Axion Inflation in Type II String Theory
Thomas W. Grimm Universit¨ at Bonn and
University of Wisconsin, Madison
with J. Louis
0710.3883 [hep-th] 0705.3253 [hep-th] JHEP 0412277, 0403067 [hep-th] Nucl.Phys.B.
Liverpool, February 2007
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Introduction and Motivation
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ê Observational Cosmology:
need theoretical models explaining data
• Inflation as a promising scenario – explains qualitative properties of universe
(flatness, homogeneity, isotropy etc.)
– growing experimental evidence from measurements of cosmological observables (e.g. WMAP data extracted from the CMB)
• Primordial gravitational waves: r = Pg /PR
<
tensor to scalar ratio r 0.3
Future experiments: (e.g. Planck satellite)
(current observational bound)
test r down to
r > 0.01
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ê Important task:
Embed Inflation into String Theory
• Challenge: Identify inflatons with sufficiently flat potentials in controlled compactification, e.g. Brane inflation
K¨ahler moduli inflation
Racetrack inflation
Dvali,Tye and many others
Conlon, Quevedo; . . .
Blanco-Pillado et al; . . .
• Explicit realizations in controlled compactifications remain hard to construct • Models reproducing known cosmological observable are not necessarily able to incorporate future observations Example: Primordial gravity waves – most string models only allow for unobservable small r Baumann,McAllister; Bean,Shandera,Tye,Xu; Kallosh,Linde
Recently:
first efforts to explore embeddings with possibly observable r Krause;Becker,Leblond,Shandera;Kallosh,Sivanandam,Soroush
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Goal of the Talk Study possibility of realizing inflation driven by a large number of axion fields within Type IIB string theory. Such models are interesting from conceptional point of view, since intrinsically stringy corrections need to be incorporated. They also can allow for a detectable amount of gravity waves r < 0.14. Outline of the Talk • Review: Axion inflation in supergravity • Axion decay constants and stringy corrections • Axion potentials: D1 instantons and D5 gaugino condensates • Axion inflation in N = 1 orientifold compactifications
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Review: Axion Inflation in supergravity
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ê Assisted inflation of N axions:
N-flation
Dimopoulos,Kachru,McGreevy,Wacker
• effective 4D Lagrangian for N axion fields ca : L =
1 2
X
fa2 ∂µ ca ∂ µ ca − Veff (ca )
a
fa are the axion decay constants • canonically normalized axions:
θa = fa ca − fa π < θa ≤ fa π
− π < ca ≤ π
• effective potential: a
Veff (θ ) = C +
N X a=1
Λ4a
1 − cos
h µa θa i fa
⇒ µa slight extension of the original proposal of Dimopoulos et al.
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• For one axion with µa = µ = 1: V(θ)
quadratic regime (chaotic inflation) 0
πf
θ
0
πf
θ
• For one axion with µa = µ < 1/3: V(θ) quadratic in whole field range of axion (chaotic inflation)
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ê Assistance effect:
Liddle,Mazumdar,Schunck; Kanti,Olive
Eqn. of motion for axions θ¨a + 3H θ˙a + ∂θa V = 0
1 H = 3MP2
1 ˙a 2 2 (θ )
2
+V
Hubble friction H: contains the whole potential V of all fields θa Downward force ∂θa V : contains only ath potential term
ê Slow roll conditions:
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ê large number N of small cycles of size: ⇒
N R-R two-form axions
small cycles of volume | -b + i v |
(keep total volume above string scale)
|t| = | − b + iv| < 1 (two-volumes below string scale)
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ê Standard example of blown-up singularity: conical singularity
Resolved conifold
resolved by small two-sphere
S2
⇒
VolS 2 < `2s
⇒
calculable α0 corrections to the quantum volume:
-
strings start to wrap on small S 2
Fcone = − 2i t2 log t + . . . ⇒
t = - b + iv
V = Vclass (R) + |t|2 log |t| + . . .
