Using a multi-messenger and multi-wavelength observational strategy to probe the nature of dark energy through direct measurements of cosmic expansion history

In the near future, the redshift drift observations in optical and radio bands will provide precise measurements on $H(z)$ covering the redshift ranges of $2<z<5$ and $0<z<0.3$. In addition, gravitational wave (GW) standard siren observations could make measurements on the dipole anisotropy of luminosity distance, which will also provide the $H(z)$ measurements in the redshift range of $0<z<3$. In this work, we propose a multi-messenger and multi-wavelength observational strategy to measure $H(z)$ based on the three next-generation projects, E-ELT, SKA, and DECIGO, and we wish to see whether the future $H(z)$ measurements could provide tight constraints on dark-energy parameters. The dark energy models we consider include $\Lambda$CDM, $w$CDM, CPL, HDE, and I$\Lambda$CDM models. It is found that E-ELT, SKA1, and DECIGO are highly complementary in constraining dark energy models. Although any one of these three data sets can only give rather weak constraints on each model we consider, the combination of them could significantly break the parameter degeneracies and give much tighter constraints on almost all the cosmological parameters. Moreover, we find that the combination of E-ELT, SKA1, DECIGO, and CMB could further improve the constraints on dark energy parameters, e.g., $\sigma(w_0)=0.024$ and $\sigma(w_a)=0.17$ in the CPL model, which means that these three promising probes will play a key role in helping reveal the nature of dark energy.