Linking the micro to macro properties of AGN feedback

# Unifying the micro and macro properties of AGN feeding and feedback

## Abstract

We unify the feeding and feedback of supermassive black holes with the global properties of galaxies, groups, and clusters, by linking for the first time the physical mechanical efficiency at the horizon and Mpc scale. The macro hot halo is tightly constrained by the absence of overheating and overcooling as probed by X-ray data and hydrodynamic simulations (). The micro flow is shaped by general relativistic effects tracked by state-of-the-art GR-RMHD simulations (). The SMBH properties are tied to the X-ray halo temperature , or related cosmic scaling relation (as ). The model is minimally based on first principles, as conservation of energy and mass recycling. The inflow occurs via chaotic cold accretion (CCA), the rain of cold clouds condensing out of the quenched cooling flow and recurrently funneled via inelastic collisions. Within 100s gravitational radii, the accretion energy is transformed into ultrafast  km s outflows (UFOs) ejecting most of the inflowing mass. At larger radii the energy-driven outflow entrains progressively more mass: at roughly kpc scale, the velocities of the hot/warm/cold outflows are a few , 1000, 500 km s, with median mass rates  10, 100, several 100  yr, respectively. The unified CCA model is consistent with the observations of nuclear UFOs, and ionized, neutral, and molecular macro outflows. We provide step-by-step implementation for subgrid simulations, (semi)analytic works, or observational interpretations which require self-regulated AGN feedback at coarse scales, avoiding the a-posteriori fine-tuning of efficiencies.

black hole accretion – ISM, IGM, ICM – methods: 3D GR-RMHD simulations, analytics
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## 1. Introduction

Last-decade observations and simulations have shown that supermassive black holes (SMBHs) and cosmic structures are not separate elements of the universe (Heckman:2014 for a review). While cosmic structures are characterized by virial radii3 ( Mpc), SMBHs have a characteristic Schwarzschild radius ( pc for ), implying a difference of 10 dex in length scale. This magnitude of separation might strike as insurmountable, however, black holes would not exist without matter feeding them, and cosmic structures would tend to a quick cold death without feedback from SMBHs (often called active galactic nuclei – AGN – to emphasize such role), thus creating a symbiotic relation.

At the present, no simulation is capable of covering simultaneously the 10 dex dynamic range involving SMBH feeding and feedback (Fig. 1), and to track the evolution from 0.1 yr to 10 Gyr. Recent attempts have been made in the direction of linking the large-scale multiphase gaseous halos of galaxies (ISM), groups (IGM), and clusters (ICM) down to the subpc accretion scale (e.g., Gaspari:2015_cca; Gaspari:2017 – G15, G17). The dark matter halos heat up the diffuse gas during gravitational collapse, creating stratified hot plasma halos ( K) filling cosmic structures, which are detected in X-ray (e.g., Anderson:2015 and refs. within). Such plasma radiatively cools in the core () through a top-down condensation cascade to dense warm gas ( -  K; optical/IR - UV) and cold gas ( K; radio), subsequently raining toward the nuclear region (). Via recurrent collisions, the condensed clouds are rapidly funneled toward the inner stable orbit (). Such process is known as chaotic cold accretion (Gaspari:2013_cca; §2.1). CCA has been independently probed by several observational works (e.g., Werner:2014; David:2014; Voit:2015_nat; Tremblay:2016 and refs. therein).

General-relativistic, radiative magneto-hydrodynamic simulations (GR-RMHD) provide crucial constraints for the last stage of feeding (e.g., Sadowski:2015; Sadowski:2016_thick; Sadowski:2017 – SG17; §2.2). Near the ISCO, the final drastic SMBH pull converts a fraction of the gravitational energy into mechanical output, ejecting most of the mass via ultra-fast outflows (UFOs). Such outflows re-heat the core, while entraining the ambient gas, in a self-regulated AGN feedback loop (Fig. 1). In the paper companion to this work (SG17), we present and discuss in-depth the GR-RMHD simulations results, including the mechanical and radiative efficiencies.

In §3, we will quantitatively link the macro and micro properties of cosmic structures and SMBHs by using first principles, as mass and energy conservation, and by preserving minimal assumptions based on last-decade observations. The final equations provide the mass outflow rates and velocities at different scales (and for different phases). In §4, we compare the predictions with recent ionized, warm, and molecular outflow samples, and discuss the limitations. In §5, we discuss how to apply our model to other studies, as subgrid simulations, semi-analytic (SAM) studies, or observational interpretations. In §6, we carefully discuss the limitations of the model and additional important features (as the duty cycle and the relation). In §7, we summarize the main points and conclude with future prospects.

## 2. Large and small scales efficiencies

We highlight here 3 key regions which are central to our study (see Fig. 1 for a full diagram).

(i) The region closest to the SMBH horizon, (a few ISCO radii), where gas is rushing toward the BH and there is no outflow. This region is fully resolved by the horizon scale GR-RMHD simulations. We denote properties in this region by a black dot, e.g., .

(ii) The ultra-fast outflow launching region, , within which the binding energy of the infalling gas is converted into mechanical outflow, not interacting yet with the ambient gas. We denote this by .

(iii) The macro region, , within which the nuclear outflow is entrained (denoted by ), slowed down, and eventually thermalized (via bubbles, shocks, and turbulent mixing). The CCA rain develops in such core, with major collisions increasing within the kpc scale (10 -100  Bondi radii4).

### 2.1. Macro efficiency: chaotic cold accretion [CCA]

We introduce now the two key property of the feeding and feedback, i.e., the mechanical efficiency which has dimension of a power divided by the rest mass energy rate, .

