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Anisotropic non-Gaussianity. Mindaugas Karčiauskas work done with Konstantinos Dimopoulos David H. Lyth. arXiv: 0812.0264. Density perturbations. Primordial curvature perturbation – a unique window to the early universe; Origin of structure <= quantum fluctuations;

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Anisotropic non-Gaussianity

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Anisotropic non gaussianity

Anisotropic non-Gaussianity

Mindaugas Karčiauskas

work done with

Konstantinos Dimopoulos

David H. Lyth


Density perturbations

Density perturbations

  • Primordial curvature perturbation – a unique window to the early universe;

  • Origin of structure <= quantum fluctuations;

  • Usually light, canonically normalized scalar fields => statistical homogeneity and isotropy;

  • Statistically anisotropic perturbations from the vacuum with a broken rotational symmetry;

  • The resulting is anisotropic and may be observable.

Statistical homogeneity and isotropy

Statistical homogeneity and isotropy

  • Density perturbations – random fields;

  • Density contrast: ;

  • Multipoint probability distribution function :

    • Homogeneous if the same under translations of all ;

    • Isotropic if the same under spatial rotation;

Statistical homogeneity and isotropy1

Statistical homogeneity and isotropy

  • Assume statistical homogeneity;

  • Two point correlation function

    • Isotropic if for ;

  • The isotropic power spectrum:

  • The isotropic bispectrum:

Statistical homogeneity and isotropy2

Statistical homogeneity and isotropy

  • Two point correlation function

    • Anisotropic if even for ;

  • The anisotropic power spectrum:

  • The anisotropic bispectrum:

Random fields with statistical anisotropy

Random Fields with Statistical Anisotropy


- preferred direction

Present observational constrains

Present Observational Constrains

  • The power spectrum of the curvature perturbation:

    & almost scale invariant;

  • Non-Gaussianity from WMAP5 (Komatsu et. al. (2008)):

  • No tight constraints on anisotropic contribution yet;

  • Anisotropic power spectrum can be parametrized as

  • Present bound(Groeneboom, Eriksen (2008));

  • We have calculated of the anisotropic curvature perturbation - new observable.

Origin of statistically anisotropic power spectrum

Origin of Statistically Anisotropic Power Spectrum

  • Homogeneous and isotropic vacuum => the statistically isotropic perturbation;

  • For the statistically anisotropic perturbation <= a vacuum with broken rotational symmetry;

  • Vector fields with non-zero expectation value;

  • Particle production => conformal invariance of massless U(1) vector fields must be broken;

  • We calculate for two examples:

    • End-of-inflation scenario;

    • Vector curvaton model.

N formalism

δN formalism

  • To calculate we use formalism

    (Sasaki, Stewart (1996); Lyth, Malik, Sasaki (2005));

  • Recently in was generalized to include vector field perturbations (Dimopoulos, Lyth, Rodriguez (2008)):

    where , , etc.

End of inflation scenario basic idea

End-of-Inflation Scenario: Basic Idea


End of inflation scenario basic idea1

End-of-Inflation Scenario: Basic Idea

- light scalar field.


Anisotropic non gaussianity

Statistical Anisotropy at the End-of-Inflation Scenario

- vector field.

Yokoyama, Soda (2008)

Anisotropic non gaussianity

Statistical Anisotropy at the End-of-Inflation Scenario

  • Physical vector field:

  • Non-canonical kinetic function ;

  • Scale invariant power spectrum => ;

  • Only the subdominant contribution;

  • Non-Gaussianity:

    where , - slow roll parameter

Curvaton mechanism basic idea




Curvaton Mechanism: Basic Idea

  • Curvaton (Lyth, Wands (2002); Enquist, Sloth (2002)):

    • light scalar field;

    • does not drive inflation.

Vector curvaton

Vector Curvaton

  • Vector field acts as the curvaton field (Dimopoulos (2006));

  • Only a smallcontribution to the perturbations generated during inflation;

  • Assuming:

    • scale invariant perturbation spectra;

    • no parity braking terms;

  • Non-Gaussianity:


Estimation of

Estimation of  

  • In principle isotropic perturbations are possible from vector fields;

  • In general power spectra will be anisotropic => must be subdominant ( );

  • For subdominant contribution can be estimated on a fairly general grounds;

  • All calculations were done in the limit ;

  • Assuming that one can show that



  • We considered anisotropic contribution to the power spectrum and

  • calculated its non-Gaussianity parameter .

  • We applied our formalism for two specific examples: end-of-inflation and vector curvaton.

  • .is anisotropic and correlated with the amount and direction of the anisotropy.

  • The produced non-Gaussianity can be observable:

  • Our formalism can be easily applied to other known scenarios.

  • If anisotropic is detected => smoking gun for vector field contribution to the curvature perturbation.

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