aa r X i v : . [ g r- q c ] S e p KCL-PH-TH/2017-14
De Sitter Stability and Coarse Graining
T. Markkanen ∗ Department of Physics, Imperial College London, SW7 2AZ, UK andDepartment of Physics, King’s College London, Strand, London WC2R 2LS, UK (Dated: September 25, 2018)We present a 4-dimensional back reaction analysis of de Sitter space for a conformally coupledscalar field in the presence of vacuum energy initialized in the Bunch-Davies vacuum. In contrastto the usual semi-classical prescription, as the source term in the Friedmann equations we useexpectation values where the unobservable information hidden by the cosmological event horizonhas been neglected i.e. coarse grained over. It is shown that in this approach the energy-momentumis precisely thermal with constant temperature despite the dilution from the expansion of spacedue to a flux of energy radiated from the horizon. This leads to a self-consistent solution for theHubble rate, which is gradually evolving and at late times deviates significantly from de Sitter. Ourresults hence imply de Sitter space to be unstable in this prescription. The solution also suggestsdynamical vacuum energy: the continuous flux of energy is balanced by the generation of negativevacuum energy, which accumulatively decreases the overall contribution. Finally, we show that ourresults admit a thermodynamic interpretation which provides a simple alternate derivation of themechanism. For very long times the solutions coincide with flat space.
I. INTRODUCTION
The de Sitter spacetime is one of the most analyticallytractable examples of a genuinely curved solution to Ein-stein’s field equation. De Sitter space is not only of aca-demic interest since in the current cosmological contextthe exponentially expanding de Sitter patch is believedto describe the evolution of the Universe soon after theBig Bang during cosmological inflation and at very latetimes when Dark Energy has begun to dominate over allother forms of energy.The potential instability of de Sitter space in quantizedtheories has been investigated in a variety of different ap-proaches and models over a span of more than 30 years[1–35], recently in [36] where we refer the reader for morereferences. To the best of our knowledge, at the momentthe issue still lies unresolved. If de Sitter space were un-stable to quantum corrections and could indeed decay,this could provide an important mechanism for alleviat-ing the cosmological constant problem and perhaps alsothe fine-tuning issues encountered in the extremely flatinflationary potentials that are required by observations.Most definitely, a de Sitter instability would have a pro-found impact on the fate of the Universe since it rulesout the possibility of an eternally exponentially expand-ing de Sitter space as classically implied by the ΛCDMconcordance model.One of the main motivations behind the original cal-culation for the evaporation of black holes in [37, 38]was the discovery of their thermodynamic characteris-tics [39, 40], in particular the connection between theblack hole horizon and entropy: the fact that black holesevaporate implies that they can also be ascribed temper-ature and understood as thermodynamic objects. Like a ∗ [email protected] black hole de Sitter space also possesses a horizon beyondwhich a local observer cannot see, which was famously in[41] shown to lead to a thermodynamic description of deSitter space analogously to a black hole. Currently, thethermodynamics of spacetime horizons is established as amature, well-studied subject [13, 42–48]. Based on ther-modynamic arguments the seminal study [41] concludedthat unlike black holes de Sitter space is stable. How-ever quite interestingly, also by invoking thermodynamicconcepts in the equally impactful work [13] it was arguedthat the de Sitter horizon in fact does evaporate. As in the original black hole evaporation calculation[37] we make use of semi-classical gravity – often referredto as quantum field theory in curved spacetime [49, 50]– in order to provide a first principle calculation of thestability of de Sitter space. Our approach allows one tostudy how the quantized matter back reacts on the clas-sical metric by using the semi-classical versions of Ein-stein’s equation. In situations where the quantum natureof gravity is subdominant this is expected to give reli-able results. Specifically we will focus on the cosmologi-cally most relevant coordinate system, the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) line element de-scribing an expanding, homogeneous and isotropic space-time. This line element results in the Friedmann equa-tions allowing a straightforward analysis of back reaction,studied for example in [33, 51–59]. The FLRW coordi-nates are rarely included in discussions of the thermody-namics of horizons, although see [47, 60, 61].The decoherence program asserts that the ubiquitousdisappearance of macroscopic quantum effects – com-monly known as the quantum-to-classical transition –stems from the observationally inaccessible environmen-tal sector that in any realistic set-up is always present[62–67]. Using this mechanism as a motivation recently The argument can be found in section 10.4 of [13]. in [36] a modification to the usual prescription for semi-classical gravity was explored where in the Einstein equa-tion one implements coarse grained expectation valuescalculated by including only those states that are observ-able. It was shown that if a part of the density matrixmay be characterized as unobservable and is neglectedfrom the quantum averaging this generically leads a qual-itatively different behaviour for the expectation value forthe energy-momentum in de Sitter space compared to theusual approach: it implies non-trivial back reaction withan evolving Hubble rate, even when as the initial condi-tion one uses the manifestly de Sitter invariant Bunch-Davies vacuum. The procedure of tracing over unobserv-able states, in addition to decoherence studies, is oftenimplemented in calculations involving spacetimes withhorizons such as black holes and Rindler space [68–70]and is a key element of the information paradox [71, 72].As a continuation of the work [36] here we explorethe gravitational implications from a particular coarsegrained density matrix: the cosmological event horizon ofde Sitter space splits the Universe into observable and un-observable patches essentially identically to a black hole,which motivates us to disregard all information containedbeyond the horizon. By using this density matrix to cal-culate the expectation values via the Friedmann equa-tions we then perform a complete 4-dimensional back re-action analysis of de Sitter space with a conformal scalarfield initialized in the Bunch-Davies vacuum.Since the event horizon of de Sitter space is anobserver-dependent concept, particle creation associatedwith the cosmological horizon was in [41] argued to leadto an observer dependence of the back reaction and henceof the metric of spacetime. There it was further con-cluded that the energy-momentum tensor sourcing thesemi-classical Einstein