5 Completeness correction

Completeness correction for faint undetected sources strongly depends on the source apparent spatial structure besides to their magnitude. To estimate completeness via simulations we must account for sub-galactic structures. These could be quite common at high redshift and detectable by HST, and for the morphology which is that of the co-moving UV and B pass-bands and hence is strongly affected by star formation episodes. These features imply that "typical'' profiles of galaxies are not able to well describe the shapes of a lot of galaxies in the HDF-S. Thus, in order to reproduce the manifold of shapes which characterizes sources in the HDF-S we generated a set of simulated frames by directly dimming the original frames themselves by various factors while keeping constant the RMS (Saracco et al. 2000). This procedure has allowed us to avoid any assumption on the source profile while providing an artificial fair dimmed sample in a real background noise. We thus define the correction factor $\bar c$ as the mean number of dimmed galaxies which should enter the fainter magnitude bin over the mean number of detected ones. It represents the inverse of the fraction of galaxies undetected in each bin. If n_i is the number of galaxies detected in the i-th bin of the original catalogue and m_i+1 is the number of sources in the i+1-th bin of the simulated catalogue, the correction factor $\bar c_{i+1}$corresponding to the ratio between the expected number of galaxies (n_i) and the number of galaxies recovered (m_i+1). The "true'' number of galaxies in the i+1-th bin of the original catalogue is then N_i+1= $n_{i+1}\cdot \bar c_{i+1}$, where $\bar c$ is the mean over different simulations for the same frame.

When dimming fluxes by a factor F=10^0.4y, magnitudes are y mag fainter, but also noise is lowered: if $\sigma$ is the sky RMS on original images, the dimmed frames have a RMS of $\sigma /F$. We then added a frame of pure Poissonian noise, with a sky RMS

$$\sigma_{{\rm noise}}^2= \sigma ^2 -(\sigma /F)^2= \sigma ^2\left(1- \frac{1}{F^2}\right ).$$

The final images have, by construction, the correct RMS. By choosing y=0.5 mag, the artificial noise added is $\sigma_{{\rm noise}}=0.6\sigma$, i.e. about one third of the final image is due to pure Poissonian noise. As showed in Figs. 5-6 and in Table 4, this choice allowed us to correct up to V₆₀₆=29, with brighter limits in the other bands.

We tried also y=1 mag, but the estimated incompleteness was catastrophic, corresponding to 96%. This result seemed caused by the high simulated noise ( $\sigma_{{\rm noise}}$=0.84 $\sigma$, i.e. almost one half of noise is artificial), which caused also brighter bins to be incomplete.

The completeness correction allowed the computation of differential number counts up to fainter magnitudes: we thus determined the slopes of the number counts relation. Our best fit gives $\gamma_U=0.47\pm0.05$, $\gamma_B\sim0.35\pm0.02$, $\gamma_V\sim0.28\pm0.01$ and $\gamma_I\sim0.28\pm0.01$ (see Volonteri et al. 2000, for a detailed discussion).

    Table 4: Correction factor $\bar c$ and error. The factor $\bar c$ accounts for the galaxies missed in the detection due to the influence of noise. The "true'' (corrected) number of galaxies in each bin is $N=\bar cn$, where n is the raw number of detections

Filter	mag	$\bar c$	$\sigma_{\rm c}$
F300W	26.75	2.12	0.07
F450W	27.25	1.11	0.05
	27.75	1.56	0.04
	28.25	3.25	0.11
F606W	28.25	1.34	0.02
	28.75	2.67	0.1
F814W	27.25	1.09	0.04
	27.75	1.81	0.11
F110W	27.25	1.24	0.14
	27.75	2.56	0.19
F160W	27.25	1.93	0.42
F222M	24.25	4.09	1.23

Figure 5: V606 and I814 number counts: raw counts are shown with empty symbols, corrected counts with filled symbols

Figure 6: B450 and U300 number counts: raw counts are shown with empty symbols, corrected counts with filled symbols