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 Non-Gaussianity - discriminator between models of the early universe - information on mechanism of generation of primordial perturbations  sensitive to second-order evolution

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

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

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

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

7. Linear theory (coordinate approach) Perturbed Friedmann universe curvature perturbation x i = const. (t)(t)  ( t+dt ) dd proper time along x i = const.: curvature perturbation on  ( t ): 

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

9. Gauge invariant quantity  F : flat  =0,  =  F  C : uniform density  t F→C  =0,  =  C Curvature perturbation on uniform density hypersurfaces [Bardeen82; Bardeen/Steinhardt/Turner ‘83]

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

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

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

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

14. Gauge invariant quantities  F : flat  =0  =0 Curvature perturbation on uniform energy density [Bardeen82; Bardeen/Steinhardt/Turner ‘83]   : uniform density   : uniform field  =0 [Sasaki86; Mukhanov88] Curvature perturbation on uniform field (comoving)

15. Large scale behavior Relativistic Poisson equation  large scale equivalence Conserved quantities large scales  =0   : uniform density   : uniform field  =0

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

17. Single scalar field  = const arbitrary

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

19. Single field inflation log a log ℓ L = H -1 t = t out  a = const. t = t in inflation Generalized nonlinear Poisson equation

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

21.  = 0  = 0  = 0 Total momentum is the gradient of a scalar

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

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

24. Covariant approach for two fields Local redefinition: adiabatic and entropy covectors: 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:  1 st order  2 nd order  1 st order  2 nd 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 Entropy field perturbation Curvature perturbation: sourced by entropy field Linear equations

28. Second order expansion Entropy: Vector term Adiabatic:

29. Total momentum cannot be the gradient of a scalar  = 0  = 0  = 0 Vector term On large scales: Second order

30. Adiabatic and entropy large scale evolution Entropy field perturbation evolves independently Curvature perturbation sourced by 1 st and 2 nd order entropy field 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.  F : flat  =0  =0   : uniform density   : uniform field  =0 Mukhanov equation  quantization

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