Quantum Field Theory for Gravity and Dark Energy

Sang Pyo KimKunsan Nat’l Univ. & APCTP

CosPA2009, U. Melbourne, 2009

Outline• Motivation • Vacuum Energy and Cosmological

Constant• QFT Method for Gravity• Conformal Anomaly• Dark Energy• Conclusion

Friedmann-Lemaitre-Robert-son-Walker Universe

• The large scale structure of the universe is homo-geneous and isotropic, described by the metric

• The theory for gravity is Einstein gravity

• Friedmann equations in terms of the redshift

)sin(

1)( 2222

2

2222 ddr

Krdrtadtds

GTgG 8

])1()1()1([)( 02

03

04

020

22

zzzHzHaa

KMR

])1(21)1([ 0

30

40

20 zzH

aa

MR

)()(1

em

obs

tataz

Hubble Parameter & Dark En-ergy

• Radiation

• Matter

• Curvature

• Cosmological con-stant

40

20

2 )1()( zHzH R

30

20

2 )1()( zHzH M

20

20

2 )1()( zHzH K

020

2 )( HzH

WMAP-5 year data

Dark Energy Models[Copeland, Sami, Tsujikawa, hep-th/0603057]

• Cosmological constant w/wo quantum gravity√• Scalar field models: what is the origin of these fields?

– Quintessence– K-essence– Tachyon field– Phantom (ghost) field– Dilatonic dark energy– Chaplygin gas

• Modified gravity: how to reconcile the QG scale with ?– f(R) gravities– DGP model

Early Universe & Inflation Models

4222

!422)(

RmVChaotic Inflation

Model

Vacuum Energy and • Vacuum energy of fundamental fields due to

quantum fluctuations (uncertainty principle):– massive scalar:

– Planck scale cut-off:

– present value:

– order of 120 difference for the Planck scale cut-off and order 40 for the QCD scale cut-off

– Casimir force from vacuum fluctuations is physical.

2

4cut22

0 3

3

vac 16)2(d

21 cut

kmk

471vac

)GeV(10

447 )GeV(108

G

Vacuum Energy and • The uncertainty principle prevents the vacuum

energy from vanishing, unless some mechanism cancels it.

• Cosmological constant problem– how to resolve the huge gap? – renormalization, for instance, spinor QED

– supersymmetry, for instance, scalar and spinor QED with the same spin multiplicity )sin(/1)cot(

8)(

0 2

/

2

2sceff

speff

2

sss

edsqELLqEsm

]3//1)[cot(8

)(charge

energyvacuum

0 2

/

2

2speff

2

ssss

edsqELqEsm

Vacuum Energy in an Ex-panding Universe

• What is the effect on the vacuum energy of the expansion of the universe?

• Unless it decays into light particles, it will fluctuate around the minimum forever!

• A systematic treat-ment next

QFT for Gravity • Charged scalar field in curved spacetime

• Effective action in the Schwinger-DeWitt proper time inte-gral

• One-loop corrections to gravity

)(,)(,0)( 2 xiqADmDDxHxH

22

][

2lnTr

2,

)2/det(

1][

iHiWiH

ede iSiW

);',()4)((

)(21

'||)(

1)(2

2/0

02

isxxFsis

eisdgxd

xexis

isdgxdiW

d

simd

isHd

RRRRRRfRf

1801

1801

121

301, 2;

;21

Nonperturbative QFT• The in- and out-state formalism

[Schwinger (51), Nikishov (70), DeWitt (75), Ambjorn et al (83)]

• The Bogoliubov transformationin0,|out0,

3

effxLdtdiiW ee

kink,kink,*

ink,ink,ink,outk,

kink,kink,*

ink,ink,ink,outk,

UbUabbUaUbaa

Nonperturbative QFT• The effective action for boson/fermion [SPK, Lee,

Yoon, PRD 78 (08)]

• Sum of all one-loops with even number of exter-nal gravitons

k

*klnin0,|out0,ln iiW

QED vs QGUnruh Effect Pair Production

Schwinger Mechanism

QED

QCD

Hawking Radiation

Black holes

De Sitter/ Expanding universe

QG Analog of QED• Naively assume the correspondence be-

tween two accelerations (Hawking-Unruh effect)

• The vacuum structure of one-loop effective action for dS may take the form [Das,Dunne(06)]

12dSRH

mqE

12//22

2

eff 32)()(Im dSRm

dS emHRL

2

20

422

2

eff 12)22)(32)(42(8)()(

n

dS

n

nn

dS mR

nnnmHRL

Effective Action for de Sitter• de Sitter space with the metric

• Bogoliubov coefficients [Mottola, PRD35 (85)]

