A&A 383, 773790 (2002)
DOI: 10.1051/00046361:20011769
Y. Ikebe^{1}  T. H. Reiprich^{2}  H. Böhringer^{1}  Y. Tanaka^{1,3}  T. Kitayama^{4}
1  MaxPlanckInstitut für extraterrestrische Physik,
Postfach 1312, 85741 Garching, Germany
2 
Department of Astronomy, University of Virginia,
PO Box 3818, 530 McCormick Road, Charlottesville,
VA 229030818, USA
3 
Institute of Space and Astronautical Science,
Yoshinodai 311, Sagamihara, Kanagawa 2298510, Japan
4 
Department of Physics, Toho University,
Miyama, Funabashi, Chiba 2748510, Japan
Received 7 June 2001 / Accepted 11 December 2001
Abstract
We present a newly measured Xray temperature function of galaxy clusters
using a complete fluxlimited 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^{2}in the 0.12.4 keV band.
Xray 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 1.411 keV.
By fitting these data with theoretically predicted temperature functions
given by the PressSchechter formalism together with a recent formation
approximation and the CDM power spectrum,
we obtained tight and individual constraints on
and .
We also employed the FormationEpoch 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 masstemperature relation are studied
and found to be as large as the statistical errors.
Key words: cosmology: observations  cosmological: parameters  Xrays: 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 PressSchechter 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 Nbody simulations, cluster masses may be presently more accurately determined from a temperature measurement and a masstemperature relation determined from detailed observations or numerical modeling. Thus alternatively, as a welldefined observational quantity, the Xray temperature function (XTF) has been measured, which can be converted to the MF by means of the masstemperature relation.
The first measurements of the XTF were reported by Edge et al. (1990) and Henry & Arnaud (1991), using an Xray fluxlimited 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 temperaturemeasurement results for each cluster with ASCA data (Tanaka et al. 1994). However, the narrow temperature ranges (310 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 powerlaw. 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 Xraycluster 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 Xray fluxlimited cluster sample with a flux limit of ergs s^{1} cm^{2} (0.12.4 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 Xray 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 1.411 keV temperature range. The Xray fluxlimited 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 fluxlimited complete sample with the fluxlimit of ergs s^{1} cm^{2} (0.12.4 keV). The 106 clusters have been derived from combining previous Xray 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} (0.12.4 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 0.12.4 keV band for each object was redetermined from the ROSAT All Sky Survey data or PSPC pointing observations whenever available by integrating all Xray 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 Xray temperature as well as the Xray flux
for each of the 106 clusters.
ASCA data are available for 88 of the 106 clusters,
and the Xray temperatures are measured by analyzing the ASCA data
(Sect. 2.2).
Using the ASCA results,
the correlation between the Xray luminosity and the temperature,
LT 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 XMMNewton 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 LT relation.
By setting a fluxlimit of
ergs s^{1} cm^{2} (0.12.4 keV),
a complete fluxlimited sample comprising 63 clusters
is then constructed (Sect. 2.4).
