non-gaussianities in general single field inflation xingang chen ctp, mit astro-ph/0507053;...
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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
• 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 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
• 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,
•
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
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)
• 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.