# Casimir effect in quadratic theories of gravity

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## Abstract

In this paper, we study the Casimir effect in a curved spacetime described by gravitational actions quadratic in the curvature. In particular, we consider the dynamics of a massless scalar field confined between two nearby plates and compute the corresponding mean vacuum energy density and pressure in the framework of quadratic theories of gravity. Since we are interested in the weak-field limit, as far as the gravitational sector is concerned we work in the linear regime. Remarkably, corrections to the flat spacetime result due to extended models of gravity (although very small) may appear at the first-order of our perturbative analysis, whereas general relativity contributions start appearing at the second order. Future experiments on the Casimir effect might represent a useful tool to test and constrain extended theories of gravity.

## 1 Introduction

Einstein’s general relativity (GR) has undergone many challenges in the last century, but it always proved its worth thanks to high precision experiments which have confirmed a plethora of its predictions [1]. Despite its extraordinary achievements, there are still open questions which need to find an answer, that is nowhere to be found in GR. For instance, in the short-distance (ultraviolet) regime, from a classical point of view Einstein’s theory turns out to be incomplete due to the presence of black holes and cosmological singularities, whereas from a quantum perspective it fails to be a renormalizable theory. On the other hand, at large scales GR is not capable of coming up with an explanation for dark matter and dark energy, even though their presence is strongly supported by observational data already available for a long time.

In the past years, all these fundamental issues stimulated a vivid investigation revolving around a plausible extension of Einstein’s GR domain. Among all the theories that popped out with the above intent, one of the straightforward approach consists of generalizing the Einstein-Hilbert action by including terms which are quadratic in the curvature, for example \({\mathscr {R}}^2,\) \({\mathscr {R}}_{\mu \nu }{\mathscr {R}}^{\mu \nu }\) and \({\mathscr {R}}_{\mu \nu \rho \sigma }{\mathscr {R}}^{\mu \nu \rho \sigma }.\) First interesting results in the context of quadratic gravity can be attributed to Stelle [2, 3], who showed that a gravitational theory described by the Einstein-Hilbert action with the addition of \({\mathscr {R}}^2\) and \({\mathscr {R}}_{\mu \nu }{\mathscr {R}}^{\mu \nu }\) turns out to be power-counting renormalizable. However, this apparatus lacks of predictability due to the presence of a massive spin-2 ghost degree of freedom which breaks unitarity at the quantum level, when the standard quantization prescription is adopted^{1}. Despite the presence of the ghost field, such a theory can still be considered predictive as an effective field theory whose validity is accurate at energy scales lower than the cut-off represented by the mass of the ghost. Another important improvement of quadratic gravity can be found in the Starobinski-model of inflation [9, 10, 11], which is able to suitably explain the current data; differently from the model of Stelle, here only the term \({\mathscr {R}}^2\) shows up in the quadratic part of the action. It is also worthwhile to highlight that gravitational actions with quadratic curvature corrections were taken into account in several different frameworks (see for example Refs. [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]).

The models so far discussed have been developed employing *local* quadratic theories of gravity, which means that the corresponding Lagrangian depends polynomially on the fields and their derivatives. In recent years, also *non-local* quadratic theories have aroused a significant interest, since the presence of non-local form-factors in the gravitational action may be useful both to solve the problem of ghosts and to considerably improve the ultraviolet behavior of the quantum theory (see Refs. [24, 25, 26, 27, 28, 29, 30, 31] for more details).

On the other hand, all the models introduced up to now can be argument of further theoretical treatment. Indeed, as already anticipated, they can be included in countless applications of the most disparate physical frameworks. For what concerns our work, we are mainly focused on the analysis of the Casimir effect when the spacetime background is described by quadratic theories of gravity. The Casimir effect is a concrete manifestation of quantum field theory (QFT), which occurs whenever a quantum field is bounded in a finite region of space. Since it was firstly introduced to the scientific community [32, 33], it has risen a constant interest and investigative efforts, due to the possibility of extrapolating substantial pieces of information from experiments. Such a statement holds true not only for the case in which the Casimir effect is analyzed in flat spacetime [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44], but also when the confinement of the quantum field occurs in a curved background [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62], and even when Lorentz symmetry is violated [63, 64, 65, 66, 67] or mixed fields are considered [68].

