Table Of Content

THE UNIVERSE EVOLUTION IN EXPONENTIAL F(R)-GRAVITY K. Bamba1, A. Lopez-Revelles2,5, R. Myrzakulov6, S. D. Odintsov2,3,4,6 and L. Sebastiani5,6 1KMI,NagoyaUniversity,Nagoya,Japan 2ICE/CSIC-IEEC, Bellaterra(Barcelona), Spain3 ICREA,Barcelona,Spain4 TSPU,Tomsk,Russia 5DipartimentodiFisica,UniversitàdiTrento,Italia 6 DepartmentofGeneral &Theoretical Physics,EurasianNationalUniversity,Astana,Kazakhstan 3 1 Agenericfeatureofviableexponential F(R)-gravity isinvestigated. Anadditional modificationtostabilizethe effective dark energyoscillationsduringmattereraisproposedandappliedtotwoviablemodels. Ananalysisonthefutureevolutionofthe 0 universeisperformed. Furthermore,aunifiedmodelforearlyandlate-timeaccelerationisproposedandstudied. 2 n a 1 Introduction reproducingthestandardcosmologyofGRwithasuit- J able correction to realize current acceleration and/or 4 The current acceleration of our universe is supported inflation one has F(R) = R+f(R) and the modifica- 1 by various observations and cosmological data [1]. tionofgravityis encodedinthe function f(R). Inthis There exist several descriptions of this acceleration case, we may use an effective fluid-like representation ] c and, among them, the most important are given by ofmodifiedgravityandinFLRWspace-timedescribed q theΛCDMModelofGeneralRelativity(GR)plusCos- by the metric ds2 = dt2+a(t)2dx2, the equations of - − r mological Constant and by the modified gravitational Motion (EOM) assume the form g theories,likeF(R)-gravity[2]. Here,westudysomeas- [ pectsandgenericfeatureofviableF(R)-gravitymodels ρDE+ρm = κ32H2, PDE+Pm =−κ12 2H˙ +3H2 . 1 which reproduce the physics of ΛCDM Model, namely (cid:16) ((cid:17)2) v power-law and exponential gravity. We show that the Here, ρ and P are the energy density and pres- 9 m m 4 behavior of higher derivatives of the Hubble parame- sure of standard matter and ρDE and PDE are the 0 termaybeinfluencedbylargefrequencyoscillationsof effective dark energy density and pressure given by 3 effective dark energy during matter era, which makes modified gravity. In order to study the dynamics of 1. solutions singular and unphysical at a high redshift. F(R) gravity models, we may introduce the variable 0 Asaconsequence,weexamineanadditionalcorrection y (z) = ρ /ρ as a function of the red shift z, H DE m(0) 3 term to the models in order to remove any instability where ρ = 3m˜2/κ2 is the matter energy density m(0) 1 with keeping the viability properties. Some comments at the present time, being m˜2 a mass scale. From the : v about the growthof matter perturbations of corrected EOM one derives an equation on the type [5] i models are given and future universe evolution is ana- X y (z)+J (z)y (z)+J (z)y (z)+J (z)=0, (3) lyzed. Wealsoproposeawaytodescribetheearlytime H′′ 1 H′ 2 H 3 r a acceleration of universe by applying exponential grav- where J1(z), J2(z) and J3(z) depend on the F(R)- ity to inflationary cosmologyand a unified description model. Inwhat follows,we will considerviable models betweeninflationandcurrentaccelerationispresented. representingarealisticscenariotoaccountfordarken- ThisworkisbasedonRefs. [3,4]. Wedenotethegrav- ergy,in particulara suitable reparameterizationof the itational constant 8πGN by κ2. Hu-Sawicki model [6], 1 F(R)=R 2Λ 1 , (4) 2 RealisticF(R)-modelsintheFLRWuniverse − − (R/Λ)4+1 (cid:26) (cid:27) and a simple example of exponential gravity [7], The action describing F(R)-gravity is given by F(R)=R 2Λ 1 e R/Λ . (5) − F(R) − − I = d4x√ g + (matter) , (1) h i − 2κ2 L Thismodelsareverycarefullyconstructedsuchthatin