Inflation and String Inflation and String CosmologyCosmology

Inflation and String Inflation and String CosmologyCosmology

Andrei Linde

Andrei Linde

The Simplest Inflationary The Simplest Inflationary ModelModel The Simplest Inflationary The Simplest Inflationary ModelModel

Eternal InflationEternal Inflation

Predictions of Predictions of Inflation:Inflation:

Predictions of Predictions of Inflation:Inflation:

1) The universe should be homogeneous, isotropic and flat, = 1 + O(10-4) [

Observations: the universe is

homogeneous, isotropic and flat, = 1 +

O(10-2)

• Inflationary perturbations should be gaussian and adiabatic, with flat spectrum, ns = 1+ O(10

-1)

Observations: perturbations are gaussian and adiabatic, with flat spectrum, ns = 1 + O(10

-2)

Any problems of Any problems of principle?principle?

Any problems of Any problems of principle?principle?

1) We must introduce a small parameter to explain a small amplitude of density perturbations .

To explain one should have m = Is it a real problem?

2) Transplanckian physics?

These effects are not expected to affect basic features of inflationary scenario. In the worst case, one may expect minor corrections to the spectrum of perturbations. These effects vanish for low-scale inflation.

3) Singularity problem

This is NOT a problem of inflation.

Moreover, inflationary predictions

practically do not depend on the existence

of the singularity.

4) Cosmological constant problem This is NOT a problem of inflation.

Moreover, the only presently known solution of this problem requires inflation, in combination with anthropic principle and string theory landscape.

5) Inflation requires initial homogeneity on scale greater than horizon

In simplest models of chaotic inflation

homogeneity is requires on the smallest

possible scale, Planck length, which is not

a problem.

In low-scale inflation, such as new or

hybrid inflation, this was a real problem.

However, this problem was solved by

considering inflation in compact

topologically nontrivial flat or open

universes: In this case homogeneity is

required on Planck scale, as in chaotic

inflation.

A.L. hep-th/0408164

Are there any Are there any realreal problems of problems of inflation?inflation?

Are there any Are there any realreal problems of problems of inflation?inflation?

The main problem is to construct realistic models of inflation in the situation when the final theory of all fundamental interactions is still absent.

Inflation in String Inflation in String TheoryTheory Inflation in String Inflation in String TheoryTheoryThe volume stabilization problem:

A potential of the theory obtained by compactification in string theory of type

IIB:

The volume stabilization problem:

A potential of the theory obtained by compactification in string theory of type

IIB:

The potential with respect to X and Y is very steep, these fields rapidly run down, and the potential energy V vanishes. We must stabilize these fields.

Volume stabilization: KKLT construction

Kachru, Kallosh, A.L., Trivedi 2003Kachru, Kallosh, A.L., Trivedi 2003Burgess, Kallosh, Quevedo, 2003Burgess, Kallosh, Quevedo, 2003

X and Y are canonically normalized field corresponding to the dilaton field and to the volume of the compactified space; is the field driving inflation

Dilaton stabilization:

Giddings, Kachru, Polchinski 2001Giddings, Kachru, Polchinski 2001

Volume stabilizationVolume stabilization Volume stabilizationVolume stabilization

Basic steps of the KKLT scenario:Basic steps of the KKLT scenario:Basic steps of the KKLT scenario:Basic steps of the KKLT scenario:

AdS minimumAdS minimum Metastable dS minimumMetastable dS minimum

Kachru, Kallosh, A.L., Trivedi 2003Kachru, Kallosh, A.L., Trivedi 2003

1) Start with a theory with runaway potential discussed above

100 150 200 250 300 350 400s

-2

-1.5

-1

-0.5

0.5V

100 150 200 250 300 350 400s

0.2

0.4

0.6

0.8

1

1.2

V

2) Bend this potential down due to (nonperturbative) quantum effects

3) Uplift the minimum to the state with positive vacuum energy by adding a positive energy of an anti-D3 brane in warped Calabi-Yau space

