aa r X i v : . [ a s t r o - ph . C O ] J u l SCG-2013-06
Construction of f ( R ) Gravity Models Jun-Qi Guo ∗ Department of Physics, Simon Fraser University8888 University Drive, Burnaby, BC Canada V5A 1S6 (Dated: July 11, 2013)In this paper, we study how to construct f ( R ) gravity models. Cosmological observations andlocal gravity tests imply that a viable f ( R ) model should be very close to the ΛCDM model. Wecreate procedures to construct viable ΛCDM-like f ( R ) models, and present multiple models ofthree types. The connections between some of these models are discussed. We also study thecosmological evolution of the ΛCDM-like f ( R ) gravity. Exact numerical integration can generateaccurate cosmological evolution, but the numerical simulation in the early-universe stage is slow dueto the oscillations of the field φ ( ≡ f ′ ) near the minimum of the effective potential V eff ( φ ). To avoidthis problem, we take the minimum of the effective potential V eff ( φ ) as an approximate solutionfor φ , and obtain the cosmological evolution. This approximate method describes the cosmologicalevolution well except in the late universe. Therefore, we use the approximate method in the early-universe evolution, and use the exact method in the late-universe one. PACS numbers: 04.25.Nx, 04.50.Kd, 11.10.Lm, 95.36.+x, 98.80.Es, 98.80.Jk
I. INTRODUCTION
The causes of cosmological acceleration remain un-known [1, 2, 3, 4, 5]. Among various approaches to ex-plain this cosmic speed-up, f ( R ) gravity is a straightfor-ward option. In the Jordan frame, one may replace theRicci curvature scalar in the Einstein-Hilbert action witha function of the scalar S JF = 116 πG Z d x √− gf ( R ) + S M [ g µν , ψ m ] , (1)where G is the Newtonian gravitational constant, and ψ m is the matter field. See Refs. [6, 7] for reviews of f ( R )theory.The f ( R ) gravity needs to confront stability, cosmolog-ical viability, and local gravity tests. In fact, the featuresof an f ( R ) model are largely determined by a poten-tial V ( φ ) as defined in the equations of motion for f ( R )theory. In this paper, we explore how to construct vi-able f ( R ) models by connecting the viability conditionswith the geometry of the potential. Cosmological obser-vations and local gravity tests imply that a viable f ( R )model should be very close to the ΛCDM model. Here wemake instructions to build viable ΛCDM-like f ( R ) mod-els, and present multiple models (three types). Some ofthem have been proposed before, but others have not.We also point out the connections between some of thesemodels.Cosmological evolution and solar system tests are twoextreme cases of the dynamics of the field φ ≡ f ′ . Inthe cosmological evolution, the universe is assumed to behomogeneous. Therefore, the cosmic dynamics only de-pend on the temporal variable. In the solar system tests,the spherically symmetric spacetime is static, then the ∗ Electronic address: [email protected] field φ only varies with respect to the spacial variable.In both cases, the field φ interacts with the matter den-sity, and the field φ can be coupled to the matter den-sity when the matter density is much greater than thecosmological constant. The solar system tests of f ( R )gravity are explored by considering an effective poten-tial constructed by the matter density and the potential V ( φ ) [8, 9, 10, 11, 12, 13, 14, 15, 16]. In this paper, weemploy the same approach to study the cosmic dynamicsof f ( R ) gravity.For a feasible f ( R ) model, in the early universe, thefield φ ( ≡ f ′ ) has a slow roll evolution so that a matterdomination phase exists. In the late universe, the field φ will be released from the coupling