Inflation from Axion Monodromy Enrico Pajer Cornell University - - PowerPoint PPT Presentation

Inflation from Axion Monodromy

Enrico Pajer

Cornell University

Stockholm June 2011

Outline

1 Minimal requirements 2 Brane inflation 3 Inflation from axion monodromy 4 Conclusions

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 2 / 42

Outline

1 Minimal requirements 2 Brane inflation 3 Inflation from axion monodromy 4 Conclusions

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 3 / 42

Minimal requirements

A successful model of inflation in string theory needs: Consistent (4D ?) effective description Something, e.g. V (φ), driving more than 60 efoldings of inflation Small, adiabatic, slightly red tilted and Gaussian curvature

• perturbations. Percent level precision!

Graceful exit and reheating into SM fields

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 4 / 42

Consistent 4D description

As long as H ≪ ms we can use SUGRA. Compactification 10D → 4D (EFT of warped compactification are poorly understood). Hierarchy: MPl ≫ ms ∼ g3/4

s

V1/2 MPl ≫ mKK ∼ gs V2/3 MPl If we see tensors at r = 0.01 then H ≃ 5 · 10−5 MPl

No one solves the 10D equations of motion

• Branes are smeared instead of localized

Time dependence of the compact metric is neglected

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 5 / 42

Driving inflation

Dimensional reduction leads to many moduli φi. The multifield potential slow-roll conditions are ǫ ≡ 1 2M2

Plgij ViVj

V 2 ≪ 1 η ≡ min e-value of M2

Plgij V;ij

V ≪ 1 where L ⊃ gij∂µφi∂µφj. Inflation is not about finding one flat

• direction. Each modulus must be at a

minimum or have a slow-roll flat potential. Moduli stabilization is essential

[see Joe’s discussion] .

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 6 / 42

Curvature perturbations: data

COBE normalization: V = (0.027)4ǫM4

Pl

Fixed by hand Red tilted: ns = 0.968 ± 0.024 Gaussian: fNL < O(100). Model dependent Adiabatic: δcγadiab < 0.02 (anti-correlated isocurvature), where δcγ

adiab ≡ 2δρc/ρc − 3δργ/(4ργ)

δρc/ρc + 3δργ/(4ργ) No tensors: r < 0.2

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 7 / 42

Curvature perturbations: reproducing the data

Relatively easy to obtain once one has 60 efoldings

• Scale invariance is a consequence of the quasi de Sitter background

The size of scalar and tensor perturbations are easily adjustable by rescaling V Adiabaticity is easily satisfied with slowly curving trajectories Non-Gaussianity can be interestingly large but generally compatible with the data Pre-inflationary remnants behave as in standard scenarios

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 8 / 42

Reheating

Not very well studied

• The embedding of the SM is

typically not explicit Inflationary sector is often separated from the SM sector in the compact space Bulk propagation induced by KK-mode and massive string modes plays a key role

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 9 / 42

Summary of minimal requirements

A successful model of inflation in string theory needs: Consistent 4D effective description Something, e.g. V (φ), driving more than 60 efoldings of inflation Small, adiabatic, slightly red tilted and Gaussian curvature perturbations Graceful exit and reheating into SM fields

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 10 / 42

What can we learn

Questions that can be answered by and explicit UV embedding Is inflation possible and natural in a UV finite theory? (Yes. Probably no) Do the constraints from string theory single out certain characteristic observables? (No) Are there correlated observables? (Yes) Can we improve on the initial condition problem? (Arguably no)

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 11 / 42

Outline

1 Minimal requirements 2 Brane inflation 3 Inflation from axion monodromy 4 Conclusions

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 12 / 42

Brane Inflation

Consider spacetime filling D3-branes in Type IIB

[Dvali & Tye ’98]

Six scalars, i.e. the position of the D3-brane. Inflation from open strings DBI like kinetic term Many contributions to the potential: background fluxes, anti-D3-branes, moduli stabilization, . . .

