Primordial Non-Gaussianities and
1 / 62

Primordial Non-Gaussianities and Quasi-Single Field Inflation - PowerPoint PPT Presentation

  • Uploaded on

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).

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
Download Presentation

PowerPoint Slideshow about 'Primordial Non-Gaussianities and Quasi-Single Field Inflation' - majed

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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


(WMAP website)

Temperature Fluctuations

(WMAP website)


(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


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 , .




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



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


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, 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



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


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


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

Periodic features

Oscillating background


Modes within horizon are oscillating


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

Other Possible Shapes and Runnings in Simple Models space

with Large non-Gaussianities


  • 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)



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.




It is scale-invariant

for constant turn case.

Perturbation Theory space

  • Field perturbations:

  • Lagrangian

Oscillating inside horizon space

Kinematic Part

  • Massless:


Constant after horizon exit

Oscillating inside horizon space

Decay as after horizon exit.

  • Massive:


, mass of order H


Oscillating inside horizon space

  • Massive:


, mass >> H


Oscillating and decay after horizon exit


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


Perturbation Method and Feynman Diagrams space

Correction to 2pt


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.



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



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


  • 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


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

  • …...