Boundary conditions and proof of the scalar, vector, tensor decomposition theorem in cosmological perturbation theory

Cosmological perturbations play a vital role in the study of the anisotropy of the cosmic microwave background radiation and in large-scale structure formation. The standard procedure for treating these perturbations is to decompose them into scalar (S), vector (V) and tensor (T) components, and assume that the three sectors independently satisfy the fluctuation equations. Ordinarily, this procedure is carried out in a convenient gauge in a background geometry with vanishing spatial 3-curvature. We have carried out a general SVT decomposition in a cosmology with an arbitrary spatial 3-curvature and expansion radius in a completely gauge invariant way that involves no choice of gauge at all. We have been able to prove the decomposition theorem in this general case, by first manipulating the fluctuation equations so that the various sectors separate out at a higher derivative level; and then finding appropriate boundary conditions under which the solving of these equations leads to the form that is required of the decomposition theorem. With these boundary conditions we thus justify the use of the decomposition theorem in the standard Einstein gravity based cosmology. In addition we establish the decomposition theorem in the alternate conformal gravity theory.