Nonlinear cosmological perturbations

1. Nonlinear cosmological perturbations Filippo Vernizzi ICTP, Trieste Astroparticles and Cosmology Workshop GGI, Florence, October 24, 2006

2. References • Second-order perturbations • Phys. Rev. D71, 061301 (2005), astro-ph/0411463 • Nonlinear perturbations • with David Langlois: • Phys. Rev. Lett. 95, 091303 (2005), astro-ph/0503416 • Phys. Rev. D72, 103501 (2005), astro-ph/0509078 • JCAP 0602, 014 (2006), astro-ph/0601271 • astro-ph/0610064 • with Kari Enqvist, Janne Högdahl and Sami Nurmi: • in preparation

3. Beyond linear theory: Motivations • Linear theory extremely useful - linearized Einstein’s eqs around an FLRW universe excellent approximation - tests of inflation based on linear theory • Nonlinear aspects: - inhomogeneities on scales larger than - backreaction of nonlinear perturbations - increase in precision of CMB data

4. super-Hubble sub-Hubble comoving wavelength conformal time inflation radiation dominated era Primordial non-Gaussianities - information on mechanism of generation of primordial perturbations - discriminator between models of the early universe • single-field inflation • multi-field inflation • non-minimal actions • curvaton (See the talks by Bartolo, Creminelli, Liguori, Lyth, Rigopoulos)

5. Linear theory (coordinate approach) • Perturbed FLRW universe curvature perturbation • Perturbed fluid • Linear theory: gauge transformation

6. =0 =0 time space Conserved linear perturbation • Gauge-invariant definition: curvature perturbation on uniform density hypersurfaces [Bardeen82; Bardeen/Steinhardt/Turner83] • For a perfect fluid, from the continuity equation [Wands/Malik/Lyth/Liddle00] Non-adiabatic pressure perturbation:  For adiabatic perturbations , is conserved on large scales

7. Nonlinear generalization Second order generalization • Malik/Wands02 Long wavelength approximation (neglect spatial gradients) • Salopek/Bond90 • Comer/Deruelle/Langlois/Parry94 • Rigopoulos/Shellard03 • Lyth/Wands03 • Lyth/Malik/Sasaki04

8. observer 4-velocity: proper time: world-line Covariant approach [Ehlers, Hawking, Ellis, 60’-70’] Work with geometrical quantities: perfect fluid Definitions: Expansion (3 x local Hubble parameter) Integrated expansion (local number of e-folds, ) • Perturbations: spatially projected gradients [Ellis/Bruni89]: spatial projection • In a coordinate system:

9. Nonlinear conserved quantity [Langlois/FV, PRL ’05, PRD ‘05] • Perturb the continuity equation • Nonlinear equation (exact at all scales): Lie derivativealong Non-perturbative generalization of Non-perturbative generalization of • conserved at all scales for adiabatic perturbations • Equation mimics linear theory

10. Interpretation [Enqvist/Hogdahl/Nurmi/FV in preparation] • Scalar quantity • Perfect fluid: continuity equation barotropic if Constant along the worldline

11. First-order expansion [Langlois/FV, PRL ’05, PRD ‘05] • Expand to 1st order in the perturbations • Reduce to linear theory

12. Second-order expansion [Langlois/FV, PRL ’05, PRD ‘05] • Expand up to 2nd order • Gauge-invariant conserved quantity (for adiabatic perturbations) at 2nd order [Malik/Wands02] • Gauge-invariant expression at arbitrary order [Enqvist/Hogdahl/Nurmi/FV in preparation]

13. Gauge-invariance [Langlois/FV06] • 2nd order coordinate transformation: [Bruni et al.97] • is gauge-invariant at 1st order but not at 2nd • However, on large scales is gauge invariant at second order

14. Nonlinear scalar fields • Rigopoulos/Shellard/vanTent05: non-Gaussianities from inflation • Lyth/Rodriguez05: -formalism (Non-Gaussianity in two-field inflation) [FV/Wands06]

15. super-Hubble sub-Hubble comoving wavelength conformal time inflation radiation dominated era Cosmological scalar fields • Scalar fields are very important in early universe models - Single-field: like a perfect fluid - Multi-fields: • richer generation of fluctuations (adiabatic and entropy) • super-Hubble nonlinear evolution during inflation • Two-field inflation: local field rotation [Gordon et al00; Nibbelink/van Tent01] Adiabatic perturbation Entropy perturbation

16. Two scalar fields [Langlois/FV06] arbitrary • Adiabatic and entropy angle: space-dependent angle • Total momentum: • Define adiabatic and entropycovectors: entropy covector is only spatial: covariant perturbation!

17. Nonlinear evolution equations [Langlois/FV06] • Homogeneous-like equations (from Klein-Gordon):  1st order  2nd order  1st order  2nd order • Linear-like equations (gradient of Klein-Gordon):

18. Linearized equations [Langlois/FV06] • Expand to 1st order • Replace by the gauge-invariant Sasaki-Mukhanov variable [Sasaki86; Mukhanov88] • First integral, sourced by entropy field [Gordon/Wands/Bassett/Maartens00] • Entropy field perturbation evolves independently • Curvature perturbation sourced by entropy field

19. Second order perturbations [Langlois/FV06] • Expand up to 2nd order: • Total momentum cannot be the gradient of a scalar

20. Adiabatic and entropy large scale evolution [Langlois/FV06] • First integral, sourced by second order and entropy field • Entropy field perturbation evolves independently • Curvature perturbation sourced by first and second order entropy field • Nonlocal term quickly decays in an expanding universe:

21. Conclusions • New approach to cosmological perturbations • - nonlinear • - covariant (geometrical formulation) • - exact at all scales • - mimics the linear theory • - easily expandable at second order • Extended to scalar fields • - fully nonlinear evolution of adiabatic and entropy components • - 2nd order large scale evolution (closed equations) of adiabatic and entropy • - new qualitative features: decaying nonlocal term

23. References • Second-order perturbations • Phys. Rev. D71, 061301 (2005), astro-ph/0411463 • Nonlinear perturbations • with David Langlois: • Phys. Rev. Lett. 95, 091303 (2005), astro-ph/0503416 • Phys. Rev. D72, 103501 (2005), astro-ph/0509078 • JCAP 0602, 014 (2006), astro-ph/0601271 • submitted to JCAP, astro-ph/0610064 • with Kari Enqvist, Janne Högdahl and Sami Nurmi: • in preparation