A&A 369, 49-56 (2001)
T. Miyaji 1,2,3 - G. Hasinger 3 - M. Schmidt 4
1 - Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
2 - Max-Planck-Inst. für Extraterrestrische Physik, Postf. 1603, 85740 Garching, Germany
3 - Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
4 - California Institute of Technology, Pasadena, CA 91125, USA
Received 14 June 2000 / Accepted 8 January 2001
This is the second paper of our investigation of the 0.5-2 keV soft X-ray luminosity function (SXLF) of active galactic nuclei (AGN) using results from ROSAT surveys of various depth. The large dynamic range of the combined sample, from shallow large-area ROSAT All-Sky Survey (RASS)-based samples to the satellite's deepest pointed observation on the Lockman Hole, enabled us to trace the behavior of the SXLF. While the first paper (Miyaji et al. 2000, hereafter Paper I) emphasized the global behavior of the SXLF, cosmological evolution and contribution to the soft X-ray background, this paper presents actual numerical values for practical use of our results. To present the binned SXLF, we have used an improved estimator, which is free from biases associated with the conventional estimator.
Key words: galaxies: active - galaxies: evolution - (galaxies:) quasars: general - X-rays: galaxies - X-rays: general
The AGN/QSO luminosity function and its evolution with
cosmic time are key observational quantities for understanding
the origin of and accretion history onto supermassive black holes,
which are now believed to occupy the centers of most galaxies.
Since X-ray emission is one of the prominent characters of
the AGN activity, X-ray surveys are efficient means of sampling
AGNs for luminosity function and evolution studies.
An X-ray selected sample of AGNs is particularly useful
because optical surveys often use point-like morphology as a
criterion for selecting AGNs (QSOs) among numerous other
objects, and thus are likely to miss moderate-luminosity
intermediate-high redshift AGNs embedded in their host galaxies.
Also, radio surveys sample only a minor population of AGNs.
The Röntgen satellite (ROSAT)
provided us with soft X-ray surveys with various
depths, ranging from the ROSAT All-Sky Survey (RASS, Voges et al. 1999)
to the ROSAT Deep Survey (RDS) on the Lockman Hole (Hasinger
et al. 1998). Various optical identification programs
of the survey fields have been conducted and the combination
of these now enabled us to construct the soft X-ray luminosity
function (SXLF) as a function of redshift.
In Paper I, we presented a number of
global representations of the 0.5-2 keV SXLF and investigated
the contribution to the soft X-ray background.
We showed that our data are not consistent with
the pure-luminosity evolution (PLE), contrary to
the suggestions of a number of previous analyses (e.g. Boyle et al.
1994; Jones et al. 1996). Instead,
we find an excess of intermediate-redshift low-luminosity
AGNs above the PLE case, some sign of which was also recognized
by Page et al. (1997). In view of this, we developed
two versions of luminosity-dependent density evolution
(LDDE1 and LDDE2) models, which represent the observed data
very well. An extrapolation of these two LDDE models below the faintest
limit of the survey (
yields significantly different predictions
for fainter fluxes, bracketing the range of a possible AGN
contribution to the soft X-ray Background.
Chandra (e.g. Mushotzky et al. 2000;
Hornschemeier et al. 2000) and XMM-Newton (2001)
are probing much fainter sources and spectroscopic identifications
of these will eventually show which of these models is closer to the
actual behavior of the AGN SXLF. However, because at least some of the
faint X-ray sources are optically too faint for spectroscopic
identification even with the largest telescopes, extending the XLF into
such faint flux level may be difficult.
In this second paper, we present practical and convenient
expressions of the observed SXLF from the ROSAT
surveys. We present our results mainly for the investigators who
are interested in particular redshift regimes
and/or comparing their models with observations.
