Viability of Arctan Model of f(R) Gravity for Late-time Cosmology Koushik Dutta∗, Theory Division, Saha Institute of Nuclear Physics, 1/AF Salt Lake, Kolkata - 700064, India. Sukanta Panda†, Avani Patel‡ Indian Institute of Science Education and Research, Bhauri, Bhopal 462066, Madhya Pradesh, India f(R) modifications of Einstein’s gravity is an interesting possibility to explain the late time acceleration of the Universe. In this work we explore the cosmological viability of one such f(R) modification proposed in [1]. We show that the model violates fifth-force constraints. The model is also plagued with the issue of curvature singularity in a spherically collapsing object, where the 6 effectivescalar field reaches to thepoint of diverging scalar curvature. 1 0 2 I. INTRODUCTION of a cosmological constant as the best fit dark energy n model. Following this line, many f(R) models are pro- a posed those behave as Λ-CDM model when the space- J Last two decades have seen substantial improvement time is sufficiently curved i.e. f R 2Λ for R Λ 7 in understanding the large-scale structures of the uni- → − ≫ and f(0) = 0 [1, 9–19]. The dynamical system analysis 2 verse from high precision measurements of cosmic mi- of f(R)theories is carriedout in categorisingmodels ac- ] crowave background radiation and distance measure- cording to their fixed points [7]. Fifth force constraints c ments of Type Ia Supernovae [2]. The later observations on these f(R) models are evaded using the chameleon q have led us to conclude that the universe we live in is mechanism [20, 21]. - expanding acceleratingly in recent times. In order to r g accommodate such evolution of the universe, the stan- One of the serious problems in these models is the [ dard model of cosmology is thought with a cosmological occurrence of singularities of various types. It was ob- constant term, Λ. This is the simplest extension to the servedthattheminimumofthescalarfieldpotentialcan 1 Einstein-Hilbert action. Moreover, it is consistent with be near to the singularity point (R ), and hence v → ∞ 8 all availablecosmologicaldata till date and is commonly it is likely that the scalar field hits the singularity if 2 knownasΛcolddarkmatter(Λ-CDM)model. However, the model parameters are not fine tuned appropriately 9 it is very difficult to explain the origin of the required [22, 23]. Since the potential well becomes shallower in 7 valueofthecosmologicalconstantfromanyfundamental the presence of matter density, the possibility of the oc- 0 physics [3]. currence of singularity increases in a matter distribution . 1 [23]. The occurrence of curvature singularitycan also be As an alternative to the Λ-CDM model, modifica- 0 seen in a collapsing astrophysical object. In this case, tions of gravity action by higher order curvature invari- 6 the singularity is analysed for suitable f(R) models ap- 1 ant terms are considered. The most promising models pliedtodenseobjectsundergoingcontractioninthepres- : in this category are f(R) theories of gravity [4]. In an v ence of linearly time-dependent collapsing mass density f(R)-theory Lagrangian, the Ricci scalar R is replaced i [24–26]. It is seen that the singularity is reached in a X by an analytical function f(R). Initially, the diverging time that is much shorter than the cosmological time r f(R) models at R = 0, e.g. inverse-power law models, scale. In [27], both static and dynamical analysis in the a f(R) Rnwithn<0,wereproposedforlate-timeaccel- ∝ contracting astrophysicalobject is carried out for a gen- eration [5]. But, the models were shown to be unviable eral f(R) model proposed in [15]. It was found that the because of matter instability [6] and failure to provide models that satisfy the fifth-force constraints are typi- matter era before the accelerating phase [7]. Addition- cally plagued with the curvature singularity issue. It is ally,the modelsalsoviolatethe fifth-force constraintsby also noted that the issue of curvature singularity