)sin)((cosh 32

22222 ddHtdtds

49,

)sinh()1(

)2/1()2/3()()1(

,)2/1()2/3(

)()1(

2

2

Hmi

kkii

Zkikik

ii

k

k

k

Effective Action for dS• Using the gamma function

and doing the contour integral, we obtain the effective action and the imaginary part:

)2/sin()1cos()2/cos()12(

64)()(

00

2

2

eff ssks

sedsPkmHHL

s

k

0

2)12(2

eff ))(ln(tanh21

122)(Imn

n

neHL

Effective Action for de Sitter• Renormalization of constants

• The effective action after renormalization

constantnalgravitatio

2

constantalcosmologic

4

term-1/R

6diveff )( dS

dSdS Rmm

RmRL

12013

)2/(6/1

)2/(1

)2/(sin)2/cos(

64)()( 24402

2

eff ssss

sedsmHHL

s

Effective Action for de Sitter• The vacuum structure of de Sitter in the

weak curvature limit (H<<m)

• The general relation holds between vac-uum persistence and mean number of produced pairs

0

1

22

eff )(n

ndS

ndSdS mRCRmRL

))(ln(tanh)1(expin0,|out0, 2

0

2)(Im22eff

k

HL ke

QFT for Gravity and • The cosmological constant from the effective ac-

tion from QFT

the cut-off from particle physics yields too large to explain the dark energy.

• QFT needs the renormalization of bare coupling constants such gravitation constant, cosmological constant and coupling constants for higher curva-ture terms.

• A caveat: the nonperturvative effect suggests a term 1/R in the action.

4offcut m

Conformal Anomaly• An anomaly in QFT is a classical symmetry which

is broken at the quantum level, such as the en-ergy momentum tensor, which is conserved due to the Bianchi identity even in curved spacetimes.

• The conformal anomaly is the anomaly under the conformal transformation:

geg 2

RbREbFbT 23

221 )

32(

2

2**

312

4

RRRRRCCF

RRRRRRRE

FLRW Universe and Confor-mal Anomaly

• The FLRW universe with the metric

has the conformal Killing vector:

• The FLRW metric in the conformal time

• The scale factor of the universe is just a conformal one, which leads to conformal anomaly.

2222 )( xdtadtds

ijijt HggL 2

))(( 2222 xddads

FLRW Universe and Confor-mal Anomaly

• At the classical level, the QCD Lagrangian is con-formally invariant for m=0:

• At the quantum level, the scale factor leads to the conformal anomaly [Crewther, PRL 28 (72)]

• The FLRW universe leads to the QCD conformal anomaly [Schultzhold, PRL 89 (02)]

)(41 mAgTiGGL a

aaa

QCD

renrenren))(1(

2)(

mmGGgT a

a

03293

ren/10)(

cmgHOT QCD

Conformal Anomaly• The conformal anomaly from the nonperturbative

renormalized effective action is

• The first term is too small to explain the dark en-ergy at the present epoch; but it may be impor-tant in the very early stage of the universe even up to the Planckian regime. The trace anomaly may drive the inflation [Hawking, Hertog, Reall PRD (01)].

2

3

22

02

6

24

0eff )(mRCRC

mHCHCHL dS

dS

Canonical QFT for Gravity• A free field has the Hamiltonian in Fourier-mode

decomposition in FLRW universe

• The quantum theory is the Schrodinger equation and the vacuum energy density is [SPK et al, PRD 56(97); 62(00); 64(01); 65(02); 68(03); JHEP0412(04)]

2

2222

232

33

3

,22

1)2(

)(akma

akdtH kk

kk

kkkkkkdatH

*2*

3

33

)2(2)(

Canonical QFT for Gravity• Assume an adiabatic expansion of the universe,

which leads to

• The vacuum energy density given by

is the same as by Schultzhold if but the re-sult is from nonequilibrium quantum field theory in FLRW universe.

• Equation of state:

32/)( aet kdti

kk

)()(32

9]89[

)2(21

B

3offcut2

2

Λbareofationrenormaliz

3

3

B

mHHHkd

kk

k

HkdH

p89

)2(21 2

3

3

BmH

Conformal Anomaly, Black Holes and de Sitter Space

Conformal Anomaly ??

Black Holes Thermodynamics = Einstein EquationJacobson, PRL (95)

Hawking temperature

Bekenstein-Hawking entropy at event horizon

First Law of Thermodynamics = Friedmann EquationCai, SPK, JHEP(05)

Hartle-Hawking temperature

Cosmological entropy at apparent horizon

Summary• The effective QFT for gravity may provide

an understanding of the dark energy.• The QCD conformal anomaly in the FLRW

universe may give the correct order of magnitude for the dark energy and explain the coincidence problem (how dark matter and dark energy has the same order of magnitude).

• The conformal anomaly may lead to a log-arithmic correction to black hole entropy and higher power of Hubble constants.