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QUANTUM FIELDS IN CURVED SPACETIME,
SEMICLASSICAL GRAVITYAND STOCHASTIC GRAVITY
E.Verdaguer
University of Barcelona
APCTP-NCTSInternational School/Workshop on
Gravitation and Cosmology
Pohang, 16-20 January, 2009
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OUTLINE
1. Introduction
2. QFT in curved spacetime
3. Bogoulibov transformations
4. Particle creation in cosmology
5. Semiclassical gravity
6. Stress tensor renormalization
7. Stress tensor fluctuations and stochastic gravity
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INTRODUCTION
BIBLIOGRAPHY :ND Birrell & PCW Davies, Quantum fields in curved space ,Cambridge University Press, 1982VF Mukhanov & S Winitzki, Introduction to quantum effectsin gravity, Cambridge University Press, 2007RM Wald, General relativity , The University of ChicagoPress, 1984RM Wald, Quantum field theory in curved spacetime and black hole thermodynamics , The University of ChicagoPress, 1994
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INTRODUCTION
HISTORY:Effects of gravity on quantum fields: Schrödinger (1932,1939)Weak gravitational field: Sexl-Urbantke (1967)Particle production in cosmology: Parker (1968),Zeldovich (1970)Stress tensor renormalization: Utiyama-DeWitt (1970),Wald (1978)Particle production by black holes: Hawking (1974)
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INTRODUCTION
LIMITS FROM QUANTUM GRAVITY :
Einstein-Hilbert action: 4
216 P
EHS d xl
gRπ
= −∫
3
33 1910 ~1.6 10Pl cm GeVc
G −= ×
spacetime region with curvature radius and 4-volumeL 4L2
~P
EH
LS
l
quantum curvature fluctuations with radius PL l≤
are unsuppressed (important) since action is ≤
[ ] [ ]1
/
2 |iS g
Dg gg e=⟨ ⟩ ∫
(the amplitude to go from a state with metric to a state w )thus gravity can be approximated as a classical field if PL l
1g 2g
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INTRODUCTIONMEANING OF PLANCK LENGTH :
r volume of size by Heinsenberg uncertainty principleexpects an energy fluctuationwith gravitational selfenergy
r
/rE
c∆ ≈
2~M
E
c cr
∆≡
2
23
2
~G
GM GE
r r c=
quantum and gravitational energies are comparablewhen~ GE E∆
3 Prc
Gl= ≡
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INTRODUCTION
Planck length is the minimum localization length :
• If a black hole shallows , iePr l< r
uncertainty principle Schwarzschild radius of : /r
Ec
∆ ≥ E∆ 2 4
2( )
2S
GM Er
c
GR
c= ∆≡
( )S P
P
r
l
R l
r≥ the uncertain is only if !
if( )SR r r<
Pr l≥
• Probe size in colliderswe need a quantum of energy : send particles with c.o.m. energy
and analyse collision products. But at energies above Planck massthe collision will create a black hole with horizon radiusShorter distances than are inaccessible!When b.h. evaporates Hawking particles
r~ /E c r∆
E∆4
2( )S
GR r
c
E∆=
SR
• In braneworld still plenty of room for QFT in CST16 17~10 10Te PVl l cm−=
( )SR r r>
~ ( )part SR rλ
( )P S Pr lr rl R >⇒ ><
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PARTICLE CREATION
( )4 2 21
2
ab
a bS d x mφ φ φη= − ∂ ∂ +∫• Free (real) scalar field in flat spacetime:infinite set of harmonic oscillators 2 2 2 2( ) | | ( ) | 0,| k k k kt t m kφ ω φ ω+ = ≡ +
• Free scalar field in cosmology:oscillators w time dep freq
( )4 2 21
2
ab
a bS d x g g mφ φ φ= − − ∂ ∂ +∫2 2 2 2 2( ) | | ( ) | (( /) )| 0,k k k kt t m k a ttφ ω φ ω+ = = +
potential
ground statewave function
2 2 / 2k kω φ
2 /2( ) ~ k k
k eω φφ −Ψ
when the frequency of the oscillator is “suddenly” changed the wavefunction no longer corresponds to the ground state. It is an excited“many-particle” state of the new potential
3·
3( , ) ( )
)(2
k x
k
ikt x
detφ φ
π= ∫
2 2 2 ( ) ·dt a t dx ds xd = − + Note: we have used: and
kφ
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PARTICLE CREATION
• The energy of a given mode in the cosmological model2 2 2/ ( )( )k kE t m a t+=
uncertainty principle gives the time a virtual pair with created at time tcan last a time
k±
t∆( ) ~
t
k
t
t
d tt E
+∆
′ ′∫ k
k−
• Take de Sitter and m=0 ( ) Hta t e= 11
() )( )
(
t t
H
t t
t t
k t
dt kE tdt
Ha tk
a et
∆+ +∆
∆ = −′′ ′ =
′ ∫ ∫
any massless virtual particle produced withcan last forever. Otherwise is finite.
