Stochastic Gravity in Conformally-flat Spacetimes

Stochastic Gravity in Conformally FlatSpacetimes

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 = e iW [J]

=

∫Dϕ e iS[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+ = e iW [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 spacelikehypersurface Σ.


Using the path integrals

e iW [J+,J−]

=

∫dϕ′

∫Dϕ+Dϕ− e iS[ϕ+]+i

∫J+ϕ+−iS[ϕ−]−i

∫J−ϕ−

=

∫CTP

Dϕ+Dϕ− e iS[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

0ds

[1

2Mx2 − V (x)

]Se [qn] =

∫ t

0ds

∑n

[1

2mnq

2n −

1

2mnω

2nq

2n

]Sint [x , qn] =

∫ t

0ds

∑n

(−Cnx qn)

(Schwinger, Feynman-Vernon, Caldeira-Leggett, Hu-Paz-Zhang,...)


The dynamics of the particle is governed by the CTP effectiveaction

e iΓ[x+,x−] = e iS[x+]−iS[x−] ×∫CTP

∏n

Dqn+Dqn−(e iSe [qn+]−iSe [qn−]

e iSint [x+,qn+]−iSint [x−,qn−])

= e iS[x+]−iS[x−]+iSIF [x+,x−]

where SIF is the influence action due to the quantum harmonicoscillators.


The influence functional SIF can be expressed in terms of theSchwinger-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 ′) = − C 2n

2mnωnsinωn(s − s ′)

νn(s − s ′) =C 2n

2mnωncosωn(s − s ′)


The influence action SIF can be written as

e iSIF = e−i∫ t0 ds

∫ s0 ds′[∆x(s)η(s−s′)Σx(s′)]

e−12

∫ t0 ds

∫ t0 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

C 2n

2mnωnsinωn(s − s ′)

ν(s − s ′) =∑n

νn(s − s ′) =∑n

C 2n

2mnωncosωn(s − s ′)

SIF is basically separated into its real and imaginary parts.


Rewriting the imaginary part of SIF as

e−12

∫∆xν∆x

= N

∫Dξe−

12

∫ξν−1ξe−

12

∫∆xν∆x

= N

∫Dξe−

12

∫(ξ−iν∆x)ν−1(ξ−iν∆x)e−

12

∫∆xν∆x

= N

∫DξP[ξ]e i

∫ξ∆x

where P[ξ] = e−12

∫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

0ds

∫ s

0ds ′∆x(s)η(s − s ′)Σx(s ′)

+

∫ t

0ds∆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 stochasticforce ξ(t),

Mx + V ′(x) +

∫ t

0ds η(t − s)x(s) = ξ(t)

The integral term is related to dissipation as one can write

η(t) =d

dtγ(t) ⇒ γ(t) =

∑n

C 2n

2mnω2n

cosωnt

and we have

Mx + V ′(x) +

∫ t

0ds 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 realand 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

πω cosωs


III. Stochastic gravity

(Hu and Verdaguer, Living Reviews in Relativity 2008)

In the stochastic gravity theory, gravity is regarded as the systemand quantum fields as the environment.

The corresponding CTP effective action is

e iΓ[g+,g−] = e iSg [g+]−iSg [g−]

∫CTP

Dϕ+Dϕ−eiSm[ϕ+,g+]−iSm[ϕ−,g−]

= e iSg [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 SIFin powers of h±.

SIF

=1

2

∫d4x

√−g(x) ⟨Tµν(x)⟩∆hµν(x)

−1

8

∫d4x d4y

√−g(x)

√−g(y)∆hµν(x)[

Kµναβ(x , y) + HµναβA (x , y) + Hµναβ

S (x , y)]Σhαβ(y)

+i

8

∫d4x d4y

√−g(x)

√−g(y)∆hµν(x)N

µναβ(x , y)∆hαβ(y)


The first term involves the expectation value of the stress energytensor as in semiclassical gravity

Gµν = κ ⟨Tµν⟩

where

⟨Tµν(x)⟩ =2√

−g(x)

δSIFδg+µν(x)

∣∣∣∣∣g+=g−


In the second term

Kµναβ(x , y) =−4√

−g(x)√

−g(y)

⟨δ2Sm[ϕ, g ]

δgµν(x)δgαβ(y)

⟩

HµναβA (x , y) = − i

2

⟨[Tµν(x),Tαβ(y)]

⟩

HµναβS (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 ] +

⟨T qµν [g ]

⟩)Stochastic gravity (including quantum fluctuations):

Gµν [g + h] = κ(Tµν [g + h] +

⟨T qµν [g + h]

⟩+ ξµν [g ]

)to linear order in h, where ξµν is the stochastic force induced bythe quantum field fluctuations.