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 13 / 42

DBI kinetic term

The DBI action gives S ⊃ √−g

• 1 − gij∂µφi∂µφj

gij is the (e.g. Kalabi-Yau) metric of the internal space Slow-roll regime: gij ˙ φi ˙ φj ≪ 1 Standard canonical kinetic term S ⊃ √−g1 2gij∂µφi∂µφj DBI regime 1 − gij ˙ φi ˙ φj ≪ 1 Happens only for highly warped metrics gij ∼ e−2A˜ gij All (infinitely many) higher derivative terms are important Maximum speed helps for prolonged inflation with steep potentials

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 14 / 42

Potentials for D3-branes

Type IIB flux compactified on a warped Calabi-Yau

[Giddings et al. 01]

Imaginary self-dual (ISD) fluxes stabilize complex structure and axio-dilaton moduli Generically warped throats are present: locally there is an explicit warped Calabi-Yau metric (essential to specify the kinetic term) Potential can be divided in four parts

[Baumann et al.]

V (φ) = const + VCoulomb + VBulk + VR VCoulomb attraction to anti D3-branes, if present. Too steep unless warping is introduced

[Kachru et al. 03]

VBulk characterizing bulk perturbations to the local solution

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 15 / 42

Dynamics

One can study the whole potential numerically

[Agarwal et al. 11]

• r find particularly simple regimes

[Baumann et al., Krause & EP 07]

V is generically too steep One can fine tune an inflection point 60 efolds of standard small-field slow-roll inflation Can be compatible with the data

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 16 / 42

DBI inflation

The EFT idea works perfectly but explicity constructions seem to fail: the warp factor does not remain strong enough for a long enough distance in field space

[Baumann & McAllister 06]

The DBI regime requires strong warping and steep potentials Only part of the 60 efolds can be in the DBI regime

[Chen et al. 07, EP 08]

Perturbations have a very small speed of sound cs ≪ 1 leading to very distinctive large non-Gaussianity Important UV input: the square root is radiatively stable and valid to all orders in α′

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 17 / 42

Summary of D-brane inflation

Consistent 4D description Can give 60 efolds of inflation but needs fine tuning to the percent level Perturbations can easily be compatible with observations Slow-roll regime has no distinctive features DBI regime has characteristic non-Gaussianity. The UV-embedding guaranties the radiative stability and consistency

• f infinite many derivatives.

Brane-anti-brane annihilation has been studied as mechanism for reheating

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 18 / 42

Outline

1 Minimal requirements 2 Brane inflation 3 Inflation from axion monodromy 4 Conclusions

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 19 / 42

Tensor modes and the Lyth bound

The detection of tensor modes would fix the scale of inflation close to the GUT scale. Measuring tensor modes puts a lower bound on the range of variation of the inflaton

[Lyth 98]

dφ MPl = dN √ 2ǫ ≃ dN r 8 ∆φ MPl > r 0.01 NCMB 30 ∆φ above the cutoff makes the use of EFT suspicious This is the main motivation to consider axion monodromy inflation

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 20 / 42

Tensor modes and the Lyth bound

The detection of tensor modes would fix the scale of inflation close to the GUT scale. Measuring tensor modes puts a lower bound on the range of variation of the inflaton

[Lyth 98]

dφ MPl = dN √ 2ǫ ≃ dN r 8 ∆φ MPl > r 0.01 NCMB 30 ∆φ above the cutoff makes the use of EFT suspicious This is the main motivation to consider axion monodromy inflation

Schematically

Tensor modes ⇒ High scale ⇒ Large field ⇒ more UV-sensitive

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 20 / 42

UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitive

• bservables.

Inflation is a UV-sensitive mechanism. Schematically V (φ) = 1 2m2φ2 +

• n

λn φn Mn−4

Pl

Most models suffer from an η-problem We need to invoke a symmetry, e.g. shift symmetry Then we need a fundamental theory (UV-finite) to ask if, how and where the symmetry is broken.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 21 / 42

Axions in field theory

Axions are scalar fields with only derivative couplings and might arise e.g. from the breaking of a U(1) symmetry

[Peccei & Quinn 77]

Hence they enjoy a continuous shift symmetry at all orders in perturbation theory φ(x) → φ(x) + constant Continuous shift symmetry is broken to a discrete shift symmetry by non-perturbative effects The axion decay constant f determines the periodicity of the canonically normalized axion L ⊃ 1 2(∂φ)2 + Λ4 cos φ f

• ⇒ φ(x) → φ(x) + 2πf

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 22 / 42

Inflation with axions

Inflate with the non-perturbative correction

[Freese et al. 90]

V (φ) = Λ4

• 1 − cos

φ f

• .

