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Whether the light originates from stellar nucleosynthesis, accretion onto compact objects, or gravitationally collapsing stellar systems, the total optical flux escaping from detected galaxies is quantified by number counts and luminosity functions. To the detection limits, number counts and luminosity functions contain exactly the same information regarding the integrated background light: the integrated flux from resolved sources is the same whether or not you know the redshift of the sources. However, in the context of predicting the EBL flux, luminosity functions contain information about the intrinsic flux distribution of the sources and thus allow us to estimate the flux from sources beyond the detection limits with better defined assumptions. In the following sections, we compare our EBL detections with the integrated flux obtained by both methods. Dust obscuration in the emitting sources will clearly reduce the UV and optical flux which escapes, but the EBL, number counts, and luminosity functions are all measurements of the escaping flux; the relative comparisons discussed in this section are therefore independent of dust extinction.

4.1. Number Counts

Using "ensemble aperture photometry" to measure the total flux from galaxies as a function of magnitude in our V555 and I814 images of the EBL field, we find that the standard photometry methods used to produce the HDF catalog systematically underestimate the flux from each source, as summarized in Section 3 (see Section 10 and Appendix B of Paper I for a thorough discussion). We use these results to derive flux corrections as a function of Deltaµ = µiso - µ0 (isophotal minus central surface brightness), which are essentially aperture corrections. These aperture corrections are similar to those found by other authors (c.f. Smail et al. 1995) and are a natural consequence of integrating an extended light profile to an insufficient radius. This effect can be quantified for exponential or de Vaucouleurs profiles, as in Dalcanton (1998). However, the corrections we show here are empirical measurements and assume nothing about the light profiles of the sources.

The corrections we derive for the two bandpasses (see Figure 2) are very similar functions of Deltaµ, which indicates that the profiles of detected galaxies are not a strong function of wavelength over the baseline of observed V to I. However, we note that a particular value of Deltaµ occurs at a brighter AB magnitude in I814 than in V606 because the limiting isophotal level (sky noise) in I814 is 0.6 AB mag brighter than in V606. The corrections are therefore larger in I814 than they are at the same AB magnitude at V606. The corrections in both bands include a correction which compensates for overestimates in the sky flux from foreground sources (the pedestal sky level described in Section 3). This correction, which accounts for errors in the local sky estimate, ranges from 0.1-0.3 mag, monatonically increasing towards fainter magnitudes. As in V606 and I814, aperture corrections for U300 band will depend on the profiles of galaxies at U300 and the surface brightness limits of the data. However, the very low signal-to-noise ratio of our F300W images prevents us from determining aperture corrections in that bandpass. The U300 photometry is discussed further below.

We have applied the aperture corrections we derive to the individual objects in the HDF V606 and I814 catalogs (Williams et al. 1996), which fractionally increases the flux of each galaxy. For example, while galaxies in the HDF catalog with V606 ~ 30 A B mag have well-detected cores, less than 30% of their total flux is recovered: the total flux of a galaxy measured to have V606 ~ 30 A B mag by standard photometry methods is actually closer to V606 ~ 28 A B mag. The corrected and uncorrected (raw) galaxy counts and corresponding integrated flux with magnitude are compared in Figure 3 and 4. The integrated flux of the corrected galaxy counts roughly corresponds to the minimum value of EBL23, as the aperture corrections were derived from the calculation of the minimum EBL23 in our data. Statistical variations in galaxy counts between fields are to be expected.

Figure 3

Figure 3. The upper panel shows the galaxy counts as published in the HDF catalog (filled circles) with sqrtN error bars and the corrected number counts (open circles), as described in Section 4.1. The solid lines show fits to the raw number counts, which change slope slightly around V606 = 26. The fit to the corrected counts is indicated by the dashed line to the detection limit and a dotted line beyond. No change in slope is apparent at the faint end for the corrected counts. All slopes are given in the text. In the lower panel, we plot the integrated flux corresponding to the galaxy counts with the same line types as in the upper panel. The data point and 1sigma error bar mark the value of EBL23 (converted to V606 from the V555 band). The corresponding ± 1sigma error range is emphasized by the hatch-marked region. For comparison, the lower limit arrow shows 2sigma lower limit of minEBL23, the integrated flux from detected sources with V606 > 23 AB mag.

Figure 4

Figure 4. The same as Figure 3, but for the I814-band. As for the V606-band, the raw I814 counts show a slight change in slope around 24-26 AB mag, while the corrected counts do not. All slopes are given in the text. The integrated flux of the raw and corrected counts are compared to our EBL23 I814 detections in the lower panel, as in Figure 3.

