Stochastic Gravity in Conformally-flat Spacetimes

Stochastic Gravity in Conformally Flat
Spacetimes
Hing-Tong Cho
Department of Physics, Tamkang University, Taiwan
(Collaboration with Bei-Lok Hu, University of Maryland, USA)
University of Witwatersrand - Feb 17, 2015
Hing-Tong Cho Stochastic Gravity in Conformally Flat Spacetimes

Outline
I. Introduction
II. Brownian motion paradigm
III. Stochastic gravity
IV. Conformal transformation and inﬂuence functional
V. Conformally ﬂat spacetimes
VI. Noise kernels of Robertson-Walker spacetimes
VII. Discussions

I. Introduction
Scattering problems:
⟨0, out|0, in⟩J = eiW [J]
=
∫
Dϕ eiS[ϕ]+i
∫
Jϕ
where W [J] is the generating function for n-point functions.

Time evolution problems or initial value problems in ﬁeld theory:
Schwinger’s in-in formalism
⟨0, in|ϕ(x1)ϕ(x2) . . . ϕ(xn)|0, in⟩
In-in generating functional
J− ⟨0, in|0, in⟩J+ = eiW [J+,J−]
=
∑
α
J− ⟨0, in|α, out⟩⟨α, out|0, in⟩J+
where |α, out⟩ is a complete set of out-states on some spacelike
hypersurface Σ.

Using the path integrals
eiW [J+,J−]
=
∫
dϕ′
∫
Dϕ+Dϕ− eiS[ϕ+]+i
∫
J+ϕ+−iS[ϕ−]−i
∫
J−ϕ−
=
∫
CTP
Dϕ+Dϕ− eiS[ϕ+]+i
∫
J+ϕ+−iS[ϕ−]−i
∫
J−ϕ−
where ϕ+ = ϕ− = ϕ′ on Σ.
CTP means Closed-Time-Path.

II. Brownian motion paradigm
A particle (system) coupled linearly to a set of harmonic oscillators
(environment):
S[x] =
∫ t
0
ds
[
1
2
M ˙x2
− V (x)
]
Se[qn] =
∫ t
0
ds
∑
n
[
1
2
mn ˙q2
n −
1
2
mnω2
nq2
n
]
Sint[x, {qn}] =
∫ t
0
ds
∑
n
(−Cnx qn)
(Schwinger, Feynman-Vernon, Caldeira-Leggett, Hu-Paz-Zhang,
...)

The dynamics of the particle is governed by the CTP eﬀective
action
eiΓ[x+,x−]
= eiS[x+]−iS[x−]
×∫
CTP
∏
n
Dqn+Dqn−
(
eiSe [{qn+}]−iSe [{qn−}]
eiSint [x+,{qn+}]−iSint [x−,{qn−}]
)
= eiS[x+]−iS[x−]+iSIF [x+,x−]
where SIF is the inﬂuence action due to the quantum harmonic
oscillators.

The inﬂuence functional SIF can be expressed in terms of the
Schwinger-Keldysh propagators
SIF [x+, x−]
=
∑
n
1
2
∫
ds ds′
[
x+(s)Gn++(s, s′
)x+(s′
) − x+(s)Gn+−(s, s′
)x−(s′
)
−x−(s)Gn−+(s, s′
)x+(s′
) + x−(s)Gn−−(s, s′
)x−(s′
)
]
due to the corresponding boundary conditions.

