**1. Introduction**

26 Quantum Gravity

Wetterich, C. (1998). Effective Nonlocal Euclidean Gravity, *General Relativity and Gravitation*,

Vol.30, pp. 159-172

In Ref. [1], Weinberg suggested that the general theory of relativity may have a non-trivial UV fixed point, with a finite dimensional critical surface in the UV limit, so that it would be asymptotically safe with an S-matrix that depends on only a finite number of observable parameters. In Refs. [2–4], strong evidence has been calculated using Wilsonian [5–8] field-space exact renormalization group methods to support asymptotic safety for the Einstein-Hilbert theory. We have shown in Refs. [9–19] that the extension of the amplitude-based, exact resummation theory of Ref. [20] to the Einstein-Hilbert theory (we call the extension resummed quantum gravity) leads to UV fixed-point behavior for the dimensionless gravitational and cosmological constants, but with the bonus that the resummed theory is actually UV finite. More evidence for asymptotic safety for quantum gravity has been calculated using causal dynamical triangulated lattice methods in Ref. [21]1. There is no known inconsistency between our analysis and Refs. [2–4, 21]. Our results are also consistent with the results on leg renormalizability of quantum gravity in Refs. [23, 24].

The reader unfamiliar with the methods of Wilson in the context of the renormalization group may consult Refs. [2, 5–8] for the details of the approach. Here we stress that in the Wilsonian formulation of the renormalization group, it does not matter whether the theory under study is actually renormalizable because the idea is to thin the degrees of freedom to those relevant to the momentum scale *k* under study. When one does this, the operators in the theory then fall into the classes of relevant, marginal and irrelevant operators as one studies the response of the theory to changes in the value of *k*. If the theory is renormalizable, then as *k* → ∞ there will be a finite number of relevant or marginal operators in the effective action, yielding an S-matrix that depends on only a finite number of parameters. If the theory is non-renormalizable, there will be an infinite number of relevant or marginal operators in the effective action as *k* → ∞. It was for this reason that the authors

<sup>\*</sup>Work supported in part by NATO grant PST.CLG.980342.

<sup>1</sup> We also note that the model in Ref. [22] realizes many aspects of the effective field theory implied by the anomalous dimension of 2 at the UV-fixed point but it does so at the expense of violating Lorentz invariance.

for *k* → ∞.

The contact with cosmology then proceeds as follows. Using a phenomenological connection between the momentum scale *k* characterizing the coarseness of the Wilsonian graininess of

<sup>29</sup> Planck Scale Cosmology and Asymptotic Safety

Refs. [25] show that the standard cosmological equations admit of the following extension:

*a*˙ *a ρ* = 0

for the density *ρ* and scale factor *a*(*t*) with the Robertson-Walker metric representation as

<sup>1</sup> <sup>−</sup> *Kr*<sup>2</sup> <sup>+</sup> *<sup>r</sup>*

so that *K* = 0, 1, −1 correspond respectively to flat, spherical and pseudo-spherical 3-spaces

Using the UV fixed points for *g*<sup>∗</sup> and *λ*∗, the authors in Refs. [25] show that the extended cosmological system given above admits, for *K* = 0, a solution in the Planck regime where 0 ≤ *t* ≤ *t*class, with *t*class a "few" times the Planck time *tPl*, which joins smoothly onto a solution in the classical regime, *t* > *t*class, which coincides with standard Friedmann-Robertson-Walker phenomenology but with the horizon, flatness, scale free Harrison-Zeldovich spectrum, and entropy problems all solved purely by Planck scale quantum physics. We now review the results in Refs. [19] for these UV limits as implied by resummed quantum gravity theory as presented in [9–18] and show how to use them to predict the current value of Λ. In this way,

We start with the prediction for *g*∗, which we already presented in Refs. [9–19]. For the sake of completeness, let us we recapitulate the main steps in the calculation. Referring to Fig. 1, we have shown in Refs. [9–18] that the large virtual IR effects in the respective loop integrals for the scalar propagator in quantum general relativity can be resummed to the *exact* result

