aa r X i v : . [ h e p - t h ] D ec USTC-ICTS-08-23
Reconstructing f ( R ) Theory from Ricci DarkEnergy
Chao-Jun Feng
Institute of Theoretical Physics, CAS,Beijing 100080, P.R.ChinaInterdisciplinary Center for Theoretical Study, USTC,Hefei, Anhui 230026, P.R.China [email protected] this letter, we regard the f ( R ) theory as an effective description for theacceleration of the universe and reconstruct the function f ( R ) from the Riccidark energy, which respects holographic principle of quantum gravity. Byusing different parameter α in RDE, we show the behaviors of reconstructed f ( R ) and find they are much different in the future.1 Introduction
The accelerating cosmic expansion first inferred from the observations ofdistant type Ia supernovae [1] has been strongly confirmed by some other in-dependent observations, such as the cosmic microwave background radiation(CMBR) [2] and Sloan Digital Sky Survey (SDSS) [3]. An exotic form of neg-ative pressure matter called dark energy is used to explain this acceleration.The simplest candidate of dark energy is the cosmological constant Λ, whoseenergy density remains constant with time ρ Λ = Λ / πG and whose equationof motion is also fixed, w Λ = P Λ /ρ Λ = − P Λ is the pressure) during theevolution of the universe. The cosmological model that consists of a mixtureof the cosmological constant and cold dark matter is called LCDM model,which provides an excellent explanation for the acceleration of the universephenomenon and other existing observational data. However, as is well know,this model faces two difficulties, namely, the ’fine-tuning’ problem and the’cosmic coincidence’ problem. The former also states: Why the cosmologicalconstant observed today is so much smaller than the Plank scale, while thelatter states: Since the energy densities of dark energy and dark matter scaleso differently during the expansion of the universe, why they are at the sameorder today?To alleviate or even solve these two problems, many dynamic dark en-ergy models were proposed such as the quintessence model relying on a scalarfield minimally interacting with Einstein gravity. Here ’dynamic’ means thatthe equation of state of the dark energy is no longer a constant but slightlyevolves with time. So far a wide variety of scalar-field dark energy models hasbeen proposed including quintessence mentioned above, k-essence, tachyons,phantoms, ghost condensates and quintom etc..Despite considerable workson understanding the dark energy have been done, we can not answer thisquestion at present, because we do not entirely understand the nature of darkenergy before a complete theory of quantum gravity is established, since thedark energy problem may be in principle a problem belonging to quantumgravity[4]. Actually, there is still a different way to face the problem of cos-mic acceleration. Since general relativity is only tested within solar systemup to now, it is possible that the observed acceleration is not the manifesta-tion of another ingredient in the cosmic pie, but rather the first signal of abreakdown of our understanding of the laws of gravitation, as stressed by Lueet al.[5]. From this point of view, one may consider the modification to the2instein-Hilbert action at larger scales with higher order curvature invariantterms such as R , R µν R µν , R µναβ R µναβ , or R (cid:3) k R as well as nonminimallycoupled terms between scalar fields and geometry (such as φ R ). These termsnaturally emerge as quantum corrections in the low energy effective actionof quantum gravity or string theory [5, 6]. The interesting models followingthis line include f ( R ) and DGP gravity, and in this letter, we will focus on f ( R ) theory where the modification is a function of the Ricci scalar only.Although we are lacking a quantum gravity theory today, we can stillmake some attempts to probe the nature of dark energy according to someprinciple of quantum gravity[7]. It is well known that the holographic prin-ciple is an important result of the recent researches for exploring the quan-tum gravity(or string theory)[4]. So that the holographic dark energy model(HDE) constructed in light of the holographic principle possesses some sig-nificant features of an underlying theory of dark energy[7]. Recently, Gaoet.al [9] proposed a holographic dark energy model in which the future eventhorizon is replaced by the inverse of the Ricci scalar curvature, and they callthis model the Ricci dark energy