Background dynamics and observational constraints of flat and non-flat $\Lambda(t)$CDM models from $H(z)$ and DESI DR2 BAO measurements

In this work, we present comprehensive observational constraints on the time-varying vacuum $\Lambda(t)$CDM cosmology using the latest baryon acoustic oscillation (BAO) data from Dark Energy Spectroscopic Instrument (DESI) Data Release 2 measurements in combination with cosmic chronometer $H(z)$ data. We explicitly quantify the impact of vacuum dynamics on the expansion history, the total effective equation of state parameter of the unified cosmic fluid, the effective dark energy equation of state parameter, and the deceleration parameter in the spatially flat $\Lambda(t)$CDM model. We perform a full Markov Chain Monte Carlo (MCMC) analysis and statistical model comparison, providing a consistent assessment of the $\Lambda(t)$CDM model relative to the standard $\Lambda$CDM framework. Our results demonstrate that $H(z)$ and BAO observations strongly constrain deviations from the $\Lambda$CDM model, driving the vacuum dynamics parameter toward $\alpha \simeq 0$, while significantly reducing parameter degeneracies and alleviating the Hubble tension.