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hep-th/0412197 UCB-PTH-04/36 Cosmology and the S-matrix Raphael Bousso∗ Center for Theoretical Physics, Department of Physics, University of California, Berkeley, CA 94720, U.S.A.; Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A. Westudyconditionsfortheexistenceofasymptoticobservablesincosmology. Withtheexception of de Sitter space, the thermal properties of accelerating universes permit arbitrarily long observations, and guarantee the production of accessible states of arbitrarily large entropy. This suggests thatsome asymptoticobservables mayexist, despitethepresenceof an eventhorizon. Comparison with decelerating universes shows surprising similarities: Neither type suffers from the limitations encountered in de Sitter space, such as thermalization and boundedness of entropy. However, we argue that no realistic cosmology permits theglobal observations associated with an S-matrix. 5 0 I. INTRODUCTION has provided overwhelming evidence that no exact ob- 0 servables exist in eternal de Sitter space—at least, none 2 that correspond to experiments that can be performed One problem with quantum gravity is that we don’t n by an observer inside the universe. This is related to know what the theory should compute. In particle a the presence of a cosmologicalevent horizon in de Sitter physics,themostpreciseobservableistheS-matrix. But J space, which limits the accessible information and emits thisquantityseemsill-suitedtocosmology,wheretheob- 5 pernicious thermal radiation. server is not outside the system, initial states cannot be 2 In this paper we use similar semi-classical reasoning set up, and experiments cannot be arbitrarily repeated to characterize constraints on exact observables in other 2 to gain statistically significant results. v This ignorance is not especially unusual or embarrass- cosmologicalsolutions. Doestheuniversecontainregions 7 where fluctuations, including those of the gravitational ing. It is rarely clear at the outset what a theory should 9 field, become arbitrarily weak? Accurate measurements compute. For example, the insight that gravity is a the- 1 take a long time1, and they require devices with a large 2 oryofasymmetric,diffeomorphism-invarianttensorfield number of states. Does the universe last long enough, 1 in itself already constituted a significant part of the de- i.e.,doesitcontaingeodesicsofinfinitepropertimeinthe 4 velopment of general relativity. But once a theory is in future? Does the causally accessible region have enough 0 its final form, the observables should be apparent. / quantum states? According to entropy bounds [1, 2, 3], h If string theory is the correctquantum theory of grav- this translates into a minimum size for the region. By t ity, then whatever it computes presumably are the ob- - asking whether such requirements are met, one can in- p servables. But string theory—perhaps because it is not vestigatewhetherexactquantummechanicalobservables e in its final form—has so far sidestepped the problem of existin a givencosmology,without knowledgeof the full h cosmologicalobservables. It defines quantum gravity for : theory. v certainclassesofgeometriescharacterizedbyasymptotic Byanobservablewemeanaquantityorlimitofquan- i conditions, such as asymptotically flat or Anti-de Sitter X tities that can actually be measured by an observer in- spacetimes. In these geometries an S-matrix happens to r side the universe, without violating laws of physics such a make sense, and string theory computes its matrix ele- as causality or entropy bounds. For example, it may ments. (In the caseofAdS, itcomputes boundarycorre- turn out that an S-matrix for de Sitter space can be for- lators, which are a close analogue of the S-matrix.) mally computed as a useful “meta-observable” [4], from However,wehaveyettolearnhowtoapplystringthe- which predictions for true, operationally defined observ- ory to cosmology or to an observer inside a black hole, ablescanbeextractedbyfurtherprocessing. Therestric- with the same level of rigor as in Anti-de Sitter space. tionsderivedbelowapplyonlytothelatter,operationally Hence, it would be premature to conclude that the S- meaningful quantities. matrix will remainthe only well-definedobject. It is too Our conclusions for different classes of universes vary earlyto know what, if anything,string theory has to say in their details, but they do strike two common chords. about cosmologicalobservables. First: Aside fromde Sitter spaceandthe obviouscaseof Fortunately, classical and quantum properties of cos- crunching universes, our necessary conditions for exact mologicalsolutionsimposesignificantconstraintsonpos- observablesaresatisfiedinalltheothercasesconsidered. sibleobservables,andmayevenhintatsomeofthe prin- Surprisingly, this includes universes with a cosmological ciplesonwhichatheorycomputingthemmustbebased. event horizon. Second: Observables that invoke a global De Sitter space is a casein point. Semi-classicalanalysis 1 Forthis reason,weshallusetheterms“asymptotic observable” ∗Electronicaddress: [email protected] and“exactobservable”interchangeably. 