The Density Perturbation in the Chaotic Inflation with Non-Minimal Coupling

We formulate a method to rigorously evaluate the density perturbation amplitude in inflation models with non-canonical gravity. As a specific example but which contains essential features of non-canonical models, the density perturbation in the λφ^4 chaotic inflation model with non-minimal coupling ξRφ^2 is carefully investigated in terms of the Hamiltonian formalism for constrained systems. We utilize a particular conformal transformation under which the original system with a non-minimally coupled scalar field is transformed into one with a minimal coupling and derive a reduced Hamiltonian for scalar-type perturbations (which describe density perturbations) in which only the dynamical degrees of freedom are contained. We then evaluate the amplitude of density perturbations arising from vacuum fluctuations of the system. We find the amplitude in the conformally transformed metric exactly agrees with that in the original metric. Thus the result obtained in the previous papers by a rather crude approach is confirmed in a more rigorous manner; i.e., the constraint on λ is weakened if ξ is large.