In this article, we study the Casimir effect in a curved spacetime emerging from a pure gravitational action quadratic in the curvature invariants. For this purpose, we closely follow the approach introduced in Ref. [69], in which the authors analyze a scalar-tensor fourth order action stemmed from a non-commutative geometric theory. In contrast, we analyze the dynamics of a massless scalar field between two nearby plates in a curved background described by quadratic theories of gravity and in the weak-field limit.

The paper is organized as follows: in Sect. 2, we briefly review the most important features of gravitational theories whose action is quadratic in the curvature invariants, with a peculiar attention to the linearized solutions. In Sect. 3, we study the dynamics of the massless scalar field in the context of the Casimir effect with a curved background. Section 4 is devoted to the calculation of the main physical quantities of the Casimir effect, namely the mean vacuum energy density and the pressure, for several quadratic theories of gravity. Section 5 contains discussions and conclusion.

Throughout the whole paper, the adopted metric signature is \(\mathrm {diag}\left( -,+,+,+\right) \) and natural units \(c=\hbar =1\) are used.

## 2 Quadratic theories of gravity

*N*is finite (namely, \(N<\infty \)), we have a local theory of gravity of order \(2N+2\) in derivatives, whereas if \(N=\infty \) and/or \(n<0\) we have a non-local theory of gravity whose form-factors \({\mathscr {F}}_i(\Box )\) are

*not*polynomials of \(\Box .\)

*slightly*curved background described by the action in Eq. (1). Thus, we can apply a weak-field approximation in order to derive the linearized regime of Eq. (1) around the Minkowski background \(\eta _{\mu \nu }\)

### 2.1 Linearized metric solutions: point-like source in a weak-field limit

^{2}:

^{3}

## 3 Massless scalar field in a curved background and the Casimir effect

We now want to study the behavior of a massless scalar field \(\psi (t,\,\mathbf {r})\) confined between two plates and embedded in a gravitational field (see Fig. 1); to this aim, we basically follow the procedure presented in Ref. [69].

^{4}For simplicity, we consider the massless scalar field \(\psi (t,\,\mathbf {r})\) confined between two parallel plates separated by a distance

*L*and with extension

*S*, placed at a distance

*R*from the gravitational non-rotating source (\(R\gg L, \sqrt{S}\)).

*R*along the

*z*direction by using the gravitational potentials \(\varPhi (\mathbf {r})\) and \(\varPsi (\mathbf {r})\) given in Eq. (12)

*z*is free to range in the interval [0,

*L*].

*t*, and

*N*is a normalization constant. The field equation in Eq. (17) thus becomes

*n*being an integer number.

### 3.1 Vacuum density energy

### 3.2 Pressure

^{5}

The relations in Eqs. (32) and (34) give the corrections to the Casimir pressure up to the second order \(\mathscr {O}(m/R^2)\). Differently from the case of GR (where contributions to the Casimir pressure appear only at the second order), for quadratic theories of gravity one obtains corrections already at \(\mathscr {O}(m/R)\) provided that \(\varPhi _0\ne \varPsi _0.\) This is an interesting outcome, since this contribution allows us to discriminate between GR and extended models of gravity. Furthermore, we also want to remark that the first-order correction increases the Casimir pressure, which is instead decreased by the second-order one.

### 3.3 Experimental constraints

The final step of our analysis consists in constraining quadratic theories of gravity by means of the present experimental sensitivity.

## 4 Application to several quadratic theories of gravity

Let us now apply the above formalism of the Casimir effect to several quadratic theories of gravity. We consider both local and non-local models and compute the corresponding metric potentials and the linearized Ricci scalar. Their knowledge is indispensable to study the dynamics of a scalar field between two plates in a slightly curved background, and thus to constrain the parameters of new physics appearing in such extended theories. The Ricci scalar does not appear in the formula for the pressure (see Eq. (32)), but we compute it for the sake of completeness.

