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Radio activity of supermassive black holes with extremely high accretion rates

Radio emission from the high- and super-Eddington accreting active galactic nuclei (AGNs) has various origins: a persistent jet, the magnetized corona and the wind-like outflows. It is now still unclear which is the leading mechanism responsible for the observed radio emission and how the radio emission is related to other characteristic parameters such as the Eddington ratio and black hole mass. In this paper, we present the 5 GHz Very Large Array (VLA) observational results of a sample of 25 extremely high Eddington accreting supermassive black holes (EESBHs, the Eddington ratio close to or above one) from narrow-line Seyfert 1 galaxies, among which 22 sources are detected. Most of EESBHs show a compact radio structure from a few hundred parsecs to one-kiloparsec scale. We estimated the lowest star formation rate surface density required for producing the observed radio emission, and found it is higher than the largest value previously detected in circumnuclear starburst galaxies, implying that the radio emission is from the AGN activity. Along with a comparison sample, we find an overall inverse radio loudness and Eddington ratio correlation ranging from sub- to super-Eddington ratios. The high-Eddington and mildly super-Eddington AGNs (-0.5<log lambda_Edd<0.6) have a radio to X-ray luminosity ratio L_R/L_X~10^-5 to 10^-4 and a steep radio spectrum, supporting that the radio emission is from transient ejecta (outflows) of corona, however, the jet contribution cannot be fully ruled out. Our highly super-Eddington sources (log lambda_Edd>~0.6) have a flatter radio spectrum, along with its low radio luminosity: L_R/L_X~10^-5, their radio emission is likely dominated by a magnetized corona, and a radiation pressure caused jet is also proposed in this paper.


INTRODUCTION
With increasing observational evidence during the past two decades, it is now widely accepted that supermassive black holes (SMBHs) with M BH ∼ 10 5−10 M reside at the centers of most galaxies (see review in Kormendy & Ho 2013). The growth of SMBHs is believed to primarily occur through gas accretion, which gives rise to a luminous active galactic nucleus (AGN) with a large energy release in the form 2016). Moreover, the violation of the Eddington limit becomes increasingly important for the various related topics of black hole growth, galaxy evolution and AGN feedback, e.g., in ultra-luminous X-ray sources (ULXs, see Kaaret et al. 2017, and references therein) with stellar-mass black hole/neutron stars, and primordial massive black holes during their early rapid growth (e.g. Volonteri & Rees 2005;Volonteri et al. 2015;Takeo et al. 2018).
The accretion states and their transitions in XRBs are now well studied by using the hardness-intensity diagrams (HIDs), which are also coupled with radio emission (and hence the jets) and the Eddington ratios. It has been proposed frequently that AGNs experience accretion state transitions (e.g. Körding et al. 2006) in a similar way with XRBs in the framework of AGN-XRB unification (Falcke et al. 2004) but with substantially longer timescales (> 10 5 yr, e.g. Schawinski et al. 2015). It is thus possible to connect these compact accreting systems by using scaling relations in parameter space with the most fruitful scheme of the triple correlation among X-ray luminosity, radio luminosity, and black hole mass (i.e., the so-called fundamental plane relation, Merloni et al. 2003). However, the fundamental plane relation is found to be successful only in unifying low luminosity AGNs and XRBs in the hard state.
The continuous jets are ubiquitous at low accretion rates (the low/hard state) in XRBs, but intermittent or entirely absent at high accretion rates (the high/soft state and the intermittent/very high state) (e.g. Meier 1996;Fender et al. 1999Fender et al. , 2004. Resembling the inverse correlation between the radio luminosity of jets and X-ray luminosity in XRBs, Ho (2002) found a similar inverse correlation between radio loudness (R ≡ L ν5 /L ν B ) and the Eddington ratio (λ Edd ≡ L bol /L Edd ) in AGNs. In this scheme, radio-loud AGNs with powerful relativistic jets often have low Eddington ratios, and vice versa. In contrast to the fundamental plane relation only valid for certain conditions, the inverse correlation between R and λ Edd is ubiquitous in both AGNs and XRBs (e.g. Broderick & Fender 2011) though with a large scatter. A global analogy between stellar-mass black holes and supermassive black holes has been established in the R and λ Edd correlation: low-luminosity AGNs are similar to XRBs in the low/hard state, and the high-or super-Eddington accreting AGNs (e.g., narrow-line Seyfert 1s) being an analogy of XRBs in the high/soft and the very high state. It should be noted here that only a few XRBs experience transitions from classical spectral states to the super-Eddington regime (e.g. Neilsen & Lee 2009). The super-Eddington accretion state is poorly understood primarily because of the extremely short timescales of the spectral state transition in XRBs. Therefore, the study of super-Eddington accretion and the associated jet formation and/or quenching in XRBs does not warrant a scaling-up to AGNs.
The dynamic timescale in different accretion states is proportional to the mass of the central black hole, and it is difficult to observe a complete burst cycle in individual AGNs resembling XRBs. Still, progress has been made by Sikora et al. (2007), providing further confirmation of R − λ Edd in-verse correlation with the Eddington ratio range from subto super-Eddington ratios in both radio-loud and radio-quiet AGNs. In this paper, the short-lived super-Eddington accreting AGNs will be studied to augment the correlation in the super-Eddington regime with robust measurements of the radio loudness and the Eddington ratio. Greene et al. (2006) have found that AGNs with super-Eddington accretion rates are predominately radio-quiet. The radio emission from radio-quiet AGNs can have various origins, including persistent jets, a magnetized corona/jet base and a wind-like outflow, in this paper, the "jets" specially refer to a collimated outflow. Furthermore, star-forming activities in the host galaxy may also contribute to the thermal and non-thermal radio emission, which typically shows hostlike extension, with a diffused and clumpy structure, and has a low surface brightness. Nuclear starbursts have been observed in some galaxies (e.g. Deo et al. 2006;Hennig et al. 2018), and a significant fraction of the narrow-line Seyfert 1 galaxies (NLS1s) tend to have circumnuclear star-forming rings (Deo et al. 2006), making them more difficult to distinguish from the AGN. These different mechanisms are fundamentally crucial in super-Eddington accreting AGNs, and in explaining the R − λ Edd correlation.
