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Trapping effects in inflation: blue spectrum at small scales

We consider the inflationary model in which the inflaton $ϕ$ couples to another scalar field $χ$ via the interaction $g^2(ϕ-ϕ_0)^2χ^2$ with a small coupling constant $g$ ($g^2 \sim 10^{-7}$). We assume that there is a sequence of "trapping points" $ϕ_{0i}$ along the inflationary trajectory where particles of $χ$-field become massless and are rather effectively produced. We calculate the power spectrum of inflaton field fluctuations originated from a backreaction of $χ$-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.