Brane-World Cosmology, Bulk Scalars and Perturbations

Brane-world cosmology, bulk scalars and perturbations

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Please note that terms and conditions apply. JHEP10(2001)026 October 8, 2001 October 22, 2001 Revised : Accepted: eorique, CEA-Saclay, F-91191 Gif/Yvette [email protected] , August 30, 2001, Received: ∗ Supergravity Models, Cosmology of Theories beyond the SM, Large We investigate aspects of cosmology in brane world theories with a bulk [email protected] [email protected] HYPER VERSION On leave of absence from Service de Physique Th´ ∗ DAMTP, Centre for Mathematical Sciences,Wilberforce Road, Cambridge Cambridge, University CB3 0WA,E-mail: UK Theoretical Physics Division, CERN CH-1211 Geneva 23, Switzerland E-mail: Scale Structure Formations, Physics of the Early Universe. scalar field. We concentrategular on spaces. a recent After modelpresent discussing motivated the the from evolution background supergravity of evolutiontained in the of in sin- the density such (second) contrast. a Randall-Sundrum scenario brane-world, We and compare we usual 4D our scalar-tensor results theories. toKeywords: those ob- Abstract: Carsten van de Bruck and Anne C. Davis Philippe Brax Brane-world cosmology, bulk scalars and perturbations cedex France JHEP10(2001)026 16 1 µν E 4.1.1 The4.1.2 high energy era The4.1.3 low-energy radiation era The matter dominated epoch4.2.1 The4.2.2 high-energy regime Radiation4.2.3 domination Matter4.2.4 domination Observational consequences 21 21 20 22 23 25 23 2.1 The background2.2 evolution Radiation dominated2.3 eras Matter dominated2.4 era Broken supergravity2.5 era The matter-supergravity transition3.1 The Raychaudhuri3.2 and Klein-Gordon equation The evolution equation for 3 13 12 8 10 9 3.3 The perturbed dynamics4.1 The Randall-Sundrum scenario4.2 Effects of the bulk scalar field on cosmological perturbations 22 A.1 BPS configurationsA.2 17 Breaking conformal 20 flatness 29 27 1. Introduction The idea that wehas sparked live a on lot aory, of hypersurface where interest recently. higher-dimensional embedded These objects, inSimilarily, theories compactifying such a are M-theory as higher (or motivated its D-branes, from dimensional effective string play low-energy space the- limit: an 11-D essential supergrav- role [1]. 3. Cosmological perturbations using the fluid flow approach 13 Contents 1. Introduction2. Brane cosmology 1 3 4. Time evolution of cosmological perturbations 19 5. ConclusionsA. Supersymmetric backgrounds 27 26 JHEP10(2001)026 2 7 is within the experimental ball-park. This accelerated expan- / 4 − -orbifold and compactifying six dimensions on a Calabi-Yau manifold, 2 = Z 0 / q 1 S Brane world theories predict that our universe was higher-dimensional in the So far, most cosmological considerations of brane worlds centered around the Cosmology may be a fruitful field where the above ideas can be tested. As such, The aim of this paper is to develop an understanding of the evolution of pertur- sion is driven by a bulk scalar field, whose parameters are constrained by the gauged ity) on a results in a five-dimensionalcated brane at world the orbifold scenario fixed with points two (see hypersurfaces, e.g. each [2]past. lo- and [3]). Becauseanswered of within this, the there contextwithin of is these the the theories. standard Furthermore, hope model cosmologyparameters of should that in be cosmology, these certain can a models. way be questions to addressed severely which constrain cannot be one-brane scenario of Randall andbrane Sundrum [4]. universe In is this embedded model,particular, in the three-dimensional the a bulk-space 5-dimensional Anti-defrom is Sitter empty, the (AdS) the negative spacetime. only cosmologicalleads to contribution In constant new to in effects the which themotivated are from curvature particle interesting bulk. physics comes for predict cosmology matter This in [5].heterotic the M-theory, simple However, bulk, for most such example, model as scenarios one scalar particular fields.of already scalar the In field Calabi-Yau 5D manifold, measures on the which deformation Other six models, other motivated small by dimensions supergravity are (SUGRA),form compactified also [3]. is predict dictated bulk by matter, the whose field theory under consideration. the study of thebecause evolution of the cosmological higher-dimensional perturbations naturetion is of of extremely the matter important world and/or [6], can anisotropiesdifferent in leave aspects the traces microwave of in sky. perturbations the A in lot distribu- of brane papers world investigated scenarios [7]. bations in brane world scenarios, intoy which model, scalar we field(s) will are use presentspaces in a [8]. the cosmological bulk. realization The of As evolution(see a a also of supergravity [11] the model for in brane alar singular world discussion four was on different brane discussed cosmological cosmology eras inabove and have depth been the bulk identified brane in scalar in tension fields). [9] this the and model.tion In cosmology epoch is particu- [10] At non-conventional where high before the energy entering scalar thefield field radia- starts is evolving frozen. in time After leadingFRW matter-radiation to cosmology equality a in the slow-down of the scalar the matterbecomes expansion dominated rate the compared era. dominant to one Eventually leadingbetween the to the scalar a matter field supergravity and era. dynamics scalarcosmological Requiring field constant that energy with coincidence densities a occursthe fine-tuning in the brane. of recent the In past supersymmetry the leadsof supergravity a breaking the era tension expansion it on has ofparemeter been the shown universe that can the be observed understood, acceleration i.e. the computed acceleration JHEP10(2001)026 (2.1) (2.2) .Our 5 x − with 5 , x )) φ lives. This scalar field ( φ U , + ) 0 2 φ ) ( B ∂φ on the brane. Indeed, the value of U (( 4 3 4 g G − − √ R x 4 3 d 5 g Z − 2 5 √ κ 3 may leave traces in the evolution of perturbations. 