5 The density of old EROs and limits on $\mathsf{{\vec z}_f}$

Since old EROs have spectra consistent with being passively evolving ellipticals, we compared their density with different model predictions.

5.1 The surface density

The total surface density of EROs with $Ks\leq19.2$ and $R-Ks\geq5.0$ in our sample is 0.88 $\pm$ 0.13 arcmin^-2, consistent with that derived by D00 and Firth et al. (2001) over larger fields. In marked contrast, the surface density of EROs (counted as old plus dusty galaxies) predicted by the hierarchical merging models at K=19.2 presented by Firth et al. (2001) and Smith et al. (2001) are below the observed density by factors of $\sim$4 and about an order of magnitude respectively.

From the minimum and maximum fractions of old EROs derived in Sect. 3, the implied surface densities of ellipticals at the same magnitude limit should be in the range 0.27-0.55 arcmin^-2. Within the uncertainties, such densities agree with those predicted by the PLE models of Daddi et al. (2000b) with $\tau=$ 0.3 Gyr (0.24 and 0.54 arcmin^-2 for $z_{\rm f}=$ 2.2 and $z_{\rm f}=$ 3.5 respectively), suggesting a minimum formation redshift of $z_{\rm f}\sim 2.2$, consistent with the age derived from the average spectrum. The result does not significantly change if the 4 old EROs with weak [OII]$\lambda$3727 emission are excluded from the sample. If $\tau=$ 0.1 Gyr is adopted, or if the minimum fraction of old EROs (31%) is applied to the surface density of EROs derived by D00 in a much wider field (0.67 arcmin^-2 over 450 arcmin²), or if the 2MASS K-band local luminosity function of early type galaxies is used (Kochanek et al. 2001), the lower limit becomes $z_{\rm f}\sim 2$.

5.2 The comoving density

Taking into account the nominal lower redshift limit imposed by the $R-Ks\geq5.0$ color cut and the upper boundary due to the sensitivity of our spectroscopy (0.85$\lesssim$z$\lesssim$1.30), we derive a comoving density of ellipticals in that redshift range of (2.16 $\pm$ 0.62) $\times 10^{-4}$ Mpc^-3. Such a density can be considered a lower limit because some ellipticals can escape the threshold of $R-Ks\geq5.0$ due to the dispersion of the observed R-Ks colors around the model color prediction. Even so, the densities expected in the Daddi et al. (2000b) PLE model with $\tau=0.3$ Gyr are in the range of (1.8, 2.7) $\times 10^{-4}$ Mpc^-3 for $z_{\rm f}\sim$ (2.2, 2.4), thus being consistent with the observed densities.