The best consistent way to solve the cooling flow problem appears to be mechanical AGN feedback self-regulated via CCA (§1). Solving the cooling flow problem means to avoid at the same time overcooling and overheating, preserving the inner structure of hot halos for  Gyr, as tightly constrained over the last decade by Chandra and XMM-Newton data (e.g., McNamara:2012; McDonald:2016). Such hot halos are continuously perturbed by subsonic turbulence (e.g., Khatri:2016). In turbulent regions where the cooling time drops below the local dynamical time, nonlinear multiphase condensation develops (Fig. 1, bottom). Such cold clouds and warm filaments collide in chaotic, inelastic way while raining on the SMBH (G15, G17; see also Pizzolato:2010), boosting the accretion rate with rapid intermittency. Massive sub-relativistic outflows are then triggered with kinetic power proportional to the large-scale inflow rate, preventing a run-away pure cooling flow (§3).

Due to self-regulation, the large-scale mechanical efficiency can be estimated by comparing the AGN energy output with the radiative energy losses, , yielding (§3.1 for the derivation)

 εBH≃10−3Tx,7.4∝c2s,xc2, (1)

where is the hot halo adiabatic sound speed and is the speed of light (the scaling shares analogy to a Mach number squared). From less massive, lower-temperature, compact galaxies to more massive, hotter, and larger galaxy clusters ( keV), the mechanical efficiency covers a range . The macro efficiency is a function of hot halo temperature (), thus total mass, decreasing for smaller halos since the cooling rate is a function (as seen later in Eq. 6). Smaller, less bound halos experience a stronger relative condensation due to the lower specific internal energy, and necessitate of less sinked material – with slightly more evacuation – in order to avoid overheating. Such quasi thermal equilibrium constraint on X-ray halos filling cosmic systems is key to set the macro efficiency.

This picture has been corroborated by self-regulated AGN simulations of CCA and massive outflows tested in clusters, groups, and isolated galaxies (e.g., Gaspari:2011a; Gaspari:2011b; Gaspari:2012b; Gaspari:2012a; Prasad:2015; Yang:2016), which independently retrieve the same range of feedback efficiencies described above in varying systems. The few available observational estimates, albeit limited by several extrapolations, are also consistent with a mechanical efficiency of the order of (Merloni:2008).

### 2.2. Horizon efficiency: GR-RMHD

Gas approaching the SMBH liberates its gravitational energy. A test particle falling straight on the BH would convert the liberated amount into kinetic energy of radial motion and, finally, take it with it below the horizon. From the point of view of the observer at infinity, no energy has been extracted. Accretion flows act in a more complex way. The liberated gravitational energy goes mostly into kinetic motion. The turbulent nature of the flow induces this energy to dissipate and heat up the gas. At the same time outflows can be generated often via the magneto-centrifugal mechanism. Only for idealized models, like advection dominated accretion flows (e.g., Narayan:1995), all the dissipated heat is advected with the flow onto the BH. In a more general case, energy is extracted from the system, and gas infalling from large radii and marginally bound, crosses the BH horizon with negative energy.

The amount of the extracted energy, i.e., the efficiency of a given accretion flow, depends solely on the energetics of the magnetized gas crossing the BH horizon. E.g., if on average gas with energy falls into the BH, then the luminosity of such a system, as seen from infinity, is . The properties of the accretion flow in the innermost region must be determined by numerical means, since the flow is highly nonlinear, strongly magnetized, and turbulent. In the companion paper, SG17, general relativistic radiative simulations of magnetized gas falling on the SMBH are carried out, testing over 5 orders of magnitude in accretion rates. SG17 show that for a non-rotating BH and standard non-saturated configuration of the magnetic field, thick accretion flows (as expected in the maintenance mode of AGN feedback) have a fairly stable extraction of the rest mass energy accreted through the horizon,

 ε∙≃0.03±0.01, (2)

Such mechanical efficiency will be the reference horizon efficiency for our model. We note that chaotic accretion (our macro scale model) will naturally lead to an average null spin configuration (e.g., King:2006). An important result from SG17 is that such value is essentially independent of the ion-electron temperature ratio, i.e., the strength of the gas cooling does not affect the mechanical efficiency value at the micro scale.

This energy outflow accelerates within the inner region () and is ejected in a quasi-spherical way (Fig. 1, top) in the form of an ultra-fast kinetic outflow of gas. The outflow is both thermally (equatorial) and magnetically driven (polar region; see also the simulations by Moller:2015). At larger distances, the outflow interacts with the ambient medium, entraining gas via shocks and mixing instabilities, finally dissipating its energy within the core region, (App. A). On top of this energy flux there might be a very thin, relativistic jet forming whenever the SMBH quickly spins and the magnetic field threads the horizon. Such a jet may be substantially energetic and could lead to larger efficiencies (e.g., Tchekhovskoy:2011). However, relativistic jets are in most cases very collimated and less likely to interact with the host. For such reason and for the null spin expected from chaotic accretion, we consider here only the wide, massive sub-relativistic outflows as dominant component of the kinetic feedback. Albeit not driving the total ram pressure, we note the jet and radio emission can still be correlated with the presence of massive AGN outflows, thus tracing some of the major AGN bubbles (§6).

The emergence of ultra-fast outflows and the connection with large-scale warm absorbers has been corroborated by other analytic studies. Fukumura:2010; Fukumura:2014 show that magnetic torques acting on the inner rotating gas can efficiently drive an outflow through the magneto-centrifugal mechanism. The MHD wind is stratified, having slower velocity at progressively larger launching radii, akin to an entrained outflow. In the radiatively efficient, Eddington regime, the spherical model by King:2014_WA suggests that radiation pressure is able to drive UFOs; the expanding, swept-up shell is decelerated by the background medium, again corroborating the key role of the entrainment action in unifying AGN outflows over a large range of scales.

