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A wealth of issues dominate our understanding of the central regions in galaxies and their role in the overall galaxy evolution on cosmological timescales - the secular evolution. But is there a dynamically distinct central kiloparsec region in galaxies? The answer appears to be positive, as a major resonance between the bar and/or spiral arm pattern speed, Omegap, on the one hand and the linear combination of the epicyclic frequency, kappa, and the angular velocity, Omega, on the other is positioned in this area, i.e.,

Equation 11 (11)

Incidentally, the right-hand side of this relation represents the precession frequency of stellar orbits. The resonance between the orbit precession frequency and the pattern speed is called the inner Lindblad resonance (ILR). A multiple number of ILRs can exist in the neighbourhood but typically their number does not exceed two. The ILR(s), if they are not saturated, dampen the propagation of waves in the stellar `fluid'. This resonance can trigger various processes in the region, e.g., gas accumulation in the form of nuclear ring(s), nuclear starbursts, nuclear bars, etc. (Shlosman 1999, and references therein). The ILR(s) can pump the kinetic orbital energy into vertical stellar motions. So while there are naturally strong interactions between the inner and outer disks, different processes dominate both regions.

The next question is whether there is a morphologically distinct central kiloparsec region in galaxies. The answer is positive again - the ILR(s) act(s) as separators between the inner and outer disk, resulting in detached bars and spiral patterns. The inner region is generally dominated by the bulge and hosts the SMBH.

A number of important issues, which also include the inner kpc directly or indirectly, are discussed by Lia Athanassoula and James Binney (this volume). We shall attempt to avoid unnecessary overlap, although some overlap is actually welcomed. In the discussion below, we shall focus, therefore, on various asymmetries in the mass distribution that drive the evolution, such as disk and halo asymmetries, large-scale stellar bars (briefly), and the dynamics of nested bars. In Section 8.2 we shall touch on the issues related to the formation and evolution of SMBHs at high redshifts. We have already reviewed, to some extent, the feedback from AGN in Section 6. The immediate environment of the SMBHs, e.g., the role of molecular tori, is beyond the scope of this discussion.

Two types of torques can have a dramatic effect on the dynamics within the central kpc, namely magnetic and gravitational torques. The former can dominate the central 1-10 pc from the SMBHs, while the latter can have an effect outside this region, on scales of gtapprox few tens of parsecs. Viscous torques can be important near the major resonances and can be neglected in other regions, in comparison with magnetic and gravitational torques.

8.1. Bars and the morphology of the central kpc

Stellar bars can be formed either as a result of a (spontaneous) break of axial symmetry - the so-called bar instability (e.g., Hohl 1971), or via tidal interaction between galaxies (e.g., Noguchi 1988) or between galaxies and DM subhaloes (e.g., Romano-Díaz et al. 2008b). Stellar bars themselves are subject to dynamical instabilities and secular evolution which affect the disk as well. Of these, we shall single out the vertical buckling instability (e.g., Combes et al. 1990). This instability has both dynamical and secular aspects. Dynamically, this instability exhibits a spontaneous break of the equatorial symmetry in the rz plane (e.g., Pfenniger & Friedli 1991; Raha et al. 1991). The action of the vertical ILR effectively converts the rotational kinetic energy of the star in the disk into vertical oscillations. This results in a vertical thickening of the stellar disk at radii smaller than the position of the vertical ILR, and in the appearance of a characteristic peanut/boxy-shaped bulge. The symmetry is always restored on the dynamical timescale (e.g., Fig. 17, note the flip-flow at ~ 2.3-2.4 Gyr). Moreover, if the equatorial symmetry is (artificially) imposed, this bulge nevertheless appears, although on a longer timescale and driven by the same resonance.

Figure 17

Figure 17. The recurrent buckling instability. Upper frames (from Martínez-Valpuesta et al. 2006): Evolution of the vertical structure in the bar: edge-on view along the bar's minor axis. The length is given in kpc and the values of the projected isodensity contours are kept unchanged. The time in Gyr is given in the upper-right corners. Note the bar flip-flop at 2.3-2.4 Gyr and the persistent vertical asymmetry at 5.2-7.5 Gyr. Lower frames: smoothed version of the above figure at 9.4 Gyr (left), and a matching galaxy from HCG 87 group of galaxies (Hubble Heritage Team), courtesy of J. H. Knapen (right).