As mentioned above, the origin of the radio emission from the super-Eddington accreting AGN itself is complicated. The slim-disk model was firstly proposed by Abramowicz et al. (1988) to describe a super-Eddington accretion flow. Recently, progress has been made in simulating the super-Eddington accretion disk (e.g. Dotan & Shaviv 2011;Begelman & Volonteri 2017;Jiang et al. 2017) and several models among them support the launching of a jet, which is driven by either the radiation-pressure (Takeuchi et al. 2009;Sądowski & Narayan 2015) or the magnetic field retrieved from the spin of the central black hole, i.e., a Blandford-Znajek jet (BZ, Blandford & Znajek 1977) (e.g. Narayan et al. 2003;Kelley et al. 2014;McKinney et al. 2015). From observations, several super-Eddington accreting systems are found to accelerate a jet, the trigger has been attributed to either a radiation-pressure (e.g. Ohsuga & Mineshige 2011;Sądowski & Narayan 2015) or the BZ magnetic field (e.g. Berger et al. 2012). Furthermore, a super-Eddington accretion disk may also drive a strong wind and accelerate relativistic electrons to produce the observed synchrotron radio emission. For example, in SS 433, such windlike radio-emitting outflows co-exist with a relativistic jet, and they have a comparable radio luminosity (e.g. Blundell et al. 2001). In AGN accretion models, a hot T b ∼ 10 9 K corona in the vicinity of the black hole accretion disk may explain the observed X-ray emission from AGNs (Haardt & Maraschi 1991). It has been suggested that a magnetically heated corona may also produce the radio emission (Laor & Behar 2008;Raginski & Laor 2016). This hypothesis is later supported by the discovery of the similar radio to X-ray luminosity ratio L R /L X ∼ 10 −5 between radio-quiet quasars and coronally active stars (Laor & Behar 2008), and by high radio frequency ( 100 GHz) observations of two nearby Seyfert galaxies (Behar et al. 2015(Behar et al. , 2018Inoue & Doi 2018).
Based on the topics mentioned above, the NLS1 galaxies may be introduced as a long-lived high Eddington ratio laboratory with an SMBH engine. NLS1s consist of a distinct class of AGNs identified by the width of their H β emission lines (< 2000 km/s, Osterbrock & Pogge 1985;Boroson & Green 1992). They are located at an extreme end of the AGN parameter space that is believed to be governed mainly by the Eddington ratio (e.g. Boroson & Green 1992). Indeed, there is growing evidence showing that NLS1s have higher Eddington ratios and lower black hole masses than normal Seyfert 1 galaxies and quasars (e.g. Boroson & Green 1992;Du et al. 2015), which implies that they are systems with rapidly accreting SMBHs. NLS1s are often radio-quiet in contrast to the broad-line AGNs (e.g, Ulvestad et al. 1995). Interestingly, NLS1s are not completely radio quiescent, and the origin of their weak radio emission holds the long controversy. Several papers have presented evidence for the presence of radio jets in a handful of radio-quiet NLS1s (e.g., Lal et al. 2004;Giroletti & Panessa 2009;Doi et al. 2013Doi et al. , 2015, which is far from a consensus. However, as a sub-class, the super-Eddington accreting NLS1s have not been comprehensively studied before. To study the super-Eddington accretion and also to investigate the origin of the radio emission in extremly high Eddington ratio accreting AGNs, we present high-resolution (sub-kpc-scale) VLA results of a sample of NLS1s that are accreting at Eddington ratios close to or exceeding one.
This paper is organized as follows: In Section 2 we describe our sample; Section 3 presents the procedure of data reduction; Sections 4 and 5 provide results from data reduction and analysis, followed with the discussion in Section 6. Throughout this paper we assume the same Λ-CDM cosmology with Sikora et al. (2007) and Broderick & Fender (2011), which is H 0 = 100 h = 70 km s −1 Mpc −1 , Ω M = 0.3, and Ω Λ = 0.7.

THE SAMPLE
Our parent sample is comprised of 60 super-Eddington accreting supermassive black hole candidates with Eddington ratios close to or above one, which was compiled by Wang et al. (2013). Given a strong relation between the Eddington ratio λ Edd and the X-ray photon index Γ 2−10 keV (e.g. Lu & Yu 1999;Brightman et al. 2013), AGNs with higher Eddington ratios are expected to have a steeper hard X-ray photon index Γ 2−10 keV according to this well-known relation. The super-Eddington AGN candidates have actually been selected indirectly by having Γ 2−10 keV > 2. This selection approach requires an accurate determination of their bolometric luminosities L bol and the knowledge of the broadband spectral energy distribution (SED, e.g. Jin et al. 2017). Unfortunately, it is difficult to measure L bol from broadband SED since much of the radiation from the accretion disk is in the form of extreme UV, which is beyond the observation windows. Still, accurate bolometric luminosity measurements (by using SED fitting) have been obtained from the literature for most of our super-Eddington AGN candidates (see col-umn 7 of Table 1 and the corresponding references) with a robust mass determination.