between the mat-ter density and the potential V ( φ ), thus generating thecosmic speed-up. In this study, the cosmic dynamics of f ( R ) gravity are explored from the early universe to thelate one. Due to the oscillations of φ in the effectivepotential V eff ( φ ), the numerical simulation of the cosmo-logical evolution can be very slow. In order to integratethe evolution more effectively, the evolution of the mini-mum point of V eff ( φ ) is taken as an approximate solutionfor φ . This approach describes the cosmological evolu-tion well until in the late universe. To supplement, theexact method is used in the late universe. Therefore, acombination of exact and approximate methods providesa complete picture of the cosmological evolution of f ( R )gravity.The paper is organized as follows. In Sec. II, we intro-duce the framework. In Sec. III, a type of non-ΛCDM-like f ( R ) model will be explored . Sec. IV discusses how toconstruct viable ΛCDM-like f ( R ) models. In Sec. V, thecosmological evolution of one example of these ΛCDM-like models is analysed. Lastly, Sec. VI summarizes ourresults. II. FRAMEWORK
A variation on the action for f ( R ) gravity with respectto the metric yields gravitational equations of motion f ′ R µν − f g µν − [ ∇ µ ∇ ν − g µν (cid:3) ] f ′ = 8 πGT µν , (2)where f ′ denotes the derivative of the function f with re-spect to its argument R , and (cid:3) is the usual notation forthe covariant D’Alembert operator (cid:3) ≡ ∇ α ∇ α . Com-pared to general relativity, f ( R ) gravity has one extrascalar degree of freedom, f ′ . The dynamics of this de-gree of freedom are determined by the trace of Eq. (2), (cid:3) f ′ = 13 (2 f − f ′ R ) + 8 πG T, (3)where T is the trace of the stress-energy tensor T µν . Iden-tifying f ′ by φ ≡ dfdR , (4)and defining a potential V ( φ ) by V ′ ( φ ) ≡ dVdφ = 13 (2 f − φR ) , (5)one can rewrite Eq. (3) as (cid:3) φ = V ′ ( φ ) + 8 πG T. (6)In order to explore how f ( R ) gravity causes cosmic speed-up, it is convenient to cast the formulation of f ( R ) grav-ity in a format similar to that of general relativity. Wecan rewrite Eq. (2) as G µν = 8 πG h T µν + T (eff) µν i , (7)where8 πGT (eff) µν = f − f ′ R g µν + ∇ µ ∇ ν f ′ − g µν (cid:3) f ′ +(1 − f ′ ) G µν . (8) T µν (eff) is the energy-momentum tensor of the effective darkenergy, and it is guaranteed to be conserved, ˜ T µν (eff); ν = 0.Equation (8) gives the definition of the equation of statefor the effective dark energy as w eff ≡ p eff ρ eff , (9)where8 πGρ eff = 3 H − πG ( ρ m + ρ r )= f ′ R − f − H ˙ f ′ + 3 H (1 − f ′ ) , (10)8 πGp eff = H − R/ − πGp r = ¨ f ′ + 2 H ˙ f ′ + f − f ′ R H − R/ − f ′ ) . (11)In order for an f ( R ) model to account for the cosmicspeed-up, w eff should be less than − / III. NON-ΛCDM-LIKE f ( R ) MODELSA. Viability conditions on f ( R ) gravity A viable f ( R ) model should be stable, mimic a cosmo-logical evolution consistent with observations, and satisfylocal gravity tests. This places some viability conditionson f ( R ) gravity as follows.1. We require f ′ to be positive to avoid anti-gravity.2. The function f ( R ) should be very close to the cur-vature scalar R at high curvature so that a matterdomination epoch can exist in the early universe.3. The f ′′ should be positive when the curvaturescalar R is greater than the cosmological constantΛ, so that the Dolgov-Kawasaki instability can beavoided and the scalaron f ′ is non-tachyonic [17].Moreover, the potential V ( φ ) should have a mini-mum such that a dark-energy domination stage anda consequent cosmic acceleration can be generatedin the late universe.4. The Big Bang nucleosynthesis, observations of theCosmic Microwave Background, and local gravitytests imply that general relativity should be recov-ered as R ≫ Λ: f ( R ) → R and f ′ →
