entropic force, running gravitational coupling and future singularities

Gen Relativ Gravit (2014) 46:1617DOI 10.1007/s10714-013-1617-7

RESEARCH ARTICLE

Entropic force, running gravitational couplingand future singularities

Maryam Aghaei Abchouyeh · Behrouz Mirza ·Zeinab Sherkatghanad

Received: 31 July 2013 / Accepted: 24 September 2013 / Published online: 13 December 2013© Springer Science+Business Media New York 2013

Abstract The effects of a running gravitational coupling and the entropic force onfuture singularities are considered. Although it is expected that the quantum correctionsremove the future singularities or change the singularity type, treating the runninggravitational coupling as a function of energy density is found to cause no change inthe type of singularity but causes a delay in the time that a singularity occurs. Theentropic force is found to replaces the singularity type I I byI I I (a = const., H =const., H → ∞, p → ∞, ρ → ∞) which differs from previously known type I I Iand to remove the w-singularity. We also consider an effective cosmological modeland show that the types I and I I are replaced by the singularity type I I I .

Keywords Cosmology · Asymptotically safe running gravitational coupling ·Entropic force · Future singularities

1 Introduction

Both the theoretical cosmology and observational data indicate that our acceleratedexpanding universe can be described by an equation of state (EOS) parameter w around

M. Aghaei Abchouyeh · B. Mirza · Z. SherkatghanadDepartment of Physics, Isfahan University of Technology,Isfahan 84156-83111, Irane-mail: [email protected]

B. Mirzae-mail: [email protected]

Z. Sherkatghanad (B)Department of Physics and Astronomy, University of Waterloo,Waterloo, ON N2L 3G1, Canadae-mail: [email protected]

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1617 of 11 M. Aghaei Abchouyeh et al.

−1 [1]. In general, when the universe passes through the �CDM epoch, w is exactly−1. However, a phantom-dominated universe can be described by w slightly less than−1 while if w is slightly more than −1, the quintessence dark epoch occurs. In theregion where barotropic index w is lower than −1, a future singularity may occur inthe form of an infinite scale factor, energy density, and pressure which has come tobe called the “big rip”. Other types of singularities have been explored, increasing thefamily of candidates [2,3]. It is known that different types of singularities may ariseduring the expansion of the universe, which may be classified as follows [4–10]

Type I (Big Rip): Infinite a, ρ, and p.Type I I (Sudden): Finite a, H , and ρ; divergent H and p.Type I I I (Big Freeze): Finite a; infinite H, ρ, and p. This singularity type is asubcase of Finite Scale Factor singularityType I V : Finite a, H, H , ρ, and p but infinite higher derivatives of H .Type V : Infinite w (barotropic index): By expanding the scale factor around thetime of the singularity, we get a(t) = c + (ts − t)n1 + (ts − t)n2 + ...., wheren1 < n2 < .... and c > 0 [11–13]. In this way, w-singularity may occure withoutany divergence in pressure or energy density with the following properties:(i) ps �= 0 : n1 = 2 and n2 = [3,∞),(ii) ps = 0 : n1 = [3,∞) and n2 = [n1 + 1,∞).w-singularity was first proposed in [13] with a different form of the scale factor.

According to Tipler’s definition [14], the big rip (Type I ) singularity is strongwhereas types I I, I I I, I V , and V are weak. Based on Krolak’s definition [15], how-ever, only types I I, I V , and V are weak [16].

Conformal anomaly effects can change the type of singularity but bulk viscositymerely decreases the time when singularities may happen [17–19].

We may adopt an approach in which the concepts of information and holographyplay the central role [20–24]. By holography is meant a situation in which all theinformation of a volume V can be encoded on its boundary screen. In this situation,the effect of entropic force can produce the generalized Friedman equations to yielda new type of future singularity. The entropic force will change the singularity fromType I I to I I I (a =const., H =const., H → ∞, p → ∞, ρ → ∞) which differsfrom previously known type I I I .

In general, the gravitational coupling is assumed as a universal constant, but theidea of variation of physical constants such as the gravitational coupling G, the chargeof the electron e, the velocity of light c has been investigated in both theoretical andexperimental physics [25–38] and might be effective to avoid singularities [39]. Inthis paper we study an asymptotically safe scenario as a quantum effect to avoid someexotic behaviors of the universe near the singularity time. Considering Gravitationalcoupling as a function of energy density in the asymptotically safe scenario [28,40–43] does not change the type of future singularity but only delay the time while asingularity appears.

