Accessing Intermediate-mass Black Holes in 728 Globular Star Clusters in NGC 4472

At the location of each GC candidate, we correct the radio-detection threshold for primary-beam attenuation at 2 cm and then convert the corrected detection threshold to a radio-luminosity threshold L_2cm at NGC 4472's distance. As Figure 2 demonstrates, the radio-luminosity thresholds reach L_{2 cm} ∼ 10^32.7–33.3 erg s⁻¹.

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To explore the prospects for deeper detections, we also wish to stack the corrected radio-detection thresholds of GC candidates with similar stellar masses. Techniques for stacking extragalactic GC populations have been developed using VLA images of nearby galaxies (Wrobel et al. 2015, 2016; Wrobel & Nyland 2020). Here, we opt to segregate NGC 4472's individual GC candidates into bins of width 0.5 dex in stellar mass M_⋆. For each bin, we evaluate its number of GC candidates, median ${M}_{\star }^{\mathrm{stack}}$, weighted-mean detection threshold ${S}_{2\,\mathrm{cm}}^{\mathrm{stack}}$, and associated radio-luminosity threshold ${L}_{2\,\mathrm{cm}}^{\mathrm{stack}}$ (Table 1). We caution that such stacking implicitly assumes a high occupation fraction of IMBHs in each bin's GCs. Meeting this condition might be easier in the high-stellar-mass bins: GC evolutionary models suggest that the more massive a GC, the more likely it is to retain a putative IMBH against gravitational wave or Newtonian recoils (e.g., Holley-Bockelmann et al. 2008; Fragione et al. 2018).

Table 1. Stacked GC Candidates in NGC 4472

Stacked Attribute	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6
Number of GC candidates	5	83	242	264	114	19
Median GC Stellar Mass, Log${}_{10}\,{M}_{\star }^{\mathrm{stack}}$ (M⊙)	4.44	4.84	5.29	5.70	6.16	6.72
Flux Density Threshold, ${S}_{2\,\mathrm{cm}}^{\mathrm{stack}}$ (μJy beam−1)	0.122	0.018	0.009	0.008	0.012	0.031
Radio-luminosity Threshold, Log${}_{10}\,{L}_{2\,\mathrm{cm}}^{\mathrm{stack}}$ (erg s−1)	32.82	32.00	31.71	31.67	31.83	32.23
IMBH Mass Threshold, Fiducial Bondi, Log${}_{10}\,{M}_{\mathrm{IMBH}}^{{\rm{B}},\mathrm{stack}}$ (M⊙)	4.91	4.61	4.50	4.48	4.54	4.69

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Regarding possible radio contaminants behind NGC 4472's GC candidates, simulated source counts near 2 cm suggest that star-forming galaxies will dominate at μJy levels (Wilman et al. 2008). As such galaxies have steep spectra and finite sizes, they will not be mistaken for IMBHs that have flat spectra and are point-like.

Regarding possible radio contaminants within NGC 4472's GC candidates, studies of stellar-mass X-ray binaries (XRBs) in the Milky Way show that their persistent radio luminosities at 6 cm top out at about 10^31.3 erg s⁻¹ for BH systems (Bahramian et al. 2020). Scaling a flat-spectrum contaminant from 6 cm to 2 cm in an individual GC, this translates to a radio luminosity L_{2 cm} ∼ 10^31.8 erg s⁻¹, well below the radio-luminosity thresholds of individual GC candidates in NGC 4472 (Figure 2). If every GC harbored such an extreme contaminant, a signal could just emerge above the deepest stacked thresholds ${L}_{2\,\mathrm{cm}}^{\mathrm{stack}}$ (Figure 2; Table 1); we dismiss such a contrived situation. Also, one radio-flaring XRB in an ultraluminous state has been identified in M31 (Middleton et al. 2013); while a radio analog of it could be detected in NGC 4472, it would fade after a few months so not be mistaken for the persistent emitters that we seek.

Only 30 of the 728 GC candidates, or 4%, have been detected with Chandra (Maccarone et al. 2003). Their X-ray luminosities ${L}_{{\rm{X}}}^{\det }$ are shown in Figure 3 after being adjusted to our assumed distance for NGC 4472. If these 30 detections arise from BH XRBs, their luminosities would correspond to systems in their high/soft state or very high state. However, the signal-to-noise ratio of the X-ray spectroscopy was insufficient to establish the emission states. Here, we assume that the 30 detections arise from BHs of unspecified mass that are emitting in the hard X-ray state. If improved X-ray spectroscopy eventually excludes hard-state emission, the ${L}_{{\rm{X}}}^{\det }$ values should be treated as upper limits on any hard-state emission.

