Finite temperature corrections to the energy-momentum tensor at one-loop in static space-times

Finite temperature corrections to the effective potential and the energy-momentum tensor of a scalar field are computed in a perturbed Minkoswki space-time. We consider the explicit mode decomposition of the field in the perturbed geometry and obtain analytical expressions in the non-relativistic and ultra-relativistic limits to first order in scalar metric perturbations. In the static case, our results are in agreement with previous calculations based on the Schwinger-De Witt expansion which indicate that thermal effects in a curved space-time can be encoded in the local Tolman temperature at leading order in perturbations and in the adiabatic expansion. We also study the shift of the effective potential minima produced by thermal corrections in the presence of static gravitational fields. Finally we discuss the dependence on the initial conditions set for the mode solutions.

covariant, did not contain contributions from the inhomogeneous gravitational fields at the leading order in metric perturbation and in the adiabatic expansion in both static and cosmological space-times.
In this work, we extend these methods to include finite temperature effects. The inclusion of the Bose-Einstein factor, accounting for the statstical distribution of the energy states, produces a smooth behavior of all quantities involved at large energies, not being necessary to apply any regularization technique (once the vacuum contribution is renormalized). As mentioned above, in order to compute the aforementioned contribution, we apply the "brute force" method described in [11,12], i.e. performing a summation over the perturbed modes of the quantum field obtained as solutions of the Klein-Gordon equation. We are able to get analytical expressions for the effective potential and the energy-momentum tensor in the non-relativistic and the ultra-relativistic limits. In the static limit, we find that local gravitational effects can be taken into account through the Tolman temperature [13]. This is in accordance with computations of the energy-momentum tensor of a scalar field at finite temperature in a static space-time using the Schwinger-DeWitt approach, [14] and [15]. However, we also obtain the explicit time dependence of the expectation values for finite times, which shows that the Tolman temperature can only be defined in the asymptotic time regions.
The work is organized as follows. Section II describes the general approach to compute an expectation value over a thermal state in a perturbed FRW metric. The particular expressions to be computed in the case of static space-times are presented in Section III. Section IV and V explain the approximations applied to obtain the final result in the non-relativitic and ultra-relativistic limits, respectively. Shifts in the minimum of the effective potential produced by thermal correction are discussed in Section VI. Our conclusions are presented in Section VII.

