Geometrical Constraint on Curvature with BAO experiments

The spatial curvature ($K$ or $\Omega_K$) is one of the most fundamental parameters of an isotropic and homogeneous universe and has a close link to the physics of the early Universe. Combining the radial and angular diameter distances measured via the baryon acoustic oscillation (BAO) experiments allows us to unambiguously constrain the curvature. The method is primarily based on the metric theory, but is less sensitive to the theory of structure formation other than the existence of the BAO scale and is free of any model of dark energy. In this paper, we estimate a best achievable accuracy of constraining the curvature with the BAO experiments. We show that an all-sky, cosmic-variance-limited galaxy survey covering the Universe up to $z> 4$ enables a precise determination of the curvature to an accuracy of $\sigma(\Omega_K)\simeq 10^{-3}$. When we assume a model of dark energy - either the cosmological constant or the $(w_0,w_a)$ model - it can achieve a precision of $\sigma(\Omega_K)\simeq \mbox{a few}\times 10^{-4}$. These forecasts require a high sampling density of galaxies, and are degraded by up to a factor of a few for a survey with a finite number density of $\sim 10^{-3}(h/{\rm Mpc})^3$.