Note that the second term in the l.h.s. of Eq. (36) is of order \(\mathscr {O}\left( m/R^2\right) \) for every analyzed theory, whereas the first one goes like \(\mathscr {O}\left( m/R\right) \). Since we assume *R* to be large, when \(\varPhi \ne \varPsi ,\) we can safely neglect the second-order contributions^{6} that stem from \(\varPsi _1\) and \(\varPhi _1.\)

We also want to stress that, in the case of Einstein’s GR, \(\varPhi =\varPsi =-Gm/r,\) and thus we recover the results for the Casimir energy and the Casimir pressure obtained in Ref. [45].

### 4.1 \(f(\mathscr {R})\)-gravity

### 4.2 Stelle’s fourth order gravity

In order to avoid tachyonic solutions, we need to require \(\beta <0\). In addition to that, the spin-2 mode is a ghost-like degree of freedom. Such an outcome is not surprising, since it is known that, for any local higher derivative theory of gravity, ghost-like degrees of freedom always appear [20].

### 4.3 Sixth order gravity

This time, tachyonic solutions are avoided for \(-3\,\alpha -\,\beta >0\), which can be satisfied by the requirement \(\alpha <0\) and \(-3\,\alpha >\,\beta \), with \(\beta >0\). The current higher derivative theory of gravity has no real ghost-modes around the Minkowski background but a pair of complex conjugate poles with equal real and imaginary parts [73, 74], and corresponds to the so called Lee-Wick gravity [75, 76]. It is worthwhile noting that in this higher derivative theory the unitarity condition is not violated, indeed the optical theorem still holds [4, 77, 78].

### 4.4 Ghost-free infinite derivative gravity

*no*extra degrees of freedom other than the massless transverse spin-2 graviton propagate around the Minkowski background.

### 4.5 Non-local gravity with non-analytic form-factors

- 1.The first model is described by the following choice of the form-factors:The two metric potentials are infrared modifications of the Newtonian one$$\begin{aligned} \mathscr {F}_1=\frac{\alpha }{\Box }\,,\,\,\,\,\mathscr {F}_2=0\,\,\,\Longrightarrow \,\,a=1\,,\,\,\,\,c=1-2\alpha \,. \end{aligned}$$(56)Such a spacetime metric has a vanishing linearized Ricci scalar,$$\begin{aligned} \begin{array}{rl} \varPhi (r)=&{}\displaystyle -\frac{Gm}{r}\left( \frac{4\alpha -1}{3\alpha -1}\right) \,,\\ \varPsi (r)=&{}\displaystyle -\frac{Gm}{r}\left( \frac{2\alpha -1}{3\alpha -1}\right) \,. \end{array} \end{aligned}$$(57)but it is not Ricci-flat, since some components of \(\mathscr {R}_{\mu \nu }\) are non-vanishing.$$\begin{aligned} \mathscr {R}=0\,, \end{aligned}$$(58)The expression for the Casimir pressure correction given by such a quadratic theory of gravity leads to the boundIn the last Equation, if we treat \(\alpha \) as a small parameter, we obtain the constraint$$\begin{aligned} \left| \frac{\alpha }{3\alpha -1}\right| \lesssim \frac{\delta \mathscr {P}}{\mathscr {P}_0}\frac{R}{6Gm}\,. \end{aligned}$$(59)Hence, differently from the other results, by performing such an expansion it is possible to have a direct access to the free parameter of the theoretical model, thus immediately obtaining a constraint on it.$$\begin{aligned} \left| \alpha \right| \lesssim \frac{\delta \mathscr {P}}{\mathscr {P}_0}\frac{R}{6Gm}\,. \end{aligned}$$(60)
- 2.The non-local form factors for the second model areIn this framework, the infrared modification is not a constant, but the metric potentials show a Yukawa-like behavior$$\begin{aligned} \mathscr {F}_1=\frac{\beta }{\Box ^2}\,,\,\,\,\,\mathscr {F}_2=0\,\,\,\Longrightarrow \,\,a=1\,,\,\,\,\,c=1-\frac{2\beta }{\Box }\,. \end{aligned}$$(61)The corresponding linearized Ricci scalar is non-vanishing and negative$$\begin{aligned} \begin{array}{rl} \varPhi (r)=&{}\displaystyle -\frac{Gm}{r}\frac{4}{3}\left( 1-\frac{1}{4}e^{-\sqrt{3\beta }r}\right) ,\\ \varPsi (r)=&{}\displaystyle -\frac{Gm}{r}\frac{2}{3}\left( 1+\frac{1}{2}e^{-\sqrt{3\beta }r}\right) . \end{array} \end{aligned}$$(62)For this second example of non-local gravity with non-analytic form-factors, we obtain the bound$$\begin{aligned} \mathscr {R}=-\frac{6\,G\,m}{r}\,e^{-\sqrt{3\beta }r}\,\beta \,. \end{aligned}$$(63)$$\begin{aligned} \left| 1-e^{-\sqrt{3\beta }R}\right| \lesssim \frac{\delta \mathscr {P}}{\mathscr {P}_0}\frac{R}{2Gm}\,. \end{aligned}$$(64)

