3. Constraints on n, b and from the temperature distribution function
In this section, we constrain the free parameters of the model by fitting the theoretical temperature distribution function to the data at z=0 . In OBB we explain why we prefer to use the temperature distribution function rather than the luminosity function to constrain the power spectrum of the density fluctuations.
Up to now, there are two observed cluster temperature distribution functions which were derived from previously existing all-sky catalogues of X-ray surveys. These two distribution functions are those of Edge et al. (1990) and Henry & Arnaud (1991), which both are described in OBB. The range of temperature covered by the data goes from 2 to almost 14 keV. In the case , these temperatures correspond to linear scales of the order of to Mpc. As a consequence, we can constrain both the normalization and the shape of the initial power spectrum on these scales.
The normalization of the power spectrum is conventionally specified through the variance, , of the mass overdensity within spheres of radius . Since the fluctuations in the galaxy distribution give (Davis & Peebles, 1983), estimates of are referred to as the bias parameter,
On the other hand, the mass within the normalization sphere depends on the value of . The free parameters of the temperature distribution function are then n which mostly determines the shape of the function, and and b which both determine the characteristic temperature, ie. the temperature of a structure of mass such that . This characteristic temperature is:
(We have neglected the factor , as it varies only slightly with . Actually, it is equal to 0.97 and 0.85 at z=0 , for and 0.2 respectively). It is interesting to notice that this expression does not depend on the Hubble constant (see OBB), and therefore that the constraint we will derive are independent of its value.
In low- models there is no motivation for a biased picture. However, there is some uncertainty in the observed value of . This implies that the unbiased case could correspond to a value of b in Eq. 4, somewhat different from one. We have then allowed b to be also a free parameter. Because both b and enter in Eq. 5, there is a degeneracy between the two parameters. Therefore, in the set of three parameters fixing the shape and the normalization of the temperature distribution function, we have preferred to vary simultaneously either ( n, ) or ( n, b ). In the latter case, is set to 0.2 which is the value obtained in usual density estimates (Peebles, 1986).
We have performed chi-square fitting of our model to each of the two existing observationally estimated temperature distribution functions. In the following, Edge et al.'s (1990) and Henry & Arnaud's (1991) temperature distributions are noted (ESFA) and (HA) respectively. The subscript 1 means that it was used in the originally published form and the subscript 2 signifies that we have added a point to the original data points. This point represents A2163, the cluster whose temperature was determined by Arnaud et al. (1992). The density at this point was calculated by assuming that there is one cluster of this temperature in the Abell survey (1958) which covers deg . The Abell catalogue is complete only between and for clusters of richness class R>0 . However, it is likely that rich clusters at higher redshifts could have been included in the Abell catalogue. We have therefore assumed completeness out to . In all cases, the error bars represent the confidence limits. In the case of HA and A2163, these errors bars correspond to Poisson statistics. We have then applied the same procedure as in OBB: the best-fit parameters are derived by chi-square minimization, contours are drawn in the two-dimensional parameter space and the uncertainties are obtained by checking by eye the goodness of fit of models lying in those contours (Fig. 1).
Table 1. Best-fit parameters
In Table 1, and for each data set, we give the best-fit parameters in the case . The values given in thick line were fixed beforehand as explained above. For a given value of , and owing to the difference in the normalization of the two observed temperature distribution functions (a factor of two higher in the case of HA), the best-fit bias parameter is higher in the case of ESFA than in the case of HA.
In Fig. 1a, is set to 0.2 and ( n, b ) are left as free parameters. The contours of constant in the n-b space are strongly elongated (Fig. 1a) which means that is correlated to the power spectrum shape: this is due to the fact that the sphere which encloses the typical mass of a cluster has a radius which is of the order of 14 Mpc in the case . Therefore, contrary to the case, clusters do not provide a unique normalisation at Mpc.
In Fig. 1b, b is set to 1 and ( n, ) are the free parameters. It appears from this figure that the lower the value of the lower the index of the power spectrum is, i.e. low models require more power on large scale in order to produce the observed clusters.
From our analysis, we can now estimate the range of which is consistent with the observed temperature distribution function at z=0 . Assuming that the rms fluctuation of galaxy density within spheres of radius is equal to 1 and that the galaxy distribution is unbiased relative to that of the mass, we infer a value of equal to from ESFA and equal to from HA. The higher value in the latter case is due to the higher normalization of the temperature distribution function as determined by Henry & Arnaud (1991). Relaxing the assumption that b is equal to 1, we fix the value of . This approach is preferable because of the degeneracy between the two parameters which both determine the degree of clustering of the mass. In the case which is the dynamical estimate of the density parameter, a value of b equal to 1 is compatible with ESFA whereas the best-fit bias parameter is close to 0.7 from HA. Recently, also using the PS formalism, White et al. (1993) computed the value of necessary to reproduce the observed density of Abell clusters in flat universes with different values of . They assumed an abundance of for clusters with masses equal to within an Abell radius and equal to within the virial radius (for h=0.5 ). From Edge et al.'s (1990) and Henry & Arnaud's (1991) cumulative temperature distribution functions, they took a mean value of at this abundance. White et al. (1993) did not attempt to extract information on the shape of the power spectrum from clusters, so we can only compare our results to the value of the bias parameter they derived. In the case they found that the value of b has to be in the range 0.6-0.8 in order to produce clusters with sufficient masses within an Abell radius at the standard abundance they assumed. This result is in agreement with the value of the bias parameter we obtain by fitting the theoretical model to HA. On the contrary, due to the lower normalization of ESFA, we find that an unbiased model is compatible with their observations. Concerning the initial power spectrum index we derive from the observed temperature distribution function, we find a value of n close to -1.5 from ESFA, and between -1.4 and -1.1 from HA. The difference is due to the scaling of the characteristic mass corresponding to the exponential cut-off in the PS mass function, which depends on both and b, and is not due to the intrinsic difference in the shape of the two observed temperature distribution functions. It is clear that because of the degeneracy between b and , the temperature distribution function of X-ray clusters at zero redshift does not allow to distinguish the open models from the critical model. Nevertheless, our analysis has revealed the interesting fact that in the case of an unbiased open model, the temperature distribution function implies a value of in the range 0.15-0.4: unbiased models with cannot reproduce the temperature distribution function. It should be noticed that although this determination is formally independent of dynamical estimates on cluster scale, the results are very similar.
© European Southern Observatory (ESO) 1997