conserved non-linear quantities in cosmology
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Conserved non-linear quantities in cosmology. David Langlois (APC, Paris). Motivations. Linear theory of relativistic cosmological perturbations extremely useful Non-linear aspects are needed in some cases: non-gaussianities [ D. Lyth’s talk ] - PowerPoint PPT PresentationTRANSCRIPT
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Conserved non-linear quantities in cosmology
David Langlois
(APC, Paris)
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Motivations• Linear theory of relativistic cosmological perturbations
extremely useful
• Non-linear aspects are needed in some cases:– non-gaussianities [ D. Lyth’s talk ]– Universe on very large scales (beyond the Hubble scales)– small scales
• Conservation laws – solve part of the equations of motion– useful to relate early universe
and ``late cosmology’’
Early Universegeometry
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Motivations• Linear theory of relativistic cosmological perturbations
extremely useful
• Non-linear aspects are needed in some cases:– non-gaussianities [ D. Lyth’s talk ]– Universe on very large scales (beyond the Hubble scales)– small scales
• Conservation laws – solve part of the equations of motion– useful to relate early universe
and ``late cosmology’’
Early Universegeometry
![Page 4: Conserved non-linear quantities in cosmology](https://reader030.vdocuments.net/reader030/viewer/2022013101/56814bd9550346895db8b340/html5/thumbnails/4.jpg)
Motivations• Linear theory of relativistic cosmological perturbations
extremely useful
• Non-linear aspects are needed in some cases:– non-gaussianities [ D. Lyth’s talk ]– Universe on very large scales (beyond the Hubble scales)– small scales
• Conservation laws – solve part of the equations of motion– useful to relate early universe
and ``late cosmology’’
Early Universegeometry
![Page 5: Conserved non-linear quantities in cosmology](https://reader030.vdocuments.net/reader030/viewer/2022013101/56814bd9550346895db8b340/html5/thumbnails/5.jpg)
Linear theory
• Perturbed metric (with only scalar perturbations)
• related to the intrinsic curvature of constant time spatial hypersurfaces
• Change of coordinates, e.g.
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Linear theory
• Perturbed metric (with only scalar perturbations)
• related to the intrinsic curvature of constant time spatial hypersurfaces
• Change of coordinates, e.g.
![Page 7: Conserved non-linear quantities in cosmology](https://reader030.vdocuments.net/reader030/viewer/2022013101/56814bd9550346895db8b340/html5/thumbnails/7.jpg)
Linear theory
• Perturbed metric (with only scalar perturbations)
• related to the intrinsic curvature of constant time spatial hypersurfaces
• Change of coordinates, e.g.
![Page 8: Conserved non-linear quantities in cosmology](https://reader030.vdocuments.net/reader030/viewer/2022013101/56814bd9550346895db8b340/html5/thumbnails/8.jpg)
Linear theory
• Curvature perturbation on uniform energy density hypersurfaces
• The time component of yields
For adiabatic perturbations, conserved on large scales
gauge-invariant
[Wands et al (2000)]
[Bardeen et al (1983)]
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Linear theory
• Curvature perturbation on uniform energy density hypersurfaces
• The time component of yields
For adiabatic perturbations, conserved on large scales
gauge-invariant
[Wands et al (2000)]
[Bardeen et al (1983)]
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Covariant formalism
• How to define unambiguously
• One needs a map
If is such that ,
then
• Idea: instead of , use its spatial gradient
[ Ellis & Bruni (1989) ]
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Covariant formalism
• How to define unambiguously
• One needs a map
If is such that ,
then
• Idea: instead of , use its spatial gradient
[ Ellis & Bruni (1989) ]
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Covariant formalism
• How to define unambiguously
• One needs a map
If is such that ,
then
• Idea: instead of , use its spatial gradient
[ Ellis & Bruni (1989) ]
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Covariant formalism
• How to define unambiguously
• One needs a map
If is such that ,
then
• Idea: instead of , use its spatial gradient
[ Ellis & Bruni (1989) ]
![Page 14: Conserved non-linear quantities in cosmology](https://reader030.vdocuments.net/reader030/viewer/2022013101/56814bd9550346895db8b340/html5/thumbnails/14.jpg)
Covariant formalism
• How to define unambiguously
• One needs a map
If is such that ,
then
• Idea: instead of , use its spatial gradient
[ Ellis & Bruni (1989) ]
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• Perfect fluid:
• Spatial projection:
• Expansion:
• Integrated expansion:
• Spatially projected gradients:
Local scale factor
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• Perfect fluid:
• Spatial projection:
• Expansion:
• Integrated expansion:
• Spatially projected gradients:
Local scale factor
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• Perfect fluid:
• Spatial projection:
• Expansion:
• Integrated expansion:
• Spatially projected gradients:
Local scale factor
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• Perfect fluid:
• Spatial projection:
• Expansion:
• Integrated expansion:
• Spatially projected gradients:
Local scale factor
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• Perfect fluid:
• Spatial projection:
• Expansion:
• Integrated expansion:
• Spatially projected gradients:
Local scale factor
![Page 20: Conserved non-linear quantities in cosmology](https://reader030.vdocuments.net/reader030/viewer/2022013101/56814bd9550346895db8b340/html5/thumbnails/20.jpg)
• Perfect fluid:
• Spatial projection:
• Expansion:
• Integrated expansion:
• Spatially projected gradients:
Local scale factor
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New approach
• Define
• Projection of along yields
• Spatial gradient
[ DL & Vernizzi, PRL ’05; PRD ’05 ]
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New approach
• Define
• Projection of along yields
• Spatial gradient
[ DL & Vernizzi, PRL ’05; PRD ’05 ]
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New approach
• Define
• Projection of along yields
• Spatial gradient
[ DL & Vernizzi, PRL ’05; PRD ’05 ]
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New approach
• Define
• Projection of along yields
• Spatial gradient
[ DL & Vernizzi, PRL ’05; PRD ’05 ]
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New approach
• Define
• Projection of along yields
• Spatial gradient
[ DL & Vernizzi, PRL ’05; PRD ’05 ]
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• One finally gets
• This is an exact equation, fully non-linear and valid at all scales !
• It “mimics” the linear equation
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• One finally gets
• This is an exact equation, fully non-linear and valid at all scales !
• It “mimics” the linear equation
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• One finally gets
• This is an exact equation, fully non-linear and valid at all scales !
• It “mimics” the linear equation
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• One finally gets
• This is an exact equation, fully non-linear and valid at all scales !
• It “mimics” the linear equation
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Comparison with the coordinate based approach
• Choose a coordinate system
• Expand quantities:
• First order
Usual linear equation !
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Comparison with the coordinate based approach
• Choose a coordinate system
• Expand quantities:
• First order
Usual linear equation !
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Comparison with the coordinate based approach
• Choose a coordinate system
• Expand quantities:
• First order
Usual linear equation !
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Comparison with the coordinate based approach
• Choose a coordinate system
• Expand quantities:
• First order
Usual linear equation !
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Comparison with the coordinate based approach
• Choose a coordinate system
• Expand quantities:
• First order
Usual linear equation !
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• Second order perturbations
After some manipulations, one finds
in agreement with previous results [ Malik & Wands ’04]
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• Second order perturbations
After some manipulations, one finds
in agreement with previous results [ Malik & Wands ’04]
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Conclusions
• New approach to study cosmological perturbations– Non linear – Purely geometric formulation (extension of the
covariant formalism)– ``Mimics’’ the linear theory equations – Get easily the second order results– Exact equations: no approximation
• Can be extended to other quantities
Cf also talk by F. Vernizzi
[ cf also Lyth, Malik & Sasaki ’05 ]
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• On super-Hubble scales: juxtaposed FLRW regions…
• Long wavelength approximation
``Separate universe’’ approach
Foliationwith unit normal vector
On super-Hubble scales,
[Sasaki & Stewart (1996)]
Salopek & Stewart; Comer, Deruelle, DL & Parry (1994)][Salopek & Bond (1990);
[Rigopoulos & Shellard (2003)]
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• Perfect fluid:
with
• Spatial projection:
• Decomposition:
• Integrated expansion:
• Spatially projected gradients:
Expansion: shear vorticity
Local scale factor