To study the radio emission from super-Eddington AGNs, we have searched our parent sample for available highresolution radio observations in the Very Large Array (VLA) data archive. We primarily collected VLA observations at C-and X-band with the A-array configuration having a resolution corresponding to a projected linear size 2 kpc at the rest frame of each object. Furthermore, we also collected L-band observations with the VLA A-array when there has no high-resolution observational data at C and X bands. Twenty-six sources meet the requirements, which also includes an intrinsic radio-loud NLS1, 1H 0323+342 (Zhou et al. 2007) that has been detected in γ-rays by Fermi-LAT (Abdo et al. 2009). Observations show that the radio emission in this source is strongly Doppler boosted (e.g. Hada et al. 2018). This source was removed from our final sample of 25 objects. The basic information of the objects used for this research is listed in Table 1. The black hole mass and 2-10 keV flux density are obtained from Wang et al. (2013), where the black hole mass was estimated by using the broad emission line width and the 5100 Å luminosity. We should note here that it's not sure that wether high-Eddington accreting systems in this paper have a similar broad line region size vs. 5100 Å luminosity relation, thus cause uncertainty of black hole mass estimation, this paper will not talk about this uncertainty. Among 25 objects in our final sample, 17 (∼ 68%) objects are at the super-Eddington regime, whereas the rest of 8 (32%) are high-Eddington sources, including four objects having lower limits. For this reason, the super-Eddington accreting AGN candidates used in this work will be referred to as extremely high Eddington ratio accreting supermassive black holes (EESBHs) with an Eddington ratio close to or above 1.
3. VLA DATA REDUCTION All the relevant raw data (visibilities) have been retrieved from the NRAO Data Archive 1 , and only the data with good quality and the correct observational configurations have been adopted. In general, each target should have been observed for more than 20 seconds and followed by one exposure of a nearby phase calibrator. Although some results have already been published (see Table 2), in order to ensure uniformity of analysis across all the parameters, we performed a manual calibration for all the datasets using the Common Astronomy Software Application (CASA v5.1.1, McMullin et al. 2007). The archive data have a time range from 1980 to 2016 and are composed of two distinct groups, the historical or ordinary VLA products and the Karl G. Jansky VLA (JVLA) products. The historical VLA datasets were scheduled as a single channel in only one or two spectral windows (SPWs), while the JVLA continuum observations are all in the multi-SPW mode with each SPW having a good bandwidth coverage. Notes: Column 1: short name; Column 2: alias; Column 3: redshift; Column 4: black hole mass estimated from broad emission line; Column 5: 2-10 keV X-ray flux density; Column 6: k-corrected B-band optical luminosity of AGNs (erg s −1 ), calculated from 5100 Å flux density f5100, fraction of AGN contribution at 5100 Å and an assumed optical spectral index αopt = −0.5; Column 7: bolometric luminosity (erg s −1 ) and references; Column 8: Eddington ratio. Our data analysis followed the standard routines described in CASA Cookbook. For the historical VLA datasets, we adopted a closest flux density standard concerning the observing date to get the absolute flux density for the primary flux calibrator, and subsequently bootstrapped onto the secondary flux density calibrators and targets. As an example, we use the flux density standards 'Perley-Butler 2013' (Perley & Butler 2013) to set the overall flux density scale for datasets from the project 15A-283 running from July to September 2015. We also determined the gain solutions by using a nearby secondary calibrator and transferred them to target sources. In contrast to using a narrow band in historical VLA datasets, the JVLA datasets were all scheduled with multiple channels and spectral windows. We thus referred to the calibrating scheme/routine as described in most recent CASA Cookbook. In addition to transferring the flux density scale and gain solutions, antenna delay and bandpass corrections were also determined by fringe-fitting the visibilities.
Deconvolution and self-calibration were performed in DIFMAP (Shepherd et al. 1994). Only for sources with a high signal-to-noise ratio (SNR> 9), self-calibration was applied using a well-established model. The self-calibration was performed initially only on phase, and subsequently on both phase and amplitude when we achieved a good model. Finally, the natural-weighting images were used for model fitting and further discussion. Two-dimensional Gaussian model was used to fit the visibility data of each target to obtain characteristic parameters, such as the integrated and peak flux density, as well as the full width at half maximum (FWHM) of the Gaussian model. The model-fitting results are listed in Table 3. We also list some special data reduction notes in Appendix A.
We estimated the uncertainty of the integrated flux density S i based on the formulae given by Hopkins et al. (2003). As our sources have an unresolved point-like nucleus, the relative errors of the integrated flux density S i were estimated from σ i /S i = 2.5σ 2 rms /S 2 p + 0.01 2 (Hopkins et al. 2003, Eq.5). The uncertainty of the peak flux density was estimated by combining the model fitting errors and the initial calibrating errors, which is σ p = σ 2 rms + (1.5σ rms ) 2 ≈ 1.8σ rms Notes: Column 1: short name; Column 2: observational bands; Column 3: project ID; Column 4: observing date; Column 5: observing time of target source; Column 6 -8: beam major axis, minor axis and position angle; Column 9: references for data sets publication. 4. RESULTS The new results for 25 EESBHs observed with the VLA are obtained from a total of 45 datasets, of which 26 were previously published (see the references in Column 9 of Table 2), and the rest of 19 datasets are analyzed in this work. All but one object (IRAS 04416+1215) has the C-band data, and 22 out of 24 objects are detected at C-band with a peak flux density above 3σ. All the sources were observed with a resolution of < 2 kpc at C-band, while 20 out of 24 sources were observed with a resolution of < 1 kpc. Table 3 lists the 5 GHz luminosity (column 8), the radio brightness temperature (column 9), and radio loudness (column 10).
