Brane-world Cosmology David Wands InstituteofCosmologyandGravitation,UniversityofPortsmouth,MercantileHouse,Portsmouth P012EG,UnitedKingdom 6 0 Abstract. Brane-worldmodels,whereobserversare restrictedtoa branein a higherdimensional 0 spacetime,offeranovelperspectiveoncosmology.Idiscusssomeapproachestocosmologyinextra 2 dimensionsandsomeinterestingaspectsofgravityandcosmologyinbrane-worldmodels. n a J COSMOLOGY AFTER EINSTEIN 9 1 A century after Einstein first proposed his theory of relativity, it has become a cor- 1 nerstone of the physical sciences. Four dimensional spacetime provides the setting for v 8 describing physical processes and in particular provides the dynamical framework for 7 cosmologicalmodelsofourexpandingUniverse. 0 It was the general theory of relativity, proposed by Einstein in 1915, that for the first 1 0 time provided equations with which to describe the dynamics of spacetime. Einstein’s 6 equation 0 G +L g =8p G T , (1) / AB AB N AB c q relates the intrinsic curvature, G , of the metric, g , to the local energy-momentum, AB AB r- TAB, while allowing for the possibility of a non-zero cosmological constant, L . Con- g sistency with Newtonian gravity in the weak-field, slow-motion limit is ensured by the : v appearanceofNewton’sconstant,G , intheconstantofproportionality. N i X In muchofmoderncosmologyEinstein’stensorequation(1)convenientlyreduces to r theFriedmann constraintequation a K 3 H2+ =L +8p G r , (2) a2 N (cid:18) (cid:19) which relates the Hubble expansion, H, and spatial curvature K/a2, of a homogeneous andisotropicFriedmann-Robertson-Walker(FRW)spacetimetothelocalenergydensity r . Homogeneous and isotropic expansion has been used to build up a remarkably suc- cessful model for the evolution of our Universe starting with a hot Big Bang at a finite timein ourpast. This model has been tested not only by qualitativefeatures such as the evolution of galaxy populations and the existence of a cosmic microwave background (CMB) radiation, but also quantitatively tested by comparing models of primordial nu- cleosynthesiswithabundanceoflightelements. One only needs to consider linear perturbations about a homogeneous and isotropic metrictobuildupacoherentpictureoftheformationofstructureinourUniverse.Small fluctuations, about one part in a hundred thousand, are observed in the temperature of the microwave background radiation and indicate the existence of small primordial perturbations in the distribution of matter and radiation in the early universe when the CMB last scattered, about 300,000 years after the Big Bang. These primordial density fluctuations provide the seeds around which the observed large-scale structure of our Universe can form simply by gravitational instability, in a cosmological model with appropriatecontributionsfromradiation,baryonicmatter,aswellascolddarkmatterand someformofdarkenergy,thatbehavesverymuchlikeEinstein’scosmologicalconstant today.Awealthofobservationaldatanowenablescosmologiststoputthisbasicpicture to the test and attempt to measure parameters such as the density of different forms of matter,thenatureoftheprimordialperturbations,and Einstein’sgravitationallaws. At the same time fundamental questions remain unanswered. Why are there 3 large spatialdimensions(not5or15)?whyisthevalueofthecosmologicalconstantsosmall? and what really happens at the initial Big Bang which represents a singular point at the start of our cosmological evolution? I cannot answer these questions in this talk, but I can show how brane-world models offer some novel and interesting perspectives on these issues. I should emphasize that this is a personal view and not intended to be a systematic review of all aspects of brane-world cosmology. For a more comprehensive reviewsee[1]. EXTRA DIMENSIONS Superstringtheoryisanattempttounifygravitywiththeotherfundamentalinteractions