GRAVITON BACKREACTION & COSMOLOGICAL CONSTANT
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GRAVITON BACKREACTION & COSMOLOGICAL CONSTANT
Tomislav Prokopec, ITP & Spinoza Institute, Utrecht University
Munchen, Oct 9 2008
Based onBased on::Tomas Janssen & Tomislav Prokopec, arXiv:0707.3919 [gr-qc] (2007) Tomas Janssen & Tomislav Prokopec, arXiv:0707.3919 [gr-qc] (2007) Tomas Janssen, Shun-Pei Miao & Tomislav Prokopec, arXiv:0807.0439 [gr-qc]Tomas Janssen, Shun-Pei Miao & Tomislav Prokopec, arXiv:0807.0439 [gr-qc]Tomas Janssen & Tomislav Prokopec, arXiv:0807.0477 (2008)Tomas Janssen & Tomislav Prokopec, arXiv:0807.0477 (2008)
˚ 1˚
THE COSMOLOGICAL CONSTANT PROBLEM
μν μν μν2 4
(vacuum matter)gravitationalgeometry energy momentumcoupling tensor
8 G ˆG (g) g = Tc c
(μ,ν =0,1,2,3)
˚ 2˚
Vacuum fluctuates and thereby contributes to the stress-energy tensor of the vacuum (Casimir 1948):
vac vac geom vacobs 2
8 G(T ) g
c
THE COSMOLOGICAL CONSTANT PROBLEM: The expected energy density of the vacuum
A finite volume V = L³ in momentum space constitutes reciprocal lattice: each point of the lattice is a harmonic oscillator with the ground state energy E/2, where E²=(cp)²+(mc²)².
Through Einstein’s equation this vacuum energy curves space-time such that it induces an accelerated expansion:
4 76 4vac Pl~m ~10 GeV
2 -46 40obs Pl~(H m ) ~10 GeV
is about 122 orders of magnitude larger than the observed value:
Q: H²Λ/3 is a classical attractor. Does it remain so in quantum theory?
Plmax ~ mk
BACKGROUND SPACE TIME
LINE ELEMENT (METRIC TENSOR):
˚ 3˚
● for power law expansion the scale factor reads:
),..1,1,1(,)()(1
22222
D
diagagorxdtadtds
aHHH
t
t01
1
0
/1
0
,)1(a
p
wwH
H .,const)1(
2
32
FRIEDMANN (FLRW) EQUATIONS (=0):
,3
82M
NGH
)(4 MMN pGH
˚ 4˚
(MASSLESS) SCALAR FIELD ACTION
SCALAR EOM
In momentum space (=0, V=0):
220
20
22
,0)(''
)2(1
)('1
VR
a
aD
aVR
g
)(
2
1
2
1 2
VRggxdS D
00ˆ,ˆ)(ˆ)()2(
)(ˆ *1
1
kk
xkikk
xkikD
D
aaeaekd
x
SCALAR THEORY
)1(2
3,)()(
4
||1)( )2()1(
kHkH
a kkk
Scalar field spectrum Pφ in de Sitter (ε=0)
sin( )ˆ ˆ0 ( , ) ( ', ) 0 ( , ) , '
dk k xx x k x x x
k k x
P
2
2
2
2
)(1
4),(
aH
kHkP
CONTAINS IR SINGULARITY
˚ 5˚
coincident 2 point function (propagator) in dS limit:
SCALAR THEORY: SINGULARITIES
►we find:
)ln(1
10)(0);( 2
0)(02 aHxxxi E
dS
● when =constant, the
1/term can be subtracted
● when =ε(t), but slowly changing in time, s.t. dε/dt<<Hε
close to matter era: =3/2+ε:
)2/3()ln(2
1
2
1
)2/3(3
10)(0 22/32
OaHx E
3/2,3
2,0,
2
3ta
tHHww MMM
22
2
~1
~)2/3(
tH
H
implying a secular growth of vacuum fluctuations that can compensate a
cosmological term
2~
3
32 ||
)2(0)(0
max
min
k
aHk
kIR
kdx
the IR singularity of a coincident 2 point function:
is IR singular for 0 ≤ ε ≤ 3/2 large quantum backreaction expected
● singularities occur when = 3/2, 4/3, 5/4,.., 1,.. 4/5, 3/4, 2/3, 1/2 & 0
CLASSICAL ATTRACTOR IN FLRW SPACES
Q: can quantum vacuum fluctuations change the late time de Sitter attractor behaviour?
