**2. The universal correlations among accretion systems**

### **2.1 X-ray variability**

X-ray emission, produced from the inner region of the accretion disk and corona, served as a proxy of accretion properties. X-ray emissions from accreting black holes have strong variability, the timing properties of X-ray emission can be explored with the power spectral densities (PSDs), *<sup>P</sup>*ð Þ*<sup>ν</sup>* <sup>∝</sup>*ν*�*<sup>α</sup>*, which is a function of timescales 1*=<sup>ν</sup>* of the variability. In both XRBs and AGNs, the PSD has a power-law spectrum. The spectral index *α*≈1 on long timescales, while it is transient to a steep spectral index *α*> 2 on short timescales. The characteristic timescale, *T*B, on where the PSD break or transitions, is a common feature in both XRBs and AGNs. Using the timescale, *T*B, as a representation of black hole accretion was established with the finding of a correlation between *T*<sup>B</sup> and black hole mass *MBH* was established by [19].

Again, it was strengthened as the break timescale is also correlated with spectral states or luminosities of both XRBs and AGNs, i.e., the low and high accretion states have different PSD profiles. Therefore, *T*<sup>B</sup> is a good reflector of black hole masses and accretion states (in terms of bolometric luminosity *Tbol*). The correlation among the critical timescale *T*B, black hole mass *MBH*, and bolometric luminosity *Lbol* is fitted by [19] (see **Figure 2**) as Eq. (1)

$$
\log T\_B = (2.10 \pm 0.15) \log M\_{\rm BH} - (0.98 \pm 0.15) \log L\_{\rm bol} - (2.32 \pm 0.2) \tag{1}
$$

They have included 10 AGNs and 2 XRBs and with a wide range of accretion rates.

Assuming *m*\_ <sup>E</sup>≈*λ*Edd, then *T*B≈*M*<sup>1</sup>*:*<sup>12</sup> BH *=m*\_ <sup>0</sup>*:*<sup>98</sup> <sup>E</sup> , where *λ*Edd ¼ *L*bol*=L*E. If the break timescale is proportional to a thermal or viscous timescale associated with the inner radius of the accretion disk, *R*disk, then from the above empirical correlation, it can be deduced *R*disk∝*m*\_ �2*=*<sup>3</sup> <sup>E</sup> . Models based on evaporation of the inner accretion disk predict *R*disk∝*m*\_ �0*:*<sup>85</sup> <sup>E</sup> and *T*B∝*M*<sup>1</sup>*:*<sup>2</sup> BH, which are roughly consistent with the empirical correlation.

Strong support or enhancement for this linkage, among characteristic timescale *T*B, black hole masses, and luminosities, comes from the correlation between *T*<sup>B</sup> and emission line region in the circumnuclear region of black holes (see **Figure 3**). As the emission line width, the full width at the half maximum *FWHM* is regulated by black hole masses and accretion rates. Therefore, the correlation between *FWHM*

#### **Figure 2.**

*The predicted break timescales, Tpredicted, derived by inserting the observed bolometric luminosities and masses into the best fit relationship (Eq. 1) to the combined sample of AGNs and GBHs [19]. Where GRS 1915*þ*1105 is presented as a filled maroon star, Cyg X-1 as blue crosses, and the 10 AGN as red circles. The low-luminosity AGN (LLAGN), NGC 4395, is shown as an open crossed red circle; the other nine AGN are filled red circles. The filled green squares are NGC 5548, and Fairall 9 and the LLAGN NGC 4258.*

#### **Figure 3.**

*Correlation of optical emission linewidth FWHM with PSD break timescale T*<sup>B</sup> *from [19]. The LLAGN NGC 4395 is shown as an open crossed red circle and the other eight AGNs are filled red circles. Filled green squares are Fairall 9 and NGC 5548, their linewidths are available but whose upper break timescales are only lower limits.*

and *T*<sup>B</sup> is a straightforward product. The correlation is explored as the permitted optical emission lines in AGN whose widths (in both broad-line AGN and narrowemission-line Seyfert 1 galaxies) correlate strongly with the characteristic X-ray timescale (see **Figure 3**),

$$
\log T\_B = \left( 4.20^{+0.71}\_{-0.56} \right) \log F \text{WHM} - 14.3 \tag{2}
$$

### **2.2 The fundamental plane of black hole activity**

The fundamental plane relation among nuclear radio luminosity, nuclear X-ray luminosity, and black hole mass unified the accretion and ejection process in the compact system. The existence of such a relationship is based on the radio emission produced in a jet/outflow and the X-ray emission produced in a disk-corona system. Both radio and X-ray power are related to black hole mass and accretion rate. Therefore, the fundamental plane relation is thought to work in any accretion system, which is in quiescent and low/hard accretion state (associated with a steady ejection, see [9, 12]).