| | ( ) ( )k Hk t a t≡ ≤
• For expansions with positive deceleration such asthe remaining time is finite but longer than in flat spacetime for m=0
1/2
0( )a t a t=
t∆
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PARTICLE CREATION
k
k−
• Gravity produces tidal forces between the virtual pair
Particles become real when work by tidal force isgreater than the energy of the particles
1
0 0
· ·k
k
El
F dl F dl E
−∆
= ≥∫ ∫
~ 1 / kl E∆
• Relative acceleration outside a spherical mass 2 2 3~
)(
GM GM GM
r r l rl∆
∆−
+1
0
21
3 3· ~ ~
2
k
k k
E
k
k
E GM EF dl E
r
M
E
G
r
−−
≥
∫
outside a black hole 2r GM≥
1c= =
3 3
2( )
k
GM
Er GM≥ ≥ 1
kEGM
≤
since horizon is a causal barrier, there should be no information in theparticles radiated: must be thermal with Hawking temptypical wavelength of radiation
1~
B
Tk GM~ SRλ
• Luminosityblack hole lifetime:
4
2~
1dML AT
dt M= =
3~ Mτ
33 8
12 12
10 ~ 10
~ ~ 10
2 ,
10 ,p
M g KT
M Tg K
−×
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QUANTUM FIELD THEORY IN CURVED SPACETIME
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FREE SCALAR FIELD
QUANTIZATION : Formally as in flat spactimeField quantization (no particle quantization as in QM)Particle concept is secondary and related to Poincareinvariance which we do not generally have in a curvedspacetime
ACTION
( )4 2 2 2 41( )
2
ab
a bS d x g g m R d xL xφ φ φ ξ φ= − − ∂ ∂ + + =∫ ∫ (1)
mass of scalar field: m metric:(-,+,+,+),a c
a ab acbR R R R= =b b b c
a a acv v v∇ = ∂ + Γ ( )1
2
c cd
ab b da a db d abg g g gΓ = ∂ + ∂ − ∂
[ ], d
a b c abcdv R v∇ ∇ =
Dimensionlessξ 1/6 conformal coupling , ex EM field in cosm backd0 minimal coupling , ex g. waves in cosm backd
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FIELD EQUATIONS
( )2 0KG a
aD m Rξ φφ ≡ ∇ ∇ − − = (2)0Sδ
δφ=
Assume (orientable) spacetime M is globally hyperbolici.e. it admits a Cauchy surface Σ
MΣ ⊂ is a closed hypersurface andnot connected by timelike curve
,p q∀ ∈Σ
p M∀ ∈ any time like curve through p intersectsin future or pasti.e. domain of dependence of
Σ
( )M D= Σ Σ
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GLOBALLY HYPERBOLIC
Spacetime topology (Geroch 70)and M can be foliated by a 1-parameter family ofCauchy surfaces
M = ×Σ
tΣt= time coordinate, defined by t=consttΣ
Theorem : if is globaly hyperbolic, KG eq.has a well posed initial value formulation.Given two smooth functions onThere is a unique solution to (2) on M such that
0
0
|
|a
an
φ φφ φ
Σ
Σ
=
= ∇
( )0 0,φ φ Σ
( ), abM g
n
tΣ
n: unit future directed normal to ΣS
( )D S
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INNER PRODUCT
If is a solution oh the KG eq. the current( )xφ( )* *
1 2 2 1
a a aj i φ φ φ φ= − ∂ − ∂
is conservedwe can define an inner product among solutions of KG eq.
0a
a j∇ =
( ) *
1 2 1 2, ( ) ( ) a
ai x x g dφ φ φ φ ΣΣ
= − ∂ − Σ∫
a ad n dΣ = Σ
* * *
1 2 1 2 2 1φ φ φ φ φ φ∂ ≡ ∂ − ∂
propeties ( ) ( )( ) ( )( ) ( )
*
1 2 2 1
1 2 2 1
*
1 2 2 1
, ,
, ,
, ,
φ φ φ φλφ φ λ φ φ
φ λφ λ φ φ
=
=
=
( )1 2,φ φ is independent of Σnot positive definite for a Hilbert product
(3)
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MODE NORMALIZATION
Assume we have a complete set of solutionsof the KG eq (2), normalized as
i,j characterize modes, if discrete okif continuous not normalizedNeed to put modes in a box(ex.plane waves) or usewave packets : discrete indices out of continuous modes
General solution of KG eq:
Fourier coefficients:
( )iu x
( )( )( )
* *
*
,
,
, 0
i j ij
i j ij
i j
u u
u u
u u
δ
δ
=
= −
=(0)iiδ δ→
( )*( ) ( ) ( )i i i i
i
x c u x d u xφ = +∑( )
( )*
,
,
i i
i i
c u
d u
φ
φ
=
= −
(4)
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CANONICAL QUANTIZATION
Canonical quantizationˆ
ˆ
i i
i i
q q
p p
→→
[ ] ˆ ˆ, ,i i i i ijPBq p i q p i δ= =
From QM to FT: ( ) ( , )iq t t xφ→
N
Nn
t
tΣ
lapse, shift vec : ,N N
0
t Nn N
n N
= +
⋅ =
Induced metric on tΣ ab ab a bh g n n= +
Adapted coordinates: tt
∂=∂
( , )it x 1
0
a
a
a i
a
t t
t x
∇ =
∇ =2 2 2 ( )( )i i j j
ijds N dt h dx N dt dx N dt= − + + +
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CANONICAL QUANTIZATION
a
att
φφ φ ∂≡ ∇ =∂
Action ( )S dtL x= ∫
( )2 2 2 31( ) ( )
2 t
a ab
a a bL x n h m R N hd xφ φ φ ξ φΣ = ∇ − ∇ ∇ − + ∫
where 1 1a a
a an NN N
φ φ φ∇ = − ∇
Canonical momentum ( )a
a
Sh n
δπ φδφ
≡ = ∇
Commutation relations [ ](3)
ˆ ˆ ˆ ˆ( , ), ( , ') ( , ), ( , ') 0
ˆ ˆ( , ), ( , ') ( ')
t x t x t x t x
t x t x i x x
φ φ π π
φ π δ
= =
= −
1=
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CANONICAL QUANTIZATION
Hamiltonian: ( )3
t
H d x LπφΣ
= −∫
dynamical eqs (Heisenberg pict)
ˆ ˆˆ,
ˆˆ ˆ,
i Ht
i Ht
π π
φ φ
∂ = ∂∂ = ∂
( )2 ˆ 0a
a m Rξ φ∇ ∇ − − =
field operator satisfies KG eq
( )† *ˆ ˆ ˆ( ) ( ) ( )i i i i
i
x a u x a u xφ = +∑Solutions (selfadj) ( )ˆˆ ,i ia uφ=
Commutation rels† †
†
ˆ ˆ ˆ ˆ, , 0
ˆ ˆ,
i j i j
i j ij
a a a a
a a δ
= =
= †ˆ ˆ,j ia a creation, annihilation op .
