  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.