2 out-state (such as the S-matrix) do not seem to describe sible resolution by a discretuum of meta-stable vacua in any experiment in cosmology. We find that the informa- string theory [17], populated by cosmological dynamics, tioncontentofagenericout-stateiscausallyinaccessible makes it all the more urgent to understand string the- even in a universe with a null infinity and no horizon. ory observables in cosmology. Explicit constructions of Wepresentanumberofintermediateresultsthatareof de Sitter vacua have been proposed (e.g., Ref. [18]), and interest in their own right: an analysis of the thermody- sophisticated counting arguments (e.g., Refs. [19, 20]) namicsandthefluctuationspectrumofquintessenceuni- broadly confirm the original estimates of the vast num- verses;anargumentdemonstratingthatanopenuniverse ber of such vacua. The present discussion does not ad- resides inside the Farhi-Guth solution; and entropic rea- dressspecificallythe developmentofatheoreticalframe- soningsuggestingthat the globalstate of anon-compact work[6,21,22]describingthis“landscape”[23]. Butthe universe is not accessible to experiment, independently question of observables is a part of this challenge,so our of event horizons. results may have some implications in this context. This paper doesnottackle the actualdefinitionofany Outline The paper is structured as follows. In the asymptotic cosmological observables (see Ref. [5, 6] for first sections we mainly consider spatially flat FRW uni- recent approaches). Even that challenge, in turn, will verses with fixed equation of state w = p/ρ. They are only be an intermediate goal. In our view, asymptotic especiallysimpleandsufficeforderivingourmainresults. observables are at best a crutch. The description of a Moreover,their late time behavior is a good approxima- real experiment involving gravity requires well-defined tion to other classes of cosmologies, including some we (but necessarily imprecise?) local observables. This is discuss at the end of the paper. a famously difficult problem in the presence of gravity. Sec. II, aside from a review of the flat FRW solutions Itis further complicated,but perhapsalsohelpfully con- andtheircausalstructure,containsourmainobservation strained, by the counter-intuitive holographic restriction about decelerating universes (w > 1/3): All observers, − on bulk degrees of freedom [1, 2, 3, 7, 8, 9, 10, 11]. This at all times, lack information about infinitely large re- task will have to be confronted eventually. gionsofthe universe,eventhoughthereis no eventhori- Relation to other work For a review of the difficul- zon. If suchregionscontainany non-redundantinforma- ties with physics in de Sitter space, see, e.g., Ref. [12]. tion, then the globalout-state computed by an S-matrix A broad discussion of the problem of observables in cos- cannot be measured. mologies with a non-positive cosmological constant was Next, we turn to eternally accelerating universes, given by Banks and Fischler [5], who noted that in a de Sitter space (w = 1) and “Q-space” ( 1 < w < non-compact universe, an S-matrix description must re- 1/3) [24]2, which h−ave a cosmological −event hori- − strict to states with a finite number of extra particles, zon [13, 14]. We show in Sec. III that Q-space exhibits andthatthosestatesareveryspecial. Whilethisrestric- thermodynamic properties similar to those of the de Sit- tion is necessary,it is not sufficient: as shownbelow, the terhorizon. ThehorizonradiusinQ-spacegrowslinearly unobserved region can have infinite entropy even if no with time, and consequently the temperature slowly de- particles are added, because of the internal states of the creases. Wefindthatthisbehaviorisconsistentwiththe matter already present. first law of thermodynamics: the temperature and entropy respond appropriately to the flux of quintessence Our analysisofthe thermodynamics ofQ-spacebuilds stress-energy across the horizon. on Refs. [13, 14], who derived its global structure and Sec. IV contains our main results for accelerating uni- pointed out that its event horizon obstructs the defini- verses. They support the existence of asymptotic ob- tion of an S-matrix. We do not question this conclusion; servables in Q-space. We study specific aspects of the indeed, we find that the difficulties with an S-matrix are thermal spectrum emitted by the horizon. The time- quite general in cosmology. We do argue, however, that dependenceofthetemperatureleadstosignificantdiffer- otherasymptoticobservablesmayexistinQ-space. This ences between de Sitter space and Q-space. In the semi- possibility wasfirstraisedby Witten [4], who notedthat classical theory, an infinite number of Hawking quanta observers will not be thermalized in Q-space. The exis- areproduced(andre-absorbed)bythehorizon. IndeSit- tence of accessible high entropy states was not demon- terspace,thetotalenergythusemitteddiverges,whereas strated there. in Q-space the energy per quantum decreases rapidly The problem of defining cosmological observables is enoughtorenderthetotalenergyfinite. Hence,observers closelyrelatedtothechallengeofdescribingphysicsfrom in Q-space will not be thermalized. the point of view of an observerfalling into a black hole. We ask whether observers will be destroyed by rare In both cases, some type of local observables will even- massive fluctuations, such as black holes. We consider tually be required, but in both cases, one can hope to make progressby asking how some of the information in the gravity-dominatedregionmay be encodedin asymptotic data [15, 16]. 2 For asubset of this range, quintessence has been proposed as a The recent discovery that the universe is accelerating modelofdarkenergy[25]. Herewestudythesesolutionssimply has turned the cosmological constant problem into the asinstructiveexamplestounderstandconditionsforasymptotic (worse)problemofsmallpositivevacuumenergy. Itspos- observables. 