Preprint numbers: TU-1092

Quantum Spacetime Instability and Breakdown of Semiclassical Gravity

Hiroki Matsui1, ∗ and Naoki Watamura2, †

1Department of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan2Department of Mathematics, Shanghai University, Shanghai, China

(Dated: February 6, 2020)

The semiclassical gravity describes gravitational back-reactions of the classical spacetime inter-acting with quantum matter fields but the quantum effects on the background is formally definedas higher derivative curvatures. These induce catastrophic instabilities and classic solutions becomeunstable under small perturbations or their evolutions. In this paper we discuss validity of thesemiclassical gravity from the perspective of the spacetime instabilities and consider cosmologicaldynamics of the Universe in this theory. We clearly show that the homogenous and isotropic flatUniverse is unstable and the solutions either grow exponentially or oscillate even in Planckian timetI = (α1GN )1/2 ≈ α110−43 sec. The subsequent curvature evolution leads to Planck-scale spacetimecurvature in a short time and causes a catastrophe of the Universe unless one takes extremely largevalues of the gravitational couplings. Furthermore, we confirm the above suggestion by comparingthe semiclassical solutions and ΛCDM with the Planck data and it is found that the semiclassicalsolutions are not consistent with the cosmological observations. Thus, the standard semiclassicalgravity using quantum energy momentum tensor 〈Tµν〉 is not appropriate to describe our Universe.

I. INTRODUCTION

There are many essential difficulties to construct aconsistent theory of quantum gravity. Almost certainly,quantum gravity where metric is also quantized togetherwith matter fields would change even fundamental con-cept of the spacetime and requires a completely differ-ent theory from classical general relativity. However,in the regime where the curvature is small, one usu-ally regard the gravity as a classic field and mattersmove on the background. Therefore, the semiclassicalapproximation is usually expected to be sufficient. Basedon this assumption, thermal Hawking radiation aroundblack holes [1] or amplification of primordial quantumfluctuations during inflation are correctly performed byquantum field theory in curved spacetime [2].

However, not only quantum fluctuations of the matterfields, but also quantum back-reaction on the spacetimemust be considered in full semiclassical approximation.The standard semiclassical gravity replaces energy mo-mentum tensor in Einstein equations by the expectationvalues of some quantum state,

Gµν + Λgµν ≡ 8πGN 〈Ψ|Tµν |Ψ〉 . (1)

The semiclassical gravity naturally includes quantum ef-fects of the matters on spacetime such as vacuum po-larization or quantum particle creation. This theory isregarded as a first-approximation to quantum gravity [3]and yields some insight into quantum nature of grav-ity. For instance, it describes evaporation of the blackholes [1, 4–7] and provides new classes of the cosmo-logical solutions [8–15]. However, the quantum energy

∗ hiroki.matsui.c6@tohoku.ac.jp† watamura@shu.edu.cn

momentum tensor has non-trivial structures which de-pend on the curvature tensor or its derivatives and in-troduce higher derivative corrections. In the semiclassi-cal gravity these higher derivative curvatures always ap-pear and are necessary to take account of the interactionof the classical gravitational field with quantum matterfields. Even in quantum gravity the higher derivativecurvatures are necessary for the renormalizability of thetheory [16]. However, these induce instabilities of classi-cal spacetime drawn from general relativity and produceunphysical massive ghosts which leads to the non-unitarygraviton S-matrix [17, 18]. Although we usually assumethat the semiclassical gravity would be applicable belowthe Planck regime, it has several undesired properties andthe validity is still unknown.

In fact, Refs [19–26] have shown that the Minkowskispacetime in the semiclassical gravity is unstable undersmall perturbations and it is not the ground state [21].The perturbations either grow exponentially or oscil-late even in the Planck time and the subsequent cur-vature evolution leads to the Planck-scale spacetime cur-vature [19, 20]. By using large N expansion of quantumgravity [21] or effective action approach [23] it was shownthat the curvature instabilities occur at the frequenciesfar below the Planck regime. The quantum de Sitterinstability for scalar fields or graviton has been also dis-cussed by Refs [27–55]. These facts strongly indicate thatthe semiclassical gravity is not a good theory of the grav-ity and the instabilities cannot be easily rescued by fullquantum theory of gravity [21, 23]. However, it has notbeen clearly shown whether the quantum instabilities areincompatible with the cosmological observations.

In this paper, we will thoroughly investigate the space-time instabilities induced by quantum back-reaction andreconsider cosmological dynamics of the Universe in thesemiclassical equations. Due to the spacetime insta-bilities, we obtain exactly nontrivial cosmological con-straints on the semiclassical gravity. We clearly show

arX

iv:1

910.

0218

6v2

[gr

-qc]

5 F

eb 2

020

2

that Minkowski spacetime or homogenous and isotropicflat spacetime are unstable and the corresponding solu-tions either grow exponentially or oscillate even in thePlanckian time tI = (α1GN )1/2 ≈ α110−43 sec where α1

is defined by the quantum energy momentum tensor orhigher derivative gravitational action. Furthermore, weconfirm the above proposition by comparing the cosmo-logical solutions of the ΛCDM and the semiclassical Ein-stein equations with the recent Planck data [56] and then,we show that the semiclassical gravity is inconsistent withthe cosmological observations unless one takes extremalvalues of the gravitational couplings. Thus, the standardsemiclassical gravity using quantum energy momentumtensor is not appropriate to describe our Universe.

The present paper is organized as follows. In Sec-tion II, we review the semiclassical gravity and intro-duce our formulation for this theory. In particular, weexplain how the higher-derivative corrections appear inthe semiclassical gravity. Furthermore, we discuss severalproblems of the semiclassical gravity such as violations ofgravitational thermodynamical laws and (averaged) nullenergy condition. In Section III, we investigate the space-time instabilities induced by quantum back-reaction.First, we consider some results of Refs [19, 24, 25] andinvestigate the instability of Minkowski spacetime undersmall perturbations. Next, we investigate quantum insta-bilities of the homogenous and isotropic FLRW Universeand obtain cosmological constraints on the semiclassicalgravity. Finally, in Section IV we discuss the validity ofthis theory and draw the conclusion of our work.

II. SEMICLASSICAL GRAVITY

In this section we review how quantum matter fieldsinteracts with the spacetime and introduce our formula-tion for the semiclassical gravity. At the quantum levelthe classical action of gravity is replaced by the effectiveaction Γeff [gµν ], that is a functional of quantum matterfields φ in the classical background metric, 1

eiΓeff [gµν ] = eiSG[gµν ]

∫Dφ eiSM[φ, gµν ], (2)

where SG [gµν ] is the gravitational action and SM [gµν ]is the classical action of matter. This procedure alsocorresponds to the large N approximation of quantumgravity [21]. When quantum corrections of graviton issmaller than the corrections of large number of matterfields, one can neglect the graviton loops. In particular,if one considers the early universe, there were certainly a

1 In theory involving gravitational fields, anomalies may existand break general covariance or local Lorentz invariance. Theseare called gravitational anomalies and generally exist when thespacetime dimension is D = 4k+ 2, k = 0, 1, 2 . . . [57]. For four-dimensional semiclassical gravity they do not appear in general.

large number of matter fields. Thus, the large N approx-imation is applicable and semiclassical gravity is valid.For the current Universe, the corresponding energy scaleis much smaller than the Planck scale and the semiclas-sical approximation should be valid.