equations cannot be defined inan observer-independent manner. Although during thetime of writing, in [41] the derivation of this energy-momentum tensor was ’in preparation’ the calculationto the best of our knowledge does not exist in literature.To a degree this gap is filled by the current work andsince the implemented coarse graining prescription is de-fined by the cosmological horizon of de Sitter space itpossesses the observer dependence put forward in [41].We will also discuss our results in the framework ofhorizon thermodynamics and provide a complete physicalpicture of particle creation in de Sitter space leading to aconsistent definition of the differential of internal energy.With this picture we are able to formulate the first lawof thermodynamics in de Sitter space, which is known tobe problematic [13], with which we show how the firstprinciple result admits an alternate derivation by usingonly thermodynamic concepts.We emphasize that as far as horizons and particle cre-ation are concerned there is no compelling reason to as-sume the arguments given not to apply also for non-conformal scalar fields, fermions and vector fields.A 2-dimensional calculation of this mechanism was ini-tially presented in [34] but here we will provide much more detail, perform also the 4-dimensional calculationand make the connection to horizon thermodynamics.Our conventions are (+,+,+) [73] and c ≡ k B ≡ ~ ≡ II. THE SET-UP In n -dimensions the matter action for a conformallycoupled scalar field is written as S m = − Z d n x √− g (cid:20) ∇ µ φ ∇ µ φ + ξ n Rφ (cid:21) , (1)with ξ n = ( n − n − , where g is the determinant of the metric and R the scalarcurvature. The equation of motion for the scalar field is (cid:0) (cid:3) − ξ n R (cid:1) φ = 0 , (2)where √− g (cid:3) = ∂ µ ( √− g ∂ µ ). The gravitational actionis given by the usual Einstein-Hilbert Lagrangian supple-mented by the cosmological constant term Λ S g = Z d n x √− g πG (cid:20) R − (cid:21) , (3)which along with (1) leads to Einstein’s equation G µν + g µν Λ = 1 M T µν , (4)where T µν is the energy-momentum tensor of the scalarfield T µν = − g µν ∇ ρ φ ∇ ρ φ + ∇ µ φ ∇ ν φ + ξ n (cid:2) G µν − ∇ µ ∇ ν + g µν (cid:3) (cid:3) φ , (5)and we have defined the reduced Planck mass M pl ≡ (8 πG ) − / .In four dimensions a spacetime that is expanding in ahomogeneous and isotropic manner and has flat spatialsections can be expressed in the form of an FLRW lineelement with ds = − dt + a ( t ) d x ≡ − ( dx ) + a (cid:2) ( dx ) + ( dx ) + ( dx ) (cid:3) , (6)which as far as we know describes the observable Universeto good accuracy. For the line element (6) the Einsteinequation (4) reduces to the Friedmann equations ( H M = ρ m + ρ Λ − (3 H + 2 ˙ H ) M = p m + p Λ , (7)where we made use of the standard definitions for theHubble rate and the energy and pressure densities forthe scalar field and the vacuum energy contributions as H ≡ ˙ aa ; ρ m ≡ T ; p m ≡ T ii /a ; ρ Λ ≡ Λ M , (8)with ρ Λ ≡ − p Λ . Summing together the Friedmann equa-tions we obtain a self-consistent evolution equation for H and the scale factor a − HM = ρ m + p m . (9)The above is a very important relation in FLRW spacesas it allows one to study the back reaction of an arbitrarymatter distribution onto the Hubble rate.The Friedmann equations (7) can also be generalizedto include quantum effects and hence be used in the diffi-cult task of determining the back reaction in a quantizedtheory. This can be done by using the expectation valuesof the renormalized quantum energy-momentum T µν ≡ h ˆ T µν i − δT µν , (10)where δT µν contains the counter terms, as the sourceterm. This approach is of course not fully quantum butrather semi-classical since the spacetime metric is notquantized. However, in cases where the curvature ofspacetime is not extreme this approach is expected togive reliable results [49, 50] and it is the framework to beadopted in this work. III. GENERAL FEATURES OF BACKREACTION IN DE SITTER SPACE
As we elaborate in section IV, when de Sitter space isparametrized in terms of an expanding FLRW metric itcan be described with an exponential scale factor a = e Ht with ˙ H = 0. From this follows an important consistencycondition: a strictly constant Hubble rate under backreaction in de Sitter space (9) implies that any classicalor quantum matter distribution must satisfy ρ m + p m = 0 , (11)i.e. it must have the same equation of state as ρ Λ . Fora conformal theory this fact alone can be shown to leadto an incompatibility of having ˙ H = 0 and any non-zeroenergy density for the matter component.For completeness we consider first the case of n -dimensions. A conformally coupled classical field withthe action (1) has a vanishing trace T µµ = 0 ⇔ ρ m + p m = nn − ρ m , (12)which can be shown from (5) with the help of the equa-tion of motion (2). So at least classically, a conformaltheory satisfies ρ m + p m = 0 only when ρ m = 0. In the quantized case the previous argument is mademore complicated by the counter term contribution δT µν ,which leads to an anomalous trace [74–77] and in de Sit-ter space in even dimensions gives T µµ = − δT µµ = 0.This however does not introduce a significant modifica-tion compared to the classical case discussed above.Any consistent prescription of renormalization of aquantum field theory one should in principle be able toexpress as a redefinition of the constants of the originalLagrangian. In curved space this means that genericallylocal curvature terms such as a term ∝ R are requiredin the Lagrangian by consistency [49]. As explained indetail in [33], the counter terms inherit the high degreeof symmetry of de Sitter background such that all al-lowed counter terms for the energy-momentum tensorsatisfy δT = − δT ii /a , which essentially means thatthe counter terms in de Sitter space may be obtained bya redefinition of the cosmological constant. From this itfollows that the counter terms and hence the conformalanomaly play no role in the dynamical equation (9) since ρ m + p m = − ( δT + δT ii /a ) | {z } =0 + h ˆ T i + h ˆ T ii i /a = nn − h ˆ T i . (13)The above equation also implies that for this argumentto hold for a conformal theory there should be no diver-gences in the energy density. When using a covariantregularisation scheme such as dimensional regularizationone may easily understand this to be true since in a con-formal theory all scales should drop out from the vac-uum terms and divergences with the correct dimensionscannot be generated. This one can easily verify withthe results of [36]. Generically, depending on the choiceof regularization some divergences may have to be sub-tracted by hand [78]. In what follows we will define thequantity ρ Sm to be the finite state dependent contribu-tion to the quantum energy density i.e. the contributionthat cannot be absorbed to a redefinition of the cosmo-logical constant. Essentially, ρ Sm contains the non-trivialphysical contribution of a given state and it is the onlyquantity needed for determining the back reaction for aconformal theory in de Sitter .Focussing on the 4-dimensional case, the semi-classicalback reaction from (9) now reads − M ˙ H = ρ m + p m = 43 ρ Sm . (14)This allows us to write a set of four conditions that can-not be simultaneously satisfied: When dimensional regularization is used one simply has h ˆ T i = ρ Sm . We note that 14 coincides with equation (3.23) of [36], where amore detailed derivation may be found. y y y FIG. 1. The complete de Sitter manifold in four dimensions,where the y and y coordinates have been suppressed. (1) A conformally coupled theory(2) A FLRW line element(3) ρ Sm = 0(4) ˙ H = 0For example, in a thermal state where ρ Sm is non-zerowith a black-body spectrum the above conditions imme-diately imply that the Hubble rate H cannot be strictlyconstant. More generally, if there exists a density of con-formal matter that may be thought to contain any en-tropy it must be in a non-vacuous state with ρ Sm = 0since the vacuum configuration is described by pure statewhich has strictly zero entropy, see [36] for more discus-sion. IV. DE SITTER SPACE IN FLRW AND STATICCOORDINATES
