Self-Protection of Massive Cosmological Gravitons
aa r X i v : . [ h e p - t h ] A ug Preprint typeset in JHEP style - HYPER VERSION
CP3-ORIGINS-2010-30LMU-ASC 61/10TUM-HEP-766/10
Self-Protection of Massive Cosmological Gravitons
Felix Berkhahn abd , Dennis D. Dietrich c , and Stefan Hofmann bd a Physik-Department T30d, Technische Universit¨at M¨unchen, James-Franck-Straße,85748 Garching, Germany b Excellence Cluster Universe, Boltzmannstraße 2, 85748 Garching, Germany c CP -Origins, Centre for Particle Physics Phenomenology, University of SouthernDenmark, Campusvej 55, 5230 Odense M, Denmark d Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universit¨at,Theresienstraße 37, 80333 Munich, GermanyE-Mails: [email protected] , [email protected] , [email protected] Abstract:
Relevant deformations of gravity present an exciting window of opportunityto probe the rigidity of gravity on cosmological scales. For a single-graviton theory, theleading relevant deformation constitutes a graviton mass term. In this paper, we investigatethe classical and quantum stability of massive cosmological gravitons on generic Friedmanbackgrounds. For a Universe expanding towards a de Sitter epoch, we find that massivecosmological gravitons are self-protected against unitarity violations by a strong couplingphenomenon.
Keywords: gravity, modified gravity, quantum field theory on curved space,cosmological perturbation theory, dark energy. ontents
1. Introduction 12. Goldstone–St¨uckelberg analysis 4
3. Discussion 9
1. Introduction
Technical naturalness is arguably one of the most promising pathfinders to physics beyondthe standard model of particle interactions and gravity, as well. It offers many excitingwindows of opportunity related to renormalisable standard model operators that share anenhanced sensitivity to the scale of new physics.Among these, the vacuum energy density is standing out in various ways. Being theunique operator with quartic sensitivity to the ultraviolet scale, it also represents the mostrelevant term in the Einstein–Hilbert action. The basic observation is that the vacuumenergy density is technically unnatural within the standard model already at energy scalesset by the lightest measured particle masses within its spectrum. In other words, thetechnical naturalness facet of this challenge is not solely tied to the quantum gravity scale,unless there is an ultraviolet-infrared conspiracy operative in the vacuum sector that alsorespects the many high-precision successes of the standard model at lower energies.Although collider experiments cannot measure the vacuum energy density, the chal-lenge it poses becomes serious once the standard model of particle physics is coupled togravity. Since gravity, being the most democratic field theory, couples to energy-momentumin a universal manner irrespective of its sources’ nature, the vacuum curves spacetime andaffects the Universe’s expansion history. As a consequence, and opening up yet anotherwindow of opportunity, it seems attractive to reconsider gravity in the deep infrared andits consistent deformations.The hunt for a fundamental completion of gravity is characterised by incorporating newdegrees of freedom in the ultraviolet with the infrared region kept untouched (in a relevantsense), its role reduced to providing the classical benchmark tests. However, this precludesthe opening up of additional gravitational degrees of freedom that could be relevant in thedeep infrared on scales that are only poorly constrained by state-of-the-art cosmologicalobservations. – 1 –istorically, this question was discouraged by a mighty no-go theorem stating the im-possibility to embed gravity in a QCD-like theory with a self-interacting graviton multipletunder the spell of Yang–Mills. More precisely, assuming locality, Poincar´e invariance, and afree-field limit consisting of massless gravitons, the only consistent deformations involvinga multiplet of gravitons are such that the deformed gauge algebra is just a direct sum ofindependent diffeomorphism algebras [1].Interestingly, by relaxing one of its conditions — allowing for new relevant degrees offreedom — this no-go theorem gave rise to a potent tool for studying consistent defor-mations of Einstein’s gravity within the effective field theory framework. For a nonzerodeformation parameter, the candidate deformations are characterised by a tremendousreduction of symmetries:diff( M ) ⊗ diff( M ) ⊗ · · · −→ diag (diff( M ) ⊗ diff( M ) ⊗ · · · ) , with M j denoting (not necessarily) different spacetime (sub-)manifolds. The resultinggauge symmetry restricts the deformation, i.e. the coupling of different geometries, to besolely constructed from invariants of the reduced symmetry [2].Under the umbrella of this framework, many proposals for relevant deformations ofgravity that have been suggested in the last decade become cousins. Even more promising,the unique ghost-free theory for a massive spin-two field propagating all degrees of freedom[3] fits under this umbrella. In the latter case, two copies of Minkowski spacetime areconsidered, one perfect the other perturbed, and the most relevant deformation becomes aspin-two mass term S deform = − m Z d x h αβ M αβµν h µν . (1.1)The mass matrix M depends only on the background geometry and is constant in thiscase. As mentioned