A.A. Coley- The dynamics of brane-world cosmological models

8/3/2019 A.A. Coley- The dynamics of brane-world cosmological models

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Einstein centennial review article / Article de

synthse commmoratif du centenaire de

lanne miraculeuse dEinstein

The dynamics of brane-world

cosmological models1

A.A. Coley

Abstract: Brane-world cosmology is motivated by recent developments in string/M-theoryand offers a new perspective on the hierarchy problem. In the brane-world scenario, ourUniverse is a four-dimensional subspace or brane embedded in a higher-dimensional bulkspacetime. Ordinary matter fields are confined to the brane while the gravitational fieldcan also propagate in the bulk, and it is not necessary for the extra dimensions to besmall, or even compact, leading to modifications of Einsteins theory of general relativityat high energies. In particular, the RandallSundrum-type models are relatively simplephenomenological models that capture some of the essential features of the dimensionalreduction of eleven-dimensional supergravity introduced by Horava and Witten. Thesecurved (or warped) models are self-consistent and simple and allow for an investigationof the essential nonlinear gravitational dynamics. The governing field equations inducedon the brane differ from the general relativistic equations in that there are nonlocal effects

from the free gravitational field in the bulk, transmitted via the projection of the bulk Weyltensor, and the local quadratic energy-momentum corrections, which are significant in thehigh-energy regime close to the initial singularity. In this review, we investigate the dynamicsof the five-dimensional warped RandallSundrum brane worlds and their generalizations,with particular emphasis on whether the currently observed high degree of homogeneity andisotropy can be explained. In particular, we discuss the asymptotic dynamical evolution ofspatially homogeneous brane-world cosmological models containing both a perfect fluid anda scalar field close to the initial singularity. Using dynamical systems techniques, it is foundthat, for models with a physically relevant equation of state, an isotropic singularity is a past-attractor in all orthogonal spatially homogeneous models (including Bianchi type IX models).In addition, we describe the dynamics in a class of inhomogeneous brane-world models, andshow that these models also have an isotropic initial singularity. These results provide supportfor the conjecture that typically the initial cosmological singularity is isotropic in brane-world

Received 24 January 2005. Accepted 19 April 2005. Published on the NRC Research Press Web site athttp://cjp.nrc.ca/ on 18 June 2005.

A.A. Coley. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS B3H 4R2, Canada(e-mail: [email protected]).

1This article is one of a series of invited papers that will be published during the year in celebration of theWorld Year of Physics 2005 WYP2005.

Can. J. Phys. 83: 475525 (2005) doi: 10.1139/P05-035 2005 NRC Canada


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cosmology. Consequently, we argue that, unlike the situation in general relativity, brane-worldcosmological models may offer a plausible solution to the initial conditions problem incosmology.

PACS Nos.: 98.89.Cq/Jk, 04.20qRsum: Le dveloppement dune cosmologie pour un monde de branes est motiv parles dveloppements rcents en thorie des cordes/M (membranes) et promet une nouvellesolution au problme hirarchique. Dans le scnario dun monde de branes, notre universest un sous-espace 4 dimensions, ou brane, enchss dans un espace-temps standard deplus haute dimension. Les champs de matire ordinaire sont confins sur la brane, alorsque le champ gravitationnel peut se propager aussi dans lespace-temps et les dimensionsadditionnelles nont tre ni petites ni compactes. En particulier, les modles du typeRandallSundrum sont des modles phnomnologiques relativement simples qui capturentcertaines des caractristiques essentielles de la rduction dimensionnelle de la supergravit onze dimensions introduite par Horava et Witten. Ces modles courbes sont auto-cohrentset simples et permettent dtudier les lments essentiels de la gravit non-linaire. Lesquations de champ induites sur la brane diffrent des quations de la relativit gnrale

cause des effets non locaux provenant du champ libre gravitationnel dans lespace-temps,transmis via la projection du tenseur de Weyl, et des corrections du tenseur nergie-impulsionquadratique local, qui sont significatifs dans le rgime des hautes nergies prs de lasingularit initiale. Dans la prsente revue, nous analysons la dynamique des mondes debranes courbes cinq dimensions de RandallSundrum ainsi que leur gnralisation, vrifiantsurtout sil est possible dexpliquer le haut niveau observ dhomognit et disotropie.Plus particulirement, nous tudions lvolution dynamique asymptotique de modlescosmologiques faits de mondes de branes homognes, contenant la fois un liquide parfaitet un champ scalaire prs de la singularit initiale. Utilisant des techniques des systmesdynamiques, nous trouvons que pour des modles avec une quation dtat physiquementsignificative, une singularit isotrope est un attracteur dans tous les modles orthogonauxspatialement homognes (incluant les modles de Bianchi de type IX). De plus, nousdcrivons la dynamique dans une classe de modles de mondes de branes inhomognes etmontrons que ces modles ont aussi une singularit initiale isotrope. Ces rsultats supportent

la conjecture qui veut que typiquement, la singularit cosmologique initiale est isotrope encosmologie des mondes de branes. Consquemment, nous avancurlcons que, contrairement la relativit gnrale, les modles cosmologiques des mondes de branes offrent une solutionplausible au problme des conditions initiales en cosmologie.

[Traduit par la Rdaction]

1. Introduction

Cosmological models in which our Universe is a four-dimensional brane embedded in a higherdimensional spacetime are of current interest. In the brane-world scenario, ordinary matter fields areconfined to the brane while the gravitational field can also propagate in the extra dimensions (i.e., inthe bulk) [15]. In this paradigm, it is not necessary for the extra dimensions to be small or compact,whichdiffers from the standard KaluzaKlein approach, and Einsteins theory of general relativity (GR)

must be modified at high energies (i.e., at early times). At low energies, gravity is also localized at thebrane (even when the extra dimensions are not small) [69].

The five 10-dimensional (10D) superstring theories and the 11D supergravity theory are believedto arise as different limits of a single theory, known as M-theory [1012]. The 11th dimension in M-theory is related to the string coupling strength, and at low energies, M-theory is approximated by11D supergravity. In HoravaWitten theory [13], gauge fields of the standard model are confined totwo 10-branes located at the end points of an S1/Z2 orbifold and the six extra dimensions on thebranes are compactified on a very small scale effectively resulting in a number of 5D moduli fields. A

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5D realization of the HoravaWitten theory and the corresponding brane-world cosmology is given inrefs. 1416, with an extra dimensionthat canbe large relative to thefundamental scale, providinga basisfor the 5D RandallSundrum (RS) models [8, 9, 17]. An important consequence of extra dimensions

is that the fundamental scale is M4+d, where d is the number of extra dimensions, rather than the 4DPlanck scale Mp M4. If the extra-dimensional volume is significantly above the Planck scale, thenthe true fundamental scale M4+d can be much less than the effective scale Mp 1019 GeV and theweakness of gravity in 4D can be understood in terms of the fact that it leaks into the extra curved orwarped dimensions.

AsintheHoravaWitten theorythe RS branes are Z2-symmetric and have a self-gravitating tension,which counteracts the influence of the negative bulk cosmological constant on the brane. The RS braneworlds and their generalizations provide phenomenological models that capture some of the features ofM-theory, they provide a self-consistent and simple 5D realization of the HoravaWitten supergravitysolutions [13], and they allow for an investigation of the essential nonlinear gravitational dynamics inthehigh-energy regime close to the initial singularity when modulieffectsfrom theextra dimensions canbe neglected. RS brane-world models have, in common, a five-dimensional spacetime (bulk) governedby the Einstein equations with a cosmological constant, in which gravity is localized at the brane dueto the curvature of the bulk (i.e., warped compactification). A negative bulk cosmological constantprevents gravity from leaking into the extra dimension at low energies. In the RS2 model there are twoZ2-symmetric branes, which have equal and opposite tensions. Standard Model fields are confined onthe negative-tension brane, while the positive-tension brane has fundamental scale M and is hidden.In the RS1 model there is only one positive-tension brane, which can be obtained in the limit as thenegative-tension branegoes to infinity. We concentrate, here,mainly on RS1 branes in which thenegativecosmological constant is offset by the positive brane tension; the RS2 brane models introduce the addedproblem of radion stabilization and complications arising from negative tension.

It has recently become important to test the astrophysical and cosmological implications of thesehigher dimensional theories derived from string theory. In particular, can these cosmological modelsexplain the high degree of currently observed homogeneity and isotropy? Cosmological observationsindicate that we live in a Universe that is remarkably uniform on very large scales. However, the spatial

homogeneity and isotropy of the Universe is difficult to explain within the standard general relativisticframework since, in the presence of matter, the class of solutions to the Einstein equations that evolvetowards a Friedmann Universe is essentially a set of measure zero [18]. In the inflationary scenario,we live in an isotropic region of a potentially highly irregular Universe as the result of an acceleratedexpansion phase in the early Universe thereby solving many of the problems of cosmology. Thus, thisscenario can successfully generate a homogeneous and isotropic Friedmann-like Universe from initialconditions, which, in the absence of inflation, would have resulted in a Universe far removed fromthe one we live in today. However, still only a restricted set of initial data will lead to smooth enoughconditions for the onset of inflation [19, 20], so the issue of homogenization and isotropization is stillnot satisfactorily solved. Indeed, the initial conditions problem (that is, to explain why the Universe isso isotropic and spatially homogeneous from generic initial conditions), is perhaps one of the centralproblems of modern theoretical cosmology. These issues were recently revisited in the context of branecosmology [21,22].