in a self-consistent quantum theory, based on strings (extended 1-dimensional objects) as the fundamentalconstituentsof matterrather than pointparticles. In particular string theoryshouldbefiniteand singularityfree. Forexample,theexistenceofaminimallengthscaleintheeffectivetheoryleadstoa “T-duality”that relates expanding and contracting cosmological solutions and has been proposedasthebasisforthepre-BigBangscenario[2]thatproposesapre-BigBangera preceding the hot Big Bang expansion. Unfortunately the nature of the transition from pre-topost-BigBangisdependentonthenaturehigher-order,possiblynon-perturbative, effects and remains elusive. This makes it hard to make robust predictions based on a pre-Big Bang phase. Itisnotfairtosaythatstringtheorydoesnotmakeanypredictions.Stringtheorydoes make a definite prediction for the number of spacetime dimensions. Spacetime should have 10 dimensions for a consistent, anomaly-free superstring theory [3]. This may not appear to be a huge success for the theory, but of we can only assert that there are four observabledimensionsand it is quite possiblethat there exist extra dimensionsthat are verysmalland/orunobservable. Only a few years after Einstein proposed his theory of dynamical four-dimensional spacetime, Kaluza began to consider the dynamical equations for a five-dimensional spacetime, realising that the degrees of freedom of the metric associated with the extra dimension could describe a vector field in our four-dimensional world [4]. If the extra dimensioniscompactandverysmall,lessthan10 19 msay,thenonlythezero-modeof − r gravity e t t a m open closed strings strings FIGURE1. Matterfieldsaredescribedbyopenstringsconfinedtobranes. themetric,orotherfields,wouldbeexcitedinterrestrialexperiments.Higherharmonics in hidden dimension(s) correspond to very massive states, requiring large energies to excitethem,and thesecan beconsistentlyset tozero inalow-energy effectiveaction. In time it was realised that the size of the extra dimension was itself a scalar field and higher-dimensional models of gravity reduce to an effective scalar-tensor theory of gravity in four-dimensions at low energies. To avoid conflict with experimental tests of gravity the size of the extra dimensions must be fixed, but there has been little progressonhowtostabliseallthemodulifields describingthesizeandshapeofhidden dimensionsin stringtheory,untilrecently. This “hosepipe view” of the extra dimensions being rolled up incredibly small and hence out of sight was almost universally adopted to deal with the embarrassment of extradimensionsin stringtheory untilthe1990’s. What changed in themid 1990’s was thatrealisationthatotherextendedobjects,higher-dimensionalmembranes,or“branes”, shouldalsoplayafundamentalroleinstringtheory[5].Branesopenedupthepossibility to related apparently different string theories, for instance string theories containing closedstringsorthosewithopenstrings. Branescansupportopenstringswhoseend-pointslieonthebrane.Theseopenstrings can describe matter fields which live on the brane. On the other hand perturbations of the higher dimensional bulk geometry are described by excitations of closed strings, such as the graviton. To a general relativist it should be clear that even if matter fields are restricted to a lower dimensional hypersurface, gravity as a dynamical theory of geometrymustexistthroughoutthespacetime. This lead several authors to consider the possibility that at least some of the extra dimensions could be far larger than had previously been imagined [6, 7]. They realised that while particle interactions are probed by high energy colliders on energies up to 1 TeV, and hence scales down to 10 19 m, gravity is barely tested on scales below − 1mm.Iftheextradimensionsweretestableonlyviagravitythentheymightberelatively large. This offers a tantalising explanation