Inflation and the origin of structure

Inflation and the origin of structure

David WandsInstitute of Cosmology and Gravitation

University of Portsmouth

Niew Views of the Universe, KICP Symposium 10th December 2005

Standard model of structure formationStandard model of structure formationprimordial perturbations

in cosmic microwave background

Inflation:Inflation: initial false vacuum state drives accelerated expansion zero-point fluctuations yield spectrum of perturbations

large-scale structure of our Universe

gravitational

instability

new observational data offers precision tests of

• cosmological parameters

• the nature of the primordial perturbations

• period of accelerated expansion in the very early universe

• requires negative pressure

e.g. self-interacting scalar field

• speculative and uncertain physics

Cosmological inflation: Starobinsky (1980)

Guth (1981)

φ

V(φ)

• just the kind of peculiar cosmological behaviour we observe today

coherent oscillations in photon-baryon plasma due to primordial density

perturbations on super-horizon scales

Wilkinson Microwave Anisotropy Probe February 2003

vacuum fluctuationsswept up by accelerated expansion 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

linking the very small to the very large!

inflation probes high energies

• cosmic expansion on large scales

• reconstruct inflaton potential• modified Friedmann equation

• quantum vacuum on small scales

• trans-Planckian effects (modified dispersion relation, Lorentz-violation...)

...)(82

2 += φπ VM

HPl

...2

22 +⎟

⎠⎞

⎜⎝⎛=

akπ

δφ

Field perturbations on a branecoupled to metric perturbations

recover 4D gravity at low energies

but probe 5D at high energies

• 5D backreaction at high energycan damp small scale oscillations

:backreaction from 5D

Koyama, Mizuno & Wands ‘05

advertisement: see talk by Andy Mennim on Monday!

primordial perturbations from scalar fields

during inflationfield perturbations φ(x,ti) on initial spatially-flat hypersurface

in radiation-dominated era curvature perturbation ζ on uniform-density hypersurface

∫=final

initialdtHN

( ) II I

initialNNN δφφ

φζ ∑ ∂∂

≈−=

on large scales, neglect spatial gradients, treat as “separate universes”

Sasaki & Stewart ’96

t

x

density perturbations from inflaton field perturbations• quantum fluctuations on spatially flat (δN=0) hypersurfaces

during inflation

• produce density perturbations in radiation-dominated era

aHk

HddN

=

⎟⎠⎞

⎜⎝⎛−== δσ

σδσ

σζ

&

222

2

2

2251

251

aHkSW

HTT

=⎟⎟⎠

⎞⎜⎜⎝

⎛≈≈⇒

σπζδ

&

2

2

2

2

3,where

126ln

ln1:tilt

Hm

HH

kdd

n

σσ

σζ

ηε

ηεζ

=−=

<<+−≈≡−

&

tensor metric perturbations

• transverse, traceless metric perturbations

• amplitude, h(t), obeys same wave equation for massless field

• remain decoupled from matter perturbations

)()(),( ),(3 xethkdxtg ijkij×+∫≈δ

2

22

264

aHkPl

HM

T=

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛≈⇒

ππ

2

2

where2ln

ln:tilt H

Hkd

TdnT

&−≡−≈≡ εε

“smoking gun” for inflation...

• inflation predicts primordial gravitational wave background

• could be

• or could be

• only detectable if inflationary scale > 10 15 GeV

aHkPlMVT

=⎟⎟⎠

⎞⎜⎜⎝

⎛≈ 4

2

12416

1010 −≈⎟⎟⎠

⎞⎜⎜⎝

⎛

PlMGeV

644

101 −≈⎟⎟⎠

⎞⎜⎜⎝

⎛

PlMTeV

Seljak et al (2004)

εζ

162

2

≈=T

r

• coupled fields during slow-roll during inflationStarobinski & Yokoyama; Sasaki & Stewart; Mukhanov & Steinhardt; Linde, Garcia-Bellido & Wands.... (1995)

•curvaton decay after inflation

weakly-coupled, late-decaying scalar fieldEnqvist & Sloth; Lyth & Wands; Moroi & Takahashi (2001)

• inhomogeneous / modulated reheating or preheating

inflaton decay-rate modulated by another light fieldDvali, Gruzinov & Zaldariaga; Kofman (2003); Kolb, Riotto & Vallinotto (2004)