What the low-resolution simulations have failed to capture, and what has been obtained by Martínez-Valpuesta et al. (2006) for the first time, is the recurrent break in the equatorial symmetry occurring on a much slower timescale of a few Gyr, around 5.2-7.5 Gyr. This slow buildup of the bar asymmetry long after the first vertical buckling occurred is rooted in the secular evolution of stellar orbits driven by the low-order vertical resonances. Unlike the first buckling, the second phase displays persistent asymmetry. The Fourier amplitude of the symmetry breaking decreases with the next stage of the instability.

About 50% of edge-on disks show peanut/boxy bulges (Martínez-Valpuesta et al. 2006 and references therein), which appear to be a clear signature of stellar bars. While we do understand the reasons for the dynamical stage of the buckling instability, we cannot predict the onset of the (second) secular stage of this instability. Models with a gas component show that the amplitude of this instability decreases with increasing gas fraction (Berentzen et al. 1998; 2007).

8.1.1. Nested bars: observational perspective

Getting rid of the angular momentum is a major issue for astrophysical systems (e.g., Sections 2 and 3). Given that a substantial mass is involved, gravitational torques appear as the most efficient mechanism for redistribution of angular momentum on various spatial scales. The formation of disks, therefore, is a reflection of the inefficiency of this process. Gravitational torques are triggered by a non-axisymmetric distribution of matter. The most frequent asymmetry in the disk is in the form of spiral arms, which lead to torques with an amplitude of ~ 1% - a quasilinear perturbation, if defined as a ratio of tangential to radial acceleration. On the other hand, bars are strongly non-linear perturbations on the level of ~ 10-100%. The importance of bars, at least after z ~ 1, is reflected in the existence of the branch of barred galaxies in the Hubble fork, although the exact statistics is still being debated (e.g., Jogee et al. 2004; Sheth et al. 2008).

But the efficiency of bars in extracting angular momentum, for example from the gas, is limited by a decade in radius, due to a strong decay in the gravitational multipole interactions (Shlosman et al. 1989, 1990). So it is only natural that bars `repeat' themselves on progressively smaller scales. In retrospect, it is not surprising that such `bizarre' systems of nested bars exist in Nature.

The first known observation of a system with nested bars has been performed on NGC 1291, classified as an SB0/a galaxy (Evans 1951). An inner twist of the optical isophote has barely been detected, and explained as an inner spiral. A much later observation of this object resulted in a large-scale bar of ~ 9.9 kpc and a nuclear bar of ~ 1.8 kpc, misaligned at ~ 30°, with the inner bar leading the outer bar (de Vaucouleurs 1975). A question has been asked on whether the presence of a second bar in NGC 1291 is an `oddity of Nature' or is a fairly common, perhaps typical, structural feature. De Vaucouleurs concluded that `the lens-bar-nucleus structure on two different scales in barred lenticular galaxies is probably not rare, and raises an interesting problem in the dynamics of stellar systems' but did not follow up on this issue. A morphological survey of 121 barred galaxies has revealed additional objects with inner structure misaligned with the outer bar, but this has been interpreted as a bulge distorted by the large-scale bar (Kormendy 1979). In a subsequent study, Kormendy (1982) has analysed bulge rotation in barred galaxies and summed up that the kinematics of these triaxial bulges is similar to those of bars while the light distribution is as in elliptical galaxies. All these bars have been stellar in origin. Their mutual interactions have not been discussed and they have not been considered as a dynamically coupled system. Shlosman et al. (1989, 1990) have suggested that nuclear bars can be of multiple types, and that nested bars form a new dynamically coupled system which redistributes angular momentum in galaxies.

In principle, nested bars can be of a few types: stellar/stellar, stellar/gaseous and gaseous/gaseous. The first two types have been observed now, while the third type has not yet been observed 4 In addition, we do not count as a separate class stellar nested bars which are gas-rich or vice versa.

The first catalogues of double-barred galaxies have been published recently. Laine et al. (2002) used an HST sample of 112 galaxies in the H-band, Erwin & Sparke (2002) analysed a sample of 38 galaxies in the optical, and Erwin (2004) considered 67 galaxies, mostly from Laine et al.. The main conclusion of these studies has been that about 25% ± 5% of all disk galaxies host nested bars, and about 1/3 of barred disks possess secondary bars. These numbers show decisively that galaxies with nested stellar bars are not a marginal phenomenon, but form an important class of dynamical systems.

