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Primordial Non-Gaussianities and Quasi-Single Field Inflation. Xingang Chen. Center for Theoretical Cosmology, DAMTP, Cambridge University. X.C., 1002.1416, a review on non-G; X.C., Yi Wang, 0909.0496; 0911.3380. CMB and WMAP. (WMAP website). Temperature Fluctuations. (WMAP website).

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Primordial Non-Gaussianities and

Quasi-Single Field Inflation

Xingang Chen

Center for Theoretical Cosmology,

DAMTP, Cambridge University

X.C., 1002.1416, a review on non-G;

X.C., Yi Wang, 0909.0496; 0911.3380


CMB and WMAP

(WMAP website)


Temperature Fluctuations

(WMAP website)



CMB and WMAP

(WMAP website)


Generic Predictions of Inflationary Scenario

Density perturbations that seed the large scale struture are

  • Primordial (seeded at super-horizon size)

  • Approximately scale-invaraint

  • Approximately Gaussian




  • But we have pixels in WMAP temperature map

Experimentally: Information is Compressed

  • Amplitude and scale-dependence of the power spectrum (2pt)

  • contain 1000 numbers for WMAP

This compression of information is justified only if

the primordial fluctuations is perfectly Gaussian.

Can learn much more from the non-Gaussian components.


Theoretically: From Paradigm to Explicit Models

  • What kind of fields drive the inflation?

  • What are the Lagrangian for these fields?

  • Alternative to inflation?

  • Quantum gravity


Non-G components: Primordial Interactions

  • Two-point correlation

Free propagation of inflaton in inflationary bkgd

  • Three or higher-point correlations (non-Gaussianities)

Interactions of inflatons or curvatons

“LHC” for Early Universe!


What we knew theoretically about the non-Gaussianities


Experimentally:

Simplest inflation models predict unobservable non-G.

(Maldacena, 02; Acquaviva et al, 02)

  • Single field

  • Canonical kinetic term

  • Always slow-roll

  • Bunch-Davies vacuum

  • Einstein gravity


Inflation Model Building

Examples of simplest slow-roll potentials:

The other conditions in the no-go theorem also needs to be satisfied.



Inflation Model Building

A landscape of potentials


Inflation Model Building

Warped Calabi-Yau


  • h-Problem in slow-roll inflation:

(Copeland, Liddle, Lyth, Stewart, Wands, 04)

  • h-Problem in DBI inflation:

(X.C., 08)

?

?

Inflation Model Building


(X.C., Sarangi, Tye, Xu, 06; Baumann, McAllsiter, 06)

  • Variation of potential:

(Lyth, 97)

?

: eg. higher dim Planck mass, string mass, warped scales etc.



Non-canonical kinetic terms: DBI inflation, k-inflation, etc

  • Always slow-roll

Features in potentials or Lagrangians: sharp, periodic, etc

  • Bunch-Davies vacuum

Non-Bunch-Davies vacuum

due to boundary condions, low new physics scales, etc

  • Single field

Multi-field: turning trajectories, curvatons, inhomogeous reheating surface, etc

Quasi-single field: massive isocurvatons

Beyond the No-Go


Bispectrum is a function, with magnitude , of three momenta:

Shape and Running of Bispectra (3pt)

  • Shape dependence:

  • (Shape of non-G)

Fix , vary , .

Squeezed

Equilateral

Folded

Fix , , vary .

  • Scale dependence:

  • (Running of non-G)


Two Well-Known Shapes of Large Bispectra (3pt) momenta:

For scale independent non-G, we draw the shape of

Local

Equilateral

In squeezed limit:


For single field, small correlation if momenta:

Physics of Large Equilateral Shape

  • Generated by interacting modes during their horizon exit

Quantum fluctuations

Interacting and exiting horizon

Frozen

So, the shape peaks at equilateral limit.

  • For example, in single field inflation with higher order derivative terms

(Inflation dynamics is no longer slow-roll)

(Alishahiha, Silverstein, Tong, 04; X.C., Huang, Kachru, Shiu, 06)


Local in position space non-local in momentum space

Physics of Large Local Shape

  • Generated by modes after horizon exit, in multifield inflation

  • Isocurvature modes curvature mode

  • Patches that are separated by horizon evolve independently (locally)

So, the shape peaks at squeezed limit.