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ESO 243-49 HLX-1 is a strong IMBH candidate, of mass ${M}_{\mathrm{IMBH}}^{\mathrm{HLX}}\sim {10}^{4-5}\,\,{M}_{\odot }$, in a star cluster which may have a GC-like stellar mass (Farrell et al. 2009; Soria et al. 2012, 2017). Cseh et al. (2015) concluded that a point-like source with a flux density of 22 ± 3.5 μJy at 6.8 GHz adequately describes the emission. For the flat spectrum appropriate to the hard X-ray state, the same flux density is adopted at 2 cm (16.4 GHz). Thus for a luminosity distance of 92 Mpc, HLX-1 is expected to have a steady 2 cm luminosity of L_{2 cm} ∼ 10^36.6 erg s⁻¹ while in its hard X-ray state (Cseh et al. 2015). If that steady emission is being Doppler boosted by a factor of about five to ten, as Cseh et al. (2015) argued, then its side-on luminosity would be about L_{2 cm} ∼ 10^35.6–35.8 erg s⁻¹. Given these various radio luminosities, Figure 2 makes it clear that radio analogs of HLX-1 would be easy to access among NGC 4472's 728 GC candidates.

Figure 3 shows that none of NGC 4472's GC candidates has an observed X-ray luminosity (Maccarone et al. 2003) as high as that of HLX-1 in its hard state, L_X ∼ 10^40.3 erg s⁻¹ (Godet et al. 2012). This absence of X-ray analogs of HLX-1 would be consistent with an absence of radio analogs of HLX-1, discussed above. However, the radio-luminosity thresholds would be more than two orders of magnitude lower than even a deboosted version of HLX-1. This would enable searches for less radio-extreme sources, even in the case where their X-ray counterparts were heavily absorbed, such that their X-ray emission was not especially remarkable.

Figure 3 shows the X-ray luminosities ${L}_{{\rm{X}}}^{\det }$ for 30 GC candidates in NGC 4472 (Maccarone et al. 2003). As Section 3.2 mentions, we assume that those 30 X-ray detections arise from BHs of unspecified mass that are emitting in the hard X-ray state. Radio-luminosity thresholds L_{2 cm} are also available from Figure 2 for those 30 GC candidates.

Armed with this information for the 30 GC candidates, we can invoke the empirical FP relation for the hard X-ray state to estimate their IMBH mass thresholds ${M}_{\mathrm{IMBH}}^{\mathrm{FP}}$, with an uncertainty of 0.44 dex (Miller-Jones et al. 2012). That FP relation adopts an X-ray range of 0.5–10 keV and a radio wavelength of 6 cm, and implicitly assumes a flat or inverted radio spectrum. The ${L}_{{\rm{X}}}^{\det }$ values from Maccarone et al. (2003) cover 0.5–8 keV which, at the signal-to-noise ratios involved, is a suitable proxy for the FP's range. The L_{2 cm} thresholds from Figure 2 were scaled to L_{6 cm} assuming a flat radio spectrum. While other FP studies (e.g., Gultekin et al. 2019) suggest larger uncertainties than those advocated by Miller-Jones et al. (2012), we follow Miller-Jones et al. (2012) for consistency with our earlier work on Milky Way GCs (Maccarone 2004; Strader et al. 2012) and extragalactic GCs (Wrobel et al. 2015, 2016, 2018; Wrobel & Nyland 2020).

The FP results are shown in Figure 4. They demonstrate that measurements of ${L}_{{\rm{X}}}^{\det }$ and thresholds for L_{6 cm} provide access to IMBH mass thresholds of ${M}_{\mathrm{IMBH}}^{\mathrm{FP}}\sim {10}^{1.7\mbox{--}3.6}\,{M}_{\odot }$. Notably, among previous attempts to apply the FP to extragalactic GCs (Miller-Jones et al. 2012; Wrobel et al. 2015, 2016), this regime has only been reached in the Local Group, for M31's G1 with its ${M}_{\mathrm{IMBH}}^{\mathrm{FP}}\lt {10}^{3.2}\,\,{M}_{\odot }$ (Miller-Jones et al. 2012).

