**4. An estimate of** Λ

To estimate the value of Λ today, we take the normal-ordered form of Einstein's equation,

$$\colon \mathcal{G}\_{\mu\nu} : + \Lambda : \mathcal{g}\_{\mu\nu} : = -\varkappa^2 : T\_{\mu\nu} : . \tag{21}$$

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 *ttr* ∼ 25*tPl* from Refs. [25], we introduce

$$\begin{split} \rho\_{\Lambda}(t\_{tr}) & \equiv \frac{\Lambda(t\_{tr})}{8\pi G\_N(t\_{tr})} \\ &= \frac{-M\_{Pl}^4(k\_{tr})}{64} \sum\_{j} \frac{(-1)^F n\_j}{\rho\_j^2} \end{split} \tag{22}$$

with

*<sup>x</sup>*<sup>±</sup> <sup>=</sup> <sup>1</sup> 2 √2 �

in Resummed Quantum Gravity: An Estimate of

⎧ ⎪⎪⎪⎪⎪⎨

*<sup>κ</sup>*<sup>2</sup>|*p*<sup>2</sup><sup>|</sup>

*<sup>κ</sup>*<sup>2</sup>|*p*<sup>2</sup><sup>|</sup>

<sup>2</sup>*κ*<sup>2</sup>|*p*<sup>2</sup><sup>|</sup> <sup>8</sup>*π*<sup>2</sup> ln � *<sup>m</sup>*<sup>2</sup>

⎪⎪⎪⎪⎪⎩

factorization arguments [40–44] to take the factorized result for *B*��

*<sup>g</sup>* (*p*)|factorized <sup>=</sup> <sup>2</sup>*κ*2|*p*2<sup>|</sup>

with *<sup>k</sup>* = (*ik*0, *<sup>k</sup>*1, *<sup>k</sup>*2, *<sup>k</sup>*3) and *<sup>k</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*k*0<sup>2</sup> <sup>−</sup> *<sup>k</sup>*1<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2<sup>2</sup> <sup>−</sup> *<sup>k</sup>*3<sup>2</sup> ≡ −*k*<sup>2</sup>

W. Israel, (Cambridge Univ. Press, Cambridge, 1979).

[4] R. Percacci and D. Perini, *Phys. Rev. D* 68 (2003) 044018.

[6] F. Wegner and A. Houghton, Phys. Rev. A 8(1973) 401.

[8] J. Polchinski, Nucl. Phys. B 231 (1984) 269. [9] B.F.L. Ward, Open Nucl.Part.Phys.Jour. 2(2009) 1.

*B*�� *<sup>g</sup>* (*p*) =

*B*��

any value of *p*2.

**7. References**

therein.

75.

exponentiate them in the exactly massless case.

<sup>Δ</sup>¯ <sup>+</sup> *<sup>λ</sup>*¯ <sup>2</sup> <sup>±</sup> ((Δ¯ <sup>+</sup> *<sup>λ</sup>*¯ <sup>2</sup>)<sup>2</sup> <sup>−</sup> <sup>4</sup>(*λ*¯ <sup>2</sup> <sup>−</sup> *<sup>i</sup>�*¯))1/2�

�

�

, *m* �= 0

, *m* = *mg* = *λ*

*<sup>g</sup>* from (26) as

, *m* = 0, *mg* = 0. (27)

*<sup>E</sup>* with *kE* the Euclidean 4-vector

, *m* = 0, *mg* = *λ*

for <sup>Δ</sup>¯ <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>m</sup>*2/*p*2, *<sup>λ</sup>*¯ <sup>2</sup> <sup>=</sup> *<sup>λ</sup>*2/*p*2and *�*¯ <sup>=</sup> *�*/*p*2. In this way, we arrive at the results, for *<sup>p</sup>*<sup>2</sup> <sup>&</sup>lt; 0,

<sup>35</sup> Planck Scale Cosmology and Asymptotic Safety

*m*<sup>2</sup>+|*p*<sup>2</sup>|

*g* |*p*<sup>2</sup>| �

where we have made more explicit the presence of the observed small mass, *mg*, of the graviton. When m=0 and one wants to use dimensional regularization for the IR regime instead of *mg*, we normalize the propagator at a Euclidean point *<sup>k</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*μ*<sup>2</sup> and use standard

<sup>8</sup>*π*<sup>2</sup> ln � <sup>|</sup>*μ*2<sup>|</sup>

In physical applications, such mass singularities are absorbed by the definition of the initial state "parton" densities and/or are canceled by the KLN theorem in the final state; we do not

