Growth of curvature perturbations for PBH formation \& detectable GWs in non-minimal curvaton scenario revisited

We revisit the growth of curvature perturbations in non-minimal curvaton scenario with a non-trivial field metric $\lambda(\phi)$ where $\phi$ is an inflaton field, and incorporate the effect from the non-uniform onset of curvaton's oscillation in terms of an axion-like potential. The field metric $\lambda(\phi)$ plays a central role in the enhancement of curvaton field perturbation $\delta\chi$, serving as an effective friction term which can be either positive or negative, depending on the first derivative $\lambda_{,\phi}$.Our analysis reveals that $\delta\chi$ undergoes the superhorizon growth when the condition $\eta_\text{eff} \equiv - 2 \sqrt{2\epsilon} M_\text{Pl} { \lambda_{,\phi} \over \lambda} < -3$ is satisfied. This is analogous to the mechanism responsible for the amplification of curvature perturbations in the context of ultra-slow-roll inflation, namely the growing modes dominate curvature perturbations. As a case study, we examine the impact of a Gaussian dip in $\lambda(\phi)$ and conduct a thorough investigation of both the analytical and numerical aspects of the inflationary dynamics.Our findings indicate that the enhancement of curvaton perturbations during inflation is not solely determined by the depth of the dip in $\lambda(\phi)$. Rather, the first derivative $\lambda_{,\phi}$ also plays a significant role, a feature that has not been previously highlighted in the literature. Utilizing the $\delta \mathcal{N}$ formalism, we derive analytical expressions for both the final curvature power spectrum and the non-linear parameter $f_\text{NL}$ in terms of an axion-like curvaton's potential leading to the non-uniform curvaton's oscillation. Additionally, the resulting primordial black hole abundance and scalar-induced gravitational waves are calculated, which provide observational windows for PBHs.