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Now consider what we know empirically about the abundance of radio AGN at high redshift, and what constraints this information may set on models of structure formation.

4.1 Observational results

No significant new datasets relevant to the luminosity function of powerful radio sources have been published since the study of the RLF published by James Dunlop and myself in 1990. This was based on nearly-complete redshift data on roughly 500 sources down to a limit of 100 mJy at 2.7 GHz, plus fainter number-count data and partial identification statistics.

The main conclusions of this study were firstly to affirm long-standing results (Longair 1966; Wall, Pearson & Longair 1980) that the RLF undergoes differential evolution: the highest luminosity sources change their comoving densities fastest. Nevertheless, because the RLF curves, the results can be described by a model of pure luminosity evolution for the high-power population, in close analogy with the situation for optically-selected quasars (Boyle et al. 1988). The characteristic luminosity in this case increases by a factor appeq 20 between the present and a redshift of 2. Similar behavior applies for both steep-spectrum and fiat-spectrum sources, which provides some comfort for those wedded to unified models for the AGN population. There is a remarkable similarity here to the evolution of `starburst' galaxies, distinguished by blue optical-UV continua and strong emission from dust which make them very bright in the IRAS 60-µm band. It has been increasingly clear since the work of Windhorst (1984) that such galaxies make up a substantial part of the radio-source population below S appeq 1 mJy. The evolution of these objects at radio wavelengths and at 60 µm is directly related because there exists an excellent correlation between output at these two wavebands. Rowan-Robinson et al. (1993) have exploited this to investigate the implications of IRAS evolution for the faint radio counts. They find good consistency with the luminosity evolution L propto (1 + z)3 reported for the complete `QDOT' sample of IRAS galaxies by Saunders et al. (1990).

Were it not for the fact that some populations of objects show little evolution (e.g. normal galaxies in the near-infrared: Glazebrook 1991), one might be tempted to suggest an incorrect cosmological model as the source of this near-universal behavior. The alternative is to look for an explanation which owes more to global changes in the Universe than in the detailed functioning of AGN. One obvious candidate, long suspected of playing a role in AGN, is galaxy mergers; Carlberg (1990) suggested that this mechanism could provide evolution at about the right rate (although see Lacey & Cole 1993). Why the evolution does not look like density evolution is still a major stumbling block, but it seems that we should be looking at this area quite intensively, given that mergers have been implicated in both AGN and starbursts, and that there may be some evidence for their operation from the general galaxy population (Broadhurst, Ellis & Glazebrook 1992).

However, it is unclear how much emphasis should be placed on this apparent universality; particularly, limited statistics make it uncertain just how well luminosity evolution is obeyed. For example, Goldschmidt et al. (1992) have produced evidence that the PG survey is very seriously incomplete at z ltapprox 1; if confirmed, this would imply that the evolution of quasars of the very highest luminosities is less than for those a few magnitudes weaker. Furthermore, the QDOT database was afflicted by an error in which 10% of the galaxies were assigned incorrectly high redshifts (Lawrence, private communication); this will probably weaken the IRAS degree of evolution. It may well be that the degree of unanimity described above will prove spurious, and that we will be left with the unsurprising situation that a complex phenomenon like AGN evolution can only be described simply when the samples are too small to show much of the detail.

4.2 Redshift cutoff and interpretation

At higher redshifts, the uncertainties increase as the data thin out, but there is evidence that the luminosity function cannot stay at its z = 2 value at all higher redshifts. The form of this `redshift cutoff' is uncertain: we cannot at present distinguish between possibilities such as a gradual decline for z > 2, or a constant RLF up to some critical redshift, followed by a more precipitous decline. We therefore present a `straw man' model designed to concentrate the minds of observers, in which the luminosity evolution goes into reverse at z appeq 2 and the characteristic luminosity retreats by a factor appeq 3 by z = 4 (Figure 2).

Figure 2

Figure 2. The evolving RLF, according to the pure luminosity evolution model of Dunlop & Peacock (1990). The main features are a break which moves to higher powers at high redshift, but which declines slightly at z gtapprox 2. the strength of the break and the rate of evolution are comparable for both radio spectral classes.

This model predicts the following fraction of objects at z > 3.5 as a function of 1.4-GHz flux-density limit: 0.5% at 100 mJy; 3% at 1 mJy. Without some form of cutoff, these numbers would be about a factor of 5 higher. The reason for the increased ease of detecting a cutoff at low flux density is that the RLF is rather flat at low powers; for rho propto P-beta and S propto nu-alpha, we expect dN / dz propto (1 + z)-beta(2+alpha)-1/2. Steep spectra and a steep RLF thus discriminate against high redshifts, but at low powers the flatter RLF helps us to see whatever high-z objects there are more easily. It should be relatively easy to test for the presence of a cutoff on the basis of these predictions. This is especially true at low flux densities (see Figure 3). Here, we still sample the flat portion of the RLF even at high redshift, and so the predicted numbers of high-redshift sources is large without a cutoff - around 15% at z > 4 for a sample at 1 mJy.

Figure 3

Figure 3. A plot of the integral redshift distributions predicted for two samples limited at 1.4-GHz flux densities of 100 mJy and 1 mJy. The upper line shows a prediction for a luminosity function which is held constant for z gtapprox 2; the lower line shows the prediction of the `negative luminosity evolution' model of Dunlop & Peacock (1990).

