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Quantum compositeness of gravity: black holes, AdS and inflation

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Please note that terms and conditions apply. JCAP01(2014)023 hysics P le ic t ar limit of quantum scattering of doi:10.1088/1475-7516/2014/01/023 N strop A soft constituent gravitons. It then follows that N a,d state. We discuss implications of this phenomenon for the . Content from this work may be used 3 osmology and and osmology C and Cesar Gomez 1312.4795 quantum eternity inflation, gravity, cosmological perturbation theory, dark energy theory a,b,c

Gravitational backgrounds, such as black holes, AdS, de Sitter and inflationary rnal of rnal Article funded byunder SCOAP the terms of the Creative Commons Attribution 3.0 License .

ou An IOP and SISSA journal An IOP and Arnold Sommerfeld Center forDepartment Theoretical Physik, f¨ur Physics, Ludwig-Maximilians-Universit¨atM¨unchen, Theresienstr. 37, 80333 Germany M¨unchen, Max-Planck-Institut f¨urPhysik, F¨ohringerRing 6, 80805 Germany M¨unchen, Center for Cosmology and4 Particle Washington Physics, Place, Department New of York, Physics,Instituto NY New de 10003, York C-XVI, F´ısicaTe´oricaUAM-CSIC, U.S.A. University, Universidad Aut´onomadeCantoblanco, Madrid, 28049 Madrid, Spain E-mail: , [email protected] [email protected] b c d a Any further distribution ofand this the work must title maintain of attribution the to work, the author(s) journal citation and DOI. constituent longitudinal gravitons. We show thatlarge this and picture small correctly black accounts for holesfor physics in cosmological of parameters. AdS, as However, it wellsemi-classical anticipates as new treatment. reproduces effects not well-known In captured inflationaryto by particular, predictions the the standard we inflationary predict history way observableon beyond corrections last the that 60 number e-foldings. are ofbe We sensitive derive e-foldings, approximated an beyond by absolute upper which anew bound neither semi-classical type de metric. of Sitter However, nor they inflationary could Universe in can principle persist in a cosmological constant problem. Keywords: ArXiv ePrint: universes, should be viewed as composite of Gia Dvali Received January 10, 2014 Accepted January 10, 2014 Published January 14, 2014 Abstract. such systems are closeGeneric to properties quantum of criticality the ofcreation ordinary graviton metric in Bose-gas description, to the including Bose-liquid background geodesic transition. motion metric, or emerge particle- as the large- J Quantum compositeness of gravity: black holes, AdS and inflation JCAP01(2014)023 25 system of many soft gravitons of similar quantum – 1 – is a composite R 6.1.1 Non-eternity versus self-reproduction 29 In [1,2] (see also [5]) we have also suggested that compositeness treatment should be 9.1 Black holes in AdS 39 6.2 Backgrounds without classical analog? 31 5.1 Quantum5.2 origin of inflationary Curvature perturbations 5.3 perturbations Tilt 5.4 Quantum corrections 6.1 Physical meaning of non-eternity bound 22 23 29 25 4.1 Compositeness and the nature of time 18 generalized to otherspace-times. gravitational systems, In particular, such we as showed the that maximally-symmetric when describing cosmological AdS space in a similar fashion, 1 Introduction In [1–4] we haveaspects developed of this a proposal quantum andblack hole some compositeness of related a theory ideas classical of were size wavelength. discussed black in We holes. [5–15]. have In discoveredEinstein The this that condensate description different this a “frozen” compositeto at system a a exhibits quantum quantum propertiesBose-liquid phase critical of transition phases. point. a between Bose- classical what The Because macroscopic we critical system of can in point thiscompositeness call reality corresponds is become proximity graviton-Bose-gas intrinsically-quantum. extremely and to The importantare graviton- quantum effects and not due criticality, must captured to even be quantum bycompositeness taken a of any into the seemingly- of account. would-be-classical the background. These standard effects analysis, since these approaches are blind to the 7 Implications for cosmological constant 8 Entanglement as quantum measure9 of de Sitter’s AdS age as graviton condensate 10 Conclusions and outlook 34 32 37 41 5 De Sitter and inflationary universe 6 Non eternity versus quantum eternity 20 28 Contents 11 Introduction 2 Black holes7 3 Emergence of the curved4 geometry9 Particle creation as non-vacuum process 16 JCAP01(2014)023 , N BH (1.1) R N..., 2 ~ R (i.e., the curvature radius) i ∼ R N | a + a | N h , are creation and annihilation operators of the – 2 – a + 1 N and √ ~ + has to be understood as the gravitational radius R a √ R limit of our picture. We show, how the motion of a probe and i ∼ ∞ N R | = + a | N + 1 N h respectively. H is the quantum state of the gravitational background of occupation number ,R i dS N | of the system. Gravitational systems, such asspaces represent black composite holes, entities AdS, ofof de microscopic wave-length quantum set Sitter constituent by or gravitons the other characteristic classical cosmological size ,R In the present paper we shall systematically apply our treatment to AdS, de Sitter We first discuss how the known aspects of the ordinary curved-metric description of As we shall see, transitions in which all initial and final constituent gravitons are off-shell In particular, we explain in this way the well-known effects of Hawking [19] and Gibbons- Obviously, for a black hole In the present paper we shallIn investigate what follows the we compositeness adopt the picture key of concept of gravity [1,2]: from AdS and inflationary spaces.phenomena as First, a particular we approximation of conductconsistency our several approach. check consistency However, and we go checks, uncoverThese beyond new reproducing new the simple effects effects known are that due are to the not quantum capturedgravity compositeness emerge by of as the standard gravitational the treatments. background. whereas for AdS/ dS orR cosmological spaces, as the corresponding curvature (Hubble) radius, source in an effective classicalthe background constituent metric off-shell is recovered gravitons.classical from its The metric, quantum entities are scattering that matrix at elements play of the the crucial form, role in recovery of the where only contribute into the recoveryprocesses of in the which some classical of metric,creation. the without final particle constituent gravitons creation. become Whereas, on-shell, amount to particle Hawking [20] radiation, as wellsky as [22]) the modes scalar of (Mukhanov-Chibisov inflationary [21])this density and phenomenon, perturbations. tensor that Moreover, (Starobin- in we explainwhy semi-classical certain treatment only aspects appears of space-times tocreation. be with rather In globally mysterious. our Namely, un-definedin description which time only the exhibit such constituenton-shell gravitons space-times the carry particles correspond phenomenon non-zero in to frequencies, their ofBose-gas critical and re-scattering. of particle- thus, quantum constituent gravitons are In systems is able static eitherhave to far space-times zero form produce with frequencies criticality and/or globally-defined and the time cannot constituent gravitons the lead to creation of on-shell particles in their interactions. constituent off-shell gravitons. Theselimit matrix elements and are recover non-vanishing the in motion the in semi-classical the background classical metric. several different perspectives. the occupation numberCFT, of computed gravitons according to coincides the AdS/CFT withindicates prescription [16–18]. that what This compositeness would remarkable coincidence be andthe underlying quantum the reason criticality central for of charge an gravitational of effective backgrounds holographic a may description. be of gravitons of wavelength JCAP01(2014)023 to (1.3) (1.2) N . Just N G a vacuum ~ = not 2 P L clock that imprints , is quantum mechanically H quantum R limit such a particle-creation process , 2 ) ∞ , as P , ~ 2 = √ /L ) approximations of the full quantum picture. ) P H N = N R – 3 – φ N /V L N 0 = ( V ( N ≡  1 background, would be a severe mistake to assume a creation of a maximal is the Planck length, which is related to Newton’s constant as N . The key point is that the inflaton-to-graviton ratio is controlled by the classical P N L The novelty, in comparison with the black hole case, is that the graviton depletion is The key novelty for inflation is that compositeness acts as a The important lesson we are distilling is that the particle-creation is Next, we shall apply our picture to differentImplementation cosmological of compositeness spaces. ideas Among to the other case things, of de Sitter and inflationary universes, The physical origin of these effects is nevertheless very transparent. In our treatment,  See [23] for a review on inflation and standard definitions. 1 φ assisted by the inflaton background,N which itself represents a Bose-gas of occupation number process, and consequently, thereact. is This no goes maximalto in entanglement which contrast created a with in particle-pair thereality a is standard the created interpretation single particles out of are emissions of Hawkingof not the radiation created existing out vacuum, according constituent and of vacuum, thus, gravitons. but is rather maximally Only they entangled. result in In from re-scattering where as in theundergoes black an hole intense case, quantumof depletion, this gravitons which and condensate inflatons gradually in is decreases the the course close of occupation to inflation number and quantum works towards criticality. emptying the reservoir. As a result, it inflationary slow-roll parameter, Most importantly, we discoverthe some consistency, new as well effects as observability, which of substantially the very change early our cosmological history view of about our Universe. we shall describe physics of both small and largeallows black holes us in to AdS in reproducetensor simple [22] the physical perturbations, well terms. as particular known (large- inflationary predictions, such as the scalar [21] and de Sitter or an inflationary Hubble patch of a classical radius can be naively interpretedthe as background a effectively vacuum becomesand, pair-creation classical. therefore, process. It becomes This gainsattributed eternal. infinite is arbitrarily, capacity because since To of in it suchany emitting this can an particles finite- limit never eternal be background measured the during entanglement any cancreate finite be a time. maximal entanglement However, among for the constituents. measurable effects into cosmological observables.information These throughout effects the are entire cumulative duration and ofpatch gather inflation. carries the quantum Hence, information in our about60 picture, the e-foldings. a entire given history Hubble This ofbe quantum inflation, read-off information way after beyond is the thewe not last are end redshifted talking of about away inflation by arethe within unaccessible the previous in the analysis. expansion semi-classical same and treatment Hubble and can have patch. been missed Obviously, in the all effects interpreted as a reservoir ofcondensate. a finite The number occupation of gravitons number and of inflatons, gravitons i.e., is, as a Bose-Einstein entanglement in a single particle emission act. It takes a number of steps of order JCAP01(2014)023 (1.4) (1.5) (1.6) = 0. This  2 = 0. cannot be arbitrarily small ~  . φ , N N ,  P √ 1 due to quantum depletion of the back- N . Thus, slower the inflaton rolls higher 1  H/ NL 1 N √ − √ − = − dV/dφ  – 4 – = 0. The fact that =  ≡  0 H V quantum = evolves exclusively due to quantum depletion, | ˙ ˙ quantum N N N | N ˙ N and the condensed gravitons have increasing number of inflaton partners is the Hubble parameter. 1 /N φ − H as compared to the black hole case. As we shall see, this factor also explains N R φ is the inflaton potential and N ≡ N V H This is a very subtle point and we would like to avoid a mis-interpretation. This state- The physics that we have just summarized can be foreseen from the following master The first term in the r.h.s. of this equation describes the purely-classical time-evolution The remarkable thing is that, since the quantum depletionIn rate particular, the is depletion rate controlled becomes by infinite for the the exact de Sitter limit The second term describes the decrease of Some interesting previous attempts, within the standard semi-classical treatment, to set a bound on the 2 number of e-foldings baseddo on assigning not finite resolve entropy quantumeffects to compositeness de we of Sitter are the can discussing.turns gravitational be out background Correspondingly, found to and our in be therefore [25–27]. bound more cannot has These severe. account approaches a for different the physical meaning and, as we shall see, ment only applies toupon de Sitter the obtained de as Sitterpure a limit without cosmological of extra term, slow-roll lightscenario or inflation scalar [28]. and a degrees it local However, does of suchite minimum not freedom, a picture, of touch “pure” such and the de as cannot potential,in Sitter in exist terms as is case eternally of also in of as severely a thetence e.g., constrained a classical eternally “Old” a by state metric. in inflationary our which ainteresting We compos- admits option quantum cannot an will state exclude approximate be that the description discussed admits possibility in that no the it approximate text. equation classical can continue description. exis- This excludes de Sitter ason a the consistent limit possible of numbersemi-classical a of description slow-roll e-foldings throughout inflation in the and history. any also theory puts of an upper inflation bound that admits an approximate where or expressing everything in terms of occupation numbers, is the ratio to re-scatter-at and jumpa out factor of the condensate. As a result the depletion is enhanced by of the graviton occupationcosmic time. number due This term to of the course classical survives increase in of the classical the limit Hubble radius with classical slow-roll parameter, therebe is and a therefore consistency the number upper of bound e-foldings on is how bounded slow from inflation above. can where the enhancement of theone scalar [22]. mode Moreover, in thisthe density depletion entire perturbations duration of [21, of the24] inflation. reservoir relative produces to a the measurable tensor effect sensitive to ground gravitons. This isinflation the and term excludes that theis puts de obvious. a Sitter consistency limit To upperbecomes understand bound irrelevant. why, on we In the can this duration regime first of take it to be small enough that the first term JCAP01(2014)023 . is   (1.9) (1.7) (1.8) on class e N as constants. For the change H . 3 / 2 and  , and we get the following consistency  4 P / 3 V . M . 3 2 3 / = 1  2 1 ~ − < N class < e is determined by the dependence of . Indeed, this is the time needed to deplete the – 5 – e  2 e N N √   > N N 0 N V P V = M . Thus, the maximal number of Hubble times, consistent   . If this time is shorter than the classically evaluated number √ / . A naive classical person would conclude that the number of N quantum 2 inflation [29], e 2 N = 1 , but in reality quantum depletion empties the condensate during /N φ 2 2 . In some sense, whenever the dynamics is dominated by quantum fin φ N m = 1 N N N  = = − V in , which gives a unique consistency bound. class N e N class ≡ e N = 1. This mismatch means that such a theory is simply inconsistent, viewed as a N It is important to stress that the bound is not an artifact due to the breakdown of some When the number of massless species is infinite, theories become inconsistent with the The precise expression in terms of In order to understand the robustness of this bound, imagine that as a would-be counter This bound excludes any potential that slopes toThe bound a translated positive as cosmological a constant constraint on and the classical potential energy, has the following This bound persists even within the composite picture of black hole, as shown in [32]. 