Constraints on primordial curvature perturbations from primordial black hole dark matter and secondary gravitational waves

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, and fitting it to observational constraints on primordial black hole dark matter, we obtain an upper bound on the power spectrum and discuss the detection of induced gravitational waves by future space based gravitational wave antenna.