Quantum Instabilities of deSitter Spacetime
National Tsing Hua
University
March 26, 2009
Larry Ford
Tufts University
also Academia Sinica and National Dong
Hwa University
deSitter Spacetime
Solution of Einstein’s equations with a positive
cosmological constant
Global deSitter spacetime is maximally symmetric
10 Killing vectors in 4 dimensions,
Same as Minkowski spacetime
A portion of deSitter spacetime describes
inflationary expansion
ds = −dt + a(t) (dx + dy + dz )
2
2
2
a(t) = e
Ht
2
2
2
Why is the stability of deSitter space important?
Instabilities might alter the predictions of
inflationary models.
Instabilities might lead to a natural resolution
of the cosmological constant problem.
decaying cosmological constant models
e.g., Dolgov, Barr, LF, ect
Part I: Gravitons in deSitter Spacetime
quantize linear perturbations active fluctuations of geometry
Write
gµν = γµν + hµν
background (deSitter) metric
metric perturbation tensor modes
Impose the transverse tracefree (TT) gauge:
h
µν
;ν
=0
h=
covariant derivative on the
background
µ
hµ
=0
hµν u = 0
µ
a timelike vector; here the
comoving obverver 4-velocity
Result: tensor modes behave as massless scalars
Lifshitz 1946
µ
!S hν
=0
scalar wave operator
Consequences:
deSitter space is classically stable
gravitons are equivalent to a pair of massless
scalar fields
!S ϕ = ϕ
;ν
;ν
=0
Infrared divergences
!ϕ(x)ϕ(y)" is not defined in the deSitter invariant state
(Bunch-Davies) vacuum, but can be defined in a
class of states which break deSitter symmetry
Linde, Starobinsky, Vilenkin & LF
Consequence: linear growth in comoving time
3
H t
!ϕ " ∼
2
4π
2
3
H t
!h hµν " ∼
2π 2
µν
Is this growth an instability of deSitter space?
One loop level: No, gauge invariant quantities do
not grow LF 1985
Two loop level: Controversial
Tsamis & Woodard (1996) claim to find
cosmological constant damping
Λef f = Λ(1 −
−2 4
π "p
H t )
6 2
Λ =3 H
2
Planck length
This result disputed by others (Garriga & Tanaka)
An alternative model: gravitons coupled to photons
J.T. Hsiang, D.S. Lee, H.L. Yu & LF
Preliminaries: stress tensor renormalization in
curved spacetime
regulator
Divergent parts:
Einstein tensor
parameter
"
gµν
Gµν !
(1)
(2)
!Tµν " ∼ A 2 + B
+ C1 Hµν
+ C2 Hµν
ln σ
σ
σ
(1)
Hµν
"
δ !√
1
1
2
ρ
2
≡√
−gR
=
2∇
∇
R
−
2g
∇
∇
R
−
g
R
+ 2RRµν
ν
µ
µν
ρ
µν
µν
−g δg
2
scalar curvature
(2)
Hµν
"
δ !√
1
1
1
αβ
α
ρ
ρ
αβ
ρ
≡√
−gR
R
=
2∇
∇
R
−
∇
∇
R
−
g
∇
∇
R
−
g
R
R
+
2R
αβ
α
ν
ρ
µν
µν
ρ
µν
αβ
µ
µ Rρν
µν
−g δg
2
2
Ricci tensor
Add R
in the gravitational
action and write Einstein’s equations as
2
αβ
R
R
and αβ
counterterms
Gµν + Λ0 gµν +
(1)
α0 Hµν
+
(2)
β0 Hµν
= 8πG0 !Tµν "
Remove the divergent parts of !Tµν "
by a renormalization of G0 , Λ0 , α0 , β0
We want the renormalized values of α0 , β0 = 0
to avoid a fourth order equation.
In general, !Tµν "ren is not expressible in terms of
geometric quantities.
An exception: conformally invariant fields (e.g., photons)
in a conformally flat spacetime (e.g. deSitter space).
(in the vacuum state)
Here
where
Bµν
Bunch, Davies,
Brown & Cassidy
1
2
1 2
αβ
ρ
= Rαβ R gµν + RRµν − Rµ Rρν − R gµν
2
3
4
31
C=
1440π 2
µ
Bµ
!Tµν "ren = C Bµν
!= 0
(photons)
conformal anomaly
In deSitter space, !Tµν "ren ∝ gµν so just shifts
Λ
Our model: assume that this form for !Tµν "ren
holds in perturbed deSitter spacetime.
Self consistent: Bµν is still a conserved tensor.
Key result: equation for the metric perturbations becomes
µ
!S hν
+ 48π C
2
"p
H
4
µ
hν
=0
tachyonic mass
and has an exponentially growing solution:
µ
hν
∝e
16πC
2
#p
3
H t
Gauge invariant quantities, such as the Ricci tensor, also grow.