3 quantum e perturbation theory andis cannot coming be from removed thenumber by enhanced of any phase inflatons re-summation. space incannot of be the the The removed Bose-condensate by depletion, reason re-summation which duethe since is catalyze the situation to number that graviton with the of the depletion. it channels excessive deof is occupation Sitter physical. This large limit number In factor is this of similar respect, massless to particle the species depletion of [30, existence black31]. holes of in black theand presence holes, this since fact thepicture cannot the number be de of saved Sitter“depletors” by limit particle-emission and is perturbative channels the inconsistent, re-summation. rate because becomes of it infinite Similarly, particle-production supplies in blows an up. infinite the number composite of assistant From here it isparticular, clear observational that constrains smaller on is time-evolution the of potential dark energy energy slower satisfy it this is bound. permitted to evolve. In quantum-mechanically, is entire reservoir! Thise-foldings number must be consistent with the classically evaluated number of This is very similar to ourenhancement black hole factor depletion equation [1,2]depletion, (see below), the apart from Hubble anhigh patch extra enters depletion into rate. a Consequently, black the hole-type standard regime, computation except of with e-foldings exceedingly in terms of bound, e-foldings is N attempt we choose description of an approximately semi-classical system. The absoluteputs bound a is severe given restriction by (1.8). on how slow theform, decrease of potential energy could be. no longer possible. Instead,deplete the order duration one of fraction inflationof of is e-foldings, determined by the theequation theory time for is that one takes inconsistent. Hubble to time, To during illustrate which the we point treat let us integrate the above For example, for we get ∆ JCAP01(2014)023 , should not be N in any space-time 1 − α soft gravitons, no matter how N )) of states corresponding to nearly- limits of various quantum scattering N ( N exp ∼ – 6 – of the constituent gravitons. It was argued [12] that ). If this is also the case for the Hubble graviton reser- N ln( limit of certain graviton condensates. The key ingredient is the R N ∼ quantum entanglement then after this time-scale it becomes an entangled quantum system. This entanglement We shall show that the depletion term in) (1.4 can contribute to a potentially-measurable Another cumulative quantum effect that is also sensitive to the entire history to the Hub- We would like to stress, that none of the correction we are talking about have to do with Finally, we would like to suggest that our pictureOur can point provide is [3] a that quantum near foundation quantum criticality of a system of This also naturally explains why a holographic description (in our sense) is only available Holographic description is subjected to corrections due to compositeness of the grav- The logic flow of the paper is the following. After briefly reviewing the composite- 4 See [33, 34] for an attempt to extend the notion of scrambling35, [ 36] to cosmology. 4 cumulative effect in the cosmologicaland observables, such tensor-to-scalar as ratios. amplitude ofscan Hence, density the perturbations by history precision of our measurement Hubble of patch these beyond parametersble the patch one last is 60 can the e-folds. either trans-Planckian gravity or otherdue UV-sensitive to regions. interactions All ofgravity our very effects effects soft are are gravitons computable taking for and place which are gravity under is an extremelywhat excellent weak control. is and sometimes quantum referred to as “holography”. for certain gravitational systems and nottum for effects others. are These very are the important,the systems systems but for that they which are the have close quan- to mistakenlycompositeness the been is quantum treated critical very point classically. important. and These for For which are example, the underlying black quantum holes,itational AdS system. and cosmological These spaces. corrections, although suppressed by fractions of a near-critical condensate oftime-scale gravitons of generate the entanglement veryvoir order efficiently, with minimal is carried by theFor realistic depleted values gravitons of ande-foldings. the in inflationary this This parameters, way the givesentanglement is entanglement an among imprinted time-scale exciting in the ismeasurements possibility within density depleted is 30 perturbations. to gravitons. an measure interesting question the How to age be to of studied fish in inflation out the by future. this detecting information from the macroscopic, exhibits an enormous degeneracygapless ( collective Bogoliubovcentral modes, charge which determined behave bygravitons. the as inverse nearly It gravitationaldimension conformal is coupling scales with of a as an the simple“holography”. area effective original fact in constituent that Plank inverse units. gravitational coupling Hence the reason for our adoption of the term ness approach to blackof holes geometry as we the sketch large the formalism underlying the quantum description confused with correctionsparticular to system ’t such Hoofts adamentally description planar is different. limit available). on Corrections Inimportant the we at particular, are gauge later talking they times theory about as have side are it a [37] fun- is cumulative (if the effect case for for and a cosmology become or black extremely hole entanglement. replacement of a classicalsate background composed curved out metric of withtional off-shell the metric (longitudinal) description quantum gravitons. are notionprocesses then of Different recovered phenomena with a as of conden- the large- the participation conven- of these constituent gravitons. For example, we identify JCAP01(2014)023 1 (2.2) (2.1) . This αN > BH 5 , satisfies, R 2 = /R λ 2 P L 1 the gravitational ≡ α αN < . . = 1 and separates, what one could call, the 2 P 2 BH 1 N L R αN = – 7 – = α N ) by the condensate state itself. The quantum depletion depends on the collectively must be viewed as a collection of constituent gravitons of wavelength One is the quantum depletion, which is very strong at the critical point. The reason for To understand the significance of this point, note that for Next, we extend the condensate formalism to the inflationary cosmology where we Finally we consider the case of AdS andThroughout black the holes paper, with irrelevant AdS factors boundary of conditions in order one will be ignored, but signs and As we shall see, the on-shell dispersion relation for resulting metric fluctuations is de- Thus, the quantum gravitational coupling among the gravitons, In a different context, the same measure of classicality is also adopted in [39–41]. 5 BH the depletion is that constituent gravitons re-scatter and are pushed out of the condensate. attraction would not be strongthe gravitational enough attraction to among the keep constituents thecollapse. would gravitons induce instability together. So towards quantum Contrary, the for enormous critical degeneracy point of isresult, collective unstable. there (Bogoliubov) are But, excitations two precisely important that because quantum are effects of nearly that criticality, determine there gapless. the is evolution an As of a the system. Bose-gas and Bose-liquid phases. automatically fixes its occupation number according to (1.2), From this relation itwhich follows exhibits that alltransition. the the This system properties critical is point of corresponds described a to as Bose-gas a at self-sustained the bound-state, critical point of a quantum phase reproduce the standardWe results work on out compositenesse-foldings. quantum corrections fluctuations These results and in motivatevery determine terms us different the perspective. to of visualize bound quantum the on cosmological depletion. maximal constant problem numberthe from of graviton a condensate formalism. important relative factors will be followed carefully. 2 Black holes Before proceeding to AdS andof Inflationary our spaces, black we wish holewell to as quantum briefly differences portrait review with some [1,2] otherR ingredients spaces. that According will to be our useful picture a later black for hole of drawing classical analogies radius as particular type of constituent gravitonsof defining the the emergent condensate metric andout sets as the the well examples time as properties particle-creation. of the de dynamics Sitter of and quantum anti particle de production. Sitter We space-times work and explain the difference in termined ( processes that reproduce theprocesses geodesic motion of in depletion a that background are classical responsible metric, for as the well as particle the creation. The notion of theintroduced above in occupation [38] number and of in soft quantum gravitons portrait as it black represents hole a constituents measure was of classicality. JCAP01(2014)023 N N  (2.4) (2.3) (2.5) m and the quantum ∞ = . N P L . 2 / 3 . But, this is a wrong intuition, N . , ) N , 1 ∼ = fixed light particle species of mass . Notice, that unlike the standard − t P N species BH ( . So the cumulative effect of the correc- species N NL O species N N √ + N P ,R P = 1 NL – 8 – t 1 NL √ = fixed √ − ~ . Obviously, in this limit − = ˙ = , BH N ˙ N /R = 0 ~ P = L H T , per emission time . BH 1 − R and naively look unimportant for large α =  /N P N NL , the depletion rate is increased by a factor √ After this time scale the black hole can no longerNotice, be that regarded — if even the approximately theory contains extra The first black hole puzzle from this compositeness point of view appears when we try to Thus, as we shall see, both equationsFinally, (2.3) we and would like ( 2.5) to have make analogs some in general the comments inflationary with the goal of putting some BH = /R tions is very important andtime-scales leads comparable to the to total the breakdown black of hole semi-classical half approximation over life-time there are extremelypressed important by corrections. 1 since These the corrections number of per emissions each is emission also are growing as sup- — as a classicalinflationary system. Universe, which As also we becomes shall fully see, quantum after there finite is time. a full analog of this phenomenon for the compositeness of the black hole is not resolved. However, as we have shown, for finite ~ In this limit, ouroration quantum depletion of process temperature indeed reproduces the Hawking’s thermal evap- of soft gravitons.constituents looks If at first we sightbe are very easy arbitrarily thinking to weakly-coupled about handle. byhard Indeed taking large to the their black expect constituents wave-length can anything holes, large. bethreshold dramatic the chosen as From to long this description as perspective in their it terms number is is of below the black hole formation understand what actually happens when you add the “last” soft constituent and you suddenly Universe. typical black hole paradoxes in thethat perspective we of can compositeness. make It a is probably black not hole surprising in pure gravity simply putting together a certain amount Thus, due to quantumλ depletion the graviton condensateHawking’s looses semi-classical one computation, graviton the ofprocess, emission wave-length in of which particles particles inthe are our particle created case in emission ispre-existing a not from in fixed a a the background vacuum condensate. geometry. blackdescriptions As Rather, hole match a in in is result, our the the due semi-classical case condensate limit, to back-reacts and depletion recoils. of The gravitons two that are already due to the existenceincreases of with increasing extra number depletion of species.in channels. the As Hence, inflationary we shall the case. see, quantumnessrate, In this of which effect particular, too explains the the finds presence the black a of excess hole counterpart inflaton of quanta scalar increases curvature the perturbations depletion over the tensor mode. At the same timegravitons the continuously condensate leak shrinks outevolution because equation of of the the quantum condensate collapse. at As the a rate result given the by the following time JCAP01(2014)023 . N notion we mean quantum small curvature effect and correspondingly the time N – 9 – = 1 at the precise moment of black hole formation. notion of a curved geometry by the and generation of entanglement takes infinite time. We αN ∞ = classical N , that appears as the coupling of the corresponding Hamiltonian also αN From the classical General Relativity point of view what appears as extraordinary after The effect of entanglement is intrinsically finite- For definiteness, we shall consider a graviton condensate that in the classical limit Before we start to apply our picture to cosmological spaces we will present the basis of Since we substitute the black hole formationthe are the nature properties ofis of space-time the the inside counterpart resulting theconnection of space black between the time, hole. the geometry and twoquantum in more descriptions Thus scattering specifically, the is of the a through compositeness nextin probe the picture. obvious the particle scattering background question at amplitude. The classical the is,this answer metric. condensate Namely, what description is the is However, equivalent becomes that it toconstituents less is the a become geodesic extremely and entangled. motion important lessan to The intense accurate understand important generation that as fact ofconstituents the is, that quantum we that time entanglement were naively at starts goesmechanically putting the taking maximally together on, to entangled. quantum place. create and critical the After point black the hole finite black become time, quantum- hole the reproduces de Sitter space.classical For limit simplicity, we reproduces shall the limit leadingto ourselves process the to and motion the the in first analysis a sub-leading that small-curvature (non-linear) in (weak correction the field) background. Under pass from a simpleclassical aggregation General Relativity of point weakly ofarena coupled view where constituents this the into is constituents a aquestion were very black is, originally special hole. what living moment happens where changes Frombuild at the dramatically. the up space-time this Thus, the point the black to obvious body hole. the system many-body The of quantum answer gravitons systemkey to becomes reason we this critical have is basic used and also questioninteraction to simple undergoes strength, is to a remarkably understand. quantum simple: phase The the transition! constituents many are The interacting, but the collective that takes the black hole constituents to become maximally entangled is larger for large of large graviton occupationappears number as on an effective a emergentscattering Minkowski description. of vacuum, a Indeed, the probe we curved shall particlenumber, metric at demonstrate reproduces the that must graviton the the only condensate, quantum geodesic in the motion limit in (2.4) the of background large occupation classical metric. a situation when the curvature radius is much larger than the characteristic wave-length of changes and reach the critical point the compositeness formalism. Torepresentation do that of we maximally will symmetric workmight out space provide explicitly times. the the clue graviton Incidentallyhole — condensate this inner from detailed geometry. the construction graviton condensate point of view — to3 unveil the black Emergence of theWe curved now geometry wish to showa how result the of concept quantum of scattering. a curved classical metric emerges in our picture as Due to this, thedescription entanglement since effect in this is case shall impossible discuss to this effect capture in infor more the the details inflationary standard below Universe. when semi-classical we shall make the study of a similar effect JCAP01(2014)023 µν g (3.7) (3.3) (3.5) (3.6) (3.2) (3.4) (3.1) into a can be . µν µν produced µν h T T µν g h, ν