We propose that this manifold is not simply topologically trivial (i.e., a 3-sphere), but may exhibit nontrivial boundary conditions or topological identifications (e.g., 3-torus, Poincar dodecahedral space) that discretize the allowable vibrational modes of fields embedded within it.
These topological boundary conditions, , effectively quantize the allowed spatial modes of scalar, vector, and tensor fields in the primordial universe. This quantization underlies the standing wave structure that seeds cosmic structure.
3.2 Wave Equation in Curved Spacetime
Consider a real scalar field propagating on a curved background spacetime. The field evolution is governed by the covariant Klein--Gordon equation:
\Box_g \Phi + V'(\Phi) = 0
where:
is the d'Alembertian operator on the curved manifold ,
is a potential term (possibly zero in the free-field case),
-
is interpreted as a metric fluctuation mode, inflaton field, or even metric-coupled informational wave.
For a compact spatial manifold with stationary boundary conditions, the solution may be expanded in eigenmodes:
\Phi(x) = \sum_n A_n \psi_n(x)
