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

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

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

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)

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

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= ≡

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 >⇒ ><

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

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∆

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

−×

QUANTUM FIELD THEORY IN CURVED SPACETIME

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

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

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

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)

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)

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= − + + +

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=

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)

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)

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= −

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=

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)

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)

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)

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

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

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→ →

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

→∑ ∫

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⟨ ⟩ − ⟨ ⟩ = ⟨ ⟩ … ==− ⟨ ⟩

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

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

− − − +

→−

−+

∞

−+ +

= −

− →

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> ∆

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∆ ↔

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−∞ < <

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:

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 =

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−∞ < ≤

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)

PRIMORDIAL PERTURBATIONS

iH ta e∼

1/ 2a t∼

1

iH const−∼

1H t−∼t

phd

RADIATION

~ DE SITTER

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

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 −∼

SEMICLASSICAL GRAVITY

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.

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⟩ = ⟩

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= − −= −

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=

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 + ≡

+

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

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

⟨ ⟩ ≡⟨ ⟩

=

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−

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

+ −

±

− +

+ − =

⟨ … …= ⟩… − …

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 + −

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 +

∫

∫

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 =

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= =

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

• 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

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= + ∇

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+

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)