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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