Spinflation. Ivonne Zavala IPPP, Durham

Transcription

1 Spinflation Ivonne Zavala IPPP, Durham Based on: JHEP04(2007)026 and arxiv: In collaboration with: R.Gregory, G. Tasinato, D.Easson and D. Mota

2 Motivation Inflation: very successful scenario in search of a theory. String theory: mathematical theory that needs experimental tests. Can string theory provide a fundamental origin for the inflaton field?

3 String Cosmology t < 1995: Modular Cosmology (e.g. Pre-Big-Bang scenario [Gasperini-Veneziano]). t > 1995: Dp-branes (more moduli!). DD-Brane and Branes at Angles (slow roll) Inflation. Natural geometrical interpretation for inflaton and end of inflation via tachyon condensation. t > 2002: Moduli stabilisation (GKP, KKLT) Flux compactifications and D-branes. New ideas for inflationary scenarios.

4 Flux Compactifications: Moduli Stabilisation (t > 2002) Fluxes: F =da (generalisation of F2=dA1) p+2 p+1 Internal fluxes: components in the internal six dimensional compact manifold. Reduce susy, warping, moduli stabilisation, hierarchies, etc. [GKP, KKLT] A large variety of SUGRA backgrounds with internal fluxes has been explored in the context of the AdS/CFT correspondence. Klebanov-Strassler (KS) solution

5 D-Brane Cosmology set up D3 Dp 4d T Q p, p p!3 Fp+2 CY 3

6 Applications to string cosmology KKLT (Kachru-Kallosh-Linde-Trivedi) scenario (concrete sugra realisation) can be used to explore possible D-brane world phenomenology and cosmology. [KKLMMT (KKLT + Maldacena-McAllister), Burgess et. al.etc.] A probe D3-brane (or anti-d3-brane) wandering in a warped flux compactification experiences a speed limit due to brane action. Thus, kinetic terms can become negligible, compared with potential terms, which then dominate and -> inflation. DBI inflation [Silverstein-Tong] (DBI=Dirac-Born-Infeld)

7 Besides potential cosmological applications of this effect, it is interesting to explore the consequences of a non-standard DBI brane action for trajectories of branes in warped backgrounds in more generality. In particular, motion of the D3-brane along the angular coordinates should have interesting effects. Angular momentum gives rise to centrifugal forces and thus, brane bounces generic along radial direction.

8 Outline Brane evolution along the radial and angular directions from brane point of view give rise to Cycling & Bouncing Universes [Easson-Gregory-Tasinato, IZ] from Mirage cosmology approach [Kehagias-Kiritsis]. Take into account gravitational backreaction by coupling the DBI action to gravity. Analyse the resulting (cosmological) brane evolution when angular motion is included: Spinflation [Easson-Gregory-Mota-Tasinato-IZ]

9 Klebanov-Strassler Geometry Type IIB solution with F3, H3, F5 internal fluxes ds 2 10 = h 1/2 (η) dx h 1/2 µ dx µ + (η) ds 2 6 ds 2 6 = Deformed Conifold h(η) = (g s Mα ) 2 2 2/3 ɛ 8/3 I(η) α = l 2 s string scale; g s string coupling; M 3 form flux units 0 UV IR D3 η uv r F 3 S 3 Calabi!Yau η r I(η) = η dx x coth x 1 sinh 2 x (sinh (2 x) 2 x) 1/3. B 2 [GKP] S 2

10 D-Brane Dynamics Brane motion described by DBI+WZ (Wess-Zumino) action S DBI = T 3 gs 1 dξ 4 e φ det(γ ab + F ab ) S W Z = q T 3 C 4 W 4 L = m { [ h 1 ]} 1 h v2 q (q = ±1) hv 2 < 1 v 2 = g ηη η 2 +g rs ẏ r ẏ s D3 m = T 3 g 1 s d 3 x; T 3 = ((2π) 3 α 2 ); 1 warped geometry

