Betti Functionals as Probes for Cosmic Topology

Abstract

The question of the global topology of the Universe (cosmic topology) is still open. In the ΛCDM concordance model, it is assumed that the space of the Universe possesses the trivial topology of R3, and thus that the Universe has an infinite volume. As an alternative, in this paper, we study one of the simplest non-trivial topologies given by a cubic 3-torus describing a universe with a finite volume. To probe cosmic topology, we analyze certain structure properties in the cosmic microwave background (CMB) using Betti functionals and the Euler characteristic evaluated on excursions sets, which possess a simple geometrical interpretation. Since the CMB temperature fluctuations δT are observed on the sphere S2 surrounding the observer, there are only three Betti functionals βk(ν), k=0,1,2. Here, ν=δT/σ0 denotes the temperature threshold normalized by the standard deviation σ0 of δT. The analytic approximations of the Gaussian expectations for the Betti functionals and an exact formula for the Euler characteristic are given. It is shown that the amplitudes of β0(ν) and β1(ν) decrease with an increasing volume V=L3 of the cubic 3-torus universe. Since the computation of the βk’s from observational sky maps is hindered due to the presence of masks, we suggest a method that yields lower and upper bounds for them and apply it to four Planck 2018 sky maps. It is found that the βk’s of the Planck maps lie between those of the torus universes with side-lengths L=2.0 and L=3.0 in units of the Hubble length and above the infinite ΛCDM case. These results give a further hint that the Universe has a non-trivial topology.