cornell university, ithaca

Inflation from axion monodromy

based on Berg, E.P. & Sjors, arXiv:0912.1341 (hep-th) andFlauger, McAllister, E.P., Westphal & Xu, arXiv:0907.2916 (hep-th)

Enrico Pajer

Cornell University, Ithaca

KITPMar 2010

Outline

1 Motivations

2 Inflation from axion monodromy

3 Dante’s Inferno

4 Conclusions

Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 2 / 43

Outline

1 Motivations


3 Dante’s Inferno

4 Conclusions


Cosmological data

We are living in the golden age ofobservational cosmology:

COBEgoes to Stockholm, WMAP hasmeasured the CMB with percentaccuracy. . .

and now Planck: the satellite, launched on May 2009, finished the firstfull sky map (95%) on Feb 14th!


Cosmological data

We are living in the golden age ofobservational cosmology: COBEgoes to Stockholm,

WMAP hasmeasured the CMB with percentaccuracy. . .



Cosmological data

We are living in the golden age ofobservational cosmology: COBEgoes to Stockholm, WMAP hasmeasured the CMB with percentaccuracy. . .



The picture emerging from the data

A mechanism whose main parameter is unknown by at least 10orders of manitude and nevertheless works?

Inflation

Inflation does not solve the horizon and flatness problem but canarguably alleviate it. It provides a mechanism that shifts in thepast the initial condition problem.Neverless it is a spectacular model to generate cosmologicalperturbations.The simplest models of inflation is compatible with the data[see e.g. WMAP7] , i.e. small, scale-invariant but slightly red tilted,Gaussian, adiabatic primordial curvature perturbations.What is left to do?



A mechanism whose main parameter is unknown by at least 10orders of manitude and nevertheless works? Inflation

Inflation does not solve the horizon and flatness problem but canarguably alleviate it. It provides a mechanism that shifts in thepast the initial condition problem.Neverless it is a spectacular model to generate cosmologicalperturbations.The simplest models of inflation is compatible with the data[see e.g. WMAP7] , i.e. small, scale-invariant but slightly red tilted,Gaussian, adiabatic primordial curvature perturbations.What is left to do?



A mechanism whose main parameter is unknown by at least 10orders of manitude and nevertheless works? InflationInflation does not solve the horizon and flatness problem but canarguably alleviate it. It provides a mechanism that shifts in thepast the initial condition problem.

Neverless it is a spectacular model to generate cosmologicalperturbations.The simplest models of inflation is compatible with the data[see e.g. WMAP7] , i.e. small, scale-invariant but slightly red tilted,Gaussian, adiabatic primordial curvature perturbations.What is left to do?



A mechanism whose main parameter is unknown by at least 10orders of manitude and nevertheless works? InflationInflation does not solve the horizon and flatness problem but canarguably alleviate it. It provides a mechanism that shifts in thepast the initial condition problem.Neverless it is a spectacular model to generate cosmologicalperturbations.

The simplest models of inflation is compatible with the data[see e.g. WMAP7] , i.e. small, scale-invariant but slightly red tilted,Gaussian, adiabatic primordial curvature perturbations.What is left to do?



A mechanism whose main parameter is unknown by at least 10orders of manitude and nevertheless works? InflationInflation does not solve the horizon and flatness problem but canarguably alleviate it. It provides a mechanism that shifts in thepast the initial condition problem.Neverless it is a spectacular model to generate cosmologicalperturbations.The simplest models of inflation is compatible with the data[see e.g. WMAP7] , i.e. small, scale-invariant but slightly red tilted,Gaussian, adiabatic primordial curvature perturbations.

What is left to do?



A mechanism whose main parameter is unknown by at least 10orders of manitude and nevertheless works? InflationInflation does not solve the horizon and flatness problem but canarguably alleviate it. It provides a mechanism that shifts in thepast the initial condition problem.Neverless it is a spectacular model to generate cosmologicalperturbations.The simplest models of inflation is compatible with the data[see e.g. WMAP7] , i.e. small, scale-invariant but slightly red tilted,Gaussian, adiabatic primordial curvature perturbations.What is left to do?


Different approaches

A lot is left to do:

Precision measurementsThe broad predictions of thesimplest model of inflation areveryfied at the percent level.

Precision measurements,e.g. Planck, will provide newtests.Departures from the simplestmodel? E.g. whatnon-Gaussian signal do welook for? How significant areanomalies?