A geometric formulation of the class of RS brane-world models is given in refs. 23 and 24. Thedynamical equations on the 3-brane differ from the equations in GR. There are local (quadratic) energy-momentum corrections that are significant only at high energies and the dynamical equations reduce tothe regular Einstein field equations of GR for late times. However, for very high energies (i.e., at earlytimes), these additional energy-momentum-correction terms will play a critical role in the evolutionarydynamics of the brane-world models. In addition to the matter field corrections, there are nonlocaleffects from the free gravitational field in the bulk, transmitted via the projection E of the bulk Weyltensor, that contribute further corrections to the Einstein equations on the brane. A particularly useful

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which reduces to

S = 112

2uu + 112

( + 2p) hfor a perfect fluid or minimally coupled scalar field (or combination thereof) and u is the 4-velocitycomoving with matter and h g +uu . The quadratic energy-momentum corrections to standardGR will be significant for42 2 (i.e., in the high-energy regime).

Dueto its symmetry properties, theprojection of thebulkWeyl tensorcan be irreduciblydecomposedwith respect to u [25,26]

E = 62

U

uu + 1

3h

+ P +Qu +Q u

where

U= 16

2E uu

is an effective nonlocal energy density on the brane (which need not be positive), arising from the freegravitational field in the bulk. There is an effective nonlocal anisotropic stress

P = 16

2

h

h 1

3h h

E

on the brane, which carries gravitational wave effects of the free gravitational field in the bulk. Theeffective nonlocal energy flux on the brane is given by

Q = 16

2 hEu

The local and nonlocal bulk modifications may be consolidated into an effective total energy-momentum tensor

G=

g+

2Ttot

(4)

where

Ttot = T +6

S 1

2E (5)

Theeffectivetotalenergydensity,pressure,anisotropicstressandenergyfluxare[25,26](foracomovingfluid the tilting case is treated later)

tot =

1 + 2

+ 6U

4(6)

ptot = p + 2

( + 2p) + 2U4

(7)

tot

=6

4

P (8)

q tot =6

4Q (9)

For an empty bulk, the brane energy-momentum tensor separately satisfies the conservation equations, T = 0. The Bianchi identities on the brane then imply that the projected Weyl tensor obeys theconstraint

E = 62

S (10)

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Consequently nonlocal bulk effects can be sourced by local bulk effects. The brane energy-momentumtensor and the consolidated effective energy-momentum tensor are both conserved separately. Theseconservation equations, as well as the brane field equations and Bianchi identities, aregiven in covariant

form in refs. 25 and 26.

2.1. Projected Weyl tensor

In general, the four independent conservation equations determine four of the nine independentcomponents ofE on the brane. There is no evolution equation for the nonlocal anisotropic stressP . Thus, in general, the projection of the 5D field equations onto the brane does not lead to a closedsystem, since there are bulk degrees of freedom whose impact on the brane cannot be predicted by braneobservers. These degrees of freedom could arise from propagating gravity waves in the bulk which aregoverned by off-brane 5D bulk dynamical equations.

If the nonlocal anisotropic stress contribution from the bulk field vanishes, i.e., ifP = 0, thenthe evolution ofE is fully determined. A special case of this arises when the brane is RobertsonWalker (RW) and in some special cases in which the source is perfect-fluid matter and branes with

isotropic 3-Ricci curvature (such as, for example, Bianchi I branes). Although these special cases cangive consistent closure on the brane, there is no guarantee that the brane is embeddable in a regularbulk, unlike the case of a Friedmann brane whose symmetries imply (together with Z2 matching) thatthe bulk is SchwarzschildAdS5 [32,33].

The nonlocal anisotropic stress terms enter into crucial dynamical equations, such as the Raychaud-huri equation and the shear propagation equation, and can lead to important changes from the GRcase. The correction terms must be consistently derived from the higher dimensional equations. In theanalysis, we shall primarily assume that the effective nonlocal anisotropic stress is zero (in the fluidcomoving frame). This is supported by a dynamical analysis [73] and the fact that since P corre-sponds to gravitational waves in higher dimensions, it is expected that the dynamics will not be affectedsignificantly at early times close to the singularity [74,75]. This is the only assumption we shall make,and it is expected that inclusion ofP will notaffect the qualitative dynamical features of the modelsclose to the initial singularity. Indeed, recent analysis provides further support for P

=0 [76, 77].

We expect that P Ug on dimensional grounds, and so for a Friedmann brane close to the initialsingularity, we expect that P a2 UC (where C is slowly varying), which is consistent withthe linear (gravitational) perturbation analysis (in a pure AdS bulk background) [60]. Hence, P isdynamically negligible close to the initial singularity. In addition, from an analysis of the evolutionequation for Q it can be shown that a small Q does not affect the dynamical evolution ofU close tothe initial singularity [21,25,26,78]. Let us discuss this in more detail.

2.1.1. Type N spacetimes

In brane-world cosmology gravitational waves can propagate in the higher dimensions (i.e., in thebulk). In some appropriate regimes, the bulk gravitational waves may be approximated by planewaves [76]. For example, there might occur thermal radiation of bulk gravitons [79]. In particular, atsufficiently high energies, particle interactions can produce 5D gravitons that are emitted into the bulk.

Conversely, in models with a bulk black hole, there may be gravitational waves hitting the brane. Atsufficiently large distances from the black hole these gravitational waves may be approximated as oftype N [76]. Alternatively, if the brane has low energy initially, energy can be transferred onto the braneby bulk particles such as gravitions; an equilibrium is expected to set in once the brane energy densityreaches a limiting value. We can study 5D gravitational waves that are algebraically special and of typeN.

Let us assume that the 5D bulk is algebraically special and of type N. This puts a constraint on the5D Weyl tensor that makes it possible to deduce the form of the nonlocal stresses from a brane point of

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view. For a 5D type-N spacetime, there exists a frame a , a , mia , i = 1, 2, and 3 such that [80,81]

Cabcd = 4C1i1j {a mibcmjd} (11)

The electric part of the Weyl tensor, Eab = Cacbdncnd, where nc is the normal vector on the brane,can for the type-N bulk be written as

Eab = C1i1j

a (micn

c) mia (cnc)

b(mjc n

c) mjb (cnc)

(12)

where Eab nb = Eaa = 0. Note that for a type-N bulk, we also have Eab b = 0, which can be rewrittenusing the projection operator on the brane, as Eab b = 0, b gbcc. Hence, the vector b is theprojection of the null vector b onto the brane, and

bb =

a na2

(13)

Inthecasethat a is time-like (a na = 0),wecanset u parallel to . In this frame, therequirementE = 0 implies that we haveU= 0 = Q, and hence we can write E = (/)4P . This caseis discussed in ref. 76. More importantly, in the case a na = 0, in which is a null vector, E can bewritten

E =

4 (14)

Hence, this case is formally equivalent to the energy-momentum tensor of a null fluid or an extremetilted perfect fluid. Using a covariant decomposition ofE , the nonlocal energy terms are given by

U= ( u )2, Q = ( u ), P = (15)The equations on the brane now close and the dynamical behaviour can be analyzed [77]. Note that

UP = QQ 13

gQQ (16)

so that in this case E is determined completely by U and Q.We can investigate the effect this type-N bulk may have on the cosmological evolution of the brane.

If we assume that we are in the regime of an isotropic past singularity and the cosmological evolutionis dominated by an isotropic perfect fluid with equation of state p = ( 1), the equations forU andQ are

U+ 43

U= 0, Q + 43

Q = 0 (17)

(see also ref. 79). For the isotropic singularity, the expansion factor is given by = 1/(t). Definingthe expansion-normalized nonlocal density, U U/2 and nonlocal energy flux, Q Q/2, weobtain

U = 23 t

(3 2)U, Q = 23 t

(3 2)Q (18)

Hence, the isotropic singularity is stable to the past with regards to these stresses if > 2/3. Similarly,we have a Friedmann Universe to the future with = 2/(t), and thus

U = 23 t

(3 4)U, Q = 23 t

(3 4)Q (19)

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This implies that if the isotropic fluid on the brane is less stiff than radiation at late times ( < 4/3),then the Friedmann Universe is stable to the future with respect to the nonlocal stresses.

These qualitative results arefurther supportedby a moredetaileddynamical systems analysis [28,82]

of the asymptotic behaviour of two tilting -law fluids in a class of Bianchi type VI0 models [77]. Inparticular, the physically relevant case of interest here, namely, that the second fluid is a null fluid ora fluid with extreme tilt was investigated. All of the equilibrium points were found and their stabilitydetermined, so that the local attractors were established. The effect a type-N bulk may have on thecosmological evolution of the brane was then investigated. It was found that if > 2/3, then the nullfluid is not dynamically important at early times; that is, the effects of the projected Weyl tensor will notaffect the dynamical behaviour close to the initial singularity. This supports the result that an isotropicsingularity is a local stable past attractor. Also, if < 4/3, the extreme tilting fluid is dynamicallynegligible to the future. Therefore, the null brane fluid is not dynamically important asymptotically atearly times for all values of of physical import, supporting the qualitative analysis above [77]. Inparticular, this implies that the effects ofE are not dynamically important (at least in this class ofmodels) in the asymptotic regime close to the singularity.