2 x 5 − d G = Z 2 5 B κ 1 S 2 = bulk and the boundary action S symmetry along the fifth dimension and identify 3 5 2 Z /M =1 2 5 κ Consider the bulk action can be seen to have changed by 37 percent since radiation/matter equality. Thus, where brane-world carries two typeslarge of energy matter, and ordinary thethat matter standard gravity and propagates model radiation incouples fields at the to at the lower bulk sufficiently standard energy. where modelthe a fields standard We living model scalar also on fields field assume transitions the have have brane-world. condensed taken At and place, low the theizes energy electro-weak coupling when the and of mechanism the hadronic proposed scalar phase section in field [12] to we the with derive brane-world a the real- ordinary self-tuning matter brane of on the cosmology the brane equations brane tension. and describing In a the this scalar coupling field in between the bulk. 2. Brane cosmology In this section we discuss theevolution. field equations on the brane and discuss the2.1 background The background evolution We consider our universe to bebedding a is boundary chosen of such a that fiveWe dimensional our impose space-time. brane-world sits The at em- a the origin of the fifth dimension. supergravity theory in theevolution of bulk. the induced On gravitational constant top of this, the model predicts a significant G the study of cosmological perturbationparticular theory the in time these evolution models of isIn rather this important, in paper webulk discuss the scalar evolution field of onequations the induced cosmological density on constrast perturbations. our and branereview the world In the effect and background section solutions discuss of found 2 the the perturbation in Friedmann we [10], equations. equation. which review In We are Einstein’s also section neededfluid in 3 flow order we to approach. derive solve the This the it perturbation approach easy equations is to using very derive the transparent thelutions for necessary of our evolution these purposes equations. equations andto and In makes the discuss section Randall-Sundrum their 4 scenario properties. weWe discuss and We conclude some usual point so- in four-dimensional out section scalar-tensor theissue differences theories. 5. of supersymmetry In breaking the and appendix conformal we flatness. discuss some details concerning the JHEP10(2001)026 (2.3) (2.8) (2.9) (2.4) (2.6) (2.10) ’s yields the i U , ab g B ab 3 8 T is the boundary contribution. 5 − x B ab δ . ) (2.7) T , 2 2 ) (2.5) + ) 0 φ B ( ab W U to the brane degrees of freedom, i.e. , ∂φ B T i ( is a characteristic scale related to the − U 0 B αφ = ab 2 ρ, p, p, p ξ W ab g (Φ) U g − ξe ab 2 1 4 = V 2 3 h = B − Rg ∂φ − ∂W = U (Φ) represents all the contributions due to the 2 1 φ W b = V 2 5 B − =diag( B κ ab = U φ∂ 2 b a ab 3 T a . In this paper we will assume that the fluid couples ∂ U τ R ωρ . The vacuum energy generated by the Φ ≡ i 4 3 = ab 12, these values arising from the parametrisation of the = p G = 2 supergravity with vector multiplets. When supergravity √ ab / T 1 N − 3, 3 in the last equation. Following the self-tuning proposal we inter- √ / ... is the bulk energy-momentum tensor and is the boundary value of the scalar field. The Einstein equations read =1 =0 as arising from a direct coupling after inclusion of condensations, phase transitions and radiative corrections. 0 ab i α φ B T a, b U We will be mainly concerned with models derived from supergravity in singular We also consider that ordinary matter lives on the brane with a diagonal energy where and the bulk potential If one considers a single vector supermultiplet then supersymmetry imposes that and the boundary term andanequationofstate to the bulk scalar field only via gravity. spaces. They involve in the bulkentirely couples specified to by the the superpotential boundary in a supersymmetric way the Lagrangian is brane tension. moduli space of the vector multiplets, and fields Φ where the dimension four potential the standard model fields Φ pret with effective coupling The bulk term is momentum tensor where where JHEP10(2001)026 in 00 (2.14) (2.17) (2.12) (2.18) (2.11) (2.15) (2.16) (2.13) E and matter φ , ab E − . ab , n ab ) 2 , correspond to supersym- n 2 ) 2 τ ab φ ) T the induced metric on the P φ ∇ , − a ( 2 ∇ ab ( ∇ B cd 2 5 n 16 τ U 5 + 16 2 ∆Φ cd ∂φ − ∂ τ ab − − . φ B π (3 b a φ B b ∂φ ab 4 ∇ ∂U ∂φ 24 ∇ τ ∂V U φ ∇ = 0 leads to the conservation equation a TW , + φ − + = = 5 a ab ∇ 0 c b ab | = N 2 1 G B ∇ τ τ φ a 4 2 1 B 2 n B ac ∂φ + ∂U ∇ τ ∂ πG U U + 4 8 a 6 τ ab = − π ab ∇ 2 + n ab + τ φ = V 8 2 ab ∆Φ τ 3 12 ab τ ∇ − E B 4 = a U = ∇ ab + π ab is defined by P ab is quadratic in the matter energy momentum tensor ab n P ab V 8 π the projected Weyl tensor of the bulk onto the brane. It appears 3 , The Bianchi identity − ab a a E τ = is the matter energy momentum tensor, = 1 is the supersymmetric case. Larger values of ab = ab ¯ τ G τ T The dynamics of the scalar field is specified by the Klein-Gordon equation which The dynamics of the brane world is specified by the four dimensional Einstein The tensor Since supersymmetry is not observed in nature, one should incorporate super- the background and attime the evolution perturbative of level. the This background and is of crucial the in scalarreads order field to and define matter the perturbations. where where we have defined the loss parameter by brane and where as an effectiveNote energy that momentum one tensor cantension on identify [13], the the i.e. four-dimensional brane, Newton’s called constant with the the Weyl brane fluid [5]. where metry breaking effects withthe a dynamics positive of energy the density coupledon on system the the comprising brane. brane. gravity, the We scalar will field analyse equations [13] where the tensor This consistency equation (2.15) will allow us to follow the time evolution of symmetry breaking. A naturalscalar way to field break to supersymmetry brane fields is by fixed coupling at the their bulk vevs. This leads to JHEP10(2001)026 (2.25) (2.24) (2.21) (2.22) (2.20) (2.23) (2.26) (2.19) . ! , ρ, ρ φ B B 4 4 ˙ ˙ φ U U 2 , + 2 + , 2 2 as a function of the brane ˙ ˙ dx ab φ 2∆Φ φ , ) 0 plays the role of a dilaton n b B | 2 B ) − . φ it is then sufficient to obtain n z,t b H H B ( V 2 B B 3 2 − 3 2 U B ∇ ab U 2 U 2 √ e + E 3 τ +2 + +∆Φ − 0 2 ln ∆Φ ) 2 − V τ ) )+ φ V V ∂ 2 − − V 8 3 8 3 ∇ b dz ≡ 2 /∂φ = further, we point out that the projected (( U + + b +2 B B 2 1 6 + 0 2 B 2 B 2 ˙ ∇ ˙ 2 b φ H ) φ U U − ∂U 6 τ φ 6 τ dt 3 3 16 φ 16 − ∇ b − +( ( − ( ) ∇ t ˙ = t b φ U ∂ φ 4 ∂ a z,t ab g H (

= = T 00 ∇ A − 4 a 2 E V e 00 +4 √ = ∇ x . Moreover we assume that we obtain a FRW induced ¨ E dta φ = 4 j ab B d 2 T Z H 6= 4 evaluated on the brane. ds Z 1 a i 0 +4 | = g = 00 S − ˙ is constant. It is remarkable that 00 E =0, √ E 2 ln ij stands for the second normal derivative of the scalar field at the brane τ is the brane expansion rate 3 E ∂ 0 . This leads to B | n ≡ φ H 2 n ∂ =0, H For the background cosmology the Klein-Gordon equation reduces to (see sec- The background cosmology is characterized by its isotropy so we consider that In order to illuminate the role of i 0 Writing the bulk metric as the differential equation for coupled to the trace of the matter energy-momentum tensor. tion 3) Klein-Gordon equation can betensor seen as an equation for the brane energy-momentum location. The effective scalar potential is defined by metric where and can be derived from the four dimensional effective action for metric on the brane. Due to the tracelessness of whenever ∆Φ which reads and E where dot stands for theby proper 4 time derivative. The bulk expansion rate is defined We derive these equations in section 3. JHEP10(2001)026 , φ =2 (2.32) (2.30) (2.28) (2.29) (2.27) N . 