Betti Functionals as a Probe for Cosmic Topology
The question of the global topology of the Universe (cosmic topology) is still open. In the $Λ$CDM concordance model it is assumed that the space of the Universe possesses the trivial topology of $\mathbb{R}^3$ and thus that the Universe has an infinite volume. As an alternative, we study in this paper 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 analyse 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 $\mathbb{S}^2$ surrounding the observer, there are only three Betti functionals $β_k(ν)$, $k=1,2,3$. Here $ν=δT/σ_0$ denotes the temperature threshold normalized by the standard deviation $σ_0$ of $δT$. 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 $β_0(ν)$ and $β_1(ν)$ decrease with increasing volume $V=L^3$ of the cubic 3-torus universe. Since the computation of the $β_k$'s from observational sky maps is hindered due to the presence of masks, we suggest a method yielding lower and upper bounds for them and apply it to four Planck 2018 sky maps. It is found that the $β_k$'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 $Λ$CDM case. These results give a further hint that the Universe has a non-trivial topology.