arXiv:hep-th/0404161v2 1 Nov 2004 hr ige oiietninbaei meddi a in embedded (AdS Sitter is de anti brane (5D) positive-tension five-dimensional par- model, single, (RS) Of a hypersurface Randall-Sundrum where the brane space. is bulk a importance higher-dimensional as ticular a viewed four-dimensional in be observable embedded may our universe that (4D) possibility the in rnucdta nteR oe [9]. model RS less the are in energies low than at pronounced [8] law braneworld Newton’s GB from the deviations gravi- in and localized the remains Moreover, mode [7]. zero order ton is spacetime next-to-leading it flat and the action, about effective as ghost-free string arise heterotic the the also in since may correction [6]. and term derivatives metric second GB linear A bulk only the contain equations gravita- in field second-order equations the to field represents leads tional it partic- that since Gauss-Bonnet of combination dimensions, The unique is five invariants in relevance curvature [5]. ular of CFT combination the (GB) to next-to-leading corrections as correspondence order terms AdS/CFT Such the higher- action. in bulk arise include the to in invariants is curvature order considerations, theory gravity string/M 4D [4]. by as (CFT) viewed theory un- is field conformal model be a RS also to the coupled correspon- can where AdS/CFT property the [3], of This dence context warped the the within bulk. to derstood due the brane of the on geometry is energies the waves, low of gravitational at zero-mode 4D localized di- the to fifth extent, corresponding graviton, the in 5D Although infinite be [2].) may Ref. mension see reviews, recent (For nrcn er,teehsbe osdrbeinterest considerable been has there years, recent In aua xeso fteR oe hti motivated is that model RS the of extension natural A omlgclprubtosfo rnwrdiflto with inflation braneworld from perturbations Cosmological otetno mltd.I diin h esrt clrra scalar to tensor the h addition, perturbations In scalar of Us amplitude. relation. amplitude tensor level. the the standard that to the show above we rise initial effects, increasing an monotonically i to after longer as leads no totically is conditions spectrum amplitude junction same The the the waves. to have change and analys Gauss-Bonnet equation exact the wave an the give same We string the case. by satisfy Randall-Sundrum motivated the correction to high-energy changes relation. a resul consistency as standard standard action, the the tensor to obeys of relative still amplitudes increased amplitude strongly The is braneworld. flation 5-dimensi observabl the type When Randall-Sundrum The spacetime. a Sitter inflation. de anti relativistic 5-dimensional general to ifications 3 rnwrdiflto sapeoeooyrltdt tigth string to related phenomenology a is inflation Braneworld nttt fCsooy&Gaiain nvriyo Portsm of University Gravitation, & Cosmology of Institute .INTRODUCTION I. enFaci Dufaux Jean-Francois 4 ne-nvriyCnr o srnm srpyis Pun Astrophysics, & Astronomy for Centre Inter-University ue ay nvriyo odn odnE N,UK 4NS, E1 London London, of University Mary, Queen 1 2 P,Uiested ai-u,945Osy France Orsay, 91405 Paris-Sud, de Universit´e LPT, srnm nt colo ahmtclSciences, Mathematical of School Unit, Astronomy 5 pctm [1]. spacetime ) 1 Dtd coe 9 2018) 29, October (Dated: ae .Lidsey E. James , otiiga4 rn,tegaiainlato is action gravitational the brane, 4D a containing vnwe h Bcretosaevr ml eaieto relative small term, very GB terms. are the Einstein-Hilbert corrections by the GB introduced significant the are that