can be carryinglongrangeforceprovidedbyextrascalard.o.fin eliminated by adding an extra curvature termto the La- the theory [8]. All these above-mentionedissues severely grangian[17, 22]. The finite-time singularityin modified constrain the allowed form of f(R), and it has steered gravity is also described in [28, 29]. It is shown that the the streamlining of f(R) models towards a new class of past singularities may be prevented for a certain range models which are analytical at R=0. of parameters. These singularities may also occur in fu- Various observational data are putting the possibility ture and can be avoided for fine-tuned initial conditions [30, 31]. Inthiswork,ourprimaryaimistoexaminetheviabil- ∗email:[email protected] ity of the model proposed in [1]. We will consider fifth- †email:[email protected] force constraint analysis, and also investigate the exis- ‡email: [email protected] tenceofcurvaturesingularityalongthelineof[27]. Addi- 2 tionally,wealsocarryoutthedynamicalsystemanalysis b x x 1 2 for the model in pointing out the differences with [1]. In 0.93 0.7304 0.9199 the above work, other than the late-time cosmology, in- 0.94 0.5776 1.0821 flationary dynamics were also investigated. In our work, 0.95 0.4852 1.1815 wewillconcentrateonlyontheviabilityofthismodelfor 0.96 0.4081 1.2624 late-time cosmology. This work is organised as follows: 0.97 0.3361 1.3334 Sec. II gives the general idea about the ArcTan model and its de Sitter points. Sec. IIA discusses the fixed 0.98 0.2630 1.3980 points of the model in understanding its proper cosmo- 0.99 0.1791 1.4582 logical evolution as a late time dark energy model. The TABLE I: deSitter points for different values of b. fifth-forceconstraintsareanalysedinSec.III.The inves- tigationofcurvaturesingularityiscarriedoutinSec.IV, with a conclusion in the final Sec. V. with b>0. We thus take 0<b<1 for all following con- siderations. Here, R is the constant curvature solution d II. f(R)=tan−1R MODEL of Eq. (2) for vacuum, and can be obtained by solving the following condition [1] As a modified theory of gravity, f(R) theory is de- bR 2b scribed by the following action 1+−(βRd )2 −Rd+ β tan−1(βRd)=0. (6) d 1 S = d4x√−g 2κ2f(R)+Lm , (1) Note that fixing the value of βRd in Eq. (6), one can fix Z (cid:20) (cid:21) the value of b. Other than the trivial solution of R0 = 0 corresponding to the Minkowski space-time, there are where f(R) is an arbitrary function of Ricci scalar R, two solutions of Eq. (6), of which one is an unstable de and it is written in the following form where we sepa- rateoutthe usualEinsteinHilbert contribution: f(R)= Sitter point x1 =βRd(1) and another is a stable de Sitter R+F(R). Lm is the matter part of the Lagrangian. As point x2 = βRd(2). If a de Sitter point Rd satisfies the mentionedearlier,thefunctionf(R)giveninEq.(1)sat- condition F (R )/F (R )>R , then it is stable and ,R d ,RR d d isfiesthe conditionf(0)=0,andf(R) R 2Λathigh candescribeprimordialandthepresentepochdominated → − curvature so that the early-time cosmology is identical by vacuum energy. For b < 0.93 the Eq. (6) has only to Λ-CDM cosmology and physics is modified at the in- trivial solution R = 0. All the de Sitter points for the 0 fraredscales(latetime)only. Byvaryingtheactionw.r.t. allowed values of b are summarised in Table. I. g we obtain the field equation whose trace is given by µν 3(cid:3)F,R(R) 2F R+RF,R(R)=κ2T , (2) A. Viability as a Dark Energy Model − − where a comma in the subscript denotes derivative w.r.t Any f(R) modification of late-time cosmology should to Ricci scalar. We note that non-vanishing (cid:3)F term ,R give rise to accelerating expansion at the present epoch gives an extra dynamical scalar degree of freedom φ = preceded by a matter dominated era. A general dy- F other than the usual graviton. ,R namical analysis of f(R) theory is carried out in [7], A new model of f(R) theory has been proposed in [1] and cosmological viability conditions are derived. The where the function F(R) is given by field equations are rewritten in terms of a set of first b order autonomous differential equations for dimension- F(R)=−βtan−1(βR) , (3) less variables y1 = f,˙R/Hf,R, y2 = f/6f,RH2, and y =R/6H2, where−following quantities−were defined 3 with β being positive and having inverse mass dimen- sion two. Note that the model considered here in Eq.