WMAP requires f > 3MPl In string theory f/MPl ∼ gs/L4 ≪ 1. Superplanckian axion decay constants seem elusive in string theory

[Bank et al. 03]

Axion monodromy

[Silverstein & Westphal (1+McAllister)] : small explicit

breaking of the shift symmetry V (φ) = const + ǫ ˜ V (φ) For small ǫ remnants of the shift symmetry protect from further corrections Is the shift symmetry violated at the Planck scale?

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 23 / 42

Axion in string theory

String theory seen from a low energy 4D observer: Model independent axions such as dualizing Bµν or Cµν Model dependent axions from integrating a p-form over a p-cycle

• f the compact manifold

c(x) =

• Σp

Cp , b(x) =

• Σ2

B2 The axion decay constant f is determined by geometrical data of the compactification. In controlled setups f < MPl

[Banks et al 03]

The shift symmetry is evident in SUGRA since only dB or dC appears in the bulk action. But gs and α′ corrections?

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 24 / 42

Shift symmetry

The 4D axion b(x) from Bij = b(x)ωij, with ω a two-form. In (bosonic) closed string theory, the vertex operator for b at zero momentum integrated over the world-sheet is V (k = 0) =

• ws

d2σǫαβ∂αXi∂βXjωijb =

• ts

B In perturbation theory the worldsheet wraps topologically trivial cycles hence V (k = 0) = 0, only derivative coplings.

[Wen & Witten, Dine & Seiberg 86]

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 25 / 42

Shift symmetry

The 4D axion b(x) from Bij = b(x)ωij, with ω a two-form. In (bosonic) closed string theory, the vertex operator for b at zero momentum integrated over the world-sheet is V (k = 0) =

• ws

d2σǫαβ∂αXi∂βXjωijb =

• ts

B In perturbation theory the worldsheet wraps topologically trivial cycles hence V (k = 0) = 0, only derivative coplings.

[Wen & Witten, Dine & Seiberg 86]

Breaking of the shift symmetry

Two ingredients can invalidate the above argument: Non-perturbative worldsheet instantons World sheet with boundaries, i.e. D-branes

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 25 / 42

The construction

The setup

[McAllister, Silverstein & Westphal 08]

Type IIB orientifolds N = 1, 4D: an axion c(x) from RR field C2 c(x) =

• Σ2

C2 Wrapping an NS5-brane over Σ2 induces a monodromy for c(x) (worldsheet with boundary) If the NS5-brane is in a warped region the potential is small Moduli stabilization ´ a la KKLT does not spoil the shift symmetry Non-perturbative corrections (e.g. to the K¨ ahler potential) induce small ripples

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 26 / 42

Linear potential for the inflaton

The shift symmetry can be broken in the presence of boundaries. Consider a D5-brane wrapped on a two-cycle Σ2. The DBI action −T5

• d6xe−Φ
• det (Gind + Bind)

The shift b(x) → b(x) + const of b(x) =

• Σ2 B2 stores

some potential energy. V (b) = T5

• L4 + b2 ∼ T5b

for large b Linear inflaton potential (and breaks SUSY). COBE normalization and control require to red-shift T5. Via S-duality, NS5 gives a monodromy for c.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 27 / 42

4D N = 1 data

Effective action of O3/O7 Calabi-Yau

• rientifolds (σΩ = −Ω).

[Grimm & Louis 04]

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 28 / 42

4D N = 1 data

Effective action of O3/O7 Calabi-Yau

• rientifolds (σΩ = −Ω).

[Grimm & Louis 04]

Assume complex structure moduli and dilaton are stabilized by fluxes at a higher scale.

