X-ray properties of reverberation-mapped AGNs with super-Eddington accreting massive black holes

The X-ray properties of Active Galactic Nuclei (AGNs) depend on their underlying physical parameters, particularly the accretion rate. We identified eight reverberation-mapped AGNs with some of the largest known accretion rates without high-quality X-ray data. We obtained new Chandra ACIS-S X-ray observations and nearly simultaneous optical spectrophotometry to investigate the properties of these AGNs with extreme super-Eddington accreting black holes (SEAMBHs). We combined our new X-ray measurements with those of other reverberation-mapped AGNs, which have the best-determined masses and accretion rates. The trend of the steepening of the spectral slope between X-ray and optical-UV, $\alpha_{\rm ox}$, with increasing optical-UV luminosity, $L_{2500\r{A}}$, holds true for even the most extreme SEAMBHs. One of our new SEAMBHs appears X-ray weak for its luminosity, perhaps due to absorption associated with orientation effects involving a slim disk thought to be present in highly accreting systems. The correlation of the $\rm 2-8~ keV$ X-ray photon index with the accretion rate also holds for the extreme SEAMBHs, which show some of the largest photon indices reported for AGNs.


INTRODUCTION
Active Galactic Nuclei (AGNs) emit radiation across the entire electromagnetic spectrum due to various physical processes ultimately powered by the gravitational potential energy of the accreting supermassive black hole (Salpeter 1964;Pringle et al. 1973;Ho 2008).The accreting matter forms a disk around the central supermassive black hole, which glows brightly in optical, UV, and soft X-rays.The spectral energy distribution of AGNs peaks in the UV emitted from the inner accretion disk.The Compton upscattering of optical-UV disk photons by the hot electrons in a region close (observations indicate three to few tens of gravitational radii, Sunyaev & Truemper 1979;McHardy et al. 2005;Wilkins et al. 2016;Chartas et al. 2016) to the black hole, called the corona, produces hard X-rays (e.g., Haardt & Maraschi 1991, 1993;Nakamura & Osaki 1993;Kawaguchi et al. 2001).Above 2 keV (in the range 2-100 keV), the X-ray spectrum is dominated by a single power-law continuum,  () ∝  −Γ , with a high-energy cut-off (Sunyaev & Titarchuk 1980).Studies have found ★ E-mail: jaya.maithil@cfa.harvard.eduthe cut-off energy is typically tens of keV or higher (Ricci et al. 2017;Molina et al. 2019;Tortosa et al. 2023).The photon index (Γ) acts as a tool to infer the energy distribution of the electrons in the corona.The two-point spectral index between 2500 Å and 2 keV ( ox , Tananbaum et al. 1979;Just et al. 2007) indicates the relative strength of the energy emitted by the disk versus the corona.
Past studies have examined the correlation between the X-ray-tooptical properties of quasars with various observed and fundamental properties.The photon index and  ox generally show no variation with redshift (e.g., Page et al. 2005;Shemmer et al. 2005Shemmer et al. , 2006;;Steffen et al. 2006;Just et al. 2007).However,  ox exhibits a strong anti-correlation with optical-UV luminosity at 2500 Å (Steffen et al. 2006;Just et al. 2007;Timlin et al. 2020).Steffen et al. (2006) find a 13.6 correlation between  ox and  2500Å using a sample of 333 AGNs with redshifts up to  ∼ 6 and spanning over five orders of magnitude in  2500Å and four orders of magnitude in X-ray luminosity.The  ox - 2500Å correlation is a by-product of the non-linear correlation between the luminosities at 2 keV and 2500 Å (Vignali et al. 2003;Strateva et al. 2005;Steffen et al. 2006;Just et al. 2007;Lusso et al. 2010;Young et al. 2010).It indicates AGNs with higher optical brightness emit relatively fewer X-rays than more optically faint AGNs (Lusso & Risaliti 2016).Through careful sample control, Lusso & Risaliti (2016) tightened the  2500Å - 2 keV relation, reducing the dispersion from ∼0.35-0.4dex (from previous studies) to ∼0.21-0.24dex seen in four orders of magnitude in luminosity.Such a tight correlation indicates that the energy emitted by the accretion disk and corona are inextricably linked.We also note that  ox shows weak to no correlation with the Eddington ratio (e.g., Young et al. 2010).
A strong anti-correlation between Γ (both soft and hard photon index) and full-width half maxima of H emitted from the broad emission-line region (Wang et al. 1996;Laor et al. 1997;Brandt et al. 1997;Wang et al. 2004) indicates the dependence of X-ray properties on the fundamental properties of black holes like mass or accretion rate.This anti-correlation is likely a secondary effect of a more intrinsic correlation between hard Γ and the Eddington ratio ( Bol / Edd ), a normalized accretion rate parameter (Shemmer et al. 2006(Shemmer et al. , 2008;;Brightman et al. 2013;Trakhtenbrot et al. 2017;Tortosa et al. 2023).As the accretion rate increases, the accretion disk heats up, glowing in soft X-rays and increasing the Compton cooling of the corona-resulting in a steepening and softening of the X-ray spectra (Shemmer et al. 2006(Shemmer et al. , 2008)).Therefore, X-rays are critical for probing the accretion process in the vicinity of black holes.
The correlation between the 2 − 8 keV photon index and Eddington ratio holds over a range of quasar luminosity and redshift.E.g., Shemmer et al. (2008) combine the Shemmer et al. (2006) sample of 25 moderate luminosity (43 < log   (5100Å) [erg s −1 ] ≤ 46) and five high luminosity (46 < log   (5100Å) [erg s −1 ] ≤ 48) type-1 radio-quiet quasars at  < 0.5 and  ∼ 2, respectively, with five high luminosity quasars at  = 1.3 − 3.2.These 35 quasars have high-quality optical spectra to derive black hole mass estimates based on the H-beta line and high-quality X-ray spectra to measure Γ.Although the sample was not complete or necessarily representative, their results established a significant correlation between Γ 2−8 keV and  Bol / Edd over four orders of magnitude in quasar luminosity.Risaliti et al. (2009) studied ∼400 AGNs with good-quality optical spectra from SDSS and XMM-Newton X-ray observations from the SDSS/XMM-Newton quasar survey (Young et al. 2009) spanning three orders of magnitude in optical luminosity.Their results show a highly significant correlation between Γ 2−8 keV and  Bol / Edd .Risaliti et al. (2009) also find that the strength and significance of the Γ 2−8 keV - Bol / Edd correlation decreases going from H to Mg ii to no correlation when Civ-based mass is used to calculate the Eddington ratio.Brightman et al. (2013) also confirmed the Γ 2−8 keV - Bol / Edd correlation using a sample of 69 type-1 radio-quiet AGNs with redshifts up to ∼ 2.1 with X-ray spectra from the Chandra Deep Field-South survey (Lehmer et al. 2005) and black hole mass estimates based on H and Mg ii measurements from the Cosmic Evolution Survey (Cappelluti et al. 2009;Elvis et al. 2009).Using 71 type-1 AGNs selected from the XMM-Newton Bright Serendipitous survey, Fanali et al. (2013) found a significant correlation between Γ 0.5−10 keV and  Bol / Edd after removing the dependence on redshift.They found a less significant correlation with Γ 2−8 keV due to a larger error in determining Γ 2−8 keV , a factor of 2.5 larger compared to the error in Γ 0.5−10 keV .