when case show even RS We the in- braneworld. to changes GB during the de- generated the in above determine perturbations flation the to tensor of important of view is properties In it standard gravity. therefore, ener- the Einstein velopments, to high 4D relative At in enhanced [13]. result is scenario amplitude RS the the gies, in evo- determined The has inflation been universe. slow-roll early during win- waves very gravitational unique the of a lution of open physics would the into detection dow a background microwave Such cosmic [12]. the imprint(CMB) of its polarization from primordial the detectable of on be spectrum could quantum- resulting waves gravitational are The scale-invariant graviton nearly [11]. the acquire fluctuations and as in- excited such During mechanically fields [10]. light expansion flation, (inflationary) epoch an accelerated underwent universe of early very the that evidence o Dbl ihEnti-as-ontgravity, Einstein-Gauss-Bonnet with bulk 5D a For rma bevtoa esetv,teei o strong now is there perspective, observational an From 2 faGusBne emi nlddi the in included is term Gauss-Bonnet a If o Maartens Roy , nvrei rnwrdebde in embedded braneworld a is universe e s lhuhtertoo esrt scalar to tensor of ratio the although ts, n napoiainta elcsbulk neglects that approximation an ing S iheeg cl,btdcessasymp- decreases but scale, energy with nlato sEnti-ibr,w have we Einstein-Hilbert, is action onal r,w hwta hr r important are there that show we ory, = oie mltd fgravitational of amplitude modified a oyta ecie iheeg mod- high-energy describes that eory i rastesadr consistency standard the breaks tio saqaiaieysmlrbehaviour similar qualitatively a as so h esrprubtos They perturbations. tensor the of is uh otmuhP12G UK 2EG, PO1 Portsmouth outh, n clrprubtosfo in- from perturbations scalar and 2 h adl-udu ae but case, Randall-Sundrum the n − κ 1 + I IL EQUATIONS FIELD II. 5 2 Z α Z brane d R 5 2 3 x d Sami M , p − ,India e, 4 as-ontterm Gauss-Bonnet a x − 4 √ R (5) − ab g , σ g R [ − ab 4 2Λ + 5 R + abcd R R abcd (1) 2
a µ (5) where x = (x ,z), gab = gab nanb is the induced where we choose in Eq. (12) the branch with an RS limit, metric, with na the unit normal to− the brane, σ (> 0) is and µ is the energy scale associated with ℓ. This reduces 2 the brane tension, and Λ5 (< 0) is the bulk cosmological to the RS relation 1/ℓ = Λ5/6 when α = 0. Note that constant. The fundamental energy scale of gravity is the there is an upper limit to the− GB coupling from Eq. (12): 2 3 5D scale M5, where κ5 =8π/M5 . The Planck scale M4 1016 TeV is an effective scale, describing gravity on the∼ ℓ2 brane at low energies, and typically M4 M5. α< , (13) The GB term may be thought of as the≫ lowest-order 4 stringy correction to the 5D Einstein-Hilbert action, with 2 which in particular ensures that Λ5 < 0. coupling constant α > 0. In this case, α , so A Friedman-Robertson-Walker (FRW) brane in an that |R | ≪ |R| AdS5 bulk is a solution to the field and junction equa- α ℓ2 , (2) tions [15]. The modified Friedman equation on the (spa- ≪ tially flat) brane is [15, 16] where ℓ is the bulk curvature scale, ℓ−2. The RS |R| ∼ 2 2 2 2 2 type models are recovered for α = 0. κ5(ρ + σ)=2 H + µ 3 4αµ +8αH . (14) The 5D field equations following from the bulk action − p are This may be rewritten in the useful form [17] α = Λ (5)g + , (3) 1 2χ Gab − 5 ab 2 Hab H2 = (1 4αµ2)cosh 1 , (15) 4α − 3 − = 2 4 cd + cdef (5)g ab cd cdef ab 1/2 H R − R R R c R 2(1 4αµ2)3 4 [ ab 2 ac b 2 κ5(ρ + σ) = − sinh χ , (16) − RR −cd R R cde