Radio Scrutiny of the X-ray-Weak Tail of Low-Mass Active Galactic Nuclei:
A Novel Signature of High-Eddington Accretion?

The G&H07 catalog consists of 174 high-confidence, unobscured low-mass AGNs identified based on the presence of broad H$\alpha$ emission in optical spectra from the Sloan Digital Sky Survey (SDSS; York et al. 2000). These AGNs are predominantly (99%) radio-quiet, tend toward high accretion rates (average Eddington ratio $\langle\ell_{\rm Edd}\rangle\approx 0.3$; median value $\ell_{\rm Edd}=0.4$), and are in the mBH range (average mass $\langle M_{\rm BH}\rangle\approx 1.2\times 10^{6}~{}M_{\odot}$; median value $M_{\rm BH}=1.3\times 10^{6}~{}M_{\odot}$). Of these objects, 65 have been observed in the X-ray by Chandra (studies performed by Greene & Ho 2007b; Desroches et al. 2009; Dong et al. 2012a; Gültekin et al. 2014).

Our sample of 18 targets was drawn from the 65 high-confidence G&H07 AGNs with Chandra X-ray coverage, which we limited to 54 objects with $\ell_{\rm Edd}>0.1$. Note that while we adopt the values of $\ell_{\rm Edd}$ reported by G&H07, Plotkin et al. (2016) explore multiple methods for applying bolometric luminosity corrections and suggest that $\ell_{\rm Edd}$ from G&H07 may be systematically underestimated by up to an order of magnitude. This has recently been corroborated by Cho et al. (2023), who showed that G&H07 mBH masses may be overestimated by up to 0.7 dex, as well as through general conclusions that typical broad line region size–luminosity relationships tend to overestimate $M_{\rm BH}$ in high-Eddington, low-luminosity (low-mass) AGNs (e.g., Du et al., 2018; Maithil et al., 2022; Woo et al., 2024). We note this to stress the likelihood that our sample is accreting at near- or super-Eddington rates above $\ell_{\rm Edd}=0.3$.

Of the 54 objects satisfying the above criteria, 7 had already been observed by VLA and examined in radio and X-ray by Gültekin et al. (2014); we designated these 7 AGNs as the “archival” portion of our sample. We found this archival sample to be predominantly X-ray normal (a result of the selection criteria used by Gültekin et al. 2014) per the following description. To define X-ray weakness, we parameterized the ratio of X-ray to UV specific luminosities following standard convention, beginning with the broadband spectral index,

where $l_{\rm 2keV}$ and $l_{\rm 2500}$ are the unabsorbed monochromatic luminosities at rest-frame 2 keV and 2500 Å, respectively (Tananbaum et al., 1979).⁴⁴4As described in Section 2.2, $l_{\rm 2500}$ is derived using a method designed to mitigate the risk of host-galaxy contamination. We also adopted the customary X-ray deviation parameter,

where $\alpha_{\rm ox,qso}$ is the value predicted by the $\alpha_{\rm ox}$–$l_{\rm 2500}$ relationship displayed by broad-line quasars.⁵⁵5For consistency with prior works, we adopt the best-fit $\alpha_{\rm ox,qso}$ relationship given by Eq. (3) of Just et al. (2007). Timlin et al. (2020) suggest an intrinsic scatter of $\pm 0.11$ dex. Finally, we adopted the X-ray weakness factor,

so that X-ray-weaker objects have larger $f_{\rm weak}$ factors. As seen in Figure 1(a), some mBH AGNs could be up to 100$\times$ weaker in X-ray than expected from the established quasar relationship (see also Dong et al. 2012a; Plotkin et al. 2016). The majority of the archival sample is X-ray normal with $f_{\rm weak}<6$ (i.e., $\Delta\alpha_{\rm ox}>-0.3$).

To examine the full range of X-ray weak low-mass AGNs, from the 47 remaining objects we finally selected those with $f_{\rm weak}>6$ (based on values of $\Delta\alpha_{\rm ox}<-0.3$ taken directly from Desroches et al. 2009 and Dong et al. 2012a). This restriction left 11 targets for new VLA radio observations (discussed in Section 2.3); we designated these AGNs as the “new” portion of our sample. Two-sample Kolmogorov–Smirnov (K-S) tests between our combined (archival + new) 18-object selection and the parent remainder show that their optically derived parameters follow similar distributions, including $l_{\rm 2500}$ ($p=0.6$), $M_{\rm BH}$ ($p=0.9$; see Figure 1(b)), and redshift ($p=0.5$; all objects are at $z\lesssim 0.15$, although we note that no explicit redshift cut was applied in our sample selection). The optical–X-ray properties of the 18 objects in our full sample are given in Table 1. Note that we do not tabulate the literature values for $l_{\rm 2keV}$, $\alpha_{\rm ox}$, $\Delta\alpha_{\rm ox}$, or $f_{\rm weak}$, but we instead give updated values derived from re-reduction of the archival X-ray observations, as described below. Throughout this paper, we refer to individual targets using their “GH ID” designation from G&H07.

