Primordial black holes and secondary gravitational waves from k/G
  inflation
  The possibility that in the mass range around $10^{-12}\ M_\odot$ most of
dark matter constitutes 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 $k\sim 10^{12}\
\text{Mpc}^{-1}$. 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 G and k inflations, we propose a new mechanism to
enhance the power spectrum at small scales by introducing a non-canonical
kinetic term $[1-2G(\phi)]X$ with the function $G(\phi)$ having a peak. Away
from the peak, $G(\phi)$ is negligible and we recover the usual slow-roll
inflation which is constrained by the cosmic microwave background anisotrpy
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 $G(\phi)$. The energy spectrum $\Omega_{GW}(f)$ of secondary
gravitational waves have the characteristic power law behaviour
$\Omega_{GW}(f)\sim f^{n}$ and is testable by pulsar timing array and space
based gravitational wave detectors.
