The Bias of the Log Power Spectrum for Discrete Surveys

A primary goal of galaxy surveys is to tighten constraints on cosmological parameters, and the power spectrum $P(k)$ is the standard means of doing so. However, at translinear scales $P(k)$ is blind to much of these surveys' information---information which the log density power spectrum recovers. For discrete fields (such as the galaxy density), $A^*$ denotes the statistic analogous to the log density: $A^*$ is a "sufficient statistic" in that its power spectrum (and mean) capture virtually all of a discrete survey's information. However, the power spectrum of $A^*$ is biased with respect to the corresponding log spectrum for continuous fields, and to use $P_{A^*}(k)$ to constrain the values of cosmological parameters, we require some means of predicting this bias. Here we present a prescription for doing so; for Euclid-like surveys (with cubical cells 16$h^{-1}$ Mpc across) our bias prescription's error is less than 3 per cent. This prediction will facilitate optimal utilization of the information in future galaxy surveys.