α 2 acbd + acde b . (4) − R R R R The junction conditions at the brane, assuming mirror where χ is a dimensionless measure of the energy density. Note that the limit in Eq. (13) is necessary for H2 to be (Z2) symmetry, are [14] non-negative. κ2 When ρ =0= H in Eq. (14) we recover the expression K Kg = 5 (T σg ) µν − µν − 2 µν − µν for the critical brane tension which achieves zero cosmo- 1 logical constant on the brane, 2α Qµν Qgµν , (5) − − 3 2 2 κ σ = 2µ(3 4αµ ) . (17) 5 − where The effective 4D Newton constant is given by [8] α αβ Qµν = 2KKµαK ν 2KµαK Kβν − 2 µ 2 αβ 2 + K K K K +2KR κ4 = 2 κ5 . (18) αβ − µν µν (1+4αµ ) + RK +2KαβR 4R Kα . (6) µν µαβν µα ν 3 2 − When Eq. (2) holds, this implies M5 M4 /ℓ. Table- Here the curvature tenors are those of the 4D induced top experiments to probe deviations from≈ Newton’s law 5 metric gµν , Kµν is the extrinsic curvature and Tµν is the currently imply ℓ < 0.1 mm, so that M5 > 10 TeV, and brane energy-momentum tensor. For a vacuum bulk, the σ > (1 TeV)4, by∼ Eqs. (2) and (17). ∼ conservation equations hold: ∼The modified Friedman equation (15), together with Eq. (16), shows that there is a characteristic GB energy ν T =0 . (7) ∇ µν scale, An AdS5 bulk satisfies the 5D field equations, with 2(1 4αµ2)3 1/8 mα = − , (19) 1 (5) (5) (5) (5) 4 ¯ = g¯ g¯ g¯ g¯ , (8) ακ5 Rabcd −ℓ2 ac bd − ad bc h i ¯ 6 (5) (5) α ¯ such that the GB high energy regime (χ 1) is ρ + ab = g¯ab = Λ5 g¯ab + ab , (9) 4 ≫ G ℓ2 − 2 H σ m . If we consider the GB term in the action as a ≫ α 24 (5) correction to RS gravity, then mα is greater than the RS ¯ab = g¯ab . (10) 1/4 H ℓ4 energy scale mσ = σ , which marks the transition to RS high-energy corrections to 4D general relativity. By It follows that Eq. (17), this requires 3β3 12β2 + 15β 2 < 0 where 6 12α β 4αµ2. Thus (to 2 significant− figures),− Λ5 = + , (11) ≡ −ℓ2 ℓ4 2 mσ 2 2Ht 2 Expanding Eq. (15) in χ, we find three regimes for the where γµν is the 4D de Sitter metric ( dt + e d~x ), dynamical history of the brane universe: and the conformal warp factor is − the GB regime, H A(z)= , (30) κ2 2/3 µ sinh Hz ρ m4 H2 5 ρ , (21) ≫ α ⇒ ≈ 16α with Z2 symmetry understood. The brane is at fixed position z = z > 0, which we can choose so that A(z )= the RS regime, 0 0 1 (i.e., sinh Hz = H/µ). The horizon is at z = . 0 ∞ κ2 Consider now the 5D spin-2 metric perturbations, 4 4 2 4 2 (5) (5) (5) (5) mα ρ σ mσ H ρ , (22) g¯ g¯ + δ g , where δ g is 5D transverse ≫ ≫ ≡ ⇒ ≈ 6σ ab ab ab ab traceless.→ For these perturbations, Eq. (4) shows that a the 4D regime, δ b = 0, so that the wave equation for the perturba- tionsH is 2 2 κ4 ρ σ H ρ . (23) a ≪ ⇒ ≈ 3 δ b =0 , (31) R The GB regime, when the GB term dominates gravity at the same as in the RS case. This means that the bulk the highest energies, above the brane tension, can use- mode solutions for metric perturbations will be the same fully be characterized as as in the RS case [13], but the GB junction conditions will introduce changes to the normalization and amplitudes H2 α−1 µ2 , H2 ρ2/3 . (24) ≫ ≫ ∝ of the modes, as discussed below. In the gauge δ (5)g = 0, we can write the perturbed The brane energy density should be limited by the az 4 metric in the form quantum gravity limit, ρ < M5 , in the high-energy regime. By Eq. (21), (5) 2 (5) 2 1/2 µ ν ds = ds¯ + A hµν dx dx , (32) 1/3 πM 1/2 ρ 2 the usual norm for α = 0. [q(z) − φ (z)] = m w(z) φ (z) , (38) −D+ D m m With the orthonormal conditions in Eq. (45), the effec- tive action for the metric perturbation, to second order, where we define the