Betti Functionals as a Probe for Cosmic Topology

The question of the global topology of the Universe (cosmic topology) is still open. In the $\Lambda$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 $\delta T$ are observed on the sphere $\mathbb{S}^2$ surrounding the observer, there are only three Betti functionals $\beta_k(\nu)$, $k=1,2,3$. Here $\nu=\delta T/\sigma_0$ denotes the temperature threshold normalized by the standard deviation $\sigma_0$ of $\delta T$. 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 $\beta_0(\nu)$ and $\beta_1(\nu)$ decrease with increasing volume $V=L^3$ of the cubic 3-torus universe. Since the computation of the $\beta_k$'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 $\beta_k$'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 $\Lambda$CDM case. These results give a further hint that the Universe has a non-trivial topology.