Distributional cosmological quantities solve the paradox of soft singularity crossing
László Á. Gergely, Zoltán Keresztes, Alexander Yu. Kamenshchik
aa r X i v : . [ g r- q c ] A p r Distributional cosmological quantities solve the paradox ofsoft singularity crossing
László Á. Gergely ∗ , Zoltán Keresztes ∗ and Alexander Yu. Kamenshchik †, ∗∗ ∗ Departments of Theoretical and Experimental Physics, University of Szeged, Tisza Lajos krt 84-86, Szeged6720, Hungary † Dipartimento di Fisica and INFN, via Irnerio 46, 40126 Bologna, Italy ∗∗ L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Kosygin street 2, 119334Moscow, Russia
Abstract.
Both dark energy models and modified gravity theories could lead to cosmological evolutions different fromeither the recollapse into a Big Crunch or exponential de Sitter expansion. The newly arising singularities may representtrue endpoints of the evolution or alternatively they can allow for the extension of geodesics through them. In the latter caseonly the components of the Riemann tensor representing tidal forces diverge. A subclass of these soft singularities, the SuddenFuture Singularity (SFS) occurs at finite time, finite scale factor and finite Hubble parameter, only the deceleration parameterbeing divergent. In a Friedmann universe evolving in the framework of general relativity they are realized by perfect fluidswith regular energy density and diverging pressure at the SFS. A particular SFS, the Big Brake occurs when the energydensity vanishes and the expansion arrives at a full stop at the singularity. Such scenarios are generated by either a particularscalar field (the tachyon field) or the anti-Chaplygin gas. By adding any matter (in particular the simplest, the dust) to thesemodels, an unwanted feature appears: at the finite scale factor of the SFS the matter energy density remains finite, implying(for a spatially flat universe) a finite Hubble parameter, hence finite expansion rate, rather then full stop. The universe wouldthen further expand through the singularity, this nevertheless seems forbidden as the energy density of the tachyonic field /anti-Chaplygin gas would become ill-defined. This paradox is relieved in the case of the anti-Chaplygin gas by redefining itsenergy density and pressure in terms of distributions peaked on the singularity. The regular cosmological quantities whichare continuous across the SFS are then the energy density and the square of the Hubble parameter; those allowing for a jumpat the SFS are the Hubble parameter and expansion rate (both being mirror-symmetric). The pressure and the decelarationparameter will contain Dirac delta-function contributions peaked on the SFS, however this is no disadvantage as they anyhowdiverge at the singularity.
Keywords: cosmology, soft singularities, anti-Chaplygin gas
PACS:
INTRODUCTION
General relativity and the Copernican principle, com-bined with observations on the Hubble redshift of thegalaxies and modelling the present baryonic content ofthe Universe by a pressureless perfect fluid (dust) to-gether with a minor contribution from radiation leads to aUniverse born from a Big Bang singularity at finite timein the past. The Big Bang is characterized by infinite val-ues of the energy density r , pressure p and temperature T . The scale factor a , characterizing the size of the Uni-verse vanishes, leading to a diverging scalar curvature, atrue singularity.The study of the rotation curves of galaxies, the sta-bility of galaxy clusters and the formation of structurein the Universe all imply the existence of a dark mattercomponent, manifesting itself only through the gravita-tional interaction and dominating over baryonic matterby approximately a factor of ten. Dark matter can be ei-ther cold or warm (but not hot), and its inclusion intothe past cosmological evolution does not eliminate the Big Bang singularity. It is also confirmed by the very ex-istence of the cosmic microwave background and lightelement abundancies in the Universe.The future can be either continued expansion (still per-sisting after infinite time or just asymptoting to a stop)or an expansion arriving to a halt after finite time, fol-lowed by a contraction phase, leading eventually to a BigCrunch singularity, which is very similar to the Big Bang.The actual scenario is selected by the amount of com-bined dark and baryonic matter densities, as compared tothe critical density. In all these scenarios the future evo-lution is decelerated due to gravitational attraction.Modern cosmological observations (distant super-novae of type Ia, the cosmic microwave background,gravitational lensing) confirm on one hand that the Uni-verse is quite close to the critical energy density ( k = r andthe pressure p of the dark energy satisfies r + p < aa = − ( r + p ) (1)(we chose units c = p G / = L , with negligible contribution to the dynamicsof the Universe in the past, however modifying its future.In the L CDM (cosmological constant and cold dark mat-ter) model the Universe asymptotes to an exponentiallyexpanding de Sitter universe. Although of appealing sim-plicity, a cosmological constant would conflict by manyorders of magnitude the outcome of all variants of calcu-lation of the vacuum expectation energy. A dynamic darkenergy model would be clearly more satisfactory and per-fectly compatible with observations (which settle but thepresent value of this field). For a review on dark energymodels see [2]. There are many dark energy candidates,their common feature being that they change the futureof the Universe in a drastic manner.In Section 2 we enlist and succinctly characterize thepossible outcomes of such dynamic dark energy dom-inated evolutions together with certain unconventionalevolutions in modified gravity theories, leading to vari-ous exotic singularities. In Section 3 we concentrate on aparticular type of evolution, leading to a Big Brake sin-gularity. By adding ordinary matter to the model, the BigBrake singularity is generalized to a Sudden Future Sin-gularity (SFS). This is still a soft and traversable sin-gularity, however the future evolution is obstructed bythe dark energy becoming ill-defined. A possible way ofovercoming this difficulty is by generalizing the cosmo-logical quantities in a distributional sense.