2.1.2. Bulk spacetimes

Mostworkonanisotropicbrane-worldUniversesusethe effective 4DEinsteinequationsonthebrane.Since the brane equations are not closed, various additional conditions on P have been proposed ina rather ad hoc way. There has been some work on determining the effects of a nonzero P on thebehaviour of brane-world cosmological models [66,8386]. For example, in ref. 87, a bulk metric with aKasner brane was presented. Since the Kasner metric is a solution of the 4D Einstein vacuum equations,the bulk metric is a simple warped extension; for the choice of bulk metric made in ref. 87 it was shownthat the brane can only be anisotropic if it contains a constant tension with = P! Other examplesinclude the Schwarzschild black-string solution [88,89], the spatially homogeneous Einstein space witha black hole [90] and the brane-wave spacetimes [76], but all of these result in Bianchi models with ananisotropic fluid source.

To have a fully consistent picture, we cannot avoid specifying the bulk geometry; i.e., we need

to construct explicit anisotropic brane cosmologies by solving the full 5D vacuum Einstein equationswith negative cosmological constant. This has nontrivial implications for the brane itself because, inbrane-world models, the Israel junction conditions (relating the extrinsic curvature to the matter onthe brane) must be satisfied. For instance, if the brane contains perfect-fluid matter, then the junctionconditions will impose constraints on the components of the extrinsic curvature.

In ref. 91 a moving brane admitting a spatially homogeneous but anisotropic Bianchi I 3D slicingin a static anisotropic bulk is considered. After solving the bulk Einstein equations, it is shown that thejunction conditions induce anisotropic stresses in the matter on the brane (i.e., the brane cannot containa perfect fluid). Indeed, from the bulk Einstein equations and the junction conditions, it follows that thisanisotropic stress can only vanish if the bulk is isotropic and hence the brane is isotropic. It is also notedthat the anisotropic stresses obtained on the brane in ref. 91 are fixed by the bulk metric. Some explicitBianchi I brane-world cosmological models using the freedom still available in embedding the 3-branewere constructed. In one example the cosmological models do not isotropize as the initial singularity

is approached, but in this example w 1 as t 0! (In other examples, the models do isotropize inthe past, but the matter content never behaves as a perfect fluid.)

This work is generalized in ref. 92 to a class of nonstatic bulks. The bulk is assumed to be empty butendowed with a negative cosmological constant and explicit analytic bulk solutions with anisotropic3D spatial slices are found. The Z2-symmetric branes were embedded in the anisotropic spacetimesand the constraints on the brane energy-momentum tensor due to the 5D anisotropic geometry arediscussed. The question of whether it is possible to find a bulk geometry and a brane with perfect-fluid matter consistent with the Israel junction conditions is studied. Einsteins equations are integrated

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explicitly in the cases in which the shear does not depend on the extra-dimension and in which themetric is separable. Remarkably, in the separable bulk solution, the analysis implies that the branecan support perfect-fluid-type matterfor certain parameter values. In a particular example, by solving

for the cosmological evolution on the brane it is found that as , converges towards the RStension and theeffective pressure is negative (but converges towards zero morerapidly than the effectiveenergy). Therefore, unlike ref. 91, it is found [92] that it is possible to find a perfect-fluid anisotropicbrane in some such bulk geometries, but again there is no flexibility in the choice of the perfect fluid incontrast with the isotropic case (i.e., only a very particular type of perfect fluid is compatible with thegiven bulk geometry).

Therefore, it doesseemdifficult to constructexamples of anisotropic branes withperfect-fluidmatter.However, these solutions [91, 92] are very special examples and say nothing about the general case. Anexact solution in the bulk gives rise to a precise relationship between the density and the pressure onthe brane. But exact closed form solutions are few and sparse; indeed, even in 4D GR there are onlya few very special exact Bianchi solutions (models are usually only specified up to a set of ordinarydifferential equations (ODE), so this approach will not shed light on the general case. Moreover, theisotropic singularity conjecture is not contradicted by these results. First, it is not necessary for P

to be zero but rather that it be dynamically negligible (i.e., some appropriate normalized quantity isnegligible) as the singularity is approached. Second, the values for the parameter is in the wrongrange in the examples of refs. 91 and 92 (the isotropic singularity theorems only apply for 4/3).

To illustrate this point further, let us consider the following Bianchi I metric:

ds2 = t2(1)

1 + t2

exp

2( 1)

2

dt2 + t4/3

1 + t2

exp

2(1 + 2cos )

3( 2)

d2

+ t4/3

1 + t2

exp

2(1 cos

3sin )

3( 2)

dy2

+ t4/3

1 + t2

exp

2(1 cos +

3sin )

3( 2)

dz2 (20)

where and are constants and > 2. For = 0, the metric is the Bintruy, Deffayet, and Langloisor BRW (see later) solution with a(T) = T1/3 [2931] (where T t1/). We note that for the metric(20), G = diag (A , B , B , B ), where B = ( 1)A and

A A0t2

1 + t2 /2

We shall show that a (anisotropic) Bianchi I perfect-fluid RS brane can be embedded in a 5D Einsteinspace using the CampbellMagaard theorem. Following Dahlia and Romero [93], we consider the 5Dmetric in Gaussian normal form (with 5th coordinate ). If we wish to embed a RS brane into a Z2symmetric bulk (with cosmologial constant ), we need to consider the equations (on the brane = 0,a constant).

T

; = + ( + p) = 0 (21)2

4

T T

13

T2

= 2

6

2 + 3p + 3p 22

= 2 + R (22)

where T is the energy-momentum tensor for a perfect fluid and R is the scalar curvature of thehypersurface = 0. If a solution of (21) and (22) can be found, then an analytic solution of the 5Dbulk equations G = g (satisfying (21) and (22) on the brane) is guaranteed by the CauchyKowalewski theorem [93].

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By a direct calculation, for metric (20) we obtain

R

=R0t

2 1 + t2 /2

(23)

= 0t

1 + t21/2

1 + 12

t2

(24)

With these expressions, (21) and (22) can always be solved to find = (t),p = p(t). To lowest orderin t, we have that R0 = R0t2(1 +O(t)), = 0t(1 +O(t)), and we can solve (21) and (22) toobtain = = 0t (1 +O(t)) and p = ( 1)(1 +O(t)). That is, to lowest order we obtain theBRW solution.

Therefore, using extended CampbellMagaard theory [93], we have demonstrated that there areexact anisotropic (of Bianchi type I) perfect-fluid brane cosmological models that can be embeddedinto a 5D bulk Einstein space (see also ref. 94). This construction does not, of course, guarenty that thebulk is regular. Note that for the example (20) constructed, as t 0 the model isotropizes (towards theBRW solution) and the singularity is isotropic.

2.1.3. Dynamical systems approach

To illustrate the possible effects ofP on the dynamics of a brane-world cosmological model, analternative approach is taken in ref. 95. Again it is assumed that the effective cosmological constant ona Bianchi I brane is zero, and the matter on the brane is of the form of a nontilted perfect fluid. Utilizingnew dependent dynamical variables, the effective Einstein field equations yield a dynamical systemdescribing the evolution of the Bianchi I brane world. In this analysis there is no evolution equation forthe quantity

P

3

2

P

H2(25)

modelling the nonlocal bulk corrections. To obtain a closed system of equations, it is then assumed that

P is a differentiable function of the remaining dynamical variables; that is, P = P(,,, U).With this assumption, the resulting brane-world spacetimes represented by the equilibrium points of thesystem are geometrically self-similar [96].

A number of models were investigated. In general, the dynamics in the case P= P0 (a constant) isquite different from that found currently in the literature. ForP= 1 (P0 = 0), none of the equilibriumpoints are isotropic, and all have some form of dark matter with U = 0. It appears that the nonlocalbulk corrections have a significant impact on the dynamical behaviour and the initial singularity is notisotropic. In the case P = P ( > 0), the BRW brane-world solution Fb may not represent thepast asymptotic state (there are parameter values of P such that the initial singularity is representedby a Kasner-like solution). In the case P= PUU ( > 0) [84,85], Fb is always a source, althoughthere will exist cosmological models that will asymptote to thepast towards anisotropic bulk-dominatedKasner-like models (i.e., the initial singularity is not necessarily isotropic). Thus, in each of these threecases,the effects of thenonlocalP couldbe significant. Unlike the standardcosmologicalbrane-world

situation in whichFb represents the generic initial behaviour in brane-world cosmologies containingordinary matter, it was found that Kasner-like bulk-dominated models can also act as sources and thatFb does not even exist as an equilibrium state in one of the cases studied. The future behaviour of themodels is determined primarily by the parameter , and is studied in ref. 95.

We note that, from (25),P is not defined in the (isotropic limit) as 0. This is problematic whenattempting to analyse isotropic equilibrium points; indeed, the isotropic limit cannot be studied and theanalysis in ref. 95 is essentially concerned with the local stability of nonisotropic equilibrium points.However, the DEs may be well defined and have an isotropic limit depending on the chosen Ansatz for

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P. In the second and third cases, a well-defined isotropic limit exists and in these cases Fb is a localattractor. In the first case (P constant), there is no isotropic limit possible. Indeed, the models do noteven allow for an isotropic solution at any time (the BRW models do not exist), and any conclusions

concerning an isotopic singularity are inappropriate in this case. Consequently the analysis in ref. 95supports the existence of a local isotropic singularity.

