Constraining Dark Matter Parameters in the Lambda-CDM Framework: A Bayesian Comparison of Planck and DES Constraints

Abstract: We present a Bayesian analysis of cosmological parameter constraints from early- and late-universe observations, focusing on the matter density parameter ($\Omega_m$) and the amplitude of matter fluctuations ($\sigma_8$) within the $\Lambda$CDM framework. Using data from the Planck 2018 satellite mission and the Dark Energy Survey (DES) Year 3, we compute theoretical predictions for angular and matter power spectra via Boltzmann solvers and perform Markov Chain Monte Carlo (MCMC) sampling using the \texttt{emcee} Python package. Our key contribution is a direct and quantitative comparison of DES and Planck constraints, assessing their consistency using chi-squared analysis and Gaussian tension metrics. We find a statistically significant $6.46\sigma$ tension in $\Omega_m$ and a $2.68\sigma$ tension in $\sigma_8$ between the two datasets. These results provide fresh evidence of persistent discrepancies in cosmological parameter estimates and suggest that simple extensions to the $\Lambda$CDM model may be insufficient to fully reconcile early- and late-time observations, motivating the need for more complex theoretical models or refined treatment of systematics.