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Astron. Astrophys. 353, 25-40 (2000)

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3.1. K-correction and AGN subclasses

In this section, we choose to present the SXLF in the observed 0.5-2 keV band, i.e., in the [FORMULA] keV range in the object's rest frame. Thus no K-correction has been appplied for our expressions presented in this section. Also we choose to include all emission-line AGNs (i.e., except BL-Lacs), including type 1's and type 2's. The primary reason for these choice is to separate the model-independent quantities, directly derived from ROSAT surveys, from model-dependent assumptions. Here we explain the philosophy behind these choices in detail.

There are a variety of AGN spectra in the X-ray regime, but the information on exact content of AGNs in various spectral classes is very limited. Currently popular models explaining the origin of the 1-100 keV CXRB involve large contribution of self-absorbed AGNs (Madau et al. 1994; Comastri et al. 1995; Miyaji et al. 1999b; Gilli et al. 1999). Although they are selected against in the ROSAT band, some of these absorbed AGNs come into our sample. These absorbed AGNs certainly have different K-correction properties than the unabsorbed ones. While these absorbed AGNs are mainly associated with those optically classified as type 2 AGNs, the correspondence between the optical classification and the X-ray absorption is not straightforward. Especially, there are many optically type-1 AGNs (with broad-permitted emission lines), which show apparent X-ray absorption of some kind. For example, a number of Broad Absorption Line (BAL) QSOs are known to have strongly absorbed X-ray spectra (e.g. Mathur et al. 1995; Gallagher et al. 1999). At the fainter/high-redshift end of our survey, there may be some broad-line QSOs of this kind or some intermediate class. Broad-line AGNs with hard X-ray spectra have been found in a number of hard surveys (Fiore et al. 1999; Akiyama et al. 1999). In Schartel et al. (1997)'s study, all except two of the 29 AGNs from the Piccinotti et al.'s (1982) catalog have been classified as type 1's, but about a half of them show X-ray absorption, some of which might be caused by warm absorbers. In view of these, using only optically-type 1 AGNs to exclude self-absorbed AGNs is not appropriate. Also optical classification of type 1 and type 2 AGNs depend strongly on quality of optical spectra. Thus classification may be biased, e.g. as a function of flux. However, the SXLF for the type 1 AGNs is of historical interest and shown in Appendix A. As shown in Appendix A., non type-1 AGNs are very small fraction of the total sample and excluding these does not change the main results significantly.

On the other hand, our sample of 691 AGNs with extremely high degree of completeness carries little uncertainties in the fluxes in the 0.5-2 keV band in the observer's frame, redshifts, and classification as AGNs. Thus, we choose to show the SXLF expression in the observed 0.5-2 keV band, or 0.5(1+z)-2(1+z) keV band at the source rest frame, in order to take full advantage of this excellent-quality sample without involving major sources of uncertainties. The expressions in the observed band may have less direct relevance for discussion on the actual AGN SXLF evolution. However they are more useful for discussing the contribution of AGNs to the Soft X-ray Background (Sect. 4), interpretation of the fluctuation of the soft CXRB, and evaluating the selection function for studying clustering properties of soft X-ray selected sample AGNs.

In practice, the expressions can also be considered a K-corrected SXLF at the zero-th approximation , since applying no K-correction is equivalent to a K-correction assuming [FORMULA]. This index has been historically used in previous works (e.g. Maccacaro et al. 1991; Jones et al. 1996), thus our expression is useful for comparisons with previous results. A [FORMULA] power-law spectrum can be considered the best-bet single spectrum characterizing the sample, because in the ROSAT sample, absorbed AGNs (including type 2 AGNs, type 1 Seyferts with warm absorbers, BAL QSOs) are highly selected against. Nearby type 1 AGNs show an underlying power-law index of [FORMULA] at [FORMULA] [keV] (e.g. George et al. 1998), which is the energy range corresponding to 0.5-2 keV for the high redshifts where K-correction becomes important. The reflection component, which makes the spectrum apparently harder, becomes important only above 10 keV. This is outside of the ROSAT band even at [FORMULA]. The above argument is consistent with the fact that the average spectra of the faintest X-ray sources, especially those indentified with broad-line AGNs, have [FORMULA] (Hasinger et al. 1993; Romero-Colmenero et al. 1996; Almaini et al. 1996) in the ROSAT band. Therefore, at the zero-th approximation, one can view our expression as a K-corrected SXLF of AGNs, especially at high luminosities. The goodness of this approximation is highly model-dependent and a discussion on further modeling beyond this zero-th approximation is given in Sect. 6.