1. This,together with the requirement of f ′′ >
0, impliesthat f ′ should be less than 1 [18]. B. Non-ΛCDM-like f ( R ) models In our previous work [16, 19, 20], we explored an R ln R model, f ( R ) = R [1 + α ln( R/R )]. In this model, themodification term causes significant deviation from gen-eral relativity at high curvature. As a result, this modelhas difficulties when it comes to developing a matterdomination stage and passing solar system tests. Thisproblem is alleviated in the modified logarithmic model f ( R ) = R a + log(
R/R )1 + log( R/R ) = R (cid:20) − b R/R ) (cid:21) , (12)where b = 1 − a . In this model, f ′ = 1 − b R/R ) + b [1 + log( R/R )] . (13)Equation (13) implies that f ′′ and V ′′ ( φ )(= ( f ′ − f ′′ R ) / f ′′ ) will change signs at some point, and the po-tential V ( φ ) is folded at that point. When b is small,the potential V ( φ ) can be folded before φ reaches the deSitter point, as shown in Fig. 1. The model is unstable atplaces where f ′′ < V ′′ ( φ ) <
0. When b is greater,the folding point of V ( φ ) can be shifted to the left sideof the de Sitter point, as shown in Fig. 2. Because ofthe “soft” logarithmic dependence in the function f ( R ),this model does not have a fully matter-dominated epoch φ V ( φ ) V ′′ ( φ ) > V ′′ ( φ ) < de Sitter point FIG. 1: The potential for the modified logarithmic model ofEq. (12) with b = 0 . R = 1. The potential V ( φ ) isfolded before φ reaches the de Sitter point. At the foldingpoint, V ′′ ( φ ) switches signs. The model is unstable at placeswhere V ′′ ( φ ) < φ V ( φ ) V ′′ ( φ ) < de Sitter point FIG. 2: The potential for the modified logarithmic modelof Eq. (12) with b = 3 and R = 1. The folding point of V ( φ ) is shifted to the left side of the de Sitter point in thisconfiguration. in the early universe. In order to explore other possi-bilities, one can generalize the function of Eq. (12) asfollows: f ( R ) = R (cid:20) − b A ( R/R ) (cid:21) , (14)where A ( R/R ) is a function of R/R . One may considera polynomial case as expressed by f ( R ) = R (cid:20) − b R/R ) n (cid:21) . (15)When n = 1, this model happens to be the simplest for-mat of the Hu-Sawicki model [12]. The complete formatof the Hu-Sawicki model is described by Eq. (23). In thecase of n = 1, f ′ = 1 − b (1 + R/R ) . (16) φ V ( φ ) de Sitter point FIG. 3: The potential for the polynomial model of Eq. (15)with n = R = 1 and b = 5. In the case of n = 1, this modelhappens to be the simplest format of the Hu-Sawicki model ofEq. (23), V ( φ ) does not have a folding point and V ′′ ( φ ) > R ≫ Λ) to the low ( R ∼ Λ) curvature regimes. φ V ( φ ) V ′′ ( φ ) < de Sitter point FIG. 4: The potential for the polynomial model of Eq. (15)with n = 2 and b = R = 1. V ( φ ) has a folding point. Forsome period of φ from the high ( φ →
1) to the low curvatureregimes, V ′′ ( φ ) is negative and, consequently, the scalaron f ′ is tachyonic. Therefore, f ′′ will not be zero, and V ( φ ) does not havea folding point from the high curvature regime ( R ≫ Λ)to the low one ( R ∼ Λ), as is shown in Fig. 3. However,as long as n > f ′′ will include more than one termof R with different signs, V ( φ ) will have a folding point,and V ′′ ( φ ) will switch signs at the folding point. Asillustrated in Fig. 4, in this case, for some period of φ from the high to the low curvature regimes, V ′′ ( φ ) isnegative and, consequently, the scalaron f ′ is tachyonic.We also considered the exponential case f ( R ) = R (cid:20) − b R/R ) (cid:21) . (17)This model has a similar problem as shown in Fig. 5.In this class of f ( R ) models of Eq. (14), there are func-tions of R in both the numerator and the denominator −0.5 0 0.5 1 1.5−0.4−0.3−0.2−0.10 φ V ( φ ) V ′′ ( φ ) < de Sitter point FIG. 5: The potential for the exponential model of Eq. (17)with b = 2 . R = 1. V ( φ ) has a folding point. Forsome period of φ from the high ( φ →
