  The standard model of cosmology describes the matter fluctuations through the
matter power spectrum, where $\sigma_{8} \equiv \sigma_{8,0} \equiv
\sigma_{8}(z = 0)$, defined at the scale of $8\,h^{-1}$\,Mpc, acts as a
normalisation parameter. Currently, there is a notable discrepancy in the
reported values of $\sigma_{8}$ obtained from cosmic microwave background and
large scale structure data analyses, which indicates a $\sim 3\,\sigma$ tension
between these measurements. This study quantifies matter fluctuations in the
Local Universe using HI extragalactic sources mapped by the ALFALFA survey to
test the standard cosmological model under extreme conditions in the highly
non-linear Local Universe, $z \approx 0$, quantifying the amplitude of the
matter fluctuations there, $\sigma_8$. Our work directly measures $\sigma_{8}$
using the ALFALFA data, where 3D distances were obtained without assuming
$H_0$, resulting in a robust model-independent analysis.