The correlations like Γ 2−8 keV - Bol / Edd and  ox −  2500Å are evident for the sub-Eddington accreting ( Bol / Edd ≪ 1) sources.Quasars with low Eddington ratios presumably have an optically thick and geometrically thin accretion disk.In the standard Shakura & Sunyaev (1973) thin disk model, the spin of the black hole defines the innermost stable orbit that dictates the radiation efficiency ().For a retrograde spin,  = 0.038, whereas for a maximally spinning black hole,  = 0.32.But in the case of a higher accretion rate, the accretion disk becomes geometrically thick or slim (Abramowicz et al. 1988;Laor & Netzer 1989;Wang & Zhou 1999;Wang et al. 2013).Due to effects like photon trapping and advection-dominated energy transport, the slim disk has significantly smaller radiation efficiency that may be independent of spin (Wang & Zhou 1999;Mineshige et al. 2000;Wang et al. 2013).The spectral energy distribution (SED) of a slim accretion disk is likely different from a thin disk with a cut-off at higher energies.Although a comparison of IR-optical-UV-X-ray SEDs of super-and sub-Eddington RM AGNs show no anomalous torus emission or a heightened ionizing continuum predicted for slim disk systems (Castelló-Mor et al. 2016, 2017).Super-Eddington accreting AGNs exhibit optical-UV SEDs that align well with the standard thin disk model.Any indications of a slim disk, may be present in the extreme ultra-violet (EUV) where data availability is limited (Castelló-Mor et al. 2016;Kubota & Done 2019).Wang et al. (2014) suggested that high accreting objects likely have slim accretion disks and coined the acronym Super-Eddington accreting massive black holes (SEAMBHs).They are characterized by large Eddington ratios ( Bol / Edd > 0.3).SEAMBHs deviate from the traditional radius-luminosity relationships established for the low-accreting AGNs of the form  ∝  ∼0.5 (e.g., Kaspi et al. 2000;Bentz et al. 2013).A dedicated RM campaign by Du et al. (2014Du et al. ( , 2015Du et al. ( , 2016Du et al. ( , 2018) ) tested the R-L relationship for the most highly accreting AGNs.They selected SEAMBH candidates based on a dimensionless accretion rate estimator, ℳ, such that a ℳ ≥ 3 implies a SEAMBH.In context of thin-disks, the Eddington ratio is related to ℳ by the equation  Bol / Edd =  ℳ (Du et al. 2014(Du et al. , 2015)).Here,  is the mass-to-radiation conversion efficiency, a parameter dependent on the black hole spin.Du et al. (2018) shows that the highest accretion rate AGNs have systematically shorter time lags, a factor of 3-8 times smaller than predicted by the canonical R-L relationship for the sub-Eddington accreting AGNs.Slim accretion disks in SEAMBHs are perhaps responsible for the shortened time lags of the broad-line region, due to anisotropic emission from the ionizing source (Wang et al. 2014;Du & Wang 2019).The canonical single-epoch black hole masses of highly accreting quasars are overestimated, on average, by a factor of two and the accretion rate parameters  Bol / Edd and ℳ consequently underestimated (Maithil et al. 2022).SEAMBHs are not so common at low redshift but their fraction is expected to be higher in the early universe (Kelly & Shen 2013).Discoveries of billion solar mass black holes at  > 6, when the universe was less than a billion years old, are suggestive of super-Eddington accretion (e.g., review by Valiante et al. (2017) and references therein).Thus, SEAMBHs are probes to understand the accretion process and disk-corona connection of the high-redshift massive highly accreting AGNs.
Recently, Liu et al. (2021) probed the disk-corona connection using a sample of 26 sub-Eddington and 21 super-Eddington accreting AGNs.They used the most accurate reverberation-mapped black hole masses eliminating a major source of uncertainty in the Γ 2−8 keV - Bol / Edd correlations coming from the black hole mass estimates used to derive the Eddington ratio.Other intrinsic differences between AGNs like black hole spin, orientation, and optical depth in the corona likely also contribute to uncertainties in such relations.Most of the past studies were limited to sub-Eddington sources and used the single-epoch black hole mass scaling relations for broademission lines like H, H, and Mg ii based on the traditional R-L relationships.Some studies looked at X-ray and optical-UV properties of super-Eddington accreting sources but they did not perform a comparative investigation with the sub-Eddington sources (e.g., Ai et al. 2011;Kamizasa et al. 2012).In this paper, we extend the investigations of Liu et al. (2021) by adding 13 extreme SEAMBHs from the SEAMBH-RM campaign that show the highest accretion rate (log ℳ > 1.5) and the largest deviation in the broad-line region (BLR) sizes from the canonical R-L relationship.We present the Chandra X-ray data of 8 SEAMBHs for the first time with simultaneous optical-UV spectra avoiding uncertainties in the derivation of  ox due to variability.Our work tests whether the extreme SEAMBHs follow the trends seen in sub-and super-Eddington accreting AGNs.In Section 2, we describe our sample and summarize the sample selection of Liu et al (2021).Section 3 presents the data reduction and analysis of our new Chandra observation and the optical-UV spectra.This is followed by an explanation of measured and derived quantities in Section 4. Section 5 presents the key results and Section 6 presents the discussion and conclusion.Throughout the paper, we adopt a cosmology with  0 = 70 kms −1 Mpc −1 , Ω Λ = 0.7 and Ω  = 0.3.

