Observational constraints on FLRW, Bianchi type I and V brane models

This study explores the compatibility of Covariant Extrinsic Gravity (CEG) with current cosmological observations. We employ the chi-square statistic and Markov Chain Monte Carlo (MCMC) methods to fit the FLRW and Bianchi type-I and V brane models to the latest datasets, including Hubble, Pantheon+ Supernova samples, Big Bang Nucleosynthesis (BBN), Baryon Acoustic Oscillations (BAO), and the structure growth rate, $f{\sigma }_8(z)$. Parameters for FLRW universe consist $({\mathrm{\Omega }}_0^{\text{(b)}},{\mathrm{\Omega }}_0^{\text{(cd)}},{\mathrm{\Omega }}_0^{\text{(k)}},H_0,\gamma ,{\sigma }_8)$, while for the Bianchi model are $({\mathrm{\Omega }}_0^{\text{(b)}},{\mathrm{\Omega }}_0^{\text{(cd)}},{\mathrm{\Omega }}_0^{(\beta )},H_0,\gamma ,{\mathrm{\Omega }}_0^{(\theta )},{\sigma }_8)$. We determine the best values for cosmological parameters. For the FLRW model, these values depend on the sign of $\gamma$: $\gamma >0$ yields $\gamma =0.00008_{-0.00011}^{+0.00015}$, and ${\mathrm{\Omega }}_0^{\text{(k)}}=0.014_{-0.022}^{+0.024}$ and $\gamma <0$ leads to $\gamma =-0.0226_{-0.0062}^{+0.0054}$, and ${\mathrm{\Omega }}_0^{\text{(k)}}=0.023_{-0.041}^{+0.039}$. In both cases ${\mathrm{\Omega }}_0^{\text{(k)}}>0$ represents a closed universe. Similarly, for the Bianchi type-V brane model, the parameter values vary with the sign of $\gamma$, resulting in $\gamma =0.00084_{-0.00021}^{+0.00019}$, ${\mathrm{\Omega }}_0^{(\beta )}=0.0258_{-0.0063}^{+0.0052}$, and ${\mathrm{\Omega }}_0^{\theta }(\times {10}^{-5})=4.19_{-0.75}^{+0.67}$ (as with the density parameter of stiff matter) for $\gamma >0$, and $\gamma =-0.00107_{-0.00020}^{+0.00019}$, ${\mathrm{\Omega }}_0^{(\beta )}=0.0259_{-0.0062}^{+0.0050}$, and ${\mathrm{\Omega }}_0^{\theta }(\times {10}^{-5})=4.17_{-0.98}^{+0.91}$ for $\gamma <0$. In both cases ${\mathrm{\Omega }}_0^{(\beta )}>0$, which represents the Bianchi type-V, because in the Bianchi type-I, $\beta =0$. Utilizing these obtained best values, we analyze the behavior of key cosmological parameters.