aa r X i v : . [ h e p - t h ] J a n Dark energy and possible alternatives
M. Sami
Center of Theoretical Physics, Jamia Millia Islamia, Jamia Nagar, Delhi-110092, India (Dated: October 25, 2018)We present a brief review of various approaches to late time acceleration of universe. The cosmo-logical relevance of scaling solutions is emphasized in case of scalar field models of dark energy. Theunderlying features of a variety of scalar field models is highlighted. Various alternatives to darkenergy are discussed including the string curvature corrections to Einstein-Hilbert action, higherdimensional effects, non-locally corrected gravity and f ( R ) theories of gravity. The recent develop-ments related to f ( R ) models with disappearing cosmological constant are reviewed. I. INTRODUCTION
The accelerated expansion has played a very important role in the history of our universe. Universe is believed tohave passed through inflationary phase at early epochs and there is a growing faith that it is accelerating at present.The late time acceleration of the universe, which is directly supported by supernovae observations, and indirectly,through observations of the microwave background, large scale structure and its dynamics, weak lensing and baryonoscillations, poses one of the most important challenges to modern cosmology.Einstein equations in their original form, with an energy-momentum tensor for standard matter on the right handside, cannot account for the observed accelerated expansion of universe. The standard lore aimed at capturing thisimportant effect is related to the introduction of the energy-momentum tensor of an exotic matter with large negativepressure dubbed dark energy in the Einstein equations. The simplest known example of dark energy (for recentreviews, see [1]) is provided by the cosmological constant Λ. It does not require adhoc assumption for its introduction,as is automatically present in the Einstein equations, by virtue of the Bianchi identities.The field theoretic understanding of Λ is far from being satisfactory. Efforts have recently been made to obtainΛ in the framework of string theory, what leads to a complicated landscape of de-Sitter vacua. It is hard to believethat we happen to live in one of the 10 vacua predicted by the theory. One might take the simplified view that,like G , the cosmological constant Λ is a fundamental constant of the classical general theory of relativity and that itshould be determined from large scale observations. It is interesting to remark that the Λ CDM model is consistentwith observations at present. Unfortunately, the non-evolving nature of Λ and its small numerical value lead to anon-acceptable fine-tuning problem. We do not know how the present scale of the cosmological constant is related toPlanck’s or the supersymmetry breaking scale; perhaps, some deep physics is at play here that escapes our presentunderstanding.The fine-tuning problem, associated with Λ, can be alleviated in scalar field models which do not disturb thethermal history of the universe and can successfully mimic Λ at late times. A variety of scalar fields have beeninvestigated to this end[1]; some of them are motivated by field/string theory and the others are introduced owing tophenomenological considerations. It is quite disappointing that a scalar field description lacks predictive power; given a priori a cosmic evolution, one can always construct a field potential that would give rise to it. These models should,however, not be written off, and should be judged by the generic features which might arise from them. For instance,the tracker models have remarkable features allowing them to alleviate the fine-tuning and coincidence problems.Present data are insufficient in order to conclude whether or not the dark energy has dynamics; thus, the quest forthe metamorphosis of dark energy continues[2]One can question the standard lore on fundamental grounds. We know that gravity is modified at small distancescales; it is quite possible that it is modified at large scales too where it has never been confronted with observationsdirectly. It is therefore perfectly legitimate to investigate the possibility of late time acceleration due to modification ofEinstein-Hilbert action. It is tempting to study the string curvature corrections to Einstein gravity amongst which theGauss-Bonnet correction enjoys special status. A large number of papers are devoted to the cosmological