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

摘要

We derive an exact expression for the correlation function in redshift shellsincluding all the relativistic contributions. This expression, which does notrely on the distant-observer or flat-sky approximation, is valid at all scalesand includes both local relativistic corrections and integrated contributions,like gravitational lensing. We present two methods to calculate thiscorrelation function, one which makes use of the angular power spectrumC_ell(z1,z2) and a second method which evades the costly calculations of theangular power spectra. The correlation function is then used to define thepower spectrum as its Fourier transform. In this work theoretical aspects ofthis procedure are presented, together with quantitative examples. Inparticular, we show that gravitational lensing modifies the multipoles of thecorrelation function and of the power spectrum by a few percents at redshiftz=1 and by up to 30% at z=2. We also point out that large-scale relativisticeffects and wide-angle corrections generate contributions of the same order ofmagnitude and have consequently to be treated in conjunction. These correctionsare 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.