for why gravity appears to be so weak when compared with theotherinteractions.The gravitationalfield ofan object could leak out into the large but hidden dimensions and gravity in our four-dimensional world seems weaker. To make this a little more precise, consider the gravitational field of a mass M in a D-dimensional spacetime. If we use Gauss’s law to calculate the gravitational field strength g at a distance r then we find g (cid:181) G M/rD 2 for distances r R, the radius D − ≪ of compactification of the hidden dimensions. But if r R then the gravitational field ≫ strengthis givenby 4p G M D g= . (3) 4p r2RD 4 − The effective value of Newton’s constant in our apparently 4-dimensional world, G , 4 can beidentifiedas G D G . (4) 4 ≡ RD 4 − Given we observe only the four-dimensional effective gravitational coupling, from which we infer a very large effective Planck scale M = 1/√G = 1019 GeV, the true 4 4 valueofthePlanckscale(thescaleatwhichquantumgravitybecomesimportant)could bemuch smallerinmodelswithlargeextradimensions. For instance, Horava and Witten in 1996 [8] proposed a supergravity model in 11- dimensions with a fundamental Planck scale close to the Grand Unified (GUT) scale of 1016 GeV where one of the extra dimensions had a size considerably larger than the conventional Planck scale of 10 35 m. But the GUT scale is still far beyond terrestrial − experiments and established particle physics models. What if quantum gravity was within reach of experiments like the LHC at CERN? If one hidden dimension was as large as 1 mm then the Planck scale could be as low as 108 GeV. With two large extra dimensions,thePlanck scalecouldbeas lowas 1 TeV [7]. RANDALL-SUNDRUM MODEL So far I have implicitly been discussing Minkowski branes in a higher dimensional Minkowskispacetime. Thisprovidesagood vacuumstateforstring theorybut weneed to go beyond flat spacetime to provide a cosmological model. Anti-de Sitter (AdS) spacetime, that is maximally symmetric space with a negative cosmological constant, L 2= 6k2,canalsoprovideausefulvacuumstateforstringtheory.Thismaynotappear − tobeverypromisingforacosmologicalmodelasanegativecosmologicalconstantleads to a cosmological collapse and big crunch in homogeneous and isotropic cosmologies. However it turns out to be a fascinating spacetime in which to consider brane-world cosmology. Randall and Sundrum produced two papers [9, 10] in 1999 which have had a huge impact in string theory and cosmology. They considered gravity on constant tension branesembedded infive-dimensionalanti-deSitterspacetime. Branescanbeembeddedatfixedy-coordinateinaGaussiannormalcoordinatesystem wheretheAdS metricis writtenas 5 ds2 =e−2k|y|h mn dxm dxn +dy2. (5) FIGURE2. RandallSundrum1,wherethehierarchybetweenthePlanckscaleandtheTeVscaleisdue tothedistancebetweentwobranesincompactAdSspacetime. FIGURE3. RandallSundrum2,wherethereisonlyonebraneembeddedinnon-compact5Dspacetime. The exponential “warp factor” means that the volume of the extra dimensional space becomes small at large y. In their first paper [9] Randall and Sundrum showed that the large hierarchy between a fundamental TeV scale and the apparent Planck scale 1019 GeV could be “explained” by a large warp factor even if the size of the extra dimension(specificallythenormaldistancebetweenbranes)wasrelativelysmall.Butin theirsecondpaper[10]theyshowedthateveniftherewasnosecondbrane,andtheextra dimension extended to infinity, gravity remained effectively localised on a single brane astheintegratedvolumeremainedfiniteasy ¥ .Thistheyproposedasan“alternative → tocompactification”. The two-brane model [9], called RS1, is not so different from earlier attempts to compactifythehiddendimensions,otherthanthatitoperatesinacurvedbulkspacetime. It is still the large volume of the hidden space that makes gravity weaker on the brane