Effective Action for Gravity
and Dark Energy
Sang Pyo Kim
Kunsan Nat’l Univ.
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
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:

vac

1

2
0
d k
( 2 )
 
m k
2
3
– Planck scale cut-off:
– present value:
 cut
4
3
 cu t


8 G
vac
2
 10
 10
 47

71
16 
2
( GeV )
( GeV )
4
4
– 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.
Why de Sitter Space in Cosmology?
• 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.
BD-Vacuum in de Sitter Spaces
• 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 dimensions,
but exhibits a thermal state: Euclidean Green
function (KMS property of thermal Green function)
has the periodicity 1 / T dS  2 / H
-The BD vacuum respects the dS symmetry in the
same way the Minkowski vacuum respects the
Lorentz symmetry.
Classical de Sitter Spaces
• Global coordinates of (D=d+1) dimensional de Sitter
ds
2
  dt  cosh ( Ht ) d  d / H
2
2
2
2
embedded into (D+1) dimensional Minkowski
a
b
2
2
a
b
spacetime
 ab X X  1 / H ,
ds   ab dX dX
have the O(D,1) symmetry.
a
b
2
2
a
• The Euclidean
 ab X X space
1 / H (Wick-rotated)
,
ds   ab dX dX
b
has the O(D+1) symmetry (maximally spacetime
symmetry).
BD-Vacuum in de Sitter Spaces
• BUT, in cosmology, an expanding (FRW) spacetime
ds
2
2

dr
2
2
2
2 
  dt  a ( t ) 
 r d  2 
2
 1  kr

does not have a Euclidean counterpart for general
a(t).
The dS spaces are an exception:
a (t ) 
1
H
e
Ht
, a (t ) 
1
cosh( Ht )
H
Further, particle production in the expanding FRW
spacetime [L. Parker, PR 183 (1969)] is a concpet
well accepted by GR community.
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
G ( x, x' ) 

P ( x , x ')
e
 imL ( P )
 dy G ( x , y ) G ( y , x ' )  
P ( x , x ')
L ( P )e
 imL ( P )


m
G ( x, x' )
Radiation in de Sitter Spaces
• QFT in dS space: the time-component equation for
a massive scalar in dS
 (t ,  )  a
( t )  u k (  ) k ( t ) ; a 
d / 2
cosh( Ht )
k
 u k (  )   k u k (  );
2
2
k  l ( l  d  1)
2
k ( t )  Q k ( t ) k ( t )  0
2
k
d ( d  2 )  a 
d a
2
Q k (t )  m  2 
  
a
4
2 a
a
2
H
Radiation in de Sitter Spaces
• The Hamilton-Jacobi equation in complex time
 k (t )  e
 iS k ( t )
; S k (t ) 
2

2

Q k ( z ) dz ; Q k ( t )  
d (d  2)
 dH 
2
 m 
 ;   l ( l  d  1) 
4
 2 
2
k   k ( t )
2
e
 2 Im S k ( t )
2

( H )
2
2
cosh ( Ht )
Stokes Phenomenon
• Four turning points
e
e
Ht ( a ) 
Ht ( b ) 
 i
 i
H

H

i
i
( H )

1
2
( H )

2
2
2
1
• Hamilton-Jacobi
action
[figure adopted from Dumlu & Dunne,
PRL 104 (2010)]
S k (t ( a )  , t (b )  )  i

H
 
Radiation in de Sitter Spaces
• One may use the phase-integral approximation and
find the mean number of produced particles [SPK,
JHEP09(2010)054].
 2 Im S ( I )
 2 Im S ( II )
 Im S ( I )  Im S ( II )
Nk  e
e
 2 cos(Re S ( I , II )) e
 4 sin ( ( l  d / 2 )) e
2
 2  / H
• The dS analog of Schwinger mechanism in QED: the
correspondence between two accelerations
(Hawking-Unruh effect)
qE
m

H 
R dS
12
Radiation in de Sitter Spaces
• 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 spaces:
-the imaginary part is absent/present in odd/even
dimensions.
2
0, out | 0, in
e
 2 Im W
e
 VT  ln( 1  N k )
k
• Does dS radiation imply the decay of vacuum energy
of the Universe?
-A solution for cosmological constant
problem[Polyakov]. Can it work?
Effective Action for Gravity
• Charged scalar field in curved spacetime

H ( x )  0,
H (x)   D D  m ,
2
D      iqA  ( x )
• Effective action in the Schwinger-DeWitt proper time integral
W 
i
2
d x  g
d


d ( is )
1
x|e
 isH
| x'
( is )
0
2

1
d x  g
d


d ( is )
e
 im s
2
is )( 4  s )
• One-loop
corrections to (gravity
f1  R , f 2 
1
30
0
R;
;

1
12
R 
2
d /2
1
180
F ( x , x ' ; is )
R  R


1
180
R  R

One-Loop Effective Action
• The in-/out-state formalism [Schwinger (51),
Nikishov (70), DeWitt (75), Ambjorn et al (83)]
e
iW
e
3
i dtd xL eff
 0, out | 0, in
• The Bogoliubov transformation between the instate and the out-state:




a k, out   k, in a k, in   k, in b k, in  U k a k, in U k
*
b k, out   k, in b k, in   k, in a k, in  U k b k, in U k
*
One-Loop Effective Action
• The effective action for boson/fermion [SPK, Lee,
Yoon, PRD 78, 105013 (`08); PRD 82, 025015,
025016 (`10); ]
W   i ln 0, out | 0, in   i  ln  k
*
k
• Sum of all one-loops with even number of external
gravitons
Effective Action for de Sitter
• de Sitter space with the metric
2
ds   dt 
2
2
cosh ( Ht )
H
2
d d
2
• Bogoliubov coefficients
l 
l 
 (1  i  )  (  i  )
 ( l  d / 2  i  )  (1  l  d / 2  i  )
 (1  i  )  ( i  )
 ( l  d / 2 )  (1  l  d / 2 )
,
 
,
lZ
0
d
2
m
2
H
2

4
Effective Action for dS
[SPK, arXiv:1008.0577]
• The Gamma-function Regularization
and the Residue Theorem
• The effective action per Hubble
volume and per Compton time
(
L eff ( H ) 
d 1
) mH
d
2
( 2 )
( d 1) / 2
2 Im L eff ( H )  ln 1  N l ,


l0
(d )
Dl

P  ds
0
e
 s
 cos(( 2 l  d  1) s / 2 )  cos( s / 2 ) 


s 
sin( s / 2 )

 sin  ( l  d / 2 ) 
2

N l  |  l |  
sinh(  )


2
Effective Action for de Sitter
• The vacuum structure of de Sitter in the weak
curvature limit (H<<m)

R 
2
L eff ( R dS )  m R dS  C n  dS2 
 m 
n0
n 1
• The general relation holds between vacuum
persistence and mean number of produced pairs
2
0, out | 0, in
e
 2 Im L eff ( H )
2
 exp   ( l  1) ln(tanh
 l  0
2
( )) 

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
L eff ( H ) 
mH
3
( 2 )

 ( l  1) P 
2
2
l0

0
ds
e
 s
 cos(( l  1) s )  cos( s / 2 ) 


s 
sin( s / 2 )

• Zeta-function regularization [Hawking, CMP 55
(1977)]

 (z) 

k 1
1
k
L eff ( H )  0
z
,

 ( 2 n )  0, n  Z ,  (0)  
1
2
QED vs QG
QED
Schwinger
Mechanism
QCD
Unruh Effect
Pair Production
Black holes
Hawking Radiation
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 vacuum
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)