The Schwinger-Keldysh propagators are
Gn++(s, s′
) = −ηn(s − s′
) sgn(s − s′
) + iνn(s − s′
)
Gn+−(s, s′
) = ηn(s − s′
) + iνn(s − s′
)
Gn−+(s, s′
) = −ηn(s − s′
) + iνn(s − s′
)
Gn−−(s, s′
) = ηn(s − s′
) sgn(s − s′
) + iνn(s − s′
)
where
ηn(s − s′
) = −
C2
n
2mnωn
sin ωn(s − s′
)
νn(s − s′
) =
C2
n
2mnωn
cos ωn(s − s′
)

The inﬂuence action SIF can be written as
eiSIF
= e−i
∫ t
0 ds
∫ s
0 ds′[∆x(s)η(s−s′)Σx(s′)]
e−1
2
∫ t
0 ds
∫ t
0 ds′ [∆x(s)ν(s−s′)∆x(s′)]
where ∆x(s) = x+(s) − x−(s) and Σx(s) = x+(s) + x−(s), and
η(s − s′
) =
∑
n
ηn(s − s′
) = −
∑
n
C2
n
2mnωn
sin ωn(s − s′
)
ν(s − s′
) =
∑
n
νn(s − s′
) =
∑
n
C2
n
2mnωn
cos ωn(s − s′
)
SIF is basically separated into its real and imaginary parts.

Rewriting the imaginary part of SIF as
e−1
2
∫
∆xν∆x
= N
∫
Dξe−1
2
∫
ξν−1ξ
e−1
2
∫
∆xν∆x
= N
∫
Dξe−1
2
∫
(ξ−iν∆x)ν−1(ξ−iν∆x)
e− 1
2
∫
∆xν∆x
= N
∫
DξP[ξ]ei
∫
ξ∆x
where P[ξ] = e− 1
2
∫
ξν−1ξ
is the Gaussian probability density of the
stochastic force ξ.
Due to this probability density one has the stochastic average
⟨ξ(s)ξ(s′)⟩s = ν(s − s′) which is called the noise kernel.

After this procedure the eﬀective action
Γ[x+, x−] = S[x+] − S[x−]
−
∫ t
0
ds
∫ s
0
ds′
∆x(s)η(s − s′
)Σx(s′
)
+
∫ t
0
ds∆x(s)ξ(s)
The equation of motion for the particle is then given by
δΓ[x+, x−]
δx+ x+=x−=x
= 0

The equation of motion is a Langevin equation with the stochastic
force ξ(t),
M¨x + V ′
(x) +
∫ t
0
ds η(t − s)x(s) = ξ(t)
The integral term is related to dissipation as one can write
η(t) =
d
dt
γ(t) ⇒ γ(t) =
∑
n
C2
n
2mnω2
n
cosωnt
and we have
M¨x + V ′
(x) +
∫ t
0
ds γ(t − s) ˙x(s) = ξ(t)
η(s − s′) is called the dissipation kernel.

The dissipation kernel and the noise kernel are respectively the real
and the imaginary parts of the same Green’s function.
They are related by the ﬂuctuation-dissipation relation (FDR)
ν(s) =
∫ ∞
−∞
ds′
K(s − s′
)γ(s′
)
where in this simple case
K(s) =
∫ ∞
0
dω
π
ω cos ωs

III. Stochastic gravity
(Hu and Verdaguer, Living Reviews in Relativity 2008)
In the stochastic gravity theory, gravity is regarded as the system
and quantum fields as the environment.
The corresponding CTP effective action is
eiΓ[g+,g−]
= eiSg [g+]−iSg [g−]
∫
CTP
Dϕ+Dϕ−eiSm[ϕ+,g+]−iSm[ϕ−,g−]
= eiSg [g+]−iSg [g−]+iSIF [g+,g−]
where Sg and Sm are the gravity and the quantum field actions,
respectively. SIF is the influence action due to the quantum field.

