The HELLAS2XMM survey. VII. The hard X-ray luminosity function of AGN up to z=4: more absorbed AGN at low luminosities and high redshifts
ABSTRACT We have determined the cosmological evolution of the density of AGN and of their Nh distribution as a function of the un-absorbed 2-10 keV luminosity up to redshift 4. We used the HELLAS2XMM sample combined with other published catalogs, yielding a total of 508 AGN. Our best fit is obtained with a luminosity-dependent density evolution (LDDE) model where low luminosity (Lx~10^43 erg s^-1) AGN peak at z~0.7, while high luminosity AGN (Lx>10^45 erg s^-1) peak at z~2.0. A pure luminosity evolution model (PLE) can instead be rejected. There is evidence that the fraction of absorbed (Nh>10^22 cm^-2) AGN decreases with the intrinsic X-ray luminosity, and increases with the redshift. Our best fit solution provides a good fit to the observed counts, the cosmic X-ray background, and to the observed fraction of absorbed AGN as a function of the flux in the 10^-15<S(2-10)<10^-10 erg s^-1 cm^-2 range. We find that the absorbed, high luminosity (Lx>10^44 erg s^-1) AGN have a density of 267 deg^-2 at fluxes S(2-10)>10^-15 erg s^-1 cm^-2. Using these results, we estimate a density of supermassive black holes in the local Universe of 3.2 (h^2_70) x 10^5 M_sol Mpc^-3, which is consistent with the recent measurements of the black hole mass function in the local galaxies.
arXiv:astro-ph/0509081v1 5 Sep 2005
Scheduled on The Astrophysical Journal, 635:???-???, 2005 December 20
Preprint typeset using LATEX style emulateapj v. 6/22/04
THE HELLAS2XMM SURVEY. VII. THE HARD X–RAY LUMINOSITY FUNCTION OF AGN UP TO Z=4:
MORE ABSORBED AGN AT LOW LUMINOSITIES AND HIGH REDSHIFTS
F. La Franca1, F. Fiore2, A. Comastri3, G.C. Perola1, N. Sacchi1, M. Brusa4, F. Cocchia2, C. Feruglio2, G.
Matt1, C. Vignali5, N. Carangelo6, P. Ciliegi3, A. Lamastra1, R. Maiolino7, M. Mignoli3, S. Molendi8, S.
Received 2005 May 26; accepted 2005 August 15
We have determined the cosmological evolution of the density of active galactic nuclei (AGN) and of
their NHdistribution as a function of the un–absorbed 2–10 keV luminosity up to redshift 4. We used
the HELLAS2XMM sample combined with other published catalogs, yielding a total of 508 AGN.
Our best fit is obtained with a luminosity-dependent density evolution (LDDE) model where low
luminosity (LX∼1043erg s−1) AGN peak at z∼0.7, while high luminosity AGN (LX>1045erg s−1)
peak at z∼2.0. A pure luminosity evolution model (PLE) can instead be rejected.
There is evidence that the fraction of absorbed (NH>1022cm−2) AGN decreases with the intrinsic
X–ray luminosity, and increases with the redshift.
Our best fit solution provides a good fit to the observed counts, the cosmic X–ray background, and
to the observed fraction of absorbed AGN as a function of the flux in the 10−15<S2−10<10−10erg
cm−2s−1range. We find that the absorbed, high luminosity (LX> 1044erg s−1) AGN have a density
of 267 deg−2at fluxes S2−10>10−15erg cm−2s−1. Using these results, we estimate a density of
supermassive black holes in the local Universe of ρBH= 3.2 h2
with the recent measurements of the black hole mass function in the local galaxies.
Subject headings: diffuse radiation — galaxies: active — galaxies: evolution — quasars: general —
surveys — X–rays: diffuse background
70×105M⊙Mpc−3, which is consistent
The understanding of the history of accretion in the
Universe and of the formation of massive black holes and
their host galaxies relies on the measurement of the active
galactic nuclei (AGN) space density and evolution.
According to the AGN unified model (Antonucci 1993)
the viewing angle between the observer and the symme-
try axis of the nuclear structure is responsible for the
different classification. In type 1 AGN the central en-
gine is directly visible. Both the broad and narrow line
emitting regions are detected in the optical spectra along
with a soft un-absorbed X–ray spectrum. On the con-
trary, a type 2 AGN classification arises when the broad
line region and the soft X–rays are obscured by a dusty
Until a few years ago the best measurements of the
cosmological evolution of the AGN luminosity function
were essentially limited to optically (e.g. La Franca &
Cristiani 1997, Croom et al. 2004), and soft X–rays (e.g.
