Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field...
Transcript of Large Primordial Non- Gaussianity from early Universe · single field inflation multi-field...
Large Primordial Non-Gaussianity from early Universe Kazuya Koyama University of Portsmouth
• Proved by CMB anisotropies nearly scale invariant nearly adiabatic nearly Gaussian • Generation mechanisms inflation curvaton collapsing universe (Ekpyrotic, cyclic)
Primordial curvature perturbations
0.960 0.013sn = ±0.16, /Sα ζ α<
local 2NL
3( ) ( ) ( )5g gx x f xζ ζ ζ= +
1119 localNL
<<− f
Komatsu etKomatsu et..alal. . 20082008
Generation of curvature perturbations • Delta N formalism curvature perturbations on superhorizon scales = fluctuations in local e-folding number
0inζ =
( , ) ' ( ', )i
ti i
tN t x dt H t x= ∫ ( , ) ( )i
BN t x N tζ = −
3( )Pδρζ
ρ= Ψ −
+
Starobinsky Starobinsky ;;8585, , StewartStewart&&Sasaki Sasaki ‘‘9595
How to generate delta N
single field inflation multi-field inflation, new Ekpyrotic isocurvature
curvaton, modulated reheating, multibrid inflation
adiabatic perturbations
entropy perturbations SasakiSasaki. . 20082008
• Suppose delta N is caused by some field fluctuations at horizon crossing
• Bispectrum
2
,
1( , ) ...2
iI I J
I J I I J
N Nt xζ δϕ δϕ δϕϕ ϕ ϕ
⎛ ⎞∂ ∂= + +⎜ ⎟∂ ∂ ∂⎝ ⎠
∑
( )1 2 3
, , ,1 2 , , , 1 2 3
, ,
( , , )
( ) ( ) perm. ( , , )I J IJI J K IJK
I I
B k k kN N N
P k P k N N N B k k kN N
ζ
ζ ζ= + +
Local type (‘classical’) (local in real space =non-local in k-space)
Equilateral type (‘quantum’) (local in k-space)
( )31 2 3 1 2 3 1 2 3( ) ( ) ( ) 2 ( ) ( , , )Dk k k k k k B k k kζζ ζ ζ π δ= + +r r r r r r
LythLyth&&Rodriguez Rodriguez ‘‘0505
Observational constraints • local type maximum signal for WMAP5 • Equilateral type maximum signal for WMAP5
1kr
2kr 3k
r
1kr
3kr
2kr
local 2NL
3( ) ( ) ( )5g gx x f xζ ζ ζ= +
3 1 2,k k k
1119 localNL
<<− f
1 2 3k k k
NL
equil151 253f− < <
Theoretical predictions Standard inflation Non-standard scenario
Single field
K-inflation, DBI inflation Features in potential Ghost inflation
Multi field
depending on the trajectory
DBI inflation curvaton new Ekpyrotic (simplest model) isocurvature perturbations (axion CDM)
local, equil ( , ) 1NLf O ε η= equil 21 / 1NL sf c=
local (1)NLf O=
( )localNL (5 / 4) / curvaton decay
f ρ ρ
equil 2 2(1 / ) / (1 ) 1NL s RSf c T= + >
1( 1)localNL sf n −> −
5 310localNLf α
RigopoulosRigopoulos, , ShellardShellard, , van van Tent Tent ’’0606 Wands and Vernizzi Wands and Vernizzi ’’0606 YokoyamaYokoyama, , Suyama and Suyama and Tanaka Tanaka ‘‘0707
Three examples for non-standard scenarios • (multi-field) K-inflation, DBI inflation • (simplest) new ekpyrotic model • (axion) CDM isocurvature model
ArrojaArroja, , MizunoMizuno, , Koyama Koyama 08060806..0619 0619 JCAPJCAP
KoyamaKoyama, , MizunoMizuno, , VernizziVernizzi, , Wands Wands 07080708..4321 4321 JCAP JCAP
HikageHikage, , KoyamaKoyama, , MatsubaraMatsubara, , TakahashiTakahashi, , YamaguchiYamaguchi ((hopefullyhopefully) ) to appear soonto appear soon
