Testing flatness of the universe with probes of cosmic distances and growth

When using distance measurements to probe spatial curvature, the geometric degeneracy between curvature and dark energy in the distance-redshift relation typically requires either making strong assumptions about the dark energy evolution or sacrificing precision in a more model-independent approach. Measurements of the redshift evolution of the linear growth of perturbations can break the geometric degeneracy, providing curvature constraints that are both precise and model independent. Future supernova, CMB, and cluster data have the potential to measure the curvature with an accuracy of $\sigma ({\Omega }_K)=0.002$, without specifying a particular dark energy phenomenology. In combination with distance measurements, the evolution of the growth function at low redshifts provides the strongest curvature constraint if the high-redshift universe is well approximated as being purely matter dominated. However, in the presence of early dark energy or massive neutrinos, the precision in curvature is reduced due to additional degeneracies, and precise normalization of the growth function relative to recombination is important for obtaining accurate constraints. Curvature limits from distances and growth compare favorably to other approaches to curvature estimation proposed in the literature, providing either greater accuracy or greater freedom from dark energy modeling assumptions, and are complementary due to the use of independent data sets. Model-independent estimates of curvature are critical both for testing inflation and for obtaining unbiased constraints on dark energy parameters.

Figure 1

Comoving distance (top panel), growth relative to $z=0$ (middle panel), and growth relative to recombination (bottom panel) vs redshift for flat, open, and closed models. For all three models, ${\Omega }_m=0.24$ and $h=0.73$. In the top panel, the three curves are indistinguishable in $H_{0}D(z)$.

Figure 2

Geometric distance degeneracy between curvature and dark energy dynamics. Curves show the dark energy fraction (top panel), relative distances (middle panel), and growth relative to $z=0$ (bottom panel) for models with distance-redshift relations that are degenerate with a fiducial flat $\Lambda \mathrm{CDM}$ model with ${\Omega }_m=0.24$ and $h=0.73$. Thick red curves correspond to open models with ${\Omega }_K=0.01$, and thin blue curves are closed models with ${\Omega }_K=-0.01$. The solid and dashed curves only match the relative distances at $z<1.7$ (shaded region) and the CMB distance at $z\sim 1000$, but not distances at intermediate redshifts where the dark energy equation of state is assumed to be constant, $w(1.7<z<z_{*})=w_{\infty }$. Dotted curves in the top panel show the dark energy evolution that would be required to match relative distances of the fiducial model at all redshifts. Parameters of each model are given in the top panel legend; for each model, $h$ is set so that ${\Omega }_m{h}^2$ is the same as in the fiducial model.

Figure 3

The first 15 principal components of $w(z)$ for SNAP and Planck (increasing variance from bottom to top), with 500 redshift bins between $z=0$ and $z_{\max }=1.7$ and $w_{\mathrm{fid}}=-1$. The vertical dashed line shows the minimum redshift of the data assumed for computing the PCs, $z_{\min }^{\mathrm{SN}}=0.03$. The PCs are offset vertically from each other for clarity with dotted lines showing the zero point for each component.

Figure 4

Growth functions of MCMC samples with general dark energy equation of state variations at $z<1.7$ within the range $-5\le w\le 3$, including either early dark energy (EDE) at $z>1.7$ ($w_{\infty }\ne -1$; top panels), curvature (${\Omega }_K\ne 0$; middle panels), or both (bottom panels). The left panels show growth relative to early times, and the right panels show growth relative to the present. Dashed red curves show growth in the fiducial flat $\Lambda \mathrm{CDM}$ model (${\Omega }_m=0.24$, $h=0.73$). Samples are selected randomly from those with likelihoods satisfying $\Delta {\chi }^2\le 4$, but for visual clarity we plot samples that are approximately evenly spaced in $G(z=0)$ (left panels) or $G_0(z=4)$ (right panels). The dotted vertical line in each panel marks the division between the low-$z$ and high-$z$ dark energy descriptions at $z=1.7$.

