Non-Gaussianities inGeneral Single Field Inflation

Xingang Chen

CTP, MIT

astro-ph/0507053;hep-th/0605045, with Minxin Huang, Shamit Kachru, Gary Shiu;astro-ph/0611645, with Eugene Lim, Richard Easther; and in progress;with Rachel Bean, Henry Tye, Jiajun Xu, in preparation.

陈新刚

Inflation Models and Observations

• Inflation mechanisms and models

Slow-roll inflation --- using flat potential; DBI inflation --- using speed-limit in warped space; K-inflation --- inflation driven by kinetic energy.

• WMAP measurement on CMBR

Spectral index:

Running of spectral index: Tensor to scalar ratio: Non-Gaussianity:

Most General Non-Gaussianities in Single Field Theory

• Motivations

Null hypothesis on specific models; Fit or constrain parameters model-independently;

Several string models has distinctive predictions on non-Gaussianities;

Straightforward evaluation of non-Gaussianities for future models in this general class.

• Single field inflation:

Inflaton is responsible for density perturbations;

Lagrangian is arbitrary function of and ;

Arbitrary sound speed and (to be defined).

• Review of several classes of models

• General formalism

• General form of non-Gaussianities

• Using non-G to probe new physics

Outline

Review of Several Classes of Models

1. Slow-roll inflation (Linde 82; Albrecht & Steinhardt 82)

V

<< 1

<< 1

Slow-roll parameters:

1. Slow-roll inflation; 2. DBI inflation; 3. K-inflation

dS inflation; Power-law inflation; Large field inflation; Small field inflation;

String models: Branes; Tachyons; Axions; Radions.

UV model (Silverstein, Tong, 03) IR model (X.C. 04)

2. DBI inflation (Silverstein, Tong & Alishahiha, 03,04; X.C. 04,05)

• Lagrangian

• Multi-throat brane inflation (X.C. 04)

Antibrane-flux annihilation (Kachru, Pearson, Verlinde, 01)

Generate branes as candidate inflatons Exit B-throat, roll through bulk, settle down in another throat Enough warping: DBI inflation; Flat potential: slow-roll inflation.

• Branes are moving ultra-relativistically

For example, in IR DBI,

Lorentz factor,

In DBI inflation, potential energy dominates,despite the fact that inflatons are ultra-relativistic.

• Pressure and Energy

3. K-inflation

• Lagrangian

For example,

• Attractor solution

• Inflation driven by kinetic energy if

• Can use a hybrid field to end the inflation

• Not realized in string theory so far

(Armendariz-Picon, Damour, Mukhanov, 99)

• Review of several classes of models

• General formalism

• General form of non-Gaussianities

• Using non-G to probe new physics

Outline

A General Formalism (Garriga & Mukhanov, 99)

• Slow variation parameters

• More general than the usual slow-roll parameters

Flat potential v.s. steep potential (DBI) or no potential (k-inflation) Non-relativistic slow-roll v.s. ultra-relativistic fast-roll

• Power spectrum

• Spectral index

• Review of several classes of models

• General formalism

• General form of non-Gaussianities

• Using non-G to probe string theory

Outline

ADM Formalism(Maldacena, 02; Seery & Lidsey 05; X.C., Huang, Kachru & Shiu, 06)

• Metric

• are Lagrangian multipliers

• Action

•

• Decomposeand expand in powers of

• Solve to , in order to expand the action to

• Plug them into the action and expand

The Quadratic Part

Only require the variation of be slow; can be arbitrary.

The Cubic Part

• The exact cubic action for scalar perturbation

The 3-Point Function

• Define

The Cubic Part

• The exact cubic action for scalar perturbation

• Various contributions 1:

The Cubic Part

• The exact cubic action for scalar perturbation

• Various contributions 2:

The Cubic Part

• The exact cubic action for scalar perturbation

• Various contributions 3:

This last term is absorbed by a redefinition:

The Cubic Part

• The exact cubic action for scalar perturbation

• Various contributions 4:

Negligible, unless there are sharp features

(X.C., Easther, Lim, 06)(Bean, X.C., Tye, Xu, in preparation)

The Cubic Part

• The exact cubic action for scalar perturbation

• Various contributions 5:

Negligible, unless there are non-trivial initial conditions

(X.C., Easther, Lim, in preparation)

• The leading contributions from each terms, in absence of sharp features and non-trivial initial conditions

Corrections terms

• The 3-pt function for a general single field inflation to

Final Results (X.C., Huang, Kachru, Shiu, 06)

• Completely specified by 5 parameters:

Size, Shape, and Running of Non-Gaussianities

• Size (magnitude) of non-Gaussianities

Large non-Gaussianity Small or large

WMAP’s ansatz

To compare, take equilateral limit in our results:

(Note: is defined in Maldacena,02; X.C.,Huang,Kachru,Shiu,06;….;

here we quote in WMAP’s convention.)

Shape of Non-Gaussianities(Babich, Creminelli, Zaldarriaga, 04; X.C., Huang, Kachru, Shiu, 06)

Equilateral shape (DBI) Local shape (Slow-roll)

Current Bound:

(WMAP team; Creminelli, Senatore, Zaldarriaga, Tegmark, 06)

CMB: Planck (Smith, Zaldarriaga, 06)

LSS: high-z galaxy surveys: similar or better resolutions. (Sefusatti, Komatsu, 06)

Slow-Roll Inflation

• In this limit, our formulae recover the slow-roll results of Maldacena, 02; Seery & Lidsey, 05.

