Linear Nash-Greene fluctuations on the evolution of $S_8$ and $H_0$ tensions

We present the perturbation equations in an embedded four space-time from the linear Nash-Greene fluctuations of background metric. In the context of a five-dimensional bulk, we show that the cosmological 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 by means of the Markov Chain Monte Carlo (MCMC) method for 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 (DESY$1$). We discuss the tensions on the Hubble constant $H_0$ and the growth amplitude factor $S_8$ of the observations from Planck 2018 Cosmic Microwave Background (CMB) and the local measurements of $H_0$ with Hubble Space Telescope (HST) photometry and Gaia EDR3. As a result, we obtain an alleviation below $\sim 1\sigma$ in the contours ($S_8$-$\Omega_m$) at $68.4\%$ confidence level (CL) when compared with DESY1 data. On the other hand, the $H_0$ tension persists with $\sim 2.6\sigma$ at $68.4\%$ CL and $1.95\sigma$ at $95.7\%$ CL, aggravated with the inclusion of BAO data.