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SpinflationIvonne ZavalaIPPP, DurhamBased on: JHEP04(2007)026 and arXiv:0709.2666In collaboration with:R.Gregory, G. Tasinato, D.Easson and D. Mota

MotivationInflation: very successful scenario insearch of a theory.String theory: mathematical theory thatneeds experimental tests.Can string theory provide a fundamental originfor the inflaton field?

String Cosmologyt < 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 andend of inflation via tachyon condensation.t > 2002: Moduli stabilisation (GKP, KKLT)Flux compactifications and D-branes.New ideas for inflationary scenarios.

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

D-Brane Cosmology set upD3Dp4dFp+2T Qp, pp!3CY 3

Applications to string cosmology• KKLT (Kachru-Kallosh-Linde-Trivedi) scenario (concrete sugrarealisation) can be used to explore possible D-braneworld phenomenology and cosmology.[KKLMMT (KKLT + Maldacena-McAllister), Burgess et. al.etc.]• A probe D3-brane (or anti-D3-brane) wandering in awarped flux compactification experiences a speedlimit due to brane action. Thus, kinetic termscan become negligible, compared with potentialterms, which then dominate and -> inflation.DBI inflation [Silverstein-Tong] (DBI=Dirac-Born-Infeld)

• Besides potential cosmological applications ofthis effect, it is interesting to explore theconsequences of a non-standard DBI braneaction for trajectories of branes in warpedbackgrounds in more generality.• In particular, motion of the D3-brane along theangular coordinates should have interestingeffects. Angular momentum gives rise tocentrifugal forces and thus, brane bouncesgeneric along radial direction.

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

Klebanov-Strassler GeometryType IIB solution with F3, H3, F5 internal fluxesds 2 10 = h −1/2 (η) dx h 1/2 µ dx µ + (η) ds 2 6ds 2 6 = Deformed Conifoldh(η) = (g s Mα ′ ) 2 2 2/3 ɛ −8/3 I(η)α ′ = l 2 sstring scale;g s string coupling;MI(η) =3 − form flux units∫ ∞ηdx x coth x − 1sinh 2 x0 UVIR(sinh (2 x) − 2 x) 1/3 .D3rB 2F 3η uv[GKP]S 3S 2Calabi!Yauηr

D-Brane DynamicsBrane 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 4L = −m{ [ h −1 √ ]}1 − h v2 − q(q = ±1)⇒hv 2 < 1v 2 = g ηη ˙η 2 +g rs ẏ r ẏ sD3m = T 3 g −1s∫d 3 x;T 3 = ((2π) 3 α ′2 );−1warped geometry

Brane trajectories (q=1, )• Conserved quantitiesy r = θE =(γ − 1)h;l θ = g θθ ˙θ γγ =√1√ = 1 − hv21 + 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

Brane trajectories (q=1, )• Conserved quantitiesy r = θE =(γ − 1)h;l θ = g θθ ˙θ γγ =√1√ = 1 − hv21 + h l 2 (η)1 − h g ηη ˙η 2 m;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) 22 ẋ2 + V (x) = E

Brane trajectories inKlebanov-Strasslerr0r UVrrminr1r2r 3r4bouncing branestF 3rS 3 S 2B2r mintcyclic branes

Induced expansion: Mirage Cosmologyds 2 4= −dτ 2 + a 2 (τ)dx i dx i ,2H ind =( h′4 h 3/4 ) 2g ηη [ E (h E + 2q) − l 2 (η) ]wherea(τ) = h −1/4 (τ)andH ind = 1 ad adηd ηdτdτ = h −1/4√ 1 − hv 2 dt• Mirage bouncing and cyclic universes

Effective 4D approach:DBI Cosmology[GKP]Calabi!YauD3KS4DInflation• Mirage approach: brane does notbackreact on the geometry.Exact in codimension one.Problematic as codimension increases.• Ideally, one would have to find afully localised solution to supergravityequations of motion.• What happens to brane trajectoriesfrom 4D point of view (backreactionis taken into account)?• Proceed in an effective 4D approach by coupling the systemto gravity as a first step to study cosmology.[Quevedo, Gibbons, Silverstein-Tong].

