The Influence of Spheroidicity on the Complexity in Compact Objects Utilizing the Vaidya-Tikekar Superdense Star Model: A Comparative Study

In this paper, we investigate how the geometrical parameter of spheroidicity, which quantifies the deviation from spherical symmetry, influences the complexity of self-gravitating, static systems. We employ the concept of complexity in self-gravitating systems as formulated by Herrera et al. (Phys. Rev. D 97, 044010, 2018) and apply it to various models of compact stellar structures, all set within the framework of the Vaidya-Tikekar (VT) background geometry. Specifically, we analyze three models: (i) the anisotropic compact stellar model by Das et al. (Gen. Relativ. Gravit. 52, 101, 2020), constrained by the Karmakar condition, (ii) the stellar model by Das et al. (Eur. Phys. J. C 84, 13, 2024) employing the VT metric ansatz under the vanishing complexity condition, and (iii) the compact stellar model using a polytropic equation of state with the VT metric ansatz (Baskey et al., New Astron. 108, 102164, 2024). These models were selected based on whether their solutions are complexity-free or not. We establish a connection between the complexity factor and the spheroidal parameter to analyze how pressure anisotropy and density inhomogeneity, as constituent components, are influenced by this geometric parameter. Our findings indicate that deviations from spherical symmetry lead to a marked increase in the complexity of the stellar structure for all the models based on VT metric ansatz. Moreover, our results indicate an inconsistent pattern in the dependence of the complexity factor (\(\varvec{Y}_{\varvec{TF}}\)) and its components on spheroidicity (\(\varvec{K}\)), demonstrating that this relationship is both model-dependent and is not generic.