Annu. Rev. Astron. Astrophys. 1988. 26: 631-86
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4. A UNIVERSAL CORRELATION FUNCTION

The increase of correlation strength with richness implies that rich, luminous systems are more strongly clustered, at a given separation, than poorer systems. The power law of the correlation functions is also observed to be identical in the various systems studied. Either initial conditions, or subsequent evolution, may be responsible for the observed phenomena. Since the observed correlation functions follow the same power law (r-1.8), the effect of increased correlation strength with richness (at a given separation) can also be expressed as a scale shift in the correlation functions (Szalay & Schramm 1985). In Figure 12 I plot the amplitude of the correlation functions of the various systems (galaxies, poor and rich clusters, superclusters) as a function of the mean separation of objects in the sample, d (see Bahcall & Burgett 1986, Bahcall 1987). The mean separation is related to the mean spatial density of objects in the sample, n, through d = n-1/3. For example, the mean separation of galaxies is about 5 Mpc, while the mean separations of R geq 1 and R geq 2 clusters are, respectively, about 50 Mpc and 70 Mpc.

It is apparent from Figure 12 that the correlation strength increases with the sample's mean separation. Moreover, a dimensionless correlation function normalized to the sample's mean separation d appears to yield a constant, universal function for nearly all the systems studied (some enhancement is required for galaxies, as described below). This universal dimensionless correlation function has the form

Equation 16 (16)

where the index i refers to the system being considered, and di is its mean separation. Relation 16 implies a universal dimensionless correlation amplitude of ~ 0.3, and, equivalently, a universal correlation scale of r0 appeq 0.5di. The correlation function of galaxies is stronger than that expressed by relation (16) by a factor of about four (Figure 12). The universality of the correlation function suggests a scale-invariant clustering process (Szalay & Schramm 1985). The stronger dimensionless galaxy correlations may imply gravitational enhancement on smaller scales. If a nonlinear process, other than gravity, participates in galaxy formation, and this process is scale-invariant, the created structure will have a single power-law correlation function, the slope of which (alpha) is related to the geometry of the structure (i.e. its fractal dimension beta). The latter is related to the correlation function slope via alpha = beta - 3 (see, e.g. Mandelbrot 1982). The fractal dimension of the universal structure implied by the above data is therefore beta appeq 1.2. Small-scale gravitational clustering may break the scale invariance and increase the dimensionless correlation amplitude for galaxies.

Figure 12

Figure 12. The dependence of the correlation function on the mean separation of objects in the system. The results are for clusters from different catalogs (Abell, Zwicky, and Shectman, as indicated by the symbols), determined by different investigators for samples of different mean densities (i.e. mean separations) (Section 3). The correlation strengths for galaxies and superclusters are also included. The solid line represents a d1.8 dependence (e.g. Szalay & Schramm 1985, Bahcall & Burgett 1986).

We do not know yet what physical process can create a scale-invariant structure with beta appeq 1.2. An innovative suggestion involves cosmic strings as the primary agent in the formation of galaxies and clusters; this model appears to create such a scale-invariant infrastructure (Turok 1985). The model yields a scale-invariant correlation function similar to that observed, with a power law of -2 (as implied by one-dimensional "string" structures with fractal dimension of unity). More detailed calculations with string models are currently being carried out by several investigators (Section 9).

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