Main conclusions after 2 years of Main conclusions after 2 years of investigation:investigation:

Main conclusions after 2 years of Main conclusions after 2 years of investigation:investigation:

It is possible to stabilize internal dimensions, and obtain an accelerating universe. Eventually, our part of the universe will decay and become ten-dimensional, but it will only happen in 1010120 years

Apparently, vacuum stabilization can be achieved in 10100 - 101000 different ways. This means that the potential energy V of string theory may have 10100 - 101000 minima where we (or somebody else) can enjoy life…

Related ideas existed long Related ideas existed long beforebefore

the stringy landscapethe stringy landscape

Related ideas existed long Related ideas existed long beforebefore

the stringy landscapethe stringy landscape

Example: Supersymmetric SU(5)

V

SU(5) SU(3)xSU(2)xU(1)

SU(4)xU(1)

Weinberg 1982: No way to tunnel from SU(5) to SU(3)xSU(2)XU(1) A.L 1983: Inflationary fluctuations bring us there

Self-reproducing Inflationary Self-reproducing Inflationary UniverseUniverse Self-reproducing Inflationary Self-reproducing Inflationary UniverseUniverse

String Theory String Theory LandscapeLandscape

String Theory String Theory LandscapeLandscape

Perhaps 10Perhaps 10100100 - 10 - 1010001000 different minimadifferent minima

Perhaps 10Perhaps 10100100 - 10 - 1010001000 different minimadifferent minima

Bousso, Polchinski; Susskind; Douglas, Denef,…Bousso, Polchinski; Susskind; Douglas, Denef,…Bousso, Polchinski; Susskind; Douglas, Denef,…Bousso, Polchinski; Susskind; Douglas, Denef,…

Lerche, Lust, Schellekens 1987Lerche, Lust, Schellekens 1987 Lerche, Lust, Schellekens 1987Lerche, Lust, Schellekens 1987

Stringy landscape provides us with a DISCRETE set of parameters corresponding to 101000 vacua of string theory.

In addition, we may have many CONTINUOUS parameters, such as the amplitude of density perturbations, the ratio of dark matter to baryons, etc., which depend on cosmological dynamics.

The simplest curvaton The simplest curvaton modelmodelThe simplest curvaton The simplest curvaton modelmodel A.L., Mukhanov, astro-ph/9610212

Consider a light field during inflation with Hubble constant H. Bunch-Davies distribution of fluctuations:

The main contribution to these fluctuations is given by exponentially large wavelengths

The main contribution to these fluctuations is given by exponentially large wavelengths

Interpretation:Interpretation:Interpretation:Interpretation:Because of the fluctuations, curvaton typically takes values of the order + H2/m or - H2/m in domains separated by walls where it vanishes (shown as a shoreline in the figure below). A typical size of these domains is

Amplitude of the curvaton Amplitude of the curvaton perturbationsperturbations Amplitude of the curvaton Amplitude of the curvaton perturbationsperturbations

The constant C shows the energy density of the curvaton particles produced during reheating. As far as we know, this contribution previously was ignored, but it can be very large.

The amplitude of density perturbations depends on our position in the curvaton landscape. In the interior of the exponentially large islands the perturbations are (locally) gaussian. On a larger scale including many domains the perturbations are nongaussian.

A.L., Mukhanov, astro-ph/0511736

Usually we assume that the amplitude of inflationary perturbations is constant, ~ 10-5 everywhere. However, in the curvaton scenario the value of is different in different exponentially large parts of the universe.

Curvaton WebCurvaton WebCurvaton WebCurvaton Web

A.L., Mukhanov, astro-ph/0511736

Consider inflaton and curvaton with masses M >> m

Curvaton-Inflaton Curvaton-Inflaton TransmutationsTransmutations Curvaton-Inflaton Curvaton-Inflaton TransmutationsTransmutations

is a curvaton only near the walls everywhere else it is an inflatonThe amplitude of perturbations is almost everywherebut it grows near the walls, forming the curvaton web.