This paper is organized as follows: Sect. 2 investigates the singularities of the gener-alized Friedman equations with a running gravitational coupling. Section 3 examinesthe effect of entropic force on future singularities. The effects of both running gravita-tional coupling and entropic force on the different types of singularities are studied in

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Sect. 4. In Sect. 5, an effective cosmological model is used to study future singularities.It is expected that a quantum correction removes the predicted future singularities ofthe universe or at least changes the singularity type.

2 Running gravitational coupling and future singularities

Following [25–27,39], we consider a Running gravitational coupling varying by time.The generalized Friedman equations are given by

3H2 = 8πG(t)ρ, (1)a(t)

a(t)= −4πG(t)

3

(ρ + 3

p

c2

), (2)

here a(t) is the scale factor and the dots is the divertive with respect to time. In thiscondition the Bianchi identity holds with respect to the covariant derivative of eachside of the Einstein field equation in this way

ρ + 3H(ρ + p

c2

)= −ρ

G(t)

G(t). (3)

A running gravitational coupling may represent some essential features of an asymp-totically safe gravitational model [28]. An asymptotically safe gravitational modelis important because of its quantum background. It is expected that some quantumcorrections on General Relativity can remove the predicted future singularities ofthe universe or at least change the singularity type. In an asymptotically safe modelthe gravitational coupling G should be a function of an energy scale, so the moreusual choice energy scale is energy density and the energy density is a functionof time. Thus the gravitational coupling can be taken as G(ρ) ≡ G(ρ(t)), whereG(ρ(t)) ∼ ρ(t)−

α2+α and α ≥ 2, which represent a kind of asymptotically safe

model [28].Therfore, the pressure p(t) and energy density ρ(t) for this form of running gravi-

tational coupling are given by (c = 1)

ρ(t) =[

3

8πH2(t)

] 11−β

, (4)

p(t) = − 1

4π

(3

2H2(t) + d H(t)

dt

) [3

8πH2(t)

] β1−β

, (5)

where, β = α2+α

and, for asymptotic safety β ≥ 0.5.Now let us consider the effect of running gravitational coupling on future singu-

larities. Since the energy density is divergent in Type I and I I I singularities, wecan consider the effect of a running gravitational coupling on these types of futuresingularities. In general, the scale factor a(t) associated with Type I for the standardFriedman equations can be expressed as follows [4]

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a(t) = a0

(t

ts − t

)n

, (6)

where, n is a positive constant and 0 < t < ts . In this situation, the scale factordiverges at a finite time (t → ts) where ts is the instant of singularity. For the scalefactor in Eq. (6), the behaviors of the energy density and pressure in Eqs. (4) and (5),respectively, show that Type I does not change in response to the effect of a runninggravitational coupling (Fig. 1). This type of singularity is so strong that it cannot beaffected by an asymptotically safe scenario; however, the larger values of β near thesingularity causes a delay in the time the singularity appears.

The scale factor related to Type I I I singularity is given as follows [8–10,24]:

a(t) = 1 −(

1 − t

ts

)n

+(

t

ts

)q

(as − 1) , (7)

For the special values n < 1 and 0 < q < 1, Type I I I singularity occurs and the strongand weak energy conditions are violated. Therefore, Type I I I singularity shouldremain intact by considering a running gravitational coupling in an asymptotically safescenario. However, our results show that larger values of β or a weaker gravitationalforce for the singularity types I and I I I correspond to a larger value of ts , whichmeans a delay in the appearance of the singularities. Thus, we may expect that anasymptotically safe running gravitational coupling (as a quantum effect) does notchange the singularity types nor remove any of the singularities. Pressure p(t) andenergy density ρ(t) are depicted in Fig. 2.