model(RDE). Of course, this model alsorespect the holographic principle.In this letter, we regard RDE as the underlying theory of dark energyand reconstruct the corresponding f ( R ) theory as an equivalent descriptionwithout resorting to any additional dark energy component, namely RDEis effectively described by f ( R ) theory. In Section II, we will briefly reviewRDE and f ( R ) models, and reconstruct function f ( R ) from RDE model inSection III. In the last section we will give some conclusions. f ( R ) theory Holographic principle [10] regards black holes as the maximally entropic ob-jects of a given region and postulates that the maximum entropy inside thisregion behaves non-extensively, growing only as its surface area. Hence thenumber of independent degrees of freedom is bounded by the surface area inPlanck units, so an effective field theory with UV cutoff Λ in a box of size L is not self consistent, if it does not satisfy the Bekenstein entropy bound [11]( L Λ) ≤ S BH = πL M pl , where M − pl ≡ G is the Planck mass and S BH is the3ntropy of a black hole of radius L which acts as an IR cutoff. Cohen et.al.[12] suggested that the total energy in a region of size L should not exceedthe mass of a black hole of the same size, namely L Λ ≤ LM p . Thereforethe maximum entropy is S / BH . Under this assumption, Li [13] proposed theholographic dark energy as follows ρ Λ = 3 c M p L − (1)where c is a dimensionless constant. Since the holographic dark energy withHubble horizon as its IR cutoff does not give an accelerating universe [14], Lisuggested to use the future event horizon instead of Hubble horizon and par-ticle horizon, then this model gives an accelerating universe and is consistentwith current observation[13, 15]. For the recent works on holographic darkenergy, see ref. [16]. In the following, we are using units 8 πG = c = ~ = 1.Recently, Gao et.al [9] proposed a holographic dark energy model in whichthe future event horizon is replaced by the inverse of the Ricci scalar curva-ture, and they call this model the Ricci dark energy model(RDE). This modeldoes not only avoid the causality problem and is phenomenologically viable,but also solve the coincidence problem of dark energy. The Ricci curvatureof FRW universe is given by R = −
6( ˙ H + 2 H + ka ) , (2)where dot denotes a derivative with respect to time t and k is the spatialcurvature. They introduced a holographic dark energy proportional to theRicci scalar ρ X = 3 α (cid:18) ˙ H + 2 H + ka (cid:19) ∝ R (3)where the dimensionless coefficient α will be determined by observations andthey call this model the Ricci dark energy model. Solving the Friedmannequation they find the result ρ X H = α − α Ω m e − x + f e − (4 − α ) x (4)where Ω m ≡ ρ m / H , x = ln a and f is an integration constant. Substi-tuting the expression of ρ X into the conservation equation of energy, p X = − ρ X − dρ X dx (5)4e get the pressure of dark energy p X = − H (cid:18) α − (cid:19) f e − (4 − α ) x (6)Taking the observation values of parameters they find the α ≃ .
46 and f ≃ .
65 [9]. The evolution of the equation of state w X ≡ p X /ρ X ofdark energy is the following. At high redshifts the value of w X is closedto zero, namely the dark energy behaves like the cold dark matter, andnowadays w X approaches − X | MeV < − ≪ . α < R , R µν R µν , R µναβ R µναβ , R (cid:3) k R or nonminimally coupled terms to the effective Lagrangianof gravitational field when quantum corrections are considered. In f ( R ) the-ory, the modification is adding a function of the Ricci scalar only, and theaction of it is as follows S = Z d x √− g [ f ( R ) + L m ] , (7)where L m is the matter Lagrangian, and f ( R ) is a function of R . Thenwe want to obtain the modified Friedmann equations by varying the gen-eralized Lagrangian. However, it is not clear how the variation has to beperformed[18]. Assuming the FRW metric, the equations governing the dy-namics of the universe are different depending on whether one varies withrespect to the metric only or with respect to the metric and the connection.These two possibilities are usually called the metric and the Palatini[19] ap-proach respectively. It is only in the case of Einstein gravity f ( R ) = R that these two methods give the same result. The problem of which methodshould be used is still a open question and a definitive answer is likely far tocome. In ref.