A (semi)-exact Hamiltonian for the curvature perturbation ζ

The total Hamiltonian in general relativity, which involves the first class and momentum constraints, weakly vanishes. However, when action is expanded around a classical solution as case of single scalar field inflationary model, there appears non-vanishing additional constraints; but this time theory becomes perturbative number fluctuation fields. We show that one can reorganize expansion solve constraint exactly, yield an explicit all order action. On other hand, be solved perturbatively tensor modes $\gamma_{ij}$ by still keeping curvature perturbation $\zeta$ dependence exact. In way, after gauge fixing, obtain semi-exact for only gets corrections from interactions with (hence exact perturbations set to zero). equations motion clearly exhibit evolution logarithmic dependence, subtle point has been debated literature. discuss long wavelength late limits, some simple non-trivial solutions zero-mode.