  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.