Molecular gas is abundant in galactic centres, but it is not clear what fraction of this gas is in a `barred' state. While there have been surveys of molecular gas within the central ~ kpc, the available resolution was insufficient for the detection of nuclear gaseous bars. Some nearby galaxies host gaseous bars, e.g., IC 342, NGC 2273, NGC 2782, Circinus, etc., but their statistical significance is not clear. Surveys with ALMA (the Atacama Large Millimetre/submillimetre Array) will probably resolve this issue.

Detecting gas-dominated nuclear bars requires surveys of gas morphology and kinematics in the central few hundred parsec of disk galaxies at a resolution of ~ 10 pc. An additional detection problem can arise from gaseous bars being very short-lived - an issue related to the stability of these objects, which we discuss below.

An analysis of the nested bars in the Laine et al. (2002) sample has clarified some of the basic properties of nested bar systems. Firstly, it has shown that the size distributions of large-scale and nuclear bars differ profoundly and exhibit a bimodal behaviour. Whether bar sizes are taken as physical or normalised by the galaxy size, r25, there is little overlap between their distributions. In physical units, this division lies at r ~ 1.6 kpc, in normalised - around r / r25 ~ 0.12. This bimodality can be explained in terms of a disk resonance, the ILR, and the above radii can be identified with the position of this resonance. The ILR acts, therefore, as a dynamical separator. The ILRs are expected to form where the gravitational potential of the inner galaxy switches from three-dimensional to two-dimensional. This normally happens at the bulge-disk interface, or alternatively, where the disk thickness becomes comparable to its radial extension. To summarise, while the sizes of large-scale bars in nested bars, as well as those of single bars, exhibit a linear correlation with disk size, nuclear bars do not show the same behaviour.

Secondly, nuclear bars almost always come in conjunction with large-scale bars - a clear signature that this is a prerequisite for their existence, although one can envisage a scenario where they form separately. Thirdly, the size distribution of nuclear rings in these galaxies peaks at the same radii of r / r25 ~ 0.12 (see above), which signals again at the crucial role the ILR plays in the dynamics of these systems. Finally, nuclear bars in nested systems have smaller ellipticities than their large-scale counterparts.

The search for stellar nuclear bars is limited to optical and NIR surface photometry - an insufficient method based on the isophote fitting which can be affected by the presence of nuclear clusters, starbursts and dust. A suggestion to look for the characteristic offset dust lanes delineating shocks in nuclear bars, similar to those observed in large-scale bars, did not work. The reason for this is a different gas-flow response in these systems (see below). Nuclear bars are not scaled-down versions of large-scale bars.

Probably the mostly intriguing aspect of nested bars is that their pattern speeds are different, at least during some particular stages of evolution, as predicted (Shlosman et al. 1989, 1990), supported by the detection of random mutual orientations of bars (Friedli et al. 1996), and confirmed in a direct measurement of their pattern speeds (Corsini et al. 2003).

8.1.2. Theoretical perspective: bars-in-bars mechanism

From a theoretical point of view, nested bar systems provide a great laboratory to study non-linear dynamics in physical systems. How do such systems form? Are they long- or short-lived? What is their role in driving the secular evolution of disk galaxies? As in the case of single bars, numerical simulations of such systems are indispensable.

The first attempts to simulate pure stellar nested bar systems have succeeded in forming both bars via the bar instability, but the lifetime of such a system was short (e.g., Friedli & Martinet 1993). The problem with the lifetime has been purely numerical - when the resolution of N-body simulations has been increased, the system of two stellar bars has lived indefinitely long (e.g., Pfenniger 2001). The issue of the initial conditions for such systems, however, is much more fundamental. It is quite revealing that in pure stellar systems one can create nested bars only by assembling the stellar disk as bar-unstable on both large and small spatial scales - naturally, because of the shorter timescale of the inner bar, it will develop first. But why would Nature create such a strongly bar-unstable stellar disk? No process known to us can create such a disk by means of a dissipationless `fluid'. Therefore, strong arguments appear in favour of a dissipational process which naturally involves the gas (Shlosman 2005, and references therein). The initial conditions necessary to create a nested bar system become nearly a trivial matter when dissipation is involved.