To see the structure of SIF we write g± = g + h± and expand SIF
in powers of h±.
SIF
=
1
2
∫
d4
x
√
−g(x) ⟨Tµν
(x)⟩ ∆hµν(x)
−
1
8
∫
d4
x d4
y
√
−g(x)
√
−g(y) ∆hµν(x)
[
Kµναβ
(x, y) + Hµναβ
A (x, y) + Hµναβ
S (x, y)
]
Σhαβ(y)
+
i
8
∫
d4
x d4
y
√
−g(x)
√
−g(y) ∆hµν(x)Nµναβ
(x, y)∆hαβ(y)

The ﬁrst term involves the expectation value of the stress energy
tensor as in semiclassical gravity
Gµν = κ ⟨Tµν⟩
where
⟨Tµν(x)⟩ =
2
√
−g(x)
δSIF
δg+µν(x)
g+=g−

In the second term
Kµναβ
(x, y) =
−4
√
−g(x)
√
−g(y)
⟨
δ2Sm[ϕ, g]
δgµν(x)δgαβ(y)
⟩
Hµναβ
A (x, y) = −
i
2
⟨
[Tµν
(x), Tαβ
(y)]
⟩
Hµναβ
S (x, y) = Im
⟨
T
(
Tµν
(x)Tαβ
(y)
)⟩
are related to the dissipation kernel

The last term, as compared to the Brownian motion model,
induces the stochastic force ξµν, with the correlation
⟨ξµν(x)ξαβ(y)⟩s = Nµναβ(x, y)
Nµναβ is the noise kernel
Nµναβ
(x, y) =
1
2
⟨{
tµν
(x), tαβ
(y)
}⟩
where tµν(x) = Tµν(x) − ⟨Tµν(x)⟩.

Einstein equation:
Gµν[g] = κTµν[g]
Semi-classical gravity (mean field):
Gµν[g] = κ
(
Tµν[g] +
⟨
Tq
µν[g]
⟩)
Stochastic gravity (including quantum fluctuations):
Gµν[g + h] = κ
(
Tµν[g + h] +
⟨
Tq
µν[g + h]
⟩
+ ξµν[g]
)
to linear order in h, where ξµν is the stochastic force induced by
the quantum field fluctuations.

IV. Conformal transformation and inﬂuence functional
Under the conformal transformation
˜gµν = Ω2
gµν ; ˜ϕ = Ω−1
ϕ
due to the conformal anomaly, one has for a conformally invariant
scalar ﬁeld (Brown and Ottewill 1985),
∫
D ˜ϕeiSm[˜ϕ,˜g]
=
∫
Dϕei(Sm[ϕ,g]+A[g,Ω]+B[g,Ω])

where
A[g, Ω] =
3
5760π2
∫
d4
x
√
−g
{(
RµναβRµναβ
− 2RµνRµν
+
R2
3
)
lnΩ
+
2R
3Ω
Ω −
2
Ω2
( Ω)2
}
B[g, Ω] =
1
5760π2
∫
d4
x
√
−g
{
−
(
RµναβRµναβ
− 4RµνRµν
+ R2
)
lnΩ
+4Rµν
(
Ω;µ
Ω;ν
Ω2
)
+
(
−R + 2
Ω
Ω
−
Ω;µΩ;µ
Ω2
) (
2
Ω;µΩ;µ
Ω2
)}

Hence, the transformation property of the inﬂuence action is
SIF [˜g+, ˜g−] = SIF [g+, g−] + A[g+, Ω+] + B[g+, Ω+]
−A[g−, Ω−] − B[g−, Ω−]
For the Robertson-Walker spacetimes ˜gµν, one only needs to
consider the inﬂuence action of Einstein universes,
δSIF [˜g+, ˜g−]
δ˜g+µν
= Ω−2 δ
δg+µν
(SIF [g+, g−] + A[g+, Ω+] + B[g+, Ω+])
to derive the Einstein-Langevin equation.

V. Conformally ﬂat spacetimes
(Candelas and Dowker 1979)
Einstein universe (R × S3) with the metric
ds2
E = −dt2
E + a2
(
dχ2
+ sin2
χ dθ2
+ sin2
χ sin2
θ dϕ2
)
.
Under a coordinate transformation,
t ± r = a tan
(
tE /a ± χ
2
)
the Einstein universe metric becomes
ds2
E = 4 cos2
(
tE /a + χ
2
)
cos2
(
tE /a − χ
2
)
ds2
M.