thanotherforces.ThereisstilladiscretespectrumofKaluza-Kleinstatescorresponding to higher harmonics on the hidden space, although the spectrum of eigenvalues is different in a curved space. And the size of the extra dimension, the distance between thetwo branes remains a scalar degree of freedom, known as the radion. It stillleads to aneffectivescalar-tensorgravityinfourdimensionsatlowenergies[11,12]whichmay bein conflict withexperimentaltestsunlesstheradionis stabilised. On the other hand, the one-brane model [10], inevitably known as RS2, offers a radically different model of dimensional reduction. The radion field in the RS1 model, decouples from gravity on the remaining brane in the limit that the second brane tends to spatial infinity. (The Brans-Dicke parameter w ¥ [11, 12].) And the discrete → spectrumofKKmodesisreplacedwithacontinuumofbulkmodes.Howeverthelightest modes are only weakly coupled to matter on the brane and gravity remains effectively four-dimensional on length scales greater than the AdS curvature scale, k 1. More − fundamentally though the single brane in AdS is an open system now where the initial state of matter on the brane (or branes) is not enough to determine the future evolution of the system. Instead one needs to specify initial data on a Cauchy hypersurface in the bulk. For example one might specify the AdS incoming vacuum state [13]. And fields onthebranecan radiateintothebulkandinformationcan escapeto futurenullinfinity. In either of the RS models there is a simple and novel interpretation of our cosmo- logical expansion. In the curved anti-de Sitter bulk spacetime (5) any motion of the brane, represented by a time-dependent trajectory y=y (t),induces an FRW metric on b the brane with scale factor a = e kyb [14, 15]. Cosmological expansion on the brane − | | correspondstomotionin acurved bulkspacetime. BRANE-WORLD GRAVITY Onewaytounderstandthegravitationaltheoryonabrane,suchastheRandall-Sundrum branes in AdS, is to use the projected Einstein equations on the brane [16, 1]. Consider a codimension-one brane with unit normal vector nA. The induced metric on the brane isthen g = (5)g n n , (6) AB AB A B − andtheextrinsiccurvatureofthebraneis K =gAC(5)(cid:209) n . (7) AB C B The4DRiemanntensoronthebranecan begivenintermsofthe5DRiemanntensorin thebulkand thebrane’sextrinsiccurvatureas [1] R =(5) R gEgFgGgH+2K K . (8) ABCD EFGH A B C D A[C D]B The higher-dimensional Einstein equations (1) determine the bulk Einstein tensor in terms of the bulk energy-momentum tensor. In the case of a vacuum bulk with only a cosmologicalconstantwehave (5)G +L (5)g =0. (9) AB 5 AB The Israel-Darmois junction conditions determine the jump in the extrinsic curvature tensor across the brane in terms of the energy-momentum tensor localised on the brane wherek 2isthegravitationalcouplingconstantin5-dimensions.IntheRandall-Sundrum 5 model the brane is a boundary of the bulk spacetime. This is equivalent to imposing a Z -symmetryacross thebranesothat K =K+ = K and hence 2 AB AB − A−B k 2 1 K = 5 Tbrane Tbraneg . (10) AB − 2 AB −3 AB (cid:20) (cid:21) This also occurs in the Horava-Witten model where the 10D boundary branes are fixed points of the orbifold S /Z . On the other hand the HW model also admits additional 1 2 branes which can move within the bulk spacetime. In this case there is an additional freedom due to the averaged extrinsic curvature, K+ +K , which is not directly AB A−B constrained by the energy-momentum tensor on the brane [17], but for simplicityI will assumeZ -symmetryacross thebraneinthefollowing. 