Curvature Singularity in f(R) Theories of Gravity
aa r X i v : . [ g r- q c ] J u l Curvature Singularity in f ( R ) Theories of Gravity
Koushik Dutta ∗ , Sukanta Panda † , Avani Patel †∗ Theory Division,Saha Institute of Nuclear Physics,1/AF Salt Lake,Kolkata - 700064, India † IISER, Bhopal,Bhauri, Bhopal 462066,Madhya Pradesh, India
Abstract
Although f ( R ) modification of late time cosmology is successful in explaining presentcosmic acceleration, it is difficult to satisfy the fifth-force constraint simultaneously. Evenwhen the fifth-force constraint is satisfied, the effective scalar degree of freedom may moveto a point (close to its potential minima) in the field space where the Ricci scalar diverges.We elucidate this point further with a specific example of f ( R ) gravity that incorporatesseveral viable f ( R ) gravity models in the literature. In particular, we show that thenonlinear evolution of the scalar field in pressureless contracting dust can easily lead tothe curvature singularity, making this theory unviable. E-mail: koushik . dutta @ saha . ac . in , sukanta @ iiserb . ac . in , avani @ iiserb . ac . in Introduction
Modern cosmological models suffer from a major theoretical difficulty, namely, the dark energyproblem. The problem stems from the high-precision observational data with strong evidencethat the universe is undergoing a phase of accelerated expansion in recent times [1]. Manymodels have been constructed for an explanation of the late time acceleration of the Universe,and those models can be mainly divided into two categories: dark energy models that changethe matter content of the Universe and modified gravity models that alter the Einstein gravity.Dark energy is generally modelled by the vacuum energy with an equation of state parameter w = − w ≃ − f ( R ) theories, where f ( R ) is an arbitrary (usuallyanalytic) function of R with R being the Ricci scalar [4, 5, 6, 7]. The equation of motion ofthe extra degree of freedom other than the graviton is of second order [8], and the models arefree from classical and quantum instabilities [9, 10]. One stringent constraint on these modelsis imposed from the avoidance of fifth force carried by the extra scalar degree of freedom[11, 12, 13], and a few successful models have been constructed [14, 15, 16]. These modelsproduce observationally consistent accelerating expansion preceded by the matter domination[17].Lately, the curvature singularity occurring at cosmological time scales have been found to bea serious problem in f ( R ) models [18]. It is well known that f ( R ) theories can be considered asequivalent scalar-tensor theories. It is easy to visualise curvature singularity in a f ( R ) theory bylooking at the form of the potential appearing in its equivalent scalar-tensor theory in the Jordanframe [19]. Due to the nonlinear motion of the scalar field, the oscillations around the potentialminimum can make the field displaced to the singular point. The presence of the matter makesthe occurrence of singularity more probable [18]. The finite-time singularity in modified gravityis described in [20, 21], whereas the singular behaviour of curvature in a contracting universe hasbeen analysed in [22]. It is shown that past singularities may be prevented for a certain rangeof parameters. These singularities may also occur in future and can be avoided for fine-tunedinitial conditions [19, 23]. It is also realised that the curvature singularities can be eliminatedby adding an extra curvature term to the Lagrangian [22, 24]. The curvature singularity canalso be seen in an astrophysical object. In this case, the singularity is analysed for suitable f ( R ) models applied to dense objects undergoing contraction in the presence of linearly time-dependent mass density [25, 26]. It is seen that the singularity is reached in a time that is muchshorter than cosmological time scale. A detailed