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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.

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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


Covariant approach

[Ehlers, Hawking, Ellis, 60’-70’]

  • Work with geometrical quantities

- perfect fluid

world-line

- volume expansion

proper time:

4-velocity

- integrated volume expansion

- “time” derivative


Covariant perturbations

[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


Conservation equation

[Langlois/FV, PRL ’05, PRD ‘05]

  • Covector:

  • “Time” derivative: Lie derivative along ub

  • Barotropic fluid


Linear theory (coordinate approach)

  • Perturbed Friedmann universe

curvature perturbation

S(t+dt)

dt

S(t)

xi = const.

  • curvature perturbation on S(t): 


Relation with linear theory

[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]


Gauge invariant quantity

SC : uniform density

d =0, =C

dtF→C

 =0, d =dF

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...


log

 = 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


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

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


Nonlinear generalization

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


Gauge invariant quantities

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]


Large scale behavior

df =0

S : uniform field

d =0

S : uniform density

  • Relativistic Poisson equation  large scale equivalence

large scales

  • Conserved quantities


New approach

[Langlois/FV, PRL ’05, PRD ‘05]

  • Integrated expansion

 Replaces curvature perturbation

  • Non-perturbative generalization of

  • Non-perturbative generalization of


f= const

Single scalar field

arbitrary


f= const

Single scalar field

 Single-field: like a perfect fluid


log

a = const.

L=H-1

inflation

log a

t=tin

t=tout

Single field inflation

  • Generalized nonlinear Poisson equation


Two-field linear perturbation

[Gordon et al00; Nibbelink/van Tent01]

  • Global field rotation: adiabatic and entropy perturbations

Adiabatic

Entropy


ds = 0

df = 0

d = 0

Total momentum is the gradient of a scalar


Evolution of perturbations

[Gordon/Wands/Bassett/Maartens00]

  • Curvature perturbation sourced by entropy field

  • Entropy field perturbation evolves independently


Two scalar fields

[Langlois/FV ‘06]

f = const

 = const

arbitrary !


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


Second order expansion

  • Entropy:

  • Adiabatic:

Vector term


Vector term

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)


Conclusions

  • 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


df =0

S : uniform field

d =0

S : uniform density

 =0

SF : flat

Mukhanov equation  quantization


Quantized variable

[Pitrou/Uzan, ‘07]

  • Nonlinear analog of

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


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