Semiclassical Equations for Weakly Inhomogeneous Cosmologies

The in-in effective action formalism is used to derive the semiclassical correction to Einstein's equations due to a massless scalar quantum field conformally coupled to small gravitational perturbations in spatially flat cosmological models. The vacuum expectation value of the stress tensor of the quantum field is directly derived from the renormalized in-in effective action. The usual in-out effective action is also discussed and it is used to compute the probability of particle creation. As one application, the stress tensor of a scalar field around a static cosmic string is derived and the backreaction effect on the gravitational field of the string is discussed.


Introduction
Our picture of the evolution of the early universe relies in the so called semiclassical theory of gravity which describes the interaction of quantum fields with the classical gravitational field. Order of magnitude arguments and Heisenberg's uncertainty principle tell us that there must be a period in the universe evolution, well after the Planck time, when the quantization of the gravitational field may be ignored but still the scale of its time variations is short enough to create elementary particles, so that matter quantization cannot be ignored. Since we lack a theory of quantum gravity it is still not known to what extent and in what sense this theory may be considered as a true semiclassical limit of quantum gravity interacting with matter fields. Plausibility arguments have been advanced by Hartle and Horowitz [1] who show that the quantum corrections to the classical action of gravity interacting with N identical non self-interacting matter fields reduce in the leading-order 1/N approximation to such semiclassical theory.
The semiclassical approach provides the framework for some realistic scenarios which may explain some of the features of the present universe. One of these scenarios is inflation [2] which may explain the homogeneity and flatness problems of the standard big bang cosmology. In the inflationary model the quantum fluctuations of the inflaton field may be the source of the small gravitational inhomogeneities which seed galaxies or gravitational waves. This may explain the universe large scale structure [3] and the presence of a hypothetical background of gravitational radiation [4]. Another scenario is the possible formation of topological defects [5] as the universe undergoes some phase transitions. Topological defects, in particular cosmic strings, may seed structure [6] and may be an alternative to inflation for the generation of structure in the universe.
In both scenarios the picture of the gravitational field that emerges is that of a conformally flat Friedmann-Robertson-Walker (FRW) background in which small gravitational perturbations are present. Large anisotropies and inhomogeneities might be present only if the universe had emerged highly inhomogeneous from the Planck era into the classical regime [7]. But since the quantum consequences of a highly inhomogeneous model are difficult to estimate one assumes that by studying small perturbations on a FRW background, a qualitative picture of the evolution of the more extreme case may result.
Here we are interested in the quantum effects produced by the presence of small perturbations in conformally flat backgrounds. Quantum effects due to small anisotropies were first considered by Zeldovich [8], Zeldovich and Starobinsky [9], Hu and Parker [10], Hartle and Hu [11], and Birrell and Davies [12], who computed the creation of conformally coupled particles interacting with the anisotropies. Conformally coupled particles are not created in conformally flat backgrounds (FRW) [13] but the anisotropies break the conformal symmetry. Different techniques were used for such computations, these techniques go from a perturbative evaluation of the Bogoliubov transformations relating two vacua of the quantum field, to the evaluation of the in-out effective action of this field in the given gravitational background. These results were extended to the presence of arbitrary perturbations, including inhomogeneities, [14] by a technique based in the perturbative evaluation of the scattering matrix which had been used in flat backgrounds by Sexl and Urbantke [15] and Zeldovich and Starobinsky [16].
Quantum effects on the geometry, the so called backreaction effect, are more difficult to evaluate because this requires, on the one hand, the computation of the renormalized stress tensor of the quantum field in order to modify the classical Einstein equations, and on the other hand, it requires the solution to these semiclassical equations. It was argued by Zeldovich [8] that the backreaction would tend to dissipate the inhomogeneities as in a sort of gravitational Lenz's law effect. This is a mechanism to homogeneize the universe, but it is usually not advocated because in the standard scenario one assumes cosmological models which cannot explain the present large scale homogeneity by any causal mechanism after the Planck era; the inflationary scenario, on the other hand, seems to solve the homogeneity problem quite naturally. It is nevertheless a mechanism for entropy production. Early work on the backreaction effect on the geometry due to anisotropies was done by Lukash and Starobinsky [17] and Lukash et al. [18], who assumed very special conditions near the Planck time, and by Hu and Parker [10] who considered a Bianchi type I anisotropic model, evaluated the stress tensor in the low frequency approximation and computed the resulting modified Einstein's equations numerically. The results of such work indicate that the dynamical mechanism of particle production achieves a rapid damping of the anisotropy if the calculations are extrapolated to the Planck era.
