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Minimalistic axion monodromy with an ultra-light throat mode Jakob Moritz DESY July 04, 2017 Theoretical Approaches to Cosmic Acceleration, Leiden A. Hebecker, JM, A. Westphal, L.T. Witkowski [arXiv:1512.04463] + ongoing work with Sascha

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SLIDE 1

Minimalistic axion monodromy with an ultra-light throat mode

Jakob Moritz

DESY

July 04, 2017 Theoretical Approaches to Cosmic Acceleration, Leiden

  • A. Hebecker, JM, A. Westphal, L.T. Witkowski [arXiv:1512.04463]

+ ongoing work with Sascha Leonhardt

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 1 / 16

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SLIDE 2

Introduction

Large field inflation and axions

Open question: Is it possible to construct models of large field inflation in string theory?

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 2 / 16

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SLIDE 3

Introduction

Large field inflation and axions

Open question: Is it possible to construct models of large field inflation in string theory? Interesting because:

1

Observable tensor modes in CMB require ∆φ & Mp. [Lyth ’96]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 2 / 16

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SLIDE 4

Introduction

Large field inflation and axions

Open question: Is it possible to construct models of large field inflation in string theory? Interesting because:

1

Observable tensor modes in CMB require ∆φ & Mp. [Lyth ’96]

2

As a matter of principle: General lessons about Quantum Gravity?

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 2 / 16

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SLIDE 5

Introduction

Large field inflation and axions

Open question: Is it possible to construct models of large field inflation in string theory? Interesting because:

1

Observable tensor modes in CMB require ∆φ & Mp. [Lyth ’96]

2

As a matter of principle: General lessons about Quantum Gravity?

To fit the CMB power spectrum, overall scale of the potential has to be kept small over large range of inflaton.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 2 / 16

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SLIDE 6

Introduction

Large field inflation and axions

Open question: Is it possible to construct models of large field inflation in string theory? Interesting because:

1

Observable tensor modes in CMB require ∆φ & Mp. [Lyth ’96]

2

As a matter of principle: General lessons about Quantum Gravity?

To fit the CMB power spectrum, overall scale of the potential has to be kept small over large range of inflaton. Ideal inflaton candidates: Axions, due to their perturbative continuous shift symmetry φ ! φ + const.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 2 / 16

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SLIDE 7

Introduction

Large field inflation and axions

Open question: Is it possible to construct models of large field inflation in string theory? Interesting because:

1

Observable tensor modes in CMB require ∆φ & Mp. [Lyth ’96]

2

As a matter of principle: General lessons about Quantum Gravity?

To fit the CMB power spectrum, overall scale of the potential has to be kept small over large range of inflaton. Ideal inflaton candidates: Axions, due to their perturbative continuous shift symmetry φ ! φ + const. Non-perturbative instanton corrections yield periodic potential V (φ) = Λ4 (1 cos(φ/f )) + ... (1)

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 2 / 16

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SLIDE 8

Introduction

Large field inflation and axions

Open question: Is it possible to construct models of large field inflation in string theory? Interesting because:

1

Observable tensor modes in CMB require ∆φ & Mp. [Lyth ’96]

2

As a matter of principle: General lessons about Quantum Gravity?

To fit the CMB power spectrum, overall scale of the potential has to be kept small over large range of inflaton. Ideal inflaton candidates: Axions, due to their perturbative continuous shift symmetry φ ! φ + const. Non-perturbative instanton corrections yield periodic potential V (φ) = Λ4 (1 cos(φ/f )) + ... (1) ! continuous shift symmetry broken to φ ! φ + 2πf .

