### 6.4 Correlation functions for gravitational perturbations

Here we solve the Einstein–Langevin equations (118) for the components 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 [169] (see also [110]) to analyze the stability of Minkowski spacetime in semiclassical gravity.

We first note that the tensors and can be written in terms of as

where we have used . Therefore, the Einstein–Langevin equation (118) can be seen as a linear integro-differential stochastic equation for the components . In order to find solutions to Equation (118), it is convenient to Fourier transform it. With the convention for a given field , one finds, from Equation (119),
The Fourier transform of the Einstein–Langevin Equation (118) now reads
where
with
In the Fourier transformed Einstein–Langevin Equation (121), , the Fourier transform of , is a Gaussian stochastic source of zero average, and
where we have introduced the Fourier transform of the noise kernel. The explicit expression for is found from Equations (101) and (102) to be
which in the massless case reduces to

#### 6.4.1 Correlation functions for the linearized Einstein tensor

In general we can write , where is a solution to Equations (118) with zero average, or Equation (121) in the Fourier transformed version. The averages must be a solution of the linearized semiclassical Einstein equations obtained by averaging Equations (118) or (121). Solutions to these equations (especially in the massless case, ) have been studied by several authors [110, 152, 169, 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

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 depends locally on . Each of these such solutions is a Gaussian stochastic field and thus can be completely characterized by the averages and the two-point correlation functions (127). For such a family of solutions, is the most general solution to Equation (121), which is linear, homogeneous, and local in . It can be written as
where 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 [169] for more details on the definition and properties of these tensor distributions). This tensor is symmetric under the interchanges of and , and is the most general solution of
In addition, we must impose the conservation condition, , where this zero must be understood as a stochastic variable, which behaves deterministically as a zero vector field. We can write , where is a particular solution to Equation (129) and is the most general solution to the homogeneous equation. Consequently, see Equation (128), we can write . To find the particular solution, we try an ansatz of the form
Substituting this ansatz into Equations (129), it is easy to see that it solves these equations if we take
with
and where the notation means that the zeros of the denominators are regulated with appropriate prescriptions in such a way that and are well-defined Lorentz-invariant scalar distributions. This yields a particular solution to the Einstein–Langevin equations,
which, since the stochastic source is conserved, satisfies the conservation condition. Note that, in the case of a massless scalar field (), the above solution has a functional form analogous to that of the solutions of linearized semiclassical gravity found in the appendix of [110]. Notice also that, for a massless conformally-coupled field ( and ), the second term on the right-hand side of Equation (130) will not contribute in the correlation functions (127), since in this case the stochastic source is traceless.

A detailed analysis given in [259] concludes that the homogeneous solution gives no contribution to the correlation functions (127). Consequently , where is the inverse Fourier transform of Equation (133), and the correlation functions (127) are

It is easy to see from the above analysis that the prescriptions in the factors are irrelevant in the last expression and thus can be suppressed. Taking into account that , with , we get from Equations (130) and (131)
This last expression is well-defined as a bi-distribution and can be easily evaluated using Equation (125). The final explicit result for the Fourier-transformed correlation function for the Einstein tensor is thus
To obtain the correlation functions in coordinate space, Equation (127), we take the inverse Fourier transform. The final result is
with
where , , are given in Equations (123) and (132). Notice that, for a massless field (), we have
with and , and where is the Fourier transform of given in Equation (117).

#### 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 (137) (or, in terms of Fourier transforms, Equation (136)), we can now solve the equations for the metric perturbations. Working in the harmonic gauge, (this zero must be understood in a statistical sense), where , the equations for the metric perturbations in terms of the Einstein tensor are

or, in terms of Fourier transforms, . Similarly to the analysis of the equation for the Einstein tensor, we can write , where is a solution to these equations with zero average, and the two-point correlation functions are defined by
We can now seek solutions of the Fourier transform of Equation (140) of the form , where is a Lorentz-invariant scalar distribution in Minkowski spacetime, which is the most general solution of . 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 , where is the most general solution to the associated homogeneous equation and, correspondingly, we have . However, since has support on the set of points for which , it is easy to see from Equation (136) (from the factor ) that and, thus, the two-point correlation functions (141) can be computed from . From Equation (136), and due to the factor , it is also easy to see that the prescription is irrelevant in this correlation function, and we obtain
where is given by Equation (136). The right-hand side of this equation is a well-defined bi-distribution, at least for (the function provides the suitable cutoff). In the massless field case, since the noise kernel is obtained as the limit of the noise kernel for a massive field, it seems that the natural prescription to avoid divergences on the lightcone 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 is

where and , with and given by Equation (138). The two-point correlation functions for the metric perturbations can be easily obtained using .

#### 6.4.3 Conformally-coupled field

For a conformally coupled field, i.e., when and , the previous correlation functions are greatly simplified and can be approximated explicitly in terms of analytic functions. The detailed results are given in [259]; here we outline the main features.

When and , we have and . Thus the two-point correlations functions for the Einstein tensor are written

where ; (see Equation (117)).

To estimate this integral, let us consider spacelike separated points , and define . We may now formally change the momentum variable by the dimensionless vector , . Then the previous integral denominator is , where we have introduced the Planck length . It is clear that we can consider two regimes: (a) when , and (b) when . In case (a) the correlation function, for the component, say, will be of the order

In case (b), when the denominator has zeros, a detailed calculation carried out in [259] shows that

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

The integrand has the same behavior as the correlation function of Equation (144), 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

and for case (b) we have

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

This expression allows a direct comparison with the graviton propagator for linearized quantum gravity in the expansion found by Tomboulis [348]. One can see that the imaginary part of the graviton propagator leads, in fact, to Equation (146). In [317] 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 expansion of quantum gravity interacting with matter fields.