Inflation from axion monodromy based on Berg, E.P. & Sj ors, - PowerPoint PPT Presentation

Exciting signatures in the sky Obervables that could deeply impact our picture of the early universe: Tensor modes: Detectable in the T anisotropies or in the polarization of the CMB. Current bound on the tensor-to-scalar ratio: r < . 20 [WMAP7+SN] . A detection would support inflation and determine the high scale (order GUT) where it took place. non-Gaussianity: Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 7 / 43

Exciting signatures in the sky Obervables that could deeply impact our picture of the early universe: Tensor modes: Detectable in the T anisotropies or in the polarization of the CMB. Current bound on the tensor-to-scalar ratio: r < . 20 [WMAP7+SN] . A detection would support inflation and determine the high scale (order GUT) where it took place. non-Gaussianity: Detectable e.g. in the three-point function of T perturbations. Current bounds are of the order a percent (shape dependent). Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 7 / 43

Exciting signatures in the sky Obervables that could deeply impact our picture of the early universe: Tensor modes: Detectable in the T anisotropies or in the polarization of the CMB. Current bound on the tensor-to-scalar ratio: r < . 20 [WMAP7+SN] . A detection would support inflation and determine the high scale (order GUT) where it took place. non-Gaussianity: Detectable e.g. in the three-point function of T perturbations. Current bounds are of the order a percent (shape dependent). A detection would rule out the simplest class of models (a slowly rolling single canonically normalized field). Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 7 / 43

Exciting signatures in the sky Obervables that could deeply impact our picture of the early universe: Tensor modes: Detectable in the T anisotropies or in the polarization of the CMB. Current bound on the tensor-to-scalar ratio: r < . 20 [WMAP7+SN] . A detection would support inflation and determine the high scale (order GUT) where it took place. non-Gaussianity: Detectable e.g. in the three-point function of T perturbations. Current bounds are of the order a percent (shape dependent). A detection would rule out the simplest class of models (a slowly rolling single canonically normalized field). It tell us about the interaction of the inflaton and give us more information than a single number. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 7 / 43

Exciting signatures in the sky Obervables that could deeply impact our picture of the early universe: Tensor modes: Detectable in the T anisotropies or in the polarization of the CMB. Current bound on the tensor-to-scalar ratio: r < . 20 [WMAP7+SN] . A detection would support inflation and determine the high scale (order GUT) where it took place. non-Gaussianity: Detectable e.g. in the three-point function of T perturbations. Current bounds are of the order a percent (shape dependent). A detection would rule out the simplest class of models (a slowly rolling single canonically normalized field). It tell us about the interaction of the inflaton and give us more information than a single number. Isocurvature modes, curvature, features in the spectrum, . . . Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 7 / 43

An example of a precious synergy And example of a synergy between theory and observation in inflation from axion mondromy Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 8 / 43

Tensor modes and the Lyth bound The detection of tensor modes, e.g. in the B-mode polarization, would fix the scale of inflation close to the GUT scale. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 9 / 43

Tensor modes and the Lyth bound The detection of tensor modes, e.g. in the B-mode polarization, 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] � r ∆ φ > M pl 0 . 01 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 9 / 43

Tensor modes and the Lyth bound The detection of tensor modes, e.g. in the B-mode polarization, 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] � r ∆ φ > M pl 0 . 01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η -problem. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 9 / 43

Tensor modes and the Lyth bound The detection of tensor modes, e.g. in the B-mode polarization, 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] � r ∆ φ > M pl 0 . 01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η -problem. This is the main motivation to consider axion monodromy inflation Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 9 / 43

Tensor modes and the Lyth bound The detection of tensor modes, e.g. in the B-mode polarization, 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] � r ∆ φ > M pl 0 . 01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η -problem. This is the main motivation to consider axion monodromy inflation Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 9 / 43

Tensor modes and the Lyth bound The detection of tensor modes, e.g. in the B-mode polarization, 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] � r ∆ φ > M pl 0 . 01 In a fundamental theory a flat potential over a superplanckian distance is hard to control, e.g. η -problem. This is the main motivation to consider axion monodromy inflation Schematically more Tensor High Large ⇒ ⇒ ⇒ UV-sensitive modes scale field Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 9 / 43