The radio brightness temperature was estimated by using the formula (e.g. Ulvestad et al. 2005): where S i is the integrated flux density of each Gaussian model component in units of mJy (column 5 of Table 3); θ is the FWHM of the Gaussian model in milli-arcsec (column 7 of Table 3); ν is the observing frequency in GHz (column 2 of Table 3), and z is the redshift. The estimated 5 GHz and 8.4 GHz radio brightness temperature are listed in Column 9 of Table 3. Because the measured component size is just the upper limit, the radio brightness temperature should be considered as a lower limit. The non-simultaneous 5 − 8 GHz spectral index α (defined as S i ∝ ν α ) based on the integrated flux densities are listed in Column 11 of Table 3. Next, we compared the radio emission at different galactic scales. For instance, the VLA A-array flux density at Lband may be taken as a measurement of the core radio emission, and the flux density derived from the Faint Images of the Radio Sky at Twenty-Centimeters (FIRST) 1.4 GHz survey (Becker et al. 1995) made with the VLA B-array may serve as a proxy of the radio emission from the entire galaxy. These two observations provide a projected spatial resolution of ∼ 4 − 8 kpc and ∼ 20 − 30 kpc, respectively. The typical size of the optical hosts of EESBHs in our sample is comparable with the beam size of VLA B-array at 1.4 GHz (the FIRST images). For those sources without the FIRST survey coverage or without detection, the NRAO VLA Sky Survey (NVSS, Condon et al. 1998) flux density has been taken instead, and the 1.4 GHz flux densities are listed in Table 4, there are 11 sources in total. Figure 1 shows the comparison between the FIRST/NVSS flux density and the VLA A-array 1.4 GHz flux density. Taking a 1 σ uncertainty, the two fluxes are in good agreement for the majority of sources. Nine out of 11 sources have a central flux density that accounts for >80% of the total radio emission. Among these sources, the average flux density difference between VLA A-array and FIRST/NVSS measurements is only 18% (Mrk 957 has a 59% VLA A-array flux density decrease by comparing it with the NVSS measurement). We have also  Figure  1 to show the comparison between the FIRST/NVSS and the VLA A-array 1.4 GHz flux density. This indicates that most sources are compact at this resolution and the radio emission is dominated by the central ∼4-8 kpc region. Furthermore, as the time lag between FIRST, NVSS and the VLA A-array 1.4 GHz observations, we can infer that our EESBHs have a quite stable radio emission.

PARAMETER CORRELATIONS
The correlation between the radio loudness R and the Eddington ratio λ Edd in the extremely high-Eddington regime (λ Edd 1) has not been well explored previously because of the limited number of super-Eddington sources and a poor estimation of their Eddington ratios. In order to form a broad parameter space for the Eddington ratio, we include 199 AGNs from Sikora et al. (2007) having a wide range of Eddington ratios from log λ Edd = −7 to 1.
An accurate estimate of the bolometric luminosity is required to get the Eddington ratio λ Edd , which is defined as λ Edd ≡ L bol /L Edd . The integral of the observed SED of the optical-ultraviolet and X-ray emission by the accretion disc and the hot corona provides a direct measurement of the bolometric luminosity L bol . One generally uses the Bband continuum luminosity L B to estimate the bolometric luminosity with a bolometric correction factor κ B expressed as L bol = κ B L B . The commonly used B-band bolometric correction factor is κ B = 10 (e.g. Marconi et al. 2004;McLure & Dunlop 2004). However, it is only suitable for sub-Eddington systems as the high-Eddington ratio sources tend to have a larger bolometric correction factor (see Vasudevan & Fabian 2007).
It was also noted by Jin et al. (2012a,b) that NLS1s often possess high Eddington ratios, having significantly Notes: Column 1: short name; Column 2: central frequency; Column 3: physical resolution in kpc; Column 4: peak flux density, '∼' means this source is only marginally detected, and for the sources have not detected we take 3σ upper limit; Column 5: integrated flux. The C-band flux density for source J0444+1221 is estimated from X-band data; Column 6: image noise root-mean-square; Column 7: F W HM of Gaussian model; Column 8: 5 GHz radio luminosity; Column 9: the lower limit of radio brightness temperature; Column 10: radio loudness; Column 11: non-simultaneous 5 to 8.4 GHz spectral index; Column 12: lower limit of SFR surface density ( * : M yr −1 kpc −2 ) estimated from radio luminosity.
higher values of the optical bolometric correction factor than other samples. Surprisingly, the average value of 5100 Å bolometric correction factor κ 5100 ≈ 70 has been found for 10 NLS1s. Wang et al. (2013) independently got κ 5100 ≈ 40 ∼ 100 for their super-Eddington accreting AGN candidates. The majority of high-Eddington sources in Sikora et al. (2007) are Palomar-Green (PG) quasars; therefore, instead of using a common bolometric correction factor, we use the bona fide bolometric luminosity which was obtained from the SED fitting to re-calculate the Eddington ratios for PG quasars and our EESBHs. The bolometric luminosities for our EESBHs and PG quasars are presented in Table 1 (column 7) and Table 5, respectively. On the other hand, the B-band bolometric correction factor κ B = 10 was used to set a lower limit of the Eddington ratios for those EESBHs without previous bolometric luminosity estimates. The radio loudness of these 199 sources are taken from Broderick & Fender (2011), measured base on the core-only radio emission. The strategy of using the core-only radio emission depends on the idea that the optical/X-ray radiation is only related to the current/recent nuclear activity, which is not directly correlated with large scale radio emission. It was found that the jet emission can exist for a timescale of ∼ 10 7−8 years (e.g., Kuźmicz et al. 2018). On the other hand, the large-scale jet emission is not only dependent on the central engine itself but also the surrounding environment. Such as radio galaxies as well as radio-loud quasars reside in denser environments than radio-quiet AGNs (e.g. Lietzen et al. 2011). Besides, the star formation in host galaxies will also contribute significant radio emission on the whole galaxy scale. Since the radio loudness and the Eddington ratio is relatively dependent on the optical or X-ray estimates, it is reasonable to adopt the core-only radio emission for the relation between radio loudness and the Eddington ratio.
The standard radio loudness parameter (Kellerman et al. 1989) is defined as R ≡ L ν5 /L ν B = 1.3 × 10 5 (L 5 /L B ), where the L 5 is the radio luminosity at 5 GHz and L B is the optical luminosity of the nucleus at λ B = 4400 Å; both are measured in erg s −1 . For our EESBHs, we have measured the core radio emission from the brightest component of the object. Nearly all of our EESBHs are unresolved with VLA A-array observation at 5 GHz (except for Akn 564), and most have a component size of a few hundred parsecs (except for four sources with relatively large redshift, i.e. z > 0.15). Therefore, the 5 GHz core-only radio emission of our EES-BHs is consistent with Broderick & Fender (2011). Here the total radio luminosity is expressed as L 5 ≡ ν 5 L ν5 and the total B-band luminosity as L B ≡ ν B L ν B . In this paper, we calculate the B-band luminosity by using the 5100 Å luminosity using the transformation L B = L 5100 × (λ B /5100) 1+αopt and assuming a constant optical spectral index α opt = −0.5 (e.g. Sikora et al. 2007). The radio luminosity is given as: where α R is the radio spectral index, assuming α R = −0.8 according to our measurement of 5 − 8.4 GHz spectral in- dex, f R is the radio flux density, and D L is the luminosity distance. Figure 2 shows the radio loudness versus the Eddington ratio for 199 comparison AGNs and our EESBHs.
