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Singularities around w = 1 − Leonardo Fernández-Jambrina ETSINavales,ArcodelaVictorias/n,E-28040-Madrid,Spain 5 Abstract. In this talk we would like to analyse the appearanceof singularities in FLRW cosmo- 1 logicalmodelswhichevolveclosetow= 1,wherewisthebarotropicindexoftheuniverse.We 0 − relatesmalltermsincosmologicaltimearoundw= 1withthecorrespondentscalefactorofthe 2 − universeandcheckfortheformationofsingularities. n a Keywords: Singularities,cosmology,BigRip,geodesiccompleteness J PACS: 98.80.-k,04.20.Dw 6 2 INTRODUCTION ] c q Accelerated expansion of our universe has been established observationally in the last - r decade from several sources of information. Even more, the barotropic index w of the g [ equation of state of the content of the universe, that is the ratio between pressure and energydensity,hasbeencheckedtobecloseto 1,valuewhichhasbeendubbedasthe 2 − v phantomdivide.Thisvaluecorrespondstoacosmologicalconstantasenergycontentof 0 theuniverse. 6 The fact of the accelerated expansion of the universe has lead to several scenarios 9 3 for a singular fate of the universe, different from eternal expansion or Big Crunch. A 0 classification of such singularities is provided in [1]. Among these new scenarios one . 1 can find Big Rip singularities [2], sudden singularities [3], Big Freeze singularities [4], 0 w-singularities[5]ordirectionalsingularities[6]. 5 1 Some of then however have been shown to be not strong enough [7] to mean the : actual end of the universe. For instance, sudden singularities [8] and w-singularities [9] v i are weak singularities. They have also been studied within the framework of modified X theoriesofgravitation[10]. r a The main idea behind the previous analysis is to assume that the scale factor of the universemaybewrittenasgeneralisedpowerexpansionofthetimecoordinate[11]and thenanalysethegeodesiccompleteness[12]oftheFLRWspacetimeendowedwithsuch scalefactor. With these ideas in mind, we would like to analyse the behaviour of cosmological models in the vicinity of w= 1. We shall show that there are scenarios which fall out − ofthepreviousclassificationand aretoberegarded separately. PERTURBED BAROTROPIC INDEX Let us assume that the barotropic index behaves approximately as a cosmological con- stantatpresent time,thatis, w= 1+h(t), h(t) 1. (1) − | |≪ Itisformallypossibletogetanexplicitexpressionforthescalefactora(t)=ef(t) for suchbarotropicindex, p 1 2aa¨ 2 dt w= = f(t)= +C. r −3−3a˙2 ⇒ 3 h(t)dt+k Z The constant C is irrelevant since it amounts to aR change of scale or time origin, a(t) eCa(t).Sincethedensityoftheenergy contentis → 2 a˙ 2 r = = f˙2 √r = , a ⇒ 3 h+k (cid:18) (cid:19) wemayuseittofix theconstantk. R Forinstance, takingtheorigin inthe nearfutureand assuminga power-lawdeviation fromunity,h(t)=a ( t)p, p>0 sothatw(0)= 1, − − 2(p+1) 1 r (t)= . − 3a k+( t)p+1 − p Two differentcases arisein thesemodels: • k=0: √r hassimplepolesoutoft =0 andr (t t0)−2, butw= 1 isregular. 6 ∼ − − • k=0: Singulardensityand scalefactorat w= 1. Namingb =2(1+p)/3a p, − 2 4 p+1 1 21+p r (t)= , f(t)= ( t) p, a(t)=e b /( t)p. 9 a t2p+2 −3 a p − − − − (cid:18) (cid:19) Weareinterestedinthelattermodels,whichhaveasingulardensityatthetimewhen w = 1 is reached. These models have a non-analytical scale factor with an essential singu−larity at t = 0 and the energy density blow up as 1/t2p+2 instead of t 2, which − is the usual power for divergencies with analytical scale factors. Two possibilitiesarise dependingon thesignoftheconstanta : • Fora >0, wehaveaGreatCrunch:a(t) 0. → • Fora <0, wehaveaGreatRip:a(t) ¥ . → GEODESICS NEAR w = 1 − Equationsforcausal geodesicsparametrised bypropertimet , dt = g dxidxj, ij − q inaflat FLRW modelreduce to dt P2 dr P = d + , = , dt s a2(t) dt ±a2(t) whereP isaconstantofmotionand d =0 fornulland d =1for timelikegeodesics. Thecaseofnullgeodesicsis simplersincetheequationsmaybeintegrated, 0 t =P 1 a(t)dt. − t Z0 ThismeansittakesaninfinitepropertimetoGreatRip(a(0)=¥ ),sothissingularity isnotaccesiblealonglightlikegeodesics. On thecontrary, fortimelikegeodesics, 0 dt t = , t 1+P2a 2(t) Z0 − allgeodesicsreach w= 1 in finiteproppertime. − Hence w = 1 becomes singular but for null geodesics. This sort of behaviour is − similarto theoneinthevicinityofaBig Rip singularity. STRONG SINGULARITIES In spite of the singular character of w= 1, it could happen that the singularity could − be not strong enough to be the end of the universe, since extended objects might avoid beingdisruptedbytidalforces on crossingthesingularity. ThisconceptofstrengthofsingularitieswasfirstcoinedbyEllisandSchmidt[12]and it was related to the cosmic censorship conjecture later on. It was further developed by otherauthors. For instance, for Tipler[13] a singularityis strong if thevolumespanned by three Jacobi fields orthogonal to the velocity of the geodesic tends to zero as it approaches its end. Królak [14] suggests a less restrictive criterion, requiring just the derivativeofthevolumewithrespect to propertimetobenegative. Since these definitions involve calculations with Jacobi fields, checking the strength of singularities may be cumbersome. Fortunately, necessary and sufficient conditions [15] have been derived involving just integrals of some components of the curvature tensoralongincompletegeodesics. Forinstance,followingTipler’sdefinition,asingularityisstrongalonganullgeodesic ofvelocityu ifand onlyiftheintegral t t dt ′dt R uiuj ′ ′′ ij 0 0 Z Z divergesas t tendsto t , wheret isthepropertimeassignedto thesingularity. 0 0 For Królak’s criterion the necessary and sufficient condition is the divergence of the integral t dt R uiuj. ′ ij 0 Z ast tends tot . 0 Such calculationsaresimpleinourcase andallowustoconcludethatsingularitiesat w = 1 are strong according to both criteria. The same result is obtained for timelike − geodesics. CONCLUSIONS Wehaveshownthattherearetwopossiblebehavioursformodelswithabarotropicindex closeto w= 1: − • Regularcrossingofw= 1. − • Essentialsingularityatw= 1:Great Crunch /Rip. − Forthelattermodelstheenergydensityblowsupatthesingularityanditsdivergence is worse than t 2. The essential singularities are strong, though null geodesics never − reach theGreat Rip. Moredetailsand references may befoundelsewhere[16]. ACKNOWLEDGMENTS The author wishes to thank the University of the Basque Country for their hospitality andfacilitiesto carry out thiswork. REFERENCES 1. NojiriS,OdintsovSDandTsujikawaS2005Phys.Rev.D71063004 2. CaldwellRR,KamionkowskiMandWeinbergNN2003Phys.Rev.Lett.91071301 3. BarrowJD2004Class.Quant.Grav.21L79 4. Bouhmadi-LópezM,Gonzalez-DíazPFandMartín-MorunoP2008Phys.Lett.B6591 5. Da¸browskiMPandDenkiewiczT2009Phys.Rev.D79063521 6. Fernández-JambrinaL2007 Phys.Lett.B6569 7. Fernández-JambrinaLandLazkozR2006Phys.Rev.D74064030 8. Fernández-JambrinaLandLazkozR2004 Phys.Rev.D70121503 9. Fernández-JambrinaL2010Phys.Rev.D82124004 10. Fernández-JambrinaLandLazkozR2009Phys.Lett.B670254 11. CattoënCandVisserM2005Class.Quant.Grav.224913 12. HawkingSWandEllisGFR1973TheLargeScaleStructureofSpace-time(Cambridge:Cambridge UniversityPress) 13. 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