Extracting H0 and rd in q(t) parametrization models

This study explores five models derived from the Pacif parameterization scheme of the Hubble parameter (H), which represent different forms of the deceleration parameter (DP). Our primary objective is to determine the parameter distributions of the Hubble constant ($H_0$) and the sound horizon ($r_d$) for each model. Additionally, we analyze the evolution of the DP in each scenario to assess its implications for late-time cosmic evolution. In our analysis, we treat $r_d$ as a free parameter, which allows us to avoid relying on specific assumptions about early-Universe physics and recombination models. This approach ensures model independence on the early-Universe side and enables late-time cosmological observations to directly constrain $r_d$ alongside other key parameters. To extract the parameter distributions for the Pacif scheme, we use a combination of Baryon Acoustic Oscillation (BAO) data, Cosmic Chronometers (CC), Type Ia Supernovae (SNe Ia), Gamma-Ray Bursts (GRBs), and Quasars. Our results yield posterior distributions for $H_0$ and $r_d$, which demonstrate notable consistency with the Planck CMB data. Using the Akaike Information Criterion (AIC), we evaluate the models, finding that all five models provide robust agreement with the latest observational data.