3. Dynamical equations

The general form of the brane energy-momentum tensor for any matter fields of perfect-fluid form(including scalar fields) can be covariantly decomposed. We adopt the notation of refs. 25 and 26.Angled brackets denote the projected, symmetric and tracefree part

V = h V , W =

h( h)

13

hh

W (26)

with round brackets denoting symmetrization. An overdot denotes u (i.e., timelike directionalderivative),

= u is the volume expansion rate of the u congruence, A

= u

=A

is

its 4-acceleration, = Du is its shear rate, and = 12 curl u = is its vorticity rate. Thecovariant spatial curl is given by [25,26]

curl V = D V , curl W = (D W ) (27)

where is the projection orthogonal to u of the brane-alternating tensor, and D is the projectedpart of the brane covariant derivative (i.e., spatial derivative), defined by

DF =

F = h h h F (28)The remaining covariant equations on the brane are the propagation and constraint equations for thekinematic quantities , A, , , and for the nonlocal gravitational field on the brane. The nonlocalgravitational field on the brane is given by the brane Weyl tensor C . This splits into the gravito-

electric and gravito-magnetic fields on the brane

E = C u u = E, H = 12

C

u = H (29)

The Ricci identity for u and the Bianchi identities C = [(R] + 16 Rg] ) give rise tothe evolution and constraint equations governing the above covariant quantities once the Ricci tensorR is replaced by the effective total energy-momentum tensor from the field equations [9799]. In thefollowing, R is the Ricci tensor for 3-surfaces orthogonal to u on the brane and R = h R .Note that in the GR limit, 1 0, we have that E 0.

3.1. Vorticity-free models with a linear barotropic equation of state

In the case of a perfect fluid with an equation of state p=

(

1), we obtain the followingequations when the vorticity is zero ( = 0) [25,26]:

A = ( 1)

1

D (30)

= (31)

U+ 43

U+ DQ + 2AQ + P = 0 (32)

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Q + 43

Q + 13

DU+ 43UA + DP + AP + Q = 1

64D (33)

+1

3 2

+

D

A + AA

+1

2 2

(3 2) = 2

2 (3 1)2

6

2U (34)

+ 23

+ E DA + AA = 32

P (35)

E + E curl H + 2

2 2A (H) 3E

= 2

22 1

2

4U + 3P + P + 3DQ + 6AQ + 3P

(36)

H + H + curl E 3H + 2A (E) = 32

curlP + ()Q

(37)

D

2

3 D = 6

2Q (38)curl H = 0 (39)

D E 13

2D [, H] = 2

3D + 1

2

2DU 2Q 3DP + 3Q

(40)

D H + [, E] = 12

3curlQ + 3[,P]

(41)

and the GaussCodazzi equations on the brane

R +1

3 E = 3

2P (42)

R +

2

32

22

2

=2

2

+12

2U (43)

3.2. Integrability conditions for P = 0, Q = 0

When P = 0 and Q = 0, these equations simplify. Using (30), (33) becomes for = 1

DU=

4( 1)

U

4

2

D (44)

Taking the directional time derivative of (44), and using the relations for interchanging space and timederivatives, e.g.,

h[D f] = h [ + A ]f

1

3h

+

D f (45)

we obtain the integrability condition4

3(3 4)U

42

22

D =

4

3(3 4)( 1)U+

4

6(3 1)2

D (46)

Taking the directional time derivative of (38) and interchanging space and time derivatives, we obtain

( 1)

[12U+ 42] 1

D = 0 (47)

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Hence, for = 1 and [12U+ 42] = 0, we haveD = 0

so that

A = DU= D = D = D2 = 0and hence, in general, the brane is spatially homogeneous. The special case = 23 , U+ 42 = 0, inwhich the integrability conditions yield no constraints, corresponds to the case in which there are nocorrections to the general relativistic equations. Finally, in the special case = 1, in which p = 0 andA = 0, we obtain analogues of (44) and (46) from which it follows that = 0, and hence = 0.

3.2.1. Spatially homogeneous branes

In the case of spatial homogeneity we have that

= 0, A = 0, D = D = Dp = DU= 0

With a general equation of state, we then obtain

+ ( + p) = 0 (48)

U+ 43

U+ DQ + P = 0 (49)

Q + 43

Q + DP + Q = 0 (50)

+ 13

2 + + 12

2( + 3p) = 2

2(2 + 3p) 6

2U (51)

+ 23

+ E + = 32

P (52)

E + E curl H + 12

2( + p) 3E

= 12

2( + p)

12

4U + 3P + P + 3DQ + 3P

(53)

H + H + curl E 3H = 32

curlP + ()Q

(54)

D = 62

Q (55)

curl H = 0 (56)

D E [, H] = 12

2Q + 3DP 3Q

(57)

D H + [, E] = 12

3curlQ + 3[,P] (58)The GaussCodazzi equations on the brane are

R + + =6

2P (59)

R + 23

2 22 2 = 2

2 + 12

2U (60)

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3.2.1.1. RW brane

It follows from the symmetries that for a RW brane Q = 0 = P . The generalized Friedmannequation (60) on a spatially homogeneous and isotropic brane is [2931]

H2 = 13

2

1 + 2

+ 1

3 k

a2+ 2Uo

2

aoa

4(61)

where k = 0, 1. The nonlocal term that arises from bulk Coulomb effects (also called the darkradiation term) is strongly limited by conventional nucleosynthesis [74, 75] (U/)nucl/2 < 0.005,where U = Uo(ao/a)4. A more stringent constraint comes from small-scale gravity experiments [79].This implies that the unconventional evolution in the high-energy regime ends, resulting in a transitionto the conventional regime, and the nonlocal term is sub-dominant during the radiation era and rapidlybecomes negligible thereafter. The generalized Raychaudhuri equation (51) (with = 0 = U) reducesto

H + H2 = 16

2

+ 3p + (2 + 3p)

(62)

The bulk metric for a flat Friedmann brane is given explicitly in refs. 2931. In natural static andspherically symmetric coordinates, the bulk metric is SchwarzchildAdS (with the Z2-symmetric RWbraneat theboundary) andis given inrefs. 32,33. TheRW brane movesradially alongthefifthdimension,with R = a(T ), where a is the RW scale factor, and the junction conditions determine the velocity viathe Friedmann equation for a. Thus, one can interpret the expansion of the Universe as motion of thebrane through the static bulk. In Gaussian normal coordinates, the brane is fixed but the bulk metric isnot manifestly static. The bulk black hole gives rise to dark radiationon thebrane via its Coulomb effect.The mass parameter of the black hole in the bulk is proportional toUo. To avoid a naked singularity, weassume that the black hole mass is non-negative.

In general, U = 0 in the Friedmann background [29,30]. Thus, for a perturbed Friedmann model,the nonlocal bulk effects are covariantly and gauge invariantly described by the first-order quantitiesDU, Q, P .

3.2.2. Static branes

Assuming spatial homogeneity (so that A = 0) with Q = P = 0, in the isotropic curvaturesubcase with R = 0, so that R = ka2 and

R = 23

R (63)

we have that

= (64)

U= 43

U (65)

+ 13

2 + 22 + 12

2(3 2) = 2

2(3 1)2 6

2U (66)

R + 23

2 22 22 2 = 2

2 + 12

2U (67)

2 = 22 (68)and the system of equations is closed.

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In the static models, we have that = 0 (and we shall assume that = 0), and1

2

2(3

2)0

= 2

2(3

1)20

6

2

U0 (69)

R0 + 220 2 =2

20 +

12

2U0 (70)

where 0, U0, and R0 are constants. These constitute two algebraic equations for four constants (and, which need not be equal to 2/3). Since U0 need not be positive, R0 need not be positive definite.There are static RW branes that can be embedded in a bulk that is not SchwarzschildAdS [100].

3.2.3. The EGS theorem in brane-world models

The isotropy of the cosmic microwave background (CMB) radiation has crucial implications forthe spatial homogeneity of the Universe. If all fundamental observers after last scattering observe anisotropic CMB, then it follows from a theorem of Ehlers, Geren, and Sachs (hereafter EGS) [101]

that in GR the Universe must have a RW geometry. More precisely, the EGS theorem states that if allobservers in a dust Universe see an isotropic radiation field, which is implicitly identified with theCMB, then that spacetime is spatially homogeneous and isotropic. This can trivially be generalizedto the case of a geodesic and barotropic perfect fluid [102, 103]. However, as has been emphasizedrecently [104,105], theresultingspacetimewill be RW only if themattercontent is that of a perfect-fluidform and the observers aregeodesic andirrotational. The EGS theorem has recently been investigated ininhomogeneous Universes models with non-geodesic observers, and inhomogeneous spacetimes havebeen found, which also allow every observer to see an isotropic CMB [104,105].

The EGS theorem is based on the collisionless Boltzmann equation and on the dynamical fieldequations. Bulk effects in brane-world models do not change the Boltzmann equation, but they dochange the dynamical field equations [2325, 33, 106], so that, in principle, the resulting geometryneed not be RW on the brane. The theorem is concerned with Universes in which all observers see anisotropic radiation field. It follows from the multipole expansions of the EinsteinBoltzman equations

for photons in a curved spacetime that even in the brane-world scenario a spacetime with an isotropicradiation field must have a velocity field of the photons that is shear-free and obeys [102,103]

A = Dq, = 3q (71)

where q is a function of the energy density of the radiation field. Any observers traveling on thiscongruence will then observe the isotropic radiation. This velocity field is also a conformal Killingvector of thespacetime, andhencea spacetimeadmitting an isotropicradiation field must be conformallystationary, in which case the metric can be given in local coordinates by

ds2 = e2q(t,x)dt2 + h dx dx

(72)

where h (x) can be diagonalised. Ifq = q(t), then the acceleration is zero.