The topic of de Sitter space in various coordinates hasbeen extensively studied in literature, for example seechapter 5 of [49] for a detailed discussion.The complete n -dimensional de Sitter manifold can beunderstood as all points contained in the n -dimensionalhyperboloid embedded in ( n + 1)-dimensional Minkowskispace. The 4-dimensional de Sitter space can then beexpressed in terms of a 5-dimensional Minkowski line el-ement ds = − ( dy ) + ( dy ) + ( dy ) + ( dy ) + ( dy ) , (15)with the constraint expressed in terms of some constant H − ( y ) + ( y ) + ( y ) + ( y ) + ( y ) = H − , (16)which is depicted in Fig. 1. The flat FLRW form ofthe de Sitter line element can be obtained by using the y y y y H − FIG. 2. Projection of the 2-dimensional de Sitter space, − ( y ) + ( y ) + ( y ) = H − , as covered by the expandingFLRW coordinates (17) denoted with blue on the ( y , y )-plane (top) and the ( y , y )-plane (bottom). parametrization y = H − sinh( Ht ) + ( H/ | x | e Ht y i = e Ht x i y = H − cosh( Ht ) − ( H/ | x | e Ht , (17)giving ds = − dt + e Ht d x , (18)where t ∈ [ −∞ , ∞ ] and x i ∈ [ −∞ , ∞ ]. The spherical po-lar coordinates are defined in the usual manner in termsof the radial, polar and azimuthal coordinates r , θ and ϕ , respectively x i = r ˆ n i ; ˆ n i = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) ; r ∈ [0 , ∞ ] , θ ∈ [0 , π ] , ϕ ∈ [0 , π ] . (19)From the relations (17) we see that in these coordinates y + y ≥
0, which is not a property of the complete deSitter manifold. This means that (17) do not cover theentire manifold, but only half of it. This is illustratedfor the 2-dimensional case in Fig. 2. The coordinatescovering the other half with y + y ≤ Ht → − Ht + iπ in (17), which also leads to aFLRW line element, but with an exponentially contract-ing scale factor.An important feature of the coordinates (17) is thatthey cover regions of spacetime a local observer would y y y y H − FIG. 3. Projection of the 2-dimensional de Sitter space, − ( y ) + ( y ) + ( y ) = H − , as covered by the static-typecoordinates (21) and (23) denoted with green and red, re-spectively, on the ( y , y )-plane (top) and the ( y , y )-plane(bottom). not be able to interact with. This can be shown by cal-culating the maximum physical distance at a time t thatcan be reached by a ray of light emanating from the origin e Ht r ∞ = e Ht Z ∞ t dt ′ e Ht ′ = H − , (20)which of course is cut-off by the cosmological event hori-zon, which from now on we will simply call the horizon.In the static parametrization of de Sitter space onlythe spacetime inside the horizon is covered y = ( H − − ¯ r A ) / sinh( H ¯ t A ) y i = ¯ r A ˆ¯ n i y = ( H − − ¯ r A ) / cosh( H ¯ t A ) , (21)which is explicitly borne out by a singularity in the lineelement ds = − (cid:2) − ( H ¯ r A ) (cid:3) d ¯ t A + (cid:2) − ( H ¯ r A ) (cid:3) − d ¯ r A + ¯ r A (cid:0) d ¯ θ A + sin ¯ θ A d ¯ ϕ (cid:1) , (22)where ¯ t A ∈ [ −∞ , ∞ ] and ¯ r A ∈ [0 , /H ], and throughoutwe will use underlines to distinguish the static coordi-nates from the FLRW ones. In particular, ¯ x i describesphysical distance and x i comoving distance. We can eas-ily verify in the static coordinates that the time it takes for a light ray to reach the horizon diverges, in agree-ment with (20). The patch covered by (21) for the 2-dimensional case is shown in Fig. 3 as the green region.The reason for the A subscript in (21) is that we alsoneed coordinates covering the spacetime outside the hori-zon. These we parametrize with y = (¯ r B − H − ) / cosh( H ¯ t B ) y i = ¯ r B ˆ¯ n i y = (¯ r B − H − ) / sinh( H ¯ t B ) , (23)where ¯ t B ∈ [ −∞ , ∞ ] and ¯ r B ∈ [1 /H, ∞ ] and the lineelement is precisely as in (22), but with A → B . Notethat in the 2-dimensional case studied in section V B onealso has a second patch beyond the horizon, which wedenote with C . The region covered by (23) is shown asthe red region in Fig. 3. It is worth pointing out thatthe coordinates (21) and (23) significantly resemble theKruskal–Szekeres parametrization of the Schwarzschildblack hole. Using the combination of the coordinates(21) and (23) one may cover the same patch as the FLRWsystem (17) as can be seen by comparing Fig. 2 and Fig.3. The regions covered separately by coordinates of type(21) and (23) we will refer to as regions A and B orthe static patches although this is not strictly speaking agood characterization for (23): since [1 − ( H ¯ r B ) (cid:3) <
0, ¯ r B is in fact the time coordinate and hence the metric (22)is explicitly time dependent beyond the horizon. V. THE COARSE GRAINEDENERGY-MOMENTUM TENSOR
In this section we assume a strictly de Sitter back-ground throughout with a = e Ht . (24)Up until now we have written the expectation valueof the energy-momentum tensor symbolically as h ˆ T µν i ,without specifying how it is to be derived. As mentionedin the introduction, our method for calculating the ex-pectation value will deviate from what is generally usedfor the semi-classical prescription [49]. Before this impor-tant topic we need to however cover some basic features.First, we will adopt the cosmologically motivatedchoice where de Sitter space is described in terms of theexpanding FLRW coordinates used for example when cal-culating the cosmological perturbations from inflation.This is a consistent approach, since for an observer atrest with the expanding FLRW coordinates the contract-ing patch is not accessible.Next we need to define the specific state to be used asthe initial condition. In the black hole context makingthe physical choice for the vacuum is an essential ingre-dient for the understanding of the evaporation process,see [68] for important pioneering work and [46] for a cleardiscussion. Again conforming to the usual choice madein inflationary cosmology, we will use the Bunch-Daviesvacuum [79, 80] as the quantum sate. A compelling moti-vation for this choice comes from the fact that the Bunch-Davies vacuum is an attractor state in de Sitter space[33], provided we make the natural assumption that theleading divergences of the