This manuscript is devoted to the investigation of the Bianchi type I universe in the context of f(R, T) gravity. For this purpose, we explore the exact solutions of locally rotationally symmetric Bianchi type I spacetime. The modified field equations are solved by assuming an expansion scalar \(\theta \) proportional to the shear scalar \(\sigma \), which gives \(A=B^n\), where A, B are the metric coefficients and n is an arbitrary constant. In particular, three solutions have been found and physical quantities are calculated in each case.

This paper is devoted to discussing compact stars in $f(\mathcal{R},\mathcal{G})$ gravity, where $\mathcal{R}$ and $\mathcal{G}$ denote the Ricci scalar and Gauss-Bonnet invariant, respectively. To meet this aim, we consider spherically symmetric space-time with an anisotropic fluid distribution. In particular, the Karmarkar condition is used to explore the compact star solutions. Furthermore, we choose two specific models of compact stars, namely LMC X-4 ($\mathrm{mass}=1.29M/M_{\odot }$ and $\mathrm{radii}=9.711\mathrm{km}$) and EXO 1785-248 ($\mathrm{mass}=1.30M/M_{\odot }$ and $\mathrm{radii}=8.849\mathrm{km}$). We develop the field equations for $f(\mathcal{R},\mathcal{G})$ gravity by employing the Karmarkar condition with a specific model already reported in literature by Lake [ Phys. Rev. D 67, 104015 (2003)]. We further consider the Schwarzschild geometry for the matching conditions at the boundary. It is important to mention here that we calculate the values of all the involved parameters by imposing the matching condition. We have provided a detailed graphical analysis to discuss the physical acceptability of parameters, i.e., energy density, pressure, anisotropy, and gradients. We have also examined the stability of compacts stars by exploring the energy conditions, equation of state, generalized Tolman-Oppenheimer-Volkoff equation, causality condition, and adiabatic index. For the present analysis, we predict some numerical values in tabular form for central gravitational metric functions, central density, and central pressures components. We have also calculated the ratio $p_{rc}/{\rho }_c$ to check the validity of Zeldovich’s condition. Conclusively, it is found that our obtained solutions are physically viable with a well-behaved nature in $f(\mathcal{R},\mathcal{G})$ modified gravity for the compact star models under discussion.

Motivated by recent proposals of possible wormhole shape functions, we construct a wormhole shape function by employing the Karmarkar condition for static traversable wormhole geometry. The proposed shape function generates wormhole geometry that connects two asymptotically flat regions of spacetime and satisfies the required conditions. Further, we discuss the embedding diagram in three-dimensional Euclidean space to present the wormhole configurations. The main feature of current study is to consider three well-known f(R) gravity models, namely exponential gravity model, Starobinsky gravity Model and Tsujikawa f(R) gravity model. Moreover, we investigate that our proposed shape function provides the wormhole solutions with less (or may be negligible) amount of exotic matter corresponding to the appropriate choice of f(R) gravity models and suitable values of free parameters. Interestingly, the solutions obtained for this shape function generate stable static spherically symmetric wormhole structure in the context of non-existence theorem in f(R) gravity. This may lead to a better analytical representation of wormhole solutions in other modified gravities for the suggested shape function.