## 5 Discussions and conclusions

In this paper, we have discussed the Casimir effect in the context of quadratic theories of gravity. In particular, we have focused our attention on the corrections to the Casimir pressure which come from the gravitational sector that makes these models different from GR. We have also described the possibility to infer several constraints on the free parameters of the above theories in a completely transparent way, by relating them to the current experimental errors. In all the analyzed contexts, we have worked in the weak field regime, in which computations related to the metric potentials and to the Casimir effect are easy to carry out. This has proved to be convenient in order to immediately separate the Casimir pressure in all of its relevant contributions, as seen in Eq. (34).

In order to enhance the attained bounds, there is the necessity either to implement the experiment of the Casimir effect near another astrophysical object (which is not the Earth) or to significantly improve the sensitivity of the experimental instruments. For what concerns the former, however, there is also the problem to adapt the solution of the Klein-Gordon equation to the nature of the spacetime that is under consideration. For instance, in the case of blackholes the ratio *R* / *Gm* is substantially lowered, but our formalism based on a linear regime is no longer valid.

As a final observation, we want to focus the attention one more time on the fact that the first-order corrections to the pressure \(\mathscr {P}\) in Eq. (34) cannot be attributable to GR, since in Einstein’s theory \(\varPhi _0=\varPsi _0\,\). Hence, any gravitational contribution to the Casimir pressure arising at this order is intimately connected to an extended theory of gravity for which \(a\ne c\) (see Eq. (12)). From an experimental point of view, such a consideration would also imply a more stringent bound on the free parameters of these models. In this perspective, future experiments of the Casimir effect in curved background might represent a powerful tool to test and constrain extended theories of gravity.

## Footnotes

- 1.
- 2.
Note that the linearized metric in Eq. (9) is expressed in isotropic coordinates, where \(dr^2+r^2d\varOmega ^2=dx^2+dy^2+dz^2.\)

- 3.
- 4.
It is worth noting that higher-order couplings in the curvature between matter and geometry can in principle be inserted in the analysis, but since we are dealing with the linearized regime only the contribution \({\mathscr {R}}\,\psi \) survives in the field equations, which means that in the action we only consider the non-minimal coupling \({\mathscr {R}}\,\psi ^2.\)

- 5.
The same computation was made in Ref. [69], but the definition of proper area was not correctly taken into account. Here, we exhibit the correct expression for the force and the pressure. Hence, Eqs. (30), (31) and (32) in Ref. [69] need to be substituted by Eqs. (32), (34) and (36) of this paper, respectively.

- 6.
These factors are also multiplied by \(L_P\), which makes our assumption even stronger, given that the proper length of the cavity is small.