The computation of the quantum stress tensor is generally difficult in practice. However, for small perturbations on a conformally flat background one may use perturbative methods to get explicit expressions. One of the most powerful and efficient methods and, one that is very well adapted to a perturbative scheme, is based on the one loop order computation of the so called in-in effective action for quantum fields interating with the gravitational perturbations. This technique is an effective action technique adapted to compute expectation values of quantum operators. It was first proposed by Schwinger [19] and Keldysh [20] and developed by Chou et al. [21]. Jordan [22] and Calzetta and Hu [23] developed the technique on a curved background, and it was then applied to derive the stress tensor of a quantum scalar field coupled to small anisotropies on a cosmological background [23].
The use of effective action methods in the backreaction context was first considered by Hartle [24], and Fischetti et al. [25], and Hartle and Hu [11] studied the effect of anisotropies. But in their formalism the basic element is the usual effective action which is related to the generating functional of the in-out vacuum persistence amplitude. This in-out formalism leads to matrix elements rather than expectation values for the quatum operators. Thus, One does not get directly from the in-out effective action the vacuum expectation value of the stress tensor of the quantum field, and one still needs to compute the Bogoliubov transformation between the in and out vacua. This method is, however, very useful for the computation of the particles created, since the probability amplitude for particle creation is directly related to the vacuum persistence amplitude.
In this paper we compute the in-in effective action to the one loop order for a massless scalar field conformally coupled to small gravitational perturbations on a spatially flat FRW background. The in-in effective action is used to derive the quantum stress tensor and the corresponding semiclassical Einstein's equations. Our results generalize the Calzetta and Hu [23] results to the case of arbitrary small perturbations including inhomogeneities, and the stress tensor we derive coincides with that obtained by Horowitz and Wald [26], who used an axiomatic approach to derive it, and by Starobinsky [27] who used a modified Pauli-Villars regularization method [9]. One should stress, however, that the stress tensor computed does not include the energy of the particles created which is a second order correction to the computed terms; it includes only first order vacuum polarization effects. One might wonder that although the energy of the particles created is small it might have a long term cumulative effect.
We should mention that the axiomatic approach to derive the stress tensor has been quite succesful in several situations. Thus Horowitz [28] obtained the stress tensor due to a scalar field minimally coupled to arbitrary gravitational linear perturbations on a flat spacetime background applying the axiomatic arguments outlined by Wald [29]. This tensor was rederived by Jordan [30] using the in-in effective action method; note that the case of conformally coupled fields may also be obtained from our cosmological model when the conformal factor is taken constant. Another approach to the quantum stress tensor based on an iteratively evaluated mode decomposition was developed by Davies and Unruh [31].
In this paper we shall not consider the solutions to the semiclassical equations, except in a simple example involving a cosmic string. The correct approach to this problem is still controversial. In fact, the semiclassical equations are known to admit runaway solutions as a consequence of the fact that they are dynamical equations with higher order derivatives. Horowitz [28] and Jordan [30] found from these solutions that flat space is unstable against quantum effects. Whether these solutions are physical and thus signal a true instability, or unphysical and thus spurious, has been the subject of some discussion in recent years. Simon [32] has argued that the semiclassical correction to Einstein's equations must be seen as analytic perturbations, in terms of the Planck constanth, to the classical Einstein's equations and that, as such, only solutions which are also analytic inh are physical. A consistent perturbative approach to find reduced equations, i.e. dynamical equations which are second order at each order of perturbation is known [33]. When this is applied it is found that flat space is perturbatively stable to first order inh [32]. Reduced semiclassical equations have been obtained also in some cosmologies [34]. Suen [35], on the other hand, has argued, on the basis of how the stress tensor is renormalized, that this tensor cannot be considered the first term of an expansion and therefore the previous reduction methods should not be applied.
In order to make this paper reasonably self-contained the in-in effective action formulation is summarized in section 2 with a view to practical applications. In section 3 the in-in effective action to the one loop order is derived for a scalar field conformally coupled to a nearly conformally flat metric. Since along this computation one derives also all the terms needed for the in-out effective action, this action is also discussed, and it is used to derive the probability for pair creation; the results agree with those obtained by other methods. In section 4 the stress tensor for the quantum field is derived from the in-in effective action and the semiclassical correction to Einstein's equations is written down. The stress tensor is seen to agree with that obtained by Horowitz and Wald [26] and Starobinsky [27]. As an exercise the semiclassical equations in two dimensional spacetime are also derived using the same formalism. In section 5 we apply the previous formula to compute the stress tensor of a quantum field around a static cosmic string and we discuss the backreaction effect on the gravitational field of the string. Note that since the gravitational field of a cosmic string can be considered a small perturbation on a flat background the above perturbative technique (in the sense of metric perturbations) can be applied. The results are in agreement with those found by other non perturbative methods [36][37] but this perturbative method opens the possibility of computing the quantum stress tensor even in time dependent situations. Work along these lines is in progress.

