Integrated perturbation theory for cosmological tensor fields. IV. Full-sky formulation

In Papers I--III [T. Matsubara, Phys. Rev. D 110, 063543 (2024); 110, 063544 (2024); 110, 063545 (2024)], we use the flat-sky and distant-observer approximations to develop a formalism with which the correlation statistics of cosmological tensor fields are calculated by the nonlinear perturbation theory, generalizing the integrated perturbation theory for scalar fields. In this work, the formalism is extended to include the full-sky and wide-angle effects in evaluating the power spectra and correlation functions of cosmological tensor fields of any rank. With the newly developed formalism, one can evaluate the nonlinear power spectra and correlation functions to arbitrary higher orders in principle. After describing the general formalism, we explicitly derive and give analytic results of the lowest-order linear theory for an illustrative purpose in this paper. The derived linear formulas with full-sky and wide-angle effects are numerically compared with the previous formulas with flat-sky and distant-observer limits in a simple model of tensor bias.