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 2 / 16

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SLIDE 9

Introduction

Axion monodromy

Problem: Such a potential requires super-Planckian decay constants f Mp to produce at least 60 e-folds of inflation.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 3 / 16

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SLIDE 10

Introduction

Axion monodromy

Problem: Such a potential requires super-Planckian decay constants f Mp to produce at least 60 e-folds of inflation. This is hard to realize in controlled string compactifications.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 3 / 16

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SLIDE 11

Introduction

Axion monodromy

Problem: Such a potential requires super-Planckian decay constants f Mp to produce at least 60 e-folds of inflation. This is hard to realize in controlled string compactifications. Possible ways out: axion-alignment [Kim,Nilles,Peloso’05/...], N-flation [Dimopoulos,Kachru,McGreevy,Wacker’05]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 3 / 16

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SLIDE 12

Introduction

Axion monodromy

Problem: Such a potential requires super-Planckian decay constants f Mp to produce at least 60 e-folds of inflation. This is hard to realize in controlled string compactifications. Possible ways out: axion-alignment [Kim,Nilles,Peloso’05/...], N-flation [Dimopoulos,Kachru,McGreevy,Wacker’05] But: These have come under pressure by arguments based on the ’weak gravity conjecture’ [Arkani-Hamed,Motl,Nicolis,Vafa’06] [Rudelius’15/Montero,Uranga,Valenzuela’15/Brown,Cottrell,Shiu,Soler’15...]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 3 / 16

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SLIDE 13

Introduction

Axion monodromy

Problem: Such a potential requires super-Planckian decay constants f Mp to produce at least 60 e-folds of inflation. This is hard to realize in controlled string compactifications. Possible ways out: axion-alignment [Kim,Nilles,Peloso’05/...], N-flation [Dimopoulos,Kachru,McGreevy,Wacker’05] But: These have come under pressure by arguments based on the ’weak gravity conjecture’ [Arkani-Hamed,Motl,Nicolis,Vafa’06] [Rudelius’15/Montero,Uranga,Valenzuela’15/Brown,Cottrell,Shiu,Soler’15...] Less constrained alternative: ’axion monodromy’ [Silverstein,Westphal ’08]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 3 / 16

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SLIDE 14

Introduction

Axion monodromy

Problem: Such a potential requires super-Planckian decay constants f Mp to produce at least 60 e-folds of inflation. This is hard to realize in controlled string compactifications. Possible ways out: axion-alignment [Kim,Nilles,Peloso’05/...], N-flation [Dimopoulos,Kachru,McGreevy,Wacker’05] But: These have come under pressure by arguments based on the ’weak gravity conjecture’ [Arkani-Hamed,Motl,Nicolis,Vafa’06] [Rudelius’15/Montero,Uranga,Valenzuela’15/Brown,Cottrell,Shiu,Soler’15...] Less constrained alternative: ’axion monodromy’ [Silverstein,Westphal ’08]

  • ! Break the periodicity of the axion weakly by introducing

’monodromic’ potential:

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 3 / 16

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SLIDE 15

Introduction

Axion monodromy

Problem: Such a potential requires super-Planckian decay constants f Mp to produce at least 60 e-folds of inflation. This is hard to realize in controlled string compactifications. Possible ways out: axion-alignment [Kim,Nilles,Peloso’05/...], N-flation [Dimopoulos,Kachru,McGreevy,Wacker’05] But: These have come under pressure by arguments based on the ’weak gravity conjecture’ [Arkani-Hamed,Motl,Nicolis,Vafa’06] [Rudelius’15/Montero,Uranga,Valenzuela’15/Brown,Cottrell,Shiu,Soler’15...] Less constrained alternative: ’axion monodromy’ [Silverstein,Westphal ’08]

  • ! Break the periodicity of the axion weakly by introducing

’monodromic’ potential: φ ! φ + 2πf ) V ! V + ∆V (2)

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 3 / 16

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SLIDE 16

Introduction

Realizations in string theory and their problems I:

type IIB string theory contains RR 2-form C2, with gauge symmetry C2 ! C2 + dΛ1 (3)

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 4 / 16

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SLIDE 17

Introduction

Realizations in string theory and their problems I:

type IIB string theory contains RR 2-form C2, with gauge symmetry C2 ! C2 + dΛ1 (3) Goal: Generate (non-periodic) potential for axions in 4d: ai = 1 2πα0 Z

Σi

2

C2 , [Σi

2] 2 H2(M)

(4)

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 4 / 16

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SLIDE 18

Introduction

Realizations in string theory and their problems I:

type IIB string theory contains RR 2-form C2, with gauge symmetry C2 ! C2 + dΛ1 (3) Goal: Generate (non-periodic) potential for axions in 4d: ai = 1 2πα0 Z