## 3. Linking the macro and micro scales

The two complementary simulations discussed above allow us to link the large-scale to small-scale properties of the feeding and feedback mechanism in a simple, coherent model. Fig. 1 illustrates the main features and characteristic scales of the model.

The large-scale outflow power can be modeled as

 POUT=εBH˙Mcoolc2, (3)

where is the quenched cooling flow rate and is the macro-scale mechanical efficiency (§2.1). The gaseous halo is losing internal energy via radiative emission (mainly via Bremsstrahlung), while the AGN feeds heating back, on average balancing the pure cooling flow. Such halos perturbed by subsonic turbulence develop local multiphase condensation within the core, as long as turbulent Taylor number (G15). As cold clouds and filaments rain down, they experience recurrent chaotic, fractal collisions, which cancel angular momentum at progressively smaller radii, in particular as they collapse within  kpc. The inflow rate can be thus considered independent of radius. In other words, during CCA rain, the cold gas condensed in the core is quickly funneled to the ISCO region with no long-term accumulation. G17 simulations showed that the CCA inflow rate is proportional to the effective viscosity of the cloud collisions, . The collisional mean free path and the ensemble velocity dispersion are directly inherited from the large-scale turbulence (for a massive galaxy,  pc,  km s) – a 3D chaotic process not tied to a radial dependence.

The inner, tiny SMBH is the actual source of energy injection with power

 Pout=ε∙˙M∙c2, (4)

where is the horizon efficiency (§2.2) and is the inflow rate through the black hole horizon. The major difference between the macro and horizon efficiency implies that the sinked mass rate is the net inflow rate surviving the ultra-fast outflow generated near the ISCO scale, before falling into the unescaping BH horizon.

### 3.1. Inflow properties

The large-scale inflow rate is the quenched cooling flow rate. The maximal pure cooling flow (CF) rate can be calculated from the enthalpy variation of the hot gaseous halo via the first law of thermodynamics (e.g., Gaspari:2015_xspec) in isobaric mode, yielding

 Lx=γγ−1kbTxμmp˙MCF=c2s,xγ−1˙MCF, (5)

where and denote the core X-ray temperature and luminosity of the hot halo (App. A), is the adiabatic index, is the average atomic weight for a fully ionized plasma with % He in mass, and and are the usual Boltzmann constant and proton mass, respectively. The last equality converts temperature into adiabatic sound speed, . From galaxies to clusters ( keV),  km s.

AGN feedback preserves the hot halos in quasi thermal equilibrium throughout the 10 Gyr evolution5. The warm filaments drop out of the hot halo just below the soft X-ray regime (G17) as the cooling curve drastically increases due to line cooling. Thereby the actual mass flux arising out of the condensation process is linked to the suppressed soft X-ray luminosity. X-ray spectroscopical observations (e.g., Peterson:2003; Kaastra:2004) show that the soft X-ray emission is on average suppressed by 2 dex compared with the pure isobaric CF tied to the core (cf. Gaspari:2015_xspec for a review of observational works and analysis of the soft X-ray spectrum quenching). For such reasons, the effective quenched cooling rate is

 ˙Mcool≃10−2˙MCF≃6.7×10−3Lxc2s,x. (6)

We note such quenched, CCA rate is typically the Bondi rate (Gaspari:2013_cca), the latter being insufficient to properly boost the AGN heating (see also Soker:2009; McNamara:2012). Since hot halos are formed via the gravitational collapse of the cosmic structures, the temperature and luminosity are interchangeable via scaling relations (Sun:2012), such as erg s (including the minor corrections due to the core radius instead of ; see App. A). We can thus rewrite Eq. 6 as

 ˙Mcool≃(1.1 M⊙yr−1)T2x,7.4=(1.1 M⊙yr−1)L2/3x,43.8, (7)

where the core and are in unit of  erg s and  K (2.2 keV), respectively. From compact galaxies to massive clusters ( keV), the inflow rate covers . Interestingly, all the below scalings can be also expressed in terms of total mass or virial radius, e.g., (App. A). It is important to note that if the core cooling time is , then the system is in a non-cool-core condition and no condensation rain, feeding, and feedback shall be applied (regardless of scaling relations), until the core cools down, igniting the self-regulated loop (see §6).

The energy conservation requirement,

 Pout=POUT, (8)

implies that the horizon inflow rate is related to the cooling rate as follows:

 ˙M∙=εBHε∙˙Mcool, (9)

where the horizon mechanical efficiency is directly provided by the GR-RMHD simulations (§2.2), . From the results and observations discussed in §2.1, hot halos must avoid at the same time overheating and overcooling, i.e., the energy lost via radiative emission in the core must be replaced by the SMBH feedback power,

 POUT≃Lx. (10)

Thereby and by using Eq. 6, the macro efficiency reduces to

 εBH=102γ−1c2sc2≃10−3Tx,7.4=10−3L1/3x,43.8 (11)

Notice that the efficiency only mildly varies with the main variable, the X-ray luminosity. We can now use both efficiencies to retrieve the horizon inflow rate relative to the macro value via Eq. 9 as

 ˙M∙≃(0.03˙Mcool)Tx,7.4=(0.03˙Mcool)L1/3x,43.8, (12)

i.e., only a few percent of the quenched cooling flow rate is actually sinked through the SMBH horizon. Substituting in Eq. 12, we consistently retrieve the accretion rate directly proportional to the X-ray luminosity,

 ˙M∙ =Lxε∙c2≃(0.04 M⊙yr−1) Lx,43.8 (13) =(0.04 M⊙yr−1) T3x,7.4.