[Kachru et al 03]

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 28 / 42

4D N = 1 data

Effective action of O3/O7 Calabi-Yau

• rientifolds (σΩ = −Ω).

[Grimm & Louis 04]

Assume complex structure moduli and dilaton are stabilized by fluxes at a higher scale.

[Kachru et al 03]

h1,1

+ orientifold-even K¨

ahler moduli from two-/four-cycle volumes complexified by

• C4

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 28 / 42

4D N = 1 data

Effective action of O3/O7 Calabi-Yau

• rientifolds (σΩ = −Ω).

[Grimm & Louis 04]

Assume complex structure moduli and dilaton are stabilized by fluxes at a higher scale.

[Kachru et al 03]

h1,1

+ orientifold-even K¨

ahler moduli from two-/four-cycle volumes complexified by

• C4

h1,1

− orientifold-odd K¨

ahler moduli from

• B2 and
• C2

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 28 / 42

4D N = 1 data

Effective action of O3/O7 Calabi-Yau

• rientifolds (σΩ = −Ω).

[Grimm & Louis 04]

Assume complex structure moduli and dilaton are stabilized by fluxes at a higher scale.

[Kachru et al 03]

h1,1

+ orientifold-even K¨

ahler moduli from two-/four-cycle volumes complexified by

• C4

h1,1

− orientifold-odd K¨

ahler moduli from

• B2 and
• C2

Supermultiplets Ga ≡ 2π

• ca − iba

gs

• ,

Tα ≡ iρα + 1 2cαβγvβvγ + gs 4 cαbcGb(G − ¯ G)c , intersection numbers cIJK =

• ωI ∧ ωJ ∧ ωK

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 28 / 42

4D N = 1 data

Tree-level K¨ ahler and super-potential

[Grimm & Louis 04]

K = −2 log V = −2 log 1 6cαβγvα(T, G)vβ(T, G)vγ(T, G)

• W

= W0 ca and ba enjoy a shift symmetry (world-sheet argument).

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 29 / 42

4D N = 1 data

Tree-level K¨ ahler and super-potential

[Grimm & Louis 04]

K = −2 log V = −2 log 1 6cαβγvα(T, G)vβ(T, G)vγ(T, G)

• W

= W0 ca and ba enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ Tα are not stabilized.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 29 / 42

4D N = 1 data

Tree-level K¨ ahler and super-potential

[Grimm & Louis 04]

K = −2 log V = −2 log 1 6cαβγvα(T, G)vβ(T, G)vγ(T, G)

• W

= W0 ca and ba enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ Tα are not stabilized. Non-perturbative corrections (gaugino condensation on D7’s) stabilize Tα

[Kachru et al. 03]

W = W0 +

h1,1

+

• α=1

Aαe−aαTα ,

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 29 / 42

4D N = 1 data

Tree-level K¨ ahler and super-potential

[Grimm & Louis 04]

K = −2 log V = −2 log 1 6cαβγvα(T, G)vβ(T, G)vγ(T, G)

• W

= W0 ca and ba enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ Tα are not stabilized. Non-perturbative corrections (gaugino condensation on D7’s) stabilize Tα

[Kachru et al. 03]

W = W0 +

h1,1

+

• α=1

Aαe−aαTα ,

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 29 / 42

4D N = 1 data

Tree-level K¨ ahler and super-potential

[Grimm & Louis 04]

K = −2 log V = −2 log 1 6cαβγvα(T, G)vβ(T, G)vγ(T, G)

• W

= W0 ca and ba enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ Tα are not stabilized. Non-perturbative corrections (gaugino condensation on D7’s) stabilize Tα

[Kachru et al. 03]

W = W0 +

h1,1

+

• α=1

Aαe−aαTα ,

Non-perturbative breaking of shift symmetry

Non-perturbative effects induce an η-problem for ba, analogous to D3-brane inflation.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 29 / 42

The axion decay constant

Which values can f take? Direct KK reduction from C2 = c(x)ω/2π gives f2 M2

Pl

= gsπ2 3VE ω ∧ ∗ω (2π)10(α′)3

• ∝ gsL2

VE .

