).

In AGNs with low Eddington ratios, the IR torus signature often weakens or
disappears, while additional mid-IR synchrotron emission associated with the
RIAF may be present (Neri-Larios et al., 2011). Begelman et al. (1984),
Narayan et al. (1995), Yuan & Narayan (2004), and Narayan & McClintock (2008)
all have anticipated the presence of RIAFs in AGNs with a
$L_{int}/L_{Edd}\lesssim 0.01$.

RIAFs are a kind of rotating accretion flow characterized by minimal radiative
losses (Ichimaru, 1977; Rees et al., 1982; Narayan & Yi, 1994). The heat
energy is instead stored as gravitational potential energy released by
turbulent strains in the accretion flow. The accreting gas is thus, very hot,
with characteristic thermal energy similar to its gravitational potential
energy; this implies $T\sim GM_{BH}m_{p}/3k_{B}R\sim 10^{12}K$ at the vicinity
of the BH, where $T$ represents the temperature of the gas, $m_{p}$ represents
the proton mass, $k_{B}$ is the Boltzmann constant, and $R$ corresponds to the
distance from the BH (Quataert, 2003).

###  3.2 Slim Accretion Disks

Slim disk model characterizes high Eddington ratio sources as having an
optically dense, geometrically not very thin, quasi-Keplerian accretion flow
onto a BH, as put out by Katz (1977), Begelman (1978), and Abramowicz et al.
(1988). Contrary to the traditional standard accretion disk model, in such a
flow, a significant fraction of the energy wasted in the disk interior is
conveyed radially with the flow rather than being re-emitted at the same
radius (Lipunova et al., 2018). Observations confirm the existence of
optically thick disks in these sources. However, much discussion remains over
the stability of these models and the required adjustments.

In Abramowicz et al. (1988), the slim disk relies on the so-called viscosity
assumption, whereas a completely self-consistent model should predict the
viscous torque (Liu & Qiao, 2022). Despite significant progress in this
regard, the 3-dimensional numerical magneto-hydrodynamical models still lack
key elements for realistically depicting the flow onto the BH. Semi-analytical
models are now a highly effective tool for comparing the models to the
observational data. The link between the limit cycle behavior expected by the
slim disk model and the occurrence of heartbeat states in specific
astronomical sources is a central concern (Yuan & Narayan, 2014).

Finally, as the accretion rate increases beyond the Eddington accretion rate
in Eq. (5), the assumptions of the standard model break down ($\dot{m}\gtrsim
1$) (Heinzeller & Duschl, 2007). Since the BHs horizon allows energy to escape
through inflow, local dissipation no longer controls the local emission. The
accretion efficiency continues to lag behind that of a standard disk as the
accretion rate rises (Bian & Zhao, 2003). These models may account for the
observed properties of Super-Eddington quasars, NLS1 galaxies (Zhou et al.,
2017), and some phases of gamma-ray bursts. This is possible with some modest
adjustments to the physics to allow for neutrino cooling and nuclear processes
(Czerny, 2019).

###  3.3 Standard Accretion Disks

The standard accretion disk serves as the fundamental framework for a
radiatively efficient, geometrically thin, and optically thick disk that emits
multi-color black-body radiation with effective temperatures between $10^{5}K$
and $10^{7}K$, as proposed by Shakura & Sunyaev (1973). Besides Shakura &
Sunyaev (1973), Novikov & Thorne (1973), and Lynden-Bell & Pringle (1974) all
contributed to the standard accretion disk model (for an accretion rate of
$10^{-2}$ $\lesssim$ $\dot{m}$ $\lesssim 1$).

In the standard representation, the accretion disk mainly emits thermal
radiation within the optical-UV wavelength range for AGNs (Liu & Qiao, 2022).
AGNs with the mentioned accretion rate are foreseen to have accretion disk
thermal characteristic time-frames spanning from a few months to a few years
(Burke et al., 2021). The central radiations from narrow accretion disks have
been the subject of several works, including Hanawa (1989), Ebisawa et al.
(1991), Li et al. (2005), Zimmerman et al. (2005), and Pereyra et al. (2006).

While several investigations use standard disks as a basis (e.g. Calvet et al.
(1999), Caroline & Terquem (2007), Lorenzin & Zampieri (2009), and Armijo
(2012)), only a limited number of studies are dedicated to evaluating the
validity of conventional accretion disk models (e.g. Beloborodov (2001) and
Khesali & Khosravi (2013)).

In BH X-ray Binaries (BHXRBs) and AGNs, it is generally agreed that gas
accretion onto BHs is the primary source of radiation power (Smith, 2021).
Viscosity causes the caught gas to spiral inward toward the center of the BH,
where it loses part of its angular momentum and converts part of its
gravitational potential energy into heat. Depending on the specifics of the
radiation mechanisms involved, a spectrum can be generated from either some or
all of the viscous heat (Liu & Qiao, 2022). Four fundamental answers describe
the accretion flows, which are the standard accretion disk (He et al., 2022),
the optically thin two-temperature disk (Shapiro et al., 1976), the slim disk
(Katz, 1977; Begelman et al., 1984; Abramowicz et al., 1988), and the
advection dominated accretion flow (Ichimaru, 1977; Rees et al., 1982). A
combination of these approaches is often used to solve problems with BHs.

##  4 The Downsizing Problem

The term ”downsizing” was first introduced by Cowie et al. (1996), where the
”downsizing problem” has been established as lower-mass galaxies continuing
star formation for long periods, whereas the most massive galaxies created the
vast bulk of their stars during the early universe (Firmani et al., 2010).

The proportion of galaxies exhibiting very low star formation appears to rise
as a transition occurs to higher masses. Meanwhile, the median star formation
rate (SFR) decreases within the high mass range. This trend is less noticeable
at lower masses, as galaxies are experiencing star formation in specific time
ranges and remain highly active overall (Villar et al., 2011).

As an example, in binary star systems, the masses measured via orbital
dynamics are often lower than the masses estimated when astronomers use
brightness alone to estimate stellar masses using theoretical models of
stellar development (Pejcha, 2020).

Hierarchical theories of galaxy formation, on the other hand, proposed that
smaller galaxies would merge into larger ones over time, and the most massive
galaxies should have had more time to form. Next, it was found that stellar
populations, average SFRs, stellar masses, and dark matter halos mass are all
shrinking as a function of cosmic time (De Lucia & Blaizot, 2007).

Moreover, in more massive systems at $z>1-2$, the peak of cosmic star
formation activity and spatial density of star-forming galaxies have occurred
earlier than in lower mass systems at $z<1$(Firmani & Avila-Reese, 2010;
Conselice et al., 2018; Somerville et al., 2018).

This suggests that as the universe expanded, the most intensive star formation
events occurred in ever less massive dark matter halos. The process of
downsizing has created challenges for scientific theories that attempt to
explain the rapid and early development of the largest galaxies, as well as
the period during which their star formation has been halted, leading to the
formation of inactive galaxies that populate the universe today (Mutch et al.,
2013).

Focusing on the SMBHs, if SMBHs are thought to originate from early
Ultracompact Dwarfs (UCD) or hyper-massive star-burst clusters, then SMBHs
cannot develop for spheroid masses below a certain threshold, leaving only the
accumulated nuclear cluster. This indicates that SMBH creation is a shrinking
process, with lower-mass galaxies creating their SMBHs on different timelines
than higher-mass galaxies (Kroupa et al., 2020).

