Measuring the Shape of the Universe

Since the dawn of civilization, humanity has grappled with the big questions of existence and creation. Modern cosmology seeks to answer some of these questions using a combination of mathematics and measurement. The questions people hope to answer include “how did the universe begin?”; “how will the universe end?”; “is space finite or infinite?”. After a century of remarkable progress, cosmologists may be on the verge of answering at least one of these questions – is space finite? Using some simple geometry and a small NASA satellite set for launch in the year 2000, the authors and their colleagues hope to measure the size and shape of space. This article explains the mathematics behind the measurement, and the cosmology behind the observations. Before setting out, let us first describe the broad picture we have in mind. Our theoretical framework is provided by Einstein’s theory of general relativity and the hot big-bang model of cosmology. General relativity describes the universe in terms of geometry, not just of space, but of space and time. Einstein’s equation relates the curvature of this space-time geometry to the matter contained in the universe. A common misconception is that the curvature of space is all one needs to answer the question “is space finite or infinite?”. While it is true that spaces of positive curvature are necessarily finite, spaces of negative or zero curvature may be either finite or infinite. In order to answer questions about the global geometry of the universe we need to know both its curvature and topology. Einstein’s equation tells us nothing about the topology of spacetime since it is a local equation relating the spacetime curvature at a point to the matter density there. To study the topology of the universe we need to measure how space is connected. In doing so we will not only discover whether space is finite, but also gain insight into physics beyond general relativity. The outline of our paper is as follows. We begin with an introduction to big bang cosmology, followed by a review of some basic concepts in geometry and topology. With these preliminaries out of the way, we go on to describe the plan to measure the size and shape of the universe using detailed observations of the afterglow from the big bang.