SAMPLE
The reverberation mapping of SEAMBHs by Du et al. (2018) leads to the identification of 14 SEAMBHs that have markedly smaller BLR sizes than expected from their luminosities and have extreme accretion rates, i.e., log ℳ > 1.5.Our core sample consists of these 14 'extreme SEAMBHs'.
Eight of these extreme SEAMBHs did not have X-ray observations above 2 keV (called 'Chandra SEAMBHs').We proposed Chandra Advanced CCD Imaging Spectrometer (ACIS-S) observations of these eight SEAMBHs in Cycle 20.We used the timed exposure ACIS operation mode with the faint telemetry format.To mitigate potential pile-up issues, we used the 1/8 sub-array with only the S3 chip turned on.The spectral coverage of Chandra allows us to measure both the 2-8 keV X-ray photon index (Γ 2−8 keV ) and, combined with optical-UV data, the  ox .To minimize the effect of variability in  ox measurements, the Chandra observations were accompanied by ground-based optical-UV spectral observations as close in time as weather permitted using facilities like the Lĳiang 2.4 m and the 2.2 m Calar Alto telescopes.We present the X-ray and optical-UV data reduction and analysis in the next section.Table 1 provides the observation log of our Chandra targets.
The remaining six extreme SEAMBHs have archival data from Chandra, XMM-Newton, and Swift.SDSSJ075051.72+245409.3 has archival ASIC-S Chandra observation from Cycle 21 (PI: Gordon Garmire, see Table 1 for details).In the absence of simultaneous optical-UV observation for this target, we use  2500Å measurement from Shen et al. (2011).SDSS J080101.41+184840.7,SDSS J093922.89+370943.9,IRAS F12397+3333 and PG 2130+099 have simultaneous X-ray observations from the XMM-Newton's European Photon Imaging Camera-PN and MOS detectors and optical-UV observations from the Optical Monitor.Lastly, Mrk 142 is observed simultaneously in the X-ray and the UV-Optical using the Swift observatory.Throughout this paper, we refer to these six extreme SEAMBHs as 'Archived SEAMBHs'.Except for SDSSJ075051.72+245409.3, the other five extreme SEAMBHs are part of the Liu et al. (2021) sample and we refer the readers to their Section 2 for details of X-ray and optical-UV analysis.We process the X-ray data of SDSSJ075051.72+245409.radio-quiet RM AGNs from Du et al. (2015Du et al. ( , 2016Du et al. ( , 2018) ) that have good signal-to-noise archival X-ray data (/ > 6 in the rest-frame > 2 keV band) and simultaneous optical-UV observations.They eliminated AGNs that are heavily absorbed in X-rays due to outflows or display broad-absorption-line features and reddening in their UV spectra.Three out of these 16 super-Eddington AGNs are part of our Chandra SEAMBHs.Liu et al. (2021) present the archival Swift (IRAS 04416+1215, SDSS J074352.02+271239.5) and XMM-Newton (SDSS J100402.61+285535.3)data of these in-common targets.As X-rays can vary, we present the archival data of these three targets as part of the Liu et al. (2021) super-Eddington sample and treat them as separate measurements.Figure 1 shows the  5100Å −  plane for our 14 extreme SEAMBHs and the Liu et al. (2021) sample, adding more data points at  > 0.16.Our extreme SEAMBHs have log  5100Å > 43.3 placing them at the luminous end of the super-Eddington AGNs studied previously.