4 C a becomes time- + φ ρ B . dt which depends on the dU ˙ 2 φ 2 dt . We can always choose the Z b 4 dt a ) 1 b ,t 12 )+6∆Φ . (0 τ ) A − p e − ) 2 2 B . + ρ ˙ = φ B H ρ (4 ( = 0 (2.31) − dt H B B a 7 b ˙ =3 U τ V H = a 3 00 (2 4 )insuchaway ∇ H 4 − ¯ b G dt ,t da = dta ˙ ρ (0 dt Z B 4 whose origin springs from the possibility of black-hole Z a 4 1 4 a )= 16 1 b 16 C/a ,t = (0 + 00 A ρ E B 12 U + 2 ρ 36 = 2 B H Using In the following we shall describe the case where the bulk theory is Notice that there are two entities which depend on the bulk. First of all there is Using the conservation of matter enters in the definition of the effective Newton’s constant (see eq. 2.13. The last dependent and either matterdoes is not present vanish. onis Notice the that that brane Newton’s or a constant the sufficient becomeson condition time-dependent this energy expression (see for loss eq. when breaking parameter 2.13). discussing conformal the We flatness will various cosmological comment eras. evolution of the scalarwhich field is in not the constrained bulk. by considerations It of specifies the brane the dynamics. part of the evolution of supergravity with vector multiplets. When no matter is present on the brane, the This completes the description ofogy, the i.e. three the equations determining Klein-Gordonequation. the equation, brane the cosmol- Friedmann equation and the conservation the dark radiation term formation in the bulk. Then there is the loss parameter ∆Φ three terms are athe combination changes of of the the energy pressure flow along onto the or fifth away dimension from [14]. the brane and This equation has alreadyφ been derived in [10]. We remember that the scalar field In particular we find that conformal flatness is broken as soon as and after one integration by parts we obtain the Friedmann equation we find that the proper time on the brane is defined by boundary condition Together with the Klein-Gordon equation this leads to JHEP10(2001)026 2 ρ (2.38) (2.37) (2.34) (2.35) (2.36) (2.33) . , 0 t 4 t / 1 . . . 0 0 = 0, i.e. a constant scalar field, for which ln φ t t =0 =0 8 β β =0 2 = 00 0 + φ C a E 0 ∆Φ φ = = a φ and find that V The usual radiation era is not modified by the presence of the scalar field We now turn to the discussion of the background evolution. The solutions given First of all we consider the high energy regime where the non-conventional dominates, i.e. for energies higherthe than the effective brane potential tension. In that case we can neglect background cosmology can bemally explicitly flat solved. implying that In particular the metric is confor- Notice that Newton’s constant does not vary in time in this case. A detailed analysisimplies is that presented the in branebrane, the dynamics we show appendix. is in the closed. appendix Inwe that When assume conformal particular flatness matter that is the not density the preserved.assume last is breaking that Nevertheless equation present of (2.33) on conformal and the flatnessthe (2.34) perturbative is are level. small still enough valid, both to for allow the one background to below as have well been as obtained on discuss in the [10], evolution but of we density need perturbations to in present2.2 section them 4. Radiation in detail dominated in eras orderThe to background cosmology can bethe solved radiation in and four different matterthe eras: dominated scalar the epochs field high and dynamics energy finallyticular dominates. era, attention the We to discuss supergravity the these era matter-supergravitycoincidence four transition where eras where now we in implies show turn thatbrane a paying requiring tension. par- fine-tuning of the supersymmetry breaking part of the while the scalar field behaves like the projected Weyl tensor vanishes altogether In the following we will focus on the case Moreover the loss parameter vanishes explicitly JHEP10(2001)026 (2.39) (2.43) (2.44) (2.40) (2.41) (2.42) , . . e t 2 , t 2 2 m = 1. This is a good approximation / , 2 ρ α H 2 1 . , t α T γ 8 α 3 4 45 ln α ≡ r t t e β approaches quickly the attractor for which − =0 ) 8 t 9 − t 15 τ φ ( + = 00 3 r 2 − a 0 N e E 3 00 a φ = = = E πG γ β = = a 8 a φ case as it leads to an accelerating universe when no matter to leading order in 3 α − a are the time and scale factors at radiation-matter equality. We are we get e α a and e τ It is useful to compare these results with general relativity, so for the moment For small =constant. Newton’s constant does not vary in time, while and which decreases like we identify in a phenomenological way Newton’s constant with the ratio 2.3 Matter dominated era The matter dominated eraus leads first to consider a the more pure interesting background sugra cosmology. case where Let as in the high energy regime.world scenario Note that is the similar solutionsee in to [15] the the and radiation [16], radiation era for in eraφ example. this solution brane The found field in Brans-Dicke theory, is present and small time deviations for Newton’s constant. interested in the small where The projected Weyl tensor is given by until coincidence where theanymore. potential The energy solution of to the the scalar evolution field equations cannot is be neglected JHEP10(2001)026 this 3 (2.48) (2.47) (2.45) 10 ∼ e z 15. As / . ) (2.49) i ! 2 6= 1 the static solution is not dx . i 5 T ) , / | dx 2 2 y α | ∂φ α + ∂W 4 4 ξ / 2 2 1 ) α | du − y = 0 (2.46) − +1 | 2 +1 + ξ e z 2 2 z W 10 12 the exponent is 1 α ln(1 / dη

− 1 SUGRA − α = 1) V )( =1 − ) ) e u − 2 z 2 this is ( z =(1 = ( 2 . This is a flat solution corresponding to a vanishing 2 ( α 5 z a N T a N φ ( G G = adx = 2 = ds V dy The time dependent background is obtained from the static solution by going to Notice that Newton’s constant, as defined in eq. (2.13), starts to decrease only As soon as supergravity is broken on the brane valid anymore. The new four dimensional potential becomes conformal coordinates leads to a decrease of the quantity (2.44) byfrom 37% the since time equality. ofradiation matter dominated and era. radiation20% Nucleosynthesis equality at constraints and the the is time variation strictlyfor of to the constant nucleosynthesis. be variation during In of less order the Newton’s than is to constant we beyond confront need our the to theory scope extractthe with the of local variation weak the tests field of paper. limit,constant the which [17]. In Newton addition, constant we assume note all that other many2.4 of masses the Broken and supergravity tests couplings era for are After coincidence matter doesdominated not by the dominate scalar anymore;pure field this sugra dynamics. case. is Let the It us is supergravity review easy era to it see briefly. that Consider the first potential the vanishes wherewehavedefined For the supergravity case with cosmological constant on the brane-world. which is now time dependent,stant. whereas In in terms general of relativity the this red-shift ratio is strictly con- leading to a static universe with JHEP10(2001)026 12 √ / (2.57) (2.53) (2.55) (2.50) (2.51) (2.52) (2.56) 1 − = α , 2 α 2 2 6 h 2 / h α 2 − ξ − ξ 3+1 α 1 (2.54) / 1 1 . 