  A new approach to cosmological perturbation theory has been recently
introduced by Bartelmann et al., relying on nonequilibrium statistical theory
of classical particles, and treating the gravitational interaction
perturbatively. They compute analytic expressions for the nonlinear matter
power spectrum, to first order in the interaction, and at one-loop order in the
linear power spectrum. The resulting power spectrum is well behaved even at
large wavenumbers and seems in good agreement with results from numerical
simulations. In this paper, we rederive their results concisely with a
different approach, starting from the implicit integral solution to particle
trajectories. We derive the matter power spectrum to first order in the
interaction, but to arbitrary order in the linear power spectrum, from which
the one-loop result follows. We also show that standard linear perturbation
theory can only be recovered at infinite order in the gravitational
interaction. At finite order in the interaction, we find that the linear power
spectrum is systematically and significantly underestimated. A comprehensive
study of the convergence of the theory with the order of the interaction for
nonlinear scales will be the subject of future work.