Therefore, assuming (71) and = 0 (as well as = 0), we immediately have that2

3D = 6

2Q (73)

H = 0 (74)From (30) and (31), we then obtain (assuming p = ( 1))

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which implies that (for = 43 )

D = 0 (76)

(When = 43 , we obtain = 0q4). Consequently, from (73) we have that

Q = 0 (77)

From (75), we have that D(q) = 0 and [D q] = 0, which then implies that A = 0. This can bestbe seen by using local coordinates (72), in which these conditions yield q(t,x ) = f(t) + g1(x ) andeq = ef( t ) +g2(x ), so that for consistency either f(t) = 0 (whence = 0) or q = q(t), and henceA = 0. Therefore, we find that

A = 0 (78)

and hence D = 0.Taking H = 0, Q = 0, D = 0, D = 0, and A = 0, which implies that

E = 32

P (79)

Equations (33) and (40) then yield

DU= 0, DP = 0, D E = 0 (80)and (36) and (42) become

E + 23

E = 0, R = 2E (81)

For the metric (72) with q = q(t), it then follows that

R = R(x

) (82)

where R is the Ricci tensor of the 3-metric h(x) [107]. Consequently,

t

R

= 0 (83)

and hence t[E] = 0 and t[P ] = 0. The remaining nontrivial equations are the conservation lawfor U, the generalized Raychaudhuri equation, and the generalized Friedmann equation.

3.3. Discussion

The 5D metric in the bulk is given in Gaussian normal coordinates by

ds2

=gAB dxA dxB = gab (t,x, y) ddx a dxb + dy2 (84)where A, B = 0, 1 3, 4 = y and a, b = 0, 1 3, the brane is located at y = 0 and the unit normalto the brane is nA = Ay . Assuming that the metric functions gab are smooth, they can be expandedin powers ofy close to the brane up to O(y 2). The metric functions are determined up to O(y 0) by(72). The extrinsic curvature on the brane, K+AB , which is discontinuous at y = 0, restricts the formof gab (t,x, y) to O(y). Assuming that there are no 5D fluxes the stress-energy of the bulk, TAB ,in the neighbourhood of the brane satisfies T0 = 0, which is valid for scalar fields, perfect fluids, acosmological constant, and combinations thereof; and we have that g0 = 0 (which then implies that

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Q = 0). The nontrivial components ofgab up to O(y 2) can then be written (with a slight change ofnotation) in the y > 0 (+-region) as

g00 = (1 + f(t)y + F ( t , x

)y2

) (85)

g = Q(t)(1 + q(t)y)h + G(t,x)y2 (86)Noting that Eab = CaCbD nC nD = Ca5b5, by a direct calculation, we find that on the brane (y = 0)

C0505 = 14Q(t)

F(t) + 112Q(t)

(6Q(t)F(t, x) 3 R(x ) 2G(t,x)) (87)

C55 = 13

3

R(x) + 1

12(3R(x ) + F(t)

+ 2Q(t)F(t, x) + 2G(t,x))h(x) 23

G(t,x) (88)

where

F(t) Qtt 1Q

(Qt)2 1

2Qf q + Qq2 1

2Qf2 (89)

G(t,x) hG (90)and C505 = 0 (so that Q = 0).

From E = R(x) and E00 = 62U(t), we then find that

G(t,x) = R(x) +

1

3(G(t, x) R(x))h(x) (91)

and

G(t,x) = 32F(t) + 36Q(t)

2U(t) + 3Q(t)F(t,x) 1

2R(x) (92)

Finally, for a bulk source consisting of a combination of scalar fields, perfect fluids, and a cosmo-logical constant, we then also have that

F ( t , x) = F (t), G(t, x) = G(t) (93)

and it follows that R(x) is constant.We can solve the full equations in the bulk iteratively. In general, it seems that the remaining bulk

field equations consist of five ODEs for the five nontrivial functions (oft) in the metric and additionalmatter fields and so there will be many possible solutions. It may be possible to integrate the bulk

equations explicitly in the case of a 5D cosmological constant.

4. Essential features

The generalized Friedmann equation, which determines the volume expansion of the Universe orthe Hubble function H = 13 , in the case of spatially homogeneous cosmological models is

H2 = 13

2

1 + 2

1

6R + 1

32 + 1

3 + 2U

2(94)

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where 22 ab ab is the shear scalar and H = a/a [2325], and the background-induced metricsatisfies

DU= Q = P = 0 (95)For a flat isotropic brane (with = 0 = U) we obtain

H2 = 2

3

1 + 2

(96)

The conservation equation is given by

+ 3H ( + p) = 0 (97)

For a minimally coupled scalar field, the energy density and pressure are, respectively,

=1

2 2

+ V (), p =1

2 2

V() (98)so that the conservation law is equivalent to the KleinGordon equation

+ 3H + V() = 0 (99)

At early times, assuming that the energy density is dominated by a scalar field and that is very large(andU= 0), the conservation equation (97) becomes

= 32 (100)

so that is monotonically decreasing and the models will eventually evolve to the low-density regimeand, hence, the usual general relativitistic scenario.

4.1. Anisotropic and curved branes

There are many reasons to consider cosmological models that are more general than RW, bothspatially homogeneous and anisotropic and spatially inhomogeneous. The 3-curvature in RW models isgiven by R = 6k/a2, where k = 0, 1 is thecurvature constant. The structure of the initial singularityin RW brane models is studied in ref. 108. An equivalent 3-curvature occurs in orthogonal spatiallyhomogeneous and isotropic curvature models, and a similar term occurs in other models such as BianchiV cosmological models. For a RW model on the brane, (95) follows, which implies that U= U(t). Inthe case of Bianchi type I, we get Q = 0 but we do not get any restriction on P . Since there is noway of fixing the dynamics of this tensor we can study the particular case in which it is zero. Thus, inRW and Bianchi I models the evolution equation for U is U = 4HU [25, 26], which integrates toU = U0/a4, which has the structure of a radiation fluid (sometimes referred to as the dark radiationterm, but where U0 can be negative).

A Bianchi I brane is covariantly characterized in ref. 66. The conservation equations reduce tothe conservation equation (97), an evolution equation for the effective nonlocal energy density on thebrane U, and a differential constraint on the effective nonlocal anisotropic stress P . In the BianchiI case, the presence of the nonlocal bulk tensor P in the GaussCodazzi equations on the branemeans that we cannot simply integrate to find the shear as in GR. However, when the nonlocal energydensity vanishes or is negligible, i.e., U = 0, then the conservation equations imply P = 0.This consistency condition implies P = 6PP /2. Since there is no evolution equation forP on the brane [25,26], this is consistent on the brane. This assumption is often made in the case of

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RW branes, and in that case it leads to a conformally flat bulk geometry [2931]. The shear evolutionequation may be integrated after contracting it with the shear, to give

= 620

a6(101)

where 0 is constant. Bianchi I models on the brane have been studied with a massive scalar field [66],with a perfect fluid with a linear equation of state [62,63,86], and a perfect fluid and scalar field [109].A similar shear term occurs in Bianchi cosmological models (such as Bianchi type V models) in whichthe hypersurfaces of homogeneity are orthogonal to the fluid velocity.

The conservation equation is given by (97). If we assume that the matter content is equivalent tothat of a nontilting perfect fluid with a linear barotropic equation of state (i.e., p = ( 1), where 0, and [0, 2]), the conservation equation then yields = 0a3, where 0 > 0. A dynamicalanalysis of scalar-field models indicates that at early times the scalar field is effectively massless. Amassless scalar field is equivalent to such a perfect fluid with a stiff equation-of-state parameter = 2.

We can, therefore, write down a phenomenological (spatially homogeneous) generalized Friedmann

equation

H2 = 20

3a3+

220

6a6 k

a2+ 1

3 +

20

a6+ U0

a4(102)

In many applications the 4D cosmological constant is assumed to be zero [8,9]; here we shall assumethat if it is nonzero it is positive; i.e., 0.

In particular, the cosmological evolution of the RS brane-world scenarios in RW and the Bianchi Iand V perfect-fluid models with a linear barotropic equation of state withU= 0 were studied in refs. 62and 63 using dynamical systems techniques. This work was generalized in ref. 63, in which the 5DWeyl tensor has a nonvanishing projection onto the three-brane where matter fields are confined, i.e.,U= 0; in the case of RW models the study was completely general whereas in the Bianchi type I casethe Weyl tensor components were neglected (the Bianchi V models were not considered).

Theparticularmodels considered sofar, inwhichthecurvatureandshearare given by theexpressionsin the above phenomenological equation, are very special. In particular, the Bianchi I and V modelsare not generic, and so the study of the dynamics of these models does not shed light on the typicalbehaviour of spatially homogeneous brane models. Clearly, more general scenarios must be consideredto obtain physical insights, and we shall study Bianchi type IX models in detail later.

4.1.1. Late times

Recent observations of distant supernovae and galaxy clusters seem to suggest that our Universeis presently undergoing a phase of accelerated expansion [110113], indicating the dominance of darkenergy with negative pressure in our present Universe. The idea that a slowlyrolling scalar field providesthe dominant contribution to the present energy density has gained prominence in recent times [114116]. In ref. 117, a potentialconsisting of a general combination of twoexponential terms wasconsideredwithin the context of brane cosmology to determine whether it is possible to obtain both early time

inflation and accelerated expansion during the present epoch through the dynamics of the same scalarfield. Such a potential, motivated by phenomenological considerations from scalar tensor theories ofgravity andhigher dimensional quantum effects or a cosmological constant together with a 4D potential,has been recently claimed to conform to all the current observational constraints on quintessence (forcertain values of the parameters) [118]. The brane-world dynamics was assumed to be governed bythe modified Friedmann equation (96) where, for reasons of simplicity, the contributions from bulkgravitons and a higher dimensional cosmological constant were set to zero [53,65,119]. It was foundthat the brane-world inflationary scenario is feasible with steep potentials (in contrast to the situation in

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standard cosmology). During inflation, the necessity of sufficient inflation and the COBE normalizedamplitude of density perturbations can be used to fix the values of the brane tension and the scale ofthe potential at this stage. The emergence of a second phase of accelerated expansion necessitates the

introduction of a second exponential term in the potential. A successful brane quintessence model witha scalar field (and a power-law potential) and a dark radiation component (U0 = 0) was discussed inrefs. [3436], with particular emphasis on how the quadratic matter term affects the evolution of thescalar field.