1 4 1 √ 7 5 − √ 1+2 − , 0 2 t − 7 4 t 2 T α 2 hη, hη, − T = 1+ − α =0 + + 11 √ 6 1 − = 00 dependence. Now for u u 0 ± 1+2 = q E ξ axis T = = SUGRA φ a 1 √ 0 ω u h q )= )= SUGRA )= ω t ( 0. The scale factor corresponds to a solution of the four u, η u, η a ( ( a φ h< and ξ 2 / 1 3 hT 2 2 α 3 = 0 t The resulting universe is characterized by the scale factor in cosmic time is accelerating. In particular we find that This is within the experimental ball-park. dimensional FRW equations with an acceleration parameter andanequationofstate with which never violates the dominant energy condition. The solution with and performing a boost along the where we have displayed the explicit we find thatequation the is Friedmann satisfied. equation Moreover is we find solved that [10]. Similarly the Klein-Gordon as the bulk metric is conformally flat. and for the equation of state JHEP10(2001)026 (2.62) (2.60) (2.61) (2.58) (2.64) (2.63) (2.59) c z . The evolution coin- α . e ρ , 2 αβ/γ , . α 4 e 3+ S . 8 V ρ − 2 M , α + 1 2 5 e c 1) 3 2 κ ρ W ρ +1 +1 − 2 3 − c e 1 ≈ z z 12 T = 2 SUGRA 2 5 4 4 S BS κ 2 H W =( 2 α M M 4 BS 1 = 1 varies slowly compared to 4 − BS 2 M φ − 1 T M H 2 the Hubble parameter derived in the unbroken super- T = and the potential energy occurs at 4 BS M . Let us now denote the supersymmetric brane tension by SUGRA 4 2 SUGRA H H = 1, then the Friedmann equation in the broken supergravity GeV T 39 − is the critical density. This is the usual extreme fine-tuning of the cosmolog- is the matter density at equality. This implies that the left-hand side is much c e ρ ρ In the matter dominated era the supergravity Hubble parameter decreases faster ical constant. Indeed it specifiesfrom that the the non-supersymmetric energy densitytions, sources, received cannot by e.g. exceed the radiative brane-world the corrections critical energy and density phase of transi- the universe. smaller that 10 Now this is anbrane extreme tension. fine-tuning of the non-supersymmetric contribution tothan the the potential contribution.gravity Coincidence contribution between the matter dominated super- and the supersymmetry breaking contribution We find that 2.5 The matter-supergravity transition Let us now investigate the transitioneras. between the If matter dominated we and denote supergravity by cides with the onefrom obtained the from potential unbroken supergravity does as not long dominate. as In the the contribution radiation dominated era this requires wherewehaveusedthefactthat where Imposing that coincidence has occurred only recently leads to case is gravity case, i.e. where JHEP10(2001)026 are (3.1) (3.2) µν π )thaninthe 2 α 6=1leadstoade T b n a , n − =0 . The off-diagonal components of j ab 13 ab g τ 6= a = i ∇ ab n = 0 for = 0, the conservation equation reads j i b π n ab n = 1 and neglecting the scalar field contribution we obtain the T =0and , see figure 1, the matter energy-momentum tensor is conserved. the normal vector to the brane and defining the induced metric by B 2 of the dominant fluid at each epoch, this approach is simpler in order =1and n 2 =0, n δρ/ρ α = The evolution of the gravitational constanttion in of the the matter era background: changes In theand the evolu- furthermore matter the era, universe the is gravitational expanding constant decreases slower (up to order FRW matter dominated era in general relativity. Perturbations in the scalar fieldversa. are the source of matter fluctuations andPerturbations vice in the projectedmatter Weyl perturbations tensor and vice act versa. as sources for the scalar and δ There are different effects which will influence the evolution of perturbations: These effects will change the growth of perturbations compared to normal 4D While discussing perturbations, we use the fluid-flow approach (see [18, 19] • • • such that formally an additional stress induced by bulk gravitons [7]. We now turn to thebrane discussion point of of cosmological view perturbations.we and will As do set we not investigate ∆Φ the intend to solve the full five-dimensional equations, 3. Cosmological perturbations using the fluid flow approach cosmological models or the Randall-Sundrumthat model. our In particular formalism it allows shouldputting be us noted to treat the Randall-Sundrum cosmology. Indeed by Denoting by Randall-Sundrum case with a flat boundary brane while putting Sitter boundary brane. and [20]) rather thanall the perturbation metric-based variables approachmetric-variables. are [6]. For expressed our The in purpose,trast main terms i.e. difference of discussing is the fluid-quantities, that evolution rather of than the density con- to obtain the evolutionequation equations. and the To Klein-Gordon do equation in so, the we comoving3.1 need frame. to The derive Raychaudhuri and the Klein-Gordon Raychaudhuri equation On the brane JHEP10(2001)026 is a (3.6) (3.7) (3.3) (3.5) (3.4) is the B . ab S and the ∩ h C , ab B ω = 1. In addition S − . The four dimen- n = 2 u is the velocity field of the , the helicity , a ab d c a .Thesurface u σ , ω u u b b + D ∇ ab b C d b c . u u σ S pn a ) and satisfies u p + = − n d c a b + + ˙ u u u a ρ a B ( n u ∇ H B ) 14 p = H B + is the induced metric on the brane , 3 b d + u B ˙ ab − u ∇ ρ c n H ab u = u =( ˙ − ρ =3 ≡ ab a = τ a is perpendicular to the normal vector u d =constant. The different metrics in the text are as follows: a ¯ u ∇ is the induced metric on the comoving hypersurface B c y =0lateron. ∇ ab ∇ ab u ,ω is orthogonal to the velocity vector and C is the brane covariant derivative and C =0 b D ab b a σ n The brane-world = a ∇ Using the relations is the full five-dimensional metric, ab Notice that no matter leaks out of the brane. Defining is the energy-momentum tensor. Notice that the vector one derives from (3.2) we have the usual decomposition in terms of the shear whereweset induced metric on brane matter and thus must be orthogonal to expansion rate where Figure 1: sional hypersurface set of comoving observersis following not matter necessarily on located theg at brane. Note that the perturbed brane JHEP10(2001)026 2 ¯ ∇ (3.8) (3.9) (3.17) (3.16) (3.12) (3.14) (3.11) (3.10) (3.13) (3.15) . Expanding b , ¯ D . b ab = 0 we find that ∂φ ∂U ¯ n . D p b a a 2 1 . = ¯ ¯ D D ∇ − b a a a p n n p ¯ = D ∇ a a b + n φ ¯ u ∇ 2 ρ a ¨ φ. + ¯ u ∇ , c − a b ¯ − , , − ˙ . D φ u ¯ 0 b c c ∇ p a B p φ n n H )=0and u D a U ) a − 4 c + = 0 leading to =0 n ¯ ab ∇ D + i · b − B ρ 00 − u K U u Hu 15 in terms of the laplacian on the comoving ab 6 one obtains τ ,u .Noticethat R − = b a φ 4 g 1 2 u u u 2 − − a ab = = ¯ H − − ¯ ab D = D u 6 τ a = K a ˙ K ,where K u = ab = ¯ + 2 a =4 B D 0 h a n φ − a H here as we will consider linear scalar perturbations u a 2 K ¯ u 00 D ˙ u a D φ ab +3 +( = D − ω − 2 a B ) ˙ = 0. Defining the covariant derivative on the hypersurface ˙ ¯ φ , see figure 1, is given by n ∇ D H b · u a leads to the Raychaudhuri equation 3 and H u = u a D ( ab ˙ 2 ab u a h +4 a ¯ σ D D ¨ φ ∇ and using . Finally we can read off the Klein-Gordon equation a p u b D = 1and +3 ab b − ρ h u (see figure 1) orthogonal to a orthogonal to − = u = 2 S C a ab = u ¯ D τ K In order to have a closed system of differential equations for the perturbations by whose divergence only. we need the Klein-Gordon equationsurface in the comoving gauge. The metric on a hyper- We have neglected C with We can now evaluate the laplacian surface is the extrinsic curvature tensor. Using ( In the comoving gauge we have leads to which is nothing but the spatial covariant derivative on the brane, one obtains where Now The junction conditions lead to JHEP10(2001)026 . 