UV-sensitivity EFT approach: learn about higher scales studying UV-sensitive observables. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 10 / 43

UV-sensitivity EFT approach: learn about higher scales studying UV-sensitive observables. Inflation is a UV-sensitive mechanism. Schematically φ n V ( φ ) = 1 2 m 2 φ 2 + � λ n M n − 4 n pl Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 10 / 43

UV-sensitivity EFT approach: learn about higher scales studying UV-sensitive observables. Inflation is a UV-sensitive mechanism. Schematically φ n V ( φ ) = 1 2 m 2 φ 2 + � λ n M n − 4 n pl Within string theory and supergravity many models suffer from an η -problem. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 10 / 43

UV-sensitivity EFT approach: learn about higher scales studying UV-sensitive observables. Inflation is a UV-sensitive mechanism. Schematically φ n V ( φ ) = 1 2 m 2 φ 2 + � λ n M n − 4 n pl Within string theory and supergravity many models suffer from an η -problem. We need to invoke a symmetry, e.g. shift symmetry. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 10 / 43

UV-sensitivity EFT approach: learn about higher scales studying UV-sensitive observables. Inflation is a UV-sensitive mechanism. Schematically φ n V ( φ ) = 1 2 m 2 φ 2 + � λ n M n − 4 n pl Within string theory and supergravity many 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 KITP Mar 2010 10 / 43

Outline 1 Motivations 2 Inflation from axion monodromy 3 Dante’s Inferno 4 Conclusions Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 11 / 43

Axion monodromy Two difficulties for large field models in a UV theory Space: ∆ φ > M pl is often impossible (e.g. brane inflation, Natural inflation) Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 12 / 43

Axion monodromy Two difficulties for large field models in a UV theory Space: ∆ φ > M pl is often impossible (e.g. brane inflation, Natural inflation) Flatness: ǫ, η ≪ 1 is rare (e.g. η problem) Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 12 / 43

Axion monodromy Two difficulties for large field models in a UV theory Space: ∆ φ > M pl is often impossible (e.g. brane inflation, Natural inflation) Flatness: ǫ, η ≪ 1 is rare (e.g. η problem) Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 12 / 43

Axion monodromy Two difficulties for large field models in a UV theory Space: ∆ φ > M pl is often impossible (e.g. brane inflation, Natural inflation) Flatness: ǫ, η ≪ 1 is rare (e.g. η problem) Axion monodromy addresses both [(Silverstein & Westphal)(1+McAllister)] Invoke a shift symmetry on an “angular” field. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 12 / 43

Axion monodromy Two difficulties for large field models in a UV theory Space: ∆ φ > M pl is often impossible (e.g. brane inflation, Natural inflation) Flatness: ǫ, η ≪ 1 is rare (e.g. η problem) Axion monodromy addresses both [(Silverstein & Westphal)(1+McAllister)] Invoke a shift symmetry on an “angular” field. The symmetry is broken in a controlled way inducing a monodromy. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 12 / 43

Axion monodromy Two difficulties for large field models in a UV theory Space: ∆ φ > M pl is often impossible (e.g. brane inflation, Natural inflation) Flatness: ǫ, η ≪ 1 is rare (e.g. η problem) Axion monodromy addresses both [(Silverstein & Westphal)(1+McAllister)] Invoke a shift symmetry on an “angular” field. The symmetry is broken in a controlled way inducing a monodromy. This enlarges the field space and provides the potential for inflation. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 12 / 43

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] Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 13 / 43

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 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 13 / 43

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 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 13 / 43

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 � φ � L ⊃ 1 2( ∂φ ) 2 + Λ 4 cos ⇒ φ ( x ) → φ ( x ) + 2 πf f Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 13 / 43

A simple example A canonically normalized axion with a shift symmetry V ( φ ) = const Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 14 / 43