study of this issue can be found in [27].1n this work, we study the issue of curvature singularity in a f ( R ) model proposed in [28].We give special emphasis on the role of external matter that makes the effective potentialshallower compared to the pure vacuum case. We find that it is impossible to find a parameterspace where both the fifth-force constraint and the curvature singularity issues are resolved.This paper is organised as follows: In Sec. 2, we review the curvature singularity and fifth forceconstraints for a general function f ( R ). The occurrence of curvature singularities and fifthforce constraint in a specific model of [28] is analysed in Sec 3. We extend the same analysisto a specific limit of the above mentioned model in Sec. 4. Finally, we summarise our resultsin Sec. 5. f ( R ) Gravity In f ( R ) theories of gravity, the Einstein-Hilbert action is modified by replacing the Ricci scalarwith an arbitrary function of R , S = 12 κ Z d x √− gf ( R ) + S M (1)with κ = 8 πG = M − P l , where S M contains the matter degrees of freedom which does not includecontributions from dark energy. For our convenience, we will write the arbitrary function inthe following form f ( R ) = R + F ( R ), so that F ( R ) captures the modifications of Einsteingravity. The effect of F ( R ) in the cosmological dynamics is relevant when local curvaturebecomes smaller than a characteristic infrared modification scale R ∗ . Above this scale, thegravity behaves approximately as Einstein gravity.By varying the action with respect to the metric tensor g µν one can obtain the field equation,and taking trace of that field equation we arrive at3 (cid:3) F ,R ( R ) − F − R + RF ,R ( R ) = κ T , (2)where T is the trace of the stress-energy tensor coming from the matter Lagrangian, and acomma with the subscript corresponds to the partial derivative with that quantity. The scalar-tensor representation of f ( R ) theory can be a found by identifying the term F ,R as a dynamicaldegree of freedom, and in literature, it is dubbed as the ‘scalaron’ field φ = F ,R = f ,R − φ .The above trace Eq. (2) can be conveniently written as (cid:3) φ = dV J dφ + κ T , (3)where dV J dφ = 13 ( R + 2 F − RF ,R ) . (4)2t may not be always possible to invert the relation φ = F ,R and obtain R ( φ ). Therefore it isconvenient to write V J in a parametric form dV J dR = dV J dφ dφdR = 13 ( R + 2 F − RF ,R ) F ,RR . (5)Here, the subscript ′ J ′ denotes the potential in the Jordan frame. The dynamics of the field φ isgoverned by its potential V J originating from the modified gravity, and a force term that is pro-portional to the stress-energy tensor T . Integrating this expression we can find V J ( R ( φ ( x, t ))).Thus, we can effectively describe the dynamics of modified gravity by the dynamics of a scalarfield whose value is uniquely determined by f ( R ). The potential V J ( φ ) will have a global minimum at φ min where cosmological evolution happens.But, as we will see, there is also a point in the field space, denoted by φ sing , where the curvaturescalar R diverges to infinity resulting into curvature singularity. Typically, the points φ min and φ sing would be separated by a finite field value with a finite energy barrier. Therefore, the fieldcan potentially reach to the singular point in the process of having small oscillations aroundits minimum, in particular when we consider objects with growing mass density [18]. Theminimum of the potential φ min can be obtained by equating dV J ( φ ) /dφ = R + 2 F − RF ,R = 0which corresponds to a constant curvature solution (cid:3) F ,R = 0 for the vacuum. Thus, φ min isalso a de Sitter point; see Eq.(2).The appearance of the singularity can be seen both in the cosmological background [18], aswell as in the astrophysical dense object going under spherical collapse [25]. The Eq.