In-in functional formalism
In this section we summarize the in-in functional formalism for the evaluation of the in-in effective action with a view to the applications of this paper. We follow, essentially, the presentations by Jordan [22], Calzetta and Hu [23], and Paz [38].
Quantum corrections to a classical field theory can be studied with the help of the effective action. For simplicity, we consider the quantization of a scalar field φ(x). The usual in-out effective action is based in the generating functional W [J] which is related to the vacuum persistence amplitude in the presence of some classical source J(x) by where T (t) is the usual time ordering operator, and (2.1) can be written as It is easy to see from the classical field equations for φ(x) in the presence of J(x) that exp iW [J] satisfies the integro differential Schwinger-Dyson equation, and that one may give a path integral represenation for its solution as, where S[φ] is the classical action of the field theory and the common shorthand notation Jφ for the integral d n xJ(x)φ(x) has been used. The functional integral is taken with the following boundary conditions: φ → e ∓iωt , where ω > 0, when the time t → ±∞, i.e. the scalar field has only negative and positive frequency modes in the in and out regions respectively; the interaction is assumed to be switched off at these asymptotic regions. By differentiating with respect to the source one generates matrix elements from W [J] , If we assume that the above expression can be reversed, the effective action is defined as the Legendre transformation of the generating functional, This functional ofφ is the generator of the one-particle-irreducible graphs (graphs that remain connected when any internal line is cut) and contains all the quantum corrections to the classical action. From (2.6) one may derive the dynamical equation for the effective mean fieldφ[0], i.e. the matrix element of the field φ in the absence of the source J(x), as which expresses the quantum corrections to the classical equation as a variational problem of the effective action.
In order to work with expectation values rather than matrix elements one can define a new generating functional whose dynamics is determined by two different external classical sources J + and J − , by letting the in vacuum evolve independentenly under these sources, (2.8) Here we have assumed that {|α, T } is a complete basis of eigenstates of the field operator φ(x) at some future time T , i.e. φ(T, x)|α, T = α(x)|α, T . Then (2.8) may be written according to (2.2) as where T (t) and T (a) mean, respectively, time and anti-time ordered operators and dα means dα = Π x dα(x) where x are the points of the hypersurface Σ defined by t = T . The generating functional has also a path integral representation, with the boundary conditions that φ + = φ − = α on Σ and that the fields φ + and φ − are pure negative and pure positive modes, respectively, in the in region i.e. φ ± → e ±iωt at t → −∞ (vacuum boundary conditions in the remote past). In a more compact form one may write, where it is understood that the sum is over all fields φ + , φ − with negative and positive frequency modes, respectively, in the remote past but which coincide at time t = T . These instead of equation (2.5). This functional generates not only the desired expectation values of time ordered field operators but also the anti-time ordered ones in the same footing In analogy with the in-out formalism the in-in effective action is defined as the Legendre transform of the new generating functional as, where the external sources are functionals of the fieldsφ + andφ − , through the definitions which we assume can be reversed. From the definitions (2.14) and (2.15) we get the equation for the expectation values and by taking J ± = 0 in (2.15) we recover the equation for the vacuum expectation value of the fieldφ[0] ≡φ ± [0, 0] = 0, in|φ(x)|in, 0 , This equation does not follow from a simple variational principle in terms of a single fieldφ: in the in-in action we have two fieldsφ + andφ − which are treated independently and only when the sources have been eliminated they become the vacuum expectation value. Note also that Γ[φ,φ] = 0 as a consequence of (2.14) and of the fact that W [J, J] = 0, which follows from (2.8) and the usual normalization for the states. Equation (2.17) is a dynamical field equation which admits an initial value formulation: the solutionφ[0] =φ ± [0, 0] is real and causal, i.e. the solution at one spacetime point depends only on data on the past of that point [22]. For a free field theory, i.e. a theory with a quadratic action, we can compute W 0 [J + , J − ] from (2.11), which becomes now a Gaussian integration for the two independent fields φ + and φ − . The corresponding propagators will be determined by the very particular boundary conditions of this problem. In fact, let us assume that the free action for φ + is and that we have an analogous action S * 0 [φ − ] for φ − , then the classical field equations are, At this stage we can introduce the compact notation, where a and b take the two values + and −, to simplify the mathematical expressions. The solutions of the classical field equations, that satisfy the boundary conditions, φ 0 ± → e ±iωt , when t → −∞ with ω ≥ 0, and φ 0 in the hypersurface Σ, which we take here at t = T → ∞, can be written as, defined with the Feynman, ∆ F , Dyson, ∆ D , and the positive, ∆ + , and negative, ∆ − , Wightman functions:

These propagators are solutions of the equations
where the operator A 0 ab is the diagonal matrix defined by Furthermore the Feynman and Dyson Green functions have a mode decomposition which reflect the boundary conditions imposed over each classical field solution φ 0 a (x), because the Green functions ∆ ± (x − y) ∼ e ±iω·(x 0 −y 0 ) correspond to negative and positive frequency modes respectively (here ω = √ p 2 + m 2 ). Note that it is also satisfied that With these propagators to guarantee the boundary conditions the Gaussian integration of (2.11) for a free field is, where a term independent of J a has been discarded to satisfy W 0 [J, J] = 0, and one can use now (2.13) to generate time ordered and anti-time ordered expectation values of field operators. In particular we have that For interacting fields one can proceed as usual by writing where we have separated the Lagrangian into a free and an interacting part, L = L 0 + L int . Then one may continue in the usual perturbative fashion; however, we are not going to consider self-interacting theories in this paper, the only interaction will be with the gravitational field.