Even if initially >> , then during eternal inflation the fluctuations of the field are generated, and it becomes much greater than almost everywhere in the universe. Then the field rolls down (its potential is more steep) and the curvaton drives inflation, i.e. it becomes the inflaton.

Bartolo, Liddle 2003, A.L., Mukhanov 2005

Dark EnergyDark Energy (Cosmological (Cosmological Constant) is about 73% of the Constant) is about 73% of the cosmic pie. Why?cosmic pie. Why?

Dark EnergyDark Energy (Cosmological (Cosmological Constant) is about 73% of the Constant) is about 73% of the cosmic pie. Why?cosmic pie. Why?

What’s about What’s about Dark MatterDark Matter, , another 23% of the pie? Why another 23% of the pie? Why there is 5 times more dark there is 5 times more dark matter than ordinary matter?matter than ordinary matter?

What’s about What’s about Dark MatterDark Matter, , another 23% of the pie? Why another 23% of the pie? Why there is 5 times more dark there is 5 times more dark matter than ordinary matter?matter than ordinary matter?

Inflation and Cosmological Inflation and Cosmological ConstantConstant

Inflation and Cosmological Inflation and Cosmological ConstantConstant

1) Anthropic solutions of the CC problem using inflation and fluxes of antisymmetric tensor fields (A.L. 1984), multiplicity of KK vacua (Sakharov 1984), and slowly evolving scalar field (Banks 1984, A.L. 1986). All of these authors took for granted that we cannot live in the universe with

2) Derivation of the anthropic constraint

(Weinberg 1987, Martel, Shapiro, Weinberg 1997)

Three crucial steps in finding the anthropic solution of the CC problem:

3) String landscape

(Bousso-Polchinski 2000, KKLT 2003, Susskind 2003, Douglas 2003,…)

Latest anthropic Latest anthropic

constraints on constraints on Latest anthropic Latest anthropic

constraints on constraints on Aguirre, Rees, Tegmark, and Wilczek, astro-ph/0511774

observed observed valuevalue

Dark matter:Dark matter: the axion the axion scenarioscenarioDark matter:Dark matter: the axion the axion scenarioscenario

Standard lore: If the axion mass is smaller than 10-5 eV, the amount of dark matter in the axion field contradicts observations, for a typical initial value of the axion field.

Anthropic argument: Due to inflationary fluctuations, the amount of the axion dark matter is a CONTINUOUS RANDOM PARAMETER. We can live only in those parts of the universe where the initial value of the axion field was sufficiently small (A.L. 1988).

Latest anthropic constraints on Latest anthropic constraints on

Dark MatterDark Matter Latest anthropic constraints on Latest anthropic constraints on

Dark MatterDark MatterAguirre, Rees, Tegmark, and Wilczek, astro-ph/0511774

Anthropic predictions for Dark Matter Anthropic predictions for Dark Matter are even better than the predictions for are even better than the predictions for the cosmological constant !the cosmological constant !

observed observed valuevalue

Why do we live in a 4D Why do we live in a 4D space?space? Why do we live in a 4D Why do we live in a 4D space?space?

Ehrenfest, 1917: Stable planetary and atomic systems are possible only in 4D space. Indeed, for D > 4 planetary system are unstable, whereas for D < 4 there is NO gravity forces between stars and planets.

If one wants to suggest an alternative solution to a problem that is solved by anthropic principle, one is free to try. But it may be more productive to concentrate on many problems that do not have an anthropic solution.

For example, there is no For example, there is no anthropic replacement for anthropic replacement for

inflationinflation

For example, there is no For example, there is no anthropic replacement for anthropic replacement for

inflationinflation

Two types of string Two types of string inflation models:inflation models:Two types of string Two types of string inflation models:inflation models:

Moduli Inflation.Moduli Inflation. The simplest class of models. They use only the fields that are already present in the KKLT model.

