Nonlinear perturbations for cosmological scalar fields

Filippo VernizziICTP, Trieste

Finnish-Japanese Workshop on Particle CosmologyHelsinki, 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

• Non-Gaussianity

- discriminator between models of the early universe

- information on mechanism of generation of primordial perturbations

sensitive to second-order evolution

Conserved nonlinear quantities

• Salopek/Bond ‘90

• Comer/Deruelle/Langlois/Parry ‘94

• Rigopoulos/Shellard ‘03

• Lyth/Wands ‘03

• Lyth/Malik/Sasaki ‘04

Long wavelength expansion (neglect spatial gradients)

Second order perturbation

• Malik/Wands ‘02

Covariant approach

• Langlois/FV ‘05• Enqvist/Hogdahl/Nurmi/FV ‘06

Covariant approach

• Work with geometrical quantities

au 4-velocity

proper time:

world-line PguuPT baa

bba

[Ehlers, Hawking, Ellis, 60’-70’]

- perfect fluid

aau

d3

1

- volume expansion

- integrated volume expansion

31

- “time” derivative

aau

Covariant perturbations

au 4-velocity

proper time:

world-line

[Ellis/Bruni ‘89]

baa

bab uugh

projector on

• Perturbations should vanish in a homogeneous universe

• Instead of , use its spatial gradient!

• Perturbations unambiguously defined

aabb hD

• In a coordinate system:

)x,t()x,t( ii

Conservation equation

aaa D

PPD

P

babab

baa uu uL

• “Time” derivative: Lie derivative along ub

aaa DD

0

aa DP

PDPP

• Barotropic fluid

[Langlois/FV, PRL ’05, PRD ‘05]

0bab

a Tu 0 P

• Covector:

Linear theory (coordinate approach)

• Perturbed Friedmann universe

curvature perturbation

jiij dxdxtadtAds 2121 222

xi = const.

(t)

(t+dt)

d

• proper time along xi = const.: (1 )d dtA

• curvature perturbation on (t): 2

3 4

aR

Relation with linear theory

iiv'

'PP

)P(

H'

3

1

aaa D

PPD

P

[Langlois/FV, PRL ’05, PRD ‘05]

• Nonlinear equation “mimics” linear theory

H

aaa DD

dtv;alnt ii

3

1

x,ttx,t

[Wands/Malik/Lyth/Liddle ‘00][Bardeen82; Bardeen/Steinhardt/Turner ‘83]

• Reduces to linear theory

a

'aH;

dt

d'

Gauge invariant quantity

F : flat=0, =F

C : uniform densitytF→C

=0, =C

'HC

Curvature perturbation on uniform density hypersurfaces[Bardeen82; Bardeen/Steinhardt/Turner ‘83]

;tH~;t'~

ttt~t

Higher order conserved quantity

22 2

11

'

H

''

'

H'

''

H

• Gauge-invariant conserved quantity at 2nd order[Malik/Wands ‘02]

3222

2

222

1

6

11

2

2

12

1

'

H

''''

H

'

'

H

'''

'

'

H

''

H

• Gauge-invariant conserved quantity at 3rd order[Enqvist/Hogdahl/Nurmi/FV ‘06]

• and so on...

Cosmological scalar fields

• Single-field

• Scalar fields are very important in early universe models

- Perturbations generated during inflation and then constanton super-Hubble scales

log a

logℓ

L=H-1

t=tout

= const

t=tin

inflation

Cosmological scalar fields

• Single-field

• Scalar fields are very important in early universe models

- richer generation of fluctuations (adiabatic and entropy)- super-Hubble nonlinear evolution during inflation

• Multi-field

- Perturbations generated during inflation and then constanton super-Hubble scales

log a

logℓ

L=H-1

t=tout

d/dt S

t=tin

inflation

Nonlinear generalization

• Rigopoulos/Shellard/Van Tent ’05/06

Long wavelength expansion (neglect spatial gradients)

Higher order generalization

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

Covariant approach

• Langlois/FV ‘06

Gauge invariant quantities

F : flat

=0

=0

'H

• Curvature perturbation on uniform energy density

[Bardeen82; Bardeen/Steinhardt/Turner ‘83]