(5)
(6)
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QUANTUM STATES
( )iu xModes with +ve def product define a Hilbert spacethe 1-particle Hilbert spaceDefine the Fock space of states of the field
H
( )( ) ...F H H H H= ⊕ ⊕ ⊗ +
( )0 1 2, , ,... ( )F Hψ ψ ψ ψ= ∈
amplitude to find field in vacuum stateprojection in space of 1-particle statesprojection in space of 2-particle states
0ψ
1ψ2ψ
( )* * *
1 2ˆ( ) , 2 ,...a u u uψ ψ ψ= ⋅ ⋅ ( )†
0 1ˆ ( ) 0, , 2 ,...a u u uψ ψ ψ= ⊗
vacuum state1-particle state in modemany particles state
( )0 1,0,0,...≡ ˆ 0 0,i ia u= ∀iu †ˆ1 0i ia=
1 21 2 1 2 1/ 2 † †ˆ ˆ, ,... ( ! !...) ( ) ( ) ... 0n n
i j i jn n n n a a−=
(7)
(8)
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PARTICLES IN MINKOWSKI ST
3
·1( )
2 (2 )
ki
i
k
t ik xu x e
ω
ω π− +=
2 22
|( , ), |a
k
k
k k
k
k k
m
ωω
≡
−
=
=
k is associated to translational isometries ontΣ
+ve frequency modes satisfy kk k
ui u
tω∂ =
∂Physical meaning of states
( )ab ab
a
LT Lη
φ∂= −
∂ ∂Hamiltonian op 3 †
3 †
( )
1ˆ ˆ ˆ ˆ2
ˆ ˆ ˆ ˆ
t
t
tt i i i
i
j j j
t i i i
i
H d xT a a
P d xT a a k
ωΣ
Σ
= = +
= =
∑∫
∑∫
Momentum op †ˆ ˆ ˆ
ˆ ˆ
í i í
í
i
N a a
N N
=
=∑ˆ ˆ ˆ ˆ, , 0jN H N P = = ˆ ˆ0 0 0 0 0
1ˆ0 02
j
i
i
i
N P
H ω
= =
=∑ 1 2 1 2 1ˆ, ,... , ,...i j i i jn n N n n n=
State particles of energy and mom1niω ( )
j
ikˆ ˆ ˆ0 0RH H H= −
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STATIONARY SPACETIMES
There is a timelike Killing vector t 0abtL g =
, KG
tiL D are selfadjoint
and commute , 0KG
tiL D =
( ) ( )†ˆ ˆ, ,A Aφ χ φ χ=†ˆ ˆA A=
there is a base of common eigenfunctions
0
i i it
KG
i
iL u u
D u
ω=
=
these are the +ve frequency modesand one has ( )3 †1 1ˆ ˆˆ ˆ ˆ....
2tt t i i i
i
H d x h a aN
φ φ ωΣ
= ∂ ∂ + = +
∑∫
One may def vacuum and particle stateswith well def energy , but not eigenstatesof momentum in general
†ˆ1 0i ia=
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NEW MODES
Assume another complete set ofnormalized KG solutions ( )iu x
( )( )( )
* *
*
,
,
, 0
i j ij
i j ij
i j
u u
u u
u u
δ
δ
=
= −
=We may write
( )† *ˆ ˆ ˆ( ) ( ) ( )i i i i
i
x a u x a u xφ = +∑
( )( ) ...F H H H H= ⊕ ⊕ ⊗ +
HOne may def a 1-particle Hilbert spacea new Fock space of statesand a new vacuum state as
ˆ 0 0,i ia u= ∀
The new creation and anihilation operators satisfy as usual† †
†
ˆ ˆ ˆ ˆ, , 0
ˆ ˆ,
i j i j
i j ij
a a a a
a a δ
= =
=
(9)
(10)
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BOGOLIUBOV COEFFICIENTS
The new modes in terms of the old modes ( )iu x ( )iu x
( )*
j ji i ji i
i
u u uα β= +∑
( )* *
ji ji ji j
i
u u uα β= −∑and also
where ( )( )*
,
,
ij i j
ij i j
u u
u u
α
β
=
= −
Bogoliubov coefficientsindependent of tΣ
(11)
(12)
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BOGOLIUBOV TRANSFORMATIONS
( )iu xSubstituting in terms of in (11),the Bogoliubov coeff satisfy
( )iu x
( )( )
* *
0
ik jk ik jk ij
k
ik jk ik jk
k
α α β β δ
α β β α
− =
− =
∑
∑† †
0T T
Iα α β βα β β α
⋅ − ⋅ =⋅ − ⋅ =
(13)
comparing (5) and (9) we get
( )( )
*
* * †
†ˆ ˆˆ
ˆ ˆ ˆ
i ji j ji
j
j ji i ji i
i
ja a a
a a a
α β
α β
= +
= −
∑
∑(14)
Note: the a’s satisfy correct commutation relations iffthe a’s bars do, provided (14)
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TWO DIFFERENT VACUA
When beta coefficient not zero the u-bar-modes contain –veu-modes and we have two different vacua
0 0≠
* † *ˆˆ 0 0 1i ji j ji j
j j
a aβ β= =∑ ∑then