(3) Rf 2bx2 ,RR m = , (7) reduces to Λ-CDM model in high curvature regime i.e. ≡ f (1+x2)(1 b+x2) ,R R 1/β. The first and second derivatives of f(R) w.r.t − ≫ R for this model are given by Rf x(1 b+x2) y ,R 3 b 2bβ2R r = − = . (8) f,R(R)=1− 1+(βR)2, f,RR(R)= (1+(βR)2)2. ≡− f −(1+x2)(x−barctan(x)) y2 (4) Here,x βR.Therearesixfixedpointscharacterisedby The condition for scalar field φ to be non-tachyonic matter d≡ensity Ωm(m) and weff(m)= 1 2H˙/(3H2). − − (f,RR >0) requiresb>0, andthe conditionfor graviton The point P5 and P6 fall on the line m(r)=−r−1, and to be of non-ghost nature (f >0) requires the Λ-CDM cosmological evolution is denoted by m= 0 ,R line. The point P (r 1,m +0) corresponds to 5 1+(βR)2 >b, for R <R< , (5) matterdominatederaw≃ith−w ≃0. ThepointsP (r = d eff 1 ∞ ≃ 3 2.0 tory correctly. Moreover, dm/dr at P has to be greater 5 than 1fortheexistenceofacceptablesaddlematterera. − From the definitions of m and r and Eq.(3), one can ob- 1.5 P6 P1 tain the expression of m′(r) [1]. Taking x 0 limit one can find that m′(r = 1) 3 < 1. T→hus the point − → − − P with x=0 is not an acceptable point being unstable 5 to its perturbations. On the other hand, in the limit of m 1.0 x 1 i.e. R 1/β, r goes to 1 with m approaching zer≫o. One can≫find that m′(r =− 1) 0.0025 > 1. − → − − 0.5 P1 Thus, in the large curvature limit the point P5 is stable under perturbations. In conclusion,the model ofEq.(3) gives saddle matter era at very large value of curvature P and the Universe moves from P to P (in the bottom 5 5 1 0-.03.0 -2.5 -2.0 -1.5 -1.0 -0.5 branch) in its cosmological evolution. According to the original reference of [7], the model belongs to the Class r II category of f(R) models. FIG. 1: Trajectory in the m−r plane for b = 0.99. Fixed points are marked with blue dots. III. LOCAL GRAVITY CONSTRAINT 2,0 < m 1) and P (m = r 1,(√3 1)/2 < We have seen in the previous section that when cos- 6 − ≤ − − − mologicalevolutionhappens from P to P via the lower m < 1) both correspond to accelerating expansion of 5 1 branch of Fig. 1, the present dark energy dominated the Universe, where the former one is a de Sitter point. epochisprecededbytheordinarymatterdominatedera. Therefore, all viable dark energy models fall into two Buttheformoff(R)shouldnotspoiltheexperimentally classes [7]: verified results of General Relativity at local scales. The ClassII:ModelsthatconnectP toP ,ClassIV:Models 5 1 fifthforceoriginatedbyanextrascalardegreeoffreedom that connectP to P . For a particularmodel, m canbe 5 6 in an f(R) theory must be attenuated on local gravita- plotted as a function of r, and its cosmological viability tional systems like earth and solar system so that the can be tested. theory can evade the local gravity tests. In the Einstein Dynamical analysis and stability of critical points for frame, the scalar field ψ (corresponding to φ in Jordon the model of Eq. (3) have been studied in [1]. Here, frame)isachameleon-likefieldwhichcouplestothemat- we reanalyse the stability of all the critical points. The terinsuchawaythattheeffectivemassofthescalarfield m vs r plot is shown in Fig.1 for b = 0.99, and all the depends on the local matter density [20]. critical points are marked. One important characteris- In the Einstein frame, the action can be rewritten as tic of this model compared to many previously analysed model is that the plot is multivalued. The upper and mlowoveerbclroacnkcwhiissesaelpoanrgattehdebcyutrhveeltihnee msca=la−r rcu−rv1a.tuArsewRe S = d4x −g˜"2Rκ˜2 − (∇˜2ψ)2 −VE(ψ)+Lm(g˜µνe−√26κψ)#, Z decreases with point P (r 1,m +0) corresponding p (9) 5 ∼− ≃ to both small and high curvature limits. where all quantities having tilde are defined in the Ein- The intersection of m(r) curve with r = 2 line steinframe. ThescalarfieldψforthemodelofEq.