A&A 383, 773-790 (2002)
DOI: 10.1051/0004-6361:20011769
A new measurement of the X-ray temperature function of clusters of galaxies
Y. Ikebe1 -
T. H. Reiprich2 -
H. Böhringer1 -
Y. Tanaka1,3 -
T. Kitayama4
1 - Max-Planck-Institut für extraterrestrische Physik,
Postfach 1312, 85741 Garching, Germany
2 -
Department of Astronomy, University of Virginia,
PO Box 3818, 530 McCormick Road, Charlottesville,
VA 22903-0818, USA
3 -
Institute of Space and Astronautical Science,
Yoshinodai 3-1-1, Sagamihara, Kanagawa 229-8510, Japan
4 -
Department of Physics, Toho University,
Miyama, Funabashi, Chiba 274-8510, Japan
Received 7 June 2001 / Accepted 11 December 2001
Abstract
We present a newly measured X-ray temperature function of galaxy clusters
using a complete flux-limited sample of 61 clusters.
The sample is constructed with the total survey area of 8.14 steradians
and the flux limit of
ergs s-1 cm-2in the 0.1-2.4 keV band.
X-ray temperatures and fluxes of the sample clusters
were accurately measured with ASCA and ROSAT data.
The derived temperature function covers an unprecedentedly wide temperature
range of 1.4-11 keV.
By fitting these data with theoretically predicted temperature functions
given by the Press-Schechter formalism together with a recent formation
approximation and the CDM power spectrum,
we obtained tight and individual constraints on
and 
.
We also employed the Formation-Epoch model
in which the distribution in the formation epoch of clusters
as well as the temperature evolution are taken into account,
showing significantly different results.
Systematics caused by the uncertainty in the mass-temperature relation are studied
and found to be as large as the statistical errors.
Key words: cosmology: observations - cosmological: parameters - X-rays: galaxies: clusters
The mass function of clusters of galaxies (MF),
the number density of the most massive virialized systems,
contains information on the structure formation history of the universe.
A theoretical framework, e.g. the Press-Schechter formalism
(Press & Schechter 1974) together with the Cold Dark Matter model,
has been established to predict the MF.
This allows us to constrain cosmological parameters
using an observationally determined MF for the present epoch
(as well as its time evolution for even tighter constraints).
In particular, ,
the amplitude of mass density fluctuations
on a scale of 8h-1 Mpc where h is the Hubble constant in
unit of 100 km/s/Mpc,
and
,
the mean matter density,
are most sensitively determined by the cluster abundance measurements
(e.g. Henry & Arnaud 1991;
White et al. 1993;
Eke et al. 1996; Kitayama & Suto 1996;
Viana & Liddle 1996; Oukbir & Blanchard 1997;
Pen 1998; Eke et al. 1998).
Observationally the local MF has been derived from
measuring masses of individual clusters
from galaxy velocity dispersions or other optical properties
by Bahcall & Cen (1993), Biviano et al. (1993), and
Girardi et al. (1998).
The estimated virial masses for individual clusters
depend rather strongly on model assumptions, however.
As argued by Evrard et al. (1997) on the basis of hydrodynamical
N-body simulations, cluster masses may be presently more accurately
determined from a temperature measurement
and a mass-temperature relation determined from detailed observations
or numerical modeling.
Thus alternatively, as a well-defined observational quantity,
the X-ray temperature function (XTF) has been measured,
which can be converted to the MF by means of the mass-temperature relation.
The first measurements of the XTF were reported
by Edge et al. (1990) and Henry & Arnaud (1991),
using an X-ray flux-limited sample of 45 clusters and 25 clusters,
respectively.
Recent observational improvement on the XTF was made
by Markevitch (1998), Henry (2000),
Blanchard et al. (2000), and Pierpaoli et al. (2001),
using more accurate temperature-measurement results
for each cluster with ASCA data (Tanaka et al. 1994).
However, the narrow temperature ranges (3-10 keV) in which
the XTF is defined so far do not allow
an investigation of a more detailed shape of the XTF
other than just fitting a single power-law.
Without better information on the actual shape of the XTF,
no independent constraints on
and
can be derived, and a wide range of combinations is still allowed.
Recently, a number of new X-ray-cluster surveys were performed,
which provide a high completeness for the brightest clusters.
This motivated us to revisit the measurement of the local XTF
and to improve its accuracy in order to derive narrower constraints
on the cosmological parameters.
Reiprich & Böhringer (2002) have compiled
a new X-ray flux-limited cluster sample
with a flux limit of
ergs s-1 cm-2 (0.1-2.4 keV).
Compared with previous samples with similar flux limits,
their catalog covers the largest volume and is the most complete.
Based on this cluster sample, Reiprich & Böhringer have measured
total masses for individual clusters and derived the X-ray mass function
for the first time.
In this paper we, using this cluster sample,
report the construction of a new measurement of the XTF.
The larger number of clusters in the sample
together with accurate temperature measurements with ASCA data
leads to a significant improvement of the XTF,
which covers 1.4-11 keV temperature range.
The X-ray flux-limited sample and the new temperature measurements
with ASCA data are presented in Sect. 2.
The derivation of the XTF is described in Sect. 3.
In Sect. 4, the XTF is used to obtain constraints on cosmological parameters,
and the results are compared with other work in Sect. 5.
Throughout the paper, the Hubble constant is given as 
km/s/Mpc,
and 
and 
denotes a decimal and natural logarithm, respectively.
For this study we use a sample of 106 galaxy clusters compiled by
Reiprich & Böhringer (2002),
which is used to construct a flux-limited complete sample
with the flux-limit
of
ergs s-1 cm-2 (0.1-2.4 keV).
The 106 clusters have been derived from combining previous X-ray cluster
or elliptical galaxy catalogs
that include Böhringer et al. (2000, 2001),
Böhringer (1999),
Retzlaff et al. in preparation,
Ebeling et al. (1996, 1998), de Grandi et al. (1999),
Beuing et al. (1999), Lahav et al. (1989),
and Edge et al. (1990).
Most of them are based on the ROSAT All Sky Survey
(Trümper 1993; Voges et al. 1999).
From these catalogs, all objects satisfying certain criteria,
mainly that the measured flux given in any of these catalogs
is above
ergs s-1 cm-2 (0.1-2.4 keV),
have been sampled.
A defined region around the center of M 87 is not included
in the survey area,
because of the large scale emission of the Virgo cluster
that compromises cluster detection and characterization in the region.
This excludes the Virgo cluster and M 86 from the master catalog.
The count rate of the PSPC (Pfeffermann et al. 1987)
in the 0.1-2.4 keV band for each object was redetermined
from the ROSAT All Sky Survey data or PSPC pointing observations
whenever available
by integrating all X-ray flux within a maximum radius where
the cluster emission is significantly detected.
See Reiprich & Böhringer (2002) for detailed descriptions
of the sample construction and data analysis.
The names and the redshifts compiled from the recent literature
of all the 106 clusters are listed in Table 1.
As described below, we newly determine the X-ray temperature as well as the X-ray flux
for each of the 106 clusters.
ASCA data are available for 88 of the 106 clusters,
and the X-ray temperatures are measured by analyzing the ASCA data
(Sect. 2.2).
Using the ASCA results,
the correlation between the X-ray luminosity and the temperature,
L-T relation, is established (Sect. 2.3).
Among the other clusters for which no ASCA data exist
(18 clusters), temperatures of 4 clusters have been measured with
Einstein, EXOSAT, or XMM-Newton in previous works
and were used in our analysis. For the remaining 14 clusters,
the temperatures are estimated from the observed ROSAT PSPC
count rate by means of the L-T relation.
By setting a flux-limit of
ergs s-1 cm-2 (0.1-2.4 keV),
a complete flux-limited sample comprising 63 clusters
is then constructed (Sect. 2.4).
*):
The Galactic latitude is lower than 20 .
 :
The values are calculated for
an open (
 ,
 )
and a flat (
,
 )
universe.
 :
The temperature is estimated with the L-T relation assuming
the open universe.
 :
The temperature is estimated with the L-T relation assuming
the flat universe.
 :
The temperature is derived with non-ASCA spectroscopy.
Einstein MPC for A2877 and A1644 (David et al. 1993),
EXOSAT LE+ME for EXO0422
(Edge & Stewart 1991),
and XMM-Newton EPIC for S1101 (Kaastra et al. 2001).
|
For our study we need to determine an average temperature
for each cluster that closely reflects the mass of the cluster,
i.e. a virial temperature.
