Radio sky maps should be very sensitive to fluctuations in the sky density of sources for the reasons given in Section 15.1. Radio sources appear to cluster about galaxies at least as much as galaxies cluster with galaxies (Longair and Seldner 1979), so radio source clustering in space should reflect large-scale inhomogeneities in the distribution of galaxies in space. However, the faint radio sources in any area of sky are spread along a line of sight almost cz / H0 3000 Mpc in length, reducing the angular sensitivity to clustering in space by averaging over many clusters in the line of sight. Consequently, existing radio surveys are sensitive only to clustering on very large scales (d > 100 Mpc) at cosmological redshifts (z 1), in contrast to optical (Bahcall and Burgett 1986, de Lapparent et al. 1986) and far-infrared (Meiksin and Davis 1986, Rowan-Robinson et al. 1986) surveys that probe nearby (z 0) clustering on scales up to d 100 Mpc.
Several different techniques have been used to search for fluctuations in the sky densities of radio sources. One is plotting the distribution of angular distances to the nearest neighbors of all sources in a survey and comparing this distribution with the expected random distribution (Maslowski et al. 1973). The nearest-neighbor test is sensitive to clustering only in the small range of angular scales between the survey resolution and the typical separation between sources. Another simple procedure is to group the sources by position on the sky, flux-density range, etc. and compare their numbers with the expected Poisson distributions (Machalski 1977). A variation of this test for confusion-limited surveys is to divide the mapped region into small areas and compare the widths W of the P(D) distributions in each (Hughes and Longair 1967). Then the effective number of sources sampled equals the number of beam areas in the whole map, potentially quite a large number. The distribution of widths W from the 480 squares, each covering 2° × 2° 91 independent beam areas, from the Green Bank 1.4-GHz sky map overlapping the north galactic pole (Condon and Broderick 1985) is shown as a histogram in Figure 15.16. It is indistinguishable from the distribution expected in the absence of clustering (filled circles). Such grouping tests are most sensitive to clustering on the grouping scale chosen, so they must be repeated on a variety of scales. But they can easily be applied to surveys with irregular boundaries. More powerful tests for clustering of discrete sources are power spectrum analysis (Webster 1976a, b) based on the Fourier transform of a map with the sources replaced by -functions, and its Fourier-transform relative, correlation function analysis (Masson 1979). A significant advantage of correlation function analysis is that confusion anticorrelation affects only the smallest correlation lags but essentially all Fourier components of the fluctuation power spectrum.
Figure 15.16. The observed widths W of the P(D) distributions in 480 2° × 2° squares (histogram) and the expected width distribution (filled circles) if sources are not clustered. Abscissa: width (mJy). Ordinate: number of maps.
No convincing detection of anisotropy in the sky distribution of extragalactic radio sources has been made. The upper limits obtained are strong enough to rule out the "local hole" interpretation of the drop in the weighted source counts at high flux densities, but they do not yet strongly constrain clustering on scales d < 100 Mpc. This limitation is primarily statistical - the source density is too low in large-scale surveys. There are (very approximately) N [cz / (H0 d )]3 / 3 clusters of comoving size d in the cone of solid angle . Only if the number n of sources in this solid angle is much greater than N can the statistical fluctuations be smaller than those caused by clustering. For P(D) analysis of confusion-limited surveys, n / b, where b is the beam area; and clustering on scales d >> (b / 3)1/3(c z / H0) might be seen. The Green Bank 1.4-GHz sky map (Condon and Broderick 1985) has b 10-5 sr and so reaches d >> 50 Mpc. Confusion-limited surveys with only moderately higher resolution may detect clustering on scales already known to exist in optical and infrared samples.
Finally, it should be noted that the speed v of the Earth relative to the extragalactic source frame produces a dipole anisotropy of amplitude [2 + ( - 1)(1 + )] / (v / c) in the differential count n(S) S- of sources with spectral index (Ellis and Baldwin 1984). This effect is just below current limits of detectability, requiring surveys covering N 2 × 105 beam areas with high gain accuracy.