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6.4 Correlation functions for gravitational perturbations

Here we solve the Einstein–Langevin equations (118View Equation) for the components G (1)μν of the linearized Einstein tensor. Then we use these solutions to compute the corresponding two-point correlation functions, which give a measure of the gravitational fluctuations predicted by the stochastic semiclassical theory of gravity in the present case. Since the linearized Einstein tensor is invariant under gauge transformations of the metric perturbations, these two-point correlation functions are also gauge invariant. Once we have computed the two-point correlation functions for the linearized Einstein tensor, we find the solutions for the metric perturbations and compute the associated two-point correlation functions. The procedure used to solve the Einstein–Langevin equation is similar to the one used by Horowitz [169Jump To The Next Citation Point] (see also [110Jump To The Next Citation Point]) to analyze the stability of Minkowski spacetime in semiclassical gravity.

We first note that the tensors A(1)μν and B (1)μν can be written in terms of G(1)μν as

2 A (1)μν = --(ℱ μνG(1)αα − ℱ ααG (1)μν), B(1)μν = 2ℱ μνG (1)αα, (119 ) 3
where we have used 3□ = ℱ αα. Therefore, the Einstein–Langevin equation (118View Equation) can be seen as a linear integro-differential stochastic equation for the components G (1)μν. In order to find solutions to Equation (118View Equation), it is convenient to Fourier transform it. With the convention f&tidle;(p) = ∫ d4xe−ipxf(x) for a given field f(x), one finds, from Equation (119View Equation),
&tidle; (1)μν 2 &tidle;(1)μν 2-2 μν &tidle; (1)α A (p) = 2p G (p) − 3p P G α(p), &tidle; (1)μν 2 μν &tidle; (1)α B (p) = − 2p P G α(p). (120 )
The Fourier transform of the Einstein–Langevin Equation (118View Equation) now reads
μν &tidle;(1)αβ &tidle;μν F αβ(p)G (p) = 8πG ξ (p), (121 )
where
μν μ ν 2 μν F αβ(p) ≡ F1 (p )δ(α δβ) + F2(p) p P ηα β, (122 )
with
[ ] 2 1 F1(p) ≡ 1 + 16πG p 4H&tidle;A (p) − 2¯α , [ ] (123 ) 16- 1-&tidle; 3-&tidle; ¯ F2(p) ≡ − 3 πG 4HA (p) + 4HB (p) − 2¯α − 6 β .
In the Fourier transformed Einstein–Langevin Equation (121View Equation), μν ξ&tidle; (p), the Fourier transform of μν ξ (x), is a Gaussian stochastic source of zero average, and
⟨ μν αβ ′ ⟩ 4 4 ′ μναβ ξ&tidle; (p)ξ&tidle; (p ) = (2π) δ (p + p )N&tidle; (p), (124 ) s
where we have introduced the Fourier transform of the noise kernel. The explicit expression for &tidle;N μναβ(p) is found from Equations (101View Equation) and (102View Equation) to be
∘ --------[ ( )2 ( )2 ] &tidle; μναβ θ(−-p2 −-4m2)- m2- 1- m2- 22 μναβ m2- 2 2 μν αβ N (p) = 720 π 1+ 4 p2 4 1+ 4 p2 (p )P +10 3Δ ξ+ p2 (p ) P P , (125 )
which in the massless case reduces to
[ ] lim N&tidle;μναβ(p) = --1-- θ(− p2) 1-(p2)2P μναβ + 60 Δ ξ2(p2)2P μνP αβ . (126 ) m→0 480 π 6

6.4.1 Correlation functions for the linearized Einstein tensor

In general we can write G(1)μν = ⟨G (1)μν⟩s + G (1f)μν, where G(f1)μν is a solution to Equations (118View Equation) with zero average, or Equation (121View Equation) in the Fourier transformed version. The averages ⟨G (1)μν⟩s must be a solution of the linearized semiclassical Einstein equations obtained by averaging Equations (118View Equation) or (121View Equation). Solutions to these equations (especially in the massless case, m = 0) have been studied by several authors [110Jump To The Next Citation Point, 152, 169Jump To The Next Citation Point, 170, 174, 223, 309, 310, 330, 344, 345], particularly in connection with the problem of the stability of the ground state of semiclassical gravity. The two-point correlation functions for the linearized Einstein tensor are defined by