operators is: d 3 A′ ± = , (39) 1 1 4 (m)µν (m) D dz ± 2 A S = d x√ γ h h 2κ2 4 − µν 5 Z and the factors h 2 (m)µν (m) 2 (m)µν (m) 2H h hµν m h hµν , (46) −4 ′2 2 2 − − q = 1 4α A A A H , (40) − − i −4 ′′ ′2 w = 1 4α A AA A . (41) where the term in braces is the standard one for 4D (mas- − − sive) gravitons on a de Sitter background. This form of the wave equation explicitly incorporates We now consider the spectrum of modes resulting from the GB junction condition. By Eq. (30), Eqs. (43) and (44). There is a normalizable bound-state − ′ zero-mode, as in the RS case: w =1 4αµ2 4αA 3[A ]δ(z z ) , (42) − − − 0 φ (z)= C A3/2(z) , (47) ′ ′ + 2 2 0 where [A ]=2A (z0 ) = 2 µ + H is the jump in ′ − A across the brane. Note from Eqs. (30), (40) and (42) where the real constant C will be determined in the fol- p 2 that, for z = z0, q = w = 1 4αµ . Thus for z>z0, lowing. The asymptotic value of the Schrodinger poten- Eq. (38) reduces6 to the Schr¨odinger-type− equation, 9 2 tial in Eq. (43), i.e., 4 H , gives the threshold between 2 9 2 2 the discrete and continuous spectra: m > 4 H , as in ′′ 15 H 9 2 + −φm = φ + + H φm the RS case [13, 23]. For the massive modes in the con- − D D − m 4 sinh2 Hz 4 tinuous tower, the two linearly independent solutions of = m2 φ , (43) Eq. (43) oscillate with constant amplitude for z . m → ∞ 5 3 The boundary condition Eq. (44) gives φm(z) as a par- a continuum of states with m> 2 H, ticular combination of these two solutions, for every m. • These modes are normalizable as plane waves and form as in the RS case α = 0 [13, 23]. the continuous spectrum of Eqs. (43) and (44). This feature is crucial for discussing stability issues 2 9 2 as well as the gravitational waves produced along the For m < 4 H on the other hand, Eq. (43) admits only one independent normalizable solution for each m. The brane. In particular, the spectrum rules out the ex- corresponding mode behaves as a decreasing exponential istence of 4D massive gravitons with m2 < 2H2 in for z . For α = 0, the only such mode which satisfies Eq. (34), which would have signalled a classical insta- the junction→ ∞ condition is the massless mode, Eq. (47). In bility of the model [19] (see also Ref. [26] for a recent GB gravity however, this issue is more subtle because discussion in the braneworld context). It has been shown of the explicit dependence of the boundary condition that a mass gap for de Sitter branes is quite generic Eq. (44) on m2. in Einstein gravity [20]. In particular it still holds if a In order to see whether the junction conditions allow second Z2-symmetric brane is introduced in the back- for discrete states other than the zero mode, it is conve- ground, Eq. (29), say at z = z2 > z0. We just note nient to introduce the new modes: here however that we can not reach the same conclusion in Gauss-Bonnet gravity. In particular, the argument Φm(z)= − φm(z) , (48) above would fail in this case, because the new boundary D term at z = z < in Eq. (51) would then be nega- 2 ∞ which are the partners of the modes φm in super- tive (while it still vanishes for α = 0). In fact, if we solve symmetric quantum mechanics [25]. They have the same Eq. (43) explicitly and impose Eq. (44), we can show that spectrum except for the zero-mode: Φ0 vanishes identi- tachyonic modes with m2 < 0 (< 2H2) may exist for the cally, by Eqs. (39) and (47). The wave equation for Φm 2-brane system with α> 0 (as well as for the 1-brane case is found by applying − to Eq. (43): with α < 0). This system may therefore suffer from the D 2 