2 Finally putting all this together we can give an expression for the Einstein tensor for theinduced metriconthebrane[16] G +L g =k 2T +k 4S E , (11) AB AB 4 AB 5 AB AB − where(i)S and(ii)E representmodificationstothestandardEinsteinequations(1) AB AB due to (i) terms quadratic in the brane energy-momentum tensor and (ii) the 5D Weyl tensorprojected onthebrane. The4Dintrinsiccurvature(8)includestermsquadraticintheextrinsiccurvatureofthe brane,and hence, via(10), theenergy-momentumonthebrane. Indeed weonlyrecover a term linear in T in Eq. (11) if the energy-momentum tensor on the brane contains a AB constantpartduetoaconstanttensionorvacuumenergydensityonthebrane,s ,sothat wesplit Tbrane =s g +T . (12) AB AB AB Theeffective4Dgravitationalcouplingconstantfortherenormalisedenergy-momentum tensor,T , inthebrane-world Einsteinequations(11)isthen givenby AB k 4s k 2 = 5 . (13) 4 6 Theeffect oftermsquadraticinthematterenergy-momentumtensorisgivenby 1 1 1 S = TT T TC+ g 3T TCD T2 . (14) AB 12 AB−4 AC B 24 AB CD − (cid:16) (cid:17) It is represents a high-energy correction to the brane-world Einstein equations and is typically unimportant when the matter density is much less than the brane tension, r s . ≪ The brane-world cosmological constant problem Theeffectivecosmologicalconstanton thebranein Eq.(11)is givenby L k 4s 2 L = 5 + 5 . (15) 2 12 In contrast to our usual 4D viewpoint that the vacuum energy density should simply vanish,orbeverysmall,inthebrane-worldwerequireinsteadthatthereisacancellation betweenthe4Dand5Dcontributionstothevacuumenergy.Anintriguingpossibilityin the brane-world is that 4D cosmological solutions might naturally seek out fixed points in an inhomogeneous 5D spacetime with small values of the cosmological constant – calledself-tuningsolutions[18]. AnoveltwistonthecosmologicalconstantproblemisprovidedbythemodelofDvali, GabadadzeandPorrati(DGP)[19]whopointedoutthatquantumloopcorrectionstoany classical model would be expected to induce terms in the effective energy-momentum tensor on the brane proportional to the brane Einstein tensor: D Tbrane = a G . In a AB AB 4Dmodelsuchcorrectionswouldsimplyrenormalisethegravitationalcouplingk 2.But 4 substitutedinto(11)thebrane-worldEinsteinequationsbecomequadraticintheEinstein tensor.ThusinadditiontotheusualvacuumsolutionwithG =0whenL =0,thereis AB a second (non-perturbative)solution with G (cid:181) a 2g . The DGP model has sparked AB − AB great interest as a novel explanation of the observed acceleration of our Universe [20], in terms of modified gravity rather than “dark energy”, but there remain questions over whethertheself-accelerating solutionsadmitunstable“ghost”modes[21]. Non-local brane gravity Equation(11)leavesonlytheprojected5D Weyltensor E = (5)C gEgFnCnD. (16) AB ECFD A B undeterminedby thelocal energy-momentumon ornearthebrane. Thisisthetidalpart ofthe5Dgravitationalfieldsoisonlydeterminedwhenonehasasolutiontothefull5D Einsteinequationswithappropriateboundary conditions. To the brane-bound observer it may be interpreted as an effective “Weyl fluid” with energy densityr˜ and 4-velocityu˜ sothat [1] A r˜ E =k 2 (g +4u˜ u˜ )+P˜ . (17) − AB 4 3 AB A B AB (cid:20) (cid:21) BecauseofthesymmetriesofthebulkWeyltensor,E istrace-freeandhencetheWeyl AB fluidistrace-free and hasbeinterpretedas“dark radiation”[22].Thisisconsistentwith the Maldacena’s AdS-CFT conjecture [23] which implies that the higher-dimensional gravitationalfield isequivalentto aconformalfield theory ontheboundary[24]. The 4D Bianchi identities, (cid:209) AG = 0, imply from Eq. (11) that the Weyl fluid’s AB energyr˜ andmomentumr˜u˜ obeylocalconservationequationsonthebrane,drivenby A thequadraticenergy-momentumtensor (cid:209) AE =k 4(cid:209) AS . (18) AB 5 AB The evolution of the Weyl anisotropic stress, P˜ , however cannot in general be deter- AB minedfrominitialconditionssetsolelyonthebrane.Thuswhiletheprojectedequations, and the Weyl fluid description in particular, may be useful for interpreting 5D gravity as seen on the brane, it may be of limited use in deriving solutions. This intrinsic non- locality of 4D gravity in the brane-world is the reason why