Can the Vacuum Energy be Dark Energy?
Sang Pyo KimKunsan Nat’l Univ.
Seminar at Yonsei Univ. Oct. 29,2010 (Talk at COSMO/CosPA, Sept. 30, 2010, U. Tokyo)
Outline• Motivation • Classical and Quantum Aspects of de
Sitter Space• Polyakov’s Cosmic Laser• Effective Action for Gravity• Conclusion
FLRW 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.• Modified gravity: how to reconcile the QG scale with ?
– f(R) gravities– DGP model
• Scalar field models: where do these fields come from?(origin)– Quintessence– K-essence– Tachyon field– Phantom (ghost) field– Dilatonic dark energy– Chaplygin gas
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 in an Ex-panding Universe
• What is the effect of the expansion of the universe on the vac-uum energy?
• Unless it decays into light particles, it will fluctuate around the minimum forever!
• The vacuum energy from the effective ac-tion in an expanding universe?
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
– SUSY, for instance, scalar and spinor QED with the same spin multiplicity (nature breaks SUSY if any) )sin(/1)cot(
8)(
0 2
/
2
2sceff
speff
2
sss
edsqELLqEsm
]3//1)[cot(8
)(chargeenergy vacuum
0 2
/
2
2speff
2
ssss
edsqELqEsm
Why de Sitter Space in Cosmol-ogy?
• The Universe dominated by dark energy is an asymptotically de Sitter space.
• CDM model is consistent with CMB data (WMAP+ACT+)
• The Universe with is a pure de Sitter space with the Hubble constant H= (/3). .
• The “cosmic laser” mechanism depletes curvature and may help solving the cosmological constant problem [Polyakov, NPB834(2010); NPB797(2008)].
• de Sitter/anti de Sitter spaces are spacetimes where quantum effects, such as IR effects and vacuum structure, may be better understood.
Classical de Sitter Spaces• Global coordinates of (D=d+1) dimensional de Sit-
ter
embedded into (D+1) dimensional Minkowski spacetime
has the O(D,1) symmetry.• The Euclidean space (Wick-rotated)
has the O(D+1) symmetry (maximally spacetime symmetry).
22222 /)(cosh HdHtdtds d
baab
baab dXdXdsHXX 22 ,/1
baab
baab dXdXdsHXX 22 ,/1
BD-Vacuum in de Sitter Spa-ces
• The quantum theory in dS spaces is still an issue of controversy and debates since Chernikov and Tagirov (1968):-The Bunch-Davies vacuum (Euclidean vacuum, in-/in-formalism) leads to the real effective action, implying no particle production in any dimen-sions, but exhibits a thermal state: Euclidean Green function (KMS property of thermal Green function) has the periodicity
-The BD vacuum respects the dS symmetry in the same way the Minkowski vacuum respects the Lorentz symmetry.
HTdS /2/1
BD-Vacuum in de Sitter Spa-ces
• BUT, in cosmology, an expanding (FRW) space-time
does not have a Euclidean counterpart for general a(t).The dS spaces are an exception:
Further, particle production in the expanding FRW spacetime [L. Parker, PR 183 (1969)] is a concept well accepted by GR community.
2
22
2
2222
1)( dr
krdrtadtds
)cosh(1)(,1)( HtH
taeH
ta Ht
Polyakov’s Cosmic Laser• Cosmic Lasers: particle production a la Schwinger
mechanism -The in-/out-formalism (t = ) predicts particle pro-duction only in even dimensions [Mottola, PRD 31 (1985); Bousso, PRD 65 (2002)].-The in-/out-formalism is consistent with the compo-sition principle [Polyakov,NPB(2008),(2008)]: the Feynman prescription for a free particle propagating on a stable manifold
)',()()',(),(
)',(
)',()(
)',()(
xxGm
ePLxyGyxGdy
exxG
xxPPimL
xxPPimL
Radiation in de Sitter Spa-ces
• QFT in dS space: the time-component equation for a massive scalar in dS
aad
aadd
akmtQ
ttQt
dllkukuHHtatutat
k
kkk
kk
kkk
d
24)2()(
0)()()(
)1();()(
)cosh(;)()()(),(
2
2
22
222
2/
Radiation in de Sitter Spaces• The Hamilton-Jacobi equation in complex
time
)(Im22
22
22
2
22)(
)(
4)2()1(;
2
)(cosh)()(;)()(;)(
tSkk
kkktiS
k
k
k
et
dddlldHm
HtHtQdzzQtSet
Stokes Phenomenon• Four turning points
• Hamilton-Jacobi ac-tion
1)(
1)(
2
2
2
2
)(
)(
HiHie
HiHie
b
a
Ht
Ht
HittS bak ),( )()(
[figure adopted from Dumlu & Dunne, PRL 104 (2010)]
Radiation in de Sitter Spaces• One may use the phase-integral approximation
and find the mean number of produced particles [SPK, JHEP09(2010)054].