3 The effect of entropic force on future singularities

In this Section, we consider the effect of entropic force on future singularities. Easson,Frampton and Smoot obtained the modified Friedman equations by considering theentropic force scenario, which introduces entropic force as a result of surface effects[21–23]. Thus, the Friedman equations are generalized to the following form [21]

Fig. 1 Pressure p and energy density ρ in an asymptotically safe scenario with respect to t for Type Isingularity for n = 3, ts = 10000, and β = 0.5 [red (dashed) line], 0.6 [blue (dashed-dotted) line], and 0.7[black (solid) line] (color figure online)

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Fig. 2 p and ρ with respect to t (asymptotically safe scenario) Type III singularity for n = 0.5, q =0.5, as = 10, ts = 10000 and β = 0.5 [red (dashed) line], 0.6 [blue (dashed-dotted) line], and 0.7 [black(solid) line] (color figure online)

a(t)

a(t)= −4πG

3(ρ + 3p) + c1 H2(t) + c2 H(t) (8)

H2(t) = 8πG

3ρ + c1 H2(t) + c2 H(t), (9)

Here, the coefficients c1 and c2 are determined by observations [22,23]. In this way,energy density ρ(t) and pressure p(t) are given by

ρ(t) = 3

8πG

[H(t)2 − c1 H(t)2 − c2 H(t)

], (10)

p(t) = 1

4πG

[−3 + 3c1

2H(t)2 + −2 + 3c2

2H(t)

]. (11)

Now let us consider the effect of entropic force on the future singularities. For thescale factor related to Type I in Eq. (6), we can plot ρ(t) and p(t) in Eqs. (10) and(11) with respect to t . The results indicate no change for Type I singularity. Also,the sudden singularity (Type I I ) occurs for the scale factor represented in Eq. (7)when 1 < n < 2 and 0 < q < 1 and the dominant energy condition is violated [17].Using the scale factor for Type I I singularity and the energy density ρ(t) and pressurep(t) in Eqs. (10) and (11) respectively, we find that the Hubble parameter H is finitealthough H , ρ, and p are infinite. ρ diverges as a result of H term in Eq. (10). Thus,for all positive values of c1 and c2 a new type of singularity is taking place which wecall it type I I I (Fig. 3). It is similar to Type I I I but only with a different behavior ofH . This new type can be characterized by finite a and H as well as infinite H , ρ, andp. Turned into Krolak’s definition, a weak singularity (Type I I ) turns into a strongsingularity (Type I I I ) [14]. Furthermore, the violated energy conditions are sensitiveto the values of coefficients c1 and c2. The results are presented in Tables 1 and 2.

For Type I V and w-singularity, the scale factor is given by [6]

a(t) = c + (ts − t)n + (ts − t)q , (12)

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Fig. 3 Pressure p and energy density ρ with respect to t (entropic force scenario) for Type I I I singularityfor ts = 10000, q = 0.5, n = 1.5, c1 = 0.1, c2 = 0.01

Table 1 Violated energyconditions for q = 0.5, n = 1.5in an entropic force scenario

c1 c2c1c2

Violatedenergyconditions

Singularity

0 < c1 < 1 0 < c2 < 1 >1 DECand DNEC

II → I I I

0 < c1 < 1 0 < c2 < 1 <1 No II → I I I

c1 > 1 c2 > 1 >1 No II → I I I

c1 > 1 c2 > 1 <1 No II → I I I

c1 > 1 0 < c2 < 1 >1 No II → I I I

Table 2 The effects of entropicforce and running gravitationalcoupling on future singularities

Entropic force c1 = 0.1, c2 = 0.01 I I → I I I

An effective c1 = 0.1, c2 = 0.01, β ≥ 0.5 I → I I I

Cosmological

Model c1 = 0.1, c2 = 0.01, β = 0 I I → I I I

where, c is a positive constant. The special values of 2 ≤ n < ∞ and n < q < ∞correspond to the singularity Type I V . Also the w-singularity occurs when: i) n = 2and q = [3,∞), ii) n = [3,∞) and q = [n + 1,∞).

Our results also show that the singularity Types I I I and I V which are describedby the scale factors in Eqs. (7) and (12), respectively, remain intact. It is interestingthat the effect of entropic force removes the w-singularity (Fig. 4). It should be notedthat the presence of H in Eqs. (10) and (11) is responsible for the removal of thew−singularity in entropic force scenario.

