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Paolo Creminelli (ICTP, Trieste). Non-Gaussianities of cosmological perturbations. with N. Arkani-Hamed, D. Babich, L. Boubekeur, S. Mukohyama, A. Nicolis, L. Senatore, M. Tegmark, and M. Zaldarriaga. Is there any correlation among modes?. Slow-roll = weak coupling. V. Friction is dominant.

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Non-Gaussianities of cosmological perturbations

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Paolo Creminelli (ICTP, Trieste)

Non-Gaussianities of cosmological perturbations

with N. Arkani-Hamed, D. Babich, L. Boubekeur, S. Mukohyama, A. Nicolis, L. Senatore, M. Tegmark, and M. Zaldarriaga


Slow-roll = weak coupling


Friction is dominant

To have ~ dS space the potential must be very flat:

The inflaton is extremely weakly coupled. Leading NG from gravity.

Completely model independent

as it comes from gravity

Unobservable (?). To see any deviation you need > 1012 data. WMAP ~ 2 x 106

Maldacena, JHEP 0305:013,2003, Acquaviva etal Nucl.Phys.B667:119-148,2003


Smoking gun for “new physics”

Any signal would be a clear signal of something non-minimal

  • Any modification enhances NG
    • Modify inflaton Lagrangian. Higher derivative terms, ghost inflation, DBI inflation…
    • Additional light fields during inflation. Curvaton, variable decay width…
  • Potential wealth of information

Translation invariance:

Scale invariance:

F contains information about the source of NG

Note. We are only considering primordial NGs. Neglect non-linear relation with observables.

Good until primordial NG > 10-5.

see P.C. + Zaldarriaga, Phys.Rev.D70:083532,2004

Bartolo, Matarrese and Riotto, JCAP 0606:024,2006


Higher derivative terms

P.C. JCAP 0310:003,2003

Change inflaton dynamics and thus density perturbations

Potential terms are strongly constrained by slow-roll.

Impose shift symmetry:

Most relevant operator:

3 point function:

In EFT regime NG < 10-5 Difficult to observe

We get large NG only if h. d. terms are important also for the classical dynamics

One can explicitly calculate the induced 3pf:


DBI inflation

Alishahiha, Silverstein and Tong, Phys.Rev.D70:123505,2004

Example where higher derivative corrections are important

A probe D3 brane moves towards IR of AdS.

Geometrically there is a speed limit


The dual description of this limit is

encoded in h.d. operators. DBI action:

The scalar is moving towards the origin of the moduli space. H.d. operators come

integrating out states becoming massless at the origin.



  • It helps inflation slowing down the scalar (potential?)
  • Generic in any warped brane model of inflation (reconstruct the shape of the throat?)
  • 3pf can be as large as you like
  • Generic 3pf for any model with:

(see e.g. S. Kecskemeti etal, hep-th/0605189)

S. Kachru etal. hep-th/0605045


General approach

X. Chen, M.x.Huang, S.Kachru and G.Shiu, hep-th/0605045

We want to calculate for a generic:

Let us rephrase the calculation in a way which emphasis symmetries: EFT around FRW

EFT around a given FRW background .

Focus on the perturbation

Goldstone of broken time-translation

  • This mode can be reabsorbed in the metric going to unitary gauge
  • In this gauge, only gravity with time-dependent spatial diff. :
  • The Goldstone can be reintroduced with Stuckelberg trick
  • ADM variables are useful:
  • Most generic action invariant under spatial diff. + reintroduce the Goldstone

At 0th order in derivative I have only:

Tadpole terms to make the given a(t) a solution:

Coefficients can be time dependent:


E.g. this is all for a

standard scalar field:

Fix the coefficients from the background:


At 0th derivative level

up to 3rd order in pert:

Only two operators are allowed by symmetries: 2 contributions (up to slow-roll)

The first operator changes the speed of sound

M4 > 0 to avoid superluminality

See P.C., M.Luty, A.Nicolis, L.Senatore,

hep-th/0606090 and Leonardo’s talk

Contribution to NGs:


Ghost inflation

with Arkani-Hamed, Mukoyama and Zaldarriaga,

JCAP 0404:001,2004

Ghost condensation:


  • Spontaneous breaking of Lorentz symmetry:
  • Consistent derivative expansion:
  • Non Lorentz-invariant action, standard spatial kinetic term NOT allowed
  • IR modification of gravity. “Higgs phase” for gravity.