Theoretical soundness

Initial conditionsIs inflation natural?EFT approach: fine tuningand symmetriesFundamental theory approach:when symmetries are brokenand fine tuning is possible?Non trivial correlationsbetween observables




Precision measurementsThe broad predictions of thesimplest model of inflation areveryfied at the percent level.Precision measurements,e.g. Planck, will provide newtests.

Departures from the simplestmodel? E.g. whatnon-Gaussian signal do welook for? How significant areanomalies?






Precision measurementsThe broad predictions of thesimplest model of inflation areveryfied at the percent level.Precision measurements,e.g. Planck, will provide newtests.Departures from the simplestmodel? E.g. whatnon-Gaussian signal do welook for? How significant areanomalies?







Theoretical soundnessInitial conditions

Is inflation natural?EFT approach: fine tuningand symmetriesFundamental theory approach:when symmetries are brokenand fine tuning is possible?Non trivial correlationsbetween observables





Theoretical soundnessInitial conditionsIs inflation natural?

EFT approach: fine tuningand symmetriesFundamental theory approach:when symmetries are brokenand fine tuning is possible?Non trivial correlationsbetween observables





Theoretical soundnessInitial conditionsIs inflation natural?EFT approach: fine tuningand symmetries

Fundamental theory approach:when symmetries are brokenand fine tuning is possible?Non trivial correlationsbetween observables





Theoretical soundnessInitial conditionsIs inflation natural?EFT approach: fine tuningand symmetriesFundamental theory approach:when symmetries are brokenand fine tuning is possible?

Non trivial correlationsbetween observables





Theoretical soundnessInitial conditionsIs inflation natural?EFT approach: fine tuningand symmetriesFundamental theory approach:when symmetries are brokenand fine tuning is possible?Non trivial correlationsbetween observables


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 slowlyrolling single canonically normalized field).It tell us about the interaction of the inflaton and give us moreinformation than a single number.

Isocurvature modes, curvature, features in the spectrum, . . .




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:







non-Gaussianity:







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 slowlyrolling single canonically normalized field).It tell us about the interaction of the inflaton and give us moreinformation than a single number.






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 slowlyrolling single canonically normalized field).

It tell us about the interaction of the inflaton and give us moreinformation than a single number.






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 slowlyrolling single canonically normalized field).It tell us about the interaction of the inflaton and give us moreinformation than a single number.



An example of a precious synergy

And example of a synergy between theory and observation in inflationfrom axion mondromy


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 ofvariation of the inflaton [Lyth 98]

∆φMpl

>

√r

0.01

In a fundamental theory a flat potential over a superplanckiandistance is hard to control, e.g. η-problem.This is the main motivation to consider axion monodromy inflation

Schematically

Tensormodes

⇒ Highscale

⇒ Largefield

⇒ moreUV-sensitive



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 ofvariation of the inflaton [Lyth 98]

∆φMpl

>

√r

0.01


Schematically

Tensormodes

⇒ Highscale

⇒ Largefield





∆φMpl

>

√r

0.01

In a fundamental theory a flat potential over a superplanckiandistance is hard to control, e.g. η-problem.

This is the main motivation to consider axion monodromy inflation

Schematically

Tensormodes

⇒ Highscale

⇒ Largefield





∆φMpl

>

√r

0.01


Schematically

Tensormodes

⇒ Highscale

⇒ Largefield



UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitiveobservables.

Inflation is a UV-sensitive mechanism. Schematically

V (φ) =12m2φ2 +

∑n

λnφn

Mn−4pl

Within string theory and supergravity many models suffer from anη-problem.We need to invoke a symmetry, e.g. shift symmetry.Then we need a fundamental theory (UV-finite) to ask if, how andwhere the symmetry is broken.


UV-sensitivity

EFT approach: learn about higher scales studying UV-sensitiveobservables.Inflation is a UV-sensitive mechanism. Schematically

V (φ) =12m2φ2 +

∑n

λnφn

Mn−4pl



UV-sensitivity


V (φ) =12m2φ2 +

∑n

λnφn

Mn−4pl

Within string theory and supergravity many models suffer from anη-problem.

We need to invoke a symmetry, e.g. shift symmetry.Then we need a fundamental theory (UV-finite) to ask if, how andwhere the symmetry is broken.


UV-sensitivity


V (φ) =12m2φ2 +

∑n

λnφn

Mn−4pl

Within string theory and supergravity many models suffer from anη-problem.We need to invoke a symmetry, e.g. shift symmetry.

Then we need a fundamental theory (UV-finite) to ask if, how andwhere the symmetry is broken.


