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Nonlinear perturbations for cosmological scalar fieldsPowerPoint Presentation

Nonlinear perturbations for cosmological scalar fields

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Nonlinear perturbations for cosmological scalar fields

Filippo Vernizzi

ICTP, Trieste

Finnish-Japanese Workshop on Particle Cosmology

Helsinki, March 09, 2007

Beyond linear theory: motivations

- Nonlinear aspects:

- effect of inhomogeneities on average expansion

- inhomogeneities on super-Hubble scales (stochastic inflation)

- increase in precision of CMB data

sensitive to second-order evolution

- Non-Gaussianity

- information on mechanism of generation of primordial perturbations

- discriminator between models of the early universe

Conserved nonlinear quantities

Second order perturbation

- Malik/Wands ‘02

Long wavelength expansion (neglect spatial gradients)

- Salopek/Bond ‘90
- Comer/Deruelle/Langlois/Parry ‘94
- Rigopoulos/Shellard ‘03
- Lyth/Wands ‘03
- Lyth/Malik/Sasaki ‘04

Covariant approach

- Langlois/FV ‘05

- Enqvist/Hogdahl/Nurmi/FV ‘06

[Ehlers, Hawking, Ellis, 60’-70’]

- Work with geometrical quantities

- perfect fluid

world-line

- volume expansion

proper time:

4-velocity

- integrated volume expansion

- “time” derivative

[Ellis/Bruni ‘89]

- Perturbations should vanish in a homogeneous universe

- Instead of , use its spatial gradient!

world-line

proper time:

- In a coordinate system:

4-velocity

projector on

- Perturbations unambiguously defined

[Langlois/FV, PRL ’05, PRD ‘05]

- Covector:

- “Time” derivative: Lie derivative along ub

- Barotropic fluid

- proper time along xi = const.:

Linear theory (coordinate approach)

- Perturbed Friedmann universe

curvature perturbation

S(t+dt)

dt

S(t)

xi = const.

- curvature perturbation on S(t):

[Langlois/FV, PRL ’05, PRD ‘05]

- Nonlinear equation “mimics” linear theory

- Reduces to linear theory

[Bardeen82; Bardeen/Steinhardt/Turner ‘83]

[Wands/Malik/Lyth/Liddle ‘00]

SC : uniform density

d =0, =C

dtF→C

=0, d =dF

SF : flat

Curvature perturbation on uniform density hypersurfaces

[Bardeen82; Bardeen/Steinhardt/Turner ‘83]

Higher order conserved quantity

- Gauge-invariant conserved quantity at 2nd order

[Malik/Wands ‘02]

- Gauge-invariant conserved quantity at 3rd order

[Enqvist/Hogdahl/Nurmi/FV ‘06]

- and so on...

logℓ

= const

L=H-1

inflation

log a

t=tin

t=tout

Cosmological scalar fields

- Scalar fields are very important in early universe models

- Single-field

- Perturbations generated during inflation and then constant

on super-Hubble scales

- Scalar fields are very important in early universe models

- Single-field

- Perturbations generated during inflation and then constant

on super-Hubble scales

logℓ

d/dt S

L=H-1

inflation

log a

t=tin

t=tout

- Multi-field

- richer generation of fluctuations (adiabatic and entropy)

- super-Hubble nonlinear evolution during inflation

Higher order generalization

- Maldacena ‘02
- FV ’04
- Lyth/Rodriguez ’05 (non-Gaussianities from N-formalism)
- FV/Wands ’05 (application of N)
- Malik ’06

Long wavelength expansion (neglect spatial gradients)

- Rigopoulos/Shellard/Van Tent ’05/06

Covariant approach

- Langlois/FV ‘06

df =0

S : uniform field

d =0

S : uniform density

=0

SF : flat

- Curvature perturbation on uniform field (comoving)

[Sasaki86; Mukhanov88]

- Curvature perturbation on uniform energy density

[Bardeen82; Bardeen/Steinhardt/Turner ‘83]

df =0

S : uniform field

d =0

S : uniform density

- Relativistic Poisson equation large scale equivalence

large scales

- Conserved quantities

[Langlois/FV, PRL ’05, PRD ‘05]

- Integrated expansion

Replaces curvature perturbation

- Non-perturbative generalization of

- Non-perturbative generalization of

logℓ

a = const.

L=H-1

inflation

log a

t=tin

t=tout

Single field inflation

- Generalized nonlinear Poisson equation

[Gordon et al00; Nibbelink/van Tent01]

- Global field rotation: adiabatic and entropy perturbations

Adiabatic

Entropy

[Gordon/Wands/Bassett/Maartens00]

- Curvature perturbation sourced by entropy field

- Entropy field perturbation evolves independently

Covariant approach for two fields

- Local redefinition: adiabatic and entropycovectors:

- Adiabatic and entropy angle:

spacetime-dependent angle

- Total momentum:

Total momentum may not be the gradient of a scalar

(Nonlinear) homogeneous-like evolution equations

- Rotation of Klein-Gordon equations:

1st order

2nd order

1st order

2nd order

- Linear equations:

(Nonlinear) linear-like evolution equations

- From spatial gradient of Klein-Gordon equations:

Adiabatic:

Entropy:

Adiabatic and entropy large scale evolution

- Curvature perturbation: sourced by entropy field

- Entropy field perturbation

- Linear equations

ds = 0

df = 0

d = 0

- On large scales:

- Total momentum cannot be the gradient of a scalar

- Second order

Adiabatic and entropy large scale evolution

- Curvature perturbation sourced by 1st and 2nd order entropy field

- Entropy field perturbation evolves independently

- Nonlocal term quickly decays in an expanding universe:

(see ex. Lidsey/Seery/Sloth)

- New approach to cosmological perturbations
- - nonlinear and covariant (geometrical formulation)
- - exact at all scales, mimics the linear theory, easily expandable

- Nonlinear cosmological scalar fields
- - single field: perfect fluid
- - two fields: entropy components evolves independently
- - on large scales closed equations with curvature perturbations
- - comoving hypersurface uniform density hypersurface
- - difference decays in expanding universe

[Pitrou/Uzan, ‘07]

- Nonlinear analog of

- At linear order converges to the “correct” variable to quantize

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