Anisotropic non-Gaussianity Anisotropic non-Gaussianity

Mindaugas Karčiauskaswork done with

Konstantinos DimopoulosDavid H. Lyth

Mindaugas Karčiauskaswork done with

Konstantinos DimopoulosDavid H. Lyth

arXiv:0812.0264arXiv:0812.0264

Density perturbationsDensity 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.

● 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

Statistical homogeneity and isotropy

Statistical homogeneity and isotropy

Statistical homogeneity and isotropy

● 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:

● 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

Statistical homogeneity and isotropy

Random Fields with Statistical

Anisotropy

Random Fields with Statistical

Anisotropy

IsotropicIsotropic

- preferred direction- 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.

● 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.

● 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 formalismδN 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.

End-of-Inflation Scenario: Basic Idea

End-of-Inflation Scenario: Basic Idea

Linde(1990)Linde(1990)

End-of-Inflation Scenario: Basic Idea

End-of-Inflation Scenario: Basic Idea

- light scalar field.- light scalar field. Lyth(2005); Lyth(2005);

- vector field.- vector field.

Statistical Anisotropy at the End-of-Inflation

Scenario

Statistical Anisotropy at the End-of-Inflation

Scenario

Yokoyama, Soda (2008)Yokoyama, Soda (2008)

Statistical Anisotropy at the End-of-Inflation

Scenario

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

Curvaton Mechanism: Basic Idea

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.

HBBIn

flatio

n

Curvaton

Vector CurvatonVector 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:

wherewhere

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

● 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

ConclusionsConclusions● 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.