II. FINITE TEMPERATURE CORRECTIONS
Given a scalar field φ, with potential V (φ), its classical action in a (D + 1)-dimensional space-time with metric tensor g µν can be written as As is well known, the solutions φ =φ of the classical equation of motion φ + V ′ (φ) = 0 (2) are those that minimize the action. On the other hand, quantum fluctuations around the classical solution δφ = φ −φ satisfy the equation of motion Let us consider a metric which can be written as a scalar perturbation around a flat Robertson-Walker background where η is the conformal time, a(η) the scale factor, and Φ and Ψ are the scalar perturbations in the longitudinal gauge. Given this geometry, the mode solutions δφ k to (3) can be found using a WKB approximation to first order in metric perturbations and to the leading adiabatic order as [12] δφ k (η, x) = δφ where are the unperturbed mode solutions with The explicit expressions for P k (η, x) and δθ k (η, x) in Fourier space are shown in Appendix A.
The effects of quantum fluctuations on the classical field configuration can be taken into account using the one-loop effective potential [12,16] where V (φ) is the tree-level potential and the expectation value of the operator δφ 2 is taken over a particular quantum state of the field. Taking into account (6) and (7) and assuming that the quantum state has a fixed number of particles per mode n k , the one-loop contribution to the effective potential reads 1 2 where we have definedP beingx = k · p/(k p) and including the general integration measure in D dimensions. From now on, we consider a thermal quantum state. Then, the number of particles per mode is given by the Bose-Einstein distribution where T is the temperature of the state, for the moment understood as a parameter of the Bose-Einstein distribution (see next section). Let us define V 1 (φ) as the one-loop quantum vacuum contribution, i.e.
and V T (φ) as the term that includes finite temperature corrections so that we can write the one-loop effective potential at finite temperature as It is important to notice that both the vacuum and the thermal contributions have a homogeneous term, corresponding to the background geometry, and an inhomogeneous one, proportional to the perturbations. Then, The homogeneous part due to vacuum effects V h 1 (φ), after applying the minimal substraction scheme MS in dimensional regularization with D = 3 + ǫ, is given by [12] A detailed analysis of the inhomogeneous part of the vacuum V i 1 (φ) was performed in [11] with a cutoff regularization, and in [12] using dimensional regularization. When a cutoff Λ is used, the result turns out to be proportional to m 2 (φ)Λ 2 Φ in the static case, i.e. only the quadratic divergence appears. In dimensional regularization we find to first order in perturbations and to the leading adiabatic order that in agreement with the absence of logarithmic divergences in the cutoff case.
In this work, we focus on the thermal contribution V T (φ). The corresponding inhomogeneous contribution can in turn be split in the terms proportional to Φ and Ψ as It is important to note that expression (9) defines the potential except for the addition of an arbitrary function which could depend on the space-time coordinates and the temperature. This function does not modify the dynamics of the field (2) since it does not introduce any dependence on m(φ).
In the same fashion, the thermal contribution to the components of the energy-momentum tensor can be obtained from the expressions given in the reference [12] including the number of particles per mode n T k , thus Let us divide the energy-momentum tensor, in the same way as for the potential case, in a vacuum contribution, which does not depend on the number of particles per mode n T k , and a thermal contribution.
each one having a homogeneous and an inhomogeneous part. It can be shown [12] that the energy-momentum tensor of the vaccum is given by T µ ν (η, p) vac = ρ vac δ µ ν , where the energy density ρ vac and pressure p vac are given in the MS renormalization scheme with D = 3 + ǫ by This implies that the inhomogeneous part of the vacuum contribution is zero when dimensional regularization is used, therefore metric perturbations do not contribute to the leading adiabatic order. In this paper, we compute the homogeneous and inhomogeneous parts of T µ ν (η, p) T .

III. STATIC SPACE-TIMES
Although the expressions for the perturbed solutions given in Appendix A are valid for general perturbed FRW space-times, in this work we focus on static space-times, i.e. we will take a = 1 and Φ = Φ(x), Ψ = Ψ(x). The general case is of great interest for cosmological scenarios, nevertheless the time dependence of the scale factor increases the complexity of the computations, making extremely difficult to obtain analytical expressions. In addition, in order to define a thermodynamic temperature, there must be a timelike Killing vector field, namely the space-time must be static or stationary.
In order to compute V T (φ) and the energy-momentum tensor T µ ν (η, p) T thermal contributions, our first step will be to expand the P k (η, p) and δθ k (η, p) functions in powers of pη [Appendix B]. These expansions, allows us to find a common structure of the integrals involved.

A. Effective potential
Taking into account (B1) and (10), it is clear that we have to deal with the following kind of integrals to compute the finite temperature correction to the effective potential. It is convenient to use the dimensionless variables u = ω k /T and x = m/T instead of k and m respectively. In terms of these new variables the integral reduces to (extracting a global factor T D+1 ) where X ≡ m(φ)/T . It is also useful to interchange the order of integration of this integral and divide it in the following way where the first part takes into account the contribution from modes with energies below the mass of the field while the second part includes the contribution from modes with energies above the mass of the field.

B. Energy-momentum tensor
To compute the energy-momentum tensor, the following integrals appear Using the same dimensionless variables u and x we get (also extracting a global factor T D+1 ) Only modes with energies above the mass of the field contribute to the energy-momentum tensor.
In the following we compute the integrals I X α,n (30) and J X α,n (32) in the non-relativistic and the ultra-relativistic limits.