carries informa- ∂ µ 0 µν ∂ h encodes the classical + µν αβ h as the result of scattering h β ∂ µν are obtained in higher order τ α ∂ 0 µν µν h η ..., .... + + , + and ) αµ , , h αβ h 0 µν ( µν h µν µν α h T µ µν h ˆ h ∂ . ∂ ν T + N ~ P ν ∂ µν N G ∂ L √ τ µν − G h αβ = µν – 10 – → h g = αν + µν h µν h 0 µν α µν h ≡ − E h ∂ η µ ) E ∂ = h ( from its coupling to the derived classical metric, − µν µν g h µν T τ µν η − in a classical gravitational field produced by the source represents a solution of the linearized Einstein’ equation µν µν τ h via virtual graviton exchanges. µν  h µν ≡ T µν represents the first correction due to cubic self-interaction of gravity, and sat- h and limit. Classically, a metric created by an energy-momentum source is the linearized Einstein’s tensor, E 0 µν stands for ensuring the canonical dimensionality. Then is the flat Minkowski metric, whereas µν N h µν τ ~ P h √ L , either exactly or order by order in weak field expansion (3.1), and then evaluate a µν E η In order to make contact with our quantum picture we should first promote One interpretation, is to first compute the classical gravitational field The second interpretation is to think about the motion of Before going into a full quantum picture, let us note that within the classical (or semi- In order to fix the language, let us first define the classical problem to which we map µν . For economy, we shall not display the full expression. One can continue this iterative T µν quantum field, that satisfies the fully non-linearchoice Einstein’s of equation. gauge, This can type be of done expansion, anywhere with in a a suitable small curvature region. between classical) description there areof two energy-momentum equivalent ways of thinking about the motion of a probe obtained by iteratively solving Einstein’s equation in weak field expansion of the metric, whereas, is the effective energyh momentum tensor of gravity evaluatedprocess on to the arbitrary first high order orders. perturbation Resuming the entire series we would recover the metric where limit of processes thattion do about not involve the graviton cubic self-interactions, self-scattering, whereas and so on. by geodesic motion of the source isfies the equation, our large- the probe. Weprobe shall of show, wave-length that shorter thanAs this the semi-classical we characteristic shall picture wave-length of fits see,classical the metric the this graviton background. quantum condensate. type scattering of of scattering a reproduces a geodesic motion in a small-curvature where where, Here iterations. Namely, JCAP01(2014)023 , can µν (3.9) (3.8) T (3.12) (3.11) (3.10) µν τ which is and µν µν in the linear h τ , given by (3.5). µν τ of gravitons with µν h k N , , ) )) x . (˜ x (˜ ) is bilinear in αβ αβ h x η ( (˜ T ) h µν ( αβ ˜ x η T 2 1 µν ) − ˜ T x x − ( = 0. Of course, this equivalence holds ) − , ~ x να ( η µν µν,αβ  . τ , since i µβ i η N | )∆ µν µν,αβ n x G + s ( | xh )∆ 4 – 11 – f µν νβ x d h ( η a graviton propagator. For example, in de Donder ˜ xτ Z i with large occupation numbers µν 4 µα η i αβ ( ˜ xd xτ k ) 1 2 4 4 x N d | (˜ ) scattering in the gravitational field of the sun (of energy xd matrix elements, = ˆ h 4 Z , µν d n ) τ s 2 P x is nonzero in the limit ~ ( Z L µν,αβ 2 P ~ µν states can describe initial and final states of an incident photon = L ∆ = ˆ h i 2 1 s ≡ f ). s | ≡ h µν N ) T G ˜ x and . − , and with the peak of the distribution being at the characteristic curvature limit of quantum processes that take into account the composition of the i N i k ) is the energy momentum tensor of the linear perturbation, x | G ( x N (˜ limit as scattering between the probe particle and the constituent gravitons. h is the solution of the linear Einstein equation (3.2). Alternatively, we can think ( µν,αβ N µν µν h T The possibility of tree-level Feynman diagram representation of the classical metric, such Although in the above formalism the gravitational field is classical, the probe The bottom line is, that any quantum picture that in the semiclassical limit reproduces For example, in first order approximation we can think of motion of as Schwarzschild, is well known [42, for43]. cosmological Our spaces, first taskas and will then the be to to large- give show an how analogous expansion this classical expansion can be understood background. first order in be either classical ormation, quantum. and the In transitions between theperturbatively the latter in initial the case and form the we of final are states working of in the semi-classical probe can approxi- be found wave-numbers radius of the system.in large- The motion in the classical background geometry is then reproduced momentum tensor the above weak fieldstrate expansion this has for a correct ourto classical composite be Einsteinian picture, viewed limit. according as We to a wish which quantum to a state demon- would-be classical geometry has For example, (of energy momentum tensor gauge it can be written as, where ∆ where Obviously, this correction is second order in to all orders inthe weak classical field metric expansion. is For given example, by second order correction to the motion in classical metric Of course, there isone nothing classical surprising and in another the quantum, give factsince the that same the the result. quantity two In seemingly fact, different both languages, languages are classical, where of the same process as the result of one-graviton exchange amplitude between JCAP01(2014)023 , j n i n (3.14) (3.13) (3.15) (3.16) ≡ ij L components is carried by shorter than satisfying the ij k λ h and µν . and annihilation, j g ) n i + L,k ∗ L,k n b ) (3.17) 0 , a ij − , k L ij + I,k  − , δ a 2 + ) k 2 Ω ( ≡ limit of a quantum state with r d is the Hubble constant. We 4 ∗ I,k j 2 δ b ij n r N I i ij H I n + ] = ( into the Fourier harmonics, 0 2 + r ikx ij 2 + L,k ij − 2 h δ , e H 2 , a 2 t dr , where ( − H 2 2 L,k ) + 1 a does not satisfy any free wave equation. Cor- µν [ H H η  L,k ij limit we need only to consider µν ) b = h = , g ij ) – 12 – ij 0 N Ht L h k µν = h + − E µν k I,k ( T b = 0 4 + cosh( δ i ij is not satisfying any massless dispersion relation. Instead, 0 I 2 ( h µ dt ] = k 0 ikx = − + I,k , is a linearlized de Sitter metric in closed FRW slicing, and we ke 00 = 2 j 4 h , a 2 x d . For such processes, we can reliably use the weak field expansion 1. ds I,k 1 q Z a − [ ≡  = H ij Ht h /r, r i x , satisfying the equation 1 and ≡ µν are expansion coefficients. The field , operators, which satisfy the usual commutation relations, i h  ∗ n L,k Let us expand the classical metric perturbation For definiteness, we shall establish this dictionary for reproducing the de Sitter geometry, It is easy to check [44], that the above equation admits several gauge-equivalent solutions We now need to re-interpret this geometry as the large- Our postulate is that the above classical solution, quantum-mechanically is represented b, b , a Hr do not satisfy any a priory dispersion relation. The relation among them is determined by I,k where have set an over-allnon-linear numerical metric, coefficient to one. This represents an approximation of a full being non-zero. which classically would amount to the motion into a background metric and treat the processmetric, perturbatively, by first considering the motion on a background linear that describe de Sitter space for short time-scales and distances. For example, Einstein equation withshall the consider source evolutionthe of Hubble short radius wave-length probes, with the wavelength and treating effects of non-linearities as higher order corrections. for large occupation numberlongitudinal of gravitons. constituent We gravitons. choseclosed the We FRW gauge slicing shall in (3.14). refer order Then, to to in reproduce these large the constituents de as Sitter geometry in Here we have introduced two orthogonal tensor projectors, with all other commutators vanishing.to Since the states states with created some by thesek occupation operators, number correspond of longitudinalthe off-shell gravitons, state the in four-momenta whichthe they state are vector. prepared. In other words, the information about a by a state vector in some Fock space, built by a set of creation, the dispersion relation is determined by the source, such as, e.g., cosmological constant. respondingly, the wave-vector and JCAP01(2014)023 = be i φ (3.19) (3.22) (3.21) (3.18) (3.20) (3.23) deSitter | a | )] . Transition to this deSitter i h limit of expectation 1 n H −→ − N ± describes the quantum ) − H i − k −→ k H ) N −→ ( . . ( 3 k N i 3 −→ δ | δ ( ) H ) i × | 3 0 limit of quantum transition 0 fixed f δ N k | ) k ( = φ , ( k 0 | δ N i )] δ a k | n ( ,H | = k 1 −→ ! δ − . This process is described by the first . 2 ( n i , where limit. However, the physics of the processes n + 1[ i ) 3 i 1 − →∞ f µν N √ δ k H H ) −→ ∞ φ h ± N H | ( ) 0 + 1[2 N N ∞ i| 3 H N X H = φ δ 6 h ( H p ) 2 N N − | N ,N N µν − 0 H H + k f i × | , k ~ a i p φ – 13 – ( | in xτ − | = e √ δ H 4 φ 0 H i | d N k ~ + 1 | = ( Z √ δ = + k i N a h | H i = ~ deSitter P N H | i | L √ + 1 H ,N + I,k H N = a in | | N ∗ k φ h | b + L,k changes by one, + 1 a | H H . 2 N + 1 = h − ) H k P N h HL = ( , with the following matrix elements, ] and reproduces the classical metric in i H ... H [ N N N | ). Propagation of a particle on a linearized classical metric perturbation on Minkowski from an initial state We are now ready to understand the quantum evolution of a probe particle interacting The contact with the classical picture is then made via large- A would-be classical weak-field de Sitter space in our picture is described as the quantum This is very similar to the process of stimulated emission of photon bySuch an emission excited atom. (absorbtion) of graviton are described by the following matrix elements √ φ We could have chosen instead a coherent state description, ~ ( 6 √ µν that we are going to discuss is better captured in the number-eigenstate representation. and In our picture this propagation should emerge ascounterpart a of large- the de Sitter state,with to wave a number final state in which the occupation number of gravitons on whichH the expectation value of the destruction operator is non-vanishing with the above graviton condensate. Let the energyvacuum is momentum given of by a the quantum coupling, particle values, where τ state Correspondingly, all subsidiary conditionsexpectation must values. be understood to be satisfied on states and The emission probability isoccupation number increased of in photons the in range the incident of wave. frequency corresponding to maximal state corresponds to a(from) the process condensate in and which evolvesFeynman a to diagram a in particle final figure1. deposits state (absorbs) a single graviton into JCAP01(2014)023 and (3.25) (3.24) tH limit, the graviton ) = 0. In this gauge h N µν η 2 1 . 2 − H µν )] 2 h , ( r 2 µ at the constituent gravitons. + ∂ H 2 φ t µν ( η ij = L  + -matrix element describing a scattering of a h 2 s t µν ij η I – 14 – [ 1 2 -corrections. Thus, the above expression recovers H − N /N µν p h H given by (3.14). Thus, in the large-  de Sitter geometry with all the usual consequences. P  L µν h ' 1, up to 1 ij h ' classical H N in a √ φ . Quantum scattering of a probe particle H P L Figure 1 The crucial point is that because of the criticality of the graviton condensate, the pref- Notice what has been the key point of the previous construction. The de Sitter con- In order to shed some more light on the physical meaning of the longitudinal graviton It is useful to do this analysis in de Donder gauge, expansions we recover the following effective classical metric, the equation (3.13) takes the form actor is one, By taking into accountrH (3.18), (3.21) and (3.22), it is clear that to leading order in the classical de Sitter metric emission (absorption) transitions generate an densate is definedsatisfying as the a equation particularpears of collection as motion the of expectation in off-shell valuecondensate the on gravitons. itself. the presence condensate The of state i.e., metric the as fluctuation cosmological a collective constant phenomenoncondensate of source it the is ap- usefulas to a visualize condensate the of condensate on-shella gravitons of small in off-shell mass, the gravitons deformed andwhere in theory, then the Einstein’s obtained taking de by theory, Sitter giving the to solutions zero-mass gravitons in limit. such deformed We theories can were directly already use analyzed. the results of [44], quantum particle JCAP01(2014)023 (3.28) (3.26) (3.27) .  2 r = 0, propagates six j 6=
i | h j . µν j n η i 6= massive bosons. At the 1 2 i is getting red-shifted by n | j 1 6 − φ n i + µν . n h 2 2 ij r δ ) H ) 2 2 on-shell t µν m 1 2 mt η −  = 2  cos( -particle, we can now summarize what  H 2 φ h 6 H = µν η ij + 1 2 , ij ) = 0 is a constraint that follows from the equation of )) δ − h – 15 – , h )) i µν mt µν interacting with gravitons, is unable to distinguish η r, i mt h 2 1 trn φ n  2 cos( ) − ) H 2 cos( − µν mt 1 3 h m − ( (1 µ = 2 − 2 sin( (1 ∂ i 2 2 2 0 m H  m ( m H 3 H − , h 2 = = = t 1. 2 i ij 1, it reproduces the de Sitter solution (3.26) of the massless theory. 0 00 H  h h h 2 1  − tm = tm 00 ), which propagates ghost. To see this equivalence, it is enough to notice that in the latter 2 h h However, they continued to form the same physical condensate as in the massless deposits a single graviton of Hubble wave-length into the condensate and increases 2 1 7 φ − The fact that this should match the classical evolution, could have been guessed by the This solution describes an oscillating massive bosonic field and quantum-mechanically The physics of this connection is the following. By introducing the mass term,The the usefulness of the above discussionComing lies back in to showing the that scattering the of off-shell the longitudinal probe µν h Notice, that the free theory defined by equation (3.27), ( 7 µν h following computation of the rate of the process described above. The process is the lowest theory a would-be gauge condition conservation of energy andboth momentum quantities are (recall conserved). in this picture the vacuum is Minkowski so motion as a resultthe of present Bianchi discussion, identity. since Existence wefrom of are the considering the point only ghost of massless inhas view gravitons the of anyway. an above We rendering effect just theory the wanted remotely-analogous issee longitudinal to however [46]. to degrees stress unimportant of the that for freedom mass. physical, the For bound-state non-linear of completions gravitons of the theories of the type (3.27) represents a Bose-Einstein condensate of zero-momentum degrees of freedom, onein of a them number being oftwo a transverse equivalent ways. ghost tensorial (withlongitudinal polarizations For the example, polarizations, of wrong the the and signas above Pauli-Fierz of thus, equation a massive the viewed is full graviton, kinetic as( the term). obtained linear a effective by theory This equation full integrating can it satisfied theory out be by is it the seen the equivalent cannot to be a ghost-free massive [45]. theory with Indeed, a viewed “wrong” (non-Pauli-Fierz) mass term, the two cases for is the underlyingparticle quantum picture ofthe this occupation evolution. number exactly Take by a one process unit. in At which the the same initial time longitudinal gravitons of the masslesson theory shell. became propagating degreestheory. of freedom In and particular got a probe particle gravitons are as legitimate constituents of the condensate as the on-shell massive bosons. which has a de Sitter solution, We now deform the theory by adding a mass term, so that (3.25) becomes, same time for Thus, the constituenttheory, gravitons are at that the are same off-shell time on-shell from from the the point point of of view view of the of massive the one. massless The de Sitter solution of the massless theory is now deformed to JCAP01(2014)023 , 2 ) P which (3.31) (3.32) (3.30) (3.29) i HL , the two H N = ( N | a α + of the actually a , | i H H N h ,N in φ | particle scatters with one , )) φ x )) (˜ x ˆ h (˜ ( h ( µν quantum depletion T µν ) T in the amplitude). The rate of the ˜ x ) ˜ x H. − H − x N = ( x ). This is