5. Evolution of luminous QSOs
In this section, we consider QSOs with high soft X-ray luminosities (), where the behavior of the SXLF can be traced up to high redshifts. Also in the high-luminosity regime, at least in the local universe, we observe very few absorbed AGNs, which could cause problems with the K-correction, in the local universe (e.g. Miyaji et al. 1999b). If this tendency extends to the high redshift universe, our ROSAT sample is a good representation of luminous QSOs and the assumed single power-law of would be a reasonable one. Thus we here investigate the evolution of the number density of the luminous QSOs using our sample. Fig. 3 shows that the SXLF at high luminosities can be approximated by a single power-law () for all redshift. Assuming this power-law with a fixed slope, we have calculated the number density of AGNs above this luminosity using the fitted normalization as described above in different redshift bins. The results are plotted in Fig. 11 for two sets of cosmological parameters. Similar curves for optically and radio-selected QSOs are discussed below.
In both cases, the number density increases up to and flattens beyond this redshift. In both cosmologies, the number density for is consistent with no evolution. The Maximum-Likelihood fits in the , region gave density evolution indices () of and for =(1.0,0.0) and (0.3,0.0) respectively. Subtle differences of the density curves seen in Fig. 11 between the two cosmologies come from two effects. Because different cosmologies give different luminosity distances, some objects which do not fall in the region for (1.0,0.0) come into the sample in lower density cosmologies. Also the comoving volume per solid angle in a certain redshift range becomes larger in lower density cosmologies, thus the number density lowers accordingly. These two effects work in the opposite sense and tend to compensate with each other, but the former effect is somewhat stronger.
It is interesting to compare this curve with similar ones from surveys in other wavelengths. In Fig. 11, we overplot number densities of optically- (Schmidt et al. 1995, hereafter SSG95) and radio-selected (Shaver et al. 1999) QSOs for (1.0,0.0). The densities of these QSOs have been normalized to match the ROSAT -selected QSOs at . This corresponds to a multiplicative factor of 7 for the SSG95 sample. Shaver et al. (1999) gave no absolute number density. In order to assess the statistical significance of the apparent difference of the behavior at between the ROSAT selected sample, we have used a Maximum-Likelihood fit to 17 QSOs in the sample in and .
where C is a constant. In above expression, corresponds to no evolution even for and is a good description of the rapid decrease of optically-selected QSO number density by SSG95. Fig. 11 shows that the radio-selected QSOs follow the SSG95 curve very well, but they do not have sufficient statistics to directly compare with the X-ray results. We have made a Maximum-Likelihood fit with only one free parameter: . Fixing at 2.3, we have obtained the best-fit value and 90% errors (corresponding to ) of . The result changed very little if we treat as a free parameter. Setting increased the value by 3.3 from the best-fit value. This change in corresponds to a 93% confidence level. The probability that exceeds the value of 1 is , considering only one side of the probability distribution.
We have also checked statistical significance of the difference using the density evolution-weighted statistics, (Avni & Bahcall 1980), which is a variant of the statistics (Schmidt 1968) for the cases where surveys in different depths are combined. The and are primed to represent that they are density evolution-weighted (comoving) volumes. If we take in Eq. (13) with as the weighting function, will give a value of 0.5 if the sample's redshift distribution follows the density evolution law of SSG95. An advantage of this method over the likelihood fitting is that one can check the consistency to an evolution law in a model-independent way, i.e., without assuming the shape of the luminosity function. Applying this statistics to 17 AGNs in , , we have obtained , where the 1 error has been estimated by (N:number of objects). The same sample has given unweighted , consistent with a constant number density. If we use a harder photon index of for the K-correction, 14 objects remain in this regime giving . The inconsistency between the SSG95 optical results and our survey is thus marginal if high-redshift QSOs have a systematically harder spectrum.
© European Southern Observatory (ESO) 2000
Online publication: December 8, 1999