Dark Sector Interactions: Probing the Hubble Parameter and the Sound Horizon

In this study, we explore the impact of the interacting parameter on dark matter in a model resulting from a parametrization of dark energy density. To ensure a model-independent approach, we treat $r_d$ as a free parameter, avoiding assumptions about the physics of the early Universe or specific recombination models. This approach allows late-time cosmological observations to directly constrain $r_d$ along with other parameters. Using recent measurements from the Dark Energy Spectroscopic Instrument (DESI) Year 1, cosmic chronometers (CC) and Pantheon$^{+}$ supernova (SNe Ia) data, we uncover a significant effect of the interacting parameter on dark matter. Our analysis reveals that while non-interacting models attribute 68.2\% of the cosmic energy density to dark energy, interacting models increase this share to 73.4\%. To further probe these differences, we evaluate the evolution of the deceleration parameter for each model, contrasting them against the $\Lambda$CDM paradigm and observational data from CC and SNe Ia measurements. Finally, we apply various statistical metrics to rigorously assess the performance of these models.