Maccacaro et al. 1991, Miyaji et al. 2000) selected type
1 AGN. While there is evidence that type 2 AGN are
1Dipartimento di Fisica, Universit` a degli Studi ”Roma Tre”,
Via della Vasca Navale 84, I-00146 Roma, Italy.
2INAF, Osservatorio Astronomico di Roma, Via , I-00100 Mon-
3INAF, Osservatorio Astronomico di Bologna, Via Ranzani 1,
I-40127 Bologna, Italy
4Max Planck Institut f¨ ur Extraterrestrische Phisik (MPE),
Giessenbachstrasse, Postfach 1312, 85741 Garching, Germany
5Dipartimento di Astronomia, Universit` a di Bologna, Via Ran-
zani 1, I-40127 Bologna, Italy
6Universit` a di Milano-Bicocca, Piazza della Scienza 3, I-20126
7INAF, Osservatorio Astrofisico di Arcetri, Largo Fermi 5, I-
50125 Firenze, Italy
8INAF, IASF, Via Bassini 15, I-20133, Milano, Italy
about a factor four more numerous than type 1 AGN (e.g.
Maiolino & Rieke 1995; Risaliti et al. 1999), their rela-
tive space density beyond the local Universe is basically
unknown. Assuming that the cosmological evolution of
type 1 and 2 AGN is the same, it was possible to simulta-
neously reproduce the X–ray background spectrum and
the X–ray counts (e.g. Setti & Woltjer 1989; Comastri et
al. 1995). This simple picture was later slightly modified
in models where the fraction of type 2 AGN was assumed
to increase towards higher redshifts (e.g. Pompilio et al.
2000, Gilli et al. 1999, 2001). The selection of complete
samples of type 2 AGN is a difficult task. In the opti-
cal they are often so dim that only the light of the host
galaxy is visible; at z >1 even the latter has usually R >
24. In the soft X–rays bands even hydrogen column den-
sities, NH, of the order of 1021−22cm−2may strongly
suppress the flux. In the hard (2–10 keV) X–rays type 2
AGN selection is less biased against, though the absorp-
tion due to large NHcolumn densities (1023−24cm−2) is
not negligible especially at low redshifts.
Early attempts to compute the hard X–ray luminosity
function, based on ASCA and Beppo-SAX observations
(Boyle et al., 1998; La Franca et al. 2002 respectively)
indicated a strong evolution for type 1 AGN, with a rate
similar to that measured in the soft X–rays. Unfortu-
nately the low spatial resolution of the X–ray detectors
prevented an unambiguous identification of the type 2
AGN optical counterparts, thus hampering a reliable de-
termination of the type 2 AGN space density.
Thanks to the high sensitivity and spatial resolution
of the hard X–ray detectors on board XMM-Newton and
Chandra, it has become possible to carry out AGN sur-
veys less biased against X–ray absorption and with more
secure optical identifications.
2 LA FRANCA ET AL.
However, already at fluxes fainter than S2−10∼10−14
erg cm−2s−1, a sizeable fraction of the X–ray sources
have optical magnitudes fainter than the spectroscopic
limit of 8–10 meter class optical telescopes, and thus the
measure of their distance has to rely on photometric red-
shifts, when it is not impossible altogether.
For these reasons, although the Chandra Deep Field
North (CDF-N; Alexander et al. 2003) and the Chan-
dra Deep Field South (CDF-S; Giacconi et al. 2002) sur-
veys have resolved a fraction of the 2–10 keV XRB as
large as 85–90% (see also Brandt and Hasinger 2005),
a clear picture of the AGN evolution able to reproduce
the whole set of observational constraints (i.e. soft and
hard X–ray counts, X–ray background, NHand redshift
distributions) is still missing.
Attempts to take into account the redshift incomplete-
ness of X–ray selected AGN have been carried out by
Cowie et al. (2003), Fiore et al. (2003), Barger et al.
(2005) combining data from deep and shallow surveys.
They independently demonstrated that the AGN number
density for luminosities lower than ∼1044erg s−1peaks
at a lower redshift than that of high luminosity objects.