(2) canalso simply be written as (cid:3) φ = dV eff J dφ , (6)where dV eff J dφ = dV J dφ + ( κ / T , i.e. V eff J incorporates the effects of matter. Now, we would like tostudy the dynamics of the scalar field in an astrophysical system whose mass density increaseswith time, i.e a collapsing object with time dependent V eff J . With the assumption of weakgravity, the covariant derivative can be replaced by the usual flat space ones [27]. Again, in theapproximation of isotropic matter distribution (spherical collapse), spatial derivatives can beneglected [25]. Thus, if we consider objects whose mass density changes with time T = T ( t ) ina homogenous way, the above equation simplifies to ∂ φ∂t + ∂V eff J ( φ, t ) ∂φ = 0 . (7)This is an oscillator equation with a time dependent nonlinear potential. Even if the initial os-cillation amplitude is small, due to the nonlinear behaviour of the motion, the above mentioned3ingular point can be reached during the evolution of the field. As we will see, in many viable f ( R ) models when a collapsing astrophysical object is considered, φ evolves to the singularpoint in a finite time which is much shorter than the cosmological time scale t U ∼ × sec.This makes the theory not viable. It is clear from the above discussion that the f ( R ) gravity theory contains a scalar degree offreedom in addition to the usual graviton. The effect of the scalar degree of freedom in thematter sector is evident when the theory is rewritten in the Einstein frame, where the gravitypart is Einstein Hilbert type. Taking the conformal transformation on the metric˜ g µν = f ,R g µν , κψ = p / f ,R , (8)the action in (1) transforms to S = Z d x p − ˜ g " ˜ R κ −
12 ( ˜ ∇ ψ ) − V E ( ψ ) + L m (˜ g µν e − √ κψ ) (9)where, V E is the scalar field potential in the Einstein frame V E = Rf ,R − f κ f ,R . (10)All the quantities having tilde are defined in Einstein frame. The mass of the scalar field ψ can be as light as the present Hubble constant, and as a result, there will appear a long rangefifth force mediated by the field. In order to satisfy the local gravity constraints one must havesome screening mechanism which can screen the fifth force on the surface of the Sun/Earth.One such mechanism is known as chameleon mechanism [12]. The scalar field will be heavierin a dense object like earth, and light when the background density is less like the interstellarspace. As long as the scalar field is heavy inside the surface of the earth, it will be frozen atthe minimum of its effective potential V eff ( ψ ) = V E ( ψ ) + e − √ κψ ρ , (11)where ρ is the energy density of the matter field and cannot contribute to the outside field.The only contribution to the outside field can be from a very thin shell around the surface ofthe earth. The thin shell parameter is given by [12]∆˜ r c ˜ r c = − ψ out − ψ in √ test , (12)where ψ in , ψ out are field values corresponding to the minimum of the effective potential insideand outside of a test body (Sun/Earth). Φ test is the gravitational potential on the surface at4he radius ˜ r c of the test body and it is given by Φ test = M test / π ˜ r c where, M test = (4 π/ r c ρ in .Here ρ in is the average density of the test body. ∆˜ r c is the width of the thin shell around thesurface of the test body. In the usual approximation of F ,R ≪ F ≪ R , it can be shownthat ψ in ≪ ψ out for all practical purposes, and the thin-shell condition reduces to [11] | ψ out | . √ test ∆˜ r c ˜ r c (13) . ( . × − (Solar system test) , . × − (Equivalence Principle test) . (14)Therefore, in practice, for a particular model of f ( R ) gravity, we just need to calculate ψ out tosee whether it satisfies the fifth-force constraints [11], [13]. In the next section, we will discussthe curvature singularity problem for a particular form of f ( R ) gravity model in combinationwith the fifth-force constraint. In this section, we will consider a model proposed in [28]: f ( R ) = R + αR ∗ β n [1 + ( R/R ∗ ) n ] − /β − o , (15)where n , β , α and R ∗ are positive parameters of the model. R ∗ is taken to be of the orderof present day average curvature of the universe. The above function satisfies the condition f ( R ) → R → f ( R ) → R + constant, for large R . This is necessary for thecorrect GR limit in early cosmological epoch [14]. Viable matter era demands n > , β > n > , β < − n [17]. In this parametrisation, Starobinsky model corresponds to n = 2 [14],whereas Hu-Sawicki model can be obtained by plugging β = 1 [15]. For the case of n > β <