theory coincide with those inflat space [81, 82].A very important feature of the Bunch-Davies vacuumis that it covers the entire FLRW de Sitter patch andhence extends also to regions that would be hidden be-hind the horizon. This is a natural requirement for aninitial condition in a case where the Universe was not al-ways dominated by vacuum energy, which from the cos-mological point of view is well-motivated: for example,the Universe may start out as radiation or matter domi-nated and only at late times asymptotically approach theexponentially expanding de Sitter space as in the ΛCDMmodel. This will turn out to be crucial for our calcula-tion.For deriving the Bunch-Davies vacuum in the FLRWcoordinates it proves convenient to make use of the con-formal time coordinate dη = dta ⇒ η = − aH ⇒ ds = a (cid:2) − dη + d x (cid:3) , (25)with which the equation of motion (2) becomes (cid:20) ∂ η + ( n − a ′ a ∂ η − a ∂ i ∂ i + n − n − a R (cid:21) ˆ φ = 0 , (26)where a ′ ≡ ∂a/∂η and a R = 2( n − a ′′ /a + ( n − n − a ′ /a ) . The solutions can be written as the mode ex-pansion ˆ φ = Z d n − k h ˆ a k u k + ˆ a † k u ∗ k i , (27)with k ≡ | k | , with the commutation relations [ˆ a k , ˆ a † k ′ ] = δ ( n − ( k − k ′ ) , [ˆ a k , ˆ a k ′ ] = [ˆ a † k , ˆ a † k ′ ] = 0 and the modes u k = 1 p (2 π ) n − a n − √ k e − i ( kη − k · x ) , (28)which define the Bunch-Davies vacuum state | i viaˆ a k | i = 0 . (29)The Klein-Gordon inner product between two solutionsto the equation of motion φ and φ can be defined interms of a spacelike hypersurface Σ, a future orientedunit normal vector n µ and the induced spatial metric γ ij as (cid:0) φ , φ (cid:1) = − i Z Σ d n − x √ γn µ φ ↔ ∇ µ φ ∗ ; ↔ ∇ µ = → ∇ µ − ← ∇ µ . (30)It is easy to show in the conformal coordinates (25)that using the vector ∂ η for normalization gives n η = a − , n i = 0 and √ γ = a n − with which the inner prod-uct takes the form (cid:0) φ , φ (cid:1) = − i Z d n − x a n − φ ↔ ∇ η φ ∗ , (31)and that the expansion (27) in terms of the Bunch-Daviesmodes (28) is properly normalized, (cid:0) u k , u k ′ (cid:1) = δ ( n − ( k − k ′ ) . (32)For completeness we also show the derivation for theBunch-Davies modes in the spherical coordinates in fourdimensions. Using an ansatz ψ ℓmk = f ℓk ( r ) Y mℓ e − ikη a , (33)where the Y mℓ are the spherical harmonics normalizedaccording to Z d Ω Y mℓ Y m ′ ℓ ′ ∗ = Z πθ =0 Z πϕ =0 dθdϕ sin θ Y mℓ Y m ′ ℓ ′ ∗ = δ ℓℓ ′ δ mm ′ , (34)the equation of motion (26) reduces to a purely radialequation (cid:26) r − ∂∂r (cid:16) r ∂∂r (cid:17) + k − ℓ ( ℓ + 1) r (cid:27) f ℓk ( r ) = 0 , (35)which has solutions expressible as linear combinations ofthe spherical Bessel functions j ν ( x ). Writing the innerproduct in spherical coordinates (cid:0) φ , φ (cid:1) = − i Z d Ω Z ∞ dr r a φ ↔ ∇ η φ ∗ , (36)and making use of the orthogonality properties of theBessel functions allows one to derive the properly nor-malized positive frequency modes ψ ℓmk = r kπ j ℓ ( kr ) Y mℓ e − ikη a . (37)The spherical modes (37) provide another representationfor the scalar field and the Bunch-Davies vacuum viaˆ φ = ∞ X ℓ =0 ℓ X m = − ℓ Z ∞ dk h ψ ℓmk ˆ a ℓmk + ψ ∗ ℓmk ˆ a † ℓmk i (38)and ˆ a ℓmk | i = 0 . (39)The equivalence of the states as defined by (29) and (39)can be demonstrated for example by showing that theWightman function as defined by the two states coin-cides, which can be easily done with the help of theRayleigh or plane wave expansion.In order to obtain well-defined quantum expectationvalues we must define a regularization and a renormaliza-tion prescription for the ultraviolet divergences. Perhapsthe most elegant way would be to analytically continuethe dimensions to n and redefine the constants of the orig-inal action to obtain physical results. Dimensional regu-larization does have a drawback however, which is thatconsistency requires one to calculate everything in n di-mensions, which is surely more difficult than to performthe calculation in 2 dimensions, for example. For ourpurposes the most convenient choice is the adiabatic sub-traction technique [83–85], which in the non-interactingcase is a consistent and a covariant approach [86], butwhere no explicit regularization is needed as the counterterms can formally be combined in the same integral asthe expectation value. This also allows us to use a strictly2- or a 4-dimensional theory. A. Tracing over the unobservable states
So far our approach has followed standard lines. If insome given state | Ψ i we were to calculate the relevant ex-pectation values h ˆ T µν i ≡ h Ψ | ˆ T µν | Ψ i and use them as thesources in the Friedmann equations we would obtain aresult that exponentially fast approaches a configurationwith ˙ H = 0 and conclude that de Sitter space is a stablesolution also when back reaction is taken into account.This is a manifestation of the de Sitter invariance and theattractor nature of the Bunch-Davies vacuum and trueas well for the non-conformal case [33]. However, as dis-cussed in section IV the horizon in de Sitter space splitsthe FLRW manifold into two patches only one of whichis visible to a local observer. This is very much analo-gous to how a black hole horizon blocks the observationalaccess of an observer outside the horizon [41, 71]. Hereis where our approach will differ from what is tradition-ally done in semi-classical gravity: following [36] whencalculating h ˆ T µν i we will use a prescription where we av-erage over only those states that are inside the horizonand thus observable. We note that quite generally coarsegraining a state is expected to bring about a qualitativechange in the results since it often leads to a violation ofde Sitter invariance [36].A configuration where a state is not completely obser-vationally accessible can be described in terms of an openquantum system. For more discussion, see for examplethe textbook [87]. If we assume that the quantum state | Ψ i can be written as a product of orthonormal states | n, A i of the observable system and | n, B i of the unob-servable environment as | Ψ i = X n p n | n, A i| n, B i , (40)we can express expectation values with a coarse grained density matrix ˆ ρ h ˆ O i ≡ Tr (cid:8) ˆ O ˆ ρ (cid:9) , (41)where the density matrix ˆ ρ is obtained by neglecting ortracing over the unobservable statesˆ ρ ≡ Tr B (cid:8) | Ψ ih Ψ | (cid:9) = X m h B, m | (cid:26)h X n p n | n, A i| n, B i i × h X n ′ p ∗ n ′ h B, n ′ |h A, n ′ | i(cid:27) | m, B i , leading to ⇔ ˆ ρ = X m | p m | | m, A ih A, m | . (42)If in the state | Ψ i there is entanglement between the ob-servable states and the unobservable states we coarsegrain over, the initially pure quantum state becomesmixed and the Von Neumann entropy of the density ma-trix will be non-zeroˆ ρ = ˆ ρ ⇔ − Tr (cid:0) ˆ ρ log ˆ ρ (cid:1) > , (43)signalling that part of the information of the initial state | Ψ i is lost or unobservable. When coarse graining leadsto entropy increase/information loss it is a generic featurethat the expectation values will not remain the same [36].For example for the energy-momentum tensor one wouldexpect to haveTr (cid:8) ˆ T µν