2 The code Z-PEG (web available)

2.1 The evolution scenarios of galaxies

The atlas of synthetic galaxies used as templates is computed with PÉGASE.2 on the basis of evolution scenarios of star formation. The synthesis method assumes that distant galaxies are similar to nearby galaxies, but look younger at high z since they are seen at more remote epochs. Respecting this constraint will make our redshift determinations more robust. In an earlier paper (Fioc & Rocca-Volmerange 1997), we investigated the selection of star formation rates able to reproduce the multi spectral stellar energy distributions of nearby galaxies. Another article by Fioc & Rocca-Volmerange (1999b) computed the statistical SEDs of about 800 nearby galaxies observed from the optical and the near-infrared for eight spectral types of galaxies and used to fit scenarios at z=0. Star formation rates (SFRs) are proportional to the gas density (with one exception, see Table 1), the astration rate $\nu$ increasing from irregular Im to elliptical E galaxies. The star formation rate of starburst SB scenario is instantaneous. Infall and galactic winds are typical gaseous exchanges with the interstellar medium. They aim to simulate the mass growth and to subtract the gas fraction (preventing any further star formation) respectively. Unlike other studies (Fernández-Soto et al. 2001a; Massarotti et al. 2001b), our scenarios do not need to add any starburst component to be consistent with $z \simeq 0$observations.

 

 

Table 1: PEGASE.2 scenarios used as template parameters. SFR= $\nu \times M_{\textrm {gas}}$, except for starbursts and irregular galaxies. $\nu$  is in units of Gyr^-1 and M _gasis the gas density. Infall time-scales are in Myrs. The dust distribution is fitted on a King profile for E and S0, while an inclinaison-averaged disk distribution is applied to spiral and irregular galaxies (see text for details). Starburst galaxies have no extinction correction. For all the scenarios, the age of the universe is an upper limit on the age.

Type	$\nu$	infall	gal. winds	age at z=0
SB	$\delta(t)$			1 Myr to 2 Gyr
E	3.33	300	1 Gyr	>13 Gyr
S0	2	100	5 Gyr	>13 Gyr
Sa	0.71	2800		>13 Gyr
Sb	0.4	3500		>13 Gyr
Sbc	0.175	6000		>13 Gyr
Sc	0.1	8000		>13 Gyr
Sd	0.07	8000		>13 Gyr
Im	0.065a	8000		>9 Gyr

^a For this scenario only, we have $SFR=\nu \times M_{\textrm{gas}}^{1.5}$.

The initial mass function (IMF) (Rana & Basu 1992), is used in our evolution scenarios. However Giallongo et al. (1998) showed that the choice of the IMF does not influence much the photometric redshift estimates of high-z  candidates (z>2.5).

PÉGASE.2 is the most recent version of PÉGASE, available by ftp and on a web site. Non-solar metallicities are implemented in stellar tracks and spectra but also a far-UV spectral library for hot stars (Clegg & Middlemass 1987) complements the Lejeune et al. (1997, 1998) library. The metal enrichment is followed through the successive generations of stars and is taken into account for spectra of the stellar library as well as for isochrones. In PÉGASE.2, a consistent treatment of the internal extinction is proposed by fitting the dust amount on metal abundances. The extinction factor depends on the respective spatial distribution of dust and stars as well as on its composition. Two patterns are modeled with either the geometry of bulges for elliptical galaxies or disks for spiral galaxies. In elliptical galaxies, the dust distribution follows a King's profile. The density of dust is described as a power of the density of stars (see Fioc & Rocca-Volmerange 1997 for details). Through such a geometry, light scattering by dust is computed using a transfer model, outputs of which are tabulated in one input-data file of the model PÉGASE. For spirals and irregulars, dust is distributed along a uniform plane-parallel slab and mixed with gas. As a direct consequence, the synthetic templates used to determine photometric redshifts at any z, as well as to fit the observational standards at z=0, are systematically reddened. We also add the IGM absorption following Madau (1995) on the hypothesis of Ly$_\alpha$, Ly$_\beta$, Ly$_\gamma$ and Ly$_\delta$ line blanketing induced by Hi clouds, Poisson-distributed along the line of sight. This line blanketing can be expressed for each order of the Lyman series by an effective optical depth $\tau_{\rm eff}=A_i \times (\lambda_{{\rm obs}}/\lambda_i)^{1+\gamma}$, with $\gamma=2.46$ and $\lambda_i=1216, 1026, 973, 950$ Å  for Ly$_\alpha$, Ly$_\beta$, Ly$_\gamma$ and Ly$_\delta$ respectively. The values of A_i are taken from Madau et al. (1996), in agreement with the Press et al. (1993) analysis on a sample of 29 quasars at z>3. We shall see below that the IGM absorption alters the visible and IR colors more than about 0.1 mag as soon as z >2, leading to a more accurate determination of photometric redshifts at these distances.

For each spectral type, a typical age of the stellar population is derived. Time scales, characteristics and ages of the scenarios are listed in Table 1.