implicationsof string curvature corrected gravity[3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. These models, however, suffer from severalproblems. Most of these models do not include tracker like solution and those which do are heavily constrained by thethermal history of universe. For instance, the Gauss-Bonnet gravity with dynamical dilaton might cause transitionfrom matter scaling regime to late time acceleration allowing to alleviate the fine tuning and coincidence problems.But it is difficult to reconcile this model with nucleosynthesis[5, 6]constraint. The large scale modification may alsoarise in extra dimensional theories like DGP model which contains self accelerating brane. Apart from the theoreticalproblems, this model is heavily constrained by observation.On purely phenomenological grounds, one could seek a modification of Einstein gravity by replacing the Ricci scalarby generic function f ( R )[13, 14, 15]. The f ( R ) gravity theories giving rise to cosmological constant in low curvatureregime are faced with difficulties which can be circumvented in f ( R ) gravity models proposed by Hu-Sawicki andStarobinsky [16, 17] (see Ref.[18] on the similar theme). These models can evade solar physics constraints by invokingthe chameleon mechanism [16, 17, 19]. An important observation has recently been made in Refs.[20, 21](see alsoRef.[22] on the related theme), namely, the minimum of scalaron potential which corresponds to dark energy can bevery near to φ = 0 or equivalently R = ∞ . As pointed out in Ref.[19], the minimum should be near the origin forsolar constraints to be evaded. Hence, it is most likely that we hit the singularity if the parameters are not properlyfine tuned. This may have serious implications for relativistic stars[23].In what follows we shall briefly review the aforesaid developments. II. LATE TIME ACCELERATION AND COSMOLOGICAL CONSTANT
Einstein equations exhibit simple analytical solutions in a homogeneous and isotropic universe. The dynamics inthis case is described by a single function of time a ( t ) dubbed scale factor , H ≡ ˙ a a = 8 πGρ − Ka ˙ ρ + 3 H ( ρ + p ) = 0 , where ρ is designates the total energy density in the universe. Three different possibilities, K = 0, K >
K < Ka = H (Ω( t ) − ρ/ρ c , ρ c = 3 H / πG Observations on CMB reveal that we live in a nearly critical universe, K = 0 or ρ = ρ c which is consistent withinflationary paradigm. The equation for acceleration has the following form,¨ aa = − πG ρ + 3 p )¨ a > ⇐⇒ p < − ρ DarkEnergy.
Thus an exotic fluid with large negative pressure is needed to fuel the accelerated expansion of universe. Let us notethat pressure corrects the energy density and positive pressure adds to deceleration where as the negative pressurecontributes towards acceleration. It might look completely opposite to our intuition that highly compressed substanceexplodes out with tremendous impact whereas in our case pressure acts in the opposite direction. It is importantto understand that our day today intuition with pressure is related to pressure force or pressure gradient. In ahomogeneous universe pressure gradients can not exist. Pressure is a relativistic effect and can only be understoodwithin the frame work of general theory of relativity. Pressure gradient might appear in Newtonian frame work inan inhomogeneous universe but pressure in FRW background can only be induced by relativistic effects. Strictlyspeaking, it should not appear in Newtonian gravity; its contribution is negligible in the non-relativistic limit. Indeed,in Newtonian cosmology, acceleration of a particle on the surface of a homogeneous sphere with density ρ and radius R is given by, ¨ R = − π GρR (1)The simplest possibility of dark energy is provided by cosmological constant which does not require an adhoc assump-tion for its introduction; it is automatically present in Einstein equations by virtue of Bian’chi identities, G µν ≡ R µν − g µν R = 8 πGT µν − Λ g µν (2)The evolution equations in this case become, H = 8 πG ρ + Λ3 (3)¨ aa = − πG ρ + 3 p ) + Λ3 (4)In case the universe is dominated by Λ, it follows from the continuity equation that p Λ = − ρ Λ and Eq.(4) tells usthat a positive cosmological constant contributes to acceleration.Observations of complimentary nature reveal that, • Ω tot ≃ • Ω m ≃ . • Ω DE ≃ . phantom dark energy with w < − a ( t ) = ( t s − t ) n , ( n = 2 / w )) (5)where t s is an integration constant. It is easy to see that phantom dominated universe will end itself in a singularityknown as big rip or cosmic doomsday , H = nt s − t (6) R = 6 " ¨ aa + (cid:18) ˙ aa (cid:19) = 6 n ( n −