⟨ ⟩ ⟨ ⟩ 𝒢μναβ(x,x′) ≡ G (1)μν(x )G (1)α β(x ′) s − ⟨G(1)μν(x)⟩s G (1)αβ(x′) s ⟨ (1)μν (1)αβ ⟩ = G f (x)G f (x′) . (127 ) s
Now we shall seek the family of solutions to the Einstein–Langevin equation, which can be written as a linear functional of the stochastic source, and whose Fourier transform G&tidle;(1)μν(p ) depends locally on &tidle;ξαβ(p). Each of these such solutions is a Gaussian stochastic field and thus can be completely characterized by the averages (1)μν ⟨G ⟩s and the two-point correlation functions (127View Equation). For such a family of solutions, G&tidle;(1)μν(p) f is the most general solution to Equation (121View Equation), which is linear, homogeneous, and local in &tidle;ξαβ(p). It can be written as
&tidle; (1)μν μν &tidle;αβ G f (p) = 8πG D αβ(p)ξ (p), (128 )
where D μναβ(p) are the components of a Lorentz-invariant tensor-field distribution in Minkowski spacetime (by “Lorentz-invariant” we mean invariant under transformations of the orthochronous Lorentz subgroup; see [169Jump To The Next Citation Point] for more details on the definition and properties of these tensor distributions). This tensor is symmetric under the interchanges of α ↔ β and μ ↔ ν, and is the most general solution of
Fμνρσ(p) Dρσαβ(p) = δμ δνβ). (129 ) (α
In addition, we must impose the conservation condition, (1)μν pνG&tidle;f (p) = 0, where this zero must be understood as a stochastic variable, which behaves deterministically as a zero vector field. We can write D μν (p) = D μν (p) + D μν (p) αβ p αβ h αβ, where D μν (p) p αβ is a particular solution to Equation (129View Equation) and μν D h αβ(p) is the most general solution to the homogeneous equation. Consequently, see Equation (128View Equation), we can write (1)μν (1)μν (1)μν G&tidle;f (p) = &tidle;Gp (p) + &tidle;G h (p). To find the particular solution, we try an ansatz of the form
μν μ ν 2 μν D p αβ(p) = d1(p)δ(αδβ) + d2(p) p P ηαβ. (130 )
Substituting this ansatz into Equations (129View Equation), it is easy to see that it solves these equations if we take
[ ] [ ] d1(p) = --1--- , d2(p) = − ---F2(p)--- , (131 ) F1 (p) r F1(p)F3 (p) r
with
[ ] F3(p) ≡ F1(p) + 3p2F2 (p ) = 1 − 48 πG p2 1H&tidle;B (p) − 2β¯ , (132 ) 4
and where the notation [ ] r means that the zeros of the denominators are regulated with appropriate prescriptions in such a way that d1(p) and d2(p) are well-defined Lorentz-invariant scalar distributions. This yields a particular solution to the Einstein–Langevin equations,
&tidle; (1)μν μν &tidle;αβ G p (p) = 8πG D p αβ(p)ξ (p), (133 )
which, since the stochastic source is conserved, satisfies the conservation condition. Note that, in the case of a massless scalar field (m = 0), the above solution has a functional form analogous to that of the solutions of linearized semiclassical gravity found in the appendix of [110Jump To The Next Citation Point]. Notice also that, for a massless conformally-coupled field (m = 0 and Δ ξ = 0), the second term on the right-hand side of Equation (130View Equation) will not contribute in the correlation functions (127View Equation), since in this case the stochastic source is traceless.

