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Astron. Astrophys. 320, 365-377 (1997)

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4. Constraints on n and b

As we have just emphasized, the relation between the ICM temperature and the cluster virial mass is, in contrast to the luminosity-mass relation, relatively well understood. For this reason, we prefer in the following to use the temperature distribution function rather than the luminosity function to draw our conclusions on the power spectrum of density fluctuations.

The two existing cluster temperature distribution functions were derived from X-ray all-sky surveys. The Edge et al. (1990) temperature function was derived from 55 clusters with fluxes greater than [FORMULA] in the 2-10 keV band. It was constructed by correcting the Piccinotti et al. (1982) survey (which is supposed to be complete down to [FORMULA]), for both mis-identification and confusion, and then by including clusters with fluxes greater than [FORMULA]. The supposed clusters with insufficient information were excluded from the analysis, but the authors estimated from the [FORMULA] - [FORMULA] and [FORMULA] distributions that their sample is [FORMULA] complete down to [FORMULA] and [FORMULA] complete down to [FORMULA].

Henry & Arnaud (1991) also used the Piccinotti et al. (1982) all sky sample with corrections for source confusion to derive the temperature distribution function. They find that a power-law with index similar to that derived by Edge et al. (1990) fits the data, but with a normalization twice as large.

Before pursuing our analysis, it is interesting to examine the luminosity function by translating it to a temperature function via the observed [FORMULA] relation. In Fig. 1 we show the results of converting various published luminosity functions by our determination of the local [FORMULA] relation (Eq. 3). The luminosity functions considered are from Edge et al. (1990), Henry & Arnaud (1991) and from the EMSS sample (Gioia et al. 1990, Henry et al. 1992). The temperature data, represented by the points in the figure, come from the two former sets of authors (Edge et al. 1990 and Henry & Arnaud 1991). Thus, only the EMSS data is entirely independent of the temperature observations. The converted temperature distribution functions from both Henry & Arnaud (1991) and the [FORMULA] EMSS give slightly flatter functions than the direct temperature observations of the temperature distribution. One can see that all of the luminosity functions are consistent with the direct determinations of the temperature distribution. Therefore, one would get similar constraints by using luminosity functions instead of the temperature distribution functions.

[FIGURE] Fig. 1. The two local determinations of the temperature distribution function, one by Henry and Arnaud (1991), shown as the squares, and the other by Edge et al. (1990), shown as the triangles, are compared to the temperature functions deduced from the EMSS (Gioia et al. 1990, Henry et al. 1992) by application of our [FORMULA] relation (see text). In this comparison, the [FORMULA] relation is assumed NOT to evolve with redshift. The different line-types display the results for different redshifts, as indicated.

In the following, we will compare the theoretical models to the observed temperature distribution functions and derive constraints on the parameters of the models. The free parameters are the index n of the power spectrum and the bias parameter b, both of which appear in the expression for the rms mass fluctuations (Eq. 1). The parameters are derived by chi-square fitting. We fit the model to each of the two temperature distribution functions, which we denote by ESFA and HA for the results of Edge et al. (1990) and Henry & Arnaud (1991), respectively. The differences will be considered as indicative of the uncertainties. We also add an additional point to each data set to represent A2163 which is the hottest known cluster. The temperature of this cluster was first determined by Arnaud et al. (1992b) who found that is is of the order of 14 keV with an uncertainty of about 1 keV. The importance of this measurement resides in the precision of the temperature determination. Indeed, the uncertainties of the cluster temperature measurements within the samples of Edge et al. (1990) and Henry & Arnaud (1991) were quite large. This has left open the possibility that the existence of such high temperature clusters were not real and that they were due to the tail of the temperature error distribution. On the contrary, although the statistical weight of A2163 is small, the precision in the temperature measurement does give us confidence in the overall shape of the distribution functions we used. We have calculated the number density by assuming that A2163 is the only such cluster in the Abell survey volume extended to a depth of [FORMULA]. The error bars represented on the data points are the [FORMULA] confidence limits. Poisson statistics were used in the case of HA and A2163. In Table 1 we give the best fit values of n and b for each data set along. The subscripts 1 and 2 refer to data sets with and without the A2163 point, respectively.