To ensure a uniform analysis, we downloaded and re-reduced all archival Chandra X-ray observations for our full 18-object sample. Our X-ray results are reported in Table 2. Data were reduced following standard procedures with the Chandra Interactive Analysis of Observations (CIAO) software v14.4, with CALDB v4.9.8 (Fruscione et al., 2006). We first re-processed the data with chandra_repro and confirmed that there were no periods of background flaring during our observations. Next we ran wavdetect on each observation to search for X-ray sources in each galaxy (setting the wavelet scales parameter to “1 2 4 6 8 12 16 24 32” and sigthresh to 10^-6). X-ray sources were found in the nuclei of 14 galaxies, with a wide range of counts reported by wavdetect, 6–1400 over 0.5–7.0 keV. For these 14 X-ray sources, we measured net full-band (0.5–7.0 keV) count rates using srcflux, adopting the default source aperture to enclose 90% of the point-spread-function (psf) at 1 keV (aperture corrections were performed in srcflux using psfmethod=arfcorr). For the four non-detections, after confirming from visual inspection that no X-ray source was present, we measured the local background near the nucleus of each galaxy. We then placed upper limits on the net count rate at the 99% confidence level, assuming Poisson statistics in the presence of background (Kraft et al., 1991).

For X-ray sources with $>$10 full-band counts (12 objects, all of which are also detected in the hard 2–7 keV band), we extracted spectra using specextract over 5$\arcsec$ apertures, and we performed spectral fitting using the Interactive Spectral Interpretation System (ISIS) v1.6.2 (Houck & Denicola, 2000). Since the majority of our spectra (9/12) have relatively few counts ($<$300), we binned each spectrum to 1 count per bin and performed our fitting using Cash statistics (Cash, 1979). We fit each spectrum using an absorbed power-law model, tbabs*ztbabs*powerlaw, where tbabs was frozen to the Galactic column densities ($N_{\rm H,gal}$) from the HI4PI survey (HI4PI Collaboration et al., 2016), and ztbabs was left as a free parameter to estimate intrinsic absorption ($N_{\rm H,fit}$) at the redshift of each object. We did not attempt joint spectral fitting because parameters like $\Gamma$ could vary on a per-object basis. This provided acceptable fits for 11/12 spectra, and we did not pursue more complicated models for these 11 spectra. For the final spectrum, GH 69 (ObsID 13858; also see Appendix A), we found the addition of a multi-temperature accretion disk was sufficient, tbabs*ztbabs*(diskbb+powerlaw). From these 12 spectral fits, we calculated unabsorbed model fluxes over the 2–10 keV hard X-ray band using the cflux convolution model in ISIS.

For objects with fewer than 10 full-band counts (including non-detections), we used the full-band count rates (or limits) to estimate unabsorbed hard-band fluxes (or upper limits) in the Portable, Interactive Multi-Mission Simulator (PIMMS).⁶⁶6We utilized the PIMMS v4.12a web tool hosted by the Chandra X-Ray Center at https://cxc.harvard.edu/toolkit/pimms.jsp. We assumed a power-law model with $\Gamma=1.9$ (from the weighted mean of the best-fit $\Gamma$ values from the 12 spectral fits), and we adopted $N_{\rm H,gal}$ from the HI4PI survey.

Finally, we re-estimated the optical–X-ray properties of our sample using the following methods. We found $l_{\rm 2keV}$ via hard-band (2–10 keV) fluxes or limits in order to characterize hard (coronal) X-ray emission and avoid possible contamination from a soft excess (e.g., Done et al., 2012; Dong et al., 2012a; Ludlam et al., 2015). Where available, we used the best-fit values of $\Gamma$ in our calculations, otherwise we used the weighted sample mean ($\Gamma=1.9$). Following prior studies (e.g., see Section 3.2 of Plotkin et al. 2016), we used H$\alpha$ line luminosities ($L_{\rm H\alpha}$) from G&H07 along with the relationship $L_{\rm H\alpha}=5.25\times 10^{42}\,(L_{\rm 5100}/10^{44}\ {\rm erg~{}s}^{-1})^{1.157}$ erg s^-1 from Greene & Ho (2005) to avoid possible host-galaxy contamination and find the AGN continuum luminosity at 5100 Å ($L_{\rm 5100}$). We then assumed a power-law continuum following the form $f_{\nu}\propto\nu^{-0.44}$ (Vanden Berk et al., 2001) to estimate $l_{\rm 2500}$. We finally calculated $\alpha_{\rm ox}$, $\Delta\alpha_{\rm ox}$, and $f_{\rm weak}$ following Eqs. (1–3), and as previously noted, we report these values in Table 1 instead of the prior literature values.⁷⁷7Our re-reduced values are generally consistent with those from the literature, with an average change in $\alpha_{\rm ox}$ of only 0.06. However, the detection status of three objects changed: two prior non-detections are now redetermined as faint detections and one prior detection is redetermined as a non-detection (see Appendix A).

The 11 new radio observations were completed in X-band (utilizing the 3-bit samplers to cover 8–12 GHz) with the VLA in B configuration under Project ID 20A-292 (PI Plotkin) from 2020 June 23 to July 11. Observation details are tabulated in Table 3. Each observation included a scan of a standard flux/bandpass calibrator, followed by scans of the science target interleaved between scans of a secondary phase calibrator.