not surprising, since in the large- √ ( H, 2 H ) ) -graviton state, this gives enhancement factor − µν,αβ H N = Hr µν,αβ – 16 – . The gravitational coupling is thus, ˙ αN ∆ ))∆ E E ˜ x x H (or ( ( 4 d φ 2 given by (3.14). Obviously, this is nothing but the first ( ) Γ = ( Z µν µν Ht h ˜ xτ ) = 4 . So overall the (3.31) is suppressed relative to the linear 2 x ) ( xd 4 µν d Ht in) (3.11 and taking the matrix element, h ( . However, this enhancement is overcompensated by an additional 2 P Z µν | H L ˆ h 2 H N ~ P H √ L ,N f limit, imitates motion in the background classical metric. φ h )) is evaluated on N in the rate (equivalently of order with x and has an energy-transfer (˜ h H µν ( N h N G µν T To summarize, we have outlined how the quantum scattering at the constituent gravi- The lowest order non-linear correction to the classical metric can be recovered by sub- limit the matrix element (3.31) reduces to a matrix element describing the scattering of -quanta in an effective classical metric 4 Particle creation asWe non-vacuum now process wish to discuss,familiar the particle quantum creation process, processparticular, which on in for a the curved the semi-classical background, deIn such limit the Sitter as recovers case case, the Hawking of evaporation. weor inflationary universe, In shall curvature analogous recover perturbations process Gibbons-Hawkingunderlying recovers [21, [20] creation physical24]. of phenomenon. particle gravity creation. In waves However,on [22] all a there curved the background exist standard is cases a avacuum vacuum [20–22, processes process. fundamental24], are difference The the crucial possible,Instead particle in difference particle creation in since the our creation our case in is vacuumexisting our that particles is no case such of is Minkowski the the with condensate. consequence globally-defined Only of time. in a unphysical limit of strictly infinite tons, in large- nonlinear correction to the classical metric in the weak field expansion. This transition corresponds toof a the quantum constituent process gravitonsout in via of which one the the virtual condensate. gravitonThe exchange, This occupation however process number without is of knocking gravitons describedmatrix it is by element not the is changing second in now Feynman this diagram bilinear process. in in So figure1. creation the relevant and graviton annihilation operators where stituting process is thus, which exactly matches the rateSitter of branch. the energy loss due to redshift on a classical expanding de amounts to a factor contribution by a factor ofN order ( φ suppression factor order in Thus, for the energy loss by the incident particle in time we get, since the graviton creation takesof place on order an JCAP01(2014)023 , H (4.1) (4.2) factors is ~ / , 2 i . H 2 H ~ N / | 4 )) H x (˜ 1) ˆ h . That is, the graviton ( − quanta is larger than that 8 H µν φ H T ∼ ) 1. This is because the matrix N by depletion. Of course this change ˜ ( x H ∼ H − N . As we shall see, this is the reason finite. This is in practice the limit where N 2 P x φ ( L N H p N 2 H is always finite, for long time-scales our i ∝ µν,αβ 1) H N − N ))∆ | limit with x H H ( N a ˆ h ∞ ( ( – 17 – H = H a µν k N N − ˜ xT p + H 4 a k xd + 4 d + H a | Z | H H N k N − k − H . Leading order process responsible for the particle creation. 1 H , 1 -particles, the new depletion channels open up, since now gravitons can k , φ + k factor is coming from the couplings, whereas one of the + H 1 ~ H h -s. Correspondingly, if the occupation number of / 1 Figure 2 φ h 2 P L is the momentum transfer in this process. For momentum transfer of order k When the graviton condensate is not pure, but contains the admixture of a second The leading contribution to a particle-creation process comes from two-in-two scattering, Both processes contribute at the same order into the particle creation.This So transition for is definite- described by the following matrix element Strictly speaking Gibbons-Hawking effect implies that a geodesic observer in de Sitter space will detect 8 Gibbons-Hawking computation is done. Bose-gas, of some re-scatter at absorbed by thecondensate propagator. leaks one Thus, gravitonThis the is of depletion the Hubble rate, physics wavelength behind is per the Γ Hubble Gibbons-Hawking particle volume creation per Hubble time. of gravitons, the depletion will be enhanced by a factor will not lead to any form of evaporation in the thermal radiation. This thermalreason radiation is however does because not Gibbons implywe Hawking a are thermality priori is describing any not in form breaking this of any evaporation. section de implies The Sitter main a symmetry. net By contrast change the of effect the value of where element on two creation and two annihilation operators is The extra behind the enhancement ofgravity the waves [22] curvature or inflationary Gibbons-Hawking density particle perturbations creation [21] [ 20]. relative to during which the two constituentout gravitons re-scatter of in the such condensate acontact and way four-graviton that becomes vertex, one propagating. becomes or Such pushed via a one process intermediate can virtual beness, graviton achieved exchange. let either by us a figure2. discuss the second process. The corresponding Feynman diagram is given in descriptions become identical. However,picture because gives dramatically different result. the above matrix element is of order JCAP01(2014)023 1  (4.3) limit N Hr in (4.3). 2 H . 2 r j n i n 2 H − = ij , h = 0 – 18 – i 0 , h 2 r 2 H − = 00 h near its center. Such a sphere is neither a black hole nor de Sitter, but the short In this respect, we should not be confused by the fact that the de Sitter also can be We must comment on the following fact. The above picture also explains why in classical The above makes clear physical sense, since a static patch of de Sitter metric for Obviously, in such a classical metric no particle creation out of vacuum should take The fact that the particle creation is not a vacuum process, has far-reaching conse- 1 − written in a static patch, which is obtained by taking the positive sign in front of The reason isby that energy the conservation, constituent theirparticles. gravitons re-scattering in cannot In such result contrast,constituent a into in gravitons creation case time-dependent of carry carry metrics positive-frequency non-zeroconstituent zero this gravitons frequencies. obstruction frequencies. is does Notice, not Thus, posite not an that arise, momenta obstacle absence even since for ofcannot the if depletion, carry momenta initial since the in gravitons opposite created the process energies, had particles since and zero can in all carry momenta. our the picture op- particles the However, are particle the real. creation final is not gravitons a vacuum static metrics with globally-definedhappen. Killing time In the a quantum condensate creation of of longitudinal particles (and/or does temporal) not gravitons that in large- place. This semi-classical factlongitudinal is gravitons explained cannot in create oura particles picture time-dependent due by metric to the leadsexplained fact lack to that in of particle the our positive creation corresponding picture frequencies. at by arbitrarily Contrary, the short possibility distances, ofquences and depletion for this into nullifying positive is the energy blackas modes. existence hole of paradoxes a as measurable well quantumde hair as Sitter, [4]. for we predicting Before would discussing newas like implications properties, vacuum for to such versus inflation confront and non-vacuum the process. two philosophies of4.1 thinking about particle-creation Compositeness andIt the is nature pretty of obviousglobal time that time Killing in vector, any thecase quantum when vacuum field we pair-creation quantize processes theory fields arethose defined in cases, forbidden. space-times on since without This a the a is notion globally-defined space-timeent not of time local possessing the vacuum Killing definitions cannot vector. a by be Bogoliubov In Typical, globally transformations although defined, that different, we eventually need lead examples to to areof connect real vacuum the pair-creation. differ- pair-creation de is Sitter aarena and semi-classical the in phenomenon black for given hole which andetry. metrics. the quantization geometrical This Geometrical is space-time sort back defined, reaction accordingly, relative to to these such processes background is geom- extremely hard to define within this reproduces such classical staticnot background take metric, place. thestatic particle AdS This metric, creation is by for depletion example does true for the graviton condensate that corresponds to But, this seemingly-innocent change makesthe a space-time dramatic is difference, no since longerthe written globally-defined time-dependence. in and this the gauge extension Thegravitons beyond corresponding same the to horizon applies these introduces by two to spaces depletion the carry follows. non-zero black frequencies holes. and particle The creation is constituent very longitudinal similar toH the metric created bywave-length particles an near uniform-density the static center sphere cannot of tell radius the less difference than from the de Sitter space. JCAP01(2014)023 from a 9 – 19 – An obvious approach to overcome these difficulties is to try to track this complicated Indeed, as already stressed, the compositeness approach to quantum geometry relies on In this approach quantization is defined relative to Minkowski space-time, while geom- Within this frame an obvious question is to understandThe the discussion graviton for condensate the coun- concrete example of de Sitter space-time in the previous section The crucial point of this result is that what in the semiclassical approximation appears For a recent discussion on this sort of puzzles see [47] and references therein. 9 dynamics using an auxiliary systemthe with goal a well-defined of global the Killinga holographic time. dual duals This field of is gravitational in theoryphenomena, essence systems, such defined where as on the black aa hope holes, globally-defined space-time where is Killing we to with time lack describe, afor (once vector. in globally-defined processes we Actually, time, include the that thedefined the touchstone semi-classically inner of Killing gravitational and holography seem time outer is (as, to“holographic space) to dual”) for be account where instance, time the is Hawkingapproach associated radiation) consequence to with in a gravity of global terms is the Killing of an vector. lack a alternative Our attempt theory compositeness of to (a a address so-called globally- this apparent conundrum representing geometry in termsquantization is of defined condensates relative ofwith to off-shell a Minkowski geometry gravitons. globally-defined where time-likeconserved In among Killing energy. other this vector things approach The andcondensates we the of only therefore count off-shell with ingredient gravitons that a neededThe collectively notion well-defined behave to of as notion “on-shellness” define solutions is of to required curvedmodes Einstein’s not equations. geometry for of the the is constituent gravitons, condensate to but state, for work the i.e., collective in with a manneretry that is is condensate-state-dependent. defined asthis collective point phenomenon of of viewof the is energy graviton that and condensate. we energy-conservation can in The use order key to standard advantage account quantization of for rules theterpart as physics of well of those as curved classical the geometry. geometries key that notions lack a global Killingalready time. contains the clueglobal to Killing the time answer.nomenon is only Indeed, the depends the on spontaneous condensate thethose quantum scattering counterpart condensates processes depletion. have of among non-vanishing the number thefrequencies. As lack of constituent gravitons off-shell already of The provided gravitons lack stressed, possessing ofby non-vanishing this the global phe- semiclassical time limit for of the the corresponding quantum classical depletion geometry rate. isas determined a consequencecomposite of picture the becomes lack simply aThis of depletion means a due to that globally-defined scattering theby among time entanglement constituent the (i.e., gravitons. of geometry, but the the insteadunderlying vacuum created depends pair the pair on creation) quantum is the particular depletion. notcurved in geometry dynamics predetermined as of Note condensates to the of again be scatteringto off-shell processes quanta that maximal a instead background the of curved defining pivotal on-shell geometry. ingredient quanta relative lies in defining semi-classical frame. Indeed, anytime, back and reaction therefore, modifies, we should among re-accommodate other the things, rules the of notion quantization of in each step. new angle. JCAP01(2014)023 . φ (5.4) (5.1) (5.2) (5.3) , and ~ m  matrix [48–50]. H , the classical S 2 φ 2 ) ~ potential controls the m/ 2 ( 2 1 φ 2 m , explicit. Classically, it is well , which we shall denote by ) =  = φ 2 ~ ˙ ( φ , and the scalar field undergoes V V ~ m , + 2 inflaton φ  = 0 . 2 2 In this case, we can consistently define φ  ) H 2 . P ~ m 1  10  − /L ~ m   H  = 2 P ~ R . In the quantum picture in which background + 2 e – 20 – L ˙ φ N = ( = H 2 N  + 3 ˙ a a ¨ φ  ≡ 2 the friction due to Hubble expansion is dominant , is easy to estimate and is given by H p 3 , which for the particular case of − 2 /L ) H ~ , the friction term is small, H √ p ≡ ~ There is no consistent limit in which one can recover eternal de Sitter ( /L 3 H /  2 ~ R φ m √ ≡   ) is the scale factor and dots stand for time derivatives with respect to the cosmic φ t ( a . For the time being we have kept the Planck constant t For Our prescription is to think about this seemingly-classical system quantum mechanically We shall now discuss howIn this order picture to comes regularize about. de Sitter we are forced to introduce other type of quanta into the Assuming the simplest form of the scalar potential This way to proceed will allow us to avoid the problems inherent to the lack of de Sitter 10 the universe inflates. The departureparameter from the de Sitternumber state of is thus e-foldings, measured by the slow-roll as the mixture ofregimes. the two Consider Bose-gases.Hubble the volume, Let oscillating us regime compute first. the occupation numbers The in occupation the number two of gravitons per evolution of the systemtions with homogeneous scalar field is described by the usual set of equa- and where time known [29] that this systems exhibits the twodamped well-understood regimes. oscillations.other In hand, this for regime the Universe expands as matter-dominated. On the game. Namely, we need tocosmic introduce a clock. cosmological fluid This that role plays is the role played of by a a homogeneous scalar field, the 5 De Sitter andIn inflationary order universe to beularize” able de to Sitter interpolatethat by between asymptotically replacing de evolves it Sitter towardsthe with and Minkowski. notion inflationary a of spaces finite constituenta we gravitons lifetime remarkable shall and fact. quasi-de “reg- applywith Sitter our well-defined cosmological reasoning. classical space In metriccompositeness doing is descritipn taken so into we accounttheory shall only that inflationary discover spaces in ofis the finite inconsistent. classical life-time can limit Not exist.mological would surprisingly Any flow constant this to problem, observation aparameters. as has state well important with as implications a for both constant post-inflationary for positive measurements cos- curvature of cosmological Thus, in our quantumquanta: picture gravitons we and inflatons. arequantum critical forced In point to order let to consider us understand a study whether state the such system composed a in of fluid more two can details. types be of at the JCAP01(2014)023 in φ (5.6) (5.7) (5.8) (5.5) ), we N φ ( relative V 1, the inflaton classical  < in the following instructive ηH N = -quanta attract gravitationally . Thus, the occupation number 2 φ . φ 2 2 class and m ~  ! φ P H = φ φ , L N R ˙ φ N N √  n .