## Notes

### Acknowledgements

The authors are thankful to Anupam Mazumdar. LB would like to thank Breno Giacchini and Shubham Maheshwari for discussions.

## References

- 1.C.M. Will, Living Rev. Rel.
**17**, 4 (2014)CrossRefGoogle Scholar - 2.K.S. Stelle, Phys. Rev. D
**16**, 953 (1977)ADSMathSciNetCrossRefGoogle Scholar - 3.K.S. Stelle, Gen. Rel. Grav.
**9**, 353–371 (1978)ADSCrossRefGoogle Scholar - 4.D. Anselmi, JHEP
**1706**, 086 (2017)ADSCrossRefGoogle Scholar - 5.D. Anselmi, JHEP
**1802**, 141 (2018)ADSCrossRefGoogle Scholar - 6.D. Anselmi, M. Piva, JHEP
**1805**, 027 (2018)ADSCrossRefGoogle Scholar - 7.D. Anselmi, M. Piva, JHEP
**1811**, 021 (2018)ADSCrossRefGoogle Scholar - 8.D. Anselmi. arXiv:1809.05037 [hep-th]
- 9.A.A. Starobinski, Phys. Lett. B
**91**, 99–102 (1980)ADSCrossRefGoogle Scholar - 10.A.A. Starobinski, Proceedings of the Second Seminar “Quantum Theory of Gravity”, Moscow, 13–15 October 1981, INR Press, Moscow, pp. 58–72 (1982)Google Scholar
- 11.A.A. Starobinsky, Sov. Astron. Lett.
**9**, 302 (1983)ADSGoogle Scholar - 12.S. Capozziello, G. Lambiase, M. Sakellariadou, An Stabile, Phys. Rev. D
**91**, 044012 (2015)ADSCrossRefGoogle Scholar - 13.G. Lambiase, M. Sakellariadou, A. Stabile, An Stabile, JCAP
**1507**, 003 (2015)ADSCrossRefGoogle Scholar - 14.G. Lambiase, M. Sakellariadou, A. Stabile, JCAP
**1312**, 020 (2013)ADSCrossRefGoogle Scholar - 15.N. Radicella, G. Lambiase, L. Parisi, G. Vilasi, JCAP
**1412**, 014 (2014)ADSCrossRefGoogle Scholar - 16.S. Capozziello, G. Lambiase, Int. J. Mod. Phys. D
**12**, 843–852 (2003)ADSCrossRefGoogle Scholar - 17.S. Calchi Novati, S. Capozziello, G. Lambiase, Grav. Cosmol.
**6**, 173–180 (2000)ADSGoogle Scholar - 18.S. Capozziello, G. Lambiase, H.J. Schmidt, Ann. Phys.
**9**, 39–48 (2000)MathSciNetCrossRefGoogle Scholar - 19.S. Capozziello, G. Lambiase, G. Papini, G. Scarpetta, Phys. Lett. A
**254**, 11–17 (1999)ADSCrossRefGoogle Scholar - 20.M. Asorey, J.L. Lopez, I.L. Shapiro, Int. J. Mod. Phys. A
**12**, 5711 (1997)ADSCrossRefGoogle Scholar - 21.A. Stabile, An Stabile, S. Capozziello, Phys. Rev. D
**88**(9), 124011 (2013)ADSCrossRefGoogle Scholar - 22.A. Stabile, An Stabile, Phys. Rev. D
**85**, 044014 (2012)ADSCrossRefGoogle Scholar - 23.G. Lambiase, S. Mohanty, An Stabile, Eur. Phys. J. C
**78**, 350 (2018)ADSCrossRefGoogle Scholar - 24.E.T. Tomboulis. arXiv:hep-th/9702146
- 25.T. Biswas, A. Mazumdar, W. Siegel, JCAP
