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Nonlinear perturbations for cosmological scalar fields

Nonlinear perturbations for cosmological scalar fields. Filippo Vernizzi ICTP, Trieste. Finnish-Japanese Workshop on Particle Cosmology Helsinki, March 09, 2007. Beyond linear theory: motivations. Nonlinear aspects:. - effect of inhomogeneities on average expansion.

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Nonlinear perturbations for cosmological scalar fields

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  1. Nonlinear perturbations for cosmological scalar fields Filippo Vernizzi ICTP, Trieste Finnish-Japanese Workshop on Particle Cosmology Helsinki, March 09, 2007

  2. Beyond linear theory: motivations • Nonlinear aspects: - effect of inhomogeneities on average expansion - inhomogeneities on super-Hubble scales (stochastic inflation) - increase in precision of CMB data  sensitive to second-order evolution • Non-Gaussianity - information on mechanism of generation of primordial perturbations - discriminator between models of the early universe

  3. Conserved nonlinear quantities Second order perturbation • Malik/Wands ‘02 Long wavelength expansion (neglect spatial gradients) • Salopek/Bond ‘90 • Comer/Deruelle/Langlois/Parry ‘94 • Rigopoulos/Shellard ‘03 • Lyth/Wands ‘03 • Lyth/Malik/Sasaki ‘04 Covariant approach • Langlois/FV ‘05 • Enqvist/Hogdahl/Nurmi/FV ‘06

  4. Covariant approach [Ehlers, Hawking, Ellis, 60’-70’] • Work with geometrical quantities - perfect fluid world-line - volume expansion proper time: 4-velocity - integrated volume expansion - “time” derivative

  5. Covariant perturbations [Ellis/Bruni ‘89] • Perturbations should vanish in a homogeneous universe • Instead of  , use its spatial gradient! world-line proper time: • In a coordinate system: 4-velocity projector on  • Perturbations unambiguously defined

  6. Conservation equation [Langlois/FV, PRL ’05, PRD ‘05] • Covector: • “Time” derivative: Lie derivative along ub • Barotropic fluid

  7. proper time along xi = const.: Linear theory (coordinate approach) • Perturbed Friedmann universe curvature perturbation S(t+dt) dt S(t) xi = const. • curvature perturbation on S(t): 

  8. Relation with linear theory [Langlois/FV, PRL ’05, PRD ‘05] • Nonlinear equation “mimics” linear theory • Reduces to linear theory [Bardeen82; Bardeen/Steinhardt/Turner ‘83] [Wands/Malik/Lyth/Liddle ‘00]

  9. Gauge invariant quantity SC : uniform density d =0, =C dtF→C  =0, d =dF SF : flat Curvature perturbation on uniform density hypersurfaces [Bardeen82; Bardeen/Steinhardt/Turner ‘83]

  10. Higher order conserved quantity • Gauge-invariant conserved quantity at 2nd order [Malik/Wands ‘02] • Gauge-invariant conserved quantity at 3rd order [Enqvist/Hogdahl/Nurmi/FV ‘06] • and so on...

  11. logℓ  = const L=H-1 inflation log a t=tin t=tout Cosmological scalar fields • Scalar fields are very important in early universe models • Single-field - Perturbations generated during inflation and then constant on super-Hubble scales

  12. Cosmological scalar fields • Scalar fields are very important in early universe models • Single-field - Perturbations generated during inflation and then constant on super-Hubble scales logℓ d/dt S L=H-1 inflation log a t=tin t=tout • Multi-field - richer generation of fluctuations (adiabatic and entropy) - super-Hubble nonlinear evolution during inflation

  13. Nonlinear generalization Higher order generalization • Maldacena ‘02 • FV ’04 • Lyth/Rodriguez ’05 (non-Gaussianities from N-formalism) • FV/Wands ’05 (application of N) • Malik ’06 Long wavelength expansion (neglect spatial gradients) • Rigopoulos/Shellard/Van Tent ’05/06 Covariant approach • Langlois/FV ‘06

  14. Gauge invariant quantities df =0 S : uniform field d =0 S : uniform density  =0 SF : flat • Curvature perturbation on uniform field (comoving) [Sasaki86; Mukhanov88] • Curvature perturbation on uniform energy density [Bardeen82; Bardeen/Steinhardt/Turner ‘83]

  15. Large scale behavior df =0 S : uniform field d =0 S : uniform density • Relativistic Poisson equation  large scale equivalence large scales • Conserved quantities

  16. New approach [Langlois/FV, PRL ’05, PRD ‘05] • Integrated expansion  Replaces curvature perturbation • Non-perturbative generalization of • Non-perturbative generalization of

  17. f= const Single scalar field arbitrary

  18. f= const Single scalar field  Single-field: like a perfect fluid

  19. logℓ a = const. L=H-1 inflation log a t=tin t=tout Single field inflation • Generalized nonlinear Poisson equation

  20. Two-field linear perturbation [Gordon et al00; Nibbelink/van Tent01] • Global field rotation: adiabatic and entropy perturbations Adiabatic Entropy

  21. ds = 0 df = 0 d = 0 Total momentum is the gradient of a scalar

  22. Evolution of perturbations [Gordon/Wands/Bassett/Maartens00] • Curvature perturbation sourced by entropy field • Entropy field perturbation evolves independently

  23. Two scalar fields [Langlois/FV ‘06] f = const  = const arbitrary !

  24. Covariant approach for two fields • Local redefinition: adiabatic and entropycovectors: • Adiabatic and entropy angle: spacetime-dependent angle • Total momentum:  Total momentum may not be the gradient of a scalar

  25. (Nonlinear) homogeneous-like evolution equations • Rotation of Klein-Gordon equations:  1st order  2nd order  1st order  2nd order • Linear equations:

  26. (Nonlinear) linear-like evolution equations • From spatial gradient of Klein-Gordon equations: Adiabatic: Entropy:

  27. Adiabatic and entropy large scale evolution • Curvature perturbation: sourced by entropy field • Entropy field perturbation • Linear equations

  28. Second order expansion • Entropy: • Adiabatic: Vector term

  29. Vector term ds = 0 df = 0 d = 0 • On large scales: • Total momentum cannot be the gradient of a scalar • Second order

  30. Adiabatic and entropy large scale evolution • Curvature perturbation sourced by 1st and 2nd order entropy field • Entropy field perturbation evolves independently • Nonlocal term quickly decays in an expanding universe: (see ex. Lidsey/Seery/Sloth)

  31. Conclusions • New approach to cosmological perturbations • - nonlinear and covariant (geometrical formulation) • - exact at all scales, mimics the linear theory, easily expandable • Nonlinear cosmological scalar fields • - single field: perfect fluid • - two fields: entropy components evolves independently • - on large scales closed equations with curvature perturbations • - comoving hypersurface  uniform density hypersurface • - difference decays in expanding universe

  32. df =0 S : uniform field d =0 S : uniform density  =0 SF : flat Mukhanov equation  quantization

  33. Quantized variable [Pitrou/Uzan, ‘07] • Nonlinear analog of • At linear order converges to the “correct” variable to quantize

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