DBI Cosmology• Warp factor and non-standard kinetic terms allow foraccelerating solutions since the potential term V candominate in strongly warped regions. CosmologicalInflationary 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 4Dcosmological expansion, in particular on acceleratingtrajectories in more concrete set up (KS)? Spinflation• What happens to cyclic/bouncing trajectories/cosmologies?

Coupled systemS 4 = M 2 P l2− g −1s∫∫d 4 x √ −g Rd 4 x √ [h√]−g−1 1 − h g ˙φm ˙φn mn − q h −1 + V (φ m )whereφ = √ T 3 ηT 3 = ((2π) 3 α ′2 ) −1MP 2 l = V 6 /κ 2 10 κ 2 10 = (2π)7 gsα 2 ′42Four dimensional metric is of FRW form:ds 2 4 = −dt 2 + a 2 (t) dx i dx i

hg φφ ˙φ2 = 1 −()1 + hl2 (φ)a 6·(q + h( H2β− V )) −2H 2 =βEH = ȧ; ;aβ =13g s (M P l /M s ) 2E =(γ − 1)h+ V P = (1 − γ−1 )− V; ; V = m 2 φ 2hγ =√1 + h l 2 (φ)/a 61 − h ˙φ 2l θ =a 3 g ˙θ θθ γl 2 (φ) = g rs l r l s

=>• Energy conditions are satisfied, thus bouncing & cycliccosmologies do not arise. However cyclic & bouncing branetrajectories survive for a while.• Angular momentum term gets dampeddue to cosmological expansion. However,it can provide a source of acceleration.=>äa = H2 (1 − ɛ)ɛ ≡ − ḢH 2ɛ =3β2H 2 { [q + h( 3H2β − V )][˙φ 2 + l2 (φ)a 6 q + h( 3H2β − V )] −1}• Near bouncing points, brane experiences kicks ofacceleration.

Consistency bounds• Backreaction:✓ acceleration of the brane has to be small instring units (validity of DBI action).⇒ ✓ SUGRA approximation g s M ≫ 1 (g s < 1)✓ curvature of the brane as it moves at speedclose to that of light.⇒γ − 1 ≪ gs−1 R 4 l −4s = g s M 2• UV scale:✓ Total 6D volume < throat volume⇒g s M

Accelerating Solutions: Spinflationaln( )5040525150302010tm 2 (g s M) 2 > 1 β ⇒inflationH 2 {→ 0 l = 01/a6l ≠ 0late timeevolution=>End of inflation:reheating throughoscillations around tip

¨ δσ Φ +(3H + 3 ˙γ γ)Perturbations in Spinflation˙ δσ Φ +(U σΦ + c 2 swherek 2 ) ( ) [ H ˙σc2a 2 δσ Φ = − S a 3 tan αa 3 (E + P ) Hc 2 SU σΦ ≡ ˙σH2 c 2 Sa 3 (E + P ))[(ḢH − ¨σ˙σ( )]˙P ˙ δs− c 2 SĖ˙σa 3 (E + P )˙σH 2 c 2 S]˙δ¨s +where(3H + ˙γ γ)U s = tan αδṡ +[(U s + k23H tan αa 2 )δs = − k2a 2( ¨σ˙σ − ˙αtan α + 3Hc2 S − cos α f ′ ) ˙σf˙σ tan α H( )˙a (E + P ) 2 P − c 2 SĖ+ ˙σξ( ˙α˙σ − f ′ cos αf tan α)˙]andd σ = cos α dφ + f(φ) sin α dθd s = f(φ) cos α dθ − sin α dφcos α =˙φ √2Xsin α =f(φ) ˙θ√2X2X ≡ ˙φ 2 + g θθ ˙θ2

General features★ Since the brane moves along different directions, variousfields can contribute to the evolution of the perturbations✓ we have a multifield inflation with non-standard DBI-kineticterms (non slow-roll) => non-Adiabatic modes appear✓ geometry of spinflation target space is strongly curved. Besidesh(η) , intrinsic metric g mn is not flat.★ One can extract some general features, without explicitlysolving the equations:✓The entropy perturbation evolves independently of the curvatureperturbation 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).

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

Conclusions (cont.)★ Inflationary solutions when angular momentum isturned on give rise to Spinflation. Angular momentumsources accelerated expansion, providing a handful ofe-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 standardslow roll multifield scenarios. Might have importantimprints.

Perspectives★ How to solve problem of getting enough inflationin 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!