Cosmographic degeneracy

We examine the dark energy and matter densities allowed by precision measurements of distances out to various redshifts, in the presence of spatial curvature and (near) arbitrary behavior of the dark energy equation of state. Degeneracies among the parameters permit a remarkably large variation in their values when using only distance measurements of the late-time universe and making no assumptions about the dark energy or curvature. Going beyond distance measurements to a lower limit on the growth of structure bounds the allowed region significantly but still leaves considerable freedom to match a flat $\Lambda$ Cold Dark Matter model with dark energy very different from a cosmological constant. The combination of distances with Hubble parameter, gravitational lensing or other large-scale structure data is essential to determining the cosmological model robustly.

Figure 1

The density parameter space ${\Omega }_m$–${\Omega }_{de}$ has forbidden regions from (1) minimum radius of curvature, (2) positivity of dark energy density, (3) minimum age of the universe, and (4) minimum total growth of structure. The dotted line passes through the points where the spatial curvature is zero. Parameter values lying in the unshaded areas are capable of exactly matching the distance-redshift relation of flat $\Lambda \mathrm{CDM}$ with ${\Omega }_m=0.28$ out to $z_{\max }=1.5$ (top panel) or $z_{\max }=4$ (bottom panel).

Figure 2

Nonexhaustive sample of the Hubble parameter $h(z)$, dark energy density ${\Omega }_{de}(z)$, and $w(z)$ for different points in the ${\Omega }_m$–${\Omega }_{de}$ parameter space that match the $\Lambda \mathrm{CDM}$ distances exactly out to $z_{\max }=1.5$ and satisfy our four conditions. Light (green) lines represent the results with curvature allowed to be nonflat. Dark (red) lines restrict to the zero-curvature case. The assumed true model is a spatially flat $\Lambda \mathrm{CDM}$ model with ${\Omega }_m=0.28$.

Figure 3

Allowed region in the density parameter space if the true model can be any flat $\Lambda \mathrm{CDM}$ model with $0.26\le {\Omega }_m\le 0.3$ and the four forbidden regions are excluded. The shaded grey area applies the distance matching up to $z_{\max }=1.5$ and the light (blue) area to $z_{\max }=4$. The black dashed line shows the flatness line, and the short (red) solid line along it represents the range of true flat $\Lambda \mathrm{CDM}$ models allowed.