Unesco July 2005 Francis Bernardeau SPhT Saclay 1

Models of inflation with primordial non-Gaussianities

Francis BernardeauSPhT Saclay

Collaboration with Collaboration with Tristan Brunier (SPhT Saclay)Tristan Brunier (SPhT Saclay)Jean-Philippe Uzan (LPT Orsay and IAP)Jean-Philippe Uzan (LPT Orsay and IAP)Lev Kofman (CITA)Lev Kofman (CITA)

astro-ph/0207295, 0209330, 0311422 and astro-ph/0207295, 0209330, 0311422 and 04033150403315

Unesco, July 2005

Unesco July 2005 Francis Bernardeau SPhT Saclay 2

Generic inflation: the generation of superhorizon fluctuations

Accelerated expansion due to the homogeneous part of the inflaton field

Inflaton fluctuations = time lapses in inflationary period

€

δt = -3HV '

δφ

Curvature fluctuations = fluctuation of the expansion factor value from a super-horizon patch to another. They are given by

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3H ˙ φ 0 = - ∂V∂φ

φ0( )

€

φ

€

V φ( )

€

Δatot

atot

= δNe = Hδt = − 3H 2

V 'δφ

Quantum fluctuations of being of the order of H

The superhorizon fluctuations are expected to be scale free and adiabatic

€

δφ

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CMB-LSS data are un-ambiguous…We have now strong evidences for initial adiabatic fluctuations

with a nearly flat power spectrum from LSS and CMB data

WMAP, first year data (Tegmark & Zaldarriaga 2002)

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• CMB = a tracer of metric fluctuations left during the inflationary period.

• An interesting window: non-Gaussian relics in primordial metric fluctuations

– To which extend can we have inflation together with non-Gaussian primordial fluctuations that are adiabatic and whose spectrum is nearly flat?

– What would be the phenomenological consequences of inflationary models with significant primordial non-Gaussianities?

A window to physics of the early Universe

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No significant NG in case of one-field inflation

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˙ ̇ δ φ + 3H ˙ δ - Δ2φ =

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-∂2V∂φ2 δφ

Klein-Gordon equation for the field fluctuations :

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-∂3V∂φ3

δφ2

2- ∂ 4V

∂φ4

δφ3

3!...

Mass term Non linear coupling terms

Non-Gaussianities are always negligible in the Slow Roll approximation (Komatsu and Spergel ‘00, Maldacena ‘02).

They are of the order of 10-5 (parameter independent)Subsequent NG effects dominate (Bartolo et al. ‘04)

Estimation of amount of NG: Super-horizon and Slow Roll approximation

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3Hδ ˙ φ = S δφ( )

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δφ t( ) = δφ0 + 13H

S t( )dt∫

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δφNG =S δφ0( )3H 2 Nefold

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Multiple field inflation• Any light scalar field is going to

experiment superhorizon fluctuations…• How can they be visible?

– Survival of isocurvature modes with adequate decay mechanisms (curvaton mechanism)

– Isocurvature to adiabatic mode transfer• bent trajectorybent trajectory• modulated inflationmodulated inflation

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Mode transfer in Multiple field inflation

It can only work if the trajectory is bent (Salopek, ‘92; Bartolo et al. ‘01, FB & Uzan ‘03), e.g.

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∂2V∂φ∂χ

≠0

Curvature fluctuations = Gaussian fluctuations of φ + non-Gaussian fluctuations of χ

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Mode transfer in Multiple field inflation (2)

Lagrangian density

The field trajectory in Slow Roll approximation is along direction θ defined by

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δ˙ ̇ φ + 3Hδ ˙ φ - δ 2φ = -∂ 2V∂φ2 δφ - ∂ 2V

∂φ∂χδχ

The field motion equations are given by

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δφ€

δχ

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δ˙ ̇ χ + 3Hδ ˙ χ - Δ2χ = -...- ∂ 3V∂χ 3

δχ 2

2- ∂ 4V

∂χ 4

δχ 3

3!...

: transverse direction, isocurvature mode

: adiabatic, direction along the trajectory

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Multiple-field inflation : phenomenological

consequences• Simultaneous isocurvature and adiabatic

fluctuations– The spectra are the same (amplitude determined by H )– Depending on the trajectory, fluctuations can be

correlated (Langlois ’99,…)– Survival of isocurvature modes depends on how the

various fields decay during reheating– Transfer of modes is possible during or at the end

of inflation• Nothing prevents the fluctuations to be significantly Nothing prevents the fluctuations to be significantly

non-Gaussian.non-Gaussian.

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A simpler working example… of hybrid inflation type (FB & Uzan ’03)

The inflaton A light transverse field

A massive transverse field that eventually undergoes a phase transition

Inflationary period stops at a time that depends on both the φ and χ values

Focusing of the classical trajectories if λ > 0

Horizon crossing

Active quantum fluctuations

φ

End of inflation

÷

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Quantum scalar field in de Sitter space

Tree order calculation of the four point correlation function for a quartic potential

• de Sitter space

• Quantification of scalar field

• The high-order cumulants

Free vacuum

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Quantum scalar field in de Sitter spaceFor a quartic coupling and in the super-horizon limit (FB, Brunier & Uzan ‘03)

efoldsN=

Classical evolution.

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Superhorizon evolution: evolution of a self-coupled stochastic field

For the non-linear evolution of a classical stochastic field a perturbation theory approach can be used (FB & Uzan ’03), KG with a non-linear source term.

Cumulant computation (at tree order) following PT techniques (Peebles, Fry, Bernardeau, Scoccimarro, etc…)

Gaussian

Leading order in λ

Depends on time!

Loop terms are ill defined (sub-horizon effects)

Filtered field

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… good up to a PDF reconstruction

The complete PDF of the curvature fluctuations can be obtained from the resolution of the motion equation for

with

Motion equation in the Slow Roll limit :

The PDF for can then be obtained from a simple nonlinear transform, or from an inverse Laplace transform of the cumulant generating function if one wants to keep only the tree order contribution. The cumulant generating function is obtained from the vertex generating function through a Legendre transform.

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Reconstructed PDF shape:

• Negative λ• Gaussian• Positive λ

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Conclusions (1)

• What are their observational signatures ? •Are such models natural in high-energy physics theories ?

In particular it is possible to build multiple-field models of inflation in which the primordial metric fluctuations can be significantly non-Gaussian. Those non-Gaussianities can be described by 3 parameters :

- “mixing angle”- amplitude of the quartic coupling parameter- and field value at survey scale (finite volume effect)

Generalized models of inflation can lead to a richer phenomenology …

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The modulated inflation• Light field fluctuations can modulate

reheating or the end of inflation processes (Dvali, Gruzinov and Zaldarriaga ’03 and Kofman ‘03)

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Modulated inflation in SuGra context

• Natural candidates for extra light fields are the moduli fields whose VEV furthermore determine the strength of the coupling constants

• An example: the hybrid type inflation (FB, Kofman, Uzan ‘04):

marks the end of inflation: can depend on χa but should not depend on χa

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Sugra D-term inflation• Super-potential

• Kähler potential of the shape (compactification with three tori)

implies that λ is independent on moduli fields and

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Phenomenological consequences

• Breaking of the relation between tensor and scalar metric fluctuations

• Possibility of having NG adiabatic fluctuations

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ConclusionsIt cannot be excluded that inflation produces significant non-Gaussianities in the metric fluctuations

A priori unlikely but would be a rich window to the physics of the early universe

• Curvaton type effects• Transfer of mode effects