For SMBHs in the local universe, such accretion rates are typically sub-Eddington, as expected for the maintenance, mechanically dominated mode of AGN feedback. As shown by Russell:2013 and corroborated by SG17, the radiative efficiency and thus power due to radiation is several dex lower than the mechanical input, and it can be neglected in terms of driver of the dynamics (albeit radiation is clearly relevant to detect and trace AGN; §6). Eq. 12-13 imply that SMBHs in lower mass halos have typically a lower absolute accretion rate. Moreover, a relatively smaller fraction of gas reaches the horizon as AGN feedback is more effective in halos with lower binding energy, which are tied to both lower and lower black holes masses (§6).

### 3.2. Outflow properties

Having assessed the inflow properties, we are now in a position to retrieve the structure of the outflows, again via minimal first principles. The power in terms of characteristic mass outflow rates6 and velocities at the launching and macro scale is

 Pout=12˙Moutv2out, (14)
 POUT=12˙MOUTv2OUT, (15)

respectively. As shown in Eq. 12, only a few percent of the total inflow is actually sinked through the SMBH horizon; most of the mass is returned as ultra-fast outflows launched within , such as

 ˙Mout =˙Mcool−˙M∙=(1−εBHε∙)˙Mcool≈˙Mcool, (16)

which leads to the inner outflow velocity via Eq. 14

 vout=√2ε∙˙M∙c2˙Mout=√2εBH1−εBH/ε∙c≃√2εBHc (17) ≃(1.4×104kms−1)T1/2x,7.4=(1.4×104kms−1)L1/6x,43.8. (18)

We note in Eq. 17 can be tied to a momentum , which satisfies .

Together with the above outflow rates, these are the typical velocities of ultra-fast outflows (UFOs) observed as blue-shifted absorption lines tracing the inner launching region near the SMBH gravitational radius (Tombesi:2012; Tombesi:2013; Fukumura:2015; more discussions and comparisons in §4). We note the outflow velocity is only weakly dependent on the halo temperature/luminosity, varying at best by a factor of 2.5. We thus expect  km s to be a fairly general attribute7 of inner launching outflows (Crenshaw:2003; Tombesi:2016 for reviews).

As the inner ultra-fast outflow propagates outward (), it will entrain the background gas (embedding the low volume-filling CCA rain8) along its way such as

 ˙MOUT=η˙Mout, (19)

where is the entrainment factor. We note at kpc scale the mechanical outflow has not yet thermalized, conserving most of the kinetic energy, as we see the formation of X-ray cavities and hot spots at larger distances. At a given radius, the entrained mass outflow rate can be retrieved via the mass flux equation

 ˙MOUT =Ωr2ρ(r)vOUT(r)=Ωr2−αρ0rα0vOUT(r) ≃Ωρ0r0rvOUT, (20)

where the inner gas density profile is typically a power-law and is the covering angle of the bipolar outflow. As shown in G17 and observational refs. within, the typical nuclear profiles for all the phases follow a slope (with  0.25 scatter), hence the last step in Eq. 20. By using Eq. 15 and 19, the entrained outflow velocity can be written as

 vOUT=√2POUT˙MOUT=η−1/2vout, (21)

which inserted in Eq. 20 yields an entrainment factor

 Missing or unrecognized delimiter for \right (22)

This implies that, while the macro velocities at a given radius are unchanged over different systems (), and are thus more robust probes, the macro outflow rate linearly increases for more massive systems . Note the mass outflow rate has much stronger relative variations than velocities (), corroborating Eq. 14-15.

Depending on the current thermodynamical background state of the system, the outflows can entrain different phases, including the hot plasma, the warm neutral/ionized gas, and the molecular gas. We use the results of the CCA simulations (G17) to retrieve the multiphase environment and profiles of the 3 phases, taking as reference macro scale  kpc. A typical plasma density  g cm at 1 kpc leads to an entrainment factor ()

 ηhot≃40 T−1x,7.4r2/31kpc. (23)

This implies median entrained mass outflow rates and velocities of 10s  yr and a few  km s, which are typical properties of observed macro ionized outflows (e.g., Nesvadba:2010; Tombesi:2013). If the halo is mainly filled with cooler gas, such as at high redshift, the entrainment can also proceed mainly via the warm ( g cm) and cold phase ( g cm)9, thus leading to more entrained outflows with

 ηwarm≃183 T−1x,7.4r2/31kpc, (24)
 ηcold≃850 T−1x,7.4r2/31kpc. (25)

Mass outflow rates with and several  yr tied to velocities and 500 km s at the kpc scale are characteristic properties found throughout observations of neutral (e.g., Morganti:2005; Morganti:2007; Teng:2013; Morganti:2015_rev) and molecular AGN outflows (Sturm:2011; Cicone:2014; Russell:2014; Combes:2015; Feruglio:2015; Morganti:2015_ALMA; Tombesi:2015), respectively (more detailed comparisons in §4).

At large radii, the outflow is halted by the external pressure, inflating a bubble and thermalizing its kinetic energy mainly via turbulent mixing (e.g., Gaspari:2012b; Soker:2016; Yang:2016). Such radius crudely corresponds to the region where the outflow ram pressure becomes equal to the hot halo pressure. Since outflow ram pressure is equal for all the phases, we can estimate the thermalization radius as , yielding via Eq. 21-23

 rth∼(55kpc)T3/2x,7.4=(55kpc)L1/2x,43.8. (26)

Above such thermalization radius, any model should simply inject thermal energy rate balancing the core . Below such radius (as resolved by most of the current MHD and cosmological simulations), any model should inject massive outflows with the above relations. Such radius roughly approaches the core radius, which is where the feedback loop is active.

