Calculations of the correlation function for a preliminary catalog (October '94) that covered a strip of sky wide in declination and in RA appear in Cress et al. (1995). It was pointed out there that the source extraction algorithms were still being developed and so some changes to the calculation would be inevitable, particularly in the case of multiple sources for which CLEANing is more difficult. An improved version of the catalog covering this wide strip was later made available at the FIRST website (January '95). Analysis of this improved version showed a significant downward shift in the correlation function. The results for the same strip of sky in the latest catalog (October '95) (the catalog used throughout this work and described in Section 2) agree well with the results for the January '95 catalog. The amplitude of the correlation function for the improved catalogs is 16% lower than that for the catalog used in Cress et al. (1995). The result for the double and multi-component sources has changed in shape and amplitude.
The power-law fits obtained here agree well with estimates of the angular correlation function for the Green Bank 4.85 GHz survey. That catalog contains 54,579 sources with and S>25 mJy. About 40% of these were used in correlation function analysis by Kooiman et al. (1995) and Sicotte (1995). As a result of the significantly larger number of sources in the FIRST survey, our random errors are, in some cases, as much as a factor of 10 smaller than those estimated from Sicotte's contour plots. Our values are within his 30% confidence intervals for the amplitude and slope of the correlation of sources with 35 mJy<S<900 mJy. The results of Kooiman et al. are also fairly consistent with ours.
In 1978, Seldner and Peebles investigated the angular correlation function
of radio sources in the 4C catalog which contains 4,836 sources with
and S>2 Jy. The correlation signal was barely
detectable but they estimated the amplitude of the correlation to be 0.02 at
. Assuming , they derived a relation between
the angular correlation function of two populations and
the spatial correlation via the luminosity functions of those populations.
This enabled them to estimate the amplitude of the spatial correlation
function of the radio sources (assuming power law behaviour). Their best
fit gave a spatial correlation scale of Mpc.
Writing their equation (27) for the
autocorrelation of radio sources and substituting for N, the number of
sources per steradian brighter than flux density S, one obtains:
where the coordinate distance is given by
and the difference in coordinate distance between two sources separated by distance r is given by y. It is assumed that y<<x; that is, that clustering is only appreciable on scales much less than the distance of the sources. The selection function is given by
where is the density of radio sources at redshift z with luminosity between and at 1.4 GHz. is given by the maximum luminosity expected () for radio sources. is the minimum luminosity required for a source to be seen at redshift z. This is given by
where is the spectral index of the source.
Dunlop & Peacock (1991) (DP) have estimated the radio luminosity function (RLF) at 2.7 GHz for steep spectrum and flat spectrum sources separately. Condon (1984) has estimated the RLF at 1.4 GHz. Their estimates are derived from source counts at various wavelengths, from redshifts of only the brighest sources (S1 Jy) and from photometry of some fainter sources. In a first attempt at comparing Peacock and Nicholson's spatial correlation (see ) with that inferred from our measurements of the angular correlation, we assume . The redshift dependence corresponds to linear growth of density perturbations. Following Seldner and Peebles, we can then estimate the value of using Condon's RLF or DP's combined steep and flat RLF's, translated to 1.4 GHz.
We calculated for and using both DP's model and Condon's model. We estimate to be Mpc but this result is rather sensitive to the assumed clustering evolution. A detailed analysis of the results will be presented in a later paper (Cress & Kamionkowski 1997).