Nonlocal Effective Field Equations for Quantum Cosmology

The possibility that the strength of gravitational interactions might slowly increase with distance, is explored by formulating a set of effective field equations, which incorporate the gravitational, vacuum-polarization induced, running of Newton’s constant G. The resulting long distance (or large time) behaviour depends on only one adjustable parameter ξ, and the implications for the Robertson-Walker universe are calculated, predicting an accelerated power-law expansion at later times t ∼ ξ ∼ 1/H. e-mail address : [email protected] e-mail address : [email protected] In the Standard Model of particle interactions, all gauge couplings are known to run with energy. Recent non-perturbative studies of quantum gravity have suggested that the gravitational coupling too may depend on a scale related to curvature, and therefore macroscopic in size. In this Letter, we investigate the effects of a running gravitational constant G at large distances. This scale dependence is assumed to be driven by gravitational vacuum polarization effects, which produce an anti-screening effect some distance away from the primary source, and therefore tend to increase the strength of the gravitational coupling. A power law running of G will be implemented via manifestly covariant nonlocal terms in the effective gravitational action and field equations. We start by assuming [1, 2] that for Newton’s constant one has G(r) = G(0) [ 1 + cξ (r/ξ) 1/ν + O((r/ξ)) ] , (1) where the exponent ν is generally related to the derivative of the beta function for pure gravity evaluated at the non-trivial ultraviolet fixed point. Recent studies have ν−1 varying between 3.0 and 1.7 [2, 3, 4, 5, 6]. These estimates rely on three different, and unrelated, nonperturbative approaches to quantum gravity, based on the lattice path integral formulation, the two plus epsilon expansion of continuum gravity, and a momentum slicing scheme combined with renormalization group methods in the vicinity of flat space, respectively. In all three approaches a non-vanishing, positive bare cosmological constant is required for the consistency of the renormalization group procedure. The mass scale m = ξ−1 in Eq. (1) is supposed to determine the magnitude of quantum deviations from the classical theory. It seems natural to identify 1/ξ2 with either some very large average spatial curvature scale, or perhaps more appropriately with the Hubble constant (as measured today) determining the macroscopic expansion rate of the universe, via the correspondence ξ = 1/H , (2) in a system of units for which the speed of light equals one. A possible concrete scenario is one in which ξ−1 = H∞ = limt→∞H(t) = √ ΩΛH0 with H 2 ∞ = Λ3 , where Λ is the observed cosmological constant, and for which the horizon radius is H−1 ∞ . As it stands, the formula for the running of G is coordinate dependent, and we therefore replace it with a manifestly covariant expression involving the covariant d’Alembertian operator 2 = g ∇μ∇ν , (3) whose Green’s function in d spatial dimensions is known to behave as < x| 1 2 |y > δ(r − d(x, y | g)) ∼ 1 rD−2 , (4)