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