Scale dependence of non-Gaussianity
from an isocurvature field
黃慶國
JCAP 11(2010)026
2011两岸粒子物理与宇宙学研讨会
中国科学院理论物理研究所
2011.4.2
1978, Arno A. Penzias and Robert W. Wilson
COBE
2006, John C. Mather and George F. Smoot
For theorists:
Why is our universe almost spatially flat,
homogeneous and isotropic?
Why not exactly homogeneous and
isotropic?
4
Inflation
Inflation is an elegant idea to solve all the puzzles in the hot big bang
model and smoothed away the initial inhomogeneities.
Nobel Prize ??
Quantum fluctuation
R = a(t)Rc
dR
v=
= HR ⇒ Rcausal = 1/H
dt
δφ = H/2π
6
Curvature perturbation
ds2 = −dt2 + a2 (t)δij dxi dxj
⇓
ds2 = −dt2 + a2 (t)e2ζ(t,x) γij (t, x)dxi dxj
ζ∼
δρ
δT
∼
ρ
T
3
Φ= ζ
5
General Relativity
Geometry = Energy
time
7
From the quantum fluctuations of fields
to the cosmic density perturbations
a ∼ eHt ∼ eN
ln ρ
δN
ρ ∼ V (φ) ∼ const.
Reheating
Reheating
φ + δφ
φ
ρ ∼ a−4
δρ
ζ∼
∼ δ ln a ∼ δN
ρ
ln a
∼ N (φi + δφi ) − N (φi )
dependens on the dynamics of inflation
8
Roughly speaking,
∆T
∼
T
�
∆T
T
�
+ fN L
L
�
∆T
T
�2
+ ...,
L
Non-Gaussianity parameter
�
∆T
T
�
L
∼ 10−5
Bispectrum (Non-Gaussianity)
3
�
ki )F (k1 , k2 , k3 )
�ζk1 ζk2 ζk3 � = (2π)3 δ (3) (
i=1
fN L
12π 4
F (k1 , k2 , k3 ) =
fN L Pζ2 · F̃ (k1 , k2 , k3 )
5
strength
shape
10
∆T
T
1.0
Non-Gaussian
Gaussian
0.8
0.6
0.4
0.2
0.0
!4
f_NL
interaction
!2
0
2
4
3 loc. 2
ζ(x) = ζL (x) + fN L ζL (x) + ...
5
�
1
F̃ loc. (k1 , k2 , k3 ) = 2 3 3 + (2 perm.)
k1 k2
�
20
1.0
15
10
5
0
0.0
0.5
0.5
1.0
Local shape
0.0
F̃
equil.
�
�
1
1
1
3
3
3
orth.
(k1 , k2 , k3 ) = 6 − 3 3 − 3 3 − 3 3
F̃
(k1 , k2 , k3 ) = 6 − 3 3 − 3 3 − 3 3
k1 k2
k2 k3
k3 k1
k1 k2
k2 k3
k3 k1
�
��
�
��
2
1
8
3
−
+
+ (5 perm.)
−
+
+ (5 perm.) .
(k1 k2 k3 )2
k1 k22 k33
(k1 k2 k3 )2
k1 k22 k33
6
5
4
1.0
1.0
0
�5
2
�10
0
0.0
0.5
0.5
0.0
0.5
0.5
1.0
0.0
Non-local shape
1.0
0.0
V (φ1 , φ2 )
Maldacena, 2002
local
Slow-roll: fN L =
DBI:
equil.
fN
L
5
(1 − ns )
12
∼
−1/c2s
κ
Alishabiha, Silverstein,
Tong, 2004
(Disfavored by WMAP7.
QGH, 2010 )
φ2
adiabatic
isocurvature
φ1
End of inflation
local
fN
L ∼ κ
QGH, 2009
In the literatures, f_NL^local is assumed to be a constant
which is scale independent.
WMAP7:
loc.
fN
L = 32 ± 21
NVSS:
loc.
fN
L
PLANCK:
local
∆fN
L ∼ 5
= 62 ± 27
(Xia, Bonaldi, Baccigalupi,
Zotti, Matarrese, Verde, Viel,
2010)
A convincing detection of local form non-Gaussianity
is the smoking-gun of multi-field inflation.
Scale dependence of f_NL^local
Hoyle, Jimenez, Verde, 2010
16
local
fN L (k)
=
local
fN L (kp )
�
k
kp
�nfN L + 12 αfN L ln(k/kp )
PLANCK:
∆nfN L
50
1
�
� 0.1
fN L fsky
Sefusatti, Liguori, Yadav, Jackson, Pajer, 2009
The spectral index and its running of f_NL^local from an isocurvature field:
n fN L
αfN L ≡
d ln |fN L |
N,σ (t∗ )
≡
= η3
d ln k
N,σσ (t∗ )
dnfN L
= (2�H − ησσ − η4 )nfN L − n2fN L
d ln k
�H
Ḣ∗
≡− 2
H∗
V ��� (σ∗ )
η3 ≡
3H∗2
ησσ
V �� (σ∗ )
≡
3H∗2
V � V ����
η4 ≡
3H∗2 V ���
Byrnes, Gerstenlauer, Nurmi, Tasinato, Wands, 2010
QGH, 2010
� σ �n
1 2 2
4
V (σ) = m σ + λm
2
m
s = 2λ
� σ �n−2
∗
m
−2/n < s < +∞
n fN L
QGH, 2010
5n(n − 1)(n − 2) m2 H∗ s
= sign(N,σ )
72π∆R
H∗2 σ∗ fN L
Heuristic estimate: typical vacuum expectation value of isocurvature field
Random walk:
Slow-roll:
Balance:
� σ �n−1
dσ
3H∗
= −m2 σ − nλm3
dt
m
d 2
H∗3
2m2 2
2nλm4 �σ 2 �n/2
�σ � =
−
�σ � −
dt
4π 2
3H∗
3H∗
mn
�
2
�
3
H
∗
σ∗ = �σ 2 � =
8π 2 (1 + ns/2) m
n fN L · f N L
QGH, 2010
3
H
∗
�σ 2 � =
t
4π 2
m 3
= sign(N,σ ) · 2.3 × 10 n(n − 1)(n − 2)(
) s
H∗
3
�
n
1+ s
2
�
�−1/3
m
n
� 0.13 n(n − 1)(n − 2)|s| 1 + s
H∗
2
�
|nfN L · fN L | � 5
0.50
detectable region !PLANCK"
0.30
0.100
n!4
0.050
m#H"
0.20
m$H"
detectable region !PLANCK"
0.500
n!6
0.15
n!8
n!8
n!6
n!4
0.010
0.10
0.005
0.001
0.001
QGH, 2010
0.005 0.010
!s! "s!0#
0.050 0.100
0.500
0.001
0.01
0.1
s !s!0"
1
10
100
Conclusions
  A convincing detection of local form non-Gaussianity will
rule out all single-field inflation models (not only the slowroll single-field models).
  A scale-independent f_NL^local is not generic prediction
of inflation.
  The scale-dependence of f_NL^local is determined by
the self-interaction of isocurvature field.
Pauli’s exclusion principle
Fermions are discovered in the US, whereas
bosons are discovered in Europe.
2012
谢 谢！