Σi

2

C2 , [Σi

2] 2 H2(M)

(4) Early idea: Wrap an NS5 brane around some 2-cycle [McAllister,Silverstein,Westphal ’08]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 4 / 16

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SLIDE 19

Introduction

Realizations in string theory and their problems I:

type IIB string theory contains RR 2-form C2, with gauge symmetry C2 ! C2 + dΛ1 (3) Goal: Generate (non-periodic) potential for axions in 4d: ai = 1 2πα0 Z

Σi

2

C2 , [Σi

2] 2 H2(M)

(4) Early idea: Wrap an NS5 brane around some 2-cycle [McAllister,Silverstein,Westphal ’08] ! Its tension increases as a ! a + 2π.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 4 / 16

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SLIDE 20

Introduction

Realizations in string theory and their problems I:

type IIB string theory contains RR 2-form C2, with gauge symmetry C2 ! C2 + dΛ1 (3) Goal: Generate (non-periodic) potential for axions in 4d: ai = 1 2πα0 Z

Σi

2

C2 , [Σi

2] 2 H2(M)

(4) Early idea: Wrap an NS5 brane around some 2-cycle [McAllister,Silverstein,Westphal ’08] ! Its tension increases as a ! a + 2π. But: For tadpole cancellation one has to introduce NS5 as well

  • ! problematic brane/antibrane back-reaction? [Conlon ’11]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 4 / 16

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SLIDE 21

Introduction

Realizations in string theory and their problems I:

type IIB string theory contains RR 2-form C2, with gauge symmetry C2 ! C2 + dΛ1 (3) Goal: Generate (non-periodic) potential for axions in 4d: ai = 1 2πα0 Z

Σi

2

C2 , [Σi

2] 2 H2(M)

(4) Early idea: Wrap an NS5 brane around some 2-cycle [McAllister,Silverstein,Westphal ’08] ! Its tension increases as a ! a + 2π. But: For tadpole cancellation one has to introduce NS5 as well

  • ! problematic brane/antibrane back-reaction? [Conlon ’11]
  • ! Required geometry: ’bifid throat’. [Retolaza,Uranga,Westphal’15]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 4 / 16

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SLIDE 22

Introduction

Realizations in string theory and their problems II:

Progress towards gaining quantitative control, i.e. effective 4d SUGRA description: ’F-term axion monodromy’ [Marchesano,Shiu,Uranga’14/Blumenhagen,Plauschinn’14/Hebecker,Kraus,Witkowski’14]

  • ! Rough idea: Going around periodic direction increases fluxes.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 5 / 16

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SLIDE 23

Introduction

Realizations in string theory and their problems II:

Progress towards gaining quantitative control, i.e. effective 4d SUGRA description: ’F-term axion monodromy’ [Marchesano,Shiu,Uranga’14/Blumenhagen,Plauschinn’14/Hebecker,Kraus,Witkowski’14]

  • ! Rough idea: Going around periodic direction increases fluxes.

However: Hard to realize mass hierarchy between inflaton and other moduli [Blumenhagen,Herschmann,Plauschinn’15/Hebecker,Mangat,Rompineve,Witkowski’15] [Blumenhagen,Valenzuela,Wolf’17]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 5 / 16

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SLIDE 24

Introduction

Realizations in string theory and their problems II:

Progress towards gaining quantitative control, i.e. effective 4d SUGRA description: ’F-term axion monodromy’ [Marchesano,Shiu,Uranga’14/Blumenhagen,Plauschinn’14/Hebecker,Kraus,Witkowski’14]

  • ! Rough idea: Going around periodic direction increases fluxes.

However: Hard to realize mass hierarchy between inflaton and other moduli [Blumenhagen,Herschmann,Plauschinn’15/Hebecker,Mangat,Rompineve,Witkowski’15] [Blumenhagen,Valenzuela,Wolf’17] This talk: Construct ’minimalistic’ model of axion monodromy as a ’playground’ for detailed studies.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 5 / 16

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The model

The model:

Instead of non-trivial 2-cycle ( ! massless axion):

  • ! Fibration of S2 over a base B: S2 trivial in H2(M).