In many clusters, the X-ray emitting hot gas is not isothermal
but a cooler gas component is often observed in the central region.
The cool component, which is due to either a cooling flow
(e.g. Fabian 1994) or the ISM of cD galaxies
(e.g. Makishima et al. 2001),
often exhibits a significant fraction of the luminosity
and has to be separated from the rest of the X-ray emission
to determine the cluster average temperature
(see also Markevitch 1998; Arnaud & Evrard 1999).
Therefore, in the ASCA data analysis,
we employed a two-temperature (2T) model,
in which isothermal plasma is filling the entire cluster region,
while, in the central region, another cooler isothermal gas component
is allowed to coexist with the hotter plasma forming a multi-phase
intra-cluster medium (ICM).
This 2T picture was established for the Centaurus cluster
(Fukazawa et al. 1994; Ikebe et al. 1999)
and also gave a good account of the ASCA data of Virgo/M 87
(Matsumoto et al. 1996),
Hydra-A (Ikebe et al. 1997),
Abell 1795 (Xu et al. 1998),
and other nearby clusters (Fukazawa 1997;
Fukazawa et al. 1998, 2000).
The hot component extends over the major part of cluster volume
as far as is measurable
(e.g. Fukazawa 1997; White 2000).
The temperature of this hot component derived from the 2T model
therefore represents the best estimation of the virial temperature
of the cluster.
Although a cooling flow spectral model also gives generally
a good description of the ASCA spectra
(e.g. Fabian et al. 1994;
White 2000; Allen et al. 2001),
the estimated temperatures from where gas starts to cool
vary considerably from author to author
and are sometimes spuriously high.
More importantly, recent XMM-Newton data clearly show
that the conventional cooling flow spectral model is not adequate
(Peterson et al. 2001; Tamura et al. 2001;
Kaastra et al. 2001).
The 2T model, on the other hand, is valid to the XMM-Newton data
and the hot component temperature well represents the temperature of
the outer main component (see Ikebe 2001).
It is known that, apart from the central cool component,
many clusters show deviations from isothermality
such as an asymmetric temperature
distribution due to merging (e.g. Briel & Henry 1994)
or a global temperature decrement towards the outside
(Markevitch et al. 1998).
For such clusters, the 2T model fit
gives insignificant flux to the cool component
and works practically as an isothermal model.
Thus the hot component temperature gives the average temperature,
which also can be a good measure of the virial temperature.
Among the 106 sample clusters,
88 clusters have been observed with ASCA,
and all the data sets have become publicly available by now.
We retrieved the ASCA data
from the ASCA archival data base provided
by NASA Goddard Space Flight Center and by Leicester University.
Here we will give a brief explanation of the analysis method
of the ASCA data.
ASCA has four focal plane instruments,
two SIS (Solid-state Imaging Spectrometer)
and two GIS (Gas Imaging Spectrometer; Ohashi et al. 1996),
which were usually used simultaneously to observe an astronomical object.
In all the observations, the GIS were operated in the normal PH mode
and the SIS were operated in FAINT mode or BRIGHT mode.
The number of active CCD chips of each SIS used was 4, 2, or 1,
corresponding to the field of view of
,
,
or
,
respectively.
We discarded the GIS and SIS data that were taken when
the elevation angle of the X-Ray Telescope (XRT) from the local horizon
was less than 
.
An additional screening requirement, that the elevation angle from
the sunlit earth be greater than
and
,
was applied to the SIS-0 and SIS-1 data, respectively.
In order to ensure a low and stable particle background,
we also discarded GIS and SIS data acquired under a geomagnetic
cutoff rigidity smaller than 6 GV.
The background is composed of cosmic X-ray background
and non-X-ray background, which were estimated for each cluster observation
from the data of blank-sky observations as well as
the data taken when the X-Ray Telescope was pointing at the dark
(night) earth.
In order to maximize the detection efficiency of the central cool component,
we accumulated spectra over two regions, a central region of 2' radius
and an outer region from 2' to
,
from each cluster data set.
,
the maximum radius, is defined as the radius
within which at least 95% of all detected source X-ray counts,
within the field of view of the GIS, are included.
The data from the two GIS sensors and two SIS sensors were combined.
Thus for each cluster, four spectra: two central and two outer spectra
for GIS and SIS, were derived.
An effective area as a function of X-ray energy
was calculated for each spectrum with an ASCA simulator (SimARF).
The point-spread-function of the ASCA XRT is so largely extended
that the X-ray flux from the central brightest region may contribute
to the outer region spectrum
(Serlemitsos et al. 1995; Takahashi et al. 1995).
We therefore, simultaneously fitted the four spectra with the 2T model,
taking into account the flux-contamination effect.
As a plasma code, we used the MEKAL model
(Mewe et al. 1985, 1986;
Kaastra 1992; Liedahl et al. 1995),
that is implemented in XSPEC version 10.0 (Arnaud 1996).
A best-fit model was derived by a chi-square minimization method.
The cool and hot component temperatures are separate free parameters.
Many clusters do not require an additional cool component to fit
the spectra, and the cool component temperatures cannot be constrained.
For each cluster we investigated the significance of adding
a cool component by comparison with a fit of a hot isothermal model
and performing an F-test.
For a cluster showing a low significance for the additional cool component
in the 2T model,
we fixed the temperature of the cool component at
,
where
is the temperature derived with the isothermal model fitting.
The choice of the fixed cool-component temperature is based on
a clear correlation found in the 2T model fitting
that, for clusters showing a high significance for the cool component,
the cool-component temperature is always very close to half of
the hot component temperature (see Ikebe 2001).
The hot component temperatures thus derived are summarized in Table 1.
A more detailed analysis procedure description and
results on the cool components will be presented in a following publication.
We compared the temperatures derived here with those of previous workers.
The temperatures agree with results of Fukazawa (1997)
within 1.5% on average (Fig. 1a),
who performed a similar analysis
on a smaller number of clusters with ASCA data,
accumulating spectra from a central region of 2-3 arcmin radius
and an outer region, and fitting them with the 2T model.
On the other hand,
temperatures given by Markevitch et al. (1998)
and White (2000),
who used different (cooling flow) models to fit the ASCA spectra,
are higher than our values respectively by a factor of 1.09 and 1.25 on average
for clusters hotter than 6 keV (Figs. 1b,c).
This difference causes a significant difference in the resulting XTFs
(see Sect. 3.2).
![\begin{figure}
\par\includegraphics[angle=-90,width=5cm,clip]{h2920f1a.ps}\par\i...
...20f1b.ps}\par\includegraphics[angle=-90,width=5cm,clip]{h2920f1c.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg72.gif) |
Figure 1:
Comparison between the cluster temperatures derived here
and those by Fukazawa (1997) a),
by Markevitch et al. (1998) b),
and by White (2000) c). |
| Open with DEXTER |
For the 88 clusters observed with ASCA,
X-ray fluxes and luminosities were calculated
from the spectral-analysis results described above
and the ROSAT PSPC count rates in 50-201 PI channel
given by Reiprich & Böhringer (2002).
Assuming isothermality in each cluster at the hot-component temperature
and the Galactic column density given in Table 1,
we determined the 0.1-2.4 keV fluxes and luminosities that reproduce
the PSPC count rates.
As summarized in Table 1, the luminosities were calculated
for two different cosmological models
with an open (
)
and a flat (
)
universe.
We then established a correlation between the luminosity
in the 0.1-2.4 keV band,
,
and the temperature of the hot component, 
,
as illustrated in Fig. 2.
The
-
relations for
the open and flat universe were fitted individually
with power-law functions by a linear regression method.
For the fit we did not use clusters with a hot component temperature
cooler than 1.4 keV,
because it is known that
a single power-law function is not a good representation
of the L-T relation over a wide temperature range.
In particular, including clusters, groups, and elliptical galaxies,
the L-T relation becomes steeper below 
1 keV
(e.g. Xue & Wu 2000).
Moreover, the theoretically predicted XTFs as given in Sect. 4 may not be
directly compared with an observed XTF (see Sect. 4.3 for more explanation).
Therefore, in the power-law fitting, we exclude 6 sample clusters
cooler than 1.4 keV and use only 82 clusters.
For the linear regression fit,
we employed the BCES(X2|X1) estimator
given by Akritas & Bershady (1996).
(In this case, the luminosity and
temperature are assigned as X2 and X1, respectively.)
The choice of the estimator is based on an argument by
Isobe et al. (1990).
