From AdS/CFT to Cosmology
Chen-Pin Yeh
Stanford University
Base on the work hep-th/0606204
“Holographic framework for eternal inflation”
NTHU, September 14
Outline
•
•
•
•
•
Motivation
Introduction to AdS/CFT
Universe created by tunneling
Correlator in CDL background
Future works
The questions
1. Can we understand the universe we observe
2. If it exists the landscape, can we say anything about
why we live in this specific vacuum
False vacuum eternal inflation
Motivation
from hep-th/0606114
1/21
3. What is the right question to ask
4. Does it have the consistent math framework
for understanding the eternal inflation
5. Does it have observational consequence of the
landscape(initial condition, bubble collision...)
.................
Motivation
2/21
AdS/CFT basic
N =4 SU(N) SYM in M4 is equivalent to IIB string
theory in AdS5 X S5
• Couplings
4
R
!2 = λ
α
λ
gs =
N
2
λ ≡ gym N
• Global symmetry
Conformal symmetry = Isometry = SO(2,4)
Introduction to AdS/CFT
3/21
• Correlation function
!
"
exp
∂B
φ0 O
#
= ZB (φ0 )
CF T
In the classical supergravity limit
ZB (φ0 ) = e
−S(φ0 )
where S(φ0 ) is the classical action
Introduction to AdS/CFT
4/21
Massless scalar example
Euclidean AdS in Poincare coordinate
d
!
1
2
ds = 2
(dxi )2
x0 i=0
Relation between bulk and boundary correlator
lim
!
x0 ,x0 →0
"−d −d
x0 x0 GB
Introduction to AdS/CFT
!
"
" !"
(x0 , x ), (x0 , !x) =
1
|x!" − !x|2d
5/21
Open universe from bubble nucleation
Coleman-De Luccia instanton in Euclidean 4D
Considering the scalar couple to gravity with the following
potential,
V/M pl
10
!1
1/2
H ~ V0 ~ 10
!10
!10
10
!120
10
0
Universe created by tunneling
1
! / M pl
6/21
Assuming the SO(4) symmetry
ds = dt + a(t)
2
where
2
2
dΩ3
2
2
dΩ3
= ds = a(X) (dX +
2
= dθ + sin
2
2
2
2
2
dΩ3 )
2
θdΩ2
The equation of motion is
1 2 ! 2
! 2
4
2
(a ) = a (Φ ) − a V (Φ) + a
2
! !
2a
Φ
!!
2 ∂V
Φ +
=a
a
∂Φ
Universe created by tunneling
7/21
• Continuation to Lorentzian signature
π
X→T + i
2
θ → iR
2
ds = a(T ) (−dT + dR + sinh R dΩ22 )
2
2
2
= a(T ) (−dT +
2
2
2
2
dH3 )
region I
(open FRW with SO(1,3) )
π
θ → − it
2
2
ds = a(X) (dX − dt + cosh t dΩ22 )
2
2
2
2
region III
(Bubble wall region )
Universe created by tunneling
8/21
I
!
IV
III
II
V
Penrose diagram for the Lorentzian
continuation of the CDL instanton
Universe created by tunneling
9/21
Two point correlator in CDL background
(for massless scalar χ )
Ĝ(X1 , X2 ; θ) = a(X1 )a(X2 )!χ(X1 , 0)χ(X2 , θ)"
The rescaled two point function satisfy
! 2
"
δ(θ)
2
−∂X1 + U (X1 ) − ∇ Ĝ(X1 , X2 ; θ) = δ(X1 − X2 ) 2
sin θ
where
U (X) ≡ a (X)/a(X) = 1 −
!!
and
!
"
1 ! 2
2
(Φ ) + 2a V (Φ)
2
U (X) → 1 as X → ±∞
Correlator in CDL background
10/21
Expanding in the complete set
! ∞
dk
∗
∗
δ(X1 − X2 ) =
uk (X1 )uk (X2 ) + uB (X1 )uB (X2 )
−∞ 2π
where
2
[−∂X
+ U (X)]uk (X) = (k 2 + 1)uk (X)
2
[−∂X
+ U (X)]uB (X) = 0
then
Ĝ(X1 , X2 ; θ) =
!
∞
−∞
dk
uk (X1 )u∗k (X2 )Gk (θ) + uB (X1 )u∗B (X2 )GB (θ)
2π
Gk (θ) and GB (θ) are the green function on S 3
Correlator in CDL background
11/21
Gk (θ) and GB (θ) satisfy the following equations separately
δ(θ)
[−∇ + (k + 1)]Gk (θ) =
2
sin θ
δ(θ)
2
−∇ GB (θ) =
2
sin θ
2
2
The solutions are
sinh k(π − θ)
Gk (θ) =
sin θ sinh kπ
!
2 1
GB (θ) = lim Gk (θ) −
k→i
π k2 + 1
Correlator in CDL background
"
1
cos θ
+
=
2π
sin θ
!
θ
1−
π
"
12/21
The complete set of orthonormal modes uk (X) with k > 0
is give by the waves coming in from the left and those coming
from the right
The modes from the left
uk (X) → e
ikX
+ R(k)e
−ikX
uk (X) → T (k)eikX
(X → −∞)
(X → ∞)
The modes from the right
u−k (X) → Tr (k)e
−ikX
u−k (X) → e
−ikX
(X → −∞)
+ Rr (k)e
Correlator in CDL background
ikX
(X → ∞)
13/21
Taking the X, X ! → −∞ limit and using the relations
R(k)∗ = R(−k), R(k)R(k)∗ + T (k)T (k)∗ = 1, we get
Ĝc (X1 , X2 ; θ) =
!