   
2.2 The $\chi ^2$ minimization procedure

A 3D-subspace of parameters (age, redshift, type) is defined by the template sets. It is used to automatically fit observational data. This subspace in the age-redshift plane is limited by the cosmology in order to avoid inconsistencies: a 10 Gyr old galaxy at z=2  cannot exist in the standard cosmology because at this redshift, the age of the universe is about 5 Gyr. Moreover the subspace is also limited by the age (redshift corrected) imposed by the adopted scenario of spectral type evolution. As an example, if elliptical and spiral galaxies must be at least 13 Gyr old at z=0, it means at least 5 Gyr at z=1 and so on.

Each point is granted a synthetic spectrum; its flux through the filter i is called $F^{\rm synth}_i$. For each point of this 3D-subspace, the fourth parameter $\alpha$ is computed with a $\chi ^2$ minimization to fit as well as possible the observed fluxes in filters:

$$\chi^2= \sum_{i=1}^{\it N}\left[\frac{F^{\rm obs}_i-\alpha \times F^{\rm synth}_i}{\sigma_i}\right]^2\cdot$$ (1)

N is the number of filters, $F^{\rm obs}_i$  and $\sigma_i$  are the observed flux and its error bar through the filter i respectively. In the case of an observed spectrum without redshift signatures, the sum can also be computed from wavelength bins. Then, each point of the 3D-subspace of parameters has a $\chi ^2$value. A projection of this 3D-$\chi ^2$ array on the redshift dimension gives the photometric redshift value $z_{\rm phot}$.

The values of $\sigma_i^2$ can be evaluated by the quadratic sum of the systematic errors and of the statistical errors. The extremely low values of observational errors, adopted as statistical, may result in anomalously high reduced $\chi ^2$ minima. In this study we consider as negligible systematic errors, keeping in mind that it maximizes the $\chi ^2$ minimum value (possibly up to 100). In such a case, statistical rules claim that the result (the photometric redshift) is not reliable and has to be excluded. Yet, with such prescriptions, most of the results would be excluded, because the photometric errors of the observations are very low. This is why all the primary solutions are often kept, including cases of very high reduced $\chi ^2$ minima. In the following, we will also adopt this philosophy. However, our error bars might appear larger than in the previous studies, that limit their results to one unique but less robust solution. Indeed, the estimation of the error bar of a photometric redshift is often estimated by the redshifts for which $\chi^2 \leq \chi^2_{\textrm{ \tiny min}}+1$. This method is only valid when the minimum reduced $\chi_{\rm r,\textrm{\tiny min}}^2 \simeq 1$ (otherwise the error bar is very underestimated).

We choose to estimate the error bar by the redshift values for which $\chi_{\rm n}^2 \leq \chi^2_{\rm n,\textrm{\tiny min}}+1$, where $\chi_{\rm n}^2$ is the $\chi ^2$ "normalized'' with $\chi^2_{\rm r,\textrm{\tiny min}}=1$ . The error bar is then much larger and may lead to secondary solutions. Fernández-Soto et al. (2001b) use another accurate estimation of the error bars which also gives secondary solutions, for the $3~\sigma$ level for instance. This is the case when the Lyman and Balmer breaks are hardly distinguished, as an example.

2.3 The Z-PEG interface

Z-PEG is an interface available on the PÉGASE web site (see footnote on page 2). Inputs are fluxes or colors of the observed galaxies and their error bars. Sets of classical filters defined in a variety of photometric systems are proposed. It is possible to use a user-defined filter, with its passband and calibration. In the general case, for which no information on the galaxy spectral type is a priori known, the minimization procedure is tested on all the template types. It is also possible to restrain the template galaxies to a given type. Types are then chosen among the 9 pre-computed synthetic spectra. The default redshift sampling is 0.25 and may be reduced on a narrower redshift range. The age and redshift axes are respectively defined by step, lower and upper limits. The user will choose the cosmological parameters (H₀, $\Omega_M$, $\Omega_\Lambda$), on which the time-redshift relation depends. The default values of cosmological parameters are those adopted in this article.

Outputs are the estimated photometric redshift $z_{\rm phot}$ and its error bar, the age and the spectral type of the best fitted synthetic galaxy. A $\chi ^2$ projection map in the age-redshift plane and its projection on the redshift and age axes (see Fig. 1) are also presented.

2.4 The coherency test

The relevance of the fit procedure is checked on a synthetic galaxy of the atlas with a known z. As an example, input data are the colors computed from the SED of the Sbc galaxy at age 5 Gyr and redshift z=0.989. The output value is, as expected, z=0.99, identical to the input galaxy redshift (z=0.989) with a precision of $\Delta z=0.01$ (no photometric error). Figure 1 shows the resulting projected $\chi ^2$ map on the age vs redshift plane for this test.

Figure 1: The test of coherency on the z=0.989 and 5 Gyr old galaxy: darker grey zones correspond to lower $\chi ^2$ values in the age - z plane. The minimum $\chi ^2$ location is the cross center of the two solid lines. The plot log $_{\textrm{\tiny 10}}$($\chi ^2$) versus redshift estimates the accuracy of the minimum and shows possible secondary minima.