Chandra X-ray reduction and spectral analysis
This paper presents Chandra data made available after CXC's fifth reprocessing campaign.We analyzed data using Chandra Interactive Analysis of Observations software (CIAO) v4.13.The level 1 event files were reprocessed using the chandra_repro script with CALDB v4.9.5 to create the level 2 event files and observationspecific bad pixel files.We extracted the net counts, i.e., source counts minus background counts from the level 2 event file.To extract source counts we used a circular aperture of 4 ′′ (2 ′′ for SDSS J074352.02+271239.5) at the source coordinate, whereas an annulus with an inner radius of 8 ′′ (5 ′′ ) and an outer radius of 13 ′′ (10 ′′ ) was used to extract the background counts.The source and background regions were saved for later use in spectral analysis.The net counts were obtained in three bands: Hard (2-8 keV), Soft (0.35-2 keV), and Broad (0.35-8 keV).Using these counts we calculated the hardness ratio (HR) defined as (Hard-Soft)/(Hard+Soft).Table 2 lists the photon counts and hardness ratio.Six out of eight Chandra SEAMBHs show a negative hardness ratio implying soft spectra, whereas two (SDSS J074352.02+271239.5 and SDSS J101000.68+300321.5)show hard spectra.One Archived SEAMBH with Chandra data, SDSSJ075051.72+245409.3,presents a soft spectrum.
We used specextract to create source and background spectra and the instrument response files with point source aperture correction to unweighted auxiliary response files.The saved circular source aperture and annulus background regions were used to extract the spectra.We used Sherpa to fit the spectral data.The source model employed is a one-dimensional power-law and an XSpec photoelectric absorption model with the hydrogen column density frozen to the Galactic value (Dickey & Lockman 1990) in the source direction (xsphabs.abs1* powlaw1d.p1).The data were grouped to 5-30 counts per bin depending on the net count of the object.We used the Levenberg-Marquardt optimization method with W-statistic for fitting the spectra of all objects.The W-statistic is equivalent of the Cash statistic (Cash 1979), appropriate when using background-subtracted source spectra of low-count sources and also works well for highcount sources (Kaastra 2017).Figure 2 shows the best-fit model for the observed-frame 0.35 to 8/(1+z) keV spectra.The best-fit parameter values for the photon index, Γ 0.35−8 keV , absorption-corrected rest-frame 0.35-8 keV flux, and the reduced statistics are presented in Table 3.
Next, we fit the observed-frame 2/(1+z) to 8/(1+z) keV X-ray band, where the underlying spectrum is mostly free of absorption or soft-excess emission (Shemmer et al. 2006(Shemmer et al. , 2008;;Risaliti et al. 2009;Brightman et al. 2013).Six Chandra SEAMBHs and the Archived SEAMBH (SDSSJ075051.72+245409.3)show soft, steep spectra well fit by using a power-law model with Galactic absorption.SDSS J074352.02+271239.5 & SDSS J101000.68+300321.5 appear as outliers and have much harder spectra than the other SEAMBHs.We fit their spectrum using a power-law model with both Galactic and intrinsic absorption (xsphabs.abs1* xszphabs.zabs1* powlaw1d.p1).SDSS J101000.68+300321.5 likely needs a more complex model, but the low photon count prevents us from uniquely doing so.For IRAS 04416+1215, we also adopted a high-energy cut-off for the power-law fixed to 44 keV from Tortosa et al. (2023).An F-test reveals that adding a high-energy cut-off doesn't improve the statistics significantly.We tested the presence of intrinsic absorption for each source.Except for SDSS J074352.02+271239.5, an F-test confirms that adding intrinsic absorption component to the Galactic-absorbed power-law model does not improve the fit significantly.Table 3 reports the 2 − 8 keV photon index (Γ 2−8 keV ), the absorption-corrected rest-frame 2-8 keV flux, reduced statistics, and the monochromatic flux density at rest-frame energy of 2 keV from the best-fit models.It should be noted that the restframe 2 − 8 keV best-fit values are the key X-ray results that we will use in our analysis.We provide the literature values of the photon index for IRAS 04416+1215, SDSS J074352.02+271239.5, SDSS J100402.61+285535.3 and SDSSJ075051.72+245409.3 in the appendix A.

