Linear Nash-Greene fluctuations on the evolution of

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 (DESY1). 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 ($$S_8$$-$$\varOmega _m$$) 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