Cluster Correlations for Scale-Free Spectra

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The cluster autocorrelation function ξ_c and the galaxy-cluster cross correlation ξ_gc are used to test the biased structure formation for scale-free spectra P(k)∝kⁿ. Following Kaiser (1984), we assume that rich clusters form only at higfr density regions with the matter distribution represented by a Gaussian random field. Then, the correlation ξ of two regions with characteristic scales R₁ and R₂ lying above the thresholds v₁ and v₂ (δ ≡ νσ), is given by the expression for the bivariate Gaussian where z(r) is the coviance function, i.e. and is the correlation of the matter distribution Gaussian filtered on the comoving scales R₁ and R₂. From the previous equations, one can obtain for P(k)∝ kⁿ, τ≪1 and large r a powers-law form either for ξ_c or ξ_gc. Moreover, the amplification factor A_{c - gc} ≡ ξ_c/ξ_gc is