The primordial black hole (PBH) formation is studied in light of the inflating curvaton. The typical scale of the PBH formation is determined by curvaton inflation, which may generate PBH with 10 14 g ≤ MPBH ≤ 10 38 g when curvaton inflation gives the number of e-foldings 5 ≤ N2 ≤ 38. The non-Gaussianity of the inflating curvaton does not prevent the PBH formation.

Recent arguments show that some curvaton field may generate the cosmological curvature perturbation. As the curvaton is independent of the inflaton field, there is a hope that the fine-tunings of inflation models can be cured by the curvaton scenario. More recently, however, D.H.Lyth discussed that there is a strong bound for the Hubble parameter during inflation even if one assumes the curvaton scenario. Although the most serious constraint was evaded, the bound seems rather crucial for many models of a low inflation scale. In this paper we try to remove the constraint. We show that the bound is drastically modified if there were multiple stages of inflation.

There are many inflationary models in which inflaton field does not satisfy the slow-roll condition. However, in such models, it is always difficult to generate the curvature perturbation during inflation. Thus, to generate the curvature perturbation, one must introduce another component to the theory. To cite a case, curvatons may generate dominant part of the curvature perturbation after inflation.However, we have a question whether it is unrealistic to consider the generation of the curvature perturbation during inflation without slow-roll. Assuming multi-field inflation, we encounter the generation of the curvature perturbation during inflation

The primordial curvature perturbation ζ may be generated by some curvaton field σ, which is negligible during inflation and has more or less negligible interactions until it decays. In the current scenario, the curvaton starts to oscillate while its energy density ρσ is negligible. We explore the opposite scenario, in which ρσ drives a few e-folds of inflation before the oscillation begins. In this scenario for generating ζ it is exceptionally easy to solve the η problem; one just has to make the curvaton a string axion, with anomaly-mediated susy breaking which may soon be tested at the LHC. The observed spectral index n can be obtained with a potential V ∝ φ p for the first inflation; p = 1 or 2 is allowed by the current uncertainty in n but the improvement in accuracy promised by Planck may rule out p = 1. The predictions include (i) running n ′ ≃ 0.0026 (0.0013) for p = 1 (2) that will probably be observed, (ii) non-gaussianity parameter fNL ∼ −1 that may be observed, (iii) tensor fraction r is probably too small to ever observed.