earlier, this matrix is uniquely determined by unitarity arguments andrequires tuning.There is a rather straightforward nonlinear completion of the leading infrared defor-mation [4], S deform = − m Z M d x √− g M αβµν ( g ) H αβ H µν ( g , g ) ,H αβ = g αβ − E µα E νβ g µν , (1.2)where E denotes the pullback from M to M . Since the spacetimes ( M , g ) and ( M , g )need not be diffeomorphic to each other (even not in the perturbative sense), H ( x ) is ingeneral not a fluctuation on M . In fact, the deformation (1.2) represents a mass term fora graviton only if ( M , g ) is a copy of ( M , g ) at the background level.Consider the case when the most relevant deformation is a mass term. Let us assumefor simplicity, that some sort of Higgs mechanism is responsible for the graviton mass gener-ation. For instance, in the Fierz–Pauli setup, a massless graviton, which has two transverse In abuse of notation and logic, we refer to metric perturbations as gravitons irrespective of the back-ground geometry. – 2 –olarisation states, combines with a Goldstone vector to a massive spin-2 field, which hassix polarisation states in general (but only five on Minkowski spacetime). The Goldstonevector carries three transverse polarisation states and one longitudinal polarisation state.When the massive graviton is at rest, its six polarisation states are completely equivalent.However, if it is moving, the longitudinal polarisation becomes increasingly parallel to thegraviton’s momentum. As a consequence, at high energies, a massive graviton might looklike the longitudinal polarisation state carried by the Goldstone vector [4]. This state-ment is known as the Goldstone equivalence theorem and rests on the underlying gaugeinvariance.The longitudinal polarisation state does not receive kinetic support from the Einstein–Hilbert term, which is precisely why it is in the focus of unitarity requirements. As amatter of fact, many distinguished features of massive gravity, like, for example the vanDam–Veltman–Zakharov discontinuity [5] [6] , the Vainshtein radius [7], or the structureof the Fierz–Pauli mass term on Minkowski spacetime [3], are captured by the longitu-dinal polarisation state of the Goldstone vector. Consistency invesitgations of massivegravity mostly focused on ghostly excitations at some finite perturbation level based on aMinkowski ground state, with the earliest exception being Higuchi’s unitarity analysis onde Sitter spacetime [8].Higuchi found a consistency relation between the deformation parameter, the gravitonmass, and the curvature scale of de Sitter, set by the cosmological constant: In order toavoid negative norm states m > H , where H denotes the Hubble constant. This boundis of great interest, since de Sitter geometry is unique in the sense that is does not requireany source specification. However, from a field theoretical point of view this makes thesetup special, because the background reference scale is constant here.In this paper, we generalise Higuchi’s bound from de Sitter to general Friedman cos-mologies by employing the Goldstone equivalence theorem outlined above. We find acompetition between classical stability, the requirement that perturbations respect thebackground, and quantum stability, the requirement that the spectrum does not containnegative norm states. For the special case of a de Sitter background, both criteria coin-cide and give rise to the unitarity bound quoted above. The situation is richer for genericFriedman cosmologies. There, the very nature of either bound is more intriguing, since itinvolves a time dependent curvature scale that is monotonously increasing or decreasing inthe past, depending on the specific sources that drive the background expansion. Generi-cally, none of the bounds can be satisfied on the entire spacetime manifold. This, however,does not imply that the theory is invalidated. Indeed, it turns out that massive gravitonsin generic Friedman universes are protected against unitarity violations. More precisely,it is not clear at all whether a unitarity bound really exists, because before entering thewould-be unitarity violating spacetime region, the theory becomes strongly coupled .We reach the following verdict: Phenomenological constraints require to choose theinitial hypersurface close to the present hypersurface. In the case of a radiation or matterdominated Friedman universe, the evolution towards future hypersurfaces is guaranteed to The connection between stability and strong coupling is clarified at the end of section 2. – 3 –e healthy (for consistent initial conditions) by the strictly monotonic background expan-sion. In the most interesting case of a matter-cosmological constant mixture, the futureevolution will be sound provided the mass is large enough and consistent initial condi-tion have been imposed. In all of the above cases, evolving backwards in time, massivecosmological gravitons will soon enter a strong coupling regime that demands a nonlinearcompletion of the theory. In other words, the fact that we have a sound theory on all pasthypersurfaces is nontrivial and ensured by a strong coupling phenomenon that is confinedto these spacetime regions. This is the advertised self-protection mechanism.Last but not least, we show that all conclusions hold for a generic Friedman source.