2ˆ0 0 | |i ji
j
N β=∑
• The vacuum contains“particles” of modes
• When there are in and out regionsasymptotically stationary we haveGRAVITATIONAL PARTICLE CREATION
iu2| |ji
j
β∑0
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PARTICLE CREATION IN COSMOLOGY
Spatially flat FRWwith static “in” and “out” regions
2 ( ) ababg η η= Ω ( )t dd η η= Ω
1 2lim limη η→−∞ →∞Ω = Ω Ω = Ω
KG eq is separable3/2
1 1( )
) ( )(2
ik x
k keu χ ηπ η
⋅=Ω
( )2
2 2 2
2
( )6 1 ( ) 0k
k
dk m
d
χ η ξ χ ηη
Ω+ Ω + − = Ω +
[ ]2
2
( )( ) ( ) 0
xE V x
d
dxx
ψ ψ− =+
where
analogous to scattering over a potential barrier in QM
2/6R = Ω Ω
ψ wave function in 1D
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QM ANALOGUE
kTkR
1
( )V x−
2 2m Ω +…
α
β1
x
η
1
2 2
x
k
ik x
ik
ik x
k
e x
e R e x
T
−
− → −∞
+ → ∞
1
2 2
2
1
1
2
1
2
i i
k k
ie
e e
ω η
ω η ω η
ηω
α β ηω
−
− → −∞
+ → ∞
kχ
kψ
IN OUT
Left traveling wave(+ve freq out mode)is partly transm. and partly reflected(-ve freq out mode)
ˆ /
ˆ /
p i t
H i t
= − ∂ ∂
= ∂ ∂
cons current( )* *
2 2| | | | 1k k
j i
T R
ψ ψ ψψ−
+⇒ =
=
corresp 2 2| | | 1| k kα β− = norm cons ) 1( ,k kuu =1 k
k k
k k
R
T Tα β→ →
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NUMBER OF PARTICLES
1
2
·
3/2
1 1
·
3
2
/2
2
1 1( )
(2 ) 2
1 1( )
(2 ) 2
iik x
k
iik x
k
eu e
eu e
ω η
ω η
ηπ ω
ηπ ω
−
−
→ −∞ =Ω
→ +∞ =Ω
IN +ve freq. mode:
OUT +ve freq. mode:
( )* *
k k kk k kl l l ll
ku u uu uα β α β− == + +∑
so that * †, k k kkl k kl kl k kl k ka a aα α δ β β δ α β− −= = = +
Number of OUT particles in mode in the IN vacuumat in volume
k
η → ∞ 3
2( )LΩ † 20 | | 0 | |k k k kN a a β⟨ ⟩ = ⟨ ⟩ =
Average particle density:
2 2 2
3 2 3 02 2
1 1| | | |
( ) 2kk
k
n N dkkL
β βπ
∞⟨ ⟩ = =
Ω Ω= ∑ ∫
3
3 3
1 1
(2 )k
d kL π
→∑ ∫
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NUMBER OF PARTICLES UNCERTAINTY• Number of particles is an adiabatic invariant: for slowexpansion WKB approx OK, well def if0kβ → 1
Hubbletω ≥
• Let A particle creation rate during intervalto measure particle number with precissionbut number of particles uncertaintotal uncertainty
t∆| 1|A t∆
~ 1E t∆ ∆ = ~E m N∆ ∆
1| |N A t
m t∆ ≥ + ∆
∆min 2 | | /N A m∆ =
• Also fluctuations in particle number are always not zero,if there is particle creation:
2 2 † † † 2 2 20 | 0 0 | 0 2 | | | || |k k k k k k k k k kN N a a a a a a α β⟨ ⟩ − ⟨ ⟩ = ⟨ ⟩ … ==− ⟨ ⟩
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CONFORMAL COUPLING• Two conformaly related metrics: 2( ) ( ) ( )ab abg x x g x= Ω
2 3 42( 1) ( ) ( 1)( 4) ( )( )ab ab
a b a bR R n g n n g− − −= Ω + − Ω ∇ ∇ Ω + − − Ω ∇ Ω ∇ Ω
• Define a conformaly related scalar field
( )1a ab
a a bggg
φ φ∇ ∇ = ∂ − ∂−
(2 )/2nφ φ−= Ω
using that
we obtain ( 2)/22 2
4( 1) 4( 1)
a n a
a aRn n
nR
nφ φ− +
∇ − −
− − + Ω ∇ ∇ +
∇ =
No particle creation for conf. fields in confly flat backgrounds
• If KG eq:2 , 0, 1/ 6a abb mg η ξΩ = == ( )310
6
a
a
a
aR φ φ− = ∇ ∇ + = Ω ∂ ∂ Ω
Modes are sols in flat st andare +ve freq modes in IN andOUT regions: NO PARTICLE CREATION
φΩ ·
3
1 1,
2| |
(2 )
t ik xi
k eu kω ωω π
+−= =Ω
1 2,Ω → Ω Ω → Ω
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SMOOTH EXPANSION2
2 2
1 2
0, 1 / 6, ( ) tanh( )
, ,
A B
const A B A B
m ξ η ρηρ
= = Ω = +/= Ω = − Ω = +
Assume (Bernard-Duncan,77 in QM)
2
1Ω
2
2Ω
η
2 ( )ηΩ
1/ ρ
( )2
2 2 2
2
( )( ) 0k
k
dk m
d
χ η χ ηη
+ Ω =+
Eq for modes:
solved in terms of hypergeometric functs.
