Konstantinos Dimopoulos

Lancaster University

Scalar vs Vector FieldsScalar vs Vector Fields Scalar fields employed to address many open issues:Scalar fields employed to address many open issues: inflationary inflationary

paradigm, dark energy (quintessence) baryogenesis (Affleck-Dine)paradigm, dark energy (quintessence) baryogenesis (Affleck-Dine) Scalar fields are ubiquitous in theories beyond the standard model Scalar fields are ubiquitous in theories beyond the standard model

such as Supersymmetry (scalar partners) or string theory (moduli)such as Supersymmetry (scalar partners) or string theory (moduli) However,However, no scalar field has ever been observedno scalar field has ever been observed Designing models using unobserved scalar fields undermines their Designing models using unobserved scalar fields undermines their

predictability and falsifiabilitypredictability and falsifiability, despite the recent precision data, despite the recent precision data The latest theoretical developments (string landscape) offer The latest theoretical developments (string landscape) offer too too

much freedommuch freedom for model-building for model-building Can we do Cosmology without scalar fields?Can we do Cosmology without scalar fields? Some topics are OK:Some topics are OK: BaryogenesisBaryogenesis , Dark Matter , Dark Matter , Dark Energy (, Dark Energy (ΛΛCDM)CDM) Inflationary expansion without scalar fields is also possible:Inflationary expansion without scalar fields is also possible:

e.g. inflation due to geometry: gravity ( - inflation)e.g. inflation due to geometry: gravity ( - inflation) However, to date,However, to date, no mechanism for the generation of the no mechanism for the generation of the

curvature/density perturbation without a scalar field existscurvature/density perturbation without a scalar field exists

Why not Vector Fields?Why not Vector Fields?

Basic Problem:Basic Problem: the generatation the generatation of a large-scale anisotropy is in of a large-scale anisotropy is in conflict with CMB observationsconflict with CMB observations

However, However, An oscillating massive An oscillating massive vector field can avoid excessive large-vector field can avoid excessive large-scale anisotropyscale anisotropy

Also, some weak large-scale Also, some weak large-scale anisotropy might be present anisotropy might be present in the CMB (“Axis of Evil”):in the CMB (“Axis of Evil”):

Inflation homogenizes Vector FieldsInflation homogenizes Vector Fields To affect / generate the curvature To affect / generate the curvature

perturbation a Vector Field needs perturbation a Vector Field needs to (nearly) dominate the Universeto (nearly) dominate the Universe

Homogeneous Vector Field Homogeneous Vector Field = in general anisotropic= in general anisotropic

l=5 in galactic coordinates

l=5 in preferred frame

Massive Abelian Vector FieldMassive Abelian Vector Field

Massive vector field:

Abelian vector field:

Equations of motion:

Flat FRW metric:

Inflation homogenises the vector field:

& Klein-Gordon

To retain isotropy the vector field must not drive inflationTo retain isotropy the vector field must not drive inflation

Vector Inflation [Golovnev et al. (2008)] uses 100s of vector fields

Vector CurvatonVector Curvaton

Pressureless and Isotropic

Vector field can be curvaton if safe domination of UniverseVector field can be curvaton if safe domination of Universe is possibleis possible

Vector field domination can occur without introducing significant anisotropy. Vector field domination can occur without introducing significant anisotropy. The curvature perturbation is imposed at dominationThe curvature perturbation is imposed at domination

&

Eq. of motion:Eq. of motion:

harmonic oscillationsharmonic oscillations

Particle Production of Vector FieldsParticle Production of Vector Fields

Conformal Invariance: vector field does not couple to metric (virtual particles not pulled outside Horizon during inflation)

Breakdown of conformality of massless vector field is necessaryBreakdown of conformality of massless vector field is necessary

Mass term not enough no scale invariance

Find eq. of motion for vector field perturbations:Find eq. of motion for vector field perturbations:

Promote to operator:

Polarization vectors:

Canonical quantization:

Fourier transform:

(e.g. , , or ) Typically, introduce Xterm :Typically, introduce Xterm :

Particle Production of Vector FieldsParticle Production of Vector Fields

Cases A&B: vector curvaton = subdominant: statistical anisotropy onlyCases A&B: vector curvaton = subdominant: statistical anisotropy only

