Confrontation of Observation and Theory: High-frequency QPOs in X-Ray Binaries, Tidal Disruption Events, and Active Galactic Nuclei

The original resonance models invoked nonlinear coupling between the orbital frequency and the radial epicyclic frequency (Abramowicz & Kluźniak 2001), or between the vertical and radial epicyclic frequencies (Abramowicz et al. 2003) in the accretion disk. In the latter case, a coupling was assumed when the frequencies were in a 3:2 ratio, in agreement with the peaks seen in a handful of X-ray power spectra known at that time.

To investigate a physical basis for this coupling, Horák (2008) analyzed weak nonlinear interactions between the epicyclic modes in slender tori, extending earlier work of Abramowicz et al. (2003). He found that the strongest resonance was between the axisymmetric modes when their frequencies were in a 3:2 ratio.

Astrophysical seismic modes are probably most familiar from studies of the Sun and stellar interiors. Acoustic pressure (p) modes have been observed in the Sun and other stars. Gravity (g) modes are predicted to exist as well, driven by buoyancy. Accretion disks are subject to the same phenomena, although the excitation methods are different: turbulence is magnetically driven rather than convective, and the g-modes are governed by inertial forces rather than buoyancy. Theoretical expectations for these modes in accretion disks are summarized by Wagoner et al. (2001), who tabulate the comparative radial locations and extents of each of the fundamental modes, which bear on their observability in that larger modes with less leakage past the ISCO will have more radiative flux to modulate.

The most visible should be the fundamental g-mode, since it has a relatively large extent (Δr ≈ GM/c²) and is located at the temperature maximum of the disk, away from leakage through the ISCO. It is trapped just below the maximum of the radial epicyclic frequency (indicated by the black dots in Figure 1). Its frequency f obeys the relation

$$\begin{eqnarray*}&&{Mf}/10{M}_{\odot }=F(a)[1-0.1L/{L}_{\mathrm{Edd}}]\end{eqnarray*}$$

for L ≲ 0.5L_Edd, with F = 71–246 Hz for a = 0 − 1 (Perez et al. 1997; Wagoner et al. 2001). This model garners further plausibility from the fact that it, alone among the diskoseismic modes, has been unambiguously detected in a hydrodynamic disk simulation (Reynolds & Miller 2009); however, we note that in this study, the QPO disappeared when a magnetic field was included, due to turbulence induced by the magnetorotational instability. Very recent work by Dewberry et al. (2020) presents a more optimistic picture: when eccentricity or warping is forced in relativistic magnetohydrodynamic simulations, the coupling of the trapped g- and p-modes to the distortion amplifies the modes even in the presence of MHD turbulence.

The pressure p-modes can be trapped between the radial epicyclic frequency and the ISCO, but have a much smaller radial extent in the disk. They may also be more prone to leakage and infall, further reducing their potential observability. However, in simulations with hydrodynamic viscosity, the g-mode leaked into outgoing p-waves of the same frequency, and propagated at the trapped-mode frequency out to larger disk radii (O'Neill et al. 2009). Such an occurrence would lead to a larger observed luminosity modulation. Furthermore, some nonaxisymmetric, low-order p-modes may be amplified by the corotation resonance between the outer and inner Lindblad resonances, or by certain magnetic field conditions in MHD disks (Lai et al. 2013); the lower limit of the frequencies of these modes is ≈0.5 × f_ISCO.

One drawback of diskoseismic models is the inability to account for the observed 3:2 ratio of HFQPO peaks seen in stellar mass black holes; no coupled modes in the proper ratio have been predicted.

An interaction between a warped disk (with vertical displacement proportional to $\sin (\varphi )$) and a normal mode of oscillation can be excited at the radius r_*, where κ = Ω/2 (Kato et al. 2008). This can produce oscillations at frequencies κ(r_*), 2κ(r_*), and 3κ(r_*). Since r_* is close to the radius of the maximum value of κ, the lowest frequency is almost equal to that of the g-mode referred to above, for all values of the black hole spin a; see Figure 12.12 of Kato et al. (2008) for the dependence of 2κ(r_*) on a, but note the factor of two error in their Equation (12.25).