UV-sensitivity


V (φ) =12m2φ2 +

∑n

λnφn

Mn−4pl



Outline

1 Motivations


3 Dante’s Inferno

4 Conclusions


Axion monodromy

Two difficulties for large field models in a UV theorySpace: ∆φ > Mpl is often impossible (e.g. brane inflation, Naturalinflation)

Flatness: ε, 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 wayinducing a monodromy.This enlarges the field space and provides thepotential for inflation.


Axion monodromy

Two difficulties for large field models in a UV theorySpace: ∆φ > Mpl is often impossible (e.g. brane inflation, Naturalinflation)Flatness: ε, η 1 is rare (e.g. η problem)

Axion monodromy addresses both [(Silverstein & Westphal)(1+McAllister)]



Axion monodromy



Invoke a shift symmetry on an “angular”field.

The symmetry is broken in a controlled wayinducing a monodromy.This enlarges the field space and provides thepotential for inflation.


Axion monodromy



Invoke a shift symmetry on an “angular”field.The symmetry is broken in a controlled wayinducing a monodromy.

This enlarges the field space and provides thepotential for inflation.


Axion monodromy





Axions in field theory

Axions are scalar fields with only derivative couplings and mightarise e.g. from the breaking of a U(1) symmetry [Peccei & Quinn 77]

Hence they enjoy a continuous shift symmetry at all orders inperturbation theory

φ(x)→ φ(x) + constant

Continuous shift symmetry is broken to a discrete shift symmetryby non-perturbative effectsThe axion decay constant f determines the periodicity of thecanonically normalized axion

L ⊃ 12

(∂φ)2 + Λ4 cos(φ

f

)⇒ φ(x)→ φ(x) + 2πf






Continuous shift symmetry is broken to a discrete shift symmetryby non-perturbative effects

The axion decay constant f determines the periodicity of thecanonically normalized axion

L ⊃ 12

(∂φ)2 + Λ4 cos(φ

f

)⇒ φ(x)→ φ(x) + 2πf






Continuous shift symmetry is broken to a discrete shift symmetryby non-perturbative effectsThe axion decay constant f determines the periodicity of thecanonically normalized axion

L ⊃ 12

(∂φ)2 + Λ4 cos(φ

f

)⇒ φ(x)→ φ(x) + 2πf


A simple example

A canonically normalized axion with a shift symmetry

V (φ) = const

+ Λ4 cos(φ

f

)+

12m2φ2

Non-perturbative effects are exponentially suppressed. They leadto 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?


A simple example


V (φ) = const + Λ4 cos(φ

f

)

+12m2φ2

Non-perturbative effects are exponentially suppressed. They leadto 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



A simple example



f

)+

12m2φ2

Non-perturbative effects are exponentially suppressed. They leadto 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



A simple example



f

)+

12m2φ2

Non-perturbative effects are exponentially suppressed. They leadto 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



The phenomenology

Observables tensor modes.

ns depends on the details of the monodromy potential.Non-pertubative effects lead to oscillations in the spectrum andlarge resonant non-Gaussianity (Raphael’s talk tomorrow)

Chaotic Inflation

0.92 0.94 0.96 0.98 1.0 1.02

ns

IIA Nil manifoldsµ10/3!2/3

N = 50 N = 60

Linear Axion Inflationµ3!

N = 50N = 60

Can we implement this idea in string theory? What do we learn?


The phenomenology

Observables tensor modes.ns depends on the details of the monodromy potential.

Non-pertubative effects lead to oscillations in the spectrum andlarge resonant non-Gaussianity (Raphael’s talk tomorrow)

Chaotic Inflation

0.92 0.94 0.96 0.98 1.0 1.02

ns


N = 50 N = 60


N = 50N = 60



The phenomenology

Observables tensor modes.ns depends on the details of the monodromy potential.Non-pertubative effects lead to oscillations in the spectrum andlarge resonant non-Gaussianity (Raphael’s talk tomorrow)

Chaotic Inflation

0.92 0.94 0.96 0.98 1.0 1.02

ns


N = 50 N = 60


N = 50N = 60



The phenomenology

Observables tensor modes.ns depends on the details of the monodromy potential.Non-pertubative effects lead to oscillations in the spectrum andlarge resonant non-Gaussianity (Raphael’s talk tomorrow)

Chaotic Inflation

0.92 0.94 0.96 0.98 1.0 1.02

ns


N = 50 N = 60


N = 50N = 60

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 Bµν or Cµν

Model dependent axions from integrating a p-form over a p-cycleof the compact manifold

c(x) =∫

Σp

Cp , b(x) =∫

Σ2

B2

The shift symmetry is valid at all order in perturbation theory butbroken non-pertubatively, e.g by world-sheet instantons or braneinstantons.The axion decay constant f is determined by geometrical data ofthe compactification.In controlled setups f < Mpl [Banks et al 03]



String theory seen from a low energy 4D observer:Model independent axions such as dualizing Bµν or CµνModel dependent axions from integrating a p-form over a p-cycleof the compact manifold

c(x) =∫

Σp

Cp , b(x) =∫

Σ2

B2





c(x) =∫

Σp

Cp , b(x) =∫

Σ2

B2

The shift symmetry is valid at all order in perturbation theory butbroken non-pertubatively, e.g by world-sheet instantons or braneinstantons.

