Renormalization-group running cosmologies: A scale-setting procedure

Abstract

For cosmologies including scale dependence of both the cosmological and the gravitational constant, an additional consistency condition dictated by the Bianchi identities emerges, even if the energy-momentum tensor of ordinary matter stays individually conserved. For renormalization-group (RG) approaches it is shown that such a consistency relation ineluctably fixes the RG scale (which may have an explicit as well as an implicit time dependence), provided that the solutions of the RG equation for both quantities are known. Hence, contrary to the procedures employed in the recent literature, we argue that there is no more freedom in identification of the RG scale in terms of the cosmic time in such cosmologies.We carefully set the RG scale for the RG evolution phrased in a quantum gravity framework based on the hypothetical existence of an infrared (IR) fixed point, for the perturbative regime within the same framework, as well as for an evolution within quantum field theory (QFT) in a curved background. In the latter case, the implications of the scale setting for the particle spectrum are also briefly discussed. Recently, indisputable evidence has been mounting to suggest that the expansion of our universe is accelerating owing to the nonvanishing value of unclustered dark energy with negative pressure, see [1]. The crucial evidence for the existence of dark energy [or the cosmological constant (CC)] relies on the CMB observations [2] which strongly support a spatially flat universe as predicted by inflationary models. By combinations of all data a current picture of the universe emerges, in which about 2/3 of the critical energy density of the present universe is made up by a background dark energy with the parameter of the equation of state −1.38 ≤ w ≤ −0.82 at 95% confidence level [3]. Pressed by these data, theorists now need explain not only why the CC is small, but also why dark-energy domination over ordinary matter density has occurred for redshifts z < ∼ 1 (the coincidence problem). Although models with a truly static CC fit these data well, they have two additional drawbacks (besides the coincidence problem): (1) they cannot theoretically explain why the CC is today small but nonvanishing; (2) they have a problem to ensure a phase of inflation, an epoch in the early universe when the CC dominated other forms of energy density. Assessing the possibility to have dynamical dark energy, rolling scalar field models (quintessence fields) [4] with generic attractor properties [5] that make the present dark energy density insensitive to the broad range of unknown initial conditions, have been aimed at dealing with the coincidence problem. Still, a quintessence potential has to be fine-tuned to yield the present ratio of ordinary matter to quintessence energy density, at the same time allowing a phase dominated by matter so that structure can form; therefore these models cannot address the coincidence problem. In addition, such models may have difficulties in achieving the current quintessence equatio...

Renormalization group and decoupling in curved space

“…which has been guessed in [19] for the cosmological constant by analogy with a similar phenomena in QED. However, we have discovered this law not for the cosmological constant but only for the higher derivative terms.…”

Density perturbations for a running cosmological constant

“…(ii) Conservation law (following from the Bianchi identity) which regulates the exchange of energy between matter and vacuum, 2) and (iii) the renormalization group equation for the CC density, assuming a soft form of decoupling for the cosmological constant [11,29,31] …”

“…At the same time, the phenomenological proposal [11,12,29,30] shows that a positive result can not be ruled out. The hypothesis of a standard quadratic form of decoupling for the quantum effects of matter fields at low energies [11,31] leads to consistent cosmological models of a running CC with potentially observable consequences [32,33,34,35,36]. Among the many possibilities one can distinguish essentially two different models of running CC in QFT in curved space-time 1 .…”

Modified dispersion relations from the renormalization group of gravity

“…Interestingly this choice has also been advocated for applications to cosmology on the basis that the Bianchiidentity has to remain valid even in the presence of running coupling constants [21,23,24]. However, we stress that the framework presented here significantly differs from the one above.…”

“…The β-functions for gravity have also been discussed in different approaches (see e.g. [19,20,21,22,23,24]). The solutions to these ordinary differential equations are…”