presence of small cycles does not increase V Compute: Large axion decay constants in resolved geometries However: fa < MP Banks,Dine,Fox,Gorbatov
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ê Generalize to: Set-ups with N vanishing cycles S2
S2
t1
t2
N axions: c A
S2 tN
ê What about other stringy corrections ? D1 branes wrapped around the S 2 ’s can become light and correct the theory Strominger; Becker,Becker,Strominger
Instanton contribution: ⇒
have to make sure that
⇒
D1 instantons are subleading in fa
ê However:
|tA | exp − + icA gs
|t|/gs > 1 ⇒
small string coupling fa is independent of the axions cA
D1 instanton corrections are the leading corrections to the scalar potential
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Axion potentials: D1 instantons and D5 gaugino condensates
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ê Recall:
Axion N -flation requires a potential of the form Veff (θa ) = C +
N X a=1
Λ4a 1 − cos
h µa θa i fa
θa = fa ca
ê To discuss potentials: ⇒
Break supersymmetry to N = 1 in four space-time dimensions
⇒
Inclusion of D-branes and orientifold planes
ê Scalar potential in N = 1 supergravity theory: 2 ¯ V = eK/MP K I J DI W DJ W − 3|W |2 /MP2 + D-terms
Axion potential only generated by non-perturbative effects: Superpotentials from D1 instantons or gaugino condensates on D5 branes
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ê D1 instantons on vanishing cycles: Ga = ca − i|ta |/gs
Couple to (instanton action):
• Type I:
orientifolds with O9 planes WD1 =
Witten
X
−iGa
Ba e
a
⇒
generalize to other orientifold scenarios (Type IIB with O5 planes)
• Type IIB / F-theory: ⇒
orientifolds with O3/O7 planes
Witten
superpotential due to D3 instantons WD3 =
X
Aα Θα (τ, Ga ) eTα
α
⇒
D1 instanton dependence - Exp (−iGa ) - through determinants Witten; Ganor; TG
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ê Gaugino condensates on space-time filling D5 branes wrapped on vanishing cycle Gauge coupling:
G = c − i|t|/gs
• gaugino condensate S of U (K) gauge group: Veneziano-Yankielowicz sup. eliminate S 1 −−−−−−−→ WD5 = Λ30 e−iG/K WVY = G S + 2πi K S log(S/Λ30 ) − 1 ⇒
µ = 1/K,
• recently:
but potential remains 2π periodic
Witten
Computation of axion potentials using geometric transition Vafa,Heckman,Seo; Aganagic,Beem,Kachru
S2
D5 brane
Geometric Transition Effective theory for gaugino condensate discribed by dual geometry Vafa; Dijkgraaf, Vafa
S3
Flux
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Axion inflation in N = 1 compactifications
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ê Embed scenario into N = 1 orientifold comapctification with O3/O7 planes Effective N = 1 theory can be compute including all N = 2 world-sheet corrections TG,Louis
projection:
O = (−)FL Ωp σ ∗
σ∗J = J
(1,1)
• split of cohomology:
H (1,1) = H−
• split of basis:
ωa−
(1,1)
⊕ H+
(1,1)
a = 1...h−
• new special coordinates associated to split basis: − B2 + iJ = ta− ωa− + tα+ ωα+
• N = 2 pre-potential:
F( ta− , tα+ )
σ ∗ Ω = −Ω
ωα+
(1,1)
α = 1...h+
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ê N = 1 complex coordinates:
complex dilaton
τ = C0 + ie−φ
Ga = ca + ie−φ Re ta−
Tα = ρα + ie−φ Re ∂tα+ F
ca
ρα
R-R two-form axions
R-R four-form axions
ê N = 1 K¨ahler potential: −2φ A A ¯ ¯ ¯ Kq (τ, G, T ) = −2 ln ie 2(F − F) − (FA + FA )(t − t ) • K¨ahler potential is complicated implicit function of N = 1 coordinates • derivatives of Kq determined by Legendre transform in N = 2 or work of Hitchin
ê N = 1 superpotential:
flux background + D-instanton corrections Z X W = G3 ∧ Ω + Θα (τ, Ga ) eiTα α
Gukov,Vafa,Witten; Witten; Ganor; TG
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ê Example:
Simplistic orientifold of the N conifold toy model
v, b
v , -b Orientifold Plane
ê general N = 1 formulas can be applied to N conifold example special coordinates:
tj+ = iv
simplistic pre-potential:
(further perturbative and non-perturbative corrections expected)
F
tj− = −b
tR = i(R/`s )2
N X i 2 i 2 1 3 (t+ ) + (t− ) = − 3! tR + tR i=1
− 2i
N X i=1
(ti+ + ti− )2 log(ti+ + ti− ) −
i 2
N X i=1
(ti+ − ti− )2 log(ti+ − ti− ) .
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ê N = 1 complex coordinates: Ti ↔ ti+
D3-coupling: TR ↔ tR
D1-coupling: Gi ↔ ti−
ê Moduli stabilization: background flux and D-instantons (1)
stabilize dilaton and complex structure moduli
(2)
effective superpotential depending on TR , Tj and Gj W = W0 +
N X
eiTj + eiTR + e−1/gs
j=1
⇒
axion potential:
Giddings,Kachru,Polchinski; KKLT
N X j=1
R-R two-form axions
Re Gj
R-R four-form axions
Re TR , Re Tj
ê numerical minimization:
j
e−iG eiTR
minimum for hTj i, hTR i and hGj i
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ê focus on R-R two-form axions in Gi :
have strong exponential suppression in W
ê numerical analysis allows to illustrate mass hierarchy: axion Re Gi = c
vs.
non-axionic partner Im Gi = −b/gs
¯ at fixed hTR i, hTj i: ê Potential Veff (G, G)
with Gi dependent W
-7.4
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-7.6 15 -7.8 10 -1.62 5
-1.6 Im G -1.58 0
Re G
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¯ at fixed hTR i, hTj i: ê Potential Veff (G, G)
without Gi dependent W
-7.4
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-7.6 15 -7.8 10 -1.62
Re G
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-1.6 Im G -1.58 0
ê non-axionic field Im Gi is stabilized by corrections to K ê axions are lighter than other bulk fields near this vacuum ⇒
axions can potentially drive inflation
(not corrections to W )
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Conclusions and outlook • Discussed specific realization of assisted axion inflation in type IIB string theory: – Axion decay constants: (a) close to Planck scale for axions from vanishing cycles (b) corrected by string world-sheets ⇒
N = 1 K¨ahler potential, K¨ahler stabilization of the saxions
– Axion potentials: from D1 instantons or gaugino condensates on D5 brane ⇒
N = 1 superpotential, periodic potential for the axions
– Embedding into N = 1 orientifolds: stabilization of all moduli keeping light axions
• Future dircections: – construction of explicit semi-realistinc models beyond N conifold scenarios – computation of axion potentials using gauge/gravity duality - geometric transitions