Optical spectral analysis
We obtained the optical spectra of the eight Chandra SEAMBHs almost simultaneously with the X-ray observations, using the Lĳiang and/or CAHA observatories.Multiple observations with two sets of spectra were taken within 2-8 days of the Chandra observations.We selected the best-quality spectra closest in time to the Chandra observation.
For the six objects observed at Lĳiang 2.4m telescope (listed in Table 1), their spectra were taken under the same settings of the grism and slit width as they were observed in Hu et al. (2015) (for IRAS 04416+1215), Du et al. (2015) (for SDSS J080131.58+354436.4), and Du et al. (2018) (for the other four objects), respectively.For the two objects observed at CAHA 2.2m telescope, the spectra were taken using the Calar Alto Faint Object Spectrograph with Grism G-200 and a slit set at a width of 3. ′′ 0, as described in Hu et al. (2021).For all the objects, the same comparison stars were taken simultaneously by rotating the slit, as they were monitored for reverberation mapping measurements.The data were reduced firstly by the standard procedures, including bias-removal, flat-field correction, wavelength calibration, and spectral extraction.Then the flux calibration was performed using a sensitivity function obtained from the simultaneously observed comparison star spectrum as in the previous reverberation mapping observations (see, e.g., Hu et al. 2021 for the details of the method).Thus the flux here can be compared directly with those in the light curves of their previous reverberation mapping observations (Hu et al. 2015;Du et al. 2015Du et al. , 2018)).
We dereddened the mean spectra and fit a slope to the continuum around the H region.The slope was extrapolated to get the monochromatic luminosities at a rest-frame wavelength of 2500 Å luminosity for each target.
(ii) We used the following equation to derive  ox where,  2 keV and  2500Å are X-ray (2 keV) and UV (2500Å) monochromatic flux densities, respectively.A highly significant anticorrelation between  ox and  2500Å provides a method to predict  ox based on  2500Å measurement.We define Δ ox as the difference between the measured  ox and predicted from Steffen et al. (2006) relation given as  ox = −(0.137± 0.008) log  2500Å + (2.638 ± 0.240). (2) (iii) The black hole mass ( BH ) is the H reverberation-mapped mass.We use two accretion rate parameters: (a) The dimensionless accretion rate parameter ℳ is defined as where,  7 =  BH /10 7  ⊙ , ℓ 44 =  5100Å /10 44 is in units of erg s −1 and  is inclination angle to the line of sight (we assumed a typical value of cos  = 0.75 for type-1 AGNs), (b) The Eddington ratio is defined as the ratio of bolometric luminosity (  Bol ) and Eddington luminosity ( Edd ), Bol / Edd = 9.26  5100Å 1.5 × 10 45  7 erg s −1 . (4) For our Chandra SEAMBHs, we used new measurements of Γ 2−8 keV ,  2 keV and  2500Å and calculated  ox using equation (1).All the X-ray and optical-UV measurements for the Archival SEAMBHs, super-Eddington, and sub-Eddington samples were adopted from Liu et al. (2021).The monochromatic luminosity at 5100 Å ( 5100Å ) and black hole properties like  BH and ℳ were taken from Du & Wang (2019) 2021) sub-Eddington sample, PG 0026+129, now has ℳ > 3, and we count it in the super-Eddington sample.We calculate the Eddington ratio for all samples using equation 4, where we make a conservative choice of using Richards et al. (2006) bolometric correction factor of 9.26 to estimate the bolometric luminosity (see Section 6 for further discussion).
Figure 4 illustrates the  ox vs  2500Å plane.The  ox values of extreme SEAMBHs range between -1.67 and -1.24, following the best-fit linear regression from Steffen et al. (2006) as expressed in equation ( 2).The mean  ox values are -1.46,-1.38 and -1.32 for the extreme SEAMBHs, super-Eddington and sub-Eddington samples, respectively. ox tends to steepen with increasing  2500Å , consistent with previous studies.We also plot the  ox −  2500Å best-fit relationship from Timlin et al. (2020), given by  ox = (−0.199± 0.011)log 10 ( 2500 ) + (4.573 ± 0.333).Notably, this slope is significantly steeper than that of Steffen et al. (2006).Timlin et al. (2020) uses a Bayesian linear regression method developed by (Kelly 2007), while Steffen et al. (2006) utilizes a bivariate data-analysis method presented in (Isobe et al. 1990) for their bestfit relations (see their respective papers for further details).Note- † For these targets extending 2-8 keV Galactic-absorbed power-law model (see Table 3) to < 2 keV energies gives the best fitting results.(as given in equation 2), plotted against  2500Å .We plot the distribution of Δ ox in Figure 5. Luo et al. (2015) defines Δ ox = −0.2 as a threshold value, below which the AGN is considered X-ray weak.Apart from one Chandra SEAMBH, Δ ox varies between −0.15 and 0.15, typical for normal X-ray emission.Only SDSS J101000.68+300321.5 appears as X-ray weak, with Δ ox = −0.23.
The limited photon count for this object hinders fitting more complex models to correct for absorption.Figure 6 plots the best-fit relation between  2 keV and  2500Å for the super-Eddington sample (log  2 keV = (0.73±0.05) log  2500Å + (4.3 ± 1.4)) and the sub-Eddington sample (log  2 keV = (0.71 ± 0.05) log  2500Å + (5.0 ± 1.5)) from Liu et al. (2021).These best-fit relationships were derived using LINMIX_ERR method (Kelly 2007), which is a linear regression method that utilizes Bayesian priors for errors (refer to Liu et al. 2021 for further details).The slope of Liu et al. 2021 best-fit relation is within the error range of the typical value of 0.6±0.1 seen in previous studies (e.g., Steffen et al. 2006;Figure 5. Histogram of the distribution of Δ ox .Except for one extreme SEAMBH, SDSS J101000.68+300321.5, all sources appear X-ray normal.Lusso et al. 2010).The trend of suppression of X-ray emission in comparison to optical-UV emission holds for extreme SEAMBHs.

Γ 2−8 keV vs accretion rate
The 2 − 8 keV photon index is known to soften or steepen with an increasing accretion rate.Shemmer et al. (2008) finds a significant correlation between Γ 2−8 keV and  Bol / Edd and derived a linear relation of the form Γ 2−8 keV =  log  Bol / Edd + , where the slope  = 0.31 and intercept  = 2.11.Subsequently, the Γ 2−8 keV - Bol / Edd correlation has been tested across AGNs spanning a wide range of luminosities and redshifts.For instance, Risaliti et al. (2009) finds  = 0.31,  = 1.97 for their full sample, Brightman et al. (2013) reports  = 0.32,  = 2.27, and Fanali et al. (2013) finds  = 0.25,  = 2.48.2021).The plot shows that the non-linear correlation between X-ray and optical-UV luminosities is also present in extreme SEAMBHs.dot-dash line.Both best-fit relationships were derived using linear regression techniques that account for measurement errors.Liu et al. (2021) employed the LINMIX_ERR method (Kelly 2007), a Bayesian approach incorporating a prior distribution for errors.In contrast, Shemmer et al. (2008) utilized the bivariate correlated errors and scatter method (Akritas & Bershady 1996), which assumes random errors with no correlation.The extreme SEAMBHs follow the trend of steeping Γ 2−8 keV as the Eddington ratio increases.The full sample shows a strong correlation between Γ 2−8 keV and  Bol / Edd , with a Pearson -coefficient of 0.70 and a probability  = 2.37 − 9. Removing the sub-Eddington sample weakens the Pearson correlation to  = 0.49,  = 0.01.We also tested the Spearman non-parametric correlation between the Γ 2−8 keV and the Eddington ratio, yielding a -coefficient = 0.70 and a probability  = 2.42 − 9 for the full sample.Without the sub-Eddington sample, the Spearman correlation decreases to  = 0.48,  = 0.01.A weak correlation between Γ 2−8 keV and  Bol / Edd is observed for the 14 extreme SEAMBHs, with Pearson (Spearman)  = 0.29(0.33), = 0.31(0.25).This may be be attributed to the shortcoming of a constant bolometric correction factor of 9.26 for estimating the bolometric luminosities in these highly accreting systems or, the small sample size, or the narrow range of  Bol / Edd and Γ 2−8 keV spanned by the sample.
Using ℳ as an accretion rate indicator is complementary to using the Eddington ratio, although these parameters are tightly correlated for AGNs.

𝛼 ox vs accretion rate and black hole mass
The physics behind the disk-corona connection responsible for the observed steepening of  ox with increasing  2500Å is not well understood (see Section 6 for further discussion).Past studies have investigated the connection between  ox and black hole mass and accretion rate.A significant correlation is evident between  ox and  BH (Done et al. 2012;Chiaraluce et al. 2018).However, there is only a weak or non-existent correlation between  ox and  Bol / Edd (Vasudevan & Fabian 2007;Shemmer et al. 2008;Fanali et al. 2013), although Grupe et al. (2010) claims a strong correlation between them.The dependence of  Bol on  2500Å might be responsible for a weak correlation between  ox and  Bol / Edd (Shemmer et al. 2008).Fanali et al. (2013) finds a significant anticorrelation between  ox and ℳ for a sample of 71 type-1 AGNs using canonical single-epoch black hole masses to estimate ℳ, which we now know suffer from accretion rate effects.Castelló-Mor et al. (2017) claims an anti-correlation between  ox and ℳ for 31 highaccretion rate sources ( ℳ>10) in their sample of 59 AGNs, using RM masses and an updated R-L relationship (Du et al. 2016) for high-accretion rate targets, although the correlation is limited by small number statistics.We examine the correlation between  ox and ℳ for our sample.Figure 9  More recently, Liu et al. (2021) argue that  ox depends on both  Bol / Edd and  BH , establishing a non-linear relationship of the form  ox = (−0.13± 0.01) log  Bol / Edd − (0.10 ± 0.01) log  BH − (0.69 ± 0.09).They employed the Python package emcee (Foreman-Mackey et al. 2013), a tool for multivariate linear regression with Bayesian inference, to obtain this best-fit relationship.More investigation is needed to decipher if this relation is fundamental or just a secondary manifestation of the  ox - 2500Å relationship.Figure 10 presents an edge-on view of this relationship and shows that extreme

