Testable dark energy predictions from current data

Given a class of dark energy models, constraints from one set of cosmic acceleration observables make predictions for other observables. Here we present the allowed ranges for the expansion rate $H(z)$, distances $D(z)$, and the linear growth function $G(z)$ (as well as other, derived growth observables) from the current combination of cosmological measurements of supernovae, the cosmic microwave background, baryon acoustic oscillations, and the Hubble constant. With a cosmological constant as the dark energy and assuming near-minimal neutrino masses, the growth function is already predicted to better than 2% precision at any redshift, with or without spatial curvature. Direct measurements of growth that match this precision offer the opportunity to stringently test and potentially rule out a cosmological constant. While predictions in the broader class of quintessence models are weaker, it is remarkable that they are typically only a factor of 2–3 less precise than forecasted predictions for future space-based supernovae and Planck CMB measurements. In particular, measurements of growth at any redshift, or the Hubble constant $H_0$, that exceed $\Lambda \mathrm{CDM}$ predictions by substantially more than 2% would rule out not only a cosmological constant but also the whole quintessence class, with or without curvature and early dark energy. Barring additional systematic errors hiding in the data, such a discovery would require more exotic explanations of cosmic acceleration such as phantom dark energy, dark energy clustering, or modifications of gravity.

Figure 1

Flat $\Lambda \mathrm{CDM}$ predictions for growth and expansion observables, showing the 68% C.L. (shading) and 95% C.L. (curves) regions allowed by current CMB, SN, BAO, and $H_0$ data. Observables include the linear growth function normalized in two different ways, $G(z)$ equal to unity at high redshift and $G_0(z)=G(z)/G(0)$; the product of the differential growth rate and the growth function $f(z)G(z)$; the growth index $\gamma (z)$ which relates $f(z)$ and ${\Omega }_{\mathrm{m}}(z)$; the expansion rate $H(z)$; and the comoving distance $D(z)$ (scaled by a factor of $1/10$ in the lower panel). Note that the separation between the 68% and 95% C.L. regions is not visible where the observables are extremely well predicted, e.g. in the $\gamma (z)$ predictions in the middle panel.

Figure 2

Predicted growth and expansion observables for nonflat (dark blue) and flat (light gray) $\Lambda \mathrm{CDM}$, plotted relative to the reference cosmology (the best fit model for flat $\Lambda \mathrm{CDM}$). Here and in subsequent figures, 68% C.L. regions are marked by shading, 95% C.L. regions are bounded by solid curves, and red curves outlined in white show the best fit model of the more general (dark blue) model class (in this case, nonflat $\Lambda \mathrm{CDM}$).

Figure 3

Flat quintessence models without early dark energy (dark blue; red curve: best fit model) vs flat $\Lambda \mathrm{CDM}$ (light gray). Other aspects here and in later figures follow Fig. 2.

Figure 4

Upper panel: Comparison of distance constraints from SN data and best fit models, plotted relative to the best fit $H_{0}D(z)$ for flat $\Lambda \mathrm{CDM}$ (dotted line). Blue points with error bars show the Union SN data in redshift bins of width $\Delta \log z=0.05$. The best fit model for flat quintessence without early dark energy is plotted as a dashed curve, and the solid curve shows how the relative distances are affected by smoothing $w(z)$ for this model by a Gaussian of width ${\sigma }_z=0.1$. The full distribution of relative distance predictions for this quintessence model class is also shown with light gray shading (68% C.L.) and curves (95% C.L.). Lower panel: $w(z)$ for each of the models from the upper panel.

Figure 5

Flat quintessence models with (dark blue; red curve: best fit model) and without (light gray) early dark energy.

Figure 6

Nonflat (dark blue; red curve: best fit model) and flat (light gray) quintessence models without early dark energy.

Figure 7

Nonflat quintessence models with (dark blue; red curve: best fit model) and without (light gray) early dark energy.

Figure 8

Predictions for ${\sigma }_8$ and ${\Omega }_{\mathrm{m}}$ for flat $\Lambda \mathrm{CDM}$ (gray contours, top), flat quintessence without early dark energy (red contours, middle), and nonflat quintessence with early dark energy (blue contours, bottom), showing 68% C.L. (light) and 95% C.L. (dark) regions.

Figure 9

Flat $w_0-w_a$ without priors on $w(z)$ (dark blue; red curve: best fit model) and with quintessence priors ($-1\le w_0\le 1$, $-1\le w_0+w_a\le 1$; light gray).

Figure 10

Nonflat $w_0-w_a$ without priors on $w(z)$ (dark blue; red curve: best fit model) and with quintessence priors ($-1\le w_0\le 1$, $-1\le w_0+w_a\le 1$; light gray).