Dark energy and QCD instanton vacuum in Friedmann-Lemaitre-Robertson-Walker universe

Abstract

The standard model of the universe, $\lambda$CDM, is based on the Friedmann-Lema\^itre-Robertson-Walker metric with a flat three-dimensional coordinate space and the Friedmann equations~\cite{ParticleDataGroup:2024cfk}. The cosmological constant $\lambda$ provides the cancellation of the matter field contributions in the flat (Minkowski) space, as was proposed long ago in 1967 by Zeldovich. The dynamical dark energy appears on the surface of the vacuum energy of matter fields at the flat (Minkowski) space. Within the Standard Model, the gluon Yang-Mills (YM) fields are playing a specific role since the properties of their vacuum, where there is the presence of the gluon condensate, provide the nonperturbative vacuum energy. It is natural to apply the successful instanton liquid model (ILM) of the QCD vacuum and its lowest excitations. Our aim is to calculate the contribution of gluon YM fields to the dark energy density. We find that the universe metric is generating the QCD vacuum excitation, which gives the contribution to the dark energy density. But this one may hardly play a central role in the dynamics of the universe, since its timescale is too small. We also find the equation-of-state parameters $w_0=-1,\,\,\,w_a=0$ in accordance with $\lambda$CDM, while the newest data, analyzed at~\cite{Shajib:2025tpd}, give them at least in the range $-0.91 <w_0< -0.73,\,\,\,\,-1.05< w_a <- 0.65$. They are requesting a contribution from an ultralight scalars such an axions, or from YM field topological configurations with the nontrivial holonomy due to the deviation from a pure de Sitter state ~\cite{VanWaerbeke:2025shm}.