In principle the momentum equation, , might be adopted instead of Eq. 14 - 15, if the outflow would immediately radiate away most of its energy. However, besides losses being likely subdominant (see Faucher:2012), the deceleration would result to be dramatic, (with reduced by a few), which would make the outflow aborted at the macro scale, inconsistently with data. Adopting the same procedure as above, the hot, warm, and molecular outflow would merely preserve 870, 280, and 90 km s at 1 kpc scale, respectively. A related crucial point to reject purely momentum-driven outflows is that self-regulation would be broken, since the macro feedback energy could not balance the core , leading to a global massive pure CF.

## 4. Comparison with observations

The proposed CCA GR-RMHD unification predicts nuclear ultra-fast outflows of the order of  km s and a progressively slower propagation of the outflow at larger radii, which are consistent with recent AGN data.

In a sample of 35 AGN, Tombesi:2013 unify the velocities of UFOs and the slower warm absorbers as a function of radial distance (see also Tombesi:2014 for analogous radio galaxy sample). Velocity is the most robust indicator (e.g., compared to mass outflow rates) since directly observed through blue-shifted absorption lines in AGN X-ray spectra. Fig. 2 shows the comparison of our model prediction (blue; §3) and the fit to the unified X-ray data. The bands denote 0.5 dex scatter, which is the typical model variation (mainly due to inner density and bipolar angle) and the range in the observed data points. The prediction of the CCA GR-RMHD model well reproduces the observed values. If the outflow would be purely driven by momentum (green line) and not energy, it would be aborted within the Bondi radius, remaining clearly below data. In other words, entrainment must occur in a gentle way, such as . In the nuclear region, the outflow tends to be slightly lower than the data, albeit within typical uncertainties. The slope of the data, -0.40, is slightly steeper than the -0.33 model. The two matches exactly if the density profile has slightly shallower (instead of 1); we did not attempt to fine-tune it, since within uncertainties of the simulated radial profiles and not granting further insight.

The mass outflow rates have very large observational uncertainties (due to the unknown geometry and projection effects) and theoretical scatter (due to the dependance, unlike the macro velocity). In the above sample, UFOs typically show , while the warm absorbers have 1.5 - 2 dex larger magnitude, which can be explained via the entrainment action ( keV). We are here not attempting to fit values of single objects; nevertheless, several X-ray studies detect nuclear  km s UFOs and ionized outflows with  km s at intermediate scale down to several 100 km s at large radii10 (see the review by Tombesi:2016 and refs. within). Follow-up observational investigations are required to better unify the radial properties of ionized outflows over a large homogeneous sample, in particular adding more low-luminosity AGN and central galaxies.

Depending on the dominant nuclear phase, the AGN ejecta can also develop into a neutral and molecular outflow. This is more common in QSOs and ULIRGs with abundant cold/warm mass with large volume filling in the core. Morganti:2005; Morganti:2007 have shown the incidence of HI outflows in several AGN, particularly radio-loud sources, via (21-cm) radio telescopes as WRST. The location of the HI outflows is 0.5 - 1.5 kpc with average velocities 1000 km s. Teng:2013 present a sample of 27 kpc-scale HI outflows detected with GBT: the average sample velocity is km s. In both samples, the mass outflow rates are uncertain (due to the dynamical time estimate), of the order of 100  yr. The above values are consistent with our median prediction of neutral outflows (Eq. 21-24) with a typical  km s and  yr at kpc scale ( keV).

In the last several years and with the advent of high-resolution radio interferometers, neutral outflows have been complemented with samples of massive molecular AGN outflows. Cicone:2014 present a sample of 19 molecular AGN outflows detected with IRAM (by using CO[1-0] emission closely tracing H gas) at the kpc scale. Averaging the peak velocity and mass outflow rates over the sample yields a velocity km s and mass rate with factor of 2 uncertainty. The sample of 6 molecular outflows in Sturm:2011 show very similar mean properties. From Eq. 25, the average molecular velocity and mass outflow rate at kpc scale is expected to be 480 km s and ( 1 keV), in agreement with the data. Other works focus on single objects, finding very similar properties at kpc scale as predicted by our model; e.g., Phoenix/A1664 BCG cores display km s and crude outflow rates (Russell:2014; Russell:2016_phoenix). A well studied multiphase outflow in both the hot and cold phase is Mrk 231 (Feruglio:2010; Feruglio:2015). IRAM data indicates a kpc-scale molecular outflow with km s and (Feruglio:2010); in the same system, Chandra and NuSTAR show the presence of a nuclear UFO with km s and . Both values are in excellent agreement with our entrainment multiphase model. Notably, the same authors remark that energy is conserved during the entrainment process, , consistently with our Eq. 8. Tombesi:2015 present another similar multiphase outflow in IRAS F11119+3257. As above, the mass outflow rates bear large uncertainties and a large sample linking the small and large radii (as done for UFOs) is currently missing; we encourage observational proposals in such unification direction. We are living a new era for multiphase AGN outflows, as the field is rapidly growing via new high-resolution ALMA cycles able to probe  km s CO outflows (as shown by Morganti:2015_ALMA).

## 5. Subgrid/SAM model for AGN feedback

Below we describe how to incorporate our model into large-scale simulations of structure formation. Let us denote the typical resolution of a given simulation by (nowadays in a typical zoom-in run). Assuming the resolution is enough to resolve the thermalization region (), we propose the following.

(i) The SMBH growth can be tracked via Eq. 12 or 13,

 ˙M∙≃(0.03˙Mcool)Tx,7.4=(0.04 M⊙yr−1)Lx,43.8,

i.e., only a few percent of the macro cold inflow rate is actually deposited into the SMBH (with coarse resolution it may be easier to estimate the cold inflow from the core , with the condition that the current central cooling time is shorter than ; see Eq. 7).

