Inflation :sources of primordial

perturbations

Inflation :sources of primordial

perturbations

David WandsInstitute of Cosmology and Gravitation

University of Portsmouth

DESY Workshop, Particle Cosmology 30th September 2004

• period of accelerated expansion in the very early universe

• requires negative pressure

e.g. self-interacting scalar field

• speculative and uncertain physics

Cosmological inflation:Cosmological inflation: Starobinsky (1980)

Guth (1981)

φ

V(φ)

• just the kind of peculiar cosmological behaviour we observe todaydark energy!

• N self-interacting scalar fields:

• Homogeneous Hubble expansion:

inflation acceleration

• inhomogeneous field perturbations, wavenumber k, coupled to metric perturbations ψ

Inflationary dynamics:Inflationary dynamics:

φ

V(φ)

03 =++ϕ

ϕϕddVH &&&

+= 22

2183 ϕπ &VGH

2ϕ&>V

...33 22

2

+=

+++ ψϕδϕϕδϕδ &&&& m

akH

Vacuum fluctuations

φ

V(φ)

• small-scale/underdamped zero-point fluctuations

• large-scale/overdamped perturbations in growing modelinear evolution ⇒ Gaussian random field

kae ik

k 2

η

δφ−

≈

2

3

232

2)2(

4

=≈

= ππ

δφπδφ Hk k

aHk

fluctuations of any scalar light fields (m<3H/2) `frozen-in’ on large scales

Hawking ’82, Starobinsky ’82, Guth & Pi ‘82

weakly scale-dependent

ε2−

η2+

• slowly changing Hubble rate• slow evolution outside horizon

• finite mass• metric back-reaction (self-gravity)

1ln

ln 2

−≡ φ

δφn

kdd

ηεφ 261 +−=−n

ε4−

2

2

2 31,

Hm

HH

≡−≡ ηε&

Slow-roll parameters:

08323

32

2

2

=

++++

⋅

QH

aa

GmakQHQ φπ &

&&&

Self-gravity of inflaton field in GR

Consistently include linear metric perturbations, ψ,using Mukhanov/Sasaki variable:

ψφδφH

Q&

−≡

obeys 4D wave equation:

used to calculate corrections, e.g., in slow-roll expansion

Sources of structure:(1) density perturbations during inflation

Inflaton field perturbations lead to adiabatic density perturbations on super-Hubble scales during inflation:

δφδρ 'V≈

Local energy conservation => conserved perturbation always exists for adiabatic perturbations on large scales

Wands, Malik, Lyth & Liddle (2001)

QQHH&&

−≈−−= ψδρρ

ζ

Conserved perturbation: Bardeen, Steinhardt & Turner (1983)

adiabatic perturbations on large scales• adiabatic perturbations e.g.,

– perturb along the background trajectory

– adiabatic perturbations stay adiabatic on large scales

tyy

xx δδδ

==&&

0=−∝

B

B

B nn

nn

nn δδ

δγ

γγ

φ

dφ/dt

slow-roll attractor

observables in inflaton scenario:• primordial density perturbation -> window onto inflation

– amplitude

– scale-dependence

– tensor-scalar ratio

aHkPlaHk MVHH

==

≈

≈ 4

222 1

2 επφζ &

( ) aHkn =+−≈− ηεζ 261

( ) TaHk nT

2162

2

−≈≈ =εζ

observational consistency test Liddle & Lyth ‘93

brane-world inflaton scenario:brane-world inflaton scenario:• 4D inflaton on our brane-world

• full 5D metric perturbations at high energies too complicated tosolve in general

• Calculate ζ during quasi-de Sitter inflation on the brane(negligible metric back-reaction for V=constant)

• ζ conserved on large scales until low energies where we can use 4D gravity Langlois, Maartens, Sasaki & Wands (2001)

• but can’t yet include metric perturbations for vacuum fluctuation on small scales, l5D,Planck < λ < lAdS

Koyama, Langlois, Maartens & Wands (2004)

• hence can’t yet calculate as slow-roll corrections for linear perturbations

Maartens, Wands, Bassett & Heard (2000)

brane-world inflaton scenario:• primordial density perturbation

– amplitude

– tensor-scalar ratio

( )AdSaHkPlaHk

HlGMVHH 2

4

222 1

2×

≈

≈

== επφζ &

( ) ( )( ) T

AdS

AdSaHk n

HlGHlFT

216 2

2

2

2

−≈×≈ =εζ

same observational consistency test Huey & Lidsey 2000

multi-field perturbationsorthogonal fields φ,χ => uncorrelated vacuum fluctuations

