Prospects for measuring the Hubble constant and dark energy using gravitational-wave dark sirens with neutron star tidal deformation

Using the measurements of tidal deformation in the binary neutron star (BNS) coalescences can obtain the information of redshifts of gravitational wave (GW) sources, and thus actually the cosmic expansion history can be investigated using solely such GW dark sirens. To do this, the key is to get a large number of accurate GW data, which can be achieved with the third-generation (3G) GW detectors. Here we wish to offer an answer to the question of whether the Hubble constant and the equation of state (EoS) of dark energy can be precisely measured using solely GW dark sirens. We find that in the era of 3G GW detectors ${\cal O}(10^5-10^6)$ dark siren data (with the NS tidal measurements) could be obtained in three-year observation if the EoS of NS is perfectly known, and thus using only dark sirens can actually achieve the precision cosmology. Based on a network of 3G detectors, we obtain the constraint precisions of $0.15\%$ and $0.95\%$ for the Hubble constant $H_0$ and the constant EoS of dark energy $w$, respectively; for a two-parameter EoS parametrization of dark energy, the precision of $w_0$ is $2.04\%$ and the error of $w_a$ is 0.13. We conclude that 3G GW detectors would lead to breakthroughs in solving the Hubble tension and revealing the nature of dark energy provided that the EoS of NS is perfectly known.