Integrated perturbation theory for cosmological tensor fields. III. Projection effects

The integrated perturbation theory (iPT) is a set of methods in nonlinear perturbation theory for the structure formation in the Universe. In Papers I and II T. Matsubara, Phys. Rev. D 110, 063543 (2024); 110, 063544 (2024)., the basic formalism and technical methods of the iPT for cosmological tensor fields are developed, generalizing the corresponding theory for scalar fields. In previous papers, methods to predict statistical quantities, such as power spectra, correlation functions, etc., of three-dimensional tensor fields are developed based on the iPT. However, observations of tensors, such as angular momenta and shapes of galaxies, etc., are only possible after the three-dimensional tensors are projected onto the two-dimensional sky. In this paper, power spectra and correlation functions of projected two-dimensional tensors are related to those of original three-dimensional tensors, so that one can make predictions for the observable statistics of projected tensor fields from the iPT. The relations are consistently represented on the basis of irreducible decomposition of both two- and three-dimensional tensors.