Observational constraints on the averaged universe

Averaging in general relativity is a complicated operation, due to the general covariance of the theory and the nonlinearity of Einstein’s equations. The latter of these ensures that smoothing spacetime over cosmological scales does not yield the same result as solving Einstein’s equations with a smooth matter distribution, and that the smooth models we fit to observations need not be simply related to the actual geometry of spacetime. One specific consequence of this is a decoupling of the geometrical spatial curvature term in the metric from the dynamical spatial curvature in the Friedmann equation. Here we investigate the consequences of this decoupling by fitting to a combination of Hubble Space Telescope (HST), CMB, type Ia supernovae (SNIa), and baryon acoustic oscillation (BAO) data sets. We find that only the geometrical spatial curvature is tightly constrained and that our ability to constrain dark energy dynamics will be severely impaired until we gain a thorough understanding of the averaging problem in cosmology.

Figure 1

Constraints on ${\Omega }_{k_{\mathrm{g}}}$ and ${\Omega }_{k_{\mathrm{d}}}$ from the CMB for $H_0=50$, 70 and $90\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, with the other parameters marginalized over. Shaded areas are 68% and 95% confidence regions.

Figure 2

Constraints on the two curvature parameters from different data sets. In the left panel we show the constraints from the CMB (the filled, thin, central contours) and SNIa (the filled, wide contours at the bottom of the plot for Union2, and unfilled and dashed for SDSS) separately, as well as the combined constraints including HST. In the right panel we show the combined constraints, with the smaller lightly shaded regions in the foreground now including the BAO. The constraints from the CMB and SNIa are extremely weak individually, but tighten when combined with HST data. Note that only the geometrical curvature is tightly constrained and not the dynamical one. The SDSS SNIa data can also be seen as inconsistent with ${\Omega }_{k_g}=0$ at greater than 95% confidence.

Figure 3

Constraints on ${\Omega }_m$ and ${\Omega }_{\Lambda }$ for the same data set combinations as in Fig. 2. The constraints from the CMB and SNIa separately are again much weaker here than in the standard case. In particular, $\Lambda =0$ is consistent with the CMB data for any ${\Omega }_m$. Union2 SNIa data, however, still clearly give $\Lambda \ne 0$. (For a discussion of bias see Ref. 56.)

Figure 4

Marginalized constraints on the dark energy parameters, with (the small, filled region at the top right) and without (the large, filled region that occupies most of the plot) assuming $k_{\mathrm{g}}=k_{\mathrm{d}}$. Allowing for effects due to averaging removes our ability to constrain the possible evolution of the dark energy equation of state.