(3),in gives the de Sitter points P1. We have two P1−points the high curvature regime (where R ≫ β1 and F,R ≪ 1) at x = 0.1791 and x = 1.4582 for b = 0.99. The is given by stability condition for the stable de Sitter point is at r = 2,0 < m 1. The de Sitter point x = 0.1791 3 3 3 belon−gs to the up≤per branch of the curve. Since, at this ψ = 2κ2 lnf,R = 2κ2 ln(1+F,R)≈ 2κ2F,R. r r r de Sitter point, m(r = 2) > 1, it is an unstable point. (10) − The point x = 1.4582 corresponding to lower branch of The potential VE(ψ) is given by the curve is a stable de Sitter point. The points P5 and P6 can be located at the intersec- V (ψ)= Rf,R(R)−f(R) =(1+(βR)2) tliinoen.sTohfethpeoimnt(rP) (ibslluoec)atceudrvaet wmith1m.54=an−drr−1 (r2e.5d4) E 2κ2f,2R(R) (cid:12)(cid:12)R=R(ψ) × 6 (cid:12) with x ≃ 0.48. As m(r ≃ −2.54)∼> 1, the poi≃nt−P6 is 1+(βR)2 βbta(cid:12)(cid:12)n−1(βR)−bR anunstablepoint. PointP representsthesaddlematter ,(11) 5 h(cid:0) 2κ2[1(cid:1) b+(βR)2]2 i(cid:12) era. TheconditionforP5 toexistism(r =−1)=0. Itis − (cid:12)(cid:12)R=R(ψ) shownin[1]thatthe pointP5 issituatedatx=βR 0. (cid:12) ∼ (cid:12) But, saddle matter point has to be at higher curvature where R = R(ψ) needs to be substituted (cid:12)by inverting than the de Sitter point P to explain the cosmic his- Eq. (10). 1 4 Let us consider a spherically symmetric body with ra- dius r˜ . The effective potential V is defined by c eff Veff(ψ)=V(ψ)+e−√26κψρ∗ , (12) where ρ∗ is a conserved quantity in the Einstein frame. We assume that the spherically symmetric body has a constant density ρ∗ = ρ inside the body (r˜< r˜ ) and in c ρ∗ =ρ ( ρ ) outside (r˜>r˜ ). ψ and ψ are the out in c in out ≪ valuesofthefieldattheminimaoftheeffectivepotential V inside and outside the object respectively. The thin eff shell parameter is given by [20] δr˜ ψ ψ c out in = − , (13) r˜c − √6Φc where Φ is the gravitational potential of the test body FIG. 2: |ψout| vs. βRd(2). c (Sun/Earth). This shows that the only thin shell having widthδr˜ aroundthesurfaceoftheobjectcontributesto c IV. CURVATURE SINGULARITY IN ARCTAN the field outside the object thus resulting into the sup- MODEL pressionofthe fifth forceonthe surfaceofthe testbody. Since ψ ψ , the above equation reduces for our | out|≫ | in| In this section, we will analyse the behaviour of the case to effectivescalardegreeoffreedominasystemofcollapsing ψ √6Φ δr˜c. (14) mass density. We will show that in finite time, the field out c | |≃ r˜ evolves to a point where the Ricci scalar diverges. c From Eq. (2), we have seen that an f(R) theory has To evade the local gravity tests, the right hand side of an extra scalar degree of freedom φ = F , compared to the above equation should be [21, 32] ,R General Relativity. The associated dynamics of the field is controlled by Eq.(2), and can be rewritten as 5.97 10−11 (Solar system test), . ( 3.43×10−15 (Equivalence Principle test).(15) (cid:3)φ= dVJ + κ2T, (18) × dφ 3 Using Rout =κρout, fromEq.(10) andEq.(12) we obtain where √6 b √6 b dVJ 1 |ψout|≈ 2κ (βκρout)2 = 2κ ((x2/Rd(2))κρout)2. (16) dφ = 3(R+2F −RF,R). (19) The Eq. (18) corresponds to an oscillator where the In the previous section, we have found that there exists energy-momentumpartbehavesasaforceterm. Wepre- stable de Sitter points only for 0.93 6 b < 1. For an fer to work in the Jordan frame since it is more conve- example let us consider b = 0.97 which has a stable de nienttoexamine the issueofcurvaturesingularityinthe Sitter point at x = βR(2) = 1.3334. From the fact 2 d Jordan frame than in the Einstein frame. that the energy density of the baryonic/dark matter in The oscillations of the scalar field φ are governed by our galaxy ρ is 10−24g/cm3 and the curvature at out ∼ the potential VJ, and the form of VJ depends on the the de Sitter minimum is roughly of the order of ρ 10−29g/cm3, ψ comes out to be c ≃ function f(R) in a given model. Inverting the relation | out| φ = F,R to write the Ricci scalar R in terms of φ, and ψ 6.682 10−11 . (17) integrating Eq. (19) w.r.t. φ, we obtain the form of the out | |≈ × potentialV . Invacuum,thefieldφoscillatesaroundthe J Since|ψout|islargecomparedtotherequiredvaluesgiven minimum φmin of the potential which is also a de Sitter in Eq.