The gravitational action is constructed by the Einstein-Hilbert term and the cosmological constant,

SEH [gµν ] ≡ − 1

16πGN

∫d4x√−g (R+ 2Λ) , (3)

and the fourth-derivative curvature terms,

SHD [gµν ] ≡∫d4x√−g(c1R

2 + c2RµνRµν

+ c3RµνκλRµνκλ + c4R

),

(4)

The fourth-derivative terms are indispensable for renor-malization to eliminate one-loop divergences in curvedspacetime. Without the higher derivative terms the semi-classical gravity becomes non-renormalizable and is notconsistent as fundamental (not effective) quantum fieldtheory. If one regard the semiclassical gravity as the fun-damental, higher derivative curvatures always exist at theclassical level. Even if these terms are not included intothe classical action, they will emerge from quantum en-ergy momentum tensors. The effective action of Eq. (2)derives the semiclassical Einstein’s equations [2],

1

8πGN

(Rµν −

1

2Rgµν + Λgµν

)+ a1H

(1)µν + a2H

(2)µν + a3Hµν = 〈Tµν〉 ,

(5)

where 〈Tµν〉 is the vacuum expectation value of the quan-tum energy momentum tensor,

〈Tµν〉 = − 2√−g

δΓeff [gµν ]

δgµν, (6)

and the geometric tensors H(1,2)µν are defined as

H(1)µν ≡

1√−g

δ

δgµν

∫d4x√−gR2

= 2∇ν∇µR− 2gµνR−1

2gµνR

2 + 2RRµν ,

H(2)µν ≡

1√−g

δ

δgµν

∫d4x√−gRµνRµν

= 2∇α∇νRαµ −Rµν −1

2gµνR

− 1

2gµνRαβR

αβ + 2RρµRρν ,

Hµν ≡1√−g

δ

δgµν

∫d4x√−gRµνκλRµνκλ

= −H(1)µν + 4H(2)

µν .

(7)

For the flat FLRW universe, the geometrical tensors H(1)µν

3

and H(2)µν has a relation H

(1)µν = 3H

(2)µν , and we get,

a1H(1)µν + a2H

(2)µν + a3Hµν

=

(a1 +

1

3a2 +

1

3a3

)H(1)µν = α1H

(1)µν ,

(8)

where we note α1 = a1 + 13a2 + 1

3a3. Furthermore, thequantum energy momentum tensor 〈Tµν〉 introduce moreadditional geometric tensors (for the detailed discussionsee Ref [2]). For instance, the renormalized vacuumenergy momentum tensor for a massless conformally-coupled scalar field is given by the conformal anomaly,

〈Tµν〉conformal =1

2880π2

(−1

6H(1)µν +H(3)

µν

)(9)

where

H(3)µν ≡

1

12R2gµν −RρσRρµσν

= RρµRρν −2

3RRµν −

1

2RρσR

ρσgµν +1

4R2gµν ,

The semiclassical gravity introduces more additional ge-ometric tensor which depends on quantum states [2]. Infact, the renormalized energy momentum tensor for amassless minimally-coupled scalar field in Bunch-Daviesvacuum state is given by [58],

〈Tµν〉ren =

(− 1

6H(1)µν +H

(3)µν

)2880π2

−H

(1)µν log

(Rµ2

)1152π2

+

(−32∇ν∇µR+ 56Rgµν − 8RRµν + 11R2gµν

)13824π2

.

(10)

which have higher derivative corrections.In the next subsection we review the renormalization

of the quantum energy momentum tensor and see howthe higher derivative curvatures appear in semiclassicalgravity.

A. Quantum Back-reaction

First, we consider adiabatic (WKB) approximation forthe conformally massless fields and derive the renormal-ized quantum energy momentum tensors in this method.It is found that the derivation of adiabatic (WKB) ap-proximation reduces the ambiguity of the UV divergencesin renormalization and it is more significant than anyother regularization in curved spacetime.

In this paper, we consider a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime,

ds2 = dt2 − a2 (t) δijdxidxj , (11)

where a (t) is the scale factor and t is the cosmic time.We introduce conformal time η defined by dη = dt/a.

Let us consider the matter action for the conformallycoupled scalar field φ with mass m,

SM =

∫d4x√−g(−1

2gµν∂µφ∂νφ−

1

2

(m2 +

R

6

)φ2

),

(12)

which leads to the Klein-Gordon equation given as

φ−(m2 +

R

6

)φ = 0. (13)

The operator φ (η, x) can be decomposed as

φ (η, x) =

∫d3k

(2π)3/2

(akeik·xϕk (η)

a (η)+ a†k

e−ik·xϕ∗k (η)

a (η)

),

(14)

where ak, a†k are the annihilation and creation operatorsrespectively. In curved spacetime quantum states are de-termined by the choice of the mode functions. The modefunction ϕk (η) should satisfy the Wronskian condition,

ϕ′∗k (η)ϕk (η)− ϕ′k (η)ϕ∗k (η) = i, (15)

which ensures the canonical commutation relations. Weadopt adiabatic (WKB) approximation to the mode func-tion φ(η, x) which is written by [59]:

ϕk (η) =1

a (η)√

2Wk (η)

(αk · e−i

∫Wk(η) dη

+ βk · ei∫Wk(η) dη

),

(16)

where the background changes slowly and must satisfythe adiabatic (WKB) conditions (ω2

k > 0 and∣∣ω′k/ω2

k

∣∣1 where ω2

k (η) = k2 +a2 (η)m2). The coefficients αk andβk satisfy the following conditions,

|αk|2 − |βk|2 = 1. (17)

The adiabatic function Wk (η) is given by

Wk (η) ' ωk −m2C

8ω3k

(D′ +D2

)+

5m4C2D2

32ω5

+m2C

32ω5k

(D′′′ + 4D′D + 3D′

2+ 6D′D2 +D4

)− m4C2

128ω7k

(28D′′D + 19D′

2+ 122D′

2+ 47D4

)+

221m6C3

256ω9k

(D′D2 +D4

)− 1105m8C4D4

2048ω11k

+ · · · ,

(18)

where C(η) = a2(η) and D = C ′/C. The mode functionϕk (η) with αk = 1 and βk = 0 is a reasonable choice fora sufficiently slow and smooth background [59],

ϕk (η) =1√

2Wk (η)C (η)· e−i

∫Wk(η) dη, (19)

4

which defines the adiabatic vacuum state |ΨA〉 which isannihilated by all the operators ak. Here we assume thatthe adiabatic vacuum state |ΨA〉 is adequate initial vac-uum in flat FLRW spacetime.

Let us consider the quantum energy momentum tensor.The classical energy momentum tensor is given by [60]

Tµν =−2√−g

δS

δgµν=

2

3∂µφ∂νφ−

1

6gµνg

ρσ∂ρφ∂σφ (20)

− 1

3φ∇µ∇νφ+

1

3gµνφφ−

1

6Gµνφ

2 +1

2m2gµνφ

2,

and the corresponding trace

Tµµ = m2φ2, (21)

which is exactly zero when m → 0. Thus, the confor-mally massless scalar field has the vanishing trace of thestress tensor classically. It is found that the conformal

invariance is broken in quantum field theory. The vac-uum expectation values of the energy momentum tensor〈Tµν〉 for ϕk (η) are given by

〈T00〉 =1

4π2C (η)

∫dkk2

[|ϕ′k (η)|2 + ω2

k|ϕk (η)|2],

⟨Tµµ

⟩=

1

2π2C2 (η)

∫dkk2

[Cm2|ϕk (η)|2

],

(22)

where 〈T00〉 is the time component of the energy mo-mentum tensor and

⟨Tµµ

⟩is the trace. These quantum

energy momentum tensor has UV divergences and onemust proceed the renormalization.