= = 1 separates the inflationary regime from √ H m ) ~  = H m φ = ~ and ( – 21 – N 3 H = 1 is that at this point the occupation numbers φ = R α  φ φ H n N N 1. Implying that, in the matter-dominated phase, the = = , we have the relation, φ  2 ) N class φ ~ / ! N P ˙ φ N N α

mL ) is the second standard slow-roll parameter. We see that understand- = ( ~ φ /V α 2 P M 00 phase transition for the inflaton Bose-gas to Bose-liquid. It is obvious that V ( ≡ 1. Thus, classically, the critical value η The second quantum meaning of Let us now see what happens in the inflationary phase. Classically, inflation takesQuantum-mechanically place this critical value hasFirst, two from very eq. interesting meanings. (5.7) we observe that this value is at the same time the critical point Viewed from this quantum perspective inflation reveals a new identity. What is usually Generalizing the counting of the occupation numbers for arbitrary potential  < quantum -condensate towards clamping. form the system cannot maintainof itself the at redshift thisφ due point to for the any significant expansion, time as scale, well bothof as because inflatons because and of gravitons the are quantum equal. instability of This the is clear from eq. (5.6). For for of seen as a classical exponentialwhenever expansion can the be inflaton regarded occupation as a number quantum dominates! reaction of As the system we shall see, the system also tries is measured by thewith slow-roll the parameter. strength Notice that since Thus, the criticality parameteroscillation of phase the we inflaton have Bose-gas of Bose-gas scalars is is thewell-known over-critical instability and slow-roll towards must gravitational parameter. be clamping of unstable.language In non-relativistic of This Bose-Einstein matter, the is condensate which amounts in indeed toinvariance the the instability due towards case. breaking to of This formation translational is of the localized lumps (so-called bright solitons) that collapse. matter domination. occupation number starts to dominate and the system reacts by storingcan the access write of the classical time evolution equations for where ing inflation in termsthe of slow-roll occupation parameters. numbers,change sheds As of a we graviton new see, and light inflatonof these on occupation the parameters the numbers rates control respectively. physical becomes the Inflation meaning order rate ends of one. when of either On the other hand theper number Hubble density volume of scalars is is given by form of the vacuum energy which drives inflation. Therefore we observe that the ratio of the occupation numbers, JCAP01(2014)023 number = 0, whereas , which as we finite P /N ,L whereas the Hubble ), and one considers = 0 t ( ∞ ~ a = N are finite and the background ) and t φ ( is still infinite at the expense of N φ N and N . In such a case, the background is still → ∞ 2 – 22 – φ 2 ) , but do not capture corrections in 1 ~ ~ m/ is kept non-zero, but ~ Finally, the fully quantum picture is achieved when all the Now are kept finite. In this case = 0). Notice that Hubble is still fixed to be finite at the expense m N Classical description corresponds to the limit: G and N ,G ~ = 0 (i.e., P = are kept fixed. In this limit the number of constituents L It is very important to understand that option 3) cannot be reproduced within 2) even In our approach, the would-be classical background is resolved and treated as a Bose- In order to understand how our approach incorporates the standard picture as a par- In order to build-up confidence in the power of our picture we shall go step by step, , m, N if one takes into accountamount all to the the higher expansion order in non-linear powers effects of of perturbations. Such effects of constituents. shall see are extremely important. is no longer a classical field, but becomes a fully quantum entity with large but classical, but the quantum fluctuationsthe on standard top analysis of has it been are done. permitted.3) This is Fully the quantum limit picture. parameters, in which Einstein condensate. This impliesof that small perturbations what from inthe the the Bose-condensate. vacuum, standard in During treatment this our appearsto depletion, case as occupy some is the creation particles represented higher are bystrong excited pushed quantum when quantum out depletion levels. of the of the condensate Asresolve condensate explained is the above, at the the compositeness depletion criticalparticles is of point. out especially the of Specifically the for condensate,classical vacuum. an result. the Of observer depletion course, that in cannot looks this as approximation, we quantum recoverticular creation the limit original it of semi- would be convenient to distinguish1) the Classical following limit. sequence of limits. of taking energy density large, by ( small quantum perturbations on top ofthe this quantum-compositeness classical of background. the Therefore, backgroundquantum in remains effects this unresolved treatment are and unseen. alla the As special corresponding we limit shall of see, our this full standard quantum computation picture. can be recovered as G radius stays finite and the system becomes2) classical. Semi-classical limit. taking by first showingapproximation. that it Then, we reproducesnot shall all captured go by the beyond the known this standard inflationary approximation treatment. and predictions discover as5.1 effects a that particular are Quantum origin ofAs inflationary a perturbations firstinflationary consistency predictions on check the spectrum ofthe of density perturbations. our perturbations In is quantum the computedtreatment portrait standard within picture the the we of semi-classical background inflation treatment shall [21, is22, reproduce described24]. by the In this classical well-known fields, to restore the balance, bythat depleting limits the the quanta duration stored of in inflation. the background. It is this depletion JCAP01(2014)023 (5.9) (5.13) (5.10) (5.11) (5.12) , whereas and corre- H R H ~ = g ∼ λ . The similar coupling in the counts the number of graviton- 2 P L N 2 φ H N = 1, the above rate can be rewritten as = H ~ N, . φ, R φ α ~ φ λ . The rate of such a depletion process is, αN + N N λ 3 H N 3 H 2 N g H R ), the depletion is dominated by two-particle H ~ R δN → = Hα = ∼ /N = – 23 – φ 1 = V gφ + δρ Γ ∼ = gφ g Γ ρ α the dominant contribution into the depletion comes from N  φ N a vacuum process in which the background only plays the role of an and where the combinatoric factor 2 ) not H P L is the number of the depleted particles (both inflatons and gravitons) of wave- is counted as the de Broglie wave-length of constituent gravitons. The density λ = ( H . This quantum depletion takes place because the background gravitons and inflatons R λ δN α Hence we shall start from 3) and recover 2) as an approximation in which background The background energy density rewritten in terms of graviton occupation number is Thus, the dominant contribution is given by the re-scattering of two constituent particles Notice that the de Broglie wave-length of the background gravitons is Observe, that the gaviton-graviton and graviton-inflaton scattering rates only differ by spondingly the effective gravitational coupling is inflaton-inflaton scattering is negligible. a multiplicative combinatoric factorin that the is two counting cases.the the graviton-inflaton Since number scattering, of available particle pairs with the characteristic momentum transfer where inflaton pairs. Using the near-criticality relation, quantum compositeness effects arelogical neglected. observables. Next we shall compute5.2 the standard cosmo- Curvature perturbations We shall first computeframework the the source inflationary of density density perturbations perturbations in is our quantumgiven language. depletion by of Within the condensate. our where perturbation then is given by, where length re-scatter and areperturbations pushed is out ofexternal the catalyzer condensate. ofare Thus, particle-creation actually in out emptying our the ofthe picture, background constituents the reservoir. generation is vacuum. of extremely Since the weak the Instead, ( gravitational the interaction among created particles in the condensate during which onehas gets two components, pushed to there a are higher threeinflaton energy types and level. of graviton-inflaton. possible Since scatterings: the condensate graviton-graviton, inflaton- scatterings. for inflatons it isprocesses essentially with infinite. graviton participation, Because in of which this the the momentum re-scattering transfer is is dominated by the JCAP01(2014)023 2 2 φ N N (5.17) (5.18) (5.19) (5.20) (5.21) (5.22) (5.14) (5.16) (5.15) . φ N N inflation, we can express P 2 φ 1 2 . NL φ m √ N N − . . , H e φ = . P = N N N  L . 2 2 gφ φ H. H √ p 3 H e H. 1 Γ = P N N 1 P R N H L − = P L P = H L = = = L ~ – 24 – gg = 2 2 T Φ 2 H r Γ T δ δ = Φ = δ , with the corresponding energy density δ φ Φ ≡ quant δρR N δ N | δρ r φ = ˙ = . Thus, we have, N Φ φ φ δ = δN . Such enhancement factor for the black hole case counts the φ N/N N = N quant | ˙ N δN by the factor gφ The tensor mode of density perturbations comes from the graviton depletion but this Applying the above results for a particular case of From (5.14) it is clear that the number of depleted quanta per Hubble time with Hubble in terms of theand number (5.20) of turn e-foldings intoin via the terms (5.3). of well-known the semi-classical In number expressions these of for e-foldings. conditions cosmological equations Namely, observables (5.16), (5.19) time due to graviton-graviton re-scattering andrelative therefore to the Γ corresponding rate is suppressed Correspondingly, for the power-spectrum of the tensor modes we have, Thus, for the ratio of tensor to scalar perturbations we obtain, and Notice the similarity within the the depletion enhancement equation factor ) (2.3 for a black hole. The difference lies Expressing the classicalter values) (5.6 of we recover the the occupation well-known expression numbers for through the density the perturbations slow [51], roll parame- wave-length is given by number of available lightaccording particle species, to which (2.5). increasesquanta the enhances Similarly, number the in of number case depletion of channels of depletion channels. inflation the large occupation number of inflaton This depletion rate can benumbers translated of into gravitons the quantum and evolution inflatons equation as, for the occupation Notice, that since neithercarry tensor inflaton helicity, quanta the above norturbations. expression the contributes In only longitudinal order to to gravitons thepotential evaluate in scalar this produced mode the contribution, by of we condensate the simply density per- have above to energy compute at the Hubble Newtonian scale. The answer is given by, JCAP01(2014)023 . H clock (5.26) (5.23) (5.24) (5.25) accord- (equiva- φ N , which has /N 2 Φ φ quantum N and dlnk dlnδ N 1 = − s n , clock. These clock works . ! φ φ corrections and show that these ˙ N N N quantum

− , but keeping the ratio η. . . ! e ˙ P ∞ − 2 N N N L  φ 2 =

3 2 / N ' 3 2 3 s = N N , clock, set by a slow-rolling scalar field, which n = s – 25 – n − Γ ˙ Γ − 1 Γ = H 1 − classical = . Of course, this clock continues to be present in our case s according to (5.8), due to the classical time-evolution of H n φ − N , this gives 1 2 φ 2 and m = 0 limit of our picture. In our full quantum treatment, this is the N ~ = . In order to evaluate this quantity let us rewrite the rate (5.13) in terms ratio. All these effects have the semi-classical counterparts, as we have Γ dt V d 1 /N − φ Γ 1 N − ) finite. Next we shall take into account finite H e N Novelty in our case is the existence of the second, 1 = − s and corresponds to clock that increases Consequently we recover all thetum standard picture. results In as this abution computation, particular of although approximation depletion we of as havewe our taken the have quan- into source ignored account of the the effect particleignored quantum of excitations contri- the this above depletion finiteness the on of background,Thus, the the nevertheless, background we itself. reservoir, effectively treating In workedlently other it in words, as we the an have limit infinitecorrections capacity of reveal source something extremely of important particles. about inflation. 5.4 Quantum corrections The key new ingredient thatoriginating our from picture the brings compositeness intoment of inflation of the is inflation background. an there additional leads In to is the a standard a classical semi-classical single, decrease treat- of demonstrated by explicitly recovering semi-classical observables in the previous section. to be evaluated atn the point of horizon crossing. This definition for our case translates as Correspondingly, this classical clock isdue also to responsible the for classical producingdecrease decrease the of of tilt the in momentum the transfer spectrum in the depletion process and also the differently from the classicaling one, to and (5.14) unlikeunheard due the off to in latter, quantum the decreases depletion conventional both treatment, of since the in background semiclassical approach condensate. the production This of is something of occupation numbers according to (5.14), 5.3 Tilt The tilt can be computed according to the standard definition, Taking the time derivative, we get Accordingly, if we takenumbers through into (5.8) account and onlythe ignore standard the result, their purely quantum classical depletion evolution (5.14), of we the immediately occupation recover In particular, for JCAP01(2014)023 , the (5.30) (5.29) (5.27) (5.28) N = 0. In this limit = 0 is inconsistent. ~  , depends on the number of e N . , 2 3 , e   N 1 N 1 ∆ N  − 1 √ − = < N. η − φ   N  – 26 – 3 2 H Total H N = = ∆ = φ φ ˙ N N N ˙ N N ∆ inflation the decrease of the occupation number in the background arbitrarily small, since eventually the depletion rate will blow up. 2  φ 2 m , since the beginning of inflation! e = N V . This equation, already gives the first warning, that for finite number number of e-folding since the onset of inflation is given by, e ∞ N = N We can distinguish two types of new quantum contributions. The second terms in the r.h.s. of equations (5.27) and (5.28) come from the quantum We can identify at least two distinct sources through which the prior expansion history is In this section we shall focus on the effect of quantum depletion. The story is nicely This equation shows that the depletion of gravitons and inflatons is a cumulative effect. We shall comecosmological back observables to assuming this that the implication, bound but is in satisfied. the meantime let us discuss the effect on Notice that since the depletionbe is removed a by cumulative effect any over re-summation.turbative many treatment. e-foldings, It the is In bounds a particular cannot consistency it bound implies that and eternal not de an Sitter artifact limit of some per- This immediately implies the upper consistency bound on the total number of e-foldings as, depletion and have the oppositelimit effect. These terms vanish in the classical limit, since in this perturbations is a vacuumthe story process is and very there different,subjected since is our to no background depletion. is notion ais The of finite a most reservoir background cumulative of interesting effect depletion. gravitons and thingThat and thus For is is, inflatons encodes us that the information quantum the aboutprior state depletion the e-foldings, of of entire ∆ an history the of inflationary background the patch Universe. at some and describing the timeinflatons evolution respectively. of The the firstevolutions occupation terms due of number to the slow-roll. ofpation r.h.s. These the number of evolution of background both increases gravitons. equations gravitons thealso The stands and Hubble occupation increases, for radius number but the of and classical up. slower the thus than inflaton the These field gravitons, occu- classical-evolution per sothe terms Hubble equations that are volume (5.27) the of and graviton course (5.28) non-zero occupation evolve into in number (5.8). the catches limit encoded in this quantum statediscussed of later the and background. the Oneinto second source account one is these entanglement, is whichpatch, two the will way depletion effects, be beyond of an the the last observer background 60 can gravitons. e-foldings. scan Bysummarized the taking by entire the history following master of equations the inflationary during ∆ number of e-fodings cannotit be is arbitrarily inconsistent to large make without running into inconsistency. Indeed, For example, for JCAP01(2014)023 (5.34) (5.31) (5.32) (5.33) coming inflation e via (5.27) 2 N φ 2 N on m φ and N φ N and N , e-foldings since the beginning ) this change can be expressed N 1 e φ √ ( . N inflation this gives, . e − V dt. 2  N φ = N N 1 H 2 √ √ φ √ f m N t in − N t − e Z 1 N 1 η , which makes the above correction negligible N − − 12 − = =  = 2 3 φ – 27 – accumulated throughout the inflationary history. = 10 N total = φ ) s s N N n n = ∆ quantum ) − − s N and 1 n (1 ∆ − N ∆(1 GeV, corresponds to 13 10 ∼ H ~ Given the explicit form of the inflationary potential The change in the occupation numbers (5.34) must be taken into account when comput- Perhaps from the observational perspective more interesting is the second type of contri- The first one arises due to additional functional dependence of Using a concrete form of the scalar potential, the above correction can be translated