A detailed analysis given in [259Jump To The Next Citation Point] concludes that the homogeneous solution (1)μν G&tidle;h (p) gives no contribution to the correlation functions (127View Equation). Consequently 𝒢μναβ(x, x′) = ⟨G (p1)μν(x)G(p1)αβ(x ′)⟩s, where G (p1)μν(x) is the inverse Fourier transform of Equation (133View Equation), and the correlation functions (127View Equation) are

⟨ ⟩ &tidle;G (1)μν(p) &tidle;G(1)αβ(p′) = 64(2π)6G2 δ4(p + p′)D μν (p)D αβ (− p)N&tidle;ρσλγ(p). (134 ) p p s p ρσ p λγ
It is easy to see from the above analysis that the prescriptions [ ] r in the factors D p are irrelevant in the last expression and thus can be suppressed. Taking into account that ∗ Fl(− p) = F l (p), with l = 1,2,3, we get from Equations (130View Equation) and (131View Equation)
[ ⟨ ⟩ δ4(p + p ′) F (p) &tidle;G(p1)μν(p)G&tidle;(p1)αβ(p′) = 64(2π)6G2 -------2-- &tidle;N μνα β(p ) −--2---p2P μν &tidle;N αβρρ(p ) s |F1 (p)| F3 (p) ∗ 2 ] − F2-(p)p2P αβN&tidle;μνρ (p) + |F2-(p-)|-p2P μνp2P αβ &tidle;N ρ σ (p) . (135 ) F3∗(p) ρ |F3 (p )|2 ρ σ
This last expression is well-defined as a bi-distribution and can be easily evaluated using Equation (125View Equation). The final explicit result for the Fourier-transformed correlation function for the Einstein tensor is thus
⟨ ⟩ 4 ′ ∘ -------2- &tidle;G (1)μν(p)G&tidle;(1)αβ(p′) = 2-(2π)5 G2 δ-(p-+-p)-θ(− p2 − 4m2 ) 1 + 4m-- p p s 45 |F1(p)|2 p2 [ ( )2 1- m2- 2 2 μναβ × 4 1 + 4 p2 (p ) P ] ( m2 )2 || F2(p)||2 +10 3Δ ξ + --2 (p2)2P μνP αβ ||1 − 3p2-----|| . (136 ) p F3(p)
To obtain the correlation functions in coordinate space, Equation (127View Equation), we take the inverse Fourier transform. The final result is
μναβ ′ π-- 2 μναβ ′ 8π- 2 μν αβ ′ 𝒢 (x,x ) = 45 G ℱx 𝒢A (x − x) + 9 G ℱ x ℱ x 𝒢B(x − x ), (137 )
with
∘ --------- m2 ( m2 )2 1 &tidle;𝒢A (p ) ≡ θ (− p2 − 4m2 ) 1 + 4-2- 1 + 4--2 -------2, p p |F1(p)| ∘ --------( )2 | |2 (138 ) &tidle; 2 2 m2- m2- ---1----|| 2F2(p)|| 𝒢B (p ) ≡ θ (− p − 4m ) 1 + 4 p2 3Δ ξ + p2 |F (p)|2 |1 − 3p F3(p)| , 1
where Fl(p), l = 1,2,3, are given in Equations (123View Equation) and (132View Equation). Notice that, for a massless field (m = 0), we have
2 &tidle; 2 F1(p) = 1 + 4πGp H (p; ¯μ ), [ ] F (p) = − 16-πG (1 + 180 Δ ξ2)1H&tidle;(p; ¯μ2) − 6Υ , 2 3 4 (139 ) [ ] F (p) = 1 − 48πGp2 15Δ ξ2H&tidle;(p; ¯μ2) − 2Υ , 3
with μ¯≡ μ exp(1920 π2¯α) and Υ ≡ β¯− 60 Δ ξ2¯α, and where H&tidle;(p;μ2 ) is the Fourier transform of 2 H (x;μ ) given in Equation (117View Equation).