We downloaded the uncalibrated Science Data Model-Binary Data Format (SDM-BDF) dataset for each target using the National Radio Astronomy Observatory (NRAO) Archive Access Tool and performed standard automated flagging, calibration, and weighting using the Common Astronomy Software Applications (CASA; CASA Team et al. 2022) package version 6.4.1 with VLA calibration pipeline version 2022.2.0.64. After reviewing the calibrated measurement set, we performed additional flagging as necessary.

Data were imaged using the CASA task tclean with a typical resolution (synthesized beamwidth) $\approx$ 0$\farcs$50. We utilized the Multi-Scale, Multi-Term Multi-Frequency Synthesis deconvolution algorithm (deconvolver = ‘mtmfs’; Rau & Cornwell 2011) with nterms = 2 and, for objects that appeared to exhibit extended emission, scales = [0,5,15]. To suppress sidelobes, we used Briggs weighting (Briggs, 1995) with robust parameter values ranging from 0.5 to 2.0 depending on field crowding and sensitivity needs for fainter targets. We stopped the cleaning process at a threshold of 2$\times$ the background rms noise ($\sigma_{\rm rms}$) using nsigma = 2.0.

The 7 archival radio observations were gathered from Project ID SD0129, completed 2012 December 13–16, and Project ID 12B-064, completed 2012 October 25 through 2013 January 6 (see Gültekin et al. 2014). These observations were performed in X-band (utilizing the 8-bit samplers to cover 8–10 GHz) with the VLA in A configuration. For consistency with the new VLA sample, we downloaded uncalibrated SDM-BDF datasets and re-imaged the archival objects (including flagging, calibration, reduction, and cleaning) using the same CASA version and methods described above. We found a typical resolution $\approx$ 0$\farcs$20.

In each new and archival image, we estimated $\sigma_{\rm rms}$ from a source-free sky region in each image. $\sigma_{\rm rms}$ was found to be consistent with noise predicted by the VLA Exposure Calculator tool. We then used the CASA task imfit to measure flux densities at the center of each target galaxy by fitting at least one two-dimensional Gaussian. For target objects with $>3\sigma_{\rm rms}$ detections, we extracted peak and integrated flux densities, central frequencies, detection coordinates, and beam-deconvolved component sizes (if extended) from imfit. We adopted the associated uncertainties as reported by imfit. For non-detections, we report upper flux density limits at the $3\sigma_{\rm rms}$ level. The radio properties of our sample are listed in Table 3. We provide the full set of VLA images in Appendix A.

We also estimated in-band spectral indices for all detected objects. First, we divided the full frequency bandpass of each target evenly into four spectral window bins ($\approx$ 1 GHz per bin for the new targets and $\approx$ 0.5 GHz per bin for the archival targets). Following the procedures outlined above, we produced four new partial-bandwidth images and emission fits per target, and measured the flux density of the target in each image. We then weighted the central-frequency peak flux densities of these four new measurements by their $1\sigma$ errors and used them to fit (via a $\chi^{2}$ minimization routine) a model power-law continuum of the form $S_{\nu}\propto\nu^{\alpha_{\rm r}}$, where $S_{\nu}$ is the flux density and $\alpha_{\rm r}$ is the reported X-band spectral index. To estimate uncertainty in the spectral indices, we used a Monte Carlo algorithm to generate a set of 1,000 mock spectra. For each spectral window bin, we varied the flux density via random sampling of a normal distribution (where the mean and standard deviation were set respectively to the observed flux density and uncertainty), and the central frequency via random sampling of a uniform distribution (defined by the bin bandwidth). Finally, we assigned $\pm\,1\sigma$ errors to $\alpha_{\rm r}$ by re-fitting a power-law model to each simulated spectrum and finding the standard deviation of the resulting spectral index distributions.

To provide a visual basis for comparison, we include in some figures a sample of radio-quiet, high-Eddington ($\ell_{\rm Edd}\gtrsim 0.3$) narrow-line Seyfert 1 (NLS1) galaxies from Yang et al. (2020), with a comparable redshift range ($z<0.17$) but a slightly higher average mass ($\langle\log M_{\rm BH}\rangle\approx 6.8$). For this comparison sample, we adopt published measurements: 5 GHz radio luminosities from Yang et al. (2020) along with 2–10 keV X-ray fluxes and host-subtracted AGN continuum fluxes at 5100 Å as reported by Wang et al. (2013) (see additional references therein). We calculate $\Delta\alpha_{\rm ox}$ as described in Section 2.2. This comparison sample is almost entirely X-ray normal and is shown by Yang et al. (2020) to display $L_{\rm R}/L_{\rm X}\sim 10^{-5}$–$10^{-4}$.

