Primordial Non-Gaussianities and Quasi-Single Field Inflation Xingang Chen Center for Theoretical Cosmology, DAMTP, Cambridge University X.C., , a review on non-G; X.C., Yi Wang, ;

CMB and WMAP (WMAP website)

Temperature Fluctuations (WMAP website)

What sources these fluctuations?

CMB and WMAP (WMAP website)

Generic Predictions of Inflationary Scenario Primordial (seeded at super-horizon size) Approximately scale-invaraint Approximately Gaussian Density perturbations that seed the large scale struture are

Two-point Correlation Function (Power Spectrum) (WMAP5) Non-Gaussianities

Is this enough?

Experimentally: Information is Compressed Amplitude and scale-dependence of the power spectrum (2pt) contain 1000 numbers for WMAP But we have pixels in WMAP temperature map This compression of information is justified only if the primordial fluctuations is perfectly Gaussian. Can learn much more from the non-Gaussian components.

What kind of fields drive the inflation? Alternative to inflation? What are the Lagrangian for these fields? Theoretically: From Paradigm to Explicit Models Quantum gravity

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! Non-G components: Primordial Interactions

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 Experimentally:  Einstein gravity

Inflation Model Building The other conditions in the no-go theorem also needs to be satisfied. Examples of simplest slow-roll potentials:

Much more complicated in realistic model building ……

Inflation Model Building A landscape of potentials

Inflation Model Building Warped Calabi-Yau

Inflation Model Building   -Problem in slow-roll inflation: ? (Copeland, Liddle, Lyth, Stewart, Wands, 04)  h-Problem in DBI inflation: ? (X.C., 08)

 Variation of potential: (Lyth, 97) : eg. higher dim Planck mass, string mass, warped scales etc.  Field range bound: (X.C., Sarangi, Tye, Xu, 06; Baumann, McAllsiter, 06) ?

Algebraic simplicity may not mean simplicity in nature.

 Canonical kinetic term 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

Shape and Running of Bispectra (3pt) Shape dependence: (Shape of non-G) Squeezed EquilateralFolded Scale dependence: (Running of non-G) Fix, vary,. Fix,, vary. Bispectrum is a function, with magnitude, of three momenta:

Two Well-Known Shapes of Large Bispectra (3pt) Equilateral Local In squeezed limit: For scale independent non-G, we draw the shape of

Physics of Large Equilateral Shape Generated by interacting modes during their horizon exit Quantum fluctuationsInteracting and exiting horizonFrozen For single field, small correlation if For example, in single field inflation with higher order derivative terms (Alishahiha, Silverstein, Tong, 04; X.C., Huang, Kachru, Shiu, 06) So, the shape peaks at equilateral limit. (Inflation dynamics is no longer slow-roll)

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) Local in position space non-local in momentum space So, the shape peaks at squeezed limit. For example, in curvaton models; (Vernizzi, Wands, 06; Rigopoulos, Shellard, van Tent, 06) (Lyth, Ungarelli, Wands, 02) multifield inflation models with turning trajectory, (very difficult to get observable nonG.)

What we knew experimentally about the non-Gaussianities

WMAP5 Data, 08 Experimental Results on Bispectra ; (WMAP group, 10) (Yadav, Wandelt, 07) Large Scale Structure (Slosar et al, 08) (Rudjord et.al., 09)

The Planck Satellite, sucessfully launched last year (Smith, Zaldarriaga, 06) ; ; (Planck bluebook)

Other Experiments Ground based CMB telescope: ACBAR, BICEP, ACT, …. High-z galaxy survey: SDSS, CIP, EUCLID, LSST … 21-cm tomography: LOFAR, MWA, FFTT, … 21cm: FFTT (Mao, Tegmark, McQuinn, Zaldarriaga, Zahn, 08) For example:

Looking for Other Shapes and Runnings of Non-Gaussianities in Simple Models Why? Data analyses Theoretical template Construct estimator for example Fit data to constrain for example  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. Underlying physics Different dynamics in inflation predict different non-G.