in The value of the inflationary Hubble parameter favored by the cosmological measure- as the function of e-foldings since the beginning of inflation. For example, for ing the cosmological parameters. In particular, taking into account (5.20) the accumulated the occupation numbers ofof gravitons inflation and changes inflatons according afterbackground to ∆ the contains equation less (5.29). gravitonseven and In if other inflatons the words, as classical the compared evolution would-be to of de the the Sitter Hubble beginning is of frozen. inflation, bution that has a cumulativethe nature and total thus number can have ofables, a due e-foldings stronger measurable to is effect the large. provided depletion This of contribution affects the cosmological observ- for last 60 e-foldings.e-foldings and Nevertheless, it eliminates haspotential the a as eternal well fundamental de as value eternal as Sitter inflation. it as restrict a the consistent number limit of of a slow-roll inflaton The cumulative quantum change ofmoments occupation is numbers given between by somegiven integrating initial by, the and second final terms time in the equations (5.27) and) (5.28 and is terms of the number of e-foldings. For example, for ments, from the secondtreatment. term This in dependence gives the an r.h.s. additional, quantum, of contribution (5.27) into the and tilt, (5.28), which is absent inBy taking the into classical account in (5.24) the full quantum time dependence of This correction into the tiltfoldings is towards local the in end time ofthe and inflation. standard it There contribution is are measured given fewSecondly, by by it remarkable the the things reinforces remaining first about the number term, it. ofexceeds upper e- it the First, bound semi-classical is unlike (5.30) contribution more as onpossibility soon important the of as at number eternal the the of inflation bound [52–57] is e-foldings, earlierplateau violated. or since times. region. any it Again scalar it starts potential excludes classically to the permitting a constant and (5.28) we obtain the quantum-corrected tilt JCAP01(2014)023 (6.1) (5.35) (5.36) (5.37) -parameter r = 0 (equivalently e-foldings will give P 8 L − 7 ) is given by φ = 0, the occupation number ~ N/N . . is thus, =fixed, but take    r 3 2 ~ e , N N, ) finite. Classically, such a field becomes N N N ∞ φ ∆ inflation we get ∆ ( ∆ 2 = φ 2 − ) there is nothing wrong in taking the limit V N φ − N ~ , there exist no possible consistent choice of 1 2 1 N , φ  m m/ – 28 –   G 2 2 φ = 2 e = 0 ≡ N N 1 r N ω ω we are discovering that a de Sitter Universe (defined as ∆ and = = ~ r r . This is impossible in the standard treatment of inflation (where 2 r/r is the number of e-foldings since the beginning of inflation till φ 2 e ω N parameter (up to terms suppressed by ) = φ r . ( e V N is the remaining number of e-foldings at the moment when the becomes infinite and number of e-foldings can be arbitrarily large. Thus, a e N + ∆ N is given by (5.34). The corrected value of e N eternal de Sitter is fully compatible with our bound (5.30). N = The expression (5.37) is remarkable in two respects. First it reinforces the same up- This discovery may sound very surprising if one judged from the standard (semi)classical In this part of the paperWhat we we are would learning like is to that understand in a the quantum physical world meaning with of gravity number this of e-foldings is Similarly, we can consistently obtain de Sitter as a limit of the slow-roll in the semi- Total per bound (5.30) onremarkable possibility the of number probing ofments pre-60 e-foldings. of e-folding cosmological Secondly, history parameters.order-one if by For this the contribution example, post-inflationary number to an is measure- ∆ where inflation large, background with depletion it 10 is gives not a taken into account. 6 Non eternity versusOne quantum very eternity important conclusionof that Planck emerges and from Newton our constants, picture is that for the finite values metric entity) as a limiting case of slow-roll inflation isintuition. inconsistent with Indeed, quantum classically, mechanics. weFor can example, make for a scalar potential arbitrarily close to a constant. parameters that could reproduce eternal de Sitter asobservation a and limit its of possible slow-roll connection inflation. to the cosmologicalfinite constant and problem. very much limited. In short, while at the samefrozen time and keeping energy the density our system findings? evolves as Ofof an course gravitons eternal not. declassical In Sitter. the classical Is theory, there since any incompatibilityclassical approximation. with In this case, we have to keep quantum change of where ∆ Applying this result to a particular case of Notice that is evaluated, whereas ∆ that moment. SoN the entire duration of inflation would be the sum of the two numbers, JCAP01(2014)023  , , so 3 H (6.3) (6.2) / 2 − . Given P N L  are non-zero.  P and L , to be considered ~ and . ~ 3 1 ) as well as the quantum  η P H M and  quantum eternity  P , 0, in such a way that M H →  ,  φ 1 N = and and . , rec . – 29 – can be reliably deduced through their dependence φ self → ∞  to violate the bound (1.7). In such a case, any region → ∞ N for the quantum and classical changes derived from  N P ) (or equivalently and t = 0. At the same time the Hubble radius has to be kept = 1 units, semi-classical de Sitter limit corresponds to the class ( | ,M a ~ N N N G ∆  = fixed ) and . We have established certain relations between the quantum and t φ ( H N φ . Our framework enables us to quantify this breakdown of the semi- quant | φ and is finite and the bound (1.7) prevents us from ever reaching de Sitter as a N N N N 1, the quantum contribution to the change must be still negligible. In particular, and ∼ ) at the expense of N H δH ∞ However, Nature is not semi-classical but quantum and both An inflationary system can be treated as approximately semi-classical if the time evolu- Therefore, the consistency of the description requires that over the time-scale of classical stops to make sense and the only possible description is in terms of quantum entities, = η P later. 6.1.1 Non-eternity versusNotice self-reproduction that the bound (1.7)tion becomes [52], saturated before regime. one This can regime reach a in so-called our self-reproduc- quantum language would take place for of the classicalconsequences. potential In violating particular, this it excludescould bound the take regime is where place self-reproduction within excluded. or themetric. eternal inflation validity This However, of this conclusion does approximate not semi-classical has exclude description some important in new notion terms of of the equations (5.14) and (5.8) respectively.quantum If this and is the not description satisfied, in theany terms system inflationary of becomes potential intrinsically classical that entities makes allows no sense. This is the case for we must have ∆ It is obvious that with such a choice M finite and constant. In familiar that our limit on slow-roll (1.7) is alwaysConsequently, satisfied. consistent limit of slow-roll. We shall now6.1 discuss the physical Physical meaning meaning ofThe this of inflationary bound. non-eternity system bound cal is entities, characterized by such a as, set ofclassical parameters. parameters through This the set quantum includes constants classi- of Nature, such as, choice, classical description in termsthat of it the is bound telling (1.7).trans-Planckian us energy The that densities, remarkable but the thing rathersystem. breakdown it about of comes this semi-classical in bound from description is the has macroscopic nothing life-time to of the dotion of with the occupation numbers change ones, such as, the classical characteristics, thesethe relations equations enable (1.2) us and tothe (1.3) read-off classical and parameters. the subsequently quantum But, deriveclassical ones their since one, through time-evolution the inevitably (5.8) quantum thereor in description are is terms situations more of when fundamentalsuch the than as, classical the description in terms of on classical characteristics. Thatwords is, the as quantum long term as in equation equations (5.8) (5.27) is and a (5.28) good must description. be In sub-dominant. other JCAP01(2014)023 (6.5) (6.8) (6.9) (6.4) (6.7) (6.6) (6.10) is subject to φ by an amount, φ on the Hubble scale per Hubble φ , = 0 . . 2 will be prevented from rolling down the 0 H. e-foldings. This is inconsistent with the P . H φ V m class 0 ∼ 3 M . N − H H. 1 V 3 − √ r ∼ ∼ − semi P class but otherwise the space-time is assumed to be = , which means that such an Universe, viewed as ∼ 0 2 − Φ – 30 – M = δφ φ δ N V H ˙ φ = √ class + semi φ δφ δφ , but do not ignore the role of the quantum fluctuations class , this slow roll causes the decrease of 1 H δφ − H and = φ -inflation the above conditions imply, H t 2 φ 2 m = V It is useful to see wereThe the idea standard of self-reproduction semi-classical is as reasoning follows about [52]. self-reproduction Let us continue to parameterize inflation slope and the domainthis shall approach continue to it inflate isthat without assumed counteracts decreasing that the the the Hubble classicalwell-characterized role rate. by slow of classical Thus, roll quantum entities, in of such mechanics as is horizon. limited to being a force On top ofThese this fluctuations are classical assumed time-evolution,time to cause given we random by, shall jumps of super-impose the quantum fluctuations. Essentially, this conditionperturbations is (5.17) equivalent be to order one, demanding that the scalarFor mode example, in of curvature It is then assumed that in such domains the field becomes incompatible with the quantum treatment. by classical entities, such as very idea of self-reproduction.in According to pushing this the idea, inflaton thebut quantum field the fluctuations up classical play the the geometricvalid. role potential description (or of at the least background stopping space its time classical is slide-down), assumed to be still for changing these entities. In these conditions, the classical inflaton field but this is excludedby by the our quantum bound. depletion Indepleted (5.14). the quanta per above In regime, one thea the Hubble would-be evolution classical time is self-reproduction is space entirely regime, time, dominated the can number only of survive for During one Hubble time, we obtain the the following condition on the classical parameters, time-dependence because of theis following the two classical sources. slow-roll, The first source of time-dependence Demanding that the above two contributions cancel each other in some domains, JCAP01(2014)023 (6.15) (6.11) (6.12) (6.13) (6.14) where  by N . In these conditions 3 / 2 − N =  . . 1 P .  P L N. 2 m H mHL N, e-foldings looses any classical meaning. In √ H √ & . once they are super-imposed to the classical = = H /H N P – 31 – = class class the above interpretation is inconsistent. Not sur- quant − − M ˙ quant N quant N ∼ δN semi semi δH δH e N δH δH should saturate the bound (1.7),  for self-reproduction. Translating equations (6.8) and) (6.9 in terms of φ . The latter change is at most, φ N in a Hubble time, lead to the existence of domains where the average value . Namely, . By taking this impact into account, we see that the space-time that enters N N and H N does not change. As a final remark let us just mention that the above quantum difficulties on the pos- The solution to this exercise is obvious from the master equation (5.27) governing the Let us now show, that for finite N sibility of eternal inflationeternal inflation can as have the somelandscape mechanism relevant itself. to consequences fill up on the the string very landscape6.2 idea as of well as using Backgrounds on without theThe notion classical previous of analog? argument onparameter self-reproduction space was where based theperimposed on quantum on fluctuations looking the classical of forHubble change the a size in inflaton region where a field effectively innutshell Hubble — the the classical time once problem classical — with value they lead this of to are form the the su- of existence inflaton self-reproduction of field is domains has that of not it changed. requires values In of a particular, after thisdefined time classical there Hubble is radius. no possibility to characterize the Universe with a well- quantum depletion effects are dominant.argument once However, we we can have developregime a resolved where similar the the self-reproducing background quantum into fluctuations constituents, of namely looking for the evolution of the self-reproduction regime after of This shows thatimpact the on semi-classical argument leading to self-reproduction neglects a huge evolution of which means that within the Hubble time the quantum depletion reduces prisingly, this already followscorrect quantum from picture our is. boundvalues In of (1.7), order to but see we this, wish let to us translate understand what what would the be the required occupation numbers we getevolution the is relation fully (6.3). dominated Since by this the value quantum violates depletion the (5.14). bound Thus, (1.7), we the have, Let us now evaluateto the talk corresponding about change the offor classical Hubble, the horizon. assuming change that Then of it taking Hubble, still into makes account sense (6.12) in relationWe (1.2), have we to get compareclassical this jump change, of to the would be change of Hubble induced by the semi- Taking the ratio we get JCAP01(2014)023 - N /N case = 1 ∞  does not saturates =  N N = 0 and the ˙ N and what creates N e-folds. e is by no means “rolling” in N N ). However, in the case of is infinite. Only in φ ( , the value of N φ V is constant, the change of the classical N to evolve, without sacrificing the relation (1.2) φ – 32 – = 0 should be interpreted as defining a particular , is determined by the energy of the source, whereas ˙ N N depends on the time-dependence of its own energy density and φ N is much larger than the would-be self-reproduction threshold -s blows up in this limit.  φ constant, while allowing N and the classical quantities. = 0. We could interpret this result as indicating that for this extreme value of reservoir of gravitons and quantum perturbations are the result of their depletion. N ˙ N The conflict now is similar to the one we have described above, but this time the quantity The essence of this inconsistency is very clear. In the case of the semi-classical self- The key difference of our approach with respect to the standard semi-classical treatment Once we accept the finiteness of the background reservoir, the physical meaning of the Thus, we are observing that in any slow-roll inflation there is an inevitable conflict of gravitons and finite ) in a “ready made” external background potential gφ φ that is not changing is, instead of the classical value of obtained from the semi-classical reasoning.conclude Thus, that if despite this the reasoning factvalue were that of still in the valid, such we inflaton domains were field to is by no means vanishing. semiclassical variation of the classicalthis inflaton state field simply of indicatesclassical the that pre-existing we background cannot potential. condensate interpret as In representing other the words, “ball” the frozen graviton somewhere background in when a between reproduction argument we are( thinking on how quantum effects affect the motion of a ball the occupation number 7 Implications for cosmological constant In our picture a Universecomposed filled out with of any form two ingredients. ofthe potential These potential energy are: energy; represents and a 1) quantum 2) Thethe entity The quanta occupation of gravitons number the that inflaton of are field sourced gravitons, that by make this up energy. The point is that Such a background is impossible to keep constant unless we get the slow-roll parameter wechange. can This find, value of after a Hubble time, some domains where the conflict is thewe semiclassical cannot result keep of the change of the value of the inflaton field. Indeed, critical point of the background condensate itself. The mismatch between the bound (1.7) does not admit any semiclassical analog. becomes infinite in theΓ limit of frozen energy density. Correspondingly,of the inflation lies depletion in rate instead relating of the interpreting quantum them fluctuations asis to the a result the of sub-structure a of vacuum process. the background, In our picture the background bound (5.30) becomes very clear.that The the bound condensate simply can comes survive from the the quantum consistency depletion requirement during portrait we arequantum directly picture addressing there the isany quantum no external effects fixed potential external of and background. the therefore background So itself. In this the semi-classical treatment becomes valid for an unlimited time. between the classicalreservoir, and whereas the the quantum quantum clock depletesto clocks. it. run Whenever slower The we than try classicalemptied to a before make clock the certain the classical is classical critical clock clock has rate, trying any the chance to to quantum refill fill it. clock up speeds The system the up becomes and inconsistent. the reservoir is JCAP01(2014)023 H = 0 with ˙ H 2 P to zero. in time. /L P N 2 H L R = = 0 we need to limit. Therefore ˙ N N ∞ = N actually counts. N = 0. The general lesson we learn = 0. It looks that the classical and only in the ˙ ˙ H N H with have completely different origin. Indeed, the H ˙ N finite and – 33 – , fully encodes quantum gravity effects as well as a N and ˙ N ˙ H . This identification has far-reaching consequences. In finite and at the same time to have H P , is evolving according to classical general relativity, while R L . ˙ H and = 0 requires turning-off this interaction, i.e., sending N ˙ N the quantum compositeness of gravity is incompatible with the classical definition quantum degrees of freedom as well as their dynamics is left unanswered. The = 0, while the quantum description is given in terms of number N ˙ H We would like to comment that we are not the first to notice some potential issues with In many approaches to the cosmological constant problem it is customary to identify In this approach the dynamics of the graviton system is governed by graviton-graviton As we have seen the speed-up of the quantum clock is caused by the increase of the in- But, what about a “dead” cosmological constant, which isThe not obviously clue resolvable for into understanding the difficulties underlying the notion of dead cosmologi- with the de Sitter entropy [20] and in that sense to set the physical Hilbert space of = 0. The two relevant quantities ˙ quantum characterizations of aother dead words cosmological constant areof positive mutually cosmological inconsistent, constant or in the de Sitter space.have observed effects The that can authorsthe be of interpreted precise as [58] connection instabilities and ofdirection. with the [ 59, our de60], Sitter The findings space. withinaddressing is difference Although, the the not is semi-classical questions fully that treatment, eternal raised clear, our in in both the framework, [58–60]. results ordinarily-used beingan sense point unlimited microscopic, of What in time. a has we theevolves space We observe the same into see that an is precisely potential admits entity what a that of thatbecome goes classical looses de maximally-entangled metric wrong (see half Sitter description here. the of cannot next for its De section). be constituents Sitter, and Such within a the a state remaining can finite half no time, presumably longer be described if we want togive keep up both thefrom classical this description discussion in isappears terms that to of the be classical inconsistent characterization with of keeping cosmological constant as Consequently, getting approach we have pursuedgravitons in of this typical note wave-length simply identifies these degrees of freedom with soft N the system asof finite-dimensional the [61]. In those approaches the question about the nature scattering, which among other things induces depletion and sets the change of particular, it means thatavailable these set degrees of of quantum freedomstates states interact corresponding defines to gravitationally a a and small thereforecondensate standard nearly-gapless spans the infinite Bogoliubov a collective Hilbert excitations finite space. of dimensional the portion Only critical of a the sub-set Hilbert of space. But, this is consistent with having a finite value of concrete identification of what degrees of freedom the quantity flaton occupation number due tothe slow-down of classical the potential classical energy clock.This density This in that is particular inevitable drives whenever inflation meansinflation. can that be de resolved Sitter into cannot constituents. be reached asquantum a constituents? consistent limit of slow-roll cal constant isincompatible to descriptions, recognize one that classicalclassical such and description a another on intrinsically-quantum-mechanical. which notion geometry The is is based, normally is defined givenN using in terms two of potentially- the Hubble constant with one setting the classicalthe clock, one governing the quantum clock, JCAP01(2014)023 by and the 66 10 ∼ N sec. Irrespectively what happens after, this 56 – 34 – sec. In order to evolve into a non-metric state, the 10 − 60 e-foldings and all the pre-existing history is erased by inflation. − . Quantum instability with respect to depletion Near quantum criticality of the condensate On the other hand, if the consistent quantum state with half-emptied de Sitter cannot The situation in our quantum picture is fundamentally-different, because of the exis- Since within our picture both black holes and inflationary spaces are represented as The generation of the entanglement in the composite picture of a black hole was studied What is the implication of this finding for the cosmological constant problem? The • • consistency 8 Entanglement as quantumWe measure now wish of to discuss de oneinflationary more Sitter’s backgrounds phenomenon qualitatively that age different makes from quantum treatment thethe of standard de latter semi-classical Sitter approach. case, and since In about the composition the of expansion thebackground. history background This is is information unresolved, the carriedthe is only imprints very by of information limited. the last In 50 semi-classical particular, perturbations an observer on can top measure of only the be defined, thetime-scale story for will reaching change such dramatically. a state, In the this positive case, cosmological irrespective term of could the be required rejected For example, it is impossible to detect thetence total of number quantum of clocks. e-foldings. entity and Unlike the the information classical aboutparticular, case, the the the expansion history background history is is isstituent encoded imprinted a gravitons in in composite (and its form inflatons). quantum quantum ofin state. Thus, the reality what In depletion is is and an usuallyated entangled the regarded entanglement multi-particle entanglement as is quantum of a carried system. the classical byof This con- depleted background, the constituents. information about depleted Thus, the by quantameasure measuring gener- at the the entanglement some absolute moment agestandard of treatment of time of de inflation. a Sitter. hypothetical observer This could iscritical in something graviton principle condensates, fundamentally-impossible we in expectgenerating entanglement the that during they their time-evolution. share closethen This acts gradually-increasing similarities as entanglement also a in second quantum efficiencyand clock, of on which on the one other hand handscale. allows to makes scan sure the that inflationary classical history description breaks down withinin a finite [11, time 12],behind using the the generation of prototype entanglement models. are: According to this study, the underlying reasons time scale is so long that the cosmological constant problem is essentially the same. system would require the time-scale of order 10 entire story hangs uponcan a be done, possibility then ofillustrate the defining this, cosmological such let constant a us problemis assume quantum is one that essentially state the unperturbed. cm. consistently. curvature In The radius If order of initial this to an occupation initially-classical number de of Sitter gravitons state in such a state is classically, even approximately. However, thisnon-eternal. does At not the prove that momentstate it there of is is quantum either no eternity. inconsistent evidence or that such a system cannot last longer in a system depletes one graviton per 10 JCAP01(2014)023 (8.2) (8.4) (8.1) (8.3) ) energy gap. P , works against N NL ( / should not be counted Ht e . Thus, for black holes, the . . ∝ ) ) ) . N t N ) quant ( t a ln( ln( N H ln( . R BH ) H species R √ R N N ln( ) = ) = 1 ) = − N N N ln( ln( – 35 – = Γ ln( N N N √ √ P P √ quant t species L P L L N = √ = = dS t inf BH t t Bogoliubov modes of the condensate are crowded within 1 N , existing in the theory. Since the black hole depletion rate (2.5) is sensitive to the Analogous reasoning applies to de Sitter and inflationary spaces, since they satisfy both Notice, that the classical expansion of the scale factor Notice, that the composite picture of black holes allows to deduce some new properties Of course, the same arguments must apply to inflation, except there is an enhancement The roles of these items in generating entanglement can be described as follows.The The instability due to quantum depletion triggers the effect of a “quantum roulette” In [12] by analyzing a simple prototype it was shown that the quantum break time for species conditions necessary for fastand entanglement. is The unstable graviton with condensate respect is to at depletion. the critical point as an instability ofthat are the excited condensate. abovethe the This classical background, exponential time but redshift not evolution ofwith the the only the condensate probe redshifts graviton itself. particles the is condensate. Inin a probe fact, The result pure particles in of only de their our instability Sitter quantumalso picture of scattering is subjected the the to condensate quantum a thattime depletion. quantum must given entanglement be by Hence, clock, counted we with expect the that characteristic quantum pure breaking de Sitter must be What we are discovering iseternity that the of entanglement inflation. clock, just as depletion of number of extra light particle species, so must be the time of the scrambling time.N In particular the dependence on the number of light particle species, in depletion due to excessby of inflaton the quanta. slow-roll As parameter, a result the quantum break time is shortened This quantum break timelogarithmic sets dependence the nicely matches minimalto a time-scale so-called which for fast-scrambling the generation conjecture black of [36],mically holes according on entanglement. should the The scramble entropy. information within a time that depends logarith- enhanced depletion ratetime, should effectively shorten the entanglement (i.e., the scrambling) This quantum clock ofthe entanglement de works Sitter against is no eternity longer of describable de as Sitter. a well-defined After classical finite background. time, proximity to the quantum criticalwhich the point system ensures can a evolve. hugepoint These density order micro of states available originate micro from states the into fact that nearfacilitating the the critical exploration ofinstability these is large set density bythe of the graviton states. Liapunov condensate. exponent, The which characteristic is time determined scale by of the the critical depletion condensate rate is Γ given of by, JCAP01(2014)023 (8.5) is an 0 ψ , where can correspond- j j  ), correspond to the for any ,...N 0 2 ψ , × j = 1 ψ . That is, by the number of steps + 1) that can be labeled by a set ; and 2) the time-scale of maximal j 1 ( − N -particle state was representable (in , Γ j p.ent , N σ − N N ψ = cannot exceed the the quantum breaking one t . mass gap above the condensate. As we have × ent ent . p . ... /N − 2 – 36 – ψ max one t t × 1 is the time after which the representation in form of a ψ . . ent . H -particle tensor product, R max N t √ depletion steps, = 1 + 1) group. Notice, that from the finite dimensionality of the spinor N . − N max ent it can no longer be represented in the form t 1-particle state. However, it could still be represented as some tensor product -particle state in mean-field approximation is well-described by a single-particle − ent . N p N − one t The second time-scale, We must also stress that the entanglement properties of graviton condensate are inde- In this map, the Cartan sub-algebra generators, These time-scales can be visualized by the following group-theoretic parameterization The two time scales emerge. The first is the time during which the system becomes We wish to focus here on the second process. As a convenient unit of measurement we In order to gain a better understanding on the entanglement phenomenon, let us think The two important time scales of the graviton condensate are: 1) The time-scale during 1. Undergoes depletion loosing its constituent gravitons2. (andGenerates inflatons); entanglement. can take a one-gravitonquantum emission instability time, time. which This alsoof time-scale defines slow is the parameter set Liapunov is by exponent, Γ the and depletion thus, rate), (5.13 which in terms arbitrary pendent of this group-theoreticsentation description. as an A useful skepticalof bookkeeping entanglement reader tool in can which group-theoretic allows simply terms. to view visualize this the repre- depletion and creation of multi-particle states.time This (8.4). time scale tensor product is simplyscale is impossible given and by system order becomes maximally entangled. This time- irrep by no meansparameterization only follows applies that to a thedegenerate small Hilbert Bogoliubov portion levels of space located the of within Hilbertstressed 1 space de above, that Sitter the accounts for entire in nearly- Hilbert our space picture of is the finite. system is This of course infinite. introduced in [63],spinor by irrep of labeling SO(2 the black hole (in the present case de Sitter) state by a required to deplete abutplays a the half role of of Page’s the time graviton [62]. reservoir. For the black hole case, this time occupation number operators ofingly Bogoliubov assume modes, two and possible their values,be eigenvalues 0 represented or by 1. a Then basis an vector initial of non-entangled state spinor of irrep de of Sitter SO(2 can at least one-particle entangled. That is, if the initial entanglement, some appropriate basis) as a after of an inflationarywhich universe is as well a describedthe graviton/inflaton by initial condensate a starting classical metric. inwave-function and thus, some Within is our initial non-entangled. quantum Subsequently,evolves state the picture, in near-critical two graviton this condensate ways: means that which the system becomes one-particle entangled, JCAP01(2014)023 (8.6) = 70. Then, gravitons and e N 1) spinor irrep, creating ) steps the system should . − i N ) N m − =ln( steps the initial state will evolve . . .  physically-existent P 2 N m − e-foldings would imply that the total ,  1  90 e-foldings, a hypothetical de Sitter one ent m − N 1) + 1). During this map it is no longer . i ≡ | − + 30 N 2. – 37 – ent N/ N 6 = basic vectors of SO(2( -long sequence of zeros and ones, total m N Total m entangled deSitter N − non | GeV, the minimal entanglement time corresponds to roughly 30 e-foldings. 13 10 fixing the typical wave length as well as the occupation number of the gravitons ∼ H AdS R Then, during one elementary depletion step, the sequence gets shortened by one unit This group theoretic picture accounts that after Thus, expressed in terms of the Hubble parameter, the time-scale for which the infla- It is extremely important that in our quantum picture, generation of density pertur- To be more concrete, let us assume that the total duration of inflation is Observing increase of entanglement within last In particular, the created quanta cannot stay unentangled within the entireOf last course, in 60 this e- discussion we are only addressing matter-of-principle questions, with- -s. This can be visualized as j  into a superposition of minimum 2 i.e., is mapped onto a spinor irrep of SO(2( of represented by a singlethat basic differ by vector one butvectors random in by eigenvalue. the a Thus, superposition superposition after roughly every doubles. of depletion minimum Thus, step, after twoentanglement. the basic number vectors of basic number of e-foldings is at least defining the condensate. More specifically we shall consider a homogeneous condensate of tionary Hubble patch becomesor at pure least de one Sitter particleHubble, patches entangled respectively. is Notice, (8.3) that and for (8.4) the for realistic values inflation of the inflationary bations is not a vacuum process, but rather a depletion of folds. Also, observingwould imply the that maximal total entanglement duration from of the inflation is beginningout larger touching of than the last 90 observational e-folds. issues 60measurements of of e-foldings detecting the entanglement density imprints fluctuationcate in spectrum. fundamental post-inflationary difference But, between thisindicate matter-of-principle our that points and inflation indi- is standard past-transparent semi-classicalpatch. for the treatments, physical since observations within they a single Hubble 9 AdS as gravitonThe condensate compositeness approach canstant be i.e also to extended AdS to spacewith time. the case The of recipe negative is cosmological to con- represent these spaces as a graviton condensate become minimum one particlethe entangled, total whereas entanglement the must maximal be number of steps for creating observer must be ablewithin to last 60 measure e-foldings. an increase in entanglement among the depleted quanta inflatons of thequanta condensate. carry information Because about of the this entanglement of crucial the difference,the background. quanta in created our during the case, firstWhereas, few these e-foldings the will depleted subsequent exhibit noneparticle quanta or entanglement very become after little 30 entanglement. more e-foldings. and The more emerging conclusions entangled are: reaching the full one Thus, if the total duration of inflation is less than 80 JCAP01(2014)023 = — (9.2) (9.4) (9.1) (9.3) (9.5) AdS AdS α R and let us r . ) 2 P , L 2 ) AdS AdS R equal to the central charge of CFT ( / 2 3 r/R AdS − r AdS α D ) , N =  r ( ) = ( P /r AdS AdS L AdS ) = 1 sets the number of nearly gapless Bogoliubov 2 P R r N N ( ). We are thus, observing an interesting fact, 3 L  ) – 38 – )( ≡ AdS r = AdS AdS ) ( Ads N r λ ( /N AdS AdS r/R AdS N N λ ≡ ) = ( and occupation number — per unit of space of size ) r r ( ( V