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The scale of homogeneity in the local Universe with the ALFALFA catalogue

We use the scaled counts in spherical caps 𝒩(<θ) and the fractal correlation dimension 𝒟2(θ) 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 θH = 16.49̂ ± 0.29̂, 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 𝒩(<θ) 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 θH estimates by using segment Cox processes of different projected areas, confirming the suitability of the ALFALFA surveyed area for the current analyses.