Instability of Quantum de Sitter Spacetime
aa r X i v : . [ h e p - t h ] A p r Instability of Quantum de Sitter Spacetime
Chiu Man Ho ∗ and Stephen D. H. Hsu † Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA (Dated: June 21, 2018)Quantized fields (e.g., the graviton itself) in de Sitter (dS) spacetime lead to particle production:specifically, we consider a thermal spectrum resulting from the dS (horizon) temperature. The energyrequired to excite these particles reduces slightly the rate of expansion and eventually modifies thesemiclassical spacetime geometry. The resulting manifold no longer has constant curvature nor timereversal invariance, and back-reaction renders the classical dS background unstable to perturbations.In the case of AdS, there exists a global static vacuum state; in this state there is no particleproduction and the analogous instability does not arise.
I. INTRODUCTION
In classical general relativity, a cosmological constantΛ has the special property that the equation of state mustsatisfy w = pρ = − p is the associated pressure and ρ the en-ergy density. This is equivalent to the energy-momentumtensor T µν satisfying T µν = Λ g µν , (2)where g µν is the metric tensor. For Λ >
0, negative pres-sure does negative work as the universe expands, and pro-vides exactly enough energy to produce new spacetimevolume filled with more cosmological constant. Thus,expansion can continue forever, leading to a highly sym-metric constant curvature spacetime known as de Sitter(dS) spacetime.A ( d + 1)-dimensional dS spacetime is a hyperboloid ina ( d + 2)-dimensional Minkowski spacetime described by x − x − x − · · · − x d +1 = − / Λ ≡ − R . (3)In the global coordinates, the dS metric is ds = dt − R cosh ( t/R ) d Ω d , (4)while in the cosmological or Friedmann-Robertson-Walker (FRW) coordinates (which cover only a portionof the global dS manifold (3)), it is given by ds = dt − a ( t ) d Ω d , (5)where a ( t ) = e Ht with H = p Λ /
3. Our discussion belowis focused on the case d = 3, or four spacetime dimen-sions.Quantum excitations, including, inevitably, gravitons(quanta of the gravitational field itself), modify at least ∗ Electronic address: [email protected] † Electronic address: [email protected] slightly the relation between pressure and energy density.Multi-particle quantum states typically have positive en-ergy density and pressure, leading to a value of w slightlylarger than −
1, and a pressure insufficiently negative tosupport dS expansion.The calculation of quantum contributions to the stress-energy tensor T µν in dS spacetime is complex, dependingon choices of regularization and vacuum state. For recentprogress on this problem, see [1–6]. In this note, we ex-plore the consequences of the relatively well-understooddS temperature on the macroscopic dynamics of space-time. II. BACK-REACTION FROM DE SITTERTHERMAL SPECTRUM
In dS spacetime, inertial observers see a thermal distri-bution of particles and a dS temperature [7]. Observerswho detect thermal particles will dispute the notion thatthe physical (renormalized) T µν is proportional to g µν (i.e., that the equation of state describes pure vacuumenergy or cosmological constant). This deviation fromthe classical form of T µν violates dS symmetry. It is dueto a subtle infrared effect (dS temperature) and will notnecessarily appear in calculations of UV contributions tothe renormalization of T µν .In [7], Gibbons and Hawking note that detector ab-sorption of thermal radiation from the dS horizon leads,via back-reaction, to shrinkage of the horizon. We are in-terested in an averaged, semiclassical T µν that appears onthe right hand side of the Einstein equations, and resultsfrom the steady occurrence of such events throughoutspacetime. Interactions will eventually equilibrate eachparticle species with the dS horizon temperature.The fact that inertial observers in dS spacetime seea thermal distribution of particles can also be under-stood in terms of the Unruh effect [8]. One can considerdS spacetime as a timelike hyperboloid embedded in aMinkowski spacetime of one higher (spatial) dimension[9, 10]. Inertial dS observers, viewed from the perspec-tive of the embedding spacetime, are uniformly acceler-ated, and hence their detectors register a thermal bath.From the Unruh perspective, it is clear that the energyof absorbed thermal particles comes from work done bythe accelerating force on the detector [8, 11]. From thedS perspective, this energy comes from work that oth-erwise would have been performed by the negative pres-sure. Thus, it clearly reduces the amount of expansionthat would otherwise occur (i.e., in the absence of quan-tum mechanics) in dS spacetime.The dS temperature is T = R − / π , where R isthe dS radius. The ratio of the thermal energy den-sity to cosmological constant is of order the latter inPlanck units, henceforth parametrized by ǫ . The localenergy density at late times in the expanding phase ofdS spacetime is therefore slightly larger than in the clas-sical case: ρ = Λ(1 + ǫ ). The corresponding pressure is p ≈ − Λ(1 − ξ ǫ ) where ξ = 1 / ξ = 0 correspondto relativistic and non-relativistic thermal particles re-spectively. Thus w = − aa = − πG ρ + 3 p ) , (6)it follows that as long as ρ > w < − /
