Subhorizon linear Nash–Greene perturbations with constraints on H ( z ) and the deceleration parameter q ( z )

Highlights: • Subhorizon linear perturbations in the embedding context. • Statistical equivalence of the present model with the $\Lambda$CDM and $w$CDM cosmologies at 1-$\sigma$ level. • Cosmography analysis on the evolution of $H\left(z\right)$ and the deceleration parameter $q\left(z\right)$ • Mild alleviation of the ${\sigma }_8$ discrepancy. The Hubble parameter $H\left(z\right)$, as a function of redshift, is modified by the presence of a new term originated from the extrinsic curvature in a four embedded space–time. Assuming an asymptotic expansion factor $a\sim 0$ or, equivalently $z\to \infty$, it is obtained a nearly resemblance of the present model ($\beta$-model) with the $w$CDM model with $H\left(z\right)\sim H_{wCDM}\left(z\right)$ at background level. We test the models using a pack of recent datasets as the "Gold 2018" growth-rate data, the best-fit Planck2018/$\Lambda$CDM parameters on the Cosmic Microwave Background (CMB), the Baryon Acoustic Oscillations (BAO) measurements, the Pantheon Supernovae type Ia and the Hubble parameter data with redshift ranging from $0.01<z<2.3$. Performing the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to ascertain the statistical viability of the model from Jeffreys' scale, we apply a joint likelihood analysis to the data with the Markov Chain Monte Carlo (MCMC) method. We find that the present model is in very good agreement with observations with a close statistical equivalence with the $\Lambda$CDM and $w$CDM cosmologies at 1-$\sigma$ level. We also show that a mild alleviation of the ${\sigma }_8$ discrepancy between the growth amplitude factor and the matter content (${\sigma }_8$-${\Omega }_m$) of the observations from CMB and Large Scale Structure (LSS) probes. A comparison of the aforementioned full pack of data in MCMC chains is made with the resulting MCMC from Pantheon SNIa+$H\left(z\right)$ in order to analyse the sensitivity of the models and how they respond to a cosmography analysis on the evolution of $H\left(z\right)$ and the deceleration parameter $q\left(z\right)$. As a result, we find that only the $\beta$-model can be unaffected to the variations of the previous datasets due to its several minima in the likelihood (degeneracies) of the related parameters with an overall percentage relative difference up to 8%.