Consistency conditions and primordial black holes in single field inflation

Abstract

We discuss new consistency relations for single field models of inflation capable of generating primordial black holes (PBH), and their observational implications for CMB $\ensuremath{\mu}$-space distortions. These inflationary models include a short period of nonattractor evolution; the scale-dependent profile of curvature perturbation is characterized by a pronounced dip, followed by a rapid growth leading to a peak responsible for PBH formation. We investigate the squeezed and the collapsed limits of three- and four-point functions of curvature perturbation around the dip, showing that they satisfy consistency relations connecting their values to the total amplification of the curvature spectrum, and to the duration of the nonattractor era. Moreover, the corresponding non-Gaussian parameters are scale dependent in proximity of the dip, with features that again depend on the amplification of the spectrum. For typical PBH scenarios requiring an order $\mathcal{O}({10}^{7})$ enhancement of the spectrum from large towards small scales, we generally find values ${f}_{\mathrm{NL}}^{\mathrm{sq}}=\mathcal{O}(10)$ and ${\ensuremath{\tau}}_{\mathrm{NL}}^{\mathrm{col}}=\mathcal{O}({10}^{3})$ in a range of scales that can be probed by CMB $\ensuremath{\mu}$-space distortions. Using these consistency relations, we carefully analyze how the scale-dependence of non-Gaussian parameters leads to characteristic features in $⟨\ensuremath{\mu}T⟩$ and $⟨\ensuremath{\mu}\ensuremath{\mu}⟩$ correlators, providing distinctive probes of inflationary PBH scenarios that can be tested using well-understood CMB physics.