• The dS analog of Schwinger mechanism in QED: the correspondence between two accelerations (Hawking-Unruh effect)
H
IISISIISISk
edl
eIIISeeN/22
)(Im)(Im)(Im2)(Im2
))2/((sin4
)),(cos(Re2
12dSRH
mqE
Radiation in de Sitter Spa-ces
• The Stokes phenomenon explains why there is NO particle production in odd di-mensional de Sitter spaces- destructive interference between two Stokes’s lines-Polyakov intepreted this as reflectionless scattering of KdV equation [NPB797(2008)].
• In even dimensional de Sitter spaces, two Stokes lines contribute constructively, thus leading to de Sitter radiation.
Vacuum Persistence• Consistent with the one-loop effective action from
the in-/out-formalism in de Sitter spaces:-the imaginary part is absent/present in odd/even dimensions.
• Does dS radiation imply the decay of vacuum en-ergy of the Universe?-A solution for cosmological constant problem[Polyakov]. Can it work?
k
)1ln(Im22
in0,|out0,kNVT
W ee
Effective Action 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
);',()4)((
)(21
'||)(
1)(2
2/0
02
isxxFsis
eisdgxd
xexis
isdgxdiW
d
simd
isHd
RRRRRRfRf
1801
1801
121
301, 2;
;21
One-Loop Effective Action• The in-/out-state formalism [Schwinger (51), Nik-
ishov (70), DeWitt (75), Ambjorn et al (83)]
• The Bogoliubov transformation between the in-state and the out-state:
in0,|out0,3
effxLdtdiiW ee
kink,kink,*
ink,ink,ink,outk,
kink,kink,*
ink,ink,ink,outk,
UbUabbUaUbaa
One-Loop Effective Action• The effective action for boson/fermion [SPK, Lee,
Yoon, PRD 78, 105013 (`08); PRD 82, 025015, 025016 (`10); ]
• Sum of all one-loops with even number of exter-nal gravitons
k
*klnin0,|out0,ln iiW
Effective Action for de Sitter• de Sitter space with the metric
• Bogoliubov coefficients for a massive scalar
22
222 )(cosh
ddH
Htdtds
4,
)2/1()2/()()1(
,)2/1()2/(
)()1(
2
2
2
0
dHm
dldlii
Zlidlidl
ii
l
l
Effective Action for dS [SPK, arXiv:1008.0577]
• The Gamma-function Regularizationand the Residue Theorem
• The effective action per Hubble volume and per Compton time
2
2eff
00
)(2/)1(eff
)sinh()2/(sin||,1ln)(Im2
)2/sin()2/cos()2/)12cos((
)2(
)2
1()(
dlNNHL
sssdl
sedsPD
mHd
HL
lll
s
l
dld
d
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
l
HL le
No Quantum Hair for dS Space?
[SPK, arXiv:1008.0577]• The effective action per Hubble volume and per
Compton time, for instance, in D=4
• Zeta-function regularization [Hawking, CMP 55 (1977)]
)2/sin()2/cos())1cos(()1(
)2()(
00
22
3
eff sssl
sedsPlmHHL
s
l
0)(21)0(,,0)2(,1)(
eff
1
HL
Znnk
zk
z
Effective Action of Spinor [W-Y.Pauchy Hwang, SPK, in preparation]
• The Bogoliubov coefficients
• The effective action
2
2eff
2
0
/3
2eff
)/cosh(sin||,1ln)(Im2
)2/sin()2/(sin
)2(2)(
HmNNHL
ss
sedsPDmHHL
jjjsp
Hms
jj
sp
21,
)1()()/2/1()/2/1(
/2/1
,)1(,)/2/1()/2/1(
)/2/1()/2/1(
0
NjHimHimHim
jnHimHim
HimHim
j
j
QED vs QGUnruh Effect Pair Production
Schwinger Mechanism
QED
QCD
Hawking Radiation
Black holes
De Sitter/ Expanding universe
Conformal Anomaly, Black Holes and de Sitter Space
Conformal Anomaly ??
Black Holes Thermodynamics = Einstein EquationJacobson, PRL (95)
Hawking temperature
Bekenstein-Hawking entropy
First Law of Thermodynamics = Friedmann EquationCai, SPK, JHEP(05)
Hartle-Hawking temperature
Cosmological entropy
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 [Schutzhold, 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 Schutzhold 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
Conclusion• The effective action for gravity may pro-
vide a clue for the origin of .• Does dS radiation imply the decay of vac-
uum energy of the Universe? And is it a solver for cosmological constant problem? [Polyakov]
• dS may not have a quantum hair at one-loop level and be stable for linear pertur-bations.
• What is the vacuum structure at higher loops and/or with interactions? (challeng-ing question)