Here we keep the traditional labels of the source sample for 199 comparison AGNs, similarly hereinafter. The linear regression slope between logR and logλ Edd is −0.73 ± 0.08 within the 95% confidence interval, the blue belt shows the upper and lower boundaries corresponding to the slope of −0.73. Interestingly, we can see that there are a lot of vacancy both at upper left and lower right in Figure 2, which imply that our sample is incomplete for larger radio loudness at the lower boundary of the Eddington ratios, and also for both larger and lower radio loudness at the higher boundary of Eddington ratios. However, the trend may also hint a real features of radio loudness at higher and lower Eddington ratios. The other problem is the regression slope between logR and logλ Edd is close to −1 and possibly governed by the mutual dependence of R and λ Edd on optical luminosity, i.e. according to the definitions, R ∝ L −1 B and λ Edd ∝ L bol , and the trouble spot is we take the relation of L bol = 10L B for the samples in Figure 2 except for PG Quasars and EESBHs. However we find that R·λ Edd = 1.3×10 6 (L R /L Edd ) (being free of the parameter L B ), if there is no correlation between L R and L Edd or they are identical, then logR vs. logλ Edd naturally has a slope of −1 caused by the mutual dependence on L B . We have therefore explored the correlation between the 5 GHz core radio luminosity L R and the Eddington luminosity L Edd , here we use M BH to replace L Edd due to the equation L Edd = 3.2 × 10 4 (M BH /M )L , see Figure 3. There is a clear linear correlation between L R and M BH and 4.9 ± 0.1 6.7 ± 0.1 5.5 ± 0.1 2.9 ± 0.1 0.9 ± 0.1 Notes: 1.4 GHz flux density in mJy. N : NVSS; F : FIRST the linear regression gives a slope of 4.86 ± 0.05 at significance level of > 95%, which differs from 1 and for that reason the correlation between logR and logλ Edd is not caused by the mutual dependence on L B . Figure 4 shows the radio luminosity versus the optical luminosity expressed in Eddington units. There is no clear boundary between the sequence of radio-loud and radio-quiet sources in our plot in comparison with the results in Sikora et al. (2007). Our EESBHs located in the high end of optical luminosity and have an intermediate distribution of radio luminosity that lies between radio-loud quasars sample and Palomar-Green quasars sample (see the histogram in Figure  4). 6. DISCUSSION 6.1. The origin of radio emission: AGN or star-forming activity?
In the central kpc region of the radio-quiet AGN host galaxies, the AGN is often not the only source that can produce the observed radio emission. Recalling that the radio emission from galaxies may have various origins in Section 1, e.g. from stellar activities such as thermal free-free emission from HII regions (e.g., Lacey et al. 1997;Ulvestad & Antonucci 1997), non-thermal synchrotron radio emission (Lisenfeld & Völk 2000) from young supernovae (SNe, e.g., Kronberg & Sramek 1985, 1992Colina et al. 2001;Alberdi et al. 2006) and supernova remnants (SNRs, e.g., Muxlow et al. 1994;Ulvestad & Antonucci 1997), or from the jets of AGNs (see Ho 2008, and references therein). It's quite possible that the star-forming activities will dominate observed radio emission of the central kpc region in starburst galaxies (e.g. Ulvestad & Antonucci 1997;Tarchi et al. 2000;Fenech et al. 2008;Batejat et al. 2011). Several observational characteristics can be used to discriminate between the radio emissions from AGNs and star-forming activities, such as the brightness temperature, the spectral index and the radio morphology (see Panessa et al. 2019, and references therein). In the following sections, we explore the radio emission of our EESBHs by using these indicators.
6.1.1. Radio morphology, spectral index and brightness temperature Previous studies show that an AGN is still the dominating radio-emitting source at radio flux densities above 0.1 mJy in radio-quiet AGNs (Padovani et al. 2011;Bonzini et al. 2013). This indicates that the radio emission in most of our EESBHs could be from AGN activities and only three sources (Mrk 684, SDSS J010712+140845 and SDSS J114008+030711) are below this threshold. In our EESBH sample, Akn 564 is the only source showing evident AGN activity with a collimated linear structure extending ∼ 1 kpc towards the north, and three resolved components in the VLA 5 GHz and 8.4 GHz images (see Figure 5). Furthermore, two sources (PG 2233+134 and Mrk 493) show flat or inverted spectra, suggesting the radio emission is from AGNs.
Our EESBHs show a moderate brightness temperature range of T B ∼ 10 2 -10 5 K at both 5 GHz and 8.4 GHz as compared with radio-loud AGNs. Only two sources (B3 1702+457 and Mrk 486) have radio brightness temperatures above 10 5 K, i.e. 10 5.36 K and 10 5.10 K, respectively. Since a star-forming region cannot reach T B ∼ 10 5 K at frequency ν > 1 GHz (Condon et al. 1991Condon 1992), we conclude that the radio emission observed in these two sources are preferentially from AGN activities.
