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, (2005), astro-ph/
Nonlinear perturbations
with David Langlois:
Phys. Rev. Lett. 95, (2005), astro-ph/
Phys. Rev. D72, (2005), astro-ph/
JCAP 0602, 014 (2006), astro-ph/
astro-ph/
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. 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)

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

6. 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

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. 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:

9. [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

10. [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

11. [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

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 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

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

15. 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

16. 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!

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 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

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, (2005), astro-ph/
Nonlinear perturbations
with David Langlois:
Phys. Rev. Lett. 95, (2005), astro-ph/
Phys. Rev. D72, (2005), astro-ph/
JCAP 0602, 014 (2006), astro-ph/
submitted to JCAP, astro-ph/
with Kari Enqvist, Janne Högdahl and Sami Nurmi:
in preparation