(ii) The AGN mechanical feedback is injected on the scales defined by a few with velocity given by Eq. 21,

 vOUT=η−1/2vout,

where is the entrainment factor at the resolved radial distance (Eq. 22) and is the nuclear velocity of the outflow set by Eq. 17.

(iii) The rate at which such outflow carries mass results from the entrainment mechanism given by Eq. 19,

 ˙MOUT=η˙Mout≃η˙Mcool,

where reflects the magnitude of the quenched cooling flow, which should self-consistently arise from the AGN feedback loop as a central cold inflow. The outflow can be injected as a mass flux through the boundary (e.g., sink the inflow rate and inject it back boosted by a factor ). If resolution does not permit to resolve the CCA inflow, it is better to not sink the gas and kick the gas mass per timestep over the most inner number of cells/particles (reaching ) directly in the domain (checking for stability). Such inner active mass per timestep is naturally a fair representation of the entrained mass outflow rate (as tested in Gaspari:2011b; Gaspari:2012b). A remark is that for very coarse resolutions (Eq. 26), injecting massive outflows loses physical meaning, and the average radiative energy losses should be simply balanced via thermal energy injection, since the outflows are expected to be thermalized.

Such prescription is perfectly suited to be used also in semi-analytic models (SAM), e.g., of galaxy and cluster evolution, as well as in the interpretation of observational data (limited by the instrumental – instead of numerical – resolution). Furthermore, the injected properties, in particular the efficiency, are known a priori, regardless of numerics, implying that the fine-tuning loop plaguing current cosmological runs can be avoided (typically fitting one mass range, but overheating or overcooling the opposite regime due to keeping a constant macro efficiency). In other words, there is no main free parameter involved, except for the scatter intrinsic in observations. A sanity check is to retrieve the observed X-ray properties, e.g., X-ray luminosity and temperature profiles of the group or cluster. If not, the implementation of AGN feedback is numerically flawed and shall be modified accordingly, not retuning the parameters, but changing the injection implementation and carefully assessing which hydrodynamic solver and discretization to use. In other words, retuning some parameters to counteract the numerical flaws must be avoided, and can be avoided with the above a-priori prescription, thus preserving predictability.

## 6. Discussion

We now discuss some details of the proposed model, together with the limitations and possible improvements.

The approach of this work differs from typical analytic modeling considering a perfect steady state solution (e.g., Bondi:1952) in which inflow and outflow coexist at exactly the same time (setting in the hydro equations). As indicated by X-ray observations and simulations (§1), the detailed self-regulated AGN feedback loop is time varying. We have instead considered a nearly stationary case over a feedback cycle, which is typically of the order of the central cooling time , where is the plasma cooling function (Sutherland:1993); from isolated galaxies to massive clusters the typical central cooling time of the hot gas varies from tens to several 100 Myr (Gaspari:2014_scalings). Within one cycle the process is time varying, with energy and mass changing form and phase. Specifically, the inflow acts first via the self-similar CCA rain, then the SMBH reacts to the feeding via nuclear ultra-fast outflows (Fig. 1). The propagating UFOs entrain the diffuse phase and thermalize in the core, such that , as shown by X-ray data (e.g., Main:2017). The background halo is recurrently contracting and expanding in a gentle manner, and is never evacuated; in other words the core oscillates near HSE. Over the whole core region and one loop time the mass and energy are conserved (the small mass loss onto the BH is replenished from the virial hot halo). Note that if central , the system is in a non-cool-core condition and the feeding/feedback is not currently active. A key observational evidence for a variable feeding mechanism, is the ubiquitous variability of AGN light curves. As discussed in G17 (Sec. 5.1) and King:2015_flicker, chaotic accretion drives a ‘flicker’ noise with major accretion events having Myr duration.

Needless to say a full, time-dependent treatment of the feeding and feedback process requires 3D (GR)MHD simulations covering the whole dynamical spatial and temporal range. However, until we will be able to break such computational barrier, we can rely on key properties of the inflows and outflows set by the multiwavelength constraints, which must be satisfied even in the advanced numerical runs. We remark X-ray data show that the feedback must be gentle and kinetically driven (with large-scale thermalization up to 100s kpc for massive clusters). Notice that the details of the energy conserving outflow are in our macro model not relevant. On the other hand, the momentum flux boost of the swept-up material due to the hot shocked gas and entrainment via hydro instabilities (e.g., Kelvin-Helmholtz and Rayleigh-Taylor) requires numerical simulations to be robustly understood. In addition to direct uplift, an interesting possibility to form molecular outflows is the in-situ condensation of the massive galaxy-scale hot wind via thermal instability – as discussed by Zubovas:2014 – which may further promote the subsequent precipitation phase.

In this work, we decided to aim for minimal assumptions and rely on first principles as much as possible. Further sophistications to the model are possible and can be easily incorporated to fit more specific objects, at the expense of an increased number of parameters. For instance, the inner background density profile can be modified with a more complex functional form than a single power law and/or assigning different volume filling profiles to the warm/cold phases. The configuration of the inner outflows can be modified by reducing , in order to accommodate for a thinner bipolar setup. We note, in one loop, the cold inflow can occur along one direction, while the entrained outflow may occur in the perpendicular direction, further corroborating the separation of the large-scale CCA inflow and outflow mass rate, instead of a perfectly radial steady-state solution. A time delay in the loop can be introduced by tracking turbulent Taylor number: if , then a rotating structure (disc, ring, torus) can momentarily reduce accretion. We did not aim to fit one particular system or AGN outflow in this study, discussing only mean values. As noted in §4, considering the scatter in cooling system properties, the outflow variations are  0.5 dex over a large sample. Fitting and interpreting single object data can be easily refined, e.g., by analyzing the core and nuclear X-ray spectrum both in terms of cooling rate (soft X-ray) and outflow line absorption features (hard X-ray).