This indicates that the Einstein universe is conformally related to
the Minkowski spacetime with the conformal factor
Ω(x) = 2 cos
(
tE /a + χ
2
)
cos
(
tE /a − χ
2
)
Indeed, the Wightman function G+
E (x, x′) = ⟨ϕ(x)ϕ(x′)⟩ of a
conformally coupled scalar in the Einstein universe is related to the
corresponding G+
M(x, x′) by
G+
E (x, x′
) = Ω−1
(x)G+
M(x, x′
)Ω−1
(x′
)

Using the Wightman function of a conformally coupled scalar ﬁeld
in Minkowski spacetime
G+
M(x, x′
) =
1
4π2(−∆t2 + ∆⃗r 2)
one has
G+
E (x, x′
) =
1
8π2a2
[
cos
(
∆tE
a
)
− cos
(
∆s
a
)]−1
where ∆s is the geodesic distance between x and x′ on S3.

For the open Einstein universe R1 × H3,
ds2
O = −dt2
O + a2
(
dχ2
+ sinh2
χ dθ2
+ sinh2
χ sin2
θ dϕ2
)
The corresponding Wightman function (Bunch 1978)
G+
O (x, x′
) =
∆s/a
4π2 sinh(∆s/a)(−∆t2
O + ∆s2)
Not
G+
E a→ia
= −
1
8π2a2
[
cosh
(
∆tE
a
)
− cosh
(
∆s
a
)]−1

Although both the closed and the open Einstein universes are
conformally ﬂat, their conformal vacua are not the same. The
conformal vacuum of the Einstein universe is the Minkowski
vacuum, while that of the open Einstein universe is the Rindler
vacuum.
Like the Minkowski and the Rindler vacua, the vacua of the
Einstein and open Einstein universes are related by thermalization.
G+
O (x, x′
)thermal =
∞∑
n=−∞
∆s/a
4π2 sinh(∆s/a)[−(∆tO + inβ)2 + ∆s2]
= G+
E a→ia
with β = 1/T = 1/2πa.

Conformally flat spacetimes with Minkowski conformal vacuum:
Spatially flat de Sitter, flat Robertson-Walker, Einstein universe,
global de Sitter, closed Robertson-Walker
Conformally flat spacetimes with Rindler conformal vacuum:
Open Einstein universe, Milne universe, open Robertson-Walker,
static de Sitter

VI. Noise kernels of Robertson-Walker spacetimes
(HTC and B.-L. Hu, CQG 32 (2015) 055006)
The new ingredient in stochastic gravity is the stochastic force ξµν
induced by the fluctuations of the quantum fields.
The correlation function of the stochastic force is the noise kernel,
Nµνα′β′ (x, x′
) =
⟨
ξµν(x)ξα′β′ (x′
)
⟩
s
The noise kernel is also the correlation function of the stress
energy tensor Tµν of the quantum field
) =
⟨
{tµν(x), tα′β′ (x′
)}
⟩
q
where tµν = Tµν − ⟨Tµν⟩.

Hence, the noise kernel can be obtained from the second derivative
on the imaginary part of the inﬂuence action
) = g+µρ(x)g+νσ(x)g+α′ξ′ (x′
)g+β′ζ′ (x′
)
4
√
g+(x)g+(x′)
δ2ImSIF [g+, g−]
δg+ρσ(x)δg+ξ′ζ′ (x′) g+=g−=g
Between two conformally related spacetimes ˜gµν = Ω2gµν, one
therefore has
˜Nµνα′β′ (x, x′
) = Ω−2
(x)Nµνα′β′ (x, x′
)Ω−2
(x′
)