1) + n ( t s − t ) (7)The big rip singularity is characterized by the divergence of H and consequently of the curvature after a finite time infuture. It should be noted that when curvature becomes large, we should incorporate the higher curvature correctionsto Einstein-Hilbert action which can modify the structure of the singularity[24]. Big rip can also be avoided in specificmodels of phantom energy[25]. A. Issues associated with Λ There are important theoretical issues related to cosmological constant. Cosmological constant can be associatedwith vacuum fluctuations in the quantum field theoretic context. Though the arguments are still at the level ofnumerology but may have far reaching consequences. Unlike the classical theory the cosmological constant in thisscheme is no longer a free parameter of the theory. Broadly the line of thinking takes the following route.The quantum effects in GR become important when the Einstein Hilbert action becomes of the order of Planck’sconstant; this happens at the Planck’s length Lp = 10 − cm corresponding to Planck energy which is of the orderof M p ≃ GeV . The ground state energy dubbed zero point energy or vacuum energy of a free quantum fieldis infinite. This contribution is related the ordering ambiguity of fields in the classical Lagrangian and disappearswhen normal ordering is adopted. Since this procedure of throwing out the vacuum energy is adhoc , one mighttry to cancel it by introducing the counter terms. The later, however requires fine tuning and may be regarded asunsatisfactory. The divergence is related to the modes of very small wave length. As we are ignorant of physics aroundthe Planck scale, we might be tempted to introduce a cut off at L p and associate with this a fundamental scale. Thuswe arrive at an estimate of vacuum energy ρ V ∼ M p (corresponding mass scale- M V ∼ ρ / V ) which is away by 120orders of magnitudes from the observed value of this quantity. The vacuum energy may not be felt in the laboratorybut plays important role in GR through its contribution to the energy momentum tensor as < T µν > = − ρ V g µν ( ρ V = Λ / πG, M P = 1 / πG )and appears on the right hand side of Einstein equations.The problem of zero point energy is naturally resolved by invoking supersymmetry which has many other remarkablefeatures. In the supersymmetric description, every bosonic degree of freedom has its Fermi counter part whichcontributes zero point energy with opposite sign compared to the bosonic degree of freedom thereby doing away withthe vacuum energy. It is in this sense the supersymmetric theories do not admit a non-zero cosmological constant.However, we know that we do not leave in supersymmetric vacuum state and hence it should be broken. For a viablesupersymmetric scenario, for instance if it is to be relevant to hierarchy problem, the suppersymmetry breaking scaleshould be around M susy ≃ GeV. We are still away from the observed value by many orders of magnitudes. Atpresent we do not know how Planck scale or SUSY breaking scales is related to the observed vacuum scale.At present there is no satisfactory solution to cosmological constant problem. One might assume that there issome way to cancel the vacuum energy. One can then treat Λ as a free parameter of classical gravity similar toNewton constant G . However, the small value of cosmological constant leads to several puzzles including the finetuning and coincidence problems. The energy density in radiation at the Planck scale is of the order of 10 GeV or ρ Λ /ρ r ∼ − Thus the vacuum energy density needs to be fine tuned at the level of one part in 10 − aroundthe Plank epoch, in order to match the current universe. Such an extreme fine tuning is absolutely unacceptable log( ρ) log(a) ρ ρ ϕΒ ϕ ρ FIG. 1: Desired evolution of field energy density ρ φ ( ρ B is the background energy density). The field energy density in caseof undershoot and overshoot joins the scaling solution for different initial conditions. At late times, the scalar field exits thescaling regime to become the dominant component. log(a) ρ ρ B ρ ϕ ρ ϕ log( ) FIG. 2: Evolution of ρ φ and ρ B in absence of the scaling solution. The scalar field after its energy density overshoots thebackground gets into locking regime and waits till its energy density becomes