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Texas Symposium on Rel. Astrophysics, Stanford University, 13. Dec - 17 Dec 2004 1 Prospects of Non-Riemannian Cosmology DirkPuetzfeld DepartmentofPhysicsandAstronomy,IowaStateUniversity,Ames,IA50011,USA∗ Inthisworkweprovidethemotivationforconsideringnon-Riemannianmodelsincosmology. Non-Riemannian extensions of general relativity theory have been studied for a long time. In such theories the spacetime continuumisnolongerdescribedbythemetricalonebutendowedwithadditionalgeometricquantities. These newquantities canbecoupledtotheintrinsicpropertiesofmatterinaverynaturalwayandthereforeprovide a richer gravitational theory, which might be necessary in view of the recent cosmological evidence for dark matter and dark energy. In this work we mainly focus on the concepts in metric-affine gravity and point out 5 theirpossiblesignificanceintheprocessofcosmologicalmodelbuilding. 0 0 2 1. Introduction 2. Theoretical landscape n a J Withthe rightamountofcrudenessonecouldsum- 2 marize the reasons to consider a drastic step, like the 1 change of the gravity theory which underlies cosmology, as follows: 1 v Cosmology, especially its observational sector, is • Largeamountsofdarkmatter/energynecessary 1 currently a thriving field of physics. On the theoreti- to fit current observations within the CCM. 3 cal side opinions have convergedto what is nowadays 2 dubbed cosmological concordance model (CCM). But, • No direct observation of a dark matter particle 1 despite ofallthe successesof this modelin describing in the laboratory. 0 5 differentcosmologicalobservations,weshouldnotfool • No theoretical explanation for the smallness of 0 ourselves to believe that the grand picture of cosmol- the dark energy component when compared to / ogystands onafirmbasis. Thereasonforthis issim- h quantum field theory. ple: Interpretationofthedatawithintheconcordance p - modelleads inevitably to the introductionofthe con- • NoreasontobelievethatGRisvalidintheearly o cepts of dark matter and dark energy1. We surely r universe, i.e. at high energies. t couldlivewith suchconcepts bystating thatthey de- s a pend on some peculiar details which yet have to be • NotestofNewtonian/generalrelativisticgravity : added to the description of our universe. Unfortu- on cosmological scales. v nately,darkmatterandenergymakeupthecomplete i X energybudgetwithinoursimplepictureandtherefore In the following we have a glance at some of the r cannotbetreatedassomeminordetailswhichremains proposed remedies for this situation. a to be workedout. This is clearlyanembarrassingsit- uation which needs to be addressed by cosmologists. 2.1. Alternatives Inthefollowingwewillhaveaglanceatthetheoret- AlthoughweknowforsurethatGRhastobemod- icallandscape ofcosmologyand pay specialattention ified in order to make it compatible with quantum to the non-Riemannian approach. Non-Riemannian theory [12], we do not have any final form of this extensions of our current gravity theory, i.e. General new gravitational theory. Additionally, we do not Relativity (GR), represent a well motivated frame- knowhowpossiblelow-energymodifications,andthus work and have been discussed extensively in the lit- modificationsthatmayplayanimportantroleforthe erature. In this short review we only address ques- aforementioned observational problems in cosmology, tions which are related to the possible cosmologi- caused by such a new theory will look like. cal significance of such an approach. Readers who In table I we provided a very rough overview over want to learn more about the fundamentals of non- current theoretical approaches to extend/replace our Riemanniangravitytheories,thegaugetheoreticalap- current gravity theory and thereby also our cosmo- proach to gravity, and metric-affine gravity (MAG) logical model. The separation into different model should consult the excellent reviews [3, 6, 7, 9, 10]. classes is sometimes not unique. For example, one couldalsocountthenon-symmetricgravitymodelsas non-Riemannian models and all of the models listed ∗Electronicaddress: [email protected] in table I could in principle also have a non-trivial 1See[4]foraninventoryofcosmological parameters. topology. PSN 1221 2 Texas Symposium on Rel. Astrophysics, Stanford University, 13. Dec - 17 Dec 2004 Table I Some examples for different classes of models recently used to explain observations of cosmological significance. Model type Description Scalar-tensor theories Modified Lagrangian, additional scalar field (maybe a leftover from some higher theory) non-minimally coupled to theRicci scalar. Higherdimensions Our universe represents only a 4-d brane in a 5-d bulk, gravity assumed to be the only interaction which propagates in the bulk. Figure 1: Classification of different spacetime types f(R,Rαβ,...) models Modified gravitational La- according to thenon-Riemannian scheme. By switching grangian in terms of the off torsion and nonmetricity we arrive at theusual curvature. Riemannian spacetime as encountered in GR. Topological models Universe assumed to have non- trivial topology, i.e. impose some additional global proper- be viewed as the natural prolongation of the line of ties of spacetime which GR, as reasoning which led to the formulation of the so far a local theory, makes no state- most successful gravity theory, namely GR. mentsabout. Non-symmetricgravity Theories in which the metric gαβ is nolonger symmetric. 