Additionally, eight sources have a radio brightness temperature T B 10 4 K (but < 10 5 K, see column 9 of Table  3) at 5 GHz and 8.4 GHz would also imply non-thermal radio emission, because a star-forming region can rarely exceed 10 4 K at such high frequencies (Condon 1992). The highest brightness temperatures from star-forming regions have been observed in luminous and ultra-luminous infrared galaxies (LIRGs and ULIRGs, respectively), which is 10 4 − 10 5 K (e.g. Condon et al. 1991;Pérez-Torres et al. 2009;Varenius et al. 2014). As none of these eight sources has been found to be a LIRGs/ULIRGs, thus strongly indicats the AGN dominated radio emission (e.g. Ulvestad & Antonucci 1997). Indeed, among these 8 objects, high radio brightness temperature nuclei (T B > 10 6 K) have been detected for Mrk 335 and IZw1 by our Very Long Baseline Array (VLBA) observations at 1.4 GHz (Obs. ID: BY145, in preparation) and for Akn 564 (Lal et al. 2004).
We note that more than half of the sources in our sample have a very low radio brightness temperature as compared with an AGN. We have checked the infrared and far-infrared data and found that 8 sources were detected by the Infrared Astronomical Satellite (IRAS). Among them, four sources (B3 1702+457, Mrk 957, Mrk 478 and PG 1402+261) have the infrared and far-infrared luminosity L IR/FIR > 10 11 L , i.e. the so-called type-1 (ultra-) luminous infrared galaxies (e.g. Ulvestad & Antonucci 1997). Among this four objects, the radio emission from B3 1702+457 has already been proven to be from AGN (Gu & Chen 2010), while the radio emission from the remaining sources are likely dominated by the star-forming activity. It should be emphasized here that the radio brightness temperature estimated in this paper is only the lower limit, which is constrained by the limited resolution of VLA A-array. Therefore, high-resolution very long baseline interferometry (VLBI) observation is crucial to distinguish the various origins of the radio emission from radio-quiet AGNs.
6.1.2. Estimating star formation rate surface density with radio emission In order to further distinguish the radio emission from the star-forming activities and AGN, we can compare the star formation rates (SFRs) measured by using direct indicators with the radio emission estimates. Here we estimate the SFR requirements for the observed radio emission by using the empirical relation derived by Condon (1992). It was proven in star-forming galaxies that the SFRs estimated with this relation are well consistent with those estimated from the infrared radiation (e.g., see Rabidoux et al. 2014). Here we only estimate the SFRs from the non-thermal process since there are higher SFRs requirements for thermal processes than for non-thermal to produce equal radio luminosity. This will give us the lowest SFR requirement in producing the observed radio emission. The formula is: (3) where L N is the non-thermal radio luminosities, SFR N (M 5 M ) is the star formation rate of stars more massive than 5 M , and estimated from the non-thermal process (i.e. nonthermal radio flux density). The extended Miller-Scalo initial mass function (Miller & Scalo 1979) with an exponent of −2.5 was used, and it was assumed that all stars with mass greater than 8 M become supernovae, and that the dust absorption is negligible in deriving these formulae (Condon 1992). Here we assume a typical spectral index of α N = −0.8 for the non-thermal radio emission. The final SFR N (M 5 M ) is subsequently scaled to the total SFRs (M 0.1 M ) by a scaling factor of 5.6 (Rabidoux et al. 2014). We calculated the disk-averaged surface densities of the star formation rates (Σ SF R ) by dividing the size of emission regions, derived from the major and minor axis of the synthesis beam.
The results are listed in the last column of Table 3. Most of our EESBHs (20 out of 25) have a lower limit for the SFR surface density Σ SFR 10 M yr −1 kpc −2 estimated at 5 GHz or 8.4 GHz. Among 20 sources, ten have Σ SFR 100 M yr −1 kpc −2 . It was noted that NLS1s tend to have a high fraction of circumnuclear star-forming rings at more or less 1 kpc from the core (Deo et al. 2006;Hennig et al. 2018), while it is still very difficult to reach a star formation surface density of 10 M yr −1 kpc −2 (see Kormendy & Kennicutt 2004;Kennicutt & Evans 2012). Five objects in our sample have the lowest requirement of Σ SFR < 10 M yr −1 kpc −2 in producing the observed radio emission, which are Mrk 42, Mrk 684, PG 1115+407, SDSS J010712.04+140845.0 and SDSS J114008.71+030711.4. Mrk 42 and Mrk 684 have a far-infrared luminosity L FIR ∼ 10 10 L , suggesting a high SFRs on the scale of the whole galaxy, that does not result in a high star formation rate surface density. Particularly, Mrk 42 was found to have a starburst ring at ∼ 300 pc (Muñoz Marín et al. 2007) and have a total ring SFR of 1.38 M yr −1 (Hennig et al. 2018), which is still below the lower limit required to produce the observed radio emission. Moreover, the VLA A-array has a resolution of ∼ 200 pc at C-band for Mrk 42, which suggests that the radio emission from Mrk 42 is less likely to result from the star-forming activity. We should note here that the SFR surface density required to produce the observed radio emission will be even    Jin et al. (2012b) larger than the value given here if we take thermal fractions into account and use the intrinsic size of the emission regions. We thus conclude that the radio emission from most of our EESBHs (at least 21 out of 25) is from AGN activities, otherwise strong nuclear star formation is required to give the observed radio emission.

AGNs with super-Eddington accretion rates: comparing with XRBs
With the population of EESBHs, as presented in this paper, we found that super-Eddington accreting AGNs have a low radio luminosity (L 5 ∼ 10 38 erg s −1 ) and most of them ( 90%) are radio quiet (R 10). There is only one super-Eddington source (KUG 1031+398) in our sam-ple marginally located in the radio-loud region (R = 12). This confirms the radio quiescence when Eddington ratios approach or exceed one. More generally, there is an inverse correlation between radio loudness and the Eddington ratio from the sub-Eddington to the super-Eddington regime. In this work, we have extended the inverse R − λ Edd correlation to the super-Eddington regime (see Figure 2), supporting a continuous jet suppression with an increasing Eddington ratio as proposed by Ho (2002) when overcoming the Eddington limit.