3 objects of fixed energy and compute the rate at which + I they are emitted, according to standard statistical me- chanics. If the energy is much larger than the temper- I+ ature, the rate will be miniscule. However, in de Sitter spacetherateisconstant,soallfluctuationsthatarenot =0 i0 =0 completely forbidden will occur. This guarantees that r r any observer who survives the thermal radiation long t=const _ enoughwilleventuallybeswallowedbyalargeblackhole I emitted by the horizon. In Q-space, the rate of such vi- big bang i olent processes decreases exponentially with time. The 0 integrated probability is therefore finite and can be ex- ceedingly small. It follows that experiments in Q-space can last for an arbitrarily long time. FIG. 1: Conformal diagrams of Minkowski space (left) and But the classical supply of matter in Q-space is a decelerating flat FRW universe (right). In the FRW case, bounded,seeminglyrulingoutexactmeasurements. Yet, anyinfinitesimalneighborhoodofspatialinfinity(circle)con- we show in Sec. IVE that arbitrarily complex matter tainsan infiniteamount ofmatterand potentially aninfinite amount of information, whereas the observer’s causal past is configurations are quantum mechanically produced by a finiteregion. the Q-space horizon: The rate for a fluctuation of a given fixed entropy—no matter how large—is constant and non-vanishing at late times. This contrasts pleas- Aquickwaytoobtainitscausalstructureistotransform antly with de Sitter space, where the entropy is strictly to conformal time, defined by dη = dt/a(t). This shows bounded by the inverse of the (fixed) cosmological con- that the metric is conformal to Minkowsi space: ds2 = stant. a(η)2ds˜2, where InSec.Vwedrawconclusionsonthenatureofobserv- ables in the universes we have studied. In particular, we ds˜2 = dη2+dr2+r2dΩ2 . (2) argue that no direct analogue of an S-matrix can be de- − fined in any flat FRW universe unless the set of allowed Hence, the conformal diagram is a subset of the states is severely restricted. Minkowski space Penrose diagram (Fig. 1), selected by InSec.VIweextendthediscussiontoopenandclosed the range of η. The existence of horizons is determined FRW solutions. We also study composite universes that as follows. feature an asymptotically flat region on the far side of a Ifandonlyifη isboundedfromabove(η η < max black hole. We show that the Farhi-Guth solution can as t ), then an observer at r = 0 is su→rrounded b∞y be regarded as an example of this setup in which the a fut→ure∞event horizon.3 The horizon is located at r = black hole resides inside an open universe produced by η η. Light-raysoriginatingbeyondthishypersurface max the decay of meta-stable de Sitter space. Because an never−reachthe observer. Similarly, if η is bounded from open universe has infinite entropy, one would not expect below, then there exists a past horizon.4 Events beyond generic microstates to be represented on the far side of this horizon cannot be influenced by the observer. the black hole, or on the asymptotic boundary. The dynamical evolution of the scale factor and the matter density is determined by the equations II. SPATIALLY FLAT UNIVERSES a˙2 8πρ = , (3) a2 3 In this section we review various classical proper- a¨ 4π = (ρ+3p) . (4) ties of flat FRW universes—in particular, the results of a − 3 Ref. [13, 14] on the causal structure of accelerating cos- We will assume that the energy density, ρ, and pressure, mologies. We ask how much matter and information is p, obey the equation of state causally accessible to an observer in the classical evolution. We find that this amount is finite in accelerating p=wρ (5) universesandunboundedindeceleratinguniverses. How- ever, even in the latter case, no more than an infinitely small fraction of the matter is ever observable. 3 By homogeneity, all comoving observers are equivalent, so we consideranobserveratr=0. Anynon-comovingobserverwhose A. Metric and causal structure spatialpositionremainsfiniteatlatetimeshasthesamehorizon as a comoving observer located at the same asymptotic spatial position. ThemetricofaspatiallyflatFRWuniverseisgivenby 4 This assumes that the FRW solution in question is past inex- tendible. Hencethisanalysisdoesnotapplytotheflatslicingof ds2 = dt2+a(t)2(dr2+r2dΩ2) . (1) deSitter space. − 4 withconstantw. Itwillbemoreconvenienttoworkwith I+ I+ the parameter f ut Thus one obtains a faǫm=ily23(owf s+ol1u)ti.ons parameterized(b6y) r=0 horizon bang r=0 ure horizon r=0 ǫ, big orizon a(t) = t1/ǫ , (7) past h 3 ρ(t) = . (8) 8πǫ2t2 I− except for ǫ = 0, which corresponds to a cosmological constant Λ. In that case a solution is given by a(t) = FIG. 2: Conformal diagrams of flat Q-space [13, 14] (left) exp[(Λ/3)1/2t], ρ=Λ/8π. and de Sitter space (right). Past and future cosmological event horizons are shown. The area of the de Sitter horizon We assume the dominant energy condition, which re- is constant, whereas the area of the Q-space horizon grows stricts ǫ to the range 0 ǫ 3. From Eq. (4) we can ≤ ≤ without bound at late times [13]. The Q-space initial singu- see directly that for ǫ>1, the expansion of the universe larity is not really null, since the curvature already becomes decelerates: a¨ < 0. This includes the familiar cases of Planckian on a nearby spacelike slice (see Fig. 3). matter domination (ǫ = 3/2) and radiation domination (ǫ = 2). For ǫ < 1, on the other hand, the scale factor grows increasingly rapidly: a¨ > 0. The degenerate case defined as the causal past of the future endpoint of the ǫ=1 will not be considered here. observer’sworldline,intersectedwiththecausalfutureof As discussed more generally above, we transform to thepastendpoint. (Notethatthelatteriscrucial: events conformal time, lying in the observer’s past but outside the bottom cone ǫ ǫ−1 cannot be probed directly and may not send any signals η = t ǫ , (9) ǫ 1 in the observer’s direction. If a signal is sent, then what − information can be gleaned about the event is precisely torevealthe causalstructure. Fordeceleratinguniverses what passes through the bottom cone.) (ǫ > 1), this expression shows that conformal time is How much matter enters an observer’s causal dia- boundedbelowbutunboundedabove. Thereisnofuture mond? We restrict for now to the classical evolution of event horizon. The conformal diagram is given by the the cosmologicalfluid, and postpone the inclusion of the upper half (η >0) of the Penrose diagram of Minkowski thermal properties of the horizon until Sec. IV. space (Fig. 1). de