Brane inflation.Brane inflation. The inflaton field corresponds to the distance between branes in Calabi-Yau space. Historically, this was the first class of string inflation models.

Moduli Inflation.Moduli Inflation. The simplest class of models. They use only the fields that are already present in the KKLT model.

Brane inflation.Brane inflation. The inflaton field corresponds to the distance between branes in Calabi-Yau space. Historically, this was the first class of string inflation models.

Inflation in string theoryInflation in string theory Inflation in string theoryInflation in string theory

KKLMMT brane-anti-brane inflation

Racetrack modular inflation

D3/D7 brane inflation

Kahler modular inflation

D3/D7 inflationD3/D7 inflationD3/D7 inflationD3/D7 inflation

Unlike in the brane-antibrane Unlike in the brane-antibrane scenario, inflation in D3/D7 model scenario, inflation in D3/D7 model does not require fine-tuningdoes not require fine-tuning because because

of the shift symmetryof the shift symmetry

Unlike in the brane-antibrane Unlike in the brane-antibrane scenario, inflation in D3/D7 model scenario, inflation in D3/D7 model does not require fine-tuningdoes not require fine-tuning because because

of the shift symmetryof the shift symmetry

Herdeiro, Hirano, Kallosh, Dasgupta 2001, 2002

Let 10Let 10500 500 flowers flowers blossomblossom Let 10Let 10500 500 flowers flowers blossomblossom

< < 00< < 00

= = 00= = 00

> 0> 0> 0> 0

In the beginning one has eternal inflation when the fields jumped from one de Sitter minimum to another. However, at some point the fields must stop jumping, as in old inflation, and start rolling, as in new or chaotic inflation: the last stage of inflation must be of the slow-roll type. Otherwise we would live in an empty open universe with << 1.

How can we create initial How can we create initial conditions for a slow-roll conditions for a slow-roll

inflation after the tunneling?inflation after the tunneling?

How can we create initial How can we create initial conditions for a slow-roll conditions for a slow-roll

inflation after the tunneling?inflation after the tunneling?

V

Initial Conditions for D3/D7 Initial Conditions for D3/D7 InflationInflation

Initial Conditions for D3/D7 Initial Conditions for D3/D7 InflationInflation

Slow roll inflation

Eternal inflation in a valley with different fluxes

The field drifts in the upper valley due to quantum fluctuations and then tunneling occurs

due to change of fluxes inside a bubble

H >> m

H >>> m

s

In D3/D7 scenario flatness of the inflaton direction does not depend on fluxes

The resulting The resulting scenario:scenario: The resulting The resulting scenario:scenario:1) 1) The universe The universe eternally jumpseternally jumps from one dS from one dS

vacuum to another due to formation of bubbles.vacuum to another due to formation of bubbles. Each bubble contains a new dS vacuum. The Each bubble contains a new dS vacuum. The bubbles contain no particles unless this bubbles contain no particles unless this process ends by a stage of a slow-roll process ends by a stage of a slow-roll inflation. Here is how:inflation. Here is how:2) 2) At some stage the universe appears in dS At some stage the universe appears in dS state with a large potential but with a flat state with a large potential but with a flat inflaton direction, as in D3/D7 model.inflaton direction, as in D3/D7 model. Quantum Quantum fluctuationsfluctuations during eternal inflation in this during eternal inflation in this state state push the inflaton field S in all push the inflaton field S in all directionsdirections along the inflaton valley. along the inflaton valley.3) 3) Eventually this state decays, and bubbles Eventually this state decays, and bubbles are produced. Each of these bubbles may are produced. Each of these bubbles may contain contain anyany possible value of the inflaton possible value of the inflaton field S, prepared by the previous stage.field S, prepared by the previous stage. A A slow-roll inflation begins and makes the slow-roll inflation begins and makes the universe flat.universe flat. It produces particles, It produces particles, galaxies, and the participants of this galaxies, and the participants of this conference:)conference:)