Let us now proceed to the main objective, namely, the evaluation of the effective action Γ[φ a ] up to the one loop order, which corresponds to the first order expansion of W [J a ] in powers ofh. As usual [40], if we assume that the action is bounded from above then we can go to Euclidean space and solve (2.11) by the steepest descent method; we keep, however, the Minkowskian notation. Let us denote by φ and let us expand the exponent in (2.11) about these background fields: and, of course, A +− (x, y) = A −+ (x, y) ≡ 0. Substituting this into (2.11) the integration is now Gaussian and we can write to this one loop order, In terms of the propagator G, which is a functional of the background fields φ (0) a (x) and takes a 2 × 2 matrix form , i.e. G(x, y) = A −1 (x, y), we can write (2.32) as The effective action, which is a functional ofφ a , can now be explicitly found to the same order. Using (2.14), (2.15), and the fact thatφ a differs from φ (0) a by a term of order h we can show that Now the equations forφ a can be deduced from (2.16) using the explicit functional dependence on the fields given by (2.34). However we should note from (2.17) that in order to get the field equations for the expectation value of φ(x) we only need the explicit dependence of the effective action on one of the fields,φ + orφ − . Therfore we are only interested in the dependence of (2.34) onφ + , say. Following Paz [38] we can write where we have used that G −1 ab ≡ A ab is diagonal, see (2.31). Thus we have, where F includes all the terms which do not contribute to the variation of the fieldφ + , i.e. δF δφ + (x) φ ± =φ = 0. This expression is very similar to the one loop in-out effective action which is given by the above equation (2.36) where G ++ is substituted by the Feynman propagator ∆ F , the main difference is in the boundary conditions: the propagator G ++ is defined as an expectation value and not as an in-out matrix element. This formalism can be extended to curved spacetimes without difficulties assuming that the spacetime is globally hyperbolic [22]. The hypersufaces of constant time are now Cauchy hypersurfaces and the in and out states are defined in the Cauchy hypersurfaces corresponding to the far past and far future respectively. Now the spacetime integrals must be performed with the volume element d n x √ −g where g µν is the spacetime metric. The above expressions (2.20), (2.23) and (2.36) are still valid except that now the Feynman, Dyson and Wightman functions have a different representation to that of (2.22). If the spacetime is asymptotically flat in the in and out regions the previous boundary conditions for the fields φ 0 ± (x) will also apply; if not, in order to be able to define physically meaningful in and out vacua we must assume that we are still able to define positive and negative frequency solutions in the asymptotic regions. This is always possible, for instance, if the asymptotic regions admit approximate timelike Killing fields. But, generally, in a curved spacetime the in and out vacua are not equivalent. Jordan [22] has shown that for quantum scalar fields in a curved spacetime the field equations are real and causal up to the two loop order and he has also checked the unitarity of the formalism restricted to vacuum states.