Other Possible Shapes and Runnings in Simple Models with Large non-Gaussianities (X.C., Huang, Kachru, Shiu, 06; X.C., Easther, Lim, 06,08)

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 In 3pt: Peaks at folded triangle limit

Other Possible Shapes and Runnings in Simple Models with Large non-Gaussianities Folded Shape: Boundary conditons “Trans-Planckian” effect Low new physics scales

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 A feature local in timeOscillatory running in momentum space 3pt:

Other Possible Shapes and Runnings in Simple Models 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 with Large non-Gaussianities Resonance: (X.C., Easther, Lim, 08; Flauger, Pajer, 10) Modes within horizon are oscillating Oscillating background Periodic features Resonance 3pt: Periodic-scale-invariance: Rescale all momenta by a discrete efold:

Other Possible Shapes and Runnings in Simple Models with Large non-Gaussianities Resonance: Periodic features from duality cascade in brane inflation Periodic features from instantons in monodromy inflation (Hailu, Tye, 06; Bean, Chen, Hailu, Tye, Xu, 08) (Silverstein, Westphal, 08; Flauger, Mcallister, Pajer, Westphal, Xu, 09)

Other Possible Shapes and Runnings in Simple Models with Large non-Gaussianities (X.C., Wang, 09)

Quasi-Single Field Inflation

Motivation for Quasi-Single Field Inflation Fine-tuning problem in slow-roll inflation In the inflationary background, the mass of light particle is typically of order H (the Hubble parameter) is needed for slow-roll inflationC.f. Generally, multiple light fields exist (Copeland, Liddle, Lyth, Stewart, Wands, 94) E.g.  One field has the mass  Others have mass Quasi-single field inflation (X.C., Wang, 09) (Ignored previously for den. pert.)

A Simple Model of Quasi-Single Field Inflation  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). Lagrangian in polar coordinates: slow-roll potential potential for massive field Here study the constant turn case

Difference Between and but etc is the main source of the large non-Gaussianities. It is scale-invariant for constant turn case.

Perturbation Theory Field perturbations: Lagrangian

Kinematic Part Massless: Constant after horizon exit Solution: Oscillating inside horizon

Massive: Oscillating inside horizon Solution: E.g., mass of order H Decay as after horizon exit.

Massive: Oscillating inside horizon Solution: E.g., mass >> H Oscillating and decay after horizon exit

Massive: Solution: We will consider the case:

Interaction Part Transfer vertex Interaction vertex We use this transfer-vertex to compute the isocurvature-curvature conversion Source of the large non-Gaussianities

Perturbation Method and Feynman Diagrams To use the perturbation theory, we need Correction to 2pt 3pt These conditions are not necessary for the model building, but non-perturbative method remains a challenge.

In-In Formalism Mixed form “Factorized form” for UV part to avoid spurious UV divergence; “Commutator form” for IR part to avoid spurious IR divergence. Introduce a cutoff. Mixed form + Wick rotation for UV part A very efficient way to numerically integrate the 3pt. (X.C., Wang, 09) (Weinberg, 05)

Numerical Results

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. Using the asymptotic behavior of Hankel functions, we get the shape for

A Shape Ansatz

Compare Numerical Ansatz

Size of Bispectrum Definition of : We get For perturbative method: Non-perturbative case is also very interesting.

Physics of Large Intermediate Shapes Quasi-equilateral: for heavier isocurvaton Fluctuations decay faster after horizon-exit, so large interactions happen during the horizon-exit. Modes have comparable wavelengths: Closer to equilateral shape. Quasi-local: for lighter isocurvaton 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. In limit, recover the local shape behavior.

Effect of the Transfer Vertex 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.

Trispectra (4pt) For the perturbative case: C.f.

Conclusions 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 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 Quasi-single field inflation Non-Gaussianities with intermediate shapes Transfer vertex in “in-in” formalism Effects of massive modes on density perturbations Compute isocurvature-curvature transition perturbatively Numerical, analytical and ansatz Using non-Gaussianities to probe early universe Different inflationary dynamics can imprint distinctive signatures in non-G

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 Non-constant turn: running power spectrum and nonG Future Directions …... Compare Intermediate Shapes, Resonant running, etc, with data and constrain and