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Astron. Astrophys. 353, 440-446 (2000)

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3. The XRB synthesis model

The intensity of the XRB has been calculated as follows:


where [FORMULA] is the AGN1 XLF, [FORMULA] the luminosity distance, [FORMULA] the AGN spectrum. Introducing the comoving volume [FORMULA] and the spectrum normalization K corresponding to the considered luminosity, i.e. [FORMULA] ([FORMULA]=0.3-3.5 keV), the above relation can be written as:


where the integration is performed in the range [FORMULA], according to the B94 0.3-3.5 keV XLF.

The XRB synthesis has been used to explore the evolution of the AGNs. The model was fitted to the data discussed by Gruber (1992), largely based on HEAO-1 A2 3-50 keV data 1.

Best fit parameters were determined by a [FORMULA]-minimization procedure based on the CERN MINUIT software package.

3.1. The baseline model

The first analysis concerns the AGN1 high-redshift XLF. We tried to model the decrease of AGN1 at high redshift by introducing a new parameter [FORMULA] corresponding to the onset of the space density decrease. For redshifts in the range [FORMULA], the density stays constant. We used three different shapes for [FORMULA]: I ) an exponential ([FORMULA]); II ) a polinomial ([FORMULA]); III ) a sharp cut-off ([FORMULA]). We allowed [FORMULA] to vary, and fixed [FORMULA]=4.5 (which is a simple but still good representation of the space density of AGN1 according to the present, sparse observations). For any of the laws described above, the XRB is overproduced. We then allowed also [FORMULA] to vary, always obtaining values of [FORMULA]. This is a value much lower than observed as discussed in Sect. 2.2, and therefore all these models can be considered unacceptable.

We thus decided to adopt a sharp cut-off (model III above) with [FORMULA] and to fit the XRB changing a different parameter. We first tried to vary the number ratio R, obtaining a value of [FORMULA]; the error corresponds to 68% confidence level, following Lampton et al. (1976). The value is still in agreement with the observations. The fit was not very satisfactory from a statistical point of view, [FORMULA]. Part of the high value of the [FORMULA] is certainly due to the fact that we have not included other classes of sources, notably starburst galaxies and clusters of galaxies, which are likely to contribute at low energies. However, the fit is bad also at energies where we expect the AGN to dominate. The fit shows also an excess at higher energies ([FORMULA] keV), probably due to the value assumed for the intrinsic cut-off energy; however, as underlined in Sect. 2.1, the choice was motivated by the aim of reproducing well the [FORMULA] 30-40 keV peak. Even if we know that a really good fit is impossible with any smooth model, because of the significantly large fluctuations of the data (mostly due to the fact that measurements from different instruments have been used simultaneously), we decided to try to improve the quality of the fit by further changing our baseline model.

3.2. The R(z) model

Because the evolution of type 2 AGNs is probably the least known ingredient of the model, we tried to introduce a z-dependence of the AGN2/AGN1 ratio, R. Different analytical models have been tried for [FORMULA]: monotonic (both exponential and power law) shapes and King profile ([FORMULA]), but none of them gave a [FORMULA] less than [FORMULA].

We found, instead, a better result when a two parameter function of the form:


is used, with a significant improvement of the fit ([FORMULA]) (see Figs. 2 and 3). According to the F-test, this corresponds to a 95% confidence level with respect to the [FORMULA]=constant fit described above. The best-fit parameters are: [FORMULA], [FORMULA]. The decrease of the [FORMULA] is mainly due to a better reproduction of the data in the 5-10 keV energy range and around the XRB energy density peak ([FORMULA]30 keV).

[FIGURE] Fig. 2. The number ratio as a function of redshift [FORMULA] in unity of the local value [FORMULA], fixing [FORMULA] (solid line). The dashed lines enclose the 68% confidence region.

[FIGURE] Fig. 3. The fit to the XRB data (solid line) with the [FORMULA] model; ASCA data are also reported, even though they have not been considered in the fit.

We then checked how much dependent the above result is on the adopted shape of the local AGN spectrum, and in particular on our choice for the soft excess which, as discussed in Sect. 2.1, may be rather extreme. To do that, we adopted the opposite assumption, i.e. absence of a soft excess. The result gives an even quicker decrease of AGN2 with redshift. This is due to the fact that in absence of the soft excess the hard power law has an higher normalization in order to produce the same 0.5-3.5 keV luminosity, which implies higher luminosity at harder energies. Thus, in order not to overproduce the XRB, less AGN2 are required.