IV. NON-RELATIVISTIC LIMIT
A. Effective potential In the non-relativistic limit m(φ)/T → ∞ (or X → ∞), the contribution from modes with energies above the mass of the field is exponentially damped because of the Bose-Einstein factor, hence the leading contribution in the non-relativistic limit is given by the first part of (30) when taking X = ∞ where ζ(x) is the Riemann Zeta function. Therefore, using expression (10) together with the result for the integral (33) and the expansion ofP k (η, p) (B1), we obtain (assuming D = 3) for the leading contributions after resummation of the series in pη Note that there is no dependence on the field (which may appear through mass terms). Therefore these expressions do not affect the field dynamics and they can be neglected. On the other hand, even though we are considering static backgrounds, there is an explicit time dependence of the result. This can be traced back to the particular mode choice in (A1) and (A2). In particular, taking the η → ∞ limit, which corresponds to setting initial conditions for the modes in the remote past, we recover static results for the effective potential.
In the static limit η → ∞, the following expression is obtained It can be shown that the leading inhomogeneous effect in the static limit only depends on the Φ potential and in fact it can be obtained from the homogeneous result replacing the temperature by the local Tolman temperature [13] Notice however that in the results for finite time given in (35) and (36), the explicit time dependence of the effective potential prevents the introduction of a Tolman temperature.
The next-to-leading correction, V , including terms O(T /m(φ)), can be obtained by applying a modified version of the Laplace's method to the following integral 1 where we have replaced the Bose-Einstein factor by the Boltzmann factor. B z (a, b) is the incomplete Beta function. When u/X ≫ 1, the integrand is exponentially damped as e −u/X . Then, we Taylor expand the expression inside the brackets around X 2 /u 2 = 1 to obtain The expression inside the exponential has a maximum at u ∼ X when X → ∞. 2 Taylor expanding the argument of the exponential around u = X up to order O(u) [including the logarithmic divergence] the integration in u can be performed to get the following result which does not depend on n. Because of the factor X D/2−α in the last expression, the expansion in pη mixes with the expansion in X(= m(φ)/T ). Finally, the next to leading contribution to the potential for pη ≪ 1 is given by A better approximation for smaller values of X is obtained if we do not drop α in the denominator in (41). This improved approximation is shown in Figure 1 (right panel). It is important to note that each order in (pη) is suppressed by a factor (T /m(φ)) with respect to the previous order, because of the mixing discussed above. For instance, the 1 The symbol ≃ stands for an approximation in the Taylor sense, while ∼ stands for an asymptotic approximation, namely the quotient between both results equals 1 in the appropriate limit. 2 Here we are dropping a term linear in α in the expression for the maximum. This means that we cannot allow α → ∞. Since α is related with the order of the expansion in p η, the results are only valid if the series appearing in (B1) is truncated at some order such that α ≪ X. Although it could be done for arbitrary α, it would not be very useful if the expression cannot be ressummed. Nevertheless, it will be shown that the l-term is supressed by a factor 1/X l , thus only the first terms are relevant in this limit (X → ∞). correction proportional to Ψ does not depend on (pη) to leading order in (T /m(φ)) [see eq. 42], then the dependence on (pη) 2 proportional to Ψ is suppressed by a factor (T /m(φ)) with respect to the (pη) 2 correction proportional to Φ, as shown in Figure 1 (right panel).
Because of the mixing between the expansion in X(= m(φ)/T ) and pη we cannot obtain a result valid for arbitrary scales p and times η. However, it is possible to obtain the static result by taking the limit η → ∞ directly on (10). According to this procedure, we get As can be checked in a straightforward way from (43), also for the next to leading contribution in the static limit, the inhomogeneous correction can be obtained from the homogeneous result by replacing the temperature with the Tolman temperature (38).