The role of stellar bars in angular momentum redistribution is most important when the gaseous component is involved. Bars are very efficient in extracting of the angular momentum from the gas (and during the bar instability also from the stars) and depositing it in the outer disk, beyond the corotation radius. The ability of the DM halo to absorb the angular momentum has been noticed long ago (e.g., Sellwood 1980; Debattista & Sellwood 1998) and quantified in terms of the lower resonances recently (Athanassoula 2002, 2003; Martínez-Valpuesta et al. 2006).

The outward flow of the angular momentum in barred galaxies is associated with the inward flow of the gas. Skipping the details, this gas accumulates within the central kpc, where the gravitational torques from the large-scale bar are minimised (e.g., Shlosman 2005). The star formation in the bar is largely dampened because of the substantial shear behind the large-scale shocks. So one should not be concerned with gas depletion before it crosses the ILR. Hence, the action of the large bar leads to a radial inflow - the rate of this inflow depends on a number of factors. In the cosmological setting which is the subject of this review, the outer bar will interact with and can in fact be triggered by the asymmetry (e.g., triaxiality) of the DM halo (Heller et al. 2007a), or by interaction with DM substructure (Romano-Díaz et al. 2008b).

The accumulation of gas within the ILR can and probably does lead to a nuclear starburst in nuclear rings. If, however, the gas inflow rate across the ILR is high enough, or the star formation is dampened by a local turbulent field, the stellar/gaseous fluid can trigger the formation of a nuclear bar (Shlosman et al. 1989), when the gas becomes dynamically important and triggers an additional responce from the stellar fluid, as confirmed by numerical simulations (Friedli & Martinet 1993; Englmaier & Shlosman 2004). In this respect, the large-scale bar is a primary and the nuclear bar is a secondary feature (Fig. 18a). This is the so-called bars-in-bars mechanism.

Figure 18

Figure 18. Formation and evolution of nested bars with dissipation (Englmaier & Shlosman 2004): (a) Schematic structure of a nested bar system; (b) Specific angular momentum, ellipticity, and angle between the primary and secondary bars (top to bottom). Note the correlation between the shape and the angle between the bars; (c) Evolution of ellipticity (dotted line) and semimajor axis (thick dots) of the secondary (nuclear) bar during decoupling; (d) Pattern speed of the primary (dotted line) and secondary (solid line) bars during decoupling, in the inertial frames.

From a dynamical point of view, it is challenging to explain how two gravitational quadrupoles tumble with different pattern speeds, Omegap (primary) and Omegas (secondary), without exerting a braking effect on each other. Such a system can serve as an astrophysical counterpart of a system of coupled oscillators which is a familiar tool to study non-linear behaviour. Essentially, three dynamical states exist for such a system, but only two of them appear dynamically long-lived. The first state involves two bars with equal pattern speeds - this is a stable state and the bars stay perpendicular to each other, which is energetically advantageous. The second state is made out of two bars which tumble with different pattern speeds, but their ratio is fixed, Omegas / Omegap ~ constant. In a way, the first state is a special case of the more general second one, and is also long-lived. The third case consists of two bars which tumble with different pattern speeds, where their ratio is time-dependent and evolves on a dynamical timescale - this is a transition state between two stable states with the fixed ratio of pattern speeds. Naturally, this state is a short-lived one. We call the first two states coupled 5 and the third one decoupled.

The explanation for the coupled state is based on a non-linear mode coupling: these modes exchange energies and angular momentum and, therefore, support their pattern speeds which otherwise would decay exponentially due to the gravitational torques. When the bars are locked in Omegas / Omegap ~ constant, a local minimum should exist in the efficiency of energy transfer between the bars. A strong resonance which can `capture' the bar is necessary. One would expect the low resonances to play a major role in the locking process, especially the ILR (e.g., Lichtenberg & Lieberman 1995; Tagger et al. 1987; Shlosman 2005).

Numerical simulations enable one to follow the decoupling process in nested bars (Englmaier & Shlosman 2004). Figures 18c,d provide an example of this process when a secondary bar, which forms within the ILR of the primary bar and obeys Omegas / Omegap ~ 1, speeds up in a short time, until its corotation radius moves to the ILR position of the primary bar, and Omegas / Omegap → 3.6. The shape of the nuclear bar depends on their mutual orientation (Fig. 18b), and is dynamically important - the bar axial ratio is one of the measures of its strength and, therefore, determines the fraction of chaotic orbits there, which is a measure of stochasticity within the bar and its possible demise. Another example in the cosmological setting describes the evolution of a nested bar system which is locked in two different coupled states and transits between them (Heller et al. 2007a).

