The observer power spectrum for lightcone statistics, integrated relativistic observables and wide angle effects

Abstract

The statistics of large-scale structure are naturally described by power spectra in Fourier space. For fields on spatial hypersurfaces, translational invariance makes different Fourier modes uncorrelated and the power spectrum diagonal. Cosmological observables, however, are measured on our past lightcone, where wide-angle effects, radial evolution and integrated effects such as lensing break this symmetry: Fourier-space statistics become non-diagonal, with mode-mixing generated by the geometry of the lightcone itself. We define a more natural observer power spectrum by Fourier transforming over observer positions on a spatial hypersurface with fixed lightcone coordinates, rather than over source positions on a single lightcone. This forms a field on the observer hypersurface with a moveable light-ray leg. Statistical homogeneity of the observer hypersurface implies that this spectrum is diagonal for any observable, whether local or integrated and does not suffer from mode-mixing. We show how the various two-point statistics used in large-scale structure analysis are each recovered as projections of the observer spectrum. This extends directly to higher-order statistics. We illustrate it by constructing the relativistic kernel for the observed galaxy number count fluctuation, including density, redshift-space distortions, Doppler, lensing magnification, and integrated Sachs-Wolfe contributions.