  • For example, in curvaton models;

(Lyth, Ungarelli, Wands, 02)

multifield inflation models with turning trajectory, (very difficult to get observable nonG.)

(Vernizzi, Wands, 06; Rigopoulos, Shellard, van Tent, 06)


What we knew spaceexperimentally about the non-Gaussianities


Experimental Results on Bispectra space

  • WMAP5 Data, 08

(Yadav, Wandelt, 07)

(Rudjord et.al., 09)

(WMAP group, 10)

;

  • Large Scale Structure

(Slosar et al, 08)


The Planck Satellite, sucessfully launched last year space

;

(Planck bluebook)

(Smith, Zaldarriaga, 06)

;


21cm: FFTT space

(Mao, Tegmark, McQuinn,

Zaldarriaga, Zahn, 08)

Other Experiments

  • Ground based CMB telescope: ACBAR, BICEP, ACT, ….

  • High-z galaxy survey: SDSS, CIP, EUCLID, LSST …

  • 21-cm tomography: LOFAR, MWA, FFTT, …

For example:


Fit data to constrain space

for example

Theoretical

template

Construct estimator

for example

Data analyses

Underlying physics

Different dynamics in inflation predict different non-G.

Looking for Other Shapes and Runnings

of Non-Gaussianities in Simple Models

  • Why?

  • So possible signals in data may not have been picked up,

  • if we are not using the right theoretical models.

  • A positive detection with one ansatz does not mean that

    we have found the right form.


Other Possible Shapes and Runnings in Simple Models space

with Large non-Gaussianities

(X.C., Huang, Kachru, Shiu, 06; X.C., Easther, Lim, 06,08)


In 3pt: space

Peaks at folded triangle limit

Other Possible Shapes and Runnings in Simple Models

with Large non-Gaussianities

Folded Shape:

(X.C., Huang, Kachru, Shiu, 06; Meerburg, van de Schaar, Corasaniti, Jackson, 09)

The Bunch-Davis vacuum:

Non-Bunch-Davis vacuum:

For example, a small


Other Possible Shapes and Runnings in Simple Models space

with Large non-Gaussianities

Folded Shape:

  • Boundary conditons

  • “Trans-Planckian” effect

  • Low new physics scales


A feature local in time space

Oscillatory running in momentum space

3pt:

Other Possible Shapes and Runnings in Simple Models

with Large non-Gaussianities

Sharp features:

(X.C., Easther, Lim, 06,08)

Steps or bumps in potential, a sudden turning trajectory, etc


Other Possible Shapes and Runnings in Simple Models space

with Large non-Gaussianities

Sharp features:

  • Consistency check for glitches in power spectrum

  • Models (brane inflation) that are very sensitive to features


Other Possible Shapes and Runnings in Simple Models space

with Large non-Gaussianities

Resonance:

(X.C., Easther, Lim, 08; Flauger, Pajer, 10)

Periodic features

Oscillating background

Resonance

Modes within horizon are oscillating

3pt:

Periodic-scale-invariance: Rescale all momenta by a discrete efold:


Other Possible Shapes and Runnings in Simple Models space

with Large non-Gaussianities

Resonance:

  • Periodic features from duality cascade in brane inflation

(Hailu, Tye, 06; Bean, Chen, Hailu, Tye, Xu, 08)

  • Periodic features from instantons in monodromy inflation

(Silverstein, Westphal, 08; Flauger, Mcallister, Pajer, Westphal, Xu, 09)


Other Possible Shapes and Runnings in Simple Models space

with Large non-Gaussianities

(X.C., Wang, 09)



  • Others have mass

(Ignored previously for den. pert.)

Quasi-single field inflation

(X.C., Wang, 09)

Motivation for Quasi-Single Field Inflation

  • Fine-tuning problem in slow-roll inflation

(Copeland, Liddle, Lyth, Stewart, Wands, 94)

In the inflationary background, the mass of light particle

is typically of order H (the Hubble parameter)

E.g.

C.f.

is needed for slow-roll inflation

  • Generally, multiple light fields exist


A Simple Model of Quasi-Single Field Inflation space

  • Straight trajectory:

    Equivalent to single field inflation

  • Turning trajectory:

    Important consequence on

    density perturbations.

