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

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2. Models for X-ray spectra and evolution of AGN

2.1. AGN1 and AGN2 spectra

The local AGN spectrum is assumed to be the sum of AGN1 and AGN2 spectra ([FORMULA], [FORMULA] respectively) weighted by the number ratio of AGN2 to AGN1, R:

[EQUATION]

Following Comastri et al. (1995), a double power-law with a Compton reflection component [FORMULA] has been adopted for the AGN1 spectrum:

[EQUATION]

with [FORMULA]=0.9 (Matsuoka et al. 1990; Pounds et al. 1990; Nandra & Pounds 1994), and [FORMULA]=1.3.

The steeper, low-energy ([FORMULA] keV) spectrum represents the so-called soft excess. The shape and contribution of this component is not well known, and in many sources evidence for its very existence is lacking altogether. As a baseline, we have adopted the same prescription as Comastri et al. (1995), but as it now appears to be rather extreme, we have also explored the opposite case, i.e. no soft excess at all (see Sect. 3.1).

The adopted value of the cut-off energy [FORMULA]=400 keV is also similar to those used in previous models (Comastri et al. 1995; Celotti et al. 1995) even if recent BeppoSAX results (Matt et al. 1999a and references therein) seem to suggest somewhat smaller values, but with a rather large spread. The number of sources with reliably measured values is so low, however, that we preferred to still use the values adopted in previous models, to make easier the comparison. In any case, we tested the effect of adopting lower values for the cut-off, and found that the peak in the XRB spectrum at [FORMULA] 30-40 keV is less well fitted.

The term [FORMULA] represents the Compton reflection component by the accretion disk and by the torus inner surface and has been evaluated following Magdziarz & Zdziarski (1995), assuming an inclination angle of 60o.

According to unified schemes, AGN2 spectra are obtained as AGN1 spectra seen through absorbing matter. The distribution of equivalent hydrogen column density ([FORMULA]) is chosen to be logarithmic, i.e. [FORMULA], that is a reasonable analytical approximation to the recent data on Seyfert galaxies (Maiolino et al. 1998; Risaliti et al. 1999).

As described in a previous paper (Matt et al. 1999b), we developed a transmitted spectrum model by means of Monte Carlo simulations, assuming a spherical geometry with the X-ray source in the centre and considering photoelectric absorption, Compton scattering and fluorescence (for iron atoms only), fixing element abundances as tabulated in Morrison & McCammon (1983).

This transmitted component, which is relevant for [FORMULA], has been so far included in XRB synthesis models only by a handful of authors (Madau et al. 1994; Celotti et al. 1995; Matt et al. 1999b; Wilman & Fabian 1999). The final spectrum [FORMULA] has been then averaged over the [FORMULA]-distribution to obtain the total AGN2 spectrum:

[EQUATION]

where the [FORMULA]-distribution has been considered in the range [FORMULA].

The AGN1 and AGN2 spectra are shown in Fig. 1. The AGN1 spectrum is flattened by the reflection component, whose contribution reaches [FORMULA] 1/3 of the total at its maximum ([FORMULA] keV). The addition of a transmitted component significantly affects the total AGN2 spectrum, again with a flattening around 30-40 keV. The Compton scattered photons in AGN2 increases the total spectrum by [FORMULA] 20% at [FORMULA] 30-40 keV with respect to a model involving only absorption.

[FIGURE] Fig. 1. The AGN1 spectrum (solid line) and the AGN2 average spectrum (dashed line) as produced by the model.

The last step in evaluating the overall local spectrum concerns the choice of the number ratio R. Maiolino & Rieke (1995) find [FORMULA], if type 1.8, 1.9 and type 1.2,1.5 Seyfert galaxies are respectively classified as AGN2 and AGN1: the estimate agrees with previous (Huchra & Burg 1992; Goodrich et al. 1994) and more recent (Ho et al. 1997) results, and so [FORMULA] has been adopted.

For the sake of simplicity, iron emission line has not been included, even if it is a common feature in AGN. However, the contribution of the line to the 1.5-7 keV XRB is expected to be less then 7% (Gilli et al. 1999a) and it is smeared out by the integration over the redshift range, so that XRB retains its characteristic smoothness (Schwartz 1992), unless the emission is dominated by a small range of redshifts (Matt & Fabian 1994).

2.2. Cosmological evolution

Hard X-rays ([FORMULA] keV) are well suited for the selection of type 1 and, especially, type 2 AGNs as they are less affected by absorption. Until recent past little was known about the evolution of the AGNs in this band. The data obtained from the ASCA satellite have allowed a first determination of the 2-10 keV AGN XLF (Boyle et al. 1997). However the statistics was still poor. On the contrary the AGN1 XLF in the soft X-ray band is retained to be well-known at low and intermediate redshift (Boyle et al. 1994; Page et al. 1996; Jones et al. 1997), while at higher redshift ([FORMULA]) insufficient sampling and lack of statistics prevent the XLF from being firmly evaluated. Anyhow, X-ray AGN1 are detected up to [FORMULA] (Miyaji et al. 1998) and no evidence for a space density turn-over is found up to [FORMULA], likewise in the optical (e.g. Kennefick et al. 1996) and radio surveys (Shaver et al. 1999).

For these reasons we chose to tie the AGN evolution to the soft X-ray XLF of AGN1. We used the pure luminosity evolution (PLE) scenario which fits the combined ROSAT and EMSS data on AGN1 space density (Boyle et al. 1994). This corresponds to a local luminosity function that can be represented by a double power-law where the break-luminosity [FORMULA], i.e. the luminosity value corresponding to the slope change, evolves as [FORMULA]. In the following, we adopt the PLE H-model of Boyle et al. (1994) (hereafter B94) in the 0.3-3.5 keV band:

[EQUATION]

with [FORMULA], [FORMULA], [FORMULA], [FORMULA] Mpc-3 [FORMULA] erg s[FORMULA]-1 and [FORMULA], [FORMULA] in unity of [FORMULA] erg s-1. The break-luminosity evolution follows:

[EQUATION]

where [FORMULA] and [FORMULA]. We limited the analysis of AGN1 XLF to a PLE model because the most recent attempts with pure density evolution models, in which the space density [FORMULA] directly evolves in redshift, overproduce the soft XRB (Hasinger 1998). We have also to introduce an AGN2 XLF, which is a matter of strong debate. In the framework of the unification scheme, we assumed the density of AGN2 to be R=4 times that of the corresponding unobscured AGN1. We assumed the [FORMULA]-distribution to be independent of the AGN1 source luminosity. A different approach may consist in setting a completely unrelated XLF, but it would involve too many parameters and there are not enough data to yield a reliable estimate.

Boyle et al. (1997) directly measured the AGN1 and AGN2 XLF on a sample of 26 2-10 keV ASCA sources at a flux limit of [FORMULA] erg cm-2 s- 1, combined with the HEAO-1 AGN. The analysis gave a result consistent with the 0.3-3.5 keV AGN1 XLF, albeit the evolution seems to be slower ([FORMULA]).

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Online publication: December 17, 1999
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