The Statistically Anisotropic Curvature Perturbation from Vector Fields Mindaugas Karčiauskas Dimopoulos, MK, JHEP 07 (2008) Dimopoulos, MK, Lyth, Rodriguez, JCAP 13 (2009) MK, Dimopoulos, Lyth, PRD 80 (2009) Dimopoulos, MK, Wagstaff, arXiv: Dimopoulos, MK, Wagstaff, Phys. Lett. B 683 (2010) Dimopoulos, MK, JHEP 07 (2008) Dimopoulos, MK, Lyth, Rodriguez, JCAP 13 (2009) MK, Dimopoulos, Lyth, PRD 80 (2009) Dimopoulos, MK, Wagstaff, arXiv: Dimopoulos, MK, Wagstaff, Phys. Lett. B 683 (2010)

Outline ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions; ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions;

Outline ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions; ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions;

Density perturbations ● The primordial curvature perturbation – a unique window to the physics of the early Universe; ● Origin of structure <= quantum fluctuations; ● Scalar fields - the simplest case; ● Why vector fields: ● Theoretical side: ● No fundamental scalar field has been discovered; ● The possible contribution from gauge fields is neglected; ● Observational side: ● Axis of Evil – the alignment of low multipoles of CMB; ● New observable – statistical anisotropy; ● The primordial curvature perturbation – a unique window to the physics of the early Universe; ● Origin of structure <= quantum fluctuations; ● Scalar fields - the simplest case; ● Why vector fields: ● Theoretical side: ● No fundamental scalar field has been discovered; ● The possible contribution from gauge fields is neglected; ● Observational side: ● Axis of Evil – the alignment of low multipoles of CMB; ● New observable – statistical anisotropy; Land & Magueijo (2005)

Outline ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions; ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions;

Scalar Field Perturbations ● (Quasi) de Sitter expansion ● The light scalar field with ● Equation of motion ● Subhorizon ● Superhorizon ● (Quasi) de Sitter expansion ● The light scalar field with ● Equation of motion ● Subhorizon ● Superhorizon Flat spacetime & no particles: Classical perturbations:

Generating the Curvature Perturbation ● The curvature perturbation: ● The formula ● The curvature perturbation: ● The formula

in Fourier Space ● The power spectrum ● The bispectrum ● The power spectrum ● The bispectrum Lyth & Rodriguez (2005)

Outline ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions; ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions;

Difficulties with Vector Fields 1.Excessive large scale anisotropy The energy-momentum tensor has anisotropic stress: 2.No particle production ● Massless U(1) vector fields are conformally invariant ● A known problem in the primordial magnetic fields literature 1.Excessive large scale anisotropy The energy-momentum tensor has anisotropic stress: 2.No particle production ● Massless U(1) vector fields are conformally invariant ● A known problem in the primordial magnetic fields literature

Avoiding excessive anisotropy ● Orthogonal triad of vector fields Armendariz-Picon (2004) ● Large number of identical vector fields Golovnev, Mukhanov, Vanchurin (2008) ● Modulation of scalar field dynamics Yokoyama, Soda (2008) ● Vector curvaton Dimopoulos (2006) ● Orthogonal triad of vector fields Armendariz-Picon (2004) ● Large number of identical vector fields Golovnev, Mukhanov, Vanchurin (2008) ● Modulation of scalar field dynamics Yokoyama, Soda (2008) ● Vector curvaton Dimopoulos (2006)

The Vector Curvaton Scenario Dimopoulos (2006) ● Massive vector field ● Energy momentum tensor ● Light vector field ● Massive vector field ● Energy momentum tensor ● Light vector field Like a pressureless isotropic matter! Like a pressureless isotropic matter! ● Heavy vector field Anisotropic EMT!

The Vector Curvaton Scenario I.Inflation ● Particle production ● Scale invariant spectrum II.Light Vector Field III.Heavy Vector Field Vector field oscillates. Behaves as preasureless isotropic matter. IV.Vector Field Decay. ● Generation of I.Inflation ● Particle production ● Scale invariant spectrum II.Light Vector Field III.Heavy Vector Field Vector field oscillates. Behaves as preasureless isotropic matter. IV.Vector Field Decay. ● Generation of Dimopoulos (2006)

Breaking Conformal Invariance ● Add a potential term, e.g. ● Modify kinetic term, e.g. ● Add a potential term, e.g. ● Modify kinetic term, e.g. E.g. electromagnetic field:

Physical Vector Field ● The comoving vector field ● The physical vector field: ● The comoving vector field ● The physical vector field:

Vector Field Perturbations ● Massive => 3 degrees of vector field freedom; ● No particles state for subhorizon modes ● Massive => 3 degrees of vector field freedom; ● No particles state for subhorizon modes Lorentz boost factor:

Vector Field Perturbations ● Classical perturbations for superhorizon modes ● The power spectra ● The anisotropy parameters of particle production : ● Classical perturbations for superhorizon modes ● The power spectra ● The anisotropy parameters of particle production : e.g.

