Constructions of $f(R,G,\mathcal{T})$ Gravity from Some Expansions of the Universe

Here we propose the extended modified gravity theory named as $f(R,G,\mathcal{T})$ gravity where $R$ is the Ricci scalar, $G$ is the Gauss-Bonnet invariant and $\mathcal{T}$ is the trace of the stress-energy tensor. We derive the gravitational field equations in $f(R,G,\mathcal{T})$ gravity by taking least action principle. Next we construct the $f(R,G,\mathcal{T})$ in terms of $R$, $G$ and $\mathcal{T}$ in de Sitter as well as power law expansion. We also construct $f(R,G,\mathcal{T})$ if the expansion follows the finite time future singulary (big rip singularity). We investigate the energy conditions in this modified theory of gravity and examine the validities of all energy conditions. Finally, we analyze the stability of the constructed modified gravity.