NAME  redshift  Flux(0.12.4 keV)  (0.12.4 keV) (h^{2} ergs s^{1} cm^{2})  
(10^{21} cm^{2})  (keV)  (10^{11} ergs s^{1} cm^{2})  Open  Flat  
^{*}PERSEUS  0.0183  1.569  6.42^{+ 0.06}_{0.06}  114.33 1.50  (4.12 0.05)e+44  (4.19 0.05)e+44 
^{*}OPHIUCHUS  0.0280  2.014  10.25^{+ 0.30}_{0.36}  36.25 1.16  (3.07 0.10)e+44  (3.14 0.10)e+44 
COMA  0.0232  0.089  8.07^{+ 0.29}_{0.27}  35.13 0.79  (2.04 0.05)e+44  (2.08 0.05)e+44 
^{*}A3627  0.0163  2.083  5.62^{+ 0.12}_{0.11}  31.60 1.57  (9.04 0.45)e+43  (9.16 0.45)e+43 
A3526  0.0103  0.825  3.69^{+ 0.05}_{0.04}  26.20 0.95  (2.99 0.11)e+43  (3.02 0.11)e+43 
^{*}AWM7  0.0172  0.921  3.70^{+ 0.08}_{0.05}  16.60 0.55  (5.30 0.18)e+43  (5.37 0.18)e+43 
^{*}A2319  0.0564  0.877  8.84^{+ 0.29}_{0.24}  12.15 0.21  (4.20 0.07)e+44  (4.39 0.07)e+44 
A3571  0.0397  0.393  6.80^{+ 0.21}_{0.18}  12.00 0.15  (2.05 0.03)e+44  (2.12 0.03)e+44 
^{*}TRIANGUL  0.0510  1.229  9.06^{+ 0.33}_{0.31}  11.26 0.13  (3.18 0.04)e+44  (3.31 0.04)e+44 
A2199  0.0302  0.084  4.28^{+ 0.10}_{0.10}  10.73 0.32  (1.06 0.03)e+44  (1.09 0.03)e+44 
^{*}3C129  0.0223  6.789  5.57^{+ 0.16}_{0.15}  10.65 0.98  (5.72 0.53)e+43  (5.82 0.54)e+43 
A1060  0.0114  0.492  3.15^{+ 0.05}_{0.05}  10.04 0.54  (1.41 0.07)e+43  (1.42 0.08)e+43 
2A0335  0.0349  1.864  3.64^{+ 0.09}_{0.08}  8.87 0.11  (1.17 0.01)e+44  (1.21 0.01)e+44 
A0262  0.0161  0.552  2.25^{+ 0.06}_{0.06}  8.78 0.55  (2.46 0.15)e+43  (2.49 0.16)e+43 
FORNAX  0.0046  0.145  1.56^{+ 0.05}_{0.07}  8.68 0.66  (1.97 0.15)e+42  (1.98 0.15)e+42 
A0496  0.0328  0.568  4.59^{+ 0.10}_{0.10}  8.20 0.10  (9.56 0.12)e+43  (9.81 0.12)e+43 
A0085  0.0556  0.358  6.51^{+ 0.16}_{0.23}  7.37 0.07  (2.48 0.03)e+44  (2.59 0.03)e+44 
A3667  0.0560  0.459  6.28^{+ 0.27}_{0.26}  7.12 0.08  (2.44 0.03)e+44  (2.54 0.03)e+44 
A2029  0.0767  0.307  7.93^{+ 0.39}_{0.36}  6.88 0.07  (4.43 0.04)e+44  (4.70 0.05)e+44 
A3558  0.0480  0.363  5.37^{+ 0.17}_{0.15}  6.68 0.05  (1.68 0.01)e+44  (1.74 0.01)e+44 
^{*}S636  0.0116  0.642  2.06^{+ 0.07}_{0.06}  6.65 0.54  (9.65 0.79)e+42  (9.75 0.79)e+42 
A2142  0.0899  0.405  8.46^{+ 0.53}_{0.49}  6.21 0.09  (5.52 0.08)e+44  (5.90 0.08)e+44 
A1795  0.0616  0.120  6.17^{+ 0.26}_{0.25}  6.21 0.03  (2.58 0.01)e+44  (2.70 0.02)e+44 
^{*}PKS0745  0.1028  4.349  6.37^{+ 0.21}_{0.20}  6.13 0.10  (7.18 0.11)e+44  (7.74 0.12)e+44 
A2256  0.0601  0.402  6.83^{+ 0.23}_{0.21}  6.03 0.14  (2.38 0.05)e+44  (2.49 0.06)e+44 
A1367  0.0216  0.255  3.55^{+ 0.08}_{0.08}  5.98 0.07  (3.02 0.04)e+43  (3.07 0.04)e+43 
A3266  0.0594  0.148  7.72^{+ 0.35}_{0.28}  5.77 0.06  (2.22 0.02)e+44  (2.32 0.03)e+44 
A4038  0.0283  0.155  3.22^{+ 0.10}_{0.10}  5.61 0.12  (4.86 0.10)e+43  (4.97 0.10)e+43 
A2147  0.0351  0.329  4.34^{+ 0.12}_{0.13}  5.45 0.29  (7.28 0.39)e+43  (7.49 0.40)e+43 