For a spatially homogeneous spacetime one can write
Nηηη′η′ (x, x′
) = C11
Nηηη′i′ (x, x′
) = C21si′
Nηηi′j′ (x, x′
) = C31si′ sj′ + C32gi′j′
Nηiηj′ (x, x′
) = C41si sj′ + C42gij′
Nηij′k′ (x, x′
) = C51si sj′ sk′ + C52si gj′k′ + C53(gij′ sk′ + gik′ sj′ )
Nijk′l′ (x, x′
) = C61si sj sk′ sl′ + C62(gij sk′ sl′ + si sj gk′l′ )
+C63(gik′ sj sl′ + gil′ sj sk′ + gjk′ si sl′ + gjl′ si sk′ )
+C64(gik′ gjl′ + gil′ gjk′ ) + C65gij gk′l′
where si = ∇i (∆s) and sj′ = ∇j′ (∆s) are the derivatives on the
spatial geodesic distance ∆s between x and x′. Also, gij′ is the
parallel transport bivector such that si = −gi
j′
sj′ .

From the traceless condition of the noise kernel is given by
C11 − C31 − 3C32 = 0
C21 + C51 + 3C52 − 2C53 = 0
C31 − C61 − 3C62 + 4C63 = 0
C32 − C62 − 2C64 − 3C65 = 0.

The conservation condition of the noise kernel is given by
∂C11
∂∆η
+
∂C21
∂∆s
+ 2AC21
∂C21
∂∆η
+
∂C31
∂∆s
+
∂C32
∂∆s
+ 2AC31 = 0
∂C21
∂∆η
−
∂C41
∂∆s
+
C42
∂∆s
− 2AC41 + 2(A + C)C42 = 0
∂C31
∂∆η
−
∂C51
∂∆s
+ 2
C53
∂∆s
− 2AC51 + 2(2A + 3C)C53 = 0
∂C32
∂∆η
−
C52
∂∆s
− 2AC52 − 2CC53 = 0
∂C41
∂∆η
+
∂C51
∂∆s
+
∂C52
∂∆s
−
∂C53
∂∆s
+ 2AC51 + CC52 − (A + 2C)C53 = 0

∂C42
∂∆η
+
∂C53
∂∆s
+ CC52 + 3AC53 = 0
∂C51
∂∆η
−
∂C61
∂∆s
−
∂C62
∂∆s
+ 2
∂C63
∂∆s
−2AC61 − 2CC62 + 2(A + 3C)C63 = 0
∂C52
∂∆η
−
∂C62
∂∆s
−
∂C65
∂∆s
− 2AC62 − 2CC63 + 2(A + C)C64 = 0
∂C53
∂∆η
−
∂C63
∂∆s
+
∂C64
∂∆s
− CC62 − 3AC63 + 3(A + C)C64 = 0
where A = 1/∆s and C = −1/∆s for R3, A = cot(∆s) and
C = − csc(∆s) for S3, and A = coth(∆s) and C = −csch(∆s) for
H3

From the Wightman function one can calculate the noise kernel
(Phillips and Hu 2001).
For the Minkowski spacetime we have
(C11)M =
3∆s4 + 10∆s2∆η2 + 3∆η4
12π4(−∆η2 + ∆s2)6
; (C21)M =
2∆s∆η(∆s2 + ∆η2)
3π4(−∆η2 + ∆s2)6
(C31)M =
4∆η2∆s2
3π4(−∆η2 + ∆s2)6
; (C32)M =
1
12π4(−∆η2 + ∆s2)4
(C41)M = −
∆s2(3∆η2 + ∆s2)
3π4(−∆η2 + ∆s2)6
; (C42)M = −
∆η2 + ∆s2
6π4(−∆η2 + ∆s2)5
(C51)M = −
4∆η ∆s3
3π4(−∆η2 + ∆s2)6
; (C52)M = 0 ; (C53)M = −
∆η∆s
3π4(−∆η2 + ∆s2)5
(C61)M =
4∆s4
3π4(−∆η2 + ∆s2)6
; (C62)M = 0 ; (C63)M =
∆s2
3π4(−∆η2 + ∆s2)5
(C64)M =
1
6π4(−∆η2 + ∆s2)4
; (C65)M = −
1
12π4(−∆η2 + ∆s2)4

For the flat Robertson-Walker spacetime,
ds2
= a2
(η)
(
−dη2
+ dr2
+ r2
dθ2
+ r2
sin2
θdϕ2
)
the corresponding coefficients for the noise kernel are given by
(Cij )fFRW = a−2
(η) (Cij )M a−2
(η′
)
For a(η) = −1/Hη one has the de Sitter in spatially flat
coordinates.

