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The Statistically Anisotropic Curvature Perturbation from Vector Fields
The Statistically Anisotropic Curvature Perturbation from Vector Fields
Mindaugas KarčiauskasMindaugas Karčiauskas
Dimopoulos, MK, JHEP 07 (2008)
Dimopoulos, MK, Lyth, Rodriguez, JCAP 13 (2009)
MK, Dimopoulos, Lyth, PRD 80 (2009)
Dimopoulos, MK, Wagstaff, arXiv:0907.1838
Dimopoulos, MK, Wagstaff, Phys. Lett. B 683 (2010)
Dimopoulos, MK, JHEP 07 (2008)
Dimopoulos, MK, Lyth, Rodriguez, JCAP 13 (2009)
MK, Dimopoulos, Lyth, PRD 80 (2009)
Dimopoulos, MK, Wagstaff, arXiv:0907.1838
Dimopoulos, MK, Wagstaff, Phys. Lett. B 683 (2010)
OutlineOutline● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
OutlineOutline● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
Density perturbationsDensity perturbations● The primordial curvature perturbation – a
unique window to the physics of 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 – the alignment of low multipoles of CMB;
● New observable – statistical anisotropy;
● The primordial curvature perturbation – a unique window to the physics of 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 – the alignment of low multipoles of CMB;
● New observable – statistical anisotropy;
Land & Magueijo (2005)Land & Magueijo (2005)
OutlineOutline● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
Scalar Field PerturbationsScalar Field
Perturbations● (Quasi) de Sitter expansion
● The light scalar field with
● Equation of motion
● Subhorizon
● Superhorizon
● (Quasi) de Sitter expansion
● The light scalar field with
● Equation of motion
● Subhorizon
● Superhorizon
Flat spacetime & no particles:Flat spacetime & no particles:
Classical perturbations:Classical perturbations:
Generating the Curvature Perturbation
Generating the Curvature Perturbation
● The curvature perturbation:
● The formula
● The curvature perturbation:
● The formula
in Fourier Space in Fourier Space
● The power spectrum
● The bispectrum
● The power spectrum
● The bispectrum
Lyth & Rodriguez (2005)Lyth & Rodriguez (2005)
OutlineOutline● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
Difficulties with Vector Fields
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 anisotropyAvoiding excessive anisotropy
● Orthogonal triad of vector fields Armendariz-Picon (2004)
● Large number of identical vector fields Golovnev, Mukhanov, Vanchurin (2008)
● Modulation of scalar field dynamics Yokoyama, Soda (2008)
● Vector curvaton Dimopoulos (2006)
● Orthogonal triad of vector fields Armendariz-Picon (2004)
● Large number of identical vector fields Golovnev, Mukhanov, Vanchurin (2008)
● Modulation of scalar field dynamics Yokoyama, Soda (2008)
● Vector curvaton Dimopoulos (2006)
The Vector Curvaton Scenario
The Vector Curvaton Scenario Dimopoulos (2006)Dimopoulos (2006)
● Massive vector field
● Energy momentum tensor
● Light vector field
● Massive vector field
● Energy momentum tensor
● Light vector field
Like a pressureless isotropic matter! Like a pressureless isotropic matter!
● Heavy vector field● Heavy vector field
Anisotropic EMT! Anisotropic EMT!
The Vector Curvaton Scenario
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.
● 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.
● Generation of
Dimopoulos (2006)Dimopoulos (2006)
Breaking Conformal Invariance
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:E.g. electromagnetic field:
Physical Vector FieldPhysical Vector Field
● The comoving vector field
● The physical vector field:
● The comoving vector field
● The physical vector field:
Vector Field PerturbationsVector Field
Perturbations● Massive => 3 degrees of
vector field freedom;
● No particles state for subhorizon modes
● Massive => 3 degrees of vector field freedom;
● No particles state for subhorizon modes
Lorentz boost factor:Lorentz boost factor:
Vector Field PerturbationsVector Field
Perturbations● Classical perturbations for superhorizon
modes
● The power spectra
● The anisotropy parameters of particle production :
● Classical perturbations for superhorizon modes
● The power spectra
● The anisotropy parameters of particle production :
e.g.e.g.