Consistently with the observational results by Russell:2013 (Fig. 12), the GR-RMHD simulations (SG17) show that for accretion rates below the Eddington rate, the nuclear SMBH power is dominated by kinetic energy over the SMBH radiative output, . The mechanical, sub-Eddington mode is the long-term maintenance mode of AGN feedback (McNamara:2012 for a review) preserving hot halos and cool-core systems in quasi thermal equilibrium at least for 9 -10 Gyr (McDonald:2014; McDonald:2016; McDonald:2017). At high redshift (), the Eddington rate can be approached triggering a brief ‘quasar’ phase (seeding part of the SMBH mass). The wind may be thus radiatively driven, although its coupling with the gas is matter of ongoing debate. Moreover, there is no physical reason to think that the mechanical power from AGN is erased in this regime, as corroborated by our GR-RMHD run covering the quasar transition (see SG17). Even in such short-lived radiative regime, the outflow is still expected to be energy conserving11 (Faucher:2012) although it may be more appropriate to use a slightly larger (Novikov:1973; Merloni:2008) and rescale Eq. 9. As long as , the outflow properties are however not significantly altered. A few quasar blasts may evacuate the system, but these anomalously powerful outbursts – which are much easier to detect – must be outliers (increasing the high-redshift population scatter) otherwise the majority of systems would later remain non-cool-core, which is not observed (e.g., Gaspari:2014_scalings). Overall, regardless of the details of the driving mechanism (e.g., magnetic versus radiative), if self-regulation is on average preserved, the proposed model applies in similar way throughout cosmic time.

In the current interpretation, the micro and macro mechanical driver is a sub-relativistic outflow. Given the BH null spin expectation from chaotic accretion (King:2006) and the high piercing collimation, a radio jet is expected to be subdominant, albeit it can coexist and trace the large-scale features, as bubbles. Observationally, radio synchrotron (electron) power is less than a percent of the cavity internal power (McNamara:2012), so only relativistic ions are left to inflate a bubble; however, this would produce strong Gamma emission in all systems, which Fermi does not typically observe. Moreover, several AGN bubbles are ghost cavities devoid of radio emission. Having said that, our model is general and the radio jet interpretation can be trivially implemented, e.g., by replacing the related micro efficiency and opening angle.

A current observational limitation which is worth discussing is the low-mass end regime. While hot, X-ray halos are well detected above stellar masses , in particular massive galaxies, galaxy groups and clusters, the precise level of the X-ray luminosity due to the diffuse component in the opposite regime ( keV) is still uncertain due to the contamination of X-ray binaries (e.g., Anderson:2015). The X-ray luminosity in such regime may be lower than our adopted scaling, and the relative cooling rate (Eq. 7) should be properly rescaled if necessary. While the outflow velocities are overall unaffected (Eq. 18-21), the mass outflow rate may be lower than the expected value. Conversely, while more massive systems have better constrained core X-ray luminosity, the stronger and harder diffuse emission substantially hinders the nuclear X-ray spectral features, making UFO detection challenging. If is not available (e.g., for proto-galaxies), we suggest to use a core temperature in lower energy bands, as condensation occurs throughout the warm and cold phase regime. Finally, supernova feedback due to star formation (e.g., with rate a few precent of the galaxy cooling rate) can also become energetically important in low mass galaxies and shall be investigated in the future.

While here we have investigated the instantaneous properties as the SMBH accretion rates, , in a separate work, we will focus on the integrated properties of the proposed unified model, as the total black hole masses and related scalings (e.g., the Magorrian relation). We anticipate some important considerations. As discussed above, the CCA self-regulation has a characteristic frequency related to the cooling time, , as the hot halo requires such time to promote condensation, rain down, and then activate the ultra-fast outflow feedback. One loop requires (the outflow active time is always shorter than the condensation time). In other words, the duty cycle increases from clusters to galaxies, as corroborated by long-term AGN feedback simulations (e.g., Gaspari:2011a; Gaspari:2011b; Gaspari:2012b) and X-ray shocks/cavities observations (e.g., Randall:2015). The number of cycles over the Hubble time is thus , with an active time . The black hole masses are expected to grow as , hence with a temperature scaling given by , as core temperature is a measure of the (stellar) velocity variance in virialized structures. This is valid in the galactic regime ( - 2 keV) as remains essentially constant for solar metallicity. For clusters, due to Bremsstrahlung, thus . Observations show a very similar scaling, with ultramassive black holes found predominantly in more massive halos which are consistent with our self-regulated CCA model inducing a steepening of the Magorrian relation (e.g., Gultekin:2009; McConnell:2013; Kormendy:2013).

## 7. Summary and Conclusions

We linked for the first time the physical micro and macro mechanical efficiency of SMBHs, the latter based on key X-ray data and hydrodynamical simulations, the former retrieved by state-of-the-art GR-RMHD horizon simulations, such that and , respectively (§2). By using minimally first principles, as conservation of energy (, where the latter is the core luminosity of the hot halo), we unified the macro and micro properties of self-regulated AGN feedback from the galactic to the cluster regime (§3).

The inflow mechanism occurs via chaotic cold accretion (CCA) – probed during the last years – i.e., the rain of cold clouds condensing out of the quenched cooling flow (), which are recurrently funneled via fractal, inelastic collisions. Near hundreds gravitational radii, the binding energy of accreting gas is strongly transformed into ultrafast outflows (UFOs) with characteristic velocity of a few  km s () ejecting most of the inflowing gas mass as ( yr for intermediate systems).