**0603**, 009 (2006)ADSCrossRefGoogle Scholar - 26.L. Modesto, Phys. Rev. D
**86**, 044005 (2012)ADSCrossRefGoogle Scholar - 27.T. Biswas, E. Gerwick, T. Koivisto, A. Mazumdar, Phys. Rev. Lett.
**108**, 031101 (2012)ADSCrossRefGoogle Scholar - 28.T. Biswas, A. Conroy, A.S. Koshelev, A. Mazumdar, Class. Quant. Grav.
**31**, 015022 (2014). (Erratum: Class. Quant. Grav.**31**, 159501 (2014))ADSCrossRefGoogle Scholar - 29.T. Biswas, A.S. Koshelev, A. Mazumdar, Fundam. Theor. Phys.
**183**, 97 (2016)CrossRefGoogle Scholar - 30.T. Biswas, A.S. Koshelev, A. Mazumdar, Phys. Rev. D
**95**, 043533 (2017)ADSMathSciNetCrossRefGoogle Scholar - 31.A.S. Koshelev, A. Mazumdar, Phys. Rev. D
**96**(8), 084069 (2017)ADSMathSciNetCrossRefGoogle Scholar - 32.H. Casimir, Proc. K. Ned. Akad. Wet.
**51**, 793 (1948)Google Scholar - 33.H. Casimir, D. Polder, Phys. Rev.
**73**, 360 (1948)ADSCrossRefGoogle Scholar - 34.K.A. Milton,
*The Casimir effect: Physical Manifestations of Zero-Point Energy*(World Scientific, River edge, 2001)zbMATHCrossRefGoogle Scholar - 35.V.V. Nesterenko, G. Lambiase, G. Scarpetta, Riv. Nuovo Cim.
**27**(6), 1–74 (2004)Google Scholar - 36.V.V. Nesterenko, G. Lambiase, G. Scarpetta, Ann. Phys.
**298**, 403 (2002)ADSCrossRefGoogle Scholar - 37.V.V. Nesterenko, G. Lambiase, G. Scarpetta, Int. J. Mod. Phys. A
**17**, 790 (2002)ADSCrossRefGoogle Scholar - 38.V.V. Nesterenko, G. Lambiase, G. Scarpetta, Phys. Rev. D
**64**, 025013 (2001)ADSCrossRefGoogle Scholar - 39.V.V. Nesterenko, G. Lambiase, G. Scarpetta, J. Math. Phys.
**42**, 1974 (2001)ADSMathSciNetCrossRefGoogle Scholar - 40.G. Lambiase, G. Scarpetta, V.V. Nesterenko, Mod. Phys. Lett. A
**16**, 1983 (2001)ADSCrossRefGoogle Scholar - 41.G. Lambiase, V.V. Nesterenko, M. Bordag, J. Math. Phys.
**40**, 6254 (1999)ADSMathSciNetCrossRefGoogle Scholar - 42.M. Bordag, U. Mohideen, V.M. Mostepanenko, Phys. Rep.
**353**, 1 (2001)ADSMathSciNetCrossRefGoogle Scholar - 43.C. Genet, A. Lambrecht, S. Reynaud, On the nature of dark energy. In: P. Brax, J. Martin, J. P. Uzan (Fronter Group) (eds.) 18th IAP Coll. on the Nature of Dark Energy: Observations and Theoretical Results in the Accelerating Universe, Paris, France, 1–5 July 2002, pp. 121–30Google Scholar
- 44.G. Bressi, G. Carugno, R. Onofrio, G. Ruoso, Phys. Rev. Lett.