3.2. The binned SXLF of AGNs

The SXLF is the number density of soft X-ray-selected AGNs per unit comoving volume per [FORMULA] as a function of [FORMULA] and z. We write the SXLF as:


Fig. 3 shows the binned SXLF in different redshift shells estimated using the [FORMULA] estimator:


where the [FORMULA] bins are indexed by j and AGNs in the sample falling into the j-th bin are indexed by i, [FORMULA] is the available comoving volume in the redshift range of the [FORMULA] bin where an AGN with luminosity [FORMULA] would be in the sample. The luminosity function is estimated at ([FORMULA],[FORMULA]), where a bar represents the [FORMULA] weighted average over the AGNs falling into the j-th bin. Also [FORMULA] is the size of the [FORMULA]th bin in [FORMULA].

[FIGURE] Fig. 3a and b. The [FORMULA] estimates of the SXLFs are plotted with estimated 1[FORMULA] errors. Different symbols correspond to different redshift bins as indicated in the panel a and data points belonging to the same redshift bin are connected. The position of the symbol attached to a downward arrow indicates the 90% upper limit (corresponding to 2.3 objects), where there is no AGN detected in the bin.

Rough 1[FORMULA] errors have been estimated by:


In case there is only one AGN in the bin, we have plotted error bars which correspond to the exact Poisson errors corresponding to the confidence range of Gaussian 1[FORMULA]. In this way, we can also avoid infinitely extending error bars in the logarithmic plot.

Fig. 3a,b shows the binned SXLF calculated for [FORMULA] and [FORMULA] respectively. In Fig. 3, we have also plotted some interesting upper-limits, in case there is no object in the bin. In the figure, we show upper limits corresponding to 2.3 objects (90% upper-limit). See caption for details.

We note that the binned [FORMULA] estimate can cause a significant bias, especially because the size of the bins tend to be large. For example, at low luminosity bins with corresponding fluxes close to the survey limit, the value of [FORMULA] can vary by a large factor within one bin. Also the choice of the point in [FORMULA] space representative of the bin, at which the SXLF values are plotted, may change the impression of the plot significantly. Thus the SXLF estimates based on the binned [FORMULA] can be used to obtain a rough overview of the behavior, but should not be used for statistical tests or a comparison with models. Full numerical values of the binned SXLF including [FORMULA] values, improved estimations by a method similar to that discussed by Page & Carrera (1999), and the numbers of AGNs in each bin will be presented in paper II.

A number of features can be seen in the SXLF. As found previously, our SXLF at low z is not consistent with a single power-law, but turns over at around [FORMULA]. The SXLF drops rapidly with luminosity beyond the break. We see a strong evolution of the SXLF up to the [FORMULA] bin, but the SXLF does not seem to show significant evolution between the two highest redshift bins. Figs. 3a,b show that these basic tendencies hold for the two extreme sets of cosmological parameters.

3.3. Analytical expression - statistical method

It is often convenient to express the SXLF and its evolution in terms of a simple analytical formula, in particular, when using as basic starting point of further theoretical models.

Here we explain the statistical methods of parameter estimations and evaluating the acceptance of the models. A minimum [FORMULA] fitting to the binned [FORMULA] estimate is not appropriate in this case because it can only be applied to binned datasets with Gaussian errors and at least 20-25 objects per bin are required to achieve this. In our case, such a bin is typically as large as a factor of 10 in [FORMULA] and a factor of two in z, thus the results would change depending where in the [FORMULA] bin the comparison model is evaluated.

The Maximum-Likelihood method, where we exploit the full information from each object without binning, is a useful method for parameter estimations (e.g. Marshall et al. 1983), while, unlike [FORMULA], it does not give absolute goodness of fit. The absolute goodness of fit can be evaluated using the one-dimensional and two-dimensional Kolgomorov-Smirnov tests (hereafter, 1D-KS and 2D-KS tests respectively; Press et al. 1992; Fasano & Franceschini 1987) to the best-fit models.