Different methods have been developed to quantify nested bar systems, especially to measure the amount and the effect of multi-periodic and chaotic orbits. As we are interested in the secular evolution of these systems, we only mention that the orbital structure associated with long-lived nested bars has been investigated. A counterpart of periodic orbits in single bars is based on the fruitful concepts of a loop (Maciejewski & Sparke 2000) and on that of the Liapunov exponents (El-Zant & Shlosman 2003). Orbit analysis based on these two concepts has demonstrated that orbits are dominated by the potentials of single bars with the possibility of migration from bar to bar across the interface between the bars (El-Zant & Shlosman 2003; Maciejewski & Athanassoula 2007).

The gas dynamics at the bar-bar interface is determined by the already irregular gas flow perturbed by the secondary bar with Omegas / Omegap > 1. For such bars, the inflow is chaotic and dominated by shocks. Strong dissipation in the gas will not allow it to settle on stable orbits there. Instead the gas will fall to smaller radii (e.g., Shlosman & Heller 2002). The immediate corollary is that one should not expect starbursts throughout secondary bars, except in the central regions. Shocks and shear within the bar would slash molecular clouds reducing the SFRs. In particular, the mode where star formation occurs in massive stellar clusters should be absent in secondary bars, except (maybe) in the central region, although even there it can be dampened if the gas can be maintained in a strongly turbulent state.

The situation is very different for bars with Omegas / Omegap = 1. In this case one should observe a relaxed flow, and a `grand-design' shock system, but no random shocks. The dissipation is decreased compared to other cases.

8.1.3. Nested bars: evolutionary corollaries

Bars are known to channel the gas to the central kpc. Over a Hubble time, bars are capable of affecting the angular momentum profile in the stellar component as well. This process acts slowly but relentlessly in changing the mass distribution in galaxies. Occasionally, during various instabilities, bars are capable of increasing the central mass concentration in galaxies on short dynamical timescales (e.g., Dubinski et al. 2009). Nested bars are the result of this evolution when the amount of gas moved by the primary bar is able to change the stellar dynamics inside the central kpc. Generally, when the gas has reached ~ 10% of the mass fraction in the central regions, it can affect the stellar dynamics there. When a secondary bar forms, the local conditions for star formation will be altered as well. The decrease in the SFR will eliminate the ISM sink and facilitate further radial infall of gas. Further surveys of gas kinematics in the central region should answer the question of whether these flows fuel the AGN.

The fuelling of AGN is of course one of the outstanding issues in galaxy evolution. Are they fuelled locally, say by a `neighbourhood' stellar cluster or by the main body of the host galaxy? Diverging views prevail on this subject. The duty cycle of AGN is not known at present. Does it depend on the class of AGN, say QSOs versus Seyferts? If the AGN are fuelled by the bars-in-bars mechanism, that characteristic timescale of the process can be as short as ~ 106 yr, as gaseous bars are short-lived. What fraction of the gas ends up in the accretion disk around the SMBH?

What is clear is that preferring local mechanisms (e.g., star clusters) in fuelling the AGN does not solve the issue, as it begs the question of what fuelled the formation of local stars. Rather, one can argue that star formation in the vicinity of the SMBH is a by-product of the overall gas inflow to the galactic centre. Such an inflow will always be associated with compression and star formation along the AGN fuelling process. The following scenario can actually lead to an anti-correlation of gaseous bars with AGN activity: if the gaseous bar activates the AGN cycle and as a by-product the local star formation, the AGN and stellar feedback would disperse the gas in the next stage.

What is the fate of gaseous bars? The central feedback can drive the local azimuthal mixing of the `barred' gas component if the energy is deposited in the turbulent motions in the gas, and ultimately contribute to the growth of the stellar bulge, disky or classical. Strong winds driven by the nuclear starburst can be a by-product. Alternatively, these winds can be driven via hydromagnetic winds, as discussed earlier. Such extensive outflows from the centres of AGN host galaxies have recently been detected.

The study of nuclear bars, especially gaseous ones, will proceed quickly when ALMA comes online. We have omitted interesting options which can in fact be detected by upcoming observations of these objects. One of these is the occasional injection of the gaseous component in the nuclear stellar bars. How does this influence the evolution of the system, and especially the gas inflow toward the central SMBH?

