Asymptotic Safety, Fractals, and Cosmology
These lecture notes introduce the basic ideas of the asymptotic safety approach to quantum Einstein gravity (QEG). In particular they provide the background for recent work on the possibly multi-fractal structure of the QEG space-times. Implications of asymptotic safety for the cosmology of the early Universe are also discussed.
A class of nonlocal truncations in quantum Einstein gravity and its renormalization group behavior
Motivated by the conjecture that the cosmological constant problem could be solved by strong quantum effects in the infrared we use the exact flow equation of Quantum Einstein Gravity to determine the renormalization group behavior of a class of nonlocal effective actions. They consist of the Einstein-Hilbert term and a general nonlinear function F(k, V) of the Euclidean space-time volume V. A partial differential equation governing its dependence on the scale k is derived and its fixed point is analyzed. For the more restrictive truncation of theory space where F(k, V) is of the form V+V ln V, V+V^2, and V+\sqrt{V}, respectively, the renormalization group equations for the running coupling…
From Big Bang to Asymptotic de Sitter: Complete Cosmologies in a Quantum Gravity Framework
Using the Einstein-Hilbert approximation of asymptotically safe quantum gravity we present a consistent renormalization group based framework for the inclusion of quantum gravitational effects into the cosmological field equations. Relating the renormalization group scale to cosmological time via a dynamical cutoff identification this framework applies to all stages of the cosmological evolution. The very early universe is found to contain a period of ``oscillatory inflation'' with an infinite sequence of time intervals during which the expansion alternates between acceleration and deceleration. For asymptotically late times we identify a mechanism which prevents the universe from leaving t…
Critical reflections on asymptotically safe gravity
Asymptotic safety is a theoretical proposal for the ultraviolet completion of quantum field theories, in particular for quantum gravity. Significant progress on this program has led to a first characterization of the Reuter fixed point. Further advancement in our understanding of the nature of quantum spacetime requires addressing a number of open questions and challenges. Here, we aim at providing a critical reflection on the state of the art in the asymptotic safety program, specifying and elaborating on open questions of both technical and conceptual nature. We also point out systematic pathways, in various stages of practical implementation, towards answering them. Finally, we also take…
Focus on quantum Einstein gravity
The gravitational asymptotic safety program summarizes the attempts to construct a consistent and predictive quantum theory of gravity within Wilson's generalized framework of renormalization. Its key ingredient is a non-Gaussian fixed point of the renormalization group flow which controls the behavior of the theory at trans-Planckian energies and renders gravity safe from unphysical divergences. Provided that the fixed point comes with a finite number of ultraviolet-attractive (relevant) directions, this construction gives rise to a consistent quantum field theory which is as predictive as an ordinary, perturbatively renormalizable one. This opens up the exciting possibility of establishin…
Asymptotically safe Lorentzian gravity.
The gravitational asymptotic safety program strives for a consistent and predictive quantum theory of gravity based on a non-trivial ultraviolet fixed point of the renormalization group (RG) flow. We investigate this scenario by employing a novel functional renormalization group equation which takes the causal structure of space-time into account and connects the RG flows for Euclidean and Lorentzian signature by a Wick-rotation. Within the Einstein-Hilbert approximation, the $\beta$-functions of both signatures exhibit ultraviolet fixed points in agreement with asymptotic safety. Surprisingly, the two fixed points have strikingly similar characteristics, suggesting that Euclidean and Loren…
R2phase diagram of quantum Einstein gravity and its spectral dimension
Within the gravitational asymptotic safety program, the renormalization group (RG) flow of the ${R}^{2}$ truncation in three and four spacetime dimensions is analyzed in detail. In particular, we construct RG trajectories which emanate from the non-Gaussian UV fixed point and possess long classical regimes where the effective average action is well approximated by the classical Einstein-Hilbert action. As an application we study the spectral dimension of the effective quantum Einstein gravity spacetimes resulting from these trajectories, establishing that the picture of a multifractal spacetime is robust under the extension of the truncated theory space. We demonstrate that regimes of const…
RG flows of Quantum Einstein Gravity in the linear-geometric approximation
We construct a novel Wetterich-type functional renormalization group equation for gravity which encodes the gravitational degrees of freedom in terms of gauge-invariant fluctuation fields. Applying a linear-geometric approximation the structure of the new flow equation is considerably simpler than the standard Quantum Einstein Gravity construction since only transverse-traceless and trace part of the metric fluctuations propagate in loops. The geometric flow reproduces the phase-diagram of the Einstein-Hilbert truncation including the non-Gaussian fixed point essential for Asymptotic Safety. Extending the analysis to the polynomial $f(R)$-approximation establishes that this fixed point come…
Finite Quantum Gravity Amplitudes: No Strings Attached
We study the gravity-mediated scattering of scalar fields based on a parameterisation of the Lorentzian quantum effective action. We demonstrate that the interplay of infinite towers of spin zero and spin two poles at imaginary squared momentum leads to scattering amplitudes that are compatible with unitarity bounds, causal, and scale-free at trans-Planckian energy. Our construction avoids introducing non-localities or the massive higher-spin particles that are characteristic in string theory.
Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation
The exact renormalization group equation for pure quantum gravity is used to derive the non-perturbative $\Fbeta$-functions for the dimensionless Newton constant and cosmological constant on the theory space spanned by the Einstein-Hilbert truncation. The resulting coupled differential equations are evaluated for a sharp cutoff function. The features of these flow equations are compared to those found when using a smooth cutoff. The system of equations with sharp cutoff is then solved numerically, deriving the complete renormalization group flow of the Einstein-Hilbert truncation in $d=4$. The resulting renormalization group trajectories are classified and their physical relevance is discus…
Bimetric Renormalization Group Flows in Quantum Einstein Gravity
The formulation of an exact functional renormalization group equation for Quantum Einstein Gravity necessitates that the underlying effective average action depends on two metrics, a dynamical metric giving the vacuum expectation value of the quantum field, and a background metric supplying the coarse graining scale. The central requirement of "background independence" is met by leaving the background metric completely arbitrary. This bimetric structure entails that the effective average action may contain three classes of interactions: those built from the dynamical metric only, terms which are purely background, and those involving a mixture of both metrics. This work initiates the first …
Matter Induced Bimetric Actions for Gravity
The gravitational effective average action is studied in a bimetric truncation with a nontrivial background field dependence, and its renormalization group flow due to a scalar multiplet coupled to gravity is derived. Neglecting the metric contributions to the corresponding beta functions, the analysis of its fixed points reveals that, even on the new enlarged theory space which includes bimetric action functionals, the theory is asymptotically safe in the large $N$ expansion.