Unesco July 2005Francis Bernardeau SPhT Saclay1 Models of inflation with primordial non-Gaussianities Francis Bernardeau SPhT Saclay Collaboration with Tristan Brunier (SPhT Saclay) Jean-Philippe Uzan (LPT Orsay and IAP) Lev Kofman (CITA) astro-ph/ , , and Unesco, July 2005

Unesco July 2005Francis Bernardeau SPhT Saclay2 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 Curvature fluctuations = fluctuation of the expansion factor value from a super-horizon patch to another. They are given by Quantum fluctuations of  being of the order of H The superhorizon fluctuations are expected to be scale free and adiabatic

Unesco July 2005Francis Bernardeau SPhT Saclay3 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)

Unesco July 2005Francis Bernardeau SPhT Saclay4 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

Unesco July 2005Francis Bernardeau SPhT Saclay5 No significant NG in case of one-field inflation Klein-Gordon equation for the field fluctuations : Mass termNon 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 (parameter independent) Subsequent NG effects dominate (Bartolo et al. ‘04) Estimation of amount of NG: Super-horizon and Slow Roll approximation

Unesco July 2005Francis Bernardeau SPhT Saclay6 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

Unesco July 2005Francis Bernardeau SPhT Saclay7 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. Curvature fluctuations = Gaussian fluctuations of φ + non-Gaussian fluctuations of χ

Unesco July 2005Francis Bernardeau SPhT Saclay8 Mode transfer in Multiple field inflation (2) Lagrangian density The field trajectory in Slow Roll approximation is along direction θ defined by The field motion equations are given by : transverse direction, isocurvature mode : adiabatic, direction along the trajectory

Unesco July 2005Francis Bernardeau SPhT Saclay9 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 non- Gaussian.Nothing prevents the fluctuations to be significantly non- Gaussian.

Unesco July 2005Francis Bernardeau SPhT Saclay10 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

Unesco July 2005Francis Bernardeau SPhT Saclay11 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

Unesco July 2005Francis Bernardeau SPhT Saclay12 Quantum scalar field in de Sitter space For a quartic coupling and in the super-horizon limit (FB, Brunier & Uzan ‘03) Classical evolution.

Unesco July 2005Francis Bernardeau SPhT Saclay13 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

Unesco July 2005Francis Bernardeau SPhT Saclay14 … 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.

Unesco July 2005Francis Bernardeau SPhT Saclay15 Reconstructed PDF shape: Negative λ Gaussian Positive λ

Unesco July 2005Francis Bernardeau SPhT Saclay16 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 …

Unesco July 2005Francis Bernardeau SPhT Saclay17 The modulated inflation Light field fluctuations can modulate reheating or the end of inflation processes (Dvali, Gruzinov and Zaldarriaga ’03 and Kofman ‘03)

Unesco July 2005Francis Bernardeau SPhT Saclay18 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  a but should not depend on  a

Unesco July 2005Francis Bernardeau SPhT Saclay19 Sugra D-term inflation Super-potential Kähler potential of the shape (compactification with three tori) implies that λ is independent on moduli fields and

Unesco July 2005Francis Bernardeau SPhT Saclay20 Phenomenological consequences Breaking of the relation between tensor and scalar metric fluctuations Possibility of having NG adiabatic fluctuations

Unesco July 2005Francis Bernardeau SPhT Saclay21 Conclusions It 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