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Astron. Astrophys. 331, L1-L4 (1998)

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2. Model

2.1. Early Growth

We suppose that the initial black holes form via a coherent collapse. This probably implies [FORMULA] 'gsim; [FORMULA], with [FORMULA] Formation of lower mass holes would be less efficient, for at least two reasons. Primordial clouds of mass less than [FORMULA] are readily disrupted by supernova-driven winds (Dekel and Silk 1986). Given the observed efficiency of black hole formation, the formation of black holes of mass below [FORMULA] is likely to be inhibited. Moreover if the precursor object forms a supermassive star, there would be substantial mass loss. Since typical first generation clouds of primordial CDM have masses of order [FORMULA] where [FORMULA] the total mass going into black holes would be significant even if they formed in only a small proportion of clouds. Of course it is also possible that primordial clouds form smaller black holes which subsequently merge as the hierarchy develops. However this process involves two additional stages of inefficiency (via formation and merging), and we regard it as an improbable pathway. For a typical [FORMULA] galaxy to form from hierarchical merging of primordial clouds and contain a supermassive black hole, we require that the efficiency f at which the supermassive black hole formed (or equivalently, the inefficiency of fragmentation in primordial clouds) satisfies f 'gsim; [FORMULA].

How small can f be in order for the consequences to be of interest? One observes today that many, if not all, galaxies contain central supermassive black holes, and that [FORMULA] where [FORMULA] is the black hole mass and [FORMULA] is the mass of the spheroidal component (Magorrian et al. 1997; Ford et al. 1997; van der Marel 1997). Once supermassive black holes are formed, the final black hole mass is enhanced during the hierarchical merging process, when dynamical friction and dissipative drag on gas can drive supermassive black holes into the center of the developing protogalaxy. Mergers provide a continuing supply of gas, and gas dissipation and accretion feed the central black hole. The most detailed numerical simulations of protogalaxy collapse with cosmological initial conditions that have hitherto been performed demonstrate that angular momentum transfer is highly effective (cf. Navarro and Steinmetz 1997). Baryons collapse to form a dense, massive central clump at the resolution limit of the simulations, rather than a centrifugally-supported disc on a galactic scale. This line of thought at least supplies a motivation for exploring (and constraining) the hypothesis that black hole formation and growth, rather than star formation, characterizes the earliest stages of galaxy formation. We will argue that the value of f self-regulates so as to approximately satisfy the observed correlation.

2.2. Quasar Winds

Massive black holes, whenever fuelled at a sufficient rate, would display quasar-like activity. For brevity we term such objects `quasars' - noting, however, that the events we are discussing may occur at higher redshifts than the typical observed quasars. An explosion model whereby outflows from early quasar-like objects led to cooled shells which fragmented into galaxies was originally developed by Ikeuchi (1981; cf. also Ostriker and Cowie 1981); this idea has, however, fallen from favour as the principal mode for galaxy formation because post-shock Compton cooling would lead to excessive spectral distortions of the cosmic microwave background spectrum. We consider here the (more localised) effects on the gas within the protogalaxy in which the quasar is embedded.

The effect of a protogalactic wind may be estimated as follows. We model a protogalaxy as an isothermal sphere of cold dark matter that contains gas fraction [FORMULA] with density [FORMULA], constant velocity dispersion [FORMULA], and mass [FORMULA] In massive halos, [FORMULA] A sufficiently intense wind from the central quasar can sweep up the gas into a shell, and push it outwards at constant velocity

[EQUATION]

In this expression, the mechanical (i.e. wind) luminosity is taken to be a fraction [FORMULA] of the Eddington luminosity [FORMULA] Note that [FORMULA] where [FORMULA] is the radiation efficiency.

Expulsion of this shell requires that its velocity should exceed the escape velocity from the protogalaxy: i.e. [FORMULA] The condition for this to be the case is that

[EQUATION]

where [FORMULA] and [FORMULA] If for example [FORMULA], black holes could in principle eject all the material from their host galaxies when their masses exceed [FORMULA] If the situation were indeed spherical, then, even if the central source switched off, the outflow would continue to expand into the intergalactic medium for up to a Hubble time before stalling due to the ambient pressure. The shell velocity decreases via an explosive outflow according to

[EQUATION]

for an explosion energy, equal to the kinetic energy at breakout, of [FORMULA] ergs and an intergalactic medium density equal to 5 percent of the Einstein de Sitter density (with [FORMULA]), as compared to the binding energy of the gas in a massive protogalaxy of [FORMULA] erg. The ambient pressure is not high enough to halt the shell, initially moving at breakout at a velocity of [FORMULA], until z 1. The shell radius is [FORMULA] and the fragment mass is (Ostriker and Cowie 1981)

[EQUATION]

where [FORMULA] is the sound velocity within the shell.

