Spinflation

Ivonne ZavalaIPPP, Durham

In collaboration with: Based on: JHEP04(2007)026 and arXiv:0709.2666

R.Gregory, G. Tasinato, D.Easson and D. Mota

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?

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.

Flux Compactifications: Moduli Stabilisation

• A large variety of SUGRA backgrounds with internal fluxes has been explored in the context of the AdS/CFT correspondence.

Klebanov-Strassler (KS) solution

• Fluxes: F =dA (generalisation of F2=dA1)✴Internal fluxes: components in the internal six dimensional compact manifold. Reduce susy, warping, moduli stabilisation, hierarchies, etc. [GKP, KKLT]

(t > 2002)

p+2 p+1

D-Brane Cosmology set up

F

T Q4d

D3

p!3

CY3

p, p

Dp

p+2

Applications to string cosmology

• 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)

• 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.]

• Besides potential cosmological applications of

• In particular, motion of the D3-brane along the

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.

angular coordinates should have interesting effects. Angular momentum gives rise to centrifugal forces and thus, brane bounces generic along radial direction.

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]

Klebanov-Strassler Geometry

Type IIB solution with F3, H3, F5 internal fluxes

[GKP]

r0 UV

IR

B2

F3

D3

r

Calabi!Yau

α′= "

2

s

M

;

gs string coupling

string scale

3 − form flux units

;

h1/2(η)h−1/2(η)

h(η) = (gsMα′)222/3 ε−8/3I(η)

I(η) =

∫∞

ηdx

x coth x − 1

sinh2 x(sinh (2x) − 2x)1/3

.

ηηuv

ds2

10 = dxµdxµ

+ ds2

6

S3

S2

ds2

6 = Deformed Conifold

D-Brane Dynamics Brane motion described by DBI+WZ (Wess-Zumino) action

m = T3 g−1

s

∫d3x

v2

= gηηη2+grsy

ry

sD3

warped geometry

(q = ±1)

T3 = ((2π)3α′2)−1

L = −m{

h−1

[

√

1 − h v2 − q]}

hv2

< 1⇒

;

SDBI = −T3g−1s

∫

dξ4e−φ√

−det(γab + Fab) SWZ = q T3

∫W4

C4

;

Brane trajectories (q=1, )

• Conserved quantities

;

;

!2(η) = grslrls

E =(γ − 1)

hlθ = gθθ θ γ

yr= θ

η2 =gηη

[

E(h E + 2) − "2(η)]

(h E + 1)2

• Brane trajectories described by equation of motion:

γ =1

√

1 − hv2=

√

1 + h "2(η)

1 − h gηη η2

Brane trajectories (q=1, )

• Conserved quantities

;

;

!2(η) = grslrls

E =(γ − 1)

hlθ = gθθ θ γ

yr= θ

η2 =gηη

[

E(h E + 2) − "2(η)]

(h E + 1)2

• Brane trajectories described by equation of motion:

γ =1

√

1 − hv2=

√

1 + h "2(η)

1 − h gηη η2

m

2x

2 + V (x) = E

Brane trajectories (q=1, )

• Conserved quantities

;

;

!2(η) = grslrls

E =(γ − 1)

hlθ = gθθ θ γ

yr= θ

η2 =gηη

[

E(h E + 2) − "2(η)]

(h E + 1)2

• Brane trajectories described by equation of motion:

γ =1

√

1 − hv2=

√

1 + h "2(η)

1 − h gηη η2

cyclic branes

Klebanov-Strassler

t

S3

S2

bouncing branes

r

r

t

min

r

r

t

min

r

2

F3

0

1

2

r

rUV

4

r3

rr

B

Brane trajectories in

Induced expansion: Mirage Cosmology

= −dτ2 + a2(τ)dxidxi ,

Hind =

(

h′

4 h3/4

)2

gηη[

E (h E + 2q) − !2(η)]

a(τ) = h−1/4(τ) Hind =

1

a

d a

dη

d η

dτwhere and

• Mirage bouncing and cyclic universes

2

ds2

4

dτ = h−1/4

√

1 − hv2 dt

Effective 4D approach:

• Mirage approach: brane does not backreact on the geometry.Exact in codimension one. Problematic as codimension increases.

• What happens to brane trajectories from 4D point of view (backreaction is taken into account)?

DBI Cosmology

• Proceed in an effective 4D approach by coupling the system to gravity as a first step to study cosmology. [Quevedo, Gibbons, Silverstein-Tong].

[GKP]

KS

D3

Calabi!Yau

Inflation

4D

• Ideally, one would have to find a fully localised solution to supergravity equations of motion.

• 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?

• 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.]

