Download Can the vacuum energy be dark matter?

Can the Vacuum Energy be
Dark Energy?
Sang Pyo Kim
Kunsan 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
homogeneous and isotropic, described by the metric
2

dr
2
2
2
2
2
2
2 
ds  dt  a (t ) 
 r (d  sin d )
2
1  Kr

• The theory for gravity is Einstein gravity
G  g   8GT
a(tobs )
• Friedmann equations in terms of the redshift 1  z 
a(tem )
2
 a 
2
2
4
3
2
   H ( z )  H 0 [ R 0 (1  z )   M 0 (1  z )   K 0 (1  z )    0 ]
a
a
1
2
4
  H 0 [ R 0 (1  z )   M 0 (1  z ) 3    0 ]
a
2
Hubble Parameter & Dark Energy
• Radiation
H 2 ( z )  H 02  R 0 (1  z ) 4
• Matter
H 2 ( z )  H 02  M 0 (1  z )3
• Curvature
H 2 ( z )  H 02  K 0 (1  z ) 2
• Cosmological
constant
H 2 ( z )  H 02   0
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:
4

1 cut d 3k
2
2
cut
 vac  
m

k

2 0 (2 )3
16 2
– Planck scale cut-off:

– present value:

 
 10  47 (GeV ) 4
8G
71
4

10
(
GeV
)
vac
– 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.
Vacuum Energy in an
Expanding Universe
• What is the effect of the
expansion of the
universe on the vacuum
energy?
• Unless it decays into
light particles, it will
fluctuate around the
minimum forever!
• The vacuum energy from
the effective action 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
2
 m 2 s / qE

(qE )
e
sp
Leff 
ds
[cot( s)  1
/ s  s
/ 3]
2 0
2
8
s
vacuum energy charge
– SUSY, for instance, scalar and spinor QED with the
same spin multiplicity (nature breaks SUSY if any)
L L
sp
eff
sc
eff
(qE )

8 2
2


0
ds
e
 m 2 s / qE
s
2
cot( s)  1 / sin( s)
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.
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
 ab X a X b  1 / H 2 ,
ds 2   ab dX a dX b
has the O(D,1) symmetry.
• The Euclidean space (Wick-rotated)
has the O(D+1) symmetry (maximally spacetime
symmetry).
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-/informalism) 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.
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 concept 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
cosh( Ht )
d / 2
(t , )  a (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 (HawkingUnruh 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 d
e
  d x  g  d (is )
F ( x, x' ; is )
d /2
0
2
(is )( 4s)
• One-loop corrections to gravity
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 in-state
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 for a massive scalar
(1  i )(i )
, 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
l 
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
Effective Action of Spinor
[W-Y.Pauchy Hwang, SPK, in preparation]
• The Bogoliubov coefficients
(1 / 2  im / H )(1 / 2  im / H )
j 
,
  (n  j  1) ,
(1 / 2    im / H )(1 / 2    im / H )

(1 / 2  im / H )(1 / 2  im / H )
1
0
j 
, jN 
1 / 2  im / H
( )(1   )
2
• The effective action
Lsp
eff ( H )  
2
(2 ) 2

mH 3  D j P 
j
2 Im L ( H )   ln 1  N j ,
sp
eff

0
e  ms / H sin 2 (s / 2)
ds
s
sin( s / 2)


sin 

N j |  j |  
 cosh( m / H ) 
2
2
QED vs QG
QED
Schwinger
Mechanism
QCD
Unruh Effect
Pair Production
Black holes
Hawking Radiation
De Sitter/
Expanding universe
Conformal Anomaly, Black
Holes and de Sitter Space
Hawking temperature
Black Holes Thermodynamics
= Einstein Equation
Jacobson, PRL (95)
Bekenstein-Hawking entropy
Conformal Anomaly ??
Hartle-Hawking temperature
First Law of Thermodynamics
= Friedmann Equation
Cai, SPK, JHEP(05)
Cosmological entropy
Conformal Anomaly
• An anomaly in QFT is a classical symmetry which is
broken at the quantum level, such as the energy
momentum tensor, which is conserved due to the
Bianchi identity even in curved spacetimes.
• The conformal anomaly is the anomaly under the
2
g

e
g 
conformal transformation: 
2 2
T  b1 F  b2 ( E   R)  b3 2 R
3
E *R * R   R R   4 R R   R 2


F  C C

 R R

 2 R R

1 2
 R
3
FLRW Universe and Conformal
Anomaly
• The FLRW universe with the metric
2
2
2
2
ds  dt  a (t )dx
has the conformal Killing vector:
Lt g ij  2 Hg ij
• The FLRW metric in the conformal time
2
2
2
2
ds  a ( )(d  dx )
• The scale factor of the universe is just a
conformal one, which leads to conformal
anomaly.
FLRW Universe and Conformal
Anomaly
• At the classical level, the QCD Lagrangian is
conformally invariant for m=0:
LQCD
1 a 
  G Ga  (i     gTa  Aa  m)
4
• At the quantum level, the scale factor leads to the
conformal anomaly [Crewther, PRL 28 (72)]

 ren
T

 (g)
2
a
G
Ga
ren
 (1   (m)) m  
ren
• The FLRW universe leads to the QCD conformal
anomaly [Schutzhold, PRL 89 (02)]
T
ren
 O( H3QCD )  1029 g / cm3  0
Conformal Anomaly
• The conformal anomaly from the nonperturbative
renormalized effective action is
3
6
R
H
2
Leff ( H )  C0 H 4  C2 2    C0 RdS
 C2 dS2  
m
m
• The first term is too small to explain the dark energy
at the present epoch; but it may be important 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)].
Canonical QFT for Gravity
• A free field has the Hamiltonian in Fourier-mode
decomposition in FLRW universe
d 3k
H (t )  
(2 )3
 1 2 a 3k2 2 
k ,
 3 k 
2
 2a

2
 2

k
2
 k  m  2 
a 

• 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)]

a 3 d 3k *
2 *


  H (t )  




k k
k  k k
3
2 (2 )

Canonical QFT for Gravity
• Assume an adiabatic expansion of the universe, which
leads to
 k (t )  e 
 i dt k
/ 2k a 3
• The vacuum energy density given by
1 d 3k
9H 2
9
H
3
  
[ k

]
( H cut off )  ( )
3
2

2 (2 ) renormalization 8k
32
mB
of bare Λ B
is the same as by Schutzhold if H  mB but the result
is from nonequilibrium quantum field theory in FLRW
universe.
1
d 3k 9 H 2
• Equation of state: p 
 
3
2 H  (2 ) 8k
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)