Figure 5

Effect of early dark energy on distance and growth. Top panel: Difference between the distance to recombination and the distance to $z_{\max }=1.5$. Bottom panel: Growth at $z_{\max }$ relative to recombination. Early dark energy models are parametrized by the fraction of dark energy at $z_{\max }$ and $w_{\infty }=w(z>z_{\max })$. From top to bottom in each panel, $w_{\infty }=-1$, $-0.7$, $-0.5$, $-0.3$, $-0.2$, $-0.1$, and 0. Flat $\Lambda \mathrm{CDM}$ is assumed here with ${\Omega }_m=0.24$ and $h=0.73$.

Figure 6

Contours of the growth rate $f$ at $z_{\max }=1.5$ as a function of ${\Omega }_K$ and ${\Omega }_{\mathrm{DE}}(z_{\max })$, for three choices of $w_{\infty }$ at fixed ${\Omega }_m=0.24$ (top panel) and three choices of ${\Omega }_m$ at fixed $w_{\infty }=-1$ (bottom panel).

Figure 7

Growth reconstruction for ${\Omega }_K=0$ and $w_{\infty }=-1$, where the fiducial cosmology is flat $\Lambda \mathrm{CDM}$ with ${\Omega }_m=0.24$ and $h=0.73$. The shaded band is the 68% C.L. region for $G_0(z)$ reconstructed from simulated SNAP and Planck data, and the true growth evolution is plotted as a solid curve. The redshift bin width for the reconstruction is $\Delta z=0.05$.

Figure 8

Comparison of MCMC and growth reconstruction curvature forecasts for the flat $\Lambda \mathrm{CDM}$ fiducial cosmology. Top panel: Combining current priors on $H_0$, $D(z=0.35)$, and ${\Omega }_{\mathrm{DE}}(z_{*})$ with SN and CMB forecasts based on SNAP and Planck, respectively. Bottom panel: Combining SNAP and Planck forecasts with cluster forecasts based on IXO as a probe of the growth evolution. For comparison, the dotted curve shows a forecast for SNAP, Planck, and future 1% measurements of $H_0$ and ${\Omega }_{\mathrm{DE}}(z_{*})$ (without growth constraints).

Figure 9

Forecasts in the ${\Omega }_K-w_{\infty }$ plane for the flat $\Lambda \mathrm{CDM}$ fiducial model, including various combinations of growth information in addition to SN and CMB distance constraints: $G(z_{\max })$ (dashed contours), $G_0(z)$ at $z\le 1.5$ (solid contours), or both (shaded blue contours). The combined growth constraint without the covariance between the predicted $G(z_{\max })$ and $G_0(z)$ from the growth reconstruction is shown as shaded red contours. Contours are plotted at 68% C.L. (thick curves/dark shading) and 95% C.L. (thin curves/light shading).

Figure 10

Growth reconstruction forecast of the marginalized probability of ${\Omega }_K$ for models with ${\Omega }_K=0$ (solid line), ${\Omega }_K=-0.01$ (short dashed line), and ${\Omega }_K=0.01$ (long dashed line). Each model is a $\Lambda \mathrm{CDM}$ cosmology with ${\Omega }_m=0.24$ and $h=0.73$.

Figure 11

Forecast in the ${\Omega }_K-w_{\infty }$ plane for an open universe with early dark energy: ${\Omega }_K=0.025$, $w_0=-1.08$, $w_a=1.02$ (shaded blue contours, right). The mean value of $G(z_{\max })$ for the growth function reconstructed from the fiducial distances of this model is plotted with dashed, unshaded contours at $G(z_{\max })=0.5$, 0.6, 0.7, 0.8, and 0.9, from top to bottom. The flat $\Lambda \mathrm{CDM}$ forecast from Fig. 11 is plotted for comparison (shaded gray contours, left). Crosses mark the best fit point for each model. Shaded contours are plotted at 68% and 95% C.L.