7 Discussion

Peacock & Nicholson (1991) have shown that apparently bright radio sources (S>0.5Jy) in the redshift range 0.01<z<0.1 are spatially correlated, following a power-law behaviour $\xi=(r/11$ h^-1Mpc)^-1.8. Loan et al. (1997), using two large-area 4.85GHz radio surveys covering $70\%$ of the sky, conclude that the 2-D distribution of radio sources brighter than 50mJy is consistent with a correlation length in the range 13<r₀<18 h^-1Mpc and an evolution parameter $-1.2<\epsilon <0$. This result is based on the RLF models developed by Dunlop & Peacock (1990) and a value of $\gamma$ = 1.8. In a separate study, Cress et al. (1996) estimated the angular correlation function of the FIRST radio survey for sources with flux density S_1.4>1mJy. Assuming $\epsilon=-1.0$ and $\gamma=2.1$, they found r₀ = 6-8h^-1Mpc (Cress et al. 1997).

The r₀ upper limits estimated in this study at different flux density cutoffs agree with the estimates derived from brighter radio samples (Cress et al. 1997; Loan et al. 1997). Because of the small number of sources in each of the independent flux-limited sub-samples ($\approx$318) the uncertainties are large and thus we cannot conclude from the present sample if there is a change in the clustering properties of radio sources with flux density. Figure 10 compares the correlation amplitude upper limits from this study with those calculated for the radio sources detected in the FIRST radio survey with $1\le S_{1.4}\le 2$mJy (Cress et al. 1996). The agreement is good, even within the Poisson errors.

In recent years, deep radio surveys, (e.g. Windhorst et al. 1985; Fomalont et al. 1997; Hopkins et al. 1998) have shown a flattening in the slope of the normalised number counts at sub-mJy levels, revealing an excess of faint radio sources over the "normal'' radio population of giant ellipticals and QSOs. To address this problem, models invoking strong evolution of either spiral galaxies (Condon 1989) or star-forming IRAS galaxies (Rowan-Robinson et al. 1993; Hopkins et al. 1998) have been combined with RLF models of the "normal'' radio population. This scenario is supported by photometric and spectroscopic studies revealing that the sub-mJy radio sources can be identified with galaxies exhibiting evidence of increased star formation (Benn et al. 1993; Windhorst et al. 1985). Furthermore, studies of local galaxies with enhanced star formation (late spirals, IRAS galaxies, HII galaxies), has shown that these objects are more weakly clustered ( $r_0\approx 2-4$h^-1Mpc, for $\gamma\approx1.5-1.7$; Davis & Geller 1976; Giovanelli et al. 1986; Saunders et al. 1992) than E/S0 galaxies. This implies that the sub-mJy population could be more weakly clustered than the E/S0 objects that host the majority of apparently brighter radio sources. Support for this view was advanced by Cress et al. (1996) who found that the slope of the angular correlation function of the sources with 1 <S_1.4<2mJy is flatter compared to that found for S_1.4>3mJy. They interpret this result as being due to the increased contribution from starburst galaxies at lower flux density limits, which have flatter angular correlation functions compared to ellipticals, which dominate at brighter flux densities. A similar argument is used by Peacock (1997) to explain the apparent conflict between the value of r₀=6.5h^-1Mpc found for a sample of radio sources brighter than 2.5mJy and the value r₀=11h^-1Mpc found by Peacock & Nicholson (1991) for sources brighter than 500mJy and redshifts in the range 0.01<z<0.1.

To explore further the implications of the two scales of clustering, and in particular to explore the potential to eliminate competing models by observations, the amplitude of the angular correlation function was estimated, by adopting a simple model (Model A) in which the radio population consists of two components, one dominating at brighter (>1mJy) fluxes, with correlation length r₀=11h^-1Mpc ( $\gamma=1.8$) and the other dominating at sub-mJy levels with r₀=5h^-1Mpc ( $\gamma=1.8$) similar to that found for local starburst galaxies. Any cross-correlation between the two radio populations is ignored. The clustering evolution parameter is taken to be $\epsilon =-1.2$ and our RLF model 2 is employed to predict the redshift distribution of the two radio populations at faint flux densities. In Fig. 10 we plot the flux density cutoff against the amplitude of the angular correlation function calculated from model A. For comparison, the expected relation assuming the same value of r₀=11h^-1Mpc ( $\gamma=1.8$) for the two populations (Model B), is also plotted, along with the upper limits for the angular correlation amplitudes, A_w, calculated in the present study for different flux density cutoffs. The uncertainties are too large to allow discrimination between the two models.

It is an interesting exercise to predict the depth and the solid angle subtended by a radio survey that would reveal at a 3$\sigma$ significance level if the clustering properties of faint radio sources were significantly different from those of the brighter ones as a result of the changing population. This is accomplished by estimating the uncertainty in A_w for a survey of a given solid angle and completeness limit, as described in Appendix A. This then is compared to the difference $\Delta A_{w}$between the correlation amplitudes predicted at the same flux density cutoff from Models A and B. To discriminate between the two models at a 3$\sigma$ confidence level, the uncertainty in A_w should be 3 times smaller than $\Delta A_{w}$. The results are shown in Fig. 11, where the flux density cutoff is plotted against the uncertainty in A_w for different survey areas. The solid line delimits the area in the parameter space in which $\delta A_{w}$ is at least 3 times smaller than $\Delta A_{w}$ and hence defines the locus of 3$\sigma$confidence level discrimination between the simplified models A and B. We conclude that either a very deep survey ( $S_{\rm lim}=40$$\mu$Jy) over $\approx1$deg² or a survey over a larger area at a brighter limit (e.g. at $S_{\rm lim}=0.2$mJy over $\approx27$deg²) is required to discriminate between models A and B and thus reveal if there exists a weakly clustered radio population at faint flux densities. Our analysis shows that the FIRST radio survey, ( $S_{\rm lim}=1$mJy) covering, at the moment, an area of $\approx1500$deg², is also appropriate for this purpose.

Figure 11: Flux density cutoff versus the estimated uncertainty in the amplitude of the angular correlation function. Dashed lines correspond to surveys of circular geometry with different radii. The Phoenix survey has a radius of 1$^{\circ }$, while the FIRST radio survey is subtending, at the moment, an area corresponding to a radius of $\approx$22$^{\circ }$. The solid line represents the locus of 3$\sigma$ confidence level discrimination between models A and B described in the text