4 The effects of both the entropic force and running gravitational couplingon future singularities

In the previous sections, we showed that the running gravitational coupling does notchange the type of singularity. However, the effect of entropic force changes thesingularity type from I I to I I I . Now, we account for the effects of both runninggravitational coupling and entropic force on Friedman equations

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Fig. 4 w with respect to t (entropic force scenario) for n = 2, q = 3.5, c = 4, c1 = 0.2, c2 = 0.01, andts = 10000

a(t)

a(t)= −4π

3ρ−β(ρ + 3p) + c1 H2(t) + c2 H(t) (13)

H2(t) = 8π

3ρ1−β + c1 H2(t) + c2 H(t). (14)

Thus, Eqs. (10), (11) reduce to

ρ(t) =[

3

8π(H(t)2 − c1 H(t)2 − c2 H(t))

] 11−β

, (15)

p(t) = 1

4π

(−3 + 3c1

2H(t)2 + −2 + 3c2

2H(t)

)

×[

3

8π

(H(t)2 − c1 H(t)2 − c2 H(t)

)] β1−β

. (16)

Our asymptotically safe scenario is not relevant to the singularity types I I, I V , andV as the energy density remains finite at the time the singularity appears. Using thescale factor in Eq. (6) for the singularity type I and Eq. (7) for type I I I , we can plotthe energy density ρ(t) and pressure p(t) in Eqs. (15) and (16) with respect to time.In this case, neither Type I nor Type I I I change as a result of the effects of eitherrunning gravitational coupling or entropic force. These types are classified as strongsingularities along the lines of Krolak’s definition [14,15].

Our results indicate that larger values of β that represent a weaker gravitational forcefor the singularity types I and I I I still correspond to a larger value of ts or a delayin the time that the singularities appear and further that the entropic force increasesthis effect of running gravitational coupling (Fig. 5). This is a novel behavior andcharacterizes the effects of both running gravitational coupling and entropic force onsingularity types I and I I I .

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Fig. 5 p and ρ with respect to t for the singularity Type I ts = 10000, n = 3 by considering the effectof a running gravitational coupling β = 0.6, c1 = c2 = 0 [blue (dashed-dotted) line] and by the effectsof both the entropic force and running gravitational coupling β = 0.6, c1 = 0.1, c2 = 0.01 [red (dashed)line] (color figure online)

5 An effective cosmological model

In this section, we consider an effective cosmological model which is a map from thegeneralized equations to the standard Friedman equations. Thus, Eqs. (13) and (14)can be expressed as

H2e f f (t) = 8π

3ρ, (17)

in compare with Friedmann Equations in general relativity, where,

H2e f f (t) = 8π

3

[3

8π

(H(t)2 − c1 H(t)2 − c2 H(t)

)] 11−β

. (18)

For the effective scale factor, we have

aef f (t) = exp

[∫Hef f (t)

]dt, (19)

Aditionally, using Eq. (17) and the conservation law ρe f f +3Hef f (t)(ρe f f + pef f

) =0, we have

ae f f (t)

aef f (t)= −4π

3(ρe f f + 3pef f ), (20)

where

ρe f f = 3H2e f f (t)

8π(21)

pef f = − 1

4π

(3

2H2

e f f (t) + He f f (t)

), (22)

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Here, effective energy density and pressure have their original definitions ρ(t) ≡ρe f f (t) and p(t) ≡ pef f (t), respectively in Eqs. (21)and (22). This means that by aredefinition for Hubble parameter the original form of Friedmann Equations remainunchanged. In order to study which singularities may change under this new defin-itions of cosmological parameters we consider Eqs. (19), (21) and (22). Using, thescale factor associated with Type I singularity in Eq. (6), we have shown that, forc1, c2 ≥ 0, the effective scale factor in Eq. (19) is finite although Hef f , ρ and pare infinite. Thus, Type I singularity is replaced by Type I I I . The singularity typeI I I is defined as finite aef f and infinite Hef f , ρ and p. In other words, considering anasymptotically safe gravity and the entropic force lead to the replacement of the strongsingularity of Type I with the weaker one of type I I I in the effective cosmologicalmodel.