Sitter space In eternalde Sitter space (ǫ=0), the Foracceleratinguniverseswith0<ǫ<1,thesituation causal diamond is the region limited by both the past isreversed. Conformaltimerangesfrom to0,andso −∞ and future event horizons. The maximum amount of is bounded above but not below. Hence, the conformal matter that can enter is the largest black hole allowed diagramisthelowerhalfoftheMinkowskiwedge(Fig.2). in asymptotically de Sitter space, the Nariai black hole. There is a future eventhorizonat r+η =0, whose area, Its entropy is one third that of the empty de Sitter hori- A , grows quadratically with time. The proper horizon E area-radius,R =(A /4π)1/2, is given by zon. But to arrange for matter to enter, one must either E E include thermaleffects, orset upappropriateinitial con- ǫ R = t . (10) ditions in the infinite past. E −ǫ 1 It is more interesting to consider a universe such as − ours, which contains an era of matter- or radiation- In the case of de Sitter space, ǫ = 0, the metric (1) domination before the cosmological constant takes over. is geodesically incomplete and extendible. The maximal The Penrose diagram for this type of solution is shown extension has closed spatial slices and is given by in Fig. 3. In that case, the bottom cone of the causal 3/Λ diamond is the future light-cone of a point at the big ds2 = ( dη2+dχ2+sin2χdΩ2) . (11) bang (usually called the particle horizon). Its structure sin2η − depends on the details of the matter content. But as Hence, the conformal diagram is a square (Fig. 2), and longasthe universeisasymptoticallyde Sitter inthe fu- de Sitter space has both past and future event horizons ture, the amount of information inside the causal patch of constant radius 3/Λ. is bounded by the entropy at late times, which is that of p empty de Sitter space. To summarize,anobserverin asymptoticallyde Sitter B. Classical observable matter content space can access at most an entropy of order the inverse cosmological constant [27]. This conclusion is indepen- The maximum spacetime region probed by an exper- dent of whether thermal effects are included, and may iment is called the causal diamond [26]. It is generally extendtoa largerclassofuniverseswithpositive cosmo- 5 + I quite large: about 10123 in Planck units. Still, like in de Sitter space, and unlike the decelerating universes, only T a finite amountof matter andinformationeverenter the causaldiamondbyconventionalevolution[31,32,33]. To t=const show that Q-space exhibits unbounded complexity, one 0 = needs to include thermal fluctuations (Sec. IV). r Decelerating FRW In a decelerating universe, the B bottom cone extends all the way to future infinity and big bang has infinite maximal area. There is no bound on the entropy that can enter. Indeed any comoving particle will enter it sooner or later. Thus, in decelerating universes FIG. 3: This conformal diagram can be interpreted in three any observer has access to arbitrarily large amounts of ways. ItrepresentspureQ-space,withaspacelikesingularity matter and entropy. reflecting a Planck scale cutoff of the classical metric (see However, there is an important order-of-limits issue. Fig. 2). It also corresponds to a big bang universe initially Letusaskhowmuchoftheuniverseisseenbyanobserver dominated by matter or radiation, which asymptotes to Q- space or de Sitter space at late times.—The causal diamond at some finite time t. One finds that the sphere at the of the observer at r =0 is shown. The bottom cone (B) has edge of the causal diamond has area finite maximal area, indicating that only a finite amount of entropy enters the observable region by classical evolution. ǫt 2 A =π . (13) In asymptotically Q-space, however, the top cone (T) allows edge (cid:18)ǫ 1(cid:19) arbitrarily large entropy. Indeed, an unbounded number of − statescanbeaccessedbyquantumfluctuationsofthehorizon This area is a bound on the amount of entropy that has (Sec. 4.4). enteredtheregionobservedbythetimet. Notethatthis doesnotdivergeatfinitet. But finitet isallanobserver can ever attain. logical constant [28].5 Hence, the number of accessible degrees of freedom is, Q-space In an accelerating universe with ǫ > 0, the atalltimes, aninfinitely smallfractionofthe totalnum- largestpossible causaldiamond is the intersection of the ber of degrees of freedom in the universe. This is shown past of the point t = t , r = 0 with the future light- late in Fig. 1; only the past light-cone is shown (rather than cone of the point t = 1, r = 0, in the limit t late → thestrongerrestrictiontothecausaldiamond),sincethis (Fig. 3). (We follow Ref. [30] in excising the high ∞ already suffices to illustrate the problem. In Sec. V we curvature region prior to the Planck time. This replaces will argue that this limitation is an important criterion the null singularity with a more standard, spacelike big distinguishing the observations made in a decelerating bang.) The lowerconeis againthe particle horizon. The FRW universe from the S-matrix of asymptotically flat uppercone,inthelimittaken,isthefutureeventhorizon space. (and so is a cone only conformally). The amount of information entering the causal diamond from the past, S , is bounded by the maximal in III. TEMPERATURE AND ENTROPY OF area of the lower cone [3, 8]. One thus finds that ACCELERATING UNIVERSES 2 ǫ Sin π 21−ǫǫ . (12) In this section we obtain the basic thermodynamic ≤ (cid:18)1 ǫ (cid:19) propertiesofQ-space: entropy,energy,andtemperature. − Wedemonstratethattheysatisfythefirstlawofthermo- Unless ǫ is very close to 1, this is at most of order unity, dynamics. We begin by reviewing the thermodynamics indicating that virtually no information enters the ob- of de Sitter space. server’s causal diamond. Note that this result applies strictly to an acceleratingǫ>0 fluid with no other matter present. A. Thermodynamics in de Sitter space