Impact of Tolman–Kuchowicz solution on dark energy compact stars in f(R) theory

In this study, we model a new relativistic dark energy star by enforcing a static and isotropic spherically symmetric fluid distribution within the framework of $f\left(R\right)$ gravity, utilizing the Starobinsky model in the field equations. The analysis is conducted using the Tolman–Kuchowicz metric potentials and by employing the equation of state which models dark energy with its density proportional to the isotropic perfect fluid density. By applying junction conditions and considering the Schwarzschild solution as the exterior geometry, we compute the unknown parameters of the constructed model. The physical plausibility of the model is thoroughly examined, ensuring the regularity and consistency of metric functions and matter variables. We analyze energy conditions to confirm the system’s physical realism and derive the mass–radius relationship along with gravitational redshift, providing significant insights into the structure of the compact star candidate 4U 1820-30. The hydrostatic equilibrium is evaluated using the TOV equation, while stability is determined through the causality condition and adiabatic index. The results demonstrate that the proposed model, for the chosen values of the model parameter, is physically consistent and satisfies all necessary stability criteria, providing a viable and realistic description of a compact stellar structure influenced by dark energy.