Large non-Gaussianities

  • Use the “ghost” field as the inflaton. It triggers the end of dS.
  • Non Lorentz-invariant scaling
  • The leading non-linear operator:


has dimension 1/4

Quite large. Close to exp. bound.

Derivative interactions are enhanced wrt standard case by NR relativistic scaling


Perturbations generated by a second field

  • Curvaton
  • Variable inflaton decay
  • 2 field inflation
  • Perturbation of parameters
  • relevant for cosmo evolution

Every light scalar is perturbed during inflation.

Its perturbations may become relevant in various ways:

Example: variable decay of right-handed neutrinos

with L.Boubekeur, hep-ph/0602052

  • Parallel Universes:
  • NG is generated by inefficiency: RHN neutrinos will not be completely dominant.

To match 10-5 we need larger fluctuations and thus larger NGs

  • In general: NG > 10-5, but model dependent. Possible isocurvature contributions.

The RHN goes out of equilibrium and

decay in ≠ way in ≠ regions of the Universe


The shape of non-Gaussianities

Babich, P.C., Zaldarriaga, JCAP 0408:009,2004


Typical for NG produced outside the horizon. 2 field models, curvaton, variable decay…


Derivative interactions irrelevant after crossing.

Correlation among modes of comparable .

F is quite complicated in the various models. But in general

Quite similar in different models


Shape comparison

The NG signal is concentrated on different configurations.

  • They can be easily distinguished (once NG is detected!)
  • They need a dedicated analysis

Consistency relation for 3-p.f.

J. Maldacena, JHEP 0305:013,2003

P.C. + M. Zaldarriaga, JCAP 0410:006, 2004

Underthe usual “adiabatic” assumption (a single field is relevant),

INDEPENDENTLY of the inflaton Lagrangian

The long wavelength mode is a frozen background for the other two: it redefines spatial coordinates.

In the squeezed limit the 3pf is small and probably undetectable

  • Models with a second field have a large 3pf in this limit.
  • Violation of this relation is a clear, model independent evidence for a second field
  • (same implications as detecting isocurvature).
  • This is experimentally achievable if NG is detected.

Analysis of WMAP data

with Nicolis, Senatore, Tegmark, Zaldarriaga, astro-ph/0509029

WMAP alone gives almost all we know about NG. Large data sample + simple.

Not completely straightforward!

It scales like Npixels5/2 ~ 1016for WMAP!!! Too much…

But if F is “factorizable” the computation time scales as Npixels3/2 ~ 109. Doable!

Use a fact. shape with equilateral properties


Real space VS Fourier space

CMB signal diagonal in Fourier space (without NG!!). Foreground and noise in real space.

Non-diagonal error matrix + linear term in the estimator

Minimum variance estimator:

Reduces variance wrt WMAP coll. analysis.

N_obs varies across the sky.

Smaller power in more observed regions.

On a given realization it looks like a NG

signal. Bigger variance.



No detection! Limits (1yr):

-27 < fNLlocal < 121 at 95% C.L.

(WMAP 3yr: -54 < fNLlocal < 114 )

-366 < fNLequil. < 238 at 95% C.L.

3 yr data. 1 sigma error for local 37 ----> 33 only 15% improvement

The quite different limits is just a consequence of the definition of fNL



  • Non-Gaussianities are enhanced in:

1. Single field models with modified Lagrangian

(single operator, DBI inflation, ghost inflation…)

2. Models with a second field

(curvaton, variable decay width, variable RHN decay…)

  • Shape of the 3-point function:
  • Local (more than 1 field) VS Equilateral (higher derivative terms)
  • Consistency relation for 3pf for any single field model
  • WMAP data analysis for different shapes