6.4.2 Correlation functions for the metric perturbations

Starting from the solutions found for the linearized Einstein tensor, which are characterized by the two-point correlation functions (137View Equation) (or, in terms of Fourier transforms, Equation (136View Equation)), we can now solve the equations for the metric perturbations. Working in the harmonic gauge, ∂ ν¯hμν = 0 (this zero must be understood in a statistical sense), where ¯hμν ≡ hμν − 1 ημνhαα 2, the equations for the metric perturbations in terms of the Einstein tensor are

μν (1)μν □ ¯h (x) = − 2G (x), (140 )
or, in terms of Fourier transforms, p2&tidle;¯hμν(p) = 2G&tidle;(1)μν(p). Similarly to the analysis of the equation for the Einstein tensor, we can write ¯μν ¯ μν ¯μν h = ⟨h ⟩s + hf, where ¯ μν h f is a solution to these equations with zero average, and the two-point correlation functions are defined by
⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ℋ μναβ(x,x′) ≡ ¯hμν(x)¯hαβ(x ′) − ¯hμν(x) ¯h αβ(x′) ⟨ ⟩s s s = ¯hμfν(x)¯h αfβ(x′) . (141 ) s
We can now seek solutions of the Fourier transform of Equation (140View Equation) of the form &tidle;¯hμfν(p) = 2D (p) &tidle;G (1f)μν(p), where D (p ) is a Lorentz-invariant scalar distribution in Minkowski spacetime, which is the most general solution of 2 p D (p) = 1. Note that, since the linearized Einstein tensor is conserved, solutions of this form automatically satisfy the harmonic gauge condition. As in Section 6.4.1 we can write D (p) = [1∕p2] + D (p) r h, where D (p) h is the most general solution to the associated homogeneous equation and, correspondingly, we have &tidle; μν &tidle; &tidle; μν ¯h f (p) = ¯h μpν(p ) + ¯h h (p). However, since Dh (p) has support on the set of points for which p2 = 0, it is easy to see from Equation (136View Equation) (from the factor θ(− p2 − 4m2 )) that ⟨&tidle;¯hμν(p)G&tidle;(1)αβ(p′)⟩ = 0 h f s and, thus, the two-point correlation functions (141View Equation) can be computed from μν αβ ⟨&tidle;¯hf (p)&tidle;¯h f (p′)⟩s = ⟨&tidle;¯hμpν(p)&tidle;¯hαpβ(p′)⟩s. From Equation (136View Equation), and due to the factor θ(− p2 − 4m2 ), it is also easy to see that the prescription [ ]r is irrelevant in this correlation function, and we obtain
⟨ ⟩ 4 ⟨ ⟩ &tidle;¯hμpν(p)&tidle;¯hαpβ(p′) = --2-2 &tidle;G (1p)μν(p) &tidle;G(p1)αβ(p′) , (142 ) s (p ) s
where ⟨ &tidle;G(1)μν(p)G&tidle;(1)αβ(p′)⟩s p p is given by Equation (136View Equation). The right-hand side of this equation is a well-defined bi-distribution, at least for m ⁄= 0 (the θ function provides the suitable cutoff). In the massless field case, since the noise kernel is obtained as the limit m → 0 of the noise kernel for a massive field, it seems that the natural prescription to avoid divergences on the lightcone p2 = 0 is a Hadamard finite part (see [322, 388] for its definition). Taking this prescription, we also get a well-defined bi-distribution for the massless limit of the last expression.

The final result for the two-point correlation function for the field μν ¯h is

μναβ ′ 4π- 2 μναβ ′ 32π- 2 μν αβ ′ ℋ (x,x ) = 45 G ℱ x ℋA (x − x ) + 9 G ℱ x ℱ x ℋB (x − x), (143 )
where &tidle; 22 &tidle; ℋA (p) ≡ [1 ∕(p )]𝒢A (p) and &tidle; 2 2 &tidle; ℋB (p) ≡ [1∕(p ) ]𝒢B(p), with &tidle; 𝒢A (p) and &tidle; 𝒢B(p) given by Equation (138View Equation). The two-point correlation functions for the metric perturbations can be easily obtained using h = ¯h − 1η ¯hα μν μν 2 μν α.

