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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

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:

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
  • 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

(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
Solve with vacuum boundary conditions:


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)
  • 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

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

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