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.