The conclusion changes somewhat if other types of matterdominate atearlytimes. Ifquintessencewerethe De Sitter space has an event horizon of radius R = source of the vacuum energy in our universe, for exam- 0 3/Λ. ItsareaisA=4πR2 =12π/Λ. ItisalsoaKilling ple, ourparticlehorizonwouldintersectourfuture event 0 hporizonwithsurfacegravityκ=R−1,withrespecttothe horizon about now (Fig. 3). Its maximal area would be 0 the usual timelike Killing vector field normalized at the origin. Consider an object of mass M in an otherwise empty 5 If the requirement of a future asymptotic region dominated by asymptoticallyde Sitter universe. Inthe presenceofthis object,thecosmologicalhorizonwillbesmallerthanthat the vacuum energy is dropped, examples with greater entropy areknowninmorethanfourspacetimedimensions[29]. of empty de Sitter space. One way to estimate its size is 6 to model the object as a small black hole. For this case B. Thermodynamics in Q-space an exact solution is known: the Schwarzschild-de Sitter black holes, with metric Slow-roll inflation can be thought of as a de Sitter- dr2 likeerawithslowlydecreasingeffectivecosmologicalcon- ds2 = V(r)dt2+ +r2dΩ2 , (14) stant. It is well-known that the apparent cosmological − V(r) 2 horizon during inflation has thermodynamic properties where akin to those of de Sitter space [35]. Indeed, this temperature is the origin of density fluctuations and so is 2M Λ V(r)=1 r2 . (15) responsible for all structure in the universe. − r − 3 Onewouldexpectsimilarconsiderationstoapplytoan For 0 < M < 1/(3√Λ), V(r) has two positive roots. eternally accelerating universe with w sufficiently close (The maximal case M = 1/(3√Λ) is known as the Nar- to 1. In this case,the universe is also locally similar to − iai solution; larger black holes do not exist in de Sitter deSitterspace,withslowlydecreasingvacuumenergy. In space.) The smaller root is the black hole horizon; it fact,sinceaw > 1fluidcanbemodeledbyascalarfield − obeys R 2M for small M. The larger root is the withcustom-designedpotential[24],itcanbethoughtof B ≈ cosmologicalevent horizon. For small M, it obeys as a special case of slow-roll inflation. Thus, horizon thermodynamics should apply in Q-space. R2 R2 2R M , (16) C ≈ 0− 0 We will now verify this expectation. Our arguments will be rigorous only for w very close to a cosmological and it decreases monotonically over the whole range of constant: M. Now suppose that a black hole, or any other object 0<w+1 1 , (20) of small mass M, falls across the cosmological horizon, ≪ restoring the observer’s patch to empty de Sitter space. though we expect our results to be qualitatively correct (This can be achieved simply by the observer moving at least in the range 1 < w < 2/3. The parameter awayfromtheobject.) ByEq.(16),thisprocessincreases ǫ= 3(w+1)willbesm−allandposit−ivefortheaccelerating thecosmologicalhorizonareaby∆A= 8πR M. Thus, 2 − 0 universes studied here. However, all classical formulas the cosmological horizon satisfies the usual first law of below are exact in ǫ. horizon dynamics [34]: The radius of the event horizonwas given in Eq. (10). κdA We will also be interested in the apparent horizon.7 In dE = , (17) any FRW universe, its proper radius is directly related − 8π to the energy density: wherewehavedefineddE tobethechangeinthemassof thematterpresentontheobserver’ssideofthehorizon.6 1/2 3 As in the case black holes, this classical relation betrays R = . (21) A (cid:18)8πρ(cid:19) the semiclassicalthermodynamicpropertiesexhibitedby thedeSitterhorizon. Analysisofquantumfieldtheoryin For a flat universe, the apparent horizon radius is thus adeSitterbackground[35,36]showsthatafreelyfalling equal to the Hubble scale, t =a/a˙, and is given by H detector will measure a temperature proportional to the surface gravity R =t =ǫt . (22) A H κ T = . (18) The two horizons satisfy the following key properties. 2π First, they are approximately equal in the regime we Moreover, in order to avoid a decrease of observable en- study. More precisely, the apparent horizon is smaller tropy in the above process, it is natural to propose that than the event horizon by a fixed ratio close to unity: the horizon area represents a true contribution to the totalentropy,asoriginallysuggestedforblackholes[37]: R A =1 ǫ , (23) A RE − S = . (19) 4 Second, neither horizon changes significantly over one Consistencyrequiresthatthesequantitiessatisfythefirst Hubble time: law of thermodynamics, which is ensured by Eq. (17). t R˙ H X =ǫ 1; X=A,E . (24) R ≪ X 6 Inblackholemechanics[34],dE thuscorrespondstothechange in mass of matter remaining outside the black hole, which is minusthechangeofblackholemass,andhenceisnegativewhen 7 Oneachconstanttimeslice,theapparenthorizonofanobserver matterisaddedtotheblackhole. HenceEq.(17)takesthesame at r =0is the sphere whose orthogonal ingoing future-directed formforblackholesandfordeSitterspace. light-rayshavevanishingexpansion. 7 Hence, a thermodynamic description of the horizon will wavelengthoforderR ),perHubbletimeR . IndeSit- A A be approximately valid, and it will not matter much terspace,R =R isaconstant,andaninfinitenumber A 0 whether we use the apparent or the event horizon for of quanta are emitted in total. (No observer will last this purpose. long enough to notice more than a finite number, how- We will workwith the apparenthorizon,since this ap- ever, as we shall see shortly.) In Q-space, we see from proach is more general. (For example, in slow-roll in- Eq.