  We propose that the merger rate of primordial black hole (PBH) binaries can
be a probe of Hubble parameter by constraining PBH mass function in the
redshifted mass distribution of PBH binaries. In next-generation gravitational
wave (GW) detectors, the GWs from PBH binaries would be detected at high
redshifts, which gives their redshifted mass and luminosity distances. From a
number of detected events, the redshifted mass distribution of PBH binaries can
be statistically obtained, and it depends on PBH mass function and redshift
distribution of detected PBH binaries. The PBH mass function can be inversely
solved by applying the gradient descent method in the relation between
redshifted mass distribution and redshift distribution. However, the
construction of redshift distribution requires an assumed Hubble parameter in a
background cosmology to extract redshift from luminosity distances, which
causes solved PBH mass function also depends on assumed Hubble parameter. To
determine the Hubble parameter, the merger rate of PBH binaries constrains on
this Hubble parameter-dependent PBH mass function by comparing calculated
merger rate distribution with observed one, and the best-fit result produces an
approximate mass distribution of the physical PBH mass function and pins down
the Hubble parameter.
