Causes of Energy Density Inhomogeneity in Energy Momentum Squared Gravity

In the presence of an anisotropic fluid, we examine the irregularity factors for a spherically symmetric relativistic matter. In \(f(\mathcal{G},T^{2})\) gravity, we investigate the equations of motion and dynamical relations using a systematic construction, where \(T\) stands for the trace of the energy-momentum tensor, and \(\mathcal{G}\) is the Gauss–Bonnet term. With the use of the Weyl tensor, we examine two well-known differential equations that would lead to an analysis of the sources of inhomogeneities. In \(f(\mathcal{G},T^{2})\) gravity, the irregularity factors are investigated by taking specific cases in the adiabatic and non-adiabatic regimes. We find that the conformal tensor and additional curvature terms compromise inhomogeneity for a pressureless nonradiating fluid and an isotropic fluid. In contrast to other cases, for a nonradiating anisotropic fluid, we observe that the term \((\Pi+\mathcal{E})\) now accounts for the survival of density inhomogeneity, rather than just the Weyl tensor and the modified terms. The last case clearly illustrates how several components, namely, radiating terms, the fluid shear and the expansion scalar in the \(f(\mathcal{G},T^{2})\) framework, are accountable for the formation of inhomogeneities from a homogeneous state of the structure. In the case \(f(\mathcal{G},T^{2})=0\), all our results reduce to those of GR.