When a positive cosmological constant is present ( 0), the de Sitter model is always the globalattractor for U 0. For U < 0, models can (re)collapse (even without a positive curvature); i.e., thereexist (re)collapsing models both for RW and Bianchi type I models for any values of . This meansthat for U < 0 the de Sitter model is only a local attractor [62, 63] (and the cosmic no-hair theorem isconsequently violated). In ref. 64, a set of sufficient conditions, which must be satisfied by the branematter and bulk metric so that a homogeneous and anisotropic brane asymptotically evolves to a deSitter spacetime in the presence of a positive cosmological constant on the brane, is derived. It is shownthat from violations of these sufficient conditions, a negative nonlocal energy density or the presence ofstrong anisotropic stress (i.e., a magnetic field) may lead the brane to collapse. ForU

0 and positive

curvature (as in the k = 1 RW models), not only can models recollapse, but there exist oscillatingUniverses in which the physical variables oscillate periodically between a minimum and a maximumvalue without reaching any spacelike singularity [63]. Indeed, when there are bulk effects, such as, forexample, in case of a negative contribution of the dark-energy density termU, the singularity theoremsare violated and a singularity can be avoided.

4.1.2. Intermediate times: inflation

Thequalitative properties of Bianchi I brane modelswith a perfect fluid and a linear equation of state(U= 0 and U = 0) are studied in refs. 62 and 63. A variety of intermediate behaviours can occur, andthe various bifurcations in these models are discussed in some detail. In addition, the maximum valueof the shear in Bianchi I models is studied in ref. 86. In particular, models with a positive curvature canrecollapse, although thecondition forrecollapse in thebraneworld canbe differentto that in GR [62,63].

Recent measurements of the power spectrum of the CMB anisotropy [120122] provide strongjustificationfor inflation[123125].Sincethe Friedmannequation is modifiedby an extra termquadraticin energy density at high energies, it is of interest to study the implications of such a modificationon the inflationary paradigm. The issue of inflation on the brane was first investigated in ref. 53, whereit was shown that on a RW brane in 5D anti de Sitter space, extra-dimensional effects are conduciveto the advent of inflation. It has also been realized that the brane-world scenario is more suitable forinflation with steep potentials because the quadratic term in increases friction in the inflaton fieldequation [53]. This feature has been exploited to construct inflationary models using both large inversepower law [119] as well as steep exponential [65] potentials for the scalar field.

In particular, for a RW brane inflation at high energies ( > ) proceeds at a higher rate thanthe corresponding rate in GR. As noted earlier, this introduces important changes to the dynamics ofthe early Universe, and accounts for an increase in the amplitude of scalar [53] and tensor [74, 75]fluctuations at Hubble crossing, and for a change to the evolution of large-scale density perturbations

during inflation [78]. The condition that radiation domination sets in before nucleosynthesis can be usedto impose constraints on theparameters of brane inflation models[65,119].A definite prediction of thesemodels is the parameter independence of the spectral index of scalar density perturbations [53,65,119].The behaviour of an anisotropic brane world in the presence of inflationary scalar fields is examinedin ref. 66. The evolution equations on the brane (with 0 = 0) for a minimally coupled massive scalarfield with V = 12 m22 is studied and it is shown that, contrary to expectations, a large anisotropy doesnot adversely affect inflation [66]. In fact, a large initial anisotropy introduces more damping into thescalar-field equation of motion, resulting in greater inflation. After the kinetic term and the anisotropy

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have dropped to a very small value, a transient anisotropy-dominated regime begins and the anisotropythenrises, graduallyapproaching its asymptotic slow-roll value for the simple chaoticpotential. Thekinetic energy does not remain constant but gradually increases as the field amplitude decreases during

slow-roll. Since the decay of anisotropy is found to be generically accompanied by a correspondingdecrease in the kinetic energy of the scalar field [66], this effect leads to greater inflation. Subsequently,the anisotropy dissipates.

In RS brane worlds, where the bulk has only a vacuum energy, inflation on the brane must bedriven by a 4D scalar field trapped on the brane [126]. In more general brane worlds, where the bulkcontains a 5D scalar field, it is possible that the 5D field induces inflation on the brane via its effectiveprojection [42, 127136]. More exotic possibilities arise from the interaction between two branes,including possible collisions, which is mediated by a 5D scalar field and which can induce eitherinflation [137,138] or a hot big-bang radiation era, as in the ekpyrotic or cyclic scenario [139141],or colliding bubble scenarios [142144] (for colliding branes in an M theory approach see refs. 145 and146). In general, high-energy brane-world modifications to the dynamics of inflation on the brane havebeen investigated [48, 69, 147151]. Essentially, the high-energy corrections provide increased Hubbledamping making slow-roll inflation possible even for potentials that would be too steep in standard

cosmology [65,119, 152156].

4.1.3. Early times: initial singularity

An important question is how thehigher dimensional bulk effects modify the picture of gravitationalcollapse and singularities. The generalized Raychaudhuri equation governs gravitational collapse andthe initial singularity on the brane. The local energy density and pressure corrections, 1124(2 + 3p),further enhance the tendency to collapse [53] if 2 + 3p > 0 (which is satisfied in thermal collapse).The nonlocal term, which is proportional toU, can act either way depending on its sign; the effect of anegativeU is to counteract gravitational collapse, and hence the singularity can be avoided in this case.

A unique feature of brane cosmology is that the effective equation of state at high densities canbecome ultra stiff. Consequently, matter can overwhelm shear for equations of state that are stifferthan dust, leading to quasi-isotropic early expansion of the Universe. For example, in Bianchi I models

[62, 63, 66], the approach to the initial singularity is matter-dominated and not shear-dominated, dueto the predominance of the matter term 2/22 relative to the shear term 20 /a6 (and 2/H2 0 as

t 0). The fact that the density effectively grows faster than 1/a6 for > 1 is a uniquely brane effect(i.e., it is not possible in GR).

Indeed, a spatially homogeneous and isotropic non-general-relativistic brane-world (without branetension; U = 0) solution, first discussed in refs. 2931, is always a source or repeller for 1 inthe presence of nonzero shear. This solution is self-similar, and is sometimes referred to as the brane-RobertsonWalker model (BRW) [22]. The equilibrium point corresponding to the BRW model is oftendenoted by Fb. The BRW model, in which

a(t) t1/3 (103)is valid at very high energies ( ) as the initial singularity is approached (t 0). In thebrane-worldscenario anisotropy dominates only for < 1 (whereas in GR it dominates for < 2), and, therefore,

for all physically relevant values of the singularity is isotropic.Iftot > 0 and tot + 3ptot > 0 for all t < t0, then from the generalized Raychaudhuri equation

(51) and using the generalized Friedmann equation (60) and the conservation equations, it follows thatfor a0 > 0 (where a0 a(t0)) there exists a time tb with tb < t0 such that a(tb) = 0, and there exists asingularity at tb, where we can rescale time so that tb = 0 and the singularity occurs at the origin. Wecan find the precise constraints on andU0 in terms ofa0 for these conditions to be satisfied at t = t0. Itfollows that if these conditions aresatisfied at t = t0 theyaresatisfiedforall0 < t < t0,andasingularitynecessarily results. These conditions are indeed satisfied for regular matter undergoing thermal collapse

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in which the the local energy density and pressure satisfy (2 + 3p) > 0 (and is certainly satisfiedfor perfect-fluid matter satisfying the weak-energy condition 0 and a linear barotropic equation ofstate with

1). On the other hand, it is known that a large positive cosmological constant or a

significant negative nonlocal term U counteracts gravitational collapse and can lead to the singularitybeing avoided in exceptional circumstances.

In more detail, we shall see later that since the dimensionless variables are bounded close to thesingularity, it follows from (175) that 0 < q < 2, where q is the deceleration parameter. Hence, from(176), H diverges as the initial singularity is approached. At an equilibrium point q = q, where qis a constant with 0 < q < 2, so that from (176), we have that H (1 + q)1t1 as t 0+( ). From (175), (177), and the conservation laws it then follows that as t 0+. Itthen follows directly from the conservation laws (180) that b = 22/6H2 dominates as t 0+(and that all of the other i are negligable dynamically as the singularity is approached). The fact thatthe effective equation of state at high densities becomes ultra stiff, so that the matter can dominate theshear dynamically, is a unique feature of brane cosmology. Hence, close to the singularity the mattercontribution is effectively given by tot = 2/2 b, ptot = (2 + 2p)/2 = (2 1)b.Therefore, as the initial singularity is approached, the model is approximated by an RW model in some

appropriately defined mathematical sense. Goode et al. [157,158] introduced the rigorous mathematicalconcept of an isotropic singularity into cosmology to define precisely the notion of a Friedmann-likesingularity. Although a number of perfect-fluid cosmologies are known to admit an isotropic singularity,in GR a cosmological model will not admit an isotropic singularity in general.

