Primordial black holes and secondary gravitational waves can be used to probe the small scale physics at very early time. For secondary gravitational waves produced after the horizon reentry, we derive an analytical formula for the time integral of the source and analytical behavior of the time dependence of the energy density of induced gravitational waves is obtained. By proposing a piecewise power law parametrization for the power spectrum of primordial curvature perturbations, we use the observational constraints on primordial black hole dark matter to obtain an upper bound on the power spectrum, and discuss the test of the model with future space based gravitational wave antenna.

The possibility that in the mass range around 10 −12 M 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 ∼ 10 12 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(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). The energy spectrum ΩGW (f ) of secondary gravitational waves have the characteristic power law behaviour ΩGW (f ) ∼ f n and is testable by pulsar timing array and space based gravitational wave detectors.

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