Making use of an almost complete sample of 247 AGN
from Chandra, ASCA and HEAO1 surveys above a lim-
iting flux of S2−10>3.8 × 10−15erg cm−2s−1Ueda et
al. (2003) were able to estimate the hard X–ray luminos-
ity function (HXLF) up to z = 3. They found that the
fraction of the X–ray absorbed AGNs decreases with the
intrinsic luminosity and that the evolution of the AGN
HXLF is best described by a luminosity-dependent den-
sity evolution (LDDE). Very similar results were also ob-
tained by Hasinger et al. (2005) using an almost complete
sample of soft X–ray selected type 1 AGN.
In this paper we expand the study carried out by Fiore
et al. (2003) with the aim to compute the shape and
evolution of the HXLF and NHdistribution of all AGN
with NH<1025cm−2up to z ∼ 4. To reach such a goal it
is necessary to cover the widest possible range in the LX-
z-NHspace, and to take into account all possible selection
effects. For these reasons we have used a large AGN
sample (about 500 objects) four times deeper than the
Ueda et al. (2003) sample. A new method to correct for
the spectroscopic incompleteness of faint X–ray sources
is presented and discussed in detail. The selection effects
due to X–ray absorption are also specifically discussed
and estimated by an appropriate X–ray “K–correction”
The paper is structured as follows: in Section 2 we
describe the adopted X–ray samples; in Section 3 the
method to compute the HXLF is discussed. The results
are presented in Section 4, discussed in Section 5 and
summarized in the last section.
Throughout this paper we call AGN all objects with
an intrinsic (corrected for NHabsorption) 2–10 keV X–
ray luminosity larger than 1042erg s−1. In the last few
years evidence for a mismatch between optical (Type
1/2) and X–ray (un–absorbed/absorbed) classification
has emerged (e.g.Fiore et al.
we refer to AGN1 if broad emission lines (rest frame
FWHM>2000 km s−1) are present, while all remaining
objects (with or without narrow emission lines in the
optical spectrum) are called AGN2. If the rest frame
column density is larger than 1022cm−2the AGN is
classified as absorbed. The adopted limit is well above
2000). In this paper
103 ( 93)
41 ( 39)
102 ( 98)
Note. — In column (2) we give the flux limit of the
samples in units erg cm−2s−1. In column (3) ve give
the total number of sources.
the number of sources brighter than the spectroscopic
limit (in parenthesis those having redshift). In column
(5) we give the spectroscopic completeness magnitude.
In column (4) we give
a1df sample (Fiore et al. 2003).
b0.5df sample (Cocchia et al. 2005).
the typical X–ray absorption by host galaxy gas (disk,
starburst regions, etc.) thus ensuring that the measured
column is most likely related to nuclear obscuration. Un-
less otherwise stated, all quoted errors are at the 68%
confidence level. We assume H0 = 70 km s−1Mpc−1,
Ωm= 0.3 and ΩΛ= 0.7.
In order to cover the widest possible range of lumi-
nosities and redshifts we combined the HELLAS2XMM
sample (Fiore et al. 2003) with other existing flux limited
samples which allowed the estimates of the rest frame NH
column density of each AGN. Whenever possible, the col-
umn density and the photon index (Γ) were determined
with a proper spectral analysis. Otherwise, we assumed
Γ = 1.8, and used the hardness ratio to measure the z=0
column density (NH0, see also the discussion about the
uncertainties of this approach in §4.1.1). The rest frame
column density (NH) was then estimated by the relation
Log(NH) = Log(NH0) + 2.42Log(1 + z), which makes
use of the Morrison & McCammon (1983) cross sections,
including also the effects of the absorption edges, and as-
sumes solar abundances from Anders & Grevesse (1989).
For those samples whose optical spectroscopic iden-
tifications are incomplete, we chose the optical magni-
tude limit at which the samples are almost spectroscop-
ically complete. The incompleteness is 6% in the HEL-
LAS2XMM, Lockman, CDF-N and CDF-S samples). In
these cases (as the X–ray–optical flux distribution of the
sources without redshift is almost similar to that of the
spectroscopically identified sources, and the fraction of
the unidentified sources is small) the sky coverage has
been reduced according to the fraction of spectra avail-
able. Table 1 contains a summary of the characteristics
of each sample. The distribution in the LX-z space of all
AGN from the spectroscopically complete sub-samples
used in our analysis are shown in Figure 1, while Figure
2 shows their distribution in the SX-R plane.
