Consistency Tests for the Cosmological Constant

We propose consistency tests for the cosmological constant which provide a direct observational signal if $\Lambda$ is wrong, regardless of the densities of matter and curvature. As an example of its utility, our flat case test can warn of a small transition of the equation of state $w(z)$ from $w(z)=-1$ of 20% from SNAP (Supernova Acceleration Probe) quality data at 4-$\sigma$, even when direct reconstruction techniques see virtually no evidence for deviation from $\Lambda$. It is shown to successfully rule out a wide range of non-$\Lambda$ dark energy models with no reliance on knowledge of ${\Omega }_m$ using SNAP quality data and a large range for using ${10}^5$ supernovae as forecasted for the Large Synoptic Survey Telescope.

Figure 1

Consistency test for $\Lambda$. We show $\mathcal{L}(z)$ for various dark energy models that we could expect to reconstruct from SNAP (middle row) and LSST (bottom row) data, which is independent of ${\Omega }_m$. (Note the different redshift ranges used for SNAP and LSST.) Also shown (top row) is the reconstructed $w(z)$ using the Padè ansatz, Eq. (3), together with 1-$\sigma$ errors; this reconstruction assumes the correct choice of ${\Omega }_m$, and it considerably worse without this. For SNAP data, we see that $\mathcal{L}(z)$ always provides evidence for evolving dark energy, except in the case when it is evolving slowly, as in the middle column. In the third row, we see that $\mathcal{L}(z)$ will be able to signal evolving dark energy to high confidence using LSST data. When both $w$ and $\mathcal{L}$ are reconstructed, $\mathcal{L}(z)$ provides additional information to $w(z)$: a given choice of $D(z)$ may be good for $w(z)$ and bad for $\mathcal{L}(z)$ (e.g., column 3) or vise versa (column 5). Indeed, in the transition model, $w(z)$ provides little evidence for $w\ne -1$ while $\mathcal{L}(z)$ signals something is wrong with $\Lambda$ at 4-$\sigma$ independently of ${\Omega }_m$.

Figure 2

Reconstructed $w(z)$ for $w=-1$ when the correct ${\Omega }_m=0.275$ is assumed (dark green) and when the incorrect value ${\Omega }_m=0.2$ is assumed (light green), with $1\sigma$ error bars, for LSST. Reconstruction methods rely heavily on precise measurements of ${\Omega }_m$. Our consistency test is immune to such pitfalls.