A0401  0.0748  1.019  7.19^{+ 0.28}_{0.24}  5.26 0.09  (3.23 0.06)e+44  (3.41 0.06)e+44 
N5044  0.0090  0.491  1.22^{+ 0.04}_{0.04}  5.20 0.04  (4.54 0.04)e+42  (4.58 0.04)e+42 
A0478  0.0900  1.527  6.91^{+ 0.40}_{0.36}  5.12 0.05  (4.58 0.04)e+44  (4.89 0.05)e+44 
HYDRAA  0.0538  0.486  3.82^{+ 0.20}_{0.17}  4.71 0.04  (1.49 0.01)e+44  (1.56 0.01)e+44 
A2052  0.0348  0.290  3.12^{+ 0.10}_{0.09}  4.57 0.08  (6.01 0.10)e+43  (6.18 0.10)e+43 
N1550  0.0123  1.159  1.44^{+ 0.03}_{0.02}  4.24 0.37  (6.93 0.61)e+42  (7.00 0.62)e+42 
A2063  0.0354  0.292  3.56^{+ 0.16}_{0.12}  4.23 0.09  (5.74 0.12)e+43  (5.90 0.12)e+43 
A1644  0.0474  0.533  4.09 0.34  (1.00 0.08)e+44  (1.04 0.09)e+44  
A0119  0.0440  0.310  5.69^{+ 0.24}_{0.28}  4.05 0.06  (8.53 0.13)e+43  (8.82 0.13)e+43 
^{*}A0644  0.0704  0.514  6.54^{+ 0.27}_{0.26}  3.97 0.07  (2.16 0.04)e+44  (2.28 0.04)e+44 
N4636  0.0037  0.175  0.66^{+ 0.03}_{0.01}  3.97 0.47  (5.84 0.69)e+41  (5.86 0.69)e+41 
A3158  0.0590  0.106  5.41^{+ 0.26}_{0.24}  3.79 0.09  (1.44 0.04)e+44  (1.51 0.04)e+44 
A1736  0.0461  0.536  3.68^{+ 0.22}_{0.17}  3.54 0.36  (8.21 0.84)e+43  (8.51 0.87)e+43 
A0754  0.0528  0.459  9.00^{+ 0.35}_{0.34}  3.34 0.09  (1.01 0.03)e+44  (1.06 0.03)e+44 
A0399  0.0715  1.058  6.46^{+ 0.38}_{0.36}  3.30 0.29  (1.85 0.16)e+44  (1.95 0.17)e+44 
MKW3S  0.0450  0.315  3.45^{+ 0.13}_{0.10}  3.22 0.05  (7.13 0.12)e+43  (7.38 0.12)e+43 
^{*}A0539  0.0288  1.206  3.04^{+ 0.11}_{0.10}  3.12 0.07  (2.80 0.06)e+43  (2.87 0.06)e+43 
EXO0422  0.0390  0.640  3.12 0.32  (5.16 0.52)e+43  (5.32 0.54)e+43  
A4059  0.0460  0.110  3.94^{+ 0.15}_{0.15}  3.10 0.07  (7.16 0.16)e+43  (7.42 0.16)e+43 
A3581  0.0214  0.426  1.83^{+ 0.04}_{0.02}  3.08 0.16  (1.53 0.08)e+43  (1.55 0.08)e+43 
A3112  0.0750  0.253  4.72^{+ 0.37}_{0.25}  3.07 0.06  (1.90 0.03)e+44  (2.01 0.04)e+44 
A0576  0.0381  0.569  3.83^{+ 0.16}_{0.15}  2.99 0.33  (4.73 0.53)e+43  (4.88 0.54)e+43 
A3562  0.0499  0.391  4.47^{+ 0.23}_{0.21}  2.89 0.04  (7.87 0.12)e+43  (8.18 0.13)e+43 
A2204  0.1523  0.594  6.38^{+ 0.23}_{0.23}  2.73 0.07  (7.13 0.19)e+44  (7.92 0.21)e+44 
A0400  0.0240  0.938  2.43^{+ 0.13}_{0.12}  2.64 0.05  (1.65 0.03)e+43  (1.68 0.03)e+43 
A2065  0.0721  0.284  5.37^{+ 0.34}_{0.30}  2.55 0.26  (1.46 0.15)e+44  (1.54 0.15)e+44 
A1651  0.0860  0.171  6.22^{+ 0.45}_{0.41}  2.52 0.05  (2.06 0.04)e+44  (2.19 0.04)e+44 
A2589  0.0416  0.439  3.38^{+ 0.13}_{0.13}  2.52 0.05  (4.75 0.10)e+43  (4.91 0.11)e+43 
MKW8  0.0270  0.260  3.29^{+ 0.23}_{0.22}  2.51 0.35  (1.99 0.27)e+43  (2.03 0.28)e+43 
A2657  0.0404  0.527  3.53^{+ 0.12}_{0.12}  2.49 0.04  (4.44 0.06)e+43  (4.58 0.07)e+43 
A3376  0.0455  0.501  4.43^{+ 0.39}_{0.38}  2.44 0.06  (5.52 0.13)e+43  (5.72 0.14)e+43 
S1101  0.0580  0.185  2.44 0.04  (9.05 0.14)e+43  (9.46 0.14)e+43  