: uniform density

: uniform field=0

[Sasaki86; Mukhanov88]

'H

R

• Curvature perturbation on uniform field (comoving)

Large scale behavior

0'R

0 R

• Relativistic Poisson equation large scale equivalence

• Conserved quantities

042

2

a

G

large scales Hak

=0 : uniform density

: uniform field=0

0'

New approach [Langlois/FV, PRL ’05, PRD ‘05]

d3

1• Integrated expansion

Replaces curvature perturbation

aaa DD

• Non-perturbative generalization of

31

aaa DD

RR• Non-perturbative generalization of

aaaa DD

R

Single scalar field

)(Vaa

2

1

2

1L

= const

au

auarbitrary

ab)ba(abbaab uqPguuPT

0 aaa

a, DuDDV

aau

Single scalar field

= const

au

aau

Single-field: like a perfect fluid

)(Vaa

2

1

2

1L

PguuPT abbaab

0 ,V

abaq 0

03 ,V'H''

Single field inflation

log a

logℓ

L=H-1

t=tout

a = const.

t=tin

inflation

03

23

a

,a D

V

0 aa R

• Generalized nonlinear Poisson equation 0aD

Two-field linear perturbation

s

• Global field rotation: adiabatic and entropy perturbations

[Gordon et al00; Nibbelink/van Tent01]

Adiabatic

Entropy

sincos cossin s

'

'

tg

= 0

= 0

= 0

iiii '''q

Total momentum is the gradient of a scalar

sincos

'

HR

s'

'H'

2 0R

Evolution of perturbations

• Curvature perturbation sourced by entropy field

[Gordon/Wands/Bassett/Maartens00]

033 2 sV'sH''s ss,

• Entropy field perturbation evolves independently

arbitrary !

Two scalar fields

),(Vaa

aa

2

1

2

1

2

1L

= const

au

= const au

aaa DDq

ab)ba(abbaab uqPguuPT

[Langlois/FV ‘06]

Covariant approach for two fields

aaa sincos

aaas cossin

• Local redefinition: adiabatic and entropy covectors:

• Adiabatic and entropy angle:

22

spacetime-dependent angle

aaaa DDDq

• Total momentum:

bbaa hq

tg

Total momentum may not be the gradient of a scalar

(Nonlinear) homogeneous-like evolution equations

• Rotation of Klein-Gordon equations:

aaaa

a, ssVH

13

aaaa

as, ssV

1

1st order 2nd order

1st order 2nd order

03 ,V'H''

0'

V' s,

• Linear equations:

(Nonlinear) linear-like evolution equations

• From spatial gradient of Klein-Gordon equations:

as,as,c

cca

acc

aas,

,aa

sVsVss

VV

31

33

as,,bc

bcca

acc

aass,a,a

VVhs

ssVsVs

1

21 2

Adiabatic:

Entropy:

Adiabatic and entropy large scale evolution

• Entropy field perturbation

• Curvature perturbation: sourced by entropy field

033 2 ass,aa sVss

aa s 2

033 2 sV'sH''s ss,

s'

H''

2

• Linear equations

0 aRa aaD

aR

Second order expansion

2cossin

''s

's

's'

ss iii

'ss'

2

1sincos

iiii V'

s'

'

1

s's'ssV iii 2

1

• Entropy:

Vector term

• Adiabatic:

iii Vs''q • Total momentum cannot be the gradient of a scalar

= 0

= 0

= 0

Vector term

• On large scales:

iiii V''

RRR 1

• Second order

iiV

H

22

R

Adiabatic and entropy large scale evolution

ii

ss,sss,

,ss,

VHsVV

ssHV

ssVsHs

223

22

6952

1

2

3233

ii,

ss, VV

sVsH 222

2

242

• Entropy field perturbation evolves independently

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

03 ii HV'V 3

1

aVi

• 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

F : flat

=0

=0 : uniform density

: uniform field=0

R

Q

02

2

23

RR'R

aH

'Haln

dt

d

Mukhanov equation quantization

Quantized variable [Pitrou/Uzan, ‘07]

aa DR

dDD aaa R

0RRR

a

aa DDelnd

d 2

3

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

• Nonlinear analog of R