Vector Field Perturbations Statistically isotropic Statistically anisotropic From observations, statistically anisotropic contribution <30%. and and/or

The Curvature Perturbation ● The curvature perturbation (δN formula) ● The anisotropic power spectrum: ● For vector field perturbations ● The non-Gaussianity ● The curvature perturbation (δN formula) ● The anisotropic power spectrum: ● For vector field perturbations ● The non-Gaussianity

Vector Field Projection

Observational Constraints ● The anisotropic power spectrum: ● Preferred direction close to the ecliptic pole ● The bound on of cosmological origin ● Detectable value by Planck ● The non-Gaussianity ● No observational constraints ● The anisotropic power spectrum: ● Preferred direction close to the ecliptic pole ● The bound on of cosmological origin ● Detectable value by Planck ● The non-Gaussianity ● No observational constraints Groeneboom et al (2009) Rudjord et al (2010) Pullen, Kamionkowski (2009)

Anisotropy Parameters ● Anisotropy in the particle production of the vector field: Depends on the conformal invariance braking mechanism ● Statistical anisotropy in the curvature perturbation : ● Anisotropy in the particle production of the vector field: Depends on the conformal invariance braking mechanism ● Statistical anisotropy in the curvature perturbation :

Outline ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions; ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions;

General Predictions 1.Anisotropic 2.The magnitude 3.Isotropic part: 1.Anisotropic 2.The magnitude 3.Isotropic part: 4.Same preferred direction 5.Anisotropic part: 6.In general not subdominant: 4.Same preferred direction 5.Anisotropic part: 6.In general not subdominant:

Outline ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions; ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions;

Two Models ● Non-minimal coupling ● Time varying kinetic function ● Non-minimal coupling ● Time varying kinetic function

Two Models ● Non-minimal coupling ● Time varying kinetic function ● Non-minimal coupling ● Time varying kinetic function Parity conserving Parity conserving

● Scale invariant power spectra => ● The vector field power spectra: ● The anisotropy in the power spectrum: ● Scale invariant power spectra => ● The vector field power spectra: ● The anisotropy in the power spectrum: Non-minimal Vector Curvaton =>

● Non-Gaussianity: ● The parameter space: ● Non-Gaussianity: ● The parameter space: Non-minimal Vector Curvaton 1.Anisotropic 2.Same preferred direction. 3.Isotropic parts are equal 4. 5.Configuration dependent modulation. 6.Modulation is not subdominant 1.Anisotropic 2.Same preferred direction. 3.Isotropic parts are equal 4. 5.Configuration dependent modulation. 6.Modulation is not subdominant

Stability of the Model ● Suspected instabilities for longitudinal mode: 1.Ghost; 2.Horizon crossing; 3.Zero effective mass; ● Suspected instabilities for longitudinal mode: 1.Ghost; 2.Horizon crossing; 3.Zero effective mass; Himmetoglu et al. (2009)

Stability of the Model ● Only for subhorizon modes ● Initially no particles & negligible coupling to other fields; ● Only for subhorizon modes ● Initially no particles & negligible coupling to other fields; Cline et al. (2004) ● Ghost

● Horizon crossing ● During inflation ● Exact solution ● Zero effective mass ● After inflation ● Linear perturbation theory breaks down at ● Horizon crossing ● During inflation ● Exact solution ● Zero effective mass ● After inflation ● Linear perturbation theory breaks down at Stability of the Model Independent constants:

● No issues of instabilities! ● At the end of inflation: and. ● Scale invariance => ● 2 nd case: ● Small coupling => can be a gauge field; ● Richest phenomenology; ● Might be an attractor solution; ● No issues of instabilities! ● At the end of inflation: and. ● Scale invariance => ● 2 nd case: ● Small coupling => can be a gauge field; ● Richest phenomenology; ● Might be an attractor solution; Varying Kinetic Function Dimopoulos & Wagstaff, in preparation

Field perturbations

Anisotropic particle production Anisotropic particle production Isotropic particle production Isotropic particle production Light vector field Light vector field Heavy vector field Heavy vector field At the end of inflation

● The anisotropy in the power spectrum: ● The non-Gaussianity: ● The parameter space & ● The anisotropy in the power spectrum: ● The non-Gaussianity: ● The parameter space & The Anisotropic Case, 1.Anisotropic 2.Same preferred direction. 3.Isotropic parts are equal 4. 5.Configuration dependent modulation. 6.Modulation is not subdominant 1.Anisotropic 2.Same preferred direction. 3.Isotropic parts are equal 4. 5.Configuration dependent modulation. 6.Modulation is not subdominant

● No scalar fields needed! ● Standard predictions of the curvaton scenario: ● The parameter space: ● No scalar fields needed! ● Standard predictions of the curvaton scenario: ● The parameter space: The Isotropic Case,

Outline ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions; ● Motivation; ● The curvature perturbation from scalar fields; ● The curvature perturbation form vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Conclusions;

● Vector fields can affect or even generate the curvature perturbation; ● If anisotropic particle production ( and/or ): ● If isotropic particle production => no need for scalar fields ● Two examples: ● Vector fields can affect or even generate the curvature perturbation; ● If anisotropic particle production ( and/or ): ● If isotropic particle production => no need for scalar fields ● Two examples: Conclusions 1.Anisotropic and 2.The same preferred direction in and 3.Isotropic parts 4. 5.Configuration dependent modulation: 6.In general modulation is not subdominant 1.Anisotropic and 2.The same preferred direction in and 3.Isotropic parts 4. 5.Configuration dependent modulation: 6.In general modulation is not subdominant

Dimopoulos, Karčiauskas, JHEP 07, 119 (2008) Dimopoulos, Karčiauskas, Lyth, Rodriguez, JCAP 13 (2009) Karčiauskas, Dimopoulos, Lyth, PRD 80 (2009) Dimopoulos, Karčiauskas, Wagstaff, arXiv: Dimopoulos, Karčiauskas, Wagstaff, arXiv: Dimopoulos, Karčiauskas, JHEP 07, 119 (2008) Dimopoulos, Karčiauskas, Lyth, Rodriguez, JCAP 13 (2009) Karčiauskas, Dimopoulos, Lyth, PRD 80 (2009) Dimopoulos, Karčiauskas, Wagstaff, arXiv: Dimopoulos, Karčiauskas, Wagstaff, arXiv: