Primordial black holes and secondary gravitational waves from $k$ and $G$ inflation

The possibility that in the mass range around 10-12  M⊙ most of dark matter is constituted of primordial black holes (PBHs) is a very interesting topic. To produce PBHs with this mass, the primordial scalar power spectrum needs to be enhanced to the order of 0.01 at the scale k∼1012  Mpc-1. The enhanced power spectrum also produces large secondary gravitational waves at the mHz band. A phenomenological delta function power spectrum is usually used to discuss the production of PBHs and secondary gravitational waves. Based on k and G inflations, we propose a new mechanism to enhance the power spectrum at small scales by introducing a noncanonical kinetic term [1-2G(ϕ)]X with the function −G(ϕ) having a peak. Away from the peak, G(ϕ) is negligible and we recover the usual slow-roll inflation which is constrained by the cosmic microwave background anisotropy observations. Around the peak, the slow-roll inflation transiently turns to ultra slow-roll inflation. The enhancement of the power spectrum can be obtained with generic potentials, and there is no need to fine tune the parameters in G(ϕ) to several significant digits. The energy spectrum ΩGW(f) of secondary gravitational waves produced by the model have the characteristic power law behavior ΩGW(f)∼fn and is testable by pulsar timing array and space based gravitational wave detectors.