As our maximum-likelihood estimator, we define


where i goes through each AGN in the sample and [FORMULA] is the expected number density of AGNs in the sample per logarithmic luminosity per redshift, calculated from a parameterized analytic model of the SXLF:


where [FORMULA] is the angular distance, [FORMULA] is the differential look back time per unit z (e.g. Boldt 1987) and [FORMULA] is the survey area as a function of limiting X-ray flux (Fig. 1). Minimizing [FORMULA] with respect to model parameters gives the best-fit model. Since [FORMULA] from the best-fit point varies as [FORMULA], we determine the 90% errors of the model parameters corresponding to [FORMULA]. The minimizations have been made using the MINUIT Package from the CERN Program Library (James 1994).

Since the likelihood function Eq. (4) used normalized number density, the normalization of the model cannot be determined from minimizing [FORMULA], but must be determined independently. We have determined the model normalization (expressed by a parameter A in the next subsections) such that the total number of expected objects (the denominator of the right-hand side of Eq. (4)) is equal to the number of AGNs in the sample ([FORMULA]).

Except for the global normalization A, we have made use of the MINUIT command MINOS (see James 1994) to serach for errors. The command searches for the parameter range corresponding to [FORMULA], where all other free parameters have been re-fitted to minimize [FORMULA] during the search. The estimated 90% confidence error for A is taken to be [FORMULA] and does not include the correlations of errors with other parameters.

The 1D-KS tests have been applied to the sample distributions on the [FORMULA] and z space respectively. The 2D-KS test has been made to the function [FORMULA]. We have shown the probability that the fitted model is correct based on the 1D- and 2D-KS tests. For the 2D-KS test, calculated probability corresponding to the D value from the analytical formula is accurate when there are [FORMULA] objects and the probabilities [FORMULA]. If we obtain a probability [FORMULA], the exact value does not have much meaning but implies that the model and data are not significantly different and we can consider the model acceptable. We have searched for models which have acceptance probabilities greater than 20% in all of the KS tests. Strictly speaking, the analytical probability from the KS-test D values are only correct for models given a priori . If we use paramters fitted to the data, this would overestimate the confidence level. A full treatment should be made with large Monte-Carlo simulations (Wisotzki 1998), where each simulated sample is re-fitted and the D-value is calculated. However, making such large simulations just to obtain formally-correct probability of goodness of fit is not worth the required computational task. Instead, we choose to use the analytical probability and set rather strict acceptance criteria.

3.4. Analytical expression - overall AGN SXLF

Using the method described above, we have searched for an analytical expression of the overall SXLF. The overall fit has been made for the redshift range [FORMULA]. Also for the fits, we have limited the luminosity range to [FORMULA].

As described in Sect. 2, the lower redshift cutoff is imposed to avoid effects of local large scale structures, which may cause a deviation from the mean density of the present epoch and thus can cause significant bias to the low luminosity behavior of the SXLF. At the lowest luminosities ([FORMULA]), there is a significant excess of the SXLF from the extrapolation from higher luminosities. This excess connects well with the nearby galaxy SXLF by Schmidt et al. (1996) (see also e.g. Hasinger et al. 1999) and may well contain contamination from star formation activity (see also Lehmann et al. 1999a). For finding an analytical overall expression, we have not included the AGNs belonging to this regime.

As an analytical expression of the present-day ([FORMULA]) SXLF, we use the smoothly-connected two power-law form:


As a description of evolution laws, the following models have been considered:

3.4.1. Pure-luminosity and pure-density evolutions

As some previous works (e.g. Della Ceca et al. 1992; Boyle et al. 1994; Jones et al. 1996; Page et al. 1996), we have first tried to fit the SXLF with a pure-luminosity evolution (PLE) model.


For the evolution factor, we have used a power-law form:


The best-fit values are listed in the upper part of Table 2 along with 1D-KS and 2D-KS probabilities using the analytical formula. In Table 2 and later tables, the three values of [FORMULA] represent the probabilities that the model is acceptable for the 1D-KS test in the [FORMULA] distribution, 1D-KS test in the z distribution, and 2D-KS test in the ([FORMULA],z) distribution respectively. Note that there are cases which are accepted by 1D-KS tests in both distributions but fail in the 2D-KS test. The results of the fit show that the PLE model is certainly rejected with a 2D-KS probability of [FORMULA] and [FORMULA] for the [FORMULA]=1 and 0.3 ([FORMULA]) cosmologies respectively.

