The variance of the CMB temperature gradient: a new signature of a
  multiply connected Universe
  In this work we investigate the standard deviation of the Cosmic Microwave
Background (CMB) temperature gradient field as a signature for a multiply
connected nature of the Universe. CMB simulations of a spatially infinite
universe model within the paradigm of the standard cosmological model present
non-zero two-point correlations at any angular scale. This is in contradiction
with the extreme suppression of correlations at scales above $60^{\circ}$ in
the observed CMB maps. Universe models with spatially multiply connected
topology contain typically a discrete spectrum of the Laplacian with a specific
wave-length cut-off and thus lead to a suppression of the correlations at large
angular scales, as observed in the CMB (in general there can be also an
additional continuous spectrum). Among the simplest examples are 3-dimensional
tori which possess only a discrete spectrum. To date, the universe models with
non-trivial topology such as the toroidal space are the only models that
possess a two-point correlation function showing a similar behaviour as the one
derived from the observed Planck CMB maps. In this work it is shown that the
normalized standard deviation of the CMB temperature gradient field does
hierarchically detect the change in size of the cubic 3-torus, if the volume of
the Universe is smaller than $\simeq 2.5 \cdot 10^3$ Gpc$^3$. It is also shown
that the variance of the temperature gradient of the Planck maps is consistent
with the median value of simulations within the standard cosmological model.
All flat tori are globally homogeneous, but are globally anisotropic. However,
this study also presents a test showing a level of homogeneity and isotropy of
all the CMB map ensembles for the different torus sizes considered that are
nearly at the same weak level of anisotropy revealed by the CMB in the standard
cosmological model.