00 (3.24) (3.18) (3.26) . (3.22) (3.23) (3.19) (3.21) (3.25) (3.20) be the R and or- ab a =0.We u . 2 ab ∇ b u u ad 1. The induced . E b − . ac u 2 u = ) ad . d a ) E u u a c ac a u +∆Φ u ∇ u . B ) ( . d ab H c − ) ab ∂φ ∂V E ω ) ) a ac E . ˙ − +3 c u u + ad ad d c a c d ∇ E E = − c u ¯ c =0and b b ∇ E . (˙ ) σ ( u c u u φ a ˙ a is the brane metric. Let d 2 u cd ab ab ab u ac u u . ¯ +( ∇ u E E E E ab ab − ( ac d a b B − u c b n d u − which enters in the evaluation of u − − u c u ( ( ∂φ ∇ ) ab ) ) c ¯ c c ∂U ad E d a p 00 16 c E ab ab u u 3 ∇ E c E ¯ c = ∇ E E ad 0 ∇ u −∇ − µν − = ac ac on the brane with induced metric u φ ) = ) ) ac E ρ ,where u u = B ab b ( the covariant derivative with respect to u ( ( d ad ad ad such that S c ¯ c c u E H a ab B E a ¯ E E a = E ∇ b ∇ b b ¯ u c u E d d d ∇ = ∂φ c ∂U ab n u u b = = ∇ + ∇ 6 1 E u ac ac ac c a ab u u u ab u ad ( ( ( + n a ∇ E c c c : E ˙ c u φ ∇ ∇ ∇ ab ac = ∇ H u E ac ) = = a ab d u )= u +4 u ∇ c ab ¨ φ ∇ E ( ac u ( c ∇ Consider now thogonal to the velocity vector Let us further define the projected tensor We find that the Klein-Gordon equation reduces to metric is given by 3.2 The evolution equation for brane covariant derivative and now turn to the evolution equation for As a next step we consider Let us consider the hypersurface where prime stands for the normal derivative. Now the junction conditions lead to Using (3.4) Notice that it involves only brane derivatives when the loss parameter ∆Φ Thefirstterminequation(3.20)canberewrittenas Then The last term can be written as JHEP10(2001)026 − .We (3.27) (3.32) (3.29) (3.30) (3.31) (3.33) (3.28) H ab E a u − B , b is a comoving H u ad 3 ad S E − E . ac b u ad 00 ) 00 ˙ u d c ˙ ¯ P π Eu B ω B − − . For the background we H + H a a 0 0 00 d c − π P − E a σ ) a . . ˙ b ˙ u u ) u = 00 +( ab − a 0 − ac R ¯ a 0 E π E ¯ π ¯ E = 0 and we explicitly assume that a a P E − ˙ B i u u ac 0 a B ( ˙ ,the spatial trace, we end up with p u u H p E H ( − 2 . c − + ab 0 − b − to get c − ∇ ) ∇ u ρ E b π d 00 00 ab c u 00 ab ˙ ad − u π 17 E ¯ E u P ∇ ad E B = − B − E ac =0and = H b = 2 c H u 3 ac a 00 ) ¯ 3 ˜ E u H ab u E d E ( c − c π . c − ˙ B u a 0 b ω c 0 +3 ∇ u c H ∇ π d − + ˙ 4 c P ad u b H c d c ¯ c ∇ 3 E − ¯ ¯ ∇ − E σ c ac 0 c ( b = c u = E ¯ − ∇ ) ¯ c ab E d 0 ), and similar for the pressure and the expansion rate c c ¯ a π ∇ ˙ u a ω P = 0. Notice that the first term of the last expression vanishes. )= x, t a = ∇ i ( − + ab ∇ 0 b d a c δρ E c ,u ¯ E E σ ac c a ( u )+ ¯ ( t − ∇ ∇ =1 c ( b 0 ∇ = 0 for the background. = ρ u ab ,ω E )= a Let us now specialize to the comoving gauge, i.e. the surface We begin with the Raychaudhuri equation and energy conservation equation We need further to evaluate (see eq. (2.14)) ∇ x, t =0 ( one and We obtain Collecting everything and defining Collecting all these ingredients leadsthe to comoving the frame. consistency equation (2.15) expressed in 3.3 The perturbed dynamics We have now obtained all theequations necessary up equations in to order linear toaverage derive order. (over the perturbation In the doing comovingρ so hypersurface) we of decompose the all quantity fluid plus quantities a into perturbation, an i.e. where we have used the tracelessness of which simplifies to and finally we need (see eq. (2.16)) retrieve the previous expression as ˙ So, from eq. (3.23) we get σ insert these expressions forsubtract all the quantities average. into the equations we obtained aboveevaluated and in a comoving basis. The Raychaudhuri equation reads JHEP10(2001)026 and . (3.34) (3.35) (3.39) (3.37) (3.40) (3.38) (3.36) =const . is the time-time 00 δ, 2 2 p/ρ E 00 a k the energy density. R − = ω ρ 00 1 2 ˙ − φ w δR , 7 00 16 1 3 ) ω δR − − . ) ) p ¨ ) δ ω p ω + δp ˙ . 3(1 + +3 δ ρ 2 2 p ρ − a k =(1+ 3(1 + (2 ) + δp 1 3 the pressure and . ρ δ ρ ˙ ) − p + Hδ ρ 1 ˙ δρ 12 δ − H 18 + ω = ˙ + δ ω )+ 2 δ =1 HδH p H 1+ 2 ( H pr ( ω H dt − +3 ω dt ω 6 ˙ ρ = H is ( 1+ − p b 00 ˙ 8 δ U δH = + δp R H ˙ ρ + ) H δ ω = V 3 8 3 ˙ is the expansion rate, H − − δ H = +(2 00 ¨ δ R . We deduce that w This observer is, of course, confined onto the brane world. . For the FRW metric, 1 = 00 2 s All quantites are measured with respect to a comoving observer. component of the Ricciof tensor. the The comoving dot observer representsjust (i.e. the the proper time-derivative behaviour time). with ofthe respect matter The field under Raychaudhuri the equation. equation influence expresses usual In of 4D particular, geometry case. and for The is matterR five-dimensional independent on character of the of the brane, spacetime it enters is only the through same as in the The proper time is notvary in a space. unique label Therefore, we ofcomoving need the hypersurfaces. to comoving transform hypersurfaces, to This because coordinate transformationis it time, given from which can by labels proper (see these time [19] to or coordinate [20]) time This gives the variation ofvation equation proper we time obtain due to the perturbations. Using the conser- leads to an equation for the density constrast In this expression c Intherestofthepaperwedenotebydotthecoordinatetimederivativeofthe perturbations. In this paper we concentrate on the case of which combined with the perturbed Raychaudhuri equation where we have defined the density contrast JHEP10(2001)026 (3.41) (3.44) (3.43) (3.42) . . ) but rather 00 δ . , δP ] ( = 00 δρ ˙ − E δφ a 0 − ρ P a 2 + δp δp V ˙ u 2 p [3 ∂φ ∂ − B − ¯ + P ∂φ 00 δ ∂U ˙ B 2 B E 6 1 U δ a 0 2 consistency equation. From the H ∂φ π + ∂ − a − 00 ˙ i ) u ) ¯ 00 E P ˙ w φ − 3 B δE 0 H c − B δH δπ H +4 (1 c − 4 19 ¨ play a role in the perturbed dynamics. As 6 ρ ¯ φ ∇ 00 ( − 0 δ a + = δP 00 E 2 2 ˙ + 0 B E a a k ˙ δ B H ˙ . Notice that the left-hand side coincide with the φ δπ 3 2 h a (small wavelength), we present solutions only for the + δH − . /a ω ∇ 4 k ) 2 2 s 00 k c − P δφ 0 1+ ( B c = H = 2 δE δH ¯ c 3 ∇ +4 ¯ ∇ − ... 