Statistics of a single sky: constrained random fields and the imprint of Bardeen potentials on galaxy clustering

Abstract

We explore the implications of a single observer's viewpoint on measurements of galaxy clustering statistics. We focus on the Bardeen potentials, which imprint characteristic scale-dependent signatures in the observed galaxy power spectrum. The existence of an observer breaks homogeneity as it singles out particular field values at her/his position, like a spontaneous symmetry breaking. As a result, spatial averaging of the data must be performed while holding the Bardeen potentials fixed at the observer's position. In practice, this can be implemented with the formalism of constrained random fields. In the traditional Cartesian Fourier decomposition, this constraint imprints a signature in the observed galaxy power spectrum at wavenumber comparable to the fundamental mode of the survey. This effect, which is well within the cosmic variance, is the same for all observers regardless of their local environment because differences of potential solely are observable. In a spherical Bessel Fourier decomposition, this constraint affects the monopole of the observed galaxy distribution solely, like in CMB data. As a corollary, the scale-dependence of the non-Gaussian bias induced by a local primordial non-Gaussianity is not significantly affected by the observer's viewpoint.