B. Energy-momentum tensor
The leading order of the energy-momentum tensor is already exponentially damped, since only modes with energies above the mass of the field contribute. We write the integral (32) as where the Bose-Einstein factor has been replaced by the Boltzmann factor. Applying the Laplace's method again we get Then, taking into account the expressions given in Section II and the result (45), the energy-momentum tensor for pη ≪ 1 is given by where ρ T and p T are the energy density and pressure produced by the thermal corrections. We have only retained the leading order in m(φ)/T . Further corrections O((pη) 2l ) are suppressed by a factor (m(φ)/T ) l . In the non-relativistic case, it is not possible to take the static limit in the final expressions since we only have the results for pη ≪ 1 as discussed before. However, the static expression can be obtained by taking the static limit in the original expressions (25) Once again, in the static limit, the inhomogeneous corrections, depending only on the Φ potential and can be obtained from the homogenous one by introducing the Tolman temperature.

A. Effective potential
In the ultra-relativistic limit, m(φ)/T → 0 (or X → 0), the dominant contribution comes from modes with energies higher than the mass of the field. Therefore, the second part of (30) gives where we have expanded the incomplete Beta function B z (a, b) for X ≪ 1 in the last line. The leading contribution comes from n = 0. Replacing the lower limit of integration by 0 we get in that limit where Li n (z) is the polylogarithm function.
Therefore, from (30) and using the expansion ofP k (η, p) in (B1) and the result (53), we can resum this contribution to get the leading contribution The explicit time dependence of the general results obtained in a static metric can be traced back to the initial conditions of the modes. Taking the limit η → ∞ in (55) and (56), the initial conditions are washed out and the remaining correction in Fourier space is In this case we can also obtain the inhomogeneous result by replacing the temperature by the local Tolman temperature (38) in the homogeneous result.
To get the real space result in the static limit, one has to compute the Fourier transform of the complete expression and then take the static limit, η → ∞. Following this procedure, it is possible to get the real space result for arbitrary perturbation (see Appendix C) which reads Therefore, as expected, the static limit and the Fourier transform commute (compare (57) and (58)). This is a general conclusion for the functions in Fourier space appearing in this paper due to the results of Appendix C.
In real space, the corrections due to Newtonian perturbations Φ N (p) and Ψ N (p) given by inside the lightcone (r < |η|) are while on and outside the lightcone (r ≥ |η|) are The next-to-leading order corrections can be obtained by computing the first part of equation (30) plus next-toleading terms coming from equation (52) (see Appendix D). Finally, after resummation of the the series expansion (B1) we get for V T (N L) , (up to O((m/T ) 5 )) where γ is Euler's constant and J n (x) Bessel functions. Considering Newtonian perturbations Φ N and Ψ N , in real space we get for the region inside the lightcone (r < |η|) and outside and on the lightcone (r ≥ |η|) Here for simplicity we have only shown the O(m/T ) 3 contributions. In the static limit η → ∞ one gets which is also valid in real space replacing Φ(p) by Φ(r) (see Appendix C). Here again we find that the inhomogeneous result can be obtained by replacing the temperature in the homogeneous contribution by the local Tolman temperature.

B. Energy-momentum tensor
The leading contribution is given by the integral (32) when n = 0 where we have replaced the lower limit of integration by 0 and expanded the integrand around X = 0 in the second line. Therefore, we get for the energy-momentum tensor which does not correspond to a perfect fluid 3 .
In real space, we have for Newtonian perturbations inside the lightcone (r < |η|) Outside the lightcone (r > |η|), we get and on the lightcone (r = |η|) the results are In the static limit, the energy density and pressure are the non-diagonal terms being zero. Once again, these results can be interpreted as being the corresponding energy density and pressure for a classical gas at the local Tolman temperature (38) in agreement with [14] and [15]. The same expressions for the static limit apply in real space (see Appendix C).