The best-fit functions are
 |
|
|
(1) |
for the open universe, and
 |
|
|
(2) |
for the flat universe.
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f2.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg79.gif) |
Figure 2:
The luminosities in the energy range of 0.1-2.4 keV in the cluster
rest frame in a flat universe
(
,
)
as a function of the hot component temperature
obtained from the two-temperature model fitted to the ASCA data.
The best-fit power-law function (Eq. (2)) is overlayed. |
| Open with DEXTER |
We have made a correction
to the derived
relation.
Since the less luminous clusters are less likely to be sampled,
the apparent mean luminosity at a given temperature,
which is estimated by the simple regression fit,
is biased towards higher values than the true mean value.
As will be shown in Sect. 4,
we studied this effect and corrected the normalization
to derive the "true''
relation,
as
 |
|
|
(3) |
for the open universe, and
 |
|
|
(4) |
for the flat universe.
The correction reduces the luminosity at a given temperature by 30%.
The corrected
relation
will also be used to calculate
in Sect. 3.
The 30% reduction of the mean temperature
results in a reduction of
(see Sect. 3) by 40%.
Using the corrected
relations,
we estimated the temperatures of 14 clusters for
which no ASCA data nor other spectroscopic measurement are available.
An
and
combination was chosen
from the
relation
such that the predicted PSPC count rate equals the observed value.
Scatter around the best-fit power-law in the L-T relation
was assigned as error.
The estimated temperatures and luminosities for the open and flat universe
cases as well as the corresponding 0.1-2.4 keV fluxes which are the same
for the open and flat universe cases are summarized in Table 1.
We set a flux-limit of
ergs s-1 cm-2in 0.1-2.4 keV band to construct a flux-limited complete sample
from the master catalog of 106 clusters.
As in Reiprich & Böhringer (2002) for
,
additional selection criteria applied are that
the absolute Galactic latitude is greater than
,
and the cluster is located neither in the Magellanic Clouds
nor in the Virgo cluster region.
The total number of clusters thus selected is 63,
whose temperatures and redshifts are shown in Fig. 3.
Since the temperatures of the individual clusters were newly determined
in this paper,
the fluxes estimated from the PSPC count rate
are slightly revised from the value in Reiprich & Böhringer (2002)
where the temperatures are compiled from literature.
All of the 63 clusters selected here, however, still correspond with
the 63
clusters.
Tests performed in Reiprich & Böhringer (2002)
showed no indication of incompleteness of the sample.
In the following analysis, we exclude 2 clusters cooler than 1.4 keV
(see Sects. 2.3 and 4.3) from the flux-limited complete sample,
and use 61 clusters to study the XTF and make constraints on
cosmological parameters.
This is the largest complete sample of clusters up to now
with temperature measurements or reliable temperature estimates.
We also stress the accuracy of the temperature measurements for
the sample clusters:
Among the 61 clusters used for constructing the XTF,
the temperatures of 56 clusters (90% of the sample)
were measured with ASCA data with 90% errors of 6-10%,
other spectroscopic measurements were used for 3 clusters,
and there are only two clusters for which the temperature is
estimated with the L-T relation.
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f3.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg85.gif) |
Figure 3:
X-ray temperature vs. redshift for the clusters of the flux-limited
sample. |
| Open with DEXTER |
Using the X-ray flux-limited complete sample of 61 clusters
(
),
we address in this section the XTF describing
the number density of clusters of galaxies in comoving
coordinates at present as a function of the X-ray temperature.
A full description of the analysis method can be found in the Appendix.
However, the key points are presented here.
The differential temperature function, 
,
is defined as
the number density of clusters in the temperature range
from
to
.
From a finite number of sample clusters,
it can be evaluated as
 |
(5) |
where i denotes a sequential number of clusters in a temperature
range from
to
,
and
is a maximum comoving volume in which
a cluster having a temperature of Ti could have been detected
under the selection criteria given above.
Following Markevitch (1998),
we assume, for a given temperature, the logarithmic luminosity, log L,
follows a Gaussian distribution around the mean
which is given with the L-T relation, i.e.
,
and with a constant standard deviation of
.
then can be evaluated as
 |
|
|
(6) |
where
is the maximum search volume for a cluster
having the luminosity of L and the temperature of Tand is given by Eq. (A.9).
Recall that
depends on the cosmological parameters
and was calculated for an open (
)
and a flat (
)
universe.
The results are listed for individual clusters in Table 2.
The XTF thus derived with 1 keV bin widths is illustrated
in Fig. 4.
The improvement of the new result over previous work
is mainly characterized by a significant broadening of the temperature range
towards lower temperature systems down to 1.4 keV.
A single power law is no longer a good description to the observationally
measured XTF and an exponential cut off towards the higher temperature
is clearly seen, as a theoretical model predicts.
The detection of such a curvature brings tight constraints
on cosmological parameters as will be shown in the next section.
Conventionally,
in Eq. (5)
is often simply replaced by
to evaluate the XTF (e.g. Henry 2000).
For comparison, the XTF evaluated with
is also overlayed in Fig. 4,
which overall shows good agreement
with the one with
,
but exhibits larger variance
due to the intrinsic scatter in the L-T relation.
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f4.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg107.gif) |
Figure 4:
The X-ray temperature function derived with
the flux-limited complete sample of 61 clusters
for the case of a flat universe
(
).
Filled circles show the XTF evaluated with
 ,
while
open stars show the one obtained with
 .
The temperature bin widths are 1 keV, except for the lowest temperature
bin that includes kT=1.4-2.5 keV.
The vertical error bars for filled circles indicate Poisson errors.
Open stars should have the same Poisson errors, which are omitted
for a clear display.
The solid curve is the best-fit Press-Schechter function derived in Sect. 4.1. |
| Open with DEXTER |
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f5.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg108.gif) |
Figure 5:
The X-ray temperature function derived with
the flux-limited complete sample of 61 clusters
for the case of a flat universe
(
).
The three lines indicate previous measurements in the form of the best-fit
power-law functions. Solid, dotted, and dashed lines
correspond to the results by
Markevitch (1998),
Edge et al. (1990),
and
Henry (2000), respectively. |
| Open with DEXTER |
In Fig. 5, we compare our result to previously obtained XTFs overlayed
in the form of best-fit power law functions.
In the 3-8 keV range, our XTF shows good agreement with
the previous results.
We have derived a cumulative XTF, which is the number density
of clusters that are hotter than a certain temperature, by
 |
(7) |
The resulting cumulative XTF is illustrated in Fig. 6
together with other recent results by Henry (2000) and
Markevitch (1998).
Our result shows agreement with Henry's XTF constructed
from his sample of 25 clusters.
The individual temperatures of the 25 clusters were evaluated
by averaging the previous temperature measurements with ASCA
and other missions based on single-temperature fits (see Henry 2000),
which agree with the hot component temperatures obtained in this work
on average within 1%.
Above 6 keV, there is apparent disagreement between our result and
the result by Markevitch (1998) showing a higher amplitude.
This is due to a larger number of clusters in the highest temperature bins
in the Markevitch sample,
despite the fact that smaller
values apply to
his sample.
(Markevitch selected his sample with the same flux limit as ours
but also limited the redshift range to 0.04 
z
0.09.)
The larger number of very hot clusters in Markevitch (1998) is
due to higher temperature estimates compared to our values
for temperatures above 6 keV (Fig. 1b).
Blanchard et al. (2000),
using a sample and temperatures mainly from the XBACS
(Ebeling et al. 1996)
and the ASCA measurement by Markevitch et al. (1998),
respectively,
obtained the XTF, which virtually perfectly agrees with that
by Markevitch (1998) above 4 keV, and has larger abundance below 4 keV.
Pierpaoli et al. (2001), who have made another attempt to derive
the XTF, used the same sample as Markevitch (1998)
but the temperatures given by White (2000).
The Pierpaoli cumulative XTF has the highest amplitude among other works.
This is because of the systematically higher temperatures by
White (2000) (see Fig. 1).
![\begin{figure}
\par\includegraphics[width=8cm,clip]{h2920f6NEW.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg111.gif) |
Figure 6:
The cumulative temperature function for the case of a flat universe
(
)
is shown
in grey solid line with its 68% error band.
The errors are estimated by
 .
Previous results from Markevitch (1998) and Henry (2000)
are also shown by the solid and dotted black lines,
respectively. |
| Open with DEXTER |
In the following, using the flux-limited complete sample of 61 clusters
(
)
and theoretical XTF models,
we obtain constraints on cosmological parameters.
