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

This study explores the compatibility of Covariant Extrinsic Gravity (CEG), a braneworld scenario with an arbitrary number of non-compact extra dimensions, with current cosmological observations. We employ the chi-square statistic and Markov Chain Monte Carlo (MCMC) methods to fit the Friedmann–Lemaître–Robertson–Walker (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, $f{\sigma }_8\left(z\right)$. Parameters for FLRW universe consist $\left({\Omega }_0^{\text{(b)}},{\Omega }_0^{\text{(cd)}},{\Omega }_0^{\text{(k)}},H_0,\gamma ,{\sigma }_8\right)$, while for the Bianchi model are $\left({\Omega }_0^{\text{(b)}},{\Omega }_0^{\text{(cd)}},{\Omega }_0^{\left(\beta \right)},H_0,\gamma ,{\Omega }_0^{\left(\theta \right)},{\sigma }_8\right)$. By comparing our models to observational data, we determine the best values for cosmological parameters. For the FLRW model, these values depend on the sign of $\gamma$ (which gives the time variation of gravitational constant in Hubble time unit): $\gamma >0$ yields $\gamma =0.00008_{-0.00011}^{+0.00015}$, and ${\Omega }_0^{\text{(k)}}=0.014_{-0.022}^{+0.024}$ and $\gamma <0$ leads to $\gamma =-0.0226_{-0.0062}^{+0.0054}$, and ${\Omega }_0^{\text{(k)}}=0.023_{-0.041}^{+0.039}$. It should be noted that in both cases ${\Omega }_0^{\text{(k)}}>0$, which represents a closed universe. Similarly, for the Bianchi type-V brane model, the parameter values vary with the sign of $\gamma$, resulting in $\gamma =0.00084_{-0.00021}^{+0.00019}$, ${\Omega }_0^{\left(\beta \right)}=0.0258_{-0.0063}^{+0.0052}$, and ${\Omega }_0^{\theta }(\times 10^{-5})=4.19_{-0.75}^{+0.67}$ (as with the density parameter of stiff matter) for $\gamma >0$, and $\gamma =-0.00107_{-0.00020}^{+0.00019}$, ${\Omega }_0^{\left(\beta \right)}=0.0259_{-0.0062}^{+0.0050}$, and ${\Omega }_0^{\theta }(\times 10^{-5})=4.17_{-0.98}^{+0.91}$ for $\gamma <0$. In both cases ${\Omega }_0^{\left(\beta \right)}>0$, which represents the Bianchi type-V, because in the Bianchi type-I, $\beta =0$. Subsequently, utilizing these obtained best values, we analyze the behavior of key cosmological parameters such as the Hubble parameter, deceleration parameter, distance modulus, equation of state, and density parameters that characterize both matter and the geometric component of dark energy, as functions of redshift. Our results notably show that the FLRW model with $\gamma <0$ is more compatible with observational data than the Bianchi model, based on various statistical criteria.