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It would be a mistake to get bogged down in definitions right at the start, but one can perhaps make a useful distinction between a cluster and a supercluster on the basis of simple dynamics. Figure 1 depicts how the radii of overdense spheres behave during the expansion of the universe, assuming that pressure effects can be neglected. If the initial overdensity is large, expansion is halted at an early stage, and there is ample time for the system to recollapse and establish a virial equilibrium. If the initial amplitude is rather smaller, the sphere may by now have stopped expanding, and commenced its infall, without yet having virialised. And a sphere with sufficiently small initial perturbation will still be expanding, though it would have suffered an excess deceleration, and its constituent particles would therefore not be moving exactly with the mean Hubble flow. In the simple case of spherical perturbations in an Einstein-de Sitter universe, any system which is already virialised must have a present density more than 200 times the mean. A system which has halted its collapse and is now displaying infall must have more than 5 times the mean density.

Figure 1

Figure 1. The dynamics of overdense spheres with p = 0 in the expanding universe. The larger the initial overdensity, the earlier the sphere's expansion halts. Systems which are already virialised must have rho / rhocrit gtapprox 200; those which are now infalling must have rho / rhocrit gtapprox 5. Note that the overdensity factor of bound or collapsing systems must be even larger in a universe with Omega < 1 and mean density below rhocrit.

The simple dynamics depicted in the figure is relevant in two different, but related, contexts. If we imagine a single sphere, which condenses around a central high density peak, the dashed lines can represent different shells: the inner ones feel a large fractional overdensity, and collapse early; the outer ones feel only a small perturbation, and therefore are merely slightly decelerated. If the early universe contained a spectrum of initial fluctuations, such that the amplitude fell off towards larger scales, then we can also use the same figure to infer that smaller mass systems will tend to have already virialised, whereas larger scales, which initially had much smaller amplitude, would be dynamically younger, and would not yet have achieved dynamical equilibrium. I shall return later to discuss this process in the context of specific cosmogonic models which postulate random fluctuations with a specific spectrum, and can be simulated by N-body calculations.

The general features of three contrasting cosmogonies are summarized in Figure 2 and its caption. The first bound systems to condense are those for which the fractional density perturbation is largest at recombination. In some models, for instance the neutrino-dominated adiabatic scheme, the first bound systems would have a cluster (or even supercluster) mass, and galaxies would form in a 'top down' way. In other models the build up of structure is hierarchical, with small systems condensing first. I shall come back later to discuss the popular Cold Dark Matter model, which is depicted by the central panel of Figure 2. This is a hierarchical scheme, with the feature that the spectrum at low masses is very flat, implying that structure builds up quickly and recently.

Figure 2

Figure 2. The cosmogonic processes after (re)combination depend on the spectrum of fluctuations which have survived damping processes, etc., at earlier times. After trec, linear growth proceeds roughly according to the law (delta rho / rho) propto t2/3, and the first gravitationally bound systems to form will have the mass for which the density contrasts at trec are biggest. In case (1), (super)clusters are the first systems to condense, and they form quite recently. Indeed, models of this kind run into difficulties because we observe quasars with redshifts z as large as 5, whereas if superclusters had collapsed as early as this, they would now be denser, and display a higher density contrast than is seen. In case (2), baryonic systems of ~ 106 Msun condense in potential wells produced by 'cold dark matter' or 'inos', which are presumed to be slow moving, so that they are not homogenized on small scales as neutrinos are (this is the 'CDM' model). The third case shows the spectrum that might arise from primordial entropy perturbations. In cases (2) and (3), subgalactic systems would form before galaxies; if these subgalactic systems provided an energy input, they could in principle generate 'secondary' perturbations on larger scales which could swamp the genuinely primordial ones.

It is perhaps helpful to define a cluster as a gravitationally-bound system which, at least at its centre, has achieved virial equilibrium; and a supercluster as a larger system which, despite having virialised substructure, is overall in a dynamically younger state, perhaps even still expanding with the universe, albeit at a decelerated rate. Most galaxies are not in large clusters. The scale on which the correlation function is unity is about 8 megaparsecs, and the typical number of galaxies in a sphere of that radius is only a few. Rich clusters, in any gravitational instability scenario, would have evolved from regions where the initial fluctuation amplitude on the relevant scale was exceptionally large.

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