Let us consider the effective scale factor in Eq. (19) for the other future singularities.For the singularity type I I the scale factor a(t) described by Eq. (7) and the effectivescale factor by Eq. (19). our results indicate that Hef f , ρ and p are infinite for β = 0and c1, c2 > 0. So, Type I I singularity is replaced by Type I I I . Furthermore, thesingularity type I I I in Eq. (7) for β ≥ 0.5 as well as types I V and V in Eq. (12) forβ = 0 remain the same as before. The results are summarized in Table 2.

6 Conclusion

In this paper, we studied the effects of running gravitational coupling and entropicforce on future singularities. Since the energy density is divergent for types I andI I I in the singularity time, we can consider the effect of asymptotically safe runninggravitational coupling for these types of future singularities. The effect of runninggravitational coupling does not change the type of singularity. However, for singularitytypes I and I I I , a weaker gravitational force (larger values of β) corresponds to adelay in the time that the singularity appear.

Furthermore, the entropic force changes the singularity from Type I I to a new typecalled Type I I I (a = const., H = const., H → ∞, p → ∞, ρ → ∞) which differsfrom previously known type I I I . The type of singularity is similar to type I I I butonly with a different behavior of H . Although the singularity types I, I I I and I V donot change as a result of considering the effect of entropic force, Type V , however, isremoved in this case.

Our results indicated that both the entropic force and a running gravitational cou-pling cause a delay in the singularity types I and I I I . Finally, by introducing a dualcosmological model, we investigated different types of singularity. It was shown thattypes I and I I change to type I I I . It is interesting that Type I is replaced by a weakerone along the lines defined by Tipler. It should be noted that the singularity typesI I I, I V and V do not change in this dual cosmological model.

References

1. Hannestad, S., Mortsell, E.: Probing the dark side: constraints on the dark energy equation of statefrom CMB, large scale structure, and type Ia supernovae. Phys. Rev. D 66, 063508 (2002)

123


2. Caldwell, R.R., Kamionkowski, M., Weinberg, N.N.: Phantom energy: dark energy with w-1 causes acosmic doomsday e. Phys. Rev. Lett. 91, 071301 (2003)

3. Nojiri, S., Odintsov, S.D.: Final state and thermodynamics of a dark energy universe. Phys. Rev. D 70,103522 (2004)

4. Nojiri, S., Odintsov, S.D., Tsujikawa, S.: Properties of singularities in the (phantom) dark energyuniverse. Phys. Rev. D 71, 063004 (2005)

5. Jambrina, L.F., Lazkoz, R.: Cosmological singularities and modified theories of gravity. J. Phys.: Conf.Ser. 314, 012061 (2011), arXiv:0903.4775 [gr-qc]

6. Jambrina, L.F.: w-singularities in cosmological models. Int. J. Mod. Phys. A, 27, (2012),arXiv:1012.3159 [gr-qc]

7. Dabrowski, M.P., Denkiewicz, T.: AIP Conf. Proc. 1241, 561 (2010). arXiv: 0910.00238. Denkiewicz, T., Dabrowski, M.P., Ghodsi, H., Hendry, M.A.: Cosmological tests of sudden future

singularities. Phys. Rev. D 85, 083527 (2012), arXiv:1201.66619. Balcerzak, T., Denkiewicz, M.P..: Density perturbations in a finite scale factor singularity universe.

Phys. Rev. D 86, 023522 (2012). arXiv:1202.328010. Denkiewicz, T.: Observational constraints on finite scale factor singularities. J. Cosmol. Astropart.

Phys. 07, 036 (2012), arXiv:1112.544711. Jambrina, L.F.: w-cosmological singularities. Phys. Rev. D 82, 124004 (2010)12. Jambrina, L.F., Lazkoz, R.: Geodesic behavior of sudden future singularities. Phys. Rev. D 70, 121503

(2004)13. Dabrowski, M.P., Denkiewicz, T.: Barotropic index w-singularities in cosmology. Phys. Rev. D 79,

063521 (2009)14. Tipler, F.J.: Singularities in conformally flat spacetimes. Phys. Lett. A 64, 8 (1977)15. Krolak, A.: Towards the proof of the cosmic censorship hypothesis. Class. Quantum Grav. 3, 267

(1986)16. Jambrina, L.F., Lazkoz, R.: Classification of cosmological milestones. Phys. Rev. D 74, 064030 (2006)17. Houndjo, S.J.M.: Conformal anomaly around the sudden singularity. Europhys. Lett., 92, 10004 (2010),

arXiv:1008.0664 [hep-th]18. Misner, C.W.: The Isotropy of the Universe. Astrophys. J. 151, 431 (1968)19. Brevik, I., Gorbunova, O.: Viscous Dark Cosmology with Account of Quantum Effects. Eur. Phys. J.