A different important candidate for exciting nonaxisymmetric modes in accretion flows around black holes is fluid instability between the magnetic field surrounding the accreting object and the inflowing material. Li & Narayan (2004) evaluated the interface between an accretion disk and the magnetosphere of the central object, and found that the boundary gives rise to both Rayleigh–Taylor and Kelvin–Helmholtz instabilities. In turn, these enable unstable modes to grow in the disk, resulting in nonaxisymmetric patterns that may translate into quasi-periodic brightness variations. This model is also interesting in that it naturally describes the frequency width and finite lifetimes of QPOs, and may even explain some frequency ratios between QPO peaks, since the low wavenumber modes will dominate due to the radial dependence of the perturbation.

McKinney et al. (2012) expanded this model in MHD simulations, and found that a Rayleigh–Taylor instability developed at the interface of the disk and jet regions in magnetically choked accretion flows. Oscillations were indeed induced at the magnetospheric interface. At this location, the rotational frequencies of the plasma, Ω, and the field lines, Ω_F, are forced to be similar by the strength of the magnetic field. The QPO frequency is therefore set by the black hole spin, which drags the field lines with Ω_F ∼ Ω_BH/4, where the spin frequency ${{\rm{\Omega }}}_{\mathrm{BH}}=({c}^{3}/{GM})a/[2(1+\sqrt{1-{a}^{2}})]$. A robust detection of a HFQPO was reported. This result is likely to be valid only for black hole spins a ≳ 0.4, below which the plasma disk rotation may dominate over the spin frequency.

This class of theories produces relations of the form Mf = F(a, r), where r is the radial position of the pressure maximum of the torus (Rezzolla et al. 2003) or the center of the orbiting "hot spot." However, a hot spot overdensity would be subject to destruction by the shear of the disk. The relativistic precession model (Stella et al. 1999) involves the nodal precession frequency Ω − Ω_⊥ and the periastron precession frequency Ω − κ, but no resonance is involved.

A general analysis of orbital resonance and relativistic precession models within the context of tori has been very recently presented by Kotrlová et al. (2020). They focus on sources in which a 3:2 (or some other) frequency ratio is observed (which constrains the value of r). Then their predictions are of the form Mf = F(a, β), where β is proportional to the thickness of the torus (β = 0 corresponds to a geodesic particle orbit). Essentially all of their well-studied torus models produce results for Mf, which are comparable to or greater than that of the epicyclic orbital resonance model (plotted in Figures 2 and 3). A problem may arise for models such as these, since time dependent properties of the source, such as β, would produce a time dependence of the frequencies. For these reasons, we do not include these models in our figures. The hot spot model is also not included, since the predicted frequency depends on the radial location of the hot spot, which is unconstrained.

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The predictions in the phase space of black hole mass, spin, and QPO frequency of the resonance, diskoseismic g-mode, nonaxisymmetric p-modes, and disk–jet coupling models are plotted in Figures 2 and 3, where we also plot the orbital frequency of the ISCO. We do not include the warp model, since the plot of its lowest frequency is very close to that of the g-mode.

In Table 1 we list observations of HFQPOs in the literature in black hole X-ray binaries with detection significance above 3σ and for which masses and spins have been determined, excluding any sources that have been formally refuted or retracted since they were first reported. Most observations were taken with Rossi X-ray Timing Explorer, launched in 1995 and decommissioned in 2012.

Table 1.  Observations of HFQPOs around Stellar Black Holes

Name	MBH	BH Spin Range	fQPO	MBH References	Spin References	QPO References
	M⊙	a	Hz
XTE J1550-564	${9.1}_{-0.6}^{+0.6}$	0.29 < a < 0.62r,c	184	1	7	14
		0.75 < a < 0.77r	184		8
XTE J1650-500	${5.0}_{-2.0}^{+2.0}$	0.78 < a < 0.8r	250	2	8	15
GROJ1655-40	${5.9}_{-0.8}^{+0.8}$	0.65 < a < 0.75c	300	3, 4	9	14, 16
		0.9 < a < 0.998r	300		10
		0.97 < a < 0.99r	300		8
GRS1915+105	${10.1}_{-0.6}^{+0.6}$	0.97 < a < 0.99r,c	41	5	11, 12, 13	17
	${12.5}_{-1.9}^{+1.9}$	0.97 < a < 0.99r,c	41	6	11, 12, 13	17
	${10.1}_{-0.6}^{+0.6}$	0.97 < a < 0.99r,c	67	5	11, 12, 13	18
	${12.5}_{-1.9}^{+1.9}$	0.97 < a < 0.99r,c	67	6	11, 12, 13	18
	${10.1}_{-0.6}^{+0.6}$	0.97 < a < 0.99r,c	166	5	11, 12, 13	19
	${12.5}_{-1.9}^{+1.9}$	0.97 < a < 0.99r,c	166	6	11, 12, 13	19

