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Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept....

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Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/CosPA, Sept. 30, 2010 U. Tokyo
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Page 1: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

Effective Action for Gravity and Dark Energy

Sang Pyo KimKunsan Nat’l Univ.

COSMO/CosPA, Sept. 30, 2010U. Tokyo

Page 2: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

Outline

• Motivation • Classical and Quantum Aspects of de

Sitter Space• Polyakov’s Cosmic Laser• Effective Action for Gravity• Conclusion

Page 3: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 4: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 5: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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.

Page 6: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 7: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 8: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 9: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 10: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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/

Page 11: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 12: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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)]

Page 13: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 14: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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.

Page 15: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 16: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 17: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 18: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 19: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 20: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 21: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 22: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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

Page 23: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

QED vs QG

Unruh Effect Pair Pro-

duction

Schwinger Mechanism

QED

QCD

Hawking Radiation

Black holes

De Sitter/ Expanding universe

Page 24: Effective Action for Gravity and Dark Energy Sang Pyo Kim Kunsan Nat’l Univ. COSMO/Co sPA, Sept. 30, 2010 U. Tokyo.

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)


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