Two-Fluid Model for Anisotropic Fluid Spacetime with Specific Stress–Energy Tensor Constraints and f(R)-Gravity

A two-fluid model can be described by an anisotropic fluid matter, and we introduced the notion of an anisotropic fluid spacetime. The algebraic and differential properties of an anisotropic fluid spacetime equipped with several forms of the stress–energy tensor is the focus of this research. We show that an anisotropic fluid spacetime with a radial pressure p‖, transverse pressure p⊥, and the energy density ρ is a generalized quasi-Einstein spacetime. We prove that a dark matter era or an anisotropic fluid spacetime with vanishing vorticity is represented by an anisotropic fluid spacetime endowed with a covariant constant stress–energy tensor; on the contrary, a dark matter era or the expansion scalar vanishes is represented by an anisotropic fluid spacetime endowed with a Codazzi type of stress–energy tensor, as long as A stays invariant under the velocity vector field ζ. Furthermore, we use the Killing velocity vector field, parallel vector fields to characterize Ricci Semi-Symmetric, T-recurrent, Pseudo-Ricci symmetric, and R^-harmonic anisotropic fluid spacetime. We find that the anisotropic fluid spacetime reflect a stiff matter and a radiation era with these geometric symmetries. Finally, we provide findings for an anisotropic fluid spacetime with a divergence-free matter tensor and the vanishing space-matter tensor and explore the dynamical aspects of cosmological epoch of an anisotropic fluid spacetime coupled with f(R)-gravity.