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Primordial black holes and secondary gravitational waves can be used to probethe small scale physics at very early time. For secondary gravitational wavesproduced after the horizon reentry, we derive an analytical formula for thetime integral of the source and analytical behavior of the time dependence ofthe energy density of induced gravitational waves is obtained. By proposing apiecewise power law parametrization for the power spectrum of primordialcurvature perturbations, we use the observational constraints on primordialblack hole dark matter to obtain an upper bound on the power spectrum, anddiscuss the test of the model with future space based gravitational waveantenna.

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