∞
−∞
"
#
dk ikδX
+ R(k)e−ikX̄ Gk (θ)
e
2π
where δX = X1 − X2 and X̄ = X1 + X2
Thus for a given instanton geometry, we can get the
Euclidean two point function by calculating the
!!
corresponding potential U = a /a , then solving the
scattering problem to get the reflection coefficient R
And analytically continue to Lorentzian FRW by
X → T + πi/2,
θ → iR
Correlator in CDL background
14/21
Correlation function (a thin-wall example)
a(X) = e
X
(X ≤ 0)
1
a(X) =
(X ≥ 0)
cosh X
The potential is
2
U (X) = 1 −
Θ(X) − δ(X)
2
cosh X
and the reflection coefficient is
i(k + i)
R(k) =
(2k + i)(k − i)
Correlator in CDL background
15/21
Continuous modes
Ĝ(cX̄) (X1 , X2 ; θ)
=
!
∞
dk
i(k + i)
−ikX̄ sinh k(π − θ)
e
2π (2k + i)(k − i)
sin θ sinh kπ
−∞
∞
"
nX
1 2πi
=
e
2 sin θ 2π n=2
$
n+1
1 # −inθ
inθ
e
−e
(2n + 1)(n − 1) π
%
&'
1 2πi (−i)
k + i −ikX̄ kπ−kθ
'
+
∂k
e
(e
− e−kπ+kθ ) '
2 sin θ 2π π
2k + i
k=i
(
(X̄+iθ)/2
(X̄−iθ)/2
1
+
e
1
+
e
1
−(X̄+iθ)/2
+
ie
log
− ie(−X̄+iθ)/2 log
=
(
X̄−iθ)/2
12π sin θ
1−e
1 − e(X̄+iθ)/2
−4ie
X̄−iθ
log(1 − e
X̄−iθ
) + 4ie
X̄+iθ
log(1 − e
X̄+iθ
2 X̄−iθ 2 X̄+iθ
) + ie
− ie
3
3
)
*
+
1
i X̄ −iθ
2
2
iθ
X̄−iθ
+
− e (e
− e ) − (−iX̄ + π − θ)e
+ (−iX̄ − π + θ)eX̄+iθ
2π sin θ
9
3
3
Bound state
2 X̄ cos θ
ĜB (X1 , X2 ; θ) = uB (X1 )uB (X2 )GB (θ) = e
3
sin θ
Correlator in CDL background
!
θ
1−
π
"
16/21
Analytically continue to Lorentzian space by X → T + πi/2, θ → iR
and define δT ≡ T1 − T2 , T̄ ≡ T1 + T2
we get the following T̄ -dependent part of the correlator !χ(T, 0)χ(T ! , R)"(T̄ )
(T̄ )
Ĝ
(X1 , X2 ; θ)
!
(T̄ )
!χ(T, 0)χ(T , R)"
=
a(T1 )a(T2 )
!
(T̄ +R)/2
(T̄ −R)/2
2
1
+
ie
ReR
e−T̄
1
+
ie
−(T̄ −R)/2
=− i+
+
ie−(T̄ +R)/2 log
−
ie
log
(
T̄
+R)/2
3
3π sinh R 12π sinh R
1 − ie
1 − ie(T̄ −R)/2
"
+4e(T̄ +R) log(1 + e−(T̄ +R) ) − 4e(T̄ −R) log(1 + e(T̄ −R) ) + 4T̄ e(T̄ −R)
The δT = T1 − T2 dependent part is
!χ(T1 , 0)χ(T2 , R)"(δT )
e−T̄
1
=−
2π cosh δT − cosh R
Correlator in CDL background
17/21
Large separation behavior
The geodesic distance ! between the two points on H3 located at
(R1 , 0) and (R2 , α), where α is an angle in S 2 , is given by
cosh ! = cosh R1 cosh R2 − sinh R1 sinh R2 cos α
In the large separation
! ∼ R1 + R2 + log(1 − cos α)
We take the late time limit as well as the large distance limit.
Namely, we first take the T̄ + ! → ∞ limit and drop the terms
with positive powers of e−(T̄ +!) . The rest of the terms are given as
an expansion in powers of eT̄ −!
Correlator in CDL background
18/21
Two interesting things coming out of this limit
1. Discrete sum
G1 =
∞
!
G∆ e(∆−2)(T1 +T2 )
∆=2
e−(∆−1)!
sinh !
−(∆−1)!
e
(∆−2)(T1 +T2 )
(∆−2)(T1 +T2 ) −∆R1 −∆R2
−∆
e
∼e
e
e
(1 − cos α)
sinh !
2. Nonnormalizable mode
!e!
G2 =
∼ R1 + R2 + log(1 − cos α)
3π sinh !
Correlator in CDL background
19/21
Summery
• AdS/CFT gives us the concrete holographic framework
to describe the gravity in AdS
• It’s likely that our universe is described by eternal
inflation. In particular the universe was tunneling from
false vacuum with higher cosmological constant
• We gave a circumstantial evidence for the existence of
the holographic dual of this tunneling backgound in the
spirit of AdS/CFT
Summery
20/21
Future works
• What is the boundary theory ?
• If there is the conformal theory, what is the central charge?
• What does the nonnormalizable mode imply? Liouville
theory?
• How to include bubble collision in the picture?
• Does it have observational consequence of the tunneling
event?
• It will be interesting to know if boundary theory can tell
us the measure of the eternal inflation
Future works
21/21