0 ) c )= δφ 0 δP ( a c E ¯ ∇ a ∇ = ( 0 δ a 0 limit. This limit enables us to deduce the effects of the bulk scalar field and . As the Friedmann equation and its time-derivative are given by complicated on density perturbations. Finally we need the perturbed version of the Notice that the components of In the same fashion, we derive the perturbed Klein-Gordon equation, using δP a 00 → µν ∇ where the coupling between the matter and scalar perturbations is explicit. equations we derived in the previous section we obtain expressions, we do not derive a general equation for the evolution of This allows us to write down the necessary perturbed equation. and usual four dimensionalδR expression. The new physical ingredients all spring from 4. Time evolution of cosmologicalIn perturbations this section we discussare very some difficult solutions to to solve theical in eras. perturbed general so dynamics. We start we The restrict inby ourselves the equations to high-energy relativistic regime, the particles. assuming different cosmolog- thenated Then universe to epoch, we be discuss followed dominated by theclosed the on normal the matter (low-energy) brane dominated radiation for domi- epoch. large Because the system is not wherewehaveused can be seen, theyinfluence are of not the bulk constrainedperturbed perturbations by dynamics on the is the brane not brane closeddeal dynamics perturbations. on with and the long For lead brane. this wave-length to phenomena In reason the where a the rest the direct of dynamics this is paper closed. we will only where the shear term vanishes automatically at first order. Similarly we have analyse each regime separately and derive the corresponding equation. eq. (3.35): k E JHEP10(2001)026 + (4.3) (4.6) (4.2) (4.1) (4.7) (4.4) (4.5) 00 P B δH +3 00 δP B H . 2 +3 ˙ φ , read 16 00 t. ˙ 00 15 δ 0 P p + δE δE 3 4 + and . equation in that limit as δP V 51 16 16 ρ − 9 . 00 ˙ ˙ µν φ φ − + 0 + = δ E ˙ δE 4 t 0 φδ B )=0 δ / = 00 δE 3 2 8 3 ˙ = U 00 P ¯ − P δ t H 20 = 00 = = 1 , δE δ + ( 18 E 2 00 B 00 ˙ B ˙ δ φ + H U 16 − 3 ρ δ δP δE 2 ˙ δ / B 4 δH 3 + ˙ +4 t U H . 0 ) δ V +4 8 + − 3 00 ¨ δ 00 = ρ = B 4 δ δE ˙ ( δE U ¯ P. 00 B p B P H + δp δH ρ +4 + − 00 ¯ ˙ P ρ E δ B p B ˙ 4 U + H δ δp ρ − + + = 00 As a first step we will first analyse cosmological versions of the Randall-Sundum In particular we can write the perturbed ˙ E δ and The overall solution of theof equation the for homogeneous the equation densityas as contrast is well a as sum the of general the solution. solution It can easily be found The perturbed quantities are then Explicitly we need as we neglect theexplicit potential in in each the era. different eras. This equationscenario with will a be single made boundarySitter brane, more bulk i.e. spacetime. we considerorder The a leading brane differential to embedded equation the in for appearance Anti-de two the of FRW three density cosmology modes. contrast modes is Thisalready is of derived mentioned to third from two be of compared fourthird to the mode dimensional the modes is usual general entirely will relativity. dueof coincide to conformal with As perturbations invariance in FRW in cosmology the the bulk. while Weyl tensor the signalling4.1 a The breaking Randall-Sundrum scenario 4.1.1 The high energy era In the high energy regime, the equations for This leads to JHEP10(2001)026 (4.9) (4.8) (4.12) (4.10) (4.13) (4.11) read δ and 00 δE , . , 3 00 / . 00 2 0 − δE t δE . 3 4 0 . δE , − 3 8 − 3 . / δE 8 = 4 9 0 + = − 2 )=0 δ t 2 t )=0 δ 2 0 / + 00 δE 2 1 00 1 H − read H δE − t 21 = δE 3 2 t 2 1 δE ( 1 δ δ ( = 00 δ − − H H ˙ + ˙ δ 00 δ + δE t and 3 0 H +4 H / δE +4 δ . 2 00 . ) + ) δt = +2 00 ¨ δ 00 δE ¨ δ δ = δE δE ( δ ( , which is decaying rapidly. 00 We now turn to the case with a scalar field in the bulk. The previous Randall- δE The full solution to the first equation is found to be Again, the solution to the last equation is 4.1.2 The low-energy radiation era In this era, the equations for and the solution to the first equation can then found to be We do, therefore, find theconstant normal mode, growing which and is decaying absent modes in in FRW cosmology. this regime4.1.3 and a The matter dominated epoch Finally, in the matter dominated epoch the equation for We recover thus the normalby growing and decaying mode as well as a mode sourced Sundrum modes will belead modified to in slight two deviationsscale ways. of factor First exponent the in of mode the allto matter exponents the the era. similar brane fluctuations Then to potential of therewill the will will the now also modification scalar analyse appear of each field new of the governed modes the by due different the regimes Klein-Gordon in equation. turn. We The solution to the last equation is JHEP10(2001)026 (4.15) (4.16) (4.18) (4.14) (4.17) (4.19) .Noticethat t ) t , and is therefore . 1 4 / δφ 7 dominate both in the ) and (ln 00 , t p 2 ( / mode is a solution of the 00 3 , 00 t . . ) , δE /t and t δE ) 4 3 . t 00 t δφ ρ ( ln ( 00 2 dt − 1 ρ . δ 0 t, t δ B ) δE δφ 2 4 t + ln ) Z B t 1 4 H αU 4 , / . / 8 3 − δφ αU − 13 t 3 ((ln − 1 + =18 =0 = ) Vδφ 22 δ 0 O 0 . − ˙ t δ ) ( 00 0 + 0 δφ H . t 2 δφ )= dt / ) ( t 3. δE t = 3 + ( t HδE 2 H ¨ δφ δ 0 ρ/ δ Z = ( δ δφ 2 = / H +4 00 +4 = 3 . t p .. ) δ ) 3 4 δE 00 − δφ ( δE ( )= t ( 2 δ is a constant mode. We can now find 0 δφ Considering the large wavelength limit is sufficient because all cosmologically We can now deduce the density constrast To obtain the solutions to these equations, we first consider the Klein-Gordon background and in the perturbationdominate equations. the We assume, expansion, that i.e. relativistic particles relevant scales are far outside theare horizon. The perturbation equations in this regime absent in the Randall-Sundrum case. Moreover for very long times we see that where Notice that the solutionhomogeneous comprises equation two while parts.In the the logarithmic The following mode 1 wehomogeneous solves will modes the always and find modes complete solving that equation. the the complete solutions differential are equations. expressible as a sum of equation. The solutions are 4.2 Effects of the bulk4.2.1 scalar The field high-energy on regime cosmological perturbations In this regime, the terms which are quadratic in In the long time regime,imating we focus on the leading growing mode, obtained by approx- wherewehaveassumedthat3 where the complete solution reads the logarithmic mode is triggered by the scalar fluctuation, This implies that we find two leading growing modes in JHEP10(2001)026 ) δ 2 H (4.20) (4.22) (4.21) (4.23) )whichis . t ln 00 − δφ δE ( 2 / 3 O ) , 00 t 00 , ( 4 3 00 δE − . δE − . . 4 3 ) δφ t δ 2 2 − δφ / t , 1 α ln 1 ( 1 ) 2 α − . δφ 00 − + t 2 t /t ( δφ − αH . In addition, there is the contribution − δφ δ, 3 α δ , 4 αH 2 0 δφ 3 t δφ m − 2 + + Z 23 αH =0 = =4 0 2 0 2 αH . / πGρ δ 2 ) 5 00 2 3 2 t − δφ − δE ) δφ H , which is much larger than in FRW cosmology. 