The aperture corrections we apply clearly have a significant impact on the slope of faint number counts. To quantify this, we fit both the raw and corrected number counts with the usual relationship between apparent magnitude and surface number density, N propto 10alpham, where N is the number of galaxies per magnitude per square degree. For the raw V606 counts, we find that the data exhibit a change in slope around 24-26 AB mag. A single fit over the range 22 < V606 < 29.5 AB mag gives alpha = 0.24 ± 0.01 with a chi2 per degree of freedom (chi2 / dof) of 1.5. Fitting the counts brighter and fainter than 26 AB mag, respectively, we find alphab = 0.28 ± 0.02 with chi2 / dof = 0.9 and alphaf = 0.21 ± 0.01 with chi2 / dof = 1.2 (solid lines in the upper panel of Figure 3). We ascribe this change in slope to the onset of significant photometry errors.

For the corrected V606, counts we find that the full 22 < V606 < 27.5 AB mag range is well fit by a slope of alpha = 0.33 ± 0.01 with chi2 / dof = 0.60 (dashed line in the upper panel of Figure 3). This result suggests that photometry errors are responsible for the change in slope at the faint end of the HDF counts, and that N(m) does not, in fact, significantly decline before the detection limit of the HDF at V606. In addition, while the integrated flux in the raw galaxy counts has converged by the apparent detection limit of the HDF, the flux from the corrected galaxy counts has not (see the lower panel of Figure 3).

We find similar results for the I814 counts (see Figure 4). As for V606, the raw I814 counts display a slight change in slope around 24-26 AB mag. We find a slope of alphab = 0.25 ± 0.01 with chi2 / dof = 0.6, and alphaf = 0.19 ± 0.02 with chi2 / dof = 2.0, brighter and fainter than 26 AB mag, respectively. For the full range 22 < I814 < 29.5 AB mag, we find alpha = 0.22 ± 0.01 with chi2 / dof = 2.1. For the corrected I814 counts, we find alpha = 0.34 ± 0.01 with chi2 / dof = 0.8 at 22 < I814 < 27 AB mag.

In Figure 5, we show the raw and corrected HDF counts relative to V- and R-band counts available in the literature for V > 15 AB mag. We have converted all of the published counts to V-band AB mag by applying constant offsets consistent with those in Fukugita, Shimasaku, & Ichikawa (1995). These incorporate mean K-corrections based on the mean redshift corresponding to the apparent magnitude of the sample. Differences between filters will have some affect on the slope of counts in surveys which cover a large range of redshift (apparent magnitude) due to changing galaxy colors and K-corrections with increasing redshift, but these effects will average out between the multiple surveys shown. This plot shows that the aperture corrections we have applied to the HDF sources produce number counts which have a slope consistent with the slope found at brighter magnitudes.

Figure 5

Figure 5. The raw and corrected number counts from Figure 3 compared to number counts from the literature, labeled by first author. The lines indicate fits to the data using the relation N propto 10alpham; alpha = 0.48 ± 0.1 and alpha = 0.33 ± 0.01 at the bright and faint ends, respectively. Note that the slope of the corrected HDF counts is well matched to that at brighter magnitudes.

In Figure 6, we show the same plot for the I-band. Again, the corrected I814 counts display a slope which is similar to that found at magnitudes brighter than 23 AB mag. Note also that slope of the counts at < 25 AB mag in V and I are the same to within the statistical errors. The aperture corrections we apply to the HDF counts at V606 and I814 extend this agreement to the current detection limits. The corrected counts imply that the faintest galaxies detected do not exhibit a significantly steeper slope in V606 than in I814, in contrast with the raw galaxy counts. This is an important constraint on galaxy evolution models.

Figure 6

Figure 6. Same as Figure 5 but for the I -band counts. The lines indicate fits to the data using the relation N propto 10alpham; alpha = 0.52 ± 0.1 and alpha = 0.34 ± 0.01 at the bright and faint ends, respectively. Note that slope of the corrected HDF counts is well matched to the slope at brighter magnitudes, and that the slope of the I and V band counts are similar at all magnitudes.

Although the signal to noise in the U300 data is too low to allow us to obtain accurate aperture corrections at that wavelength, the minimum EBL23 at U300 implies consistent colors for faint and bright galaxies at U - V, as in V - I (see Figure 1 and Table 2). We note, also, that the color of the integrated flux from galaxies is consistent with the color of the total background light within 2sigma. In other words, no exotic population of sources is required to produce the detected background.

