JHEP04(2015)086 Springer April 20, 2015 March 31, 2015 : : January 21, 2015 : ) spacetime lead Published Accepted 10.1007/JHEP04(2015)086 Received se particles reduces slightly doi: spectrum resulting from the cal spacetime geometry. The sity, e reversal invariance, and back- e is no particle production and rturbations. In the case of AdS, Published for SISSA by [email protected] , . 3 1501.00708 The Authors. c Models of Quantum Gravity, Classical Theories of Gravity

Quantized fields (e.g., the graviton itself) in de Sitter (dS , [email protected] Department of Physics andEast Astronomy, Lansing, Michigan MI State 48824, Univer U.S.A. E-mail: ArXiv ePrint: there exists a globalthe static analogous vacuum instability state; does in not this arise. state ther Keywords: to particle production:dS specifically, (horizon) we temperature. considerthe The a rate energy thermal of required expansion toresulting and manifold excite eventually no the longer modifies has thereaction constant semiclassi renders curvature nor the tim classical dS background unstable to pe Instability of quantum de Sitter spacetime Open Access Article funded by SCOAP Abstract: Chiu Man Ho and Stephen D.H. Hsu JHEP04(2015)086 1 2 3 5 (1.5) (1.3) (1.4) (1.2) = 3, or d . 2 itter (dS) spacetime. +2)-dimensional Minkowski duce new spacetime volume R continue forever, leading to a d , FRW) coordinates (which cover 2 d ≡ − has the special property that the Ω , d Λ / ) 2 d 3 Ω − , d t/R the energy density. This is equivalent to 1 (1.1) ( ) = t 2 ρ − ( µν 2 g +1 = )), it is given by a 2 d cosh x p ρ – 1 – 0, negative pressure does negative work as the − 1.3 2 = Λ 2 = R > dt µν w − T = 2 − · · · − 2 satisfying dt 2 2 3. Our discussion below is focused on the case ds x / = µν Λ T 2 − 2 1 p ds x = − 2 0 H x with is the associated pressure and p Ht e is the metric tensor. For Λ ) = +1)-dimensional dS spacetime is a hyperboloid in a ( t µν ( d g a A( where four spacetime dimensions. 1 Introduction In classical general relativity, aequation cosmological of constant state Λ must satisfy 3 Anti-de Sitter spacetime 4 Remarks Contents 1 Introduction 2 Back-reaction from de Sitter thermal spectrum exactly, where highly symmetric constant curvature spacetime known as de S universe expands, and providesfilled exactly with enough more energy cosmological to constant. pro Thus, expansion can In the global coordinates, the dS metric is only a portion of the global dS manifold ( while in the cosmological or Friedmann-Robertson-Walker ( spacetime described by where the energy-momentum tensor JHEP04(2015)086 , ], ). µν ǫ 12 T – 5 . µν ] for some T 13 embedding = Λ(1 + in dS space- ρ µν T mal radiation from otion that the physical of one higher (spatial) rformed by the negative ] use the formalism of [ l distribution of particles 4 temperature) and will not – vacuum state. For recent ]. From the dS perspective, hat would otherwise occur rgy tensor the latter in Planck units, 1 hout spacetime. Interactions om the classical form of 19 sity and pressure, leading to orbed thermal particles comes e classical case: uanta of the gravitational field ctors register a thermal bath. , pic dynamics of spacetime. he renormalization of o contain excitations of the vacuum horizon. We are interested in an sure and energy density. Multi- ion of particles and a dS temper- 14 dS horizon temperature. ]. One can consider dS spacetime ndition that they are annihilated by the 14 is the dS radius. The ratio of the ansion in [ considered in this paper. See [ alculations (and related results for the energy R articular time slice. This suffices to demonstrate (i.e., that the equation of state describes pure – 2 – , where π In this note, we explore the consequences of the 2 µν g / 1 1 1, and a pressure insufficiently negative to support − ]. − R 12 – = 1 . The local energy density at late times in the expanding ǫ T that appears on the right hand side of the Einstein equations µν T is proportional to ]. Inertial dS observers, viewed from the perspective of the µν T 18 slightly larger than – 15 w ], Gibbons and Hawking note that detector absorption of ther ]. Observers who detect thermal particles will dispute the n 13 13 In [ The calculation of quantum contributions to the stress-ene The dS temperature is The fact that inertial observers in dS spacetime see a therma Quantum excitations, including, inevitably, gravitons (q Calculations of particle production due to cosmological exp 1 time is complex,progress depending on on this choices problem, of see regularization [ and phase of dS spacetime is therefore slightly larger than in th thermal energy densityhenceforth to parametrized cosmological by constant is of order vacuum energy or cosmological constant). This deviation fr will eventually equilibrate each particle species with the can also be understood in terms of the Unruh effect [ and results from the steady occurrence of such events throug dS expansion. necessarily appear in calculations of UV contributions to t violates dS symmetry. It is due to a subtle infrared effect (dS in which adefined vacuum on state a defined laterdestruction on slice. operators an associated The with early vacuum modeparticle states functions time production, of are slice a but defined p it is bymomentum is the shown tensor) not co t are clear sensitive torelevant to us discussion. the whether dS these temperature c effects this energy comes frompressure. work that Thus, otherwise it would(i.e., clearly have in been reduces the absence pe the of amount quantum of mechanics) in expansion dS t spacetime. as a timelikedimension hyperboloid [ embedded in a Minkowski spacetime the dS horizon leads, via back-reaction,averaged, to semiclassical shrinkage of the In dS spacetime, inertial observersature see [ a thermal distribut (renormalized) relatively well-understood dS temperature on the macrosco 2 Back-reaction from de Sitter thermal spectrum itself), modify at leastparticle slightly quantum the states relation typicallya between have value pres positive of energy den spacetime, are uniformly accelerated,From the and Unruh perspective, hence it their isfrom clear dete work that done the by energy the of accelerating abs force on the detector [ JHEP04(2015)086 is V (2.2) (2.3) (2.4) (2.5) (3.1) (2.1) 1 and the − 6= = 0 correspond w ξ . (The Bunch-Davies , al invariant. If the late as also been emphasized ) 3 and lume of the universe 2 he classical and quantum ǫ / stant (negative) curvature, to account quantum effects, R e to quantum effects should the simple equations above. − ount. Earlier work has found reaction of these quantum ef- he Friedmann equation ≡ , during the contracting phase hibit IR instabilities, since dS = 1 particles would lead to radical sing the equation of continuity ) (1 f the dS temperature. Expan- 4 way, new particles are produced. Λ · ξ S / ) 3 spectively. Thus , . − Ht ) t . = p 3 . 2 = 0 ǫ/ ) ) where Z + 3 ξ ˙ a a − ξ ǫ ρ − ) 3, acceleration is still positive. Therefore, an ( ξ 2 / H 3(1+ − ) = exp(3 1 – 3 – ǫ Y − ≈ 3 a − πG − − 4 Λ(1 V ∼ 2 t − (1 ǫ 3 X ], although the relation to our results is not clear. w < · + 3 (1 + log = ≈ − − ˙ 26 ǫ ǫ ¨ a a – 2 p 21 W 0 and classical + V 2 ρ > ≈ T V ) = 0, one can show that p ], or an approximate version of it, is an attractor.) + 20 ρ ( ˙ a a ].) Therefore, the early and late time geometries, taking in Conservation of energy implies that the resulting proper vo The resulting quantum spacetime also cannot be time-revers 1 + 3 ˙ slightly smaller than in the classical case: The above equation can be solved to give be approximately thatvacuum of [ a thermal bath at the dS temperature