Proposed space-based gravitational-wave (GW) detectors such as DECIGO and BBO will detect -10[6] neutron-star (NS) binaries and determine the luminosity distances to the binaries with high precision. Combining the luminosity distances with cosmologically induced phase corrections on the GWs, cosmological expansion out to high redshift can be measured without the redshift determinations of host galaxies by electromagnetic observation and be a unique probe for dark energy. On the other hand, such a NS-binary foreground should be subtracted to detect primordial GWs produced during inflation. Thus, the constraining power on dark energy and the detectability of the primordial gravitational waves strongly depend on the detector sensitivity and are in close relation with one another. In this paper, we investigate the constraints on the equation of state of dark energy with future space-based GW detectors with/without identifying the redshifts of host galaxies. We also study the sensitivity to the primordial GWs, properly dealing with the residual of the NS-binary foreground. Based on the results, we discuss the detector sensitivity required to achieve the forementioned targeted study of cosmology.
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