Thus, large angular scale measurements are not sensitive to topology. A fair estimate of the number of cells, needed to tile this hyperbolic ball is given by the volume ratio, where Vol is the volume of the fundamental domain in curvature units. In a universe with this is a whopping ratio of Vol . Considering that there are an infinite number of hyperbolic three manifolds with volumes less that 3, we have no shortage of model universes that can be seen by this method. Even if was as high as 0.95, we would still have an infinite number of manifolds to chose from that would produce at least some matched circle pairs.
Lets consider an infinite hyperbolic universe tiled with copies of our fundamental domain.
We can draw the last scattering sphere centered not only around our own location, but also around all of our images. The circles arise at the intersections of these spheres. Even at a resolution of 0.5 where we can use the three highest frequency maps, there are still over 125 independent points along each circle. Once we have detected the circles, J. Weeks showed us that we can use them to determine the topology of the universe. Each pair of circles lie on matched faces of the fundamental cell, thus, each pair of circles gives us an element of the fundamental group .