New Test of the Friedmann-Lemaître-Robertson-Walker Metric Using the Distance Sum Rule

We present a new test of the validity of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, based on comparing the distance from redshift 0 to $z_1$ and from $z_1$ to $z_2$ to the distance from 0 to $z_2$. If the Universe is described by the FLRW metric, the comparison provides a model-independent measurement of spatial curvature. The test relies on geometrical optics, it is independent of the matter content of the Universe and the applicability of the Einstein equation on cosmological scales. We apply the test to observations, using the Union2.1 compilation of supernova distances and Sloan Lens ACS Survey galaxy strong lensing data. The FLRW metric is consistent with the data, and the spatial curvature parameter is constrained to be $-1.22<{\mathrm{\Omega }}_{K0}<0.63$, or $-0.08<{\mathrm{\Omega }}_{K0}<0.97$ with a prior from the cosmic microwave background and the local Hubble constant, though modeling of the lenses is a source of significant systematic uncertainty.

Figure 1

Probability distribution of $k$, using the SIE model (black, solid line), SIE model with additional 20% error (blue, dotted-dashed line) and the model of [19] (green, dotted line). The inset shows the case with the prior $k<0.1$.