Probing cosmic homogeneity in the Local Universe

We investigate the transition scale to homogeneity, $R_H$, using as cosmic tracer the spectroscopic sample of blue galaxies from the Sloan Digital Sky Survey (SDSS). Considering the spatial distribution of the galaxy sample we compute the two point correlation function $\xi(r)$, the scaled counts in spheres $\mathcal{N}(<r)$, and the fractal dimension $\mathcal{D}_2(r)$ to quantify the homogeneity scale in the Local Universe ($0.04 < z < 0.20$). The sample in analysis is compared with {\it random} and {\it mock} catalogues with the same geometry, and the same number of synthetic cosmic objects as the dataset, to calculate the covariance matrix for the errors determination. The criteria adopted for the transition-to-homogeneity follows the literature, it is attained when $\mathcal{D}_2(r)$ reaches the $1$ per cent level of the limit value $3$ (i.e., where it reaches $2.97$) as the scale increases. We obtain $R_H = 70.33 \pm 10.74$ Mpc$/h$, at the effective redshift $z_{\text{eff}}=0.128$, for a sample containing $150\,302$ SDSS blue galaxies with $0.04 < z < 0.20$. Additionally, we perform robustness tests by analysing the homogeneity scale in sub-volumes of the original one, obtaining coherent results; we also check for a possible artefact in our procedure examining a homogeneous synthetic dataset as a pseudo-data, verifying that such systematic is absent. Because our analyses concentrate in data at low redshifts, $z < 0.20$, we find interesting to use cosmography to calculate the radial comoving distances; therefore in this subject our analyses do not use fiducial cosmological model. For completeness, we evaluate the difference of the comoving distances estimation using cosmography and fiducial cosmology.