NLS1s have been found to occupy extreme positions in the Boroson & Green (1992) Eigenvector 1 and other related parameter spaces (e.g., Sulentic et al., 2000; Grupe, 2004). They are thought to possess low-mass SMBHs ($M_{\rm BH}\sim 10^{6}$–$10^{8}~{}M_{\odot}$) at high (near- or super-Eddington) accretion rates, and they often display extreme X-ray properties such as weakness, variability, and spectral complexity (see, e.g., Gallo 2018 and references therein). The overlap of G&H07 AGNs with some NLS1 characteristics has been investigated to moderate extent (see, e.g., G&H07; Greene & Ho 2007b; Desroches et al. 2009; Dong et al. 2012a), but it is still unclear how distinct the two populations are. In summary, some features of classical NLS1s are shared with the G&H07 sample, such as relatively narrow broad-line emission, high Eddington ratio (although for G&H07 this is likely a selection effect), and predominant radio-quietness. However, G&H07 objects also tend to have different optical Fe II and [O III] strength distributions and flatter $\alpha_{\rm ox}$ on average (although this may be similar specifically to super-Eddington NLS1s). While NLS1s are not key to this paper, we discuss them in the following sections when relevant.

Our primary interest lies in evaluating the relationship between radio and X-ray emission. Following Terashima & Wilson (2003), we define the ratio of radio to X-ray luminosities as $\mathcal{R}_{\rm X}=L_{\rm R}/L_{\rm X}$, where $L_{\rm R}$ is the radio luminosity $\nu L_{\nu}$ at $5$ GHz and $L_{\rm X}$ is the 2–10 keV X-ray luminosity. These quantities are listed in Table 4 along with $\mathcal{R}_{\rm O}$ and, for convenience, $\Delta\alpha_{\rm ox}$. Figure 2(a) shows where our sample resides in $L_{\rm R}$–$L_{\rm X}$ space. We find that the mean value of $\mathcal{R}_{\rm X}$ in our sample of G&H07 AGNs ($\langle\log\mathcal{R}_{\rm X}\rangle=-4.1\pm 0.7$; the quoted error is the standard deviation) is higher than the comparison sample ($\langle\log\mathcal{R}_{\rm X}\rangle=-4.7\pm 0.8$) by $\sim$ 0.6 dex, although this does not appear particularly significant given the range of uncertainty.

Comparison with $\Delta\alpha_{\rm ox}$ allows us to better evaluate $\mathcal{R}_{\rm X}$ in terms of X-ray weakness, as shown in Figure 2(b). Following the Monte Carlo approach described below (in order to account for error and non-detections),¹⁰¹⁰10The tests available in the Astronomical SURVival Statistics package (ASURV; Isobe et al. 1986; Lavalley et al. 1992) assume that censored data follow intrinsic distributions consistent with the non-censored data. This may not be a realistic assumption in our case, so we adopt a Monte Carlo approach instead. we observe a tentative anticorrelation between $\log\mathcal{R}_{\rm X}$ and $\Delta\alpha_{\rm ox}$ in our sample, finding a Kendall’s Tau rank correlation coefficient $\tau=-0.48\pm 0.05$ with a null-hypothesis probability $p_{\rm null}=0.008\pm 0.006$. A linear regression gives slope and intercept coefficients $m=-2.5\pm 0.3$, $b=-4.8\pm 0.1$ (where it is assumed $\log\mathcal{R}_{\rm X}=m\Delta\alpha_{\rm ox}+b$; solid orange line in Figure 2(b)).

To find the above result, we use the following Monte Carlo scheme to generate $10^{4}$ iterations of mock $\log\mathcal{R}_{\rm X}$ and $\Delta\alpha_{\rm ox}$ measurements for each object. For radio, we randomly vary the X-band radio flux density and spectral index $\alpha_{\rm r}$ via normal distributions (the mean and standard deviation for the former are the observed flux density and uncertainty, while for the latter we use either the measured slope and error or sample weighted mean and error according to the rule defined in Section 3) before recalculating $L_{\rm R}$. For the X-ray, we similarly vary the flux and photon index $\Gamma$ via normal distributions (the mean and standard deviation are again the observed flux or photon index and the associated uncertainty) before recalculating $L_{\rm X}$ and $\Delta\alpha_{\rm ox}$. For radio or X-ray non-detections, flux values are drawn randomly from a uniform distribution between the reported upper limit and 50% of the upper limit, and we incorporate uncertainty on our assumed $\alpha_{\rm r}$ or $\Gamma$ values by using the sample weighted mean and standard deviation to define the normal distribution for their random variation. For each Monte Carlo iteration, we perform a linear fit via orthogonal distance regression (weighted by uncertainty) and Kendall’s Tau correlation test; the results reported above give the mean and $\pm\,1\sigma$ error (found from the 16^th and 84^th percentile values) of the distributions from these tests. We also assign $\pm\,1\sigma$ error bars to each data point in Figure 2(b) by finding the 16^th and 84^th percentile values of each object’s mock $\mathcal{R}_{\rm X}$ and $\Delta\alpha_{\rm ox}$ distributions.