Additionally, there can be a conflict between the measured SFRs and the high
SFRs predicted by shrinking. According to the Integrated Galactic Initial Mass
Function (IGIMF) idea, this stress might be reduced if shrinking time frames
are somewhat longer (Pflamm-Altenburg et al., 2007). In conclusion, downsizing
is the general tendency towards lower mass galaxies having a shorter duration
for the creation of SMBHs than higher mass galaxies (Fontanot et al., 2009).

Better said, the downsizing problem is the discrepancy between the masses of
stars measured directly and those inferred from their luminosity. Thus, to
understand how SMBHs and their host galaxies have evolved, the downsizing
problem can be recognized (Li et al., 2012; Izumi, 2018). It is anticipated
that the same peak that can be seen in the SFR in terms of redshift plots
(Neistein et al., 2006; Fontanot et al., 2009), which depends on the mass of
the host structures, can appear in the plot of the SMBHs mass in terms of
redshift. This is expected because the mass of the host structures and the
mass of their central SMBHs are directly correlated.

Therefore, it is beneficial to investigate this phenomenon for the mass
evolution of SMBHs. Tabasi et al. (2023) have reported a similar occurrence,
reporting a peak for mass in terms of redshift, which is comparable to the
peak associated with different downsizing plots. In the following Sections, we
aim to apply the same approach to determine the accretion efficiency in terms
of redshift. This measure takes into account not only the central mass of
quasars but also factors, such as the bolometric and optical luminosity to
calculate the accretion efficiency. Thus, if the results indicate a peak, it
suggests that accretion efficiency should also be addressed regarding the
downsizing problem. In the next Section, we introduce the main dataset we have
used in this work.

##  5 Dataset

QUOTAS dataset is a revolutionary dataset allowing data-driven analysis of the
SMBH population. The importance of the QUOTAS dataset is emphasized when
comparing correlations between observed SDSS quasars and their hosts to models
like Schneider et al. (2010); it is demonstrated that with the use of Machine
Learning (ML), a template can be provided to have a more acceptable match of
expanded simulated samples of quasars with the observational survey volumes
(Natarajan et al., 2023). Another reason for the significance of the QUOTAS
dataset is that with aggregating and co-locating the high redshift ($z\geq$ 3)
quasars population with simulated data spanning at the same cosmic epochs,
there is a possibility to examine and study SMBHs.

To create the QUOTAS dataset, the NASA-IPAC Extragalactic Database (NED; Cook
et al. (2023)) has been used to gather information. However, some data values
are missing from the NED repository due to inaccurate photometric redshifts.
In this study, the aim is to investigate the accretion efficiency of the
central SMBHs, as introduced by the QUOTAS dataset, using the data provided by
Piotrovich et al. (2023b).

The theoretical models that attempt to depict the evolution of the populations
of the BH assembly history over time rely on two functions that have been
determined by observations: the BH Mass Function (BHMF) and the Quasar
Luminosity Function (QLF). The BHMF represents the evolution of mass as a
statistical measure of the distributed BH mass via redshifts. Statistically
measuring the distribution of quasar luminosities across redshift, the BHMF is
comparable to the QLF in that it represents the accretion history, referring
to the QUOTAS dataset (Natarajan et al., 2023).

On the other hand, the QuasarNET dataset is presented as a deep Convolutional
Neural Network (CNN) that can accurately classify astrophysical spectra and
estimate their redshift. These two activities are framed as a feature
detection problem, where the presence or absence of spectral characteristics
determines the class and their wavelength identifies the redshift (Busca &
Balland, 2018). The QuasarNET dataset establishes a sample purity and coverage
of $99.51\pm 0.03\%$ when run through the BOSS data (Alam et al., 2015) for
quasar identification by emission lines, exceeding the requirements of many
analyses.

With the QuasarNET dataset, the line confusion problem that causes
catastrophic redshift failures has been reduced to below 0.2%. To clarify, the
problem of line-confusion happens when spectral lines from varying redshifts
may coincide in the same observed frequency, leading to potential confusion
(Cheng et al., 2020). It can occur in large-scale spectroscopic surveys, which
may cause a shift and broadening of the BAO peak, potentially leading to
errors in determining the position of the BAO peak (Massara et al., 2021).

The BOSS database is used to train the QuasarNET dataset. This database has
about 500,000 quasar-target spectra visually reviewed and annotated by human
specialists (Pâris et al., 2017). For the classification of spectra, including
Broad Absorption Line (BAL) features, the QuasarNET dataset will also be
expanded, reaching an accuracy of $98.0\pm 0.4\%$ for recognizing BAL and
$97.0\pm 0.2\%$ for rejecting non-BAL quasars. Spectra with a BAL from ongoing
and prospective astrophysical surveys like eBOSS (Dawson et al., 2016), DESI
(Aghamousa et al., 2016a, b), and 4MOST (de Jong et al., 2012; Alonso et al.,
2016) might be easily classified using the QuasarNET dataset. Since it was
trained on data with low signal-to-noise and medium resolution, redshifts may
also be reliably calculated using neural networks (Busca & Balland, 2018).

A separate line finder in the output layer identifies each emission line.
These line-finders are specific to the emission line they detect. To implement
this feature identification, the latest strategies for object detection in
photos have been used as inspiration for a classification and regression
problem. Seven line-finders covering wavelengths have been used from
Ly$\alpha$, CIV, CIII, MgII, H$\alpha$, H$\beta$, and BAL CIV in a range of
[121.6 nm–656.3 nm]. These lines are chosen for quasars with redshifts between
0 and 5.45 so that at least two of them are visible in the optical
spectrograph of BOSS (Redmon & Farhadi, 2017).

To achieve a comprehensive model for quasars that aligns with both high and
low redshift observations, data from datasets that include both is required.
Combining the QUOTAS+QuasarNET dataset with a set of PG quasars enables us to
attain the desired outcome. In contrast to the QUOTAS+QuasarNET dataset, which
uses higher redshift data, DL11 has also been utilized, with low redshift data
values less than 1 ($z\sim 0.025–0.5$).

In Davis & Laor (2011), the absolute accretion rate may be inferred from
standard thin accretion disk model spectral fits in individual AGNs, given the
SMBH mass. The ratio of the bolometric luminosity to the accretion rate is
used to determine the accretion efficiency. This technique is used to find
$\epsilon$ in 80 PG quasars (Schmidt & Green, 1983) with known bolometric
luminosity (Neugebauer et al., 1987). The accretion rate is defined by a
standard accretion disk model fitted to the optical luminosity density, and
SMBH mass is either given by the bulge star velocity dispersion or the wide
line area.

DL11 includes 80 PG quasars and has a wide range of bands being covered,
including optical to far-UV (Scott et al., 2004) and X-rays (Brandt et al.,
2000). Although the unobservable EUV emission has a significant error, this
selection has obtained a robust estimate of the bolometric luminosity. Aside
from the accretion rate estimates, these luminosities allow us to determine
the accretion efficiency of each source in the population.

##  6 Creating Flux and volume-limited samples

To work on the QUOTAS+QuasarNET+DL11 dataset, model a parameter through
redshift, and eliminate biases, one needs to apply flux- and volume-limit
correcting methods. The correction of datasets, especially those including
higher redshift objects, is crucial to ensuring the accuracy and reliability
of the analyses and results. Implementing these methods allows relevant
information to be accurately captured while avoiding overestimation or
underestimation due to different biases. In the following, the biases and
problems that arise when working with higher redshift data are explained.