*<sup>g</sup>* (*k*)

*<sup>s</sup>* <sup>+</sup> *<sup>i</sup>�*) (5)

(*k*<sup>2</sup> − *<sup>m</sup>*<sup>2</sup> − <sup>Σ</sup>�

*<sup>F</sup>*(*k*)|resummed <sup>=</sup> *ieB*��

*dr*<sup>2</sup>

*<sup>N</sup>* = 0

*GN*(*t*) = *GN*(*k*(*t*))

*<sup>t</sup>* for *ξ* > 0, the authors in

(3)

Λ(*t*) = Λ(*k*(*t*)) (2)

*p*(*t*) = *ωρ*(*t*), (4)

<sup>2</sup>(*dθ*<sup>2</sup> + sin2 *θdφ*2)

the average effective action and the cosmological time *t*, *k*(*t*) = *<sup>ξ</sup>*

in Resummed Quantum Gravity: An Estimate of

( *a*˙ *a* )<sup>2</sup> + *K <sup>a</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> 3 Λ + 8*π* <sup>3</sup> *GN<sup>ρ</sup>*

*ρ*˙ + 3(1 + *ω*)

*ds*<sup>2</sup> <sup>=</sup> *dt*<sup>2</sup> <sup>−</sup> *<sup>a</sup>*(*t*)<sup>2</sup>

we put the arguments in Refs. [25] on a more rigorous theoretical basis.

for constant time t. The equation of state is

**3.** *g*<sup>∗</sup> **and** *λ*<sup>∗</sup> **in resummed quantum gravity**

*i*Δ�

where *p* is the pressure.

Λ˙ + 8*πρG*˙

in Ref. [2–4] have chosen to use Wilsonian methods to study the Einstein-Hilbert theory, which is naively non-renormalizable by the standard power-counting arguments. What they find is that there are only a finite number of relevant or marginal operators in the effective action as *k* → ∞, asymptotic safety. There is no contradiction with the naive expectation because the Wilsonian methods take into the account the non-perturbative changes in the scale dimensions of the theory's operators due to interactions. Unlike the methods in Refs. [2–4] which have unphysical cut-off dependence from thinning the degrees of freedom procedures and unphysical gauge dependence, our results have no such dependence on cut-offs or gauge choice. That we agree with the findings of Refs. [2–4] then strengthens these results. Contact with experiment is now in order.

Specifically, in Ref. [25], it has been argued that the approach in Refs. [2–4] to quantum gravity may provide a realization2 of the successful inflationary model [27, 28] of cosmology without the need of the inflaton scalar field: the attendant UV fixed point solution allows one to develop Planck scale cosmology that joins smoothly onto the standard Friedmann-Walker-Robertson classical descriptions so that one arrives at a quantum mechanical solution to the horizon, flatness, entropy and scale free spectrum problems. In Ref. [19], using the resummed quantum gravity theory [9–18], we recover the properties as used in Refs. [25] for the UV fixed point with "first principles" predictions for the fixed point values of the respective dimensionless gravitational and cosmological constants. Here, we carry the analysis one step further and arrive at a prediction for the observed cosmological constant Λ in the context of the Planck scale cosmology of Refs. [25]. We comment on the reliability of the result as well, as it will be seen already to be relatively close to the observed value [29–31]. More such reflections, as they relate to an experimentally testable union of the original ideas of Bohr and Einstein, will be taken up elsewhere [32].

The discussion is organized as follows. In the next section we review the Planck scale cosmology presented in Refs. [25]. In Section 3 we review our results [19] for the dimensionless gravitational and cosmological constants at the UV fixed point. In Section 4, we combine the Planck scale cosmology scenario [25] with our results to predict the observed value of the cosmological constant Λ. Appendix 1 contains the evaluation of our gravitational resummation exponent.