IV. Conformal transformation and influence functional

Under the conformal transformation

gµν = Ω2gµν ; ϕ = Ω−1ϕ

due to the conformal anomaly, one has for a conformally invariantscalar field (Brown and Ottewill 1985),∫

Dϕe iSm[ϕ,g ] =

∫Dϕe i(Sm[ϕ,g ]+A[g ,Ω]+B[g ,Ω])


where

A[g ,Ω] =3

5760π2

∫d4x

√−g

(RµναβR

µναβ − 2RµνRµν +

R2

3

)lnΩ

+2R

3ΩΩ− 2

Ω2(Ω)2

B[g ,Ω] =

1

5760π2

∫d4x

√−g

−(RµναβR

µναβ − 4RµνRµν + R2

)lnΩ

+4Rµν

(Ω;µΩ;ν

Ω2

)+

(−R + 2

Ω

Ω− Ω;µΩ

;µ

Ω2

)(2Ω;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 toconsider the influence action of Einstein universes,

δSIF [g+, g−]

δg+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

ds2E = −dt2E + a2(dχ2 + sin2χ dθ2 + sin2χ sin2θ dϕ2

).

Under a coordinate transformation,

t ± r = a tan

(tE/a± χ

2

)the Einstein universe metric becomes

ds2E = 4 cos2(tE/a+ χ

2

)cos2

(tE/a− χ

2

)ds2M .


This indicates that the Einstein universe is conformally related tothe 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 aconformally coupled scalar in the Einstein universe is related to thecorresponding 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 fieldin Minkowski spacetime

G+M(x , x ′) =

1

4π2(−∆t2 +∆r 2)

one has

G+E (x , x ′) =

1

8π2a2

[cos

(∆tEa

)− cos

(∆s

a

)]−1

where ∆s is the geodesic distance between x and x ′ on S3.


For the open Einstein universe R1 × H3,

ds2O = −dt2O + a2(dχ2 + sinh2χ dθ2 + sinh2χ sin2θ dϕ2

)The corresponding Wightman function (Bunch 1978)

G+O (x , x ′) =

∆s/a

4π2 sinh(∆s/a)(−∆t2O +∆s2)

Not

G+E

∣∣a→ia

= − 1

8π2a2

[cosh

(∆tEa

)− cosh

(∆s

a

)]−1


Although both the closed and the open Einstein universes areconformally flat, their conformal vacua are not the same. Theconformal vacuum of the Einstein universe is the Minkowskivacuum, while that of the open Einstein universe is the Rindlervacuum.

Like the Minkowski and the Rindler vacua, the vacua of theEinstein 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 ξµνinduced 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 stressenergy tensor Tµν of the quantum field

Nµνα′β′(x , x ′) =⟨tµν(x), tα′β′(x ′)

⟩q

where tµν = Tµν − ⟨Tµν⟩.


Hence, the noise kernel can be obtained from the second derivativeon the imaginary part of the influence action

Nµνα′β′(x , x ′) = 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 , onetherefore 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′ + C52sigj′k′ + C53(gij′sk′ + gik′sj′)

Nijk′ l′(x , x′) = C61si sjsk′sl′ + C62(gijsk′sl′ + si sjgk′ l′)

+C63(gik′sjsl′ + gil′sjsk′ + gjk′si sl′ + gjl′si sk′)

+C64(gik′gjl′ + gil′gjk′) + C65gijgk′ l′

where si = ∇i (∆s) and sj ′ = ∇j ′(∆s) are the derivatives on thespatial geodesic distance ∆s between x and x ′. Also, gij ′ is theparallel 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) andC = − csc(∆s) for S3, and A = coth(∆s) and C = −csch(∆s) forH3


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 + r2dθ2 + 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 flatcoordinates.


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 globalcoordinates.