HELLAS2XMM: THE HARD X–RAY LUMINOSITY FUNCTION OF AGN3
Fig. 1.— LX–z plane for all AGN used in this analysis. Different
symbols corresponds to different surveys, as labeled in the top left
corner. Absorbed sources are also highlighted by a cross.
2.1. The HELLAS2XMM sample
We used the HELLAS2XMM 1df (1 degree field) sam-
ple (Fiore et al. 2003) plus the recently available ex-
tension of 0.5 deg2(HELLAS2XMM 0.5df Cocchia et
al. 2005). The HELLAS2XMM 1df sample contains 122
sources, serendipitously detected in five XMM-Newton
fields with SX(2–10 keV)> 0.8×10−14erg cm−2s−1. In
our analysis we used the fluxes and the column densi-
ties measured by X–ray spectral analysis (Perola et al.
2004). Among the 122 sources we discarded one star
(object n. 0537006) and one extended source (object n.
26900013). For three sources with low signal-to-noise the
hardness ratio and redshift were used to estimate the rest
frame NH. In summary, the sample contains 120 sources,
115 optically identified, and 95 with measured redshift
and optically classified. We restricted our analysis to the
sources brighter than R=23.65. Down to this limit 93
out of 103 sources have been spectroscopically identified.
The HELLAS2XMM 0.5df sample consists of 110 ob-
jects brighter than SX(2–10 keV)= 8 × 10−15erg cm−2
s−1. Among them, 44 sources brighter than R=21.4 (but
otherwise randomly selected) have been spectroscopically
2.2. The Piccinotti sample
The Piccinotti sample is the brightest included in our
analysis. It has been obtained through observations car-
ried out by the HEAO1 satellite, and contains 31 sources
selected over an area of 26919 deg2down to SX(2–10
keV)= 2.9 × 10−11erg cm−2s−1(Piccinotti et al. 1982).
The column densities have been taken from the litera-
ture, and are derived from X–ray spectral analyses.
Fig. 2.— R-band magnitude versus the 2–10 keV X–ray flux
for all sources of the samples used in this analysis. The dashed
lines are the spectroscopic limits of completeness adopted in our
2.3. The AMSSn sample
The AMSSn sample consists of 74 AGN at fluxes
brighter than SX(2–10 keV)= 3 × 10−13erg cm−2
s−1(Akiyama et al. 2003). The total area covered is
45 deg2at the fainter fluxes and rises up to ∼69 deg2
at bright fluxes. The NH column densities have been
derived from the hardness ratios values.
2.4. The HBS28 sample
The HBS28 sample (Caccianiga et al. 2004) consists
of 27 AGN and 1 star selected in the 4.5-7.5 keV band.
The sources are brighter than SX(2–10 keV)= 2.2×10−13
erg cm−2s−1(assuming Γ=1.8) and have been selected
over 82 XMM-Newton pointed fields, corresponding to a
total of 9.756 deg2. All sources have been spectroscop-
ically identified, and their column densities have been
measured through X–ray spectral fits.
2.5. The Lockman Hole sample
The Lockman Hole sample consists of 55 sources se-
lected within the 12 arcmin radius of the XMM-Newton
observation. The sources are brighter than SX(2–10
keV)= 2.6 × 10−15erg cm−2s−1(Baldi et al. 2002).
Optical identifications and X–ray spectral fits are from
Mainieri et al. (2002). Spectroscopic redshifts and clas-
sifications have been obtained for 41 objects, while 3
sources have photometric redshifts. We restricted our
analysis to the sources brighter than R=23.50. Down to
this limit 39 out of 41 sources have been spectroscopically
4 LA FRANCA ET AL.
2.6. The CDF-N sample
In order to reach almost spectroscopic completeness we
have selected an X–ray bright subsample in the CDF-N.
The subsample consists of 146 sources (see Table 1) se-
lected within the 10 arcmin radius of the Chandra ob-
servation (Alexander et al. 2003).
reaches SX(2–10 keV)> 10−15erg cm−2s−1in the in-
ner 5.85 arcmin radius, SX(2–10 keV)> 2.49 × 10−15
erg cm−2s−1in the annulus between 5.85 and 6.5 ar-
cmin radii, and SX(2–10 keV)> 3.61 × 10−15erg cm−2
s−1in the annulus between 6.5 and 10.0 arcmin radii.