comparable to ρ B . It then begin to evolve andover takes the background. similar picture holds in case of the overshoot. at theoretical grounds. Secondly, the energy density in cosmological constant is of the same order of matter energydensity at the present epoch. The question what causes this coincidence has no satisfactory answer.Efforts have recently been made to understand Λ within the frame work of string theory using flux compactification.String theory predicts a very complicated landscape of about 10 de-Sitter vacua. Using Anthropic principal, weare led to believe that we live in one of these vacua! It is easier to believe in God than in these vacua! III. SCALAR FIELD DYNAMICS RELEVANT TO COSMOLOGY
The fine tuning problem associate with cosmological constant led to the investigation of cosmological dynamics of avariety of scalar field systems such as quientessence, phantoms, tachyons and K-essence. In past years, the underlyingdynamics of these systems has been studied in great detail. It is worthwhile to bring out the broad features that makesa particular scalar field system viable to cosmology. The scalar field model aiming to describe dark energy shouldpossess important properties allowing it to alleviate the fine tuning and coincidence problems without interfering withthe thermal history of universe. The nucleosynthesis puts an stringent constraint on any relativistic degree of freedomover and above that of the standard model of particle physics. Thus a scalar field has to satisfy several importantconstraints if it is to be relevant to cosmology. Let us now spell out some of these features in detail. In case thescalar field energy density ρ φ dominates the background (radiation/matter) energy ρ B , the former should redshiftfaster than the later allowing radiation domination to commence which in tern requires a steep potential. In this case,the field energy density overshoots the background and becomes subdominant to it. This leads to the locking regimefor the scalar field which unlocks the moment the ρ φ is comparable to ρ B . The further course of evolution cruciallydepends upon the form the potential assumes at late times. For the non-interference with thermal history, we requirethat the scalar field remains unimportant during radiation and matter dominated eras and emerges out from thehiding at late times to account for late time acceleration. To address the issues related to fine tuning, it is importantto investigate the cosmological scenarios in which the energy density of the scalar field mimics the background energydensity. The cosmological solution which satisfy this condition are known as scaling solutions , ρ φ ρ B = const. (8)The steep exponential potential V ( φ ) ∼ exp ( λφ/M P ) with λ > w B ) in the frame work of standard GR gives riseto scaling solutions. Nucleosynthesis further constraints λ . The introduction of a new relativistic degree of freedom ata given temperature changes the Hubble rate which crucially effects the neutron to proton for temperature of the orderof one MeV when weak interactions freeze out. This results into a bound on λ , namely, Ω φ ≡ w B ) /λ < . − . λ > ∼ .
5. In this case, for generic initial conditions, the field ultimately enters into the scaling regime, the attractorof the dynamics and this allows to alleviate the fine tuning problem to a considerable extent. The same holds for thecase of undershoot, see Fig.1.Scaling solutions, however, are not accelerating as they mimic the background (radiation/matter). One thereforeneeds some late time feature in the potential. There are several ways of achieving this: (1) The potential that mimicsa steep exponential at early epochs and reduces to power law type V ∼ φ p at late times gives rise to acceleratedexpansion for p < / < w φ > = ( p − / ( p + 1) < − / tracker solutions. For a viable cosmic evolutionwe need a tracker like solution.Recently, a variety of scalar field models such as tachyon and phantom were investigated as candidates of darkenergy. In case of tachyon with equation of state parameter ranging from − IV. MODIFIED THEORIES OF GRAVITY AND LATE TIME ACCELERATION
In view of the above discussion, it is perfectly legitimate to investigate the possibility of late time acceleration dueto modification of Einstein-Hilbert action. Some of these modifications are inspired by fundamental theories of highenergy physics where as the others are based upon phenomenological considerations.