[18], a method was proposed to reconstruct the form of f ( R )from a given Hubble parameter H ( z ) from observational data such as SN’sGold data[1] in the metric formulation. What is needed to reconstruct f ( R )in their approach is an expression for H ( z ), so we can use a H ( z ) predicted by5 given dark energy model to determine what is the f ( R ) theory which giverise to the same dynamics[18]. In the following, we will follow this methodto reconstruct f ( R ) from RDE, so we use the metric formulation, althoughthe dynamical equations are more simply in Palatini formulation.Variation with respect to the metric leads to the modified Einstein equa-tion [20] G µν = R µν − g µν R = T ( curv ) µν + T ( m ) µν , (8)where G µν is the Einstein tensor and T ( curv ) µν = 1 f ′ ( R ) (cid:26) g µν [ f ( R ) − Rf ′ ( R )] + f ′ ( R ) ; αβ ( g µα g νβ − g µν g αβ ) (cid:27) (9)and the stress-energy tensor of matter T ( m ) µν = ˜ T ( m ) µν /f ′ ( R ) (10)with ˜ T mµν the standard minimally coupled matter stress-energy tensor. Hereand in the following, we denote with a prime the derivative with respect to R . With the FRW metric, we obtain the modified Friedmann equations H + ka = 13 (cid:20) ρ curv + ρ m f ′ ( R ) (cid:21) (11)2 ¨ aa + H + ka = − ( p curv + p m ) , (12)where ρ m and p m are the matter-energy density and pressure respectively,and we have defined the same quantities for the effective curvature fluid as: ρ curv = 1 f ′ ( R ) (cid:26)
12 [ f ( R ) − Rf ′ ( R )] − H ˙ Rf ′′ ( R ) (cid:27) (13)and p curv = 1 f ′ ( R ) (cid:26) aa ˙ Rf ′′ ( R ) + ¨ Rf ′′ ( R ) + ˙ R f ′′′ ( R ) −
12 [ f ( R ) − Rf ′ ( R )] (cid:27) . (14)Applying the Bianchi identity to eq.(8), we obtain the conservation law forthe total energy density ρ tot = ρ curv + ρ m /f ′ ( R ) as follows˙ ρ tot + 3 H ( ρ tot + p tot ) = 0 . (15)6n the following, we will study the case of flat universe, i.e. the spatialcurvature k = 0 with the matter as dust, namely p m = 0 and we do notconsider the interaction between the matter and the curvature fluid. Thus,the matter energy density is conserved so that ρ m = 3 H Ω m e − x .In fact, eq.(11),(12) and (15) are not independent, so we will consideronly eq.(11) and (15). Combine these two equations we obtain˙ H = − f ′ ( R ) (cid:26) H Ω m e − x + ¨ Rf ′′ ( R ) + ˙ R h ˙ Rf ′′′ ( R ) − Hf ′′ ( R ) i(cid:27) . (16)Using the relation d/dt = Hd/dx to replace the variable t by x , eq.(16) canbe rewritten as a third order differential equation of f ( x ) C ( x ) d fdx + C ( x ) d fdx + C ( x ) dfdx = − m e − x , (17)where C n ( x ) consists of h ( x ) ≡ H ( x ) /H and its derivatives, where H isthe present Hubble parameter,see Appendix A. From eq.(17), one can seethat what is needed to reconstruct f ( R ) is an expression for h ( x ). As aconsequence, one could adopt for h ( x ) predicted by a given dark energymodel and determine what is the f ( R ) theory[18]. In the following, we willreconstruct f ( R ) according to RDE. f ( R ) from RDE From eq.(8) in [9], we get h ( x ) = 22 − α Ω m e − x + f e − ( − α ) x , (18)in RDE with h ( x = 0) = 1 as definition. Thus22 − α Ω m + f = 1 . (19)The differential equation (17) is so complicated that we will solve it nu-merically. According to ref.[18], the boundary conditions, i.e. the values of7 and its first and second derivatives with respect to x evaluated at x = 0are as follows (cid:18) dfdx (cid:19) z =0 = (cid:18) dRdx (cid:19) z =0 (20) (cid:18) d fdx (cid:19) z =0 = (cid:18) d Rdx (cid:19) z =0 (21) f ( x = 0) = f ( R ) = 6 H (1 − Ω m ) + R . (22)These conditions are chosen on the basis of physical consideration only.Rewrite eq.(11) explicitly with 8 πG and k = 0 as H = 8 πG (cid:20) ρ curv + ρ m f ′ ( R ) (cid:21) . (23)This equation shows that the function f ′ ( R ) is equivalent to redefine theNewton gravitational constant G as G/f ′ ( R ), that is time dependent in f ( R )theory. In order to be consistent with solar system experiments at x = 0, theeffective gravitational constant G/f ′ ( R ) must equal to G , thus f ′ ( R ) = 1,so we get f ′ ( R ) = 1 → "(cid:18) dRdx (cid:19) − dfdx z =0 = 1 (24)which leads to eq.(20). A second condition comes from that any f ( R ) theorymust fulfill the condition f ′′ ( R ) = 0 in order to not contradict solar systemtests [21], then it gives rise to eq.(21). Finally, the present value of ρ curv ineq.(13) is ρ curv ( x = 0) = f ( R ) − R . (25)By using the eq.(11) evaluated at present ( x = 0), and eq.(25), we obtainthe final boundary condition eq.(22).Given α = 0 .