C. 56, 425 (2008)20. Verlinde, E.: On the origin of gravity and the laws of Newton. JHEP 1104, 029, (2011), arXiv:1001.0785

[hep-th]21. Easson, D.A., Frampton, P.H., Smoot, G.F.: Entropic accelerating universe. Phys. Lett. B 696, 273

(2011), arXiv:1002.4278 [hep-th]22. Easson, D.A., Frampton P.H., Smoot, G.F.: Entropic Inflation, arXiv:1003.1528 [hep-th]23. Casadio, R., Gruppuso, A.: CMB acoustic scale in the entropic-like accelerating universe. Phys. Rev.

D 84, 023503 (2011), arXiv:1005.0790 [gr-qc]24. Barrow, J.D.: Sudden future singularities. Class. Quantum Grav. 21, L79 (2004)25. Barrow, J.D.: Cosmologies with varying light-speed. Phy. Rev. D. 59, 043515 (1998). arXiv:astro-

ph/981102226. Barrow, J.D., Magueijo, J.: Solving the flatness and quasi-flatness problems in Brans-Dicke cosmolo-

gies with a varying light speed. Class. Quantum Grav. 16, 1435 (1999)27. Gopakumar, P., Vijayagovindan, G.V.: Solutions to cosmological problems with energy conservation

and varying C, G and L. Mod. Phys. Lett. A 16, 957 (2001)28. Casadio, R., Hsu, S.D.H., Mirza, B.: Asymptotic safety, singularities, and gravitational collapse. Phys.

Lett. B 695, 317 (2011), arXiv:1008.2768 [gr-qc]29. Albrecht, A., Magueijo, J.: A time varying speed of light as a solution to cosmological puzzles. Phy.

Rev. D. 59, 043516 (1998). arXiv:astro-ph/981101830. Barrow, J.D.: Varying G and other constants, arXiv:gr-qc/971108431. Barrow, J.D., Parsons, P.: Phys. Rev. D 55 (1997)32. Hindmarsh, M., Litim, D., Rahmede, Ch.: Asymptotically safe cosmology. JCAP 1107, (2011)

arXiv:1101.540133. Contillo, A., Hindmarsh, M., Rahmede, Ch.: Renormalisation group improvement of scalar field infla-

tion. Phys. Rev. D 85, 043501 (2012). arXiv:1108.042234. Bonanno, A., Reuter, M.: Cosmology of the planck Era from a renormalization group for quantum

gravity. Phys. Rev. D 65, 043508 (2002). arXiv:hep-th/0106133

123


35. Reuter, M., Weyer, H.: Renormalization group improved gravitational actions: a Brans-Dicke approach.Phys. Rev. D 69, 104022 (2004)

36. Dirac, P.A.M.: The cosmological constants. Nature 139, 323 (1937)37. Barrow, J.D., Magueijo, J.: Varying-theories and solutions to the cosmological problems. Phys. Lett.

B 443, 104 (1998)38. Barrow, J.D.: Varying Alpha. Ann. Phys. 19, 202, Berlin (2010)39. Dabrowski, M.P., Marosek, K.: Regularizing cosmological singularities by varying physical constants.

JCAP, 1302, 012 (2013), arXiv:1207.4038 [hep-th]40. Wetterich, C.: Exact evolution equation for the effective potential. Phys. Lett. B 301, 90 (1993)41. Reuter, M.: Nonperturbative evolution equation for quantum gravity. Phys. Rev. D 57, 971 (1998),

arXiv:9605030 [hep-th]42. Souma, W.: Non-trivial ultraviolet fixed point in quantum gravity. Prog. Theor. Phys. 102, 181 (1999),

arXiv:9907027 [hep-th]43. Weinberg, S.: Asymptotically safe inflation. Phys. Rev. D 81, 083535 (2010), arXiv:0911.3165 [hep-th]

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