Cosmological Topology

Opening

At the largest scale, topology is mostly a set of constraints on what has not been seen. The global topology of space—whether it is simply connected or wraps back on itself—cannot be inferred from curvature alone; it must be searched for directly. The topology of structure within space—the filaments, voids, loops, and possible defects of the early universe—is measured, but its global counterpart remains bounded and null. This chapter separates the two questions and records the instances where topology has been used as a test rather than an image.

Two different questions, often conflated

Cosmology asks two topological questions and popular accounts routinely merge them. The first is about global topology: is space simply connected, or is it a finite space identified under some group of translations or rotations, so that light can return by more than one path? Curvature — measured, and very close to flat — does not answer this. A flat space can be infinite or a torus.

The second is about the topology of structure within space: the connectivity of the cosmic web, the loops and voids in the galaxy distribution, and the defects a symmetry-breaking early universe would have left behind. These are questions about a field defined on space, not about space itself, and they are where the measurements are.

The instances, cited

Matched-circle searches. A finite space would print the same temperature pattern along pairs of circles on opposite sides of the CMB sky. Searches in WMAP and Planck data have found no such pairs above noise, setting a lower bound on the size of any fundamental domain comparable to the diameter of the observable universe.

Planck constraints on global topology. The 2015 and 2018 analyses report no detection of multiply connected topology and quote limits in terms of the smallest allowed identification scale. This is a genuine measurement with a null result, and it is the honest headline of the chapter.

Persistent homology of the cosmic web. Betti numbers and persistence diagrams computed on galaxy surveys and on simulations quantify the number of independent loops and enclosed voids as a function of density threshold. The technique distinguishes cosmological models that share the same two-point correlation function, which is its practical justification.

Genus statistics. The genus of iso-density surfaces in the galaxy distribution has been used since the 1980s as a shape statistic; its near-Gaussian form remains a consistency check on primordial statistics.

Topological defects. Cosmic strings, monopoles, and textures follow from plausible symmetry-breaking histories. Constraints from CMB anisotropy, pulsar timing, and lensing searches bound their contribution to well under a per cent of structure. None has been detected.

What the null results are worth

It would be easy to file this chapter as disappointment. It is the opposite: bounds are the most transferable products in cosmology, and the matched-circle limits genuinely close off a family of otherwise attractive models. A theory that predicts a small torus universe is now in conflict with data. That is what a measurement is for.

The chapter also serves as the volume's discipline check at the largest scale. Chapter 13 warned that spacetime defects remain unobserved; this chapter states the observational reason and quantifies it. When earlier books in the corpus used cosmic filaments and the web as imagery, they were using structure topology, which is measured, and not global topology, which is bounded and unseen. Keeping those two apart is most of the work here.