Observational Constraints and Cosmic Growth Index of Realistic $f(G)$ Gravity Frameworks using MCMC Analysis

Abstract

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, ${\Omega }_{m0}$. For Model~II, the full analysis yields $H_0=64.02^{+4.13}\ast -3.08,\mathrm{km},{\mathrm{s}}^{-1},\mathrm{Mp}{\mathrm{c}}^{-1}$ and $\Omega \ast m0=0.249_{-0.038}^{+0.047}$. We further assess the statistical performance of the models relative to $\Lambda$CDM using $\Delta {\mathrm{AIC}}_c$, $\Delta \mathrm{BIC}$, and $\Delta \mathrm{DIC}$. For Model~II, the full dataset gives $\Delta {\mathrm{AIC}}_c=4.233$ and $\Delta \mathrm{DIC}=5.971$, favoring the $\Lambda$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 $z\approx 0.5005$ and $z\approx 0.6086$ for Models~I and II, respectively, highlighting differences in their cosmological evolution.

28 pages, 7 figures