IR quantum gravity solves naturally cosmic acceleration and its coincidence problem
Fotios K. Anagnostopoulos, Georgios Kofinas, Vasilios Zarikas
aa r X i v : . [ g r- q c ] F e b IR Quantum Gravity solves naturally cosmic acceleration and its coincidence problem ∗ Fotios K. Anagnostopoulos, † Georgios Kofinas, ‡ and Vasilios Zarikas
3, 4, § National and Kapodistrian University of Athens, Physics Department,Panepistimioupoli Zografou, 15772, Athens, Greece Research Group of Geometry, Dynamical Systems and Cosmology,Department of Information and Communication Systems Engineering,University of the Aegean, Karlovassi 83200, Samos, Greece School of Engineering, Nazarbayev University, Nur-Sultan (Astana), Kazakhstan General Department, University of Thessaly, Lamia, Greece (Dated: February 16, 2021)The novel idea is that the undergoing accelerated expansion of the universe happens due to infraredquantum gravity modifications at intermediate astrophysical scales of galaxies or galaxy clusters,within the framework of Asymptotically Safe gravity. The reason is that structures of matter areassociated with a scale-dependent positive cosmological constant of quantum origin. In this contextno extra unproven energy scales or fine-tuning are used. Furthermore, this model was confrontedwith the most recent observational data from a variety of probes, and with aid of Bayesian analysis,the most probable values of the free parameters were extracted. Finally, the model proved to bestatistically equivalent with ΛCDM, and thus being able to resolve naturally the concept of darkenergy and its associated cosmic coincidence problem.
I. INTRODUCTION
The idea that Quantum Gravity effects can be important at astrophysical and cosmological distances has recentlyattracted much attention. In particular the framework of Exact Renormalization Group (RG) approach for quantumgravity [1], also found in the literature under the names Asymptotic Safety (AS) or Quantum Einstein Gravity, hasopened the possibility of investigating both the ultraviolet (UV) and the infrared (IR) sector of gravity in a systematicmanner.The key element is the Effective Average Action Γ k [ g µν ], a Wilsonian coarse-grained free energy, which defines aneffective field theory appropriate for the momentum scale k . This Γ k , evaluated at tree level, describes appropriatelyall gravitational phenomena, including all loop effects. The application to Einstein-Hilbert action generates RG flowequations [2], which have made possible the consistent study of the scaling behavior of Newton constant G andcosmological constant Λ at high energies [3], [4]. The initial idea was first demonstrated by Weinberg [5], where hesuggested that a pertubatively divergent theory could be consistently defined in four dimensions at a nontrivial UVfixed point with the dimensionless g ( k ) = G ( k ) k non-vanishing in the k → ∞ limit. In the framework of AS, oneshould also include Λ, which becomes energy dependent and receives quantum contribution form vacuum fluctuations.Recent works have also considered matter fields or a growing number of purely gravitational operators in the action.In particular, truncations involving quadratic terms in the curvature or higher powers of the Ricci scalar have beenstudied [6], [7]. In all the investigations the UV critical surface has turned out to be finite dimensional ( d UV = 3),implying that the theory is nonperturbatively renormalizable.A weakly coupled gravity at high energies is expected to generate important consequences in several astrophysicaland cosmological contexts and in fact the RG flow of Γ k , obtained by different truncations of theory space, has beenthe basis of various investigations of “RG improved” black hole spacetimes [8], [9] and early Universe models [10]-[12]. II. IR POINT FROM AS APPROACH
The behavior of AS theory is more complicated at low energies, corresponding at cosmological or astrophysicalscales. The problem arises because the β -functions of any local operator of the type √ gR n are singular in the IR ∗ Essay has been awarded an Honorable Mention in the Gravity Essay Competition 2019 with the Essay published in a Special issue ofIJMPD. † Electronic address: [email protected] ‡ Corresponding author; Electronic address: gkofi[email protected] § Electronic address: [email protected] due to the presence of a pole at λ ( k ) = Λ( k ) /k = 1 /
2. This pole indicates that the Einstein-Hilbert truncation isnot trustable approximation and new relevant operators emerge in the k → h µν [4], quantumeffects dynamically drive Λ along the RG trajectories generated by the unstable infrared modes of the gravitationalsector. Close to the IR fixed point, Λ runs proportional to k to avoid the singularity,Λ( k ) k = λ IR ∗ + h k θ , k → , (2.1)where λ IR ∗ < / λ -evolution, h is a constant related to the eigenvalues of the stabilitymatrix, and θ > k →
0) = 0,regardless of its bare value. Furthermore, recent investigations based on a conformal reduction of Einstein gravitydiscovered a new IR fixed point suggesting the existence of the counterpart of the physical IR fixed point present inthe full theory [21].
