Renormalizing the vacuum energy in cosmological spacetime: implications for the cosmological constant problem

Abstract

The renormalization of the vacuum energy in quantum field theory (QFT) is usually plagued with theoretical conundrums related not only with the renormalization procedure itself, but also with the fact that the final result leads usually to very large (finite) contributions incompatible with the measured value of

$$\Lambda$$ in cosmology. As a consequence, one is bound to extreme fine-tuning of the parameters and so to sheer unnaturalness of the result and of the entire approach. We may however get over this adversity using a different perspective. Herein, we compute the zero-point energy (ZPE) for a nonminimally coupled (massive) scalar field in FLRW spacetime using the off-shell adiabatic renormalization technique employed in previous work. The on-shell renormalized result first appears at sixth adiabatic order, so the calculation is rather cumbersome. The general off-shell result yields a smooth function $$\rho _{\mathrm{vac}}(H)$$ made out of powers of the Hubble rate and/or of its time derivatives involving different (even) adiabatic orders $$\sim H^N$$ ( $$N=0, 2,4,6,\ldots )$$ , i.e. it leads, remarkably enough, to the running vacuum model (RVM) structure. We have verified the same result from the effective action formalism and used it to find the $$\beta$$ -function of the running quantum vacuum. No undesired contributions $$\sim m^4$$ from particle masses appear and hence no fine-tuning of the parameters is needed in $$\rho _{\mathrm{vac}}(H)$$ . Furthermore, we find that the higher power $$\sim H^6$$ could naturally drive RVM-inflation in the early universe. Our calculation also elucidates in detail the equation of state of the quantum vacuum: it proves to be not exactly $$-1$$ and is moderately dynamical. The form of $$\rho _{\mathrm{vac}}(H)$$ at low energies is also characteristic of the RVM and consists of an additive term (the so-called ‘cosmological constant’) together with a small dynamical component $$\sim \nu H^2$$ ( $$|\nu |\ll 1$$ ). Finally, we predict a slow ( $$\sim \ln H$$ ) running of Newton’s gravitational coupling G( H). The physical outcome of our semiclassical QFT calculation is revealing: today’s cosmic vacuum and the gravitational strength should be both mildly dynamical.