A COMPENDIUM OF EXOTICCOSMOLOGICAL SINGULARITIES
The common characteristic of all dark energy induced,novel type of singularities is that they occur in fi-nite time. Despite certain components of the Riemanncurvature tensor diverging some of these singularitiesremain traversable. The classification below based ontraversability is consistent with Królak’s definition of thestrongness of a singularity [3].
Strong singularities
These are the singularities of type I. and III. in theclassification of Ref. [4].Singulatities of type I. occur for phantom dark energymodels (with barotropic index w = p / r slightly smallerthan − a and is characterized by diverging Hubble parameter H = ˙ a / a and a diverging ˙ H . Due to the Raychaudhuriequation (1) and the Friedmann equation (cid:18) ˙ aa (cid:19) = r (2)then both r and p also diverge. The energy density andpressure thus behave similarly as in the Big Bang orBig Crunch, however this happens at infinite, rather thanvanishing scale factor.Singularities of type III. are very similar, H , ˙ H , r and p diverge, however this occurs at finite scale factor.Therefore the singularities of type III. are also knownas Finite Scale Factor singularities. Note that althoughthis singularity is strong according to Królak’s definition,it shows up as weak according to Tipler’s definition[6], which seems then less adequate to characterize thestrongness of a singularity. The singularities of type III.are compatible with available cosmological observations[7].A particular singularity of type III. is the Big Freeze,occurring in the evolution of the generalized phantomChaplygin gas [8].
Weak singularities
Pure kinematical investigations of evolutions in aFriedmann universe lead to the possibility of Sudden Fu-ture Singularity (SFS) occurrence [9]. Such singularitiesare of type II in the classification of Ref. [4] and are char-acterized by finite scale factor a and finite Hubble pa-rameter H , while ˙ H diverges. Hence at these singulatitiesthe energy density is finite, while the pressure diverges.As the metric contains only the scale factor, the geodesicequations will contain but H , hence point particles maypass through this singularity, generating afterwards thenew geometry. The diverging ˙ H appears only in the devi-ation equation, generating infinite tidal forces at the SFS Nevertheless, a finite scale factor is also characteristic for othersingularities. In fact all weak (soft) singularities to be mentioned in thispaper occur not only at finite time, but also finite (but non-vanishing)scale factor. rossing, but only for an infinitesimally short time [10].SFS are weak in both the Królak and Tippler’s defini-tions.A particular SFS occurs when a full stop is realizedat the singularity. Such Big Brake singularities could beproduced by the dynamics of an anti-Chaplygin gas or bya particular tachyonic scalar field showing superluminalevolution over certain periods of its existence [11]. Thetachyonic model does not violate causality due to thecontinued homogeneity and isotropy of the Universe andwas shown to be in agreement with observations on thesupernovae of type Ia [12]. The Big Brake occurs aftera time comparable with the present age of the Universeand was also shown explicitly to be traversable and toeventually evolve into a Big Crunch [13]. In Refs. [15]the solutions of the Wheeler-DeWitt equation for thequantum state of the universe in the presence of the BigBrake singularity was studied.A time-reversed version of the Big Brake singularityis the Big Démarrage [8], when the Universe starts ex-panding from a state of infinite pressure but finite energydensity.There are also weak singularities characterized by van-ishing pressure and vanishing energy density, howevertheir ratio, the barotropic index w being divergent . Boththe singularities of type IV. from the classification of Ref.[4], where the time derivatives of rank three or higherof the scale factor diverge; and the w-singularities withcompletely regular scale factor introduced in Ref. [14]belong here. These singularities are quite soft, they donot harm in any way the evolution of the Universe orstandard matter, rather manifest themselves only in thedark energy model, possibly signaling its breakdown.