(22) thatR growslinearlywithtime. However,the A flation, there may be no event horizon, but one would integrated number of quanta still diverges (though only still like to describe the approximate thermal state dur- logarithmically, not linearly as in the de Sitter case): ing inflation.) Hence, an observer at r = 0 will perceive athermalheatbathwithslowlytime-dependenttemper- dt logt . (30) ature Z R ∼ →∞ A 1 T = (25) What is the total energy radiated? In de Sitter space, 2πR A the typical energy of each quantum is fixed, so the radi- and will ascribe to the apparent horizon a Bekenstein- ated power integrates to infinite energy, suggesting that Hawking entropy it will erode any physical structure. Any observer in de Sitter space will be thermalized by the steady stream of S =πR2 . (26) radiation from the horizon. A In Q-space,the rate of emission of quanta and the en- Asaconsistencycheck,letusverifythatthefirstlawof ergy per quantum eachgo like R−1. Hence, the radiated thermodynamicsissatisfied. WefollowRef.[38],wherea A powerdropsoff like the inversesquare oftime, andit in- similar check was performed for slow-rollinflation. Con- tegrates to a finite total radiated energy. Quantitatively sider an infinitesimal time interval dt. The amount of the total energy radiated after the time t=t is 0 energy crossing the horizon during this time is obtained byintegratingthefluxofthestresstensoracrossthesur- ∞ dt 1 ∞ dR 1 A face,contractedwiththe(approximate)generatorsofthe E = = . (31) ≈Z R2 ǫ Z R2 ǫR (t ) horizon, the future directed ingoing null vector field ka: t0 A RA(t0) A A 0 −dE =4πRA2 Tabkakbdt=4πRA2 ρ(1+w)dt=ǫd(t2.7) eFrogryebxeagmapnleto, tdaokminignatt0ettohebeevtohleuttiiomneofatouwrhuicnhivderasrek,tehne- In the last equality we have used Eq. (21). By Eqs. (26) totalenergy(tobe)radiatedbythecosmologicalhorizon and (22), the horizon entropy increases by would be comparable to that of a single quantum with wavelength of order the present Hubble scale. Thus, the dS =(2πR )R˙ dt=(2πR )ǫdt . (28) Q-spacehorizonfallsfarshortofthermalizingthematter A A A it contains, in stark contrast with the de Sitter horizon. The term in parentheses is the inverse temperature, Eq. (25). Thus we confirm the first law, B. Large energy fluctuations in de Sitter space dE =T dS . (29) − What is the probability for a state of specified energy IV. THERMAL FLUCTUATIONS IN Etoberadiatedbythehorizon? Asidefromaslowdeath ACCELERATING UNIVERSES by thermalization, observers in de Sitter space also face thethreatofcollisionswithobjectsofgreaterenergythan Both in Q-space and in de Sitter space, the thermal the typical Hawking quanta. Though exponentially sup- horizonproducesfluctuations—butasweshallseeinthis pressed, such objects will eventually appear as rare fluc- section, their implications are quite different in the two tuationsinthethermalspectrum. Aparticularlydestruc- cases. Fluctuations in de Sitter space are fatal to exper- tiveexampleisthatofanearlymaximalSchwarzschild-de iments. We show, however, that in Q-space fluctuations Sitter black hole, which will swallow the observer. arebenign: entropicenoughtoproducecomplexsystems, For small energy, E R , the problem is approxi- 0 ≪ but notenergetic enoughto destroyanobservermeasur- mately equivalent to that of a hot cavity at temperature ing them. T =(2πR )−1. The horizon provides the heat bath. For 0 largerenergies,gravitationalbackreactioncanchangethe volume of the cavity and the temperature of the horizon A. Typical quanta by factors of order unity. In particular,there is a largest possible energy, corresponding to a black hole that just We begin by asking: What is the total number of fits inside the cosmological horizon. We will take these quanta emitted by the horizon? For de Sitter as for Q- finitenesseffectsintoaccountbutwebeginbyconsidering space, the expected rate is one quantum (typically with small energies. 8 The probability to find in the cavity a particular state will have radius R 2E. For larger energy, however, B ≈ i , of energy E , is given by the backreaction on the cosmological horizon is signifi- i | i cant,andthe definitionofenergyitself becomesambigu- 1 1 P(i )= e−Ei/T = exp( 2πE R ) . (32) ous. We will simply use the black hole radius, RB, as an i 0 | i Q Q − energy-like parameter and abandon the estimate (34) in favor of a direct computation of the rate of black hole Foracavitywithradiusoforderthe inversetemperature nucleation [40, 41]: (andareasonablenumberofspecies),wecanneglectfac- tors of the partition function, P B =exp[S (R ) S ] . (37) SdS B dS R − Q= exp( E /T) , (33) 0 i − X|ii S (R ) is the total entropy of a Schwarzschild-de Sit- SdS B ter geometry with a black hole of radius R . It is given B sinceitisdominatedbyafewstatesofenergyT andsois by a quarter of the sum of the black hole and the cos- of order unity. Note that the probability P(i ) is really a rate per time interval of order the interact|iion time of mological horizon area. SdS = πR02 is the entropy of the empty de Sitter solution with the same cosmologi- the heat bath, R . 0 cal constant. Einstein’s equation implies for any static The probability to find an arbitrary state with energy spherically symmetric vacuum solution [41]: E is larger than (32) by a factor of the number of such states, N(E)=eS(E): R2 +R2 +R R =R2 . (38) B C B C 0 P E =exp[S(E) 2πER0] . (34) Here,RCistheradiusofthecosmologicalhorizon. Hence R − 0 the creation rate (37) is simply AdeSitterspacevariantoftheBekensteinbound[7,39], P B the D-bound [26], guarantees that the exponent will be =exp[ πR R ] . (39) B C R − non-positive. For highenergies comparedto the thermal 0 energy R−1, the second term in the exponent is large. 0 The exponent agrees well with Eq. (36) in a large region Thus the rate of the corresponding fluctuations will be of overlap: For R R , one can take R 2E. More- B 0 B exponentially suppressed, unless the entropy enhance- ≪ ≈ over, the contribution S(E) from the black hole entropy ment factor eS(E) nearly cancels the suppression term, is subleading. leaving an exponent of order unity. We now estimate Already the smallest black holes, with R 1 and B S(E) to argue that this is not the case. ≈ P exp( πR ),areexponentially suppressedandthus B 0 In a quantum field theory coupled to gravity (a de- ∼ − very unlikely to arise in the thermal spectrum. At scriptionwhichshouldbelocallyvalidatlatetimes),the fixedcosmologicalconstant,onefinds fromEq.