Planck Scale Cosmology and Asymptotic Safety in Resummed Quantum Gravity

In Weinberg's asymptotic safety approach, a finite dimensional critical surface for a UV stable fixed point generates a theory of quantum gravity with a finite number of physical parameters. We argue that, in an extension of Feynman's original formulation of the theory, we recover this fixed-point UV behavior from an exact re-arrangement of the respective perturbative series. Our results are consistent with the exact field space Wilsonian renormalization group results of Reuter {\it et al.} and with recent Hopf- algebraic Dyson-Schwinger renormalization theory results of Kreimer. We obtain the first"first principles"predictions of the dimensionless gravitational and cosmological constants and our results support the Planck scale cosmology of Bonanno and Reuter. We conclude with an estimate for the currently observed value of the cosmological constant.


Introduction
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][3][4], strong evidence has been calculated using Wilsonian [5] field-space exact renormalization group methods to support asymptotic safety for the Einstein-Hilbert theory. We have shown in Refs. [6,7] that the extension of the amplitude-based, exact resummation theory of Ref. [8] to the Einstein-Hilbert theory (we call the extension resummed quantum gravity) leads to UV fixedpoint 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. [9] 1 . There is no known inconsistency between our analysis and Refs. [2][3][4]9]. Our results are also consistent with the results on leg renormalizability of quantum gravity in Refs. [11]. Contact with experiment is now in order.
Specifically, in Ref. [12], it has been argued that the approach in Refs. [2][3][4] to quantum gravity may provide a realization 2 of the successful inflationary model [14,15] 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. [7], using the resummed quantum gravity theory [6], we recover the properties as used in Refs. [12] 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. [12]. We comment on the reliability of the result as well, as it will be seen already to be relatively close to the observed value [16]. More such reflections, as they relate to an experimentally testable union of the original ideas of Bohr and Einstein, will be taken up elsewhere [17].
The discussion is organized as follows. In the next section we review the Planck scale cosmology presented in Refs. [12]. In Section 3 we review our results [7] for the dimensionless gravitational and cosmological constants at the UV fixed point. In Section 4, we combine the Planck scale cosmology scenario [12] with our results to predict the observed value of the cosmological constant Λ.

Planck Scale Cosmology
More precisely, we recall the Einstein-Hilbert theory 1 We also note that the model in Ref. [10] 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. 2 The attendant scale choice k ∼ 1/t used in Refs. [12] was also proposed in Ref. [13].
where R is the curvature scalar, g is the determinant of the metric of space-time g µν , Λ is the cosmological constant and κ = √ 8πG N for Newton's constant G N . Using the phenomenological exact renormalization group for the Wilsonian [5] coarse grained effective average action in field space, the authors in Ref. [12] have argued that the attendant running Newton constant G N (k) and running cosmological constant Λ(k) approach UV fixed points as k goes to infinity in the deep Euclidean regime: 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 the average effective action and the cosmological time t, k(t) = ξ t for ξ > 0, the authors in Refs. [12] show that the standard cosmological equations admit of the following extension: , and Λ(t) = Λ(k(t)) for the density ρ and scale factor a(t) with the Robertson-Walker metric representation as ds 2 = dt 2 − a(t) 2 dr 2 1−Kr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ) so that K = 0, 1, −1 correspond respectively to flat, spherical and pseudo-spherical 3-spaces for constant time t. The equation of state is p(t) = ωρ(t) where p is the pressure.
Using the UV fixed points for g * and λ * , the authors in Refs. [12] 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 t Pl , 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. [7] for these UV limits and show how to use them to predict the current value of Λ.

g * and λ * in Resummed Quantum Gravity
We start with the prediction for g * , which we already presented in Refs. [6,7]. We have shown in Refs. [6] 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 i∆ ′ for B ′′ g (k) = κ 2 |k 2 | 8π 2 ln m 2 m 2 +|k 2 | , where this form holds for the UV regime, so that the resummed scalar propagator falls faster than any power of |k 2 |. An analogous result [6] holds for m=0. As Σ ′ s , the residual self-energy function, starts in O(κ 2 ), we may drop it in calculating one-loop effects. It follows that, when the respective analogs of i∆ ′ F (k)| resummed are used for the elementary particles, all quantum gravity loop corrections are UV finite [6].
When we use our resummed propagator results, as extended to all the particles in the SM Lagrangian and to the graviton itself, the denominator of the graviton propagator becomes [6] (M Pl is the Planck mass) q 2 + Σ T (q 2 ) + iε ∼ = q 2 − q 4 c 2,e f f 360πM 2

An Estimate of Λ
To estimate the value of Λ today, we take the normal-ordered form of Einstein's equation, : G µν : +Λ : g µν := −κ 2 : T µν :. The coherent state representation of the thermal density matrix then gives the Einstein equation in the form of thermally averaged quantities with Λ given by our result above in lowest order. Taking the transition time between the Planck regime and the classical Friedmann-Robertson-Walker regime at t tr ∼ 25t Pl from Refs. [12], we introduceρ Λ (t tr ) ≡ and use the arguments in Refs. [23] (t eq is the time of matter-radiation equality) to get the first principles estimate, from the method of the operator field, ρ Λ (t 0 ) ∼ = −M 4 Pl (1+c 2,e f f k 2 tr /(360πM 2 Pl )) 2 64 4 where we take the age of the universe to be t 0 ∼ = 13.7 × 10 9 yrs. In the latter estimate, the first factor in the square bracket comes from the period from t tr to t eq (radiation dominated) and the second factor comes from the period from t eq to t 0 (matter dominated) 4 . This estimate should be compared with the experimental result [16] 5 ρ Λ (t 0 )| expt ∼ = (2.368 × 10 −3 eV (1 ± 0.023)) 4 .
To sum up, our estimate, while it is definitely encouraging, is not a precision prediction, as possible hitherto unseen degrees of freedom have not been included and t tr is not precise, yet. -We thank Profs. L. Alvarez-Gaume and W. Hollik for the support and kind hospitality of the CERN TH Division and the Werner-Heisenberg-Institut, MPI, Munich, respectively, where a part of this work was done. 3 We note the use here in the integrand of 2k 2 0 rather than the 2( k 2 +m 2 ) in Ref. [7], to be consistent with ω = −1 [22] for the vacuum stress-energy tensor. 4 The method of the operator field forces the vacuum energies to follow the same scaling as the non-vacuum excitations. 5 See also Ref. [24] for an analysis that suggests a value for ρ Λ (t 0 ) that is qualitatively similar to this experimental result.