A simple example A canonically normalized axion with a shift symmetry � φ � V ( φ ) = const + Λ 4 cos f Non-perturbative effects are exponentially suppressed. They lead to a very exciting phenomenology, see Raphael Flauger’s talk! Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 14 / 43

A simple example A canonically normalized axion with a shift symmetry � φ � + 1 V ( φ ) = const + Λ 4 cos 2 m 2 φ 2 f Non-perturbative effects are exponentially suppressed. They lead to a very exciting phenomenology, see Raphael Flauger’s talk! Break the shift symmetry explicitely in a controlled way. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 14 / 43

A simple example A canonically normalized axion with a shift symmetry � φ � + 1 V ( φ ) = const + Λ 4 cos 2 m 2 φ 2 f Non-perturbative effects are exponentially suppressed. They lead to a very exciting phenomenology, see Raphael Flauger’s talk! Break the shift symmetry explicitely in a controlled way. m controls the breaking: in the limit m → 0 the potential is flat. Higher corrections are suppressed in m/ Λ for some cutoff Λ Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 14 / 43

A simple example A canonically normalized axion with a shift symmetry � φ � + 1 V ( φ ) = const + Λ 4 cos 2 m 2 φ 2 f Non-perturbative effects are exponentially suppressed. They lead to a very exciting phenomenology, see Raphael Flauger’s talk! Break the shift symmetry explicitely in a controlled way. m controls the breaking: in the limit m → 0 the potential is flat. Higher corrections are suppressed in m/ Λ for some cutoff Λ Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 14 / 43

A simple example A canonically normalized axion with a shift symmetry � φ � + 1 V ( φ ) = const + Λ 4 cos 2 m 2 φ 2 f Non-perturbative effects are exponentially suppressed. They lead to a very exciting phenomenology, see Raphael Flauger’s talk! Break the shift symmetry explicitely in a controlled way. m controls the breaking: in the limit m → 0 the potential is flat. Higher corrections are suppressed in m/ Λ for some cutoff Λ What happens beyond the effective description? Is the shift symmetry broken above the cutoff? Are non-perturbative effects always negligible? Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 14 / 43

The phenomenology Observables tensor modes. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 15 / 43

The phenomenology Observables tensor modes. n s depends on the details of the monodromy potential. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 15 / 43

The phenomenology Observables tensor modes. n s depends on the details of the monodromy potential. Non-pertubative effects lead to oscillations in the spectrum and large resonant non-Gaussianity (Raphael’s talk tomorrow) Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 15 / 43

The phenomenology Observables tensor modes. n s depends on the details of the monodromy potential. Non-pertubative effects lead to oscillations in the spectrum and large resonant non-Gaussianity (Raphael’s talk tomorrow) Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 15 / 43

The phenomenology Observables tensor modes. n s depends on the details of the monodromy potential. Non-pertubative effects lead to oscillations in the spectrum and large resonant non-Gaussianity (Raphael’s talk tomorrow) Chaotic Inflation Linear Axion Inflation µ 3 � N = 50 N = 60 IIA Nil manifolds µ 10/3 � 2/3 N = 50 N = 60 0.92 0.94 0.96 0.98 1.0 1.02 n s Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 15 / 43

The phenomenology Observables tensor modes. n s depends on the details of the monodromy potential. Non-pertubative effects lead to oscillations in the spectrum and large resonant non-Gaussianity (Raphael’s talk tomorrow) Chaotic Inflation Linear Axion Inflation µ 3 � N = 50 N = 60 IIA Nil manifolds µ 10/3 � 2/3 N = 50 N = 60 0.92 0.94 0.96 0.98 1.0 1.02 n s Can we implement this idea in string theory? What do we learn? Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 15 / 43

Axion in string theory String theory seen from a low energy 4D observer: Model independent axions such as dualizing B µν or C µν Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 16 / 43

Axion in string theory String theory seen from a low energy 4D observer: Model independent axions such as dualizing B µν or C µν Model dependent axions from integrating a p-form over a p-cycle of the compact manifold � � c ( x ) = C p , b ( x ) = B 2 Σ p Σ 2 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 16 / 43