(38)that objects of highest entropy for a given energy E are ei- R (R ) is a monotonicallydecreasing function: the cos- C B ther a black hole, or a radiation gas with temperature mologicalhorizongetssmallerforlargerblackholes. But τ and radius χ such that E χ3τ4. (We assume that ≈ RBRC(RB) grows monotonically, so larger black holes the number of species with mass less than τ is not sig- are more and more unlikely. The biggest black hole al- nificant, i.e., less than 104). The entropy of the black lowedbyEq.(38)hasR =R /√3andissuppressedby holeisoforderE2. Theentropyofthethermalradiation exp( πR2/3). B 0 is χ3τ3 (Eχ)3/4. This is maximized by choosing the − 0 ≈ However,nomatterhowsmalltherateofsuchfluctua- radius occupied by the gas as large as possible, χ = R . 0 tions,indeSitterspaceitisindependentoftime. Hence, Thus the maximal entropy of thermal radiation is even the most unlikely fluctuation will eventually occur, S (ER )3/4 . (35) on a timescale of order R0/P. therm 0 ≈ Whether this is larger than the black hole entropy E2 depends on the size of the horizon. C. Large energy fluctuations in Q-space For R−1 < E < R3/5, thermal radiation wins. Hence, 0 0 in this regim∼e w∼e obtain the following upper bound for Inaw> 1acceleratinguniverse,Eqs.(32),(34),and − the logarithm of the production rate: (37) still describe the probability for the corresponding fluctuations, if we substitute R for R . But as we shall A 0 S(E) 2πER 2π(1 δ)ER (36) see now, violent events of a specified energy E are not 0 0 − ≤− − likely to ever occur at late times, no matter how long for some small number δ (ER0)−1/4 1. At the level one waits. ≈ ≪ ofaccuracyrequiredbelow,theS(E)termcanclearlybe Firstconsiderafluctuationoflessthanthe Plancken- dropped altogether (δ 0). ergy, E 1. Its rate is given by Eq. (34). Let t be For R3/5 < E < R≈, a black hole dominates the en- a sufficie≤ntly late time so that the temperature of0the 0 0 semble. Nea∼r the∼lower end of this range, the black hole horizonhas become small comparedto the energy of the 9 fluctuation: R (t ) = ǫt E−1. What is the total (Here α is a numerical coefficient involving Stefan’s con- A 0 0 ≫ probability (E) for the fluctuation of energy E to oc- stantandtheeffectivenumberofspecieswithmassbelow P cur after the time t ? τ;forsmallspeciesnumber,itwillbeontheorderof10.) 0 ∞ P (t) For S >R06/5 the lightest object is a black hole with ra- (E) = dt E (40) dius R∼=(S/π)1/2. Thus Eq. (39) implies P Z R (t) B t0 A 1 ∞ P dt exp[S(E) 2πER (t)]. (41) S =exp[ (πS)1/2R ] . (45) ≤ R Z − A R − C 0 t0 0 1 ∞ dR exp[ (2π δ)ER ] (42) From Eq. (38) it follows that the suppression becomes A A ≤ ǫR Z − − 0 RA(t0) stronger if S is increased at fixed cosmological constant. 1 Since these rates are constant, the situation is similar to = exp[ (2π δ)ER (t ()4].3) (2π δ)ǫER (t ) − − A 0 the caseoflarge-energyfluctuations: All events that can A 0 − occur in de Sitter space, will occur. (Here δ [ERA(t0)]−1/4.) Since ERA(t0) 1,the total However,thereisanabsoluteentropyboundindeSit- ≈ ≫ probability is exponentially small. ter space [27, 28]. There are no states with entropy Fluctuations greater than the Planck energy cannot greaterthanthatofthe horizonofemptyde Sitter,πR2. 0 be considered until the horizon has grown large enough Thisboundreferstothecombinedentropyofthe cosmo- to contain a black hole of energy E. During the period logical horizon and of the matter it encloses. If we ask E/ǫ < t < E5/3/ǫ, a fluctuation of energy E is most about the entropy only of systems contained within the likely∼to o∼ccur in the form of a black hole. But this horizon, the limit is more stringent: there can be no ob- power-law time interval is insufficient to overcome the jects with entropy greater than that of the Nariai black exponentialsuppressionin Eq.(37), so the fluctuation is hole (πR2/3). Fluctuations with greater entropy cannot 0 extremely unlikely to occur during this period. There- occur; their probability is exactly zero. This fundamen- after, the thermal ensemble begins to dominate, and the tally limits the complexity and accuracy of experiments fluctuation rate is given by Eq. (34). Then the analy- in de Sitter space. sis of the previous paragraph applies, with t = E5/3/ǫ. 0 Since ER (t ) is again large, the integrated probability A 0 remains negligible for all times. E. Large entropy fluctuations in Q-space We conclude that large energy fluctuations inevitably occur in de Sitter space (if only after an exponentially InQ-spacethehorizongrowslinearly. Afluctuationof largetime),guaranteeingthedestructionofanyobserver. entropy S first becomes possible (in the form of a maxi- In Q-space, however, the temperature falls monotoni- malblackhole)whenthehorizonreachesaradiusoforder cally. AfteritdropsbelowagivenenergyE,fluctuations S1/2. Thus, the rate begins at e−S, the suppression of a of that energy become virtually impossible. Nariai black hole. Thereafter the required black hole radius remains constant. But the corresponding radius of the cosmological horizon, R , increases as the effective D. Large entropy fluctuations in de Sitter space C cosmological constant decreases, according to Eq. (45). Hence the fluctuation becomes more and more unlikely. Whatisthe probabilityforastate ofspecifiedentropy Eventually the horizon radius satisfies S to be radiated by the de Sitter horizon? We have seen in the previous subsection that the probability of R >S5/6 . (46) A fluctuations is mainly determined by their energy; the ∼ entropy factor in Eq. (34) turned out to be negligible. In this regime, the lightest object of entropy S is an or- Hence the question is, what is the lightest