1) to the low curvatureregimes, V ′′ ( φ ) is negative and, consequently, the scalaron f ′ is tachyonic. of the modification term, which results in V ( φ ) havinga folding point. At that folding point, V ′′ ( φ ) switchessigns, and the scalaron f ′ is tachyonic at places where V ′′ ( φ ) <
0. To avoid this problem, one may replace thefunction of R in the numerator with a constant, and ob-tain ΛCDM-like models. In these models, the function f ( R ) is approximately equal to ( R − IV. CONSTRUCTION OF ΛCDM-LIKE f ( R )MODELSA. Procedures In this study, we construct three types of ΛCDM-like f ( R ) models as expressed by Eqs. (20), (27), and (34),respectively. For simplicity, we take the Type II f ( R )gravity described by Eq. (27) as an example to explainhow to construct viable ΛCDM-like f ( R ) models. Thefunction for the Type II f ( R ) gravity is f ( R ) = R − b [ c − A ( R/R )] . (18)The procedures are as follows.1. The parameters b and R have the same energyscale as the cosmological constant. For simplicity, R is set to be equal to 1.2. Generally, A ( R/R ) goes to zero at high curvature,such that the f ( R ) model reduces to the ΛCDMmodel at high curvature. This will make the f ( R )model have a matter domination epoch in the earlyuniverse, and avoid the solar system tests as well. 3. At high curvature, f ′ should be positive to avoidanti-gravity. This can be guaranteed without diffi-culty, because at high curvature regime the modi-fication term b [ c + A ( R/R )] is much less than themain term R in the function f ( R ).4. The f ′′ should be positive when R >
Λ, such thatthe Dolgov-Kawasaki instability can be avoided[17] and the scalaron f ′ is non-tachyonic. The po-tential V ( φ ) should have a minimum, such that themodel is stable and can mimic the later cosmic ac-celeration. These can be obtained by the followingmeasures.(a) Note that in f ( R ) theory V ′′ ( φ ) = f ′ − f ′′ R f ′′ . (19)Generally, f ′ > f ′′ R and f ′ ∼ V ′′ ( φ ) ismainly determined by f ′′ , and f ′′ is deter-mined by the modification term in the func-tion f ( R ). Thus we make sure that f ′′ > V ′′ > A ( R/R ).(b) Let V ′ ( φ ) = (2 f − f ′ R ) / > f ( R ) → R and f ′ → b and c to make sure that V ′ ( φ ) = (2 f − f ′ R ) / < R ∼ Λ.5. The requirement of f ′ < f ′′ > B. Type I ΛCDM-like f ( R ) models One obtains the first type of ΛCDM-like f ( R ) modelby replacing the term b · R in the numerator of the mod-ification term in Eq. (14) with the parameter b , so that f ( R ) = R − bc + A ( R/R ) , (20)where c is another parameter. One can construct someviable models as follows by letting A ( R/R ) take someelementary functions and implementing the proceduresdiscussed in Sec. IV A. • Logarithmic format 1: f ( R ) = R − bc + log(1 + R /R ) . (21)Example parameters for this model are b = 1 . c = R = 1. The potential for this model with theseparameters is plotted in Fig. 6. As expected, the po-tential has a minimum, and V ′′ ( φ ) is positive from the −1 −0.5 0 0.5 1−0.4−0.3−0.2−0.10 φ V ( φ ) FIG. 6: The potential for the model of Eq. (21) with b = 1 . c = R = 1. The potential has a minimum, and V ′′ ( φ )is positive from the high ( R ≫ Λ, φ →
1) to the low ( R ∼ Λ, φ ∼ .
8) curvature regimes. Therefore, this model should bestable, and should also have a sensible cosmological evolutionas verified in Sec. V. high ( R ≫ Λ, φ →
1) to the low ( R ∼ Λ, φ ∼ .
