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Astron. Astrophys. 363, 476-492 (2000)

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5. Expected accuracy on real data

In order to discuss on the expected accuracy and possible systematic errors when exploring real data, we have performed a complete set of simulated catalogues, with a more realistic (non uniform) redshift distribution and [FORMULA] ratio along the SEDs. We have adopted the simple PLE model proposed by Pozzetti et al. (1996, 1998), with minimal changes, to derive the redshift distributions and to assign a magnitude to each object in the different filters. Four galaxy types (exponentially decaying SFR with characteristic time [FORMULA] Gyr and 10 Gyr, constant SFR with evolution in time and at a fixed age of 0.1 Gyr) and their corresponding luminosity functions are used to reproduce the number of galaxies expected at a given redshift and absolute magnitude [FORMULA]. Apparent magnitudes are computed from the evolved SEDs, with ages depending on the redshift considered, the formation redshift, set to [FORMULA], and the cosmological parameters. Photometric errors are scaled to apparent magnitudes assuming the approximate relation [FORMULA], where [FORMULA] is the signal to noise ratio, which is given as a function of the apparent magnitude through [FORMULA], [FORMULA] being the signal to noise ratio at a given reference magnitude [FORMULA]. For simplicity, the photometric error is assigned to the apparent magnitude m according to a Gaussian distribution of [FORMULA] fixed [FORMULA]. This relation is set to reproduce the rapid increase of uncertainties when approaching the limiting magnitudes. According to these equations, a value of [FORMULA], corresponding to [FORMULA], is reached 2.5 magnitudes brighter than the magnitude corresponding to [FORMULA]. An object with [FORMULA] is non-detected in the involved filter ([FORMULA]). An object is included in the final catalogue if it is detected in the filter I (assuming that this is the selection filter), and in at least two other filters. The last requirement is needed to compute [FORMULA].

The same filter combinations discussed in Sect. 3 have been used to produce the new simulated catalogues. The simulations in Sect. 3 represent an ideal case, with an infinite depth and a fixed photometric error, disregarding the dependence on errors versus magnitudes. However, the relevant quantities [FORMULA], l% and g% strongly depend on the number of objects in each redshift bin and then on the limiting magnitudes.

To give a qualitative idea of the accuracy expected with different observational configurations, we consider two representative cases.

5.1. Deep pencil beam surveys

Firstly, we focus on simulations obtained in the case of a pencil beam-like survey, i.e. a very deep observation, covering a small area. From the photometric point of view, the main improvement with respect to the uniform distributions presented above is that we can introduce, for each object, a realistic [FORMULA] in the different filters, with different values from filter to filter. We assume that the detection limit is reached ([FORMULA]) at magnitudes similar to the limiting magnitudes of the HDF, as reported in the column [FORMULA](d) of Table 4 and in the right part of the same table for the HDF-N filters. To obtain approximately the same number of galaxies observed in the HDF, a field of 5 arcmin2 has been simulated. The percentages of spectral types included in the simulated catalogue are [FORMULA]% for E, Sb, Im, and Im([FORMULA] Gyr) respectively. In order to reproduce the observed number counts at faint magnitudes (Williams et al. 1996) we assume an open cosmological model, with [FORMULA] and [FORMULA]. In this case, the peak of the redshift distribution is at [FORMULA] and very few objects are seen at low-z, in particular at redshifts between [FORMULA] and [FORMULA]. Moreover, the PLE model is known to overestimate the population of high redshift galaxies.


Table 4. Limiting magnitudes at [FORMULA]: (d) deep pencil beam-like survey, (s) shallow ground-based survey.

In Table 5 we display the computed quantities [FORMULA], l% and g% for the set of filters of the HDF-N, and for all the other deep survey combinations considered in Sect. 3 (marked by (d) in the second column). The table contains the dispersion and the percentages of spurious and catastrophic objects, computed from a set of 10 independent simulations for each configuration. The interpretation of data in Table 5 must take into account that the definition of g% depends on the dispersion [FORMULA] computed using the correctly assigned objects, and this quantity is quite sensitive to the different filter sets and redshift bins. Nevertheless, these simulations take properly into account the observed properties of galaxies in deep surveys, such as the presence of faint objects with huge photometric errors, and the lack of detection in some filters leading to an uncertain [FORMULA] estimate (that is, increasing the probability of misidentifications, enlarging the error bars and the dispersion around the true value). In particular, this effect is evident when looking at the trend of the l% values. At higher redshift we find an increasing number of faint objects that are non detected in some filters. This leads to an increase of l%. Because of the depth of limiting magnitudes, we adopt the non detection law number 1 for optical filters and 2 for the near infrared ones. In the highest redshift bins, the increase of the dispersion value tends to mitigate the effect of the [FORMULA] deterioration in the value of g%.


Table 5. The dispersion [FORMULA] and the percentage of catastrophic and spurious objects, l% and g%, with errors, in five redshift bins, computed from 10 realizations of simulated catalogues with a redshift distribution derived from a PLE model. (s) and (d) refer to shallow and deep surveys respectively. The data are replaced by a dash when there are not enough data to compute the statistics. For the three examples in Fig. 9, Fig. 10 and Fig. 11 we also present the quantities mentioned above as a function of the limiting signal to noise ratio considered for the detection.