ZM (cid:20) (cid:21) the high curvature regime F(R Λ)=R 2Λ. Here, ≫ − where F(R) is a generic function of the Ricci scalar R Λ plays the role of the Cosmological Constant which only, g is the determinantofthe metric tensorg and decreases in the flat limit (F(R 0) 0) where the µν → → (matter) is the matter Lagrangian. In realistic models Minkowski solution is recovered. L 1 j 2.0 j 1.4 1.5 1.2 1.0 1.0 0.5 0.8 0.0 0.5 1.0 1.5 2.0 2.5 z 1 2 3 4 5 6 z (a) (b) Figure1: Evolutionofthejerkparameterinpureexponentialmodel(a)andinexponentialmodelwithcorrection term (b) overlapped with the ones in ΛCDM Model. 3 DE-oscillations in the matter dominated era inthemodels(4)-(5)theapproachestotheCosmologi- calConstantaredifferentfromeachother,wemaysay Since in matter dominated era R = 3m˜2(z +1)3, one thatdarkenergyoscillationsinmattereraareageneric maylocallysolveEq. (3)aroundz+δz,where δz z. feature of realistic F(R) gravity, in which the cosmo- | |≪ The solution reads logical evolution is similar to those in ΛCDM Model. In order to remove the divergences in the derivatives yH(z+δz)=a0+b0δz+C0e2(z01+1)(cid:16)−α±√α2−4β(cid:17)δz, of the Hubble parameter, we can introduce a function (6) g(R)forwhichtheoscillationfrequencyofthedarken- where C is a generic constant and a , b , α and ergy density acquires a constant value (z) = 1/√δ, 0 0 0 F β are constants depending on the model. It turns δ > 0, during matter era, stabilizing the oscillations out that the dark energy perturbations remain small with the use of a correction term. This term has been around z as soon as the exponential term does not derived as diverge in expanding universe, when δz < 0. The ex- 1/3 R plicitconditions(1 F (R))F (R)/(2F (R)2)> 7/2 g(R)= γ˜Λ , (7) − ′ ′′′ ′′ − − 3m˜2 and 1/(RF (R)) > 12 are found for stable matter (cid:18) (cid:19) ′′ era [3]. In the case of realistic gravity described by where γ˜ = 1.702δ, so that (R g′′(R))−1/2/(z + 1) the models (4)-(5), where a0 = Λ/(3m˜2), it is easy to when R = 3m˜2(z + 1)3. We∗explore the models see that such conditions are well satisfied. Further- (4)-(5) with adding these correction. The effects of more, since in both of the models F′′(R) 0+, we the compensating term vanish in the de Sitter epoch, → understand that the discriminant of Eq. (6) is nega- when R = 4L and the models resemble to a model tive and dark energy oscillates with frequency (z) with an effective cosmological constant, provided that F ≃ (R F′′(R))−1/2/(z+1). Since by increasing the cur- γ˜ (m˜2/Λ)1/3. For example, a reasonable choice is ∗ ≪ vatureF (R)tendstozero,forlargevaluesofthered- to put γ˜ = 1/1000, which corresponds to the oscilla- ′′ shift the darkenergydensity oscillateswith ahighfre- tions frequency of the dark energy during matter era quency and also its derivatives become large and may = 41.255 with a period T 0.152. In order to F ≃ bedominantinsomederivativesoftheHubbleparame- appreciate the role of the proposed correction, let us ter whichapproachaneffective singularitymaking un- consider the jerk parameter j(z), which depends on physical the solution. Moreover,it may be stated that the second derivative of H(z)1, for the exponential the closer the model is to the ΛCDM Model (i.e., as model (5), whose numerical extrapolation2 is plotted muchF (R)isclosetozero),thebiggerthe oscillation and overlapped with its behavior in ΛCDM Model on ′′ frequency of dark energy becomes. As a consequence, the left side of Fig. 1. Despite the fact that the dark despite the fact that the dynamics of the universe de- energy