8) cur-vature regimes. Therefore, this model should be stable,and should also have a sensible cosmological evolution asverified in Sec. V. The potentials for all the ΛCDM-like f ( R ) models presented in this paper have been plotted,with the parameters taking appropriate values. They alllook similar to Fig. 6 and therefore are not individuallyshown here. • Logarithmic format 2: f ( R ) = R − bc + 1 / log(1 + R/R ) . (22)For this model, example parameters, which can generatea potential similar to the one plotted in Fig. 6, are b = c = 5 and R = 1. • Polynomial format: f ( R ) = R − bc + ( R /R ) n = R − b ( R/R ) n c ( R/R ) n + 1 , (23)where n is a positive integer number. Example param-eters for this model are b = 2, c = R = 1, and n = 3.This is the Hu-Sawicki model [12]. • Exponential format 1: f ( R ) = R − bc + exp( − R/R ) . (24)Example parameters for this model are b = 2 and c = R = 1. This model is almost the same as the one dis-cussed in Ref. [21]: f ( R ) = R − CA + B exp( − R/D ) + CA + B , (25) where
A, B, C, and D are parameters. • Exponential format 2: f ( R ) = R − bc + exp( R /R ) . (26)Example parameters for this model are b = 5, and c = R = 1. C. Type II ΛCDM-like f ( R ) models In the Type I models described by Eq. (20), the mod-ification term, c + A ( R/R A ( R/R
0) switched, so that f ( R ) = R − b [ c − A ( R/R )] . (27)Some models of this type can be constructed as follows. • Logarithmic format 1: f ( R ) = R − b (cid:20) c − log (cid:18) R R (cid:19)(cid:21) . (28)Example parameters for this model are b = c = R = 1.When c is equal to zero, one obtains f ( R ) = R − αR log (cid:18) RR (cid:19) , (29)where α is a positive parameter. This model is discussedin Ref. [22]. The model in Eq. (28) reduces to the ΛCDMmodel faster than the one in Eq. (29). • Logarithmic format 2: f ( R ) = R − b (cid:20) c − R/R ) (cid:21) . (30)Example parameters for this model are b = R = 1 and c = 5. • Polynomial format: f ( R ) = R − b (cid:20) c − (cid:18) R R (cid:19) n (cid:21) . (31)Example parameters for this model are b = R = 1 and c = n = 2. This model and the Hu-Sawicki model inEq. (23) can be considered to be modifications of the1 /R model, in which f ( R ) = R − µ /R , where µ is aparameter with units of mass [23]. In the 1 /R model, f ′′ and V ′′ ( φ ) are negative, so the scalaron f ′ is tachy-onic. This problem is avoided in the modified versions inEqs. (23) and (31). • Exponential format 1: f ( R ) = R − b [ c − exp ( − R/R )] . (32)Example parameters for this model are b = R = 1 and c = 2. This model is explored in Refs. [24, 25, 26, 27]. • Exponential format 2: f ( R ) = R − b [ c − exp( R /R )] . (33)Example parameters for this model are b = 5, c = 2, and R = 1. D. Type III ΛCDM-like f ( R ) models One can combine Eqs. (20) and (27), and obtain thethird type of ΛCDM-like f ( R ) models f ( R ) = R − b c − A ( R/R ) d + A ( R/R
0) = R + b (cid:20) − c + dd + A ( R/R (cid:21) . (34)Some models of this type can be constructed as follows. • Exponential format: f ( R ) = R − b tanh( R/R )= R − b − exp( − R/R )1 + exp( − R/R R + b (cid:20) −
21 + exp( − R/R ) (cid:21) . (35)Example parameters for this model are b = R = 1. Thismodel is presented in Ref. [28]. The model described byEq. (25) is almost the same as this model. • Logarithmic format: f ( R ) = R − b c − log(1 + R /R ) d + log(1 + R /R ) . (36)Example parameters for this model are b = 6 and c = d = R = 1. • Polynomial format: f ( R ) = R − b c − ( R /R ) n d + ( R /R ) n . (37)Example parameters for this model are b = 6, c = d = R = 1, and n = 2. V. COSMOLOGICAL EVOLUTION
In this section, we will explore the cosmological evolu-tion of the ΛCDM-like f ( R ) gravity by taking the modelin Eq. (21) as an example. A. Formalism