8.2. The origin of SMBHs: the by-product of galaxy evolution?

QSOs have been detected so far up to z ~ 7, when the age of the Universe was substantially less than 1 Gyr. Even more intriguing is the inferred mass of their SMBHs, gtapprox Modot. Within the framework of hierarchical buildup of mass, these objects must originate in rare, highly over-dense regions of the Universe.

Indeed, recent studies of high-z QSOs have indicated that they reside in very rare over-dense regions at z ~ 6 with a comoving space density of ~ (2.2 ± 0.73)h3 Gpc-3. Numerical simulations give a similar comoving density of massive DM haloes, of a few × 1012 Modot. The caveat, however, is that this similarity depends on the QSO duty cycle - the fraction of time the QSO is actually active (Romano-Díaz et al. 2011a). For a duty cycle which is less than unity, QSOs will appear less rare and reside in less massive haloes. On the other hand, these QSOs are metal-rich and are therefore plausibly located at the centres of massive galaxies.

The causal connection between AGN and their host galaxies is a long-debated issue, and substantial evidence has accumulated in favour of this relation. SMBHs are ubiquitous. If the formation and growth of SMBHs is somehow correlated with their host galaxy growth within the cosmological framework, what are the possible constraints on the formation time of these objects? What are the possible seeds of SMBHs?

8.2.1. SMBH seeds: population III versus direct collapse

Population III (Pop III) stars can form BH seeds of M ~ 102 Modot (e.g., Bromm & Loeb 2003). An alternative to this is the so-called `monolithic' (or direct) collapse to a SMBH (Begelman et al. 2006). A gas with a primordial composition, which has a cooling floor of Tgas ~ 104 K, can collapse into DM haloes with a virial temperature of Tvir gtapprox Tgas ~ 104 K. This corresponds roughly to Mvir ~ 108 Modot. In a WMAP7 (Wilkinson Microwave Anisotropy Probe) universe, this can involve about ~ 2 × 107 Modot baryons. If the gas has been enriched previously by Pop III stars, the halo mass can be smaller and the amount of baryons involved can be smaller as well.

The Pop III SMBH seeds will be required to grow from ~ 102 Modot to gtapprox 109 Modot in less than ~ 6 × 108 yr, i.e., from z ~ 20 to z ~ 7. The e-folding time for SMBH growth is (Salpeter 1964):

Equation 12 (12)

where LE is the Eddington luminosity and epsilon ~ 0.1 is the accretion efficiency. The growth time to the SMBH masses estimated for high-z QSOs is

Equation 13 (13)

which is uncomfortably long and close to the age of the Universe. If the SMBH growth is via mergers, this requires frequent merger events. When the merging rate is too high, SMBHs can be ejected via slingshots, which should limit the efficiency of this process. If the dominant mode is accretion, it will be limited by ~ LE and maybe by the feedback, which again would limit the efficiency, or even cut off the accretion process.

Alternatively, monolithic (direct) collapse will require a growth from ~ 106-7 Modot to ~ 109 Modot, from z ~ 12-20 to z ~ 6-7. This will involve fewer mergers, but growth by accretion may lead to fragmentation of the accretion flows, star formation and depletion of the gas supply for the seed SMBH.

The collapse rates can be estimated from dot{M} ~ v3 / G (Shlosman & Begelman 1989), where v is the characteristic infall velocity, which results in mass accretion rates of ~ 10-4 to 10-5 Modot yr-1 for a stellar collapse with virial temperatures ~ 101-2 K, ~ 10-2 to 10-4 Modot yr-1 for a Pop III collapse with ~ 102-3 K, and gtapprox 10-1 Modot yr-1 for a direct collapse to the seed SMBH with gtapprox 104 K.

However, a spherical collapse over decades in r is improbable - the angular momentum barrier will stop it sufficiently quickly. Numerical simulations of a baryon collapse into 108 Modot DM haloes have shown explicitly that the angular momentum plays an important role (e.g., Wise et al. 2008), as expected of course. The baryon collapse in the presence of a typical halo angular momentum, lambda ~ 0.05, will develop virial turbulent velocities, driven by the gravitational potential energy. The turbulence is supersonic. The collapse can proceed until the angular momentum barrier stops it. If the system reaches equilibrium, the turbulent velocities will decay and the disk will fragment forming stars. However, the ability of the flow to fragment depends on its equation of state, and should not be taken for granted (e.g., Paczynski 1978; Shlosman et al. 1990). Begelman & Shlosman (2009) find that such a decay in the turbulent support will trigger a bar instability in the gaseous disk, before it fragments. In other words, the global instability sets in before the local instability develops. The bar instability will create intrinsic shocks in the gas and the collapse will resume - again pumping energy into turbulent motions. This is the same bars-in-bars runaway scenario as discussed in the previous section.

