Equation of state of the running vacuum

Abstract Recent studies of quantum field theory in FLRW spacetime suggest that the cause of the speeding up of the universe is the running vacuum (RV), see Moreno-Pulido and Solà Peracaula (Eur Phys J C 82 (6): 551, 2022; Eur Phys J C 80 (8): 692, 2020). Appropriate renormalization of the energy-momentum tensor shows that the vacuum energy density is a smooth function of the Hubble rate and its derivatives: ᵥac= ᵥac (H, Ḣ, H, ) ρ vac = ρ vac (H, H ˙, H ¨, …). This is because in QFT the quantum scaling of ᵥac ρ vac with the renormalization point turns into cosmic evolution with H. As a result, any two nearby points of the cosmic expansion during the standard FLRW epoch are smoothly related through ᵥac O (H²) δ ρ vac ∼ O (H 2). In our approach, what we call the ‘cosmological constant’ Λ is just the nearly sustained value of 8 G (H) ᵥac (H) 8 π G (H) ρ vac (H) around (any) given epoch, where G (H) is the running gravitational coupling. In the present study, after summarizing the main QFT calculations supporting the RV approach, we focus on the calculation of the equation of state (EoS) of the RV for the entire cosmic history within such a QFT framework. In particular, in the very early universe, where higher (even) powers ᵥac O (HN) ρ vac ∼ O (H N) (N=4, 6, N = 4, 6, ⋯) triggered inflation during a short period in which H= H = const, the vacuum EoS is very close to wᵥac=-1 w vac = - 1. This ceases to be true during the FLRW era, where it adopts the EoS of matter during the relativistic (wᵥac=1/3 w vac = 1 / 3) and non-relativistic (wᵥac=0 w vac = 0) epochs. Interestingly enough, we find that in the late universe the EoS becomes mildly dynamical and mimics quintessence, wᵥac -1 w vac ≳ - 1. It finally asymptotes to -1 - 1 in the remote future, but in the transit the RV helps alleviating the H₀ H 0 </