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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;
- 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

- 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 isotropy

- Assume statistical homogeneity;
- Two point correlation function
- Isotropic if for ;
- The isotropic power spectrum:
- The isotropic bispectrum:

Statistical homogeneity and isotropy

- Two point correlation function
- Anisotropic if even for ;
- The anisotropic power spectrum:
- The anisotropic bispectrum:

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

- 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.

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

Linde(1990)

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

Inflation

HBB

Curvaton Mechanism: Basic Idea- Curvaton (Lyth, Wands (2002); Enquist, Sloth (2002)):
- light scalar field;
- does not drive inflation.

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:

where

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

Conclusions

- 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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