Linear Growth of Matter Perturbations Probed by Redshift-Space Distortions in Interacting $\Lambda(t)$CDM Cosmologies

In the context of a spatially flat $\Lambda(t)$CDM cosmology, we investigate interacting dark energy (IDE) scenarios characterized by phenomenological interaction terms proportional to the Hubble expansion rate and the dark energy density. Our analysis is performed at both the background and linear perturbation levels, with particular emphasis on the evolution of dark matter density fluctuations. Cosmological constraints are derived from a joint analysis of CMB distance priors, Baryon Acoustic Oscillations (BAO), Type Ia supernovae (SNe Ia) from Pantheon+, Redshift-Space Distortions (RSD), and $H(z)$ data from Cosmic Chronometers (CC). Using the linear growth of matter perturbations, we estimate the clustering parameter $S_8$ within IDE extensions of the flat $\Lambda(t)$CDM framework. At the perturbative level, we consider interaction terms of the form $Q_{\text{I}}=\varepsilon a H\bar{\rho}_{\Lambda(t)}$ (Model I) and $Q_{\text{II}}=\varepsilon H\bar{\rho}_{\Lambda(t)}$ (Model II). From the combined dataset, we obtain the constraints $S_8 = 0.870 \pm 0.026$ for Model I and $S_8 = 0.872 \pm 0.026$ for Model II. Finally, we discuss the implications for the coupling parameter $\varepsilon$, taking into account the semi-analytical approximations and observational data employed in this study.