We have used soft X-ray sources identified with AGNs with redshift information from a combination of ROSAT surveys in various depths/areas from a number of already published and unpublished sources. Detailed description of the definition of the sample, ROSAT countrate-to-flux conversion, and survey area are shown in Paper I. The summary of the samples, which is a duplicate of Table 1. of Paper I with updated references, is shown in Table 1. The details of the nature and completeness of each sample were discussed in Paper I. The limiting flux versus survey area relation were also shown in Paper I.
As in Paper I, we present the SXLF in the observed 0.5-2 keV band, i.e., in the 0.5(1+z)-2(1+z)keV range in the object's rest frame. This is equivalent to assuming an energy index of 1. Thus no K-correction was applied for our expressions presented here. The reasons for this choice are explained in detail in Paper I. This choice is particularly important for this paper, which is intended to be used as observational constraints for population-synthesis-type models (e.g. Madau et al. 1994; Comastri et al. 1995; Gilli et al. 1999, 2000; Miyaji et al. 2000) with various spectral assumptions. Because of that, it is more useful to provide quantities in a model-independent form rather than applying a particular version of model-dependent K-corrections. By presenting the data in this manner, one can avoid the difficulty of reverse K-correcting and re-applying new K-corrections when the new results from Chandra and XMM provide better knowledge of the X-ray spectra of the population.
|0.015-0.2||0.1||269||0.60 +0.68-0.32||0.59 +0.23-0.29||2.1 +0.4-0.3||4.22 +2.53-2.61||0.99, 0.68, 0.61|
|0.2-0.4||0.3||113||0.89 +1.20-0.46||0.67 +0.30-0.38||2.5 +0.4-0.3||5.25 +3.48-3.51||0.93, 0.56, 0.72|
|0.4-0.8||0.6||99||0.54 +0.85-0.29||0.33 +0.52-0.87||2.2 +0.3-0.2||5.95 +2.29-2.29||0.85, 0.85, 0.57|
|0.8-1.6||1.2||135||1.48 +1.14-0.56||0.40 +0.41-0.53||2.4 +0.2-0.2||4.07 +1.33-1.34||0.99, 0.96, 0.83|
|1.6-2.3||2.2||44||1.2(*)||0.0(*)||2.1 +0.2-0.1||0(*)||0.27, 0.36, 0.19|
|2.3-4.6||3.0||25||1.0(*)||...||1.9 +0.2-0.2||0(*)||0.72, 0.99, 0.64|
|0.015-0.2||0.1||269||0.59 +0.71-0.32||0.59 +0.23-0.30||2.1 +0.4-0.3||4.13 +2.56-2.63||0.99, 0.60, 0.54|
|0.2-0.4||0.3||113||0.93 +1.30-0.49||0.67 +0.31-0.39||2.4 +0.4-0.3||5.31 +3.49-3.51||0.99, 0.53, 0.79|
|0.4-0.8||0.6||99||0.69 +1.07-0.36||0.37 +0.50-0.80||2.3 +0.3-0.2||5.90 +2.28-2.28||0.97, 0.81, 0.57|
|0.8-1.6||1.2||135||2.14 +1.67-0.83||0.42 +0.40-0.52||2.4 +0.2-0.2||4.13 +1.34-1.34||0.98, 0.97, 0.72|
|1.6-2.3||2.2||44||1.8(*)||0.0(*)||2.1 +0.2-0.1||0(*)||0.19, 0.41, 0.15|
|2.3-4.6||3.0||25||1.0(*)||...||1.9 +0.2-0.2||0(*)||0.64, 0.96, 0.75|
|0.015-0.2||0.1||269||0.71 +0.86-0.39||0.62 +0.22-0.29||2.1 +0.4-0.3||3.79 +2.56-2.64||0.99, 0.54, 0.61|
|0.2-0.4||0.3||113||1.09 +1.53-0.58||0.67 +0.30-0.39||2.4 +0.4-0.3||4.95 +3.49-3.51||0.97, 0.56, 0.82|
|0.4-0.8||0.6||99||0.85 +1.41-0.46||0.36 +0.51-0.87||2.2 +0.3-0.2||5.69 +2.28-2.27||0.96, 0.81, 0.60|
|0.8-1.6||1.2||135||2.69 +2.07-1.04||0.43 +0.39-0.51||2.4 +0.2-0.2||4.10 +1.33-1.34||0.98, 0.97, 0.74|
|1.6-2.3||2.2||44||2.0(*)||0.0(*)||2.1 +0.1-0.1||0(*)||0.14, 0.40, 0.16|
|2.3-4.6||3.0||25||1.0(*)||...||1.9 +0.2-0.2||0(*)||0.78, 0.98, 0.76|
First, we find a smooth analytical function for each redshift bin using a Maximum-likelihood fitting. The absolute goodness of the resulting expression can then be tested by one- and two-dimensional Kolgomorov-Smirnov tests (hereafter, 1D-KS and 2D-KS tests respectively; Press et al. 1992; Fasano & Franceschini 1987). See Paper I for detailed description of these methods. These fittings and tests can be applied to unbinned data sets thus are free from artifacts and biases from binning.
