Circles in the Sky: Finding Topology with the Microwave Background Radiation

TL;DR

This paper proposes a new statistical method to detect the topology of a finite universe by identifying circle intersections in the cosmic microwave background, applicable to all small homogeneous topologies.

Contribution

It introduces a topology detection statistic based on circle intersections, extending analysis beyond the commonly studied toroidal models.

Findings

The statistic can identify all small finite homogeneous topologies.

Previous models like the toroidal universe are unreliable for topology detection.

The method is sensitive to universe sizes smaller than the horizon.

Abstract

If the universe is finite and smaller than the distance to the surface of last scatter, then the signature of the topology of the universe is writ on the microwave background sky. Previous efforts to search for this topology have focused on one particular model: a toroidal flat universe. We show how both the high degree of spatial symmetry of this topology and the integrability of its geodesics make it unreliable as a paradigmatic example, and discuss why topology on scales significantly smaller than the horizon are not ruled out by previous analyses focussing on this special case. We show that in these small universes the microwave background will be identified at the intersections of the surface of last scattering as seen by different ``copies'' of the observer. Since the surface of last scattering is a sphere, these intersections will be circles, regardless of the background geometry or topology. We therefore propose a statistic that is sensitive to all small finite homogeneous topologies. Here, small means that the distance to the surface of last scatter is smaller than the ``periodicity scale'' of the universe.