We used both spectroscopic and spectro-photometric
identifications and redshifts available from the literature
(Barger et al. 2003). We restricted our analysis to sources
brighter than R=24.65. Down to this limit 102 out of 108
sources have been spectroscopically identified. The NH
column densities have been derived from the hardness
The sky coverage
2.7. The CDF-S sample
Altough the CDF-S has been observed for 1 Ms in-
stead of the 2 Ms spent in the CDF-N, we selected a
spectroscopically complete X–ray bright subsample with
the same sky coverage as for the CDF-N. Indeed, at our
adopted flux limits, the difference in the exposure time
does not affect the sky coverage. The sample consists
of 127 sources (see Table 1; Giacconi et al. 2002 and
Alexander et al. 2003). We used both spectroscopic and
spectro-photometric redshifts available from the litera-
ture (Szokoly et al. 2004; Zheng et al. 2004). Moreover,
given that both Szokoly et al. (2004) and Zheng et al.
(2004) identifications are based on the X–ray source cat-
alogue of Giacconi et al. (2002), we have revised some
optical/X–ray associations according to the improved as-
trometry provided by Alexander et al. (2003). We re-
stricted our analysis to sources brighter than R=25.00.
Down to this limit 98 out of 102 sources have been spec-
troscopically identified. The NH column densities have
been derived from the hardness ratios.
We searched for a functional fit to the density of the
AGN as a function of the un–absorbed 2–10 keV lumi-
nosity (LX), the rest frame absorbing column density
(NH), and the redshift (z). The method is based on the
comparison, through χ2estimators, of the observed and
expected numbers of AGN (in the LX-z space) and of
the NHdistributions, obtained from computations which
take into account all the observational selection effects of
Once a HXLF evolution model is assumed, the number
of expected AGNs (E) in a given bin of the LX-z-NH
space is the result of the sum, over the number of samples
Nsamp, of the expected number of AGN in each sample
taking into account the area coverage of each ith sample
Ωi(L,NH,z), the NH distribution f(LX,z;NH), and a
completeness function g(LX,z,NH,Ri), where Riis the
spectroscopic limit of completeness of the ith sample:
? ? ?
3.1. The shape of the Luminosity Function
In order to describe the evolution of the AGN, we used
standard functional forms, such as the pure luminosity
evolution (PLE) model and a luminosity-dependent den-
sity evolution (LDDE) model (see next Section and, e.g.,
Boyle et al. 1998; Miyaji et al. 2000; La Franca et al.
2002; Ueda et al. 2003). The HXLF, representing the
number density per unit comoving volume and per unit
Log LX, as a function of LXand z, was expressed as:
We adopted a smoothly-connected two power-law form
to describe the present-day HXLF,
dΦ(LX,z = 0)
= A[(LX/L∗)γ1+ (LX/L∗)γ2]−1. (3)
3.2. The K-correction
In order to convert the observed 2–10 keV fluxes (SX)
to the intrinsic 2–10 keV luminosities (LX) and vice-
versa, for each observed or “expected” AGN with a given
NH, a K-correction has been computed by assuming a
photon index Γ = 1.8, an exponential cutoff (e−E/EC) at
EC= 200 keV, and the corresponding photoelectric ab-
sorption (see §4.1.1 for a discussion on the use of different
3.3. The completeness function
All the faint samples used in our analysis (HEL-
LAS2XMM, Lockmann, CDF-S, CDF-N) are nearly
spectroscopicallycomplete down to a certain optical limit
magnitude (R=21.4−25, see Table 1). In order to com-
pute the number of expected AGN in a certain bin of the
LX-z-NH space, we introduced the completeness func-
tion g(LX,z,NH,R) which provides the probability that
a given AGN with luminosity LX, redshift z and column
density NH, had an apparent R-band magnitude brighter
than the spectroscopic limits of completeness R of each
For this reason we derived an empirical relationship
between the un–absorbed X–ray luminosity LXand the
optical luminosity LR9for AGN1 and AGN2, and mea-
sured their spread (see Figure 3). For AGN1 we found:
LogLR= 0.959(±.025)× LogLX+ 2.2(±1.1),(4)
with a 1σ dispersion of 0.48 (in LogLR units) around
the best fit solution. The linear correlation coefficient is
9The LRluminosity is in erg s−1(νLν), computed at 660 nm,
where the flux is f[erg s−1cm−2Hz−1] = 2.84×10−20×10−0.4R
HELLAS2XMM: THE HARD X–RAY LUMINOSITY FUNCTION OF AGN5
Fig. 3.— LogLX–LogLRrelation for optical AGN1 and AGN2.