How effective this expulsion would actually be, depends on the geometry of the outflow and on the degree of inhomogeneity of the protogalactic gas. Realistically, the outflow may be directional (probably bipolar) rather than spherically-symmetric. Some gas may survive and even be overpressured by double radio lobes to form stars (Begelman and Cioffi 1989)

It is even possible (Natarajan et al 1997) that swept-up material, after cooling down into a dense shell, may fragment into a class of dwarf galaxies; these would differ from normal dwarfs in not being embedded in dark halos, and consequently be more liable to disruption by massive star formation. Hence a high-mass black hole has two contrasting effects. It inhibits star formation in its host halo by blowing gas out; on the other hand, the ejected gas may eventually pile up in a cool shell that breaks up into small galaxies.

2.3. Protogalaxy Core

The implications of (1) are that a massive hole, if it continues to emit at close to the Eddington rate, could expel gas completely from its host galaxy. (The amount of gas that has to be accreted is in itself negligible in this context: each unit mass of gas accreted into the hole can release enough energy to expel of order [FORMULA] times its own mass from the far shallower potential well of the halo.) However, the fuelling demands a continuing accumulation of gas in the centre (probably supplied by hierarchical merging). We can therefore interpret (1) as setting an upper limit to the mass of a hole that can exist in a galaxy where star formation can proceed efficiently.

Expression (1) gives a relation between a black hole and its surrounding halo. If a significant fraction of the Eddington luminosity emerged in `mechanical' form, this gives the criterion that the energy liberated on the dynamical timescale of the halo is equal to the gravitational binding energy. The same expression can be derived by a different argument. The maximum rate at which gas can be fed towards the centre of a galaxy (as the outcome of mergers, etc) is [FORMULA] where [FORMULA] is the velocity dispersion in the merged protogalactic system. A quasar could expel all this gas from the galactic potential well on a dynamical timescale if its mass exceeded a critical value obtained by requiring that [FORMULA]

This is indistinguishable from the observed relation between the masses of central holes and those of their host spheroids (e.g. Magorrian et al. 1997). The observed relation has considerable scatter but hints at a dependence of black hole on spheroid mass that rises more rapidly than linearly: e.g. the Milky Way has a central black hole mass of [FORMULA] whereas M87, with a spheroid mass that is only [FORMULA] larger than that of the Milky Way, has a central black hole of mass [FORMULA].

We therefore hypothesize that, in the merger process leading to the formation of typical galaxies, the hole mass stabilises near the critical value. Hierarchical merging, augmented by continuing black hole growth, helps maintain the black hole-to-spheroid mass ratio at the self-regulation level. We suggested in Sect. 1, however, that in the first bound systems gas may accumulate into a single compact unit and evolve into a supermassive hole. If single black holes are indeed favoured over star formation in this way, the holes in these first systems would be far above the `critical' value appropriate to such small halos with low velocity dispersions: in other words [FORMULA] Star formation would then be inhibited (except in a disc close to the central quasar) until, via mergers, the halos had become large enough to bring the mass of the actual central hole below the critical mass (which, as we have seen, grows faster than linearly with halo mass). Thereafter, the scaling would be maintained.

If. after mergers had formed high-mass halos, the central hole were still above the critical mass, the implications would be ominous for galaxy formation. Such a system would end up as a supermassive black hole embedded in a low surface brightness galaxy. Around the central host quasar, one would expect to find a cavity of hot gas. Compton cooling at high redshift will tend to quench any associated extended radio emission. If large radio sources were responsible for the hot gas bubbles, then the geometry is not a sensitive issue. Indeed, double lobes are as effective as spherical outflows and are more reminiscent of the geometry of the associated hot gas bubbles with the similar energetics of reported Sunyaev-Zeldovich decrements that have no apparent associated galaxy cluster but with possibly associated quasars (Jones et al. 1997; Partridge et al. 1997). Natarajan et al. 1997) argue that one might expect to detect a shell of newly formed Magellanic irregular-type galaxies at the periphery of such bubbles.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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