5 Simulations

Because of the relatively small Phoenix field size, one should be cautious about the interpretation of the correlation amplitude upper limits, since field-to-field fluctuations may affect our results. For example Oort (1987a) carried out the angular correlation function analysis using sub-mJy radio surveys smaller than Phoenix and found evidence for anisotropic source distribution in only some of them.

Firstly, to test the significance of our results we construct catalogues of randomly distributed points and calculate the correlation function from a region with the same geometry as the Phoenix field. In all trials, the estimated amplitudes are found to be consistent with zero within the Poisson standard deviation, $\sigma_{\rm Poisson}$ (Fig. 5). Therefore, the amplitudes derived here (non-zero at the 2 $\sigma_{\rm Poisson}$ significance level), imply a non-uniform distribution of the faint radio population, albeit at the $2\sigma$ confidence level. The exceptions are the $S_{1.4}\ge 0.4$mJy and $0.4\le S_{1.4} \le 0.9$mJy sub-samples. This might indicate a low correlation amplitude at faint flux densities (e.g. $A_{w}\le 3\ 10^{-3}$, $\gamma=1.8$, $S_{1.4}\ge 0.4$mJy), that cannot be detected by the small size of the Phoenix field. However, one should be cautious about this interpretation, since at 0.4mJy the sample is also likely to be affected by incompleteness.

Figure 5: The angular correlation function for a random sample of points. The error bars are Poisson estimates

To further investigate the sensitivity of our results to cosmic variance, artificial galaxy catalogues are constructed with a built-in correlation function of the form $A_{w}\,\theta^{-\delta}$. Then, we attempt to recover the correlation amplitude from a field having the same geometry as the Phoenix survey.

The algorithm used to generate simulated catalogues with projected hierarchical power-law clustering is described by Infante (1994) and is similar to that employed by Soneira & Peebles (1978). Firstly, two points separated by an angle $\theta_{1}$ are randomly placed within a 50$\times$50 degrees area. Each of these points serves as the center of a new pair of randomly oriented points with angular separations $\theta_{2}=\theta_{1}/\lambda$, where $\lambda$ is a real number. The four points generated at the previous step are the new centers for pairs of points separated by $\theta_{3}=\theta_{2}/\lambda$. Therefore the L-th step produces 2^L-1 pairs with angular distance $\theta_{L}$ between the points of the pair.

Figure 6: The angular correlation function for the simulations B and D calculated from an area $\approx 10$ times larger than that of the Phoenix field. The error bars are Poisson estimates

Adopting a value $\lambda=\theta_{l}/\theta_{l+1}=1.8$ for the ratio of separations between successive levels, produces a power-law distribution of points with exponent $\delta\approx0.8$ (Soneira & Peebles 1978). The angular distance between points at the first level is set $\theta_{1}$ = 1.5deg. Additionally, the number of hierarchical levels, L, is drawn from a uniform probability distribution in the interval $[L_{\rm min},~L_{\rm max}]$ = [6, 9]. The minimum and maximum number of levels are such that the smallest angular separation of the points of a pair is less than $\approx$1arcmin. However, these parameters produce too many points at each level, resulting in large angular correlation amplitudes, A_w, compared to those found in bright radio samples (Loan et al. 1997; Cress et al. 1996). Therefore, each point is assigned a fractional survival probability, f. This step does not change the form of the built-in angular correlation function (Bernstein 1994), but allows tuning of A_w. Values of f between 0.01 and 0.05 produce amplitudes in the range $1\ 10^{-3}$ to $20\ 10^{-3}$ respectively. Here we only consider amplitudes in the interval $1\ 10^{-3}<A_{w}<10\ 10^{-3}$, similar to those found in brighter radio samples (Cress et al. 1996). The parameters used for four such simulations (trials A-D) are listed in Table 3. The estimated $w(\theta )$ for trials B, D are plotted in Fig. 6.

Figure 7: Cumulative distribution of $\Delta$ defined in Eq. (11)

The procedure described above is repeated until the surface density of points within the 50$\times$50 degrees area exceeds the surface density of sources of the present radio sample, at a given flux density cutoff. The angular correlation function is then calculated from the central 36deg²region, to ensure a uniform surface density of points. The amplitude of the correlation function is estimated by fitting the following function to the observations

  

Table 3: Parameters used to construct simulated catalogues. The densities of 145, 170 and 200 sources per deg², correspond to the surface densities of the present sample at flux density cutoffs of 0.6, 0.5 and 0.4mJy respectively

Trial	$\lambda$	$L_{\rm min}$	$L_{\rm max}$	f	density (deg-2)
A	1.8	6	9	0.03	145
B	1.8	6	9	0.04	170
C	1.8	6	9	0.01	170
D	1.8	6	9	0.02	200

$$w(\theta)=A^{\rm sim}_{w} \times \theta ^{-0.8}-\omega_{\Omega},$$ (10)

where the integral constraint, $\omega_{\Omega}$, is 0.54 $A^{\rm sim}_{w}$ for $\delta=-0.8$. The fitting is performed for angular separations between 3 to 50arcmin. Additionally, for a given mock catalogue, the angular correlation function is also calculated for nine fields, distributed within the central region of the 50$\times$50 degree area, each having the geometry of the Phoenix survey. The amplitude, $A^{\rm est}_{w}$, and the Poissonian uncertainty, $\sigma_{\rm Poisson}$, are calculated (see Sect. 3) for each of these smaller fields and then compared to $A^{\rm sim}_{w}$. The difference, $\Delta$, between $A^{\rm est}_{w}$ and $A^{\rm sim}_{w}$ normalised to the Poisson standard deviation is defined

$$\Delta=\frac{\vert A^{\rm sim}_{w}-A^{\rm est}_{w}\vert}{\sigma_{\rm Poisson}}.$$ (11)

For a given survival probability, f, and surface density of objects (corresponding to a flux density cutoff) six artificial catalogues are generated. The cumulative distribution of $\Delta$ values (Fig. 7), shows that in $\approx70\%$ of the trials, $A^{\rm est}_{w}$ lies within $\approx 2\sigma _{\rm Poisson}$ from $A^{\rm sim}_{w}$. For a Gaussian distribution, this is the definition of one standard deviation. Therefore, the simulations carried out here indicate that the angular correlation amplitudes determined from a radio survey with similar characteristics (i.e. geometry, surface density of objects) to those of the Phoenix survey, are representative of the faint radio population amplitudes within two Poisson standard deviations. This error is smaller than that estimated by the bootstrap resampling technique ( $\approx3\,\sigma_{\rm Poisson}$; see Sect. 3), suggesting that cosmic variance is not significantly affecting the present study. However, it is likely that the simulations presented here do not correctly model the higher order correlations produced by gravitational instabilities, on which the $w(\theta )$ errors depend. Nevertheless, the Poisson errors estimated here are independent of higher order correlations and are only meaningful when compared relative to the Poisson uncertainties of the real data-set. In the rest of this study we retain the bootstrap uncertainties as the formal errors, since this method has been widely used to assess the accuracy of $w(\theta )$.