What can the detection of a single pair of circles-in-the-sky tell us about the geometry and topology of the Universe?

In a Universe with a detectable nontrivial spatial topology, the last scattering surface contains pairs of matching circles with the same distribution of temperature fluctuations—the so-called circles-in-the-sky. Searches undertaken for nearly antipodal pairs of such circles in cosmic microwave background maps have so far been unsuccessful. Previously, we had shown that the negative outcome of such searches, if confirmed, should in principle be sufficient to exclude a detectable nontrivial spatial topology for most observers in very nearly flat ($0<\mid {\Omega }_{\mathrm{tot}}-1\mid \lesssim {10}^{-5}$) (curved) universes. More recently, however, we have shown that this picture is fundamentally changed if the universe turns out to be exactly flat. In this case, there are many potential pairs of circles with large deviations from antipodicity that have not yet been probed by existing searches. Here, we study under what conditions the detection of a single pair of circles-in-the-sky can be used to uniquely specify the topology and the geometry of the spatial section of the Universe. We show that from the detection of a single pair of matching circles one can infer whether the spatial geometry is flat or not, and if so we show how to determine the topology (apart from one case) of the Universe using this information. An important additional outcome of our results is that the dimensionality of the circles-in-the-sky parameter space that needs to be spanned in searches for matching pairs of circles is reduced from six to 5 degrees of freedom, with a significant reduction in the necessary computational time.

Figure 1

This figure depicts the circles at the intersection of the last scattering surface with its images. The parameter $\theta$ is a measure of the deviation of the circles from being antipodal, and $\varphi$ measures the relative phase between the circles of radius $\nu$.

Figure 2

The allowed combinations of the circles-in-the-sky parameters $\theta$ (the deviation from antipodicity) and $\varphi$ (the relative phase between circles) for compact orientable flat manifolds. Each thick contour line corresponds to one allowed class of holonomies, which take the form of a screw motion with twist parameter $\alpha$ given in Eq. (1). The different orientable compact flat manifolds $E_i$ ($i=1\cdots 6$) that include each holonomy class are indicated; those for which this holonomy may generate the most readily detectable circle pair are in bold. The dotted part in each countour line corresponds to configurations where the circle pairs with the largest radii are always translational, and thus antipodal. In all cases, the compactification length $L$ is not fixed, but the precise combination of $\theta$ and $\varphi$ will depend on the position of the observer, as depicted in the light contours corresponding to different distances $r$ from the axis of the screw motion (in units of $L$). The shaded area corresponds to the parameter values that have been probed in the recent searches for antipodal and nearly antipodal circles in CMB maps [6, 7].