same spin-2 tachyonic instability present for Minkowski ′′ 3 H 9 2 branes [27]. (Note that the tachyonic instability in the − +Φm = Φ + + H Φm − D D − m 4 sinh2 Hz 4 case of two de Sitter branes with Einstein gravity [22, 28] 2 = m Φm . (49) is a spin-0 radion mode.) The boundary condition follows from Eqs. (43) and (44), V. AMPLITUDE OF THE ZERO-MODE 2 ′ + 1 4αµ 3 2 2 Φ (z )= − + µ + H Φm(z0) , (50) We can now estimate the spectrum of graviton fluctu- m 0 2 2 2 4α µ + H ations generated in de Sitter inflation on the brane, by p treating each mode as a quantum field in four dimen- for α = 0, whilep Φ (z )=0 for α = 0. m 0 sions, as in the RS case [13, 20] (see Refs. [29, 30] for a In particular,6 this boundary condition no longer in- five-dimensional approach). volves the mass of the modes (and reduces to Dirichlet- For m2 > 9 H2, the massive modes are strongly sup- type for α = 0). This is the essential property we need. 4 pressed on large scales and remain in their vacuum Multiplying Eq. (49) by Φm and integrating by parts, we find that state [13, 20]. These modes can therefore be neglected in the following. However, the zero-mode is over-damped ∞ ∞ 2 2 9 2 2 3 2 Φm and acquires a spectrum of classical perturbations on m H dz Φm = H dz 2 super-horizon scales. For m2 = 0, the effective action − 4 z0 4 z0 sinh Hz ∞ Z ∞ Z Eq. (46) has the standard form of 4D general relativ- ′2 ′ ity, except for the overall factor κ2 instead of κ2, which + dz Φm + ΦmΦm . (51) 5 4 z0 − (0) Z z0 rescales the amplitude of quantum fluctuations in hµν ac- cordingly [13]. Thus the amplitude of gravitational waves Consider now a would-be normalizable mode φ with m produced on super-horizon scales on the brane is given m < 3 H. Its partner Φ must decrease exponentially 2 m by when z , as does φm. The corresponding upper boundary→ term ∞ at infinity in Eq. (51) therefore vanishes. H 2 The lower boundary term on the brane, by Eq. (50), is A2 = κ2 φ2(z ) . (52) T 5 0 0 2π positive for the minus branch of solutions, defined in Eq. (12), and for α 0. It vanishes for α = 0. Thus in this case, the right-hand≥ side of Eq. (51) is positive, The normalization of the discrete zero-mode, φ0(z0)= C, while the left-hand side is negative. This can be satisfied introduces further rescaling relative to the 4D result. By Eqs. (45) and (47), the condition (φ , φ ) = 1 gives: only for Φm = 0, i.e., for m = 0. We therefore conclude 0 0 that the spectrum of KK modes consists only of (1+4αµ2) C−2 = F −2(H/µ) , (53) the massless bound-state zero-mode, µ α • 6 where we used Eq. (18), and where 2 − 1 4αµ − 1 F 2(x)= 1+ x2 − x2 sinh 1 . (54) α − 1+4αµ2 x p This generalizes the function F0(x) found for the RS 12 case [13]. When x H/µ 0, we have Fα 1; the amplitude of the normalized≡ → zero-mode on a Minkowski→ brane measures the ratio between the effective 4D New- 10 2 2 ton constant κ4, and the 5D constant κ5. The modified tensor amplitude is therefore 8 H 2 A2 = κ2 F 2(H/µ) , (55) amplitude T 4 2π α 6 and the correction to standard 4D general relativity lies in the last factor: 4 2 2 AT Fα = 2 . (56) 2 [AT ]4D This correction depends on the GB coupling α and on the 20 40 60 80 100 120 140 160 energy scale at which inflation occurs, relative to the 5D energy scale curvature scale µ, and it reduces to the result of Ref. [13] 2 for the RS case α = 0. (The correction to the 4D result FIG. 1: The dimensionless amplitude Fα of the tensor zero- may also be expressed via an effective Planck mass during mode relative to the 4D general relativity