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Each IMBH mass threshold in Figure 4 suffers from a considerable scatter of 0.44 dex. Still, such thresholds do open up the prospect of searching for accretion signatures from IMBHs in tens of GC candidates at a distance of 16.7 Mpc, in the GC-rich Virgo Cluster. Deeper X-ray imaging of NGC 4472, with either Chandra or its possible successor, Lynx, could reveal sources with lower values of ${L}_{{\rm{X}}}^{\det }$ in its 698 other GC candidates. Each such new datum added to Figure 4 would have a rightward shift on the abscissa and an upward shift on the ordinate, thereby giving access to a higher ${M}_{\mathrm{IMBH}}^{\mathrm{FP}}$ value.

This scenario involves predicting the mass of an IMBH that, if accreting at the Bondi rate from the tenuous gas supplied to the GC from its evolving stars, is consistent with the synchrotron radio luminosity of the GC (Maccarone 2004; Strader et al. 2012). Classic Bondi flows involve perfect gases, with spatially infinite distributions, accreting onto isolated central masses; IMBHs in our case. This scenario may be overly simplistic for GC settings but can still offer useful guidance.

Some systematic uncertainties need to be recognized. First, the thermodynamic state—isothermal, adiabatic, or intermediate between the two—of the gas flow feeding the Bondi accretion is unknown in GCs. To quantify the importance of this shortcoming, we note that the accretion rate for the isothermal Bondi case is almost a factor of ten higher than for the adiabatic case (e.g., Pellegrini 2005). For a given IMBH mass, the accretion signature from a Bondi flow will thus be stronger for the isothermal case.

However, the study by Pepe & Pellizza (2013) raises a second systematic uncertainty: for four Milky Way GCs, that study numerically solved the accretion rates for isothermal Bondi flows toward IMBHs embedded in realistic GC potentials. They reported that the flows appeared to achieve higher accretion rates than classic isothermal Bondi flows.

Recent hydrodynamical simulations of gas flows in early-type galaxies favor an adiabatic behavior (Inayoshi et al. 2020), with lower accretion rates in comparison to the isothermal case (Pellegrini 2005). Future hydrodynamical simulations of gas flows in GCs will eventually establish their true physical properties. In the interim, our aim here is to simply benchmark the accretion signatures. Being mindful of the systematic uncertainties mentioned above, we follow our earlier work on Milky Way GCs (Maccarone 2004; Strader et al. 2012) and extragalactic GCs (Wrobel et al. 2015, 2016, 2018; Wrobel & Nyland 2020), and assume fiducial Bondi flows where the thermodynamic state of the gas is intermediate between the isothermal and adiabatic cases. Specifically, we assume (Strader et al. 2012):

• i.  
  For a given IMBH mass ${M}_{\mathrm{IMBH}}^{{\rm{B}}}$, gas is captured at 3% of the fiducial Bondi rate for a density of 0.2 particles cm⁻¹ (Freire et al. 2001; Abbate et al. 2018) and a temperature of 10⁴ K (Scott & Rose 1975). We caution that the density estimate hinges on measurements of only two objects, namely the Milky Way GCs 47 Tuc and M15.
• ii.  
  An X-ray luminosity given by ${L}_{{\rm{X}}}^{{\rm{B}}}=\epsilon \dot{M}{c}^{2}$, for a given radiative efficiency and mass accretion rate $\dot{M}$. To generalize beyond the standard _sta = 0.1, it is assumed that at low accretion rates the efficiency scales linearly with the accretion rate. A low accretion rate is defined as being less than 2% of the Eddington rate, typical for where XRBs transition to a state with an inner advection-dominated accretion flow with a persistent X-ray luminosity. Continuity with _sta = 0.1 is required at the transition, leading to $\epsilon =0.1((\dot{M}/\dot{{M}_{\mathrm{edd}}})/0.02)$ and thus a prediction for ${L}_{{\rm{X}}}^{{\rm{B}}}$.
• iii.  
  Given ${M}_{\mathrm{IMBH}}^{{\rm{B}}}$ and ${L}_{{\rm{X}}}^{{\rm{B}}}$, a radio luminosity L_{6 cm} can be estimated from the empirical FP relation. A flat radio spectrum is used to scale from L_{6 cm} to L_{2 cm}.