density ρ Λ/κ2 is negligible at high red DE ≃ pends on the matter and the dark energy density re- shift, the effects of its oscillations become relevant in mains very small, some divergences in the derivatives thejerkparameter,whichgrowsupwithanoscillatory oftheHubbleparametercanoccur,showingadifferent behavior and diverges at z 2.8. On the right side of ≃ featureofthiskindofmodelswithrespecttothecaseof the figure, we plot the graphic of the jerk parameter GR with an undynamic CosmologicalConstant. Since for the same model with the addition ofthe correction 1Theformofthejerkparameterasafunctionoftimeisj(t)=a′′′(t)/(a(t)H(t)), a(t) beingthescalefactoroftheuniverse. 2WehaveusedMathematica 7(cid:13)c. 2 term g(R) previously discussed. Also in this case the sity perturbation δ = δρ /ρ satisfies the equation m m jerk parameter oscillates with the same frequency of δ¨ + 2Hδ˙ 4πG (a,k)ρ δ = 0 with k being the co- eff m − the dark energy, but here such oscillations have a con- moving wavenumber and G (a,k) being the effective eff stant frequency and do not diverge at high red shift. gravitational “constant” depending on the model and It is also important to stress that the term added expressed as a function of k and the scale factor a, to stabilize the dark energy oscillations in the matter such that k2/a2 H. The equation for matter per- ≫ dominated epoch does not cause any problem to the turbationhasanequivalentdescriptionintermsofthe good proprieties of the models (4)-(5), which continue growth rate f dlnδ/dlna. The growth index γ(z) g ≡ to satisfy all the cosmological and local gravity con- is defined as is defined as the quantity satisfying the straints. It has been demonstrated in several papers relation f (z)=Ω (z)γ(z) with Ω (z)=8πGρ /3H2 g m m m that in order to evade the solar system tests [8] or beingthematterdensityparameter. Thegrowthindex avoidlargecorrectionsto NewtonLaw atlargecosmo- cannot be observed directly, but it can be determined logicalscales[9],amodifiedgravitytheorymustsatisfy from the observational data of both the growth factor the following relation andthe matterdensity parameteratthe same redshift and this is the reason for which in the last period its 1 F (R) ′ R 1, (8) calculation in different theories has become of a great 3R F (R) − ≫ (cid:18) ′′ (cid:19) interest. In Ref. [4] various parameterizations for the when R assumes the typical curvature value of the growth index γ(z) have been studied for the models constant background around a matter source. For (4)-(5) with correction term g(R). It has been found the models (4)-(5) without correction terms, this con- that the Ansatz which better fits the (numerical) so- dition is always well satisfied, due to the fact that lution of the growth rate for the considered models is F′′(R Λ) 0+. Next, if we consider the addition of given by γ =γ0+γ1z, where γ0,1 are constants. ≫ ≃ g(R),whosesecondderivativeisdominant,theleftside of the letter equation reads 34/3m˜2/(2Λγ˜) R/m˜2 2/3. 5 Future universe evolution Despite the fact that this quantity is smaller than the (cid:0) (cid:1) The confrontation of the models (4)-(5) with SNIa, one obtained for the pure models (4)-(5), it still re- BAO, CMB radiation and gravitational lensing has mains sufficiently largeandthe correctionto the New- been executed in the past in several works and all ton Law is very small. For example, at the typical the numerical analysis have demonstrated that they value of the curvature in the solar system, namely R 10 61eV2, it corresponds to 2 106, which is are consistent