The continuous lines correspond to eq. 4 and 5.
r=0.773, corresponding to a negligible (<10−13) proba-
bility that the data are consistent with the null hypoth-
esis of zero correlation. For AGN2 a flatter relation was
LogLR= 0.462(±.026)× LogLX+ 23.7(±1.1), (5)
with a 1σ dispersion of 0.40 (in LogLRunits), and a lin-
ear correlation coefficient r=0.462, again corresponding
to a negligible (< 2×10−13) probability that the data are
consistent with the null hypothesis of zero correlation. In
order to compute the above relationships a linear least
squared method with errors (assumed 0.2 dex) in both
axes has been used. The difference between the two rela-
tions should be attributed to the dominance in the opti-
cal of the AGN component in the AGN1, which produces
an almost linear relationship between X–ray and optical
luminosity (see La Franca et al., 1995 for similar results
in the soft X–rays). In AGN2, where the nucleus is ob-
scured, the optical luminosity is instead dominated by
the host galaxy (see also Fiore et al. 2003).
For each pair of un-absorbed X–ray luminosity and red-
shift, the above relationships (with their spreads) can be
used to compute the probability of an AGN to appear
brighter than a certain optical magnitude, and thus be
spectroscopically identified. The observed spreads of the
two relationships are due to a combination of the intrin-
sic spread with the observational uncertainties. Given
our aims, both effects should be taken into account, and
we have thus not subtracted the contribution of the ob-
servational uncertainties from the spread estimates. To
choose which LX-LR relationship to use (eq. 4 or eq.
5), we need also to know the probability of an AGN to
appear as an AGN1 (or, its complement, an AGN2) as a
function of LX, NHand z: Q1(LX, z, NH). This proba-
bility was estimated from the sample itself as described
Figure 4 shows the distribution of the observer frame
column density NH0as a function of LXfor AGN1 and
Fig. 4.— LogLX–LogNH0plane at z ≤ 1 (bottom), 1 < z ≤
2 (middle), z > 2 (up). Open squares are optical AGN1, filled
circles are optical AGN2. On the right side the dashed lines are the
cumulative distribution functions of the NH0values for AGN1 with
LX>1043.5erg s−1, while the continuous lines are the complement
of the cumulative distribution function of the NH0values for AGN2
with LX>1043.5erg s−1.
AGN2, in three redshift intervals. Here we do not use the
rest frame NH, but instead the observer frame NH0which
is equivalent to an hardness ratio (see also Hasinger
2003). As can be seen in Figure 5, the probability to
find an AGN1 is not only dependent on NH0, but de-
pends also on the luminosity. The probability to find an
AGN1 increases with increasing luminosities, and there
is a relevant fraction of low luminosity (LX<1043erg s−1)
un–absorbed objects which are AGN2, while a fraction of
the high luminosity (LX>1045erg s−1) absorbed objects
are AGN1. This result, if it is not due to the contamina-
tion by the galaxy light in the lower luminosity AGN2, is
against the simplest version of the AGN unified model.
The analysis of this issue is beyond the scope of this pa-
per (see Panessa & Bassani 2002, Page et al. 2003, Steffen
et al. 2003, Ueda et al. 2003, Brusa et al. 2003, Perola
et al. 2004 and Barger et al. 2005 for similar results and
discussions. See also §4.6)
As Figure 4 shows, there is no evidence of a dependence
on redshift of the distribution of AGN1 and AGN2 as a
function of LX and NH0. We have thus estimated the
probability of an AGN to appear as an AGN1 as a func-
tion of LXand NH0only, by assuming no dependence on
redshift. This probability has been estimated as a func-
tion of LX in two bins of NH010: at NH0≤1021.5cm−2
10We chose to use, here, the observed column densities (NH0)
instead of the intrinsic ones (NH) in order to eliminate the de-
pendencies on the redshift.A constant (with z) NH0=1021.5
cm−2separation limit corresponds to a shifts of this limit to-
wards higher values of NH with increasing redshift (as the in-
trinsic and the z=0 column densities are related by the equation
log(NH) = log(NH0) + 2.42log(1 + z)). We will come back to this
point in the next Sections. However, we wish to stress here that the
above relationships have been derived only in order to correct the