Observed power spectrum and frequency-angular power spectrum

The two-point summary statistics is one of the most commonly used tools in the study of cosmological structure. Starting from the theoretical power spectrum defined in the 3D volume and obtained via the process of ensemble averaging, we establish the construction of the “observed 3D power spectrum,” folding the unequal-time information around the average position into the wave modes along the line of sight. We show how these unequal-time cross-correlation effects give rise to scale-dependent corrections in the observable 3D power spectrum. We also introduce a new dimensionless observable, the “frequency-angular power spectrum,” which is a function of dimensionless and directly observable quantities corresponding to Fourier counterparts of angles and redshifts. While inheriting many useful characteristics of the canonical observed power spectrum, this newly introduced statistic does not depend on physical distances and is hence free of so-called Alcock-Paczyński effects. Such observable thus presents a clear advantage and simplification over the traditional power spectrum. Moreover, relying on linear theory calculations, we estimate that unequal-time corrections, while generally small, can amount to a few percent on large scales and high redshifts. Interestingly, such corrections depend on the bias of the tracers and the growth rate, but also their time derivatives, opening up the possibility of new tests of cosmological models. These radial mode effects also introduce anisotropies in the observed power spectrum, in addition to the ones arising from redshift-space distortions, generating nonvanishing odd multiples and imaginary contributions. Finally, we investigate the effects of unequal-time corrections in resumming long displacements (IR resummation) of the observed power spectrum.