The full-sky relativistic correlation function and power spectrum of galaxy number counts: I. Theoretical aspects

TL;DR

This paper derives an exact, all-scale relativistic correlation function for galaxy counts, including lensing and other effects, and discusses its impact on power spectrum measurements at different redshifts.

Contribution

It provides a novel exact expression for the relativistic galaxy correlation function without approximations, applicable at all scales.

Findings

Gravitational lensing alters correlation multipoles by up to 30% at z=2.

Relativistic effects and wide-angle corrections are of similar magnitude and must be combined.

Flat-sky approximation is inadequate for relativistic galaxy analyses.

Abstract

We derive an exact expression for the correlation function in redshift shells including all the relativistic contributions. This expression, which does not rely on the distant-observer or flat-sky approximation, is valid at all scales and includes both local relativistic corrections and integrated contributions, like gravitational lensing. We present two methods to calculate this correlation function, one which makes use of the angular power spectrum C_ell(z1,z2) and a second method which evades the costly calculations of the angular power spectra. The correlation function is then used to define the power spectrum as its Fourier transform. In this work theoretical aspects of this procedure are presented, together with quantitative examples. In particular, we show that gravitational lensing modifies the multipoles of the correlation function and of the power spectrum by a few percent at redshift z=1 and by up to 30% and more at z=2. We also point out that large-scale relativistic effects and wide-angle corrections generate contributions of the same order of magnitude and have consequently to be treated in conjunction. These corrections are particularly important at small redshift, z=0.1, where they can reach 10%. This means in particular that a flat-sky treatment of relativistic effects, using for example the power spectrum, is not consistent.