The axion decay constant f is determined by geometrical data ofthe compactification.In controlled setups f < Mpl [Banks et al 03]




c(x) =∫

Σp

Cp , b(x) =∫

Σ2

B2

The shift symmetry is valid at all order in perturbation theory butbroken non-pertubatively, e.g by world-sheet instantons or braneinstantons.The axion decay constant f is determined by geometrical data ofthe compactification.

In controlled setups f < Mpl [Banks et al 03]




c(x) =∫

Σp

Cp , b(x) =∫

Σ2

B2



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 momentumintegrated over the world-sheet is

V (k = 0) =∫wsd2σεαβ∂αX

i∂βXjωijb =

∫tsB

In perturbation theory the world-sheet wraps topologically trivialcycles 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 effectsWorld sheet with boundaries, i.e. D-branes


Shift symmetry



i∂βXjωijb =

∫tsB



Two ingredients can invalidate the above argument:Non-perturbative effects

World sheet with boundaries, i.e. D-branes


Shift symmetry



i∂βXjωijb =

∫tsB



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


The ingredients

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 a 5-brane over Σ2 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 Kahler potential) inducesmall ripples


The ingredients


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

c(x) =∫

Σ2

C2 .



The ingredients



c(x) =∫

Σ2

C2 .

Wrapping a 5-brane over Σ2 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 Kahler potential) inducesmall ripples


The ingredients



c(x) =∫

Σ2

C2 .

Wrapping a 5-brane over Σ2 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 Kahler potential) inducesmall ripples


The ingredients



c(x) =∫

Σ2

C2 .

Wrapping a 5-brane over Σ2 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 Kahler potential) inducesmall ripples


The ingredients



c(x) =∫

Σ2

C2 .



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. TheDBI action

−T5

∫d6xe−Φ

√det (Gind +Bind)

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

Σ2B2 stores

some potential energy.

V (b) = T5

√L4 + b2 ∼ T5b for large b

Linear inflaton potential (and breaks SUSY). COBEnormalization and control require to red-shift T5.

Via S-duality, NS5 gives a monodromy for c.


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. TheDBI action

−T5

∫d6xe−Φ

√det (Gind +Bind)

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

Σ2B2 stores

some potential energy.

V (b) = T5

√L4 + b2 ∼ T5b for large b

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


4D N = 1 data

Effective action of O3/O7 Calabi-Yauorientifolds (σΩ = −Ω). [Grimm & Louis 04]

Assume complex structure moduliand dilaton are stabilized by fluxes ata higher scale. [Kachru et al 03]

h1,1+ orientifold-even Kahler moduli from two-/four-cycle volumes

complexified by∫C4

h1,1− orientifold-odd Kahler moduli from

∫B2 and

∫C2

Supermultiplets

Ga ≡ 2π(ca − ib

a

gs

),

Tα ≡ iρα +12cαβγv

βvγ +gs4cαbcG

b(G− G)c ,

intersection numbers cIJK =∫ωI ∧ ωJ ∧ ωK


4D N = 1 data

Tree-level Kahler and super-potential [Grimm & Louis 04]

K = −2 logVE = −2 log[

16cαβγv

α(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 (ED3 or gaugino condensation onD7’s) stabilize Tα [Kachru et al. 03]

W = W0 +h1,1

+∑α=1

Aαe−aαTα ,

Non-perturbative breaking of shift symmetry

Non-perturbative effects could spoil the shift symmetry. In fact theyinduce an η-problem for ba, analogous to D3-brane inflation.


