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, $fσ_8(z)$. Parameters for FLRW universe consist $\left(Ω^{\text{(b)}}_0, Ω^{\text{(cd)}}_0, Ω^{\text{(k)}}_0, H_0, γ, σ_8\right)$, while for the Bianchi model are $\left(Ω^{\text{(b)}}_0, Ω^{\text{(cd)}}_0, Ω^{(β)}_0, H_0, γ, Ω^{(θ)}_0, σ_8\right)$. We determine the best values for cosmological parameters. For the FLRW model, these values depend on the sign of $γ$: $γ> 0$ yields $γ=0.00008^{+0.00015}_{-0.00011}$, and $Ω^{\text{(k)}}_0=0.014^{+0.024}_{-0.022}$ and $γ< 0$ leads to $γ=-0.0226^{+0.0054}_{-0.0062}$, and $Ω^{\text{(k)}}_0=0.023^{+0.039}_{-0.041}$. In both cases $Ω^{\text{(k)}}_0>0$ represents a closed universe. Similarly, for the Bianchi type-V brane model, the parameter values vary with the sign of $γ$, resulting in $γ= 0.00084^{+0.00019}_{-0.00021}$, $Ω^{(β)}_0 =0.0258^{+0.0052}_{-0.0063} $, and $Ω^θ_0(\times 10^{-5} ) = 4.19^{+0.67}_{-0.75}$ (as with the density parameter of stiff matter) for $γ> 0$, and $γ= -0.00107^{+0.00019}_{-0.00020}$, $Ω^{(β)}_0 = 0.0259^{+0.0050}_{-0.0062} $, and $Ω^θ_0(\times 10^{-5} ) = 4.17^{+0.91}_{-0.98}$ for $γ< 0$. In both cases $Ω^{(β)}_0>0$, which represents the Bianchi type-V, because in the Bianchi type-I, $β=0$. Utilizing these obtained best values, we analyze the behavior of key cosmological parameters.