Date post: | 30-Dec-2015 |
Category: |
Documents |
Upload: | moses-griffin |
View: | 215 times |
Download: | 1 times |
Effective Action for Gravity and Dark Energy
Sang Pyo KimKunsan Nat’l Univ.
COSMO/CosPA, Sept. 30, 2010U. Tokyo
Outline
• Motivation • Classical and Quantum Aspects of de
Sitter Space• Polyakov’s Cosmic Laser• Effective Action for Gravity• Conclusion
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 quan-
tum 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 physi-cal.
2
4cut22
0 3
3
vac 16)2(
d
2
1 cut
kmk
471
vac)GeV(10
447 )GeV(108
G
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 cur-vature 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.
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 vac-uum, in-/in-formalism) leads to the real ef-fective action, implying no particle produc-tion in any dimensions, but exhibits a ther-mal 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 re-spects the Lorentz symmetry.
HTdS /2/1
Classical de Sitter Spaces
• Global coordinates of (D=d+1) dimensional de Sitter
embedded into (D+1) dimensional Minkowski spacetime
have the O(D,1) symmetry.• The Euclidean space (Wick-rotated)
has the O(D+1) symmetry (maximally space-time symmetry).
22222 /)(cosh HdHtdtds d
baab
baab dXdXdsHXX 22 ,/1
baab
baab dXdXdsHXX 22 ,/1
BD-Vacuum in de Sitter Spa-ces
• BUT, in cosmology, an expanding (FRW) spacetime
does not have a Euclidean counterpart for general a(t).The dS spaces are an exception:
Further, particle production in the expand-ing FRW spacetime [L. Parker, PR 183 (1969)] is a concpet 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 production only in even dimensions [Mottola, PRD 31 (1985); Bousso, PRD 65 (2002)].-The in-/out-formalism is consistent with the composition 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 equa-tion 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 approxima-tion and find the mean number of produced particles [SPK, JHEP09(2010)054].
• The dS analog of Schwinger mechanism in QED: the correspondence between two ac-celerations (Hawking-Unruh effect)
H
IISISIISISk
edl
eIIISeeN/22
)(Im)(Im)(Im2)(Im2
))2/((sin4
)),(cos(Re2
12dSRH
m
qE
Radiation in de Sitter Spa-ces
• The Stokes phenomenon explains why there is NO particle production in odd dimensional 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 spa-ces:-the imaginary part is absent/present in odd/even dimensions.
• Does dS radiation imply the decay of vacuum energy of the Universe?-A solution for cosmological constant prob-lem[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 integral
• 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), Nikishov (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 external gravitons
k
*klnin0,|out0,ln iiW
Effective Action for de Sitter
• de Sitter space with the metric
• Bogoliubov coefficients
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(
)21
()(
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 vacuum 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
QED vs QG
Unruh Effect Pair Pro-
duction
Schwinger Mechanism
QED
QCD
Hawking Radiation
Black holes
De Sitter/ Expanding universe
Conclusion
• The effective action for gravity may provide 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 perturbations.
• What is the vacuum structure at higher loops and/or with interactions? (challenging question)