Next, we rewrite the vacuum expectation values 〈Tµν〉of the energy momentum tensor by the adiabatic approxi-mation [60] and renormalize the quantum energy momen-tum tensors as follows,

〈T00〉 =1

8π2C (η)

∫dkk2

[2ωk +

C2m4D2

16ω5k

− C2m4

64ω7k

(2D′′D −D′2 + 4D′D2 +D4

)+

7C3m6

64ω9k

(D′D2 +D4

)− 105C4m8D4

1024ω11k

],

⟨Tµµ

⟩=

1

4π2C2 (η)

∫dkk2

[Cm2

ωk+C2m4

8ω5k

(D′ +D2

)− 5C3m6D2

32ω7k

− C2m4

32ω7k

(D′′′ + 4D′′D + 3D′2 + 6D′D2 +D4

)+C3m6

128ω9k

(28D′′D + 21D′2 + 126D′D2 + 49D4

)− 231C4m8

256ω11k

(D′D2 +D4

)+

1155C5m10D4

2048ω13k

],

(23)

where the lowest-order term of the quantum energy mo-mentum tensor 〈Tµν〉 actually diverges,

〈T00〉diverge =1

4π2C (η)

∫dkk2ωk →∞. (24)

By adopting the dimensional regularization, the quantumenergy momentum tensor 〈Tµν〉 can be regularized,

〈T00〉reg = −m4C

64π2

[1

ε+

3

2− γ + ln 4π + ln

µ2

m2

]+m2D2

384π2− 1

2880π2C

(3

2D′′D − 3

4D′

2 − 3

8D4

),

⟨Tµµ

⟩reg

= − m4

32π2C

[1

ε+ 1− γ + ln 4π + ln

µ2

m2

]+

m2D2

192π2C

(2D′ +D2

)− 1

960π2C2

(D′′′ −D′D2

).

(25)

where µ is the renormalization parameter and γ is theEuler-Mascheroni constant. The 1/ε terms representthe UV divergences and they must be absorbed by thecounter-terms of the gravitational action.

Hence, the renormalized energy momentum tensor forflat spacetime is

〈T00〉ren =m4C

64π2

(lnm2

µ2− 3

2

)+m2D2

384π2

− 1

2880π2C

(3

2D′′D − 3

4D′

2 − 3

8D4

),

⟨Tµµ

⟩ren

=m4

32π2C

(lnm2

µ2− 1

)+

m2D2

192π2C

(2D′ +D2

)− 1

960π2C2

(D′′′ −D′D2

).

(26)

where the first terms are the running cosmological con-stant corrections which originates from the lowest adi-abatic term. On the other hand, the latter parts ex-press vacuum polarization or quantum particle creationin curved spacetime. The anomaly term of Eq. (26) isconsistent with using dimensional regularization [61, 62]and it is equal to a2(x)/16π2 [3] where a2(x) is a coeffi-cient of the DeWitt-Schwinger formalism. The conformal

5

anomaly is given by the massless limit of Eq. (26),⟨Tµµ

⟩anomaly

= limm→0

⟨Tµµ

⟩ren

= − 1

960π2C2

(D′′′ −D′D2

)= − 1

2880π2

[(RµνR

µν − 1

3R2

)+R

]=

1

360(4π)2E − 1

180(4π)2R,

(27)

By using the adiabatic approximation we obtain the fol-lowing expression for a massless fermion [63, 64],⟨Tµµ

⟩fermion

anomaly= − 1

2880π2

[11

(RµνR

µν − 1

3R2

)+ 6R

]=

11

360(4π)2E − 6

180(4π)2R.

(28)

The conformal anomaly for the gauge field in adiabaticexpansion is given by [65]:⟨Tµµ

⟩gauge boson

anomaly= − 1

2880π2

[62

(RµνR

µν − 1

3R2

)− (18 + 15 log ξ)R

]=

62

360(4π)2E +

(18 + 15 log ξ)

180(4π)2R.

(29)

where ξ is a gauge fixing parameter defined by the co-variant gauge fixing term [65]:

Lgf = −√−g2ξ

(∇µAµ)2. (30)

The gauge dependence of Eq. (29) also exists in theDeWitt-Schwinger expansion formalism [66–68]. Theadiabatic approximation reproduces the gauge depen-dence of the R term which has also the regularization-scheme dependence. However, the gauge fixing param-eter can be removed by the gravitational coupling con-stants in the Einstein equation and we can drop the gaugefixing parameter ξ. It is found out that the adiabatic ex-pressions for the conformal anomaly precisely matchesthe expression derived by effective action using the di-mensional regularization [3].

The renormalized energy momentum tensor 〈Tµν〉renare generally given as follows,

〈Tµν〉ren = α1H(1)µν + α3H

(3)µν + α4H

(4)µν (31)

where the geometric tensor H(4)µν depends on quantum

states [2] and the above equations are defined as fourth-order derivative equations. The dimensionless parame-ters α1,3 for massless fields are given by [2],

α1 =−1

2880π2

(NS6

+NF − 3NG

), (32a)

α3 =1

2880π2

(NS +

11

2NF + 62NG

), (32b)

where we considerNS scalars (spin-0), NF Dirac fermions(spin-1/2) and NG abelian gauge fields (spin-1). Forinstance, the Minimal Supersymmetric Standard Model(MSSM), takes the following values: NS = 104, NF = 32and NG = 12. On the other hand, the Standard Model(SM) takes the following values: NS = 4, NF = 24 andNG = 12 where the right-handed neutrinos are assumed.Finally, the current universe has only photon, NS = 0,NF = 0 and NG = 1. It is found that the large numberof the scalar fields or fermions lead to the negative α1

and induce the runaway solutions.

B. Covariant Conservation Laws

Briefly, we comment covariant conservation laws forthe quantum energy momentum tensor [8]. The energymomentum tensor in both classical and quantum casesmust satisfy the covariant conservation laws,

∇µ 〈Tµν〉ren = 0, (33)

which means energy and momentum conservations. Werewrite the trace of Eq. (31) for massless conformally-invariant fields in terms of the scale factor a(t),⟨Tµµ

⟩ren

= −36αa−3[a2a(4) + 3aaa(3) + aa2 − 5a2a

]+ 12βa−3aa2, (34)

where the dots and bracketed superscripts denote differ-entiation with respect to t. Using the above expression,Eq. (33) derives the renormalized vacuum energy density,

ρren = −36αa−4

[a2aa(3) + aa2a− 1

2a2a2 − 3

2a4

]+ 3βa−4a4 + Ca−4, (35)

where C is a constant from integration and the last termcorresponds to the thermal radiation ρ ∝ a−4. Hence, therenormalized expression of the energy momentum ten-sor of Eq. (31) satisfies the covariant conservation laws.From here, we drop the constant C for simplicity.

