Annu. Rev. Astron. Astrophys. 1991. 29: 499-541
Copyright © 1991 by Annual Reviews Inc. All rights reserved

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5.1 Topology and Fair Samples

A common experience to those who analyze the properties of redshift samples has been that of verifying that the largest traceable inhomogeneities in the galaxian distribution escape through the boundaries of the sampled region, raising the question of what is a fair sample of the universe. To some degree, the measure of fairness for a sample depends on the statistical tools that are to be applied to determine its properties. While the scale length of the galaxy-galaxy correlation function xi (r) is about the same in any sample (hovering near r0 = 5 h-1 Mpc), the sizes of the largest structures in 2- and 3-D surveys, be they voids, or connected high density regions, are nonetheless comparable to those of the sampled volumes, typically 20 h-1 to 60 h-1 Mpc. How much power is there at the large scales, or up to what scale is there any measurable power? A useful analogy to illustrate the ambiguities associated with these measurements might be the determination of the size of oceans on Earth: Although a single body of water can be continuously traced around the planet's surface, we characterize the oceanic structure by a scale length related to the mean separation of continents. Thus, the definition of structure and the measurement of the ocean's size are strongly linked to topology. Topological decriptions of the galaxian distribution have evolved from the visual, highly subjective appreciation of morphologies in poorly sampled data (see Peebles' (1984) criticism], to more objective numerical approaches applied to progressively richer samples. Panoramic surveys have played an important role in forming our views of the cosmic fabric, as in establishing the prejudices that permeate the numerical approaches that strive for its objective description.

The concept of ``supercluster'' gained acceptance during the late 1970s. The early studies of the Coma (Chincarini & Rood 1975, Gregory & Thompson 1978), Hercules (Tarenghi et al 1980), and Perseus regions (Gregory et al 1981, Einasto et al 1980) identified features that extend well beyond the boundaries of clusters (see the review by Oort 1983). Kirshner et al (1981) and Davis et al (1982) brought attention to the existence of large, underdense volumes with typical sizes of tens of Mpc that the first CfA slice suggested (de Lapparent et al 1986), might be outlined by a bubblelike galaxian distribution. The Arecibo Pisces-Perseus supercluster survey, on the other hand, revealed the existence of very extensive, linear, filamentlike structures (Haynes & Giovanelli 1986). Topology-discriminating algorithms have been developed (e.g. Gott et al 1987, Ryden et al 1989) and applied to a variety of redshift catalogs (Gott et al 1989): they indicate that, when the galaxian distribution is smoothed to scales as large as the correlation length (r0), the topology appears spongelike in all samples, an appearance consistent with the standard model in which today's structure has grown from small, random noise fluctuations in the early universe. However, when the galaxian distribution is smoothed to lengths smaller than r0, the character of the topology shifts towards a meatball rather than a bubble character (i.e. one where volumes are dominated by voids surrounding the enhancements in the galaxian distribution, rather than one where voids are surrounded by closed surfaces of enhanced galaxian density). Reservations on the adequacy of this type of analysis have been expressed by Geller & Huchra (1988).

The ability of a survey to trace a structure from one end of the volume sampled to the other does not exclude that the volume has approximated a fair sample. In a cellular network, for example, connected structures that stretch across the sampled region would be seen, no matter how large the sampled volume is, yet a fair sample will be approximated once the volume spans a few cells. Has such a point been reached with nearby, wide-area redshift surveys, or have we not yet run the gamut of large-scale inhomogeneities? The open-ended aspect of the Pisces-Perseus supercluster as surveyed over nearly two radians, and the galaxian distribution on a 360° display that led to the popularization of the feature dubbed ``the Great Wall'' (Geller and Huchra 1989) suggests that the answer to that question might be negative. Even more impressively, the finding of Broadhurst et al (1990) of a strong clustering feature with a characteristic scale of 128 h-1 Mpc indicates that structure may exist on scales much larger than could have been found by wide-area surveys. Karachentsev (1984) pointed out a similar feature in the clustering spectrum, with a periodicity of about 70 h-1 Mpc, although his result, based on only 92 redshifts, was less convincing than that of Broadhurst et al. Given the Karachentsev finding and the pencil-beam nature and huge redshift coverage of the Broadhurst et al survey, the specific value of the periodicity remains statistically weak. Kaiser & Peacock (1990) warn that in pencil-beam surveys, apparent clustering on large scales might result from aliasing of 3-D clustering on small scales, a form of clustering noise. In addition, one should maintain clear perspective that, over volumes of size comparable with those discussed here, several aspects of the galaxian distribution suggest that mean properties have been measured: e.g. the identity of slopes of galaxy counts log N (m) in all directions, known already from the work of Hubble, pose a strong case in favor of fairness over sizes on the order of 100 h-1 Mpc. Nonetheless, the perspective that the large-scale structure of the galaxian distribution might be dominated by a network of structures with characteristic scale on the order of 100 h-1 Mpc is most tantalizing, and it remains to be established whether the diminishing amplitude of density perturbations with the increasing size of inhomogeneities (found and suggested) can be accommodated with constant slope of the galaxy counts. Much debate will also be focused on the issue of the adequacy of previously favored theoretical and modeling schemes, in the description of the largest scale clustering features. Weinberg & Gunn (1990) have examined the structures expected to be seen when varying magnitude limits are imposed on the numerical results of biased cold dark matter simulations. They propose that wide angle surveys to limiting magnitudes fainter than 16.5 should reveal an even greater wealth of structure. They also underscore the resilience of current theoretical schemes to the threats of, at first sight, hostile observational results: ``. . . the existence of 50-150 h-1 Mpc structures is not in itself an argument against gravitational instability, Gaussian fluctuations, and cold dark matter''.

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