11 Brane trajectories (q=1, ) Conserved quantities y r = θ E = (γ 1) h ; l θ = g θθ θ γ γ = 1 = 1 hv h l 2 (η) 1 h g ηη η 2 ; l 2 (η) = g rs l r l s Brane trajectories described by equation of motion: η 2 = gηη [ E(h E + 2) l 2 (η) ] (h E + 1) 2

12 Brane trajectories (q=1, ) Conserved quantities y r = θ E = (γ 1) h ; l θ = g θθ θ γ γ = 1 = 1 hv h l 2 (η) 1 h g ηη η 2 ; l 2 (η) = g rs l r l s Brane trajectories described by equation of motion: η 2 = gηη [ E(h E + 2) l 2 (η) ] (h E + 1) 2 m 2 ẋ2 + V (x) = E

13 Brane trajectories (q=1, ) Conserved quantities y r = θ E = (γ 1) h ; l θ = g θθ θ γ γ = 1 = 1 hv h l 2 (η) 1 h g ηη η 2 ; l 2 (η) = g rs l r l s Brane trajectories described by equation of motion: η 2 = gηη [ E(h E + 2) l 2 (η) ] (h E + 1) 2

14 Brane trajectories in Klebanov-Strassler r 0 r UV r r min r 1 r 2 r 3 r 4 S 3 bouncing branes t F 3 r B 2 S 2 t cyclic branes

15 Induced expansion: Mirage Cosmology ds 2 4 = dτ 2 + a 2 (τ)dx i dx i, 2 H ind = ( h 4 h 3/4 ) 2 g ηη [ E (h E + 2q) l 2 (η) ] where a(τ) = h 1/4 (τ) and H ind = 1 a d a dη d η dτ dτ = h 1/4 1 hv 2 dt Mirage bouncing and cyclic universes

16 Effective 4D approach: DBI Cosmology [GKP] Calabi!Yau D3 KS 4D Inflation Mirage approach: brane does not backreact on the geometry. Exact in codimension one. Problematic as codimension increases. Ideally, one would have to find a fully localised solution to supergravity equations of motion. What happens to brane trajectories from 4D point of view (backreaction is taken into account)? Proceed in an effective 4D approach by coupling the system to gravity as a first step to study cosmology. [Quevedo, Gibbons, Silverstein-Tong].

17 DBI Cosmology Warp factor and non-standard kinetic terms allow for accelerating solutions since the potential term V can dominate in strongly warped regions. Cosmological Inflationary solutions when brane moves only along a radial coordinate in simple throat AdS5xS5: DBI Inflation [Silverstein-Tong, Chen, etc.] What is the effect of brane angular motion (spin) on 4D cosmological expansion, in particular on accelerating trajectories in more concrete set up (KS)? Spinflation What happens to cyclic/bouncing trajectories/ cosmologies?

18 Coupled system S 4 = M 2 P l 2 g 1 s d 4 x g R d 4 x [h ] g 1 1 h g φm φn mn q h 1 + V (φ m ) where φ = T 3 η T 3 = ((2π) 3 α 2 ) 1 MP 2 l = V 6 /κ 2 10 κ 2 10 = (2π)7 gsα Four dimensional metric is of FRW form: ds 2 4 = dt 2 + a 2 (t) dx i dx i

19 hg φφ φ2 = 1 ( 1 + hl2 (φ) a 6 ) ( q + h ( H 2 β V )) 2 H 2 = β E H = ȧ ; ; a β = 1 3g s (M P l /M s ) 2 E = (γ 1) h + V P = (1 γ 1 ) V ; ; V = m 2 φ 2 h γ = 1 + h l 2 (φ)/a 6 1 h φ 2 l θ = a 3 g θ θθ γ l 2 (φ) = g rs l r l s

20 => Energy conditions are satisfied, thus bouncing & cyclic cosmologies do not arise. However cyclic & bouncing brane trajectories survive for a while. Angular momentum term gets damped due to cosmological expansion. However, it can provide a source of acceleration. => ä a = H2 (1 ɛ) ɛ Ḣ H 2 ɛ = 3β 2H 2 { [ q + h ( 3H 2 β V )] φ 2 + l2 (φ) a 6 [ q + h ( 3H 2 β V )] 1 } Near bouncing points, brane experiences kicks of acceleration.

