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)
What sources these fluctuations?
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
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.
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
Experimentally:
Inflation Model Building
Examples of simplest slow-roll potentials:
The other conditions in the no-go theorem also needs to be satisfied.
Much more complicated in realistic model building ……
Inflation Model Building
A landscape of potentials
Inflation Model Building
Warped Calabi-Yau
Inflation Model Building
 h-Problem in slow-roll inflation: (Copeland, Liddle, Lyth, Stewart, Wands, 04)
?
 h-Problem in DBI inflation:
(X.C., 08)
?
 Field range bound:
(X.C., Sarangi, Tye, Xu, 06; Baumann, McAllsiter, 06)
?
 Variation of potential:
(Lyth, 97)
: eg. higher dim Planck mass, string mass, warped scales etc.
Algebraic simplicity may not mean simplicity in nature.
Beyond the No-Go
 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
Shape and Running of Bispectra (3pt)
Bispectrum is a function, with magnitude
• Shape dependence:
(Shape of non-G)
Fix
Squeezed
• Scale dependence:
(Running of non-G)
, vary
Equilateral
Fix
, of three momenta:
,
,
.
Folded
, vary
.
Two Well-Known Shapes of Large Bispectra (3pt)
For scale independent non-G, we draw the shape of
Equilateral
In squeezed limit:
Local
Physics of Large Equilateral Shape
• Generated by interacting modes during their horizon exit
Quantum fluctuations
Interacting and exiting horizon
Frozen
For single field, small correlation if
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)
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;
(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 experimentally about the non-Gaussianities
Experimental Results on Bispectra
• 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
;
(Planck bluebook)
;
(Smith, Zaldarriaga, 06)
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:
21cm: FFTT
(Mao, Tegmark, McQuinn,
Zaldarriaga, Zahn, 08)
Looking for Other Shapes and Runnings
of Non-Gaussianities in Simple Models
• Why?
Underlying physics
Different dynamics in inflation predict different non-G.
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.
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:
In 3pt:
For example, a small
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 time
Oscillatory 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:
Periodic features
(X.C., Easther, Lim, 08; Flauger, Pajer, 10)
Oscillating background
Modes within horizon are oscillating
3pt:
Periodic-scale-invariance: Rescale all momenta by a discrete efold:
Resonance
Other Possible Shapes and Runnings in Simple Models
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
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
(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
 One field has the mass
 Others have mass
Quasi-single field inflation
(Ignored previously for den. pert.)
(X.C., Wang, 09)
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).
Here study the constant turn case
Lagrangian in polar coordinates:
slow-roll potential
potential for massive field
Difference Between
and
but
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:
Solution:
Constant after horizon exit
Oscillating inside horizon
1.5
20
1.0
10
0.5
30
25
20
15
10
5
10
3.0
2.5
2.0
1.5
1.0
0.5
20
0.5
30
• Massive:
Solution:
, mass of order H
E.g.
Oscillating inside horizon
Decay as
after horizon exit.
2.0
20
1.5
10
1.0
30
25
20
15
10
5
0.5
10
20
3.0
2.5
2.0
1.5
1.0
0.5
• Massive:
Solution:
, mass >> H
E.g.
Oscillating inside horizon
Oscillating and decay after horizon exit
0.4
10
0.2
5
3.0
30
25
20
15
10
2.5
2.0
1.5
1.0
0.5
0.2
5
5
0.4
0.6
10
0.8
• Massive:
Solution:
We will consider the case:
Interaction Part
• 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
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
(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.
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.
• Using the asymptotic behavior of Hankel functions, we get the shape
for
for
• Lying between the equilateral form
We call them Intermediate Shapes.
, and local form
(X.C., Wang, 09)
Not superposition of previously known shapes.
.
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)
C.f.
For the perturbative case:
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
• 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
Future Directions
• Compare Intermediate Shapes, Resonant running, etc, with data
and constrain
and
• 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
• …...