• In slow-roll inflation, the non-Gaussianity is unobservable,

•

DBI Inflation

(Alishahiha, Silverstein & Tong, 04)

K-Inflation

• Another leading shape (Gruzinov, 04)

• Potentially observable in K-inflation

Remind:

• Sound speed is constant, non-G does not run

• Review of several classes of models

• General formalism

• General form of non-Gaussianities

• Using non-G to probe new physics

Outline

1) Constraining String models; 2) Probing compactification geometry;

3) Probing sharp features; 4) Probing inflationary vacuum;

5) Measuring stringy correlation-functions.

Constraining String Models

• In GKP-type warp compactification, is restricted by the size of the throat

Excessive non-Gaussianities(X.C., 05; Baumann, Mcallister, 06; Bean, Shandera, Tye & Xu, 07)

• In the UV DBI model (Silverstein, Tong & Alishahiha, 03,04)

Viable only if

• In fact, before data comparison is made, probe brane back-reaction is already too large.

Require: But:

(Bean, X.C., Peiris, Xu, 07)

(see last week talk)

• In the IR DBI model (X.C. 04,05)

Large non-G can also be small enough to satisfy current observations

Testable in the future experiments:

In future experiments:on CMB scales, Planck can achieve

on LSS scales, high-z galaxy surveyscan reach similar or better resolutions.

(Smith, Zaldarriaga, 06; Sefusatti, Komatsu, 07)

Constraining microscopic parameters:

For example, the upper bound in the result:

(Bean, X.C., Peiris, Xu, 07)

Probing Geometry in String Compactification

• Running of non-Gaussianity

Shape of geometry in extra dimension (X.C. 05)

• Combining with the correlated feature in 2-pt function(Shiu, Underwood, 06)

Radius dependence of warp factor time dependence of sound speed

Scale dependence (running) of non-G

•

Probing Inflationary Vacuum

• General vacuum state for inflaton fluctuations:

The Bunch-Davis vacuum:

• Consider corrections

Replace one of with

(X.C.,Huang,Kachru,Shiu,06)

(Martin, Brandenberger, 00)

• The size of the non-Gaussianities

• The shape of the non-Gaussianities

Peak in the folded triangle limit,

Divergence is artificial: if non-standard vacuum exits only up to M, divergence is replaced within

Probing Sharp Features

Glitches in CMB power spectrum:

Cosmic variance, or new physics?

Sharp features in slow-roll potential

• Consider a small but sharp step (Adams, Cresswell, Easther, 01)

• Without the step,

with the step,

Cause a dip in density perturbations with ratio:

(Covi, et al, 06)

• As the inflaton falls down the step,

within

results in abrupt changes in:

• The contribution becomes important

(X.C., Easther, Lim, 06)

Calculate the associated large non-G

Choose c and d to fit the power spectrum Predict the non-G

• Distinctive features: 1) localized around the location of feature; 2) characteristic oscillatory running, c.f. mild running in DBI.

• Since running dominates, shape dependence varies a lot

• Experimental bound for such non-Gaussianities has not been done.

Sharp features in DBI inflation(Bean, X.C., Tye, Xu, in preparation)

• Duality cascade can cause sharp features in warp factor (Hailu, Tye, 06)

Abrupt change in sound speed

• Associated with non-Gaussianities features, on top of the original nearly-scale-invariant large non-G.

• In IR DBI inflation, at earlier times, i.e. larger scales, Hubble energy > redshifted string scale. (Phase transition)

Not only scalar fluctuations, but also stringy fluctuations.

• Happens at

• Warped spaceProvides speed limit

Redshifts string scale (Randall, Sundrum, 99)

• IR DBI mode predicts large, but regional, running of spectral index(X.C., 05, 06; Bean, X.C., Peiris, Xu, 07)

Measuring Stringy Correlation functions

(1)(2)(3)(4)

2) Hubble-expansion-induced stringy phase1) Field theory regime

Density perturbations:

1) : Field theory applies;

2) : Open string creation (Stringy quantum fluctuations);

3) : Closed string creation starts;

4) : Closed strings smooth out background (de Sitter back-reaction cuts off the throat).

Stringy phase transition – the reminder 1(from the last week talk)

• Stringy phase transition:

Hubble scale < string scale:

Fluctuation speed < speed of light:

Density perturbations:

Spectrum index:

• Field theory regime

Phase transition at:

if

Stringy phase transition – the reminder 2

• Large non-Gaussianities are stringy near larger scales

Stringy 2-pt is only estimated; buteven estimation of stringy non-Gis currently unavailable.

Experiments ahead of string theory!

• Compare IR model with data

stringy phase transition happens near largest CMB scales

(Bean, X.C., Peiris, Xu, 07)

Conclusions

• A full non-Gaussianity in general single field inflation specified by 5 parameters;

• Explicit form of momentum dependence, including a few potentially observable;

• Recovered all previously known results, explore unknown regions.

• Probing new physics and string theory models, including field theoretic with strong string motivations and completely stringy physics.