with the last observational data com- − ≃ × ing from our universe. In particular, they repro- very large effectively. Concerning the matter instabil- duce the correct amount of dark energy today Ω = ity [10], this might also occur when the curvature is DE rather large, as on a planet (R 10 38eV2), as com- 0.721 0.015 [5, 12]. Moreover, the addition of cor- − ± ≃ rection term (7) does not have any influence at the paredwiththeaveragecurvatureoftheuniversetoday (R 10 66eV2). This instability is avoided as soon present day range of detection. Starting from Eq. (3), − ≃ we are able to study perturbations around the de Sit- as 1/(RF (R)) is much larger than one on the planet ′′ ter solution R = 4Λ of our universe today in the surfaces. For our correction term g(R), this quantity dS evaluatedatR 10 38eV2correspondsto4 1021and considered models. Performing the variation with re- − ≃ × spect to y (z) = y +y (z) with y = R /12m˜2 and alsointhiscasewedonothaveanyparticularproblem. H 0 1 0 dS y (z) 1 and assuming the contributions of radia- 1 | | ≪ tion and matter to be much smaller than y , one finds 4 Growth index 0 the following solution for y (z), 1 After the discoveryofcosmicacceleration,alotofthe- 4ζ ories for the dark energy have been proposed, and it y1(z)=C0(z+1)21(3±√9−4β)+ (z+1)3, (9) β has become very important to find a way to discrimi- nating among these theories, most of them exhibit the where C is a generic constant and ζ = 1 + (1 0 − same expansion history. In this context, the charac- F (R))/(RF (R)) and β = 4 +4F (R)/(RF (R)), ′ ′′ ′ ′′ − terization of growth of the matter density perturba- the both evaluated at R = R . The de dS tions isverysignificantinmodified gravity,since mod- Sitter solution is stable provided by condition els with the same cosmological evolution have a dif- F (R )/((R )F (R )) > 1, which is a well know ′ dS dS ′′ dS ferent cosmic growth history. In order to execute it, result. Ithasalsobeendemonstratedthatsinceinreal- the so-called growth index γ(z) is useful [11]. Un- isticF(R)-gravitymodelsforthedeSitteruniverse0< der the subhorizon approximation, the matter den- R F (R ) 1, one has F (R )/(R F (R )) > dS ′′ dS ′ dS dS ′′ dS ≪ 3 25/16,giving the negative discriminant of Eq. (9) and 6 Exponential gravity for early and late-time anoscillatorybehaviortothedarkenergydensitydur- cosmic acceleration ing this phase. Thus, in this case the dark energy Equation of State (EoS) parameter ω P /ρ One of the most important goals for modified grav- DE DE DE oscillates infinitely often around the line o≡f the phan- ity theories is that all the processes in the expan- tom divide ω = 1 [13]. These models possess one sion history of the universe are realized, from infla- DE crossingin the recen−tpast, after the end of the matter tion to the late cosmic acceleration epoch, without in- dominatedera,andinfinite crossingsinthefuture,but voking the presence of dark components. Models of the amplitude of such crossingsdecreasesas (z+1)3/2 the type (5) may be combined in a natural way to andit does not cause anyseriousproblemto the accu- obtain the phenomenological description of the infla- racy of the cosmologicalevolution during the de Sitter tionary epoch, mimicking a smooth versionof F(R)= epochwhichisingeneralthe finalattractorofthe sys- R 2Λ 1 e−R/Λ Λiθ(R Ri),whereRiisthetran- − − − − tem. However,since the existence of a phantom phase sitionscalarcurvatureforwhichasuitablecosmological (cid:2) (cid:3) cangivesomeundesirableeffectssuchasthepossibility constantΛi switchesonproducinginflation. Following