A1650  0.0845  0.154  5.68^{+ 0.30}_{0.27}  2.43 0.26  (1.92 0.21)e+44  (2.04 0.22)e+44 
A2634  0.0312  0.517  3.45^{+ 0.16}_{0.16}  2.38 0.06  (2.51 0.06)e+43  (2.58 0.07)e+43 
A3391  0.0531  0.542  5.89^{+ 0.45}_{0.33}  2.20 0.07  (6.78 0.21)e+43  (7.06 0.22)e+43 
A2597  0.0852  0.250  4.20^{+ 0.49}_{0.41}  2.20 0.04  (1.77 0.04)e+44  (1.88 0.04)e+44 
ZwCl1215  0.0750  0.164  2.17 0.05  (1.34 0.03)e+44  (1.42 0.03)e+44  
A2244  0.0970  0.207  5.77^{+ 0.61}_{0.44}  2.10 0.07  (2.19 0.08)e+44  (2.35 0.08)e+44 
A0133  0.0569  0.160  3.97^{+ 0.28}_{0.27}  2.06 0.03  (7.33 0.10)e+43  (7.66 0.11)e+43 
A2163  0.2010  1.227  10.55^{+ 1.01}_{0.68}  2.04 0.05  (9.28 0.23)e+44  (10.59 0.27)e+44 
A2255  0.0800  0.251  5.92^{+ 0.40}_{0.26}  2.02 0.04  (1.42 0.03)e+44  (1.51 0.03)e+44 
IIIZw54  0.0311  1.668  2.01 0.25  (2.11 0.27)e+43  (2.16 0.27)e+43  
A3395s  0.0498  0.849  5.55^{+ 0.89}_{0.65}  2.01 0.13  (5.42 0.34)e+43  (5.63 0.35)e+43 
N507  0.0165  0.525  1.40^{+ 0.04}_{0.07}  2.00 0.04  (5.88 0.12)e+42  (5.96 0.12)e+42 
MKW4  0.0200  0.186  1.84^{+ 0.05}_{0.03}  2.00 0.05  (8.65 0.24)e+42  (8.79 0.24)e+42 
UGC03957  0.0340  0.459  1.98 0.19  (2.49 0.24)e+43  (2.56 0.25e+43  
A3827  0.0980  0.284  1.98 0.19  (2.10 0.20)e+44  (2.25 0.22)e+44  
A3822  0.0760  0.212  5.12^{+ 0.43}_{0.31}  1.98 0.24  (1.26 0.15)e+44  (1.33 0.16)e+44 
IIZw108  0.0494  0.663  1.90 0.23  (5.06 0.61)e+43  (5.26 0.63)e+43  
M49  0.0044  0.159  1.33^{+ 0.03}_{0.03}  1.89 0.03  (3.93 0.07)e+41  (3.94 0.07)e+41 
ZwCl1742  0.0757  0.356  1.87 0.13  (1.18 0.08)e+44  (1.25 0.09)e+44  
S405  0.0613  0.765  1.82 0.24  (7.48 1.00)e+43  (7.84 1.05)e+43  
A3532  0.0539  0.596  4.41^{+ 0.19}_{0.18}  1.78 0.05  (5.65 0.17)e+43  (5.88 0.17)e+43 
A3695  0.0890  0.356  1.76 0.26  (1.54 0.23)e+44  (1.65 0.25)e+44  
HCG94  0.0417  0.455  3.30^{+ 0.17}_{0.16}  1.72 0.03  (3.25 0.05)e+43  (3.36 0.06)e+43 
A3528s  0.0551  0.610  4.60^{+ 0.49}_{0.27}  1.70 0.04  (5.65 0.15)e+43  (5.89 0.15)e+43 
S540  0.0358  0.353  1.62 0.13  (2.26 0.18)e+43  (2.32 0.19)e+43  
A2877  0.0241  0.210  1.61 0.03  (1.01 0.02)e+43  (1.03 0.02)e+43  
A3395n  0.0498  0.542  5.11^{+ 0.47}_{0.43}  1.54 0.10  (4.16 0.27)e+43  (4.33 0.28)e+43 
A2151w  0.0369  0.336  2.58^{+ 0.19}_{0.20}  1.53 0.05  (2.26 0.07)e+43  (2.33 0.07)e+43 
A3560  0.0495  0.392  1.50 0.06  (4.02 0.17)e+43  (4.17 0.17)e+43  
A2734  0.0620  0.184  5.07^{+ 0.36}_{0.42}  1.47 0.06  (6.19 0.25)e+43  (6.49 0.26)e+43 
A0548e  0.0410  0.188  2.93^{+ 0.17}_{0.15}  1.46 0.04  (2.68 0.08)e+43  (2.77 0.08)e+43 
A1689  0.1840  0.180  8.58^{+ 0.84}_{0.40}  1.45 0.03  (5.53 0.10)e+44  (6.25 0.11)e+44 
A1914  0.1712  0.097  8.41^{+ 0.60}_{0.58}  1.45 0.03  (4.77 0.11)e+44  (5.36 0.12)e+44 