0, the model becomes indistinguishable from the ΛCDM model [13]. In the original paperof [28], the model was analysed in the limit β → ∞ and for n = 1, and it was shown that themodel is free from curvature singularity issue. We will discuss about the model in this limitin the next section. In this section, we will analyse the model in its full generality given byEq. (15).First, we briefly discuss the fifth-force constraints before moving into the curvature singu-larity issue which is the main focus of the article. In the limit of large curvature R ≫ R ∗ , theform of f ( R ) is reduced to f ( R ) = R + αR ∗ β n ( R/R ∗ ) − n/β − o . (16)As discussed in the previous section, to evade the fifth-force, the canonical scalar field in theEinstein frame outside the test body must satisfy the constraint of Eq. (14). In the large5 R R * n Ψ out < ´ - for Β= Figure 1: parameter space of n and R /R ∗ for different values of β . Pink region shows allowedregion for β = 1, purple region for β = 5 and yellow region for β = 10.curvature limit, the scalar field at its minimum of the potential outside the test body is givenby [23] ψ out = r κ ln(1 + F ,R ) ∼ √ κ F ,R = − √ κ nα (cid:18) κ ρ out R ∗ (cid:19) − nβ − , (17)where we have used the minimisation condition of V eff in evaluating R = κ ρ out at the minimumof the effective potential. If R is the curvature at the de Sitter minimum R = κ ρ crit , wedefine a dimensionless variable x = R /R ∗ to obtain | ψ out | ∼ √ κ nα (cid:18) x ρ out ρ crit (cid:19) − nβ − . (18)Now ρ critical ≃ − g m/cm , and ρ out ≃ − gm/cm (typical baryonic/dark matter density).Because of F ( R ) ≪ f ( R ), α should be of order of unity. In this case, the fifth-force constraintof Eq. (14) can be easily translated to the condition n/β & x ∼ O (1). For general valuesof parameters the allowed regions are shown in Fig. 1, where we have used the condition ofde Sitter minimum in writing α in terms of other parameters. It is clear that the fifth forceconstraint is satisfied when n/β & β → ∞ ,the model immediately violates the fifth-force constraints [23], even though it can evade thecurvature singularity problem [28]. 6 V J / R * φ n=2, β =1n=2, β =5n=2, β =10 -1.2-1-0.8-0.6-0.4-0.2 -1 -0.8 -0.6 -0.4 -0.2 0 V J / R * φ n=1, β =1n=2, β =1n=3, β =1 Figure 2: V J /R ∗ vs. φ . In both figures α = 2 .
0. In the left figure, n = 2 and β = 1 , ,
10 and inthe right figure n = 1 , , β = 1. The minimum of a potential is marked with a dot sign. In this section, we will discuss the issue of curvature singularity in the Jordan frame for themodel described by Eq. (15). Looking at the form of the potential V J , we argue for the presenceof curvature singularity in this model. A preliminary analysis was done in [23], but in theEinstein frame. Moreover, the true nature of the singularity can be understood when theeffects of the matter is taken care properly via V eff J [18]. Typically, the existence of matterpushes the minimum point close to the singular point. We will show it for ‘Log model’ in thenext section - See Fig. 4. In the next subsection, we will analyse the dynamics of the field bysolving the equation of motion to show that the field indeed reaches to the singular point in atime scale much smaller than the age of the Universe.For our case, the scalar degree of freedom φ in the Jordan frame can be identified as φ = − nα ( R/R ∗ ) n − [1 + ( R/R ∗ ) n ] − /β − . (19)Note that R → ∞ when φ → i.e φ sing = 0 in this model. At the same time, we also notethat positive values to φ correspond to negative curvature except for n = 1 case. In Fig. 2,we plot the Jordon frame potential V J . The potential is multivalued which is common tomany f ( R ) gravity models, and in this case the physically relevant potential is the lower one.As described in the previous section, the de Sitter minimum of the potential, about whichscalar field oscillates, can be obtained from the condition dV J /dφ = 0 which is equivalent tothe condition dV E /dψ = 0. We see from the plot that as we increase the value of β , the heightof the potential barrier between the de Sitter point and the singularity point increases, andalso the de Sitter minimum of the potential shifts away from the singularity in the field space. The analytical expression is complicated, and is not very illuminating. Thus, we do not show it explicitlyhere. n , larger values of β , the de Sitter minimum is at safer distance from thecurvature singularity. This is in accordance with the findings of [23] (in Einstein frame) wherein the limit of β → ∞ , the potential barrier becomes infinitely large, thus inaccessible by thefield and effectively hiding the singularity [23, 28]. On the other hand, looking at the rightpanel plot of Fig. 2 we see that making n large pushes the minimum close to the singular point. We consider a self gravitating system (e.g astronomical objects of dark matter cloud) whosemass density is changing with time. This is in contrast to the static configurations in f ( R )theories [29, 30, 31]. We solve Eq. (7) for the contracting homogeneous and isotropic cloud ofpressureless dust whose density ρ m is much greater than the critical density ρ crit . For example, ρ m ∼ − gm/cm for a dust cloud in a galaxy. We parametrise the trace of energy-momentumtensor T as T ( t ) = − T (1 + t/t ch ) , (20)where, T = ρ m is the energy-momentum density at t = 0, and t ch is the characteristic time fordensity variations. It can be estimated by t ch ∼ d/v , where d would be the typical dimensionof the collapsing object and v is its velocity during collapse. Here, we assume that the objectcollpases only under the effects of gravity and therefore velocity v is nothing but the escapevelocity on the surface of the object. Consequently, t ch can be calculated by t ch ∼ p / (8 πGρ m )which comes out to be ∼ . × sec for ρ m ∼ − gm/cm . The time within which thesystem meets curvature singularity is denoted by t sing . For a physical system, t sing must besmaller than the age of the Universe denoted by t U ∼ × sec. We have assumed that thegravitational field of the object is weak and hence the background metric is a flat Minkowskimetric, and because of the homogeneity and isotropy D’Alembertian has been replaced withordinary time derivative [27].With the above mentioned variation of matter density, the governing equation for the scalarfield looks like 3 ∂ t F ,R + 2 F + R − RF ,R − κ T (1 + t/t ch ) = 0 . (21)Changing the variables from R and t to y = η ( R ∗ /R ) and τ = t/t ch we obtain y ′′ + nβ y ′ y = τ ch (cid:20) − y − n/β (cid:18) (1 + τ ) − y − + 2 αβη (cid:19) + nαη − − n/β (cid:18) βn (cid:19)(cid:21) , (22)where, η = κ T /R ∗ , and τ ch is given in terms of model parameters and characteristic time t ch τ ch = (cid:18)(cid:0) κ T /R ∗ (cid:1) nβ βn + β R ∗ nα (cid:19) t ch . (23)In deriving the above equation of motion, we have used the Eq. (16). The singularity is reachedwhen y = κ T /R becomes zero. The nonlinear behaviour of Eq. (22) can amplify the initial8 κ T / R t/t ch n=2, β =10 n=2, β =100n=2, β =10 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 κ T / R t/t ch n=4, β =5n=3, β =5n=2, β =5n=1, β =5 Figure 3: The oscillation amplitudes are plotted against physical time for several parameterchoices of the model. κ T /R vs. t . In both the figures, t ch = 1 . × sec , η = 10 and α = 2. n β n/β t sing (Sec)2 1 2 8 . × . × . × . × . × × − . × Table 1: t sing for different parameter values of the model defined by Eq. (15). We have chosen α = 2. For all these choices of parameters, the time scale of singularity is much smaller thanthe age of the Universe.oscillation amplitude around the potential minimum and the field can reach to the singularpoint in a time scale that is smaller than the age of the Universe. The equation has been solvedfor y ′ (0) = 0 and y (0) = 1 initial values, and it is consistent with R/R ∗ ≫ t ch = 1 . × Sec and different values of n and β is givenin Fig. 3.It can be seen that the amplitude of the oscillations increase with time, and after a finitetime the value of y approaches zero which is a singular point. It is necessary that the timescale within which y goes to the singular point must be shorter than t U . As an example, taking R ∗ ∼ /t U , t sing = 2 . × Sec ≪ t U for n = 4, β = 5 and α = 2. It is clear from the plotsthat higher values of β and lower n values are needed for the considered model to remain freefrom curvature singularity. This is consistent with the primary idea that we had obtained bylooking at the potential V J ( φ ) for the scalar field. The time t sing has been listed in Table. 1 for9ifferent values of parameters n and β . It is evident that the relevant parameter for the timescale of singularity t sing is the ratio between n and β . The singularity time t sing increases withsmaller values of n/β .Thus, it can be concluded that singularity can be avoided in principle by taking very largevalues of β [28] and/or small values of n . On the other hand, as discussed earlier, local gravitytests constrain the values of the parameters n/β &
2. Therefore, it is nearly impossible prac-tically to find a region in the parameter space where the model of Eq. (15) can be free fromcurvature singularity, as well as it can evade the local gravity tests.
In the last section, we have seen that the curvature singularity can be avoided in principle bytaking large β and small n limit of the general model given by (15). In [28], it has been claimedthat if we take n = 1 and β → ∞ limit of Eq. (15), the generalised model is reduced to thefollowing logarithmic function: f ( R ) = R − αR ∗ ln (1 + R/R ∗ ) . (24)Subsequently, this particular form of the model has been analysed and it is shown that thecurvature singularity is absent [28], even though it violates the fifth force constraint of n/β & R ≪ R ∗ and it breaks downimmediately for the analysis of curvature singularity. Therefore, we consider Eq. (24) as anindependent model and revisit the issue of curvature singularity in the presence of matter.The scalar field φ in the Jordan frame for the model can be expressed as φ = − α R/R ∗ , (25)and we see that φ → R corresponding to the curvature singularity. Toinvestigate the issue, we first find out the potential V J ( φ ) /R ∗ = − φ − αφ + 16 φ − αφ ln ( − α/φ ) + 13 α ln ( − /φ ) , (26)and note that V J ( φ ) → ∞ as φ reaches the singularity point. Naively, the presence of the infinitepotential barrier was the argument in claiming the absence of curvature singularity [23], [28].But including the matter source in term of a collapsing cloud of Eq. (20), the effective potentialcan be written as V eff J ( φ, t ) R ∗ = − φ − αφ + 16 φ − αφ ln (cid:18) − αφ (cid:19) + 13 α ln (cid:18) − φ (cid:19) − κ T R ∗ φ (cid:18) tt ch (cid:19) . (27)For different values of κ T /R ∗ at t = t ch , the plot of V eff J vs. φ has been shown in Fig. [4].The de Sitter points are marked by stars in the plot. From the figure, it can be seen that10 V J e ff / R * φ κ T /R * =0 κ T /R * =10 κ T /R*=50 κ T / R t/t ch κ T /R * =10 κ T /R * =10 κ T /R * =10 Figure 4: Left: V eff J vs. φ where dots represent the potential minimum. In the figure, κ T /R ∗ =0 , ,
50 and α = 1 .
