The question of the global topology of the Universe (cosmic topology) is still open. In the
CDM concordance model, it is assumed that the space of the Universe possesses the trivial topology of
, and thus that the Universe has
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The question of the global topology of the Universe (cosmic topology) is still open. In the
CDM concordance model, it is assumed that the space of the Universe possesses the trivial topology of
, 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
are observed on the sphere
surrounding the observer, there are only three Betti functionals
,
. Here,
denotes the temperature threshold normalized by the standard deviation
of
. 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
and
decrease with an increasing volume
of the cubic 3-torus universe. Since the computation of the
’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
’s of the Planck maps lie between those of the torus universes with side-lengths
and
in units of the Hubble length and above the infinite
CDM case. These results give a further hint that the Universe has a non-trivial topology.
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