Cosmological Perturbations from Braneworld Inflation with a Gauss

Cosmological perturbations from braneworld inflation with a Gauss-Bonnet term Jean-Francois Dufaux1, James E. Lidsey2, Roy Maartens3, M Sami4 1LPT, Universitéde Paris-Sud, 91405 Orsay, France 2Astronomy Unit, School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK 3Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 2EG, UK 4Inter-University Centre for Astronomy & Astrophysics, Pune, India (Dated: October 29, 2018) Braneworld inflation is a phenomenology related to string theory that describes high-energy mod- ifications to general relativistic inflation. The observable universe is a braneworld embedded in 5-dimensional anti de Sitter spacetime. When the 5-dimensional action is Einstein-Hilbert, we have a Randall-Sundrum type braneworld. The amplitude of tensor and scalar perturbations from inflation is strongly increased relative to the standard results, although the ratio of tensor to scalar amplitudes still obeys the standard consistency relation. If a Gauss-Bonnet term is included in the action, as a high-energy correction motivated by string theory, we show that there are important changes to the Randall-Sundrum case. We give an exact analysis of the tensor perturbations. They satisfy the same wave equation and have the same spectrum as in the Randall-Sundrum case, but the Gauss-Bonnet change to the junction conditions leads to a modified amplitude of gravitational waves. The amplitude is no longer monotonically increasing with energy scale, but decreases asymp- totically after an initial rise above the standard level. Using an approximation that neglects bulk effects, we show that the amplitude of scalar perturbations has a qualitatively similar behaviour to the tensor amplitude. In addition, the tensor to scalar ratio breaks the standard consistency relation. I. INTRODUCTION From an observational perspective, there is now strong evidence that the very early universe underwent an epoch In recent years, there has been considerable interest of accelerated (inflationary) expansion [10]. During in- in the possibility that our observable four-dimensional flation, light fields such as the graviton are quantum- (4D) universe may be viewed as a brane hypersurface mechanically excited and acquire nearly scale-invariant embedded in a higher-dimensional bulk space. Of par- fluctuations [11]. The resulting spectrum of primordial ticular importance is the Randall-Sundrum (RS) model, gravitational waves could be detectable from its imprint where a single, positive-tension brane is embedded in a on the polarization of the cosmic microwave background (CMB) [12]. Such a detection would open a unique win- five-dimensional (5D) anti de Sitter (AdS5) spacetime [1]. (For recent reviews, see Ref. [2].) Although the fifth di- dow into the physics of the very early universe. The evo- mension may be infinite in extent, the zero-mode of the lution of gravitational waves during slow-roll inflation has 5D graviton, corresponding to 4D gravitational waves, is been determined in the RS scenario [13]. At high ener- localized at low energies on the brane due to the warped gies, the amplitude is enhanced relative to the standard geometry of the bulk. This property can also be un- result in 4D Einstein gravity. In view of the above de- arXiv:hep-th/0404161v2 1 Nov 2004 derstood within the context of the AdS/CFT correspon- velopments, therefore, it is important to determine the dence [3], where the RS model is viewed as 4D gravity properties of tensor perturbations generated during in- coupled to a conformal field theory (CFT) [4]. flation in the GB braneworld. We show that significant A natural extension of the RS model that is motivated changes to the RS case are introduced by the GB term, by string/M theory considerations, is to include higher- even when the GB corrections are very small relative to order curvature invariants in the bulk action. Such terms the Einstein-Hilbert terms. arise in the AdS/CFT correspondence as next-to-leading order corrections to the CFT [5]. The Gauss-Bonnet (GB) combination of curvature invariants is of partic- II. FIELD EQUATIONS ular relevance in five dimensions, since it represents the unique combination that leads to second-order gravita- For a 5D bulk with Einstein-Gauss-Bonnet gravity, tional field equations in the bulk metric and since the containing a 4D brane, the gravitational action is field equations contain only linear second derivatives [6]. A GB term may also arise as the next-to-leading order 1 5 (5) correction in the heterotic string effective action, and it is = 2 d x g [ 2Λ5 + S 2κ5 − − R ghost-free about flat spacetime [7]. Moreover, the gravi- Z p + α 2 4 ab + abcd ton zero mode remains localized in the GB braneworld [8] R − RabR RabcdR and deviations from Newton’s law at low energies are less d4x √ g σ , (1) pronounced than in the RS model [9]. − − Zbrane 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 <mα αµ < 0.038 , (20) 1 2 1 4 ⇒ 2 µ = 1 1+ αΛ5 , (12) ℓ ≡ 4α " − r 3 # which is consistent with Eq. (2). 3 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).

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