DISCUSSION & CONCLUSIONS
One version of the disk-corona model assumes a thin Shakura & Sunyaev (1973) disk tightly coupled with a plane-parallel corona.The magnetic field in the accretion disk is produced by dynamo action, and the buoyancy of the magnetic fields generates magnetic loops that emerge into the corona.The loops reconnect with other loops, transferring the magnetic energy of the disk to thermal energy, thereby heating the corona.A stable corona is formed when the density of the corona reaches a certain value, allowing equilibrium between the heating by magnetic flux loops and cooling by Compton scattering.The coupling between the optical-UV emission from the accretion disk and the hard X-ray emission from the corona is often explained via such magnetic reconnection-heated model (Liu et al. 2003;Cao 2009).A fraction of gravitational energy is transferred from the accretion disk to the corona through magnetic fields that inhibit the fast cooling of the corona (Merloni & Fabian 2001).Magnetohydrodynamic simulations predict a decrease in the fraction of energy dissipated from the accretion disk as the disk transitions to radiation-pressure dominated in case of higher accretion rate (higher  Bol / Edd ) (Jiang et al. 2014(Jiang et al. , 2019)).As a result, the corona becomes relatively more compact and weaker with an increasing Eddington ratio, resulting in a steeper  ox .An increase in  Bol / Edd implies more soft photons from the accretion disk to cool the corona by Compton scattering, resulting in a steeper/softer Γ 2−8 keV .In SEAMBHs, the inner accretion disk is perhaps geometrically thick, according to the slim disk model (Abramowicz et al. 1988;Laor & Netzer 1989;Wang & Zhou 1999;Wang et al. 2013), leading to differences in the accretion disk-corona connection.However, our analysis shows that the Γ 2−8 keV - Bol / Edd ( ℳ) and  ox - 2500Å (also  2 keV - 2500Å ) shows no dichotomy between the sub and super-Eddington sources.This could mean that the transition from geometrically thin to a slim disk is not abrupt, and the disk-corona connection remains intact in super-Eddington AGNs.Another possibility is that there is no structural in the accretion disk of sub-and super-Eddington sources.
The correlations between X-ray properties and Eddington ratios studied in this paper are influenced by the choice of the bolometric correction used to estimate the bolometric luminosity.We used a correction factor of 9.26 to estimate the bolometric luminosity from  5100Å , consistent with literature values ranging between 7 and 13 (e.g., Elvis et al. 1994;Richards et al. 2006;Runnoe et al. 2012, andreferences within). However, Jin et al. (2012) find the bolometric correction factor of 15 for their full sample and 20 for the sample of 12 narrow-line Seyfert 1 galaxies, using a color temperaturecorrected SED fitting model.Additionally, there are theoretical bolometric corrections based on the thin disk model that yield substantial differences, but they are inconsistent with empirically determined bolometric luminosies (e.g., Kubota & Done 2019).
We compared our bolometric luminosity measurements with those obtained through SED fitting by Liu et al. (2021) and the bolometric correction factor described in Netzer (2019).Liu et al. (2021) derived bolometric luminosities by integrating the infrared-to-X-ray SED (see their Section 2.7 for details).For the three AGNs in our Chandra sample, i.e., IRAS04416+1215, SDSSJ074352.02+271239.5, SDSSJ100402.61+285535.3, the differences in the logarithm of bolometric luminosity measured by Liu et al. (2021) compared to our method are 0.083, -0.017, -0.147, respectively.Notably, we found a higher level of agreement for the super-Eddington sample, with a median difference of 0.183, ranging between -0.197 and 0.413.In contrast, the median difference for the sub-Eddington sample was 0.25, ranging from -0.187 to 0.683.It is important to emphasize that the determination of bolometric luminosities through multiwavelength SEDs may have considerable uncertainties, especially for super-Eddington accreting quasars.This is due to various factors such as host-galaxy contamination and variability effects due to non-simultaneous SED data.Furthermore, super-Eddington accreting quasars may have significantly enhanced EUV emission compared to typical quasars, but there is no clear observational constraint on this due to the lack of data (Jin et al. 2012;Castelló-Mor et al. 2016;Kubota & Done 2019).Next, we test our choice of bolometric correction against the correction factor given in Netzer (2019).Instead of a constant of 9.26, the Netzer (2019) bolometric correction factor is a function of monochromatic luminosity at 5100Å and is expressed as 40 [ 5100Å (observed)/10 42 erg s −1 ] −0.2 .The two sets of estimates give very similar values.The agreement is slightly better for the sub-Eddington sample, with the median value of logarithmic difference being -0.0016, while it is -0.0576 for the super-Eddington sample.
Our choice of bolometric correction is conservative and does not make any enhancement in the bolometric luminosity, hence Eddington ratio, preferentially for the SEAMBHs even though that may be the case (Jin et al. 2012).
One of our targets, SDSS J101000.68+300321.5, exhibits a flatter/harder Γ 2−8 keV and appears as X-ray weak (Δ ox < −0.2).It could be that the X-ray data for this object is not good enough, as indicated by large error bars, or the possibility that the X-ray emission is shielded by a puffed-up slim accretion disk.Luo et al. (2015) explains X-ray weakness in highly accreting AGNs through an orientation effect in a slim accretion disk.When viewed at larger inclination angles, X-rays emitted from the central region may be absorbed by the puffed-up inner accretion disk (Luo et al. 2015;Ni et al. 2018).Liu et al. (2019) estimate that 15-24% of super-Eddington AGNs should exhibit extreme X-ray variability.Our comparison sample from Liu et al. (2021) selectively chooses the high X-ray state for the sub-and super-Eddington samples with multiple X-ray observations, thereby excluding any X-ray weak object from the sample.One recent X-ray spectroscopic study of highly accreting AGNs by Laurenti et al. (2022) reports 29% of the sample as X-ray weak.A dramatic change in X-ray flux is seen in some weak emission-line quasars with high accretion rates whose X-ray variability can be explained due to changes in the thickness of the accretion disk (Liu et al. 2019;Ni et al. 2020).Our target SDSS J101000.68+300321.5 could be an X-ray-weak weak-emission line quasar that has weak high-ionization lines like C iv 1549.The lack of UV spectra for this target inhibits us from testing this hypothesis.Alternatively, the X-ray weakness may be caused by absorption from outflows, which also manifest UV absorption troughs in some AGNs (e.g., Kaastra et al. 2014, and references therein).
It is worth noting that the extreme SEAMBHs studied in this paper are primarily narrow-line Seyfert 1 galaxies (NLS1s).NLS1s are believed to be accreting material at rates approaching the Eddington limit, as supported by various studies (Komossa et al. 2006;Komossa 2018;Gallo 2018;Foschini 2020).The optical spectra of NLS1s are characterized by several distinctive features, such as relatively narrow H emission lines, strong Fe ii lines and weak [O iii] lines (Boroson 2002).These characteristics align with the Eigenvector 1 trends, known to correlate with accretion rate (Boroson & Green 1992;Marziani et al. 2001;Boroson 2002;Yuan & Wills 2003;Shen & Ho 2014;Sun & Shen 2015).In X-rays, NLS1s exhibit steep 2-10 keV spectra and a pronounced soft X-ray excess (e.g.Boller et al. 1996;Brandt et al. 1997;Véron-Cetty et al. 2001).The question arises: is the steepness of the 2-8 keV X-ray spectrum solely due to the extreme accretion rate, or can it be attributed to factors like coronal geometry or distinct cooling mechanisms within the corona specific to NLS1s?This intriguing question presents a potential avenue for future investigations using broad-band X-ray spectra.Recent research has shown a certain subjectivity in these classifications.For instance, Jin et al. ( 2023) indicated a connection between NLS1s with extreme accretion rates and weak-line quasars (WLQs).Additionally, Ha et al. (2023) demonstrated that WLQs follow the same trend of C iv versus Eddington ratio as other type-1 quasars.
In conclusion, we present new Chandra X-ray data of nine SEAMBHs.Our core sample consists of 14 SEAMBHs that have extremely high accretion rates (log ℳ > 1.5) and show the largest offset between the radius of BLR from the RM measurement and the one estimated from the canonical R-L relationship.We investigated the X-ray and optical-UV properties of these 14 extreme SEAMBHs and compared them to the sub-and super-Eddington quasars from Liu et al. (2021).To mitigate errors due to variability, we took almost simultaneous X-ray and optical-UV observations for the Chandra SEAMBHs.However, it should be noted that there is an intrinsic delay between the X-ray and the optical-UV variability due to the spatial difference in the regions emitting these radiations.We used Du & Wang (2019) for the black hole properties.Our results indicate that extreme SEAMBHs indeed have a steep 2-8 keV X-ray photon index and demonstrate a steeper power-law slope.They are consistent with correlation between Γ 2−8 keV and  Bol / Edd (also ℳ) seen in sub-and super-Eddington accreting sources.We show that the  ox - 2500Å (also  2 keV - 2500Å ) correlation extends to the extreme SEAMBHs.The  ox -ℳ correlation remains weak after the inclusion of eight extreme SEAMBHs; however, the bivariate relationship established by Liu et al. (2021) between  ox ,  Bol / Edd and  BH holds for the extreme SEAMBHs.
3 along with the eight Chandra SEAMBHs and present the details in Section 3.1.We use the remaining Liu et al. (2021) sample of 16 super-Eddington (0.47 < log ℳ < 1.5) and 26 sub-Eddington accreting AGNs to complement our sample.Liu et al. (2021) selected