I) The selection of quasars based on their brightness or location can lead to
a bias towards more luminous objects. As a result, the selection process is
influenced by a preference for brighter objects (Green et al., 1986; Hewett et
al., 1995). II) Redshift bias occurs when quasars are either over-represented
or under-represented in specific redshifts, which can compromise analyses due
to their varied luminosity. Their low luminosity and faintness cause under-
representation, and high luminosity or brightness cause over-representation
(York et al., 2000). III) The distribution of varying luminosities is a
sensitive matter; therefore, the luminosity function needs to be properly
estimated. Estimating the luminosity function causes a bias due to its varied
luminosity distribution through cosmic time (Schmidt & Green, 1983; Boyle et
al., 2000). IV) It is possible that rare objects and quasars might not be
detected in deep surveys or when using narrow selection criteria. However,
these rare objects can provide vital information about the physical processes
or beneficial details. Thus, failure to detect rare objects can cause a
significant bias (Fan et al., 2003). V) Dust and gas can obscure quasars,
leading to a bias toward brighter objects. This bias can ruin identifying
quasars with different levels of extinction. This bias is also known as
galactic extinction bias (Blakeslee et al., 2002). VI) Quasars may be observed
in more than one survey, leading to a bias because of the difference in
selection criteria or coverage of surveys, thus leading to the survey bias
(York et al., 2000). VII) The efficiency and completeness of the quasar sample
near high redshift quasars are significantly reduced since their colors are
similar to regular stars (Fan, 1999; Skrutskie et al., 2006). In other words,
an issue of incompleteness is observed at the brighter end of galaxy
magnitudes, which is attributed to the saturation of bright galaxies and
blending by saturated stars (Tago et al., 2010).

Flux-limiting the samples of quasars can significantly improve the results and
eliminate most of the mentioned biases. By using the flux-limiting method, the
selection bias can be resolved because it selects quasars based on their
observed luminosity without any bias concerning brighter objects. Therefore,
the bias that results from the preferred selection can be eliminated (York et
al., 2000; Croom et al., 2004). Furthermore, to address the issue of redshift
distribution bias, it is important to use a flux-limited sample. Such a sample
helps to make the data more representative across different redshift ranges by
including more distant quasars that may be under-represented in the sample due
to their faintness and lower luminosity. This helps to reduce any bias in the
redshift distribution by including quasars through varied redshifts (Pâris et
al., 2017). Moreover, the bias towards estimating the luminosity function can
be mitigated by flux-limiting a sample of quasars with varying luminosity
distributed uniformly and balanced over cosmic time due to the uniform and
balanced distribution of quasars in the flux-limited sample (Ilbert et al.,
2004; Cole, 2011). Ultimately, The flux-limited sample can clear bias for
detecting rare objects by including rare objects such as high-redshift quasars
and identifying them accurately (Banados et al., 2018).

Therefore, limiting the utilized dataset with flux constraints is necessary
for accurate results. The QUOTAS+QuasarNET dataset contains 23301 quasars from
the BOSS SDSS-III dataset. Considering that the minimum absolute magnitude of
the BOSS dataset is approximately $M_{i}$ $\approx$ -24.5 at the redshift of
$z=2.2$, this information can be used to estimate the detectable absolute
magnitude for quasars located at further distances (Ross et al., 2013).

To calculate the mean absolute magnitude for each co-moving distance, first,
the minimum absolute magnitude at each redshift needs to be found. A plot can
then be illustrated by fitting the minimums of absolute magnitudes, and any
quasar with an absolute magnitude below the minimum is removed. To convert the
apparent magnitude, ”$m$”, to the absolute magnitude, ”$M$”, at redshift
$z=0$, the mean redshift, ”$z_{cl}$”, and mean co-moving distance,
”$d_{com}$”, are utilized. For this purpose, the following equation can be
used (Tempel et al., 2014)

$M_{\lambda}=m_{\lambda}-25-5log_{10}(d_{L})-K_{\lambda},$ (9)

where $d_{L}=d_{com}(1+z_{cl})$ is the luminosity distance in units
$h^{-1}$Mpc, $K_{\lambda}$ is the correction factor, and the index $\lambda$
refers to each of the $ugriz$ filters. For distance calculations in this work,
a flat universe with zero curvature is considered.

One advantage of this type of sample is that it utilizes as much observational
data as possible. However, many research projects require creating volume-
limited samples, which means that a large amount of data must be rejected. A
flux-limited sample naturally has negative selection effects, even though
attempts have been made to minimize them. An extra issue comes up when it
comes to Quasi-stellar objects (QSOs) (Ghirlanda et al., 2012). QSOs must have
a certain absolute magnitude to be included in the volume-limited sampling.
The latter is reliant on the distance of a QSO. Initially, without group
formation, the redshift of a QSO must be used to calculate its absolute
magnitude. Once a group has been formed, the mean radial velocity (redshift)
of the group determines the distance of the QSOs from one another (Best et
al., 2024).

Moreover, after creating the flux-limited sample, some biases remain unsolved,
including the bias regarding galactic extinction, the survey bias, and the
bias of incompleteness. Therefore, an additional method is pursued called
volume-limiting. After volume-limiting and considering varied extinction
levels, the bias regarding galactic extinction can be solved. This reduces the
effect of quasars less affected by extinction (Skrutskie et al., 2006).
Volume-limited samples can also solve the survey bias caused by selecting
quasars based on distance, and they can be tailored to specific surveys (York
et al., 2000; Ross et al., 2012). Additionally, the volume-limited sample
addresses the incompleteness bias by including a greater number of quasars in
a selected cosmic volume of space (Croom et al., 2004). Therefore, it is
necessary to use the volume-limiting method to solve biases.

To apply the volume-limiting method on a dataset, one should get familiar with
the Friends of Friends (FoF) method. The group discovery is based on the FoF
method, which was proposed by Turner & Gott III (1976). The FoF algorithm has
been the most widely used method for detecting clusters and groups in galactic
redshift data. This approach uses a certain neighborhood radius, the Linking
Length (LL), to connect objects into systems. It is assumed that all objects
within the LL radius are part of the same system for any given object. The
selected LL significantly impacts the richness and quantity of the identified
groupings. Typically, LL is not fixed but rather permitted to change as a
function of distance and other variables. It has been noted that the
objectives of the particular investigation dictate the LL to be used. Finding
a large number of groups with uniform distance-dependent general group
attributes is the goal here (Cunningham et al., 2020).

The best estimate for the LL is (Tago et al., 2010)

$\frac{LL}{LL_{0}}=1+a\arctan{(\frac{z}{z_{*}})},$ (10)

where $a=1.00$ and $z_{*}=0.050$ are free parameters, and $LL_{0}$ is the
value of LL at the initial redshift of $z=0$. Flux-limited samples can be made
with constant or varied LL, and the size of the groups depends on the LL. The
constant value for $LL_{0}$ is considered to be 250 km$s^{-1}$ ($\sim 0.25$
$h^{-1}$Mpc).

Only the brightest items may be seen while looking at distant objects. The
parameters of a set of objects are found by calculating their absolute
magnitude from their mean distance. If an object or a group member is too
faint to be seen, this will delete them automatically. Applying the flux-
limited approach and fixing the data are both made possible by this procedure.
The downside of this procedure is that the remaining absolute magnitudes of
the objects can exceed the range of magnitudes covered by the sample. In such
a situation, the sample has to have those objects deleted and the groupings
recreated. After this is completed, the revised group data must be used to
recalculate the absolute magnitudes of the objects. It is important to note
that there is no assurance for the end of this cycle (Tempel et al., 2014).

The separation of galaxies along the LoS or the Plane of the Sky (PoS) is used
to group them in the FoF percolation technique. All galaxies that are
connected in pairs based on their distances are grouped. The two connecting
lengths that identify FoF are the LoS and the PoS. The average distance
between field galaxies is used to normalize these lengths (Duarte & Mamon,
2014).

