Konstantinos Dimopoulos

1. Vector Curvaton without istabilities Konstantinos Dimopoulos Lancaster University Work done with M. Karciauskas and J.M. Wagstaff 0907.1838, 0909.0475

2. 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: Baryogenesis , 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

3. 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”):

4. Massive vector field: Abelian vector field: Equations of motion: Flat FRW metric: Inflation homogenises the vector field: & Klein-Gordon Massive Abelian Vector Field Renormalisable • To retain isotropy the vector field must not drive inflation Vector Inflation [Golovnev et al. (2008)] uses 100s of vector fields

5. & • Eq. of motion: harmonic oscillations Vector Curvaton • Safe domination of Universe required [KD, PRD 74 (2006) 083502] Pressureless and Isotropic • Vector field domination can occur without introducing significant anisotropy. The curvature perturbation is imposed at domination

6. 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: Polarization vectors: 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)

7. 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

8. (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 (2008) • 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

9. 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) -4 • 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: 6 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

10. 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 0907.1838 0909.0475