III. COSMOLOGY BY MATCHING LOCAL ASTROPHYSICAL METRICS WITH GLOBALCOSMOLOGICAL METRIC
In [19], the recent cosmic acceleration naturally emanated from the recent formation of structure. A Swiss cheese(Einstein-Strauss) model was constructed to derive the cosmology. The interior static spherically symmetric metric,modeling a galaxy or a galaxy cluster, matches smoothly to a cosmological exterior across a spherical boundary. Aquantum improved Schwarzschild-de Sitter (SdS) interior metric was used, which contains the appropriate antigravityeffect. This model uses dimensionless order one parameters of AS, the conventional Newton constant G N and theastrophysical length scale. It provides a recent passage to the sufficient acceleration, while the freedom of the orderone parameters has to be constrained by observational data. To the best of our knowledge, this is the first solutionof the dark energy problem without using fine-tuning or introducing add-hoc energy scales.The exterior FRW metric is ds = − dt + a ( t ) (cid:20) dr − κr + r (cid:0) dθ + sin θ dϕ (cid:1)(cid:21) (3.1)with a ( t ) the scale factor. The interior metric has the form ds = − (cid:16) − G k MR −
13 Λ k R (cid:17) dT + dR − G k MR − Λ k R + R (cid:0) dθ + sin θdϕ (cid:1) (3.2)with the functions G k = G ( k ) , Λ k = Λ( k ) determined by AS. The matching of the two patches occurs at the boundarywith constant r = r Σ , which at the same time experiences the universal expansion. Since astrophysical structures arestill large compared to the cosmological scales, Λ( k ) is expected to differ slightly from its IR form k , so a power lawΛ k = γk b , with γ > b close to the value 2, is a fair approximation of the running behavior (2.1). Additionally,it is set G k = G N at observable macroscopic distances, in agreement with standard Newton law.The energy scale k is expected to be associated with a characteristic length scale L , k = ξ/L , where ξ is an orderone dimensionless number. In the Swiss cheese approach, only the value R S = ar Σ of the Schucking radius (or k S )enters the cosmic evolution. Although a simple option is L = R , a more natural one is to set as L the proper distance D ( R ) = Z R d R (cid:16) − G N M R −
13 Λ k R (cid:17) − / . (3.3) z w D E ( z ) a s t r o p h . z w D E ( z ) a ll FIG. 1: The evolution of w DE ( z ) with the corresponding 1 σ and 2 σ uncertainties. In the upper panel we use the best fit valuesfrom the combination H ( z )/Pantheon/QSO, while in the lower panel we utilize those of H ( z )/Pantheon/QSO/CMB shift . Thedashed line corresponds to ΛCDM value w = − The new cosmological constant term is Λ k R ∼ G N (cid:0) √ G N r Σ (cid:1) b (cid:0) r Σ D (cid:1) b R , where ˜ γ = γG − b N a dimensionless order onenumber. For b close to the value 2.1, the quantity G N (cid:0) √ G N r Σ (cid:1) b is very close to the order of magnitude of the standardcosmological constant Λ ≃ . × − GeV of the concordance ΛCDM model, while the factor (cid:0) r Σ D (cid:1) b contributes onlya small distance-dependent deformation since the today value of D S should be of order r Σ in order to have the correctamount of dark energy. Therefore, the hard coincidence of the standard Λ ∼ H , has been exchanged with a mildadjustment of the index b close to 2.1.The Hubble evolution in terms of the redshift z arises by the integration of the cosmological equations of the modelwhich consider the Israel-Darmois matching conditions on the boundary radius, H ( z ) H = Ω m (1+ z ) + h Ω − b DE − b ξ ˜ γ b ( G N H ) b r Σ a √ G N z z i − b + Ω κ (1+ z ) . (3.4)In addition, a preliminary analysis has shown that there is no obvious contradiction between the model discussedand the internal dynamics of the astrophysical object. The potential and the force due to the varying cosmologicalconstant term are small percentages of the corresponding Newtonian potential and force, where the precise valuesdepend on the considered structure and the considered point at the boundary of the object or inside. IV. THOROUGH TEST OF THE MODEL USING OBSERVATIONAL DATA
