Observational Constraints and Cosmic Growth Index of Realistic $f(G)$ Gravity Frameworks using MCMC Analysis
We present a comprehensive observational analysis of modified Gauss--Bonnet, or $f(G)$, gravity by investigating both the background cosmological expansion and sub-horizon linear matter perturbations. We consider two viable functional forms: an arctangent parameterization (Model~I) and a generalized polynomial power-law model (Model~II). Using a Markov Chain Monte Carlo (MCMC) ensemble sampler, we constrain their parameter spaces through background and structure-growth observations. Our analysis employs cumulative combinations of Cosmic Chronometers (CC), the Pantheon+ Type Ia Supernovae compilation (PP), Redshift-Space Distortions (RSD), and the Year-1 Baryon Acoustic Oscillation measurements from the Dark Energy Spectroscopic Instrument (DESI BAO). The inclusion of DESI BAO data produces a noticeable downward shift in the preferred values of the Hubble constant, $H_0$, and present matter density parameter, $Ω_{m0}$. For Model~II, the full analysis yields $H_0=64.02^{+4.13}*{-3.08},\mathrm{km,s^{-1},Mpc^{-1}}$ and $Ω*{m0}=0.249^{+0.047}_{-0.038}$. We further assess the statistical performance of the models relative to $Λ$CDM using $Δ\mathrm{AIC}_c$, $Δ\mathrm{BIC}$, and $Δ\mathrm{DIC}$. For Model~II, the full dataset gives $Δ\mathrm{AIC}_c=4.233$ and $Δ\mathrm{DIC}=5.971$, favoring the $Λ$CDM baseline. At the perturbation level, both models exhibit stable growth histories compatible with large-scale structure observations. The models predict transitions in the late-time expansion dynamics at $z\approx0.5005$ and $z\approx0.6086$ for Models~I and II, respectively, highlighting differences in their cosmological evolution.