2. Goldstone–St¨uckelberg analysis
In this section we derive the classical and quantum stability requirements for massivecosmological gravitons.Consider two copies of a generic (background) spacetime, ( M B , γ ) and ( M B , g ) with γ ≡ g + h , where h denotes the metric perturbation obeying | g ( t ) | ≫ | h ( t, x ) | . In this case, H = h is a perturbation (where, for simplicity, we have chosen the same coordinate systemon both manifolds) under the spell of diff( M B ) for vanishing deformation parameter.Turning on the most relevant deformation (1.2), the gauge symmetry is deformed to thediagonal subgroup of diff( M B ) ⊗ diff( M B ). In other words, the deformation removes thefreedom to gauge h relative to the background geometry. The massive graviton, however,still carries six degrees of freedom, due to the second Bianchi identity ∇ µ h µν = ∇ ν h . (2.1)In general, this is not a gauge choice. For instance, in the undeformed theory for masslessgravitons on Minkowski spacetime the constraint (2.1) is not a legitimate gauge, becausethe corresponding gauge shifts become singular for this choice (as a testimony of the vanDam–Veltman–Zakharov discontinuity on this background).As is well known, in view of this explicit symmetry deformation, there are two equiv-alent state descriptions. In the first case, the metric perturbation is split according to h µν = h ⊥ µν + ∇ ( µ V ν ) , (2.2)where h ⊥ is covariantly conserved and carries two transverse degrees of freedom, while V isunconstrained and carries four degrees of freedom. The latter can be decomposed further, V = V ⊥ + ∂ Ψ. Here ∇ · V ⊥ = 0 and Ψ carries one degree of freedom. Using this statedescription, the theory can be considered as a gauge fixed theory.The second and equivalent state description, called the Goldstone–St¨uckelberg com-pletion, is based on adding four degrees of freedom, carried by a vector field π in order torestore the original gauge symmetry, diff( M B ) ⊗ diff( M B ). In this case, the completion isgiven by H µν ≡ h µν + ∇ ( µ π ν ) , (2.3)– 4 –here H has ten degrees of freedom, of which six are carried by h and four by the Goldstone–St¨uckelberg vector π , which can be further decomposed as π = π ⊥ + ∂φ . Here ∇ · π ⊥ = 0and φ carries one scalar degree of freedom.As mentioned above, the crucial point of this construction is the restored gauge sym-metry that allows to shift h and π relative to the background such that the Goldstone–St¨uckelberg completed H itself is rendered gauge invariant. It is clear that four degrees offreedom represent gauge redundancies, leaving us with six physical degrees of freedom.We choose to work with the Goldstone–St¨uckelberg completed state, for which theleading relevant deformation is exactly the celebrated Fierz–Pauli mass term, S mass = − m Z M B d x √− g H αβ M αβµν ( g ) H µν , M αβµν ( g ) = g αµ g βν − g αβ g µν . (2.4)The mode of the metric fluctuation corresponding to the Goldstone–St¨uckelberg scalar φ dominates scattering processes at high momenta. This is tantamount to the Goldstoneboson equivalence theorem and most easily understood from the observation that this modeenters processes with at least two derivatives, ∇ µ ∂ ν φ , which, therefore, grows fastest in thehigh momentum limit. This is precisely the regime for which we are interested in studyingthe stability of the deformed theory.The field φ mixes with the metric perturbation h through the mass term (2.4),2 m ( (cid:3) h − ∇ µ ∇ ν h µν ) φ = 2 m (cid:16) R (0) µν ( g ) h µν − R (1) ( g, h ) (cid:17) φ , (2.5)where we have integrated by parts and introduced the Ricci tensor R (0) ≡ R evaluated onthe background configuration g , as well as the Ricci scalar R (1) expanded to first order in h . In order to eliminate the kinetic mixing term R (1) ( g, h ) φ , we carry out a conformaltransformation ˆ g µν = Ω g µν ≡ (1 + ω ) g µν , (2.6)which, at the linear level, is equivalent to ˆ h µν = h µν + 2 ω η µν . The Einstein–Hilbert termtransforms as √− g Ω R = p − ˆ g (cid:16) ˆ R − − ˆ g ab ∂ a Ω ∂ b Ω (cid:17) . (2.7)In order to eliminate the mixing between h µν and φ we must choose Ω = 1 − m φ or,equivalently (since φ is a first order quantity in the expansion of H µν ), ω = − m φ .The conformal transformation (2.7) contributes a standard kinetic term for the Goldstone–St¨uckelberg scalar, while the massive deformation (2.4) gives rise to a non-standard kineticcontribution with the metric field replaced by the background Ricci tensor,2 m (cid:16) ( (cid:3) φ ) − ∇ µ ∇ ν φ ∇ µ ∇ ν φ (cid:17) = 2 m R µν ∂ µ φ∂ ν φ . (2.8)The action for φ is given by S = Z d x p − ˆ g (cid:16) A ˙ φ + B ij ( ∂ i φ )( ∂ j φ ) + ˙ φD i ∂ i φ (cid:17) , (2.9)– 5 –ith A = 2 m ( − m g + R ), B ij = 2 m ( − m g ij + R ij ), and D i = 4 m ( − m g i + R i ).In general, these coefficients are spacetime dependent functions. Note that in (2.9) wehave not displayed any potential terms (self-couplings) such as R ( g ) hφ , since quantumstability refers to the free evolution, and classical stability relies on the kinetic terms athigh momenta. The corresponding Hamilton density reads H = π √− ˆ gA − p − ˆ gB ij ( ∂ i φ )( ∂ j φ ) , (2.10)where π ≡ δ L / ˙ φ . The Hamiltonian is unbounded from below for A < B ij positivedefinite.In the case of a generic Friedman background geometry the action for the Goldstone–St¨uckelberg scalar reduces to S = Z d x p − ˆ g (cid:18) A ( t ) ˙ φ + B ( t ) (cid:16) ~ ∇ φ/a (cid:17) (cid:19) , (2.11)where A ( t ) = 6 m ( m − ˙ H − H ), B ( t ) = − m (3 m − ˙ H − H ), H = H ( t ) denotesthe Hubble parameter, and a = a ( t ) the scale factor. These coefficients depend on theenergy-momentum source curving the Friedman background and, in particular, can changesigns during the background evolution. Classical stability requires A >
B < φ → f φ with ˙ f = − Cf / (2 A ), where C ≡ ˙ A − HB . TheLagrangian transforms as f − L = A ˙ φ − C φ ˙ φ + B (cid:16) ~ ∇ φ/a (cid:17) + C / (4 A ) φ . (2.12)The coefficient A controls the sign between ˙ φ and f − π = 2 A ˙ φ − Cφ and has thereforean important impact on the quantum stability of (2.11). This can be worked out along thecanonical quantisation prescription. As usual, we postulate the equal time commutationrelations (cid:2) φ ( t, x ) , π ( t ′ , x ′ ) (cid:3) t = t ′ = iδ (3) (cid:0) x − x ′ (cid:1) , . . . , (2.13)and decompose the field φ ( x ) into modes U ( t, k ) ≡ u ( t, k ) exp (i k · x ), where u ( t, k ) satisfies A ¨ u − " B (cid:18) k a (cid:19) + A (cid:18) dd t + C A (cid:19) C A u = 0 . (2.14)The fact that the collection of U ( t, k ) represents a complete orthonormal set of solutions(with respect to a spatial hypersurface-independent scalar product) results in a simplecondition on the Wronskian of the solutions,( u ∗ ˙ u − u ˙ u ∗ ) ( t, k ) = 1 . (2.15)As a consequence, the Goldstone–St¨uckelberg scalar may be expanded as φ ( t, x ) = Z d k (2 π ) / p | A | f (cid:16) u ( t, k ) exp (i k · x ) a ( k ) + u ∗ ( t, k ) exp ( − i k · x ) a † ( k ) (cid:17) . (2.16)– 6 –nserting this expansion and the corresponding one for π into the canonical commutationrelations (2.13) yields h a ( k ) , a † ( k ′ ) i = sign( A ) δ (3) ( k − k ′ ) . (2.17)The construction of a vacuum state and Fock space can now proceed as usual. However,whenever A <