The XTF at the present epoch is most sensitive to the amplitude of
density fluctuations on a scale of 8 h-1 Mpc, ,
and least sensitive to the cosmological constant, 
.
We therefore concentrate on two different
model
families including open (
,
)
and flat (
)
universes.
Recent measurements of high-redshift SN Ia indicate a positive cosmological
constant (Perlmutter et al. 1999; Riess et al. 1998).
Moreover, the cosmic microwave background (CMB) temperature-fluctuation
map recently derived with the Boomerang experiment strongly supports
a flat universe model (de Bernardis et al. 2000).
Therefore, a flat universe model with a positive cosmological constant
is currently the most plausible description of our universe.
For the purpose of comparison with previous works,
we also consider the case of an open universe.
For the theoretical prediction of the XTF,
we employ two different models,
a conventional Press-Schechter mass function (Sect. 4.1)
and the Formation-Epoch model by Kitayama & Suto (1996)
that is a modification of the former theory (Sect. 4.2).
The model XTF depends also on the Hubble constant,
which is here assumed to be 71 km/s/Mpc (i.e. h=0.71)
based on the latest results of the HST Hubble constant project
(see Mould et al. 2000 and references therein).
According to the Press-Schechter formalism,
the number density of clusters with a mass in the range of M to
that have collapsed before a redshift z is given as
 |
|
|
(8) |
where 
is the mean density of the universe at present,
is the linear overdensity of a system collapsed at a redshift
z and evaluated at present
(for which we use the formula given in Kitayama & Suto 1996),
and 
is the variance of mass extracted from the density fluctuation
field within a spherical region that includes a mass M on average.
For the mass variance, we employed an analytical formula derived
by Kitayama & Suto (1996) based on the CDM power spectrum,
which is given as
 |
(9) |
where
p = 0.0873,
.
is the shape parameter and is given as
![\begin{displaymath}%
\Gamma = \Omega_{\rm m,0} h (T_0/2.7 \;{\rm K})^{-2}
\exp [ - \Omega_{\rm B} (1 + \sqrt{2h} \Omega_{\rm m,0}^{-1} ) ],
\end{displaymath}](/articles/aa/full/2002/09/aah2920/img124.gif) |
(10) |
where T0 and
are
the temperature of the cosmic microwave background
and the baryon density,
which are assumed to be 2.726 K (Mather et al. 1994)
and
(Burles & Tytler 1998),
respectively.
The mass function given above is converted to the XTF by means of
a mass-temperature relation.
We assume that the ICM in a cluster is described by an isothermal
model (e.g. Forman & Jones 1982),
in which the density profile is given as
![$ n_{\rm gas} = n_{\rm gas,0} [ 1 + (R/R_ {\rm c})^{2} ]^{-3\beta/2} $](/articles/aa/full/2002/09/aah2920/img128.gif)
,
and that hydrostatic equilibrium is achieved.
The virial mass is defined as
 |
(11) |
where T is the temperature of ICM,
is the virial radius,
is the core radius,
is the mean molecular weight of the ICM (set to 0.6),
is the proton mass, G is the gravitational constant,
and 
is the ratio between a cluster mean density
and the mean density of the universe at the cluster-formation redshift z.
For
we use the formula given in Kitayama & Suto (1996).
Since
,
the virial mass is approximately given as,
 |
(12) |
Note that when
,
(
),
this equation becomes identical to that derived from a modified singular
isothermal sphere by Bryan & Norman (1998).
Although an additional modification to the M-T relation is often applied
by multiplying a factor accounting for departures from virial equilibrium
(e.g. Henry 2000), we did not introduce this factor here.
The observed values of
are distributed in a relatively
narrow range around 2/3 and
may show a weakly positive correlation with temperatures
(Arnaud & Evrard 1999; Vikhlinin et al. 1999;
Horner et al. 1999).
Vikhlinin et al. measured
values systematically
with ROSAT PSPC data excluding a central region inside 30% of
the virial radius and found a correlation of
with X-ray temperature,
which can be fitted with the linear function
(Henry 2000).
We employ this relation and substitute it into Eq. (12).
The effect of using different M-T relations
will be studied in Sect. 5.
We further assume that a cluster collapsed just before it is observed.
This recent formation approximation, which has been extensively
applied in previous conventional modelings, is relaxed
in the Formation-Epoch model described in the next section.
Combining all the formulae given above,
the model XTF at a given redshift is given by
 |
(13) |
Hereafter we refer to this model XTF as the PS model.
In the model XTF given in the previous section,
the formation redshift, zf, of an observed cluster
is assumed to be the same as the observed redshift
(i.e. recent formation approximation).
However, this assumption is not reasonable in particular
in a low density universe.
Kitayama & Suto (1996) constructed a model XTF
taking into account the distribution of the dark-matter-halo formation epoch
as well as the evolution of temperature (hereafter FE model).
They showed that the XTF predicted by the FE model can be significantly
different from the PS model described in Sect. 4.1,
in particular when the temperature evolves after the collapse.
The FE model is given as,
![$\displaystyle %
\phi_{\rm FE}(T_{\rm obs},z) {\rm d}T_{\rm obs}
= \int^{\infty}...
...},z)}
\frac{{\rm d}M_{\rm v}}{{\rm d}T}(T,z_{\rm f})\right] {\rm d}T_{\rm obs},$](/articles/aa/full/2002/09/aah2920/img141.gif) |
|
|
(14) |
where
and 
are given in Eqs. (8)
and (12), respectively, and
given by Lacey & Cole (1993)
is a differential distribution function of halo formation epochs
of a cluster that has a mass of
at the observed redshift z.
At the collapse redshift 
,
the cluster is assumed to be formed
with a mass of ,
increasing in mass after its formation.
gives
the evolution in temperature via
,
where
is the virial temperature
at the collapse redshift ,
while
is the observed temperature
at the redshift z.
s=0 corresponds to no temperature evolution,
while s>0 and s<0 indicate positive and negative temperature evolution
after the collapse, respectively.
Instead of fitting the model XTF to the XTF derived in Sect. 3,
we use the predicted XTF and the L-T relation
to calculate the expected luminosity and temperature distribution
for the given survey selection function which is given
in a form of
,
and perform a fit to the observed luminosity-temperature distribution
of the 61 clusters selected in Sect. 2.4.
In this analysis,
the intrinsic luminosity distribution for a given temperature,
which in other words is the conditional luminosity function
at a given temperature,
is assumed to follow a Gaussian function in logarithmic scale
with a constant standard deviation
and a mean luminosity given by a power-law function of the temperature.
This assumption can be justified by observational results as shown
in Sect. 3.1.
In order to simplify a model calculation,
we assume that the XTF does not change within the search volume
of our flux limit
and use a model XTF at the median redshift of our cluster sample
at z=0.046 as a representative value.
Systematic errors on the final results caused from this assumption
are found to be negligibly small compared with statistical errors.
Incorporated by another Gaussian that represents temperature
measurement errors,
the expected cluster number density
in a unit logarithmic temperature and a unit logarithmic luminosity
is given as
 |
|
|
|
![$\displaystyle \frac{1}{\sqrt{2\pi \sigma_{\log L}^2}}
\exp \left\{ -\frac{(\log...
...2}\right\}
\frac{{\rm d}T}{{\rm d}\log T} \right] {\rm d}\log L\ {\rm d}\log T,$](/articles/aa/full/2002/09/aah2920/img153.gif) |
|
|
(15) |
where 
is calculated for h=0.71 and normalized with the h-3factor to a case of the h=1 universe.
is a maximum comoving volume calculated for h=1,
in which a cluster having a temperature of Tand a luminosity of L could have been detected
under our sample selection criteria,
for which the calculation method is given by Eq. (A.9).
The temperature measurement errors are
=
from our ASCA results.
Free parameters to a fit are;
and
specifying ;
A and 
specifying the power-law function of the L-T relation
that gives the mean of the luminosity;
and
that is the constant standard deviation
of the luminosity from the L-T relation in logarithmic scale.
Although
varies with the assumed model universe,
i.e. with the values of
as well as which are the fitting parameters,
we used a specific
value calculated with
(
) = (0.2, 0), or (0.2, 0.8),
in the case of an open or flat universe, respectively.
Since all the clusters in our sample are nearby,
is virtually constant for different
values,
using the fixed
does not introduce any considerable
systematic error on the final constraints on
and
.
