1. Anisotropic non-Gaussianity Mindaugas Karčiauskas work done with Konstantinos Dimopoulos David H. Lyth 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. ● 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. ● 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; ● 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

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

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

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. ● 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. ● 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. ● 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. - vector field. Statistical Anisotropy at the End-of-Inflation Scenario 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 ● Physical vector field: ● Non-canonical kinetic function ; ● Scale invariant power spectrum => ; ● Only the subdominant contribution; ● Non-Gaussianity: where, - slow roll parameter

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

15. Vector Curvaton ● Vector field acts as the curvaton field (Dimopoulos (2006)) ; ● Only a small contribution to the perturbations generated during inflation; ● Assuming: ● scale invariant perturbation spectra; ● no parity braking terms; ● Non-Gaussianity: ● Vector field acts as the curvaton field (Dimopoulos (2006)) ; ● Only a small contribution 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 ● 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. ● 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.