Inflation, String Theory, Inflation, String Theory,

Andrei Linde

Andrei Linde

and Origins of Symmetryand Origins of Symmetry

Contents: Contents:

Inflation as a theory of a harmonic oscillator Inflation and observations Inflation in supergravity String theory and cosmology Eternal inflation and string theory landscape Origins of symmetry: moduli trapping

Inflation as a theory of a harmonic oscillator Inflation and observations Inflation in supergravity String theory and cosmology Eternal inflation and string theory landscape Origins of symmetry: moduli trapping

Inflation as a theory of a harmonic oscillator Inflation as a theory of a harmonic oscillator

Einstein:

Klein-Gordon:

Einstein:

Klein-Gordon:

Equations of motion:Equations of motion:

Compare with equation for the harmonic oscillator with friction:Compare with equation for the harmonic oscillator with friction:

Logic of Inflation: Logic of Inflation:

Large φLarge φ large H large H large friction large friction

field φ moves very slowly, so that its potential energy for a long time remains nearly constantfield φ moves very slowly, so that its potential energy for a long time remains nearly constant

No need for false vacuum, supercooling, phase transitions, etc.No need for false vacuum, supercooling, phase transitions, etc.

Add a constant to the inflationary potential

- obtain two stages of inflation

Add a constant to the inflationary potential

- obtain two stages of inflation

A photographic image of quantum fluctuations blown up to the size of the universe

A photographic image of quantum fluctuations blown up to the size of the universe

How important is the gravitational wave

contribution?

How important is the gravitational wave

contribution?

For these two theories the ordinary scalar perturbations coincide: For these two theories the ordinary scalar perturbations coincide:

Is the simplest chaotic inflation natural? Is the simplest chaotic inflation natural?

Often repeated (but incorrect) argument: Often repeated (but incorrect) argument:

Thus one could expect that the theory is ill-defined at Thus one could expect that the theory is ill-defined at

However, quantum corrections are in fact proportional to However, quantum corrections are in fact proportional to

and to and to

These terms are harmless for sub-Planckian masses and densities, even if the scalar field itself is very large.

These terms are harmless for sub-Planckian masses and densities, even if the scalar field itself is very large.

Chaotic inflation in supergravity Chaotic inflation in supergravity

Main problem:Main problem:

Canonical Kahler potential isCanonical Kahler potential is..

Therefore the potential blows up at large |φ|, and slow-roll inflation is

impossible:

Therefore the potential blows up at large |φ|, and slow-roll inflation is

impossible:

Too steep, no inflation…Too steep, no inflation…

A solution: shift symmetry A solution: shift symmetryKawasaki, Yamaguchi, Yanagida 2000

Equally legitimate Kahler potential Equally legitimate Kahler potential

and superpotentialand superpotential

The potential is very curved with respect to X and Re φ, so these fields vanishThe potential is very curved with respect to X and Re φ, so these fields vanish

But Kahler potential does not depend onBut Kahler potential does not depend on

The potential of this field has the simplest form, without any exponential terms:The potential of this field has the simplest form, without any exponential terms:

Inflation in String Theory Inflation in String Theory

The volume stabilization problem:

Consider a potential of the 4d theory obtained by compactification in string theory of type IIB

The volume stabilization problem:

Consider a potential of the 4d theory obtained by compactification in string theory of type IIB

Here Here is the dilaton field, and is the dilaton field, and describes volume of the compactified spacedescribes volume of the compactified space

The potential with respect to these two fields is very steep, they run down, and V vanishesThe potential with respect to these two fields is very steep, they run down, and V vanishes

The problem of the dilaton stabilization was solved in 2001,The problem of the dilaton stabilization was solved in 2001,Giddings, Kachru and Polchinski 2001Giddings, Kachru and Polchinski 2001

but the volume stabilization problem was most difficult and was solved only recently (KKLT construction)but the volume stabilization problem was most difficult and was solved only recently (KKLT construction)

Kachru, Kallosh, Linde, Trivedi 2003Kachru, Kallosh, Linde, Trivedi 2003

Burgess, Kallosh, Quevedo, 2003Burgess, Kallosh, Quevedo, 2003

Volume stabilization Volume stabilizationBasic steps:Basic steps:

Warped geometry of the compactified space and nonperturbative effects AdS space (negative vacuum energy) with unbroken SUSY and stabilized volume

Uplifting AdS space to a metastable dS space (positive vacuum energy) by adding anti-D3 brane (or D7 brane with fluxes)

Warped geometry of the compactified space and nonperturbative effects AdS space (negative vacuum energy) with unbroken SUSY and stabilized volume

Uplifting AdS space to a metastable dS space (positive vacuum energy) by adding anti-D3 brane (or D7 brane with fluxes)

AdS minimum Metastable dS minimum

Inflation with stabilized volume Inflation with stabilized volume Use KKLT volume stabilizationUse KKLT volume stabilization Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi 2003Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi 2003

Introduce the inflaton field with the potential which is flat Introduce the inflaton field with the potential which is flat due to shift symmetrydue to shift symmetry

Break shift symmetry either due to superpotential or due to Break shift symmetry either due to superpotential or due to radiative correctionsradiative corrections

Hsu, Kallosh , Prokushkin 2003Koyama, Tachikawa, Watari 2003Firouzjahi, Tye 2003Hsu, Kallosh 2004

Alternative approach: Modifications of kinetic terms in the strong coupling Alternative approach: Modifications of kinetic terms in the strong coupling regimeregime Silverstein and Tong, 2003

Alternative approach: Modifications of kinetic terms in the strong coupling Alternative approach: Modifications of kinetic terms in the strong coupling regimeregime Silverstein and Tong, 2003

String inflation and shift symmetry

Hsu, Kallosh , Prokushkin 2003

Why shift symmetry? Why shift symmetry?