For an analytical expression, we use the smoothed two-
power-law formula, as we did in Paper I. Here,
we fit the data in narrow redshift bins and thus evolution
in each redshift bin is assumed to be a pure density evolution
|Figure 1: The binned XLFs from a simulated sample using three different estimators (symbols with error bars as labeled) are compared with the underlying "true'' XLF represented by dashed lines. The vertical positions of the three different estimators have been shifted vertically for display|
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The luminosity range of the fit is from Log to the maximum available luminosity in the sample. As shown below and in Paper I, the SXLF below the minimum luminosity has a significant excess above the smooth extrapolation. This excess smoothly connects with the SXLF of the non-AGN population (e.g. Hasinger et al. 1999) and the X-ray emission may be significantly contaminated by non-AGN activities. The set of parameters which give the best fit for each redshift bin are shown in Table 2 along with the results of the 1D- and 2D- KS tests (see the notes of the table). The parameter errors correspond to a likelihood change of 2.7 (90% confidence errors). In any case, Eq. (1) gives a statistically satisfactory expression for all redshift bins.
The estimator, which is a generalized version of the original estimator (Schmidt 1968) applied to a sample composed of subsamples of different depths (see Paper I; Avni & Bahcall 1980), has been widely used for binned luminosity functions (LF; we use the acronym LF when the discussion is not limited to the luminosity function in the X-ray band) in the literature. However, as discussed in Paper I (see also Wisotzki 1998; Page & Carrera 1999), using it for a binned LF estimator can cause significant biases, especially if the bin covers the flux range where the available solid angle of the survey changes rapidly as a function of flux. Also, the choice of the location in a bin with a non-negligible width at which the data point is plotted significantly changes the impression of the plot. In Fig. 3 of Paper I, however, we plotted the estimates, because of the lack of a reasonable alternative at the time of writing that paper, with caveats on biases associated with the method. We note that the estimator can be used in an unbinned manner by considering a set of delta-functions weighted by (or ) at the positions of sample objects in the luminosity space (Schmidt & Green 1983), and this method is free from biases mentioned above. While this unbinned method is a powerful tool to predict, e.g., the source counts, it does not provide practical means of plotting. In this paper, we have developed an improved estimator, which is explained in the next subsection.
As an alternative to the
we have developed the following estimator for the binned
LF, which is free from most biases unavoidable
method. In Sect. 3.1, we have
found a smooth analytical function which describes the behavior of the
SXLF in a given redshift range. Having the best-fit smooth function,
the estimated numerical value for the SXLF in a given
bin in the
Another advantage of this estimator over is that exact errors at a given significance can be evaluated using Poisson statistics. One disadvantage of this estimator is that it is model-dependent, at least in principle. Since our analytical expressions are satisfactory representations in any case and the estimator is not sensitive to the details of the underlying model, the uncertainties due to the model dependence are practically negligible.