4D N = 1 data



16cαβγv


W = W0

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

Non-perturbative corrections (ED3 or gaugino condensation onD7’s) stabilize Tα [Kachru et al. 03]

W = W0 +h1,1

+∑α=1

Aαe−aαTα ,




4D N = 1 data



16cαβγv


W = W0

ca and ba enjoy a shift symmetry (world-sheet argument).No-scale structure of K ⇒ Tα are not stabilized.Non-perturbative corrections (ED3 or gaugino condensation onD7’s) stabilize Tα [Kachru et al. 03]

W = W0 +h1,1

+∑α=1

Aαe−aαTα ,




Moduli stabilization

The supersymmetric conditions ensuring a minimum are

0 = DαW = −Aαaαe−aαTα −Wvα

2VE,

0 = DaW = Wπcαacv

αbc

VE

DαW = 0 fixes Tα (complex equation)DaW = 0 fixes ony ba = 0ca still enjoys a shift symmetry [Lust et al 06, Grimm 07]


It is crucial to know how the shift symmetry is broken. Modulistabilization a la KKLT is incompatible with ba shift symmetry.





2VE,

0 = DaW = Wπcαacv

αbc

VE

DαW = 0 fixes Tα (complex equation)

DaW = 0 fixes ony ba = 0ca still enjoys a shift symmetry [Lust et al 06, Grimm 07]







2VE,

0 = DaW = Wπcαacv

αbc

VE

DαW = 0 fixes Tα (complex equation)DaW = 0 fixes ony ba = 0

ca still enjoys a shift symmetry [Lust et al 06, Grimm 07]







2VE,

0 = DaW = Wπcαacv

αbc

VE

DαW = 0 fixes Tα (complex equation)DaW = 0 fixes ony ba = 0ca still enjoys a shift symmetry [Lust et al 06, Grimm 07]




The axion decay constant

Which values can f take? Direct KK reduction from C2 = c(x)ω/2πgives

f2

M2pl

=gsπ

2

3VE

( ∫ω ∧ ∗ω

(2π)10(α′)3

)∝ L2

c

VE.

Using N = 1 4D data one finds

−12f2 (∂c)2 = ⊂M2

plKGG |∂G|2 ,

f2

M2pl

=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]


Constraints from the moduli stabilization

A series of constraints follow from consistency and computability

small coupling ⇒ gs 1

small world-sheet instantons ⇒ vα > 1π√gs

no higher instantons ⇒ Tα >Nαπ , with Nα . 50 D7-branes

no destabilization ⇒ V (φCMB) < UmodHigh scale inflation and KKLT stabilization lead an upper bound onthe volume (lower bound on ms/Mpl)

τα 73− 8 log(vαπ√gs

2gs

),

VE < h(1,1)+

√gs 1.8 · 104 ,







no destabilization ⇒ V (φCMB) < Umod

High scale inflation and KKLT stabilization lead an upper bound onthe volume (lower bound on ms/Mpl)


2gs

),

VE < h(1,1)+

√gs 1.8 · 104 ,







no destabilization ⇒ V (φCMB) < UmodHigh scale inflation and KKLT stabilization lead an upper bound onthe volume (lower bound on ms/Mpl)


2gs

),

VE < h(1,1)+

√gs 1.8 · 104 ,


The amplitude of modulations

Non-perturbative corrections leads to ripples on the linear potential

V (φ) = µ3φ+ bµ3f cos(φ

f

)

F-term corrections need instantons with four fermionic zeromodes, e.g. non-BPS instantons. Few is known due to the lack ofholomorphicity. Educated guess:

K = −2 log[VE + e−SED1 cos(c)

]= −2 log

[VE + e

− 2πv+√gs cos(c)

].

D-term is protected by holomorphicity, corrections should besmall.



Expanding as x = x0 + e−SED1x1 + . . . , the moduli stabilization gives

VSUGRA = Umod,0[1 + e−SED1 cos c

(K(1) + 2Re

W(1)

W(0)

)]

Hence the estimate size of the ripples is

bf =Umod,0 φµ3φ

e−SED1

(K(1) + 2Re

W(1)

W(0)

)< 2c0 · 109M4

pl

gsV2E

e−2/gs

Exponentially suppressed in v+/√gs > 1/(πgs).

Enhanced by high moduli stabilization barrier.



Expanding as x = x0 + e−SED1x1 + . . . , the moduli stabilization gives

VSUGRA = Umod,0[1 + e−SED1 cos c

(K(1) + 2Re

W(1)

W(0)

)]Hence the estimate size of the ripples is

bf =Umod,0 φµ3φ

e−SED1

(K(1) + 2Re

W(1)

W(0)

)< 2c0 · 109M4

pl

gsV2E

e−2/gs

Exponentially suppressed in v+/√gs > 1/(πgs).

Enhanced by high moduli stabilization barrier.


