Observational constraints on dark energy and cosmic curvature

Current observational bounds on dark energy depend on our assumptions about the curvature of the universe. We present a simple and efficient method for incorporating constraints from cosmic microwave background (CMB) anisotropy data and use it to derive constraints on cosmic curvature and dark energy density as a free function of cosmic time using current CMB, Type Ia supernova (SN Ia), and baryon acoustic oscillation data. We show that there are two CMB shift parameters, $R\equiv \sqrt{{\Omega }_m{H}_0^2}r(z_{\mathrm{CMB}})$ (the scaled distance to recombination) and $l_a\equiv \pi r(z_{\mathrm{CMB}})/r_s(z_{\mathrm{CMB}})$ (the angular scale of the sound horizon at recombination), with measured values that are nearly uncorrelated with each other. Allowing nonzero cosmic curvature, the three-year WMAP (Wilkinson Microwave Anisotropy Probe) data give $R=1.71\pm 0.03$, $l_a=302.5\pm 1.2$, and ${\Omega }_b{h}^2=0.02173\pm 0.00082$, independent of the dark energy model. The corresponding bounds for a flat universe are $R=1.70\pm 0.03$, $l_a=302.2\pm 1.2$, and ${\Omega }_b{h}^2=0.022\pm 0.00082$. We give the covariance matrix of $(R,l_a,{\Omega }_b{h}^2)$ from the three-year WMAP data. We find that $(R,l_a,{\Omega }_b{h}^2)$ provide an efficient and intuitive summary of CMB data as far as dark energy constraints are concerned. Assuming the Hubble Space Telescope (HST) prior of $H_0=72\pm 8(\mathrm{km}/\mathrm{s}){\mathrm{Mpc}}^{-1}$, using 182 SNe Ia (from the HST/GOODS program, the first year Supernova Legacy Survey, and nearby SN Ia surveys), $(R,l_a,{\Omega }_b{h}^2)$ from WMAP three-year data, and SDSS (Sloan Digital Sky Survey) measurement of the baryon acoustic oscillation scale, we find that dark energy density is consistent with a constant in cosmic time, with marginal deviations from a cosmological constant that may reflect current systematic uncertainties or true evolution in dark energy. A flat universe is allowed by current data: ${\Omega }_k=-{0.006}_{-0.012-0.025}^{+0.013+0.025}$ for assuming that the dark energy equation of state $w_X(z)$ is constant, and ${\Omega }_k=-{0.002}_{-0.018-0.032}^{+0.018+0.041}$ for $w_X(z)=w_0+w_a(1-a)$ (68% and 95% confidence levels). The bounds on cosmic curvature are less stringent if dark energy density is allowed to be a free function of cosmic time, and are also dependent on the assumption about the early time property of dark energy. We demonstrate this by studying two examples. Significant improvement in dark energy and cosmic curvature constraints is expected as a result of future dark energy and CMB experiments.

Figure 1

Ratio of the dark energy density $X(z)\equiv {\rho }_X(z)/{\rho }_X(0)$ and the matter density ${\rho }_m(z)/{\rho }_m(0)=(1+z{)}^3$ for dark energy models with dark energy equation of state $w_X(z)=w_0+w_a(1-a)$.

Figure 2

The scaled distance to recombination $R$, the angular scale of the sound horizon at recombination $l_a$, and the baryon density ${\Omega }_b{h}^2$ from the three-year WMAP data.

Figure 3

CMB angular power spectra for dark energy models that give the same values of $R$ or $l_a$.

Figure 4

The expected $R$ and $l_a$ as functions of curvature for five simple dark energy models (with the same line types as in Fig. 3).

Figure 5

The expected $R$ and $l_a$ as functions of curvature for the dark energy models from Fig. 1 (with the same line types). Both $R$ and $l_a$ are needed to constrain cosmic curvature.

Figure 6

Constraints on dark energy density using $R$, $l_a$, and ${\Omega }_b{h}^2$ from the three-year WMAP data, together with 182 SNe Ia and SDSS BAO measurement. The shaded areas indicate the 68% confidence regions, while the outside contours bound the 95% confidence regions.

Figure 7

Constraints on the expansion history of the universe $H(z)$ that corresponds to Fig. 6, using $R$, $l_a$, and ${\Omega }_b{h}^2$ from the three-year WMAP data, together with 182 SNe Ia and SDSS BAO measurement. The error bars indicate the 68% confidence intervals.

Figure 8

Constraints on $(w_0,w_a)$ using $R$, $l_a$, and ${\Omega }_b{h}^2$ from the three-year WMAP data, together with 182 SNe Ia and SDSS BAO measurement. The 68% and 95% confidence contours are shown.

Figure 9

The probability distribution function of cosmic curvature for different assumptions about dark energy: the model-independent dark energy density ${\rho }_X(z)$ reconstructed in the last subsection, the two parameter dark energy model $w_X(z)=w_0+w_a(1-a)$, and a constant dark energy equation of state.