Quantum Field Theory for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. & APCTP CosPA2009, U. Melbourne, 2009
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
Kr
drtadtds
GTgG 8
])1()1()1([)( 02
03
04
020
22
zzzHzHa
aKMR
])1(2
1)1([ 0
30
40
20 zzH
a
aMR
)(
)(1
em
obs
ta
taz
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)(
R
mVChaotic 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
2
1 cut
kmk
471
vac)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
eds
qELL
qEsm
]3//1)[cot(8
)(
chargeenergyvacuum
0 2
/
2
2speff
2
ssss
eds
qEL
qEsm
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][
iHi
WiH
ede iSiW
);',()4)((
)(2
1
'||)(
1)(
2
2/0
0
2
isxxFsis
eisdgxd
xexis
isdgxdi
W
d
simd
isHd
RRRRRRfRf
180
1
180
1
12
1
30
1, 2;
;21
Nonperturbative QFT
• The in- and out-state formalism [Schwinger (51), Nikishov (70), DeWitt (75), Ambjorn et al (83)]
• The Bogoliubov transformation
in0,|out0,3
effxLdtdiiW ee
kink,kink,*
ink,ink,ink,outk,
kink,kink,*
ink,ink,ink,outk,
UbUabb
UaUbaa
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)]
12dSR
Hm
qE
12//2
2
2
eff 32
)()(Im dSRm
dS emH
RL
2
20
422
2
eff 12)22)(32)(42(8
)()(
n
dS
n
nn
dS m
R
nnn
mHRL
Effective Action for de Sitter
• de Sitter space with the metric
• Bogoliubov coefficients [Mottola, PRD35 (85)]
)sin)((cosh 32
22222 ddHtdtds
4
9,
)sinh(
)1(
)2/1()2/3(
)()1(
,)2/1()2/3(
)()1(
2
2
H
mi
kk
ii
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 s
sks
s
edsPk
mHHL
s
k
0
2)12(2
eff ))(ln(tanh2
1
122)(Imn
n
n
eHL
Effective Action for de Sitter
• Renormalization of constants
• The effective action after renormalization
constantnalgravitatio
2
constantalcosmologic
4
term-1/R
6diveff )( dS
dSdS Rmm
R
mRL
120
13
)2/(
6/1
)2/(
1
)2/(sin
)2/cos(
64
)()(
24402
2
eff sss
s
s
eds
mHHL
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
n
dSndSdS m
RCRmRL
))(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 )
3
2(
2
2**
3
12
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)]
)(4
1mAgTiGGL a
aaa
QCD
renrenren))(1(
2
)(
mmGG
gT 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 )(m
RCRC
m
HCHCHL 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()(
a
km
a
a
kdtH kk
kk
kkkkk
kdatH
*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 k
dti
kk
)()(32
9]
8
9[
)2(2
1
B
3offcut2
2
Λbareofationrenormaliz
3
3
B
m
HH
Hkd
kk
k
Hkd
Hp
8
9
)2(2
1 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.