To better determine the impact of certain choices we have made in defining $\mathcal{R}_{\rm X}$, we produce variants of the $\log\mathcal{R}_{\rm X}$–$\Delta\alpha_{\rm ox}$ fit as follows. While these alternative results are largely consistent with the above error analysis, we describe them briefly here and report them along with the primary result in Table 5 for completeness. In the primary fit reported above (v.1 in Table 5), $L_{\rm R}$ is found using estimated 5 GHz flux densities; to eliminate the error possible from extrapolating along an uncertain spectral index, we produce an alternative and slightly more robust fit (v.2) with $L_{\rm R}$ taken at the observed central frequency ($\nu_{\rm cent}\sim$ 9–10 GHz depending on target; see Table 3). We also produce a version of this fit (v.3) using integrated core-component flux densities to show that our results are not significantly influenced by our primary choice of peak flux densities, as well as a version (v.4) that excludes all 7 objects with radio and/or X-ray non-detections (although for this last we point out the small remaining sample size and high uncertainty).

While the above simulations suggest that none of the data points in Figure 2(b) has substantial error (apart from non-detections), our ability to fully account for uncertainty is likely complicated by other phenomena or relationships (e.g., co-dependence of $\Gamma$ and $\Delta\alpha_{\rm ox}$). The influence of X-ray variability is particularly challenging to assess, as variability estimates are only available for a few of our targets (discussed in Section 4.2.1). However, we do attempt a rudimentary illustration of the effect of X-ray variability. We follow the Monte Carlo scheme described above for X-ray uncertainty, but incorporate an additional random variation via normal distribution where both the mean and the standard deviation are defined by the mock flux value. The green point and diagonal bars in the upper right of Figure 2(b) illustrate the average effect X-ray variability might have on individual uncertainties.

Finally, we comment briefly on whether there may be a redshift or evolutionary dependence in our results. Rankine et al. (2024) show that the X-ray luminosity distribution across redshift may have an impact on $\alpha_{\rm ox}$ (at least in SDSS quasars, with $M_{\rm BH}$ potentially being the driver of the evolution). While this may be the reason for finding generally flatter values of $\alpha_{\rm ox}$ in mBHs (e.g., Desroches et al., 2009; Baldassare et al., 2017), we stress that our sample covers a fairly narrow range in both mass and redshift. We find no correlation or dependency between combinations of redshift or mass and $\mathcal{R}_{\rm X}$, $\mathcal{R}_{\rm O}$, or $\Delta\alpha_{\rm ox}$ in our sample ($p_{\rm null}\sim 0.6$).

Before discussing possible sources of X-ray weakness, we investigate how our results may be affected by the source of radio emission. While $\mathcal{R}_{\rm O}$-loud AGNs are typically seen to be dominated by jets and diffuse radio lobes, $\mathcal{R}_{\rm O}$-quiet AGNs often show signatures of a number of alternative radio components at various physical scales (e.g., Panessa et al. 2019 and references therein; McCaffrey et al. 2022; Yang et al. 2022, 2024). Diagnosis of the dominant emission mechanism can sometimes be informed by the radio spectral index $\alpha_{\rm r}$. At the physical scale and frequency pertinent to our observations, a compact corona or jet base is expected to have $\alpha_{\rm r}\gtrsim-0.5$ from optically thick, self-absorbed synchrotron emission; non-collimated outflows such as disk winds may produce $\alpha_{\rm r}\approx-0.1$ from optically thin free-free emission; shocks related to jet lobes or disk winds will show $\alpha_{\rm r}\lesssim-0.5$ from optically thin synchrotron emission; and star formation will also often result in $\alpha_{\rm r}\lesssim-0.5$. For our purposes, we define X-band spectral slopes $\alpha_{\rm r}>0$ as “inverted,” $-0.5\leq\alpha_{\rm r}\leq 0$ as “flat,” and $\alpha_{\rm r}<-0.5$ as “steep.”

For $\mathcal{R}_{\rm O}$-quiet, supermassive AGNs, the approximation that $\mathcal{R}_{\rm X}\sim$ constant seems to hold even at kpc scales (e.g., from their results at 1.4 GHz, Panessa et al. 2015 suggest that even the total, large-scale radio emission is linked to accretion-related X-ray emission), yet the exact value of the relationship and the extent of its scatter will be at least partly dependent on the dominant radio emission mechanism being observed (see, e.g., Laor & Behar, 2008; Behar et al., 2015; Yang et al., 2020; Panessa et al., 2022). We also reiterate that the exact nature of the relationship is not yet understood. Chen et al. (2023) examined Very Long Baseline Array observations of a sample of $\mathcal{R}_{\rm O}$-quiet quasars in which radio emission at physical scales $\lesssim 0.4$ pc displays generally flat/inverted spectral slopes and follows $\log\mathcal{R}_{\rm X}\sim-6$ with a tight correlation, and they noted that VLA observations of the same sample (at larger physical scales) show generally steeper slopes and follow $\log\mathcal{R}_{\rm X}\sim-5$ with more scatter in the correlation. As they explain, this is consistent with small-scale (sub-pc) nuclear emission being produced by a compact, optically thick core component (e.g., corona or jet base), and larger-scale (sub-kpc) nuclear emission being associated with a variety of possible extended components. Similar results recently obtained by Wang et al. (2023) and Shuvo et al. (2024) appear to corroborate this interpretation, suggesting that an anticorrelation or bimodal relationship between $\alpha_{\rm r}$ and $\mathcal{R}_{\rm X}$ might be observable for X-ray-normal objects. Additionally, Laor et al. (2019) and Alhosani et al. (2022) show that radio slope and compactness are anticorrelated with Eddington ratio (i.e., they find compact emission with flat/inverted slopes at $\ell_{\rm Edd}<0.3$, and less-compact emission with steep slopes at $\ell_{\rm Edd}>0.3$) in $\mathcal{R}_{\rm O}$-quiet AGNs. These observations are consistent with a unified accretion model in which nuclear outflows are more likely at high $\ell_{\rm Edd}$ (e.g., Giustini & Proga, 2019).