ˆ ρ (cid:9) = h Ψ | ˆ T µν | Ψ i , (44)implying that the coarse grained system has a differentgravitational response compared to the un-coarse grainedcase.Importantly, the Bunch-Davies vacuum in de Sitterspace before coarse graining is a zero entropy state, butas explained covers also regions that are hidden from alocal observer. If tracing over the unobservable statesleads to a non-zero entropy it also suggests the presenceof a non-zero energy density, which in light of the argu-ments given in section III implies ˙ H = 0 and gives animportant link between loss of information from coarsegraining and a potentially non-trivial back reaction inour prescription.Our choice of neglecting the unobservable states fromthe expectation values can be motivated as follows. Firstof all it is a standard procedure in branches of physicswhere having only partial observable access to a quantumstate is a typical feature. An important example is thedecoherence program: without an unobservable environ-ment the quantum-to-classical transition does not takeplace [88]. Neglecting unobservable information is crucialalso for the inflationary paradigm: in order to obtain thecorrect evolution of large scale structure as seeded by theinherently quantum fluctuations from inflation one mustcalculate the gravitational dynamics from the classical-ized i.e. coarse grained energy-momentum tensor [89].Perhaps most importantly, the energy-momentum ten-sor one obtains after neglecting the unobservable statescorresponds to what an observer would actually measureand in this sense has clear physical significance.A profound feature of our prescription is that sincethe horizon in de Sitter space is an observer dependentquantity, so is then the back reaction itself. Although arather radical proposition, this does not imply an imme-diate inconsistency. After all, observer dependence is aubiquitous feature in general – and even special – relativ-ity. Furthermore, the well-known observer dependence ofthe concept of a particle in quantum theories on curvedbackgrounds was argued to lead to such a conclusion al-ready in the seminal work [41].Although our prescription of using a coarse grainedenergy-momentum tensor as the source term for semi-classical gravity deviates from the standard approachmaking use of h ˆ T µν i ≡ h Ψ | ˆ T µν | Ψ i , we would like to em-phasize that at the moment there is no method for con-clusively determining precisely which object is the correctone [49]. This stems from the fact that semi-classicalgravity is not a complete first principle approach, butrather an approximation for describing some of the gravi-tational implications from the quantum nature of matter.Before a full description of quantum gravity is obtainedit is likely that this state of affairs will remain.Tracing over inaccessible environmental states thatare separated by a sharp boundary from the accessibleones generically leads to divergent behaviour close to theboundary. This is encountered for example in the con-text of black hole entropy [90] and entanglement entropyin general [91, 92]. Although by introducing a cut-off ora smoothing prescription well-defined results can be ob-tained [72], there is valid suspicion of the applicabilityof the semi-classical approach when close to the horizon.However, we can expect reliable results at the limit whenthe horizon is far away. At this limit there exists a nat-ural expansion in terms of physical distance in units ofthe horizon radius, or more specifically in terms of thedimensionless quantities H ¯ x i , in the notation of sectionIV. The neglected terms we will throughout denote as O ( H ¯ x ). This limit can be expressed equivalently as be-ing far away from the horizon or close to the center of theHubble sphere and can equally well be satisfied when H is large such as during primordial inflation or when it isvery small as it is during the late time Dark Energy dom-inated phase we are currently entering. The limit wherethe observer is far from the horizon is also the limit takenin the standard black hole analysis [37]. B. Two dimensions
For completeness we first go through the steps of the 2-dimensional argument presented in [34], before proceed- ing to the full 4-dimensional derivation.The various coordinate systems in de Sitter space intwo dimensions can be expressed analogously to whatwas discussed in four dimensions in section IV. The mainmodification is that since there is only one spatial coor-dinate there are now two horizons, at ± /H . For thepatch inside the horizon in two dimensions one may usethe FLRW y = H − sinh( Ht ) + ( H/ x e Ht y = e Ht xy = H − cosh( Ht ) − ( H/ x e Ht , (45)or static coordinates y = ( H − − ¯ x A ) / sinh( H ¯ t A ) y = ¯ x A y = ( H − − ¯ x A ) / cosh( H ¯ t A ) , (46)giving ds = − dt + e Ht dx , (47)and ds = − (cid:2) − ( H ¯ x A ) (cid:3) d ¯ t A + (cid:2) − ( H ¯ x A ) (cid:3) − d ¯ x A , (48)respectively, with t, ¯ t A , x ∈ [ −∞ , ∞ ] and ¯ x A ∈ [ − /H, /H ]. Since in two dimensions there are two hori-zons there are also two patches beyond the horizon thatcan be covered with the FLRW coordinates or with y = (¯ x B − H − ) / cosh( H ¯ t B ) y = ¯ x B y = (¯ x B − H − ) / sinh( H ¯ t B ) , (49)where ¯ t B ∈ [ −∞ , ∞ ] and ¯ x B ∈ [1 /H, ∞ ], and y = (¯ x C − H − ) / cosh( H ¯ t C ) y = ¯ x C y = (¯ x C − H − ) / sinh( H ¯ t C ) , (50)where ¯ t C ∈ [ −∞ , ∞ ] and ¯ x C ∈ [ −∞ , − /H ].The relations between the various coordinate systemsbecome quite simple when using the light-cone coordi-nates defined in terms of conformal time (25) as ( V = η + x = + e − Ht (cid:0) ¯ x − H − (cid:1) ,U = η − x = − e − Ht (cid:0) ¯ x + H − (cid:1) , (51)where the notation implies the same definition in all threeregions A, B and C . As is clear from the definitions (51)the V and U coordinates can also be conveniently usedto split the FLRW patch in terms of the regions A, B and C since they vanish at the horizons 1 /H and − /H ,respectively. This is illustrated in Fig. 4. Furthermore,in the static patches we define the tortoise coordinates d ¯ x = (cid:2) − ( H ¯ x ) (cid:3) d ¯ x ∗ ⇒ ¯ x ∗ = (2 H ) − log (cid:12)(cid:12)(cid:12)(cid:12) H ¯ x − H ¯ x (cid:12)(cid:12)(cid:12)(cid:12) , (52)with ¯ x ∗ A ∈ [ −∞ , ∞ ], ¯ x ∗ B ∈ [0 , ∞ ] and ¯ x ∗ C ∈ [ −∞ , U and V in terms of the staticones. The results can be summarized as ( V = − H − e − Hv A U = − H − e − Hu A , Reg . A , (53) ( V = + H − e − Hv B U = − H − e − Hu B , Reg . B , (54)and ( V = − H − e − Hv C U = + H − e − Hu C , Reg . C , (55)and it is also convenient to define light-cone coordinateswith respect to the static coordinates ( v = ¯ t + ¯ x ∗ u = ¯ t − ¯ x ∗ . (56)The core of the calculation is finding an expression forthe coarse grained density matrix (42), from which theunobservable information related to states beyond thehorizon is removed. If we assume that any possible en-tanglement occurs only between modes with the samemomentum, the density matrix where the hidden statesare traced over can be written as the product in momen-tum spaceˆ ρ ≡ Tr BC (cid:8) | ih | (cid:9) = Y k Tr BC (cid:8) | k ih k | (cid:9) ≡ Y