so many outstanding prob- lems remain, including the nature of black hole solutions on the brane or anisotropic cosmologies,and requirehigher-dimensionalsolutions. FRW COSMOLOGY ON A BRANE One,important,caseinwhichtheprojectedfieldequationsaresufficientisthebehaviour of4Dhomogeneousandisotropic(FRW)cosmologies.Inthiscasethemaximalsymme- try of 3D space requires that the Weyl anisotropic stress vanishes and the general form ofthemodifiedFriedmann equation(2)onthebraneis K r m 3 H2+ =L +k 2r 1+ + , (19) a2 4 2s a4 (cid:18) (cid:19) (cid:16) (cid:17) where m is an integration constant on the brane set by the initial density of the Weyl fluidordark radiationon thebrane. In fact a generalisation of Birkhoff theorem implies that the general 5D vacuum spacetime admitting an FRW brane cosmology is Schwarzschild-Anti-de Sitter (SAdS) [14, 15]. The integration constant m in (19) represents the mass of the black hole in the SAdS spacetime (though in a compact RS1 model the singularity may lie outside the physicalregionofthespacetimebetween thebranes). The cosmological expansion described by the modified Friedmann equation (19) can be interpreted as motion of the brane in a static, but curved bulk. The staticbulk metric can bewrittenas dR2 ds2 = f(R)dT2+ +R2dW 2 , (20) − f(R) K wheredW 2 is thelineelementon amaximallysymmetric3-space, curvatureK, and K 2 R m f(R)=K+ . (21) ℓ −R2 (cid:18) (cid:19) On theotherhand if wechoosea Gaussian normalcoordinatein which thebrane is at a fixedlocationy=y thenthelineelementbecomes b ds2 = n2(t ,c )dt 2+dc 2+a2(t ,c )dW 2 , (22) K − where the explicit forms of n and a are given in Ref. [25]. The two coordinate systems arerelated by apseudo-Lorentztransformationat thebrane[27] ndt √fdT =L (q ) , (23) dc dR/√f (cid:18) (cid:19) (cid:18) (cid:19) where coshq sinhq L (q ) (24) ≡ sinhq coshq (cid:18) (cid:19) andtheLorentzfactorduetothemotionofthebraneinthebulkcoordinatesis R2H2 coshq = 1+ , (25) s f andH istheHubbleexpansionrate(19). This offers a novel perspective on 4D cosmology, not least the cosmological singu- larity problem. For instance, Garriga and Sasaki showed that an inflating brane and its SAdS bulk can be created “out of nothing” by a de Sitter-brane instanton [28]. Others have tried to describe the big bang singularity on the brane as a singular event within a regularhigher-dimensionalbulk,seeforexampleRefs. [29, 30,31, 32, 33]. Colliding FRW branes One scenario that has attracted much attention is the ekpyrotic model [29, 30] where the Big Bang on the brane-world is identified as a collision between branes. In the original version a hidden brane traverses the bulk and when it hits the boundary brane its kinetic energy is released, heating our observable universeand initiating the hot Big Bang. It turns out to be possible to give a complete description in general relativity of the collision of maximally symmetric codimension-one branes (or shells) in vacuum [26], similar to that envisaged in the original ekpyrotic model. Consider the simplest case of two incoming FRW brane-worlds (a and b) coalescing at a 3D collision surface to give one outgoing brane (c) as shown in Figure 4. The intervening regions (I, II and III) are necessarily SAdS in vacua. Thus the coordinate system on each brane (20) and in each bulkregion(22)arerelatedbypseudo-Lorentztransformationsoftheformgivenin(24). There is a simple geometrical constraint that the product of all the Lorentz transfor- mations(24), as onecompletesacircuit aroundthecollisionsurface, isunity[34,26] P L (q )=1. (26) i i Thejunctionconditions(10)enableonetorelatethecorrespondingjumpintheextrinsic curvature across each brane to the energy density on the brane, and hence one obtains a general relativistic version of the local conservation of energy-momentum at the collision[26] r coshq =r coshq +r coshq . (27) c c a a b b