The model cluster distribution was then fitted to the observed
temperature and luminosity distribution.
Assuming a Poisson distribution of cluster number counts
at a given luminosity and temperature,
we defined a logarithmic likelihood function as
 |
(16) |
where i denotes each sample cluster and the integration
is performed over the 
(ergs s-1) range from 41.5 to 45.5
and the 
(keV) range from 0.14 to 1.05
(corresponding to T = 1.4-11.2 keV).
By maximizing the likelihood function,
we derived the best-fit model XTF as well as the L-T relation.
There are two reasons for setting the low temperature cut-off at 1.4 keV.
First, as discussed in Sect. 2.3,
the L-T relation breaks down for elliptical galaxies which have
a lower gas mass fraction, and actually gets steeper below 1 keV
as shown by e.g. Xue & Wu (2000).
Thus the simple power-law modeling for the L-T relation is not appropriate
for the whole temperature range.
The second reason is based on the more essential problem concerned
with the comparison between the theoretical XTFs and the observation.
The systems at the low temperature end are mostly single elliptical galaxies
and some of them may not be isolated but surrounded by larger scale dark
matter halos.
As has been shown by e.g. Matsushita (1998),
X-ray brightness profiles of elliptical galaxies have large variety
suggesting a significant variation in total mass for a given temperature,
i.e. a considerable scatter in the M-T relation would be expected.
Therefore, a cluster abundance at the low temperature end probably
does not directly correspond to an abundance of dark matter halos
at a certain mass scale.
We first present the results from the PS model fitting.
The best-fit values and errors of the fitting parameters
are summarized in Table 3.
The best-fit distribution function in the
space
in the flat universe case is illustrated in Fig. 7.
The corresponding model XTF is also illustrated in Fig. 4.
The constraint in
space is shown in
Fig. 8 for each open and flat universe case.
The contours represent the 90% confidence range for two parameters
of interest, and the constraint is notably tight.
In order to demonstrate an improvement on the constraint
by the widest temperature range and the largest sample ever achieved,
we also performed the fitting with the PS model
only using 51 sample clusters above 3 keV.
As shown in Fig. 8, the improvement is significant,
in particular, for
.
Since
is tightly related to the overall shape of
the mass variance, i.e. the power spectrum,
a larger mass range to measure the mass fluctuation amplitude
is essential.
The "true'' L-T relation employed in Sects. 2.3 and 3.1
was estimated by this fitting.
Fixing the
value as obtained in Sect. 2.3,
to 2.47 and 2.50 for the open and flat universe case, respectively,
we performed the PS model fitting and derived
,
for the open universe case,
and
,
for the flat universe case,
which are given in Eqs. (3) and (4)
and are applied in Eq. (6).
![\begin{figure}
\par\includegraphics[width=7cm,clip]{h2920f7.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg167.gif) |
Figure 7:
The best-fit model function in the luminosity-temperature space
for the case of the flat universe is shown with contours.
The 63 clusters from the flux-limited complete sample
are overlayed with open circles.
The vertical dotted lines indicate the temperature range used for the fitting.
Therefore, 61 clusters are used to make constraints. |
| Open with DEXTER |
![\begin{figure}
\par\includegraphics[angle=-90,width=6cm,clip]{h2920f8a.ps}\par\includegraphics[angle=-90,width=6cm,clip]{h2920f8b.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg168.gif) |
Figure 8:
Constraints on
and
with the PS model
in case of the open (upper panel) and flat (lower panel) universe.
Bold contours are results including all sample clusters hotter than 1.4 keV,
while dotted contours are derived when only clusters hotter than 3.0 keV
are used. |
| Open with DEXTER |
![\begin{figure}
\par\includegraphics[angle=-90,width=6cm,clip]{h2920f9a.ps}\par\includegraphics[angle=-90,width=6cm,clip]{h2920f9b.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg169.gif) |
Figure 9:
Constraints on
and
in case of
the open (upper panel) and flat (lower panel) universe.
Thin closed contours show results with the PS model.
Bold dotted and solid closed contours are derived with
the s=1 and s=0 FE models,
with and without a temperature evolution effect, respectively.
The line shows the constraints from the COBE normalization
by Bunn & While (1997) for h=0.71. |
| Open with DEXTER |
The FE model gives somewhat different results.
In addition to
and ,
the FE model introduces another parameter, s, that models
a temperature evolution, which can be a free parameter in a fit.
However, following Kitayama & Suto (1996), in the following analysis,
we rather fixed s at s=0 and s=1 such that they represent no evolution
and a maximum possible positive evolution of the temperature, respectively.
Since there is no observational evidence for the evolution of the temperature,
e.g. no redshift dependency in the L-T relation is found
(Mushotzky & Scharf 1997;
Donahue et al. 1999; Della Ceca et al. 2000),
the no evolution model may be more plausible.
However,
the effect of the temperature evolution is still interesting to study here.
The fitting results are summarized in Table 3,
and the constraints on
and
are illustrated
in Fig. 9.
The FE model in the no temperature evolution case (s=0)
shows similar results with those of the PS model,
since the XTF predicted by the s=0 FE model
is very close to that of the PS model
(see Fig. 8 in Kitayama & Suto 1996).
On the other hand,
in the maximum temperature evolution cases (s=1),
the FE model gives significantly smaller
values
than those with the PS model.
The M-T relation used in our analysis may be subject to uncertainties
introducing a significant systematic error in the final results
on the cosmological parameters.
Theoretical arguments yield a scaling law, that is
(e.g. Kaiser 1986).
A number of numerical simulations support this scaling law,
but different normalizations are derived from different simulations.
Henry (2000) summarized such differences in the normalization
of the M-T relation derived from different hydrodynamical simulations.
A cluster mass may vary from 62% to 119% of the mean value.
Table 5:
Results with different M-T relations.
The parameter ranges correspond to 90% statistical error
for one parameter of interest
plus systematic errors with different M-T relations applied.
| Model XTF |
|
|
| PS(Open) |
0.09-0.37 |
0.55-1.13 |
| FE(Open, s= 0) |
0.07-0.32 |
0.61-1.15 |
| FE(Open, s= 1) |
0.03-0.23 |
0.52-0.80 |
| PS(Flat) |
0.10-0.38 |
0.57-1.19 |
| FE(Flat, s= 0) |
0.10-0.35 |
0.65-1.33 |
| FE(Flat, s= 1) |
0.06-0.18 |
0.66-0.97 |
The M-T relations derived observationally from X-ray mass measurements
give somewhat different results.
Horner et al. (1999) derived an M-T relation which has
a similar exponent as the scaling law but a lower normalization than
the numerical simulations.
On the other hand, Nevalainen et al. (2000)
and Finoguenov et al. (2001) obtained steeper relations
of T1.79 and T1.78, respectively,
and lower normalizations.
In order to investigate the effects of the different M-T relations,
we repeated the fitting performed in Sect. 4 with the PS model
for the flat universe case.
The six M-T relations we have examined are summarized in Table 4.
Each M-T relation can be expressed with Eq. (12),
replacing the
value with the formula given in Table 4.
In Fig. 10, the best-fit parameters derived with the different
M-T relations are compared with the result presented in Sect. 4.4.
The effect of using different M-T relations is significant
and the systematic errors introduced are comparable to, or even larger than,
the current statistical errors.
As seen in Fig. 10, a change in the index of the M-T relation
moves the best-fit point along the elongation direction
of the confidence contour, while an amplitude change moves the best-fit
point along the perpendicular direction.
Taking into account these systematics from the different M-T relations,
the constraints on
and
are revised
and summarized in Table 5.
With the FE model, where the recent formation approximation is not valid,
it is not appropriate to use the empirical M-T relation obtained based
on the observations of nearby clusters.
This is because
the current M-T relation should be a result of accumulation
of different M-T relations from different formation redshifts which
depend on the cluster masses.
Therefore, the M-T relations obtained from the Hydrodynamic simulations
may be more appropriately applied with the FE model.
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f10.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/Timg174.gif) |
Figure 10:
The best-fit values of
and derived with the PS model in the case of the flat universe,
for which different M-T relations are applied.
A 90% confidence contour derived in Sect. 4 is also overlayed. |
| Open with DEXTER |
We compared our constraints on the cosmological parameters
with previous works.
Bahcall (2000) summarized the cluster constraints
on
and
as 0.2 
0.1 and
1.2 0.2 (68% confidence level), respectively.
These values are consistent with our results.