As an alternative, we have also tried the Pure-Density Evolution model (PDE), which seemed to fit well in our preliminary analysis for the [FORMULA]=1 ([FORMULA]) universe (Hasinger 1998).


where [FORMULA] has the same form as Eq. (8). The 2D-KS probabilities are [FORMULA] and 0.1 for the [FORMULA]=1 and 0.3 ([FORMULA]) respectively. Thus the acceptance of the overall fit is marginal, especially for [FORMULA]=1. However the PDE model has a serious problem of overproducing the soft X-ray background (Sect. 4). For a further check, we have made separate fits to high luminosity ([FORMULA]) and low luminosity ([FORMULA]) samples to compare the evolution index [FORMULA] in [FORMULA] for [FORMULA]. We have obtained [FORMULA] and [FORMULA] (90% errors) for the high and low luminosity samples respectively. Thus the density evolution rate is somewhat slower at low luminosities. Of course at the low luminosity regime, the fit was weighted towards nearby objects. If the evolution does not exactly follow the power-law form ([FORMULA]), spurious difference in evolution rate can arise. Visual inspection of Fig. 3 might suggest that at [FORMULA], the evolution rate seems larger at low luminosities, as opposed to the results shown above for [FORMULA]. However, performing the same experiment for the [FORMULA] AGNs showed [FORMULA] and [FORMULA] for the high and low luminosity samples respectively, indicating no difference within relatively large errors. For the [FORMULA] sample, the results are [FORMULA] and [FORMULA], again, for the high and low luminosity samples respectively. This difference and the soft CXRB overproduction problem lead us to explore a more sophisticated form of the overall SXLF expression as described in the next section.

3.4.2. Luminosity-dependent density evolution

We have tried a more complicated description by modifying the PDE model such that the evolution rate depends on luminosity (the Luminosity-Dependent Density Evolution model). In particular, as shown above, it seems that lower evolution rate at low luminosities than the PDE case would fit the data well. This tendency is also seen in the optical luminosity function of QSOs (Schmidt & Green 1983; Wisotzki 1998). The particular form we have first tried (the LDDE1 model) replaces [FORMULA] in Eq. (9) by [FORMULA], where


In Eq. (10), The parameter [FORMULA] represents the degree of luminosity dependence on the density evolution rate for [FORMULA]. The PDE case is [FORMULA] and a greater value indicates lower density evolution rates at low luminosities.

The best-fit LDDE1 parameters and the results of the KS tests are shown in Table 3. Table 3 shows that considering the luminosity dependence to the density evolution law has significantly improved the fit. The 2D-KS probabilities (analytical) are more than 30% for all sets of cosmological parameters.


Table 3. Best-fit LDDE1 parameters.
a) Units - A: [[FORMULA]], [FORMULA]: [[FORMULA]], Parameter errors correspond to the 90% confidence level (see Sect. 3.3).

We have considerd another form of the LDDE model (designated as LDDE2), which was made to produce 90% of the estimated 0.5-2 keV extragalactic background. The details of the construction of the LDDE2 is discussed in Sect. 4, where the contribution to the Soft Cosmic X-ray Background is discussed. In figures in the following discussions, the LDDE2 model is also plotted.

For an illustration, in Fig. 4 we show the behavior of the density evolution index for [FORMULA] as a function of luminosity for our PDE, LDDE1 and LDDE2 models. Fig. 5 shows the behavior of the model SXLFs at z=0.1 and 1.2. In this figure, only the part drawn in thick lines is constrained by data and thin lines are model extrapolations. These figures are only meant for illustrative purposes and thus are only shown for the [FORMULA]=[FORMULA] cosmology, where differences among models are more pronounced.

[FIGURE] Fig. 4. The behavior of the evolution indices at [FORMULA] are shown as a function of luminosity for various density evolution models: PDE (short-dashed, Sect. 3.4.1), LDDE1 (long-dashed 3.4.2), and LDDE2 (dot-dashed, Sect. 4). The lines for the [FORMULA] [FORMULA] case are shown.

[FIGURE] Fig. 5. The behavior of the model SXLFs at z=0.1 and 1.2 are shown respectively for the PLE (dotted), PDE (short-dashed), LDDE1 (long-dashed), and LDDE2 (dot-dashed) models. For the z=1.2 curves, thick-line parts show the portion covered by the sample ([FORMULA]) and the thin-line parts are extrapolations to fainter fluxes. The lines are for [FORMULA] [FORMULA].