Before ending this section let us rewrite (2.36) in a more convenient form for us. In general the propagator G ++ cannot be found exactly and has to be evaluated perturbatively. For instance, in the next section we will take perturbations h µν (x) to a given background metric and only the exact propagator corresponding to the background is known. Thus we write where A 0 ab is the unperturbed (diagonal) operator whose propagator, G 0 ab , is known, 38) and the diagonal operators V ab + · · · contain the perturbative terms (in the next section they will correspond to perturbations of order |h µν | and |h µν | 2 respectively). We can write where the products are operator products. In particular, expanding the logarithmic term in (2.36) and using that A 0 ++ G 0 +− = 0, we finally get, (2.31). Note that if it were not for the last term which involves the propagator G 0 +− this expression for the in-in effective action would agree with the in-out effective action which involves only one field φ(x); see, for instance, Hartle and Hu [11]. Therefore the term containing the propagator G 0 +− is the only new term that contributes to the field equation forφ + (x). It can be seen [38] that the effect of the last term in (2.41) is to make the field equation forφ(x) causal: if one takes the derivative of the in-in effective action with respect toφ + (x) and puts φ + =φ − =φ, the resulting field equation is causal.
Notice that in the case of a free scalar field on a nearly flat background the propagator G 0 ab which solves (2.38), corresponding to the flat background, and which provides the boundary conditions of the in-in problem is simply given by G 0

In-in effective action
In this section we compute the in-in effective action (2.41) for a conformal field in a nearly conformally flat spacetime. The cosmological background consists of a spatially flat homogeneous FRW with small perturbations as, where a(η) = exp(ω(η)) is the conformal factor, η is the conformal time dη = dt/a, t is the cosmological time, and h µν (x) is a symmetric tensor representing arbitrary small perturbations; we take the metric signature (− + · · · +). The classical action for a free (i.e. with no self-interactions) massless conformally coupled scalar field Φ(x) is given by where ξ(n) = n−2 4(n−1) ,R is the Ricci scalar for the metricg µν , and we take the spacetime dimensions, n, arbitrary for the moment in view of dimensional regularization. Because of the conformal coupling one may simplify the problem by defining a new matter field φ(x) and a new metric g µν (x) as, Then the action (3.2), after integration by parts and assuming no contributions of the surface integrals, is equivalent to the action for the field φ(x) in the nearly flat metric g µν , where R is the Ricci scalar for the metric g µν (x). Therefore the problem has been reduced to that of a scalar field φ(x) in a nearly flat background. We must keep in mind that the physical field is Φ(x), but the fact that the two fields differ by just a power of the conformal factor, which is a function of time only, simplifies considerable the connection between the vacua of the two fields. For instance, a positive frequency mode in flat space will correspond to a positive frequency mode in the conformally related space. Since for a free field in flat space the in and out vacua are equivalent (there is no particle creation) the same is true for the vacua of the conformal field in the conformally flat background. As a consequence non trivial quantum effects can be produced only by the breaking of conformal flatness which in this case is due to the coupling of the quantum field with the gravitational perturbations. The above action can be expanded in terms of these perturbations as, where the first term is simply the action for the field φ in flat spacetime, and the higher perturbative terms carry all the information on the interaction with the perturbations. Since we are interested in deriving the semiclassical correction to Einstein's equations due to the quantum effect of the scalar field but keeping the gravitational field as classical, we have to add to the effective action Γ m (m stands for matter fields), the classical action of the gravitational field S g [g µν ]. We should also add the action of any other classical source but we shall ignore this for simplicity; note that its effects on the semiclassical equations may be taken into account by simply adding the corresponding classical stress tensor to the quantum stress tensor. Furthermore, in order to renormalize the effective action it is sufficient to add to the usual Einstein's action, terms quadratic in the Riemann tensor, where µ is an arbitrary mass scale which will be useful in dimensional regularization. The quadratic terms with poles at n = 4 are those which are necessary to cancel the divergencies of Γ m ; notice that with this election one obtains the correct trace anomaly.