K-inflation • Non-canonical kinetic term • Field perturbations (leading order in slow-roll)
4 1( , ),2
S d x gP X X μμφ φ φ= − = ∂ ∂∫
2 ,
, ,2X
sX XX
Pc
P XP=
+
21 , 16T
ss pl
H PP r cc M Pζ
ζ
εε
⎛ ⎞= =⎜ ⎟⎜ ⎟
⎝ ⎠
sound speed
AmendarizAmendariz--Picon etPicon et..al al ‘‘9999
GarrigaGarriga&&Mukhanov Mukhanov ‘‘9999
Bispectrum • DBI inflation cf local-type
2
1 2 3 1 2 3 3 3 31 2 3
( , , ) ( , , ) ,P
B k k k F k k kk k k
ζ=
local local 3 3 31 2 3 NL 1 2 3
3( , , ) ( )10
F k k k f k k k= + +
( )1( ) 1 2 ( ) 1 ( )( )
P X f X Vf
φ φφ
= − − − −
1 2 3 21( , , )s
F k k kc
1kr
3kr
2kr
1kr
2kr 3k
r
LoVerde etLoVerde et..alal. . ‘‘0707
Aishahiha etAishahiha et..alal. . ‘‘0404
Observational constraints • (too) large non-Gaussianity • Lyth bound
eff2
35 1108NL
s
fc
= − for equilateral configurations
2
21 6
p
rM N
φΔ⎛ ⎞ =⎜ ⎟Δ⎝ ⎠716 10sr c ε −= <
1 4 0.04 0.013sn ε− ±
D3
anti-D3
inflaton
UVφ φ< eff 300NLf >Huston etHuston et..al al ’’0707, , Kobayashi etKobayashi et..al al ‘‘0808
Multi-field model • Adiabatic and entropy decomposition adiabatic sound speed entropy sound speed
4 1( , ),2
IJ IJ I JS d x gP X X μμφ φ φ= − = ∂ ∂∫
( )2( ) ( ),2
IJ J II J
bP X P X X X X X X= = + −% % %0 0X X=%
2 ,
0, ,2X
adX XX
Pc
P X P=
+%
% % %
%
% %
201enc bX= + δσ
sδ
adiabatic
entropy
ArrojaArroja, , MizunoMizuno, , Koyama Koyama ’’0808 RenauxRenaux--PetelPetel, , SteerSteer, , Langlois Langlois TanakaTanaka‘‘0808
• Multi-field k-inflation • Multi-field DBI inflation Final curvature perturbation
2 1enc =
( )1( ) det( ( ) ) 1 ( )( )
IJ I JIJP X g f G V
f μν μ νφ φ φ φφ
= − − − ∂ ∂ − −
2 2en adc c=
( ) ( )IJP X P X=
,H HS ss
ζ δσ δσ
= =& &* *RST Sζ ζ= +
LangloisLanglois&&RenauxRenaux--PetelPetel’’0808
RenauxRenaux--PetelPetel, , SteerSteer, , Langlois TanakaLanglois Tanaka‘‘0808
Transfer from entropy mode • Tensor to scalar ratio • Bispectrum k-dependence is the same as single field case! large transfer from entropy mode eases constraints • Trispectrum different k-dependence from single field case?
RST
2116
1sRS
r cT
ε=+
eff2 2
35 1 1108 1NL
s RS
fc T
= −+
3eff
22,NLf
ζ
ζ
2 2
3 2
1
1
RS
RS
T
T
ζ
ζ
∝ +
∝ +
MizunoMizuno, , KoyamaKoyama, , Arroja Arroja ’’0909
RenauxRenaux--PetelPetel, , SteerSteer, , Langlois TanakaLanglois Tanaka‘‘0808
New ekpyrotic models • Collapsing universe • Ekpyrotic collapse
1H −
ak
t t− ↔bounce
( ) ( ) , 1na t t n= −Khoury etKhoury et..alal. . ‘‘0101
• Old ekpyrotic model spectrum index for is the bounce may be able to creat a scale invariant spectrum but
it depends on physics at singularity • New ekpyrotic model
0cV V e ϕ−= −
22/( ) ( ) ca t t= −
ζ 3sn =
1 1 2 21 2
c cV V e V eϕ ϕ− −= − −
212/( ) ( ) ca t t= −
222/( ) ( ) ca t t= −
22/( ) ( ) ca t t= −
22/2 2 2
1 2
1 1 1( ) ( ) ,ca t tc c c
= − = +
Khoury etKhoury et..alal. . ‘‘0101
Lyth Lyth ‘‘0303
Lehners etLehners et..al al , , Buchbinder etBuchbinder et..alal. . ‘‘0707
B
B2