6.4.3 Conformally-coupled field

For a conformally coupled field, i.e., when m = 0 and Δ ξ = 0, the previous correlation functions are greatly simplified and can be approximated explicitly in terms of analytic functions. The detailed results are given in [259Jump To The Next Citation Point]; here we outline the main features.

When m = 0 and Δ ξ = 0, we have 𝒢B (x) = 0 and &tidle; 2 −2 𝒢A (p) = θ(− p )|F1(p)|. Thus the two-point correlations functions for the Einstein tensor are written

∫ ′ μναβ ′ π-- 2 μνα β -d4p-----eip(x−x)-θ(− p2)--- 𝒢 (x,x ) = 45 G ℱx (2π)4|| 2 &tidle; 2 ||2, (144 ) |1 + 4π Gp H (p; ¯μ )|
where &tidle; 2 2 −1 0 2 i 2 H (p; ¯μ ) = (480π ) ln [− ((p + iε) + p pi)∕μ ]; (see Equation (117View Equation)).

To estimate this integral, let us consider spacelike separated points (x − x′)μ = (0,x − x ′), and define y = x − x′. We may now formally change the momentum variable pμ by the dimensionless vector sμ, pμ = s μ∕|y |. Then the previous integral denominator is |1 + 16 π(LP ∕|y |)2s2H&tidle;(s)|2, where we have introduced the Planck length √ -- LP = G. It is clear that we can consider two regimes: (a) when LP ≪ |y |, and (b) when |y | ∼ LP. In case (a) the correlation function, for the 0000 component, say, will be of the order

4 𝒢0000(y) ∼ L-P-. |y|8

In case (b), when the denominator has zeros, a detailed calculation carried out in [259Jump To The Next Citation Point] shows that

( ) 𝒢0000(y ) ∼ e−|y|∕LP -LP- + ⋅⋅⋅ +---1--- , |y|5 L2P|y|2

which indicates an exponential decay at distances around the Planck scale. Thus short scale fluctuations are strongly suppressed.

For the two-point metric correlation the results are similar. In this case we have

∫ ′ μναβ ′ 4-π 2 μναβ -d4p--------eip(x−x)θ(−-p2)------- ℋ (x,x ) = 45 G ℱx (2π)4 || ||2. (145 ) (p2)2 |1 + 4π Gp2 H&tidle;(p; ¯μ2)|
The integrand has the same behavior as the correlation function of Equation (144View Equation), thus matter fields tends to suppress the short-scale metric perturbations. In this case we find, as for the correlation of the Einstein tensor, that for case (a) above we have
4 0000 -LP- ℋ (y) ∼ |y|4,

and for case (b) we have

(LP ) ℋ0000 (y) ∼ e−|y|∕LP --- + ... . |y |

It is interesting to write expression (145View Equation) in an alternative way. If we use the dimensionless tensor P μναβ introduced in Equation (107View Equation), which accounts for the effect of the operator ℱ μxναβ, we can write

∫ ′ μναβ ′ 4π- 2 -d4p--eip(x−-x)P-μναβ θ(−-p2) ℋ (x,x ) = 45 G (2π)4 || 2 2||2. (146 ) |1 + 4π Gp H&tidle;(p;μ¯ )|
This expression allows a direct comparison with the graviton propagator for linearized quantum gravity in the 1 ∕N expansion found by Tomboulis [348Jump To The Next Citation Point]. One can see that the imaginary part of the graviton propagator leads, in fact, to Equation (146View Equation). In [317Jump To The Next Citation Point] it is shown that the two-point correlation functions for the metric perturbations derived from the Einstein–Langevin equation are equivalent to the symmetrized quantum two-point correlation functions for the metric fluctuations in the large N expansion of quantum gravity interacting with N matter fields.
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