AGNs with extremely high Eddington ratios and Galactic accreting black holes in a very high state (e.g., Greene et al. 2006), hold the same tendency of the radio loudness versus the Eddington ratio (e.g., Broderick & Fender 2011). Furthermore, with the state transition in XRBs from the low/hard state to the high/soft state, a radio outburst is always associated with the soft X-ray peak at the end of the transition phase, i.e., the very high state. Some of our extremely high Eddington accreting AGNs indeed show kpcscale emission or kpc-scale radio structures (KSRs) in their FIRST and NVSS images, implying a past ejecting activity. The kpc-scale radio emission in B3 1702+457 was already noted by Giroletti et al. (2017). Mrk 684 shows an extension of ∼ 60 kpc to the north-west in the NVSS 1.4 GHz image (left panel of Figure 6). The NVSS image reveals a twocomponent system where the second component is consistent in position with FIRST. A visual inspection of the SDSSi image confirms that there is no obvious optical counterpart associated with the second component that also shows a structure of an AGN hotspot, implying a physical connection with the radio nucleus. PG 1011-040 shows a slight bipolar extension in the FIRST image towards the north-east and the south-west, perpendicular to the major axis of the host galaxy (middle panel of Figure 6). These characteristics more likely imply that the central AGN is responsible for this radio emission (e.g. Yang et al. 2018), whereas a strong starburst nucleus is also possible (e.g. Seaquist & Odegard 1991). In SDSS 114008.71+030711.4, we find a strong radio source offset north-west from the galaxy center in the FIRST 1.4 GHz image at a distance of 4.5 arcsec (7 kpc) with a peak flux density of 0.84 ± 0.15 mJy (see Figure 6). There is no optical counterpart coinciding with this radio source in the HST/WFPC2 F814W image, implying that the central AGN or an off nuclear massive black hole (see Reines et al. 2020) is responsible for this emission. Previous studies showed that kpc-scale radio structures might be a common feature in radio-loud Seyfert galaxies. There are about 10 NLS1s so far that show kpc-scale radio structures, including one radio-quiet source (Gliozzi et al. 2010;Doi et al. 2012Doi et al. , 2015Richards & Lister 2015). If AGNs have the similar state transition with XRBs, then the kpc-scale radio structures found in our sample might imply the past ejecting ac-tivities during the transition state. Similarly, the Galactic Xray binary SS 433 has super-Eddington ratio accretion at all times during the outbursts, and the episodes of jet activity are responsible for the formation and shaping of the surrounding elongated radio structures (e.g. Broderick et al. 2018).
The very high state in Galactic black holes possesses the highest Eddington ratio but is not super-Eddington. It was suggested that super-Eddington accretion in ULXs and some XRBs may indicate a new 'ultra-luminous' accretion state (Gladstone et al. 2009;Sutton et al. 2013;Brorby et al. 2015;Roberts et al. 2016), which is supported by the finding of a few micro-quasars with super-Eddington accretion that can transit between classical states and the ultra-luminous state (e.g., GRS 1915+105, see Fender et al. 2004). The same scenario was also proposed to explain the powerful radioemitting outflows in tidal disruption events (e.g., . We propose that a few (if not all) of the super-Eddington accreting AGNs can be a better analogy of stellar-mass accreting systems in the ultra-luminous state, but with longer timescales.

The Origin of Radio Emission: Jet-driven versus
Outflow-driven and Corona-driven The extremely high Eddington ratio accreting AGNs are predominately radio-quiet, whereas the radio emission is not completely absent or is dramatically reduced in comparison with the sub-Eddington ratio AGNs (see Figure 2). In radioquiet AGNs, several radio-emitting mechanisms are still in competition with each other, such as low-power jets, radioemitting wind-like outflows, and magnetized corona/jet base. In this paragraph, we will further discuss the origin of radio emission among several AGN dominated activities.
We have explored the correlation between the radio and Xray luminosities, and between the radio spectral index and the Eddington ratio. Figure 7 shows the radio luminosity versus the X-ray luminosity with markings of the log L R /L X = −5 and log L R /L X = −4 lines. Most of our extremely high-Eddington accreting AGNs have a radio to X-ray luminosity Figure 7. The 5 GHz radio luminosity versus the 2-10 keV X-ray luminosity for our EESBH sources. The grey dashed lines correspond to LR/LX = 10 −5 and LR/LX = 10 −4 . ratio in the range log L R /L X = −5 to −4. Sources with a radio to X-ray luminosity ratio log L R /L X ∼ −5 are found to be similar to coronally active stars, which would imply magnetized corona/jet base dominated radio emission. Because the radio emission from radio-loud AGNs, and hence from jets, have a radio to X-ray luminosity ratio of log L R /L X ∼ 10 −2 to 1, therefore, sources having a radio to X-ray luminosity ratio close to or slightly exceeding log L R /L X ∼ −4 would imply a combination of corona and jets (Laor & Behar 2008;Behar et al. 2015;Laor et al. 2019). Figure 8 shows the distribution of the non-simultaneous radio spectral index α along the Eddington ratio λ Edd , the spectral index uncertainties are estimated from the integrated flux density errors. The non-simultaneity of the 1.4, 5 and 8.4 GHz observations in these objects may induce spectral index errors, whereas it may not lead to a large uncertainty in spectral index due to the stability of radio emission of our EESBHs (see Section 4). Most of our (9 out of 11) sources have a steep radio spectrum, implying that the radio emission from our EESBHs is dominated by the ejecta rather than the core. In Figure 8, we also plot the 5 − 8.4 GHz spectral index and the Eddington ratio distribution for radio-quiet quasars from Laor et al. (2019) as a comparison. Laor et al. (2019) discovered an inverse correlation between α and λ Edd using a sample of 25 radio-quiet quasars, spanning a large range of logλ Edd from −1.6 to 0.6. Note that the high-Eddington (log λ Edd > −0.5) quasars tend to have a steep spectrum (α < −0.5, i.e., below the horizontal dotted line in Figure  8), and it was interpreted as that the radio emission is from optically thin outflows (see Laor et al. 2019). It is clear from Figure 8 that most of our high-Eddington sources follow a similar trend. Furthermore, the 11 EESBHs having a radio to Figure 8. The radio spectral index α versus Eddington ratio λ Edd . The spectral index is measured between 1.4 and 5 GHz (red filled squares) and between 5 and 8.5 GHz (red unfilled squares) for our EESBHs. 5 ∼ 8.4 GHz spectral index and the Eddington ratio distribution for radio-quiet quasars from Laor et al. (2019) are plotted as blue dots, where the blue line and belt are linear regression and 95% confidence interval, respectively. The horizontal dotted line at α = −0.5 is the division of flat (above) and steep (below) spectrum radio sources, the vertical dotted line designates λ Edd = 1.