5. Right: κ T /R vs. t/t ch for η = 10 , , with α = 1 . φ = 0) as the density of matterincreases [18]. One can conclude that the possibility that the scalar field φ may hit the curvaturesingularity is more likely in the presence of matter than in the vacuum. But, it requires theunderstanding of dynamics of the field which we will analyse now.We investigate the presence of curvature singularity by solving Eq. (7) numerically in con-tracting astrophysical objects as we have done in the previous section. The equation of motionin this case reads y ′′ − τ ch (cid:20) ln ( y ) − ln ( η ) + ηα y + 1 − ηα (1 + τ ) (cid:21) = 0 , (28)where the approximations at large curvature limit F ( R ) ≃ − αR ∗ ln ( R/R ∗ ) and F ,R ≃ − ( αR ∗ ) /R have been used. Here, τ ch is given by τ ch = r ηR ∗ t ch . (29)The solution of the above equation y ( τ ) is plotted in Fig. 4 for ρ m = 10 − , − and 10 − gm/cm i.e. for η = 10 , and 10 . The characteristic time t ch for these densities are 1 . × , . × and 1 . × sec correspondingly. The initial conditions have been chosen as y (0) = 1and y ′ (0) = 0. It is seen that the oscillations of y grows with time and eventually meets thesingularity. It is evident from the figure that the singularity is reached earlier for larger η .It was noted earlier that the model is singularity free in the absence of matter because of itslarge potential barrier at the singular point. But we show that in the presence of astrophysicaldensities in a collapsing object the system indeed can reach to the singular point.11 Discussion and Conclusion
In this work, we have investigated the existence of curvature singularity problem in f ( R ) modelgiven by Eq. (15) and by Eq. (24) [28]. For our analysis we have chosen this particular model asthis form of f ( R ) incorporates several f ( R ) models well known in the literature. We have madethe dynamical analysis of the fluctuations of the associated scalar field around the minimum ofthe potential to show that the field indeed reaches to the singular point when the effects of thematter are considered.We have carried out our analysis of curvature singularity in the Jordan frame where thetheory has been rewritten in terms of a scalar field φ . The singularity is reached when φ = φ sing ,and in our case, φ sing = 0. The scalar field oscillates around its de Sitter minimum of thepotential V J . The potential barrier at the singularity point, and the distance between the deSitter point and the singular point are finite. Therefore the scalar field can reach to φ sing during its cosmological evolution or during the collapse of dense astrophysical objects. In thiswork, we have shown this effect by solving the nonlinear evolution equation of the field. Withincreasing value of n/β in the considered model, the system reaches to the singularity earlier.In particular, we have solved the stress equation for collapsing astrophysical object and shownthat the Ricci scalar R oscillates and after a time t ∼ t sing , it reaches to a divergent value.The time within which singularity is reached, i.e. t sing , is less than the age of the Universe forgeneric values of n and β . On the other hand, local gravity tests put the constraint on themodel parameters such that n/β &
2. Taking large enough values of β can in principle help usavoiding the singularity in cosmological time scale, but it immediately violates the fifth forceconstraints. Therefore, we conclude that the model can not satisfy constraints coming from thecurvature singularity and fifth force simultaneously.On the other hand, the ‘Log Model’ as discussed in Sec. 4 seems to be free from curvaturesingularity [23],[28]. But, this conclusion was drawn in the absence of matter. When weincorporate the effects of matter in the calculations, we see that the minimum of the effectivepotential moves closer to the singular point and the height of the barrier at φ sing becomessmaller. In fact, the solutions of the scalar field dynamics show that the singularity is reachedin a time scale smaller than the age of the Universe.Looking at the present analysis in conjunction with previous understanding, it is temptingto think that it is very difficult, if not impossible, to construct any function f ( R ) that simul-taneously satisfies both fifth force constraint and resolves the singularity problem. As we haveseen, and also has been noted earlier [23], the height of the potential barrier goes opposite tosatisfying the fifth force constraint. For example, in the considered model, the height of the bar-rier (thus avoidance of singularity) is proportional to β , but having large β immediately violatesthe fifth force constraints. Therefore, it would be very interesting in finding generic structureof the function f ( R ) that can simultaneously satisfy both the above mentioned constraints. Wehope to look into this issue in future. 12 cknowledgements KD is partially supported by a Ramanujan Fellowship and Max Planck Society-DST VisitingFellowship. SP and AP would like to thank Theory Division of SINP where part of the workhas been carried out as visitors.
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