The Statistically Anisotropic Curvature Perturbation from Vector Fields Mindaugas Karčiauskas 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:

Outline ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary; ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary;

Density perturbations ● Primordial curvature perturbation – a unique window to 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: alignment of spherical harmonics of CMB; ● Large cold spot, radio galaxy void; ● Primordial curvature perturbation – a unique window to 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: alignment of spherical harmonics of CMB; ● Large cold spot, radio galaxy void; Land & Magueijo (2005)

Outline ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary; ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary;

Scalar Field Perturbations ● Assume de Sitter expansion; ● The light scalar field with modes ● Fourier modes ● Equation of motion ● Assume de Sitter expansion; ● The light scalar field with modes ● Fourier modes ● Equation of motion

Scalar Field Perturbations ● The power spectrum 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 & & WMAP

Outline ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary; ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary;

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 1.Orthogonal triad of vector fields Armendariz-Picon (2004) 2.Large number of identical vector fields Golovnev, Mukhanov, Vanchurin (2008) 3.Modulation of scalar field dynamics Yokoyama, Soda (2008) 4.Vector curvaton Dimopoulos (2006) 1.Orthogonal triad of vector fields Armendariz-Picon (2004) 2.Large number of identical vector fields Golovnev, Mukhanov, Vanchurin (2008) 3.Modulation of scalar field dynamics Yokoyama, Soda (2008) 4.Vector curvaton Dimopoulos (2006)

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. ● Onset of the Hot Big Bang ● 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. ● Onset of the Hot Big Bang ● 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:

Outline ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary; ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary;

Vector Field Perturbations ● Massive => 3 degrees of vector field freedom; ● The power spectra ● The anisotropy parameters of particle production : ● Massive => 3 degrees of vector field freedom; ● The power spectra ● The anisotropy parameters of particle production :

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 Groeneboom et al. (2009) No constraints yet! Rudjord et al. (2009) No constraints yet! Rudjord et al. (2009)

Vector Field Projection

● 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 : Anisotropy Parameters

Outline ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary; ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary;

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; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary; ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary;

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: 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 ● Two suspected instabilities for longitudinal mode: 1. Ghost; 2. Horizon crossing; 3. Zero effective mass; 1.Ghost. ● Only for subhorizon modes: ● Initially no particles & negligible coupling to other fields; 2.Horizon crossing. ● Exact solution: ● Two suspected instabilities for longitudinal mode: 1. Ghost; 2. Horizon crossing; 3. Zero effective mass; 1.Ghost. ● Only for subhorizon modes: ● Initially no particles & negligible coupling to other fields; 2.Horizon crossing. ● Exact solution: 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; ● No issues of instabilities! ● At the end of inflation: and. ● Scale invariance => ● 2 nd case: ● Small coupling => can be a gauge field; ● Richest phenomenology; Varying Kinetic Function

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; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary; ● Motivation; ● Curvature perturbation from scalar fields; ● Difficulties with vector fields; ● Curvature perturbation from vector fields; ● Predictions for vector curvaton scenario; ● Two models; ● Summary;

● Vector fields can affect or generate the curvature perturbation; ● If anisotropic particle production ( and/or ): ● New observable => smoking gun for vector field contribution; ● If isotropic particle production => no need for scalar fields ● Two examples: ● Vector fields can affect or generate the curvature perturbation; ● If anisotropic particle production ( and/or ): ● New observable => smoking gun for vector field contribution; ● 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: