**2. Planck scale cosmology**

More precisely, we recall the Einstein-Hilbert theory

$$\mathcal{L}(\mathbf{x}) = \frac{1}{2\kappa^2} \sqrt{-g} \left(\mathbf{R} - 2\Lambda\right) \tag{1}$$

where *R* is the curvature scalar, *g* is the determinant of the metric of space-time *gμν*, <sup>Λ</sup> is the cosmological constant and *<sup>κ</sup>* <sup>=</sup> <sup>√</sup>8*πGN* for Newton's constant *GN*. Using the phenomenological exact renormalization group for the Wilsonian [5–8] coarse grained effective average action in field space, the authors in Ref. [25] have argued that the attendant running Newton constant *GN*(*k*) and running cosmological constant Λ(*k*) approach UV fixed points as *k* goes to infinity in the deep Euclidean regime:

$$k^2 G\_N(k) \to \mathcal{g}\_{\ast \prime} \Lambda(k) \to \lambda\_\ast k^2$$

<sup>2</sup> The attendant scale choice *<sup>k</sup>* <sup>∼</sup> 1/*<sup>t</sup>* used in Refs. [25] was also proposed in Ref. [26].

for *k* → ∞.

2 Will-be-set-by-IN-TECH

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

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

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

original ideas of Bohr and Einstein, will be taken up elsewhere [32].

<sup>L</sup>(*x*) = <sup>1</sup>

<sup>2</sup> The attendant scale choice *<sup>k</sup>* <sup>∼</sup> 1/*<sup>t</sup>* used in Refs. [25] was also proposed in Ref. [26].

2*κ*<sup>2</sup>

where *R* is the curvature scalar, *g* is the determinant of the metric of space-time *gμν*, <sup>Λ</sup> is the cosmological constant and *<sup>κ</sup>* <sup>=</sup> <sup>√</sup>8*πGN* for Newton's constant *GN*. Using the phenomenological exact renormalization group for the Wilsonian [5–8] coarse grained effective average action in field space, the authors in Ref. [25] have argued that the attendant running Newton constant *GN*(*k*) and running cosmological constant Λ(*k*) approach UV fixed

*<sup>k</sup>*<sup>2</sup>*GN*(*k*) <sup>→</sup> *<sup>g</sup>*∗, <sup>Λ</sup>(*k*) <sup>→</sup> *<sup>λ</sup>*∗*k*<sup>2</sup>

<sup>−</sup>*<sup>g</sup>* (*<sup>R</sup>* <sup>−</sup> <sup>2</sup>Λ) (1)

with experiment is now in order.

resummation exponent.

**2. Planck scale cosmology**

More precisely, we recall the Einstein-Hilbert theory

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*) = *<sup>ξ</sup> <sup>t</sup>* for *ξ* > 0, the authors in Refs. [25] show that the standard cosmological equations admit of the following extension:

$$(\frac{\dot{a}}{a})^2 + \frac{K}{a^2} = \frac{1}{3}\Lambda + \frac{8\pi}{3}G\_N\rho$$

$$\dot{\rho} + 3(1+\omega)\frac{\dot{a}}{a}\rho = 0$$

$$\dot{\Lambda} + 8\pi\rho \dot{G}\_N = 0$$

$$G\_N(t) = G\_N(k(t))$$

$$\Lambda(t) = \Lambda(k(t))\tag{2}$$

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

$$ds^2 = dt^2 - a(t)^2 \left( \frac{dr^2}{1 - Kr^2} + r^2(d\theta^2 + \sin^2\theta d\phi^2) \right) \tag{3}$$

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

$$p(t) = \omega \rho(t),\tag{4}$$

where *p* is the pressure.

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