Interaction of polytropic dark energy in cosmological model: Constraints from observational data

We investigate an interacting polytropic dark energy (PDE) model characterised by the equation of state $p_{d} = \alpha \rho_{d}^{\,1+\frac{1}{\beta}}$, where the interaction between dark energy and pressureless matter is modelled via a linear coupling term $Q = 3\eta H\rho_{d}$. The background dynamics are formulated by deriving the Hubble parameter in the interacting scenario, and the model parameters are constrained through a Markov Chain Monte Carlo (MCMC) analysis using three joint observational data sets: Hubble77+BAO26, Hubble77+Pantheon$^+$, and Hubble77+BAO26+DESI DR2. The resulting best-fit values of $(H_0, \Omega_{d0}, \eta)$ are $(69.22^{+1.27}_{-1.24},\,0.73^{+0.02}_{-0.02},\,-0.22^{+0.10}_{-0.12})$, $(69.23^{+1.27}_{-1.22},\,0.73^{+0.02}_{-0.02},\,-0.34^{+0.15}_{-0.17})$, and $(67.77^{+1.26}_{-1.24},\,0.73^{+0.02}_{-0.02},\,\\-0.02^{+0.10}_{-0.11})$ respectively for the respective data combinations. Our results indicate a positive energy density and negative pressure over the full redshift range, with the evolution of the equation-of-state parameter and state finder parameters placing the model firmly within the Quintessence regime. The study of the deceleration parameter also reveals a shift from a decelerating to an accelerating cosmic expansion. The estimated present age of the Universe is $14\,\mathrm{Gyr}$, consistent with recent observational data. Furthermore, the sign of $Q$ implies a current energy transfer from dark energy to matter. These findings support the interacting PDE framework as a viable candidate for explaining late-time cosmic acceleration and related large-scale dynamics.