As mentioned, objects that do not meet volume-limited sample requirements are
removed from groups after calculating their absolute magnitudes and distances.
Afterward, the FoF technique is applied to the identified sub-sample of QSOs,
keeping the LL constant. The process of group finding is finished with this
step (Etherington & Thomas, 2015). In this work, after applying both flux- and
volume-limiting methods, 12359 data from the QUOTAS+QuasarNET dataset remain
for calculation. Moreover, the effect of flux- and volume-limiting on low
redshift data is little to none. Therefore, after removing the accretion
efficiencies higher than ten from the DL11 dataset, 79 quasars remain.

##  7 $\epsilon$ – $M_{BH}$ relationship

Previous works have demonstrated how luminosity, redshift, and mass of quasars
can significantly impact the relationship between the SMBH accretion
efficiency and mass. Therefore, the QUOTAS+QuasarNET dataset is employed to
calculate the accretion efficiency of each object using Eq. (7) and try to
plot the accretion efficiency parameter in terms of SMBHs mass. The
flexibility to find a more precise and general result is provided by the fact
that the SMBHs mass in the QUOTAS+QuasarNET dataset spans from $10^{8}$ to
$10^{11}M_{\odot}$. This work aims to expand the results of Raimundo et al.
(2012) by modeling the accretion efficiency of SMBHs in low and high redshifts
altogether.

A related figure can be plotted to investigate the relationship between the
SMBH accretion efficiency and mass. By drawing lines that represent the
maximum and minimum luminosity, the boundaries for the distribution of SMBH
accretion efficiency can be defined. Moreover, it is possible to plot the
Eddington ratio, which includes maximums and minimums, as well as to surround
the data with the four lines of $L_{min}$, $L_{max}$, $\lambda_{min}$, and
$\lambda_{min}$.

In Sec. 5, the corrections required in the considered dataset have been
discussed. The first step in the data cleaning process has been to remove the
unrealistic values for data points, including zero mass or negative numbers.
Then, the accretion efficiency for each wavelength contained in the dataset is
calculated, which includes $1350\AA$, $1700\AA$, and $3000\AA$. To determine
the accretion efficiencies for the QUOTAS+QuasarNET dataset, Eq. (7) is used.
As previously stated, the DL11 dataset is included in order to create a
general and comprehensive model for low and high redshift values.

The smallest value from the three calculated accretion efficiencies needs to
be selected to identify the maximum Eddington ratio leading to the minimum
accretion efficiency. This important step allows the data to be filtered and
the most relevant information to be selected. The required data for satisfying
all parameters of Eq. (7) by flux and volume-limiting the QUOTAS+QuasarNET
dataset are presented in Table LABEL:tab:qn-tab-fin.

In Fig. 1, the relationship between SMBHs accretion efficiency and mass is
plotted using the flux- and volume-limited QUOTAS+QuasarNET. The original data
contained about 37648 quasars, and about 76 final quasars had the required
information, including the optical luminosity, the bolometric luminosity, the
Eddington ratio, redshift, and the SMBH mass of each object after the data
pre-processing.

As shown in the plot, there is a positive correlation between SMBH mass and
accretion efficiency. This means that the more mass of SMBHs there is, the
more efficient the accretion is. Eq. (3) has been used to calculate the
maximum and minimum Eddington ratio. Bolometric luminosity reaches its
maximum, Eddington luminosity reaches its minimum, and the Eddington ratio
reaches its maximum, and vice versa. This way, the related lines to
$\lambda_{min}$ and $\lambda_{max}$ can be plotted.

Figure 1:  This plot represents the relationship between the SMBH accretion
efficiency and mass of the flux- and volume-limited QUOTAS+QuasarNET dataset.
In the plot, the solid green lines represent $L^{\prime}_{max}$,
$L^{\prime}_{min}$, $\lambda^{\prime}_{min}$, and $\lambda^{\prime}_{min}$. As
depicted in the plot, there is a positive correlation between SMBH mass and
accretion efficiency. This indicates that higher SMBH masses are associated
with greater accretion efficiency.  Figure 2:  This graph illustrates the
relationship between SMBH accretion efficiency and mass for the flux- and
volume-limited QUOTAS+QuasarNET+DL11 dataset. In the plot, the solid green
lines represent $L^{\prime}_{max}$, $L^{\prime}_{min}$,
$\lambda^{\prime}_{min}$, and $\lambda^{\prime}_{min}$ for the
QUOTAS+QuasarNET+DL11 dataset; where the solid grey lines represent $L_{max}$,
$L_{min}$, $\lambda_{min}$, and $\lambda_{min}$ for the DL11 dataset.

Lastly, the maximum and minimum values of bolometric luminosity are gathered
to calculate the values for accretion efficiency using Eq. (7). This equation
provides a way to estimate the efficiency of accretion based on the luminosity
produced by the accretion process. Insights into the physical processes that
govern SMBH growth and accretion are gained by using these equations. A
specific bond is created by the lines evident in the graphs.

The QUOTAS+QuasarNET dataset is designed for quasars with high redshift. It is
indicated that adding the DL11 dataset of the 79 PG quasars to the flux- and
volume-limited QUOTAS+QuasarNET data collection and conducting a comparison
between the two datasets could be an exciting discovery. It has been noted
that after removing the nonphysical and duplicated data, the combined dataset
of flux- and volume-limited QUOTAS+QuasarNET and the PG quasars consist of
approximately 155 quasars.

The same filtering and correction technique as Fig. 1 has been applied to the
QUOTAS+QuasarNET+DL11 dataset as well, as shown in Fig. 2. It is important to
note that all the data used in Fig. 2 are in the valid range of estimation for
the standard accretion disk efficiency of $0.038<\epsilon<0.42$ (Zhang & Lu,
2020). It is evident that due to the low redshift data of DL11, the accretion
efficiency is found in the lower SMBH mass range. An intriguing finding is
that the average range of accretion efficiency decreases when the SMBH mass
range is lowered. The accretion efficiency is projected to decrease for SMBHs
with lower mass in low redshift, just as it rises with SMBHs with higher mass
of the high redshift data. Thus, the idea can be put forth that there is a
correlation between SMBH accretion efficiency and mass independent of
redshift.

##  8 Accretion efficiency history

In the last Section, a direct correlation between the SMBH accretion
efficiency and mass was found, independent of redshift. Now, it must be
checked whether redshift is directly correlated with accretion efficiency and,
if so, to what extent. In this Section, the possibility of a correlation
between redshift and accretion efficiency is explored. As a result, a color
bar illustrates the central SMBH mass, and the overall plots show the
accretion efficiency in terms of redshift.

As seen in Fig. 3, it is illustrated that an interesting correlation between
the growth of redshift and the increase in accretion efficiency is present for
low redshift SMBHs. It is clear that in redshifts lower than $z\sim 0.5$, the
mass of the observed SMBHs decreases. According to the flux- and volume-
limited QUOTAS+QuasarNET dataset, which has a redshift range of $5.844\lesssim
z\lesssim 7.1$, the population of higher mass SMBHs increases with the
decrease of redshift as shown in Fig. 4. Furthermore, with decreasing the
redshift, SMBHs mass in the order of $10^{7}M_{\odot}<M_{BH}<10^{9}M_{\odot}$
can be observed as well.

