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)