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 30 / 42

The axion decay constant

Which values can f take? Direct KK reduction from C2 = c(x)ω/2π gives f2 M2

Pl

= gsπ2 3VE ω ∧ ∗ω (2π)10(α′)3

• ∝ gsL2

VE . Using N = 1 4D data one finds −1 2f2 (∂c)2 = ⊂ M2

PlKG ¯ G |∂G|2 ,

f2 M2

Pl

= gs 8π2 cα−−vα VE .

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 30 / 42

The axion decay constant

Which values can f take? Direct KK reduction from C2 = c(x)ω/2π gives f2 M2

Pl

= gsπ2 3VE ω ∧ ∗ω (2π)10(α′)3

• ∝ gsL2

VE . Using N = 1 4D data one finds −1 2f2 (∂c)2 = ⊂ M2

PlKG ¯ G |∂G|2 ,

f2 M2

Pl

= gs 8π2 cα−−vα VE .

Axion decay constant in string theory

In controlled setups gs ≪ 1 and L ≫ α′, hence f ≪ MPl.

[Banks et al. 03]

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 30 / 42

The effective potential

Inflaton potential V (φ) = µ3φ + bµ3f cos φ f

• b < 1 ⇒ monotonic potential

Large-field: φ ≫ MPl. COBE: µ = 6 · 10−4MPl and φin ≃ 11MPl. f ≪ MPl many short ripples. Different from the superplanckian case that seems to be hard to achieve in string theory.

[Banks et al. 03]

Two frequencies: expansion H and oscillations ω ≡ ˙ φ/f α ≡ ω H = √ 2ǫ f Oscillations in the CMB #osci ≃ α

The effective potential

Inflaton potential V (φ) = µ3φ + bµ3f cos φ f

• b < 1 ⇒ monotonic potential

Large-field: φ ≫ MPl. COBE: µ = 6 · 10−4MPl and φin ≃ 11MPl. f ≪ MPl many short ripples. Different from the superplanckian case that seems to be hard to achieve in string theory.

[Banks et al. 03]

Two frequencies: expansion H and oscillations ω ≡ ˙ φ/f α ≡ ω H = √ 2ǫ f Oscillations in the CMB #osci ≃ α

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 31 / 42

Tensor modes

Detectably large tensor modes: r = 0.07. We will know soon! Right red tilted spectrum. Non-perturbative corrections can generate characteristic oscillations. Smoking gun of axion inflation

Chaotic Inflation

0.92 0.94 0.96 0.98 1.0 1.02

ns

IIA Nil manifolds µ10/32/3 N = 50 N = 60 Linear Axion Inflation µ3 N = 50 N = 60

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 32 / 42

Spectrum of scalar perturbations

The oscillations in the potential induce oscillations in the spectrum: Ps(k) = As k k∗ ns−1 1 + δns cos φk f

• = As

k k∗ ns−1+

δns ln(k/k∗) cos

φk

f

• Enrico Pajer

(Cornell) Inflation from Axion Monodromy Stockholm June 2011 33 / 42

Spectrum of scalar perturbations

The oscillations in the potential induce oscillations in the spectrum: Ps(k) = As k k∗ ns−1 1 + δns cos φk f

• = As

k k∗ ns−1+

δns ln(k/k∗) cos

φk

f

• The frequency is simply

φk f = √ 2ǫ f log k We have computed δns analytically at leading order in b

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 33 / 42

Observational constraints on the spectrum

The best fit next to the unbinned WMAP5 data looks like

5 10 50 100 500 1000

• 2000

2000 4000 6000 8000 1C 2Π ΜK2 5 10 50 100 500 1000

• 2000

2000 4000 6000 8000 1C 2Π ΜK2

The improvement of the fit is not statistically significant. The bound of WMAP5 data on the parameters of the model is roughly fb < 10−4

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 34 / 42

Resonant non-Gaussianity

Large non-Gaussianity from modulations

Modulations on the potential violate slow roll and can induce large non-Gaussianity. Resonant non-Gaussianity can be computed analytically

[Flauger & EP 10] .