k ˆ ρ k , (57)where we define k to be a scalar going from −∞ to ∞ and where | k i is the k ’th Fock space contribution to theBunch-Davies vacuum, | i = Q k | k i .Next we need to find an expression for the Bunch-Davies vacuum in terms of observable and unobservablestates. This is obtained by relating the Bunch-Daviesmodes to the ones defined in the 2-dimensional static co-ordinates found in (46), (49) and (50). From (26 – 29)we see that in two dimensions the mode expansion in deSitter space coincides with the flat space result and canbe written in the light-cone coordinates (51) asˆ φ ≡ ˆ φ V + ˆ φ U = Z ∞ dk √ πk h e − ikV ˆ a − k + e ikV ˆ a †− k + e − ikU ˆ a k + e ikU ˆ a † k i . (58) Region A Region B Region C VU xη U = , h o r i z o n V = , h o r i z o n FIG. 4. The expanding FLRW patch in de Sitter space isdescribed by η ≤ −∞ < x < ∞ , and can be covered bythe A, B and C coordinate systems defined in (46), (49) and(50). In (58) we have split the quantum field to two contribu-tions according their dependence on U or V since thesenewer mix and can be thought as separate sectors, as isevident by taking into account [ˆ a − k , ˆ a † k ] = 0, V = V ( v )and U = U ( u ) from (53 – 56). Since ˆ φ V is expressed onlyin terms of the ˆ a − k operators it consists solely of parti-cles moving towards the left and similarly for ˆ φ U and theright-moving particles.In two dimensions also the static coordinates give riseto a trivial equation of motion (cid:3) ˆ φ = 0 ⇔ (cid:0) ∂ t − ∂ x ∗ (cid:1) ˆ φ = 0 , (59)but much like for the Unruh effect, we must carefullydetermine the correct normalization for the modes in thestatic patches. Namely, we need to make sure that weare consistent in terms of defining a positive frequencymode.As discussed after equation (30), the modes in (58) aredefined to be positive frequency in terms of the vector ∂ η and we need to respect this definition also in the staticpatches A, B and C . If we choose ∂ ¯ t A , − ∂ ¯ x ∗ B and ∂ ¯ x ∗ C for A, B and C respectively, it is a simple matter of usingthe vector transformations ∂ µ = ∂x ˜ α ∂x µ ∂ ˜ α with (53 – 56) toshow that ∂ ¯ t A = − H (cid:0) η∂ η + x∂ x (cid:1) (60) − ∂ ¯ x ∗ B = + H (cid:0) x∂ η + η∂ x (cid:1) (61) ∂ ¯ x ∗ C = − H (cid:0) x∂ η + η∂ x (cid:1) , (62)which with the help of Fig. 4 one may see to be time-likein terms of conformal time and future-oriented in theirrespective regions . Here we neglect the k = 0 zero mode, whose quantization is anon-trivial issue in 2-dimensional field theory [93], but does notpose problems in four dimensions. For example, from Fig. 4 we see that in region C we have − x ≥ UV ≤ ⇔ x ≥ η . φ , φ ) A = − i Z d ¯ x ∗ A φ ↔ ∇ ¯ t A φ ∗ ; Reg . A , (63)( φ , φ ) B = + i Z d ¯ t B φ ↔ ∇ ¯ x ∗ B φ ∗ ; Reg . B , (64)( φ , φ ) C = − i Z d ¯ t C φ ↔ ∇ ¯ x ∗ C φ ∗ ; Reg . C , (65)with which the expression for the scalar field in the staticcoordinates becomesˆ φ = Z ∞ dk √ πk × (cid:16) e − ikv A ˆ a A − k + H . C (cid:17) + (cid:16) e − iku A ˆ a A k + H . C (cid:17) , Reg . A , (cid:16) e + ikv B ˆ a B − k + H . C (cid:17) + (cid:16) e − iku B ˆ a B k + H . C (cid:17) , Reg . B , (cid:16) e − ikv C ˆ a C − k + H . C (cid:17) + (cid:16) e + iku C ˆ a C k + H . C (cid:17) , Reg . C . (66)where ’H . C . ’ stands for ’hermitian conjugate’. The formin (66) can be used to define the scalar field ˆ φ on theentire expanding FLRW patch, precisely as (58). Thecrucial point is that in general the Bunch-Davies vacuumas defined by (29) is an entangled combination of statesinside and outside the horizon, which leads to an increasein entropy once the hidden states are traced over.We will first perform the entire calculation for ˆ φ V , afterwhich writing the results for ˆ φ U becomes trivial. We canfirst focus only on the regions A and B , since for ˆ φ V onlythe horizon at V = 0 is relevant, which we elaborate morebelow. From (66) we then getˆ φ V = Z ∞ dk √ πk h e − ikv A ˆ a A − k + e ikv A ˆ a A †− k + e ikv B ˆ a B − k + e − ikv B ˆ a B †− k i , (67)where the modes defined in the A region are to be under-stood to vanish in region B and vice versa for the modesin B .The approach we will use was originally presented in[68] and is based on the fact that any linear combina-tion of positive modes defines the same vacuum as a sin-gle positive mode [49]. The main constraint is that theBunch-Davies modes (58) are continuous across the hori-zon. Making use of (53 – 56) we can write e − iv A k = e ikH ln( − HV ) ; Reg . A , (68) e − πkH (cid:0) e ikv B ) ∗ = e − πkH e ikH ln( HV ) = e ikH ln( − HV ) ; Reg . B , (69)where in the last line we chose the complex logarithm to have a branch cut as ln( −
1) = iπ . Because of thischoice the sum of (68) and (69) is continuous across thehorizon and analytic when ℑ [ V ] <
0, so it must be ex-pressible as a linear combination of e − ikV i.e the positivefrequency Bunch-Davies modes from (58). In a similarfashion starting from e + iv A k it is straightforward to find asecond continuous and well-behaved linear combination.With such linear combinations we have yet another rep-resentation for the scalar field in addition to (58) and(67)ˆ φ V = Z ∞ dk √ πk p − γ (cid:26)h e − ikv A + γ (cid:0) e ikv B ) ∗ i ˆ d (1) − k + h γ (cid:0) e − ikv A ) ∗ + e ikv B i ˆ d (2) − k + H . C . (cid:27) , (70)where we have defined γ ≡ e − πk/H and used (63 – 64)to get properly normalized modes. An important resultmay be derived by realizing that the operators ˆ d (1) − k andˆ d (2) − k annihilate the Bunch-Davies vacuum since the modesin the square brackets of (70) are continuous and mustbe linear combinations of the positive frequency Bunch-Davies modes and (67) allows us to express them in termsof ˆ a A − k and ˆ a B − k ((cid:0) ˆ a A − k − γ ˆ a B †− k (cid:1) | − k i = 0 (cid:0) ˆ a B − k − γ ˆ a A †− k (cid:1) | − k i = 0 . (71)The above relations are identical to what is found in the2-dimensional Unruh effect, and black hole evaporationand imply that | − k i is an entangled state in terms thenumber bases as defined by ˆ a A − k and ˆ a B − k . Following [72]the normalized solution to (71) can be written as | − k i = p − γ ∞ X n − k =0 γ n − k | n − k , A i| n − k , B i , (72)where | n − k , A i and | n − k , B i are particle number eigen-states as defined by ˆ a A − k and ˆ a B − k . If as in (42) we traceover the unobservable states the density matrix becomesprecisely thermalˆ ρ − k = Tr B (cid:8) | − k ih − k | (cid:9) = (1 − γ ) ∞ X n − k =0 γ n − k | n − k , A ih A, n − k | . (73)A physical argument can also be used to rule out oneof the two possible choices for the branch cut. Had wemade the different choice the result would have given aninfinite number of produced particles at the ultravioletlimit, which is an unphysical solution as the ultravioletmodes should be indifferent to the global structure ofspacetime and experience no particle creation.As mentioned, only the regions A and B are relevantfor ˆ φ V . The reason why one may neglect the contribution1from region C is apparent from the relations (53 – 56)and (66): ˆ φ V has no dependence on U so no mixing ofmodes is needed