Einstein’s equations in FLRW spaces (0):
˚ 6˚
► CLASSICAL SOLUTION
M
MMM
pw
Hw
H
H
,3
1)1(2
322
,
33
82 M
NGH
twH M 3
)1(2
3coth
3
Classical (de Sitter) attractor
Quantum behaviour (?)
3/H
t3/
0
3)1(
23
cosh
)1(23
2
tw
w
M
M
SCALAR PROPAGATOR IN FLRW SPACES
This propagator allows for determination of the quantum backreaction and more generally effects of quantum scalar fields in dynamical FLRW spaces
˚ 7˚
SCALAR PROPAGATOR
Janssen & Prokopec 2007Janssen, Miao & Prokopec 2008
aHHyDDD
FD
DDHH
xxi DD
DD
D
D
D0122/
12
2 ,4
1;2
;2
1,
2
1
2/2
12
1
)4(
)'(|1|)';(
,)1(
)2(41
)2)(1(21
)2)(1(
2
12
22
2
DDDDDDD
D
HOPE: THAT THIS SCALAR PROPAGATOR RESUMS THE LOGS OF a:
2sin4,
'
||||)|'(|)';( 2
22 lHy
xxixxy
1const.)],()ln(1[2
200
H
HOaHaHH
)'(),;( xxixxiRgg D ► EOM
► Ansatz: ),()'(),;( 2/1 yaaxxi D
►
l = geodesic distance in de Sitter space
LAGRANGIAN FOR PERTURBATIONS˚ 8˚
Graviton: lagrangian to second order in h
► PERTURBATIONS ,16,)()(ˆ22
N
aa
Gggxg
►GAUGE: graviton propagator in exact gauge is not known. We added a gauge fixing term:
)()()(ˆ xx
► GRAVITON-SCALAR MIXING
shellon0
2200
2)2( )(')2(''2
1''
VaaHDaaL DD
● lagrangian must be diagonalized w.r.t. the scalar fields 00 &
02 '2
1,
2
1
gaFFFggLGF
● upon a suitable rotation tensor, vector and 2 scalar fields decouple on shell
GRAVITON PROPAGATOR IN FLRW SPACES˚ 9˚
Janssen, Miao & Prokopec 2008
aHHyDDD
FD
DDHH
xxi nDnD
nDnD
D
D
Dn 0,,12
,,
2/
12
2 ,4
1;2
;2
1,
2
1
2/2
12
1
)4(
)'(|1|)';(
,)1(
4)2(
2)2)(1(
)1(2
)1(1
2
12
22
2,
DDDDnnnDn
DnD
EOM (symbolic) DiiD
GRAVITON PROPAGATORS
► VECTOR DOFs: 1vector1vector , iiDD lj
lj
ijij
► GHOST DOFs: 001
0001
000shellon
ghost ,,
iiiDDD ghost
)2,1,0(,)1(2
)1(1 2
nHnn
nDngggDn
Dnn iiD
GRAVITON PROPAGATORS˚10˚
► SCALAR AND TENSOR DOFs (G=3x3 operator matrix):
,
)()(0
)()(0
00
22
02
20
2022
022
isiciisc
iiscisic
i
iMklrs
DiIiMG
0)( 3
22
i
Di klrsslkrklrs
,2
)3(2
D
D )sin(),cos(,3
)3(2)2tan(
scD
D
SCALAR 1 LOOP EFFECTIVE ACTION
When the determinant is evaluated in a FLRW space, it leads to a backreaction that can compensate Λ.
ONE LOOP (MASSLESS) SCALAR FIELD EFFECTIVE ACTION:
˚11˚
DIAGRAMMATICALLY 1 LOOP(vacuum bubble):
NB: Can be calculated by knowing the relevant propagator.
NB2: Propagators are not known for general spaces; now known for FLRW spaces with constant ε.
Janssen, Miao & Prokopec 2008
[ ] [ ] [ ]
1/ 2
1loop contribution
1[ ] [ ] ln
2[ ]
i iS iS ie D e e S Tr g
Det g
..
