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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 investigatedusing solely such GW dark sirens. To do this, the key is to get a large numberof accurate GW data, which can be achieved with the third-generation (3G) GWdetectors. Here we wish to offer an answer to the question of whether theHubble constant and the equation of state (EoS) of dark energy can be preciselymeasured using solely GW dark sirens. We find that in the era of 3G GWdetectors ${\cal O}(10^5-10^6)$ dark siren data (with the NS tidalmeasurements) could be obtained in three-year observation if the EoS of NS isperfectly known, and thus using only dark sirens can actually achieve theprecision cosmology. Based on a network of 3G detectors, we obtain theconstraint 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 EoSparametrization of dark energy, the precision of $w_0$ is $2.04\%$ and theerror of $w_a$ is 0.13. We conclude that 3G GW detectors would lead tobreakthroughs in solving the Hubble tension and revealing the nature of darkenergy provided that the EoS of NS is perfectly known.

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