**88**, 041804 (2002)ADSCrossRefGoogle Scholar - 45.F. Sorge, Class. Quant. Grav.
**22**, 5109 (2005)ADSCrossRefGoogle Scholar - 46.V.G. Bagrov, I.L. Buchbinder, S.D. Odintsov, Phys. Lett. B
**184**, 202 (1987)ADSMathSciNetCrossRefGoogle Scholar - 47.V.G. Bagrov, I.L. Buchbinder, S.D. Odintsov, Yad. Fiz.
**45**, 1192 (1987)Google Scholar - 48.I.L. Buchbinder, P.M. Lavrov, S.D. Odintsov, Nucl. Phys. B
**308**, 191 (1988)ADSCrossRefGoogle Scholar - 49.S.D. Odintsov, Sov. Phys. J.
**32**, 458 (1989)CrossRefGoogle Scholar - 50.M.R. Setare, Class. Quant. Grav.
**18**, 2097 (2001)ADSCrossRefGoogle Scholar - 51.E. Calloni, L. di Fiore, G. Esposito, L. Milano, L. Rosa, Int. J. Mod. Phys. A
**17**, 804 (2002)ADSCrossRefGoogle Scholar - 52.G. Esposito, G.M. Napolitano, L. Rosa, Phys. Rev. D
**77**, 105011 (2008)ADSMathSciNetCrossRefGoogle Scholar - 53.G. Bimonte, G. Esposito, L. Rosa, Phys. Rev. D
**78**, 024010 (2008)ADSCrossRefGoogle Scholar - 54.E. Calloni, M. De Laurentis, R. De Rosa, F. Garufi, L. Rosa, L. Di Fiore, G. Esposito, C. Rovelli, P. Ruggi, F. Tafuri, Phys. Rev. D
**90**, 022002 (2014)ADSCrossRefGoogle Scholar - 55.B. Nazari, Eur. Phys. J. C
**75**, 501 (2015)ADSCrossRefGoogle Scholar - 56.P. Bueno, P.A. Cano, V.S. Min, M.R. Visser, Phys. Rev. D
**95**, 044010 (2017)ADSMathSciNetCrossRefGoogle Scholar - 57.M.R. Tanhayi, R. Pirmoradian, R. Int, J. Theor. Phys.
**55**, 766 (2016)CrossRefGoogle Scholar - 58.S.A. Fulling, K.A. Milton, P. Parashar, A. Romeo, K.V. Shajesh, J. Wagner, Phys. Rev. D
**76**, 025004 (2007)ADSMathSciNetCrossRefGoogle Scholar - 59.K.A. Milton, P. Parashar, K.V. Shajesh, J. Wagner, J. Phys. A
**40**, 10935 (2007)ADSMathSciNetCrossRefGoogle Scholar - 60.K.V. Shajesh, K.A. Milton, P. Parashar, J.A. Wagner, J. Phys. A
**41**, 164058 (2008)ADSMathSciNetCrossRefGoogle Scholar - 61.K.A. Milton, K.V. Shajesh, S.A. Fulling, P. Parashar, Phys. Rev. D
**89**, 064027 (2014)ADSCrossRefGoogle Scholar - 62.K.A. Milton, S.A. Fulling, P. Parashar, A. Romeo, K.V. Shajesh, J.A. Wagner, J. Phys. A
**41**, 164052 (2008)ADSMathSciNetCrossRefGoogle Scholar - 63.M. Blasone, G. Lambiase, L. Petruzziello, A. Stabile, Eur. Phys. J. C
**78**(11), 976 (2018)ADSCrossRefGoogle Scholar - 64.M.B. Cruz, E.R. Bezerra de Mello, A.Y. Petrov, Phys. Rev. D
**96**(4), 045019 (2017)ADSMathSciNetCrossRefGoogle Scholar - 65.M.B. Cruz, E.R. Bezerra De Mello, A.Y. Petrov, Mod. Phys. Lett. A
**33**, 1850115 (2018)ADSCrossRefGoogle Scholar - 66.M. Frank, I. Turan, Phys. Rev. D
**74**, 033016 (2006)ADSCrossRefGoogle Scholar - 67.A. Martin-Ruiz, C.A. Escobar, Phys. Rev. D
**94**, 076010 (2016)ADSCrossRefGoogle Scholar - 68.M. Blasone, G.G. Luciano, L. Petruzziello, L. Smaldone, Phys. Lett. B
**786**, 278 (2018)ADSCrossRefGoogle Scholar - 69.G. Lambiase, A. Stabile, An Stabile, Phys. Rev. D
**95**, 084019 (2017)ADSMathSciNetCrossRefGoogle Scholar - 70.N.D. Birrell, P.C.W. Davies,
*Quantum Fields in Curved Space*(Cambridge University Press, Cambridge, 1982)zbMATHCrossRefGoogle Scholar - 71.I.S. Gradshteyn, I.M. Ryzhik,