In order to compare the goodness of the estimators, we performed simulations. Using the actual best-fit model for the 0.2<z<0.4 bin for , we generated a set of simulated AGNs. The number of simulated AGNs are 10 times those of the actual sample in order to reduce the Poisson errors. Using the simulated AGNs and the actual flux-area relation of our combined sample, we estimated binned SXLFs using three different estimators: , (Eq. (2)) and that of PC. The results are compared with the underlying SXLF, which was used to generate the simulated AGNs, in Fig. 1. For the models to evaluate , we used the re-fitted model using the simulated sample rather than the original model. The 1 errors for the (Eq. (2)) and the PC estimators are Poisson errors calculated using Eqs. (7) and (12) of Gehrels (1986). On the other hand, the errors for the estimator are from Eq. (3) of Paper I and are inaccurate for bins with a small number of AGNs.
As shown in Fig. 1, the estimator best represents the original model and no estimated point deviates from the underlying model by more than 2. The estimator underestimates the XLF in the lowest luminosity bin as found in PC. We note that the PC estimator also systematically underestimates the LF in this particular case of the underlying model and the flux-area relation. This is expected because their estimator implicitly builds in the assumption as the underlying LF shape. This is much more weighted towards higher luminosities than any part of the realistic AGN XLF. Since the amount of this bias depends on the underlying model and the flux-area relation, as well as the points in the bin where the data are plotted, it is not surprising that the bias is not apparent in Fig. 2 of PC. They have also compensated for this bias upon comparing the estimated LF with a model. Instead of correcting the estimated binned LF using a good model (which our estimator does), they calculated the "model-expectated value of the estimator'' to compare with the estimated value from the data. Detailed investigation and comparison of these different estimators in various cases are beyond the scope of this paper. Judging from this simulation, the above discussion on biases, and that the exact Poisson errors can be used for errors, we choose to use the estimator for our plots and tabulation.
|Figure 2: The binned SXLFs estimated by Eq. (2) are plotted with Poisson errors corresponding to the significance range of Gaussian 1 from Eqs. (7) and (12) by Gehrels (1986). The data points and error estimates are more accurate than Fig. 3 of Paper I. Different symbols correspond to different redshift bins as indicated in the lower-left part of panel a). The symbol attached to a downward arrow indicates the 90% upper limit (corresponding to 2.3 objects) for the bin with no AGN in the sample. The best-fit analytical model for each redshift bin in the luminosity range used for the fit is overplotted in dashed lines|
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The resulting binned SXLF are listed in Tables 3, 4, and 5 for different sets of cosmological
Figure 2 shows that the lowest-redshift, lowest-luminosity bin has a significant excess over the two power-law analytical expression (AGNs belonging to this bin have not been used for the two power-law fit), thus the actual underlying SXLF has a much steeper slope than that used for . In order to evaluate the bias caused by this, we have made an estimate of this particular bin using the local slope of instead of from the two power-law model. This gave a value about 10% lower, thus, the difference is much smaller than the statistical errors for this bin.
We have shown the tables of observed SXLF values for a number of standard set of cosmological parameters. These values are intended for direct comparison with models and plotting with realistic error bars. However, we list a number of caveats and sources of uncertainties, and related issues.
This work is based on a combination of extensive ROSAT surveys from a number of groups. Our work is indebted to the effort of the ROSAT team and the optical followup teams in producing data and the catalogs used in the analysis. In particular, we thank K. Mason, A. Schwope, G. Zamorani, I. Appenzeller, and I. McHardy for providing us with and allowing us to use their data prior to publication of the catalogs. TM was supported by a fellowship from the Max-Planck-Society during his appointment at MPE. GH acknowledges DLR grant FKZ 50 OR 9403 5. We thank the referee, T. Shanks, for useful comments.