to have the Big Rip [14] or Big Rip scenario with the the first proposal of Ref. [16], we consider the form of disintegrationof bound structures [15], it is important F(R)withanaturalpossibilityofaunifieddescription to observe that in the models (4)-(5) with the adding of our universe modification (7), the effective EoS parameter of the R R n universe F(R) = R 2Λ 1 e−Λ Λi 1 e−(cid:16)Ri(cid:17) − − − − (cid:16) (cid:17) (cid:20) (cid:21) ρ +ρ 2(z+1)dH(z) 1 ω DE m = 1+ , (10) +γ¯ Rα, (12) eff ≡ PDE+Pm − 3H(z) dz R˜iα−1! never crosses the phantom divide line in the past, and where Ri and Λi are the typical values of transition thatonlyintheveryfarfuture,whenz isverycloseto curvature and expected cosmological constant during 1,itcoincides withω andthe crossingsoccur. Let inflation,respectively,andnisanaturalnumberlarger DE u−s consider for example the exponential case (5) plus than unity (here, we do not write the correction term term (7). The numericalevaluation of Eq. (3) leads to forthestabilityofoscillationsinthematterdominated Hubble parameter era). In Eq. (12), the last term γ¯(1/R˜iα−1)Rα, where γ¯ is a positive dimensional constant and α is a real number, works at the inflation scale R˜ and is actu- H(z)= m˜2[y (z)+(z+1)3], (11) i H ally necessary in order to realize an exit from infla- p tion. An accurate analysis of this model shows that if andthereforeω (z). Wedepictthecorrespondingcos- eff α > 1 and n > 1, we avoid the effects of inflation at mologicalevolutionof ω (z)for 1<z <2 in Fig.2. eff − small curvatures and it does not influence the stabil- We can see that ω (z) starts from zero in the matter eff ity of the matter dominated era. The de Sitter point dominatederaandasymptoticallyapproaches-1with- R which describes the early time acceleration exists oωut(azn)ycaropspsreescitahbelelidneevioaftipohna.nOtonmlydwihveidne,zh≃ow−0i.t90is, ifdRS˜i = RdS and it reads as RdS = 2/(γ¯(2−α)+1) eff under the condition (R /R)n 1. Thus, the infla- shown on the right. By considering the scale factor dS i ≫ tion is unstable if αγ¯(α 2) > 1. We may evaluate a(t) = exp(H0t), where H0 ≃ 6.3×10−34eV−1 is the the characteristic numbe−r of e-folds during inflation Hubble parameter of the de Sitter universe, we may N = log[(z + 1)/(z + 1)], where z and z are the conclude that the crossings appear at 1026 years. i e i e redshifts at the beginning andat the end of earlytime Ω_eff -1.0 -0.5 0.5 1.0 1.5 2.0 z cosmic acceleration. Given a small cosmological perturbation y (z) at the redshift z to the effective dark -0.2 1 i i energy density of inflation y = R /(12m˜2) , so that 0 dS -0.4 y(z) = y +y (z), we have from Eq. (3) avoiding the Ω_eff 0 1 -0.6 -0.9985 matter contribution -0.9990 -0.8 -0.9995 y1(zi)=C0(zi+1)x, (13) -1.0 -0.96 -0.94 -0.92 -0.90 z with Figure 2: Effective EoS parameter for 1 < z < 2 in exponential model. On the right, the re−gion −1<z < x= 1 3 25 16F′(RdS) . (14) 0.9 is stressed. 2 −s − RdSF′′(RdS)! − 4 Here, x < 0 if the de Sitter point is unstable. As a effective modified gravity energy density and the cur- consequence, the perturbation y (z) grows up in ex- vature decrease,we may conclude that it is possible to 1 panding universe as y (z) = y (z)[(z+1)/(z +1)]x. acquire a gravitational alternative scenario for a uni- 1 1 i i When y (z) is on the same order of the effective mod- fied description of inflation in the early universe with 1 ified gravityenergy density y (z) of the de Sitter solu- the late-time cosmic acceleration. 