The scale of homogeneity in the local Universe with the ALFALFA catalogue
We use the scaled counts in spherical caps $\mathcal{N}(<\theta)$ and the fractal correlation dimension $ \mathcal{D}_{2}(\theta) $ procedures to search for a transition scale to homogeneity in the local universe as given by the ALFALFA catalogue (a sample of extragalactic HI line sources, in the redshift range $0 < z < 0.06$). Our analyses, in the 2-dimensional sky projected data, show a transition to homogeneity at $\theta_H = 16.49^{\circ} \pm 0.29^{\circ}$, in remarkable accordance with the angular scale expected from simulations, a result that strengthens the validity of the cosmological principle in the local universe. We test the robustness of our results by analysing the data sample using three versions of the $\mathcal{N}(<\theta)$ estimator, which show a well agreement between them. These statistical estimators were validated using mock realizations generated assuming a fractal distribution of points, successfully recovering the input information. In addition, we perform further analyses showing that our approach is also able to indicate the presence of under- and over-densities in the data. Finally, we verify the influence of the sample size to the $\theta_H$ estimates by using segment Cox processes of different projected areas, confirming the suitability of the ALFALFA surveyed area for the current analyses.