No matter what are the theoretical merits of a given model, it needs to be confronted with the observational data.In [20], authors used the most recent observational data sets, namely direct measurements of the Hubble rate H ( z )[22], Supernovae Ia (Pantheon data set [23]), Quasi-Stellar-Objects (QSO), Baryonic Acoustic Oscillations and directmeasurements of the CMB shift parameters, to constraint the free parameters of the AS cosmological model [19].It was found that this model is very efficient and in excellent agreement with observations. The energy density ofmatter that was calculated is compatible with the same quantity imposed by concordance cosmology from the CMBangular power spectrum. Another result is that the AS model supports a lower value of Hubble constant than thevalue derived from Cepheids, so the best fit value for H is closer to the Planck value.In addition, after reconstructing the effective dark energy equation-of-state parameter w DE ( z ) using the derivedvalues of the free parameters, it was found that the today’s value w DE is close to w = −
1. Both the transitionredshift and the current value of the deceleration parameter are in perfect agreement with the corresponding valuescalculated with model-independent techniques.Finally, the comparison of ΛCDM model with the considered AS model in terms of the fitting properties, usinga variety of information criteria, revealed that the AS model is statistically equivalent with that of ΛCDM. This isa significant conclusion since the AS model, unlike the majority of the cosmological models, does not include newfields in nature or has fine-tuning problems. Therefore, it must be considered as a viable and efficient alternative
Model Ω m h Ω b h χ AIC ∆AIC H ( z ) /Pantheon/QSO AS model 0 . +0 . − . . +0 . − . - 84 .
463 92.889 0.937ΛCDM 0 . +0 . − . . ± .
013 - 85.700 91.952 0 H ( z ) /Pantheon/QSO/CMB shift AS model 0 . ± .
001 0 . ± .
009 0 . ± . . +0 . − . . ± .
006 0 . ± . m , Ω b and the statistical criteria values χ , AIC,∆AIC for the AS cosmology and ΛCDM. cosmological scenario towards explaining the recent accelerated expansion of the universe. Interestingly, the twomodels are expected to have differences at the perturbation level. Acknowledgments
G. Kofinas and V. Zarikas acknowledge the support of Orau Grant Number 110119FD4534, “Quantum gravity atastrophysical scales”. [1] M. Reuter, “Nonperturbative evolution equation for quantum gravity”, Phys. Rev. D , 971 (1998),doi:10.1103/PhysRevD.57.971 [hep-th/9605030].[2] D. Dou and R. Percacci, “The running gravitational couplings”, Class. Quant. Grav. , 3449 (1998), doi:10.1088/0264-9381/15/11/011 [hep-th/9707239].[3] D. F. Litim, “Fixed points of quantum gravity”, Phys. Rev. Lett. , 201301 (2004), doi:10.1103/PhysRevLett.92.201301[hep-th/0312114].[4] A. Bonanno and M. Reuter, “Proper time flow equation for gravity”, JHEP , 035 (2005), doi:10.1088/1126-6708/2005/02/035 [hep-th/0410191].[5] S. Weinberg, “Ultraviolet divergences in quantum theories of gravitation”, in S.W. Hawking and W. Israel, editors, GeneralRelativity: an Einstein Centenary Survey , Cambridge University Press, 1979.