Measuring the matter fluctuations in the Local Universe with the ALFALFA catalog
The standard model of cosmology describes the matter fluctuations through the matter power spectrum, where $σ_{8} \equiv σ_{8,0} \equiv σ_{8}(z = 0)$, defined at the scale of $8 h^{-1}$ Mpc, acts as a normalisation parameter. Currently, the literature reports measurements of $σ_{8}$ analysing different cosmic tracers, where some of these results were obtained assuming a fiducial cosmology. In this study we measure, in a model-independent approach, the matter fluctuations in the Local Universe using HI extragalactic sources mapped by the ALFALFA survey. Our analyses allow us to test the standard cosmological model under extreme conditions in the highly non-linear Local Universe, quantifying the amplitude of the matter fluctuations there. Our work directly measures $σ_{8}$ using the 3-dimensional distances of the HI sources determined by the ALFALFA survey without assuming a fiducial cosmology, resulting in a robust model-independent measurement of $σ_{8}$. Our methodology involves the construction of suitable mock catalogues to simulate the large scale structure features observed in the data, applying the 2-point correlation function, and making use of Markov Chain Monte Carlo methods to estimate the parameters. Analysing these data we measure $σ_8 = 0.78 \pm 0.04$ for $h = 0.6727$, $σ_8 = 0.80 \pm 0.05$ for $h = 0.698$, and $σ_8 = 0.83 \pm 0.05$ for $h = 0.7304$. Considering the data pairs $(σ_8, H_0)$ from the Planck CMB and Atacama Cosmology Telescope (ACT) CMB-lensing analyses, our measurement agrees with them within $1\,σ$ confidence level. From a model-independent perspective, we find that the scale where the matter fluctuation is $1$ is $R = 7.2 \pm 1.5~\text{Mpc}$.