Figure 1 .
Figure 1.Monochromatic luminosity at 5100 Å as a function of redshift.The plot shows the redshift-luminosity space spanned by our sample.

Figure 3 .
Figure 3. Histogram of the distribution of Γ 2−8 keV for all samples.The sub-Eddington sources primarily have Γ 2−8 keV < 2 with a mean value of 1.81, whereas, the super-Eddington and extreme SEAMBHs samples span a wider range in Γ 2−8 keV and have a mean value of 2.11.
Figure 4 shows that the Timlin et al. (2020) relationship does not fit our sample very well, particularly at the low luminosity end.Timlin et al. (2020) derived their relation from a sample of 753 quasars with 29.3 ≲ log  2500Å ≲ 31.6,spanning only two orders of magnitude.In contrast, the Steffen et al. (2006) relation is based on a sample of 293 quasars with 28 ≲ log  2500Å ≲ 33, covering four orders of magnitude.Given that the luminosity of our sample spans three orders of magnitude, 27.2 < log  2500Å < 30.7, it is more appropriate to use the Steffen et al. (2006) relationship to calculate Δ ox .The bottom panel of Figure 4 shows the residual between the measured  ox and the expected value from Steffen et al. (2006) relation

Figure 4 .
Figure 4.The top panel shows that  ox becomes more negative with increasing  2500Å following the Steffen et al. (2006) relation represented by the black solid line.Our sample shows a large deviation from Timlin et al. (2020) relation, represented by the green dash-dot line, especially at the low luminosity end.The bottom panel plots Δ ox as a function of  2500Å .A Δ ox less than −0.2 implies significant X-ray weakness.

Figure 6 .
Figure 6. 2keV vs  2500Å .The black dashed (green dashed-dotted) line represents the best-fit relation for super-Eddington (sub-Eddington) sources Liu et al. (2021).The plot shows that the non-linear correlation between X-ray and optical-UV luminosities is also present in extreme SEAMBHs.

Figure 9 .
Figure 9.  ox vs ℳ.The plot shows a weak anti-correlation with Spearman r = -0.45,p = 5.1E-4 and weakens further if the sample is excluded.

Figure 10 .
Figure 10. ox as a function of both ℳ and  BH .The solid black line represents the best-fit relation from Liu et al. (2021) and the dashed line represents the 0.07 scatter around it.

Table 1 .
Dickey & Lockman (1990)bservation log of the eight extreme SEAMBHs selected for Chandra observation and one Archived SEAMBH.Redshift, black hole mass, and dimensionless accretion rate measurements are fromDu & Wang (2019).Galactic neutral hydrogen column density ( H ) is fromDickey & Lockman (1990).Chandra exposure time (or LIVETIME) is the ONTIME corrected for average dead time corrections.† This target was observed by Chandra (PI: Gordon Garmire) which is analyzed in this paper and is part of the Archived SEAMBHs sub-sample.

Table 2 .
Photon counts in broad, hard, and soft energy bands, and the hardness ratio.