From all of the above, we can infer that the scatter of $\mathcal{R}_{\rm X}$ in our sample may be partly due to varied nuclear radio emission mechanisms. Indeed, as with many of the above-noted studies, our sample spans a range of roughly 2 dex in $\mathcal{R}_{\rm X}$ (see Figure 2(a)). While it may be possible for X-ray-normal populations to show an $\alpha_{\rm r}$–$\mathcal{R}_{\rm X}$ relationship as described above, in an X-ray shielding scenario it is likely that the X-ray-weak objects would deviate from such a relationship, with some compact, flat/inverted-slope radio sources moving into the higher $\mathcal{R}_{\rm X}$ range. Unfortunately, we lack a sufficient number of observations with the sensitivity or resolution needed to make statistical statements regarding the radio emission mechanisms of our sample. As our sample spans a fairly narrow range of high $\ell_{\rm Edd}$, it is also probably not meaningful to attempt to use $\ell_{\rm Edd}$ as a surrogate for $\alpha_{\rm r}$ (and for completeness we note that a correlation between $\mathcal{R}_{\rm X}$ and $\ell_{\rm Edd}$ is not found; $p_{\rm null}=0.5\pm 0.2$). More sensitive and higher resolution follow-up observations of the nuclear radio emission (from, for example, a next-generation VLA; e.g., Selina et al. 2018; Plotkin & Reines 2018) will allow us to better understand the dominant radio emission mechanism and its relationship with X-ray emission (e.g., Behar et al., 2015, 2020; Chen et al., 2022).

However, the radio emission mechanism is unlikely to be directly responsible for the observed $\mathcal{R}_{\rm X}$–$\Delta\alpha_{\rm ox}$ anticorrelation because, as we discuss below, it does not appear that $\mathcal{R}_{\rm X}$ loudness in our sample can be explained simply by excess or boosted radio emission. First, we note that the $\log\mathcal{R}_{\rm X}$–$\Delta\alpha_{\rm ox}$ fit produced using integrated radio flux densities (v.3 in Table 5) is consistent with the versions derived from peak flux densities, which implies that while a generally larger observed radio scale for the X-ray-weak objects (from lower resolving power in VLA B configuration) may introduce a risk of observational bias in radio emission mechanisms, such a bias is unlikely to be responsible for the anticorrelation.

Next, the histograms shown in Figure 3(a) compare measures of radio loudness via the distributions of $\mathcal{R}_{\rm X}$ and $\mathcal{R}_{\rm O}$, and they suggest that a majority of our sample are comparatively weak in X-rays. Expanding on this observation, Figure 3(b) explores our sample on the $\log\mathcal{R}_{\rm O}$–$\log\mathcal{R}_{\rm X}$ plane, again with the $\mathcal{R}_{\rm O}$-quiet NLS1s from Yang et al. (2020) for visual comparison. The vertical dash-dot lines denote the conventional $\mathcal{R}_{\rm O}$-quiet (teal; $\log\mathcal{R}_{\rm O}=1$) and $\mathcal{R}_{\rm O}$-loud (orange; $\log\mathcal{R}_{\rm O}=2.48$) demarcations, and we refer to the range between as $\mathcal{R}_{\rm O}$-intermediate. The horizontal dashed lines similarly show our adopted $\mathcal{R}_{\rm X}$-quiet (teal; $\log\mathcal{R}_{\rm X}=-4.5$) and $\mathcal{R}_{\rm X}$-loud (orange; $\log\mathcal{R}_{\rm X}=-2.75$) demarcations (based on empirical boundaries suggested by Terashima & Wilson 2003 and Panessa et al. 2007), and we refer to the range between as $\mathcal{R}_{\rm X}$-intermediate. Much of our sample resides in a clearly $\mathcal{R}_{\rm X}$-intermediate space, and, importantly, the X-ray weakest ($f_{\rm weak}\gtrsim 10$, shown by pink symbols) appear to have $\mathcal{R}_{\rm X}$ values higher than typical compared to their $\mathcal{R}_{\rm O}$ (see Figure 3(b) caption text pertaining to the yellow shaded band). We also find that excluding the two $\mathcal{R}_{\rm O}$-intermediate objects from our $\mathcal{R}_{\rm X}$–$\Delta\alpha_{\rm ox}$ analysis still gives a consistent fit result (v.5 in Table 5).

Furthermore, we do not find evidence of a relationship between $\Delta\alpha_{\rm ox}$ and any other parameter that, in the context of the above discussion, could be linked to the radio emission mechanism, including $r_{\rm maj}$, $\mathcal{R}_{\rm O}$, $\ell_{\rm Edd}$, or $\alpha_{\rm r}$ (acknowledging limited dynamic range in several parameters, and also very small number statistics for the $\alpha_{\rm r}$ subsample).