in order to obtain analytic behaviouracross the horizon U = 0. Thus, including all regions A, B and C in (70) would still give (72), which we havealso explicitly checked.So far we have only studied ˆ φ V i.e. the particles mov-ing to the left. By using (53 – 56) and (66) the cal-culation involving ˆ φ U proceeds in an identical mannerresulting also in a thermal density matrix, but in termsof the right-moving particles | n k , A i .Putting everything together, the density matrix (57)obtained by tracing over the states beyond the horizonin the Bunch-Davies vacuum is precisely thermal with theGibbons-Hawking de Sitter temperature T = H/ (2 π )ˆ ρ = Y k (cid:0) − e − πkH (cid:1) ∞ X n k =0 e − πkH n k | n k , A ih A, n k | . (74)Since in (74) all states except the ones belonging toregion A are neglected we can write the expectation valueof the energy-momentum tensor from (5) by expressingˆ φ as the top line from (66) h ˆ T vv i = h ˆ T uu i = Z ∞ dk π (cid:20) k ke πk/H − (cid:21) ; h ˆ T uv i = 0 , (75)where for simplicity we have dropped the A labels. Thefinal unrenormalized expression in the FLRW coordinates(47) can be obtained by using the tensor transformationlaw T µν = ∂x ˜ α ∂x µ ∂x ˜ β ∂x ν T ˜ α ˜ β with (53 – 56) and (51). Thisgives h ˆ T i = h ˆ T ii i /a = (cid:20) − H ¯ x ) + 1(1 + H ¯ x ) (cid:21) × Z ∞ dk π (cid:20) k ke πk/H − (cid:21) , (76)and h ˆ T i i /a = (cid:20) − H ¯ x ) − H ¯ x ) (cid:21) × Z ∞ dk π (cid:20) k ke πk/H − (cid:21) . (77)As the discussion after equation (42) addressed, (76)and (77) have divergent behaviour on the horizons ¯ x = ± H − . This is distinct to the usual ultraviolet diver-gences encountered in quantum field theory, which arealso present in (76) and (77) as the divergent integrals.For our purposes the relevant limit of being close to theorigin is obtained with an expansion in terms of H ¯ x giv- ing h ˆ T i = h ˆ T ii i /a = Z d k π (cid:20) k ke πk/H − (cid:21) + O ( H ¯ x ) (78) h ˆ T i i /a = H ¯ x (cid:2) h ˆ T i + h ˆ T ii i /a (cid:3) + O ( H ¯ x ) . (79)The last step in the calculation is renormalization.When we neglect the O ( H ¯ x ) contributions i.e. studyonly the region far from the horizon the result is pre-cisely homogeneous and isotropic for which the counterterms can be found by calculating the energy-momentumtensor as an expansion in terms of derivatives the scalefactor. This is the adiabatic subtraction technique [83–85], with which the 2-dimensional counter terms werefirst calculated in [94] giving coinciding results to [95].This technique gives the counter terms for the energyand pressure components as formally divergent integrals δT = Z d k π k H π ; δT ii /a = Z d k π k − H π . (80)Note that the apparent divergence in the flux contribu-tion (79) is an artefact of our use of non-regulated in-tegrals. When dimensionally regulated the sum of theenergy and pressure density divergences cancels, also formassive particles. Physically one can understand thisfrom the requirement that Minkowski space must be sta-ble under back reaction. This issue is discussed morein section VI, see also the equation (113). For now wecan simply neglect the divergences in (79) or following[34] formally derive the flux counter terms by demand-ing covariant conservation of δT µν . The renormalizedenergy-momentum is then ρ m = Z d k π ke πk/H − − H π + O ( H ¯ x ) , (81) p m = Z d k π ke πk/H − H π + O ( H ¯ x ) , (82) T i /a ≡ f m = H ¯ x (cid:0) ρ m + p m (cid:1) + O ( H ¯ x ) . (83)We can clearly see that far away from the horizonthe energy-momentum describes a homogeneous andisotropic distribution of thermal particles. It can bechecked to be covariantly conserved and to have the usualconformal anomaly [49], although in de Sitter space theconformal anomaly is not important and one may cancelthe ± H π contributions by a redefinition of the cosmolog-ical constant. Contrary to what one usually encountersin cosmology, the energy density in (81) is constant de-spite the fact that space is expanding and genericallydiluting any existing particle density. This is explainedby the additional term (83) representing a continuous in-coming flux of particles, which replenishes the energy lost2by dilution. This naturally raises an important questionconcerning the source of the flux, which will be addressedin section VI where we write down solutions that are con-sistent in terms of semi-classical back reaction.In the language of section III the ± H / (24 π ) terms arestate independent contributions resulting from renormal-ization and the integral over the thermal distribution isthe state dependent contribution ρ Sm . The results (81 –83) can be seen to be in agreement with the argumentsof section III, in particular the 2-dimensional version of(13) when taking into account of the modifications arisingfrom our choice of not to implement dimensional regular-ization.Before ending this subsection we comment on a tech-nical detail regarding the divergences generated in thecoarse grained state. A general – or certainly a desir-able – feature of quantum field theory is the universalityof the generated divergences and renormalization. Fora quantum field on a de Sitter background, one shouldbe able to absorb all divergences in the redefinition ofthe cosmological constant, preferably also in the coarsegrained state (74). But from (76) and (77) we can seethat this does not hold due to the ¯ x -dependence of thegenerated divergences, likely related to coarse grainingand the additional divergence when approaching the hori-zon. We emphasize however that even if in a carefullydefined coarse graining divergences ∝ ¯ x are not gener-ated, the renormalized result would coincide with (81 –83), because up to the accuracy we are interested in allthe needed counter terms could be derived via adiabaticsubtraction, which satisfies δT = − δT ii /a [33]. C. Four dimensions
The calculation in four dimensions proceeds in princi-ple precisely as the 2-dimensional derivation of the pre-vious subsection. The main differences are that the so-lutions in four dimensions are analytically more involvedand that there is only one horizon. Quantization of ascalar field in the static de Sitter patch has been stud-ied in [96–98] to which we refer the reader for more de-tails. Quite interestingly, although the line element fora Schwarzschild black hole and de Sitter space in staticcoordinates are very similar, the latter has an analyticsolution for the modes while the former does not.We begin by writing useful coordinate transformationsbetween the 4-dimensional FLRW coordinates and staticcoordinates of section IV, in region A ( e Ht = e H ¯ t A p − ( H ¯ r A ) r = e − Ht ¯ r A , Reg . A , (84)and region B ( e Ht = e H ¯ t B p ( H ¯ r B ) − r = e − Ht ¯ r B , Reg . B . (85) The tortoise coordinates and light-cone coordinates canbe obtained trivially from the 2-dimensional results (51),(52) and (53 – 56) with the replacements x → r and¯ x → ¯ r .Due to spherical symmetry and time-independence ofthe metric (22) in the coordinates (21) we introduce asimilar ansatz to the equation of motion (2) as in thespherical