Noether Symmetry Approach in Energy-Momentum Squared Gravity

In this paper, we investigate the newly developed $f(R,\mathbf{T}^2)$ theory ($R$ is the Ricci scalar and $\mathbf{T}^2=T_{\alpha\beta}T^{\alpha\beta},~T _{\alpha\beta}$ demonstrates the energy-momentum tensor) to explore some viable cosmological models. For this purpose, we use the Noether symmetry approach in the context of flat Friedmann-Robertson-Walker (FRW) universe. We solve the Noether equations of this modified theory for two types of models and obtain the symmetry generators as well as corresponding conserved quantities. We also evaluate exact solutions and investigate their physical behavior via different cosmological parameters. For the prospective models, the graphical behavior of these parameters indicate consistency with recent observations representing accelerated expansion of the universe. In the first case, we take a special model of this theory and obtain new class of exact solutions with the help of conserved quantities. Secondly, we consider minimal and non-minimal coupling models of $f(R,\mathbf{T} ^{2})$ gravity. We conclude that conserved quantities are very useful to derive the exact solutions that are used to study the cosmic accelerated expansion.