The integrated perturbation theory for cosmological tensor fields I: Basic formulation

In order to extract maximal information about cosmology from the large-scale structure of the Universe, one needs to use every bit of signals that can be observed. Beyond the spatial distributions of astronomical objects, the spatial correlations of tensor fields, such as galaxy spins and shapes, are ones of promising sources that one can access in the era of large surveys in near future. The perturbation theory is a powerful tool to analytically describe the behaviors and evolutions of correlation statistics on large scales for a given cosmology. In this paper, we formulate a nonlinear perturbation theory of tensor fields in general, based on the formulation of integrated perturbation theory for the scalar-valued bias, generalizing it to include the tensor-valued bias. To take advantage of rotational symmetry, the formalism is constructed on the basis of irreducible decomposition of tensors, identifying physical variables which are invariant under the rotation of the coordinates system.