[6] D. Benedetti, P. F. Machado and F. Saueressig, “Taming perturbative divergences in asymptotically safe gravity”, Nucl.Phys. B , 168 (2010), doi:10.1016/j.nuclphysb.2009.08.023 [arXiv:0902.4630 [hep-th]].[7] A. Codello, R. Percacci and C. Rahmede, “Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormal-ization Group Equation”, Annals Phys. , 414 (2009), doi:10.1016/j.aop.2008.08.008 [arXiv:0805.2909 [hep-th]].[8] A. Bonanno and M. Reuter, “Spacetime structure of an evaporating black hole in quantum gravity”, Phys. Rev. D ,083005 (2006), doi:10.1103/PhysRevD.73.083005 [hep-th/0602159].[9] G. Kofinas and V. Zarikas, “Avoidance of singularities in asymptotically safe Quantum Einstein Gravity”, JCAP ,no. 10, 069 (2015), doi:10.1088/1475-7516/2015/10/069 [arXiv:1506.02965 [hep-th]].[10] A. Bonanno and M. Reuter, “A cosmology of the Planck era from the renormalization group for quantum gravity”, inI. Ciufolini, E. Coccia, M. Colpi, V. Gorini, and R. Peron, editors, Recent Developments in Gravitational Physics , p. 461,2006.[11] G. Kofinas and V. Zarikas, “Asymptotically Safe gravity and non-singular inflationary Big Bang with vacuum birth”, Phys.Rev. D , no. 10, 103514 (2016), doi:10.1103/PhysRevD.94.103514 [arXiv:1605.02241 [gr-qc]].[12] V. Zarikas and G. Kofinas, “Singularities and Phenomenological aspects of Asymptotic Safe Gravity”, J. Phys. Conf. Ser. , no. 1, 012028 (2018), doi:10.1088/1742-6596/1051/1/012028[13] M. Reuter and H. Weyer, “Quantum gravity at astrophysical distances?”, JCAP , 001 (2004), doi:10.1088/1475-7516/2004/12/001 [hep-th/0410119].[14] J. Alexandre, V. Branchina and J. Polonyi, “Instability induced renormalization”, Phys. Lett. B , 351 (1999),doi:10.1016/S0370-2693(98)01491-9 [cond-mat/9803007]. [15] O. Lauscher, M. Reuter and C. Wetterich, “Rotation symmetry breaking condensate in a scalar theory”, Phys. Rev. D ,125021 (2000), doi:10.1103/PhysRevD.62.125021 [hep-th/0006099].[16] A. Bonanno and G. Lacagnina, “Spontaneous symmetry breaking and proper time flow equations”, Nucl. Phys. B , 36(2004), doi:10.1016/j.nuclphysb.2004.06.003 [hep-th/0403176].[17] M. Reuter and H. Weyer, “Renormalization group improved gravitational actions: A Brans-Dicke approach”, Phys. Rev.D , 104022 (2004), doi:10.1103/PhysRevD.69.104022 [hep-th/0311196].[18] M. Reuter and H. Weyer, “Running Newton constant, improved gravitational actions, and galaxy rotation curves”, Phys.Rev. D , 124028 (2004), doi:10.1103/PhysRevD.70.124028 [hep-th/0410117].[19] G. Kofinas and V. Zarikas, “Solution of the dark energy and its coincidence problem based on local antigravity sources with-out fine-tuning or new scales”, Phys. Rev. D , no. 12, 123542 (2018), doi:10.1103/PhysRevD.97.123542 [arXiv:1706.08779[gr-qc]].[20] F. K. Anagnostopoulos, S. Basilakos, G. Kofinas and V. Zarikas, “Constraining the Asymptotically Safe Cosmology: cosmicacceleration without dark energy”, JCAP , 053 (2019), doi:10.1088/1475-7516/2019/02/053 [arXiv:1806.10580 [astro-ph.CO]].[21] E. Manrique, M. Reuter and F. Saueressig, “Bimetric Renormalization Group Flows in Quantum Einstein Gravity”, AnnalsPhys. , 463 (2011), doi:10.1016/j.aop.2010.11.006 [arXiv:1006.0099 [hep-th]].[22] R. Jimenez and A. Loeb, “Constraining cosmological parameters based on relative galaxy ages”, Astrophys. J. , 37(2002), doi:10.1086/340549 [astro-ph/0106145].[23] D. M. Scolnic et al. , “The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 andCosmological Constraints from the Combined Pantheon Sample”, Astrophys. J.859