( ) 2 1
·1 12 1 , ;1 ;
2
tanh( ,
4 4) exp · ln cosh
inin i ik x
kin in
iu x ik x
i i ei
iF
ω η
η
ω ω ω ω ρηη ω η ρηρ ρ ρ ρπω πω
− − − ++
→−∞
− ++ −
−
=
− →
IN
OUT
where ( )2 2 2 2 2 2
1 2
1
2, ,in out out ink km mω ω ω ω ω±≡ + Ω ≡ + Ω ≡ ±
then writeand determine and
*
k kk k ku u uα β −= +
kα kβ
( ) 2
·
1
tanh( , ) exp · ln cosh
1 12 1 , ;1 ;
24 4
out k x
kou
out i i
t out
i i i i eFu x ik x i
ω η
η
ω ω ω ω ρηη ω η ρηρ ρ ρ ρπω πω
− − − +
→−
−+
∞
−+ +
= −
− →
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SMOOTH EXPANSION
one getsfrom wherenumber of particlescreated in mode k
( ) ( )( ) ( )
( ) ( )( ) ( )
1/2 1/21 1
,/ 1 1
/ / / /
/ / /
in out in outout out
k kin in
i i i i
i i i i
ω ρ ω ρ ω ρ ω ρω ωα βω ωω ρ ω ρ ω ρ ω ρ+ − −+
−Γ Γ Γ Γ= =
Γ Γ Γ
− − − − + Γ
22 sinh ( / )
| |sinh( / )sinh( / )
k k in outN
πω ρβπω ρ πω ρ
−
⟨ ⟩ = =
SLOW EXPANSION/
0
2 in
k eN πω ρ
ρ→
−⟨ ⟩→When expansion rate we are in the adiabaticapproximation and exponential suppression for freq
/ ~ 0ρΩ Ω →
exp1 / η> ∆
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STEP EXPANSION
2
1Ω
2
2Ω
η
2 ( )ηΩ
Corresponds to1 0 2 0, ( ) ( ) ( )ρ η θ η η θ η η→ ∞ Ω = Ω − + Ω −
Solutions are easy, imposing continuityand “current” conservation, we get:
0 0
0
2 2
0
( )
( )
,
,
in
out out
k
i i
k out ou
i i
t
ie
e e e e
ω η
ω η ω ηω η ω η
χ η η ηω ωχ η η ηω ω
+−+
−−
−
−
−
= <
= >−
from where 0 02 2,k k
i i
out in out ine e
ω η ω ηω ωα βω ω ω ω
+−−+
= =
0η
22 ( )
| |k k out inN
ωβω ω
−
⟨ ⟩ = =
( )2 2 2 2 2 2
1 2
1
2, ,in out out ink km mω ω ω ω ω±≡ + Ω ≡ + Ω ≡ ±
High freq modesnot well repres.need cut off , since from smooth
4 4
2 21 1~1/ , ~ , ~energyk N k dk k dk
kk
kω ρ− ⟨ ⟩ ⟨ ⟩ → ∞∫ ∫
~kN e πω η− ∆ ~1/ ( )cω π η∆
1/η ρ∆ ↔
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DE SITTER
• Inflationary cosmology is well described by de Sitter:• Maximally symmetric. Hypersurface orthogonal
timelike Killing field with Killing horizons
2
0
2
4
1 1sinh( )
2
1 1cosh( )
2
; 1,2,3; ,
Ht
Ht
Ht
i i i
z Ht He xH
z Ht He xH
z e x i t x
= +
= −
= = −∞ < < ∞
oz
4z
1z
0 4 0z z+ > t cnt=
x cnt=Hyperboloidin 5D MinkowskiIn spatially flat coord (cover ½):
2 2 2 2 2 2
0 1 2 3 4ds dz dz dz dz dz= − + + + +
( )2 2 2 2 2( )Ht i j i j
ij ijds dt e dx dx d dx dxδ η η δ= − + = Ω − + 1( )
Hη
η−Ω = 1 Hte
Hη −−=
/ 1/ 0η−∞
Ω Ω = − → 2/ 2 /ηΩ Ω =
2 2 2 2 2 2
0 1 2 3 4z z z z z H −− + + + + =
0η−∞ < <
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DE SITTERCorresponds to a FRW model with a cosmological constantthis can be seen as a perfect fluid with p ε= −
)(ab a b ab abp u u pgT gεε Λ+ − ≡=
Λ
For a flat FRW model00 Einstein eqs lead to
2 2 2 ( ) i j
ijdt a t dds dx xδ= − +2
2 8
3H
a
a Gπ εΛ
≡ =
0,a
abT constεΛ= ⇒∇ =
1( ) Hta t H e−=
The metric describes a static maximally symmetrixc spacetimein expanding coordinates. These coordinates do not cover thedS spacetime. Introduce conformal timethen
1 e (( p) x )ta t d Ht tη −∞
= − −= −∫( )2 2 2 2 2 2 2
2 2( sin 0,
10) ,dr r dd dd
Hrs η θ θ ϕ η
η= + + + − ∞ < < ≤− < ∞
change to new coord ,sin sin
cos cos cos cosr
η χηη χ η χ
= =+ +
( ( ))1
aH
ηη
η −Ω =≡scale factor:
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DE SITTER( )2 2 2 2 2 2 2
2 2sin ( sin 0,)
s
1, 0
ind dds d d
Hη χ χ θ θ ϕ π η χ π
η= + + + < <− − < ≤
covers entire dS. Closed univ. contracts forexpands for flat, closed, open dS describe samespacetime in diff coord. Any hypersurface is hypersurface ofconstant energy density (unlike other FRW).