Axion in string theory String theory seen from a low energy 4D observer: Model independent axions such as dualizing B µν or C µν Model dependent axions from integrating a p-form over a p-cycle of the compact manifold � � c ( x ) = C p , b ( x ) = B 2 Σ p Σ 2 The shift symmetry is valid at all order in perturbation theory but broken non-pertubatively, e.g by world-sheet instantons or brane instantons. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 16 / 43

Axion in string theory String theory seen from a low energy 4D observer: Model independent axions such as dualizing B µν or C µν Model dependent axions from integrating a p-form over a p-cycle of the compact manifold � � c ( x ) = C p , b ( x ) = B 2 Σ p Σ 2 The shift symmetry is valid at all order in perturbation theory but broken non-pertubatively, e.g by world-sheet instantons or brane instantons. The axion decay constant f is determined by geometrical data of the compactification. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 16 / 43

Axion in string theory String theory seen from a low energy 4D observer: Model independent axions such as dualizing B µν or C µν Model dependent axions from integrating a p-form over a p-cycle of the compact manifold � � c ( x ) = C p , b ( x ) = B 2 Σ p Σ 2 The shift symmetry is valid at all order in perturbation theory but broken non-pertubatively, e.g by world-sheet instantons or brane instantons. The axion decay constant f is determined by geometrical data of the compactification. In controlled setups f < M pl [Banks et al 03] Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 16 / 43

Shift symmetry The 4D axion b ( x ) from B ij = b ( x ) ω ij , with ω a two-form. In (bosonic) closed string theory, the vertex operator for b at zero momentum integrated over the world-sheet is � � d 2 σǫ αβ ∂ α X i ∂ β X j ω ij b = V ( k = 0) = B ws ts In perturbation theory the world-sheet wraps topologically trivial cycles hence V ( k = 0) = 0, only derivative coplings. [Wen & Witten, Dine & Seiberg 86] Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 17 / 43

Shift symmetry The 4D axion b ( x ) from B ij = b ( x ) ω ij , with ω a two-form. In (bosonic) closed string theory, the vertex operator for b at zero momentum integrated over the world-sheet is � � d 2 σǫ αβ ∂ α X i ∂ β X j ω ij b = V ( k = 0) = B ws ts In perturbation theory the world-sheet 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 effects Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 17 / 43

Shift symmetry The 4D axion b ( x ) from B ij = b ( x ) ω ij , with ω a two-form. In (bosonic) closed string theory, the vertex operator for b at zero momentum integrated over the world-sheet is � � d 2 σǫ αβ ∂ α X i ∂ β X j ω ij b = V ( k = 0) = B ws ts In perturbation theory the world-sheet 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 effects World sheet with boundaries, i.e. D-branes Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 17 / 43

The ingredients The setup [McAllister, Silverstein & Westphal 08] Type IIB orientifolds. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 18 / 43

The ingredients The setup [McAllister, Silverstein & Westphal 08] Type IIB orientifolds. N = 1, 4D: an axion c ( x ) from RR field C 2 � c ( x ) = C 2 . Σ 2 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 18 / 43

The ingredients The setup [McAllister, Silverstein & Westphal 08] Type IIB orientifolds. N = 1, 4D: an axion c ( x ) from RR field C 2 � c ( x ) = C 2 . Σ 2 Wrapping a 5-brane over Σ 2 induces a monodromy for c ( x ) (world-sheets with boundary). Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 18 / 43

The ingredients The setup [McAllister, Silverstein & Westphal 08] Type IIB orientifolds. N = 1, 4D: an axion c ( x ) from RR field C 2 � c ( x ) = C 2 . Σ 2 Wrapping a 5-brane over Σ 2 induces a monodromy for c ( x ) (world-sheets with boundary). If the 5-brane is in a warped region the potential is flat. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 18 / 43

The ingredients The setup [McAllister, Silverstein & Westphal 08] Type IIB orientifolds. N = 1, 4D: an axion c ( x ) from RR field C 2 � c ( x ) = C 2 . Σ 2 Wrapping a 5-brane over Σ 2 induces a monodromy for c ( x ) (world-sheets with boundary). If the 5-brane is in a warped region the potential is flat. Moduli stabilization ´ a la KKLT does not spoil the shift symmetry. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 18 / 43