The fundamental plane of black hole activity explored by [12] is

$$\log L\_{\mathbb{R}} = \left( \mathbf{0.60}\_{-0.11}^{+0.11} \right) \log L\_{X} + \left( \mathbf{0.78}\_{-0.09}^{+0.11} \right) \log M\_{\text{BH}} + \mathbf{7.33}\_{-4.07}^{+4.05},\tag{3}$$

While the Merloni's fundamental plane has a large dispersion when uses it to estimate black hole masses from radio and X-ray luminosities. The recent updates include correlation from [21]

$$\log L\_{\mathbb{R}} = \left( \mathbf{0}.60^{+0.11}\_{-0.11} \right) \log L\_{X} + \left( \mathbf{0}.78^{+0.11}\_{-0.09} \right) \log M\_{\text{BH}} + \mathbf{7.33}^{+4.05}\_{-4.07}, \tag{4}$$

and the most recent version [22]

$$\log\left(\frac{M\_{\rm BH}}{10^8 M\_{\odot}}\right) = (1.09 \pm 0.10) \log\left(\frac{L\_R}{10^{38} \text{erg} \,\text{s}^{-1}}\right) \tag{5}$$

$$+ (-0.59^{+0.16}\_{-0.15}) \log\left(\frac{L\_X}{10^{40} \text{erg} \,\text{s}^{-1}}\right)$$

$$+ (0.55 \pm 0.22)$$

Additionally, the very high/intermediate state may also produce radio ejection that can follow the same trend [see 4, they also include transient sources]. However, including sources with a very high/intermediate state induces a dispersion in the fundamental plane relation. This is primarily due to the evolution of individual radio blobs, as the radio ejecting process is episodic in this state. Compact symmetric objects (CSOs) are thought the episodic ejection produced by AGNs in a very high/transient state. There are two types of known contamination in the fundamental plane of black hole activity: (1) radio emissions from lobes will be enhanced when they propagate through a dense medium [23]; (2) X-ray emission contains a contribution from the jet, e.g., through synchrotron or inverse Compton mechanisms [24]. Furthermore, taking the radio emissions from lobes of CSOs is unmatched by the X-ray observations, because the radio emissions from lobes are substantially produced in different epochs from the core X-ray emission.

There are several works exploring the fundamental plane relation on CSOs [25–27]. Most of the results suggest CSOs do deviate from the classical trends. To be specific, an exploration of fundamental plane relation on a sample of CSOs (with radio flux density from lobes) indicates that they can follow the trend, while their radio luminosity is �1 dex higher than the original fitting of the fundamental plane relation [see 26], which is consistent with the transient state in XRBs. Another point is the vacuum region between stellar-mass black holes and supermassive black holes in the fundamental plane of black hole activity. This region can be filled by either intermediatemass black holes or extremely low luminosity AGNs. We will discuss it in the following sections.

#### **2.3 The inverse correlation between radio loudness and Eddington ratio**

The persistent 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., [3, 28, 29]). Resembling the inverse correlation between the radio luminosity of jets and X-ray luminosity in XRBs, Ho [16] found a similar inverse correlation between radio-loudness (ℛ ¼ *L<sup>ν</sup>*5*=L<sup>ν</sup>B*) and the Eddington ratio (*λEdd* ¼ *Lbol=LEdd*) 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 being valid for certain conditions, the inverse correlation between ℛ and *λEdd* is ubiquitous in both AGNs and XRBs [18] although with a large scatter. A global analogy between stellar-mass black holes and SMBHs has been established in the ℛ and *λEdd* correlation: low-luminosity AGNs are similar to XRBs in the low/hard state, and with the high- or super-Eddington accreting AGNs (e.g., NLS1s) 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 [30]. The super-Eddington accretion state is poorly understood primarily because of the extremely short timescales of the

#### **Figure 4.**

*Radio loudness* ℛ *vs Eddington ratio λEdd. The markers are designated in the left-bottom corner and the error bars in some extremely high Eddington ratio accreting SMBHs come from the uncertainty of the BH spin. The vertical dotted line is λEdd* ¼ 1 *and the horizontal dotted line at* ℛ ¼ 10 *represents the division between radio-loud (above) and radio-quiet (below) sources.*

spectral state transition in XRBs. Therefore, the study of the production and quenching of AGN jets in super-Eddington accretion systems will shed light on the physical properties of AGNs during this short-lived spectral state (**Figure 4**).

