Approaches to Quantum Gravity

126 R. PercacciEinstein–Hilbert action, which describes GR at macroscopic distances, may playan important role also in the UV limit, as the results of section 8.4 indicate. Withthis in mind, one can explore the consequences of a RG running of gravitationalcouplings also in other regimes.Motivated in part by possible applications to the hierarchy problem, Percacci[35] considered a theory with an action of the form (8.25), in the intermediateregime between the scalar mass and the Planck mass. Working in cutoffunits (8.13), it was shown that the warped geometry of the Randall–Sundrummodel can be seen as a geometrical manifestation of the quadratic running of themass.For applications to black hole physics, Bonanno & Reuter [6] have includedQuantum Gravity effects by substituting G with G(k) in the Schwarzschild metric,where k = 1/r and r is the proper distance from the origin. This is a gravitationalanalogue of the Ühling approximation of QED. There is a softening of the singularityat r = 0, and it is predicted that the Hawking temperature goes to zero forPlanck mass black holes, so that the evaporation stops at that point.In a cosmological context, it would be natural to identify the scale k with afunction of the cosmic time. Then, in order to take into account the RG evolutionof the couplings, Newton’s constant and the cosmological constant can bereplaced in Friedman’s equations by the effective Newton’s constant and the effectivecosmological constant calculated from the RG flow. With the identificationk = 1/t, where t is cosmic time, Bonanno & Reuter [7] have applied this idea tothe Planck era, finding significant modifications to the cosmological evolution; amore complete picture extending over all of cosmic history has been given in [38].It has also been suggested that an RG running of gravitational couplings may beresponsible for several astrophysical or cosmological effects. There is clearly scopefor various interesting speculations, which may even become testable against newcosmological data.Returning to the UV limit, it can be said that asymptotic safety has so farreceived relatively little attention, when compared to other approaches to QuantumGravity. Establishing this property is obviously only the first step: derivingtestable consequences is equally important and may prove an even greater challenge.Ultimately, one may hope that asymptotic safety will play a similar rolein the development of a QFT of gravity as asymptotic freedom played in thedevelopment of QCD.8.6 AcknowledgementsI wish to thank R. Floreanini, D. Dou, D. Perini, A. Codello and C. Rahmede forpast and present collaborations, and M. Reuter for many discussions.