The possibility that in the mass range around ${10}^{-12}{M}_{\odot }$ 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 $k\sim {10}^{12}{\mathrm{Mpc}}^{-1}$. 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 $[1-2G(\varphi )]X$ with the function $-G(\varphi )$ having a peak. Away from the peak, $G(\varphi )$ 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 $G(\varphi )$ to several significant digits. The energy spectrum ${\mathrm{\Omega }}_{\mathrm{GW}}(f)$ of secondary gravitational waves produced by the model have the characteristic power law behavior ${\mathrm{\Omega }}_{\mathrm{GW}}(f)\sim f^n$ and is testable by pulsar timing array and space based gravitational wave detectors.

With the enhancement mechanism provided by a noncanonical kinetic term with a peak, the amplitude of primordial curvature perturbations can be enhanced by seven orders of magnitude at small scales while keeping to be consistent with observations at large scales. The peak function and inflationary potential are not restricted in this mechanism. We use the Higgs model and T-model as examples to show how abundant primordial black hole dark matter with different mass and scalar induced secondary gravitational waves with different peak frequency are generated. We also show that the enhanced power spectrum for the primordial curvature perturbations and the energy density of the scalar induced secondary gravitational waves can have either a sharp peak or a broad peak. The primordial black holes with the mass around ${10}^{-14}-{10}^{-12}{M}_{\odot }$ produced with the enhancement mechanism can make up almost all dark matter, and the scalar induced secondary gravitational waves accompanied with the production of primordial black holes can be tested by the pulsar timing arrays and spaced based gravitational wave observatory. Therefore, the mechanism can be tested by primordial black hole dark matter and gravitational wave observations.

We derive the general formulae for the scalar and tensor spectral tilts to the second order for the inflationary models with non-minimally derivative coupling without taking the high friction limit. The non-minimally kinetic coupling to Einstein tensor brings the energy scale in the inflationary models down to be sub-Planckian. In the high friction limit, the Lyth bound is modified with an extra suppression factor, so that the field excursion of the inflaton is sub-Planckian. The inflationary models with non-minimally derivative coupling are more consistent with observations in the high friction limit. In particular, with the help of the non-minimally derivative coupling, the quartic power law potential is consistent with the observational constraint at 95% CL.