0 = = ( 2 2 t = / ( ˙ )= δ HδE 0 3 H t − δφ 00 H dt δφ ˙ δ ( t +4 +4 2 δE . H 00 .. +2 ) Z H ) t ¨ 2 + δ 00 δE α ¨ δφ δ + ( δE 2 ( / +2 1 . − ) t 2 / δφ 1 ( − δ H + +4 t 1 .. δ ) = δφ ( δ of bulk gravity.gravity In is simply this not regime four-dimensional. we cannot4.2.2 expect Radiation the domination normal behaviour,In because this regime thenot Newton’s appear constant in is the notfor Klein-Gordon varying the in equation. scalar time. This field leads Moreover and matter to the the does density perturbation contrast equations triggered by the decreasing scalar mode in 4.2.3 Matter domination In this regime Newton’s constantdependent. varies in time This as the leadsThe background to scalar perturbation field a equations is read very time rich structure of modes for the density contrast. As before we find homogeneous and complete solutions. More specifically which is much larger than the normal 4 Notice that the firstfrom mode the springs complete from equation. the homogeneous The equation density and contrast the can last then one be deduced: Interestingly, this growth of fluctuations isof anomalous, the in growing the mode sense is that larger the than one. exponent The reason for this is the source term (18 the leading mode increases like leading to the perturbed There are two homogeneous modes asfrom in the FRW cosmology. scalar New field. contributions emerge In particular there is a growing mode in JHEP10(2001)026 1 − − = (4.25) (4.28) (4.27) (4.24) . ) a is rapidly δφ ( . t 00 β 2 α H δE 3and / 225 808 +3 i =2 + .. c . ) t + 3 2 2 a i 00 δ δφ α − ( t β 1 δE 2 = − 64 225 t 3 = / 1 decaying mode. We obtain 2 + αH 00 . − 3 − 1+ ) − − = δφ . ( ) ,a = 2 a in the brane world model. Therefore, 2 ˙ − δ. 1 + 2 (4.26) δφ t α 2 − ( i ,δE − 2 c ˜ a − 45. i t . b 2 t 1 = i 375 β β 24 1036 αH 8 slower i 2 1 3 3 b δφ , − + − − ≈− 2 = 1 = 55. Thus, the growth of fluctuations is smaller, 2 α i . = − a 0 − 00 16 = 225 a 67 very close to the FRW value. That the growing ≈ . E 1 ) 0 3 37. − = + / . 2 ,δφ c 0 3 2 ≈ a δH i − a 3 t ,a = are deformations of the FRW modes due to the scalar field. / i 2 2 = δ 3 1 a / α a 2 − )+4( = δ ˜ a 00 δ 7875 δE 11624 and ( 3 / − H 2 δ 3 2 +4 = . ) 23. Thus, the mode is decaying even faster than normal. The other decaying 3 . / 00 1 2 a − In addition to the growing mode, there are two decaying modes. One of these On top of these three modes there are seven homogeneous modes corresponding δE ( . Defining each mode by the ansatz = 2 mode corresponds to decaying modes is aa modification of the usual to zero modes ofequations. the Let various us differentialThe discuss operators homogeneous them modes appearing of in the in some perturbedexponents the Raychaudhuri detail equation three have as characteristic they coupled have remarkable properties. modes. We find that ˜ decaying with an exponent than in the normalfield matter is growing dominated with epoch. the same The exponent perturbation as the of density the constrast bulk and scalar mode is larger than thematter dominated one universe found is in expanding generaloverdensities, relativity which comes want from the toexpansion, fact can contract that grow but the more have easily to in the compete brane with world model. the cosmological In particular we find that we find that for all modes.solutions of There the are coupledcorresponding two differential to types equations. the of exponents We find modes. that there Let are us three first modes discuss the complete Notice that these modesfield are two potential deformations lifts of the the usual degeneracy FRW modes. between the The scalar doublets of The modes This system of equations possessesα power law solutions, which we derive up to order JHEP10(2001)026 67 . 0 (4.31) (4.29) (4.32) (4.30) (4.33) ≈ 3 / 2 a , , ω 1 2 3 . α 2 − α 45 56 1 + − 512 135 . 3 5 = 2 − α − a 10 + = 45 32 . 3 − / + 5 27 10 = − 3 2 25 = 10 − and leads to two modes 0 − ¯ a = leads to 3 δφ ,a / δ 2 2 − ω α 2 ,a a 3 leads to a single mode with 8 15 + 00 − =0 3 2 67 is not generated and the power spectrum is normalized . 0 δE = 0 ˜ a = 0 ≈ a + a a follow the density contrast. It is instructive to compare these φ 55. The 00-component of the Weyl-tensor is always decaying, whereas equations comprises three differential operators. The first order differ- . 0 00 is the Brans-Dicke parameter. Note, that there are more modes in these ≈ E 3 ω / 2 a The In conclusion, we have found two the growing modes, with exponent ˜ The homogeneous modes of the Klein-Gordon equation are characterized by the and Eventually the operator acting on perturbations in results to the onesand found decaying in modes Brans-Dicke are theory. modified Here as the follows exponent [21]: of the growing corresponding to an almost constant mode and a mode which decreased very fast. ential operator acting on Notice that in thehave epoch thus of described accelerated the perturbation expansion modes the in structures the get four frozen cosmological in. eras. We The differential operator acting on exponents where theories as well. Hereto we the find presence that of the aBrans-Dicke growing theory. scalar and field decaying which modes couples are to shifted gravity in due 4.2.4 a way Observational which consequences differs from Although we have focused(qualitative) on conclusions the from our largemodes findings. wavelength in limit First the only,imposed of matter we all, in can dominated which the draw epoch of earlymode some appear, with the universe. exponent two depends growing It on is the likely initial that conditions both modes are generated. If the to the observations today (suchhave as much more to power galaxy in clusters), thequintessence the matter models, brane perturbations world for than model one example. would would expect Thus, in in normal this case there should be more galaxy JHEP10(2001)026 26 67 contributes significantly to perturbations, then . 0 ≈ a -limit, which we considered in this paper. This involves a study of the k Our findings indicate that there are considerable differences to the Randall- There are other open questions, which our work leaves. As discussed in [22], in Another consequence is a modification of the spectrum of anisotropies in the 5. Conclusions We have presented anarios discussion with of a cosmological bulkspaces, perturbations we scalar were in able field. brane tolimit. find world For solutions sce- As of a the is model perturbation thepertrubation equations motivated equations case in on from with the the large SUGRA brane brane scale bulk are world in equations, not scenarios singular closed. which of is Instead thebations a one Randall-Sundrum where very has type, also difficult analysed to from task the solve themetric in the full general. based five approach In dimensional or [7], pointanother the cosmological of formalism, covariant pertur- approach. view namely the using Here, fluid eitherour in flow the this purposes approach paper introduced this we by formalismthe have Hawking allowed density used [18]. contrast. us For to derive easily theSundrum evolution scenario equation as for well asgrowth of to structures usual is scalar-tensor different thanof theories in perturbations, in these in four models. particular dimensions. A ofto more the The elaborate bulk make investigation perturbations, definitive would be predictionsand necessary the concerning