The turbulent support is especially helpful in suppressing fragmentation in ~ 108 Modot DM haloes when the baryons are metal-rich. Previously, it was suggested that the fragmentation in metal-rich flows in such haloes would suppress the formation of a SMBH, and lead to star formation instead.

Supersonic turbulence requires continuous driving. A reliable diagnostic of this flow is the density PDF, discussed in Section 6. The log-normal PDF depends on the density fluctuations around some average rho0, or x ident rho / rho0 when normalised (Fig. 19). The supersonic turbulence developing during the overall collapse will generate gas clumps. Two spatial scales characterise this turbulence - the Jeans scale and the so-called transition scale, below which the flow becomes supersonic, i.e., in essence the clump size.

Figure 19

Figure 19. Simulation of direct collapse to the SMBH (from Choi et al. 2012). Left: the projected gas density along the rotation axis at an intermediate time of t ~ 4.6 Myr shows the central disk-like configuration. The frame captures the central runaway collapse forming one of the gaseous bars on a scale of ~ 1 pc. The collapse proceeds from the initial scale of ~ 3 kpc - the DM halo virial radius. The overall gas density profile at this time is propto r-2. Right: a representative log(PDF) of the gas density field in the left frame (blue histogram line). The red solid line is the analytical counterpart of the log-normal PDF (see text). The close match between the two distributions confirms the fully developed turbulence field in the collapsing gas.

A fraction of these clumps will be Jeans-unstable. However, only the fraction of these clumps that contract on a timescale shorter than the free-fall time for the overall collapse can actually contribute to the star formation. These `active' clumps should have x gtapprox xcrit ~ (vturb / vK)2 M2, where vturb is the typical turbulent velocity, vK is the Keplerian velocity and M is the Mach number of the supersonic flow. Other clumps, even when Jeans-unstable, will be swept away by the next crossing shock. To estimate the fraction of star-forming clumps, one should integrate the PDF from xcrit to ∞. For M gtapprox 3, this fraction is ltapprox 0.01. Hence in a flow with a mildly supersonic turbulence the SFR is heavily dampened.

8.2.2. Direct collapse: two alternatives

Present numerical simulations of the direct collapse to the SMBH cannot answer the ultimate question about how and when the SMBH will form. At small spatial scales new physical processes should become important, such as radiative transfer, as the matter becomes opaque. This can be estimated to happen at ~ 1013 alpha sigma103(kappa / kappaes) cm, where sigma10 ident sigma / 10 km s-1, and kappaes is the electron scattering opacity. At what stage will the bars-in-bars cycle be broken? What are the intermediate configurations that lead to the SMBH?

The fate of the direct collapse has been analysed by Begelman et al. (2006) under the assumption that the angular momentum is unimportant. Under these conditions, there is no preferential channel for energy release. The photon trapping in the collapsing matter results in the formation of a single accreting massive object in a pressure equilibrium, termed a quasistar. The follow-up evolution leads to thermonuclear reactions within the object for ~ 106 yr. At some point, neutrino cooling becomes important, which defines the quasistar core, ~ 10 Modot. Neutrino-cooled core collapse will determine the seed SMBH mass. The subsequent rapid super-Eddington growth of the seed will result in M ~ 104-6 Modot.

What is the possible alternative to this scenario? Choi et al. (2012) have assumed that the angular momentum dominates at some spatial scale of the inflow and it does indeed define the preferential channel for energy release. Under these conditions, photon trapping may not be important, and an optically-thick disk/torus will form instead of a quasistar. This is a way to bypass the thermonuclear reaction stage as well, but at the price of additional dynamical instabilities. The SMBH seed which forms will be more massive than in the previous case, M ~ 106-7 Modot.

4 One can envisage also the existence of gaseous/stellar bars, where the large-scale bar is gaseous and the inner bar is stellar. Back.

5 Note that various definitions of coupled/decoupled states exist in the literature, and are refined occasionally by the same authors, including in our own work. Back.

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