21 Consistency bounds Backreaction: acceleration of the brane has to be small in string units (validity of DBI action). SUGRA approximation g s M 1 (g s < 1) curvature of the brane as it moves at speed close to that of light. γ 1 gs 1 R 4 l 4 s = g s M 2 UV scale: Total 6D volume < throat volume g s M < ( 4 η UV ) 3/2 l s ɛ 2/3 3gs π 2 M pl M s

22 Angular momentum can help! 4D Inflation σ σ = φ 2 + g θθ θ2 Calabi!Yau D3 φ N(φ) = Hdt = H(φ) φ dφ h(!) KS! Warp factor ! Inflation Brane trajectory t

23 Accelerating Solutions: Spinflation a ln( ) t m 2 (g s M) 2 > 1 β inflation H 2 { 0 l = 0 1/a 6 l 0 late time evolution => End of inflation: reheating through oscillations around tip

24 Perturbations in Spinflation δσ Φ + ( 3H + 3 γ γ ) δσ Φ + ( U σφ + c 2 s k 2 a 2 ) ( ) [ H σc 2 δσ Φ = S a 3 tan α a 3 (E + P ) Hc 2 S ( ) ] P δs c 2 SĖ σ where U σφ σh2 c 2 S a 3 (E + P ) ) [(Ḣ H σ σ a 3 (E + P ) σh 2 c 2 S ] δ s + ( 3H + γ γ ) δṡ + (U s + k2 a 2 ) δs = k2 a 2 σ tan α H ( ) a (E + P ) 2 P c 2 SĖ ξ where U s = tan α [ 3H tan α ( σ σ α tan α + 3Hc2 S cos α f ) σ f + σ ( ) ] α σ f cos α f tan α and d σ = cos α dφ + f(φ) sin α dθ d s = f(φ) cos α dθ sin α dφ cos α = φ 2X sin α = f(φ) θ 2X 2X φ 2 + g θθ θ2

25 General features Since the brane moves along different directions, various fields can contribute to the evolution of the perturbations we have a multifield inflation with non-standard DBI-kinetic terms (non slow-roll) => non-adiabatic modes appear geometry of spinflation target space is strongly curved. Besides h(η), intrinsic metric g mn is not flat. One can extract some general features, without explicitly solving the equations: The entropy perturbation evolves independently of the curvature perturbation at large scales. Curvature and entropy perturbations evolve at different speeds. (Curvature perturbations move with a speed c 2 S = γ 2 1. Entropy perturbations move at speed of light).

26 Conclusions Explored consequences of putting a D3 probe to spin in a warped flux SUGRA background (KS). Both from a mirage perspective and effective 4D. From a mirage approach, radial open and closed trajectories induce cyclic & bouncing universes. When gravity is taken into account cyclic & bouncing cosmologies disappear, but cyclic & bouncing trajectories persist.

27 Conclusions (cont.) Inflationary solutions when angular momentum is turned on give rise to Spinflation. Angular momentum sources accelerated expansion, providing a handful of e-folds at beginning of inflation. DBI inflation is in tension with consistency bounds. Spinflation can then help at begining of inflaiton. Series of bounces could help with this tension. Cosmological perturbations in Spinflation are naturally, multifield with important differences to standard slow roll multifield scenarios. Might have important imprints.

28 Perspectives How to solve problem of getting enough inflation in agreement with approximations and observations? (larger V6, wrapped branes, etc.) Non-Gaussianities: are they too large in spinflation? More general spinflation analysis: V (φ, θ) Concrete calculation of reheating mechanism Still a lot of open issues for Spinflation!