2.3. Metric-affine gravity Tensor-vector-scalarthe- Additional vector field intro- ory duced by hand into the defini- In metric-affine gravity the spacetime continuum tionofthemetric,extendedLa- which contains matter carries both stresses σαβ (or grangianwhichcontainsthead- momentum currents) and hyperstresses ∆αβγ (or hy- ditional vectorial quantity and permomentum currents). The geometry of spacetime an extra scalar field. isdescribedbymeansofametricg andanindepen- Non-Riemannian models Spacetime no longer Rieman- αβ dent affine connection Γγ . The metric is still sym- nian,newfieldstrengthstorsion αβ Tαβγ and nonmetricity Qαβγ metric,i.e.gαβ =gβα,buttheconnectionisno-longer couple to intrinsic properties of given by the metric compatible connection α 2 nβγo mattersuch as thespin. known from GR and may be asymmetric Γγ 6=Γγ . αβ βα IfwedefineCartan’storsiontensorTα :=Γα and βγ [βγ] the nonmetricity tensor Q := −∇ g then, cf. αβγ α βγ 2.2. Non-Riemannian gravity [22],the affineconnectionmightbe splitupasfollows One ofthe generalframeworksfor non-Riemannian Γαβγ =(cid:8)αβγ(cid:9) + Tβγα−Tγαβ +Tαβγ gravity theories is metric-affine gravity (MAG) as re- 1 + (Q α+Q α −Qα ). (1) viewedbyHehletal.in[10]. Inthefollowingwefocus 2 βγ γ β βγ on the new geometrical notions in metric-affine gravity and try to explain their possible impact on cos- Furthermore, one assumes that the momentum cur- mology. The reasons to pick MAG as starting point rent Σαβ of matter essentially couples to the metric arevaried: (i)Amongthedifferentalternativegravity gαβ whereas the hypermomentum current ∆αβγ cou- theories MAG represents a very well motivated and ples to the affine connection Γα of the spacetime. βγ naturalgeneralization,c.f. the introductionof[10] for From the last assumption and the splitting of the a list of arguments, and (ii) there exists a general connection as given in eq. (1) it becomes clear that Lagrangian formulation of MAG according to which MAG incorporates several other alternative gravita- many other non-Riemanniantheories may be system- tional theories, as well as GR itself. For example it atically classified. (iii) There exist several exact (also is well known from Einstein-Cartan (EC) theory that non-cosmological) solutions for MAG which rank it among the best studied alternative gravity theories duringthelastyears. (iv)Theideatocoupleintrinsic features of matter to new geometrical quantities can 2(cid:8)αβγ(cid:9):= 12gαµ(cid:0)∂βgγµ+∂γgβµ−∂µgβγ(cid:1). PSN 1221 Texas Symposium on Rel. Astrophysics, Stanford University, 13. Dec - 17 Dec 2004 3 thetorsionofspacetimecouplestotheintrinsicspinof particles. In figure 1 we sketchedhow differentspace- times,andtherebydifferentalternativetheorieswhich make use of these richer spacetime concepts, maybe classified with respect to the torsion and nonmetricity. The gravitational field Lagrangian of MAG is ex- pected to be of the form L = L(g,∂g,Γ,∂Γ) and a matterLagrangianminimallycoupledtothenewgeo- metricalfieldsL =L (ψ,∇ψ,g). Thegravitational m m fieldequationsaregivenbythevariationalderivatives Figure 2: In more complex fluid models matter may be with respect to the metric δL/δgαβ ∼ σαβ and the represented by a medium with dislocations or finely connection δL/δΓα ∼ ∆ βγ. We only mention here dispersed voids. Non-Riemannian models allow for a βγ α that a very general suggestion for the dynamics of natural coupling of thenew geometrical quantities like this theory, which makes use of a slightly different torsion and nonmetricity to such kind of fluid properties. but equivalent notation than the one used here, has been made in [11]. Since the field equations of the FLRW model are extremely simple and the main evidence (see [2, 23] for the latest SNIa samples) for the new dark energy 3. Cosmology component comes from cosmological tests which are related to the expansion history of the universe, it is In[17]weprovidedabriefchronologicalguidetothe naturalto study the impactofchangesin this history literature on non-Riemannian cosmological models. andtheirpossibleorigin. Therehavebeenseveralsug- Therein the developments in cosmology were traced gestionsformodificationsduringthelastyearscoming back to the early seventies and were given in table fromdifferentdirections,cf.tableI. Non-Riemannian form. Most of the early non-Riemannian cosmologi- models,especiallyMAG,provideaverygoodstarting cal models were basedon Einstein-Cartantheory. In- pointforthestudyofchangesofthecosmologicalfield vestigations mainly revolved around the construction equations which are justified by a grander theory, in of exact solutions and the question of whether or not the case of MAG by the general Lagrangianprovided an initial singularity can be