Following the previous section we introduce now two fields φ + (x) and φ − (x) which at some future hypersurface Σ coincide: φ + (x) = φ − (x). Since the scalar field is proved by the gravitational field we must assume that the two fields evolve in two different geometries g + µν (x) and g − µν (x) respectively where g ± µν (x) ≡ η µν + h ± µν (x), and since we assume that the fields have no interaction other than the gravitational we need not introduce the classical external currents J ± (x). Thus we shall write the total in-in effective action for the gravitational and the matter fields as, where Γ m [h + µν , φ + ; h − µν , φ − ] contains the quantum effects of the scalar field. Following eq. (2.41) we write Γ m in a perturbative expansion in h µν as Γ m = Γ (0) m + Γ (1) m + Γ (2) m + ... and write only the terms which contribute to the variation of h + µν , which are also those which contribute to the variation of φ + (x). Thus we can write, Γ To compute V (1) and V (2) we use (2.31) and (2.37) and expand S m [φ] as in (3.5), h αβ h αβ η µν and R (1) and R (2) are the first and second order terms, respectively, of the scalar curvature (see equation B.9 from the appendix). From these expressions one gets the operators A 0 , V (1) and V (2) by diffenciation with respect to the field φ(x), The explicit form of the operator V (2) will not be needed, however. Now we can write the propagator, from (3.12), as, recall that this is the propagator for the field φ(x), the propagator for the physical field Φ(x) is related to this by the conformal factor [12], but we do not need it here. All we need to know is that the boundary conditions for the physical field are determined by the boundary conditions of φ(x) in flat space. The effective action depends on the fieldsφ ± and the metric perturbations h ± µν . We may obtain the equation for the fieldφ(x) (field equation) by functional derivation with respect toφ + , i.e. equation (2.17), and the equations for the metric perturbations (backreaction equations) by functional derivation with respect to h + µν . Our primary interest is to obtain the backreaction equations. From equation (2.17) one can see that the vacuum expectation valueφ ≡ 0, in|φ|in, 0 = O(h µν ) (it would be zero in flat space), and thus the contribution to the backreaction equations coming from the matter action term will be at least of second order in the perturbations when the field equations are substituted. Formally, one can compute the effective action as a functional of h ± µν only, and thus, the terms involving the fieldφ + (S (i) m [φ + ]) are not necessary. We can now compute each of the (divergent) terms (3.8). As it is well known the first term Γ (0) m is easily renormalized by adding a suitable counterterm that cancels the divergencies which lead to the conformal anomaly [12] [25], but, this term does not play any role in the stress tensor of the field because it is independent of h µν .
We can now go to the next term in (3.8), Γ (1) m , but this formally divergent (tadpole) term has no contribution, since T r(V (1) + G 0 ++ ) involves n-dimensional integrals of the form 1/k 2 , k α /k 2 , and k α k β /k 2 (where k α is the integration momentum variable) which are identically zero in dimensional regularization [41]. Therefore there is no term linear in h µν in the effective action, i.e. we have Γ (1) m = 0. The first non trivial quantum contributions to the stress tensor coming from Γ m are quadratic in h µν and we thus need to compute Γ (2) m . Here again the second (tadpole) term in (3.8), T r(V gives no contribution in dimensional regularization, since the typical integrals are of the same type that those of T r(V . For this reason we do not need the explicit form of the operator V (2) .
The problem is thus reduced to the evaluation of the third and fourth terms in (3.8). As we have stressed in section 2 the third term also appears in the evaluation of the in-out effective action and the fourth term is typical of the in-in contribution to this order. Let us evaluate the third term, i.e. T 1 ≡ − i 4 T r(V We now introduce the projector P µν = η µν − p µ p ν /p 2 , the symbol η µναβ ,h µν ≡ η µανβ h αβ , change the p integration by p ′ ≡ q − p, rename p ′ as p again, and write T 1 as, The momentum integrals can be computed in the standard way, see appendix D, and expanding around n = 4 we get after a rather long calculation, 19) and the parameter µ 0 is a fixed parameter involving Euler's constant γ. Now using the expressions of the Riemann components in terms of the projectors P µν of appendix B.2 we can write, 21) and α ≡ (2880π 2 ) −1 . Note that the divegent terms with a pole at n = 4, are local and quadratic in the curvature. They may be compensated by counterterms in the gravitational part of the action S (2) g coming from (3.6). Recall that the curvature terms here depend on the metric g µν rather than the physical metricg µν .
Let us compute now the fourth term of Γ (2) m , i.e. T 2 ≡ i 2 T r(V , which depends on the propagators G 0 +− = −∆ + and G 0 −+ = ∆ − , represented in (2.22) with m = 0. We can write changing the integration variable from p to q − p as in the previous case we can write, After performing the phase space integrals and expanding around n = 4, see appendix D, we obtainL µναβ (p) = p 4 I 2 (p) 1440 (3P µβ P να − P µν P αβ ) + O(n − 4) , (3.24) where which has no poles at n = 4, therefore the term T 2 requires no counterterms in the action to be renormalized. Using the expressions of the Riemann components in terms of P µν (see appendix B.2) we get, Here again the Riemann components refer to the metric g µν . We now have Γ (2) m = T 1 + T 2 which must be renormalized by adding the gravitational action up to the second order in h µν . The explicit expansion of S g , in (3.6), up to this order in terms of the curvature components of the metric g µν is: +O(n − 4), (3.28) where we have dropped the + sign on the fields for simplicity. Finally adding these terms to Γ m and including only the terms which contribute to the variation of h + µν we get the renormalized effective action, whereμ ≡ µµ 0 and we have substituted 1 by the volume densities √ −g in all the integrals involving quadratic curvature terms of the metric g µν in order to facilitate the identification of the exact variational formulae of the appendix E needed in the computations of the next section.