δφφχδχδ &&

−≡s

additional source for density perturbationsSasaki & Stewart ’96; Gordon et al ‘01

δχφχδφ &&

+≡Q“adiabatic” “entropy”

wave equations:

QH

aa

Gd

VdakQHQ

++++

⋅23

32

2

2

2 83 φπφ

&&&&

03 2

2

2

2

≈

+++

≈

sds

VdaksHs

sddO

δδδ

δσθ

&&&

Scale dependence of isocurvature fluctuations

ε2−

ssη2+

• slowly changing Hubble rate• slow evolution outside horizon

• finite mass• metric back-reaction (self-gravity)

1ln

ln 2

−≡ snkdsd

δ

δ

sssn ηεδ 221 +−=−

ε4−

2

2

2 31,

Hm

HH s

ss ≡−≡ ηε&

Slow-roll parameters:

entropy perturbations-> density perturbations on large scales

sddH δσθζ 2≈&

φ

χ

slow-roll attractor

θδs

φ

χ

a unique late-time attractor gives only adiabatic

perturbations at late times

θδs

sddH δσθζ 2≈&

entropy perturbations-> adiabatic density perturbations on large scales

two-field scenario: Wands, Bartolo, Matarrese & Riotto 2002

• primordial density perturbation enhanced after Hubble exit

– amplitude

– scale-dependence

– gravitational waves

– plus • residual isocurvature modes? correlation angle cos∆• non-Gaussianity?

aHkPlaHk MVHH

==

∆

≈

∆

≈ 42

22

22 1

sin1

2sin1

επφζ &

( )( )∆+∆∆+∆+

∆−−≈−22

2

coscossin2sin2

cos461

sss

n

ηηη

ε

φφφ

ζ

( ) ∆−≈∆≈ =22

2

2

sin2sin16 TaHk nT

εζ

Sources of structure:(2) inhomogeneous reheating at end of inflation

Inflaton decay rate, Γ, sensitive to moduli fields

δχδ 'Γ≈Γ

earlier decay changes local radiation density afterKofman (2003); Dvali, Gruzinov & Zaldarriaga (2003)

ΓΓ

−=

=−−=

=

δρ

δρψδρ

ρζ

ψγ

γ

61

40

&

H

Sources of structure:(3) late-decaying scalars after the end of inflation

Polonyi/moduli problem• weakly-coupled massive scalar fields

• displaced from true vacuum

• begin to oscillate when H<m, with ρχ α a-3

• come to dominate over radiation, ργ α a-4

• must decay into radiation before nucleosynthesis

Enqvist & Sloth; Lyth & Wands; Moroi & Takahashi (2001)

χχ

ψγ

γ ζρδρ

ζ decay,

04

Ω≈

=

=

curvaton mechanism

observational signature? non-Gaussianity

constraints on fNL from WMAP fNL < 134

hence Ωχ,decay > 0.01 and 10 -5 < δχ/χ < 10 -3

simplest kind of non-Gaussianity:Komatsu & Spergel (2001)

Wang & Kamiokowski (2000)

211 ζζζ NLf+≈

+Ω≈Ω≈

2

decay,decay, χδχ

χδχζζ χχχ

recall that for curvaton

Lyth, Ungarelli & Wands ‘02

decay,decay,1

1,χ

χ χδχζ

Ω≈

Ω≈ NLf

corresponds to

c.f. non-Gaussianity from inflaton scenario

single-field consistency relation:

Maldacena (2002)

Gruzinov; Creminelli & Zaldarriaga (2004)

12

34

1<<

+−≈

−=

ηεsNL

nf

“easy” to disprove the (simplest) inflaton scenario

Conclusions:Conclusions:

1. Precise data allows/requires us to study more detailed models of inflation and cosmological perturbations

2. Several mechanisms can produce primordial density perturbations on large-scales:

• inflaton perturbations during inflation

• inhomogeneous rehating at end of inflation

• late-decaying scalars (“curvaton”) after end of inflation

3. Distinctive observational tests:

• gravitational waves

• (non-)Gaussianity

• residual (correlated) entropy/isocurvature perturbations