Solve with vacuum boundary conditions:Solve with vacuum boundary conditions:

&

Obtain power spectra:Obtain power spectra: expansion = isotropic

Vector Curvaton = solely responsible for only in Case CVector Curvaton = solely responsible for only in Case C

Case C:Case C: isotropic particle production

Case B:Case B: parity conserving (most generic)

Case A:Case A: parity violating

Observations: weak bound

Statistical Anisotropy:Statistical Anisotropy: anisotropic patterns in CMB anisotropic patterns in CMB

Lorentz boost factor: from frame with

Non-minimally coupled Vector CurvatonNon-minimally coupled Vector Curvaton

Perturb &Fourier XformEq. of motion:

Transverse component:Transverse component: (Parity conserving)

Scale invariance if: &

&

Non-minimally coupled Vector CurvatonNon-minimally coupled Vector Curvaton Longitudinal component:Longitudinal component:

The vector curvaton can be the The vector curvaton can be the cause of statistical anisotropycause of statistical anisotropy

Case B: The vector curvaton Case B: The vector curvaton contribution to must be contribution to must be subdominant subdominant

saturates observational bound

Statistical Anisotropy and non-GaussianityStatistical Anisotropy and non-Gaussianity

Observations: Observations:

The Planck satellite will increase precision to:

Non Gaussianity in vector curvaton scenario:Non Gaussianity in vector curvaton scenario:

Non-Gaussianity = correlated with statistical anisotropy:Non-Gaussianity = correlated with statistical anisotropy: Smoking gun

Non-minimally coupled case: Non-minimally coupled case:

Non-Gaussianity in scalar curvaton scenario: Non-Gaussianity in scalar curvaton scenario:

: projection of unit vector onto the - plane

& &

Reduction to scalar curvaton case if: Reduction to scalar curvaton case if: &

ConclusionsConclusions A vector field can contribute to the curvature perturbationA vector field can contribute to the curvature perturbation

In this case, the vector field undergoes rapid harmonic oscillations In this case, the vector field undergoes rapid harmonic oscillations during which it acts as a during which it acts as a pressureless isotropic fluidpressureless isotropic fluid

Hence, when the oscillating vector field dominates, it introduces Hence, when the oscillating vector field dominates, it introduces negligible anisotropy (“Axis of Evil”?) negligible anisotropy (“Axis of Evil”?)

The challenge is to obtain candidates in theories beyond the The challenge is to obtain candidates in theories beyond the standard model, which can play the role of the vector curvatonstandard model, which can play the role of the vector curvaton

The vector field can act as a curvaton if, after inflation, its mass becomes:The vector field can act as a curvaton if, after inflation, its mass becomes: ( zero VEV: ( zero VEV: vacuum = Lorentz invariant vacuum = Lorentz invariant ))

Physical Review D Physical Review D 7474 (2006) 083502 : (2006) 083502 : hep-ph/0607229hep-ph/0607229

Physical Review D Physical Review D 7676 (2007) 063506 : 0705.3334 [ (2007) 063506 : 0705.3334 [hep-ph]hep-ph]Journal of High Energy Physics 07 (2008) 119 : 0803.3041 [Journal of High Energy Physics 07 (2008) 119 : 0803.3041 [hep-th]hep-th]

If particle production is If particle production is isotropicisotropic then the vector curvaton can then the vector curvaton can alone generate the curvature perturbation in the Universealone generate the curvature perturbation in the Universe

If particle production is If particle production is anisotropicanisotropic then the vector curvaton can then the vector curvaton can give rise to statistical anisotropy, potentially observable by Planckgive rise to statistical anisotropy, potentially observable by Planck

Correlation of statistical anisotropy and non-Gaussianity in the Correlation of statistical anisotropy and non-Gaussianity in the CMB is the smoking gun for the vector curvaton scenarioCMB is the smoking gun for the vector curvaton scenario

arXiv:0806.4680 [hep-ph]arXiv:0806.4680 [hep-ph]

arXiv:0809.1055 [astro-ph]arXiv:0809.1055 [astro-ph]

arXiv:0812.0264 [astro-ph]arXiv:0812.0264 [astro-ph]