With assumptions i–iii, the radio-luminosity thresholds L_{2 cm} shown in Figure 2 can be converted to the IMBH mass thresholds ${M}_{\mathrm{IMBH}}^{{\rm{B}}}$ displayed in Figure 5, for both individual GC candidates and radio-stacked GC candidates. The statistical uncertainties in the parameters adopted for the conversion cause each IMBH mass threshold to have a statistical error of 0.39 dex (Strader et al. 2012). The fiducial Bondi model suggests access to IMBH masses of ${M}_{\mathrm{IMBH}}^{{\rm{B}}}\sim {10}^{4.9-5.1}\,{M}_{\odot }$ for all 728 individual GC candidates and of ${M}_{\mathrm{IMBH}}^{{\rm{B}},\mathrm{stack}}\sim {10}^{4.5}\,{M}_{\odot }$ for radio stacks of about 100–200 GC candidates (Table 1). Encouragingly, that stack level at a distance of 16.7 Mpc resembles the stack level reported from VLA observations of 49 massive GCs in M81 at a distance of 3.6 Mpc (Wrobel et al. 2016).

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For context, Fragione et al. (2018) offer three examples of how a star cluster's stellar mass and IMBH mass could evolve to yield a massive, present-day GC. Those examples, plotted in Figure 5, achieve present-day values of ${M}_{\star }^{\mathrm{Fra}}\sim {10}^{5.9-6.4}\,{M}_{\odot }$ and ${M}_{\mathrm{IMBH}}^{\mathrm{Fra}}\sim {10}^{5.0-5.2}\,\,{M}_{\odot }$. At similar stellar masses, the ${M}_{\mathrm{IMBH}}^{\mathrm{Fra}}$ values are formally within twice the statistical error of the ${M}_{\mathrm{IMBH}}^{{\rm{B}}}$ values, whether for individual GCs or radio-stacked GCs. But this is a weak statement, due to the substantial statistical uncertainties of the fiducial Bondi model as well as to the aforementioned systematic uncertainties.

For the 728 GC candidates, the fiducial Bondi model also yields predictions of ${L}_{{\rm{X}}}^{{\rm{B}}}$, the X-ray-luminosity thresholds for hard-state emission in the 0.5–10 keV band. These predictions are shown in Figure 6, along with an indication of their statistical uncertainty of 0.39 dex (Strader et al. 2012). The ${L}_{{\rm{X}}}^{{\rm{B}}}$ thresholds appear to cluster below the ${L}_{{\rm{X}}}^{\det }$ values of the 30 Chandra-detected GC candidates. As discussed in Section 3.2, those 30 X-ray detections could correspond to XRB systems in their high/soft state or very high state, with no relation at all to the presence or absence of accreting IMBHs. But if some of the 30 X-ray detections are hard-state emitters involving accreting IMBHs, Figure 6 suggests that the fiducial Bondi model could be underpredicting their ${L}_{{\rm{X}}}^{{\rm{B}}}$ thresholds.

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Figure 6 also indicates that the predicted ${L}_{{\rm{X}}}^{{\rm{B}}}$ thresholds for all 728 GC candidates appear to cluster below the maximum persistent luminosity of stellar-mass XRBs (Maccarone et al. 2003). While deeper X-ray imaging could reveal additional sources with lower values of ${L}_{{\rm{X}}}^{\det }$ in the GC candidates, such imaging on its own would not be able to say if any detections of hard-state X-ray emission arose from stellar-mass XRBs or from fiducial Bondi flows onto IMBHs. This represents a serious X-ray contamination issue, and is the reason why we opted to avoid reporting in Table 1 the X-ray luminosities from the radio-stacked GCs.

We do not mean to discourage deeper X-ray imaging of NGC 4472 with either Chandra or its possible successor, Lynx: such imaging, when combined with ngVLA imaging and an FP analysis, can be used to separate X-ray detections into bins for X-ray binaries and for IMBHs (Wrobel et al. 2018). Also, X-ray binaries are known to be time-variable in both the radio and X-ray bands, so such radio–X-ray synergy would be strengthened by simultaneous observations with the ngVLA and the X-ray mission.