(15), we can say that this model does not evade point [27]. Cosmological evolution happens around this thelocalgravityconstraintsforb=0.97. Wecangetthe point. There is also a point φ where Ricci scalar sing sameresultforallthe values ofb inthe acceptablerange divergesR , anditis finite field distance awayfrom given in Table I. It can be seen from Eq. (6) that b is the minimu→m∞φ . While the scalar field φ oscillates min notindependentbutvariesasthevalueofβRd varies. In aroundφmin,itisenergeticallypossibleforthefieldtohit Fig. 2 we show ψ with respect to x (= βR(2)). It is the singularity if the potential difference between φ clearlyseenthat| ψout| isgreaterthan02.5 10−d11. From and φ is finite. min out sing | | × this result, one can draw the conclusion that the model We first analyse whether the curvature singularity given by Eq. (3) hardly satisfies the Solar System con- point exists in the model of Eq. (3), and secondly we straint and does not satisfy Equivalence Principle con- investigate the evolution of the Ricci scalar in a collaps- straint. ing object whose energy density is linearly growing. In 5 presence of matter-energy density, the oscillations of the 0.14 field φ are governed by the effective potential Veff. In b=0.93 J 0.12 b=0.97 this case, the Eqns.(18) and (19) can be rewitten as b=1.0 0.1 ∂Veff 0.08 (cid:3)φ= J , (20) ∂φ VJ 0.06 β 0.04 ∂Veff 1 0.02 J = (R+2F RF +κ2T). (21) ,R ∂φ 3 − 0 In fact, the minimum of the potential Veff shifts from -0.02 J -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 thedeSitterpoint,andmovesclosertothecurvaturesin- φ gularity point. Thus the effects of matter are necessary to be investigated even when a model is well behaved in FIG.3: βVJ vs. φfordifferentvaluesofparameterb. Thede vacuum [23, 27]. Sitterpoints are marked by∗ in theplot. Let us first examine the profile of the potential V for J vacuum i.e, κ2T =0. For our model, the scalar field φ is given by scalar R indeed diverges. For this, we study the issue in an astro-physical object like galactic cloud of dust b φ= . (22) which collapses under its own gravity. Acknowledging −1+x2 the fact that its density increases with time, the energy- momentum tensor for such a system can be empirically Integrating Eq. (19) w.r.t. φ, we obtain the potential in taken as terms of φ as T = κ2T (1+t/t ). (25) 0 ch φ2 1 b/φ (4+3b)φ 1 b/φ − βV = − − + − − J Here, t is the characteristic time of the collapsing − 6 12 ch p p object. The value of typical characteristic time for a 2b 1 ( 4+3b)+φ tan−1( 1 b/φ) . (23) collapsing galactic cloud is t 1.34 1015sec [27]. − 3 8 − − − ch ∼ × (cid:18) (cid:19) Though the above form of T is not exact, it can give p qualitativelycorrectscenario,providedthatthe contrac- From Eq. (22), it is clear that as φ 0, x , leading → →∞ tion is slow enough. Since the density of dust cloud, to a curvature singularity. ρ =10−24g/cm3 is muchgreaterthan the averageden- We now probe the height of the potential barrier be- m sity of the universe i.e. ρ =10−29g/cm3, we can take tweenthedeSitterminimumφ andthesingularpoint crit min the limit R 1/β in Eq. (2). Nevertheless the astro- φ . Intheregionbetweenφ andφ , b/φvaries sing min sing ≫ from1+(βR )2to ,andthereforewecanta−kethelimit physical density ρm is low enough to consider the back- d ∞ groundmetricasaMinkowskimetric[26]. Therefore,we b/φ 1 for this region in Eq. (23). It can be easily − ≫ can replace covariantderivatives with partial derivatives seenthatthe thirdtermofthepotentialinEq.(23)dom- in Eq.(2). Moreover,spatialderivativesare alsoignored inatesoverothertermsinthislimit,andtheEq.