Quantum Gravity Slows Inflation

We consider the quantum gravitational back-reaction on an initially inflating, homogeneous and isotropic universe whose topology is T 3 × R. Although there is no secular effect at one loop, an explicit calculation shows that two-loop processes act to slow the rate of expansion by an amount which becomes non-perturbatively large at late times. By exploiting Feynman’s tree theorem we show that all higher loops act in the same sense. ∗ e-mail: [email protected] and [email protected] † e-mail: [email protected] Inflation provides a wonderful explanation for the fact that the cosmic microwave background is observed to be in thermal equilibrium to about one part in 105, even for regions of the universe which are only now coming into contact with one another [1]. However, inflationary cosmology has no pretensions of explaining why the dimensionless product of the cosmological constant and Newton’s constant is observed to be zero to within about one part in 10120 [2]. Indeed, the assumption of inflation imposes severe restrictions on any proposal for understanding this grotesque hierarchy. For if we assume that the responsible mechanism can screen a cosmological constant of any size, and that it makes no distinction between the bare cosmological constant and contributions from the matter potential, then one has to explain why inflation was ever able to start. One must also understand why the screening mechanism operates slowly enough to permit the ≈ 55 e-foldings of inflation needed to produce the isotropy of the cosmic microwave background. We have proposed that the cosmological constant isn’t unreasonably small but only appears so due to the screening effect of infrared processes in quantum gravity [3,4]. * These infrared processes can become strong because the graviton is massless and because a non-zero cosmological constant endows it with a self-interaction of dimension three. (By way of contrast, the massless gluons of QCD possess only dimension four couplings.) Since dimension three couplings between positive norm bosons lower the vacuum energy, the effect is to screen the bare cosmological constant. Like all infrared effects, our process derives from the causal and coherent superposition of interactions throughout the past lightcone. The effect is absent before the onset of inflation because thermal fluctuations disrupt the coherent superposition of interactions from different regions of the very early universe; it is only after inflation has redshifted the temperature that a coherent effect * It has also been suggested that screening can occur due to Hawking radiation [5], or from the existence of an infrared fixed point in various effective theories of gravitation [6]. Our mechanism is nearest to Ford’s proposal [7], which was based on the assumption that the coincidence limit of the graviton propagator grows in time. Note, however, that our formalism does not show temporal growth for the coincident propagator. It is also significant that — if such growth had been present — Ford’s effect would emerge from an entirely different set of diagrams than ours.