0, the construction results in negative norm states, which violate unitarity.For an arbitrary spacetime, this quantisation procedure bears conceptual challenges,because there may be no Killing vectors at all to define positive frequency modes. Thesituation is simpler for a Friedman background since it accommodates a restricted set ofisometries, i.e. invariance under spatial rotations. Then, together with the correspondingKilling vectors there exist associated (natural) coordinates. Of course, coordinate systemsare physically irrelevant — a fact that renders the particle concept somewhat arbitrary oncurved spacetimes. However, this concerns the interpretation of the theory. The unitarityrequirement
A <
A >
B <
A >
B <
A >
0, to have astable solution, requires
B <
0. Further, in order to have a damped solution at late (early)times demands
C > C <
A, B, C in a spatially flat FriedmanUniverse, consider the Friedman equations3 H = 8 πGρ + Λ , (2.18)3 (cid:16) ˙ H + H (cid:17) = − πG ( ρ + 3 p ) + Λ , (2.19)where ρ is the density, p the pressure, and Λ the cosmological constant. An expandinguniverse is characterised by H > ρ <
0, so ˙
H <
0. Furthermore, ¨ H = − H ˙ H (1+ c ),where c s denotes the isentropic sound speed. As a consequence, ¨ H >
0, for an equation ofstate governing an arbitrary mixture of matter and radiation.In the absence of a cosmological constant, H and all its time derivatives vanish atlate times. The late time asymptotics of the coefficients are therefore given solely by thegravitons mass, A = 6 m = − B > C = 0. Consequently, the modes (2.14) arewell-behaved. In a de Sitter universe, A = 6 m ( m − H ) = − B and C ∝ A . The absenceof negative norm states requires m > H , which is nothing else but the famous Higuchibound. We see that in this case, classical and quantum stability collapses to a singlecriterion between the graviton mass and the constant de Sitter curvature scale.For a generic Friedman universe, at any moment in time, A > − B and C = 6 m H (3 m − ¨ H/H − H − H ) < HA . As a consequence, evolving the modes (2.14) backward in time, B and C change signs before A does (i.e. at later cosmological times). In the case of C this– 7 –ign change stabilises the modes since it leads to mode damping. This stabilisation, how-ever, is only marginal and nullified by the change of sign of B which triggers an exponentialinstability that dominates the large proper momentum regime.As a consequence, evolving the modes backwards in cosmological times, the systementers first a strong coupling regime (at later cosmological time) before it would violateunitarity (at earlier cosmological times). In this sense, the strong coupling regime protectsmassive cosmological gravitons from unitarity violations. In other words, whenever massivegravitons in an expanding Friedman universe experience unitarity violations, then, for sure,they are already in a strong coupling regime that demands a nonlinear completion, but notvice versa. In this respect, de Sitter is a borderline geometry since both inconsistenciescoincide.It is important to appreciate that once a generic mode enters the classical instabilityregion B >
0, it does not destabilise the background instantaneously. Instead, the charac-teristic time scale for this to happen is T ( k ) ∝ a/k . As a consequence, modes with arbitrarylarge proper momentum become strongly coupled without further delay once B >
0. Moreprecisely, for arbitrary initial conditions, there will always be a critical proper momentum k ∗ such that all modes with momenta k > k ∗ enter the nonlinear regime before they wouldviolate the unitarity bound.Strictly speaking, the self-protection mechanism is confined to and efficient for B ≤ B becoming positive, they hit a strong coupling regime. Indeed, for B < B →
0, the proper kinetic energy density ∝ B ( ∇ φ/a ) becomes subdominant onany scale as compared to the potential energy. Hence, the exponential instability discussedabove, characterized by B >
0, is a testimony of this strong coupling regime and canlegitimately be used to identify it.