2
1][ 4
ggxdS
GRAVITON 1 LOOP EFFECTIVE ACTION
☀ When renormalized, one gets the one loop effective action:
˚12˚
Janssen, Miao & Prokopec 2008
]det[]det[
]det[(fields)
tensort
ghost)0()2()0(
ijklorvecij
iSiSiSi
DD
DeeDe
L
ijklij DiTrDTr
iDTr
iS
1
ghosttensor
vector)0( )][ln()][ln(2
)][ln(2
)3(
1
11
1
1ln2)1(ln)1011149186(
16
11
0
23244
33
221023
QQ p
H
H
aVa
► i: renormalization dependent constants
► H0: a Hubble parameter scale
► (z)=dln[(z)]/dz: digamma function
► can be expanded around the poles of (z):
0,2
1,
3
2,
4
3,
5
4,..1,..,
4
5,
3
4,
2
3,2p
►EFFECTIVE ACTION:
● the poles 0, 1, 2 (dS, curv, rad) are not relevant.
NB: Q & pQ can be obtained from the conservation law: )3(44QQQ pHaa
dt
d
DYNAMICS NEAR THE DE SITTER POLE
►near the de Sitter pole (ε=0): small quantum effect
˚13˚
● Late time dynamics: asymptotes a nearly-classical de Sitter attractor:
0ln2
93
1
23 2
0
42
HHG
H
HB
w
HHAGH N
MN
(A,B: undetermined constants)
Classical (de Sitter) attractor
Quantum corrected attractor
3/H
t3/
t3/
DYNAMICS NEAR THE MATTER ERA POLE
►near the matter era pole (ε=3/2): secular growth and large quantum effects
˚14˚
● Late time dynamics: asymptotes the near-pole classical attractor:
)1(341011149186|1|,0)1(241
23 32
42
MpppppppM
N
M
ww
HG
w
HH
Classical branch
1. Quantum branch
3/H
INSET
3/H
2
3
DYNAMICS NEAR THE MATTER ERA POLE II
►Hubble parameter vs time (εp~3/2, <0):
˚15˚
● Late time dynamics: asymptotes the near-pole classical attractor:
Classical (de Sitter) attractor Quantum corrected attractor
3/H
t3/
tH
3
2
DYNAMICS NEAR THE 5/4 POLE
►near the ε=5/4 pole (>0): secular growth and large quantum effects
˚16˚
● Late time dynamics: asymptotes the near-pole classical attractor:
0)(
)1(241
23
42
p
p
M
N
M w
HG
w
HH
Classical branch
Quantum branch
3/H
4
5
INSET
3/H
DYNAMICS NEAR THE 5/4 POLE II
►Hubble parameter vs time (εp~5/4, >0):
˚17˚
● Late time dynamics: asymptotes the near-pole classical attractor:
Classical (de Sitter) attractor Quantum attractor
3/H
t3/
tH
5
4
DARK ENERGY AND COSMOLOGICAL CONSTANT
Dark energy has the characteristics of a cosmological constant Λeff, yet its origin is not known
˚18˚
But why is Λeff so small?
UNKNOWN SYMMETRY?
GRAVITATIONAL BACKREACTION!?
EXPLANATION?
This work suggests that it may be the gravitational backreaction of gravitons (plus matter).
SUMMARY AND DISCUSSION
Scalar matter and graviton VACUUM fluctuations in a near de Sitter universe induce a weak quantum backreaction at 1 loop order (also at 2 loops?).
˚19˚
We considered the quantum backreaction from massless scalar and graviton 1 loop vacuum fluctuations in expanding backgrounds
► What is the quantum backreaction of other quantum fields (fermions, photons)?
1 loop backreaction can be strong when = 3/2, 4/3, 5/4,.., 1,.. 4/5, 3/4, 2/3, 1/2 (-2/3≤w≤0)
OPEN QUESTIONS:► we calculated in the approximation ε=(dH/dt)/H²=const. What is the effect of dε/dt 0 (mode mixing)? ► is the backreaction gauge dependent? (Exact gauge?)
Janssen & Prokopec 2008
Koksma & Prokopec 2008
Miao & Woodard 2008, ..► what happens at 2 loop?
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