Download Effective Action for Gravity and Dark Energy

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:
4

1 cut d 3k
2
2
cut
 vac  
m

k

2 0 (2 )3
16 2
– Planck scale cut-off:

71
4

10
(
GeV
)
vac

 10  47 (GeV ) 4
– present value:   
8G
– 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 / TdS  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 2  cosh 2 ( Ht )d 2d / H 2
embedded into (D+1) dimensional Minkowski
spacetime
 ab X a X b  1 / H 2 ,
ds 2   ab dX a dX b
have the O(D,1) symmetry.
• The Euclidean
 ab X a X b space
1 / H 2(Wick-rotated)
,
ds 2   ab dX a dX b
has the O(D+1) symmetry (maximally spacetime
symmetry).
BD-Vacuum in de Sitter Spaces
• BUT, in cosmology, an expanding (FRW) spacetime
2

dr
2
2
2
2
2
ds  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:
1 Ht
1
a(t )  e , a(t )  cosh( Ht )
H
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 )
 dyG( x, y)G( y, x' )  
P ( x , x ')
L ( P )e
imL( P )


G ( x, x ' )
m
Radiation in de Sitter Spaces
• QFT in dS space: the time-component equation for
a massive scalar in dS
(t , )  a
d / 2
cosh( Ht )
(t ) uk ()k (t ); a 
H
k
 2uk ()  k 2uk ();
 (t )  Q (t ) (t )  0
k
k
k 2  l (l  d  1)
k
k
d (d  2)  a  d a
2
Qk (t )  m  2 
  
a
4
a 2 a
2
2
Radiation in de Sitter Spaces
• The Hamilton-Jacobi equation in complex time
k (t )  e iS
k
(t )
; S k (t )  
2
(

H
)
Qk ( z )dz; Qk (t )   2 
cosh 2 ( Ht )
d (d  2)
 dH 
2
  m 
 ;   l (l  d  1) 
4
 2 
2
2
2
k  k (t )  e
2
 2 Im S k ( t )
Stokes Phenomenon
• Four turning points
[figure adopted from Dumlu & Dunne,
PRL 104 (2010)]
e
Ht( a )
H
(H ) 2
 i
i
1
2


e
Ht( b )
H
(H ) 2
 i
i
1
2


• Hamilton-Jacobi
action

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].
Nk  e
2 Im S ( I )
e
2 Im S ( II )
 2 cos(Re S ( I , II ))e
 Im S ( I )  Im S ( II )
 4 sin 2 ( (l  d / 2))e 2 / H
• The dS analog of Schwinger mechanism in QED: the
correspondence between two accelerations
(Hawking-Unruh effect)
qE
 H
m
RdS
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.
0, out | 0, in
2
e
2 ImW
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  m2 , D     iqA ( x)
• Effective action in the Schwinger-DeWitt proper time integral

i
1
d
W    d x  g  d (is )
x | e isH | x'
0
2
(is )
im2 s

1
e
  d d x  g  d (is )
F ( x, x' ; is )
d /2
0
2 corrections to (gravity
is )( 4s)
• One-loop
1
1 2
1
1
;

f1  R, f 2 
R;  R 
R R

R R 
30
12
180
180
One-Loop Effective Action
• The in-/out-state formalism [Schwinger (51),
Nikishov (70), DeWitt (75), Ambjorn et al (83)]
eiW  e 
i dtd 3 xLeff
 0, out | 0, in
• The Bogoliubov transformation between the instate and the out-state:
ak,out   k,in ak,in   k,* inbk,in  U k ak,inU k
bk,out   k,inbk,in   k,* in ak, in  U k bk,inU 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
cosh
( Ht )
2
2
2
ds  dt 
d

d
H2
• Bogoliubov coefficients
(1  i )(i )
l 
, lZ0
(l  d / 2  i )(1  l  d / 2  i )
(1  i )(i )
m2 d 2
l 
, 

2
(l  d / 2)(1  l  d / 2)
H
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
d 1
)mH d
2
Leff ( H ) 
(2 ) ( d 1) / 2
(

(d )
D
 l P
2 Im Leff ( H )  ln 1  N l ,
l 0

0
e s  cos(( 2l  d  1) s / 2)  cos( s / 2) 
ds

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 
Leff ( RdS )  m 2 RdS  Cn  dS2 
m 
n 0
n 1
• The general relation holds between vacuum
persistence and mean number of produced pairs
0, out | 0, in
2
 e2 Im Leff ( H )  exp   (l  1) 2 ln(tanh 2 ( ))
l 0

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
mH 3
Leff ( H ) 
(2 ) 2

 (l  1) P 
2
l 0

0
e s  cos((l  1) s)  cos( s / 2) 
ds

s 
sin( s / 2)

• Zeta-function regularization [Hawking, CMP 55
(1977)]

1
1
 ( z )   z ,  (2n)  0, n  Z  ,  (0)  
2
k 1 k
Leff ( H )  0
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)