DBI Cosmology

where

Coupled system

M2

Pl = V6/κ2

10

φ =

√

T3 η T3 = ((2π)3α′2)−1

Four dimensional metric is of FRW form:

κ210 =

(2π)7

2g2sα′4

S4 =M2

Pl

2

∫d4x

√

−g R

− g−1

s

∫

d4x√

−g

[

h−1

√

1 − h gmnφmφn− q h−1 + V (φm)

]

ds2

4 = −dt2 + a

2(t) dxidxi

E =(γ − 1)

h+ V P =

(1 − γ−1)

h− V

!2(φ) = grslrls

lθ = gθθ θ γa3

hgφφ φ2 = 1 −

(

1 +h"2(φ)

a6

)

·

(

q + h

(

H2

β− V

))

−2

; ;

; ; V = m2φ2

β =1

3gs(MPl/Ms)2H =

a

aH

2= Eβ

γ =

√

1 + h "2(φ)/a6

1 − h φ2

• Angular momentum term gets damped due to cosmological expansion. However, it can provide a source of acceleration.

• Energy conditions are satisfied, thus bouncing & cyclic cosmologies do not arise. However cyclic & bouncing brane trajectories survive for a while.

a

a= H

2(1 − ε)

ε ≡ −

H

H2

ε =3β

2H2

{

[

q + h

(

3H2

β− V

)]

φ2 +$2(φ)

a6

[

q + h

(

3H2

β− V

)]

−1}

• Near bouncing points, brane experiences kicks of acceleration.

=>

=>

Consistency bounds

• UV scale:

• Backreaction: ✓ acceleration of the brane has to be small in string units (validity of DBI action).

✓ curvature of the brane as it moves at speed close to that of light.

✓ Total 6D volume < throat volume

gsM <

(

4

ηUV

)3/2"s

ε2/3

√

3gsπ2Mpl

Ms

γ − 1 " g−1

sR4"−4

s= gs M2

⇒

⇒

✓ SUGRA approximation gsM ! 1⇒ (gs < 1)

t

!

Inflation

!

2 4 6 8 10

100000

200000

300000

400000

500000

600000

h( )

!

Warp factor

Brane trajectory

KS

D3

Calabi!Yau

Inflation

4D

Angular momentum can help!

N(φ) =

∫Hdt =

∫H(φ)

φdφ

σ =

√

φ2 + gθθ θ2σ

φ

Accelerating Solutions: Spinflation

H2→

0 ! = 0

! != 0{

1/a6

late time evolution

m2(gsM)2 >1

β⇒ inflation

a

50

52

51

t

10

20

30

40

50

ln( )

End of inflation:reheating through oscillations around tip

=>

Perturbations in Spinflation

¨δσΦ +

(

3H + 3γ

γ

)

˙δσΦ +

(

UσΦ+ c2

s

k2

a2

)

δσΦ =−

(

H σc2S

a3 (E + P )

) [

a3 tanα

Hc2S

(

P − c2SE

)

δs

σ

]˙

UσΦ≡

σH2c2S

a3 (E + P )

[(

H

H−

σ

σ

)

a3(E + P )

σH2c2S

]˙

δs +

(

3H +γ

γ

)

δs +

(

Us +k2

a2

)

δs = −

k2

a2

σ tanα H

a (E + P )2

(

P − c2

SE)

ξ

Us = tanα

[

3H tanα

(

σ

σ−

α

tanα+ 3Hc2

S − cos αf ′ σ

f

)

+ σ

(

α

σ−

f ′ cos α

f tan α

)˙]

where

where

and

cos α =φ

√

2Xsin α =

f(φ) θ√

2X

dσ = cos α dφ + f(φ) sin α dθ d s = f(φ) cos α dθ − sinα dφ

2X ≡ φ2+ gθθ θ2

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

✓ Curvature and entropy perturbations evolve at different speeds. (Curvature perturbations move with a speed . Entropy perturbations move at speed of light).

★ One can extract some general features, without explicitly solving the equations:

✓The entropy perturbation evolves independently of the curvature perturbation at large scales.

c2

S = γ−2

! 1

✓ geometry of spinflation target space is strongly curved. Besides , intrinsic metric is not flat.gmnh(η)

★ Explored consequences of putting a D3 probe to spin in a warped flux SUGRA background (KS). Both from a mirage perspective and effective 4D.

★ When gravity is taken into account cyclic & bouncing cosmologies disappear, but cyclic & bouncing

trajectories persist.

★ From a mirage approach, radial open and closed trajectories induce cyclic & bouncing universes.

Conclusions

Conclusions (cont.)

★ 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.

★ 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.

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!