The Homogeneity Scale of the universe

In this study, we probe the cosmic homogeneity with the BOSS CMASS galaxy sample in the redshift region of $0.43<z<0.7$. We use the normalised counts-in-spheres estimator $\mathcal{N}(<r)$ and the fractal correlation dimension ${\mathcal{D}}_2(r)$ to assess the homogeneity scale of the universe. We verify that the universe becomes homogenous on scales greater than ${\mathcal{R}}_H\simeq 64.3\pm 1.6h^{-1}Mpc$, consolidating the Cosmological Principle with a consistency test of $\mathrm{\Lambda }$CDM model at the percentage level. Finally, we explore the evolution of the homogeneity scale in redshift.