The effect of using a pure photoelectric absorption model for AGN2, as done in some previous papers, has also been tested. In this case the spectrum intensity is higher than the photoelectric absorption + Compton scattering model (Matt et al. 1999b), and again a steeper [FORMULA] emerges.

It is important to note that the introduction of [FORMULA] implies the presence of a density evolution component for AGN2. Density evolution has been taken into account by the ROSAT All Sky Survey (RASS) 0.5-2 keV data analysis of Miyaji et al. (1998), who introduce a luminosity dependent density evolution (LDDE) basically corresponding to a sudden drop in the evolution rate at lower luminosity. However, the RASS sample contains both AGN1 and AGN2, so the authors cannot outline any differences between the two kind of sources and estimate their evolutionary properties separately.

So far we derived the AGN evolution only by fitting the XRB. Then we checked the consistency of our results with the soft (ROSAT and Einstein, 0.3-3.5 keV) and hard (BeppoSAX, 5-10 keV) counts. In Fig. 4 the source counts predicted by the model in the two bands are compared to the ROSAT (Georgantopoulos et al. 1996) + Einstein/EMSS (Gioia et al. 1990) and BeppoSAX/HELLAS (Fiore et al. 1999) data.

[FIGURE] Fig. 4. (Upper panel) The integral soft counts (0.3-3.5 keV) compared to ROSAT + EMSS data. (Lower panel) The integral hard counts (5-10 keV) compared to HELLAS data. The source counts have been evaluated including the [FORMULA] term.

A good agreement in the soft band is found, while the hard counts are underestimated by a factor of [FORMULA] 1.5 at [FORMULA] [FORMULA] erg cm- 2 s-1. A possible solution of this discrepancy is discussed in the next section.

3.3. The normalization

Recently, new XRB data up to 10 keV have been obtained with imaging instruments on-board ASCA (Gendreau et al. 1995, Miyaji et al. 1998) and BeppoSAX (Molendi et al. 1997, Parmar et al. 1999). While there is still some disagreement between the XRB normalizations obtained by BeppoSAX and ASCA (Parmar et al. 1999), both of them are higher than that obtained by HEAO-1. For instance, BeppoSAX/MECS result is 30% higher than the HEAO-1 (Vecchi et al. 1999).

So, we have fitted our model to the XRB introducing a 30% higher normalization to the data, but retaining the same spectral shape. According to the procedure discussed in Sect. 3.2, we first tried with a constant [FORMULA] obtaining [FORMULA] ([FORMULA]). Again a better fit is obtained with a redshift dependence of R. The fit to [FORMULA] yields [FORMULA], [FORMULA] ([FORMULA]). [FORMULA] is shown in Fig. 5, while the corresponding fit is shown in Fig. 6. With respect to the fit with the old normalization, there is a more pronounced increase of the fraction of AGN2 for z between 0.5 and 2.

[FIGURE] Fig. 5. Same as in Fig. 2, but introducing a 30% higher normalization to XRB data.

[FIGURE] Fig. 6. Same as in Fig. 3, but introducing a 30% higher normalization to the Gruber (1992) data.

The improvement is significant at the 98% confidence level, according to the F-test. It is worth noting than now ASCA data are well reproduced, even if not used in the fit. This supports the higher normalization hypothesis. Both [FORMULA] are significantly higher than the fits obtained with the old normalization, mainly due to a clear deficiency of the model below [FORMULA]5 keV (where, however, there may be a contribution from other classes of sources). In this range the contribution of the AGN1, which cannot change in our model, is most relevant, and it was already deficient with the lower normalization. At higher energies, instead, the data are well fitted thanks to the increased number of AGN2. However, we remind that a disagreement between the model and the data starts to emerge for [FORMULA] keV. This is unavoidable, as mentioned in Sect. 3.1, when the cut-off energy value is fixed in order to have the XRB energy density peak at [FORMULA]30-40 keV.

The most interesting result is shown in Fig. 7, illustrating soft and hard counts. The higher normalization enables to reproduce both. Even if the normalization problem must still be considered an open issue, it cannot help being noted that with the high normalization a global solution is found.

[FIGURE] Fig. 7. Same as in Fig. 5, but introducing a 30% higher normalization to the Gruber (1992) data.

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999