It is not just a requirement which is desirable for inflation model builders, but, in a certain class of string theories, it is an unavoidable consequence of the mathematical structure of the theory

It is not just a requirement which is desirable for inflation model builders, but, in a certain class of string theories, it is an unavoidable consequence of the mathematical structure of the theory

Hsu, Kallosh, 2004

The Potential of the Hybrid D3/D7 Inflation Model

The Potential of the Hybrid D3/D7 Inflation Model

is a hypermultiplet

is an FI triplet

In many F and D-term models the contribution of cosmic strings to

CMB anisotropy is too large

This problem disappears for very small coupling g

Another solution is to add a new hypermultiplet, and a new global symmetry, which makes the strings semilocal and topologically unstable

Semilocal Strings are Topologically

Unstable

Semilocal Strings are Topologically

Unstable Achucarro, Borill, Liddle, 98Achucarro, Borill, Liddle, 98

D3/D7 with two hypers D3/D7 with two hypers

Detailed brane construction - D-term inflation dictionary Detailed brane construction - D-term inflation dictionary

Dasgupta, Hsu, R.K., A. L., Zagermann, hep-th/0405247Dasgupta, Hsu, R.K., A. L., Zagermann, hep-th/0405247

Resolving the problem of cosmic string production: additional global symmetry, no topologically stable strings, only semilocal strings, no danger

Confirmation of Urrestilla, Achucarro, Davis; Binetruy, Dvali, R. K.,Van Proeyen

Resolving the problem of cosmic string production: additional global symmetry, no topologically stable strings, only semilocal strings, no danger

Confirmation of Urrestilla, Achucarro, Davis; Binetruy, Dvali, R. K.,Van Proeyen

Brane construction of generalized D-term inflation models with additional global or local symmetries due to extra branes and hypermultiplets.

Brane construction of generalized D-term inflation models with additional global or local symmetries due to extra branes and hypermultiplets.

Bringing it all together: Double Uplifting

Bringing it all together: Double Uplifting

First uplifting: KKLTFirst uplifting: KKLT

KKL, in progressKKL, in progress

Second uplifting in D3/D7 model Second uplifting in D3/D7 model

Inflationary potential at as a function of S andInflationary potential at as a function of S and

Shift symmetry is broken only by quantum effectsShift symmetry is broken only by quantum effects

Potential of hybrid inflation with a stabilized volume modulusPotential of hybrid inflation with a stabilized volume modulus

For two hypers: For two hypers:

Inflaton potential:Inflaton potential:

Symmetry breaking potential:Symmetry breaking potential:

Can we have eternal inflation in such models?

Can we have eternal inflation in such models?

Yes, by combining these models with the ideas of string theory landscape

Yes, by combining these models with the ideas of string theory landscape

String Theory Landscape String Theory Landscape

Perhaps 10 different vacua Perhaps 10 different vacua

100100

de Sitter expansion in these vacua is de Sitter expansion in these vacua is eternal.eternal. It creates quantum fluctuations It creates quantum fluctuations

along all possible flat directions and along all possible flat directions and provides necessary initial conditions provides necessary initial conditions

for the low-scale inflationfor the low-scale inflation

Landscape of eternal inflation Landscape of eternal inflation

Self-reproducing Inflationary Universe

Finding the way in the landscape Finding the way in the landscape

• Anthropic Principle: Love it or hate it but use it

• Vacua counting: Know where you can go

• Moduli trapping: Live in the most beautiful valleys

• Anthropic Principle: Love it or hate it but use it

• Vacua counting: Know where you can go

• Moduli trapping: Live in the most beautiful valleys

Quantum effects lead to particle production which result in moduli trapping near enhanced symmetry points

These effects are stronger near the points with greater symmetry, where many particles become massless

This may explain why we live in a state with a large number of light particles and (spontaneously broken) symmetries

Quantum effects lead to particle production which result in moduli trapping near enhanced symmetry points

These effects are stronger near the points with greater symmetry, where many particles become massless

This may explain why we live in a state with a large number of light particles and (spontaneously broken) symmetries

Beauty is AttractiveBeauty is Attractive

Kofman, A.L., Liu, McAllister, Maloney, Silverstein: hep-th/0403001

also Silverstein and Tong, hep-th/0310221

Kofman, A.L., Liu, McAllister, Maloney, Silverstein: hep-th/0403001

also Silverstein and Tong, hep-th/0310221

Basic Idea Basic Idea

Consider two interacting moduli with potential

Suppose the field φ moves to the right with

velocity . Can it create particles ? Nonadiabaticity condition:

is related to the theory of preheating after inflation

Kofman, A.L., Starobinsky 1997

It can be represented by two intersecting valleys

V

φ

When the field φ passes the (red) nonadiabaticity region near the point of

enhanced symmetry, it created particles χ with energy density proportional to φ. Therefore the rolling field slows down and stops at the point when

Then the field falls down and reaches the nonadiabaticity region again…

V

φ

When the field passes the nonadiabaticity region again, the number of particles (approximately) doubles, and the potential becomes two times more steep. As a result, the field becomes trapped at distance that is two times smaller than before.

Trapping of a real scalar field

Thus anthropic and statistical considerations are supplemented by a dynamical selection mechanism, which may help us to understand the origin of symmetries in our world.