5. Bianchi IX brane-world cosmologies

The brane energy-momentum tensor, including both a perfect fluid and a minimally coupled scalarfield, is given by

T = Tpf + Tsf (104)where

Tpf = uu + ph (105)

Tsf = ;; g

1

2; ; + V()

(106)

and u is the fluid 4-velocity and is the minimally coupled scalar field having potential V(). If; istimelike, then a scalar field with potential V() is equivalent to a perfect fluid having an energy densityand pressure

sf = 12

;; + V() (107)

psf = 12

;; V() (108)

The local matter corrections S to the Einstein equations on the brane are given by (3), which isequivalent to

Spf = 1

122uu + 1

12 ( + 2p) h (109)

for a perfect fluid and

Ssf =1

6

1

2; ; + V()

;; +

1

12

1

2; ; + V()

3

2; ; V()

g (110)

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for a minimally coupled scalar field. If we have both a perfect fluid and a scalar field and we assume thatthe gradient of the scalar field, ; , is aligned with the fluid 4-velocity, u (that is, ;/

; ; =

u), the local brane effects due to a combination of a perfect fluid and a scalar field are then

S = 112

1

2; ; + V()

2uu

+ 112

1

2; ; + V

+ 2p 3

2; ; V()

h (111)

The nonlocal effects from the free gravitational field in the bulk are given by E [25, 26]. Wewill assume here that DU = Q = P = 0. Since P = 0, in this case the evolution ofE isfully determined. In general, U= U(t) = 0 (and can be negative) [2931]. All of the bulk correctionsmentioned above may be consolidated into an effective total energy density and pressure as follows. Themodified Einstein equations take the standard Einstein form with a redefined energy-momentum tensor(assuming = 0) according to (4) and (5), where the redefined total-energy density and pressure dueto both a perfect fluid and a scalar field are given by

total = +

12

; ; + V()

+ 46

4

12

1

2; ; + V()

2+U

(112)

ptotal = p +

12

; ; V()

+ 46

4

12

1

2; ; + V()

+ 2p 3

2; a ; a V()

+ 1

3U

(113)

As a consequence of the form of the bulk energy-momentum tensor and ofZ2 symmetry, it follows[23, 24] that the brane energy-momentum tensor separately satisfies the conservation equations (where

we assume that the scalar field and the matter are noninteracting); i.e., + 3H ( + p) = 0 (114)

+ 3H + V

= 0 (115)

whence the Bianchi identities on the brane imply that the projected Weyl tensor obeys the constraint

U+ 4HU= 0 (116)

5.1. Bianchi IX models: setting up the dynamical system

The source term (restricted to the brane) is a noninteracting mixture of nontilting perfect-fluidordinary matter with p = ( 1), and a minimally coupled scalar field with an exponential potentialoftheform V()

=V

0ek (where

=(t)) [159162].Thevariablesarethesameasthoseintroduced

in ref. 28, with the addition of

= 2

3D2, =

3D2, =

2V

3D2, =

3

2

3D, U =

U

3D2(117)

where

D

H2 + 14

(n1n2n3)2/3

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The total equivalent dimensionless energy density due to all sources and bulk corrections is

total

2total

3D2 =

+

+

+ 2

+c2

U

+c2

4

D2 + + 22

(118)

where c2 4/4.The governing differential equations for X [D, H , 1, 2, N1, N2, N3, , , , , U] are

as follows [163]:

D = (1 + q)H D (119)

H = q(H2 1) (120)+ = (q 2)H+ S+ (121) = (q 2)H S (122)N1

=N1(H

q

4+) (123)

N2 = N2(Hq + 2+ + 23) (124)N3 = N3(Hq + 2+ 2

3) (125)

= H (2(q + 1) 3 ) (126) = 2H (q + 1) (127)

= (q 2)H

6

2k (128)

= 2

(1 + q)H +

6

2k

(129)

U =2

H

U

(q

1) (130)

where +, are the shear variables, N1, N2, and N3 are the curvature variables (relative to a group-invariant orthonormal frame), and a logarithmic (dimensionless) time variable, , has been defined.The quantity q is the deceleration parameter and S+ and S are curvature terms that are defined by thefollowing expressions:

q 22+ + 22 +(3 2)

2 + 22 + c2U+

c2

4D2

+ 2 +

(3 1) + 52

(131)

S+ 16 N2 N3

2 16

N1 2 N1 N2 N3 (132)S 1

6

3

N2 N3

N1 + N2 + N3

(133)

In addition, there are two constraint equations that must also be satisfied:

G1(X) H2 + 14

(N1N2N3)2/3 1 = 0 (134)

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where

V

=1

12 ( N21

+ N22

+ N23

2(

N1

N2

+ N1

N3

+ N2

N3)

+3(

N1

N2

N3)

2/3Equation (134) follows from the definition ofD, and (135) is the generalized Friedmann equation. Theresulting Bianchi type IX brane-world equations are now closed and suitable for a qualitative analysisusing techniques from dynamical systems theory. The system of equations X = F(X) are subject tothe two constraint equations G1(X) = 0 and G2(X) = 0. These constraint equations essentially restrictthe dynamics of the dynamical system X = F(X) to lower dimensional surfaces in the state space. Inprinciple, these constraint equations may be used to eliminate two of the twelve variables provided theconstraints are not singular.

5.1.1. Comments

Since the dynamical system is invariant under the transformation , we can restrict ourstate space to D

0. Note that the dynamical system is also invariant under the transformation

(+, , N1, N2, N3) (12 + 32 ,

32 + 12 , N2, N3, N1). This symmetry implies that

any equilibrium point with a nonzero term, will have two equivalent copies of that point located atpositions that are rotated through an angle of 2/3 and centered along a different axis of the N .

If we assume the weak-energy condition for a perfect fluid (i.e., 0), then we restrict the statespace S to the set 0. Since we are investigating the behaviour of the Bianchi type IX brane-worldmodels, we can restrict the state space to N 0 without loss of generality. There are six matter-invariant sets in addition to the invariant sets associated with the geometry of the spacetime [163]. Theevolution equation (130) for U implies that the surface U = 0 divides the state space into threedistinct regions, U+ = {X S|U > 0}, U0 = {X S|U = 0}, andU = {X S|U < 0}.

From (134), we have that 1 H 1 and N1N2N3 8. Furthermore, in the invariant sets U+andU0, using (225), it can be shown that

0 2+, 2, , , 2, , V 1

However, knowing that 0 V 1 and 0 N1N2N3 8 is not sufficient to place any bounds on theNs or D. Furthermore, in the invariant setU we cannot place upper bounds on any of the variableswithout some redefinition of the dimensionless variables (117). It is possible to show that the function

W =

H2 1[23 ]

[+(32)]

2 U (136)

(where , are arbitraryparameters) is a first integral of thedynamical system; that is, W = 0 [163]. Inthe invariant setU+ U0, it is possible to show that q 1, which implies that 0 as for those models that expand for all time. Using invariants, we find that H2 1 as , so thatfor ever-expanding models

U

0 when > 4/3. We also note that the function defined by

Z (N1N2N3)2 (137)

satisfies the evolution equation Z = 6qZ , and is consequently monotone close to the singularity [28].As mentioned earlier, generically the Bianchi type IX models have a cosmological initial singularity

in which , and consequently b dominates as t 0+. This can be proven by more rigorousmethods [164166]. It canalso be shown by qualitativemethods that thespatial 3-curvature is negligableat the initial singularity.

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5.2. Initial singularity

We include a general potential for the scalar field, V, and study what happens as D at theinitial singularity. We also include normal matter with 1

< 2. Equations (128) and (129) become

= (q 2)H (138) = 2(q + 1)H + 2 (139)where (related to the usual inflationary slow roll parameter ) is defined by 3V /2V.

From the Friedmann equation we have that

c2

4D2

+ + 22 1 (140)

and hence each term D, D, and D2 is bounded (since the left-hand side is the sum of positivedefinite terms). Hence, as D , , 2, and 0. It is easy to show that , U 0 asD . Hence, (131) becomes

q = 2 2+ + 2+AD2 (141)where

A c2

4

+ 2 +

(3 1) + 52

(142)

Assuming H > 0, (119) and (120) imply that as for D , either q 0 or q is positivein a neighbourhood of the singularity (q can oscillate around zero; indeed, it is the possible oscillatorynature of the variables that causes potential problems). However, ifq 0, (126) implies = H(2 3 ) (143)

which implies a contradiction for >2

3 (i.e., 0 as ). Hence, as , q > 0,where D andH = q

1 H2

(144)

and hence H 1 (assuming positive expansion) monotonically. (Note that this implies the existenceof a monotonic function, and hence there are no periodic orbits close to the singularity near the setH = 1 all orbits approach H = 1.) In addition, (134) gives (N1N2N3) 0.

From (126), (127), (128), and (129) we have that as

2 0 (145)

that is, the scalar field becomes effectively massless, and

2 0 if < 2 (146)

This follows directly from the evolution equation in the case of an exponential scalar field potential,

=

32 k, and follows for any physical potential for which is bounded as . Hence, we obtain

q = 2

2+ + 2

+ (3 1)C2 (147)

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(where = 2, C2 = c24 D24 if < 2; = 2, C2 = c2

4 D2

+ 22

if = 2; = andC2 = c24 D22 if there is no scalar field), where C2 as , and the Friedmann equationbecomes1

2+ + 2

C2 = 1

12

N21 + N22 + N23

2

N1N2 + N1N3 + N2N3

0 (148)

From (126) and (128), we then obtainC2

= 2C2Hq (3 1) (149)We still have the possibility of or Ni oscillating as the singularity is approached (as in the

Mixmaster models). A rigorous proof that oscillatory behaviour does not occur can be presented usingthe techniques of Rendall and Ringstrm[164,165] (using analytic approximations to the brane-Einsteinequations for , N; i.e., estimates for these quantities that hold uniformly in an open neighbourhoodof the initial singularity). We can then prove that 0 as , 1, and we obtain theBRW source.