C. Gravitational Thermodynamics and AveragedNull Energy Condition

In this section let us briefly discuss several problems ofthe semiclassical gravity such as violations of the gravi-tational thermodynamical law which is characterized bythermodynamical entropy S and the averaged null energycondition. Famously, the black hole thermodynamics as-sumes that black holes have the entropy, quantified bythe area of the event horizon,

SBH =A

4GN, (36)

6

where the horizon area A is quantified by the surfacegravity κ and the mass MBH of stationary black holes,

dMBH =κdA

8πGN+ (rotation & charge terms). (37)

Classically, the black holes acquire mass from other mas-sive objects and SBH always increases. This fact matchesthe thermodynamical interpretation of the entropy. How-ever, the black holes may lose its mass due to the Hawk-ing radiation with the temperature,

TH =κ

2π, (38)

and thus SBH decreases. On the other hand, thermalcharacter of the event horizon in de Sitter space formallydefines de Sitter entropy,

SdS =πH−2

GN, (39)

where the horizon area is given by A = 4πH−2 and thetime-evolution is written as follows,

dSdS

dt= −2πH−3H

GN. (40)

By using the de Sitter entropy SdS one can get inter-esting consequences such as a no-go theorem for slow-rolleternal inflation. Let us consider slow-roll inflation drivenby a inflaton field φ. For the slow-roll inflation, we have

H = −4πGN φ2 . (41)

Hence, the de Sitter entropy SdS is rewritten as,

dSdS

dN= − 2πH

GNH4=

8π2φ2

H4∼(δρ

ρ

)−2

& 1, (42)

where N is the number of e-foldings defined by dN =Hdt, ρ is the energy density and δρ is the energy densityperturbation satisfying |δρ/ρ| . 1. The total number ofe-folding Ntot is bounded as follows [69],

Ntot . ∆S = Send − Sini, (43)

where Send and Sini are the de Sitter entropy at the endand the beginning of the inflation, respectively. For thelarge field inflation, the entropy at the beginning is muchsmaller than that at the end, and then one get ∆S ∼ Send

but ∆S Send for the small field inflation. In any casethe total e-folding number Ntot is strictly restricted.

Let’s discuss a more general case. For flat FLRW uni-verse, the Friedmann equations yield a simple equation,

H = −4πGN (ρ+ P ) . (44)

Hence, the de Sitter entropy can be written as follows,

dSdS

dt= 8π2H−3 (ρ+ P ) ≥ 0, (45)

which always increases and matches gravitational ther-modynamical laws when the null energy condition (NEC)is satisfied,

Tµνkµkν ≥ 0 =⇒ ρ+ P ≥ 0, (46)

where kµ is the null (light-like) vector. It is known thatthe NEC and the gravitational thermodynamical lawsare closely related with each other [69]. In general rel-ativity, the NEC is a necessary condition to eliminateany pathological spacetime or unphysical consequencessuch as wormhole, geometric instability and superlumi-nal propagation. It is well known that the classical mat-ters always satisfy the NEC and in this sense the classicalgeneral relativity does not violate any gravitational prin-ciples. However, these classical conditions can be easilyviolated in quantum field theory (QFT) and the NECis broken even for quantum fields in Minkowski space-time [70] (e.g. squeezed vacuum states [71]). More gen-erally, the averaged null energy condition (ANEC) [72],which is satisfied in Minkowski spacetime [73] and pro-hibits a traversable wormhole [74] have been proposed,∫

γ

Tµνkµkνdl ≥ 0 =⇒

∫ ∞−∞

1

a(ρ+ P ) dt ≥ 0, (47)

where the integral is taken over a null geodesic γ, kµ isthe parameterized tangent vector to the geodesic and lis the affine parameter [75]. However, it has been knownthat curved spacetime, i.e., semiclassical gravity violatesthe NEC or ANEC [76–79] and the de Sitter entropy de-creases [80]. This seems very plausible because the energydensity undergoes quantum vacuum fluctuations and thevariances of the energy density would be allowed to beboth negative and positive in QFT. Although so-called“quantum inequalities (QIs) [81] has been proposed, inprinciple, there is no lower limit for the negative vacuumenergy and one can take any non-physical spacetime. Ina nutshell, such quantum effects on gravity require care-ful discussion and theory should not break these basicgravitational principles.

Finally, let us briefly see the violations of the NEC,ANEC and gravitational thermodynamical laws in semi-classical gravity and simply consider conformal anomaly⟨Tµµ

⟩anomaly

in Eq. (67). For the semiclassical gravity,

the NEC, ANEC and de Sitter entropy can be violatedwith various conditions as follows;

ρ+ P = 12α1

(6H2 + 3HH +

...H)− 4α3H

2H 0,

(48)∫ ∞−∞

1

a(ρ+ P ) dt =

∫ ∞−∞

12α1

a

(6H2 + 3HH +

...H)

− 4α3

aH2H

dt 0, (49)

dSdS

dt= 96π2α1

(6H−3H2 + 3H−2H +H−3 ...

H)

− 32π2α3H−1H 0, (50)

7

which suggests that semiclassical gravity is incompatiblewith basic gravitational principles [69].

III. QUANTUM SPACETIME INSTABILITY

We now turn to a more quantitative discussion of thevalidity of the semiclassical gravity. The quantum energymomentum tensor holding higher derivative terms modi-fies the Einstein’s equations and destabilizes the classicalsolutions. In this section, we investigate the quantumspacetime instabilities in semiclassical gravity and con-sider the cosmological dynamics of the Universe.

First, we will revisit some results of Refs [19, 24, 25]and discuss the instability of the Minkowski spacetimeunder small perturbations. Next, we discuss the instabil-ities of the homogenous and isotropic FLRW Universe,and consider the cosmological dynamics. Our results sug-gest that the corresponding solutions of the semiclassicalgravity can not be incompatible with the cosmologicalobservations.

For simplicity, we consider the semiclassical Einstein’sequations for the massless conformally-invariant fieldsand the classical radiation or non-relativistic matters,

1

8πGN

(Rµν −

1

2Rgµν + Λgµν

)= 〈Tµν〉ren + T c

µν

= α1H(1)µν + α3H

(3)µν + T c

µν , (51)

where T cµν is the energy momentum tensor for ordinary

radiation or non-relativistic matters.

A. Instability of Minkowski spacetime underconformally flat perturbations

Let us consider the Minkowski spacetime with no mat-ter fields and investigate whether the spacetime is stableunder the perturbations. Although in classical generalrelativity, the Minkowski spacetime should be completelystable, semiclassical gravity does not ensure this impor-tant fact. Refs [19–26] have shown that the Minkowskispacetime is unstable under small perturbations and theperturbations either grow exponentially or oscillate evenin Planck time. This leads to disaster. Before studyingthe FRLW instabilities and considering the influence onthe current Universe, let us reconsider some results of theMinkowski instability of Ref [19, 24, 25].

In the Minkowski spacetime, the geometric tensors insemiclassical Einstein’s equations satisfy,

Gµν = H(1)µν = H(3)

µν = 0. (52)

Now, we consider only conformally flat perturbations andwrite the metric as follows,

gµν = Ω2ηµν , (53)

where ηµν is the Minkowski metric (+−−− convention)and Ω is the conformal parameter.

