Inflation with vector fields revisited: heavy entropy perturbations and primordial black holes

We revisit inflation coupled with vector fields employing kinetic coupling in the comoving gauge. It is known that there is a cumulative effect $IN^2$ on the curvature power spectrum. For a large number of e-foldings $N$, this contribution is so significant that it could violate observational constraints when the ratio of kinetic energy between the vector fields and the inflaton $I$ is not extremely small. In this paper, we explore a regime where $I\gg 1$. This regime has not been extensively explored due to the limitations of perturbative methods. We found that the entropy perturbation becomes heavy in this regime and the cumulative effect decays away on super-horizon scales. Consequently, the power spectrum retains its scale invariance in the decoupling limit. By straightforwardly integrating out the heavy modes near horizon-crossing, we derive a low-energy effective field theory describing a massless adiabatic perturbation with an imaginary speed of sound $c_s^2= -1/3$. Namely, the inflation with vector fields presents a potential mechanism for generating primordial black holes.