Noether Symmetry Analysis of Anisotropic Universe in Modified Gravity

In this paper, we study anisotropic universe using Noether symmetries in modified gravity. In particular, we choose locally rotationally symmetric Bianchi type-$I$ universe for the analysis in $f(R,\mathcal{G})$ gravity, where $R$ is the Ricci scalar and $\mathcal{G}$ is the Gauss-Bonnet invariant. Firstly, a model $f(R,\mathcal{G})=f_0R^l+f_1\mathcal{G}^n$ is proposed and the corresponding Noether symmetries are investigated. Further, we have also recovered the Noether symmetries for $f(R)$ and $f(\mathcal{G})$ theories of gravity. Secondly, some important cosmological solutions are reconstructed. Exponential and power law solutions are reported for a well-known $f(R,\mathcal{G})$ model, i.e., $f(R,\mathcal{G})=f_0R^n\mathcal{G}^{1-n}$. Especially, the Kasner's solution is recovered and it is anticipated that the familiar de-Sitter spacetime giving $\Lambda CDM$ cosmology may be reconstructed for some suitable value of $n$.