Primordial black holes and secondary gravitational waves from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>k</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>G</mml:mi></mml:math> inflation

Abstract

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

Topics & Concepts