The ingredients The setup [McAllister, Silverstein & Westphal 08] Type IIB orientifolds. N = 1, 4D: an axion c ( x ) from RR field C 2 � c ( x ) = C 2 . Σ 2 Wrapping a 5-brane over Σ 2 induces a monodromy for c ( x ) (world-sheets with boundary). If the 5-brane is in a warped region the potential is flat. 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 KITP Mar 2010 18 / 43

Linear potential for the inflaton The shift symmetry can be broken in the presence of boundaries. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 19 / 43

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 Σ 2 . The DBI action � � d 6 xe − Φ det ( G ind + B ind ) − T 5 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 19 / 43

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 Σ 2 . The DBI action � � d 6 xe − Φ det ( G ind + B ind ) − T 5 � The shift b ( x ) → b ( x ) + const of b ( x ) = Σ 2 B 2 stores some potential energy. � L 4 + b 2 ∼ T 5 b V ( b ) = T 5 for large b Linear inflaton potential (and breaks SUSY). COBE normalization and control require to red-shift T 5 . Via S-duality, NS5 gives a monodromy for c . Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 19 / 43

4D N = 1 data Effective action of O 3 /O 7 Calabi-Yau orientifolds ( σ Ω = − Ω). [Grimm & Louis 04] Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 20 / 43

4D N = 1 data Effective action of O 3 /O 7 Calabi-Yau orientifolds ( σ Ω = − 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 KITP Mar 2010 20 / 43

4D N = 1 data Effective action of O 3 /O 7 Calabi-Yau orientifolds ( σ Ω = − Ω). [Grimm & Louis 04] Assume complex structure moduli and dilaton are stabilized by fluxes at a higher scale. [Kachru et al 03] h 1 , 1 + orientifold-even K¨ ahler moduli from two-/four-cycle volumes � complexified by C 4 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 20 / 43

4D N = 1 data Effective action of O 3 /O 7 Calabi-Yau orientifolds ( σ Ω = − Ω). [Grimm & Louis 04] Assume complex structure moduli and dilaton are stabilized by fluxes at a higher scale. [Kachru et al 03] h 1 , 1 + orientifold-even K¨ ahler moduli from two-/four-cycle volumes � complexified by C 4 h 1 , 1 � � − orientifold-odd K¨ ahler moduli from B 2 and C 2 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 20 / 43

4D N = 1 data Effective action of O 3 /O 7 Calabi-Yau orientifolds ( σ Ω = − Ω). [Grimm & Louis 04] Assume complex structure moduli and dilaton are stabilized by fluxes at a higher scale. [Kachru et al 03] h 1 , 1 + orientifold-even K¨ ahler moduli from two-/four-cycle volumes � complexified by C 4 h 1 , 1 � � − orientifold-odd K¨ ahler moduli from B 2 and C 2 Supermultiplets c a − ib a � � G a ≡ 2 π , g s iρ α + 1 2 c αβγ v β v γ + g s G ) c , 4 c αbc G b ( G − ¯ T α ≡ � ω I ∧ ω J ∧ ω K intersection numbers c IJK = Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 20 / 43

4D N = 1 data Tree-level K¨ ahler and super-potential [Grimm & Louis 04] � 1 � 6 c αβγ v α ( T, G ) v β ( T, G ) v γ ( T, G ) K = − 2 log V E = − 2 log W = W 0 c a and b a enjoy a shift symmetry (world-sheet argument). Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 21 / 43

4D N = 1 data Tree-level K¨ ahler and super-potential [Grimm & Louis 04] � 1 � 6 c αβγ v α ( T, G ) v β ( T, G ) v γ ( T, G ) K = − 2 log V E = − 2 log W = W 0 c a and b a enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ T α are not stabilized. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 21 / 43

