Proper Fourier decomposition formalism for cosmological fields in spherical shells

Abstract: Cosmological random fields are often analysed in spherical Fourier-Bessel basis. Compared to the Cartesian Fourier basis this has an advantage of properly taking into account some of the relevant physical processes (redshift-space distortions, redshift evolution). The observations usually come in redshift slices and have a partial sky coverage. These masking effects strongly correlate Fourier-Bessel modes that are meant for a perfect spherical geometry and result in a lot of redundant measurements. This work proposes a new Fourier basis that is better suited for measurements in redshift shells and results in fewer Fourier modes, with the radial modes strictly uncorrelated on large scales and the angular modes with significantly reduced redundancy. I argue that the spherical Fourier analysis of cosmological fields should always use these new modes instead of the historically established Fourier-Bessel eigenfunctions. The new angular modes on the other hand have number of practical advantages and disadvantages and whether or not to adopt them for a particular analysis should be made on a case by case basis.