AdS AdS R N . This means that the AdS condensate reduced to a cell of space of size AdS R = r when viewed quantum-mechanically, AdS space is a critical graviton condensate . The criticality condition in this mean field approximation is determined by, the coupling strength between two of the constituent gravitons, i.e., 2 ) AdS AdS the space-time dimension. is critical and therefore — according to our previous characterization of holography — α As we did in the case of black holes, we can define in the mean field approximation the One of the main points of the graviton condensate model is to map classical geometry It is natural to think that this equality is not just an extraordinary coincidence, and The criticality of the condensate representing AdS can be easily understood in the /R D P AdS L The first thing thatof this AdS. counting reveals Indeed is the the mean well-known field confining potential gravitational created potential by these gravitons is simply given by, with effective coupling, equal to for into a quantum state characterizedthis by works the in graviton the occupationcompute particular numbers. the case In number of order of AdS to gravitonsFor let see the inside how us case this consider region of a for four region the dimensions of AdS we space condensate get, of defined size by9.1). ( which agrees with the confining gravitational potential in classical AdS. that leads to can be fully described into terms notice of [1,2] the that gapless the Bogoliubovthe number modes. central of In extension gravitons this per of sensehand, cell the it at of is CFT space the interesting in at critical criticality the point indeed standard agrees the AdS/CFT with number correspondence. On the other R modes of themodes graviton condensate. is conformal Because (at of least criticality, up the to physics 1 of these Bogoliubov gravitons of wave length ( namely that with number of nearly-conformal Bogoliubov modes that the two CFT’s areswallow, since actually would one imply and that we thecharge are same. discovering that a An second miraculously independent opposite coincides CFT conclusionAlthough, with wit the would we the central be one cannot hard suggested to disprovelead by such to the the an AdS/CFT conclusion optionof correspondence. that the we black the consider hole quantum it holography,the foundation is highly physics in of of unlikely. quantum-criticality AdS collective of holography, excitations the We just graviton approximately are as Bose-gas, conformal. thus which in makes spirit the of case the Wilson-Kadanoff approach to renormalization group. Indeed, if we consider the according to AdS/CFT conjecture. JCAP01(2014)023 2 P M . In /L (9.7) (9.6) (that 2 r r 2 /R gravitons. 2 r 2 P 4 L r 2 R ) gravitons of wave r ( AdS N . 2 AdS gravitons of wave length the condensate always looks, for 2 P and of r . r/R r /L M 2 i.e., the holographic counting , can be easily estimated to be r r ) = , R r 2 P L AdS 2 M /R 2 P = – 39 – L )( M , and we try to describe it using gravitons of wave- that makes the previously-defined combined system r N ( is equivalent to adding a certain amount of gravitons r the gravitons sourced by the AdS cosmological constant r AdS AdS R N , i.e., of wave length AdS gravitons of wave length ∼ R M ) ) can acquire enough energy to escape. The dominant effect , without affecting the AdS boundary conditions. r M r (  N . Let us now look more carefully into the corresponding graviton AdS r . More specifically R r , larger than  r larger than r r . For localized in a region of size AdS , we shall need (in order to preserve the same amount of energy) r R M Using this condensate model we can even figure out how soft gravitons (we are con- Before describing the criticality condition we can, already at this qualitative level, under- As a simple consistency check of this idea, we shall show that it correctly describes the Again, we shall treat this case in terms of the graviton occupation numbers. Adding a Again, according to the condensate portrait of black holes, we shall characterize a AdS , i.e., the system has effectively positive specific heat. Indeed the previous expression for sidering the case Note, that what dominatessourced in by this the expression cosmologicaltemperature, is i.e., constant. the with dragging the If created typicalr we energy by of identify the depleted this hard gravitons, we gravitons the escape observe escape that energy energy it with of scales softin the with gravitons AdS expected agrees [64]. with the Hawking temperature of large black holes corresponds to scattering processes of soft gravitons with hard gravitons of wave length stand some main featuresflat of black black holes, holes evaporation in is AdS.the understood order In in of the terms magnitude blackthe of of hole constituent quantum portrait the gravitons. depletion. of black Let hole asymptotically Morefor us specifically, temperature a apply is soft this determined logic graviton by to sourced the our by escape AdS energy condensate. of The escape energy of gravitons critical. physics of small and large black holes in9.1 AdS. Black holesIn in order AdS to move on let us now add to AdS a gravitational field produced by some mass corresponding to maximal packing, provided weis, scale once the we area introduce of the theimplements boundary scale the by factor invariance of defining the AdS gravitongroup condensate geometry). under transformations In Wilson-Kadanoff in renormalization other the words, sense AdS that geometry at arbitrarythis scale sense we can saycondensate. that AdS geometry is the manifestation of the criticality of the graviton localized in a region of size length mass black hole of mass M by the value of are harder than the ones sourced by the external mass This is exactly the same counting we did at scale the AdS geometry, holographic, i.e., as composed of gravitons of wave length system. It is composed of of typical wave length condensate at scale length JCAP01(2014)023 . AdS (9.9) (9.8) (9.12) (9.10) (9.13) (9.14) R  : scattering r M r/R . Hence, in the and hard gravitons if M r are blueshifted and r/R M are blue shifted by scattering , r corresponding to 2 P ) (9.11) /L /r . , . 2 ) 2 AdS 2 overlaps with 2 P r . r M , R ) ( L ) r N 2 2 P = α r r/R 4 4 ( L ( ) ) M M α ) M /r 3 N M ) for the soft gravitons as we did for the hard 2 AdS N 4 AdS r ) 4 N /r R ( ) AdS ( /r / ≡ M R /r r/R – 40 – 3 2 AdS ( λ ( ) r r R AdS ( ≡ AdS → = R ) M ( R ) = ( r given by (9.6), becomes is equal to the escape energy. In other words the system λ r ( M ( ≡ = ( N M M α M ) r N ( M N ) constant allows us to estimate the number of soft gravitons after r ( transfer momentum. In other words, the effect of the scattering with M the strength of gravitational coupling among soft gravitons. After λ , we expect that the soft gravitons sourced by 2 AdS /r AdS /R 2 P R L of the soft gravitons into a wave length r  r ) = r . In average a soft graviton of wave length ( (notice the similarity with the depletion process we encountered in case of inflationary α In this regime the criticality condition will be defined by imposing that the typical Now we can understand the deep meaning of the number defined in (9.10). It is simply This scattering is dominated with processes of momentum-transfers of the order of AdS AdS /R that, after plugging the value of which nicely leads to the mass-to-size relation energy of the gravitons sourced by the Bekenstein-Hawking entropy ofexpression the for large the black mass hole. in (9.10) Namely, we after get plugging the above This is indeedBefore a we need very to interesting studyThe the number corresponding criticality whose conditions critical for meaning point the will will condensate define in become the the regime clear large black in hole a in moment. AdS. achieves criticality when thewith the soft hard gravitons ones and ofimmediately their wave-length translates number is into reduced the according following to self-consistent (9.10). relation This between criticality condition This is a well-known mass-to-size relation for a large black hole in AdS! which is nothing else but the Bekenstein Hawking entropy. Keeping the value of blueshifting, namely: blueshifting we change, with consequently that its totalintroduce number the is mean reduced. field effective To coupling estimate its number after blueshifting we ones sourced by the cosmological constant, namely regime of universe due to re-scattering of soft inflaton quanta1 at hard gravitons). R processes with 1 the hard gravitons is to blueshift the soft gravitons sourced by the mass they are distributed homogeneously.wave length Thus the net effect of hard gravitons is to change the JCAP01(2014)023 = i ≡ N N is the (9.15) | d quantum ) where d log( ∼ S , the entropy for maximal limit gives similar results. AdS ∞ . = i M N N < N Ψ i ⊗ | curved background metric by the are blue shifted and its number is reduced to AdS is physically simpler. In this case the gravitons Ψ M | – 41 – . This fact fundamentally changes the physics of AdS ∞ limit from transition matrix elements of the type: X R classical , i.e., at the large black hole Hawking temperature. = = N is negligible. In other words, the small black holes in  i N MS r recover the classical geometry. In this way, a curved back- Ψ | ∼ AdS ∞ R T In particular we have illustrated this emergence for the exam- =  N . Since for large black holes ,... r i . Therefore the reduced density matrix approximately describes a N N AdS | a N ∼ + a | S and N are harder than those sourced by the cosmological constant and the effect of h N , M i of states that in N | i a | 0 | 1 One important aspect of the composite picture is that particle-creation is not a vac- The graviton portrait of large black holes in AdS also sheds some interesting light on the For maximally-symmetric spaces, such as black holes, de Sitter or AdS,The the known physics of phenomena available in the metric description, such as the geodesic mo- The case of small black holes For concrete computations, we have adopted the particle number representation N ) − limit of quantum scattering processes of the constituent gravitons. The geometric in- as defined in equation). (9.10 In the spirit of thermofield formalism [ 65–67] we can use + N a entanglement-generation in the blackparadox hole discussion. radiation giving a new twist to the information At criticality we expecttakes this place state when to be the gravitons maximally sourced entangled. by As discussed above criticality ∞ terpretation emerges inh the infinite- ples of de Sitter and AdS spaces. uum process, butduring rather which the some result ofwith of the them scattering vacuum become of process on-shell the only final initially-existing for states. off-shell gravitons This process, can be “confused” such representation ispoint. very similar to a Bose-Einstein condensatetion at and/or the the quantum critical particle-creation in a background metric, can be recovered as the N this maximally entangled puremaximally-entangled state the to Von define Neumanndimension a entropy of reduced will the thermal be smaller density givenrespectively Hilbert matrix. by space. If At state criticality is the number of degrees of freedom are sourced by the latter inAdS the do regime not differbeen able significantly to from describe asymptoticallygraviton a flat condensates great at black amount criticality. of holes. the known In physics summary, of we black10 have holes in AdS in Conclusions terms and of outlook In this paper we have pushed3]. further the compositeness The approach to key gravity concept put forward is in [1– to replace the ( ground is represented by awe quantum state can in use a the Fock space coherent of state off-shell representation, gravitons. which Alternatively, in thermal state at temperature entanglement is notion of aquantum composite constituents multi-particle is statecharacteristic played wave-length built by and/or frequencies the on set longitudinal Minkowski by (usually vacuum. a would-be off-shell) classical The gravitons, curvature radius. with role the of the corresponding quantum wave function.system Indeed of the gravitons should naturaltwo be quantum types a state of pure for gravitons state the in combined the tensor product of the Hilbert spaces of the JCAP01(2014)023 – 42 – For instance, proving inconsistency of such a quantum state, would of course provide Finally, our picture provides a quantum foundation of what can be called the emergence Applying this framework to inflationary spaces, we have recovered the known predictions Next, compositeness imposes a severe bound on the duration of the inflationary and/or The quantum physics of de Sitter that we are uncovering could provide an answer to Indeed, in our description de Sitter, within a finite time, becomes a fully quantum entity, = 0. Understanding the underlying nature and consistency of such a state is crucial for ˙ a strong evidencephysics. of On incompatibility the other ofwill hand, become the if very such positive subtle acase, and quantum cosmological state we this constant is will is proved need an with to to interesting be quantum reconsider problem consistent, the for the way future situation weof investigation. pose holography. the problem. In Inclassical particular, any gravitational it backgrounds establishes anddescription an approximately the explicit two conformal appear connection quantum astum theories. between parts of critical the In the point. (seemingly) same our entity:the The the classical Bose-gas mean-field metric, of description gravitons whereasubov at of the the modes this quan- approximate of Bose-Einstein CFT thelogic, is condensate treating condensate a represents AdS that space theory become aseffective governing a nearly description collective graviton Bogoli- gapless of Bose-gas at at the the the Bogoliubov critical modes critical point, the point. of CFT the emerges With graviton as this condensate. an The coincidence addressing the cosmological constantevident problem. that the Whatever cosmological is constant the problem outcome will of acquire this a study, new it appearance. is in simple terms ofparticle-creation scattering is of not a the vacuumof constituent process, the gravitons but and background, rather it inflatons. originatesscan from imprints However, the the some since composite entire cumulative the structure history effectsdetecting of in the the imprints the of inflationary cosmological theto patch. observables Universe’s the that history This end dating creates of way a before inflation. the (so last far 60 theoretical) e-foldingsde hope prior Sitter of states, beyond whichThe none bound of these is spaces non-perturbative canthe be and question treated cannot as about be approximately-classical. and the removed inflationary by eternity some spaces of re-summation. cannotapproximate such be classical states. This eternal description raises What in ofcould in we the the principle are usually-used background. persist sense discovering eternallymetric However, in is of description. as a eternity-valid that new we in de quantum have the state, Sitter discussed, no they longer subjectedthe to issues a classical raised ininstabilities [58] appearing and [ 59, within60].could the Some be semi-classical time interpreted treatment agoclassical of as these treatment authors de the it have Sitter problems pointed isthey background, out could, with impossible certain which for to the example,picture fully eternal be provides clarify de attributed a the to Sitter. concreteprinciple) meaning a be framework of breakdown answered. However, in these of In within whichof particular, instabilities, the we the semi- perturbative perturbation since see issues expansion. that theory the raisedphysical and Our effects by meaning. cannot are be these real. blamed authors on can breakdown Most (in importantly, wein can which order identify half theiris of microscopic the not constituent excluded gravitonssuch that have a been the maximally-entangled depleted system and andpresumably can half-depleted red-shifted what get quantum away. replaces It in state. stabilized suchN (or a As state we at the have classical least discussed characteristics, become is above, the very quantum long relation lived) in JCAP01(2014)023 ]. 30 , SPIRE IN , ][ ]. (2013) 742 (2013) 1 61 SPIRE IN (2013) 025 A 28 Class. Quant. Grav. ][ , 08 (2013) 419 arXiv:1203.3372 JHEP , B 719 Fortsch. Phys. Microscopic Picture of , ]. Int. J. Mod. Phys. (2012) 240[ arXiv:1302.6581 , Phys. Lett. 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Orlandi, particular, as suggested in [1,2, non-gravitational13, classical38], entities, it such is asrations. natural solitons to For and example, generalize other compositeness as non-perturbativefield approach shown field to of in configu- [1,2] magnetic the monopoleshole, counting is of with very constituent the similar softOther role gauge to field bosons counting of configurations in of the the can longitudinalfects Planck be than gravitons mass treated will in taken in show a up up exactly black as the by new same the 1 spirit. scalar The vacuum compositeness expectation ef- value. between the parameters of thispoint system we and cannot the claimstriking central charge and that of deserve this CFT further is is investigation. remarkable. the At origin this of AdS/CFT conjecture, but similarities are analysis. Acknowledgments It is a pleasure toRug, thank Alex Jose Pritzel Barbon, and Danielpart Nico Flassig, by Wintergerst Valentino for Humboldt Foit, Andre discussions. 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