• Multi-field scaling solution is unstable entropy perturbation which has a scale invariant spectrum is converted to adiabatic perturbations
2
2 20
c Hsδπ
ζ
= 1
22 2 2
1 22
T
sc H
c c
δ δϕ
ζ δϕπ
∝
∝ =+
B
B
B2
B: B2:
sδsδ
B2
KoyamaKoyama, , Mizuno Mizuno & & WandsWands’’ 0707
• Delta-N formalism
22
212
dN d NN s sds ds
δ δ δ= +
sδB
B2
sδ
logN a=
log H
Nδ
B
B2
• Predictions • Generalizations changing potentials conversion to adiabatic perturbations in kinetic
domination
2 21 2
local 2 1NL 1
1 11 4 0
05 5 ( 1)
12 3
s
s
nc c
r
f c n −
⎛ ⎞− = + >⎜ ⎟
⎝ ⎠=
= > −
KoyamaKoyama, , MizunoMizuno, , Vernizzi Vernizzi & & WandsWands’’ 0707
Buchbinder etBuchbinder et..al al 0707
Lehners Lehners &&Steinhardt Steinhardt ‘‘0808
Non-Gaussianity from isocurvature perturbations
• (CDM) Isocurvature perturbations are subdominant • Non-Gaussianity can be large
0.067 ( 1)
0.008 ( 1.5) <0.001 ( 2)
SS
S
S
S
P nP P
nn
ζ
α = < =+
< ==
BoubekeurBoubekeur& & Lyth Lyth ‘‘0606, , Kawasaki etKawasaki et..al al ’’0808, , Langlois etLanglois et..al al ‘‘0808
Axion CDM • Massive scalar fields without mean • Entropy perturbations • Dominant nG may come from isocurvature perturbations
2 2CDM mδρ σ
2 234
CDM rg g
CDM r
S δρ δρ η ηρ ρ
= − ≡ −3
3/22(1)
SO
S
32
3 3eff 52
2 1/22 2
1 10NL
Sf α α
ζ ζ
LindeLinde&&Mukhanov Mukhanov ‘‘9797, , Peebles Peebles ’’9898, , Kawasaki etKawasaki et..alal‘‘0808
Bispectrum of CMB from the isocurvature perturbation
Adiabatic (fNL=10)
Equilateral triangle
SN
⎛ ⎝ ⎜
⎞ ⎠ ⎟
2
= Il1l2l3
2 bl1l2l3
2
Cl1Cl2
Cl3Δ l1l2l32≤ l1 ≤ l2 ≤ l3
∑
Noise: WMAP 5-year
The isocurvature perturbations can generate large non-Gaussianity (fNL~40)
adi iso
S SN N
⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
3/2eff 41
0.067NLf α⎛ ⎞= ⎜ ⎟⎝ ⎠
• Minkowski functional measures the topology of CMB map (=weighted some of bispectrum)
no detection of non-G from isocurvature perturbations and get constraints
comparable to constraints from power spectrum
WMAP5 constraints -Minkowski functional
0.070 ( 1)
0.042 ( 1.5)
0.0064 ( 2)
n
n
n
η
η
η
α
α
α
< =
< =
< =
HikageHikage, , KoyamaKoyama, , MatsubaraMatsubara, , TakahashiTakahashi, , Yamaguchi Yamaguchi ‘‘0808
Theoretical predictions Standard inflation Non-standard scenario
Single field
K-inflation, DBI inflation Features in potential Ghost inflation
Multi field
depending on the trajectory
DBI inflation curvaton new Ekpyrotic (simplest model) isocurvature perturbations (axion CDM)
local, equil ( , ) 1NLf O ε η= equil 21 / 1NL sf c=
local (1)NLf O=
( )localNL (5 / 4) / curvaton decay
f ρ ρ
1( 1)localNL sf n −> −
5 310localNLf α
equil 2 2(1 / ) / (1 ) 1NL s RSf c T= + >
Conclusions • Power spectrum from pre-WMAP to post-WMAP • bispectrum WMAP 8year Planck Do everything you can now!
Axion CDM
Classical mean of axion quantum fluctuations if
a aa f θ=
inf / 2a af Hθ π<
2inf / 2a Hδ π=
2
inf*
*
,2
a
a
a Haa
δρ δρ π
⎛ ⎞= =⎜ ⎟
⎝ ⎠
cdm0.82 2
2 *12 11 2
cdm
,
0.111.8 1010 GeV 10 GeV
a a
a
a
S r r
F arh
δρρ
−−
Ω= =
Ω
⎛ ⎞⎛ ⎞ ⎛ ⎞= × ⎜ ⎟⎜ ⎟ ⎜ ⎟ Ω⎝ ⎠ ⎝ ⎠ ⎝ ⎠