RXJ2344  0.0786  0.354  1.37 0.03  (9.34 0.21)e+43  (9.91 0.23)e+43  
A3921  0.0936  0.280  5.39^{+ 0.38}_{0.35}  1.31 0.04  (1.27 0.03)e+44  (1.36 0.04)e+44 
A1413  0.1427  0.162  6.56^{+ 0.65}_{0.44}  1.28 0.03  (2.92 0.08)e+44  (3.23 0.09)e+44 
N5813  0.0064  0.419  0.76^{+ 0.19}_{0.19}  1.27 0.14  (5.62 0.62)e+41  (5.66 0.62)e+41 
A1775  0.0757  0.100  3.66^{+ 0.34}_{0.20}  1.26 0.04  (8.02 0.23)e+43  (8.49 0.25)e+43 
A3528n  0.0540  0.610  4.79^{+ 0.50}_{0.44}  1.26 0.05  (4.00 0.15)e+43  (4.17 0.16)e+43 
A1800  0.0748  0.118  1.20 0.15  (7.39 0.95)e+43  (7.82 1.01)e+43  
A3888  0.1510  0.120  1.09 0.04  (2.77 0.11)e+44  (3.08 0.12)e+44  
A3530  0.0544  0.600  4.05^{+ 0.32}_{0.30}  0.96 0.04  (3.11 0.15)e+43  (3.25 0.15)e+43 
N5846  0.0061  0.425  0.64^{+ 0.04}_{0.03}  0.83 0.03  (3.35 0.12)e+41  (3.36 0.12)e+41 
N499  0.0147  0.525  0.66^{+ 0.02}_{0.03}  0.45 0.02  (1.07 0.04)e+42  (1.08 0.05)e+42 
A0548w  0.0424  0.179  0.25 0.02  (4.84 0.43)e+42  (5.03 0.45)e+42  
^{*)}:
The Galactic latitude is lower than 20.
(Edge & Stewart 1991), and XMMNewton 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 Xray 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 Xray emission to determine the cluster average temperature (see also Markevitch 1998; Arnaud & Evrard 1999).
Therefore, in the ASCA data analysis, we employed a twotemperature (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 multiphase intracluster 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), HydraA (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 XMMNewton 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 XMMNewton 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 (Solidstate 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 XRay 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 SIS0 and SIS1 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 Xray background and nonXray background, which were estimated for each cluster observation from the data of blanksky observations as well as the data taken when the XRay 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 Xray 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 Xray energy was calculated for each spectrum with an ASCA simulator (SimARF). The pointspreadfunction of the ASCA XRT is so largely extended that the Xray 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 fluxcontamination 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 bestfit model was derived by a chisquare 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 Ftest. 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 coolcomponent 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 coolcomponent 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 23 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).