In this paper, we consider the homogeneous uni-verse in the flat Friedmann-Robertson-Walker metric, ds = − dt + a ( t ) d x . In this case, the universe canbe modeled by a four-dimensional dynamical system of { φ, π, H, a } , where π ≡ ˙ φ, (38) H is the Hubble parameter, and the dot ( · ) denotes thederivative with respect to time. Equation (3) providesthe dynamical equation for π ,˙ π = − Hπ − V ′ ( φ ) + 8 πG ρ m . (39)The equation of motion for H is˙ H = R − H . (40)The definition of the Hubble parameter gives˙ a = aH. (41)The system is constrained by H + πφ H + 16 f − φRφ − πG φ ( ρ m + ρ r ) = 0 , (42)where ρ m and ρ r are the densities of matter and radia-tion, respectively. Equations (38)-(42) provide a closeddescription of the dynamical system of { φ, π, H, a } . B. The evolution from the exact method
A straightforward way to simulate cosmological evo-lution is to integrate the equations of motion (39)-(41).However, from the point of view of numerics, at highredshift, the field φ evolves very slowly and oscillates be-tween V ′ ( φ ) and 8 πGρ m /
3. The behaviors of the termsin Eq. (39) are shown in Fig. 7, revealing the relation be-tween the four terms in Eq. (39): | H ˙ φ | < | ¨ φ | ≪ V ′ ( φ ) ≈ πGρ m /
3. The field φ oscillates near the minimum of theeffective potential V eff ( φ ), which is defined by V ′ eff ( φ ) = V ′ ( φ ) + V ′ m ( φ ) , (43)with V ′ m ( φ ) = − πGρ m / . (44)These oscillations produce particles, and could be a pos-sible source of energetic cosmic rays [29]. The oscillationsof φ also make it inconvenient to numerically integratethe evolution in the early universe. Therefore, in thenext sub-section, we consider employing an approxima-tion method instead. As a consequence of the oscillationsof the field φ , the equation of state w eff also oscillates near −
1, as shown in Fig. 8.The behaviors of the field φ and the terms in the equa-tion of motion for φ (39) at low redshift are shown inFigs. 9 and 10, respectively. Figure 9 reveals that the ( z − z ) T e r m s i n E . o . M . f o r φ · − V ′ · − · π G ρ m / ¨ φ − H ˙ φ FIG. 7: The numerical evolution at high redshift for themodel of Eq. (21) with b = 1 . c = R = 1. z =20256 . φ evolves veryslowly and oscillates between V ′ ( φ ) and 8 πGρ m /
3. The fig-ure shows the relation between the four terms in Eq. (39): | H ˙ φ | < | ¨ φ | ≪ V ′ ( φ ) ≈ πGρ m / ( z − z ) w e ff FIG. 8: The oscillations of w eff in the numerical evolutionat high redshift for the model of Eq. (21) with b = 1 . c = R = 1. z = 20256 . φ , the equation of state w eff alsooscillates near − field φ has a slow roll when the matter density is greaterthan the cosmological constant. The field φ drops signif-icantly as the matter density comes to the cosmologicalconstant scale, oscillates at the minimum of the poten-tial V ( φ ), and then eventually stops due to friction force − H ˙ φ . As demonstrated in Fig. 10, in the late universe,compared to other terms in Eq. (39), the matter forceterm 8 πGρ m / φ ≈ − H ˙ φ − V ′ ( φ ) . (45)These results can also be interpreted via the effectivepotential V eff ( φ ) as plotted in Figs. 11 and 12. The ef-fective potential V eff is very flat at high redshift and hasa drop at low redshift, such that the field φ has a slowroll at high redshift and then a drop at low redshift. The −1 z + 1 φ FIG. 9: The evolution for φ at low redshift for the modelof Eq. (21) with b = 1 . c = R = 1. The field φ hasa slow roll when 8 πGρ m > Λ. After that, the field dropssignificantly, oscillates, and eventually stops at the minimumof the potential V ( φ ) due to friction force − H ˙ φ . −1 −1−0.500.51 z + 1 T e r m s i n E . o . M . f o r φ − V ′ − · π G ρ m / ¨ φ − H ˙ φ −1 −0.02−0.0100.010.02 V ′ · π G ρ m / ¨ φ − H ˙ φ FIG. 10: The terms in the equation of motion (E.o.M.) (39)for φ at low redshift for the model of Eq. (21) with b = 1 . c = R = 1. In the late universe, compared to other terms inEq. (39), the matter force term 8 πGρ m / φ ≈ − H ˙ φ − V ′ ( φ ). evolutions of Ω i ′ s and w eff at low redshift are shown inFig. 13. The Ω i ′ s are defined as Ω i = 8 πGρ i / (3 H ),where i refers to the indexing of radiation, matter or ef-fective dark energy. The ΛCDM-like models can mimic acosmological evolution, fitting the observations withoutdifficulty. Comparison of Figs. 8 and 13 demonstratesthat the phantom behavior of w eff ( w eff crosses −
1) atlow redshift is nothing but an extension of that behaviorat high redshift.