3, accelera-tion is still positive. Therefore, an accelerating expansionof dS spacetime is still expected. Using the equation ofcontinuity ˙ ρ + 3 ˙ aa ( ρ + p ) = 0, one can show that˙ ǫǫ + 3 (1 + ξ ) ˙ aa = 0 . (7)The above equation can be solved to give ǫ ∼ a − ξ ) . (8)In fact, the situation is more complicated than suggestedby the simple equations above. As particles produced byearlier expansion are redshifted away, new particles areproduced. After many Hubble timescales, the averageenergy density due to quantum effects should be approx-imately that of a thermal bath at the dS temperature.(The Bunch-Davies vacuum [12], or an approximate ver-sion of it, is an attractor.)Conservation of energy implies that the resultingproper volume of the universe V is slightly smaller thanin the classical case: V ≈ V classical · (1 − ǫ ) = exp(3 Ht ) · (1 − ǫ ) , (9)and so log V t ≈ H − ǫ/ t . (10)Thus, the spacetime which results from incorporatingback-reaction of these quantum effects is no longer one ofconstant curvature. At late times, the classical and quan-tum spacetimes differ macroscopically, despite the small-ness of the dS temperature. Expansions about the orig-inal (classical) dS spacetime should exhibit IR instabili-ties, since dS is not an exact solution once back-reactionis taken into account. Earlier work has found evidence of instabilities in dS [13, 14], although the relation to ourresults is not clear.The resulting quantum spacetime also cannot be time-reversal invariant. If the late time thermal particle den-sity were also found at early times, during the contract-ing phase of global dS spacetime, the resulting blue-shiftof thermal particles would lead to radical departure fromthe vacuum Einstein equations. (This point has also beenemphasized in [1].) Therefore, the early and late time ge-ometries, taking into account quantum effects, cannot bethe same. III. ANTI-DE SITTER SPACETIME
Anti-de Sitter (AdS) spacetime is similarly a surface ofconstant (negative) curvature, satisfying the constraint(for simplicity we restrict to AdS ) T + W − X − Y − Z = − / Λ ≡ R (11)in five-dimensional Minkowski space with metric ds = dT + dW − dX − dY − dZ . (12)Some AdS worldlines correspond to uniform accelera-tion in the embedding space [10], again suggesting thepresence of thermal (Unruh) radiation and modificationof the semiclassical geometry. However, AdS differs fromdS in an important way: one can define global static co-ordinates in AdS, T = R p r /R cos( t/R ) W = R p r /R sin( t/R ) X = r cos θY = r sin θ cos φZ = r sin θ sin φ (13)with metric ds = (cid:18) r R (cid:19) dt − (cid:18) r R (cid:19) − dr − r d Ω . (14)Although this metric violates spatial translation invari-ance, the fact that it is static implies that there is a quan-tum vacuum state that is time-independent: it does notexhibit particle production or thermal radiation. For thisspecial choice of vacuum state, AdS is stable to the dSinstability discussed above. This result is a consequenceof the existence of a global timelike Killing vector. Otherchoices of AdS vacuum state, such as the one appropriateto the “cosmological” (non-static) coordinates (coveringonly a portion of global AdS) T = R cos( t/R ) W = R sin( t/R ) cosh χX = R sin( t/R ) sinh χ cos θY = R sin( t/R ) sinh χ sin θ cos φZ = R sin( t/R ) sinh χ sin θ sin φ (15)with metric ds = dt − R sin ( t/R ) (cid:2) dχ + sinh χ d Ω (cid:3) (16)do in fact lead to particle production [15] and consequentmodification of the spacetime geometry. The differencebetween the two cases is the choice of quantum vacuumstate. The global vacuum state is defined on a spacelikeslice (e.g., at fixed t ) in coordinates (13), but this coversthe range −∞ < T < ∞ in the embedding space. Incontrast, a fixed t slice in the cosmological coordinatescorresponds to fixed T . Therefore, the global vacuumstate can impose conditions on past and future quantumstates in the cosmological coordinates, which lead to thecancelation of otherwise expected particle production dueto acceleration. Stability of AdS depends on choice ofvacuum state, and the spacelike surface on which it isdefined. IV. REMARKS
The argument presented here is the simplest one weknow of that indicates the instability of dS spacetime once quantum effects are considered. The effect we iden-tified is small, but does break the dS symmetries even inthe asymptotic regions of the manifold. We do not ex-clude the possibility of more dramatic quantum effects,such as significant decay of the cosmological constant it-self [13, 16]. Note that our effect specifically dependson the back-reaction, via the Einstein equations, of thespacetime geometry to modification of the equation ofstate. We do not address the possibility that a quantumfield (e.g., massive scalar) propagating on a fixed dS back-ground could have some intrinsic instability [13, 14, 17].
Acknowledgements.
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