46, Ω m = 0 .
27 and h ( x ) in eq.(18), with relation eq.(19)and three boundary condition eq.(20)-(22), the differential equation eq.(17)can be solved numerically. We plot the function f ( R ) with respect to R inFig.1. 8 - - - - - -
200 0 - - - - - f H R L Reconstruct f H R L Figure 1.
Reconstructed f(R) with ≤ z ≤ , where the redshift z = e − x − , Ω m = 0 . and α = 0 . (solid), α = 0 . (dashed) and α = 0 . (dash-dotted). From Fig.1 one can see that for small | R | (small z also), the functions aredistinguishable for different parameter α . Differences between these function f ( R ) become significant when | R | (or z ) increases. In order to compare withthe results in [18], we also show our results on a lf − lR plane in Fig.2, where lf ≡ ln( − f ) and lR ≡ ln( − R ) used in [18], and our results are consistentwith theirs. Moreover, Fig.1 and Fig.2 indicate the parameter α plays a im-portant role in the remote past. Actually, the value of α also determines thefuture evolution of R . To illustrate this, we plot the evolution of R in thefuture in Fig.3. As is expected, for α < .
5, the curves indicate | R | → ∞ inthe future, which is the behavior of phantom with equation of state smallerthan − α = 0 . | R | varies a little, and the dark energy becomes more and more like a cos-mological constant. For α > . R will vanish in the future.9 H - R L L og H - f L Reconstruct lf - lR Figure 2.
Reconstructed f(R) in lf − lR plane with ≤ z ≤ , where theredshift z = e − x − , Ω m = 0 . and α = 0 . (solid), α = 0 . (dashed) and α = 0 . (dash-dotted). lf ≡ ln( − f ) and lR ≡ ln( − R ) . - - - - - -
50 Z R R H z L Figure 3.
The future evolution of R with ≤ z ≤ , where the redshift z = e − x − , Ω m = 0 . and α = 0 . (solid), α = 0 . (dashed) and α = 0 . (dash-dotted). - - - - - - - - - - - f H R L Reconstruct f H R L Α= Figure 4.
Reconstructed f(R) with the redshift z = e − x − from around down to − , Ω m = 0 . and α = 0 . . The arrow denotes the decreasingdirection of z , and the point corresponds to the current value at z = 0 . - - - - - - - - - - - - -
50 R f H R L Reconstruct f H R L Α= Figure 5.
Reconstructed f(R) with the redshift z = e − x − from around down to − , Ω m = 0 . and α = 0 . . The arrow denotes the decreasingdirection of z , and the point corresponds to the current value at z = 0 . - - -
10 0 - - - - f H R L Reconstruct f H R L Α= Figure 6.
Reconstructed f(R) with the redshift z = e − x − from around down to − , Ω m = 0 . and α = 0 . . The arrow denotes the decreasingdirection of z , and the point corresponds to the current value at z = 0 . In Fig.4, Fig.5 and Fig.6, we can see that the difference is more distinc-tively reflected by the function f ( R ) reconstructed according to the futureevolution of RDE. For α = 0 .
