Analysis on the evolution of $sigma_8(z)$ from Linear Nash-Greene fluctuations

From the linear Nash-Greene fluctuations of background metric, we present the perturbation equations in an embedded four space-time. In the context of a five-dimensional bulk, we show that the perturbations are only propagated by the gravitational tensorial field equation. In Newtonian conformal gauge, we study the matter density evolution in sub-horizon regime and on how such scale may be affected by the extrinsic curvature. We apply a joint likelihood analysis to the data with the Markov Chain Monte Carlo (MCMC) method for cosmological parameter estimation using a pack of recent datasets as the Pantheon Supernovae type Ia, the Baryon Acoustic Oscillations (BAO) from DR12 galaxy sample and Dark Energy Survey (DES) Y$1$. The constrained parameters are tested in the cosmography analysis on the evolution of Hubble function $H(z)$ and the deceleration parameter $q(z)$, as well as on the evolution of the growth rate $fsigma_8(z)$ of the extended Gold 2018 growth-rate dataset with 25 datapoints. Moreover, we obtain an alleviation of the $sigma$ tension in the contours between ($sigma_8$-$Omega_m$) of the observations from Cosmic Microwave Background (CMB) and large scale structure (LSS) probes.