Figure 3:  This plot shows the relationship between accretion efficiency,
redshift, and SMBH mass of the DL11 low redshift PG quasars. The visualization
reveals an intriguing correlation between increasing redshift and accretion
efficiency. As redshift values rise, a corresponding increase in accretion
efficiency is expected. Figure 4:  This plot illustrates the relationship
between accretion efficiency, redshift, and SMBH mass using the flux- and
volume-limited samples from the QUOTAS+QuasarNET dataset. At redshifts
exceeding 5 ($z>5$), the correlation between SMBH mass and redshift is
apparent, showing a decrease in SMBH mass as redshift increases. Consequently,
consistent results are obtained.  Figure 5:  This graph illustrates the
relationship between accretion efficiency, redshift, and SMBH mass using the
flux- and volume-limited samples from the QUOTAS+QuasarNET+DL11 datasets,
along with a trend line indicating the peak of accretion efficiency. A
regression fit has been applied to the flux- and volume-limited datasets,
clearly showing the peaks of the data, with observed values around $z\sim
2.675$.

What can be understood from the said graphs is that in low redshifts, the mass
of the central SMBHs of quasars in the local universe tends to decrease as
$z\sim 0.025$. Therefore, SMBHs with closer distance have less mass compared
to farther SMBHs.

The situation for the high redshift data is inversely correlated. Therefore,
investigating the two datasets of DL11 and QUOTAS+QuasarNET together, the
QUOTAS+QuasarNET+DL11 datasets can anticipate a peak where the two trends
meet, as can be seen in Fig. 5.

To understand how accretion efficiency behaves over time, it is necessary to
compare new and corrected data that have been subjected to flux- and volume-
limitations. These restrictions are prominent to ensure that the data is as
precise and reliable as possible. In Sec. 6, a deeper delve is made into why
these limitations are deemed essential in the context of this study. Despite a
more than 50% reduction in the abundance of the data, accretion efficiency
shows growth as a function of growing mass. Therefore, the correlation between
SMBHs accretion efficiency and SMBHs mass is shown independent of flux- and
volume-limitations.

It should be mentioned again that since flux- and volume-limit corrections
have little to no effect on data with low redshifts, the plot of flux- and
volume-limited for the DL11 dataset is identical to Fig. 3.

Visualizing the results is also essential to understand the data better. In
Fig. 5, a regression fit has been applied to the flux- and volume-limited
data. This results in a clear indication of the peaks of the data, with values
of approximately $z\sim 2.675$ being observed. This peak can serve as a
reminder for the redshift range, $z_{SFR}\sim 1-3$, in which the peak of SFR
occurs.

As discussed before, SFR shows the amount of mass produced per year in a
galaxy. Using the low-mass X-ray binaries (LMXBs) data, White & Ghosh (1998)
has reported the SFR peak to be $z\sim 1.5$. Moreover, multiple works such as
Atek et al. (2014), Zolotov et al. (2015), and Driver et al. (2018) show that
the peak of SFR is at $z\sim 2$, about 3.5 billion years after the Big Bang.
To go further into detail of the redshift range of the peak of SFR, Hopkins &
Beacom (2006) produced an extensive collection of observations of luminosity
density, covering a wide range of redshifts up to $z=6$. Their findings
indicate a significant increase in the SFR at low redshifts, reaching its
highest point at around $z\simeq 2$ and declining at higher redshifts
(Vangioni et al., 2015). Cucciati et al. (2012) reports the Star Formation
Rate Density (SFRD) peak to be $z\sim 2$, using the VIMOS-VLT Deep Survey
(VVDS). In Jo et al. (2021), for the 1752 objects of the Spitzer COSMOS Legacy
survey, the SFRD peak is approximated between $z\sim 2.2-2.4$. Moreso, in
Schinnerer et al. (2016), Scoville et al. (2017), and Tacconi et al. (2018)
using the data from Main Sequence (MS) galaxies, the peak of SFR is located in
$z\sim 1.5-3$. Additionally, according to Reddy et al. (2007), Madau &
Dickinson (2014), and Hodge & da Cunha (2020), the redshift in which SFR is at
its maximum is thought to be around $z\sim 1-3$. If the target data is
increased and more accurate surveys are used, with the condition that they
will include all the necessary parameters for calculating the accretion
efficiency, the final result will be more precisely finalized. By analyzing
this data, a better understanding of how accretion efficiency behaves over
time can be obtained, and more accurate predictions about future trends can be
made.

By fitting a trend line to the corrected flux- and volume-limited
QUOTAS+QuasarNET+DL11 dataset, a new equation is attained,

$log(\epsilon_{2\sigma})=-0.01621z^{2}+0.08671z-1.07704\pm 0.8505,$ (11)

As mentioned before, the peak nears the SFR peak much more closely. Note that
the $4\sigma$ error bar for the accretion efficiency of the mentioned data is
$\epsilon_{4\sigma}\sim 1.7011$. Based on Fig. 5, and the observed peak, the
accretion efficiency should be part of the parameters where the downsizing
problem must be investigated for. Thus, future works could delve deeper into
this concept, focusing on the downsizing process and considering accretion
efficiency.

##  9 Conclusion

In this paper, we have employed datasets containing SMBHs in the center of
quasars, using the information of QUOTAS+QuasarNET+DL11 datasets. The
QUOTAS+QuasarNET datasets used by projects, such as the QUOTAS and QuasarNET
datasets, encompass 37648 high redshift objects, and the DL11 dataset includes
80 PG low redshift quasars. These datasets include essential parameters, such
as SMBH mass, bolometric luminosity, optical luminosity, and Eddington
luminosity.

Using these parameters, numerous models have been developed to estimate the
accretion efficiency for SMBHs, and various previous calculations and
simulations have been conducted. However, whether we can accurately make a
model to find the dependency of accretion efficiency on redshift and SMBH mass
has been questioned. Different methods have been proposed for calculating this
parameter, and in this work, we have opted for an approach based on optical
luminosity because other methods yield values beyond the feasible range
($0.038<\epsilon<0.42$) for accretion efficiency.

Since there is a need for a comprehensive model to calculate the accretion
efficiency in all redshifts, the DL11 low redshift ($z<0.5$) dataset has been
used in addition to the QUOTAS+QuasarNET. By fitting the spectra of individual
AGNs with the standard thin accretion disk model, it is possible to deduce the
absolute accretion efficiency. This dataset provides a robust estimate of the
bolometric luminosity as it includes a wide variety of bands, from optical to
far-UV and X-rays. These luminosities allow one to ascertain the accretion
efficiency of each source.

As a part of the data pre-processing, we have excluded AGNs with RIAF and slim
disks. Additionally, in a subsequent data refinement step, we have applied the
flux- and volume-limiting corrections to be able to work on the dataset
through redshift. The primary focus has been on investigating the evolution of
the accretion efficiency over redshift or cosmic time to determine if it
follows a discernible pattern.

Given the research conducted by Tabasi et al. (2023) on SMBHs mass, we were
intrigued by the possibility of modeling based on redshift. In the mentioned
work, they found a peak for SMBH mass evolution through redshift; therefore,
we were curious to see whether the accretion efficiency had the same peak or
not.

In the QUOTAS+QuasarNET dataset, there is a visible pattern where accretion
efficiency rises as the redshift decreases, along with an increase in mass
growth. On the other hand, in the DL11 dataset, as redshift increased, both
accretion efficiency and mass exhibited growth, leading us to identify a
redshift peak between the trends of these two datasets. Subsequently, we have
recognized the optimal fit for the flux- and volume-limited
QUOTAS+QuasarNET+DL11 dataset. The calculations of this paper begin by
examining the correlation between the mass of SMBHs and the accretion
efficiency. It is evident from Fig. 2 that SMBHs with a higher mass also have
higher accretion efficiency. Nonetheless, the fact that the central SMBHs of
quasars are heavier at higher redshifts is evident. Then, an examination is
conducted into the correlation between accretion efficiency and redshift. The
peak of the accretion efficiency factor is observed with the information of 79
quasars from the DL11 dataset and 76 quasars from the QUOTAS+QuasarNET
dataset, after flux- and volume-limiting corrections, as illustrated in Fig.
5.