/ 2π η π< < −−
/ 2 0π η<− <
We get
0χ = χ π=
0η =
η π= −
constη =
constη =
constχ =
CONFORMAL D. OF DSeχ
pχ
r const=
0r =
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BUNCH-DAVIES VACUUM
Klein-Gordon eq ( )2 0a
a m Rξ∇ ∇ − − Φ =
metric: 2 ( )ab abg η η= Ω
modes:3/ 2
1 1( )
(2 ) ( )
ik x
k ku e χ ηπ η
⋅=Ω
( )2
2 2 2
26 1 0k
k
dk m
d
χ ξ χη
Ω+ + Ω + − = Ω
1( )
Hη
η−Ω =
2
2
ηΩ =Ω
in de Sitter:
36 /R = Ω Ω
• At there is an initial vacuum state def by +ve freq modesη → −∞1
2
ini
k
in
eω ηχ
ω−→
2 2
in kω =
( )†ˆ ˆ ˆ( ) ( ) ( )k k k k
k
x a u x a u x∗Φ = +∑ †
'ˆ ˆ, ( ')k ka a k kδ = −
ˆ 0 0;ka k= ∀
0η−∞ < ≤
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MECHANISM FOR PERTURBATIONS
• In inflation : Quantum fluctuations of scalar and tensor metricmodes. Scalar modes couple to matter
• Basic mechanism: scalar field, 0m ξ= =
22
20k
k
dk
d
χ χη
Ω+ − = Ω
2 2 2 2 2( ) 2k k Hω η Ω≡ − = − Ω
Ω
2 2 22ph phk Hω = −
, Ht
ph ph
kk ke
ωω −≡ ≡ =Ω Ω
• If initially field oscillates, but redshifts exponentiallyin time, eventually stops oscillating and latter :
phk H phk
2phk H=phk H
2
2 2
20k
k
d
d
χ χη η
− =21
k g dc cχ ηη
= +
for growing solution: 3/ 2
1 1
(2 )
ik x
k ku e frozenχπ
⋅=Ω
∼
grows
decays
grows
• Cosmological per (Lifshitz 46, Bardeen 80, Kodama-Sasaki 84,Mukhanov et al 92)
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PRIMORDIAL PERTURBATIONS
iH ta e∼
1/ 2a t∼
1
iH const−∼
1H t−∼t
phd
RADIATION
~ DE SITTER
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PRIMORDIAL PERTURBATIONS
ph Hω
• As universe expands, redshifts and , oscillation stopsand field amplitude approaches a constant value
ph Hω ∼
• Before oscillation stops the field has quantum zero pointfluctuations which are preserved by further expansion
• A mode with is probably in its ground state, asprevious expansion redshifted away any initial excitation
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PRIMORDIAL PERTURBATIONS
• Since amplitude frozen when mode had it is thesame for all modes, apart from volume factor in normalizationwhich depends on cosmological time at freeze out
ph Hω ∼
• After inflation ends expansion rate drops faster thanwavenumber, when mode oscillates again: seeds fordensity perturbations that grow by gravitational instabilities
1
ph Hλ −∼
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SEMICLASSICAL GRAVITY
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THE STRESS TENSOR
• Gravity couples to the stress energy tensor of matter. To couple the classical metric to quantum matter fields weneed a classical stress tensor made out of quantum fields.The only reasonable one is the expectation value ofthe quantum stress tensor in some quantum state of thefields.
• But this stress tensor is formally a bilinear product of thefield operators and has divergences due to the divergencesin the two point function in the coincidence limit.
• In flat spacetime these divergences are easily dealt withby substraction of the e.v. in the vacuum state. In curvedspacetime we need a more subtle covariantrenormalization procedure.
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EFFECTIVE ACTION• Classicaly Einstein eqscan be obtained from the action i.e.where
8ab B ab abBg TG Gπ− = −Λ
g mS SS= + 0ab
S
g
δδ
=
2ab ab
m
gT
g
Sδδ
=−
• In semiclassical gravity we want on rhs:in QFT this can be obtained from the effective action
ˆ abT⟨ ⟩
W2
ab ab
W
gT
g
δδ
=−
⟨ ⟩
( ) ( ),
2 2
,
1 12 , (
16 2)n n
g B m
ab
a b
B
S d g R dx S RG
g mx g φ φ ξ φπ
− − −= +Λ =−∫ ∫
• To find consider the generating functionalW
[ ] [ ] [ ] ( ) ( )
,0 | 0,n
miS i d x gJ x x
Jout in D eZ Jφ φφ + −∫≡ ⟨ ⟩ = ∫
in flat space because[ ]0 1Z = | 0, | 0,out in⟩ = ⟩
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EFFECTIVE ACTION[ ] [ ] [ ]
, ,0 0 | | 0m
m m
iSi D iZ S e out S in
φδ φ δ δ ⟩= ⟨=∫
[ ]0 iWZ e≡def effective actionnot exp value but same ultraviolet div
take| 0,2
0,
,0 |
,0 |
ab
ab
inout TW
outg ing
δδ
⟨ ⟩
⟨=
− ⟩W
Effective action in terms of the propagator
[ ] [ ]exp2
n n n
x x xy y x x xyxd y g g Ki
Z J D d i d x g Jφ φ φ φ = − +
− − −∫ ∫ ∫
changing the integration variable a Gaussian integration