in anisotropies order cluster in abundance, the CMB. large scale structure models based on SUGRAat in some singular distance spaces fromthis we our mirror expect brane brane a universe. second on We brane the have in not structures the addressed on bulk the our influence brane of universe. Similarly, we haven’t clusters at high redshiftwith in comparable matter the density brane (withIfthemodewithexponent world a cosmological model constant than or quintessence inthe field). normal power Einstein in gravity themore perturbations work is in needed both in theories order would to be make thiscosmic similar. more microwave background concrete. However, radiation much (CMB).last This scattering is because surface thematter (LSS) distance dominated will to epoch. the be Asscales. modified a result, due the However, to first wethis peak the believe will effect, slower this be it expansion effect shifted to isradiation, in to larger conceivable for be the angular example) that small might isocurvature toimprint survive in modes be the quite (between CMB. detected. long To the makethe in more In Weyl the small concrete fluid addition predictions, radiation it to and era, isbulk necessary leaving gravitational to their field, go beyond asthe well brane as [7]. perturbations The calculations inequations are the is very beyond bulk difficult and the scalar a scope field detailed of away discussion this from of the paper bulk and will be presented elsewhere. JHEP10(2001)026 (A.4) (A.5) (A.6) (A.7) (A.1) (A.3) such that a p , , belongs to the vector multiplet =0 x i = 0 (A.2) . =0 λ j j is fixed by the boundary condition. j x x i ± t ij φ − i λ ij z p 2 a Q ij ∂ p γ a ,x + a Q p p p WQ 2) and Γ + i W 27 i µ , = γ ,δ ± + x i = 8 i Γ= φ i =1 = a + x =0 i ∂ ± i i φ µ µi µ ∂ = 1. Define the vector ψ µ D a δ . This leads to the first order equations iγ x Q a φ Q and j i can be split into positive and negative chiralities ) i . The unknown component a t σ p ( a = 1. Rotation invariance implies that the only non-zero components Q 2 is the gravitino spinor ( = j i µi ψ =1and Q z p Now the spinors where where A. Supersymmetric backgrounds A.1 BPS configurations In this appendix wethe will bulk. consider To do configurations so which we preserve will supersymmetry look for in Killing spinors satisfying the identities We are grateful to Brandon Carter,Andrew Jai-Chan Mennim Hwang, Zygmunt and Lalak, David Roy Wands Maartens, forby discussions. the Deutsche This Forschungsgemeinschaft work (DFG) was and supported PPARC in (UK). part Acknowledgments discussed the dynamics of theencompassed radion in in the these Weyl tensor. models.work. We The will, effects however, of turn these to should these be questions in future and the matrix such that Γ satisfying comprising the scalar field and are JHEP10(2001)026 (A.8) (A.9) (A.12) (A.13) (A.14) (A.10) (A.16) (A.11) . ) ] c p d + i [ a g b Γ) p ,x − W ] . c ,x ) p i d [ W b = 0 is automatically satisfied a dx g i γ . z a b + i p p dx 2 ( . p cd ,x 8 =0 − + ,x γ p b j 2 W . γ W . ,x a ij 2 dz abcd p p 1 W R = 0 (A.15) + =1 8 1 2 =0 = 0 (A.17) p 2 2 WQ p 1 28 = 2 + p µ 4 ± ab dt abcd γ + i ab E p − i 8 = ] W ∆Φ γ ( b T ) 2 )+ x + bc φ z,t i ( W z g ,D a A ∂ µ 1 ad 2 64 D g e D [ − − = = 2 bd . More generally we find that the background geometry g + i 5 ds ac ] b g ( 2 AdS ,D a W D 1 [ 16 − = = 0, i.e. at the critical points of the superpotential one recognizes the abcd ,x R W Let us come to the bulk evolution of the Killing spinors. They satisfy provided which preserves supersymmetry in the bulk is such that the Weyl tensor vanishes as the Weyl tensorsantisymmetrized product form, of in the metric theThis tensor set implies with that of itself or the curvature with bulk tensors, a geometry symmetric the is tensor. conformally complement flat to and the can be written as is fulfilled. Now one gets from which we deduce that Riemann tensor of At each point thissupersymmetries. select the Now (A.3) chirality is defined satisfied by provided Γ, i.e. we preserve one half of the This leads to the vanishing For This is thepositive BPS sign, we equation find for that the supersymmetric boundary condition configurations. at Now choosing the This leads to the vanishing of the projected Weyl tensor This is a firstbility order condition differential equation which can be solved provided the integra- JHEP10(2001)026 (A.24) (A.20) (A.23) (A.21) (A.18) (A.22) ) (A.19) b . The only solution is 5 p a γ p 6= . +3 ,x ab g W 2 x j 0 ab p ± . T ( W B ,x ij ξ 4 = = 1 case satisfy the projection equation U 2 Q W H. − +2 p 5 ab T − ,x 4 g x 2 = iγ 29 2 = R W 4 B = ± W 5 A ˜ − H + x = z − ∂ ab ab ± i g R = 2 4 R W 2 a − = ab R 6= 1. Notice that the breaking of supersymmetry is global due to the T = 0, i.e. supersymmetry is completely broken by the non-supersymmetric = 0. This is not compatible with (A.7) as Γ t So we have shown that breaking supersymmetry on the brane leads to broken Let us perform a small change of coordinates in the bulk Let us now consider the second brane where supersymmetry is not broken. Now p = 0 background configurations which still satisfy a system of two first order BPS N conditions. A.2 Breaking conformal flatness Let us now considerWe the will case where show matter thatboundary and one conditions radiation are cannot are satisfied. deform presentconformal This on the flatness implies the of bulk that brane. the matter geometry bulk. on in the brane such breaks a the way that the presence of two boundaries respecting incompatible supersymmetries. therefore brane with This implies that Einstein equations and the curvature scalar as are satisfied with in the bulk. Moreover the bulk and brane expansion rates coincide Notice that (A.14) leads to the Ricci tensor Notice that (A.22) anduniverse. (A.8) are Solutions the of these BPSGordon equations equations equation. satisfy which lead the Einstein to the equations and accelerating the Klein- the Killing spinors for the supersymmetric JHEP10(2001)026 , B (A.30) (A.31) (A.25) (A.29) (A.26) (A.32) (A.27) (A.28) =0.This ρ Nucl. Phys. , . c ξ a . φ∂ 0 ca | b g ∂ z c aa aa ∂ ∂ g c δg ξ 0 0 + b | | c The universe as a domain wall a ∂ 0 ξ ξ aa | b aa g + aa ]. . z aa g g a φ∂ cb g ∂ . a a ξ z g b ∂ 6 ρ z ∂ ∂ − =0 c ∂ a ∂ B t 0 c 2 ∂ ξ = | U ξ ξ a + ) z 0 a ∂ b | − ∂ ∂ i ξ aa t )= ξ a g 30 p aa φ = + ii z ∂ =2 b g g ∂ 0 + ∂ i 0 | ( ab boundary condition then reads = | a ∂ 0 δ g ) | ∂ c ii ]. z hep-th/9803235 B ab ( aa g ∂ 2 ξ = c aa U g z δg =0leadsto ξ aa z g 0 ,δ ∂ | z z g ∂ c 0 , vols. 1 and 2, Cambridge University Press, Cambridge | ∂ ( ∂ ξ ) δ a φ aa z δ φ∂ )= aa g ∂ c independent this cannot be satisfied unless z g ( ab ∂ ∂ δ i g z x ∂ δ (1999) 086001 [ ( )= hep-th/9602070 δ φ a String Theory ∂ ( δ Strong coupling expansion of Calabi-Yau compactification D59 (1996) 135 [ 1999. 471 Phys. Rev. [1] J. Polchinski, [2] E. Witten, [3] A. Lukas, B.A. Ovrut, K.S. Stelle and D. 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