Time scale for the onset of the instability is
1
τ=
≈
16πC #2p H 3
Ep = Planck energy
2
E
4 p
10 2
EI
H
−1
EI = energy scale of inflation
Implications:
Allows adequate inflation to solve horizon &
flatness problems
Does not allow for eternal inflation.
Part II: Quantum Stress Tensor Fluctuations in deSitter
Spacetime
C.H. Wu, K.W. Ng & LF; also work in progress with
S.P. Miao & R. Woodard
Basic idea look at the effects of fluctuations of the
vacuum electromagnetic field stress tensor.
Stress tensor and expansion fluctuations
α
u = 4-velocity of a congruence of timelike geodesics
θ=
α
u;α =
expansion of the congruence
Raychaudhuri equation
Rµν = 8π(Tµν
1
− gµν T )
2
Ordinary matter: focussing
Tµν Fluctuations
θ
Fluctuations
Conservation law for a perfect fluid:
ρ̇ + θ(ρ + p) = 0
E.g., Robertson-Walker universe:
Stress tensor fluctuations imply density fluctuations
(Not necessarily for the same field)
Robertson-Walker Spacetime
dt = a(η)dη
dt = comoving time
dη = conformal time
Conformally invariant fields:
RW
Cµναβ
(x, x! )
=a
−4
(η) a
−4
(η
!
f lat
) Cµναβ (x, x! )
Stress tensor correlation function
Cµναβ (x, x! ) = !Tµν (x) Tαβ (x! )" − !Tµν (x)"!Tαβ (x! )"
(conformal anomaly cancels)
θ fluctuations
Assume σµν = ωµν = 0, so that
dθ
1 2
µ ν
= −Rµν u u − θ
dλ
3
Let θ = θ0 + θ1 , where θ0 = 3ȧ/a, and
dθ1
2
µ ν
= − (Rµν u u )q − θ0 θ1
dt
3
θ1 (t) = −a
−2
(t)
!
t
t0
dt a (t ) (Rµν u u )q
"
2
"
µ ν
Expansion correlation function:
!θ(η1 ) θ(η2 )" − !θ(η1 )"!θ(η2 )" = (8π)2 ×
! η1
! η2
−2
−2
−1
" −1 "
a (η1 ) a (η2 )
dη a (η)
dη a (η ) E(∆η, r)
η0
η0
E(∆η, r) = flat space energy density
correlation function
(r + 3∆η )
=
4π 4 (r2 − ∆η 2 )6
2
Eem
2 2
Treat as distributions - integrate by parts
Inflationary expansion followed by reheating
and a radiation dominated universe
1
a(η) =
,
1 − Hη
η0 < η < 0,
a(η) = 1 + H η,
η0 = conformal time
when inflation begins
η > 0,
tR = reheating time in comoving time
a(t) = e
H(t−tR )
,
t ≤ tR ,
!
a(t) = 1 + 2H(t − tR ),
t ≥ tR .
Effect of expansion fluctuations on
redshifting after reheating
Let p = wρ and integrate the conservation law
to find the density fluctuations:
!"
δρ
ρ
#2 $
= (1 + w)2
%
ηs
dη1 a(η1 )
%
ηs
dη2 a(η2 )
0
0
(!θ(η1 ) θ(η2 )" − !θ(η1 )"!θ(η2 )")
ηs = conformal time of last scattering
Power spectrum of the density fluctuations:
!"
δρ
ρ
#2 $
=
%
3
i!
k·∆!
x
d ke
(Non-Gaussian fluctuations)
Pk (ηs )
!
Result:
Pk =
4
−!p (1
4
3
k |η0 |
2
+ w)
ln
[a(η
)]
s
480π 2
2
!
3
Pk ∝ |η0 | Grow as the duration of inflation increases- a
different instability of deSitter space.
Interpret as due to non-cancellation of anti-correlated
θ fluctuations.
Interpret the sign as telling us whether the density
fluctuations on a given scale are correlated or
anti-correlated.
density perturbation:
a quantum gravity effect
! "3/2
δρ
S
2
∝ #p
ρ
λ
S = H|η0 | = expansion factor during inflation
λ=
length scale of the perturbation
Constraint on the duration of inflation:
! 12
"5/3
δρ
10 GeV
−4
37
S < 10
< 10
ER
ρ
reheating energy
Allows enough inflation to solve the horizon and
flatness problems
Opens the possibility of observing quantum gravity
effects as a non-Gaussian, non-scale invariant
component in the large scale structure.
Summary
1) deSitter space is classically stable.
2) In pure quantum gravity, it is stable at the one loop level.
3) This may not hold in higher orders?
4) In a simple model with gravitons + photons, there
is a homogeneous instablity. No eternal inflation.
5) When stress tensor fluctuations are included,
there is also an inhomogeneous instability.
6) The latter could produce observable features in
large scale structure.