X-ray luminosity ratio of log L R /L X ∼ −5, suggesting the contribution from the radio-emitting corona (Laor & Behar 2008;Behar et al. 2015;Laor et al. 2019). To sum up, a possible interpretation for the radio emission in our EESBHs is the transient ejecta caused by the optically thin outflow (outflowing corona), which enhance the comparison between the high Eddington AGNs and XRBs in the soft state since the transient ejections were generally observed in the soft state of XRBs.
Interestingly, few of our highly super-Eddington systems with log λ Edd 0.6 do not follow the above trend (see Figure 8). The flatter radio spectra in such cases might be likely due to the jet activity. However, the observed radio to X-ray luminosity ratio for the three highly super-Eddington ratio AGNs is L R /L X 10 −5 , which rules out the possibility of traditional ('BP' or 'BZ') jets (e.g. Blandford & Znajek 1977;Blandford & Payne 1982) and the radio emission is suggested to be dominated by a magnetized corona (aka jet base). The other possibility is the radiation-pressure-driven jets start to dominate in this regime (e.g. Takeuchi et al. 2009;Sądowski & Narayan 2015;Fabrika et al. 2015), where a weak and short jet produces optically thick radio emission and appears like a scaled-down version of radio-loud AGNs. Interestingly, there is a slight positive αλ Edd correlation has been found in radio-loud quasars (see Laor et al. 2019), which may likely support this idea. Extended radio emission was observed in the Galactic microquasar SS 433, which was attributed to a mixture of radiation-pressure-driven jets and radio-emitting outflow (e.g. Okuda et al. 2005). There is only one source (Akn 564) in our sample that shows such quasi-continuous radio ejection, and VLBA images show that B3 1702+457 in our EESBHs sample has a bipolar emission (Doi et al. 2011). More such super-Eddington accreting AGNs with simultaneous multi-band observations are required to confirm these spectral characteristics. Furthermore, future VLBI observation of super-Eddington accreting AGNs will provide more evidence on the different radio-emitting mechanisms as proposed here. 7. SUMMARY The main results of this paper are summarized as follow: 1. The compact and core-dominated radio emission is found in our sample of EESBHs. Most of the radio emission originates within the central few hundred parsecs to one kilo-parsec region, implying that the sources are compact at these scales.
2. In our extremely high Eddington accreting systems, we have estimated the lowest star formation rate surface density required to produce the observed radio emission. This surface density is higher than the maximum value that has been detected in circumnuclear starburst galaxies suggesting that the radio emission comes from AGN activity.
3. A global inverse correlation has been established between the radio loudness R and the Eddington ratio λ Edd from the sub-to the super-Eddington regime. There is no clear demarcation found in this distribution for EESBHs, indicating a continuous suppression of radio emission from the high-to the super-Eddington regime. Super-Eddington accreting AGNs are predominately radio-quiet, but not completely radio quenched.
4. The high-Eddington and mildly super-Eddington AGNs (−0.5 <log λ Edd < 0.6) have a radio to X-ray luminosity ratio L R /L X ∼ 10 −5 -10 −4 and a steep radio spectrum, implying the radio emission was dominated by the transient ejecta where the outflowing corona may be at work. This work was supported by the National Key R&D Program of China (2016YFA0400702) and the National Science Foundation of China (11721303, 11991052). SY thanks for the support from the KIAA-CAS fellowship, which is jointly supported by Peking University and Chinese Academy of Sciences. SY is supported by the Boya fellowship. MFG acknowledges support from the National Science Foundation of China (11873073). This work makes use of public data from NSF's Karl G. Jansky Very Large Array (VLA), the VLA facility is operated by National Radio Astronomy Observatory (NRAO). The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This work makes use of SDSS data, Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. APPENDIX A. NOTES ON VLA RESULTS FOR A FEW SOURCES Akn 564: We have measured a peak flux density of 5.69 ± 0.03 mJy beam −1 at C band, which is consistent with the peak flux density of 5.68 ± 0.02 mJy beam −1 obtained by Berton et al. (2018). The source is resolved into three components at C and X bands, we only take account of the integrated flux density of the central component, which is 5.84 ± 0.03 mJy at C band and 3.18 ± 0.07 mJy at X band. Schmitt et al. (2001) also fit the X-band image with three components, and the inferred integrated flux density of the central component is 3.1 mJy, and being consistent with our results. The size of the radio structure is ∼ 2 arcsec (corresponding to physical linear size of ∼ 4 kpc).
Mrk 684: The VLA A-array observation at C band has the baseline range from 0 to 0.6 Mλ. Here we constrain the baseline to be > 0.04 Mλ, this procedure rejects some diffused emission and results in detection of the central source with a peak flux density of ∼ 0.05 mJy beam −1 , but the signal-to-noise ratio is only 3.3.
IRASF 12397+3333: This source shows diffused emission with a linear extension along the major axis of its host galaxy. This radio structure can be modeled with four Gaussian components. Here we only take into account the flux density from the central compact component as the integrated flux of this source.
IRAS 04416+1215: There is no available VLA data at C band for this source. Alternatively, we estimated a C-band flux density by using the L-band to X-band spectral index (α 8.4 1.4 = −0.91), the resulted 5 GHz flux density is 4.07 mJy. SDSS J010712.04+140845.0 and PG 1115+407: These two sources are not detected at C band. We take the three times rms noise as the upper limit of the flux density.