[TABLE]

Table 1. Best-fit parameters


It is difficult to assess the real nature of the detection statistics for clusters and Poisson statistics may not be relevant. Accordingly, the error bars should be treated with some caution. In the absence of further knowledge we assume that the errors are normally distributed and that the given error bars correspond to the rms deviation. We then draw the confidence contours which would contain 68.3%, 90% and 95.4% of the normally distributed data in the [FORMULA] plane (Fig. 2). Since the error distribution function is unlikely to be gaussian, the actual probability associated with these contours cannot be evaluated. However, we estimated the reliability of the models by checking by eye the goodness of fit for different values of (b,n) on the 1 [FORMULA] contour.

[FIGURE] Fig. 2c-d. The left hand side column shows observed temperature distribution functions (points) with the corresponding best fitting theoretical functions (solid lines). From top to bottom: a Henry & Arnaud (1991) data. b Edge et al. (1990) data. Temperature distribution functions fitting the data at 1 [FORMULA] level are also represented: a The dashed lines represent [FORMULA] and 1.75 in the case [FORMULA] and the dotted lines represent [FORMULA] and [FORMULA] in the case [FORMULA]. b The dashed lines represent [FORMULA] and 1.9 in the case [FORMULA] and the dotted lines represent [FORMULA] and [FORMULA] in the case [FORMULA]. The right hand side column shows confidence region ellipses corresponding to [FORMULA]. These contours correspond to [FORMULA], [FORMULA] and [FORMULA] respectively, for normally distributed data.

From the ESFA data set we find:

[EQUATION]

[EQUATION]

while with the HA data we find:

[EQUATION]

[EQUATION]

These constraints are similar to those drawn by Blanchard & Silk (1991), Henry & Arnaud (1991) or more recently by Bartlett et al. (1995). Because Henry & Arnaud (1991) use a smaller number of clusters spread over a shorter range of temperatures, their data set provides less stringent constraints on the parameters. We finally consider as robust the following intervals:

[EQUATION]

[EQUATION]

The temperature distribution does a good job in constraining both the shape and the amplitude of the power spectrum. This constraints were derived using clusters with temperatures between 2 and 14 keV, so the shape of the fluctuation spectrum is actually constrained over the range 5 and 10 [FORMULA] Mpc. One important implication is that the CDM model cannot explain the shape of the temperature distribution function: its power spectrum is too steep on galaxy cluster scales. Instead of the CDM value of [FORMULA], the data suggest that n is closer to -2 over these scales. This is in agreement with analyses based on other methods, for example the power spectrum determination of Hamilton et al. (1991) and Peacock and Dodds (1994). This conclusion applies independently of the normalization of the spectrum. In considering the cosmic microwave background temperature fluctuations, we may make the additional statement that the normalization required by the clusters does not conform to the normalization demanded of CDM by the COBE measurements: given the CDM spectrum, the latter favors bias factors of order unity or less.

Our conclusions mainly rely on the validity of the temperature-mass relation. It should be emphasized that any error in this relation enters the exponential of the Gaussian in the mass function. As discussed above, the [FORMULA] relation may be derived from the assumptions of isothermality and hydrostatic equilibrium and has been checked further by numerical simulations. However, standard mass estimates from hydrostatic equilibrium may underestimate the actual masses of clusters (Balland & Blanchard 1996). One may make several remarks here concerning future studies on the relation between the state of the ICM and the underlying dark matter. One will eventually be able to measure the temperature profile of the gas using the spatially resolved spectroscopic data of XMM and AXAF. For the present, one may attempt to constrain this profile by combining X-ray images and radio maps of the Sunyaev-Zel'dovich effect (assuming sphericity). Once given a temperature profile, the temperature-mass relation may in principle be deduced only from the assumption of hydrostatic equilibrium. From our point of view, the most exciting prospect employs the weak distortion of gravitationally lensed background galaxies to probe the cluster binding mass. By examining a sample of clusters with lensing data, X-ray images and even maps of the Sunyaev-Zel'dovich effect, one can directly constrain the temperature-mass relation.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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