Note. Observational data and references for HFQPOs around stellar mass black holes from the literature, in order of increasing R.A. In objects where multiple HFQPOs are reported in small integer ratios, only the lowest frequency is given. Superscripts on the spin ranges indicate the measurement methods: Fe Kα reflection spectroscopy (r) or continuum fitting (c). References are as follows: (1) Orosz et al. (2011), (2) Orosz et al. (2004), (3) Beer & Podsiadlowski (2002), (4) Shahbaz (2003), (5) Steeghs et al. (2013), (6) Reid et al. (2014), (7) Steiner et al. (2011), (8) Miller et al. (2009), (9) Shafee et al. (2006), (10) Reis et al. (2009), (11) Blum et al. (2009), (12) Miller et al. (2013), (13) McClintock et al. (2006), (14) Remillard et al. (2002), (15) Homan et al. (2003), (16) Remillard et al. (1999), (17) Strohmayer (2001), (18) Morgan et al. (1997), (19) Belloni et al. (2006).

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In Figure 2, we plot these observations. We show the stellar mass black holes both on the same scale as the upcoming supermassive plot, for easier comparison, and on a scale more appropriate for their own values.

The observed frequencies of HFQPOs do not shift freely with large luminosity changes (Remillard & McClintock 2006). The quality factors Q = f/FWHM of these HFQPOs range between 2 and 15, with fractional rms ranging from ∼1% to 7% (Belloni et al. 2012).

Two of the plotted sources exhibit an approximate 3:2 ratio in frequency of the HFQPO peaks: XTE J1550-564 (184, 276 Hz) and GRO J1655-40 (300, 450 Hz). GRS 1915+105 also has several detected HFQPOs that are not in a 3:2 ratio (41, 67, 166 Hz). While the 3:2 pairs have appeared simultaneously, most often they are detected singly, in separate observations (Remillard & McClintock 2006). In Figure 2 only the lower frequency of the pair is plotted for clarity; in the case of GRS 1915+105, all three QPOs are plotted, since no 3:2 harmonic ratios are present.

We note that HFQPOs in a 3:2 ratio have also been discovered in H 1743-322 (Homan et al. 2005; Remillard & McClintock 2006), but the mass estimates for this black hole are model-dependent and rely upon the QPO frequency itself. Such mass estimates cannot be used to constrain the various models without circular reasoning, so this object is excluded from our analysis. We further note that the definite 250 Hz QPO in XTE J1650-500 was reported along with two additional marginal detections at ∼110 and ∼170 Hz that form an approximate 1:2:3 harmonic set with the 250 Hz detection. Because these detections are not highly significant and the harmonic ratio is speculative, we include only the 250 Hz oscillation in our analysis. Inclusion of the lower-frequency oscillations in Figure 2 would not change the conclusions of this paper, only shifting the green data point downwards by an amount encompassed by the measurement error.

Objects may appear more than once due to different mass or spin measurements in the literature. Spin is determined by one of two methods: thermal continuum fitting and iron-line spectroscopy, see Table 1. The continuum fitting method effectively determines the spin by fitting the thermal spectrum with a fully relativistic accretion disk model in a Kerr spacetime (e.g., Shafee et al. 2006). This is only possible when many of the other parameters in the fit, such as the black hole mass, inclination, and distance to the source are constrained by independent means. The iron-line spectroscopic method models the Fe Kα fluorescence line generated through the irradiation of the disk by hard X-rays (e.g., Brenneman & Reynolds 2006). The extent of the line's redshifted wing increases as the ISCO approaches the event horizon (i.e., with increasing spin, see Figure 1).

We note that in the case of GRS 1915+105, the maximal spin that we report here is found by both the reflection and continuum modeling methods in separate studies (McClintock et al. 2006; Miller et al. 2013). However, Blum et al. (2009), while finding the same maximal spin value if using the entire broadband spectrum, finds a much lower spin of a ∼ 0.56 if only the soft X-ray spectrum is used in the modeling.

