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Cosmological back-reaction in modified gravity and its implications for dark energy

We study the effective stress-energy tensor induced by cosmological inhomogeneity in $f(R)=R+cR^2$ and equivalent scalar-tensor theories, motivated both by models of early universe inflation and by phenomenological alternative cosmologies to the standard $Λ$-CDM. We use Green and Wald's framework for averaging over classical fluctuations of short-wavelength $λ$. By ensuring that the leading non-linear terms from the fluctuations of the Einstein terms and the corrections both contribute in the formal limit as $λ\to0$, we derive a diffeomorphism invariant effective stress-energy tensor whose trace is non-vanishing and of the right sign to potentially account for the current acceleration of the universe. However a more phenomenologically acceptable dark energy model would be required if this effect were to fully account for the current acceleration.