Non-Singular Bouncing Model in Energy Momentum Squared Gravity

This work is concerned to study the bouncing nature of the universe for an isotropic configuration of fluid $\mathcal{T}_{\alpha\beta}$ and Friedmann-Lema\^{i}tre-Robertson-Walker metric scheme. This work is carried out under the novel $f(\mathcal{G},\mathcal{T}_{\alpha \beta} \mathcal{T}^{\alpha \beta})$ gravitation by assuming a specific model i.e, $f(\mathcal{G},\mathcal{T}^2)=\mathcal{G}+\alpha \mathcal{G}^2+2\lambda \mathcal{T}^2$ with $\alpha$ and $\lambda$ are constants, serving as free parameters. {The terms $\mathcal{G}$ and $\mathcal{T}^2$ served as an Gauss-Bonnet invariant and square of the energy-momentum trace term as an inclusion in the gravitational action respectively, and is proportional to $\mathcal{T}^2=\mathcal{T}_{\alpha \beta} \mathcal{T}^{\alpha \beta}$.} A specific functional form of the Hubble parameter is taken to provide the evolution of cosmographic parameters. A well known equation of state parameter, $\omega(t)=-\frac{k \log (t+\epsilon )}{t}-1$ is used to represent the dynamical behavior of energy density, matter pressure and energy conditions. A detailed graphical analysis is also provided to review the bounce. Furthermore, all free parameters are set in a way, to make the supposed Hubble parameter act as the bouncing solution and ensure the viability of energy conditions. Conclusively, all necessary conditions for a bouncing model are checked.