The power spectrum of matter distribution in the present universe
has been measured from the spatial distribution of galaxies
and galaxy clusters,
and found to be generally well reproduced by the CDM power spectrum
(Peacock & West 1992; Einasto et al. 1993;
Jing & Valadarnini 1993; Einasto et al. 1997;
Retzlaff et al. 1998;
Tadros et al. 1998; Miller & Batuski 2001).
An important feature of the power spectra is a turnover,
where the scale is closely linked to the horizon scale at matter-radiation
equality.
Most recently Schuecker et al. (2001),
using a sample of 452 X-ray selected clusters of galaxies,
determined the location of the maximum of the power spectrum
in the range of
h Mpc-1.
According to the CDM power spectrum,
is well approximated by
 |
(17) |
where ,
given in Eq. (10),
is a function of
.
Therefore one finds
.
The constraints on
we have derived in Sect. 5.1
with the PS model yield
h Mpc-1,
which overlaps with the result by Schuecker et al.
Our results with the FE model give
h Mpc-1in the case of the open universe,
and
h Mpc-1in the case of the flat universe,
both of which are also consistent with the result by Schuecker et al.
A measurement of the matter density fluctuation at the largest
spatial scale was performed using the COBE DMR experiment
(e.g. Bennett et al. 1996).
Bunn & White (1997), using the CDM power spectrum model,
extrapolated the COBE results to cluster scales
and gave constraints in
-
space,
which are shown in Fig. 9.
In the case of the open universe,
our results with any of the model XTFs
do not overlap the COBE constraints,
even if the effects of the uncertainty in the M-T relation
are taken into account.
On the other hand, in the case of the flat universe,
our results with the effects of the uncertainty in the M-T relation,
are consistent with the COBE constraints
regardless of the choice of the model XTF.
The most important result concerning the XTF derived here
is that it covers a large enough range of temperatures
to reveal the shape of the function beyond a simple power law
representation.
As expected from theoretical considerations,
there should be a very sharp cut-off at the high mass end
and the function should turn into a shallower slope
at the low mass end as shown by our observations.
We have demonstrated that these constraints on the actual slope
of the XTF allow us to derive independent constraints on the two parameters
which are most important in cosmic structure formation models:
and .
Leaving out only 16% of the sample clusters from the low temperature end
almost doubles the uncertainty for these independent constraints.
Therefore, while these independent constraints are an important achievement
of the present work on one hand,
this demonstration also shows that
these constraints are sensitive to any systematic error
introduced into the XTF.
For this reason we have studied the stability of our results to
variations introduced to the observed data and also variations
of the theoretical modeling.
We have fully incorporated the measurement errors
in fluxes and temperatures of the sample clusters.
With the typical errors of 5% for the flux and 6-10% for the temperature,
the biases were found to be relatively small and not to cause a drastic change
on the final results in our analysis.
On the other hand,
we have shown that there are considerable variations
and uncertainties in the theoretical modeling of the XTF,
which introduce significant systematic errors in the final constraints
on
and .
Various M-T relations that are employed to model the XTF
have resulted in considerably different values of
and .
Another source of variation in the theoretical modeling
that we have studied is
a recent formation approximation which is employed with
the Press-Schechter formalism to build the PS model here.
Using the FE model,
in which the distribution of cluster formation redshifts
and the evolution of the mass as well as temperature of a cluster after
the collapse are analytically formulated,
we have shown that a considerable difference could be introduced
in the derived cosmological parameters, in particular in .
This further motivates us to establish a more sophisticated model
to the XTF incorporating new X-ray observations
of the high redshift clusters with Chandra and XMM-Newton.
Finally, taking into account all the uncertainties,
we put our general constraints on
and as
0.03-0.37, 
0.52-1.15 in an open universe
and
0.06-0.38,
0.57-1.33 in a flat universe,
respectively.
From the comparison with the COBE constraints,
a flat universe is more preferable than an open universe.
Acknowledgements
We thank Peter Schuecker, Makoto Hattori, Patrick Henry,
and Paul Lynam for valuable discussions and comments.
We have made use of the ASCA archival database
at Goddard Space Flight Center/NASA, U.S.A.
and the Leicester Database and Archive Service
at the Department of Physics and Astronomy,
Leicester University, UK.
We acknowledge FTOOLS.
Here we present our method, used in Sect. 3,
to evaluate the X-ray temperature function in more detail.
A flux limited sample has been extensively used for constructing
the X-ray luminosity function (XLF) by many authors
(e.g. Piccinotti et al. 1982;
Edge et al. 1990; Gioia et al. 1990;
Henry et al. 1992; Ebeling et al. 1997;
Rosati et al. 1998; de Grandi et al. 1999;
Ledlow et al. 1999).
Here we present the first description of how to construct
this function in the presence of measurement
uncertainties, correlation scatter and selection biases.
For a sample with a finite number of objects,
the differential XLF, i.e. the number density of clusters
having the rest frame luminosity in a range of
to
defined in a comoving space, can be evaluated as
 |
(A.1) |
where i denotes individual clusters in the luminosity range of a bin.
is a maximum search volume measured
in comoving space where a cluster having the rest frame luminosity
of L and the temperature of T could have been detected
under the flux limited condition.
Introducing
,
the maximum redshift at which such a cluster
could have been detected,
and
,
a total volume in the comoving coordinate
from us to a given redshift,
we can derive
as
 |
(A.2) |
where 
is the total sky coverage in steradians,
and
is, in the case of a flat universe, simply given by
 |
(A.3) |
where 
is a luminosity distance,
while, in the case of an open universe,
it is given with a more complicated form (see e.g. Sandage 1988).
is given by solving the following equation
 |
(A.4) |
where
is the flux limit defined in a certain
energy range in the observed frame,
and
is the same observed-frame luminosity
that is expected if the cluster is located at
.
In order to solve this equation with the K-correction,
the spectral energy distribution, i.e. the temperature, T, is required,
although the dependency of
on T is generally weak.
Using
,
the differential XTF can be given as
 |
(A.5) |
This estimator is used in e.g. Henry (2000).
However, since there is a distribution of luminosities
for a given temperature, this method introduces a scatter in a derived XTF,
unless there are large numbers of clusters in each temperature bin.
If the distribution of the luminosity for a given temperature
is known, we can obtain an average search volume by summing
in a whole range of luminosities
weighted with the probability.
Assuming that luminosities for a given temperature are distributed
like a Gaussian with a mean given by the power-law function,
,
and with a constant standard deviation, in logarithmic scale,
we can obtain the average search volume at a given temperature as
![$\displaystyle %
V_{\rm max}(T) =
\int^{\infty}_{-\infty}
\frac{v_{\rm max}(L,T)...
...frac{(\log AT^{\alpha} - \log L)^2}{2\sigma_{\log L}^2}}\right]
{\rm d}\log L .$](/articles/aa/full/2002/09/aah2920/img197.gif) |
|
|
(A.6) |
We can then evaluate the temperature function as
 |
(A.7) |
Performing Monte-Carlo simulations,
we confirmed that both estimators, the
method
and
method are consistent and unbiased,
that is they reproduce the temperature function fed to the simulations
in the limit of an infinite number of simulated clusters,
and the mean value of the temperature functions obtained
from a number of independent measurements with a finite number of clusters
is also equal to the true value in the limit of an infinite number of
measurements.
However, the
method gives smaller variance
in a measurement of the temperature function
and therefore it is more efficient than the
method.
This is well demonstrated in Fig. 4.
When we apply this analysis method to our cluster sample,
we further consider the following two points:
our first concern is that the flux measurement may have a redshift bias
such that for clusters with the same true luminosity,
different luminosities are derived for different redshifts.
In order to study this effect we produced a sample of cluster images
at various redshifts
as observed by the ROSAT PSPC
and determined their fluxes
with the growth curve analysis method (Böhringer et al. 2000)
as performed for the actual observed data of the present sample
by Reiprich & Böhringer (2002).
The simulations show that the resulting luminosities are kept constant
regardless of the redshift - at least in a range that we are considering
here, i.e. z<0.2,
despite the overall signal to noise ratio worsening
with a dimming of the surface brightness that follows
(1+z)-4.
Secondly, the effect from the measurement errors on X-ray fluxes
is considered.
If there are considerable measurement errors in the flux measurements,
a cluster that should be in the flux-limited sample
could be found to be fainter than the flux limit
and to be missed from the sample, and vice-versa.