For the closed Robertson-Walker spacetime,
ds2
= a2
(η)
(
−dη2
+ dχ2
+ sin2
χdθ2
+ sin2
χ sin2
θdϕ2
)
which is conformal to the Einstein universe.
In a similar manner, the coeﬃcients of the noise kernels are
(Cij )cFRW = a−2
(η) (Cij )E a−2
(η′
)
For a(η) = α/ sin η one has the de Sitter spacetime in the global
coordinates.

The coeﬃcients of the Einstein universe noise kernel are
(C11)E =
4 − cos2
∆η − 6 cos ∆η cos ∆s − cos2
∆s + 4 cos2
∆η cos2
∆s
192π4(cos ∆η − cos ∆s)6
(C21)E =
sin ∆η sin ∆s(1 − cos ∆η cos ∆s)
48π4(cos ∆η − cos ∆s)6
(C31)E =
sin2
∆η sin2
∆s
(C32)E =
1
192π4(cos ∆η − cos ∆s)4
(C41)E = −
(1 + cos ∆η)(1 − cos ∆s)(2 − cos ∆η + cos ∆s − 2 cos ∆η cos ∆s)
(C42)E = −
1 − cos ∆η cos ∆s

(C51)E = −
sin ∆η sin ∆s(1 + cos ∆η)(1 − cos ∆s)
; (C52)E = 0
(C53)E = −
sin ∆η sin ∆s
(C61)E =
(1 + cos ∆η)2
(1 − cos ∆s)2
; (C62)E = 0
(C63)E =
(1 + cos ∆η)(1 − cos ∆s)
(C64)E =
1
; (C65)E = −
1
192π4(cos ∆η − cos ∆s)4

For the open Robertson-Walker spacetime,
ds2
= a2
(η)
(
−dη2
+ dχ2
+ sinh2
χdθ2
+ sinh2
χ sin2
θdϕ2
)
which is conformal to the open Einstein universe.
Again the coeﬃcients of the noise kernels are
(Cij )oFRW = a−2
(η) (Cij )O a−2
(η′
)
For a(η) = αeη one has the Milne universe.

The coeﬃcients of the noise kernel in the open Einstein universe
are
(C11)O
=
[(−∆η2
+ ∆s2
)2
(∆η4
+ 8∆η2
∆s2
+ 3∆s4
) + 12a2
∆s2
(∆η2
+ 3∆s2
)(3∆η2
+ ∆s2
)]
144a4π4(−∆η2 + ∆s2)6
csch
2
(
∆s
a
)
−
∆s(∆η2
+ ∆s2
)
12a3π4(−∆η2 + ∆s2)4
coth
(
∆s
a
)
csch
2
(
∆s
a
)
+
[2∆s2
(−∆η2
+ ∆s2
)2
+ 3a2
(∆η4
+ 6∆η2
∆s2
+ ∆s4
)]
144a6π4(−∆η2 + ∆s2)4
csch
4
(
∆s
a
)
+
∆s(∆η2
+ ∆s2
)
24a5π4(−∆η2 + ∆s2)3
coth
(
∆s
a
)
csch
4
(
∆s
a
)
+
∆s2
48a6π4(−∆η2 + ∆s2)2
csch
6
(
∆s
a
)
etc

One can obtain the noise kernel for the static de Sitter case if ﬁrst
a conformal transformation with
Ω(χ) =
1
α cosh χ
and a further coordinate transformation with
tanh χ = αr
are made on the open Einstein noise kernel.

