Trapping effects in inflation: blue spectrum at small scales

We consider the inflationary model in which the inflaton $\phi$ couples to another scalar field $\chi$ via the interaction $g^2(\phi-\phi_0)^2\chi^2$ with a small coupling constant $g$ ($g^2 \sim 10^{-7}$). We assume that there is a sequence of "trapping points" $\phi_{0i}$ along the inflationary trajectory where particles of $\chi$-field become massless and are rather effectively produced. We calculate the power spectrum of inflaton field fluctuations originated from a backreaction of $\chi$-particles produced, using the Schwinger's "in-in" formalism. We show that the primary curvature power spectrum produced by these backreaction effects is blue, which leads to a strong overproduction of primordial black holes (PBHs) in subsequent radiation era.
Comment: v2: 14 pages, 5 figures, version accepted by Phys. Rev. D