In Table 2 we list observations of HFQPOs in the literature in AGN and tidal disruption events with detection significance of above 3σ and for which a reasonable mass estimate has been determined. Note that some HFQPOs have been detected in sources where there is no mass or spin determination.

Table 2.  Observations of QPOs around Supermassive Black Holes

Name	BH Spin	log MBH	fQPO	QPO Band	Object Type	QPO References	MBH References	Spin References
	a	M⊙	Hz
TON S 180	<0.4	${6.85}_{-0.5}^{+0.5}$	5.56 × 10−6	EUV	NLS1	1	11	21
ESO 113-G010	0.998	${6.85}_{-0.24}^{+0.15}$	1.24 × 10−4	X	NLS1	2	12	12
ESO 113-G010	0.998	${6.85}_{-0.24}^{+0.15}$	6.79 × 10−5	X	NLS1	2	12	12
1H0419-577	>0.98	${8.11}_{-0.50}^{+0.50}$	2.0 × 10−6	EUV	Sy1	1	13	22
RXJ 0437.4-4711	⋯	${7.77}_{-0.5}^{+0.5}$	1.27 × 10−5	EUV	Sy1	1	13	⋯
1H0707-495	>0.976	${6.36}_{-0.06}^{+0.24}$	2.6 × 10−4	X	NLS1	3	14	23
RE J1034+396	0.998	${6.0}_{-3.49}^{+1.0}$	2.7 × 10−4	X	NLS1	4	15	15
Mrk 766	>0.92	${6.82}_{-0.06}^{+0.05}$	1.55 × 10−4	X	NLS1	5	16	24
ASASSN-14li	>0.7	${6.23}_{-0.35}^{+0.35}$	7.7 × 10−3	X	TDE	6	17	6
MCG-06-30-15	>0.917	${6.20}_{-0.12}^{+0.09}$	2.73 × 10−4	X	NLS1	7	18	25
XMMU J134736.6+173403	⋯	${6.99}_{-0.20}^{+0.46}$	1.16 × 10−5	X	Sy2	8	8	⋯
Sw J164449.3+573451	⋯	${7.0}_{-0.35}^{+0.30}$	5.01 × 10−3	X	TDE	9	19, 20	⋯
MS 2254.9-3712	⋯	${6.6}_{-0.60}^{+0.39}$	1.5 × 10−4	X	NLS1	10	13	⋯

Note. Observational data and references for QPOs around supermassive black holes from the literature, in order of increasing R.A. References are as follows: (1) Halpern et al. (2003), (2) Zhang et al. (2020), (3) Pan et al. (2016), (4) Gierliński et al. (2008), (5) Zhang et al. (2017), (6) Pasham et al. (2019), (7) Gupta et al. (2018), (8) Carpano & Jin (2018), (9) Reis et al. (2012), (10) Alston et al. (2015), (11) Mathur et al. (2012), (12) Cackett et al. (2013), (13) Grupe et al. (2010), (14) Zhou & Wang (2005), (15) Czerny et al. (2016), (16) Bentz & Katz (2015), (17) van Velzen et al. (2016), (18) Bentz et al. (2016), (19) Mïller & Gültekin (2011), (20) Seifina et al. (2017), (21) Parker et al. (2018), (22) Jiang et al. (2019), (23) Zoghbi et al. (2010), (24) Buisson et al. (2018), (25) Miniutti et al. (2007).

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The identification of these oscillations as HF- rather than LFQPOs is achieved by several factors, which are typically the same from paper to paper and are based on similarities with better-determined objects. The first AGN QPO detection in RE J1034+396 (Gierliński et al. 2008) has a high coherence and a measured black hole mass consistent with the known mass–frequency scaling relation for HFQPOs in XRBs. As in several of the other cases, assuming the detected QPO frequency was an LFQPO results in mass upper limits that are inconsistent with M_BH measurements from independent sources, and are often unfeasibly low for AGN (∼10⁵M_⊙). Additionally, the high accretion rate and strong soft excess in the X-ray energy spectrum and the shape of the broadband noise component in the temporal power spectrum of all of the other claimed detections are similar to RE J1034+396, while a number of the other papers cite also the high coherence and low fractional variability of their QPOs as consistent with those observed in BHB HFQPOs. In short, each paper makes the case for their detection being HFQPO, which is our criterion for including them.