If one is interested in the production of particles, the in-in effective action is not suitable because the probability of particle creation is related to the transition amplitude from the in to the out vacua, and this amplitude is directely related to the in-out effective action [24]. Let us write the vacuum persistence amplitude in the presence of an external source, i.e. 0, out|in, 0 J = exp(iW [J]). The probability of pair creation is proportional to the imaginary part of W P = 2ImW, but if we take J = 0 and consider the quantum fields propagating in the gravitational background, then W is just the in-out effective action: Γ (io) . The calculations leading to such an action are similar to those for the in-in case, although they are simpler because we do not need to introduce two fields. The main work has already been done: consider just a single field in (3.8) and ignore the term T 2 . The renormalized action is obtained again by adding the gravitational action (3.28), the final result can be read directly from (3.29): ignore the plus indices and the term involving K 2 (x − y). The non local term now includes K 1 (x − y;μ) only, this term is complex and it is responsible for the particle creation effect. It turns out that the pair creation probability is given, however, by a local term as is well known [24] [14]. In fact, the imaginary part of the kernel K 1 (x − y;μ) is, From the expression for Γ (io) and (3.30), after performing the x and y integrations which lead to the Fourier transform of the curvature tensor, R µναβ (p), we get Finally, using that the Gauss-Bonnet topological invariant is zero, the above relation can be written in terms of the Fourier transform of the Weyl tensor of the physical metricg µν as, (3.32) in agreement with the expressions computed by other means [14].

Semiclassical equations
In this section we obtain the quantum mechanically corrected Einstein's equations due to the presence of a massless conformal scalar quantum field. The semiclassical equations for the metric perturbation can be found by functional differentiation of the in-in effective action (3.29) with respect to h + µν (x) and then restricting h + From equation (4.1) it is easy to derive the equations of motion to first order; we use that for an arbitrary functional A[g µν ], to find the variation of the first two terms of (3.29). Notice that we assume that ω(x) is a scalar function independent of the metric (in general we will assume that ω(x) depends on the spacetime point only, in particular, in the flat FRW case it will be a function of the cosmological time t only).
Using the expressions listed in the appendix E one can show that the semiclassical equations can be written as where G µν (x) is the Einstein's tensor, C µανβ (x) the Weyl's tensor, B µν (x) and A µν (x) are the exact spacetime tensors given by the variation of d 4 xR 2 (x) and d 4 xC µανβ C µανβ , respectively (see appendix E.2 and use that C µανβ C µανβ = R µανβ R µανβ − 2R µν R µν + 1 3 R 2 ), with respect to an arbitrary metric g µν (x) and H µν (x) is the spacetime tensor defined by In equations (4.3) we are only interested in the expressions of these tensors up to first order in the perturbations, as indicated by the bracketed subindices and we recall that an over tilde on the tensors refer to the physical metric. The non-local part H(x − y;μ) is the sum of the integrals (3.21) and (3.27), which can be simplified to by using ln(±i) = ±i π 2 and lim ǫ→0 ln(ǫ + ix) = ln |x| + sign(x)i π 2 . Note that this equation is real, in spite of appearences, because the imaginary part of the integrand is an odd term with respect to the integration variable p µ . One can also notice that H(x − y;μ) differs from that defined by Horowitz [28] by a factor (1/4π).
Equations (4.3) are dynamical equations with higher order derivative terms. When the background is flat they reduce to the field equations studied by Horowitz [28] and Jordan [30]. As we have noted in the previous section one can add a classical stress matter source to these field equations.
To compare the functional method used in this paper with other techniques and, in particular, to compare equations (3.29) and (4.3) with previous results one can give, for exemple the energy-momentum tensor of the quantum field and the expression of the semiclassical equations in two dimensions.

Stress tensor to first order
From (4.3) one can read the zero and first order vacuum expectation value of the energymomentum tensor of the scalar field, The stress tensor to first order in h µν , T µν (1) , is in agreement with that obtained by Horowitz and Wald [26] and Starobinsky [27]. On the other hand, the zeroth order tensor T µν (0) , which gives the exact stress tensor for a conformal scalar field in a conformally flat spacetime, agrees with that found by other techniques [42] [12]. Note also that we recover the trace anomaly result to this order in h µν ,

Two dimensional gravity
As an exercise we will derive here the semiclassical corrections to the stress tensor of the scalar field in two spacetime dimensions. At the classical level there is no dynamics for the gravitational field in two dimensions because the Einstein tensor is identically zero, but the semiclassical corrections due to the presence of a quantum field lead to non trivial effects. Since we have worked in n dimensions in order to use dimensional regularization, we can use most of the work done in the previous section to analize the two dimensional case. From equations (3.17) and appendix D it is not difficult to show that, Then, the in-out contribution to the effective action T 1 becomes, From equation (3.23) it is straighforward to see that there is no contribution to the effective action due to the typical in-in term T 2 . Quantum corrections appear to be non-local and quadratic in the scalar curvature, but contrary to the four dimensional case there are no divergent terms in the regularization process and counteraction terms are not needed. Finally one can express the effective action to one loop as, The stress tensor of the quantum field can be obtained from the second term of the above equation by differentiating with respect to the metric. To order zero in the perturbation the energy-momentum tensor vanishes, but to first order there is a non-local contribution, (4.14) and the trace is local T µ (1)µ (x) (2D) = R/(24π) + O(h 2 µν ) as expected [12]. Note that in two dimensions there is no difference between the in-out and the in-in effective actions, thus the use of the usual in-out effective action to derive the semiclassical equations is justified.

