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 Λ 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 ρ vac (H) made out of powers of the Hubble rate and/or of its time derivatives involving different (even) adiabatic orders ∼ H N (N = 0, 2, 4, 6, ...), 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 β-function of the running quantum vacuum. No undesired contributions ∼ m 4 from particle masses appear and hence no fine-tuning of the parameters is needed in ρ vac (H). Furthermore, we find that the higher power ∼ 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 ρ 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 ∼ νH 2 (|ν| ≪ 1). Finally, we predict a slow (∼ 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.

The Cosmological Constant Problem and Running Vacuum in the Expanding Universe

“…All devised forms of DE are, in principle, plagued with the same fine tuning illness and to a degree which is no lesser than the CC itself. Paradoxically enough, it turns out that the vacuum energy can actually escape such a fate, as we will show here, after it is properly renormalized [46,47]. The vacuum is in fact the most fundamental notion of QFT.…”

“…This was first shown in the literature in Ref. [46] and further extended in [47]. In these works, the VED appears as an expansion in powers of H and its time derivatives: ρ vac = ρ vac (H, Ḣ, Ḧ, ...).…”

“…The Λ(t)-class has no obvious relation with fundamental physics, but it was useful to illustrate the possible effects implied by a time-evolving cosmological term. These models, however, should not be confused with the "running vacuum models" (RVMs), in which the dynamical dependence of Λ, or more precisely of the VED, stems from explicit computation of quantum effects in QFT in curved spacetime [46,47]. Therefore, despite some existing confusion in the literature, the RVMs are to be understood in that (much more) restricted sense.…”

“…On the theoretical side, the RVM was originally motivated by semi-qualitative RG arguments [76], which were later on further elaborated within an action functional approach [77]. However, the theoretical underpinning of the RVM within the full-fledged context of QFT in curved spacetime appeared only very recently, see [46] and also the very comprehensive exposition [47], where the topics about the renormalization of the vacuum energy in cosmological spacetime and the implications on the cosmological constant problem are taken up in extenso and in depth. For related studies about renormalization in curved spacetime, see e.g.…”

“…Conventions: We use the same geometric conventions as in [46,47]. They correspond to (+, +, +) in the standard classification by Misner-Thorn-Wheeler [84].…”