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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

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)

Linear theory (coordinate approach)

- Perturbed FLRW universe

curvature perturbation

- Perturbed fluid

- Linear theory: gauge transformation

=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

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

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:

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

Interpretation

[Enqvist/Hogdahl/Nurmi/FV in preparation]

- Scalar quantity

- Perfect fluid: continuity equation

barotropic

if

Constant along the worldline

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

[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]

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

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!

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

[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

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