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)


(C31)E =sin2 ∆η sin2 ∆s


(C32)E =1


(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)

48π4(cos∆η − cos∆s)6; (C52)E = 0

(C53)E = − sin∆η sin∆s


(C61)E =(1 + cos∆η)2(1− cos∆s)2

48π4(cos∆η − cos∆s)6; (C62)E = 0

(C63)E =(1 + cos∆η)(1− cos∆s)


(C64)E =1

96π4(cos∆η − cos∆s)4; (C65)E = − 1



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 universeare

(C11)O

=[(−∆η2 + ∆s2)2(∆η4 + 8∆η2∆s2 + 3∆s4) + 12a2∆s2(∆η2 + 3∆s2)(3∆η2 + ∆s2)]

144a4π4(−∆η2 + ∆s2)6csch

2(

∆s

a

)

−∆s(∆η2 + ∆s2)

12a3π4(−∆η2 + ∆s2)4coth

(∆s

a

)csch

2(

∆s

a

)

+[2∆s2(−∆η2 + ∆s2)2 + 3a2(∆η4 + 6∆η2∆s2 + ∆s4)]

144a6π4(−∆η2 + ∆s2)4csch

4(

∆s

a

)

+∆s(∆η2 + ∆s2)

24a5π4(−∆η2 + ∆s2)3coth

(∆s

a

)csch

4(

∆s

a

)+

∆s2

48a6π4(−∆η2 + ∆s2)2csch

6(

∆s

a

)

etc


One can obtain the noise kernel for the static de Sitter case if firsta conformal transformation with

Ω(χ) =1

α coshχ

and a further coordinate transformation with

tanhχ = αr

are made on the open Einstein noise kernel.


The coefficient functions (Cij)sdS of the noise kernel in static deSitter 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 = −ξ2dτ2 + dξ2 + dy2 + dz2

one can obtain the noise kernel after the series of conformal andcoordinate transformations.

It is however easier to work directly with the Rindler Wightmanfunction

G+R =

1

4π2

(α

ξξ′ sinhα

)(1

−(τ − τ ′)2 + α2

),

where

coshα =ξ2 + ξ′2 + (y − y ′)2 + (z − z ′)2

2ξξ′.


For example,

Nτττ′τ′ =G2

9α2ξ3ξ′3

ξ3ξ′3 + (1 − ξ

2ξ′2)(ξ2 + ξ

′2) cothαcschα

+ξξ′[(1 + 5α2) + (2 − 3α2)ξ2ξ′2 − 2α(4 − ξ

2ξ′2) cothα

]csch

2α

+α(1 − ξ2ξ′2)(ξ2 + ξ

′2)(1 − 2α cothα)csch3α

+α2ξξ

′(7 − 4ξ2ξ′2)csch4

−8π2G3 sinhα

9α3ξ2ξ′2

ξ3ξ′3

[(2 + 5α2) − 6α cothα

]−α(ξ2 + ξ

′2)(3 − ξ2ξ′2 − 2α cothα)cschα

+2α2ξξ

′[4 + ξ

2ξ′2 − α(4 − ξ

2ξ′2) cothα

]csch

2α

+α3(1 − ξ

2ξ′2)(ξ2 + ξ

′2)csch3α

+64π4G4 sinh2 α

9α2ξξ′

3ξξ′

[3 + (3 + α

2)ξ2ξ′2 − 2αξ2ξ′2 cothα

]−α(ξ2 + ξ

′2)(5 − α cothα)cschα + 5α2ξξ

′csch

2

−1024π6G5 sinh3 α

9α

6ξξ′(1 + 2ξ2ξ′2) − α(ξ2 + ξ

′2)cschα

+16384π8G6 sinh4 α

9

ξ2ξ′2(1 + 3ξ2ξ′2)


VII. Discussions

1. The noise kernel is the two point correlation function of thestress-energy tensor. It represents the backreaction of thequantum fluctuations of the matter field onto the backgroundspacetime. Therefore, it is interesting to investigate thebehaviors of the noise kernel near horizons as well as initialsingularities of various FRW spacetimes.

2. Other than the noise kernel we need to consider the conformaltransformation of the Einstein-Langevin equation in detail. Inparticular, we should also investigate the transformation of theterms 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 withthe scaling factor a(η) being a solution to the semiclassicalEinstein equation. To avoid solutions that are not physical,one might resort to consistent procedures like the orderreduction method of Parker and Simon.

4. Subsequently, one could solve for hµν using standardperturbation methods around the Robertson-Walkerbackgrounds. Here ξµν acts like an external force. Thecorrelator ⟨hµν(x)hα′β′(x ′)⟩s can therefore be evaluated withthe appropriate noise kernels.



Date post:	17-Jul-2015
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Stochastic Gravity in Conformally-flat Spacetimes

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