A. String curvature corrections
It is interesting to investigate the string curvature corrections to Einstein gravity amongst which the Gauss-Bonnetcorrection enjoys special status. These models, however, suffer from several problems. Most of these models do notinclude tracker like solution and those which do are heavily constrained by the thermal history of universe. Forinstance, the Gauss-Bonnet gravity with dynamical dilaton might cause transition from matter scaling regime to latetime acceleration allowing to alleviate the fine tuning and coincidence problems. Let us consider the low energyeffective action, S = Z d x √− g h κ R − (1 / g µν ∂ µ φ ∂ ν φ −− V ( φ ) − f ( φ ) R GB i + S m (9)where R GB is the Gauss-Bonnet term, R GB ≡ R − R µν R µν + R αβµν R αβµν (10)The dilaton potential V ( φ ) and its coupling to curvature f ( φ ) are given by, V ( φ ) ∼ e ( αφ ) , f ( φ ) ∼ e − ( µφ ) (11)The cosmological dynamics of system (9) in FRW background was investigated in Ref.[5, 6]. It was shown thatscaling solution ca be obtained in this case provided that µ = λ . In case µ = λ , we have the de-Sitter solution. Hence,the string curvature corrections under consideration can give rise to late time transition from matter scaling regime.Unfortunately, it is difficult to reconcile this model with nucleosynthesis[5, 6]constraint. B. DGP model
In DGP model, gravity behaves as four dimensional at small distances but manifests its higher dimensional effects atlarge distances. The modified Friedmann equations on the brane lead to late time acceleration. The model has serioustheoretical problems related to ghost modes superluminal fluctuations. The combined observations on backgrounddynamics and large angle anisotropies reveal that the model performs worse than Λ
CDM [29].
C. Non-local cosmology
An interesting proposal on non-locally corrected gravity involving a function of the inverse d’Almbertian of theRicci scalar, f ( (cid:3) − R )), was made in Refs.[30] For a generic function f ( (cid:3) − R ) ∼ exp( α (cid:3) − R ), the model can lead tode-Sitter solution at late times. The range of stability of the solution is given by 1 / < α < / w eff ranging as ∞ < w eff < − /
3. For 1 / < α < / / < α < /
3, the underlyingsystem is shown to exhibit phantom and non-phantom behavior respectively; the de Sitter solution corresponds to α = 1 /
2. For a wide range of initial conditions, the system mimics dust like behavior before reaching the stablefixed point at late times. The late time phantom phase is achieved without involving negative kinetic energy fields.Unfortunately, the solution becomes unstable in presence of the background radiation/matter[30].
D. f(R) theories of gravity
On purely phenomenological grounds, one could seek a modification of Einstein gravity by replacing the Ricci scalarby f ( R ). The f ( R ) gravity theories giving rise to cosmological constant in low curvature regime are plagued withinstabilities and on observational grounds they are not distinguished from cosmological constant. The recently intro-duced models of f ( R ) gravity by Hu-Sawicki and Starobinsky (referred as HSS models hereafter) with disappearingcosmological constant[16, 17] have given rise to new hopes for a viable cosmological model within the framework ofmodified gravity. The action of f ( R ) gravity is given by[13], S = Z (cid:20) f ( R )16 πG + L m (cid:21) √− g d x, (12)which leads to the following modified equations, f ′ R µν − ∇ µν f ′ + (cid:18) (cid:3) f ′ − f (cid:19) g µν = 8 πGT µν . (13)which are of fourth order for a non-linear function f(R). Here prime ( ′ ) denotes the derivatives with respect to R .The modified Eq.(13) contains de-Sitter space time as a vacuum solution provided that f (4Λ) = 2Λ f ′ (4Λ). Thus, the f ( R ) theories of gravity may provide an alternative to dark energy. The f ( R ) gravity theories apart from a spin twoobject necessarily contain a scalar degree of freedom. Taking trace of Eq.(13) gives the evolution equation for thescalar degree of freedom, (cid:3) f ′ = 13 (2 f ′ − f ′ R ) + 8 πG T. (14)It would be convenient to define scalar function φ as, φ ≡ f ′ − , (15)which is expressed through Ricci scalar once f ( R ) is specified. We can write the trace equation (Eq.