Title: The trouble with spatial curvature

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Abstract:
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The latest results from the Planck satellite 2018 CMB temperature and polarization data appear to prefer a spatially closed Universe, with curvature parameter Ωk<0, at high significance: if true, this could represent a big problem for the extremely successful inflationary paradigm. In order to obtain reliable constraints on spatial curvature it is necessary to combine Planck data with external datasets to break the so-called geometrical degeneracy. However, in most cases this is achieved at the price of significant tensions with Planck data within a non-flat Universe, bringing the reliability of the resulting dataset combination into question. I will discuss new perspectives into this “spatial curvature debate” focusing on two classes of external datasets: full-shape measurements of the galaxy power spectrum from the BOSS survey, and cosmic chronometer measurements of the expansion rate H(z) from the relative ages of passively evolving galaxies. I will argue that despite the hints for a closed Universe from Planck and the aforementioned tensions, we can still confidently assert that the Universe is spatially flat at least to the Ωk~10^-2 level.