Generating large primordial fluctuations in single-field inflation for PBH formation

In order to produce appreciable amount of primordial black holes (PBHs), the square amplitude of curvature perturbation must take a large value of $\mathcal{O}(0.01)$, namely, seven digits larger than the value observed by cosmic microwave background radiation (CMB) on large scales. Such a large fluctuation can be achieved by violating the slow-roll (SR) condition within a short duration. The best known of such possibilities is the ultraslow-roll (USR) inflation. We calculate the power spectrum of curvature perturbation in a simple single-field inflation model which evolves through the SR-USR-SR regimes so that both large-amplitude small-scale fluctuation for PBH formation and small-amplitude large-scale fluctuation as observed by CMB are realized. We further calculate the bispectrum and one-loop correction to the power spectrum induced by the third-order action of curvature perturbation as the beginning of precision cosmology on small scales. As a result, we show that single-field inflation model realizing PBH formation can be constrained by the quantum correction.