[ ] ( ) 11/2det exp ( ,
2)n n
x y x F yK xd y Gi
Z J d g x yg JJ− = − − −
∫as expected [ ] ( )
2 ln( ) ( ) ( , )
( ) ( )F
ZT x y iG x y
J x J
J
y
δφ φ
δ δ⟨ ⟩ = −= −
( ) ( ) ( )11/2det det( ) exp ln(tr )(1/ 2)F FK G G
−= =− −
[ ] ( )ln ln )0 t2
(r F
iW i Z G= − −= −
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EFFECTIVE ACTION
Using the DeWitt-Schwinger series for the Feynman propagin curved st (using normal coordinates centered at x), we maydefine the effective Lagrangian
eff
nW d gx L= −∫( ) 2
1/2
1 /2 (
' / 20
/2 )
2
0
4
/2
/20
( , )lim ( , ( )
2(4
1
))
( )( )2)2(4
j n i m
eff x nj
n
n j
s s
x j
nj
j
L a idsx x
x is e
m
x
nx m ja
σ
π
µπ
− − −∞∞
−→
=
− ∞−
=
′∆′
≈ Γ
= −
−
∑ ∫
∑
in 4D the first three terms divergeintroduce mass scale parameter to keep dim of 4D
( ) ( , )j jx a xa x=
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EFFECTIVE ACTIONusing that the first three terms are( ) 1/ )(OγΓ = − +ε ε ε
4 22
0 1
2/2 2
41 1 1
4 2
2ln
( ) ( 24 2)div n
m aa
m amL
n n n nγ
π µ
+
= − + − − − +
−
where0 1
2
2 2
2
1,
6
1 1 1 1 1 1
180 1
( ) 1, ( )
(80 2 6
)6 5
abcd ab
abcd ab
a R
R R
x a x
a x R RR R
ξ
ξ ξ
−
− − − −
= =
= ∇ +
which are purely geometrical and can absorbed into the Lag
2/2
2
2ln
2 1 1 1 1( ) ( ) ( )
16 16 4) 2(4
new
g n
B
B B
mL x A B R x a x
nGGγ
π π π µ
= − + + + − + −
Λ+
where2
/2 /
4 2 2
2 22
4 1 1 2ln ln
(1/ 6 ) 1 1,
4 2 4 2(4 ( 2) (4 ( 2)) )n n
m m m mA
n nn n nB
ξγ γπ µ π µ
−≡ + + − −− − +
+
≡
renormalized constants: 8 ,1 16
B
B B
B
GG A G
G Bπ
πΛ ≡ Λ + ≡
+
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LHS OF SEMICLASSICAL EINSTEIN EQren eff divL L L−≡
• One writes in the lhs of Einstein of equation• Define the renormalized gravitational Lag
(2)
ab ab ab ab abgG HB Hα β γ+ + + +Λ
;
(2
2
)
2 2
2 2
2
;
;
1 12
2
1 1 1
2 2
12 4 2 4 4
2 2
2
1
2
ab ab ab ab abab
cd cd
ab cd ab ab ab ab
n
n cd
n
cd cdabab
cdef cde
a
f cde c cd
ab cdef cdef acde b ab a bb b acab
x R R RRg
x R R
B d gR g g Rg
H d gR R g g R Rg
H d gR R R R R R R R Rg
R R Rg
x g R Rg
δδ
δδδ
δ
≡ = ∇ − +
≡ = ∇ −
− −−
− − −∇ +
≡ = −
−
− + − + − +−
∇
∫
∫
∫ cadbR
In 4D the toplogical invariant ( )4 24abcd ab
abcd abd g R R Rx R R− − +∫(2)4ab ab abH B H= − +
and there is a two parameter ambiguity• To obtain the ev of the stress tensor on rhs of Einstein eqneed other approaches. Ex: In-in or CTP effective action
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IN-OUT VERSUS IN-IN
The in-out formalism begins with the generatingfunctional related to the vacuum persistence amplitude[ ]W J
[ ],0 | 0,
iW J
Je out in≡ ⟨ ⟩
In the interacting picture it can be written as[ ] ( ),0 | exp | 0,
iW J
Iout T i dte H in∞
−∞≡ ⟨ ⟩∫
1 () )(n
IH d x J x xφ−= ∫
its path integral representation is[ ] [ ] [ ] ( ) ( )niS i d x Ji xW J x
D eeφ φφ + ∫= ∫
one generates the matrix elements (effective mean field)
[ ] [ ],0 | ( ) | 0,
( ) ,0 | 0,
J
J
W out x in
J x out
JJ
in
δ φ φδ
⟨ ⟩ ≡⟨ ⟩
=
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IN-OUT VERSUS IN-IN
To work with expectation values one can define a newgenerating functional determined by two external classicalsources and let the in vacuum evolve independently
[ ],,0 | , , | 0,
JiW J
J Jin T T ineα
α α+ −
+−⟨= ⟩ ⟨ ⟩∑
| ,Tα ⟩ is a complete basis of eigenstates of the field operatorat some future time ( , ) | , ( ) | ,T x T x Tφ α α α⟩ = ⟩
Its path integral representation[ ] [ ] [ ]( ) [ ] [ ]( ), i S J iiW S JJ J
d D De e eφ φ φ φα φ φ− − −+ + +− +− + −
− ++= ∫∫ ∫
with b.c. and pure +ve, pure –ve freq in past| |T Tφ φ α+ −= = φ+ φ−
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IN-OUT VERSUS IN-INIn more compact form (b.c. understood)
[ ] [ ] [ ] [ ] [ ]( ), i S Ji J SJW JD D ee
φ φ φ φφ φ + + −− + −+ − −+
−−
+= ∫the functional generates expectation values
[ ] [ ],,0 | ( ) | 0,
( )J
JJ
W Jin x in
J
JJ
x
δφ φ
δ±
+ −
+ =
⟨ ⟩= ≡
also time and anti-time ordered correlators[ ]
( ) ( ),
1 1
1 1(,0 | ( ) ( ) | 0,
) ( ) ( )J
iW J J
J
J
ein T y T x in
i J i Jx y
δ φ φδ δ
+ −
±
− +
+ − =
⟨ … …= ⟩… − …
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INFLUENCE FUNCTIONAL