C. The evolution from the approximation method
Due to the oscillations of φ , it is not convenient tonumerically integrate the cosmological evolution in theearly universe. However, since the field φ oscillates near −1 −0.06−0.04−0.0200.020.040.06 z + 1 V e ff ( φ ( z )) . V . V m V e ff −1 −8.44−8.42−8.4−8.38 x 10 −3 −1 −5051015 x 10 −6 . ( V + . ) FIG. 11: V eff ( φ ) vs. z + 1 at low redshift for the model ofEq. (21) with b = 1 . c = R = 1. The potentials V , V m ,and V eff are defined by Eqs. (5), (43), and (44), respectively. V eff = V + V m . V eff is very flat at high redshift and has a dropat low redshift, such that the field φ has a slow roll at highredshift and then a drop at low redshift. φ V e ff ( φ ) . V . V m V e ff FIG. 12: V eff ( φ ) vs. φ at low redshift for the model of Eq. (21)with b = 1 . c = R = 1. V eff = V + V m . −1 Ω i Ω m Ω r Ω e ff −1 −2−1.5−1−0.50 z + 1 w e ff FIG. 13: The evolutions of Ω i ′ s and w eff at low redshift forthe model of Eq. (21) with b = 1 . c = R = 1. −1 z + 1 φ FIG. 14: The evolution for φ in the approximation method forthe model of Eq. (21) with b = 1 . c = R = 1. The field φ has a slow roll at high redshift and a drop at low redshift. −1 Ω i Ω m Ω r Ω e ff −1 −1−0.50 z + 1 w e ff FIG. 15: The evolutions of Ω i ′ s and w eff in the approximationmethod for the model of Eq. (21) with b = 1 . c = R = 1. −1 −1−0.500.51x 10 −3 z + 1 V e ff ( φ ( z )) − V − V m V e ff FIG. 16: The effective potential V eff ( φ ) in the approximationmethod for the model of Eq. (21) with b = 1 . c = R = 1.The potentials V , V m , and V eff are defined by Eqs. (5), (43),and (44), respectively. V eff = V + V m . V eff is very flat at highredshift and has a drop at low redshift, such that the field φ has a slow roll at high redshift and a drop at low redshift. the minimum of the effective potential V eff ( φ ) in the earlyuniverse, one can take the evolution of the minimum of V eff ( φ ) approximately to be that of the field φ . The ap-proximate solution describes the evolution of φ accuratelyin the early universe. In the late universe, the deviationof the field φ from the minimum of V eff ( φ ) becomes rela-tively large, and the exact method should be used as hasbeen done in Sec. V B.Equations (38) and (39) can be combined as¨ φ = − H ˙ φ − V ′ ( φ ) + 8 πG ρ m . (46)For the ΛCDM-like models, the field φ evolves very slowlyin the early universe. From this we get | H ˙ φ | < | ¨ φ | ≪ V ′ ( φ ) ≈ πG ρ m , (47)as shown in Fig. 7. Note that ρ m = ρ m /a and ˙ a = aH ,where ρ m is the matter density of the current universe.These, together with V ′ ( φ ) ≈ πGρ m /
3, lead to˙ φ ≈ − H V ′ V ′′ . (48)Therefore, in our approximation method, we takeEq. (48) as an approximate equation of motion for φ ,we replace Eqs. (38) and (39) by Eq. (48), but keepEqs. (40)-(42). The results of the approximation methodare shown in Figs. 14-16. The field φ has a slow roll evolu-tion in the early universe due to the approximate balancebetween V ′ and 8 πGρ m /
3, and the the evolutions of Ω i ′ s are consistent with cosmological observations.The exact and the approximate methods are supple-mentary. The numerical integration in the exact methodat high redshift is very slow due to the oscillations ofthe field f ′ , but can mimic the cosmological evolution inthe late universe very easily. The approximate methoddoes not yield an accurate cosmological evolution in thelate universe, but can mimic a smooth evolution in theearly universe. Therefore, these two methods can be usedtogether to explore the cosmic dynamics of f ( R ) gravity. D. The equation of state w eff The Big Bang nucleosynthesis and the observationsof the Cosmic Microwave Background imply that gen-eral relativity should be recovered in the early universe,which means that f ( R ) → R and f ′ → R ≫ Λ. Onthe other hand, f ′′ should be positive to ensure that thescalaron f ′ is non-tachyonic. Consequently, f ′ should beless than 1 [18]. In fact, an f ′ greater than 1 in the earlyuniverse can cause a pole in the equation of state w eff ,as shown in Fig. 17 and also pointed out in Ref. [19].This situation can be explained with Eqs. (9) and (10).The first term, ( f ′ R − f ) /
2, in Eq. (10) can be positive.(This term is equal to αR/ R ln R −2 Ω i Ω m Ω r Ω e ff −2 −10−50510 z + 1 w e ff A pole
FIG. 17: The equation of state w eff for the R ln R model with α = 0 .