Exotic brane-world singularities
In brane-worlds the Einstein equation is replaced bythe effective Einstein equation. Beside the cosmolog-ical constant term and the energy-momentum tensorthis equation has additional source terms: i) a quadraticsource in the energy-momentum tensor (which becomesimportant only at high energy densities or pressures), ii) apull-back to the brane of non-standard model fields act-ing in 5 dimensions, iii) the asymmetric embedding ofthe brane. All these are reviewed in detail in [16].A singularity very similar to the SFS, dubbed quies-cent singularity arises in brane cosmology, in which r and H remain finite, but all higher derivatives of thescale factor diverge as the cosmological singularity is ap-proached [17].Brane-world dynamics however, in particular the pres-ence of the energy-momentum squared term among thesource terms allows for the appearance of even stranger singularities, which are characterized by diverging r and p , nevertheless regular evolutions of the scale factor.Such an example is provided by the collapse of a perfectfluid metamorphosing into dark energy [18].Another such singularity arises in the context of brane-world flat Swiss-cheese cosmologies, in the presence ofa huge cosmological constant [19]. At this singularity thescale factor, its first, second and all higher derivativesstay regular. This universe forever expands and decel-erates, as its general relativistic analogue, the Einstein-Straus model [20]. However after a finite time the pres-sure diverges to plus infinity. This smooth pressure sin-gularity is different from the case when both the pres-sure and the second derivative of the scale factor diverge,the latter stays regular. The accompanying energy den-sity turns negative shortly before reaching the singularityand becomes ill-defined there. The asymmetric embed-ding enhances the apparition of such a singularity. Thereis a critical value of the asymmetry in the embedding,above which these singularities necessarily appear [21].If one combines the cosmological constant, theenergy-momentum and the energy-momentum squaredsource terms into an effective fluid, it turns out thatthis is dust, following the standard evolution of anEinstein-Straus model. The effective energy densityevolves through positive values through the singularity,towards reaching asymptotically zero, as the universeexpands. In terms of the effective dust source it is quitenatural that the singularity can be passed through. Nev-ertheless the pressure of the physical fluid diverges andits energy density becomes ill-defined. The singularityis induced by the brane dynamics non-linear in theenergy-momentum, modified as compared to GR. SFS CROSSING
The Big Brake singularity is the simplest SFS and phe-nomenological models, like a tachyonic scalar field oranti-Chaplygin gas were found, which evolve into a BigBrake [11]. Although the tachyonic scalar field has a sub-luminal evolution at present and mimics well dark en-ergy [12], also displays a dust-like (dark matter like) be-haviour in the more distant past [13], a more comprehen-sive cosmological model would certainly include bary-onic matter as well, customarily modelled by dust. Theaddition of dust to the tachyonic scalar field however in-duces a paradox. Its energy density at any finite scalefactor being positive, by virtue of the Friedmann equa-tion the Hubble parameter will not vanish at the singu-larity. The Big Brake is replaced by a SFS exhibiting afinite expansion rate. The paradox arises from allowingfor further expansion: for larger scale factor than the onecharacterizing the SFS the tachyonic field becomes ill-defined. The same paradox also arises when the dust isdded to the anti-Chaplygin gas.In Ref. [22], based on certain distributional identitieswe have worked out the details of including a distribu-tional contribution to the pressure of the anti-Chaplygingas (and equivalently to ˙ H ), centered on the SFS: p ACh = s A H S | t SFS − t | + H S d ( t SFS − t ) , (3)˙ H = − H SFS d ( t SFS − t ) − s A H SFS a SFS sgn ( t SFS − t ) p | t SFS − t | . (4)Then H may have a jump (the derivative of the Heavisidefunction being a delta function). In order to keep the en-ergy density continuous, H should not have a jump, thuswhen crossing the SFS, the Hubble parameter shouldobey a Z -symmetry. If the Universe arrives to the SFSwith the expansion rate ˙ a SFS , after crossing it it will havethe expansion rate − ˙ a SFS . The respective equations are: H ( t ) = H SFS sgn ( t SFS − t )+ s A H SFS a SFS sgn ( t SFS − t ) p | t SFS − t | . (5)In order to preserve the anti-Chaplygin gas equation ofstate p = A / r a delta function also enters the denomi-nator of the energy density. Alternatively, r may be keptregular, but then the equation of state should be general-ized into a distributional relation.There is a full analogy with a ball bouncing back froma wall or a tennis / squash racquet. A simple descriptionof the process includes a sudden reversal of the normalvelocity. A detailed description instead requires to al-low for modelling the ball deformation. After being com-pressed, the ball will reach a full stop, before bouncingback. A description of the SFS crossing without distribu-tions would require to deform the equation of state in the2–component fluid, such that H = ACKNOWLEDGMENTS
LÁG is grateful for the organizers of the MultiCos-moFun12 Conference for invitaton and financial sup-port. We acknowledge useful discussions with V. Gorini,M. O. Katanaev, V. N. Lukash, U. Moschella, D. Po-larski and A. A. Starobinsky. This work was supportedor partially supported by European Union / EuropeanSocial Fund grant TÁMOP-4.2.2.A-11/1/KONV-2012-0060 (LÁG), OTKA grant no. 100216 (ZK), and theRFBR grant no. 11-02-00643 (AYK).
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