Towards optimal cluster power spectrum analysis

The power spectrum of galaxy clusters is an important probe of the cosmological model. In this paper, we develop a formalism to compute the optimal weights for the estimation of the matter power spectrum from cluster power spectrum measurements. We find a closedform analytic expression for the optimal weights, which takes into account: the cluster mass, finite survey volume effects, survey masking, and a flux limit. The optimal weights are w(M,χ ) ∝ b(M,χ )/[1+ n̄h(χ )b2(χ )P (k)], where b(M, χ ) is the bias of clusters of mass M at radial position χ (z), n̄h(χ ) and b2(χ ) are the expected space density and bias squared of all clusters, and P (k) is the matter power spectrum at wavenumber k. This result is analogous to that of Percival et al. We compare our optimal weighting scheme with mass weighting and also with the original power spectrum scheme of Feldman et al. We show that our optimal weighting scheme outperforms these approaches for both volumeand flux-limited cluster surveys. Finally, we present a new expression for the Fisher information matrix for cluster power spectrum analysis. Our expression shows that for an optimally weighted cluster survey the cosmological information content is boosted, relative to the standard approach of Tegmark.