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).  
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For the 88 clusters observed with ASCA, Xray fluxes and luminosities were calculated from the spectralanalysis results described above and the ROSAT PSPC count rates in 50201 PI channel given by Reiprich & Böhringer (2002). Assuming isothermality in each cluster at the hotcomponent temperature and the Galactic column density given in Table 1, we determined the 0.12.4 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 0.12.4 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 powerlaw 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 powerlaw function is not a good representation
of the LT relation over a wide temperature range.
In particular, including clusters, groups, and elliptical galaxies,
the LT 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 powerlaw 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(X_{2}X_{1}) estimator
given by Akritas & Bershady (1996).
(In this case, the luminosity and
temperature are assigned as X_{2} and X_{1}, respectively.)
The choice of the estimator is based on an argument by
Isobe et al. (1990).
The bestfit functions are
Figure 2: The luminosities in the energy range of 0.12.4 keV in the cluster rest frame in a flat universe ( , ) as a function of the hot component temperature obtained from the twotemperature model fitted to the ASCA data. The bestfit powerlaw function (Eq. (2)) is overlayed.  
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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
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 bestfit powerlaw in the LT relation was assigned as error. The estimated temperatures and luminosities for the open and flat universe cases as well as the corresponding 0.12.4 keV fluxes which are the same for the open and flat universe cases are summarized in Table 1.
We set a fluxlimit of ergs s^{1} cm^{2}in 0.12.4 keV band to construct a fluxlimited 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 fluxlimited 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 610%, other spectroscopic measurements were used for 3 clusters, and there are only two clusters for which the temperature is estimated with the LT relation.
Figure 3: Xray temperature vs. redshift for the clusters of the fluxlimited sample.  
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NAME  (keV)  (h^{3} Mpc^{3})  
Open  Flat  
A2163  10.55  2.39e+08  2.65e+08 
A0754  9.00  1.40e+08  1.52e+08 
A2142  8.46  1.14e+08  1.23e+08 
COMA  8.07  9.69e+07  1.04e+08 
A2029  7.93  9.12e+07  9.78e+07 
A3266  7.72  8.30e+07  8.89e+07 
A0401  7.19  6.50e+07  6.91e+07 
A0478  6.91  5.66e+07  6.00e+07 
A2256  6.83  5.45e+07  5.77e+07 
A3571  6.80  5.36e+07  5.67e+07 
A0085  6.51  4.60e+07  4.87e+07 
A0399  6.46  4.46e+07  4.72e+07 
A2204  6.38  4.28e+07  4.52e+07 
ZwCl1215  6.30 6.36  4.09e+07  4.49e+07 
A3667  6.28  4.05e+07  4.28e+07 
A1651  6.22  3.92e+07  4.14e+07 
A1795  6.17  3.81e+07  4.02e+07 
A2255  5.92  3.29e+07  3.46e+07 
A3391  5.89  3.23e+07  3.40e+07 
A2244  5.77  3.01e+07  3.17e+07 
A0119  5.69  2.86e+07  3.00e+07 
A1650  5.68  2.85e+07  2.99e+07 
A3395s  5.55  2.64e+07  2.76e+07 
A3158  5.41  2.40e+07  2.50e+07 
A2065  5.37  2.34e+07  2.44e+07 
A3558  5.37  2.35e+07  2.45e+07 
A3112  4.72  1.48e+07  1.53e+07 
A1644  4.70  1.46e+07  1.51e+07 
A0496  4.59  1.34e+07  1.39e+07 
A3562  4.47  1.22e+07  1.27e+07 
A3376  4.43  1.18e+07  1.22e+07 
A2147  4.34  1.10e+07  1.13e+07 
A2199  4.28  1.04e+07  1.08e+07 
A2597  4.20  9.75e+06  1.01e+07 
A0133  3.97  8.02e+06  8.25e+06 
A4059  3.94  7.76e+06  7.98e+06 
A0576  3.83  7.07e+06  7.26e+06 
HYDRAA  3.82  6.98e+06  7.16e+06 
A3526  3.69  6.15e+06  6.30e+06 
A1736  3.68  6.11e+06  6.26e+06 
2A0335  3.64  5.87e+06  6.00e+06 
A2063  3.56  5.41e+06  5.53e+06 
A1367  3.55  5.36e+06  5.48e+06 
A2657  3.53  5.25e+06  5.36e+06 
A2634  3.45  4.84e+06  4.94e+06 
MKW3S  3.45  4.87e+06  4.97e+06 
A2589  3.38  4.48e+06  4.57e+06 
MKW8  3.29  4.08e+06  4.16e+06 
A4038  3.22  3.78e+06  3.85e+06 
A1060  3.15  3.50e+06  3.56e+06 
A2052  3.12  3.37e+06  3.43e+06 
IIIZw54  2.98 3.00  2.87e+06  2.99e+06 
EXO0422  2.90  2.60e+06  2.63e+06 
S1101  2.60  1.76e+06  1.77e+06 
A0400  2.43  1.37e+06  1.38e+06 
A0262  2.25  1.04e+06  1.04e+06 
