Uncorrelated modes of the non-linear power spectrum

Abstract

Non-linear evolution causes the galaxy power spectrum to become broadly correlated over different wavenumbers.It is shown that pre-whitening the power spectrum ± transforming the power spectrum in such a way that the noise covariance becomes proportional to the unit matrix ± greatly narrows the covariance of power.The eigenfunctions of the covariance of the pre-whitened non-linear power spectrum provide a set of almost uncorrelated non-linear modes somewhat analogous to the Fourier modes of the power spectrum itself in the linear, Gaussian regime.These almost uncorrelated modes make it possible to construct a nearminimum variance estimator and Fisher matrix of the pre-whitened non-linear power spectrum analogous to the Feldman±Kaiser±Peacock (FKP) estimator of the linear power spectrum.The paper concludes with summary recipes, in gourmet, fine and fastfood versions, of how to measure the pre-whitened non-linear power spectrum from a galaxy survey in the FKP approximation.An appendix presents FFTLog, a code for taking the fast Fourier or Hankel transform of a periodic sequence of logarithmically spaced points, which proves useful in some of the manipulations.