Inferring S₈ (z) and (z) with cosmic growth rate measurements using machine learning

Abstract Measurements of the cosmological parameter S₈ S 8 provided by cosmic microwave background and large scale structure data reveal some tension between them, suggesting that the clustering features of matter in these early and late cosmological tracers could be different. In this work, we use a supervised learning method designed to solve Bayesian approach to regression, known as Gaussian Processes regression, to quantify the cosmic evolution of S₈ S 8 up to z 1. 5 z ∼ 1. 5. For this, we propose a novel approach to find firstly the evolution of the function ₈ (z) σ 8 (z), then we find the function S₈ (z) S 8 (z). As a sub-product we obtain a minimal cosmological model-dependent ₈ (z=0) σ 8 (z = 0) and S₈ (z=0) S 8 (z = 0) estimates. We select independent data measurements of the growth rate f (z) and of f ₈ (z) f σ 8 (z) according to criteria of non-correlated data, then we perform the Gaussian reconstruction of these data sets to obtain the cosmic evolution of ₈ (z) σ 8 (z), S₈ (z) S 8 (z), and the growth index (z) γ (z). Our statistical analyses show that S₈ (z) S 8 (z) is compatible with Planck Λ CDM cosmology; when evaluated at the present time we find ₈ (z=0) = 0. 766 0. 116 σ 8 (z = 0) = 0. 766 ± 0. 116 and S₈ (z=0) = 0. 732 0. 115 S 8 (z = 0) = 0. 732 ± 0. 115. Applying our methodology to the growth index, we find (z=0) = 0. 465 0. 140