Model independent tests of the standard cosmological model

The dark energy problem has led to speculation that not only may $\Lambda \mathrm{CDM}$ be wrong, but that the Friedmann-Lemaître-Robertson-Walker models themselves may not even provide the correct family of background models. We discuss how direct measurements of $H(z)$ can be used to formulate tests of the standard paradigm in cosmology. On their own, such measurements can be used to test for deviations from flat $\Lambda \mathrm{CDM}$. When combined with supernovae distances, Hubble rate measurements provide a test of the Copernican principle and the homogeneity assumption of the standard model, which is independent of dark energy or a metric based the theory of gravity. A modification of this test also provides a model independent observable for flatness which decorrelates curvature determination from dark energy. We investigate these tests using Hubble rate measurements from age data, as well as from a Hubble rate inferred from recent measurements of the baryon acoustic oscillations. While the current data are too weak to say anything significant, these tests are exciting prospects for the future.

Figure 1

A plot of $Om(z)$ in different models, obtained using distances. The figure shows the range of behavior from curvature in each fan of curves (key, top left). The three gray fans show how differing ${\Omega }_m$ values interact with curvature for $\Lambda \mathrm{CDM}$ (key, top right). The effect of changing $w$ by a constant is illustrated in the red and brown fans. Similar curves are found if we use $h$ rather than $D$ to form $Om(z)$, but without the crossover at high redshift. It is clear that, although $Om(z)$ deviates from a constant because of curvature, this deviation is not strong for small curvature values; therefore, $Om(z)$’s utility lies in picking up non-$\Lambda$ behavior.

Figure 2

The best fit curves to the constitution supernovae, shown here with vertical error bars reduced by a factor of 10 for clarity. The distance modulus is shown with the best fit flat $\Lambda \mathrm{CDM}$ subtracted off (with $H_0=65{\mathrm{kms}}^{-1}{\mathrm{Mpc}}^{-1}$). The curves all have a better fit to the data than $\Lambda \mathrm{CDM}$, shown by the dashed horizontal line. Note the lack of SNIa above $z\sim 1$ means that $D(z)$ is not well constrained there.

Figure 3

Tests for flat $\Lambda \mathrm{CDM}$, $Om(z)$, and $\mathcal{L}(z)$. The smoothing method allows us to construct these directly. The curves shown are all a better fit to the data than flat $\Lambda \mathrm{CDM}$, which is shown as the dashed line in the plots. These two plots are made from the same reconstruction. We have shown $\mathcal{L}(z)$ up to $z=1$ because the error-bars at the higher redshifts are so large. $\mathcal{L}(z)$ is derived by using the second derivative of the results from the smoothing method where $Om(z)$ is using the first derivative, so the error-bars for $\mathcal{L}(z)$ are larger.

Figure 4

Resultant $Om(z)$ using two values of $H_0$. The points indicated with an arrow show the SN (constitution) data combined with the recent BAO (SDSS-DR7) and CMB (WMAP) data. The other points show the same using age data. The thick vertical lines represent the spread in the constructed point arising from uncertainty in the reconstructed distances; the thin vertical lines represent the errors arising from $H(z)$ and $H_0$.

Figure 5

Resultant ${\mathcal{O}}_k(z)$ using two values of $H_0$. The points indicated with an arrow show the SN (constitution) data combined with the recent BAO (SDSS-DR7) and CMB (WMAP) data. The other points show the same using age data. The thick vertical lines represent the spread in the constructed point arising from uncertainty in the reconstructed distances; the thin vertical lines represent the errors arising from $H(z)$ and $H_0$. The faded points are two age data points which are at such high redshift they have no corresponding SNIa, so should not be taken too seriously.