Power spectrum in the cave

Forthcoming galaxy surveys will provide measurements of galaxy clustering with an unprecedented level of precision, that will require comparably good accuracy. Current models for galaxy correlations rely on approximations and idealizations that might be inadequate for ultra precise measurements. On the other hand, exact calculations have proven to be computationally too expensive to be efficiently implemented in real data analyses. We start a project to provide precise and accurate formalisms for galaxy correlations, and in this paper we investigate the 3D angular power spectrum including effects of unequal time correlations. We establish an explicit link between the full- and flat-sky spectra by performing an asymptotic expansion of the full-sky result around the equal time case. The limiting case coincides with the idealized spectrum that a meta-observer would measure if it had access to the entire 4D Universe. The leading term in the obtained flat-sky expansion is the only translationally invariant term in the plane perpendicular to the line of sight, while the higher-order terms account for the deviation from this invariance. We study the behavior of such corrections for a simplified universe where we can analytically solve the power spectrum and have full control of the equations, therefore being able to understand the exact nature of all the terms and the origin of the corrections. We highlight that the conclusions and the structure of the unequal time spectra are fully general and serve as lessons and guidance in understanding galaxy clustering in any cosmology. Finally, we show that our flat-sky unequal time expression matches the exact full-sky calculation remarkably better than commonly adopted approximations, even at the largest scales and for both shallow and deep redshift bins.