The redshift peak for the flux- and volume-limited dataset occurred at
approximately $z\sim 2.675$. This peak can be a recollection of the SFR peak,
which is between $1<z<3$, prompting us to explore potential similarities
between this peak and the SFR peak, as well as investigate whether the
accretion efficiency could mirror the SFR in AGNs. The peak we have found is
closely aligned with existing introduced values by different works, as
mentioned in Sec. 8.

It is worth noting that Tabasi et al. (2023) have created a model for SMBHs
mass in terms of redshift, and the observed peak has been approximately $z\sim
4.72$. They have also stated that they have not considered SFR in their work.
In contrast, our work has used the calculations related to the accretion
efficiency for the mentioned result, and it seems that the final peak has
automatically considered the SFR, where it has been implemented internally.

Future works can investigate the relationship between the mass of central
SMBHs of quasars, SFR, and accretion efficiency. Moreover, a comprehensive and
complete model can be achieved by incorporating data from additional surveys,
utilizing a larger dataset across all redshifts, and comparing the final
models with the newly added data from the JWST dataset. In conclusion, it is
critical to underscore the importance of priming any constructed model on
empirical evidence; relying on simulation, given the present state of
knowledge, would be impractical because an accretion model should be chosen
between various models for simulations at the beginning. Meanwhile, none of
the models are perfect for all quasars in different redshifts. Thus, more
accurate observations are needed to estimate the accretion efficiency
reliably. Therefore, the physics behind the accretion of SMBHs lacks a
flawless model, which can be improved with a significant increase in data.

##  10 Data Availability

The code behind the data pre-processing of this article is publicly available
on GitHub.

We have used the QUOTAS dataset (Natarajan et al., 2023) in this work, which
supports the findings of this study and is publicly available on the Google
Kaggle platform and can be found at https://www.kaggle.com/data
sets/quotasplatform/quotas.

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Table 1:  This table represents the flux- and volume-limited QUOTAS+QuasarNET dataset, which has been used in this work. Object | $z^{a}$ | Log $M_{BH}^{b}$ [$M_{\odot}$] | Log $L_{bol}^{c}$ [erg/s] | Log $\lambda_{edd}^{d}$ [erg/s] | Log $L_{opt}^{e}$ [erg/s] | $\epsilon_{opt}^{f}$
---|---|---|---|---|---|---
SDSS J0005-0006 | 5.844 | 8.0 | 46.67 | 0.556 | 45.958 | 0.011
SDSS J1411+1217 | 5.854 | 8.954 | 47.182 | 0.114 | 46.47 | 0.041
SDSS J1411+1217 | 5.903 | 9.204 | 47.272 | -0.046 | 46.56 | 0.062
SDSS J1306+0356 | 6.017 | 9.23 | 47.123 | -0.222 | 46.411 | 0.078
SDSS J1306+0356 | 6.018 | 9.462 | 47.053 | -0.523 | 46.342 | 0.135
SDSS J1630+4012 | 6.058 | 9.23 | 47.043 | -0.301 | 46.332 | 0.085
SDSS J0303-0019 | 6.079 | 8.699 | 46.591 | -0.222 | 45.879 | 0.049
SDSS J1623+3112 | 6.211 | 9.342 | 47.155 | -0.301 | 46.444 | 0.094
SDSS J1030+0524 | 6.302 | 9.301 | 47.114 | -0.301 | 46.402 | 0.091
SDSS J1030+0524 | 6.299 | 9.38 | 47.193 | -0.301 | 46.481 | 0.097
SDSS J0842+1218 | 6.069 | 9.462 | 47.178 | -0.398 | 46.467 | 0.117
SDSS J0005-0006 | 5.844 | 7.778 | 46.67 | 0.851 | 45.958 | 0.007
SDSS J1411+1217 | 5.854 | 8.699 | 47.182 | 0.342 | 46.47 | 0.024
SDSS J1411+1217 | 5.903 | 9.0 | 47.272 | 0.176 | 46.56 | 0.041
SDSS J1306+0356 | 6.018 | 9.23 | 47.053 | -0.222 | 46.342 | 0.084
SDSS J1630+4012 | 6.058 | 8.954 | 47.043 | -0.097 | 46.332 | 0.048
SDSS J0303-0019 | 6.079 | 8.477 | 46.591 | 0.114 | 45.879 | 0.031
SDSS J1623+3112 | 6.211 | 9.079 | 47.155 | -0.097 | 46.444 | 0.055
SDSS J1030+0524 | 6.302 | 9.041 | 47.114 | -0.046 | 46.402 | 0.053
SDSS J1030+0524 | 6.299 | 9.146 | 47.193 | -0.097 | 46.481 | 0.06
SDSS J0842+1218 | 6.069 | 9.23 | 47.178 | -0.155 | 46.467 | 0.073
SDSS J0002+2550 | 5.818 | 9.684 | 47.296 | -0.502 | 46.584 | 0.079
