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

2. The method

Photometric redshifts (hereafter ) are based on the detection of strong spectral features, such as the 4000 Å break, Balmer break, Lyman decrement or strong emission lines. In general, broad-band filters will allow to detect only "breaks", and they are not sensitive to the presence of emission lines, except when their contribution to the total flux in a given filter is higher or of the same order of photometric errors, as it happens in the case of AGNs (Hatziminaoglou et al. 2000).

The method used in this paper to compute photometric redshifts is a SED fitting through a standard minimization procedure, computed with our code hyperz . The observed SED of a given galaxy is compared to a set of template spectra:

where , and are the observed and template fluxes and their uncertainty in filter i, respectively, and b is a normalization constant.

The new Bruzual & Charlot evolutionary code (GISSEL98, Bruzual & Charlot 1993) has been used to build 8 different synthetic star-formation histories, roughly matching the observed properties of local field galaxies from E to Im type: a delta burst, a constant star-forming system, and six µ-models (exponentially decaying SFR) with characteristic time-decays chosen to match the sequence of colours from E-S0 to Sd. We use the Initial Mass Function (IMF) by Miller & Scalo (1979), but this choice has a negligible impact on the final results, as discussed in Sect. 4.6. The upper mass limit for star formation is . The basic database includes only solar metallicity SEDs, but other possibilities are discussed in Sect. 4. The library also includes a set of empirical SEDs compiled by Coleman et al. (1980) (hereafter CWW) to represent the local population of galaxies. CWW spectra were extended to wavelengths Å and Å using the equivalent GISSEL spectra. The synthetic database derived from Bruzual & Charlot includes 408 spectra (51 different ages for the stellar population and 8 star-formation regimes). In most applications, there is no sensible gain when the number of µ-models is reduced to only 3, thus including only 255 spectra.

Throughout this paper we use the same set of broad-band filters, with characteristics presented in Table 1. These filters cover all the wavelength domain under study, without major overlap or gap. We also include the HDF filters used in Sects. 4 and 5 (from Biretta et al. 1996). The hyperz filter library is an enlarged version of the original Bruzual & Charlot one, and presently includes 163 filters and detector responses. All magnitudes given in this paper refer to the Vega system.

Hyperz has been optimized to gain in efficiency when computing on large catalogues. The input data for a given catalogue are magnitudes and photometric errors. To compute a reliable estimate of , the colours and the corresponding photometric errors must be obtained with particular care, including uncertainties due to zero-points, intrinsic accuracy, etc. Magnitudes are obtained within the same aperture in all filters, after correction for seeing differences between images. For a given catalogue, the relevant parameters introduced in the calculation are:

• The set of template spectra. This point includes the SFR type, the possible link between the age and the metallicity of the stellar population, and the choice of an IMF. It is discussed in Sect. 4.

• The reddening law is usually taken from Calzetti et al. (2000), but 4 other laws are also included in the code. This is discussed in Sect. 4.4. The input value is , corresponding to a dust-screen model, with , where and are the observed and the intrinsic fluxes, respectively. The extinction at a wavelength is related to the colour excess and to the reddening curve by , with except for the Small Magellanic Cloud () and the Calzetti's law (). The normal setting for ranges between 0 and 1.5 magnitudes. The mean galactic extinction correction towards a given line of sight can be introduced in terms of , and it is applied to the whole catalogue.

• Flux decrements in the Lyman forest are computed according to Giallongo & Cristiani (1990) and Madau (1995), both of them giving similar results.

• The limiting magnitude in each filter, and the rule to be applied in the case of non detection. The rule is set for each filter independently, and there are 4 different possibilities: 0) the filter is not taken into account in the computation; 1) the flux in this filter is set to 0 with an error bar corresponding to the flux deduced from the limiting magnitude; 2) the flux in this filter is set to of the limiting flux, according to the limiting magnitude, and the associated 1 sigma error is times this value; 3) the flux and the 1 sigma error in this filter are computed from the limiting magnitude and from the error associated to the limiting magnitude (both fixed). Case 1 is the usual setting when one is dealing with a relatively deep survey in the considered filter, whereas case 0 applies to "out-of-field" objects. Case 2 and 3 are well suited for relatively shallow surveys. The idea of "shallow" and "deep" in this context refers to relative values of the limiting magnitudes associated to the different filters in the photometric catalogue.

• The cosmological parameters , and , which are only related here to the maximum age allowed to the stellar population at a given redshift. The age checking is an option.

Table 1. Characteristics of filters used in the simulations: the effective wavelength and the surface of the normalized response function.

Due to the degeneracy in the parameter space defined by the SFR type, age, metallicity and reddening, the computation for a given object is equivalent to finding the most likely solution for the redshift across this parameter space, regardless to details on the best-fit SED (see Fig. 1). Both the and the SED are obtained through hyperz , together with the best fit parameters (, spectral type, metallicity and age). Because of the degeneracy between these parameters, the relevant information shall be the redshift and the rough SED type, in the sense that a given object has a "blue" or "red" continuum at a given z, but no reliable information can be obtained about the other parameters from broad-band photometry alone.

Fig. 1. Artist view of the SED fitting procedure to compute . The figure presents a likelihood map for a representative object at . The shaded area encloses the highest confidence level region according to the associated probability. Each point on the redshift-age map corresponds to the best fit of the SED obtained across the parameter space. The degeneracy in the parameter space is shown in this example.

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
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