result, plotted against the dimensionless energy scale of inflation, H/µ. (The inflation, following Ref. [20].) ≡ 2 −3 The GB term introduces significant corrections to the Gauss-Bonnet coupling is given by β 4αµ = 10 .) RS case. In the GB regime, as characterized by Eq. (24), we have − (1+4αµ2) H 1 F 2(H/µ) , (57) α ≈ 8αµ2 µ while the RS case α = 0 yields 14 3 H F 2 (H/µ) . (58) α=0 ≈ 2 µ 12 Thus the GB term suppresses tensor perturbations rel- ative to the 4D result, at high energies, unlike the RS 10 case where the tensor amplitude is strongly enhanced. If we consider the GB term as a perturbative correction to 8 the Einstein-Hilbert 5D action, then β 4αµ2 1, and ≡ ≪ there is an RS regime as the energy drops (but remains amplitude above the brane tension). Thus we expect that the tensor 6 amplitude is enhanced at lower energies (RS regime) and suppressed at higher energies (GB regime), so that there 4 is a maximum at intermediate energies. This qualitative behaviour is confirmed in Figs. 1 and 2. 4 2 The quantum gravity limit ρ 1000 1 800 0.8 600 0.6 400 scalar amplitude tensor/ scalar ratio 0.4 200 0.2 0 20 40 60 80 100 120 140 .1 .1e2 .1e3 energy scale energy scale 2 FIG. 3: The dimensionless amplitude Gα of density pertur- FIG. 4: The ratio of tensor to scalar perturbations, R = 2 2 bations relative to the 4D general relativity result, plotted AT /AS, relative to the 4D general relativity ratio, plotted against the dimensionless energy scale of inflation, H/µ. (The against the dimensionless energy scale of inflation, log(H/µ). − 2 −3 Gauss-Bonnet coupling is given by β ≡ 4αµ2 = 10 3.) (The Gauss-Bonnet coupling is given by β ≡ 4αµ = 10 .) where tensors are enhanced less strongly than scalars, so that the tensor/ scalar ratio R = A2 /A2 is suppressed in ǫ 2(1 β)4 sinh 2 χtanh χ sinh2 χ T S = − 3 , (68) comparison with the standard case. When there is a GB 2 2 ǫRS 9(1 + β)(3 β)[(1 β)cosh χ 1] term, Eqs. (57) and (70) show that the scalars are more − − 3 − η 2(1 β)3 sinh2 χ strongly suppressed at high energies than the tensors, so = − 2 , (69) that the tensor/ scalar ratio is enhanced at high energies. ηRS 3(1 + β)(3 β)[(1 β)cosh χ 1] − − 3 − At lower and intermediate energies the ratio is a compli- ′2 2 3 ′′ 2 2 cated function of x, given the interplay between RS and and ǫRS 2σV /(κ4V ) and ηRS 2σV /(κ4V ) are the corresponding≡ RS slow–roll parameters≡ [38]. GB effects. It follows from Eqs. (54) and (66) that As in the case of tensor perturbations, the scalar per- R turbations with GB corrections behave very differently = (71) R4D compared to the RS case. At high energies, the GB term √ 2 2 3 again leads to a suppression of scalar perturbations rel- [2 1+ x (3 β +2βx )+2(β 3)] − − −1 − , ative to the standard result. In the GB regime, as char- 27(1 + β)2x6[(1 + β)√1+ x2 (1 β)x2 sinh x 1] − − acterized by Eq. (24), we have 2 2 where R4D = [AT /AS]4D. The ratio of tensor to scalar 27 1+ β 3 1 amplitudes, relative to the standard ratio, has a maxi- G2 . (70) mum at low energies, a minimum at high energies, and α ≈ 64 β x3 grows like x2 at very high energies. This is illustrated in By contrast, in the RS case, scalar perturbations are Fig. 4. strongly enhanced at high energies. Thus we have a sim- ilar qualitative behaviour to the tensor case: there is an RS regime of amplification at lower energies, and a GB B. Consistency relation regime of suppression, with a maximum at intermediate energies. The qualitative behaviour of the dimensionless The consistency relation for the GB braneworld is