In the work by Yang *et al*. [13], the inverse correlation between radio loudness and the Eddington ratio has been extended to the super-Eddington regime. It was shown that the correlation is even more fundamental than between radio and X-ray luminosity and black hole mass, as the correlation works among extremely low-power accreting black holes. However, the radio loudness-Eddington ratio correlation has a large dispersion, and it suggests there is an additional parameter at work.

#### **2.4 X-ray loudness versus Eddington ratio**

It is impossible to observe a whole state transition in AGNs due to their extremely long evolutionary timescales. While, fortunately, an unbiased sample of AGNs will naturally have a mixture of AGNs in various accretion states, e.g., an AGN sample includes low-luminosity AGNs (LLAGNs), low-excitation emission line regions (LINERs), and narrow line Seyfert I galaxies (NLS1s). The properties/structures of the accretion disk and corona are represented by the X-ray loudness or UV to X-ray spectral index *α*ox, which is defined, for example [31], as

$$a\_{\rm ox} = -\frac{\log \left(\nu L\_{\nu}\right)\_{\rm o} - \log \left(\nu L\_{\nu}\right)\_{\rm x}}{\log \nu\_{\rm o} - \log \nu\_{\rm x}} + \mathbf{1},\tag{6}$$

Where ð Þ *νL<sup>ν</sup>* <sup>o</sup> and ð Þ *νL<sup>ν</sup>* <sup>x</sup> are monochromatic immensities at the rest-frame optical/UV and X-ray energies, *λ*<sup>o</sup> ¼ *c=ν*<sup>o</sup> ¼ 2500Å and *E*<sup>x</sup> ¼ *hν*<sup>x</sup> ¼ 2keV, respectively; correlates primarily with *L<sup>ν</sup>*,*<sup>x</sup>*; and that there is a strong correlation between *L<sup>ν</sup>*,o and *L<sup>ν</sup>*,x (**Figure 5**).

**Figure 5.**

*The transition behavior along X-ray loudness α*ox *versus Eddington ratio L=L*Edd *[31]. A change of the sign of the α*ox *can be observed for L=L*Edd ¼ 0*:*01*. The transition behavior is free of black hole masses. The dotted and dashed lines correspond to the correlations found by Lusso et al. [32] and Grupe et al. [33], respectively.*

A characteristic spectral behavior was found by taking the typical galactic X-ray binary GRO J1655�40 as a template, which drives a complete state transition within 1 year. The X-ray loudness versus Eddington ratio distribution has a "V"-shaped morphology, i.e., in low Eddington ratios of *L*bol*=L*Edd <0*:*01, the X-ray loudness has an inverse correlation with Eddington ratio, while in higher Eddington ratios of *L*bol*=L*Edd > 0*:*01, XRBs show a positive correlation between X-ray loudness and Eddington ratio.

The correlation between X-ray loudness and Eddington ratio can be explained as the evolution of accretion flow along with accretion state transition. In the quiescent state, *L*bol*=L*Edd < 0*:*01, the accretion disk is described as the radiatively inefficient advection-dominated accretion flow (ADAF) and drives a compact jet, where ADAF dominates the UV emission and corona dominate X-ray emission. Furthermore, the jet may also contribute to UV and X-ray emissions in this state. With the increase of Eddington ratios in this state, (1) the inner ADAF will progressively fall to touch with the outer thin disk (SSD), leading to a decrease in UV emission; (2) the corona will extend upward from the accretion disk and enhance the X-ray emission. Eventually, the increase of the Eddington ratio in this state leads to the hardening of the UV to X-ray spectrum. With the further increase of Eddington ratios from *L*bol*=L*Edd ¼ 0*:*01 to 1, the hot accretion flow of the thin disk produces strong thermal UV emission and the corona will be suppressed into small scale. This leads to a signature that with the increase of Eddington ratios the UV to X-ray spectrum will gradually soften and result in the positive correlation between *α*ox and *L*bol*=L*Edd.

It should be expected that XRBs and AGNs have a similar accretion flow and evolution scheme, which corresponds to the straightforward XRB/AGN analogy. The correlation between *α*ox and *L*bol*=L*Edd was found in a large unbiased sample of AGNs [34–40], indicating that the different types of AGNs have an evolutionary connection with each other and the AGN accretion flow may evolve similarly with XRBs. Recently, a fascinating finding is that the changing-look AGNs evolve along the *α*ox vs. *L*bol*=L*Edd trend, which hints at the state transition in individual AGN.
