Geometrical constraint on curvature with BAO experiments

Masahiro Takada and Olivier Doré

Phys. Rev. D 92, 123518 – Published 17 December 2015

Abstract

The spatial curvature ( $K$ or $Ω_{K}$ ) 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, and 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 $σ (Ω_{K}) ≃ 1 0^{- 3}$ . When we assume a model of dark energy—either the cosmological constraint or the $(w_{0}, w_{a})$ model—it can achieve a precision of $σ (Ω_{K}) ≃ a few \times 10^{- 4}$ . 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 $\sim 10^{- 3} (h / Mpc)^{3}$ .

Received 13 August 2015

DOI:https://doi.org/10.1103/PhysRevD.92.123518

Authors & Affiliations

Masahiro Takada¹ and Olivier Doré^2,3

¹Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Chiba 277-8583, Japan
²Caltech M/C 350-17, Pasadena, California 91125, USA
³Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA

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Vol. 92, Iss. 12 — 15 December 2015

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Images

Figure 1
The shaded region in each redshift bin, with width $Δ z = 0.2$ , denotes the fractional error of the angular diameter distance determination $D_{A} (z_{i})$ , expected for a cosmic-variance-limited galaxy survey with full-sky coverage ( $f_{sky} = 1$ ) and a sufficiently high number density of galaxies in each redshift over the redshift range from $z = 0$ to $z_{\max} = 7$ . The errors include marginalization over other parameters (see text for details). For comparison, the solid curves show the fractional difference in the radial or angular diameter distances, $D_{C} (z)$ or $D_{A} (z)$ , for an open-geometry universe with $Ω_{K} = 0.001$ compared to the fiducial flat universe, with the dark matter density and the cosmological constant being fixed. The curvature parameter alters $D_{C} (z)$ and $D_{A}$ in different ways, and the ratio $D_{A} / D_{C}$ displays a characteristic redshift dependence. To be more precise, the fractional difference is defined, e.g., as $D_{A} (z; Ω_{K} = 0.001) / D_{A, fid} (z; Ω_{K} = 0) - 1$ . The dashed and dot-dashed curves show the fractional difference when changing $Ω_{m 0}$ and the dark energy equation of state $w_{0}$ , by $Δ Ω_{m 0} = 0.0001$ and $Δ w_{0} = 0.01$ , within a flat universe. In this case, the changes keep $D_{A} (z) = D_{C} (z)$ .
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Figure 2
An expected accuracy of the curvature parameter determination with the BAO measurements for a cosmic-variance-limited galaxy survey as in Fig. 1. Here we consider the galaxy survey covering the redshift range $0 \leq z \leq z_{\max}$ . The top curve shows the purely geometrical constraint on $Ω_{K}$ [Eq. (7)], giving the fundamental limit in a sense that the constraint is free of any gravity theory or dark energy model and rests on the geometrical nature of the four-dimensional spacetime of the expanding homogeneous and isotropic universe. The cosmic-variance-limited survey can achieve a precision better than 0.1 % such as $σ (Ω_{K}) ≃ 0.0008$ . If we assume Einstein gravity (GR) and employ a model of dark energy, the curvature constraint can be improved. However, the precision depends on which dark energy model we adopt. If we assume the dark energy model parametrized by its equation of state $w_{DE} (z) = w_{0} + w_{a} (1 - a)$ around the fiducial model with the cosmological constant, the curvature constraint can be significantly improved. If we restrict ourselves to the cosmological constant ( $w_{DE} = - 1$ ), the best curvature constraint is achieved (as denoted by the lower curve), but the constraint is model dependent.
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Figure 3
Similar to the previous figure, but this figure shows how the curvature constraints are degraded for a galaxy survey with a finite number density of galaxies in each redshift bin. Here we consider the same number density in different redshift bins for simplicity.
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