VI. THERMAL SHIFT OF THE EFFECTIVE POTENTIAL MINIMA
Once the effective potential is obtained, the value of the field for which determines the value attained by the classical fieldφ. The inhomogeneous contributions to the effective potential will now induce a spatial dependence onφ which can be written aŝ whereφ 0 is the minimum of the potential in the absence of metric perturbations, but including the one-loop corrections, i.e.
then to first order in metric perturbations and taking into account that V i 1 = 0 in dimensional regularization, we get Thus, the relative classical field variation is given by the temperature correction The perturbation is therefore proportional to the third derivative of the tree-level potential, so that variations in the field expectation value are only generated in theories with self-interactions. In the non-relativistic limit and in the static limit we get in Fourier space In the ultra-relativistic limit, we obtain for arbitrary η which in the static limit reduces to valid also in real space replacing Φ(p) by Φ(r). In particular, in real space we get for Newtonian potentials inside the lightcone (r < |η|) while oustide and on the lightcone (r ≥ |η|) Thus, we see that outside and on the lightcone (r ≥ |η|), the result reduces to minus the static limit result (104). Inside the lightcone (r < |η|), the thermal shift depends on time and approaches asymptotically the static case. From these results we see that there is a negligible shift in the classical fieldφ at low temperature because of the exponential suppression, however, depending on the form of the tree-level potential, the shift generated by metric perturbations in the ultra-relativistic limit could be relevant in certain cases. Now, let us focus on the critical temperature of the phase transition T c defined by [16] V eff (φ 0 + ∆φ) = V eff (0) where V eff ,φ 0 and ∆φ depend on the temperature T . Expanding equation (107) around the critical temperature in the absence of metric perturbations T 0 c , we get for the leading order which is the definition of T 0 c . Considering the next to leading order and solving for δT c = T c − T 0 c , we obtain the following expression for the shift in the critical temperature produced by metric perturbations 4 It can be shown (see Appendix E) that in the static limit therefore, in that case, the shift in the critical temperature is given by i.e. once again the curvature perturbation Ψ does not contribute to the shift.

VII. CONCLUSIONS
Considering a scalar field at finite temperature in an inhomogeneous static space-time, we have computed the one-loop corrections to the effective potential and to the energy-momentum tensor induced by static scalar metric perturbations around a Minkowski background to first order in metric perturbations. To this aim, we have applied the formalism developed in [11,12]. In particular we have used the explicit expressions for the perturbed field modes together with the assumptions of adiabatic evolution of the field. In order to obtain analytical expressions, the non-relativistic and ultra-relativistic limits have been considered.
In the non-relativistic limit, we obtained the corresponding expressions in the static limit and also the limits for large-scale perturbations (small p) or times close to the initial time. In the ultra-relativistic limit, we obtain the complete results for arbitrary p and η up to O(m/T ) 5 . In the static limit, our results agree with those in [14] and [15] which were obtained by means of the Schwinger-de Witt expansion. The energy density and pressure in the static limit are consistent with a local thermal distributions at the local Tolman temperature. Besides, our results are sensitive to the initial conditions set at the initial time for the mode solutions.
We have also discussed the space-dependent shift in the classical field induced by the metric perturbations. As expected, in the non-relativistic limit the shift is Boltzmann suppressed. However, in the ultra-relativistic case and depending on the shape of the potential, the shift could be non-negligible.
The results of the paper have shown that mode summation is a useful technique to obtain explicit expressions for one-loop quantities at zero and finite temperature. Unlike the more standard Schwinger-de Witt expansion, this method allows to calculate not only the local contributions to the effective action, but also the finite non-local ones which will appear at second order in the perturbative expansion. Future work along this line will allow to explore this possibility. P k (η, p) = η 0 e −i k·p β k (η,η ′ ) H k (η ′ , p) 2ω k (η ′ ) dη ′ + e −i k·p β k (η,0) P k (0, p) . (A1) δθ k (η, p) = η 0 e −i k·p β k (η,η ′ ) G k (η ′ , p) dη ′ + e −i k·pβ k (η,0) δθ k (0, p) .
where P k (0, p), δθ k (0, p) are the intial conditions, and P k (0, p) is fixed by the orthonormalization condition of the modes while δθ k (0, p) remains arbitrary. The arbitrariness in δθ k (0, p) can also be absorbed in a change of the lower integration limit in (A2). As we will see, only setting the time origin to η 0 → −∞, which is equivalent to taking η → ∞, corresponds to the exact static limit.
Full details about the solutions (A1) and (A2), and about the orthonormalization condition are given in [12].