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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. D 71 , 061301 (2005), astro-ph/0411463 Nonlinear perturbations with David Langlois:

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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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