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° 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 three-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 three-torus, if the volume of the Universe is smaller than ≃2.5 × 10³ Gpc³. 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.