The coeﬃcient functions (Cij )sdS of the noise kernel in static de
Sitter spacetime are given by
(Cij )sdS = (1 − r2
)−1
(Cij )O(1 − r′2
)−1
,
where the geodesic distance
∆s = cosh−1
{
(1 − r2
)−1/2
(1 − r′2
)−1/2
[
1 − rr′
(cos θ cos θ′
+ sin θ sin θ′
cos(ϕ − ϕ′
))
] }
.

For the Rindler space
ds2
= −ξ2
dτ2
+ dξ2
+ dy2
+ dz2
one can obtain the noise kernel after the series of conformal and
coordinate transformations.
It is however easier to work directly with the Rindler Wightman
function
G+
R =
1
4π2
(
α
ξξ′ sinh α
) (
1
−(τ − τ′)2 + α2
)
,
where
cosh α =
ξ2 + ξ′2 + (y − y′)2 + (z − z′)2
2ξξ′
.

For example,
Nτττ′τ′ =
G2
9α2ξ3ξ′3
{
ξ
3
ξ
′3
+ (1 − ξ
2
ξ
′2
)(ξ
2
+ ξ
′2
) coth αcschα
+ξξ
′
[
(1 + 5α
2
) + (2 − 3α
2
)ξ
2
ξ
′2
− 2α(4 − ξ
2
ξ
′2
) coth α
]
csch
2
α
+α(1 − ξ
2
ξ
′2
)(ξ
2
+ ξ
′2
)(1 − 2α coth α)csch
3
α
+α
2
ξξ
′
(7 − 4ξ
2
ξ
′2
)csch
4
}
−
8π2
G3
sinh α
9α3ξ2ξ′2
{
ξ
3
ξ
′3
[
(2 + 5α
2
) − 6α coth α
]
−α(ξ
2
+ ξ
′2
)(3 − ξ
2
ξ
′2
− 2α coth α)cschα
+2α
2
ξξ
′
[
4 + ξ
2
ξ
′2
− α(4 − ξ
2
ξ
′2
) coth α
]
csch
2
α
+α
3
(1 − ξ
2
ξ
′2
)(ξ
2
+ ξ
′2
)csch
3
α
}
+
64π4
G4
sinh2
α
9α2ξξ′
{
3ξξ
′
[
3 + (3 + α
2
)ξ
2
ξ
′2
− 2αξ
2
ξ
′2
coth α
]
−α(ξ
2
+ ξ
′2
)(5 − α coth α)cschα + 5α
2
ξξ
′
csch
2
}
−
1024π6
G5
sinh3
α
9α
{
6ξξ
′
(1 + 2ξ
2
ξ
′2
) − α(ξ
2
+ ξ
′2
)cschα
}
+
16384π8
G6
sinh4
α
9
{
ξ
2
ξ
′2
(1 + 3ξ
2
ξ
′2
)
}

VII. Discussions
1. The noise kernel is the two point correlation function of the
stress-energy tensor. It represents the backreaction of the
quantum ﬂuctuations of the matter ﬁeld onto the background
spacetime. Therefore, it is interesting to investigate the
behaviors of the noise kernel near horizons as well as initial
singularities of various FRW spacetimes.
2. Other than the noise kernel we need to consider the conformal
transformation of the Einstein-Langevin equation in detail. In
particular, we should also investigate the transformation of the
terms related to the dissipation kernel.

3. Next we shall try to solve the Einstein-Langevin equation
Gµν[g + h] = κ (⟨Tµν[g + h]⟩ + ξµν[g])
Here gµν is the background Robertson-Walker spacetime with
the scaling factor a(η) being a solution to the semiclassical
Einstein equation. To avoid solutions that are not physical,
one might resort to consistent procedures like the order
reduction method of Parker and Simon.
4. Subsequently, one could solve for hµν using standard
perturbation methods around the Robertson-Walker
backgrounds. Here ξµν acts like an external force. The
correlator ⟨hµν(x)hα′β′ (x′)⟩s can therefore be evaluated with
the appropriate noise kernels.

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