If the measured flux distribution is assumed to be following
a Gaussian function with the sigma value of
,
the probability that a cluster of the luminosity Lat the redshift of z is found to be brighter
than the flux limit
is given as
![\begin{displaymath}%
W(L,z,f_{\rm lim})
= \int^{\infty}_{f_{\rm lim}} \frac{1}{\...
...p\left[{-\frac{(f-f_0)^2}{2\sigma_{\rm f}^2}}\right] {\rm d}f,
\end{displaymath}](/articles/aa/full/2002/09/aah2920/img203.gif) |
(A.8) |
where
.
Therefore, Eq. (A.2) should be modified as
 |
(A.9) |
Setting
to 10-12 ergs s-1 cm-2,
a typical flux measurement error around the flux limit,
this modification brings a 2% and 0.5% increase to
at 1 and 10 keV, respectively,
compared with the simple estimate with Eq. (A.2).
Although the correction is still insignificant with respect to
the Poisson error
that each temperature bin would have, we use the Eq. (A.9)
in the derivation of the XTF in Sect. 3.
Our sample selection criteria specify several parameters
in Eqs. (A.5), (A.6), (A.8) and (A.9)
as 
8.139 str,
=
ergs s-1 cm-2in 0.1-2.4 keV band in the observed frame,
and then
should be the 0.1-2.4 keV band luminosity
in the observed frame, which corresponds to a cluster rest frame luminosity
in the energy range from (
to (
) 2.4 keV.
Also, from the L-T relation studied in Sects. 2.3 and 4,
we estimated the following parameters as
,
,
and
for an open universe case, and
,
,
and
for a flat universe case.
The XTF thus derived is shown in Fig. 4.
Finally, we should note that Ti in Eq. (A.5)
is the measured cluster temperature rather than the "true'' temperature.
The derived XTF should have been smeared due to
the temperature measurement error,
which brings another systematic error into the resulting XTF.
In order to investigate the possible systematic errors due to
the temperature measurement errors and to confirm our analysis method,
we performed a Monte-Carlo simulation.
With an adopted XTF of
,
a number of cluster observations are simulated taking into account
the scatter in the L-T relation as well as the flux and temperature
measurement errors, and the XTF was constructed as performed
with the real data using Eqs. (A.5), (A.6),
(A.8) and (A.9).
The input XTF and the rederived XTF from the simulations show complete
agreement except a minor systematic increase - at most 2% at 10 keV,
which is due to the temperature measurement error for very hot clusters.
These systematics (Eddington bias; Eddington 1944)
remain in the XTF derived in Sect. 3 (Fig. 4),
while, for the model fitting in Sect. 4,
the temperature measurement error is taken into account (Eq. (15)).
- Akritas, M. G., & Bershady, M. A. 1996, ApJ, 470, 706
In the text
NASA ADS
- Allen, S. W., Fabian,
A. C., Johnstone, R. M., Arnaud, K. A., & Nulsen, P. E. J. 2001, MNRAS, 322, 589
In the text
NASA ADS
- Allen, S. W. 2000, MNRAS, 315, 269
NASA ADS
- Arnaud, K. A. 1996, Astronomical Data Analysis
Software and Systems V, ed. G. Jacoby, & J. Barnes, ASP Conf. Ser., 101, 17
In the text
- Arnaud, M., & Evrard, A. E. 1999, MNRAS, 305, 631
In the text
NASA ADS
- Böhringer, H. 1999, private communication
In the text
- Böhringer, H., Voges, W., Huchra, J. P., et al. 2000, ApJS, 129, 435
In the text
NASA ADS
- Böhringer, H., Schuecker, P., & Guzzo, L. 2001, A&A, 369, 826
In the text
NASA ADS
- Bahcall, N. A., & Cen, R. 1993, ApJ, 407, L49
In the text
NASA ADS
- Bahcall, N. A. 2000, Phys. Rep., 333-334, 233
In the text
NASA ADS
- Bennett, C. L., Banday, A. J., Gorski, K. M., et al. 1996, ApJ, 464, L1
In the text
NASA ADS
- Beuing, J., Döbereiner, S., Böhringer, H., & Bender, R. 1999, MNRAS, 302, 209
In the text
NASA ADS
- Biviano, A., Girardi, M., Giuricin, G., Mardirossian, F., & Mezzetti, M. 1993, ApJ, 411, L13
In the text
NASA ADS
- Blanchard, A., Sadat., R., Bartlett., J. G., & Le Dour, M. 2000, A&A, 362, 809
In the text
NASA ADS
- Briel, U. G. & Henry, J. P. 1994, Nature, 372, 439
In the text
NASA ADS
- Bryan, G. L., & Norman, M. L. 1998, ApJ, 495, 80
In the text
NASA ADS
- Bunn, E. F., & White, M. 1997, ApJ, 480, 6
In the text
NASA ADS
- Burles, S., & Tytler, D. 1998, ApJ, 499, 699
In the text
NASA ADS
- de Bernardis, P., Ade, P. A. R., Bock, J. J., et al. 2000, Nature, 404, 955
In the text
NASA ADS
- de Grandi, S., Guzzo, L., Böhringer, H., et al. 1999, ApJ, 513, L17
In the text
NASA ADS
- de Grandi, S., Böhringer, H., Guzzo, L., et al. 1999, ApJ, 514, 148
In the text
NASA ADS
- David, L. P., Slyz, A., Jones, C., et al. 1993, ApJ, 412, 479
In the text
NASA ADS
- Della Ceca, R., Scaramella, R., Gioia, I. M., et al. 2000, A&A, 353, 498
In the text
NASA ADS
- Donahue, M., Voit, G. M., Scharf, C. A., et al. 1999, ApJ, 527, 525
In the text
NASA ADS
- Ebeling, H., Voges, W., Böhringer, H., et al. 1996, MNRAS, 281, 799
In the text
NASA ADS
- Ebeling, H., Edge, A. C., Fabian, A. C., et al. 1997, ApJ, 479, L101
In the text
NASA ADS
- Ebeling, H., Edge, A. C., Böhringer, H., et al. 1998, MNRAS, 301, 881
In the text
NASA ADS
- Eddington, A. S. 1940, MNRAS, 100, 354
In the text
NASA ADS
- Edge, A. C., Stewart, G. C., Fabian, A. C., & Arnaud, K. A. 1990, MNRAS, 245, 559
In the text
NASA ADS
- Edge, A. C., & Stewart, G. C. 1991, MNRAS, 252, 414
In the text
NASA ADS
- Einasto, J., Gramann, M., Saar, E., & Targo, E. 1993, MNRAS, 260, 705
In the text
NASA ADS
- Einasto, J, Einasto, M., Gottlöber, S., et al. 1997, Nature, 385, 139
In the text
NASA ADS
- Eke, V. R., Cole, S., Freml, C. S., & Henry, J. P. 1998, MNRAS, 298, 1145
In the text
NASA ADS
- Eke, V. R., Cole, S., & Frenk, C. S. 1996, MNRAS, 282, 263
In the text
NASA ADS
- Evrard, A. E. 1997, MNRAS, 292, 289
In the text
NASA ADS
- Fabian, A. C. 1994, ARA&A, 32, 277
In the text
NASA ADS
- Fabian, A. C., Arnaud, K. A., Bautz, M. W., & Tawara, Y. 1994, ApJ, 436, L63
In the text
NASA ADS
- Finoguenov, A., Reiprich, T. H., & Böhringer, H. 2001, A&A, 368, 749
In the text
NASA ADS
- Forman, W., & Jones, C. 1982, ARA&A
In the text
- Fukazawa, Y., Ohashi, T., Fabian, A. C., et al. 1994, PASJ, 46, L55
In the text
NASA ADS
- Fukazawa, Y. 1997, Ph.D. Thesis, the University of Tokyo
In the text
- Fukazawa, Y., Makishima, K., Tamura, T., et al. 1998, PASJ, 50, 187
In the text
NASA ADS
- Fukazawa, Y., Makishima, K., Tamura, T., et al. 2000, MNRAS, 313, 21
In the text
NASA ADS
- Gioia, I. M., Henry, J. P., Maccacaro, T., et al. 1990, ApJ, 356, L35
In the text
NASA ADS
- Girardi, M., Borgani, S., Giuricin, G., Mardirossian, F., & Mezzetti, M. 1998, ApJ, 506, 45
In the text
NASA ADS
- Henry, J. P., Gioia, I. M., Maccacaro, T., et al. 1992, ApJ, 386, 408