To determine the dynamical behaviour close to the initial singularity, we need to examine whathappens as D . Let us present a heuristic analysis (and include ordinary matter). We define a newbounded variable

d = DD + 1 , 0 d 1 (150)

and we examine what happens as d 1 (assuming H > 0). From above, we have that d 1 andH 1 monotonically, and hence we need to consider the equilibrium points in the set d = 1.

The analysis depends on whether the quantityA defined by (142) that occurs in the expression for qin equation (141) is zero or not in an open neighbourhood of the singularity. Without loss of generality,assuming that A > 0, we define a new time variable by

f = (1 d)2

A

Hf

and the remaining evolution equations (on d = 1) become

+ = +, = , N1 = N1, N2 = N2, N3 = N3, = 2, = 2,

= , = 2, U = 2U (151)Therefore, the only equilibrium point is

+ = = N1 = N2 = N3 = = = = = U = 0and this is a local source. This equilibrium point corresponds to the BRW solution [21] with

c2

4D22 = 1, D2 = 0

D2 = 0

In the next two sections, we will consider the two cases of a scalar field and a perfect fluid separately.

Due to the quadratic nature of the brane corrections to the energy momentum tensor, a rich varietyof intermediate behaviour is possible in these two fluid models. We note that a dark radiation densityUtogether with a scalarfield is similar to previous scaling models [167,168]. Here, theequivalent equationof state is pU = 13U. It is known that the bifurcation value for these scaling models is k2 = 3,which for = 4/3 corresponds to a value k2 = 4, a bifurcation value found in the analysis here.The intermediate behaviour of these multifluid models is extremely complex [163]. A more completeanalysis of the Bianchi type II models (in which some of the intermediate behaviour is outlined) wasstudied in refs. 169 and 170.

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6. Qualitative analysis

6.1. Scalar field case

In the invariant setU

+ U0

, we can show that q 1. This implies that the set = 0 is theinvariant set containing all of the past asymptotic behaviour for ever-expanding models in U+ U0. Itcan be argued that a scalar field becomes essentially massless as it evolves backwards in time, hence itwill dominate the dynamics at early times. In an effort to understand the dynamical behaviour at earlytimes we shall first assume that there is no perfect fluid ( = 0) and the 4D cosmological constant iszero.

If = = 0 (i.e., for a scalar field only), then the dynamical system simplifies (effectively theevolution equations for and are omitted). The curvature terms S+ and S are defined as before,and the deceleration parameter q becomes

q 22+ + 22 + 22 + c2

1

4D2

2 +

52

+ U

(152)

The two constraint equations are given explicitly by

1 = H2 + 14

N1N2N3

2/3(153)

H2 = 2+ + 2 + total +1

12

N21 + N22 + N23

2

N1N2 + N2N3 + N1N3

(154)

where

total 2total

3D2= + 2 + c2

1

4D2

+ 22

+ U

(155)

We first consider theequilibrium points at finite D. We define X = {D, H , +, , N1, N2, N3, ,, U} and we restrict the state space accordingly to be S = {X S| = 0, = 0}. (Note, isa discrete parameter where

=1 corresponds to expanding models, while

= 1 corresponds

to contracting models.) There are a number of saddle points [163]. The zero curvature, power-lawinflationary RobertsonWalker equilibrium point P , defined by X0 = [0, , 0, 0, 0, 0, 0, k

6

6 , 1 k2

6 , 0] is a local sink in the eight-dimensional phase space S ifk2 < 2 (if = 1, this point is a localsource). When 2 < k2 this point becomes a saddle. We note that there are models that recollapse. Thereare no sources for finite values ofD.

We considered the equilibrium points at infinity in the previous section. We now show that (for noperfect fluid and a scalar field with an exponential potential) the BRW solution is always an equilibriumpoint of the system and a local stability analysis establishes that it is a local source. To analyze thedynamical system for large values ofD, we define the following new variables:

= r2 sin2 , = r cos , = 14

c2D2

+ 2

2

= 14

c2D2r4

so that the variable D is essentially replaced by the bounded variable in the set U U0 (hence inthis set the only variables that remain unbounded are the Ns). The dynamical system becomes

H = q

H2 1

(156)

+ = (q 2) H+ S+ (157) = (q 2) H S (158)

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N1 = N1

Hq 4+

(159)

N2=

N2 Hq + 2+ + 2

3 (160)

N3 = N3

Hq + 2+ 2

3

(161)

U = 2HU (q 1) (162) = 2H

q + 1 6cos2

(163)

r = rH (q 2 + 3sin2 ) (164)

= sin

3H cos +

6

2kr

(165)

where

q 22+ + 2

2 + r

2 3cos2 1+ 6cos2 1+ c2U (166)The two constraint equations become

1 = H2 + 14

N1N2N3

2/3(167)

H2 = 2+ + 2 + + c2U+ r2 +1

12

N21 + N22 + N23

2

N1N2 + N2N3 + N1N3

(168)

Fromtheconstraintequations,wehave0 {H2, 2+, 2, U, , r} 1, 0 ,and N1N2N3 8 (assuming U 0).

The BRW solution is represented by an equilibrium point in the set = 0 (D ). Ifwe let X = {H , +, , N1, N2, N3, U, , r , }, then the relevant equilibrium points are X0 =[, 0, 0, 0, 0, 0, 0, 1, 0,

2

2 ]. Using the constraint equation to eliminate U, the eigenvalues of thelinearization at the points = 0, are(10, 10, 5, 5, 5, 3, 3, 3, 3)

A value of that satisfies = 0 in a neighbourhood of an equilibrium point corresponds to a tangentplane to an invariant surface passing through that equilibrium point. In the analysis above, the directions = 0 and = correspond to the = 0 invariant surface (i.e., the massless scalar field models). Weobserve that this equilibrium point is a source that strongly repels away from = 0 (for > 0). Thatis, when traversed in a time reverse direction, typical orbits would asymptotically approach a masslessscalar-field BRW solution.

In summary, the isotropic BRW solution is a global source and the past asymptotic behaviour of theBianchi IX brane world containing a scalar field is qualitatively different from that found in GR.

We note that assuming

U

0, the future asymptotic behaviour of the Bianchi IX brane world

containing a scalar field with an exponential potential is not significantly different from that found inGR [163]. For 0 < k

2 there no longer exists any equilibrium pointrepresenting an expanding model that is stable to the future. We, therefore, conclude that if k >

2,

then the Bianchi IX models must recollapse. In ref. 171, it was shown that ifk >

2, then a collapsingmassless scalar-field solution is a stable equilibrium point. In the brane-world scenario, we have thatthis final end-point is the BRW solution. However, if U < 0, then a variety of new behaviours arepossible, including possible oscillating cosmologies [63].

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6.2. Perfect fluid case: No chaos in brane-world cosmology

We shall now discuss the asymptotic dynamical evolution of spatially homogeneous brane-worldcosmological models with no scalar field close to the initial singularity in more detail. Due to the

existence of monotone functions, it is known that there are no periodic or recurrent orbits in orthogonalspatially homogeneous Bianchi type IX models in GR. In particular, there are no sources or sinks andgenerically Bianchi type IX models have an oscillatory behaviour with chaotic-like characteristics, withthe matter density becoming dynamically negligible as one follows the evolution into the past towardsthe initial singularity. Using qualitative techniques, Ma and Wainwright [28] show that the orbits ofthe associated cosmological dynamical system are negatively asymptotic to a lower two-dimensionalattractor. This is the union of three ellipsoids in R5 consisting of the Kasner ring joined by Taubseparatrices; the orbits spend most of the time near the self-similar Kasner vacuum equilibrium points.More rigorous global results are possible. Ringstrm has proven that a curvature invariant is unboundedin the incomplete directions of inextendible null geodesics for generic vacuum Bianchi models, and hasrigorously shown that the Mixmaster attractor is the past attractor of Bianchi type IX models with anorthogonal perfect fluid [165,166].

All spatiallyhomogeneous Bianchi models in GR expand indefinitely except for the type IX models.Bianchi type IX models obey the closed universe recollapse conjecture [172,173], whereby initiallyexpanding models enter a contracting phase and recollapse to a future Big Crunch. All orbits in theBianchi IX invariant sets are positively departing; to analyse thefuture asymptotic states of such modelsit is necessary to compactify phase-space. The description of these models in terms of conventionalHubble- or expansion-normalized variables is only valid up to the point of maximum expansion (whereH = 0). An appropriate set of alternative normalized variables leading to the compactification ofBianchi IX state space, which were suggested in ref. 28 and utilized in refs. 174 and 171, 175177, wasdiscussed in the previous section.

Let us discuss the Bianchi type IX brane-world models with a perfect-fluid source. Unlike earlier,and following refs. 62, 63 and 28, we define Hubble-normalized shear variables +, , curvaturevariables N1, N2, and N3 and matter variables (relative to a group-invariant orthonormal frame), and alogarithmic (dimensionless) time variable, , defined by d = H dt. These variables do not lead to aglobal compact phase space, but they are bounded close to the singularity [28]. The governing evolutionequations for these quantities are then

+ = (q 2)+

Documents

A.A. Coley- The dynamics of brane-world cosmological models