Vector Curvaton
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Vector Curvaton. without istabilities. Konstantinos Dimopoulos. Lancaster University. Work done with M. Karciauskas and J.M. Wagstaff. 0907.1838, 0909.0475. e.g. inflation due to geometry: gravity ( - inflation). Scalar vs Vector Fields.

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Vector Curvaton

without istabilities

Konstantinos Dimopoulos

Lancaster University

Work done with M. Karciauskas and J.M. Wagstaff

0907.1838, 0909.0475

Scalar vs vector fields

e.g. inflation due to geometry: gravity ( - inflation)

Scalar vs Vector Fields

  • Scalar fields employed to address many open issues: inflationary paradigm, dark energy (quintessence) baryogenesis (Affleck-Dine)

  • Scalar fields are ubiquitous in theories beyond the standard model such as Supersymmetry (scalar partners) or string theory (moduli)

  • However,no fundamental scalar field has ever been observed

  • Designing models using unobserved scalar fields undermines their predictability and falsifiability, despite the recent precision data

  • The latest theoretical developments (string landscape) offer too much freedom for model-building

  • Can we do Cosmology without scalar fields?

  • Some topics are OK:


, Dark Matter

, Dark Energy (ΛCDM)

  • Inflation without fundamental scalar fields is also possible:

  • However, to date,no mechanism for the generation of the curvature/density perturbation without a scalar field exists

Why not vector fields

l=5 in galactic coordinates - inflation)

l=5 in preferred frame

Why not Vector Fields?

  • Inflation homogenizes Vector Fields

  • To affect / generate the curvature perturbation a Vector Field needs to (nearly) dominate the Universe

  • Homogeneous Vector Field = in general anisotropic

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

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

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

Massive abelian vector field

Massive vector field: - inflation)

Abelian vector field:

Equations of motion:

Flat FRW metric:

Inflation homogenises the vector field:

& Klein-Gordon

Massive Abelian Vector Field


  • To retain isotropy the vector field must not drive inflation

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

Vector curvaton

& - inflation)

  • Eq. of motion:

harmonic oscillations

Vector Curvaton

  • Safe domination of Universe required

[KD, PRD 74 (2006) 083502]


and Isotropic

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

Particle production of vector fields

Mass term not enough no scale invariance - inflation)

(e.g. , , or )

  • Typically, introduce Xterm :

  • Find eq. of motion for vector field perturbations:

Fourier transform:

Promote to operator:



Canonical quantization:

Particle Production of Vector Fields

  • Breakdown of conformality of massless vector field is necessary

Conformal Invariance: vector field does not couple to metric

(virtual particles not pulled outside Horizon during inflation)

Particle production of vector fields1


Lorentz boost factor:

from frame with

  • Obtain power spectra:

expansion = isotropic

  • Case A:

parity violating

  • Case B:

parity conserving (most generic)

isotropic particle production

  • Case C:

  • Statistical Anisotropy: anisotropic patterns in CMB

Observations: weak bound

  • Vector Curvaton = solely responsible for only in Case C

Particle Production of Vector Fields

Groeneboom and Eriksen (2009)

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

Non minimal coupling to gravity

(Parity conserving) - inflation)

  • Case B: Vector curvaton contribution to must be subdominant

  • Possible instabilities:

Himmetoglu, Contaldi and Peloso (2009)

Exact solution found with no pathologies

KD, Karciauskas, Lyth and Rodriguez (2009)

Scale invariance if:


Non-minimal coupling to Gravity

KD & Karciauskas


  • The vector curvaton can cause statistical anisotropy only

  • Longitudinal component unstable at horizon crossng

  • Longitudinal component = ghost when subhorizon

Interactions with other fields = negligible

Subhorizon for limited time (from Planck length to Horizon)

Negative energy subdominant to inflation energy

Vector curvaton without instabilities

Motivates model even if vector field is not gauge boson - inflation)

at Horizon exit

-1 ± 3

  • If gauge boson then (weakly coupled during inflation)

  • In supergravity = gauge kinetic function (holomorphic)


  • Kahler corrections to the scalar potential result in masses:

  • Fast-rolling scalar fields cause significant variation to

It is natural to expect during inflation

  • Paticle production anisotropic (Case B) if:


1< < 10

  • Vector Curvaton can be naturally realised in SUGRA, without

  • Paticle production isotropic (Case C) if:


Vector Curvaton without instabilities

KD (2007)

  • Maxwell kinetic term does not suffer from instabilities (ghost-free)

Scale invariance:

No need for fundamental scalar field



  • Vetor Curvaton: the only known mechanism which can form the curvature perturbation without fundamntal scalar fields

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

  • Hence, the vector field introduces negligible anisotropy at domination

  • If particle production is isotropic then the vector curvaton can alone generate the curvature perturbation in the Universe

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

  • A Massive Abelian vector curvaton with a Maxwell kinetic term & varying kinetic function and mass can generate isotropic (anisotropic) perturbations if heavy (light) by end of inflation without giving rise to any instabilities (e.g. ghosts)

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