amplitude of scalar perturbations is shown in Fig. 3. derived by differentiating the gravitational wave ampli- Furthermore, the different scaling of the scalar and tude, Eq. (55), with respect to comoving wavenumber tensor amplitudes at high energies leads to another in- k(ϕ) = a(ϕ)H(ϕ). In the extreme slow-roll limit, varia- triguing difference from the RS case. In the RS case, tions in the Hubble parameter are negligible relative to 9 changes in the scale factor. This implies that the tensor spectral index can be expressed as − d ln (xF ) 2 a dH 2 n = α . (72) T − d ln x H da The GB braneworld correction to the gravitational am- 1.8 plitude, Eq. (54), satisfies an important first-order differ- ential equation: 2 2 d −2 2Fα[1 + β(1 + x )] 1.6 ln(xFα) = . (73) d ln x − (1 + β)√1+ x2 Furthermore, the scalar field equation can be expressed in the form 1.4 degeneracy factor dH dH V ′2 a = . (74) da − dV 3H2 1.2 Hence, substituting Eqs. (55), (64), (73) and (74) into Eq. (72) implies that the tensor to scalar ratio is given by 1 20 40 60 80 100 120 140 2 AT Q energy scale 2 = nT , (75) AS − 2 FIG. 5: The tensor/ scalar degeneracy factor Q, plotted where against the dimensionless energy scale of inflation, H/µ. (The − Gauss-Bonnet coupling is given by β ≡ 4αµ2 = 10 3.) 6 [1 + β(1 + x2)] dH Q−1 = H . (76) 2 2 κ4 (1 + β)√1+ x dV The function Q(H) determines to what extent the de- form of the degeneracy factor is Q 2. In this limit the → generacy of the consistency equation is lifted in GB consistency equation is given by braneworld inflation and we therefore refer to it as the 2 “degeneracy factor”. For our normalization conventions, AT 2 = nT . (79) it takes the value Q = 1 in the standard and also the RS AS − inflationary scenarios. The behaviour of Q as a function of energy scale is shown Differentiating Eqs. (15) and (16) with respect to χ, in Fig. 5. Eq. (76) then implies that It is interesting that Eq. (79) is independent not only (1 β)cosh χ of the specific form of the inflaton potential, but also Q = − , (77) of the parameters of the model, specifically the brane 1+2(1 β) sinh2(χ/3) cosh(χ/3) − tension, σ, the bulk cosmological constant, Λ5, and the where the result has been simplified by employing GB parameter, α. As in the standard 4D and the 5D Eq. (18). Using the identity cosh χ + cosh(χ/3) = RS scenarios, the consistency relation can be expressed 2 cosh(2χ/3) cosh(χ/3), we find that the degeneracy fac- entirely in terms of observable parameters and, in this tor takes the simple form: sense, Eq. (79) may be viewed as a model-independent observable signature of GB braneworld inflation in the 1+ β +2βx2 high energy limit, Eq. (21). Q = . (78) 1+ β + βx2 We may conclude immediately from Eq. (78) that VII. CONCLUSIONS GB effects lift the degeneracy of the consistency equa- tion, since the factor Q depends directly on the energy Brane inflation offers a phenomenology that allows us scale, x, corresponding to the time when the observ- to explore some of the cosmological implications of ideas able modes went beyond the Hubble radius during in- arising from string and M theory. The effects on inflation- flation. The standard form of the consistency equation ary perturbations from the extra dimensional nature of (Q = 1) is recovered for α = 0, and also in the limit gravity introduce new features that need to be computed βx2 4αµ2x2 1, corresponding to the regimes of and then subjected to the constraints from high-precision Eqs.