In the following we argue that both the stability and unitarity behaviour of the theoryare generically not altered by the specific choice of a matter action S matter (cid:2) Ψ i , g µν (cid:3) , whereΨ i denotes the collection of matter degrees of freedom. This is remarkable because thematter action explicitly depends on the metric field and, in particular, on the Goldstone–St¨uckelberg scalar φ . This coupling could, in principle, change the dynamics of φ in arelevant way such that the self-protection mechanism will be overridden. As we will showbelow, this is not the case.Since the matter action is invariant under general coordinate transformations, theGoldstone–St¨uckelberg scalar enters the matter sector only via the conformal transforma-tion (2.6). Let us first expand the matter action to second order in the fluctuations, S matter [Ψ i + δ Ψ i , g + h ] ⊃ Z d x d y h µν ( x ) (cid:18) h αβ ( y ) δδg αβ ( y ) + δ Ψ i ( y ) δδ Ψ i ( y ) (cid:19) T µν ( x ) . (2.20)After the conformal transformation (2.6), h = ˆ h + 2 m φg , the first term on the right-handside of (2.20) contributes only self-interaction terms of the form φ and couplings hφ . The– 8 –oefficient of this potential term can be estimated to be of order H . Hence, the potentialterm is subdominant as compared to the kinetic term at high proper momenta.The conformal transformation of the second term on the right-hand side of (2.20)gives rise to a coupling between the Goldstone–St¨uckelberg scalar and the matter degreesof freedom, − m Z d x φ ∂T∂ ( ∂ µ Ψ i ) ∂ µ δ Ψ i , (2.21)where we have again neglected potential terms of the form φδ Ψ i . A coupling like (2.21)will not modify the momentum field conjugated to φ , and, hence, the unitarity bound isrobust against its inclusion. Suppose now we integrate (2.21) by parts, thereby producinga time derivative acting on φ . This will modify the conjugated momentum field, but it willonly contribute a term proportional to δ Ψ i . Again, such a coupling respects the unitaritybound.With respect to stability, the coupling (2.20) might alter the dynamics of the Goldstone–St¨uckelberg field substantially, as it represents a derivative coupling to the matter degreesof freedom. However, the coefficient of this derivative coupling is a vector field constructedsolely from background quantities. Due to the isometries of the Friedman geometry, thisbackground vector has to be of the form (cid:18) ∂T∂ ( ∂ µ Ψ i ) (cid:19) ∝ e µ ≡ (1 , , , . (2.22)Therefore, no spatial derivative enters the coupling (2.21). Thus the modes experience anadditional source proportional to ∂ t δ Ψ i , and as long as the matter sector takes good careof itself this source cannot alter the stability bound, in particular at large proper momenta.
3. Discussion
Let us consider the concrete expressions for the coefficients A and B and discuss the possibleimplications of the corresponding bounds. For a de Sitter spacetime, A = 6 m ( m − H ) = − B . The absence of negative normstates requires the unitarity bound m > H to hold, which is just the well-known Higuchibound (in our conventions). Provided this bound has been satisfied, classical stability isestablished automatically, as A = − B . In fact, de Sitter geometry is unique within theclass of Friedman spacetimes where classical and quantum stability requirements coincide,and where stability is solely expressed in terms of two model parameters, i.e. the gravitonmass and the constant de Sitter curvature scale. Since linearisation is permissible, thestability bound, m > H , should be an important consistency condition for a nonlinearcompletion, as well. – 9 – .2 Generic Friedman spacetimes For a generic Friedman spacetime, the absence of negative norm states demands m >H + ˙ H , while classical stability requires m > H + ˙ H/
3. We were able to confirm thesestability bounds by a full-fledged perturbation analysis of massive cosmological gravitonsinvolving all degrees of freedom, as well as a complete set of couplings. (See [9].) The samebounds were also derived for the special case of scalar field matter in [12]. The findingsof [13], however, do not coincide with ours, due to the unconventional matter Lagrangianthat has been used in their paper.In the case of an expanding Universe the unitarity bound will always be satisfiedduring radiation ( m > − H ) or matter domination ( m > − H / m > H / m > H / H + ˙ H will become positive and eventually constant asthe Universe evolves towards its de Sitter fate. If the cosmological graviton’s mass is largerthan the asymptotic value for the Hubble parameter set by the de Sitter curvature scale, thetheory always respects unitarity. If, however, the mass parameter is chosen to be smallerthan the cosmological constant, it is guaranteed that the φ modes first enter the epoch ofclassical instability, before (evolving forward in time) they hit the then would-be unitaritybound (since ˙ H <
H <
0, so that the massive cosmological gravitons always first enter the strong couplingregime, rendering the would-be unitarity bound fictitious and opening an exciting windowof opportunity towards a self-protection mechanism. Finally, let us state the situation isreversed in a contracting universe, where ˙
H >
0. Here, negative norm φ states will show upin the weak coupling regime. There is no obvious self-protection mechanism in this case. Acknowledgments
The authors would like to thank Eugeny Babichev, Cliff Burgess, Stanley Deser, Gia Dvali,Justin Khoury, Michael Kopp, Florian Niedermann, and Andrew Tolley for helpful and– 10 –nspiring discussions. DDD acknowledges gratefully the hospitality of the Arnold Sommer-feld Center and the Excellence Cluster Universe. The work of DDD was supported by theDanish Natural Science Research Council. The work of SH was supported by the DFGcluster of excellence ’Origin and Structure of the Universe’ and by TRR 33 ’The DarkUniverse’.
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