We present an analysis of the 2–point correlation function , \xi ( r ) , of the X –ray Brightest Abell–type Cluster sample ( XBACs ; Ebeling et al .
1996 ) and of the cosmological constraints that it provides .
If \xi ( r ) is modelled as a power–law , \xi ( r ) = ( r _ { 0 } / r ) ^ { \gamma } , we find r _ { 0 } \simeq 26.0 \pm 4.5 h ^ { -1 } { \mathrm { Mpc } } ~ { } and \gamma \simeq 2.0 \pm 0.4 , with errors corresponding to 2 \sigma uncertainties for one significant fitting parameter .
As a general feature , \xi ( r ) is found to remain positive up to r \simeq 50 –55 h ^ { -1 } { \mathrm { Mpc } } ~ { } , after which it declines and crosses zero .
Only a marginal increase of the correlation amplitude is found as the flux limit is increased from 5 \times 10 ^ { -12 } erg s ^ { -1 } cm ^ { -2 } to 12 \times 10 ^ { -12 } erg s ^ { -1 } cm ^ { -2 } , thus indicating a weak dependence of the correlation amplitude on the cluster X –ray luminosity .
Furthermore , we present a method to predict correlation functions for flux–limited X –ray cluster samples from cosmological models .
The method is based on the analytical recipe by Mo & White ( 1996 ) and on an empirical approach to convert cluster fluxes into masses .
We use a maximum–likelihood method to place constraints on the model parameter space from the XBACs \xi ( r ) .
For scale–free primordial spectra , we find that the shape parameter of the power spectrum is determined to lie in the 2 \sigma range 0.05 \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } } % \Gamma \raise - 2.0 pt \hbox { \hbox to 0.0 pt { \hbox { $ \sim$ } } \raise 5.0 pt \hbox { $ < $ } } % 0.20 .
As for the amplitude of the power–spectrum , we find \sigma _ { 8 } \simeq 0.4 –0.8 for \Omega _ { 0 } = 1 and \sigma _ { 8 } \simeq 0.8 –2.0 for \Omega _ { 0 } = 0.3 .
The latter result is in complete agreement with , although less constraining than , results based on the local cluster abundance .