  The cosmological evolution of the binary black hole (BH) merger rate and the
energy density of the gravitational-wave (GW) background are investigated. To
evaluate the redshift dependence of the BH formation rate, BHs are assumed to
originate from low-metallicity stars, and the relations between the star
formation rate, metallicity and stellar mass of galaxies are combined with the
stellar mass function at each redshift. As a result, it is found that when the
energy density of the GW background is scaled with the merger rate at the local
Universe, the scaling factor does not depend on the critical metallicity for
the formation of BHs. Also taking into account the merger of binary neutron
stars, a simple formula to express the energy spectrum of the GW background is
constructed for the inspiral phase. The relation between the local merger rate
and the energy density of the GW background will be examined by future GW
observations.