SDSS J000825.77-062604.6 | 5.929 | 9.157 | 47.028 | 0.04 | 46.272 | 0.075
SDSS J002806.57+045725.3 | 5.982 | 9.59 | 47.004 | -0.541 | 46.292 | 0.086
SDSS J003311.40-012524.9 | 5.978 | 8.452 | 46.5 | -0.065 | 45.788 | 0.027
CFHQS J005006+344522 | 6.251 | 9.639 | 47.257 | -0.639 | 46.494 | 0.153
CFHQS J022122-080251 | 6.161 | 9.207 | 46.663 | -0.658 | 44.976 | 0.127
SDSS J0353+0104 | 6.057 | 9.066 | 47.107 | -0.25 | 46.217 | 0.056
SDSS J0810+5105 | 5.805 | 9.537 | 47.226 | -0.426 | 46.514 | 0.097
SDSS J083525.76+321752.6 | 5.902 | 8.948 | 46.481 | -0.58 | 45.77 | 0.05
SDSS J0840+5624 | 5.816 | 9.223 | 46.707 | -0.63 | 45.995 | 0.066
SDSS J0841+2905 | 5.954 | 9.468 | 47.036 | -0.234 | 46.324 | 0.115
SDSS J0842+1218 | 6.069 | 9.595 | 47.238 | -0.418 | 46.526 | 0.104
SDSS J1143+3808 | 5.8 | 9.654 | 47.103 | -0.665 | 46.391 | 0.107
SDSS J114803.28+070208.3 | 6.344 | 9.499 | 47.136 | -0.323 | 46.408 | 0.127
SDSS J120737.43+063010.1 | 6.028 | 10.038 | 46.964 | -0.922 | 46.253 | 0.26
SDSS J124340.81+252923.9 | 5.842 | 9.757 | 47.23 | -0.641 | 46.326 | 0.201
SDSS J1250+3130 | 6.138 | 9.159 | 47.117 | -0.059 | 46.405 | 0.057
SDSS J1257+6349 | 5.992 | 9.521 | 46.808 | -0.642 | 46.096 | 0.12
SDSS J1335+3533 | 5.87 | 9.777 | 47.194 | -0.697 | 46.483 | 0.143
SDSS J142516.30+325409.0 | 5.862 | 9.433 | 46.981 | -0.566 | 46.269 | 0.067
SDSS J142738.59+331242.0 | 6.118 | 8.838 | 47.056 | -0.038 | 46.151 | 0.038
CFHQS J142952.17+554717.6W | 6.119 | 9.282 | 46.926 | -0.47 | 46.214 | 0.081
SDSS J1436+5007 | 5.809 | 9.665 | 47.133 | -0.646 | 46.421 | 0.136
SDSS J1545+6028 | 5.794 | 9.213 | 46.677 | -0.65 | 45.965 | 0.082
SDSS J1602+4228 | 6.083 | 9.553 | 47.269 | -0.282 | 46.557 | 0.108
SDSS J160937.27+304147.7 | 6.146 | 9.623 | 46.784 | -0.89 | 46.073 | 0.11
SDSS J1623+3112 | 6.254 | 9.39 | 47.12 | -0.36 | 46.409 | 0.083
SDSS J1630+4012 | 6.066 | 9.349 | 46.85 | -0.665 | 46.138 | 0.093
SDSS J2307+0031 | 5.9 | 8.479 | 46.808 | -0.092 | 45.792 | 0.024
CFHQS J232914-040324 | 5.883 | 8.808 | 46.363 | -0.559 | 45.637 | 0.08
PSO J060.5529+24.8567 | 6.17 | 9.324 | 47.149 | -0.24 | 46.437 | 0.069
PSO J210.4472+27.8263 | 6.166 | 9.541 | 46.962 | -0.638 | 46.25 | 0.145
PSO J228.6871+21.2388 | 5.893 | 9.081 | 46.732 | -0.464 | 46.02 | 0.05
PSO J333.9859+26.1081 | 6.027 | 9.464 | 46.899 | -0.288 | 46.187 | 0.135
SDSS J232908.28-030158.8 | 6.417 | 8.398 | 46.563 | 0.114 | 45.851 | 0.027
CFHQS J005006+344522 | 6.253 | 9.415 | 47.246 | -0.208 | 46.534 | 0.098
CFHQS J022122-080251 | 6.161 | 8.845 | 46.411 | -0.481 | 45.699 | 0.081
CFHQS J222901+145709 | 6.152 | 8.079 | 46.49 | 0.38 | 45.778 | 0.015
CFHQS J210054-171522 | 6.087 | 8.973 | 46.703 | -0.31 | 45.991 | 0.075
SDSS J164121.64+375520.5 | 6.047 | 8.38 | 46.791 | 0.362 | 46.079 | 0.02
CFHQS J005502+014618 | 5.983 | 8.38 | 46.511 | 0.079 | 45.799 | 0.028
VIKING J010953.13-304726.3 | 6.791 | 9.124 | 46.708 | -0.538 | 46.0 | 0.1
VIKING J030516.92-315056.0 | 6.614 | 8.954 | 46.875 | -0.194 | 46.176 | 0.056
PSO J167.6415-13.4960 | 6.515 | 8.477 | 46.672 | 0.086 | 45.954 | 0.029
ULAS J1120+0641 | 7.084 | 9.393 | 47.262 | -0.244 | 46.556 | 0.09
HSC J1205-0000 | 6.73 | 9.672 | 46.556 | -1.222 | 45.845 | 0.372
PSO J231.6576-20.8335 | 6.586 | 9.484 | 47.276 | -0.319 | 46.568 | 0.108
PSO J247.2970+24.1277 | 6.476 | 8.716 | 47.248 | 0.415 | 46.531 | 0.024
PSO J323.1382+12.2986 | 6.588 | 9.143 | 46.908 | -0.357 | 46.204 | 0.081
VIKING J234833.34-305410.0 | 6.902 | 9.297 | 46.633 | -0.77 | 45.954 | 0.141
ULAS J1120+0641 | 7.1 | 9.209 | 47.174 | -0.319 | 46.462 | 0.04
VIKING J234833.34-305410.0 | 6.9 | 9.322 | 46.685 | -0.745 | 45.973 | 0.156
VIKING J010953.13-304726.3 | 6.7 | 9.031 | 46.741 | -0.62 | 46.029 | 0.055
VIKING J030516.92-315056.0 | 6.6 | 9.028 | 46.932 | -0.167 | 46.22 | 0.041
Notes: We report accretion efficiency using optical luminosity, bolometric
luminosity, and SMBH mass utilizing Eq. (7). a Redshift. b Mass of the SMBH in
units of $M_{\odot}$. c Bolometric luminosity in units of erg $s^{-1}$. d
Eddington ratio in units of erg $s^{-1}$. e Optical luminosity in units of erg
$s^{-1}$. f Accretion efficiency.

