A measurement of the scale of homogeneity in the early Universe

Abstract

We present the first measurement of the homogeneity index, ℋ, a fractal or Hausdorff dimension of the early Universe from the Planck CMB temperature variations δT in the sky. This characterization of the isotropy scale is model-free and purely geometrical,independent of the amplitude of δT.We find evidence of homogeneity (ℋ = 0) for scales larger thanθℋ = 65.9 ± 9.2 deg on the CMB sky. This finding is at odds with the ΛCDM prediction, which assumes a scale invariant infinite universe.Such anomaly is consistent with the well known low quadrupule amplitude in the angular δT spectrum, but quantified in a direct and model independent way. We estimate the significance of our finding for ℋ = 0 using a principal component analysis from the sampling variations of the observed sky. This analysis is validated with theoretical prediction of the covariance matrix and simulations, booth base purely on data or in the ΛCDM prediction.Assuming translation invariance (and flat geometry) we can convert the isotropy scale θℋ into a (comoving) homogeneity scale which is very close to the trapped surface generated by the observed cosmological constant Λ.