They are very large and are not scale invariant

[Chen et al. 08] .

The resonant shape is orthogonal to any other known shape. Hence there are almost no constraints on it.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 35 / 42

Oscillations in the spectrum and bispectrum

These are the observational constraints from the spectrum together with a contour plot of fres as functions of f and b.

• 4
• 3
• 2
• 1

log10f

• 4
• 3
• 2
• 1

log10b

TT TTT

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 36 / 42

Summary of axion monodromy inflation

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 37 / 42

Outline

1 Minimal requirements 2 Brane inflation 3 Inflation from axion monodromy 4 Conclusions

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 38 / 42

Conclusions

There exist models in string theory where inflation is technically natural The string theory embedding is important to investigate the robustness of effective models against Planck corrections Not all models have charactering features, but there are correlated

• bservables such as tensors and oscillations

Improvements

moduli stabilization and dS vacua reheating 10D description non-fine tuned small field models

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 39 / 42

Conclusions

There exist models in string theory where inflation is technically natural The string theory embedding is important to investigate the robustness of effective models against Planck corrections Not all models have charactering features, but there are correlated

• bservables such as tensors and oscillations

Improvements

moduli stabilization and dS vacua reheating 10D description non-fine tuned small field models

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 39 / 42

Backreaction

The most serious inflaton-dependent backreaction we identified:

• Σ C2 = 0 induces δND3(φ) =

φ 2πf units of D3 charge on the NS5.

D3-charge changes the warp factor and hence all warped volumes h(y) → h(y) + δh(y, φ) T α are warped 4-cycle volumes and are stabilized. The inflaton-dependent shift of T α can make the potential too steep. This correction is suppressed by δND3(φ)/N D3 but not by warping (because it comes from the CS term).

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 40 / 42

Backreaction

The most serious inflaton-dependent backreaction we identified:

• Σ C2 = 0 induces δND3(φ) =

φ 2πf units of D3 charge on the NS5.

D3-charge changes the warp factor and hence all warped volumes h(y) → h(y) + δh(y, φ) T α are warped 4-cycle volumes and are stabilized. The inflaton-dependent shift of T α can make the potential too steep. This correction is suppressed by δND3(φ)/N D3 but not by warping (because it comes from the CS term). Can we fix it?

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 40 / 42

Dipole suppression

The tadpole is canceled by the anti-NS5, so the total δND3(φ) is zero Depending on the geometry there can be a dipole suppression u/d

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 41 / 42

Dipole suppression

The tadpole is canceled by the anti-NS5, so the total δND3(φ) is zero Depending on the geometry there can be a dipole suppression u/d

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 41 / 42

Dipole suppression

The tadpole is canceled by the anti-NS5, so the total δND3(φ) is zero Depending on the geometry there can be a dipole suppression u/d We constructed a toy model that present this suppresion.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 41 / 42

Light KK modes

A large flux on the brane suppresses the KK masses. E.g. for a spacetime filling D5 wrapped on a 2-cycle of volume v2 When

• B2 ≡ b = 0, then

m2

KK,b ≃

v2 v2 + b2 m2

KK

large inflaton vev implies light KK modes. Typically mKK,b ∼ H.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 42 / 42

Light KK modes

A large flux on the brane suppresses the KK masses. E.g. for a spacetime filling D5 wrapped on a 2-cycle of volume v2 When

• B2 ≡ b = 0, then

m2

KK,b ≃

v2 v2 + b2 m2

KK

large inflaton vev implies light KK modes. Typically mKK,b ∼ H.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 42 / 42

Light KK modes

A large flux on the brane suppresses the KK masses. E.g. for a spacetime filling D5 wrapped on a 2-cycle of volume v2 When

• B2 ≡ b = 0, then

m2

KK,b ≃

v2 v2 + b2 m2

KK

large inflaton vev implies light KK modes. Typically mKK,b ∼ H. Notice that problems arise with large vevs.

Enrico Pajer (Cornell) Inflation from Axion Monodromy Stockholm June 2011 42 / 42