How well-proportioned are lens and prism spaces?

TL;DR

This paper investigates the relationship between the shape of lens and prism space topologies and their impact on cosmic microwave background anisotropies, challenging the idea that well-proportioned spaces suppress large-scale anisotropies.

Contribution

It tests the well-proportioned conjecture using lens and prism spaces, revealing that shape alone does not determine CMB anisotropy suppression and providing new insights into space topology effects.

Findings

Counter-examples to the well-proportioned conjecture found.

Inhomogeneous lens spaces show shape-dependent CMB statistics.

The Poincaré dodecahedron exhibits unique CMB properties.

Abstract

The CMB anisotropies in spherical 3-spaces with a non-trivial topology are analysed with a focus on lens and prism shaped fundamental cells. The conjecture is tested that well proportioned spaces lead to a suppression of large-scale anisotropies according to the observed cosmic microwave background (CMB). The focus is put on lens spaces L(p,q) which are supposed to be oddly proportioned. However, there are inhomogeneous lens spaces whose shape of the Voronoi domain depends on the position of the observer within the manifold. Such manifolds possess no fixed measure of well-proportioned and allow a predestined test of the well-proportioned conjecture. Topologies having the same Voronoi domain are shown to possess distinct CMB statistics which thus provide a counter-example to the well-proportioned conjecture. The CMB properties are analysed in terms of cyclic subgroups Z_p, and new point of view for the superior behaviour of the Poincar\'e dodecahedron is found.