MKW4  1.84  5.00e+05  4.96e+05 
A3581  1.83  4.88e+05  4.84e+05 
FORNAX  1.56  2.72e+05  2.68e+05 
N1550  1.44  2.05e+05  2.02e+05 
N507  1.40  1.83e+05  1.79e+05 
N5044  1.22  1.10e+05  1.08e+05 
N4636  0.66  1.16e+04  1.10e+04 
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 LT relation.
Figure 4: The Xray temperature function derived with the fluxlimited 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=1.42.5 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 bestfit PressSchechter function derived in Sect. 4.1.  
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Figure 5: The Xray temperature function derived with the fluxlimited complete sample of 61 clusters for the case of a flat universe ( ). The three lines indicate previous measurements in the form of the bestfit powerlaw functions. Solid, dotted, and dashed lines correspond to the results by Markevitch (1998), Edge et al. (1990), and Henry (2000), respectively.  
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We have derived a cumulative XTF, which is the number density
of clusters that are hotter than a certain temperature, by
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).
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.  
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In the following, using the fluxlimited 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 highredshift SN Ia indicate a positive cosmological constant (Perlmutter et al. 1999; Riess et al. 1998). Moreover, the cosmic microwave background (CMB) temperaturefluctuation 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 PressSchechter mass function (Sect. 4.1) and the FormationEpoch 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 PressSchechter 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
The mass function given above is converted to the XTF by means of
a masstemperature 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
,
and that hydrostatic equilibrium is achieved.
The virial mass is defined as
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 FormationEpoch model described in the next section.
Combining all the formulae given above,
the model XTF at a given redshift is given by
In the model XTF given in the previous section, the formation redshift, z_{f}, 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 darkmatterhalo 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,
Model XTF  A  
PS(Open)  0.18 _{0.05}^{+0.08}  0.96 _{0.09}^{+0.11}  42.15 _{0.20}^{+0.17}  2.47 _{0.24}^{+0.26}  0.24 _{0.04}^{+0.04} 
PS(Open, T>3 keV)  0.25 _{0.09}^{+0.12}  0.89 _{0.11}^{+0.13}  42.19 _{0.28}^{+0.27}  2.42 _{0.37}^{+0.38}  0.22 _{0.03}^{+0.05} 
FE(Open, s=0)  0.15 _{0.05}^{+0.07}  0.98 _{0.07}^{+0.08}  42.14 _{0.20}^{+0.17}  2.49 _{0.25}^{+0.26}  0.24 _{0.04}^{+0.04} 
FE(Open, s=1)  0.11 _{0.05}^{+0.07}  0.68 _{0.03}^{+0.04}  42.12 _{0.20}^{+0.18}  2.51 _{0.25}^{+0.27}  0.24 _{0.04}^{+0.04} 
PS(Flat)  0.19 _{0.05}^{+0.08}  1.02 _{0.11}^{+0.12}  42.15 _{0.20}^{+0.17}  2.49 _{0.25}^{+0.26}  0.24 _{0.03}^{+0.04} 
PS(Flat, T>3 keV)  0.26 _{0.09}^{+0.12}  0.94 _{0.13}^{+0.14}  42.19 _{0.29}^{+0.27}  2.44 _{0.38}^{+0.39}  0.23 _{0.04}^{+0.04} 
FE(Flat, s=0)  0.18 _{0.05}^{+0.07}  1.14 _{0.11}^{+0.13}  42.15 _{0.20}^{+0.17}  2.49 _{0.25}^{+0.26}  0.24 _{0.03}^{+0.04} 
FE(Flat, s=1)  0.14 _{0.05}^{+0.06}  0.89 _{0.08}^{+0.08}  42.14 _{0.21}^{+0.17}  2.51 _{0.25}^{+0.26}  0.24 _{0.03}^{+0.04} 
Instead of fitting the model XTF to the XTF derived in Sect. 3,
we use the predicted XTF and the LT 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 luminositytemperature 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 powerlaw 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
There are two reasons for setting the low temperature cutoff at 1.4 keV. First, as discussed in Sect. 2.3, the LT 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 powerlaw modeling for the LT 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), Xray 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 MT 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 bestfit values and errors of the fitting parameters are summarized in Table 3. The bestfit 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'' LT 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).