The corresponding Ricci tensor Rµν is given by

Rµν = 4Ω−2 (∂µΩ) (∂νΩ)− 2Ω−1∂µ∂νΩ

− Ω−1ηµν (∂α∂αΩ)− Ω−2ηµν (∂αΩ∂αΩ) ,(54)

where Ω = 1 reproduces the Minkowski spacetime solu-tion. Now, we rewrite Eq. (51) using this expression withthe conformally flat perturbations Ω = 1+γ and one canobtain [19],

−∂µ∂νγ + (γ) ηµν + 48πα1GN

× [−∂µ∂ν (γ) + (γ) ηµν ] = 0,(55)

where ≡ ηαβ∂α∂β = ∂α∂α. This perturbation equa-tion for γ can be rewritten by a simple equation,

(∂µ∂ν − ηµν) f = 0, (56)

where we define f ≡ (γ + 48πα1GNγ) and the generalsolution is clearly,

f = kαxα + const., (57)

where kα expresses a constant vector field and xα is aposition vector field in the Minkowski spacetime. Hence,we can get the following equation,

(1 + 48πα1GN) γ = kαxα + const. (58)

The most general solution to the above equation is

γ = kαxα + χ+ const., (59)

where inhomogeneous solution γ = kαxα + const. is puregauge in the Minkowski spacetime and the metic pertur-bation χ satisfies the Klein-Gordon equation,

χ+1

48πα1GNχ = 0, (60)

which admits the spatially homogeneous solutions,

χ = C1 sin (ωt) + C2 cos (ωt) , C3et/τ , (61)

where ω = (48πα1GN )1/2 and C1,2,3 are constants. Thefirst solution is for α1 > 0 and consistent with the or-dinary Klein-Gordon equation. However, if one takes48πα1 ∼ O(1), the perturbations oscillate in the Plancktime tP = (GN )1/2 = 10−43 sec and they emit the Planckenergy photons, E ∼ 1019 GeV [19] which is unreason-able for the observed Universe. The last possible solutionis given for α1 < 0. We denote τ = (48πα1GN )1/2. Thiscorrespond to the Klein-Gordon equation with a negativemass and suggests that the perturbations exponentiallygrow even in the Planck time.

Once the Minkowski spacetime is perturbed, the per-turbations lead to a catastrophe. For α1 < 0 the insta-bility time-scale can be summarized as,

tI = (48πα1GN )1/2 = (48πα1)1/2 · 10−43 sec.

= (48πα1/10118)1/2 Gyr.(62)

8

The instability time tI must be as large as the age ofthe observed Universe, tAge = 13.787 ± 0.020 Gyr. [56](Planck 2018, TT,TE,EE+lowE+lensing+BAO 68% lim-its). Otherwise, the perturbations or scale factor expo-nentially grow and our Universe is seriously destabilized.Hence, we obtain the stable condition against quantumback-reaction,

α1 & 10118, (63)

which requires a large value of the gravitational curvaturecoupling or a large number of the particles species N ∼10118 for the high energy theory. Hence, the α1 < 0 caseis trouble for the homogenous and isotropic flat Universe.We will confirm these results in different methods in thenext subsection 2.

B. Numerical analysis for Minkowski spacetimeinstability under perturbations

Let us consider the FLRW spacetime with matter fieldsand study the spacetime insatiabilities from the quantumback-reaction 3. For the Minkowski spacetime we haveno classical matter and no cosmological constant.

By using Eq. (51) we obtain the following semiclassicalequation [82],

dR

dt=

1

12HR2 −HR− H

16πGNα1+α3

α1H3, (64)

If we assume that the third term of Eq. (64) can dominatedue to the smallness of 16πGNα1, we can approximatelyrewrite,

d2H

dt2≈ − 1

16πGNα1H, (65)

where we consider H(0) ≈ 0, R(0) ≈ 0. This admitsthe exponential or oscillating solutions for the Plancktime tP for 16πα1 ∼ O(1). Thus, these are consistentwith the previous discussion. Next, we numerically con-firm the above analytical estimations. Let us assumethe Minkowski spacetime perturbed by the small Hubblevariations. Using Eq. (64) and R = 6(H + 2H2), we ob-tain the differential equation. We investigate the systemof equations starting at t = 0 with various conditions andperturbations. It is found that the numerical solutionsof the system show that the Minkowski spacetime is un-stable and they are consistent with the above analyticalestimation.

2 Taking the following conformal parameter,

gµν = Ω2ηµν = a2(η)ηµν ,

one can get a similar consequence for the scale factor a(η).3 The similar analysis was given recently by one of the au-

thors [80].

1.× 10-44 5.× 10-441.× 10-43 5.× 10-431.× 10-42 5.× 10-421.× 10-41

10-54

10-52

10-50

10-48

10-46

10-44

t [sec]

H

1.× 10-44 5.× 10-441.× 10-43 5.× 10-431.× 10-42 5.× 10-421.× 10-4110-49

10-39

10-29

10-19

10-9

10

t [sec]

H

FIG. 1. We consider the instability of the Minkowski space-time under perturbations and show that the dynamics ofthe Hubble perturbation H(t) around the Planck time tP =10−43 sec. We assume the initial conditions and the couplingsof Eq. (66). The top figure assumes α1 = 10−86 whereas thebottom figure takes α1 = −10−86. The top-to-bottom linescorresponds to H0 = 10−43,−45,−48, respectively.

In Fig.1 we demonstrate numerical results for the Hub-ble perturbation H(t) determined by Eq. (72) with thefollowing conditions and couplings,

Fig.1: H0 = 10−43,−45,−48, H0 = 0,

48πα1GN = ±10−86, 8πGNα3 = 0, 10−86,(66)

where we set these parameters to respect the Planck timetP = 10−43 and it is found that magnitudes of the per-turbation Υ(t) and values of α3 are irreverent for the dy-namics. We found that the Hubble oscillations with thePlanck frequency occur for α1 > 0 whereas for α1 < 0,the Hubble perturbations exponentially grow even in thePlanck time tP. The small values of α1 leads to the fasterdestabilization and the stability of the Minkowski space-time requires a large value of |α1|.

9

1.00000 1.00001 1.00002 1.00003 1.000040.90

0.95

1.00

1.05

1.10

τ

h

1.00000 1.00001 1.00002 1.00003 1.000040.90

0.95

1.00

1.05

1.10

τ

h

x = -10-10.2

x = -10-10

x = -10-9.8

1.0000 1.0001 1.0002 1.0003 1.0004

1

10

100

1000

104

τ

h

1.0000 1.0001 1.0002 1.0003 1.0004

1

10

100

1000

104

τ

h

FIG. 2. We show the numerical solution of Eq. (72) with the initial conditions and the higher-derivative couplings of Eq. (74).These figures show that the dynamics of the dimensionless Hubble parameter h(τ) in a few normalization time τ . The dashedline shows the de Sitter solution h(τ) = 1 from the general relativity. The right panel is y = 0 whereas the tight panel is y = 0.The top panels shows that the solutions oscillate where where the top-to-bottom lines corresponds x = 10−10.0,−12.0,−14.0.The bottom panels on the other hand shows the solutions exponentially grow where the top-to-bottom lines corresponds

x = −10−9.8,−10.0,−10.2. We found out that the time-scale τI is τI ≈ |x|1/2 ≈ 10−5.