E.g. Large non-Gaussianities with novel shapes.

Running power spectrum (non-constant case only).

Here study the constant turn case

Lagrangian in polar coordinates:

slow-roll potential

potential for massive field


Difference Between and space

is the main source of

the large non-Gaussianities.

but

but

etc

It is scale-invariant

for constant turn case.


Perturbation Theory space

  • Field perturbations:

  • Lagrangian


Oscillating inside horizon space

Kinematic Part

  • Massless:

Solution:

Constant after horizon exit


Oscillating inside horizon space

Decay as after horizon exit.

  • Massive:

Solution:

, mass of order H

E.g.


Oscillating inside horizon space

  • Massive:

Solution:

, mass >> H

E.g.

Oscillating and decay after horizon exit


Solution:

We will consider the case:


Interaction Part space

  • Transfer vertex

We use this transfer-vertex to compute

the isocurvature-curvature conversion

  • Interaction vertex

Source of the large

non-Gaussianities


Perturbation Method and Feynman Diagrams space

Correction to 2pt

3pt

To use the perturbation theory, we need

These conditions are not necessary for the model building,

but non-perturbative method remains a challenge.


In-In Formalism space

(Weinberg, 05)

  • Mixed form

(X.C., Wang, 09)

Introduce a cutoff .

“Factorized form” for UV part to avoid spurious UV divergence;

“Commutator form” for IR part to avoid spurious IR divergence.

Mixed form + Wick rotation for UV part

A very efficient way to numerically integrate the 3pt.



for

for

Squeezed Limit and Intermediate Shapes

  • In squeezed limit, simple analytical expressions are possible.

  • Squeezed limit behavior also provide clues to guess a simple shape ansatz.

  • Can be used to classify shapes of non-G.

  • Lying between the equilateral form , and local form .

We call them Intermediate Shapes.

(X.C., Wang, 09)

Not superposition of previously known shapes.


A Shape Ansatz the shape


Compare the shape

Numerical

Ansatz


For perturbative method: the shape

Non-perturbative case is also very interesting.

Size of Bispectrum

  • Definition of :

  • We get


  • Quasi-local: for lighter isocurvaton

In limit, recover the local shape behavior.

Physics of Large Intermediate Shapes

Fluctuations decay faster after horizon-exit,

so large interactions happen during the horizon-exit.

Modes have comparable wavelengths:

Closer to equilateral shape.

Fluctuations decay slower after horizon-exit,

so non-G gets generated and transferred

more locally in position space.

In momentum space, modes become more non-local:

Closer to local shape.


Effect of the Transfer Vertex the shape

A comparison of shapes before and after it is transferred

  • Before:

Squeezed limit shape is , amplitude is decaying.

  • After:

Shapes are changed during the transfer, slightly towards the local type.

Important to investgate such effects in other models,

including multi-field inflation.


C.f. the shape

For the perturbative case:

Trispectra (4pt)


Conclusions the shape

  • Using non-Gaussianities to probe early universe

Different inflationary dynamics can imprint distinctive signatures in non-G;

No matter whether nonG will turn out to be observable or not,

detecting/constraining them requires a complete classification of their profiles.


Conclusions the shape

  • Using non-Gaussianities to probe early universe

Classification:

  • Higher derivative kinetic terms: Equilateral shape

  • Sharp feature: Sinusoidal running

  • Periodic features: Resonant running

  • A non-BD vacuum: Folded shape

  • Massive isocurvatons: Intermediate shapes

  • Massless isocurvaton: Local shape


Conclusions the shape

  • Using non-Gaussianities to probe early universe

Different inflationary dynamics can imprint distinctive signatures in non-G

  • Quasi-single field inflation

Effects of massive modes on density perturbations

  • Transfer vertex in “in-in” formalism

Compute isocurvature-curvature transition perturbatively

  • Non-Gaussianities with intermediate shapes

Numerical, analytical and ansatz


  • Compare the shapeIntermediate Shapes, Resonant running, etc, with data

  • and constrain

and

Future Directions

  • Non-constant turn: running power spectrum and nonG

  • Build models from string theory, obtain natural values for parameters

  • More general concept of Quasi-Single Field Inflation:

  • massive (H) fields – inflaton coupling can be more arbitrary

  • …...


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