3.5. Comparison of the data and the models

For a demonstration of the comparison between the analytical expressions and the data, we have plotted the [FORMULA] curve (the Log N - Log S curve plotted in such a way that the Euclidean slope becomes horizontal) for AGNs in our sample with expectations from our models (Fig. 6). Also the redshift distribution of the sample has been compared with the models in Fig. 7. These two comparisons already show intersting features. As expected, the PLE underpredicts and PDE overpredicts the number counts of lowest flux sources. In the redshift distribution, the PLE overpredicts the number of [FORMULA] sources while it slightly underpredicts the [FORMULA] sources. Although the deviation in each redshift bin seems small, the deviations in the neighboring bins are consistent and these systematic deviations can be sensitively picked up by the KS test in the z distribution (see small values of the [FORMULA] in z for the PLE model in Table 2).

[FIGURE] Fig. 6. The [FORMULA] (a horizontal line corresponds to the Euclidean slope) curve for our sample AGNs is plotted with 90% errors at several locations and are compared with the best-fit PLE (dotted), PDE (shot-dashed), LDDE1 (long-dashed), LDDE2 (dot-dashed) models for the [FORMULA] [FORMULA] (upper panel) and [FORMULA] (lower panel). The thin-solid fish is from the fluctuation analysis of the Lockman Hole HRI data (including non-AGNs) by Hasinger et al. (1993)

[FIGURE] Fig. 7. The redshift distribution of the AGN sample, histogrammed in equal interval in [FORMULA], is compared with predictions from the best-fit PLE (dotted), PDE (shot-dashed), LDDE1 (long-dashed), and LDDE2 (dot-dashed) models for two sets of cosmological parameters as labeled. The assymmetric error bars correspond to approximate 1[FORMULA] Poisson errors calculated using Eqs. (7) and (11) of Gehrels (1986) with [FORMULA].

The plots in Figs. 6 and 7 are comparisons of distributions in one-dimensional projections of a two-dimensional distribution. Only with these projected plots, one can easily overlook important residuals localized at certain locations. Thus we also would like to show the comparison in the full two-dimensional space. In literature, models are often overplotted to the binned SXLF plot calculated by the [FORMULA] estimate like Fig. 3. However, given unavoidable biases associated with the binned [FORMULA] estimates (see Sect. 3.2), such a plot can cause one to pick up spurious residuals. Thus we have plotted residuals in the following unbiased manner. For each model, we have calculated the expected number of objects falling into each bin ([FORMULA]) and compared with the actual number of AGNs observed in the bin ([FORMULA]). The full residuals in term of the ratio [FORMULA] are plotted in Fig. 8 for the PDE, LDDE1 and LDDE2 models for two sets of cosmological parameters as labeled. The error bars correspond to 1[FORMULA] Poisson errors ([FORMULA]) estimated using Eqs. (7) and (11) of Gehrels (1986) with [FORMULA]. Points belonging to different redshift bins are plotted using different symbols as labeled (identical to those in Fig. 3). These residual plots show which part of the [FORMULA] space the given models are most representative of, which part is less constrained because of the poor statistics, and where there are systematic residuals. It seems that the models underpredict the number of AGNs in the highest luminosity bin at [FORMULA] by a factor of 10, but statistical significance of the excess is still poor (2 objects against the models predictions of about 0.2). These AGNs do not constrain the fit strongly and excluding them did not change the results significantly. Also there is a scatter up to a factor of 2 from the model in [FORMULA], but no points are more than 2[FORMULA] away from either of the LDDE1 and LDDE2 models in both cosmologies.

[FIGURE] Fig. 8. The full residuals of the fit are shown for the PDE, LDDE1 and LDDE2 models in two sets of cosmological parameters as labeled in each panel. The redidual in each bin has been calculated from actual number of sample AGNs falling into the bin and the model predicted number. Different symbols correspond to different redshidt bins as indicated above the top panel, which are identical to those used in Fig. 3. One sigma errors have been plotted using approximations to the Poisson errors given in Gehrels (1986). The upper limit corresponds 2.3 objects (90% upper-limit).