02 and R = 1. Since f ′ is greater than 1 in the earlyuniverse, the effective dark-energy density ρ eff described byEq. (10) can be zero at some moment, which will cause a polein the equation of state w eff . model, in which f ( R ) = R [1 + α ln( R/R )] and α is pos-itive.) The second term, − H ˙ f ′ , is positive because f ′ will roll down to the minimum of the potential V ( φ ).However, the third term, 3 H (1 − f ′ ), is negative when f ′ is greater than 1 in the early universe. Then, in thelater evolution, f ′ will decrease, cross 1, and move to theminimum of the potential V ( φ ). At some moment, theenergy density of the effective dark energy ρ eff will bezero. This will generate a pole in the equation of state w eff defined by Eq. (9). The ΛCDM-like models do nothave such a problem, since the function f ( R ) ≈ R − f ′ ≈ f ( R ) model should also pass the solarsystem tests. We studied how the Hu-Sawicki modeldeals with the solar system tests in Ref. [20]. It turnedout that the ΛCDM-like f ( R ) models have the advantageof passing the solar system tests. VI. CONCLUSIONS
In this paper, we studied how to construct f ( R ) grav-ity models which satisfy the stability and viability condi-tions. Cosmological observations and local gravity testsplace stringent requirements on the format of the func-tion f ( R ). A feasible f ( R ) model needs to be very closeto the ΛCDM model. For the ΛCDM-like models, it isnot hard to obtain the recovery of general relativity inthe early universe and a cosmic speed-up in the late uni-verse. The condition f ′ > R in the function f ( R ). The thing is howto make the potential V ( φ ) have a minimum. One way isto tune the parameters, such that V ′′ ( φ ) > V ′ ( φ ) > R ≫ Λ, and V ′ ( φ ) < R <
Λ. Once0these have been achieved, the potential V ( φ ) will havea minimum. With this method, three types of ΛCDM-like f ( R ) models expressed by Eqs. (20), (27), and (34)are constructed. In addition to these three types, a vi-able ΛCDM-like f ( R ) may also take other forms. Forexample, the Starobinsky model takes the form f ( R ) = R + λR [(1 + R /R ) − n −
1] [30], and a new exponentialmodel takes the form f ( R ) = ( R − λc ) exp[ λ ( c/R ) n ] [31].We also studied the cosmological evolution of theΛCDM-like f ( R ) models. The field φ evolves slowly inthe early universe due to the quasi-static balance between V ′ ( φ ) and 8 πGρ m /
3, and is released from the couplingbetween V ′ ( φ ) and 8 πGρ m / φ near the minimum of the ef-fective potential V eff ( φ ). To avoid this problem, we take the minimum of V eff ( φ ) as an approximate solution for φ and obtain the cosmological evolution from the earlyuniverse to the late one. This approximation method de-scribes the cosmological evolution well except in the lateuniverse when the curvature scalar is below the cosmo-logical constant scale. We use the exact method to studythe late-universe evolution. Then, a combination of theexact and the approximate methods provides a completepicture of the cosmological evolution of f ( R ) gravity. Acknowledgments
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