The full-sky Spherical Fourier-Bessel power spectrum in general relativity

We present a formalism for analyzing galaxy clustering on the lightcone with the 2-point correlation in the Spherical Fourier-Bessel (SFB) formalism, which is a natural choice to account for all wide-angle and relativistic (GR) effects. We extend previous studies by including all projection and GR effects, developing an efficient numerical implementation that avoids the use of the Limber approximation, includes multi-bins correlations and a full non-diagonal covariance. Using this formalism, we investigate the impact of neglecting GR corrections, and in particular how much this could bias measurements of the non-Gaussianity parameter $f_\mathrm{NL}$. Our results show that not including relativistic projection terms can systematically and non-negligibly bias estimates of $f_\mathrm{NL}$. The exact results depend on survey specifications and galaxy population properties, but we stress that a bias will generally be present. Finally, we develop a novel prescription for cross-bin correlations that allow to search for a clean signal of relativistic corrections, and show that this requires the use of the 3D full-sky formalism.