From the above analysis, we conclude that the increase in $\mathcal{R}_{\rm X}$ loudness observed in our sample does not appear to be an artifact of enhanced radio loudness, and is instead related to genuine X-ray weakness. We defer discussion of X-ray variability to Section 4.2.1 and assume for the moment that our sample is not particularly variable. In the case of intrinsic X-ray weakness, the previously described empirical correlations suggest that $L_{\rm R}$ and $L_{\rm X}$ should scale in tandem, and that we should still generally expect the ratio $\mathcal{R}_{\rm X}$ to remain constant, independent of $\Delta\alpha_{\rm ox}$ (although we note that this expectation remains to be confirmed, with the challenge being identification of a purely intrinsically X-ray-weak sample with radio coverage).

On the other hand, if there is a compact shielding medium that obscures X-rays along our line of sight (while other emission regions remain unobscured) then we should see $\mathcal{R}_{\rm X}$ anticorrelate with $\Delta\alpha_{\rm ox}$. To examine what might be observed in this case, we construct a simple prediction model,

where $f_{\rm weak}$ is derived from $\Delta\alpha_{\rm ox}$ as described in Section 2.2, and $\mathcal{R}_{\rm X,exp}$ is the value of $\mathcal{R}_{\rm X}$ that would otherwise be expected for the object from an X-ray-unobscured line of sight. Converting to $\log\mathcal{R}_{\rm X}$–$\Delta\alpha_{\rm ox}$ space,

We see that the predicted slope is $-2.6$, which is consistent with the best-fit slope from the linear regression to our sample ($-2.5\pm 0.3$; see Section 3.1). Our sample’s best-fit intercept value is $-4.8\pm 0.1$. However, the expected (unshielded) value of $\mathcal{R}_{\rm X,exp}$ for each AGN is likely to be partly dependent on the dominant radio emission mechanism (as discussed in Section 4.1), so we adopt $\log\mathcal{R}_{\rm X,exp}=-4.7$ from the Yang et al. (2020) sample mean and include a $\pm$ 1 dex scatter. The yellow-shaded area in Figure 2(b) illustrates this simple model, and while it is not conclusive, it appears at least qualitatively consistent with our observations. More sensitive observations of our X-ray-weakest targets and larger samples with X-ray-weak, high-Eddington AGNs will help overcome small-sample statistical limitations, improve dynamic range, and better test the prediction.

While it is possible to simply interpret the observed anticorrelation as being induced by the fact that $\Delta\alpha_{\rm ox}$ and $\mathcal{R}_{\rm X}$ are dependent on the X-ray luminosity and its inverse (respectively), we stress again that we take the $\sim$ constant $L_{\rm R}/L_{\rm X}$ empirical relationship to suggest a null hypothesis of no correlation expected between $\Delta\alpha_{\rm ox}$ and $\mathcal{R}_{\rm X}$. That is, for whatever reason, the mBH AGNs that appear X-ray weak relative to the optical do not follow the expected $L_{\rm R}/L_{\rm X}$ behavior derived from $\mathcal{R}_{\rm O}$-quiet, X-ray-normal AGNs with X-ray luminosities spanning many orders of magnitude.

Indeed, other works have previously discussed the possibility of X-ray shielding/absorption in the context of $\mathcal{R}_{\rm X}$ analysis involving higher-mass AGNs. For example, in their examination of the drivers of radio spectral slopes in quasars, Laor et al. (2019) note that a few of the objects in their sample have higher (X-ray-weaker) values of $\mathcal{R}_{\rm X}$ and also show significant signs of X-ray absorption, including one confirmed broad absorption line (BAL) quasar.¹⁴¹⁴14By convention, narrow absorption lines (NALs) typically have FWHM $<500$ km s^-1, broad absorption lines (BALs) have FWHM $>2000$ km s^-1, and mini-BALs occupy the range between (see, e.g., Weymann et al. 1981; Hamann & Sabra 2004 and references therein; Gibson et al. 2009). BAL quasars are often seen to be X-ray weak, an observation historically attributed to orientation-based obscuration (e.g., Gallagher et al., 2001, 2002, 2006), and recent studies have further linked BAL activity and X-ray weakness/variability in high-Eddington BAL quasars to strong outflows (e.g., Yi et al., 2019a, b, 2020; Rankine et al., 2020; Saez et al., 2021). Likewise, Zuther et al. (2012) find that BAL quasars which are radio-quiet per $\mathcal{R}_{\rm O}$ still have high observed $\mathcal{R}_{\rm X}$, in a manner similar to the X-ray-weak portion of our sample (see their Figure 5 in comparison to our Figure 3(b)). Still, we cannot ignore the possibility of intrinsic X-ray weakness in some objects (e.g., Liu et al., 2018a; Vito et al., 2018).