(⇠massive Wilson ’line’ [Marchesano,Shiu,Uranga’14])

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 6 / 16

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SLIDE 26

The model

The model:

Instead of non-trivial 2-cycle ( ! massless axion):

  • ! Fibration of S2 over a base B: S2 trivial in H2(M).

(⇠massive Wilson ’line’ [Marchesano,Shiu,Uranga’14]) Canonical example: S3 = S2-fibration over interval.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 6 / 16

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SLIDE 27

The model

The model:

Instead of non-trivial 2-cycle ( ! massless axion):

  • ! Fibration of S2 over a base B: S2 trivial in H2(M).

(⇠massive Wilson ’line’ [Marchesano,Shiu,Uranga’14]) Canonical example: S3 = S2-fibration over interval. In our scenario: Strong warping along ’interval’.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 6 / 16

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SLIDE 28

The model

The model:

Instead of non-trivial 2-cycle ( ! massless axion):

  • ! Fibration of S2 over a base B: S2 trivial in H2(M).

(⇠massive Wilson ’line’ [Marchesano,Shiu,Uranga’14]) Canonical example: S3 = S2-fibration over interval. In our scenario: Strong warping along ’interval’. Why? If S2 degenerates in two strongly warped regions:

  • ! we find a single exceptionally light KK-mode!

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 6 / 16

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SLIDE 29

The model

The geometry:

Crucial ingredient: family of S2’s that collapses in two strongly warped regions:

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 7 / 16

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SLIDE 30

The model

The geometry:

Crucial ingredient: family of S2’s that collapses in two strongly warped regions: double-throat.

  • !
  • !

two warped throats

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 7 / 16

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SLIDE 31

The model

The geometry:

Crucial ingredient: family of S2’s that collapses in two strongly warped regions: double-throat.

  • !
  • !

two warped throats

  • !

family of S2’s

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 7 / 16

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SLIDE 32

The model

The geometry:

Crucial ingredient: family of S2’s that collapses in two strongly warped regions: double-throat.

  • !
  • !

two warped throats

  • !

family of S2’s

  • bulk CY orientifold

with K¨ ahler moduli stabilization (e.g. KKLT, or better KL?) [Kachru,Kallosh,Linde,Trivedi’03/Kallosh,Linde’04]

  • ! see Ander’s talk

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 7 / 16

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SLIDE 33

The model

The geometry:

Crucial ingredient: family of S2’s that collapses in two strongly warped regions: double-throat.

  • !
  • !

two warped throats

  • !

family of S2’s

  • bulk CY orientifold

with K¨ ahler moduli stabilization (e.g. KKLT, or better KL?) [Kachru,Kallosh,Linde,Trivedi’03/Kallosh,Linde’04]

  • ! see Ander’s talk

Such double throats are simple generalizations of single throat geometries and have been constructed explicitly. [Aganagic,Beem,Seo,Vafa’08]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 7 / 16

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SLIDE 34

The model

The geometry:

Crucial ingredient: family of S2’s that collapses in two strongly warped regions: double-throat.

  • !
  • !

two warped throats

  • !

family of S2’s

  • bulk CY orientifold

with K¨ ahler moduli stabilization (e.g. KKLT, or better KL?) [Kachru,Kallosh,Linde,Trivedi’03/Kallosh,Linde’04]

  • ! see Ander’s talk

Such double throats are simple generalizations of single throat geometries and have been constructed explicitly. [Aganagic,Beem,Seo,Vafa’08] Local geometry of each individual throat described by the Klebanov-Strassler solution. [Klebanov,Strassler’00]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 7 / 16

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SLIDE 35

The model

The Klebanov-Strassler throat:

Recall: Warped throats are a common feature of flux

  • compactifications. [Giddings,Kachru,Polchinski’02/...]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 8 / 16

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SLIDE 36

The model

The Klebanov-Strassler throat:

Recall: Warped throats are a common feature of flux

  • compactifications. [Giddings,Kachru,Polchinski’02/...]

With fluxes M ⇠ Z

Σ3

F3 and K ⇠ Z

˜ Σ3

H3 , gsM ⌧ K , (5)

  • n dual pair of three-cycles (Σ3, ˜

Σ3),

  • btain long, strongly warped and smooth throat.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 8 / 16

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SLIDE 37

The model

The Klebanov-Strassler throat:

Recall: Warped throats are a common feature of flux

  • compactifications. [Giddings,Kachru,Polchinski’02/...]

