Constraining the Topology of the Universe

The first year data from the Wilkinson Microwave Anisotropy Probe are used to place stringent constraints on the topology of the Universe. We search for pairs of circles on the sky with similar temperature patterns along each circle. We restrict the search to back-to-back circle pairs, and to nearly back-to-back circle pairs, as this covers the majority of the topologies that one might hope to detect in a nearly flat universe. We do not find any matched circles with radius greater than $25°$. For a wide class of models, the nondetection rules out the possibility that we live in a universe with topology scale smaller than 24 Gpc.

Figure 1

The maximum value of the circle statistic as a function of radius, $\alpha$ (in degrees), for a simulated finite universe model. The peaks in the plots correspond to positions of matched circles. The dashed (blue) line corresponds to the detection threshold described below.

Figure 2

The maximum value of the circle statistic as a function of radius for back-to-back circle pairs on scrambled CMB skies. The results are shown for seven different realizations. The white line is a fit for the false positive rate. The dashed (blue) line shows the false detection threshold set so that fewer than 1 in 100 simulations yield a false event.

Figure 3

The maximum value of the circle statistic as a function of radius for the WMAP data for back-to-back circles. The dotted (red) line in the figure is for an orientable topology. The solid line is for nonoriented topologies. The white line is the expected false detection level. The dashed (blue) line is the detection threshold. The spike at $90°$ is due to a trivial match between a circle centered at one point and its copy centered around the antipodal point.

Figure 4

The maximum value of the circle statistic as a function of radius for the WMAP data for circles whose centers are separated by greater than $170°$. The solid line in the figure is for an orientable topology. The dotted (red) line is the expected false detection level and the dashed (blue) line is the detection threshold.