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 -2 G (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 Ω_GW(f ) of secondary gravitational waves produced by the model have the characteristic power law behavior Ω_GW(f )∼fⁿ and is testable by pulsar timing array and space based gravitational wave detectors.