≡ (22) and (23)≪ in the history of the GB braneworld. cosmological data. Here we have concentrated on com- However, in the GB regime of Eq. (24), the asymptotic puting the corrections to the standard results for tensor 10 and scalar perturbations that are generated during slow- Because the scalar suppression is stronger than the roll inflation at energies where brane effects become dom- tensor suppression in the GB regime, the relative ten- inant. This has previously been done for the Randall- sor contribution, as a fraction of the scalar amplitude, 2 2 Sundrum braneworld, based on 5-dimensional Einstein R = AT /AS, is enhanced in the GB regime, in compari- gravity. We have introduced a Gauss-Bonnet term, since son with the standard result. This is shown by Eq. (71): string theory arguments indicate that this term is a high-energy perturbative correction to the gravitational R = action. This correction leads to significant qualitative R changes, even in the perturbative regime β 4αµ2 1. 4D √ 2 2 3 For the tensor perturbations, we have given≡ an exact≪ [2 1+ x (3 β +2βx )+2(β 3)] − − − . analysis, including the 5D modes. The wave equation 27(1 + β)2x6[(1 + β)√1+ x2 (1 β)x2 sinh 1 x−1] and its fundamental solutions are not changed by the − − GB term. The spectrum contains a normalizable zero- By contrast, in the RS case the relative tensor contribu- mode and a continuous tower of massive modes after a tion is suppressed. 3 mass gap, m > 2 H, as in the RS case. The massive Furthermore, the consistency relation between the ten- modes are not excited during inflation, as in the RS case. sor/ scalar ratio and the tensor spectral tilt is different However, the GB term changes the boundary conditions in the GB case, i.e., at the brane, and therefore changes the normalization of the zero-mode, as shown by Eq. (54): 1+ β +2βx2 n R = T , 2 −2 − 1+ β + βx2 2 AT 2 1 β 2 −1 1 2 = 1+ x − x sinh . [AT ]4D − 1+ β x p by Eq. (78). The RS model by contrast has the same This leads in the GB regime to a suppression of tensor 1 consistency relation as the standard case, R = 2 nT . perturbations relative to the standard result, unlike the Our results provide a basis on which to confront− the enhancement that arises in the RS case β = 0. GB braneworld with observational constraints, and this For the scalar perturbations, we used an approxima- is under investigation. tion where bulk perturbations decouple from the density perturbations. We showed that the GB modifications to the Friedman equation lead to a significant change from the RS case, as given by Eq. (66): A2 3(1 + β)x2 3 Acknowledgements: S = . 2 2 2 We thank Pierre Binetruy, Christos Charmousis, [AS ]4D 2√1+ x (3 β +2βx )+2(β 3) − − Naresh Dadhich, Stephen Davis, Lev Kofman, David These perturbations are again suppressed in the GB Langlois, Jihad Mourad, Sergei Odintsov, Renaud regime relative to the standard result, unlike the RS en- Parentani, Danielle Steer and David Wands for helpful hancement. discussions. The work of RM is supported by PPARC. [1] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 jiri and S. Odintsov, Phys. Lett. B484, 119 (2000) (1999) [hep-th/9906064]. [hep-th/0004097]; S. Nojiri and S. D. Odintsov, J. [2] P. Brax and C. van de Bruck, Class. Quantum High Energy Phys. 07, 049 (2000) [hep-th/0006232]; L. Grav. 20, R201 (2003) [hep-th/0303095]; R. Maartens, Anchordoqui, C. Nunez, and K. Olsen, J. High En- gr-qc/0312059; P. Brax, C. van de Bruck, and A-C. ergy Phys. 10, 050 (2000) [hep-th/0007064]; S. No- Davis, hep-th/0404011. jiri and S. Odintsov, Phys. 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