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

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

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

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 comoving wavelength inflation radiation dominated era sub-Hubble super-Hubble conformal time (See the talks by Bartolo, Creminelli, Liguori, Lyth, Rigopoulos)

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

Conserved linear perturbation Gauge-invariant definition: curvature perturbation on uniform density hypersurfaces =0 space time =0 [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

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

Covariant approach Work with geometrical quantities: perfect fluid [Ehlers, Hawking, Ellis, 60’-70’] Work with geometrical quantities: perfect fluid observer 4-velocity: proper time: world-line 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:

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

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

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

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]

Gauge-invariance 2nd order coordinate transformation: [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

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

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 comoving wavelength inflation radiation dominated era sub-Hubble super-Hubble conformal time Two-field inflation: local field rotation [Gordon et al00; Nibbelink/van Tent01] Adiabatic perturbation Entropy perturbation

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

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):

Linearized equations Expand to 1st order [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

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

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:

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

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