4D N = 1 data Tree-level K¨ ahler and super-potential [Grimm & Louis 04] � 1 � 6 c αβγ v α ( T, G ) v β ( T, G ) v γ ( T, G ) K = − 2 log V E = − 2 log W = W 0 c a and b a enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ T α are not stabilized. Non-perturbative corrections (ED3 or gaugino condensation on D7’s) stabilize T α [Kachru et al. 03] h 1 , 1 + A α e − a α T α , � W = W 0 + α =1 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 21 / 43

4D N = 1 data Tree-level K¨ ahler and super-potential [Grimm & Louis 04] � 1 � 6 c αβγ v α ( T, G ) v β ( T, G ) v γ ( T, G ) K = − 2 log V E = − 2 log W = W 0 c a and b a enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ T α are not stabilized. Non-perturbative corrections (ED3 or gaugino condensation on D7’s) stabilize T α [Kachru et al. 03] h 1 , 1 + A α e − a α T α , � W = W 0 + α =1 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 21 / 43

4D N = 1 data Tree-level K¨ ahler and super-potential [Grimm & Louis 04] � 1 � 6 c αβγ v α ( T, G ) v β ( T, G ) v γ ( T, G ) K = − 2 log V E = − 2 log W = W 0 c a and b a enjoy a shift symmetry (world-sheet argument). No-scale structure of K ⇒ T α are not stabilized. Non-perturbative corrections (ED3 or gaugino condensation on D7’s) stabilize T α [Kachru et al. 03] h 1 , 1 + A α e − a α T α , � W = W 0 + α =1 Non-perturbative breaking of shift symmetry Non-perturbative effects could spoil the shift symmetry. In fact they induce an η -problem for b a , analogous to D3-brane inflation. Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 21 / 43

Moduli stabilization The supersymmetric conditions ensuring a minimum are D α W = − A α a α e − a α T α − W v α 0 = , 2 V E D a W = Wπc αac v α b c 0 = V E Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 22 / 43

Moduli stabilization The supersymmetric conditions ensuring a minimum are D α W = − A α a α e − a α T α − W v α 0 = , 2 V E D a W = Wπc αac v α b c 0 = V E D α W = 0 fixes T α (complex equation) Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 22 / 43

Moduli stabilization The supersymmetric conditions ensuring a minimum are D α W = − A α a α e − a α T α − W v α 0 = , 2 V E D a W = Wπc αac v α b c 0 = V E D α W = 0 fixes T α (complex equation) D a W = 0 fixes ony b a = 0 Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 22 / 43

Moduli stabilization The supersymmetric conditions ensuring a minimum are D α W = − A α a α e − a α T α − W v α 0 = , 2 V E D a W = Wπc αac v α b c 0 = V E D α W = 0 fixes T α (complex equation) D a W = 0 fixes ony b a = 0 c a still enjoys a shift symmetry [L¨ ust et al 06, Grimm 07] Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 22 / 43

Moduli stabilization The supersymmetric conditions ensuring a minimum are D α W = − A α a α e − a α T α − W v α 0 = , 2 V E D a W = Wπc αac v α b c 0 = V E D α W = 0 fixes T α (complex equation) D a W = 0 fixes ony b a = 0 c a still enjoys a shift symmetry [L¨ ust et al 06, Grimm 07] Non-perturbative breaking of shift symmetry It is crucial to know how the shift symmetry is broken. Moduli a la KKLT is incompatible with b a shift symmetry. stabilization ´ Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 22 / 43

The axion decay constant Which values can f take? Direct KK reduction from C 2 = c ( x ) ω/ 2 π gives f 2 g s π 2 � � ∝ L 2 ω ∧ ∗ ω � c = . M 2 (2 π ) 10 ( α ′ ) 3 3 V E V E pl Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 23 / 43

The axion decay constant Which values can f take? Direct KK reduction from C 2 = c ( x ) ω/ 2 π gives f 2 g s π 2 � � ∝ L 2 ω ∧ ∗ ω � c = . M 2 (2 π ) 10 ( α ′ ) 3 3 V E V E pl Using N = 1 4D data one finds − 1 2 f 2 ( ∂c ) 2 G | ∂G | 2 , ⊂ M 2 = pl K G ¯ Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 23 / 43