Figure 7: The bestfit model function in the luminositytemperature space for the case of the flat universe is shown with contours. The 63 clusters from the fluxlimited 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.  
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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.  
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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.  
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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 LT 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 MT 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 MT relation derived from different hydrodynamical simulations.
A cluster mass may vary from 62% to 119% of the mean value.
Source  
Hydrodynamic simulation (mean)  
Hydrodynamic simulation (maximum)  
Hydrodynamic simulation (minimum)  
Horner et al.  0.496 T^{0.133} 
Nevalainen et al.  0.437 T^{0.193} 
Finoguenov et al.  0.415 T^{0.187} 
Model XTF  
PS(Open)  0.090.37  0.551.13 
FE(Open, s= 0)  0.070.32  0.611.15 
FE(Open, s= 1)  0.030.23  0.520.80 
PS(Flat)  0.100.38  0.571.19 
FE(Flat, s= 0)  0.100.35  0.651.33 
FE(Flat, s= 1)  0.060.18  0.660.97 
The MT relations derived observationally from Xray mass measurements give somewhat different results. Horner et al. (1999) derived an MT 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 T^{1.79} and T^{1.78}, respectively, and lower normalizations.
In order to investigate the effects of the different MT relations, we repeated the fitting performed in Sect. 4 with the PS model for the flat universe case. The six MT relations we have examined are summarized in Table 4. Each MT relation can be expressed with Eq. (12), replacing the value with the formula given in Table 4. In Fig. 10, the bestfit parameters derived with the different MT relations are compared with the result presented in Sect. 4.4. The effect of using different MT 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 MT relation moves the bestfit point along the elongation direction of the confidence contour, while an amplitude change moves the bestfit point along the perpendicular direction. Taking into account these systematics from the different MT 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 MT relation obtained based on the observations of nearby clusters. This is because the current MT relation should be a result of accumulation of different MT relations from different formation redshifts which depend on the cluster masses. Therefore, the MT relations obtained from the Hydrodynamic simulations may be more appropriately applied with the FE model.
Figure 10: The bestfit values of and derived with the PS model in the case of the flat universe, for which different MT 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 matterradiation
equality.
Most recently Schuecker et al. (2001),
using a sample of 452 Xray 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
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 MT 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 MT 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 cutoff 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 610% 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 MT 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 PressSchechter 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 Xray observations of the high redshift clusters with Chandra and XMMNewton.
Finally, taking into account all the uncertainties, we put our general constraints on and as 0.030.37, 0.521.15 in an open universe and 0.060.38, 0.571.33 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 Xray temperature function in more detail.
A flux limited sample has been extensively used for constructing
the Xray 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
Using
,
the differential XTF can be given as
(A.7) 
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 Xray fluxes
is considered.
If there are considerable measurement errors in the flux measurements,
a cluster that should be in the fluxlimited sample
could be found to be fainter than the flux limit
and to be missed from the sample, and viceversa.
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
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^{2}in 0.12.4 keV band in the observed frame, and then should be the 0.12.4 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 LT 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 T_{i} 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 MonteCarlo simulation. With an adopted XTF of , a number of cluster observations are simulated taking into account the scatter in the LT 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)).