•inhomogeneous end of inflationLyth; Salem (2005)

but fluctuations in other fields can also perturb radiation density after inflation

primordial perturbations from isocurvature fields during inflation• quantum fluctuations on spatially flat (δN=0) hypersurfaces

during inflation

• produce density perturbations in radiation-dominated era

• amplitude depends upon physics• but spectral tilt set during inflation

δχχ

ζddN

=

22

2

2

2251

251

aHkSW

HddN

TT

=

⎟⎠⎞

⎜⎝⎛⎟⎟⎠

⎞⎜⎜⎝

⎛≈≈⇒

πχζδ

2

2

2 3,where

221:tilt

Hm

HH

n

χχ

χζ

ηε

ηε

=−=

+−≈−

&

where N(χ) dependent on physics

chaotic inflation:chaotic inflation:

(I) inflaton perturbations

16.096.0261

≈≈+−=

rn ηε

( )01.0

21

21 22222

≈==

+==

χχσσ ηηε

χσφ mmV

(II) isocurvature perturbations

16.01221

<<≈+−=

rn ηε

Seljak et al (2004)

modulated preheating?Byrnes & Wands (astro-ph)

inflaton

distinctive observational predictions

inflaton perturbationsadiabatic no isocurvature perturbationsGaussian

isocurvature field perturbationsnon-adiabaticpossible residual isocurvature modes...... correlated with curvature perturbation possible non-Gaussianity

Gaussian field perturbations to first order

give non-Gaussian metric perturbation to second order

linear evolution -> Gaussian perturbations stay Gaussian

non-linear evolution -> non-Gaussianity!

single-field inflation Maldacena (2002);

Acquaviva, Bartolo, Matarrese & Riotto (2002+);

beyond slow roll Creminelli & Zaldarriaga (2003);

Lidsey & Seery (2004);

multi-field inflation Rigopoulos, Shellard & van Tent (2003+)

Lyth & Rodriguez (2004);

Allen, Gupta & Wands (2005)

φδφ

ζ 11 ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=N

( )212

2

22 21 φδ

φφδ

φζ ⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=NN

“local” non-Gaussianity

constraints from WMAP: -58 < fNL < 134

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

Wang & Kamiokowski (2000)

211 5

3 ζζζ NLf−=

gives bispectrum:

[ ])()()()()()(),,( 133221321 kPkPkPkPkPkPfkkkB NL ++∝ζ

more data to come...

• perturbation due to inflaton field

• where can be calculated during inflation

• N,σσ must be small during slow-roll inflation• but perturbations from non-adiabatic perturbations

dependent upon subsequent expansion history

Detectable non-Gaussianity can come from non-adiabatic perturbations in non-inflaton fields

Lyth & Rodriguez (2005)

( )21,2,2

1,1

σδσδζ

σδζ

σσσ

σ

NN

N

+=

=

)/()/(,/ 2,, σσσ σσσ &&&&& HHHNHN +−==

( )21,2,2

1,1

χδχδζ

χδζ

χχχ

χ

NN

N

+=

=

non-Gaussianity from curvaton decay

constraints on fNL from WMAP - fNL < 58

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

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

Wang & Kamiokowski (2000)

211 )5/3( ζζζ NLf−≈

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+Ω≈Ω≈

2

decay,decay, χδχ

χδχζζ χχχ

recall that for curvaton

Lyth, Ungarelli & Wands ‘02

decay,decay,1 6

5,χ

χ χδχζ

Ω−≈⎟⎟

⎠

⎞⎜⎜⎝

⎛Ω≈ NLf

corresponds to

Conclusions:Conclusions:

• Inflation links very small scale vacuum fluctuations to verylarge scale structure of our universe

• Precision cosmology (especially cosmic microwave background data) offer detailed measurements of primordial density perturbations

• Gravitational waves, primordial isocurvatureperturbations and/or non-Gaussianity could provide valuable info about origin of perturbations

• Single-field slow-roll inflation predicts adiabatic density perturbations with negligible non-Gaussianity could givedetectable gravitational waves

• Multi-field inflation allows non-adiabatic perturbations during inflation, which could give detectable primordial isocurvature perturbations and/or local non-Gaussianity