C. Numerical analysis for FLRW spacetimeinstability

Let us consider the de Sitter spacetime under the Hub-ble perturbation and discuss the cosmological evolution.Now using Eq. (51) for the FLRW metric, we obtain thesemiclassical Friedmann equations,

H2 =Λ

3− 48πGNα1

(6H2H + 2HH − H2

)+ 8πGNα3H

4 +8πGN

3ρm,

(67)

where ρm is the energy density of the classical matter andsatisfies the covariant conservation law,

ρm = −3H (ρm + Pm) = −3H (1 + ω) ρm, (68)

where w = P/ρ is an equation-of-state parameter. Forthe non-relativistic matter, radiation and cosmologicalconstant one takes w = 0, 1/3,−1, respectively.

We rewrite the above semiclassical equations in terms

of the dimensionless parameters,

h2 = −x(6h2h′ + 2hh′′ − h′2

)+ yh4 + z,

z′ = −3h (1 + ω) z ,(69)

where these parameters are given by

τ = H0t, h = H/H0,

x = 48πGNα1H20 , y = 8πGNα3H

20 ,

z = Λ/3H20 + 8πGNρm/3H

20 ,

(70)

where H0 is the initial Hubble parameter. We note thefollowing relations of these parameters,

H0 ∼ 1014 GeV, MP ∼ 1018 GeV, α1,3 ∼ 10−2

=⇒ x, y ∼ 10−10,

H0 ∼ 10−42 GeV, MP ∼ 1018 GeV, α1,3 ∼ 10−2

=⇒ x, y ∼ 10−122,

(71)

where the formar and latter Hubble parameter corre-sponds to an expample consistent with typical inflationand current universe, respectively [56]. The dynamics of

10

1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

τ

h

1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

τ

h

x = -10-1.0

x = -10-1.5x = -10-2.0

1.0 1.5 2.0 2.5 3.0-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

τ

h

x = -10-1.5

x = -10-2.0x = -10-1.0

1.0 1.5 2.0 2.5 3.0

0

2

4

6

8

τ

h

FIG. 3. We compare the numerical solution of Eq. (72) with the conditions of Eq. (76) and the standard solution h(τ) = 1/2τfrom the general relativity which is presented by the dashed line. The right panel is y = 0 whereas the tight panel is y = 0.The top panels shows that the solutions oscillate for x = 10−1.0,−3.0,−5.0. The bottom panels where the top-to-bottom linescorresponds x = −10−1.0,−1.5,−2,0 shows the instabilities for the matter-dominated Universe and the solutions are inconsistentwith the standard general relativity.

the dimensionless Hubble parameter h with w = −1 forthe spacetime is determined by,

h2 = −x(6h2h′ + 2hh′′ − h′2

)+ yh4 + z , (72)

where z is constant and prime express the derivative withrespect to the dimensionless time τ . The suitable deSitter initial conditions for Eq. (72) are

τ0 = 1, h0 = 1, h′0 = 0, z0 = 1. (73)

In order to investigate the de Sitter spacetime instabil-ities, we consider the numerical solutions of the de Sittersystem starting at τ0 = 1 with various initial conditionsand perturbations. In Fig.2, we present the numerical re-sults for the dimensionless parameter h(τ) from Eq. (72)with the following conditions,

Fig.2 : h0 = 1 + 10−1.0, h′0 = 0,

x = 10−10.0,−12.0,−14.0, y = 0, 10−10.0,

h0 = 1, h′0 = 0,

x = −10−9.8,−10.0,−10.2, y = 0, 10−10.0,

(74)

where we compare these results with the de Sitter solu-tion h(τ) = 1 from the general relativity. The top pan-els show that the Hubble perturbations in the de Sitterspacetime oscillate and the oscillation time-scale becomesshorter for the small values of |x|. On the other hand, thebottom panels show that the de Sitter spacetime is desta-

bilized in τ ≈ |x|1/2 and the smallness of |x| amplifies theinstabilities. The dynamics for y > 0 and y < 0 showsthe similar results and it is found that y is irreverent forthe dynamics.

Next, we consider matter-dominated stage of the Uni-verse with w = 0. The natural initial conditions for thesystem are given by,

τ0 = 2/3, h0 = 1, h′0 = −3/2, z0 = 1, (75)

where we take z = 8πGNρm/3H20 and Λ = 0. In Fig.3 we

investigate the system of equations starting at τ0 = 2/3

11

x = -10-1.5

x = -10-1

x = -10-2

x = -10-2.5

x = -10-3x = -10-3.5

x = -10-4

0 2 4 6 8 10

58

60

62

64

66

68

t [Gyr]

H[kms-1MPc-1]

FIG. 4. In this figure we compare standard result of theΛCDM and the numerical solutions of Eq. (81) with the con-ditions of Eq. (80) and Eq. (83). The dashed line correspondsto the central value of the Planck data [56] and the green lineexpresses the allowed region of the Planck data (Planck 2018,TT,TE,EE+lowE+lensing+BAO 68% limits). The semiclas-sical gravity or conformal anomaly expect |x| ≈ 10−122 andthey are not consistent with the observations.

with the following conditions,

Fig.3: h0 = 1 + 0.3, h′0 = −3/2,

x = 10−1.0,−3.0,−5.0 y = 10−1.0,

h0 = 1, h′0 = −3/2,

x = −10−1.0,−1.5,−2,0, y = 10−1.0,

(76)

where we compare them the standard cosmic solutionh(τ) = 2/3τ . The top panels show that the matter-dominated Universe is unstable for the small perturba-tions but the solutions converge the general relativity.The bottom panels show that the matter-dominated Uni-

verse is unstable for τ ≈ |x|1/2 and inconsistent with theusual general relativity. In this case, the instability time-scale τI should be larger than of order unity,

τI ≈ |x|1/2 & O(1), (77)

and thus using the current value of the Hubble parameterH0 ∼ 10−42 GeV, we obtain the following constraint,

|α1| & 10118, (78)

which is consistent with Eq. (63). Now, we confirmed theresults of the previous subsection and found out that theFLRW spacetime is unstable under the perturbations forα1 > 0 or the evolution for α1 < 0. Since the theoreti-cally expected value of α1 is given by Eq. (32), the aboveconstraint is unacceptably large.

D. Instability of the current Universe

Finally, we consider cosmological evolution of the Uni-verse with the quantum back-reaction and derive a strict

cosmological constraint for the semiclassical gravity. Be-fore considering the detail, let us review the standard cos-mological history of the Universe based on ΛCDM withinflation. First, the Universe proceeds inflation [10, 83–86] (around 10−35 to 10−32 sec). Second, the inflationends and thermal radiation dominates up to the recom-bination with the cosmic microwave background (CMB)radiation (around 379,000 years). After the radiation-dominated stage, the Universe is dominated by non-relativistic matters, and then galaxies and clusters aregradually formed. From about 9.8 Gyrs, the expansionof the universe begins to accelerate via unknown darkenergy. The age of the current Universe is about 13.8Gyrs and the present value of the Hubble parameter isH0 ≈ 67.7 km/s.MPc.

