Stellar structure and stability in f(R) gravity: An analysis with the Karmarkar condition

We derive the EFEs for $f\left(R\right)$ gravity within a spherically symmetric static spacetime, which allows us to determine the radial pressure, density and tangential pressure, as well as the stress–energy tensor for anisotropic compact stellar objects. The metric coefficients of the spherically symmetric static spacetime are determined using the Karmarkar condition (a constraint on the Riemann curvature tensor) specifically for the Carroll-Duvvuri-Trodden (CDT) model. The unknown metric coefficients are computed by applying matching conditions that compare the interior and exterior geometric solutions at the border of the stellar objects. These unknowns are expressed in terms of the mass and radius of the compact star being studied. Using these formulations, we analyze compact stars for their density and pressure evolution, TOV equations, EoS, sound speed and anisotropy factor. If these conditions are satisfied, a compact star is considered to be in a stable configuration. We have established the stability criteria for three compact stars with known observational data. This framework offers insights into the equilibrium and structural properties of compact stars under $f\left(R\right)$ gravity. It can also be extended to analyze the stability properties of other compact stars in modified gravity theories.