object with dinary thermal state. For all remaining time, the rate of entropy S? such a fluctuation is given by Eq. (44). ForS <R6/5thelightestobjectisathermalstatewith What matters about this asymptotic rate is not its 0 temperat∼ure τ S1/3/R and energy E S4/3/R .8 It (miniscule) value, but that it is both constant and non- 0 0 is radiated with≈a probability derived from≈ Eq. (34): zero,howeverlargeonechoosesS. ItdependsonRA only in that R is the time interval for which P represents A S PS =exp(S αS4/3) . (44) the probability of one fluctuation. Hence the integrated R − probability for afluctuation ofentropyS divergeslog- 0 P arithmically: ∞ P S (S) = dt (47) 8 At least it is the lightest object that we are sure exists. If a P Z R t0 A lighter object has the same entropy, Eqs. (44), (47) and (49) ∞ dR stillprovidelower bounds onitsrate ofproduction, leavingour ǫ−1exp(S αS4/3) A . (48) conclusionsintact. ≤ − ZS5/6 RA →∞ 10 This is aremarkableresult: objects ofanycomplexity, asymptotic regions, η 0 and η π in the global met- → → no matter how large, will eventually be emitted by the ric (11), on which the global in and out-states might be horizon. The key observation is that at fixed entropy, defined. AnS-matrixbetweensuchstateswould,atbest, therearemany“scalingstates”whoseenergyisinversely bea“meta-observable”[4]: Itwouldrelateastatenoone proportional to their linear size. As the horizon grows, can set up (because of the past event horizon η = χ) to these states become energetically cheaper at the same a state no one can measure (because of the future event rate as the temperature drops, leaving their probability horizon η =π χ). This conclusion does not improve if − invariant. This restricts consideration to massless fields thepastasymptoticregionisreplacedbyabigbang;this at late times.9 only trades the past eventhorizonfor a particle horizon, Indeed, the strongerstatementholds that eachscaling and does not affect the future event horizon. microstate individually is produced with certainty: Nor are there any other asymptotic observables in asymptotically de Sitter space. The total accessible en- dR (i )=ǫ−1exp( αS4/3) A . (49) tropy is bounded (Sec. IIB), and the duration of any P | i − Z RA →∞ experiment fundamentally limited by thermal erosion (Sec.IVA)andbycollisionswithblackholes(Sec.IVB). This includes highly structured,irregularconfigurations. V. ASYMPTOTIC OBSERVABLES B. Q-space In this section we will compare the various cosmologi- LikedeSitterspace,quintessencedominateduniverses cal solutions studied above,with an eye on the complex- ( 1<w < 1/3)have a cosmologicaleventhorizon[13, − − ity and precision of measurements that can be achieved, 14](seeSec.II). Hence,theglobalstateintheasymptotic and on the possibility of defining exact asymptotic ob- future cannot be measured, and there is no S-matrix. servables or an S-matrix. However,some other asymptotic observables may well By an asymptotic observable, we mean any quantity exist. We have seenin the previoussection that Q-space that can be measured with arbitrary precision at suffi- issignificantlymorewelcomingto physiciststhande Sit- ciently late times. We expect that asymptotic observ- terspace. Thermalfluctuations11 arepresentbutaretoo ables exist only in spacetimes where experiments of ar- weak to terminate experiments by erosion(Sec. IVA) or bitrarily long duration can be made and an arbitrarily byblackholeproduction(Sec.IVC). Becausethecosmo- large amount of entropy can be accessed. An S-matrix logicalhorizonbecomes arbitrarilylarge [13], there is no is a special case of an asymptotic observable, consisting absoluteentropybound. WhatweshowedinSec.IVEis ofmatrixelementsbetweenthecompleteinitialandfinal thatanunboundednumberofdifferentstatesareactually asymptotic states of a closed isolated system. produced at late times. Thus, observers can experience The limitations on observation discussed here are im- arbitrarilycomplexevents(and,onemightimagine,store posed by fundamental aspects of the cosmological solu- large amounts of information for long times).12 tion, such as its causal and thermal properties, and its information content. This allows us to proceed without any assumptions about the nature of experiments or ob- C. Decelerating FRW servers.10 Decelerating universes ( 1/3 < w < 1) clearly satisfy − A. De Sitter important conditions for the existence of asymptotic observables, as noted by several authors [4, 5, 6, 13, 14]. As the particle horizon grows,the amount of entropy al- AsymptoticallydeSitterspacetimes(w = 1)arepar- − lowedinthecausaldiamondincreaseswithoutbound,as ticularlyhostiletoobservers. ThereisnoS-matrix,since does its actual matter content (Sec. IIB). This may in- the observer’s causal diamond misses almost all of the clude massive particles, if they are stable and if they are not converted to radiation by black holes. 9 In our estimates, this restriction is implemented by using the number of massless species in the thermodynamic formulas for theenergyandentropyofathermalcavity. Notethatothertypes 11 We should emphasize again that the thermal properties of Q- of universes will also have only massless particles at late times, space discussedinSecs. IIIand IVwererigorouslyderivedonly ifmassiveparticlesareunstableorprocessedbyblackholes. in the limitof smallǫ (w →−1). But weexpect no qualitative 10 Additional restrictions may arise, for example, from a limited transitionsatleastintherange−1<w<−2/3. supply of free energy or inability to harvest this energy for ex- 12 Whether these fluctuations, which involve only massless fields, periments(see,e.g.,Refs.[4,42,43,44]). Totheextentthatthey cangiverisetoanapparatuscapableofprecisemeasurementsis are insurmountable, they may further constrain the asymptotic another question, and wedonot claim tohave proven that this observables. willhappen.