46, there exists a turnaround point on thecurve at present epoch ( z = 0), and the decreasing | R | begin to increaseat this point, which means the phantom-like dark energy will dominate theuniverse. The existence of the turnaround point is a common feature forall the phantom-dark energy models realized in f ( R ) theory because of thecompetition between dark energy and matter [22]. For α = 0 . f ( R ) linearlydepends on R up to a constant, which corresponding to the de Sitter space,where f ( R ) = R + 2Λ. For α = 0 . f ( R ) increases from negative to positiveand seems a inverse power law dependence on R in the future. In conclusion, we have followed the method proposed in ref.[18] to recon-struct the function f ( R ) in the extended theory of gravity according tothe Ricci dark energy model, which respects holographic principle of quan-tum gravity. We show the behaviors of f ( R ) reconstructed with parameter α = 0 . , . , .
54 in RDE and find that the dependence of f ( R ) on R isdifferent for different α , and such a difference is much more distinctive in the12uture. The basic reconstruction procedure is simply: once the function h ( x )given by some dark energy model, one can solve the differential equation (17)to obtain f with relation eq.(19) and three boundary condition eq.(20)-(22).In our case, h ( x ) is given by RDE in (18) and results of reconstruction areshown in Fig.1-6. The parameter α plays a important role to determine thedependence of f on R , so we hope that the future high precision observationdata may be able to determine it and reveal some significant features of theunderlying theory of dark energy.It should be noted that RDE is obtained within the framework of generalrelativity, rather than any other extended gravity theory such as f ( R ) theory.What we done in this letter is to reconstruct the f ( R ) theory to effectivelydescribe RDE in Einstein gravity. Whether RDE can be generalized to f ( R )theories is question worth further investigation as HDE [22]. ACKNOWLEDGEMENTS
The author would like to thank Miao Li for a careful reading of the manuscriptand valuable suggestions. We are grateful to Qing-Guo Huang for useful dis-cussions.
Appendix A
The coefficients C n ( x ) in eq.(17) are C = 2 h (cid:18) d Rdx (cid:19) (cid:18) dRdx (cid:19) − − (cid:20) h d Rdx + (cid:18) dh dx − h (cid:19) d Rdx (cid:21) (cid:18) dRdx (cid:19) − + dh dx (cid:18) dRdx (cid:19) − (26) C = − h (cid:18) d Rdx (cid:19) (cid:18) dRdx (cid:19) − + (cid:18) dh dx − h (cid:19) (cid:18) dRdx (cid:19) − (27) C = h (cid:18) dRdx (cid:19) − (28)13 ( x ) in eq.(2) in the case of k = 0 is R = − (cid:18) h + 12 dh dx (cid:19) H (29) References [1] A. G. Riess et al. [Supernova Search Team Collaboration], “Observa-tional Evidence from Supernovae for an Accelerating Universe and aAstron. J. , 1009 (1998) [arXiv:astro-ph/9805201].S. Perlmutter et al. [Supernova Cosmology Project Collaboration], As-trophys. J. , 565 (1999) [arXiv:astro-ph/9812133].