0 tiondescribinginflation,themodelexitsfrominflation. Inthisway,wecanestimatethenumberofe-foldsdur- References ing inflation as [1] E. Komatsuet al. [WMAP Collaboration],Astro- 1 y (z) 1 i N log . (15) phys. J. Suppl. 192, 18 (2011). ≃ x y (cid:18) 0 (cid:19) A value demanded in most inflationary scenarios is [2] S. NojiriandS.D. Odintsov,Phys.Rept. 505,59 at least N = 50–60. A classical perturbation on the (2011). (vacuum) de Sitter solution may be given by the pres- [3] E. Elizalde, S. D. Odintsov, L. Sebastiani and ence of ultrarelativistic matter in the early universe. S. Zerbini, Eur. Phys. J. C 72, 1843 (2012). Thus, we can predict the numbers of e-folds for the model(12)bycomputingxinEq. (14),whichdepends [4] K. Bamba, A. Lopez-Revelles, R. Myrzakulov, on model parameters. Let us consider an example S.D.OdintsovandL.Sebastiani,arXiv:1207.1009 by supposing that ultrarelativistic matter/radiation is [gr-qc]. 106 times smaller than that of effective modified grav- [5] K. Bamba, C. -Q. Geng and C. -C. Lee, JCAP ity energy density( Λ) at the beginning of infla- ∼ i 1011, 001 (2010). tion. By choosing the viable model parameters n=4, Ri = 2Λi, γ¯ = 1 and α = 8/3, such that the curva- [6] W. Hu and I. Sawicki, Phys. Rev. D 76, 064004 ture at the inflation results to be RdS = 6Λi, one has (2007). x= 0.218 and, by using Eq. (15), we obtain N 64 and−z = e 64(z + 1) 1. The numerical extr≃apo- [7] G.Cognola,E.Elizalde,S.Nojiri,S.D.Odintsov, e − i − L. Sebastiani and S. Zerbini, Phys. Rev. D 77, lation of Eq. (3) gives us y (z ) = 0.88y (z) and H e H i 046009 (2008); E. V. Linder, Phys. Rev. D 80, R(z ) = 0.853R , confirming that the effective modi- e dS 123528 (2009). fied gravity energy density and the curvature decrease at the end of inflation. After that, it is reasonable [8] T. Chiba, T. L. Smith and A. L. Erickcek, Phys. to expect that the physical processes described by the Rev.D75,124014(2007);G.J.Olmo,Phys.Rev. ΛCDMModelarereproducedbythefirsttermofmod- D 75, 023511 (2007). ification to gravity. [9] S. Nojiri and S. D. Odintsov, Phys. Lett. B 652, 343 (2007). 7 Conclusions [10] A.D.DolgovandM.Kawasaki,Phys.Lett.B573, Inthiswork,wehaveanalyzedsomefeatureofrealistic 1 (2003); V. Faraoni, Phys. Rev. D 74, 104017 F(R)-gravity mimicing the physics of ΛCDM Model. (2006). Two viable models have been considered. We have seen that since the corrections to GR at small curva- [11] E. V. Linder, Phys. Rev. D 72, 043529 (2005). tures may lead to undesired effects at high curvatures, [12] H. W. Lee, K. Y. Kim and Y. S. Myung, Eur. some adding modifications are required. In particular, Phys. J. C 71, 1748 (2011). we have reconstructed a compensating term in order to stabilize the oscillationsof the effective dark energy [13] H. Motohashi, A. A. Starobinsky and at high red shifts with retaining the viability propri- J. ’i. Yokoyama,JCAP 1106, 006 (2011). eties. Furthermore,wehavebrieflydiscussedaboutthe [14] R.R.Caldwell,M.KamionkowskiandN.N.Wein- growthindex and aninvestigationon the cosmological berg, Phys. Rev. Lett. 91 071301(2003). evolution of the universe has been executed showing thattheeffectivecrossingofthephantomdivide,which [15] P.H.Frampton,K.J.LudwickandR.J.Scherrer, characterizes the de Sitter epoch, occurs in the very Phys. Rev. D 84, 063003(2011). far future. We also have considered the inflationary [16] E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebas- cosmology in exponential gravity. It has been derived tiani and S. Zerbini, Phys. Rev. D 83, 086006 thecorrelationbetweenthee-foldsduringinflationand (2011). themodelparameters. Sinceattheendofinflationthe 5