Table 2: This table represents the flux- and volume-limited DL11 dataset for low redshift PG quasars. Object | $z^{a}$ | Log $M_{BH}^{b}$ [$M_{\odot}$] | Log $L_{bol}^{c}$ [erg/s] | Log $\lambda_{edd}^{d}$ [erg/s] | Log $L_{opt}^{e}$ [erg/s] | Log $\dot{M}^{f}$ [$M_{\odot}yr^{-1}$] | $\epsilon_{obs}^{g}$
---|---|---|---|---|---|---|---
0003 + 158 | 0.45 | 9.055 | 46.92 | -0.358 | 45.87 | 0.79 | 0.302
0003 + 199 | 0.026 | 7.22 | 45.13 | -0.342 | 43.91 | -0.06 | 0.034
0007 + 106 | 0.089 | 8.561 | 45.52 | -0.972 | 44.55 | -0.42 | 0.191
0026 + 129 | 0.145 | 7.833 | 46.15 | 0.053 | 44.99 | 0.8 | 0.05
0043 + 039 | 0.386 | 8.952 | 45.98 | -0.648 | 45.47 | 0.36 | 0.091
0050 + 124 | 0.059 | 7.238 | 45.12 | 0.162 | 44.41 | 0.58 | 0.008
0052 + 251 | 0.154 | 8.745 | 46.06 | -0.822 | 45.0 | -0.04 | 0.282
0157 + 001 | 0.163 | 8.006 | 45.93 | -0.261 | 45.02 | 0.59 | 0.049
0804 + 761 | 0.1 | 8.352 | 45.82 | -0.3 | 44.79 | 0.2 | 0.093
0838 + 770 | 0.132 | 7.992 | 45.22 | -0.493 | 44.56 | 0.08 | 0.031
0844 + 349 | 0.064 | 7.759 | 45.4 | -0.479 | 44.31 | -0.01 | 0.058
0921 + 525 | 0.035 | 7.206 | 44.47 | -0.802 | 43.56 | -0.55 | 0.023
0923 + 129 | 0.029 | 7.233 | 44.53 | -0.665 | 43.58 | -0.49 | 0.023
0923 + 201 | 0.193 | 9.094 | 45.68 | -1.134 | 44.81 | -0.47 | 0.309
0947 + 396 | 0.206 | 8.53 | 46.2 | -0.909 | 45.2 | 0.19 | 0.224
1001 + 054 | 0.161 | 7.645 | 45.36 | -0.02 | 44.69 | 0.59 | 0.013
1011 - 040 | 0.058 | 7.19 | 45.02 | -0.146 | 44.08 | 0.17 | 0.015
1012 + 008 | 0.186 | 8.0 | 45.53 | 0.0 | 44.95 | 0.46 | 0.026
1022 + 519 | 0.045 | 6.94 | 45.1 | -0.6 | 43.56 | -0.36 | 0.065
1048 - 090 | 0.346 | 9.022 | 46.57 | -0.679 | 45.45 | 0.3 | 0.417
1048 + 342 | 0.167 | 8.241 | 45.7 | -0.687 | 44.74 | 0.02 | 0.105
1049 - 006 | 0.36 | 8.989 | 46.29 | -0.63 | 45.46 | 0.34 | 0.195
1100 + 772 | 0.312 | 9.112 | 46.61 | -0.749 | 45.51 | 0.29 | 0.457
1103 - 006 | 0.423 | 9.132 | 46.19 | -0.737 | 45.43 | 0.21 | 0.209
1114 + 445 | 0.144 | 8.415 | 45.92 | -0.927 | 44.75 | -0.16 | 0.263
1115 + 407 | 0.154 | 7.505 | 45.59 | -0.139 | 44.58 | 0.49 | 0.028
1116 + 215 | 0.176 | 8.425 | 46.27 | -0.139 | 45.31 | 0.69 | 0.083
1119 + 120 | 0.05 | 7.28 | 45.18 | -0.462 | 44.01 | -0.06 | 0.038
1121 + 422 | 0.225 | 7.856 | 45.87 | -0.232 | 44.8 | 0.48 | 0.054
1126 - 041 | 0.06 | 7.598 | 45.16 | -0.434 | 44.19 | -0.02 | 0.034
1149 - 110 | 0.049 | 7.729 | 44.75 | -0.916 | 43.79 | -0.66 | 0.058
1151 + 117 | 0.176 | 8.435 | 45.43 | -0.801 | 44.65 | -0.2 | 0.095
1202 + 281 | 0.165 | 8.462 | 45.39 | -1.053 | 44.58 | -0.38 | 0.129
1211 + 143 | 0.081 | 7.831 | 46.41 | 0.051 | 44.85 | 0.68 | 0.117
1216 + 069 | 0.332 | 9.0 | 46.61 | -1.0 | 45.62 | 0.51 | 0.282
1226 + 023 | 0.158 | 8.876 | 47.09 | -0.012 | 46.03 | 1.18 | 0.182
1229 + 204 | 0.064 | 8.004 | 45.06 | -0.804 | 44.24 | -0.35 | 0.056
1244 + 026 | 0.048 | 6.614 | 44.74 | 0.235 | 43.7 | 0.15 | 0.009
1259 + 593 | 0.477 | 8.738 | 47.04 | -0.085 | 45.79 | 0.99 | 0.245
1302 - 102 | 0.278 | 8.749 | 46.51 | -0.08 | 45.71 | 0.92 | 0.087
1307 + 085 | 0.154 | 8.541 | 45.93 | -0.651 | 44.92 | 0.05 | 0.166
1309 + 355 | 0.182 | 8.155 | 45.63 | -0.421 | 44.95 | 0.37 | 0.04
1310 - 108 | 0.034 | 7.759 | 44.37 | -1.183 | 43.56 | -1.0 | 0.052
1322 + 659 | 0.168 | 8.076 | 45.92 | -0.409 | 44.78 | 0.27 | 0.098
1341 + 258 | 0.086 | 7.878 | 44.94 | -0.756 | 44.13 | -0.37 | 0.045
1351 + 236 | 0.055 | 8.216 | 44.57 | -1.748 | 43.93 | -1.14 | 0.112
1351 + 640 | 0.088 | 8.656 | 45.31 | -1.058 | 44.69 | -0.38 | 0.107
1352 + 183 | 0.151 | 8.299 | 45.72 | -0.629 | 44.65 | -0.06 | 0.132
1402 + 261 | 0.164 | 7.845 | 46.07 | 0.018 | 44.82 | 0.63 | 0.06
1404 + 226 | 0.098 | 6.713 | 45.21 | 0.232 | 44.16 | 0.55 | 0.01
1411 + 442 | 0.09 | 7.874 | 45.06 | -0.535 | 44.45 | 0.02 | 0.025
1415 + 451 | 0.113 | 7.797 | 45.6 | -0.579 | 44.34 | -0.06 | 0.1
1416 - 129 | 0.129 | 9.002 | 45.82 | -0.845 | 44.94 | -0.21 | 0.234
1425 + 267 | 0.364 | 9.317 | 46.35 | -1.28 | 45.55 | 0.07 | 0.417
1426 + 015 | 0.086 | 8.921 | 45.84 | -1.117 | 44.71 | -0.49 | 0.479
1427 + 480 | 0.22 | 7.978 | 45.64 | -0.344 | 44.69 | 0.27 | 0.051
1435 - 067 | 0.129 | 8.3 | 45.6 | -0.412 | 44.9 | 0.26 | 0.048
1440 + 356 | 0.078 | 7.335 | 45.62 | -0.013 | 44.37 | 0.43 | 0.034
1444 + 407 | 0.268 | 8.158 | 46.28 | -0.122 | 45.11 | 0.66 | 0.091
1501 + 106 | 0.036 | 8.482 | 44.9 | -1.172 | 44.18 | -0.79 | 0.107
1512 + 370 | 0.371 | 9.168 | 47.11 | -0.867 | 45.48 | 0.2 | 1.82
1519 + 226 | 0.136 | 7.777 | 45.98 | -0.311 | 44.45 | 0.18 | 0.138
1534 + 580 | 0.03 | 8.047 | 44.49 | -1.565 | 43.63 | -1.24 | 0.12
1535 + 547 | 0.039 | 7.01 | 44.34 | -0.373 | 43.9 | -0.01 | 0.005
1543 + 489 | 0.401 | 7.844 | 46.43 | 0.369 | 45.27 | 1.18 | 0.039
1545 + 210 | 0.264 | 9.0 | 46.14 | -1.0 | 45.29 | 0.01 | 0.295
1552 + 085 | 0.119 | 7.364 | 45.04 | 0.04 | 44.5 | 0.56 | 0.007
1612 + 261 | 0.131 | 7.913 | 45.38 | -0.395 | 44.54 | 0.15 | 0.038
1613 + 658 | 0.129 | 8.953 | 45.89 | -1.457 | 44.75 | -0.59 | 0.676
1617 + 175 | 0.114 | 8.729 | 45.44 | -0.88 | 44.63 | -0.38 | 0.145
1626 + 554 | 0.132 | 8.371 | 45.53 | -0.94 | 44.46 | -0.4 | 0.186
1704 + 608 | 0.372 | 9.198 | 46.67 | -0.772 | 45.65 | 0.38 | 0.437
2112 + 059 | 0.46 | 8.834 | 46.47 | 0.116 | 45.92 | 1.16 | 0.046
2130 + 099 | 0.063 | 7.805 | 45.52 | -0.367 | 44.35 | 0.05 | 0.065
2209 + 184 | 0.07 | 8.601 | 46.02 | -1.353 | 44.11 | -0.98 | 2.188
2214 + 139 | 0.066 | 8.308 | 45.15 | -1.027 | 44.36 | -0.5 | 0.098
2251 + 113 | 0.326 | 8.816 | 46.13 | -0.363 | 45.6 | 0.66 | 0.066
2304 + 042 | 0.043 | 8.32 | 44.49 | -1.633 | 43.67 | -1.35 | 0.155
2308 + 098 | 0.434 | 9.372 | 46.61 | -0.936 | 45.62 | 0.22 | 0.537
Notes: We report accretion efficiency using optical luminosity, bolometric
luminosity, and SMBH mass. a Redshift. b Mass of the SMBH in units of
$M_{\odot}$. c Bolometric luminosity in units of erg $s^{-1}$. d Eddington
ratio in units of erg $s^{-1}$. e Optical luminosity in units of erg $s^{-1}$.
f Accretion rate, in units of $M_{\odot}yr^{-1}$. g Accretion efficiency.