We consider the semiclassical Friedmann equations ofEq. (67) for the conditions of ΛCDM,

H2(t0) = H20 [ΩΛ,0 + Ωm,0 + Ωr,0] ,

ΩΛ,0 + Ωm,0 + Ωr,0 ≈ 1,(79)

where ΩΛ,0 is the density parameter for the cosmologicalconstant and Ωm,0, Ωr,0 are the current density values ofnoh-reativistc matters including dark matter, and radi-ation. From the recent Planck data [56] (Planck 2018,TT,TE,EE+lowE+lensing+BAO 68% limits) we have,

H0 ≈ 67.66± 0.42 [km s−1 MPc−1]

≈ 7.25× 10−2 [Gyr−1],

ΩΛ,0 = 0.6889± 0.0056,

Ωm,0 = 0.3111± 0.0056,

(80)

where (GN )1/2 = 10−59 Gyr. For simplicity, we rewritethe semiclassical Friedmann equations as follows,

h2 = −x(6h2h′ + 2hh′′ − h′2

)+ yh4 + λ0 + z,

z′ = −3hz ,(81)

where the dimensionless parameters are given by

τ = H0t, h = H/H0,

x = 48πGNα1H20 , y = 8πGNα3H

20 ,

λ0 = Λ/3H20 , z = 8πGNρm/3H

20 .

(82)

In Fig.4 we assume the following initial conditions andvarious couplings based on the ΛCDM,

Fig.4: x = −10−1.0,−2.0,−3.0,−4.0, y = 0,

h0 = 1, h′0 = −3/2 Ωm,0,

λ0 = ΩΛ,0 z0 = Ωm,0.

(83)

The Fig.4 shows the spacetime instabilities for the small|x| in a short time and the corresponding spacetime solu-tions are inconsistent with the future or current evolutionof the Universe unless one takes |x|1/2 ≈ O(1). However,the semiclassical gravity expects the extremal small value

|x|1/2 ≈ 10−61 for the current Universe and that is notconsistent with the observations.

12

The spacetime instability of the semiclassical gravity iscertainly a serious problem and the solutions are not beconsistent with the cosmological observation. It has beenargued that the semiclassical solutions must be given bythe truncating perturbative expansions [87, 88] and thequantum higher derivative corrections can be regarded assmall perturbations from the classical solution. However,this procedure is ad hoc approach for the semiclassicalequations much below the Planck scale and ineffectivenear the Planck regime. That is a problem when oneconsider large N expansion where the semiclassical grav-ity could be adequate to describe Planckian phenomenadue to the suppression of the graviton loops [21]. Alsothe Euclidean formulation of quantum gravity imposesthe boundary condition and the curvature instability orrunaway solutions might be removed [89]. However, it isnot clear how to handle the quantum energy momentumtensor 〈Tµν〉 in these procedures and the problem of thequantum instability is still left open.

IV. CONCLUSION

Semiclassical gravity describe the interactions betweenclassical gravity and quantum matters, and the quantumback-reaction is formally defined as the higher-derivativecurvatures. These induce instabilities of the classic so-lutions and Refs [19–26] presented that the Minkowski

spacetime is unstable under small perturbations. Thespacetime instability was seen as a serious problem insemiclassical gravity. However, it has not been discussedwhether the semiclassical instabilities are inconsistent inour Universe.

In this paper we have shown that the homogenous andisotropic FLRW Universe interacting with quantum mat-ter fields are unstable under small perturbations or theevolutions. We have analytically and numerically demon-strated that the homogenous and isotropic cosmologicalsolutions either grow exponentially or oscillate even inthe Planckian time tI = α110−43 sec. For α1 > 0, thecurvature perturbations oscillate rapidly and would emitthe Planck energy photons [19] which is unacceptable forthe observed Universe. On the other hand, for α1 < 0,the evolution of the curvature perturbation leads to thePlanckian curvature or singularity. These instabilities in-duce a catastrophe unless one takes extremal values of thegravitational couplings or fundamental particle species|α1| & 10118. We have also confirmed these results basedon the cosmological evolution by comparing ΛCDM andthe semiclassical Einstein solutions using the Planck dataand it is found that these solutions of the semiclassicalgravity including conformal anomaly are not consistentwith cosmological observations.

Acknowledgements.— NW would like to thank ShingoKukita for the valuable discussions.

[1] S. W. Hawking, Euclidean quantum gravity, Commun.Math. Phys. 43, 199 (1975), [,167(1975)].

[2] N. D. Birrell and P. C. W. Davies, Quantum Fields inCurved Space, Cambridge Monographs on MathematicalPhysics (Cambridge Univ. Press, Cambridge, UK, 1984).

[3] I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro,Effective action in quantum gravity (1992).

[4] W. G. Unruh, Phys. Rev. D14, 870 (1976).[5] S. M. Christensen and S. A. Fulling, Phys. Rev. D15,

2088 (1977).[6] P. Candelas, Phys. Rev. D21, 2185 (1980).[7] P.-M. Ho, H. Kawai, Y. Matsuo, and Y. Yokokura, JHEP

11, 056 (2018), arXiv:1807.11352 [hep-th].[8] P. C. W. Davies, Phys. Lett. 68B, 402 (1977).[9] M. V. Fischetti, J. B. Hartle, and B. L. Hu, Phys. Rev.

D20, 1757 (1979).[10] A. A. Starobinsky, Phys. Lett. 91B, 99 (1980),

[,771(1980)].[11] P. Anderson, Phys. Rev. D28, 271 (1983).[12] P. R. Anderson, Phys. Rev. D29, 615 (1984).[13] T. Azuma and S. Wada, Prog. Theor. Phys. 75, 845

(1986).[14] S. Nojiri and S. D. Odintsov, Phys. Lett. B562, 147

(2003), arXiv:hep-th/0303117 [hep-th].[15] S. Nojiri and S. D. Odintsov, Phys. Lett. B595, 1 (2004),

arXiv:hep-th/0405078 [hep-th].[16] R. Utiyama and B. S. DeWitt, J. Math. Phys. 3, 608

(1962).[17] K. S. Stelle, Phys. Rev. D16, 953 (1977).

[18] F. d. O. Salles and I. L. Shapiro, Phys. Rev. D89, 084054(2014), [Erratum: Phys. Rev.D90,no.12,129903(2014)],arXiv:1401.4583 [hep-th].

[19] G. T. Horowitz and R. M. Wald, Phys. Rev. D17, 414(1978).

[20] G. T. Horowitz, Phys. Rev. D21, 1445 (1980).[21] J. B. Hartle and G. T. Horowitz, Phys. Rev. D24, 257

(1981).[22] S. Randjbar-Daemi, J. Phys. A14, L229 (1981).[23] R. D. Jordan, Phys. Rev. D36, 3593 (1987).[24] W.-M. Suen, Phys. Rev. D40, 315 (1989).[25] W. M. Suen, Phys. Rev. Lett. 62, 2217 (1989).[26] P. R. Anderson, C. Molina-Paris, and E. Mottola, Phys.

Rev. D67, 024026 (2003), arXiv:gr-qc/0209075 [gr-qc].[27] L. H. Ford, Phys. Rev. D31, 710 (1985).[28] E. Mottola, Phys. Rev. D31, 754 (1985).[29] E. Mottola, Phys. Rev. D33, 1616 (1986).[30] I. Antoniadis, J. Iliopoulos, and T. N. Tomaras, Phys.

Rev. Lett. 56, 1319 (1986).[31] I. Antoniadis and E. Mottola, J. Math. Phys. 32, 1037

(1991).[32] A. Higuchi, Class. Quant. Grav. 4, 721 (1987).[33] D. Polarski, Phys. Rev. D41, 442 (1990).[34] N. C. Tsamis and R. P. Woodard, Phys. Lett. B301, 351

(1993).[35] N. C. Tsamis and R. P. Woodard, Annals Phys. 238, 1

(1995).[36] N. C. Tsamis and R. P. Woodard, Commun. Math. Phys.