Backreaction

The most serious inflaton-dependent backreaction we identified:∫ΣC2 6= 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)

Tα are warped 4-cycle volumes and are stabilized.The inflaton-dependent shift of Tα can make the potential toosteep.

This correction is suppressed by δND3(φ)/ND3 but not by warping(because it comes from the CS term).

Can we fix it?


Backreaction


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)



Can we fix it?


Backreaction



h(y)→ h(y) + δh(y, φ)

Tα are warped 4-cycle volumes and are stabilized.

The inflaton-dependent shift of Tα can make the potential toosteep.


Can we fix it?


Backreaction



h(y)→ h(y) + δh(y, φ)



Can we fix it?


Dipole suppression

The tadpole is canceled by the anti-NS5, so the total δND3(φ) iszero

Depending on the geometry there can be a dipole suppression u/d

We constructed a toy model that present this suppresion.


Dipole suppression

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

We constructed a toy model that present this suppresion.


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 6= 0, then

m2KK,b '

v2

v2 + b2m2KK

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

Notice that problems arise with large vevs.


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


m2KK,b '

v2

v2 + b2m2KK


Notice that problems arise with large vevs.


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


m2KK,b '

v2

v2 + b2m2KK


Notice that problems arise with large vevs.Enrico Pajer (Cornell) Inflation from axion monodromy KITP Mar 2010 29 / 43

Outline

1 Motivations


3 Dante’s Inferno

4 Conclusions


Back to the Lyth bound

There is a dichotomy which becomes evident with more than oneinflaton.

The bound is on the effective inflatonφeff , i.e. the length of the inflationarytrajectory ∆φeff ≡

∫dφeff

Quantumcorrections growwith the vev’s offundamental fields.

The Lyth bound

The consequences of the Lythbound are generically differentin multi-field inflation

How complicate a potential can provide this classical trajectories?


A simple model

The potential is as simple as this:

V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]

Two (canonically normalized) axions r, θ, with respective axiondecay constants fr, fθ.The shift symmetry of r is broken by a monodromy term W (r).This could be anything. For illustration W (r) = m2r2/2.A non-perturbative effect involves a linear combination of r and θ.θ enjoys a shift symmetry to all order in perturbation theorybroken only by non-perturbative effects to θ → θ + 2πfθ.


A simple model


V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]

Two (canonically normalized) axions r, θ, with respective axiondecay constants fr, fθ.

The shift symmetry of r is broken by a monodromy term W (r).This could be anything. For illustration W (r) = m2r2/2.A non-perturbative effect involves a linear combination of r and θ.θ enjoys a shift symmetry to all order in perturbation theorybroken only by non-perturbative effects to θ → θ + 2πfθ.


A simple model


V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]

Two (canonically normalized) axions r, θ, with respective axiondecay constants fr, fθ.The shift symmetry of r is broken by a monodromy term W (r).This could be anything. For illustration W (r) = m2r2/2.

A non-perturbative effect involves a linear combination of r and θ.θ enjoys a shift symmetry to all order in perturbation theorybroken only by non-perturbative effects to θ → θ + 2πfθ.


A simple model


V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]

Two (canonically normalized) axions r, θ, with respective axiondecay constants fr, fθ.The shift symmetry of r is broken by a monodromy term W (r).This could be anything. For illustration W (r) = m2r2/2.A non-perturbative effect involves a linear combination of r and θ.

θ enjoys a shift symmetry to all order in perturbation theorybroken only by non-perturbative effects to θ → θ + 2πfθ.


A simple model


V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]

Two (canonically normalized) axions r, θ, with respective axiondecay constants fr, fθ.The shift symmetry of r is broken by a monodromy term W (r).This could be anything. For illustration W (r) = m2r2/2.A non-perturbative effect involves a linear combination of r and θ.θ enjoys a shift symmetry to all order in perturbation theorybroken only by non-perturbative effects to θ → θ + 2πfθ.


The infernal potential

The potential on the two-field space

The periodicity in θ is evident in polar coordinates.

Hence the name...


Solution of the infernal dynamics

In the regimeA. fr fθ Mpl,B. Λ4 frm

2r0,r can be integrated out (mr > H), i.e. r = r(θ) and one finds theeffective single field potential

Veff (φeff) =12m2

eff φ2eff , meff ≡ m

frfθ

where φeff ' cos(fr/fθ)θ + sin(fr/fθ)r.


The extra dial and the η-problem

The η-problem is alleviated

Since meff = m(fr/fθ),even if m ∼ H and hence r would have an η-problem, a mildhierachy fr/fθ ∼ O(.1) gives slow-roll inflation.Intuitively φeff is mostly θ which has a shift symmetry.


