Anisotropic non-Gaussianity

1. Anisotropic non-Gaussianity Mindaugas Karčiauskas work done with Konstantinos Dimopoulos David H. Lyth arXiv:0812.0264

2. 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.

3. 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;

4. Statistical homogeneity and isotropy • Assume statistical homogeneity; • Two point correlation function • Isotropic if for ; • The isotropic power spectrum: • The isotropic bispectrum:

5. Statistical homogeneity and isotropy • Two point correlation function • Anisotropic if even for ; • The anisotropic power spectrum: • The anisotropic bispectrum:

6. Random Fields with Statistical Anisotropy Isotropic - preferred direction

7. 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.

8. 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.

9. 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.

10. End-of-Inflation Scenario: Basic Idea Linde(1990)

11. End-of-Inflation Scenario: Basic Idea - light scalar field. Lyth(2005);

12. Statistical Anisotropy at the End-of-Inflation Scenario - vector field. Yokoyama, Soda (2008)

13. 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

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

15. 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

16. 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

17. 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.