Determining the spatial curvature (${\mathrm{\Omega }}_{\mathrm{k}}$) independent of cosmic microwave background observations plays a key role in revealing the physics of the early Universe. The Hubble tension is one of the most serious issues in modern cosmology. We investigate halo catalogs identified from $N$-body simulations at $z=2$ and 3, mimicking high-redshift galaxy surveys. We measure redshift-space correlation functions of halos from the two snapshots. We detect clear features of baryon acoustic oscillations and redshift-space distortions. We find that we can obtain a few percent constraints on both the geometric distances and growth of structure at the distant universe in future surveys. By taking into account the information of the underlying matter power spectrum, we demonstrate that we can also achieve constraint on the Hubble constant $H_0$ with a few percent as well as the spatial curvature with $|{\mathrm{\Omega }}_{\mathrm{k}}|\lesssim 0.1$ by observing galaxies with the number density with ${\overline{n}}_{\mathrm{g}}\simeq {10}^{-4}(h^3{\mathrm{Mpc}}^{-3})$. Our analysis provides a timely forecast for the upcoming spectroscopic surveys, which target emission line galaxy or dusty star-forming galaxy samples.

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• Cosmological parameters
• Dark energy
• Evolution of the Universe
• Large scale structure of the Universe
• Sky surveys
• Galaxies
• Distances, redshifts, & velocities
• Astrophysical & cosmological simulations