The Birth of Neutrino Astrophysics Naoko Kurahashi Neilson University of Wisconsin, Madison CosPA –…
Can the Vacuum Energy be Dark Energy? Sang Pyo Kim Kunsan Nat’l Univ. Seminar at Yonsei Univ. Oct....
-
Upload
valerie-hill -
Category
Documents
-
view
229 -
download
0
Transcript of Can the Vacuum Energy be Dark Energy? Sang Pyo Kim Kunsan Nat’l Univ. Seminar at Yonsei Univ. Oct....
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
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.• 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
2
1 cut
kmk
471
vac)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
eds
qELL
qEsm
]3//1)[cot(8
)(
chargeenergy vacuum0 2
/
2
2speff
2
ssss
eds
qEL
qEsm
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
kr
drtadtds
)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
xxP
PimL
xxP
PimL
Radiation in de Sitter Spa-ces
• QFT in dS space: the time-component equation for a massive scalar in dS
a
ad
a
add
a
kmtQ
ttQt
dllkuku
H
Htatutat
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
dddll
dHm
Ht
HtQdzzQtSet
Stokes Phenomenon
• Four turning points
• Hamilton-Jacobi ac-tion
1)(
1)(
2
2
2
2
)(
)(
Hi
Hie
Hi
Hie
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
12dSR
Hm
qE
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)((
)(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
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,
UbUabb
UaUbaa
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
d
H
m
dldl
ii
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
s
ssdl
s
edsPD
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
n
dSndSdS m
RCRmRL
))(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 s
ssl
s
edsPl
mHHL
s
l
0)(
2
1)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
s
s
s
edsPDmHHL
jjjsp
Hms
jj
sp
2
1,
)1()(
)/2/1()/2/1(
/2/1
,)1(,)/2/1()/2/1(
)/2/1()/2/1(
0
NjHimHim
Him
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 )
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 [Schutzhold, 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 Schutzhold 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
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)