NLS1s are often found to possess complex X-ray spectra along with rapid and large-amplitude variability, which may suggest dynamic, possibly multi-component coronae that fluctuate with changes in accretion flow (e.g., Gallo 2018 and references therein). Such properties imply that NLS1s which are X-ray weak are often intrinsically so. However, samples of NLS1s have been found with (albeit inconclusive) suggestions of X-ray absorption. Williams et al. (2004) examined a sample of NLS1s in which the X-ray weakest objects tend to display flatter X-ray spectra with no sign of absorption in their optical spectra (which may suggest X-rays are preferentially obscured). Likewise, Gallo (2006) divided a sample of NLS1s into objects with “simple” (i.e., power-law-consistent) and “complex” X-ray spectra; they found that the complex-spectrum objects also tend to be significantly X-ray weaker with signs of absorption or reflection (as well as stronger optical Fe II emission, which may indicate higher accretion rates in the context of Eigenvector 1).

Therefore, while we do not expect the existing X-ray data to offer a “smoking gun,” we still search for additional signs of X-ray shielding in our sample by comparing the observed X-ray spectrum and $\Delta\alpha_{\rm ox}$. The shielding medium should modify the intrinsic spectrum associated with the corona, hardening (flattening) it as soft X-rays are more likely to be absorbed, scattered, and/or reflected away (see, e.g., Gibson et al. 2009, particularly their Figure 7 and related discussion on X-ray absorption in BAL quasars). Some variance is expected in the relationship, as the extent to which the spectrum is modified will depend on the covering factor and column density of the obscuring medium. Figure 5 examines the dependence of the observed X-ray photon index, $\Gamma$, on $\Delta\alpha_{\rm ox}$. We caution that not only is there a higher degree of uncertainty for the X-ray weaker objects, but this plot also does not incorporate our X-ray weakest, non-detected objects. Additionally, we find one target (GH47, tagged by name in Figure 5) that may be an outlier: while this source does not appear to be extremely X-ray weak, the shape of its Chandra spectrum is relatively hard ($\Gamma=0.36$), which may imply a high level of absorption.¹⁵¹⁵15The Chandra observation of GH 47 has too few counts to quantify absorption directly from the data, but Ludlam et al. (2015) examined this object using multi-epoch XMM-Newton observations and found it to have a 0.3–10 keV spectrum that may suggest an obscured, Type 2 AGN (see Appendix A).

Even given the above caveats, it appears plausible that the X-ray-weaker objects in our sample show generally harder (flatter) X-ray spectra. The solid orange line in Figure 5 shows a linear fit via orthogonal distance regression (weighted by uncertainty), excluding GH 47. We find only a hint of a correlation at the $p_{\rm null}=0.087$ level from Kendall’s Tau. However, Plotkin et al. (2016) performed a photon stacking analysis of X-ray non-detected, primarily moderate- to high-Eddington G&H07 AGNs and found a flat, albeit highly uncertain, stacked spectrum ($\Gamma=1.0^{+1.1}_{-0.5}$). Following the linear fit on Figure 5 down to our X-ray-weakest, non-detected targets’ upper limits on $\Delta\alpha_{\rm ox}$ ($\lesssim-0.7$) leads to values of $\Gamma\sim 1.0$, consistent with the Plotkin et al. (2016) analysis.¹⁶¹⁶16While this stacking analysis may conflict with our assumption of the weighted sample mean for our X-ray non-detected objects ($\Gamma=1.9$; see Section 2.2), we have already included uncertainty on $\Gamma$ in our error analysis (see Section 3) and found that it is unlikely to significantly alter our main result. Furthermore, while we could use the fit found here to refine our assumed $\Gamma$ for non-detections based on their $\Delta\alpha_{\rm ox}$, the issue becomes circular as our $\Delta\alpha_{\rm ox}$ calculations are already dependent on $\Gamma$. While we caution against drawing strong conclusions from this assessment, we do consider it suggestive of the X-ray-weak objects having, on average, spectral characteristics consistent with X-ray obscuration. More sensitive observations are needed to better determine the sample’s X-ray spectral properties.

Finally, we find that the available observations already imply a relative limit on the radial extent of a physical X-ray shielding region. Ludlam et al. (2015) performed X-ray variability analysis of a subsample of G&H07 mBH AGNs (including three objects overlapping ours: GH 47, GH 211, and GH 213) and found short-term variability (timescales of $\sim$ 100–1000 s) suggesting X-ray emission on scales $\sim$ 0.2–2 au. The X-ray-obscuring medium must reside outside this range, yet the G&H07 sample was identified as optically unobscured from SDSS spectra. Furthermore, Laor & Behar (2008) (see Section 3.5.2 therein) estimate that even the smallest possible scale emission predicted at 5 GHz, e.g., from a compact radio corona or jet base, should originate from a region $\gtrsim 100$ times the extent of the X-ray emitting core. The tentative $\mathcal{R}_{\rm X}$–$\Delta\alpha_{\rm ox}$ anticorrelation itself disfavors shielding of nuclear radio emission. The putative shielding region therefore appears to be compact, affecting coronal X-ray emission but neither optical nor radio. Critically, however, no UV spectral observations exist for any significant portion of the G&H07 sample at the time of writing (to our knowledge). Obtaining such data will allow us to better constrain the spectral energy distribution of these AGNs and further eliminate suspicion of a more extended obscuring medium in the X-ray weak population.