[2] D. N. Spergel et al. [WMAP Collaboration], “Wilkinson MicrowaveAnisotropy Probe (WMAP) three year results: Astrophys. J. Suppl. , 377 (2007) [arXiv:astro-ph/0603449].E. Komatsu et al. [WMAP Collaboration], arXiv:0803.0547 [astro-ph].[3] J. K. Adelman-McCarthy et al. [SDSS Collaboration], arXiv:0707.3413[astro-ph].[4] E. Witten, hep-ph/0002297.[5] A. Lue, R. Scoccimarro and G. Starkman, Phys. Rev. D , 044005(2004) [arXiv:astro-ph/0307034].[6] H. Sandvik, M. Tegmark, M. Zaldarriaga and I. Waga, Phys. Rev. D , 123524 (2004) [arXiv:astro-ph/0212114].[7] X. Zhang, Phys. Lett. B , 1 (2007) [arXiv:astro-ph/0604484].[8] G. ’t Hooft, gr-qc/9310026;L. Susskind, J. Math. Phys. , 6377 (1995) [hep-th/9409089].[9] C. Gao, X. Chen and Y. G. Shen, [arXiv:0712.1394 astro-ph].[10] R. Bousso, Rev. Mod. Phys. , 825 (2002) [arXiv:hep-th/0203101].[11] J. D. Bekenstein, Phys. Rev. D , 2333 (1973).J. D. Bekenstein, “A Universal Upper Bound On The Entropy To EnergyRatio For Bounded Phys. Rev. D , 287 (1981).1412] A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Rev. Lett. , 4971(1999) [arXiv:hep-th/9803132].[13] M. Li, Phys. Lett. B , 1 (2004) [arXiv:hep-th/0403127].[14] S. D. H. Hsu, Phys. Lett. B , 13 (2004) [arXiv:hep-th/0403052].[15] Q. G. Huang and M. Li, JCAP , 013 (2004)[arXiv:astro-ph/0404229].Q. G. Huang and M. Li, JCAP , 001 (2005)[arXiv:hep-th/0410095].[16] X. Zhang and F. Q. Wu, Phys. Rev. D , 023502 (2007)[arXiv:astro-ph/0701405].C. Feng, B. Wang, Y. Gong and R. K. Su, JCAP , 005 (2007)[arXiv:0706.4033 astro-ph].H. Wei and S. N. Zhang, Phys. Rev. D , 063003 (2007)[arXiv:0707.2129 astro-ph].B. C. Paul, P. Thakur and A. Saha, [arXiv:0707.4625 gr-qc].J. f. Zhang, X. Zhang and H. y. Liu, Eur. Phys. J. C , 693 (2007)[arXiv:0708.3121 hep-th].C. J. Feng, [arXiv:0709.2456 hep-th].Y. Z. Ma and Y. Gong, [arXiv:0711.1641 astro-ph].H. M. Sadjadi, JCAP , 026 (2007) [arXiv:gr-qc/0701074].K. Y. Kim, H. W. Lee and Y. S. Myung, Mod. Phys. Lett. A , 2631(2007) [arXiv:0706.2444 gr-qc].E. N. Saridakis, JCAP , 020 (2008) [arXiv:0712.2672 astro-ph].E. N. Saridakis, Phys. Lett. B , 335 (2008) [arXiv:0712.3806 [gr-qc].A. A. Sen and D. Pavon, [arXiv:0801.0280 astro-ph].M. Li, C. Lin and Y. Wang, [arXiv:0801.1407 astro-ph].A. J. M. Medved, [arXiv:0802.1753 hep-th].C. J. Feng, arXiv:0806.0673 [hep-th].H. Mohseni Sadjadi and N. Vadood, arXiv:0806.2767 [gr-qc].15. Horvat, arXiv:0806.4825 [gr-qc].C. J. Feng, arXiv:0809.2502 [hep-th].B. C. Paul, P. Thakur and A. Saha, arXiv:0809.3491 [hep-th].M. R. Setare and E. N. Saridakis, arXiv:0810.0645 [hep-th].[17] C. J. Feng, arXiv:0810.2594 [hep-th].[18] S. Capozziello, V. F. Cardone and A. Troisi, Phys. Rev. D , 043503(2005) [arXiv:astro-ph/0501426].[19] M. Ferraris, M. Francaviglia, I. Volovich, Class. Quant. Grav. (1994)1505;G. Allemandi, A. Borowiec, M. Francaviglia, Phys. Rev. D, (2004)043524;G. Allemandi, A. Borowiec, M. Francaviglia, Phys. Rev. D, (2004)103503.[20] S. Capozziello, S. Carloni and A. Troisi, Recent Res. Dev. Astron. As-trophys. , 625 (2003) [arXiv:astro-ph/0303041].[21] R. Dick, Gen. Rel. Grav., (2004) 217;A.E. Dominguez, D.E. Barraco, Phys. Rev. D, (2004) 043505.[22] X. Wu and Z. H. Zhu, Phys. Lett. B660