Trapping effects on inflation

We develop a Lagrangian approach based on the influence functional method so as to derive self-consistently the Langevin equation for the inflaton field in the presence of trapping points along the inflaton trajectory. The Langevin equation exhibits the backreaction and the fluctuation-dissipation relation of the trapping. The fluctuation is induced by a multiplicative colored noise that can be identified as the particle number density fluctuations; and the dissipation is a new effect that may play a role in the trapping with a strong coupling. In the weak coupling regime, we calculate the power spectrum of the noise-driven inflaton fluctuations for a single trapping point and studied its variation with the trapping location. We also consider a case with closely spaced trapping points and find that the resulting power spectrum is blue.