Testing primordial black hole and measuring the Hubble constant with
  multiband gravitational-wave observations
  Primordial curvature perturbations with a large enough amplitude on small
scales can lead to two kinds of gravitational waves which are expected to be
detected by multiband gravitational-wave observations. One is induced due to
the nonlinear coupling of the curvature perturbation to tensor perturbation
while the other is produced by coalescences of binary primordial black holes
formed when the scalar perturbations reenter the horizon in the radiation
dominant era. In this letter, we identify the relation of peak frequency for
the spectra of such two stochastic gravitational-wave backgrounds. This peak
frequency relation offers a new criterion for the existence of primordial black
holes. Moreover, the relation provides a new method for measuring the Hubble
constant $H_0$ through multiband observations of stochastic gravitational-wave
backgrounds. Such a method does not need the redshift information which is
necessary in the standard siren method.
