Asymptotic freedom and safety in quantum gravity

Abstract

We compute non-perturbative flow equations for the couplings of quantum gravity in fourth order of a derivative expansion. The gauge invariant functional flow equation for arbitrary metrics allows us to extract β-functions for all couplings. In our truncation we find two fixed points. One corresponds to asymptotically free higher derivative gravity, the other is an extension of the asymptotically safe fixed point in the Einstein-Hilbert truncation or extensions thereof. The infrared limit of the flow equations entails only unobservably small modifications of Einstein gravity coupled to a scalar field. Quantum gravity can be asymptotically free, based on a flow trajectory from the corresponding ultraviolet fixed point to the infrared region. This flow can also be realized by a scaling solution for varying values of a scalar field. As an alternative possibility, quantum gravity can be realized by asymptotic safety at the other fixed point. There may exist a critical trajectory between the two fixed points, starting in the extreme ultraviolet from asymptotic freedom. We compute critical exponents and determine the number of relevant parameters for the two fixed points. Evaluating the flow equation for constant scalar fields yields the universal gravitational contribution to the effective potential for the scalars. CONTENTS I. Introduction II. Summary: Setup, flow equations and fixed point structure A. Effective action for gravity B. Flow equations C. Asymptotic freedom D. Asymptotic safety E. Interpolating functions F. Gravity contributions to the flow of the effective potential for scalars III. Flow generator and heat kernel method A. Scalar bosons B. Gauge bosons C. Weyl fermions IV. Flow generators from metric fluctuations A. Physical metric decomposition and gauge invariant flow B. Physical metric fluctuations C. Measure contribution V. R-fixed point and critical exponents A. Einstein-Hilbert truncation B. R 2 truncation C. Vanishing cosmological constant D. Full system VI. Infrared region VII. Conclusions A. Setup

Quantum gravity and scale symmetry in cosmology

“…Towards the beginning in the early stages of inflation terms with more than two derivatives may become more important. This holds, in particular, if quantum gravity is asymptotically free [36][37][38][39] with dominant terms quadratic in the curvature tensor involving four derivatives. It is also possible that the beginning stage is better described in terms of other degrees of freedom, for example by gauge fields and a vierbein in "pregeometry" [40].…”

“…We remain for this note with the metric. We also do not discuss the interesting possibility of asymptotically free quantum gravity which involves higher order curvature invariants [36][37][38][39].…”

The Quantum Gravity Connection between Inflation and Quintessence

“…At the ultraviolet fixed point for k → ∞ nothing changes anymore, permitting to extrapolate the model to arbitrarily high momenta or short distances and to render thereby quantum gravity complete. This is a typical scenario of asymptotic safety [67,[72][73][74][75][76][77][78][79], while asymptotic freedom [80][81][82] may also be possible [83] in the presence of higher order curvature terms. (We omit a discussion of the higher order curvature terms in this note because they play only a negligible role for the crossover cosmologies in the range relevant for observations).…”