avoided in such models. in [11]. Let us now come to the rhs, i.e. the matter In the 1980s more general types of Lagrangians were side, of the field equations. considered. The inclusion of quadratic terms in the Lagrangian,leading to dynamical degreesof freedom, was mainly motivated by the framework of Poincaré 3.1. Continua with microstructure gauge theory (PGT) and led to new classes of exact solutions. The story continued with the advent A very promising approach to construct cosmolog- of the inflationary model, which led to investigations ical models in a non-Riemannian setup is related to which tried to mimic or justify this new idea within the availabilityofmore sophisticatedfluid models. In different non-Riemannian scenarios. Till the end of thecaseofmetric-affinegravityObukhov&Tresguer- the 1990s most of the works in non-Riemannian cos- res [15, 16] devised a fluid model termed the hyper- mology (NRC) were focused on the description of the fluid 4. This kind of fluid can be used as a natural early stages of the universe. This bias can mainly be sourceforthehypermomentumcurrentwhichappears ascribed to the estimates for the new spin-spin con- ontherhsofthefieldequationsofMAG.Thenewde- tact interaction encountered in Einstein-Cartan the- greesof freedomin sucha fluid model can be coupled ory. This interaction shows up at extremely high en- to the new geometrical properties, i.e. torsion and ergydensities3 andmightthereforeplayonlyacrucial nonmetricity. In the hyperfluid picture, which can roleintheearlyuniverse. Thisfocushaschangeddur- viewed as a generalization of early spin-fluid models ingrecentyears,mostlyduetothe persistingneedfor [13, 14, 19, 20, 21, 24], the motion of the fluid is de- the large amount of dark matter and more recently scribed by usual four-velocity and a triad attached also dark energy. The requirement of dark energy at to each fluid element, which can undergo arbitrary late stages of the cosmic evolution might be taken as deformations during the motion of the fluid. This an indicator for the presence of new physics possibly is analogous to the description of continua with mi- due to some non-Riemannian relics in cosmology. crostructure [5] in the theory of elasticity. In figure 2 we sketched two examples of such media, namely 3In [8, 9] it was estimated that this may be the case at approximately1047g/cm3. 4Seealso[1]. PSN 1221 4 Texas Symposium on Rel. Astrophysics, Stanford University, 13. Dec - 17 Dec 2004 Figure 3: How to build and compare cosmological models? one withdislocations andanotherone withfinely dis- at the same time explain the effects caused by dark persed voids. The hyperfluid and special cases of it matter/energy in a purely gravitational way. Most wereusedinseveralcosmologicalmodelsinthepast5, of the models proposed so far are either not worked asystematictreatmentforafairlygeneralLagrangian out to a sufficient level of detail, fail one or more of of MAG will be published in [18]. thecosmologicaltests,orarenotdistinguishablefrom the CCM with the currently available data. But, and this cannot be stressed enough, we are clearly only 4. How to test and compare? beginning to explore the different possibilities of non- Riemannian models. This is especially true for the As we sketched in figure 3, the list of different cos- cosmologicalsectorofmetric-affinegravitywhichcur- mological tests is quite long, and still growing. Usu- rently only covers a very small region of the theoreti- allyonestartswithatheoreticalmodelfromtheupper cally permissible parameter space. portion of the figure and then compares it to some of the observations in the lower portion of the figure in ordertofalsifyit. Inviewofthesometimesverydiffer- enttheoreticalapproachesthiscanbecomequitecum- Acknowledgments bersome, i.e. one has to spend a lot of time to work out the single tests in a scenario which significantly deviates from the cosmological concordance model. The author wants to thank F.W. Hehl for constant Thereforeoneofthemostpressingtasksincosmology adviceandsupport. ThefinancialsupportbyM.Pohl is the development of a post-Newtonian framework and SLAC is gratefully acknowledged. Additionally, which allows us to compare different theoretical ap- the author wants to thank S. LeBohec for the hospi- proaches in a systematic and somewhat standardized tality during his stay at University of Utah. way, and hopefully allows for the a fast backreaction of the cosmologicaltests on the theoretical model. In figure 3 we denoted such a framework, which has yet to be developed, by the connecting middle part. References 5. Conclusion & Outlook [1] BabourovaO.V., FrolovB.N., Int. J. Mod. Phys. A13 5391-5407(1998) Up to now there seems to be no real competitor [2] Barris B.J., et al., Astrophys. J. 602 571-594 model in the non-Riemannian context which can re- (2004) placethecurrentcosmologicalconcordancemodeland [3] Blagojević M., Gravitation and gauge symme- tries. (IOP Publishing, Bristol and Philadelphia, 2002) [4] Fukugita M., Peebles P.J.E., Astrophys. 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