At larger radii, the outflow entrains progressively more mass, such as and , with . At roughly the kpc scale, the characteristic velocities of large-scale hot/warm/cold outflows are predicted to be a few , 1000, and 500 km s, respectively (depending on the inner dominant gas phase). The related average mass outflow rates (for 1 keV systems) are expected to be of the order of 10, 100, several 100  yr, respectively. Such properties are in agreement with observations of UFOs, and kpc-scale ionized, neutral, and molecular outflows (§4). Velocities are the more robust and stable indicator compared with outflow rates, both observationally and in the model. Ultimately, the outflows thermalize within the system core (), balancing the cooling losses, and allowing another self-regulated loop to reload via CCA rain and outflow feedback – with frequency .

A key aspect of the newly presented model is that the irradiated cool-core energy rate () reflects the gas flow onto the tiny SMBH, creating a symbiotic link over a 10 dex dynamical range. The tiny SMBHs are not isolated point objects where space-time diverges, but appear to be central actors in the evolution of both the micro and cosmic structures. In particular, the SMBH growth rate is linked to the large-scale halo and thus any other cosmic scaling (e.g., ), in addition to inducing a consistent relation. Despite the necessary limitations (§6), the CCAUFO model captures the essential ingredients than any more sophisticated self-regulation model and simulation should have at its core, in particular the gentle quasi-thermal equilibrium of plasma halos.

The pursued minimalism of the CCAUFO model makes it suited to be trivially implemented in subgrid modules and semi-analytic works (§5), as well as in estimates for the interpretation of observational studies, e.g., related to nuclear and entrained outflow velocities and mass rates. The proposed model presents a simple physical unification scheme upon which construct and conduct future multiwavelength investigations, e.g., selecting the systems in terms of the core X-ray luminosity (or other related macro observable). Instead of classifying a phenomenological aspect of a peculiar AGN, we encourage observational campaigns in the direction of understanding the common, unified physics of multiphase inflows/outflows (e.g., §4) and to systematically consider the connection between the AGN and the global hot halo. A larger and homogeneous X-ray, optical, and radio sample of such properties, from low-mass galaxies to massive clusters, is needed to robustly test the link of the micro and macro properties of AGN feedback.

## Acknowledgements

MG and AS acknowledge support for this work by NASA through Einstein Postdoctotral Fellowship number PF5-160137 and PF4-150126 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. Support for this work was also provided by NASA Chandra award number G07-18121X. FLASH code was in part developed by the DOE NNSA-ASC OASCR Flash center at the University of Chicago. HPC resources were provided by the PL-Grid Infrastructure and the NASA/Ames HEC Program (SMD-16-7251). We thank B. McNamara, G. Tremblay, J. Stone, M. McDonald, R. Morganti, F. Tombesi, M. Cappi, and F. Combes for insightful discussions.

## Appendix A Core luminosity and temperature

Most of the X-ray luminosity comes from the region well within due to the steep radial density profile (emissivity is ). By using the available Chandra and XMM (losing sensitivity at large radii) luminosities is thus a fair proxy for the core luminosity. More accurately, we can model the surface brightness with a profile, , where is the projected radius and is the inner normalization. Integrating over thin annuli yields

 Lx(

The cooling radius is typically equal to the core radius (; Vikhlinin:2006), since the radial breaking naturally emerges via the loss of pressure, (Ettori:2000). Cool-core systems are better fitted by a sum of two beta models for the core and the outskirt; characteristic values are and , respectively (e.g., Ettori:2000). Plugging in this values in a double model following each Eq. A1, the average correction for the core luminosity is . Notice that the outskirts, , contribute in negligible measure, . Overall, the chosen luminosity radius does not significantly alter the results presented in §3. The temperature profile shows even less variation than density, varying by a factor 2 - 3. By emission-weighting it, the core is typically 10 percent lower than the ambient (Ettori:2000; Vikhlinin:2006) – again, a minor variation.

For an idealized self-similar spherical collapse, it is well known that (Kravtsov:2012). However, observational data show that non-gravitational/feedback processes steepen such relation as (Sun:2012). In §3, we are interested in the X-ray luminosity and temperature tied to the core/cooling region, i.e., the radius within which the temperature profile slope becomes positive (, related to , where is the Hubble time). By using the above minor corrections, the core scaling relation becomes erg s. For reference, in the local universe, the scaling between radius and temperature () is (Sun:2009a), leading to a physical core radius .

### Footnotes

1. affiliationtext:   Einstein Fellow.
2. affiliationtext:   Spitzer Fellow.
3. The radius encloses times the critical overdensity ( is the Hubble parameter;  km s Mpc) giving an enclosed mass ; for the virial radius and 500 for observational constraints.
4. The Bondi radius, , is not strictly relevant for CCA but provides a known reference intermediate (pc) scale between the macro and micro region.
5. McDonald:2017 show that cool cores are observed even up to with properties identical to local ones.
6. The term due to is subdominant and can be neglected.
7. This is also similar to the characteristic nuclear (100 - 200 ) escape velocity, i.e., as the driven outflow overcomes gravity.
8. Through the feedback cycle, the underlying halo gently expands during entrainment, and contracts after dissipation, restoring quasi hydrostatic equilibrium (HSE). X-ray observations indeed show that density profiles in cool-core systems vary only by a small amount, even after strong outbursts (e.g., McNamara:2016).
9. Here we assume that the characteristic phase densities retrieved in G17 apply over the whole inner region as a background; this is more typical at high redshift, as cold flows can penetrate deep within the growing proto-galaxy.
10. In low-mass galaxies the thermalization radius is  kpc, thus the outflow can rapidly decline in velocity (and mass rate).
11. As cooling acts on electrons, this slows down inverse Compton process; free-free cooling is secondary.
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