The extra dial and the field range

What about the field range?

∆φeff ' 15Mpl, but...whole inflationary dynamics is takes place inside

0 < θ < 2πfθ , 0 < r < 15Mplfrfθ

Provided fr/fθ ∼ O(10−1 − 10−2), chaotic inflation takes place ina region subplanckian in size.




∆φeff ' 15Mpl, but...

whole inflationary dynamics is takes place inside

0 < θ < 2πfθ , 0 < r < 15Mplfrfθ





∆φeff ' 15Mpl, but...whole inflationary dynamics is takes place inside

0 < θ < 2πfθ , 0 < r < 15Mplfrfθ



Summary of the effective model

Phenomenology:Dante’s Inferno gives prediction similar to a single-field slow-rollchaotic model.

The precise numbers depend on the details of the monodromyterm W (r).Generically, observable tensor modes are generated.

Theoretical considerations:

The inflaton is mostly an axion with a shift symmetry (onlynon-perturbative corrections) which alleviates the η-problem.The whole large-field inflationary dynamics takes place within aregion subplanckian in size.Issues related to the large vev’s of the axions are alleviated



Phenomenology:Dante’s Inferno gives prediction similar to a single-field slow-rollchaotic model.The precise numbers depend on the details of the monodromyterm W (r).

Generically, observable tensor modes are generated.Theoretical considerations:




Phenomenology:Dante’s Inferno gives prediction similar to a single-field slow-rollchaotic model.The precise numbers depend on the details of the monodromyterm W (r).Generically, observable tensor modes are generated.

Theoretical considerations:





Theoretical considerations:The inflaton is mostly an axion with a shift symmetry (onlynon-perturbative corrections) which alleviates the η-problem.

The whole large-field inflationary dynamics takes place within aregion subplanckian in size.Issues related to the large vev’s of the axions are alleviated




Theoretical considerations:The inflaton is mostly an axion with a shift symmetry (onlynon-perturbative corrections) which alleviates the η-problem.The whole large-field inflationary dynamics takes place within aregion subplanckian in size.

Issues related to the large vev’s of the axions are alleviated




Theoretical considerations:The inflaton is mostly an axion with a shift symmetry (onlynon-perturbative corrections) which alleviates the η-problem.The whole large-field inflationary dynamics takes place within aregion subplanckian in size.Issues related to the large vev’s of the axions are alleviated


A cartoon of Dante’s Inferno

New ingredients: two two-cycles Σr and Σθ


The two axions

Non-perturbative effects (e.g. ED1) and the monodromy term(5-brane) can wrap two overlapping but non-identiacal two-cycles.

We can choose a basis of two-cycles such that only one axion has amonodromy, say r.We expect the effective potential of the form

V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]Big advantage: we do not need to carefully understand W (r) likein the single field caseEven if W (r) has an η-problem, inflation can work providedfr fθ.


The two axions

Non-perturbative effects (e.g. ED1) and the monodromy term(5-brane) can wrap two overlapping but non-identiacal two-cycles.We can choose a basis of two-cycles such that only one axion has amonodromy, say r.

We expect the effective potential of the form

V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ



The two axions

Non-perturbative effects (e.g. ED1) and the monodromy term(5-brane) can wrap two overlapping but non-identiacal two-cycles.We can choose a basis of two-cycles such that only one axion has amonodromy, say r.We expect the effective potential of the form

V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]

Big advantage: we do not need to carefully understand W (r) likein the single field caseEven if W (r) has an η-problem, inflation can work providedfr fθ.


The two axions


V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ

)]Big advantage: we do not need to carefully understand W (r) likein the single field case

Even if W (r) has an η-problem, inflation can work providedfr fθ.


The two axions


V (r(x), θ(x)) = W (r) + Λ4

[1− cos

(r

fr− θ

fθ



Smaller axion vev’s

Constraints from consistency and computabiility are the same asin the single field case.

The main difference is that now

∆r ' ∆φfrfθ ∆φ ∼ 15Mpl

A smaller ∆r implies less induced D3 charge, so less backreaction.KK-modes are heavier

mKK,b ∼mKK

b' mKK

∆φfθfr mKK

∆φ.

The shift of 4-cycle volumes is not a problem since it involves rwhile the inflaton is mostly θ which enjoys a shift symmetry.



Constraints from consistency and computabiility are the same asin the single field case.The main difference is that now



mKK,b ∼mKK

b' mKK

∆φfθfr mKK

∆φ.






A smaller ∆r implies less induced D3 charge, so less backreaction.