The lack of turnover in the corrected counts strongly suggests that sources do exist at apparent magnitudes beyond the present detection limit. If we impose no limit on the apparent magnitude of sources and simply extrapolate the galaxy counts beyond using alpha = 0.33 (dotted line in Figure 3), we obtain a prediction for the total integrated EBL23 of 1.3 cgs, which is 1sigma below the measured value in the EBL field. In this case, the predicted EBL23 converges around V606 ~ 50 AB mag, which is significantly fainter than a dwarf galaxy at z ~ 6. However, very little flux is obtained from the faintest bins. If we impose the limit V606 ~ 38 AB mag as the faintest apparent magnitude for a realistic source (e.g., a dwarf galaxy with MV ~ - 10 AB mag at z ~ 4), we obtain a flux of 1.2 × 10-9 cgs. The flux from sources with I814 > 23 AB mag is 1.3 × 10-9 cgs if we adopt alpha = 0.34, with the flux converging around I814 ~ 60 AB mag. Adopting a more realistic faint cut-off of ~ 38 AB mag, as discussed for V606, we obtain a total flux of 1.2 × 10-9 cgs, 1sigma below the mean detected value of EBL23 at I814 (see Figure 4).

In order to obtain a cumulative flux equal to the mean detected EBL (or the upper limit) from sources brighter than ~ 38 mag, the slope of the galaxy counts in the range 28-38 AB mag would clearly need to increase at some point beyond the current detection limit. For example, the total flux from sources 23 < V606 < 38 AB mag will produce the mean detected EBL if the sources with 23 < V606 < 28 AB mag obey a slope of alpha = 0.33 and sources with 28 < V606 < 38 AB mag obey alpha = 0.42. We stress, however, that the total flux obtained from sources with such a steep faint-end slope is critically dependent on the cut of magnitude: the total flux reaches 5.1 × 10-9 cgs if we integrate the counts to 50 AB mag, and to 9.0 × 10-9 cgs if we integrate to 60 AB mag. Recall that our 2sigma upper limit on EBL23 at V606 is 5.0 × 10-9 cgs. For alpha = 0.35 at V > 28 AB mag, the integrated flux reaches 1.37, 1.51, and 1.57 × 10-9 cgs (converged) for faint cut-off magnitudes of 40, 60, and 80 AB mag respectively. Although it is obviously impossible to place firm constraints on the number counts beyond the detection limit, as they may change slope at any magnitude, we conclude that it is very unlikely that the slope beyond V606 ~ 28 AB mag is steeper than alpha = 0.42. If the slope continues at 0.33 < alpha < 0.35, the EBL23 reaches a roughly 1.3-1.5 × 10-9 cgs by V606 ~ 40 AB mag, < 1sigma below our detected value.

Similarly, for the I-band the integrated flux from sources matches the mean detected EBL23 if the sources with 23 < I814 < 27 AB mag obey a slope of alpha = 0.34 and sources with 27 < I814 < 39 AB mag obey alpha = 0.42. For those slopes, the total flux reaches the 2sigma upper limit of the EBL23 at I814 by 50 AB mag. For alpha = 0.36, slightly above the slope we find for the corrected counts, the integrated flux reaches 1.31, 1.58, and 1.62 × 10-9 cgs (converged) for faint cut-off magnitudes of 40, 60, and 80 AB mag, respectively. As for the V-band, we conclude that it is unlikely that the I-band faint-end slope is steeper than 0.42 at any magnitude. For a slope of 0.34 < alpha < 0.36 for I > 27 AB mag, the EBL reaches 1.2-1.3 × 10-9 by I814 ~ 40 AB mag, 1sigma below our detected value.

In summary, we conclude from the corrected number counts shown in Figure 3 - 6 that sources are likely to exist beyond the detection limit of the HDF. Furthermore, if the number counts continue with the slope we measure at the faintest levels, then the predicted EBL23 is within 1sigma of the detected EBL23 at both V606 and I814. If we extrapolate beyond the detection limits assuming the slope found from the corrected number counts, we find that less than 50% of EBL23 comes from sources beyond the current detection limit at V606 or I814 - the majority of the light contributing to EBL23 comes from sources which can be individually detected.

Finally, we note that our ensemble photometry method yields a statistical correction for the light lost from the wings of galaxies beyond the detection isophote. This light cannot, by definition, be recovered by standard single-object photometry. In contrast, the ensemble photometry method effectively adds together the light beyond the detection isophote from many galaxies to enable a significant measurement of that light.

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