Impact of Interacting Dark Energy on the Growth of Matter Density Perturbations: Observational Constraints from DESI and Multi-Probe Data

We investigate the impact of a non-gravitational interaction between dark matter and dark energy on the growth rate of matter density perturbations within the framework of a $w$CDM scenario, where the coupling is proportional to the dark energy density. To incorporate the effects of this interaction in cosmological analysis, we develop a parameterization for the growth rate based on a second-order approximation for the growth index $\gamma$ that explicitly includes the coupling constant $\alpha$. This formalism reveals a theoretical degeneracy: the coupling induces a correction $\mathrm{\Delta }\gamma \simeq 1.1\alpha$, allowing an interacting dark energy model to mimic the growth index predicted by certain modified gravity theories. We confront the model with the latest multi-probe observations, including the Pantheon+ sample of Type~Ia supernovae, Baryon Acoustic Oscillation (BAO) data from the Sloan Digital Sky Survey (SDSS) and the second data release (DR2) of the Dark Energy Spectroscopic Instrument (DESI), Cosmic Microwave Background (CMB) measurements, Hubble parameter $H(z)$ data, and redshift-space distortion (RSD) measurements, to simultaneously constrain the coupling strength $\alpha$ and the dark energy equation of state $w$. Our analysis finds $\alpha$ consistent with zero and $w$ with $-1$ at the $1\sigma$ confidence level, showing no statistical evidence for a departure from the standard $\mathrm{\Lambda }$CDM cosmology. The observational constraints effectively break the theoretical degeneracy with modified gravity, limiting the possible interaction-induced shift in the growth index to $\mathrm{\mid }\mathrm{\Delta }\gamma \mathrm{\mid }\lesssim 0.03$ ($3\sigma$). This establishes the growth index as a robust diagnostic for distinguishing between a non-minimal interaction in the dark sector and a genuine modification of gravity.