• Here we have two fields : the gravitational field which istreated classically and the matter fields which are quantum.We can integrate out the matter fields as in an openquantum system (Feynman-Vernon 63)
abg jφsystem environment
[ ] [ ] ( )exp , ,IFiS
IF m mF e D D S g S gφ φ φ φ+ −+ − + − ≡ = − ∫
• In-in (CTP) effective action for dynamical eq of expectationvalues: Schwinger 61, Keldysh 64
is called the influence functional, describeseffect of matter fields on the gravitational field
,IF gF g + −
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STRESS TENSOR EXPECTATION VALUE
• The expectation value of stress tensor operator canbe obtained from the influence action
'
[ , ']2ˆ [ ]ren
IFab ren g gab
S g gT g
gg
δδ =⟨ ⟩ =
( )
( )
4 2 2 2
4
1[ ]
2
1[ ] 2 [ ]
2
ab
m a bM
c
B
B
g gM
S g d x g g m R
S g d x g R S g
φ φ φ ξ φ
κ
= − ∇ ∇ − −
= − − Λ +
∫
∫
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CTP EFFECTIVE ACTION
• The semiclassical Einstein equation can be obtained form theClosed Time Path effective action, neglecting graviton loops
[ ] [ ] [ ] [ ], ' ' , 'CTP g g IFg g S g S g S g gΓ = − +
• The influence functional has ultraviolet divergencies that canbe renormalized using counterterms in the gravitational action
[ ] [ ] [ ] [ ], ' ' , 'ren ren ren
CTP g g IFg g S g S g S g gΓ = − +
•The semiclassical Einstein equations are obtained as
'
[ , ']0CTP
g gab
g g
g
δδ =
Γ =
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SEMICLASSICAL EINSTEIN EQUATIONRenormalization introduces quadratic tensors(4 renorm coupling constants, determined by exp)
ˆ[ ] [ ] [ ] [ ]ab ab ab ab ab renG g g A g B g T gα β κ+ Λ − − = ⟨ ⟩
where 41ab cdef
cdef
ab
A d x gC Cgg
δδ
= −− ∫
4 21ab
ab
B d x gRgg
δδ
= −− ∫
28 8 / PG mκ π π= =
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LIMITS OF SEMICLASSICAL GRAVITY
• Away from Planck scales : when effects of gravity ignored(q energy flucts in Planck size vol = gravitational energyof fluctuation)
• Quantum fluctuations of stress tensor small (suitable states):Ford 82, Kuo-Ford 93, Phillips-Hu 97,00
2 2ˆ ˆ 0T T⟨ ⟩ − ⟨ ⟩ ≈
P∆ ,Pt t∆
If N (large) fields are assumed to be coupled to gravityit is zero whennext to leading is
N → ∞( )1 /O N
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• The 2-point quantum correlations of stress tensor play a role
• Noise kernel ph observable that measures quantum fluctsof stress tensor (free of ultraviolet divergencies)
(real and +ve semidefinite)It defines a Gaussian stochastic tensor
• Symmetric, divergenceless, traceless (conformal field).
1ˆ ˆ( , ) ( ), ( )
2abcd ab cdN x y t x t y= ⟨ ⟩ ˆ ˆ ˆ
ab ab abt T T I≡ − ⟨ ⟩
0ab sξ⟨ ⟩ = ( ) ( ) ( , )ab cd s abcdx y N x yξ ξ⟨ ⟩ =
[ ]ab gξ
NOISE KERNEL
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STOCHASTIC GRAVITY
• We will assume linear perturbation of semiclassical solution
ab abg h+
• Einstein-Langevin equation:
ˆabT
ˆ( )g h g hG Tκ ξ+ += ⟨ ⟩ +
(1) (1)ˆ[ ] [ ] [ ]ab ab ren abG g h T g h gκ κξ+ = ⟨ + ⟩ +
• Extend semiclassical Einstein equations to consistentlyaccount for the fluctuations of
2 2 ˆ( ) 0g h m Rξ φ+∇ − − =
it is gauge invariant ( )' 2ab ab a bh h ζ= + ∇
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SOLUTIONS OF EINSTEIN-LANGEVIN EQUATIONS
• They are stochastic equations and determine correlations
0 4( ) ( ) ' ( , ') ( ')ret cd
ab ab abcdh x h x d x gG x x xκ ξ= + −∫0 0 2( ) ( ) ( ) ( ) ( , ') ( ', ') ( ', )ret efgh ret
ab cd s ab cd s abef ghcdh x h y h x h y G x x N x y G y yκ⟨ ⟩ = ⟨ ⟩ + ∫∫
(flucts due to initial state) (due to matter field flucts)
1 ˆ ˆ( ), ( ) ( ) ( )2
ab cd ab cd sh x h y h x h y⟨ ⟩ = ⟨ ⟩
• It can be shown (Roura-EV) that q. metric correl. in 1/N:
Intrinsic fluctuations Induced fluctuations+
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SOME APPLICATIONS OF S.G.
• Structure formation in cosmology , agrees with linearperturbation approach and can go beyond, to includeone-loop matter contributions (Weinberg 05,06,Maeda-Urakawa 08, Roura-EV 08)
• Fluctuations near black hole horizons (Hu-Roura 06,08)
• Validity of semiclassical gravity (Horowitz 80, Anderson,Molina-Paris and Mottola 03, Hu-Roura-EV 04)
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