In fact, the situationAs is particles more produced by complicated earlier thanAfter expansion suggested are many redshifted by Hubble a timescales, the average energy density du expansion is no longer exactly exponential. However, from t and so to relativistic and non-relativistic thermal particles re 3 Anti-de Sitter spacetime Anti-de Sitter (AdS)satisfying spacetime the is constraint (for similarly simplicity a we restrict surface to of Ad con of global dS spacetime,departure the from resulting blue-shift the ofin vacuum thermal [ Einstein equations.cannot be (This the point same. h time thermal particle density were also found at early times spacetimes differ macroscopically, despitesions the about smallness the o originalis (classical) not dS an spacetime exactevidence solution should of once ex instabilities back-reaction in is dS taken [ into acc Thus, the spacetime whichfects results from is incorporating no back- longer one of constant curvature. At late times, t The corresponding pressure is accelerating expansion of dS spacetime is still expected. U it follows that as long as ρ JHEP04(2015)086 , ), 17 slice 3.3 (3.5) (3.6) (3.4) (3.2) (3.3) t . ) in coordinates ( t 2  Ω 2 he embedding space [ . d f quantum vacuum state. Ω acelike surface on which it 2 e cosmological coordinates, on and modification of the portant way: one can define 2 r such as the one appropriate t: it does not exhibit particle χ d φ φ − at fixed 2 dZ vacuum state, AdS is stable to ) ) ence of the existence of a global 2 e production due to acceleration. cos sin − only a portion of global AdS) dr θ 2 θ θ t/R nce, the fact that it is static implies t/R 1 . Therefore, the global vacuum state + sinh − dY sin sin cos T 2  cos( sin( 2 χ − χ χ χ 2 2 2 dχ r 2 R  /R /R φ φ ) 2 2 dX r r 1 + ) ) cosh ) sinh ) sinh ) sinh ] and consequent modification of the spacetime cos sin −  – 4 – t/R ( 2 θ 28 θ θ 1 + 1 + 2 t/R − t/R t/R t/R t/R , in the embedding space. In contrast, a fixed 2 p p cos sin sin dW 27 sin ∞ dt cos( sin( sin( sin( sin( R r r R r 2 +  R 2 R R R R R = = = = = 2 2 r − R = = = = = dT T Z Y X 2 W < T < T = Z Y dt X W 1 + 2 =  −∞ ds 2 = ds 2 ds Some AdS worldlines correspond to uniform acceleration in t ], again suggesting the presence of thermal (Unruh) radiati with metric in the cosmological coordinates corresponds to fixed can impose conditions on pastwhich lead and to future the quantum cancelation states ofStability in otherwise of expected th AdS particl depends onis choice defined. of vacuum state, and the sp do in fact lead to particle production [ timelike Killing vector. Otherto choices the “cosmological” of (non-static) AdS coordinates vacuum (covering state, Although this metric violates spatialthat translation there invaria is a quantum vacuumproduction state or that thermal is time-independen radiation.the For dS this instability special discussed choice above. of This result is a consequ geometry. The difference betweenThe global the vacuum two state is cases definedbut is on this a the covers spacelike the choice slice range o (e.g., global static coordinates in AdS, semiclassical geometry. However, AdS differs from dS in an im 18 in five-dimensional Minkowski space with metric with metric JHEP04(2015)086 ]. , , SPIRE ommons ]. IN ] [ ]. SPIRE SPIRE IN ]. [ , , IN (1972) 705 ][ (1968) 562 Attractor states and 28 es. 1. es. 2. SPIRE 21 gr-qc/0005102 IN supported by the Office [ t we identified is small, [ ns of the manifold. We ssibility that a quantum t indicates the instability ts, such as significant de- (1974) 176 ichigan State University. redited. f the spacetime geometry to ]. ]. Mottola, 87 gr-qc/9906120 [ (1974) 341 SPIRE SPIRE ]. Note that our effect specifically Phys. Rev. Lett. IN IN (2000) 124019 31 , Phys. Rev. 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