There are two AGN in which candidate low-frequency QPOs have been claimed: 2XMM J1231+1106 (Lin et al. 2013) and KIC 9650712 (Smith et al. 2018). The LFQPO identification was based on mass scaling of frequency assuming a linear relationship between LFQPO periods and black hole mass (see, for example, the Introduction of Carpano & Jin 2018), yielding consistent masses with other independent estimates, as well as low coherence and high fractional variability consistent with XRB LFQPOs. Because LFQPOs are not likely to be caused by the same mechanism as HFQPOs, we do not include them in this analysis.

Finally, we note that three QPOs have been claimed in extreme-UV light curves of AGN, which may have different physical origins and for which the authors do not explicitly speculate on the low- or high-frequency identity, although they do hypothesize that their origin is diskoseismic: TON S 180, 1H 0419-577, and RXJ 0437.4-4711 (Halpern et al. 2003). Because these are the only detections not in X-rays and with no explicit frequency identification, we use different symbols in Figure 3.

In Figure 3, we plot these observations. In the case that spin measurements are unknown, objects are plotted as colored bands that account for the error in the mass and frequency measurements (where the error in the black hole mass overwhelmingly dominates). Frequencies have been corrected for cosmological redshift, and are given in the galaxy's rest frame.

Although we only plot positive spins in Figure 3, since all current spin observations are in this regime, it is important to remember that for supermassive black holes, retrograde spin (a < 0) is a physically valid possibility. We therefore indicate the prediction for each of the theories for a = −1 on the left edge of the plot, for the curious. The spins reported here for SMBHs were measured using the Fe Kα line method, often modeled in tandem with X-ray spectroscopic fits; the details for each measurement can be found in the spin references in Table 2. Although thermal continuum fitting alone can in principle be used to determine the spin in AGN, it is not as straightforward, due mainly to the fact that the thermal disk continuum peaks in the extreme UV rather than the X-ray band, but also other complexities; see Reynolds (2019) for a helpful summary. As is the case with stellar mass black holes, many of the other parameters of the fit including mass and accretion rate are constrained by independent observations, such as the width of broad optical Balmer lines (e.g., Trakhtenbrot & Netzer 2012) or the correlation between M_BH and the stellar velocity dispersion of the bulge (M–σ_*, Ferrarese & Merritt 2000).

It is important to put the search for these SMBH QPOs into context. The variability of AGN is a red-noise process, and as such, can easily give rise to spurious periodic signals in both X-ray and UV/optical light curves (Vaughan 2005; Vaughan et al. 2016). Because of this, many QPOs reported early on, typically before 2005, have been ruled out either by more careful analyses or by longer subsequent light curves. For our analysis, because our purpose is to establish whether any observations are consistent with the same models as employed for stellar-mass black holes, we include all AGN QPO detections that have not been formally retracted or refuted. All objects included in our plot have a detection significance in excess of 3σ in their discovery papers. However, we note that how that significance is calculated varies from paper to paper; often, the significance is calculated only for regions of the light curve where the QPO is deemed present, while the significance of the QPO in the full light curve is much lower. The quality factors Q of the QPOs are typically similar for AGN and stellar mass black holes, ranging from ∼2 to 15. The rms fractional variability amplitudes span a larger range and reach higher values than in stellar mass black holes, ranging from 2% to 50%. The details for each object can be found by reading the QPO discovery references given in Table 2.

Detection of two QPOs in small integer ratios is far less common in supermassive black holes than in stellar mass black holes. Although their detection is far from ubiquitous among XRBs, being seen in only three objects (Motta 2016), there may not yet be a convincing case in AGN. When Zhang et al. (2017) reported the discovery of their 1.55 × 10⁻⁴ Hz QPO in Mrk 766, they noted that it would be in a 3:2 ratio with the previously reported 2.4 × 10⁻⁴ Hz QPO in that same object (Boller et al. 2001), although the two peaks were not present at the same time. However, the Boller et al. (2001) result is more likely to be caused by red noise than an actual periodicity (Vaughan 2005). The QPOs in ESO 113-G010 are in a 2:1 ratio (Zhang et al. 2020), and Alston et al. (2015) report the detection of a coherent soft lag in MS 2254.93712 that may indicate a harmonic component consistent with a 3:2 ratio. While such ratios may be consistent with resonance models generally, the ratios are approximate and the number of detections remains few.