KK-modes are heavier

mKK,b ∼mKK

b' mKK

∆φfθfr mKK

∆φ.







mKK,b ∼mKK

b' mKK

∆φfθfr mKK

∆φ.



Outline

1 Motivations


3 Dante’s Inferno

4 Conclusions


To do list

There are many possible further directionsConstruct a more explicit model.

Investigate perturbative moduli stabilization.Better understand the non-perturbative effects, also in light oftheir phenomenological consequences.Find a more controlled monodromy.Investigate the phenomenological consequences of the two-fieldmodel. . .


To do list

There are many possible further directionsConstruct a more explicit model.Investigate perturbative moduli stabilization.

Better understand the non-perturbative effects, also in light oftheir phenomenological consequences.Find a more controlled monodromy.Investigate the phenomenological consequences of the two-fieldmodel. . .


To do list

There are many possible further directionsConstruct a more explicit model.Investigate perturbative moduli stabilization.Better understand the non-perturbative effects, also in light oftheir phenomenological consequences.

Find a more controlled monodromy.Investigate the phenomenological consequences of the two-fieldmodel. . .


To do list

There are many possible further directionsConstruct a more explicit model.Investigate perturbative moduli stabilization.Better understand the non-perturbative effects, also in light oftheir phenomenological consequences.Find a more controlled monodromy.

Investigate the phenomenological consequences of the two-fieldmodel. . .


To do list

There are many possible further directionsConstruct a more explicit model.Investigate perturbative moduli stabilization.Better understand the non-perturbative effects, also in light oftheir phenomenological consequences.Find a more controlled monodromy.Investigate the phenomenological consequences of the two-fieldmodel

. . .


To do list

There are many possible further directionsConstruct a more explicit model.Investigate perturbative moduli stabilization.Better understand the non-perturbative effects, also in light oftheir phenomenological consequences.Find a more controlled monodromy.Investigate the phenomenological consequences of the two-fieldmodel. . .


Conclusions

Cosmology offers a handle on high scale physics.

Embedding axion monodromy inflation into string theory providessome insight into the possible origin of the flatness of thepotential, i.e. shift symmetry.The effective field theory approach hides many of the possibledifficulties that can arise in a UV-finite theory of gravity.Phenomenologically axion monodromy fits existing data.It suggests exciting signal for the near future such as tensormodes, r ' 0.07, and possibly oscillations in the scalar spectrumand resonant non-Gaussianity (see Raphael’s talk tomorrow).


Conclusions

Cosmology offers a handle on high scale physics.Embedding axion monodromy inflation into string theory providessome insight into the possible origin of the flatness of thepotential, i.e. shift symmetry.

The effective field theory approach hides many of the possibledifficulties that can arise in a UV-finite theory of gravity.Phenomenologically axion monodromy fits existing data.It suggests exciting signal for the near future such as tensormodes, r ' 0.07, and possibly oscillations in the scalar spectrumand resonant non-Gaussianity (see Raphael’s talk tomorrow).


Conclusions

Cosmology offers a handle on high scale physics.Embedding axion monodromy inflation into string theory providessome insight into the possible origin of the flatness of thepotential, i.e. shift symmetry.The effective field theory approach hides many of the possibledifficulties that can arise in a UV-finite theory of gravity.

Phenomenologically axion monodromy fits existing data.It suggests exciting signal for the near future such as tensormodes, r ' 0.07, and possibly oscillations in the scalar spectrumand resonant non-Gaussianity (see Raphael’s talk tomorrow).


Conclusions

Cosmology offers a handle on high scale physics.Embedding axion monodromy inflation into string theory providessome insight into the possible origin of the flatness of thepotential, i.e. shift symmetry.The effective field theory approach hides many of the possibledifficulties that can arise in a UV-finite theory of gravity.Phenomenologically axion monodromy fits existing data.

It suggests exciting signal for the near future such as tensormodes, r ' 0.07, and possibly oscillations in the scalar spectrumand resonant non-Gaussianity (see Raphael’s talk tomorrow).


Conclusions

Cosmology offers a handle on high scale physics.Embedding axion monodromy inflation into string theory providessome insight into the possible origin of the flatness of thepotential, i.e. shift symmetry.The effective field theory approach hides many of the possibledifficulties that can arise in a UV-finite theory of gravity.Phenomenologically axion monodromy fits existing data.It suggests exciting signal for the near future such as tensormodes, r ' 0.07, and possibly oscillations in the scalar spectrumand resonant non-Gaussianity (see Raphael’s talk tomorrow).


cornell university, ithaca

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