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Conserved non-linear quantities in cosmology

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 ]

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Conserved non-linear quantities in cosmology

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  1. Conserved non-linear quantities in cosmology David Langlois (APC, Paris)

  2. 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 Universe geometry

  3. 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 Universe geometry

  4. 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 Universe geometry

  5. Linear theory • Perturbed metric (with only scalar perturbations) • related to the intrinsic curvature of constant time spatial hypersurfaces • Change of coordinates, e.g.

  6. Linear theory • Perturbed metric (with only scalar perturbations) • related to the intrinsic curvature of constant time spatial hypersurfaces • Change of coordinates, e.g.

  7. Linear theory • Perturbed metric (with only scalar perturbations) • related to the intrinsic curvature of constant time spatial hypersurfaces • Change of coordinates, e.g.

  8. Linear theory • Curvature perturbation on uniform energy density hypersurfaces • The time component of yields For adiabatic perturbations, conserved on large scales [Bardeen et al (1983)] gauge-invariant [Wands et al (2000)]

  9. Linear theory • Curvature perturbation on uniform energy density hypersurfaces • The time component of yields For adiabatic perturbations, conserved on large scales [Bardeen et al (1983)] gauge-invariant [Wands et al (2000)]

  10. Covariant formalism [ Ellis & Bruni (1989) ] • How to define unambiguously • One needs a map If is such that , then • Idea: instead of , use its spatial gradient

  11. Covariant formalism [ Ellis & Bruni (1989) ] • How to define unambiguously • One needs a map If is such that , then • Idea: instead of , use its spatial gradient

  12. Covariant formalism [ Ellis & Bruni (1989) ] • How to define unambiguously • One needs a map If is such that , then • Idea: instead of , use its spatial gradient

  13. Covariant formalism [ Ellis & Bruni (1989) ] • How to define unambiguously • One needs a map If is such that , then • Idea: instead of , use its spatial gradient

  14. Covariant formalism [ Ellis & Bruni (1989) ] • How to define unambiguously • One needs a map If is such that , then • Idea: instead of , use its spatial gradient

  15. Perfect fluid: • Spatial projection: • Expansion: • Integrated expansion: • Spatially projected gradients: Local scale factor

  16. Perfect fluid: • Spatial projection: • Expansion: • Integrated expansion: • Spatially projected gradients: Local scale factor

  17. Perfect fluid: • Spatial projection: • Expansion: • Integrated expansion: • Spatially projected gradients: Local scale factor

  18. Perfect fluid: • Spatial projection: • Expansion: • Integrated expansion: • Spatially projected gradients: Local scale factor

  19. Perfect fluid: • Spatial projection: • Expansion: • Integrated expansion: • Spatially projected gradients: Local scale factor

  20. Perfect fluid: • Spatial projection: • Expansion: • Integrated expansion: • Spatially projected gradients: Local scale factor

  21. New approach [ DL & Vernizzi, PRL ’05; PRD ’05 ] • Define • Projection of along yields • Spatial gradient

  22. New approach [ DL & Vernizzi, PRL ’05; PRD ’05 ] • Define • Projection of along yields • Spatial gradient

  23. New approach [ DL & Vernizzi, PRL ’05; PRD ’05 ] • Define • Projection of along yields • Spatial gradient

  24. New approach [ DL & Vernizzi, PRL ’05; PRD ’05 ] • Define • Projection of along yields • Spatial gradient

  25. New approach [ DL & Vernizzi, PRL ’05; PRD ’05 ] • Define • Projection of along yields • Spatial gradient

  26. One finally gets • This is an exact equation, fully non-linear and valid at all scales ! • It “mimics” the linear equation

  27. One finally gets • This is an exact equation, fully non-linear and valid at all scales ! • It “mimics” the linear equation

  28. One finally gets • This is an exact equation, fully non-linear and valid at all scales ! • It “mimics” the linear equation

  29. One finally gets • This is an exact equation, fully non-linear and valid at all scales ! • It “mimics” the linear equation

  30. Comparison with the coordinate based approach • Choose a coordinate system • Expand quantities: • First order Usual linear equation !

  31. Comparison with the coordinate based approach • Choose a coordinate system • Expand quantities: • First order Usual linear equation !

  32. Comparison with the coordinate based approach • Choose a coordinate system • Expand quantities: • First order Usual linear equation !

  33. Comparison with the coordinate based approach • Choose a coordinate system • Expand quantities: • First order Usual linear equation !

  34. Comparison with the coordinate based approach • Choose a coordinate system • Expand quantities: • First order Usual linear equation !

  35. Second order perturbations After some manipulations, one finds in agreement with previous results [ Malik & Wands ’04]

  36. Second order perturbations After some manipulations, one finds in agreement with previous results [ Malik & Wands ’04]

  37. 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 ]

  38. Foliation with unit normal vector ``Separate universe’’ approach • On super-Hubble scales: juxtaposed FLRW regions… • Long wavelength approximation On super-Hubble scales, [Sasaki & Stewart (1996)] [Salopek & Bond (1990); Salopek & Stewart; Comer, Deruelle, DL & Parry (1994)] [Rigopoulos & Shellard (2003)]

  39. Perfect fluid: with • Spatial projection: • Decomposition: • Integrated expansion: • Spatially projected gradients: shear vorticity Expansion: Local scale factor

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