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.

Abstract. We derive a lower bound on the field excursion for the tachyon inflation, which is determined by the amplitude of the scalar perturbation and the number of e-folds before the end of inflation. Using the relation between the observables like n s and r with the slow-roll parameters, we reconstruct three classes of tachyon potentials. The model parameters are determined from the observations before the potentials are reconstructed, and the observations prefer the concave potential. We also discuss the constraints from the reheating phase preceding the radiation domination for the three classes of models by assuming the equation of state parameter w re during reheating is a constant. Depending on the model parameters and the value of w re , the constraints on N re and T re are different. As n s increases, the allowed reheating epoch becomes longer for w re = −1/3, 0 and 1/6 while the allowed reheating epoch becomes shorter for w re = 2/3.

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