*Table of Integrals, Series and Products*(Academic, New York, 1980)zbMATHGoogle Scholar - 72.B.L. Giacchini, T. de Paula Netto. arXiv:1806.05664 [gr-qc]
- 73.A. Accioly, B.L. Giacchini, I.L. Shapiro, Phys. Rev. D
**96**(10), 104004 (2017)ADSCrossRefGoogle Scholar - 74.B.L. Giacchini, Phys. Lett. B
**766**, 306 (2017)ADSCrossRefGoogle Scholar - 75.L. Modesto, I.L. Shapiro, Phys. Lett. B
**755**, 279 (2016)ADSMathSciNetCrossRefGoogle Scholar - 76.L. Modesto, Nucl. Phys. B
**909**, 584 (2016)ADSCrossRefGoogle Scholar - 77.D. Anselmi, M. Piva, JHEP
**1706**, 066 (2017)ADSCrossRefGoogle Scholar - 78.D. Anselmi, M. Piva, Phys. Rev. D
**96**(4), 045009 (2017)ADSMathSciNetCrossRefGoogle Scholar - 79.J. Edholm, A.S. Koshelev, A. Mazumdar, Phys. Rev. D
**94**(10), 104033 (2016)ADSMathSciNetCrossRefGoogle Scholar - 80.L. Buoninfante, Master’s Thesis (2016). arXiv:1610.08744v4 [gr-qc]
- 81.L. Buoninfante, A.S. Koshelev, G. Lambiase, A. Mazumdar, JCAP
**1809**(09), 034 (2018)ADSCrossRefGoogle Scholar - 82.L. Buoninfante, A.S. Koshelev, G. Lambiase, J. Marto, A. Mazumdar, JCAP
**1806**(06), 014 (2018)ADSCrossRefGoogle Scholar - 83.L. Buoninfante, G. Harmsen, S. Maheshwari, A. Mazumdar, Phys. Rev. D
**98**(8), 084009 (2018)ADSCrossRefGoogle Scholar - 84.L. Buoninfante, G. Lambiase, A. Mazumdar. arXiv:1805.03559 [hep-th]
- 85.L. Buoninfante, A.S. Cornell, G. Harmsen, A.S. Koshelev, G. Lambiase, J. Marto, A. Mazumdar, Phys. Rev. D
**98**(8), 084041 (2018)ADSCrossRefGoogle Scholar - 86.A. Mazumdar, G. Stettinger. arXiv:1811.00885 [hep-th]
- 87.A.O. Barvinsky, G.A. Vilkovisky, Phys. Rept.
**119**, 1–74 (1985)ADSCrossRefGoogle Scholar - 88.S. Deser, R.P. Woodard, Phys. Rev. Lett.
**99**, 111301 (2007)ADSMathSciNetCrossRefGoogle Scholar - 89.E. Belgacem, Y. Dirian, S. Foffa, M. Maggiore, JCAP
**1803**(03), 002 (2018)ADSCrossRefGoogle Scholar - 90.L. Tan, R.P. Woodard, JCAP
**1805**(05), 037 (2018)ADSCrossRefGoogle Scholar - 91.S. Deser, R.P. Woodard, JCAP
**1311**, 036 (2013)ADSCrossRefGoogle Scholar - 92.A. Conroy, T. Koivisto, A. Mazumdar, A. Teimouri, Class. Quant. Grav.
**32**(1), 015024 (2015)ADSCrossRefGoogle Scholar - 93.R.P. Woodard, Universe
**4**(8), 88 (2018)ADSCrossRefGoogle Scholar - 94.V.M. Mostepanenko, J. Phys. Conf. Ser.
**161**, 012003 (2009)CrossRefGoogle Scholar

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