We stress that the standard analytic properties of the 1PI 2pt functions obtain here, as we use standard Feynman rules. Wick rotation changes the Minkowski space Feynman loop integral

*kE* = (*k*0, *k*1, *k*2, *k*3) with metric *δμν* = *diag*(1, 1, 1, 1). Thus our results rigorously correspond to <sup>|</sup>*p*2<sup>|</sup> <sup>=</sup> <sup>−</sup>*p*<sup>2</sup> in (26), (27) with *<sup>m</sup>*<sup>2</sup> replaced with *<sup>m</sup>*<sup>2</sup> <sup>−</sup> *<sup>i</sup>�*, with *�* <sup>↓</sup> 0, following Feynman, for *p*<sup>2</sup> < 0; by Wick rotation this is the regime relevant to the UV behavior of the Feynman loop integral. Standard complex variables theory then uniquely specifies our exponent for

[1] S. Weinberg, in *General Relativity, an Einstein Centenary Survey*, eds. S. W. Hawking and

[2] M. Reuter, Phys. Rev. D 57 (1998) 971; O. Lauscher and M. Reuter, *ibid.* 66 (2002) 025026;

[3] D. F. Litim, Phys. Rev. Lett. 92(2004) 201301; Phys. Rev. D 64 (2001) 105007, and references

[5] K. G. Wilson, Phys. Rev. B 4 (1971) 3174, 3184; K. G. Wilson, J.Kogut, Phys. Rep. 12 (1974)

[7] S. Weinberg, "Critical Phenomena for Field Theorists", *Erice Subnucl. Phys.* (1976) 1.

A. Bonanno and M. Reuter, *ibid.* 62 (2000) 043008, and references therein.

� *<sup>d</sup>*4*<sup>k</sup>* with *<sup>k</sup>* = (*k*0, *<sup>k</sup>*1, *<sup>k</sup>*2, *<sup>k</sup>*3) for real *<sup>k</sup><sup>j</sup>* and *<sup>k</sup>*<sup>2</sup> <sup>=</sup> *<sup>k</sup>*0<sup>2</sup> <sup>−</sup> *<sup>k</sup>*1<sup>2</sup> <sup>−</sup> *<sup>k</sup>*2<sup>2</sup> <sup>−</sup> *<sup>k</sup>*3<sup>2</sup> into the integral *<sup>i</sup>*


�

*g m*<sup>2</sup> *<sup>g</sup>*+|*p*<sup>2</sup>|

<sup>8</sup>*π*<sup>2</sup> ln � *<sup>m</sup>*<sup>2</sup>

<sup>8</sup>*π*<sup>2</sup> ln � *<sup>m</sup>*<sup>2</sup>

(25)

(26)

� *<sup>d</sup>*<sup>4</sup>*kE*

and use the arguments in Refs. [38] (*teq* is the time of matter-radiation equality) to get the first principles estimate, from the method of the operator field,

$$\begin{split} \rho\_{\Lambda}(t\_{0}) &\cong \frac{-M\_{Pl}^{4}(1+c\_{2,eff}k\_{tr}^{2}/(360\pi M\_{Pl}^{2}))^{2}}{64} \sum\_{j} \frac{(-1)^{F}n\_{j}}{\rho\_{j}^{2}} \\ &\times \left[\frac{t\_{tr}^{2}}{t\_{eq}^{2}} \times \frac{t\_{eq}^{2/3}}{t\_{0}^{2/3}}\right] \\ &\cong \frac{-M\_{Pl}^{2}(1.0362)^{2}(-9.197 \times 10^{-3})}{64} \frac{(25)^{2}}{t\_{0}^{2}} \\ &\cong (2.400 \times 10^{-3} eV)^{4}. \end{split} \tag{23}$$

where we take the age of the universe to be *<sup>t</sup>*<sup>0</sup> <sup>∼</sup><sup>=</sup> 13.7 <sup>×</sup> 109 yrs. In the latter estimate, the first factor in the square bracket comes from the period from *ttr* to *teq* (radiation dominated) and the second factor comes from the period from *teq* to *t*<sup>0</sup> (matter dominated) 4. This estimate should be compared with the experimental result [29–31]5 *<sup>ρ</sup>*Λ(*t*0)|expt <sup>∼</sup><sup>=</sup> (2.368 <sup>×</sup> <sup>10</sup>−3*eV*(<sup>1</sup> <sup>±</sup> 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 *ttr* is not precise, yet.