(23)can because of the presumed homogeneity and isotropy. be rewritten as Considering new variables y = κ2T /R and τ = t/t , 0 ch b theequationofmotioncanbeobtainedintermsofy and βV = ( 4+3b)tan−1 b/φ . (24) J −12 − − τ as (cid:16)p (cid:17) Since tan−1( b/φ) goes to a finite constant value for y′′+ y′2 +τ2 y−1 1 1+ τ y−1+ large −b/φp, w−e have βVJ ∝ b. We thus conclude that y ch (cid:20)3(cid:18) τch(cid:19)− 3 the height of the potential barrier is finite and propor- 2 by p tan−1(αy−1) =0 , (26) tional to the value of b. This makes the model (3) vul- 3α − 3α2 (cid:21) nerable to curvature singularity. In Fig. [3], potential βV vs. φ has been plotted for different values of the whereα=βκ2T ,τ = α3/2bβt ,andprimedenotes J 0 ch ch parameter b. φ = 0 corresponds to the curvature singu- derivative w.r.t. τ. We solve the above equation and p larity point where curvature R diverges to infinity. The inquire what happens to y(τ) within characteristic time de Sitter points aremarkedby “ ”in the plot. It canbe as the object collapses. Here, α ρ /ρ 105 and β m crit ∗ ∼ ∼ seen that the de Sitter points for larger values of b are is average curvature of the universe at present time and atagreaterdepthfromthesingularitycausingthelesser its numericalvalue is givenby β =1/t2, where t is the U U probability of scalar field φ reaching singularity. age of the universe. To make our analysis more quantitative we need to To solve Eq. (26), we take y′(0) = 0 and y(0) = 1 solve the equation of motion and confirm that curvature as initial conditions. The solutions of Eq. (26) are plot- 6 ately violates the observational tests. Following the line b=0.93 of [27], the issue of curvature singularity is investigated b=0.97 2 in the Jordan frame. b=1.0 Tocheckthecosmologicalviabilityofthemodel,wedo 1.5 fixed point analysis and point out some differences with R /0 the results found in [1]. The stable fixed point P which T 6 2κ 1 is responsible to give rise to the acceleratedexpansionis not presentin this model. We find the stable fixed point 0.5 P at very large value of the curvature. The point P 5 5 correspondsto a saddle matter era which is at curvature 0 higherthanthedeSittereragivenbyP . Wealsoexam- 1 0 0.005 0.01 0.015 0.02 0.025 0.03 ine the viability of the model at local gravity scales by t/t ch putting fifth force constraint through chameleon mecha- nism given in [20]. FIG. 4: Oscillations of y=κ2T0/R vs. t/tch. Wefindoutthatinthefieldspacethereexistsasingu- larity point where the curvature scalar diverges to infin- ity. ThepotentialbarrierbetweenthedeSitterminimum ted in Fig. [4] for b = 0.93,0.97 and 1.0. One can no- andthe singularitypointisfiniteforallallowedvaluesof tice that R corresponds to y 0. The Fig. [4] theparameterb. Wehaveconsideredtheevolutionofthe clearly show→s t∞hat the oscillations →of y gradually in- scalar curvature in a spherically collapsing object. The creases and after a finite time it reaches to zero result- numerical solution of the Eq. (26) is plotted in Fig. 4. ing into curvature singularity. It can be seen that y From the plot, it is evident that the time taken to reach becomes zero sooner for smaller b, and later for higher thesingularityisfinite andmuchlessthanthe ageofthe values of b. This is consistent with what we obtained universe. by our previous static analysis by looking at the poten- Inconclusion,themodelofEq.(3)isplaguedwiththe tial. FromournumericalsolutionsofEq.(26),weobtain issue of fatal curvature singularity. Additionally, we also t =0.032 1015, 0.033 1015and0.035 1015secfor havefoundthatthemodeldoesnotsatisfythefifthforce sing × × × b = 0.93, 0.97 and 1.0 respectively. Thus, we find that constraint, and therefore it is not a viable model for the the collapsing object encounters the curvature singular- late time Universe showing acceleration. ity within time muchsmaller than the cosmologicaltime scale. Acknowledgements V. 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