162, 217 (1994).

13

[37] N. C. Tsamis and R. P. Woodard, Nucl. Phys. B474, 235(1996), arXiv:hep-ph/9602315 [hep-ph].

[38] N. C. Tsamis and R. P. Woodard, Phys. Rev. D54, 2621(1996), arXiv:hep-ph/9602317 [hep-ph].

[39] V. F. Mukhanov, L. R. W. Abramo, and R. H. Bran-denberger, Phys. Rev. Lett. 78, 1624 (1997), arXiv:gr-qc/9609026 [gr-qc].

[40] L. R. W. Abramo, R. H. Brandenberger, and V. F.Mukhanov, Phys. Rev. D56, 3248 (1997), arXiv:gr-qc/9704037 [gr-qc].

[41] N. Goheer, M. Kleban, and L. Susskind, JHEP 07, 056(2003), arXiv:hep-th/0212209 [hep-th].

[42] R. H. Brandenberger, in 18th IAP Colloquium on theNature of Dark Energy: Observational and TheoreticalResults on the Accelerating Universe Paris, France, July1-5, 2002 (2002) arXiv:hep-th/0210165 [hep-th].

[43] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi,Phys. Rev. D68, 046005 (2003), arXiv:hep-th/0301240[hep-th].

[44] F. Finelli, G. Marozzi, G. P. Vacca, and G. Venturi,Phys. Rev. D71, 023522 (2005), arXiv:gr-qc/0407101 [gr-qc].

[45] T. Janssen and T. Prokopec, Class. Quant. Grav. 25,055007 (2008), arXiv:0707.3919 [gr-qc].

[46] T. Janssen and T. Prokopec, Annals Phys. 325, 948(2010), arXiv:0807.0447 [gr-qc].

[47] T. Janssen, S.-P. Miao, and T. Prokopec, (2008),arXiv:0807.0439 [gr-qc].

[48] A. M. Polyakov, Nucl. Phys. B834, 316 (2010),arXiv:0912.5503 [hep-th].

[49] A. Shukla, S. P. Trivedi, and V. Vishal, JHEP 12, 102(2016), arXiv:1607.08636 [hep-th].

[50] P. R. Anderson and E. Mottola, Phys. Rev. D89, 104038(2014), arXiv:1310.0030 [gr-qc].

[51] P. R. Anderson and E. Mottola, Phys. Rev. D89, 104039(2014), arXiv:1310.1963 [gr-qc].

[52] R. Myrzakulov, S. Odintsov, and L. Sebastiani, Phys.Rev. D91, 083529 (2015), arXiv:1412.1073 [gr-qc].

[53] G. Cusin, F. de O. Salles, and I. L. Shapiro, Phys. Rev.D93, 044039 (2016), arXiv:1503.08059 [gr-qc].

[54] P. Peter, F. D. O. Salles, and I. L. Shapiro, Phys. Rev.D97, 064044 (2018), arXiv:1801.00063 [gr-qc].

[55] I. Kuntz and R. da Rocha, Eur. Phys. J. C79, 447 (2019),arXiv:1903.10642 [hep-th].

[56] N. Aghanim et al. (Planck), (2018), arXiv:1807.06209[astro-ph.CO].

[57] L. Alvarez-Gaume and E. Witten, Nucl. Phys. B234, 269(1984), [,269(1983)].

[58] T. S. Bunch and P. C. W. Davies, J. Phys. A11, 1315(1978).

[59] L. Parker and S. A. Fulling, Phys. Rev. D9, 341 (1974).[60] T. S. Bunch, J. Phys. A13, 1297 (1980).

[61] S. Deser, M. J. Duff, and C. J. Isham, Nucl. Phys. B111,45 (1976).

[62] M. J. Duff, Nucl. Phys. B125, 334 (1977).[63] A. Landete, J. Navarro-Salas, and F. Torrenti, Phys.

Rev. D89, 044030 (2014), arXiv:1311.4958 [gr-qc].[64] A. del Rio, J. Navarro-Salas, and F. Torrenti, Phys. Rev.

D90, 084017 (2014), arXiv:1407.5058 [gr-qc].[65] C.-S. Chu and Y. Koyama, Phys. Rev. D95, 065025

(2017), arXiv:1610.00464 [hep-th].[66] R. Endo, Prog. Theor. Phys. 71, 1366 (1984).[67] D. J. Toms, Phys. Rev. D90, 044072 (2014),

arXiv:1408.0636 [hep-th].[68] A. R. Vieira, J. C. C. Felipe, G. Gazzola, and M. Sam-

paio, Eur. Phys. J. C75, 338 (2015), arXiv:1505.05319[hep-th].

[69] N. Arkani-Hamed, S. Dubovsky, A. Nicolis,E. Trincherini, and G. Villadoro, JHEP 05, 055(2007), arXiv:0704.1814 [hep-th].

[70] H. Epstein, V. Glaser, and A. Jaffe, Nuovo Cim. 36,1016 (1965).

[71] C.-I. Kuo, Nuovo Cim. B112, 629 (1997), arXiv:gr-qc/9611064 [gr-qc].

[72] A. Borde, Class. Quant. Grav. 4, 343 (1987).[73] G. Klinkhammer, Phys. Rev. D43, 2542 (1991).[74] J. L. Friedman, K. Schleich, and D. M. Witt, Phys.

Rev. Lett. 71, 1486 (1993), [Erratum: Phys. Rev.Lett.75,1872(1995)], arXiv:gr-qc/9305017 [gr-qc].

[75] L.-F. Li and J.-Y. Zhu, Phys. Rev. D79, 044011 (2009),arXiv:0812.3532 [gr-qc].

[76] M. Visser, Phys. Lett. B349, 443 (1995), arXiv:gr-qc/9409043 [gr-qc].

[77] M. Visser, Phys. Rev. D54, 5116 (1996), arXiv:gr-qc/9604008 [gr-qc].

[78] D. Urban and K. D. Olum, Phys. Rev. D81, 024039(2010), arXiv:0910.5925 [gr-qc].

[79] D. Urban and K. D. Olum, Phys. Rev. D81, 124004(2010), arXiv:1002.4689 [gr-qc].

[80] H. Matsui, (2019), arXiv:1901.08785 [hep-th].[81] L. H. Ford, Proc. Roy. Soc. Lond. A364, 227 (1978).[82] J. Z. Simon, Phys. Rev. D43, 3308 (1991).[83] A. H. Guth, Phys. Rev. D23, 347 (1981), [Adv. Ser. As-

trophys. Cosmol.3,139(1987)].[84] K. Sato, Mon. Not. Roy. Astron. Soc. 195, 467 (1981).[85] A. D. Linde, QUANTUM COSMOLOGY, Phys.

Lett. 108B, 389 (1982), [Adv. Ser. Astrophys. Cos-mol.3,149(1987)].

[86] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48,1220 (1982), [Adv. Ser. Astrophys. Cosmol.3,158(1987)].

[87] J. Z. Simon, Phys. Rev. D45, 1953 (1992).[88] L. Parker and J. Z. Simon, Phys. Rev. D47, 1339 (1993),

arXiv:gr-qc/9211002 [gr-qc].[89] S. W. Hawking and T. Hertog, Phys. Rev. D65, 103515

(2002), arXiv:hep-th/0107088 [hep-th].