Vector Field PerturbationsVector Field Perturbations
Statistically isotropicStatistically isotropic
Statistically anisotropicStatistically anisotropic
From observations, statistically anisotropic contribution <30%.From observations, statistically anisotropic contribution <30%.andand
and/orand/or
The Curvature Perturbation
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
Vector Field ProjectionVector Field Projection
Observational Constraints
Observational Constraints
● The anisotropic power spectrum:
● Preferred direction close to the ecliptic pole
● The bound on of cosmological origin
● Detectable value by Planck
● The non-Gaussianity
● No observational constraints
● The anisotropic power spectrum:
● Preferred direction close to the ecliptic pole
● The bound on of cosmological origin
● Detectable value by Planck
● The non-Gaussianity
● No observational constraints
Groeneboom et al (2009)Groeneboom et al (2009)
Rudjord et al (2010)Rudjord et al (2010)
Pullen, Kamionkowski (2009)Pullen, Kamionkowski (2009)
Anisotropy ParametersAnisotropy Parameters
● 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:
OutlineOutline● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
General PredictionsGeneral Predictions
1. Anisotropic2. The magnitude3. Isotropic part:
1. Anisotropic2. The magnitude3. Isotropic part:
4. Same preferred direction5. Anisotropic part:
6. In general not subdominant:
4. Same preferred direction5. Anisotropic part:
6. In general not subdominant:
OutlineOutline● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
Two ModelsTwo Models
● Non-minimal coupling
● Time varying kinetic function
● Non-minimal coupling
● Time varying kinetic function
Two ModelsTwo 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-minimal Vector Curvaton
=>=>
● Non-Gaussianity:
● The parameter space:
● Non-Gaussianity:
● The parameter space:
Non-minimal Vector Curvaton
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 ModelStability of the Model
● Suspected instabilities for longitudinal mode:
1. Ghost;
2. Horizon crossing;
3. Zero effective mass;
● Suspected instabilities for longitudinal mode:
1. Ghost;
2. Horizon crossing;
3. Zero effective mass;
Himmetoglu et al. (2009)Himmetoglu et al. (2009)
Stability of the ModelStability of the Model
● Only for subhorizon modes
● Initially no particles & negligible coupling to other fields;
● Only for subhorizon modes
● Initially no particles & negligible coupling to other fields;
Cline et al. (2004)Cline et al. (2004)
● Ghost● Ghost
● Horizon crossing
● During inflation
● Exact solution
● Zero effective mass
● After inflation
● Linear perturbation theory breaks down at
● Horizon crossing
● During inflation
● Exact solution
● Zero effective mass
● After inflation
● Linear perturbation theory breaks down at
Stability of the ModelStability of the Model
Independent constants:Independent constants:
● No issues of instabilities!
● At the end of inflation: and .
● Scale invariance => 1. 2.
● 2nd case:
● Small coupling => can be a gauge field;
● Richest phenomenology;
● Might be an attractor solution;
● No issues of instabilities!
● At the end of inflation: and .
● Scale invariance => 1. 2.
● 2nd case:
● Small coupling => can be a gauge field;
● Richest phenomenology;
● Might be an attractor solution;
Varying Kinetic FunctionVarying Kinetic Function
Dimopoulos & Wagstaff, in preparationDimopoulos & Wagstaff, in preparation
Field perturbationsField perturbations
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 inflationAt 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,
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,The Isotropic Case,
OutlineOutline● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Motivation;
● The curvature perturbation from scalar fields;
● The curvature perturbation form vector fields;
● Predictions for vector curvaton scenario;
● Two models;
● Conclusions;
● Vector fields can affect or even generate the curvature perturbation;
● If anisotropic particle production ( and/or ):
● If isotropic particle production => no need for scalar fields
● Two examples:
● Vector fields can affect or even generate the curvature perturbation;
● If anisotropic particle production ( and/or ):
● If isotropic particle production => no need for scalar fields
● Two examples:
ConclusionsConclusions
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:0907.1838
Dimopoulos, Karčiauskas, Wagstaff,
arXiv:0909.0475
Dimopoulos, Karčiauskas, JHEP 07, 119 (2008)
Dimopoulos, Karčiauskas, Lyth, Rodriguez, JCAP 13
(2009)
Karčiauskas, Dimopoulos, Lyth, PRD 80 (2009)
Dimopoulos, Karčiauskas, Wagstaff,
arXiv:0907.1838
Dimopoulos, Karčiauskas, Wagstaff,
arXiv:0909.0475