(14)) in the termsof V and T as (cid:3) φ = dVdφ + 8 πG T. (16)The potential can be evaluated using the following relation dVdR = dVdφ dφdR = 13 (2 f − f ′ R ) f ′′ . (17)The functional form of f ( R ) should satisfy certain requirements for the consistency of the modified theory of gravity.The stability of f ( R ) theory would be ensured provided that, • f ′ ( R ) > − graviton is not ghost, • f ′′ ( R ) > − scalaron is not tachyon.The f ( R ) models which satisfy the stability requirements can broadly be classified into categories: (i) Models in which f ( R ) diverges for R → R where R finite or f ( R ) is non analytical function of the Ricci scalar. These models eithercan not be distinguishable from Λ CDM or are not viable cosmologically. (ii) Models with f ( R ) → R → CDM in high redshift regimeand give rise to cosmological constant in regions of high density and differ from the latter otherwise; in principal thesemodels can be distinguished from cosmological constant.Models belonging to the second category were proposed by Hu-Sawicki and Starobinsky [16, 17]. The functionalform of f ( R ) in Starobinsky parametrization is given by, f ( R ) = R + λR "(cid:18) R R (cid:19) − n − . (18)Here n and λ are positive. And R is of the order of presently observed cosmological constant, Λ = 8 πGρ vac . Theproperties of this model [17] can be summarized as follows:1. Stability conditions are satisfied as f ′ , f ′′ >
02. Flat space time is an unstable solution of the model.3. For | R | ≫ R , f ( R ) = R − ∞ ). The high-curvature value of the effective cosmological constant is Λ( ∞ ) = λR / φ is given by φ ( R ) = − nλRR (1 + R R ) n +1 . (19)Notice that R → ∞ for φ →
0. One can easily compute V ( R ) for a given value of n . For instance, in case of n = 1,we have VR = 124 (1 + y ) (cid:8)(cid:0) − − y − y − y (cid:1) λ + (cid:0) y + 11 y + 21 y − y (cid:1) λ (cid:9) − λ − y, (20)where y = R/R . In the FRW background, the trace equation (Eq. (14)) can be rewritten in the convenient form¨ φ + 3 H ˙ φ + dVdφ = 8 πG ρ. (21) - - Φ- - (cid:144) R FIG. 3: Plot of effective potential for n = 2 and λ = 1 .
2. The red spot marks the initial condition for evolution.
The time-time component of the equation of motion (13) gives the Hubble equation H + d (ln f ′ ) dt H + 16 f − f ′ Rf ′ = 8 πG f ′ ρ. (22)It should be noted that the stability conditions ensure that the effective gravitational constant G eff = G/f ′ appearingin Eq.(22) is positive. The simple picture of dynamics which appears here is the following: above infrared modificationscale ( R ), the expansion rate is set by the matter density and once the local curvature falls below R the expansionrate gets effect of gravity modification. For pressure less dust, the effective potential has an extremum at,2 f − Rf ′ = 8 πGρ. (23)For a viable late time cosmology, the field should be evolving near the minimum of the effective potential. The finitetime singularity inherent in the class of models under consideration severely constraints dynamics of the field. The curvature singularity and fine tuning of parameters
The effective potential has minimum which depends upon n and λ . For generic values of the parameters, theminimum of the potential is close to φ = 0 corresponding to infinitely large curvature. Thus while the field is evolvingtowards minimum, it can easily oscillate to a singular point. However, depending upon the values of parameters, wecan choose a finite range of initial conditions for which scalar field φ can evolve to the minimum of the potentialwithout hitting the singularity. Λ- - - - - Φ min FIG. 4: Plot of φ min versus λ for different values of n . With increase in n , φ min approaches zero for smaller values of λ . Thecurves from bottom to top correspond to n = 1 , , , We find that the range of initial conditions allowed for the evolution of φ to the minimum without hitting singularityshrinks as the numerical values of parameters n and λ increase. This is related to the fact that for larger values of n and λ , the minimum fast moves towards φ = 0, see figure 4. The numerical values should be accurately chosen toavoid hitting the singularity. Avoiding singularity with higher curvature corrections
We know that in case of large curvature, the quantum effects become important leading to higher curvature cor-rections. Keeping this in mind, let us consider the modification of Starobinsky’s model, f ( R ) = R + αR R + R λ " − R R ) n , (24)then φ becomes φ ( R ) = RR " α − nλ (1 + R R ) n +1 . (25)When | R | is large, the first term which comes from αR dominates. In this case, the curvature singularity R = ±∞ corresponding to φ = ±∞ , see Fig.5. Hence, in this modification, the minimum of the effective potential is separatedfrom the curvature singularity by the infinite distance in the φ, V ( φ ) plane. In case of n = 2, the expressions for φ and V ( φ ) are given by, φ ( y ) = 2 αy − λy (1 + y ) , (26) VR = − y ) (cid:8) λ y (cid:0) − − y − y + 106 y + 595 y + 105 y (cid:1)(cid:9) − y ) (cid:8) y + α (cid:0) y + 9 y + 4 y (cid:1)(cid:9) + 13 αy + 132 (32 α − λ ) tan − ( y ) . (27)For n = 2 , λ = 2 and α = 0 .
5, we have a large range of the initial condition for which the scalar field evolves tothe minimum of the potential. Though the introduction of R term formally allows to avoid the singularity butcan not alleviate the fine tuning problem as the minimum should be brought near to the origin to respect the solarconstraints. Last but not the least one could go beyond the approximation (see Eq.(23)) by iterating the trace equationand computing the corrections to R given by equation (23). As pointed by Starobinsky[17], such a correction mightbecome large in the past. This may spoil the thermal history and thus needs to be fine tuned. The aforesaid discussionmakes it clear that HSS models are indeed fine tuned and hence very delicate.In case of any large scale modification of gravity, one should worry about the local gravity constraints. The f ( R )theories belong to the class of scalar tensor theories corresponding the Brans-Dicke parameter ω = 0 or the PPNparameter γ = (1 + ω ) / (2 + ω ) = 1 / γ = 1 consistent with observation ( | γ − | < ∼ − ). Thisproblem can be circumvented by invoking the so called chameleon mechanism. In case, the scalar degree of freedomis coupled to matter, the effective mass of the field depends upon the matter density which can allow to avoid theconflict with solar physics constraints. V. SUMMARY
We have briefly summarized here various approaches to understand the late time acceleration of universe. In caseof scalar field models of dark energy, we emphasized the relevance of scaling solutions in alleviating the fine tuningand the coincidence problems. We hope that the future data would reveal the metamorphosis of dark energy.Amongst the various alternatives to dark energy, the f ( R ) gravity models have received considerable attention inpast years. There are broadly two classes of f ( R ) models, namely, those in which f ( R ) diverges as R → R ( R isfinite) or f ( R ) is non-analytic in R . And those with f ( R ) → R →
0, they reduce to Λ
CDM in the limit ofhigh redshift and give rise to cosmological constant in high density regime. These models can evade local gravityconstraints with the help of the so called chameleon mechanism and have potential capability of being distinguishedfrom Λ
CDM [31].Unfortunately, the f ( R ) models with chameleon mechanism are plagued with curvature singularity problem whichmay have important implications for relativistic stars[23]. The model could be remedied with the inclusion of higher0 - - Φ- (cid:144) R FIG. 5: Plot of the effective potential for n = 2 , λ = 2 and α = 1 / R correction. the minimum of the effectivepotential in this case is located at φ min = 3 .
952 ( R min = 3 . curvature corrections[32]. At the onset, it seems that one needs to invoke fine tunings to address the problem[33].The presence of curvature singularity certainly throws a new challenge to f ( R ) gravity models. In our opinion, theproblem requires further investigations. It would also be interesting to look for a realistic scenario of quintessentialinflation in the frame work of f ( R ) gravity. VI. ACKNOWLEDGEMENTS
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