Backreaction on the field of a static cosmic string
In this section, as a simple example, we discuss the backreaction due to one loop quantum effects on the gravitational field of a static cosmic string. First, we must compute the vacuum expectation value of the stress tensor of a conformally coupled massless scalar quantum field outside the core of a straight and static cosmic string.
In the weak field approximation the metric of a cosmic string can be seen as a small perturbation about flat space, g µν (x) = η µν + h µν (x). (5.1) Let T µν c be the stress tensor of a cosmic string (or of any other classical source), the semiclassical equations up to first order in the perturbation, can be written us, There is no zero order correction, T µν (0) , to the classical Einstein's equations because the background is flat; see (4.6). The vacuum expectation value T µν (1) is obtained from equation (4.7) when the conformal function is ω = 0. We have Note that the quantum correction term T µν (1) depends on the Einstein tensor G µν , thus one may use Einstein's equations to the classical order, which is already first order in h µν , to substitute G µν by 8πG N T µν c . This simplifies considerably the problem since the explicit gravitational field of the string (or the classical source) is not required to compute T µν (1) . The stress tensor of a static cosmic string along the z-axis can be written in the thin line approximation as, T cµ ν (x, y) = −µδ(x)δ(y)diag(1, 0, 0, 1), (5.5) where µ is the mass per unit lenght of the string. Since G N µ ∼ 10 −6 for GUT strings we can assume that G N µ ≪ 1 and the linear approximation (5.1) is justified. In fact, ignoring quantum effects the stress tensor (5.5) leads in the linear approximation to the conical metric with a deficit angle of 8πG N µ [43] Equations (5.4) can be expressed in terms of the string stress tensor and its trace T ≡ T c = −2µδ(x)δ(y), as, In this case, B (1)µ ν (x) in (5.3) gives no contribution outside the core of the string because it is proportional to partial derivatives of a delta function with support on the z-axis. The tensor components A (1)µ ν (x), in the Minkowskian coordinates (t, x, y, z) in which equation (5.1) is given, are The non-local term H(x − y;μ) in equation (5.3) can be expressed as a delta function; following Jordan [30] we can write Introducing these expressions into (5.3), we may perform the space integrals; this is easy due to the presence of the delta functions. In fact, as an example, let us compute the x y component of the stress tensor, T y x = T x y , to first order in G N µ. The only contribution to this component comes from This expression can be written as, (5.10) Following straightforward steps, The remaining non null components of the stress tensor in Minkowskian coordinates to the same order G N µ are, (5.12) Because of the cylindrical symmetry of the problem it is better to express this tensor components in polar coordinates as, which is independent of the arbitrary renormalization scaleμ, (as expected because it would contribute with delta functions with support on the core of the string which we do not consider). This tensor coincides with the first order development in G N µ of previous exact results obtained by other techniques [37]. Note that in the exact case, i.e. when the classical solution is found explicitly, the one loop quantum stress tensor is simply (5.13) where one changes 32G N µ by 2 The backreaction equations (5.2) have been solved by Hiscock [36] who found that the linear corrections to the metric outside the string are such that the spacetime is no longer flat space with a deficit angle: the two surface perpendicular to the string is an hyperboloid which asymptotically approaches the conical surface at large distances (the one loop quantum corrections to h µν (x) are of the form G N µh/r 2 ). Note that the semiclassical equations here have no higher order derivatives because we have treated the quantum terms as a perturbative correction (as Hiscock does) in line with Simon's arguments [32]. Work on the backreaction on dynamic cosmic strings [44] is in progress.

Acknowledgements
We would like to thank Bei-Lok Hu and Esteban Calzetta for stimulating discussions and helpful suggestions. This work has been partially supported by a CICYT research project number AEN90-0028.