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

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