In the case of HDF-N filter set, we considered also two subcatalogues built with more restrictive selection criteria, requiring the detection both in [FORMULA] and in at least two other filters to be [FORMULA] and [FORMULA]. Statistics concerning these simulations are tabulated in Table 5. Obviously, when considering objects with increasing [FORMULA], the accuracy of [FORMULA] estimate significantly improves. In Fig. 9 we show the results obtained on the comparison between [FORMULA] and model redshifts, and also on the [FORMULA] versus [FORMULA] reconstruction. Most of the discrepancy is due to objects with [FORMULA].

[FIGURE] Fig. 9. Top: comparison between [FORMULA] and [FORMULA] for realistic catalogue HDF-like. See the text for the considered limiting magnitudes. Small dots, crosses and circles correspond to objects brighter than [FORMULA] respectively, at least in the I filter and in two other filters. Bottom: redshift distributions for the simulation on the top with [FORMULA]. Solid line: [FORMULA]. Dashed line: [FORMULA].

5.2. Shallow wide field surveys

In the second case, the aim is to reproduce the observational conditions reached when using 8 m telescopes and a wide field detector. In particular, we consider the case of a survey in a [FORMULA] arcmin2 field, observed with all the filter sets considered in Sect. 3. The adopted limiting magnitudes are shallower and conservative with respect to the values in the previous simulations. They are shown in the column [FORMULA](s) of Table 4. The percentages of the different spectral types for catalogues with these limiting magnitudes, using the same detection criteria, are similar to the previous ones, being [FORMULA]% for E, Sb, Im, and Im([FORMULA] Gyr) respectively.

Results for [FORMULA], l% and g% are presented in Table 5 and marked by a (s). For the filter sets UBVRI and UBVRIJK we repeated the same procedure adopted for the HDF-N simulated catalogue, to build two subcatalogues with higher [FORMULA] thresholds. Fig. 10 presents the results obtained with the five optical bands only, whereas Fig. 11 displays the equivalent results with the additional photometry in two near infrared filters. Fig. 10 and Fig. 11 show the associated input and recovered redshift distributions.

[FIGURE] Fig. 10. Top: comparison between [FORMULA] and [FORMULA] for realistic catalogue for a shallow survey with 5 filters. The symbols are the same as in Fig. 9. Bottom: redshift distributions for the simulation on the top with [FORMULA]. Solid line: [FORMULA]. Dashed line: [FORMULA].

[FIGURE] Fig. 11. Top: comparison between [FORMULA] and [FORMULA] for realistic catalogue for a shallow survey with 7 filters. The symbols are the same as in Fig. 9. Bottom: redshift distributions for the simulation on the top with [FORMULA]. Solid line: [FORMULA]. Dashed line: [FORMULA].

The peak of the redshift distribution in this case is at a lower redshift compared to the HDF simulation. Wide-field surveys allow to obtain a better sampling of the bright end of the luminosity function with respect to HDF-like surveys, the later being more suited to explore the faint luminosity regime. The value of g% and l% change significantly when considering the same set of filters, but a different kind of survey. On the contrary, [FORMULA] remains similar.

An interesting feature is the opposite trend displayed by deep pencil-beam compared to shallow wide-field surveys with respect to the low and high redshift regimes for a given filter set. At low redshifts, the value of g% is larger for the deep pencil-beam survey (type (d) in the Table 5) than for the shallow (s) wide-field one. Conversely, the accuracy of deep surveys overcome that of the shallow ones at high redshifts. In this context, the separation between low and high redshift regimes is marked by the [FORMULA] - 2 bin.

This behaviour could be easily explained when we consider the different characteristics of the catalogues produced in the two cases. The deep survey catalogue contains few low redshift galaxies, and most of them derive from the faint tail of the luminosity function. These faint galaxies are much more abundant than the bright ones, than they are present in the catalogue even though the volume covered at low redshift by this survey is small. The photometric errors for these intrinsically faint objects are rather large, thus causing a poor estimate of [FORMULA]. On the contrary, the wide-field survey contains a large quantity of bright galaxies at low redshift, wich have sufficiently small photometric errors to obtain accurate [FORMULA]s. The faintest objects are lost in this case because of the shallow detection limits. The majority of galaxies in the shallow wide survey lies in the low redshift bins, around the peak of [FORMULA]. When we consider the population of galaxies beyond the peak of the redshift distribution, the photometric errors in the shallow survey become important and an increasing fraction of objects is non detected in various filters. These problems hamper a robust determination of [FORMULA]. On the contrary, the pencil beam survey take advantage of its depth, allowing to compute [FORMULA] at higher redshifts.

On the basis of these results, we caution that the kind of analysis presented here is strongly advised when a photometric survey is undertaken in view of computing [FORMULA]s. In particular, the filter configuration and the photometric depth to reach in each filter have to be determined accurately in advance, in order to optimize the survey and to study the feasibility of the project.

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Online publication: December 11, 2000