Information content in anisotropic cosmological fields: Impact of different multipole expansion scheme for galaxy density and ellipticity correlations

Multipole expansions have been often used for extracting cosmological information from anisotropic quantities in observation. However, which basis of the expansion is best suited to quantify the anisotropies is not a trivial question in any summary statistics. In this paper, using the Fisher matrix formalism, we investigate the information content in multipole moments of the power spectra of galaxy density and intrinsic ellipticity fields in linear theory from the Alcock-Paczynski effect and redshift-space distortions (RSD). We consider two expansion schemes, the associated Legendre basis as well as the standard Legendre basis conventionally used in literature. We find that the information in the multipoles of the intrinsic alignment (IA) power spectra in the associated Legendre basis converges more slowly to that in the full 2D power spectra than in the Legendre basis. This trend is particularly significant when we consider a high number density sample. In this high number density case, we show that the errors on the Hubble parameter obtained from the multipoles of the IA cross- and autopower spectra in the associated Legendre basis are respectively about 6% and 10% larger than the full 2D case even when we use multipoles up to l 1/4 6. Our results demonstrate that the choice of basis is arbitrary but changes the information content encoded in multipoles depending on the sample and statistics under consideration.