  Evolution of the rate density of cosmological gamma-ray bursts (GRBs) is
calculated and compared to the BATSE brightness distribution in the context of
binary neutron-star mergers as the source of GRBs, taking account of the
realistic star formation history in the universe and evolution of compact
binary systems. We tried two models of the evolution of cosmic star formation
rate (SFR): one is based on recent observations of SFRs at high redshifts,
while the other is based on a galaxy evolution model of stellar population
synthesis that reproduces the present-day colors of galaxies. It is shown that
the binary merger scenario of GRBs naturally results in the comoving
rate-density evolution of \propto (1+z)^{2-2.5} up to z ~ 1, that has been
suggested independently from the compatibility between the number-brightness
distribution and duration-brightness correlation. If the cosmic SFR has its
peak at z ~ 1--2 as suggested by recent observations, the effective power-index
of GRB photon spectrum, \alpha >~ 1.5$ is favored, that is softer than the
recent observational determination of \alpha = 1.1 \pm 0.3. However, high
redshift starbursts (z >~ 5) in elliptical galaxies, that have not yet been
detected, can alleviate this discrepancy. The redshift of GRB970508 is likely
about 2, just below the upper limit that is recently determined, and the
absorption system at z = 0.835 seems not to be the site of the GRB.