With fluxes M ⇠ Z

Σ3

F3 and K ⇠ Z

˜ Σ3

H3 , gsM ⌧ K , (5)

  • n dual pair of three-cycles (Σ3, ˜

Σ3),

  • btain long, strongly warped and smooth throat.

Klebanov-Strassler solution: Warped version of deformed conifold

  • solution. [Candelas,de la Ossa’89]

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 8 / 16

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SLIDE 38

The model

A toy model of the throat:

Simple toy geometry: M = S3 ⇥ X, with ’capped cylinder’ X,

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 9 / 16

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SLIDE 39

The model

A toy model of the throat:

Simple toy geometry: M = S3 ⇥ X, with ’capped cylinder’ X, y S3

1/2

S2 ⇥ R = S2 R R

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 9 / 16

slide-40
SLIDE 40

The model

A toy model of the throat:

Simple toy geometry: M = S3 ⇥ X, with ’capped cylinder’ X, y S3

1/2

S2 ⇥ R = S2 R R

1 ’Radial’ direction y runs from

tip ’upwards’.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 9 / 16

slide-41
SLIDE 41

The model

A toy model of the throat:

Simple toy geometry: M = S3 ⇥ X, with ’capped cylinder’ X, y S3

1/2

S2 ⇥ R = S2 R R

1 ’Radial’ direction y runs from

tip ’upwards’.

2 y = const. slices: S2 ⇥ S3. Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 9 / 16

slide-42
SLIDE 42

The model

A toy model of the throat:

Simple toy geometry: M = S3 ⇥ X, with ’capped cylinder’ X, y S3

1/2

S2 ⇥ R = S2 R R

1 ’Radial’ direction y runs from

tip ’upwards’.

2 y = const. slices: S2 ⇥ S3. 3 At the tip, the S2 degenerates

smoothly, while the S3 stays at finite size.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 9 / 16

slide-43
SLIDE 43

The model

A toy model of the throat:

Simple toy geometry: M = S3 ⇥ X, with ’capped cylinder’ X, y S3

1/2

S2 ⇥ R = S2 R R

1 ’Radial’ direction y runs from

tip ’upwards’.

2 y = const. slices: S2 ⇥ S3. 3 At the tip, the S2 degenerates

smoothly, while the S3 stays at finite size.

4 S3

! Σ3 carries RR flux X ! ˜ Σ3 carries NS flux.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 9 / 16

slide-44
SLIDE 44

The model

A toy model of the throat:

Simple toy geometry: M = S3 ⇥ X, with ’capped cylinder’ X, y S3

1/2

S2 ⇥ R = S2 R R

1 ’Radial’ direction y runs from

tip ’upwards’.

2 y = const. slices: S2 ⇥ S3. 3 At the tip, the S2 degenerates

smoothly, while the S3 stays at finite size.

4 S3

! Σ3 carries RR flux X ! ˜ Σ3 carries NS flux.

5 Randall-Sundrum-like metric:

ds2 = e2kyηµνdxµdxν + ds2

M

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 9 / 16

slide-45
SLIDE 45

The model

The ansatz & spectrum:

We make the ansatz C2 = φ(x, y) · ω2 , (6) for y 0 where ω2 is the harmonic two form of S2.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 10 / 16

slide-46
SLIDE 46

The model

The ansatz & spectrum:

We make the ansatz C2 = φ(x, y) · ω2 , (6) for y 0 where ω2 is the harmonic two form of S2. Kaluza-Klein reduction to 4d can be done analytically.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 10 / 16

slide-47
SLIDE 47

The model

The ansatz & spectrum:

We make the ansatz C2 = φ(x, y) · ω2 , (6) for y 0 where ω2 is the harmonic two form of S2. Kaluza-Klein reduction to 4d can be done analytically. Mass spectrum: All but lightest mode:

m2

n

M2

p ⇠ n2 m2 wKK

M2

p

⇠ n2e2kL.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 10 / 16

slide-48
SLIDE 48

The model

The ansatz & spectrum:

We make the ansatz C2 = φ(x, y) · ω2 , (6) for y 0 where ω2 is the harmonic two form of S2. Kaluza-Klein reduction to 4d can be done analytically. Mass spectrum: All but lightest mode:

m2

n

M2

p ⇠ n2 m2 wKK

M2

p

⇠ n2e2kL. Lightest mode:

m2 M2

p ⇠ m2 wKK

M2

p e2kL ⇠ e4kL

! ’ultra-light’.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 10 / 16

slide-49
SLIDE 49

The model

The ansatz & spectrum:

We make the ansatz C2 = φ(x, y) · ω2 , (6) for y 0 where ω2 is the harmonic two form of S2. Kaluza-Klein reduction to 4d can be done analytically. Mass spectrum: All but lightest mode:

m2

n

M2

p ⇠ n2 m2 wKK

M2

p

⇠ n2e2kL. Lightest mode:

m2 M2

p ⇠ m2 wKK

M2

p e2kL ⇠ e4kL

! ’ultra-light’. Crucial point The lowest lying mode is exponentially lighter than the higher KK-modes.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 10 / 16

slide-50
SLIDE 50

The model

The ansatz & spectrum:

We make the ansatz C2 = φ(x, y) · ω2 , (6) for y 0 where ω2 is the harmonic two form of S2. Kaluza-Klein reduction to 4d can be done analytically. Mass spectrum: All but lightest mode:

m2

n

M2

p ⇠ n2 m2 wKK

M2

p

⇠ n2e2kL. Lightest mode:

m2 M2

p ⇠ m2 wKK

M2

p e2kL ⇠ e4kL

! ’ultra-light’. Crucial point The lowest lying mode is exponentially lighter than the higher KK-modes. Note: We find same behavior in actual KS-background (numerically).

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 10 / 16

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SLIDE 51

The model

The field profiles:

n=0

χ0( )

χ0(y) ⇡ 1 c · e4ky

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 11 / 16

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SLIDE 52

The model

The field profiles:

n=0

χ0( )

χ0(y) ⇡ 1 c · e4ky

n=1

χ1( )

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 11 / 16

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SLIDE 53

The model

The field profiles:

n=0

χ0( )

χ0(y) ⇡ 1 c · e4ky

n=1

χ1( )

n=2

  • χ2( )

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 11 / 16

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SLIDE 54

The model

The field profiles:

n=0

χ0( )

χ0(y) ⇡ 1 c · e4ky

n=1

χ1( )

n=2

  • χ2( )

n=3

  • χ3( )

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 11 / 16

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SLIDE 55

The model

Axion monodromy with the ultra-light mode:

So: Realizing the mass hierarchy was easy.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 12 / 16

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SLIDE 56

The model

Axion monodromy with the ultra-light mode:

So: Realizing the mass hierarchy was easy. Moreover: Decay constant and ’critical field excursion’ are related:

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 12 / 16

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SLIDE 57

The model

Axion monodromy with the ultra-light mode:

So: Realizing the mass hierarchy was easy. Moreover: Decay constant and ’critical field excursion’ are related: φcritical 2πf = M π , M : #RR-flux quanta (7)

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 12 / 16

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SLIDE 58

The model

Axion monodromy with the ultra-light mode:

So: Realizing the mass hierarchy was easy. Moreover: Decay constant and ’critical field excursion’ are related: φcritical 2πf = M π , M : #RR-flux quanta (7) Therefore: Axion monodromy realized with full quantitative control:

  • 1.0
  • 0.5

0.5 1.0

  • critical

0.1 0.2 0.3 0.4 0.5 V()

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 12 / 16

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SLIDE 59

The model

What about inflation?

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 13 / 16

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SLIDE 60

The model

What about inflation?

Back-reaction?

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 13 / 16

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SLIDE 61

The model

What about inflation?

Back-reaction? Interference with K¨ ahler moduli stabilization controlled due to strong warping, but...

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 13 / 16

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SLIDE 62

The model

What about inflation?

Back-reaction? Interference with K¨ ahler moduli stabilization controlled due to strong warping, but... Local back-reaction (on warped KK modes) becomes strong at φcritical . Mp . (8)

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 13 / 16

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SLIDE 63

The model

What about inflation?

