Tyler McMaken with Glenn Starkman

Constraining the Topology of the Universe: A Generalization of Flat Manifold Parameter Spaces

An open question for cosmologists concerns whether the universe is infinite or finite. For a flat cosmology, the topology of the universe can take on any one of 18 manifolds, some of which are infinite in certain dimensions and some of which have completely finite fundamental domains. Analyses of the cosmic microwave background (CMB) have been used to constrain the sizes of possible fundamental domains for generic topologies using the circles-on-the-sky technique, however, suprisingly the more stringent topology-specific constraints that can be obtained using the correlation matrix have not been exhaustively performed for the full set of possible manifolds, even for flat space. In particular, fundamental domains which are not rectangular prisms have been omitted, as have slab spaces in which the identification of opposite planes includes anything but a 180 degree rotation. A comprehensive search through the up-to-twelve parameter space for each manifold would require much more computational time than is currently feasible, but this project aims to (a) tabulate and examine the parameter space required to fully describe each flat Euclidean manifold, (b) generalize the eigenmode solutions of the slab spaces to allow for arbitrary rotations.  The ultimate goal is to perform a likelihood analysis for the correlation matrix of at least one generalized manifold to determine if such a topology fits with CMB data.