Whether or not the redshift cutoff is real, we seem to have direct evidence that the characteristic comoving density of radio galaxies has not altered greatly between z appeq 4 and the present. Integrating to 1 power of 10 below the break in the RLF, we find

Equation 5

Is this a surprising number? In models involving hierarchical collapse, the characteristic mass of bound objects is an increasing function of time. At high mass, the abundance of objects falls exponentially if the statistics of the density field are Gaussian. Clearly, a model such as CDM (which falls in this class) will be embarrassed if the density of massive objects stays high to indefinite redshifts. The analysis of this problem, using the Press-Schechter mass-function formalism (Press & Schechter 1974) was first given by Efstathiou & Rees (1988) for optically-selected quasars.

There are two degrees of freedom in the analysis: what mass of object is under study, and what are the parameters of the fluctuation power spectrum? For the first, Efstathiou & Rees had to construct a long and uncertain chain of inference leading from quasar energy output, to black-hole mass, to baryonic galaxy mass, to total halo mass. For radio galaxies, things are much simpler, because we can see the galaxy directly. Infrared observations imply that, certainly up to z = 2, the stellar mass of radio galaxies has not changed significantly. At low redshift, there is direct evidence that the mass of radio galaxies exceeds 1012 Msun, so it seems reasonable to adopt this value at higher redshift. Figure 4 shows the Press-Schechter predictions for two COBE-normalized CDM models. The low-h model which fits the shape of the galaxy-clustering power spectrum (Peacock 1991) intersects the observed number density at low-ish redshifts (7-8), whereas the `standard' h = 0.5 model with its higher degree of small-scale power predicts many more objects. This is clearly only a suggestive coincidence at present, but it is clearly interesting that the model which most nearly describes large-scale structure also predicts that the formation of massive objects should occur near the point at which we infer a lack of high-z AGN.

Figure 4

Figure 4. The epoch dependence of the integral mass function in CDM, calculated using the Press-Schechter formalism as in Efstathiou & Rees (1988). The normalization is to the COBE detection of CMB fluctuations. Results are shown for two Hubble constants: the `standard' Omega h = 0.5 (upper panel) and Omega h = 0.3 (lower panel). Here, Omega h is merely a fitting parameter used to describe the shape of the power spectrum, and it does not presuppose a true value of the Hubble constant. The vertical scaling of density with h is given explicitly, and the mass values assume h = 0.5. The extra small-scale power in the former case means that many more massive hosts than the observed radio-galaxy number (horizontal line) are predicted, even at z gtapprox 10.

4.3 Black-Hole abundances

In the spirit of this meeting, it is probably important to concentrate on integrated properties of the radio-source population. One important feature of this sort is the relic density of black holes deposited by the work of past AGN. This is something which has been discussed extensively for radio-quiet quasars, but which has not been given so much attention in the radio waveband alone. The advantage of doing this is that, as discussed above, we have a rather good idea of which galaxies host radio-loud AGN, and therefore we know where to look for any debris from burned-out AGN. The basic analysis of this problem goes back to Soltan (1982). He showed that the relic black-hole density may be deduced observationally in a model-independent manner, as follows.

The mass deposited into black holes in time dt by an AGN of luminosity Lnu is

Equation 6

where epsilon is an efficiency, and g is a bolometric correction. To obtain the total mass density in black holes, we have to multiply the above equation by the luminosity function (which already gives the comoving density, as required) and integrate over luminosity. The integral can be converted to one over redshift and flux density, and the integrand depends of the observable distribution of redshifts and flux densities, so the answer is model dependent. Doing this for the Radio LF gives a much lower answer than for optically-selected QSOs, which have a much higher density:

Equation 7

Since we know rather well the present density of massive elliptical galaxies (e.g. Loveday et al. 1992), we may distribute half the above radio mass into ellipticals above the median radio-galaxy luminosity, with the following result for the mean hole mass:

Equation 8

What is the bolometric correction for radio galaxies? We know that the total output generally peaks in the IRAS wavelength regime, with an effective g ~ 100 (Heckman, Chambers & Postman 1992); this gives

Equation 9

which paints a rather less optimistic prospect for detection than studies based on the output of QSOs. This is because, even with such a large g, the actual energy radiated by radio galaxies is rather low, and this is not compensated for fully by the relative rareness of the host galaxies. The above figure is not easy to reconcile with large black-hole masses suggested for some radio AGN. For example, Lauer et al. (1992) suggest a central mass of Mbullet appeq 3 x 109 Msun for M87. Without suggesting that M87 is greatly atypical, this can always be made consistent by assuming a low enough efficiency. However, this would not fit well with the view that radio galaxies are powered via electrodynamic extraction of black-hole rotational energy (e.g. Blandford 1990); here the efficiency can be up to epsilon = 1 - 2-1/2. If masses of order 109 Msun are substantiated in several radio galaxies or radio-quiet massive ellipticals, this would be quite a puzzle. Probably the simplest solution would be to suggest that the total energy was higher than suggested by the above sum - perhaps because radio ellipticals spend part of their lives as QSOs, where the total energy output would be considerably higher for a given radio power.

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