Cosmological Fluctuations in Standard and Conformal Gravity

In the theory of cosmological perturbations, extensive methods of simplifying the equations of motion and eliminating non-physical gauge modes are required in order to construct the perturbative solutions. One approach, standard within modern cosmology, is to decompose the metric perturbation into a basis of scalars, vectors, and tensors defined according to their transformation behavior under three-dimensional rotations (the S.V.T. decomposition). By constructing a projector formalism to define the basis components, we show that such a decomposition is intrinsically non-local and necessarily incorporates spatially asymptotic boundary conditions. We continue application of the S.V.T. decomposition and solve the fluctuation equations exactly within standard cosmologies as applied to both Einstein gravity and conformal gravity, finding that in general the various S.V.T. gauge-invariant combinations only decouple at a higher-derivative level. To match the underlying transformation group of General Relativity and thus provide a manifestly covariant formalism, we introduce an alternate scalar, vector, tensor basis with components defined according to general four-dimensional coordinate transformations. In this basis, the fluctuation equations greatly simplify, where one can again decouple them into separate gauge-invariant sectors at the higher-derivative level. In the context of conformal gravity, we use similar constructions to solve the fluctuation equations exactly within any geometry that is conformal to flat and show that in a radiation era Robertson-Walker cosmology, fluctuations grow as $t^4$