The homogeneity scale and the growth rate of cosmic structures

We propose a novel approach to obtain the growth rate of cosmic structures, $f(z)$, from the evolution of the cosmic homogeneity scale, $R_{\text{H}}(z)$. Our methodology needs two ingredients in a specific functional form: $R_{\text{H}}(z)$ data and the matter two-point correlation function today, i.e., $\xi(r, z=0)$. We use a Gaussian Process approach to reconstruct the function $R_{\text{H}}$. In the absence of suitable observational information of the matter correlation function in the local Universe, $z \simeq 0$, we assume a fiducial cosmology to obtain $\xi(r, z=0)$. For this reason, our final result turns out to be a consistency test of the cosmological model assumed. Our results show a good agreement between: (i) the growth rate $f^{R_{\text{H}}}(z)$ obtained through our approach, (ii) the $f^{\Lambda\text{CDM}}(z)$ expected in the fiducial model, and (iii) the best-fit $f(z)$ from data compiled in the literature. Moreover, using this data compilation, we perform a Gaussian Process to reconstruct the growth rate function $f^{\text{data}}(z)$ and compare it with the function $f^{R_{\text{H}}}(z)$ finding a concordance of $< \!2 \,\sigma$, a good result considering the few data available for both reconstruction processes. With more accurate $R_{\text{H}}(z)$ data, from forthcoming surveys, the homogeneity scale function might be better determined and would have the potential to discriminate between $\Lambda$CDM and alternative scenarios as a new cosmological observable.