We investigate the evolution of supermassive binary black holes (BBHs) in galaxies with realistic property distributions and the gravitational-wave (GW) radiation from the cosmic population of these BBHs. We incorporate a comprehensive treatment of the dynamical interactions of the BBHs with their environments by including the effects of galaxy triaxial shapes and inner stellar distributions, and generate a large number of BBH evolution tracks. By combining these BBH evolution tracks, galaxy mass functions, galaxy merger rates, and supermassive black hole-host galaxy relations into our model, we obtain the statistical distributions of surviving BBHs, BBH coalescence rates, the strength of their GW radiation, and the stochastic GW background (GWB) contributed by the cosmic BBH population. About ~1%-3% (or ~10%) of supermassive BHs at nearby galactic centers are expected to be binaries with mass ratio >1/3 (or >1/100). The characteristic strain amplitude of the GWB at frequency 1/yr is estimated to be ~$2.0^{+1.4}_{-0.8}times 10^{-16}$, and the upper bound of its results obtained with the different BH-host galaxy relations can be up to $5.4times 10^{-16}$, which await testing by future experiments (e.g., the Square Kilometer Array, FAST, Next-Generation Very Large Array). The turnover frequency of the GWB spectrum is at ~0.25nHz. The uncertainties on the above estimates and prospects for detecting individual sources are also discussed. The application of the cosmic BBH population to the Laser Interferometer Space Antenna (LISA) band provides a lower limit to the detection rate of BBHs by LISA, ~0.9/yr.
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