The only data point which is more than 2[FORMULA] away from LDDE1 or LDDE2 model is the lowest luminosity bin at [FORMULA] (filled triangle), i.e., [FORMULA] for ([FORMULA]) = (1.0,0.0) or [FORMULA] for ([FORMULA]) = (0.3,0.0). Both LDDE1 and LDDE2 models overpredict the number of AGNs by a factor of [FORMULA] in both cosmologies, which are [FORMULA] away. However, this location corresponds to the faintest end of the deep surveys with a certain amount of incompleteness in the identifications. Our incompleteness correction method (Sect. 2) is valid only if the unidentified source are random selections of the X-ray sources in the similar flux range. However, these sources have remained unidentified because of the difficulty of obtaining good optical spectra and not by a random cause. Thus it is possible that the incompleteness preferentially affects a certain redshift range. Actually the deficiencies were much larger in the previous version (see Fig. 8 of Hasinger et al. 1999). The discrepancies decreased after the February 1999 Keck observations of the faintest Lockman Hole sources with rather long exposures, where three of the four newly identified source turned out to be concentrated in this regime. Thus it is quite possible that the remaining four unidentified sources are also concentrated in this regime. In that case, the LDDE models can also fit to this bin within 2[FORMULA]. Actually the newly identified and unidentified sources typically have very red [FORMULA] colors (Hasinger et al. 1999; Lehmann et al. 1999b), which probably belong to a similar class to those found by Newsam et al. (1998). If the red [FORMULA] color comes from the stellar population of underlying galaxy, they are likely to be in a concentrated redshift regime. On the other hand, if it represents obscured AGN component, they can be in a variety of redshift range. At this moment, it is not clear whether the deficiencies in this location is due to incompleteness or indicate an actual behavior of the SXLF.

Based on the results of the 1-D and 2-D KS tests, we have rejected the PLE model. We favor the LDDE1 and LDDE2 models over the PDE model based on the KS tests and as well as the CXRB constraints (see below). It may be interesting to show the exact location where the largest discrepancies are for these models, as compared to the LDDE models. This can be most clearly shown by plotting residuals in the [FORMULA] space. We have shown the [FORMULA] residuals for redshift bins where there are notable differences among these models, i.e., [FORMULA] and [FORMULA]. These are shown in Fig. 9. For both cosmologies, the PLE model systematically shifts from overprediction to underprediction with increasing luminosity at the lowest redshift bin. At the higher resdhift bin, the opposite shift can be seen. The curve converges closer to zero at both high and low luminosity ends just because there are only small numbers of objects in these bins causing poor statistics. More apparently in the [FORMULA] universe, the PDE model also shows a significant scatter around zero.

[FIGURE] Fig. 9. Residuals in the -[FORMULA] space (see text) are shown for two resdhift bins, i.e., 0.015[FORMULA] and 0.8[FORMULA], where differences among different models are apparent. Different line styles correspond to different models. See caption for Fig. 5 for the line styles. The luminosity bins are shown as horizontal bars bordered by ticks.

The data in the lower luminosity part [FORMULA] in the lowest redshift bin ([FORMULA]) are crucial in rejecting the PLE model, as seen in Figs. 3 and 9. This regime, consisting of [FORMULA] AGNs, has low SXLF values compared with the PLE extrapolation from the higher redshift data. Actually we cannot discriminate between the PLE and LDDE models for the sample of AGNs with [FORMULA] excluded. For the [FORMULA] sample, we could find good fits (with all of the KS probabilities in [FORMULA], z, and 2D exceeding 0.2) in any of the PLE and LDDE models. The acceptance of the PDE model was marginal ([FORMULA]). The [FORMULA] regime is mainly contributed by AGNs in the RASS-based RBS and SA-N surveys, whose flux-area space have not been explored previously. Since these samples are completely identified (see Sect. 2) and we have included all emission-line AGNs, the relatively low value in this regime is not because of the incompleteness or sampling effects. The only source of possible systematic errors which could affect the analysis would be in the flux measurements, because of the differences in details of the source detection methods among different samples. Some systematic shift of flux measurements might have occured between measurements in, e.g., the pointed and RASS data (for which there is no evidence). Thus we have made a sensitivity check by shifting the fluxes of all RBS and SA-N AGNs by [FORMULA] and [FORMULA]. The flux-area relation (Fig. 1) has been modified accordingly. In either case in either value of [FORMULA], the basic results did not change and especially the PLE model has been rejected with a large significance (with [FORMULA] ranging [FORMULA]).

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Online publication: December 8, 1999