Back-reaction? Interference with K¨ ahler moduli stabilization controlled due to strong warping, but... Local back-reaction (on warped KK modes) becomes strong at φcritical . Mp . (8) fundamental problem? [Blumenhagen,Valenzuela,Wolf’17] No: H ⌧ mwKK.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 13 / 16

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SLIDE 64

The model

What about inflation?

Back-reaction? Interference with K¨ ahler moduli stabilization controlled due to strong warping, but... Local back-reaction (on warped KK modes) becomes strong at φcritical . Mp . (8) fundamental problem? [Blumenhagen,Valenzuela,Wolf’17] No: H ⌧ mwKK.

  • ! Usefulness for inflation becomes issue of detailed numerical

analysis.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 13 / 16

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SLIDE 65

The model

What about inflation?

Back-reaction? Interference with K¨ ahler moduli stabilization controlled due to strong warping, but... Local back-reaction (on warped KK modes) becomes strong at φcritical . Mp . (8) fundamental problem? [Blumenhagen,Valenzuela,Wolf’17] No: H ⌧ mwKK.

  • ! Usefulness for inflation becomes issue of detailed numerical

analysis. Luckily: Since throat metric is known, this is difficult but doable!

  • ! ongoing work...

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 13 / 16

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SLIDE 66

Conclusion

Summary

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 67

Conclusion

Summary

Have identified ultra-light axion mode.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 68

Conclusion

Summary

Have identified ultra-light axion mode. Can realize axion monodromy over many axion ’windings’.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 69

Conclusion

Summary

Have identified ultra-light axion mode. Can realize axion monodromy over many axion ’windings’. No brane/antibrane pair required ! no explicit SUSY breaking.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 70

Conclusion

Summary

Have identified ultra-light axion mode. Can realize axion monodromy over many axion ’windings’. No brane/antibrane pair required ! no explicit SUSY breaking. For inflation: Need numerical back-reaction study (ongoing work).

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 71

Conclusion

Summary

Have identified ultra-light axion mode. Can realize axion monodromy over many axion ’windings’. No brane/antibrane pair required ! no explicit SUSY breaking. For inflation: Need numerical back-reaction study (ongoing work). Back-reaction effects are surprisingly ’local’ and therefore tractable.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 72

Conclusion

Summary

Have identified ultra-light axion mode. Can realize axion monodromy over many axion ’windings’. No brane/antibrane pair required ! no explicit SUSY breaking. For inflation: Need numerical back-reaction study (ongoing work). Back-reaction effects are surprisingly ’local’ and therefore tractable.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 73

Conclusion

Summary

Have identified ultra-light axion mode. Can realize axion monodromy over many axion ’windings’. No brane/antibrane pair required ! no explicit SUSY breaking. For inflation: Need numerical back-reaction study (ongoing work). Back-reaction effects are surprisingly ’local’ and therefore tractable. The end!

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 14 / 16

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SLIDE 74

Appendix

The ultra-light mode in the actual KS-throat:

In the actual KS background, one cannot derive an analytical solution. But: Can use most general ansatz compatible with SU(2) ⇥ SU(2) symmetry group of the deformed conifold. [Minasian,Tsimpis’99] (Preliminary) solution to type IIB linearized equations of motion:

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 15 / 16

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SLIDE 75

Appendix

Construction of double throat [Aganagic,Beem,Seo,Vafa’08]

Conifold can be described by subset of C4 that satisfies u · v = y2 + x2 . (9)

1

Replace x by polynomial W 0(x) = g(x x1)(x x2).

2

For g = 0: Curve of A1 singularities.

3

Blowing up singularity ! curve of P1’s.

4

For g 6= 0: Still, curve of S2’s.

5

Geometric transition: u · v = y 2 + W 0(x)2 + f1(x) , (10) with f1 a degree one polynomial.

Jakob Moritz (DESY) Minimalistic axion monodromy with an ultra-light throat mode July 04, 2017 16 / 16

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SLIDE 76

Funding acknowledgement: This work is supported by the ERC Consolidator Grant STRINGFLATION under the HORIZON 2020 grant agreement

  • no. 647995.