Theoretical Cosmology and Particle Astrophysics at Caltech Marc Kamionkowski July 21, 2004.
DESY Workshop, Particle Cosmology 30th September 2004 … · 2004-10-07 · Inflation : sources of...
Transcript of DESY Workshop, Particle Cosmology 30th September 2004 … · 2004-10-07 · Inflation : sources of...
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