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We present a (hopefully) novel calculation of the vacuum energy in expandingFLRW spacetimes based on the renormalization of quantum field theory innon-zero backgrounds. We compute the renormalized effective action up to the$2-$point function and then apply the formalism to the cosmological backgroundsof interest. As an example we calculate for quasi de Sitter spacetimes theleading correction to the vacuum energy given by the tadpole diagram and showthat it behaves as $\sim H_0^2 \Lambda_{\rm pl}$ where $H_0$ is the Hubbleconstant and $\Lambda_{\rm pl}$ is the Planck constant. This is of the sameorder of magnitude as the observed dark energy density in the universe.

On the Problem of Vacuum Energy in FLRW Universes and Dark Energy