In the text
NASA ADS
- Henry, J. P. 2000, ApJ, 534, 565
In the text
NASA ADS
- Henry, J. P., & Arnaud, K. A. 1991, ApJ, 372, 410
In the text
NASA ADS
- Horner, D. J., Mushotzky, R. F., & Scharf, C. A. 1999, ApJ, 520, 78
In the text
NASA ADS
- Ikebe, Y., Makishima, K., Ezawa, H., et al. 1997, ApJ, 481, 660
In the text
NASA ADS
- Ikebe, Y., Makishima, K., Fukazawa, Y., et al. 1999, ApJ, 525, 58
In the text
NASA ADS
- Ikebe, Y. 2001, Proc. of Tracing Cosmic Evolution with Clusters of Galaxies, Sesto Pusteria, Bolzano, Italy, July 3-6, 2001 [astro-ph/0112132]
In the text
- Isobe, T., Feigelson, E. D., Akritas, M. G., & Babu, G. J. 1990, ApJ, 364, 104
In the text
NASA ADS
- Jing, Y. P., & Valadarnini, R. 1993, ApJ, 406, 6
In the text
NASA ADS
- Kaastra, J. S. 1992, An X-Ray Spectral Code for Optically Thin Plasmas, Internal SRON-Leiden Report, updated version 2.0
In the text
- Kaastra, J. S., Ferrigno, C., Tamura, T., et al. 2001, A&A, 365, L99
In the text
NASA ADS
- Kaiser, N. 1986, MNRAS, 222, 323
In the text
NASA ADS
- Kitayama, T., & Suto, Y. 1996, ApJ, 469, 480
In the text
NASA ADS
- Lacey, C. G., & Cole, S. 1993, MNRAS, 262, 627
In the text
NASA ADS
- Lahav, O., Edge, A. C., Fabian, A. C., & Putney, A. 1989, MNRAS, 238, 881
In the text
NASA ADS
- Ledlow, M. J., Loken, C., Burns, J. O., Owen, F. N., & Voges, W. 1999, ApJ, 516, L53
In the text
NASA ADS
- Liedahl, D. A., Osterheld, A. L., & Goldstein, W. H. 1995, ApJ, 438, L115
In the text
NASA ADS
- Makishima, K., Ezawa, H., Fukazawa, Y., et al. 2000, PASJ, in press
In the text
- Markevitch, M. 1998, ApJ, 504, 27
In the text
NASA ADS
- Markevitch, M., Forman, W. R., Sarazin, C. L., & Vikhlinin, A. 1998, ApJ, 503, 77
In the text
NASA ADS
- Mather, J. C., Cheng, E. S., Cottingham, D. A., et al. 1994, ApJ, 420, 439
In the text
NASA ADS
- Matsumoto, H., Koyama, K., Awaki, H., et al. 1996, PASJ, 48, 201
In the text
NASA ADS
- Matsushita, K. 1998, Ph.D. Thesis, the University of Tokyo
In the text
- Mewe, R., Gronenschild, E. H. B. M., & van den Oord, G. H. J. 1985, A&AS, 62, 197
In the text
NASA ADS
- Mewe, R., Lemen, J. R., & van den Oord, G. H. J. 1986, A&AS, 65, 511
In the text
NASA ADS
- Miller, C. J., & Batuski, D. J. 2001, ApJ, 551, 635
In the text
NASA ADS
- Mould, J. R., Huchra, J. P., Freedman, W. L., et al. 2000, ApJ, 529, 786
In the text
NASA ADS
- Mushotzky, R. F., & Scharf, C. A. 1997, ApJ, 482, L13
In the text
NASA ADS
- Nevalainen, J., Markevitch, M., & Forman, W. 2000, ApJ, 536, 73
In the text
NASA ADS
- Ohashi, T., Ebisawa, K., Fukazawa, Y., et al. 1996, PASJ, 48, 157
In the text
NASA ADS
- Oukbir, J., & Blanchard, A. 1997, A&A, 317, 1
In the text
NASA ADS
- Peacock, J. A., & West, M. 1992, MNRAS, 259, 494
In the text
NASA ADS
- Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565
In the text
NASA ADS
- Pen, U. 1998, ApJ, 498, 60
In the text
NASA ADS
- Peterson, J. R., Paerels, F. B. S., Kaastra, J. S., et al. 2001, A&A, 365, L104
In the text
NASA ADS
- Pfeffermann, E., Briel, U. G., Hippmann, H., et al. 1987, Proc. of SPIE, 733, 519
In the text
- Piccinotti, G., Mushotzky, R. F., Boldt, E. A., et al. 1982, ApJ, 253, 485
In the text
NASA ADS
- Pierpaoli, E., Scott, D., & White, M. 2001, MNRAS, 325, 77
In the text
NASA ADS
- Press, W. H., & Schechter, P. 1974, ApJ, 187, 425
In the text
NASA ADS
- Reiprich, T. H., & Böhringer, H. 2002, ApJ, in press
[astro-ph/0111285]
In the text
- Retzlaff, J., Borgani, S., Gottlöber, S., Klypin, A., & Müller, V. 1998, New Astron., 3, 631
In the text
NASA ADS
- Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, AJ, 116, 1009
In the text
NASA ADS
- Rosati, P., Della Ceca, R., Norman, C., & Giacconi, R. 1998, ApJ, 492, L21
In the text
NASA ADS
- Sandage, A. 1988, ARA&A, 26, 561
In the text
NASA ADS
- Schuecker, P., Böhringer, H., Guzzo, L., et al. 2001, A&A, 368, 86
In the text
NASA ADS
- Serlemitsos, P. J., Jalota, L., Soong, Y., et al. 1995, PASJ, 47, 105
In the text
NASA ADS
- Tadros, H., Efstathiou, G., & Dalton, G. 1998, MNRAS, 296, 995
In the text
NASA ADS
- Takahashi, T., Markevitch, M., Fukazawa, Y., et al. 1995, ASCA Newsl. (NASA/GSFC), 3, 25
In the text
- Tamura, T., Kaastra, J. S., Peterson, J. R., et al. 2001, A&A, 365, L87
In the text
NASA ADS
- Tanaka, T., Inoue, H., & Holt, S. S. 1994, PASJ, 46, L37
In the text
NASA ADS
- Trümper, J. 1993, Science, 260, 1769
In the text
NASA ADS
- Viana, P. T. P., & Liddle, A. R. 1996, MNRAS, 281, 323
In the text
NASA ADS
- Vikhlinin, A., Forman, W., & Jones, C. 1999, ApJ, 525, 47
In the text
NASA ADS
- Voges, W., Aschenbach, B., Boller, T., et al. 1999, A&A, 349, 389
In the text
NASA ADS
- White, S. D. M., Efstathiou, G., & Frenk, C. S. 1993, MNRAS, 262, 1023
In the text
NASA ADS
- White, D. A. 2000, MNRAS, 312, 663
In the text
NASA ADS
- Xu, H., Makishima, K., Fukazawa, Y., et al. 1998, ApJ, 500, 738
In the text
NASA ADS
- Xue, Y., & Wu, X. 2000, ApJ, 538, 65
In the text
NASA ADS
Copyright ESO 2002
(0.1-2.4 keV) (h-2 ergs s-1 cm-2)
![\begin{figure}
\par\includegraphics[angle=-90,width=5cm,clip]{h2920f1a.ps}\par\i...
...20f1b.ps}\par\includegraphics[angle=-90,width=5cm,clip]{h2920f1c.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img72.gif)
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f2.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img79.gif)
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f3.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img85.gif)
(h-3 Mpc-3)
10.55
6.30 
6.36
4.70![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f4.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img107.gif)
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f5.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img108.gif)
![\begin{figure}
\par\includegraphics[width=8cm,clip]{h2920f6NEW.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img111.gif)
![\begin{figure}
\par\includegraphics[width=7cm,clip]{h2920f7.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img167.gif)
![\begin{figure}
\par\includegraphics[angle=-90,width=6cm,clip]{h2920f8a.ps}\par\includegraphics[angle=-90,width=6cm,clip]{h2920f8b.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img168.gif)
![\begin{figure}
\par\includegraphics[angle=-90,width=6cm,clip]{h2920f9a.ps}\par\includegraphics[angle=-90,width=6cm,clip]{h2920f9b.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img169.gif)
![\begin{figure}
\par\includegraphics[angle=-90,width=8cm,clip]{h2920f10.ps}\end{figure}](/articles/aa/full/2002/09/aah2920/img174.gif)
