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-Lemaître-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.