form of the FLRW metric in (33) ψ A ℓmk = f A ℓk (¯ r A ) Y mℓ e − ik ¯ t A , (86)leading to the radial equation (cid:26) ¯ r − A ∂∂ ¯ r A (cid:16) ¯ r A (cid:2) − ( H ¯ r A ) (cid:3) ∂∂ ¯ r A (cid:17) + k − ( H ¯ r A ) − ℓ ( ℓ + 1)¯ r A − H (cid:27) f A ℓk (¯ r A ) = 0 , (87)where we have used ξ R = 2 H . With a suitable ansatzthe above may be reduced to a hypergeometric equationand has the solution in terms of the Gaussian hypergeo-metric function f A ℓk (¯ r A ) = D A ℓk ( H ¯ r A ) ℓ (cid:2) − ( H ¯ r A ) (cid:3) ik H × F (cid:20) ℓ ik H + 12 , ℓ ik H + 1; ℓ + 32 ; ( H ¯ r A ) (cid:21) , (88)where D A ℓk is a normalization constant. With the help of(84) and (85) we can show that ∂ ¯ t A = e − Ht (cid:2) ∂ η − H ¯ r A ∂ r (cid:3) , (89)so inside the horizon in the static coordinates we can use ∂ ¯ t A for defining positive frequency modes and the innerproduct via (30), which reads( φ , φ (cid:1) A = − i Z d Ω Z /H d ¯ r A ¯ r A − ( H ¯ r A ) φ ↔ ∇ ¯ t A φ ∗ . (90)Using (86) and (88) in the above allows one to solve forthe normalization constant D ℓk . For details we refer thereader to [97, 98] and appendix A, but here we simplywrite the result1 D A ℓk = √ πkH F (cid:20) ℓ ik H + 12 , ℓ ik H + 1; ℓ + 32 ; 1 (cid:21) = √ πk Γ (cid:2) ℓ + (cid:3) Γ (cid:2) − ikH (cid:3) H Γ (cid:2) (cid:0) ℓ − ikH + 1 (cid:1)(cid:3) Γ (cid:2) (cid:0) ℓ − ikH + 2 (cid:1)(cid:3) , (91)up to factors of modulus one.Having derived the solutions in the static patch insidethe horizon (21), the solutions in coordinates coveringthe outside of the horizon (23) follow trivially from thefact that the line element and thus the radial equation(87) have identical form. Again however, we must care-fully determine the correct normalization for the positive3frequency mode beyond the horizon. With (84) and (85)we can show ∂ ¯ r B = e − Ht ( H ¯ r B ) − (cid:2) H ¯ r B ∂ η − ∂ r (cid:3) , (92)which, similarly to the 2-dimensional derivation, impliesthat ¯ r B plays the role of time. The inner product in the B region then becomes( φ , φ (cid:1) B = − i Z d Ω Z ∞−∞ d ¯ t B ¯ r B (cid:2) ( H ¯ r B ) − (cid:3) φ ↔ ∇ ¯ r B φ ∗ , (93)Since the inner product (93) is independent of the choiceof hypersurface we can evaluate it at the limit ¯ r B → /H .It is then a simple matter of using (91) to show that ψ B ℓmk = f B ℓk (¯ r B ) (cid:0) Y mℓ (cid:1) ∗ e ik ¯ t B , (94)in region B is the correctly normalized positive frequencymode provided that f B ℓk (¯ r B ) = D B ℓk ( H ¯ r B ) ℓ (cid:2) ( H ¯ r B ) − (cid:3) − ik H × F (cid:20) ℓ − ik H + 12 , ℓ − ik H + 1; ℓ + 32 ; ( H ¯ r B ) (cid:21) , (95)with D B ℓk = ( D A ℓk ) ∗ , see appendix A.In the 4-dimensional expanding FLRW patch the scalarfield can then be written in the coordinates (21) and (23)asˆ φ = ∞ X ℓ =0 ℓ X m = − ℓ Z ∞ dk h ψ A ℓmk ˆ a A ℓmk + H . C . i Reg . A , ∞ X ℓ =0 ℓ X m = − ℓ Z ∞ dk h ψ B ℓmk ˆ a B ℓmk + H . C . i Reg . B . (96)Despite the analytically involved structure of themodes the arguments we used in the 2-dimensional caseapply practically identically in four dimensions. This isdue to the simplification that occurs when approachingthe horizon. At this limit with the help of (91) and thetortoise coordinates (52) we get for the mode in region Aψ A ℓmk H ¯ r A → −→ H Y mℓ √ πk n(cid:2) cosh( H ¯ r ∗ A ) (cid:3) − o + ik H e − ik ¯ t A ∼ Y mℓ √ πk ¯ r A e − ikv A , (97)and for the mode in region Bψ B ℓmk H ¯ r B → −→ H (cid:0) Y mℓ (cid:1) ∗ √ πk n(cid:2) sinh( H ¯ r ∗ B ) (cid:3) − o − ik H e + ik ¯ t B ∼ (cid:0) Y mℓ (cid:1) ∗ √ πk ¯ r B e + ikv B , (98) where ∼ denotes an equality up to constant factors ofmodulus one. Comparing the above to (66) reveals thatthe discontinuity is precisely of the same form as in twodimensions, leading to two non-trivial linear combina-tions that are continuous across the horizon ψ A ℓmk + γ ( ψ B ℓmk ) ∗ and γ ( ψ A ℓmk ) ∗ + ψ B ℓmk , (99)where again γ ≡ e − πk/H . Expressing the scalar fieldin terms of two representations, as was done in two di-mensions in (70), one may reproduce the steps of the2-dimensional derivation and deduce the thermality ofthe 4-dimensional coarse grained density matrixˆ ρ = Y ℓmk (cid:0) − e − πkH (cid:1) ∞ X n ℓmk =0 e − πkH n ℓmk | n ℓmk , A ih A, n ℓmk | . (100)The energy density in four dimensions is of course moredifficult to write in a clear form than the 2-dimensionalresult due to the presence of the hypergeometric func-tions in the mode solution (88). However for our purposesonly the limiting case of a being far from the horizonis relevant, for which the static line element (22) coin-cides with flat space up to small terms O ( H ¯ x ). Thisand the fact that our theory is conformal imply thatup to O ( H ¯ x ) the energy density in the coarse grainedstate (100) should coincide with that of a black-bodywith T = H/ (2 π ). In appendix B we verify this assertionwith an explicit calculation.Tracelessness of the unrenormalized energy-momentum tensor immediately fixes the ratio ofthe energy and pressure densities, again dropping the A labels as irrelevant h ˆ T ¯0¯0 i = 3 h ˆ T ¯ i ¯ i i = Z d k (2 π ) (cid:20) k ke πk/H − (cid:21) + O ( H ¯ x ) , (101)where the off-diagonal components cancel due to thelack of flux and shear in a static spherically symmet-ric case. From the results expressed in the static coor-dinates in (101) we already see a very important out-come from coarse graining the energy-momentum tensorwith respect to states beyond the horizon: the sum ofthe pressure- and energy-density does not vanish, a rela-tion which is satisfied by the ’00’ and ’ ii ’ components ofEinstein tensor in the static coordinates at the centre ofthe Hubble sphere. This implies that there is non-trivialback reaction since Einstein’s equation with (101) as thesource is not solved by the static line element describingde Sitter space (22). We can conclude that coarse grain-ing such that only observable states are left leads to aviolation of de Sitter invariance.Following the 2-dimensional procedure of the previ-ous subsection, the usual tensor transformations with thehelp of (84) and (85) allow us to write the result in theFLRW form, which we can renormalize by using the 4-dimensional adiabatic counter terms that can be found4for example in [36]. The final result is ρ m = Z d k (2 π ) ke πk/H − H π , (102) p m = 13 Z d k (2 π ) ke πk/H − − H π , (103) f m, i = H ¯ x i (cid:0) ρ m + p m (cid:1) , (104)where f m, i ≡ T i /a and we have neglected terms of O ( H ¯ x ) . So quite naturally, the 4-dimensional resulthas exactly the same thermal characteristics to the 2-dimensional one in (81 – 83). Far away from the horizon(102 – 104) is homogeneous and isotropic with ρ m + p m = 43 ρ Sm = 43 Z d k (2 π ) ke πk/H − H π , (105)and again the terms ± H π responsible for the confor-mal anomaly are irrelevant and could have been removedwith a redefinition of Λ. The above can be seen to be inagreement with the discussion of section III, specificallythe right-hand side of equation (14).For clarity we summarize the arguments of this sectionhere once more: in de Sitter space as described by the ex-panding FLRW coordinates (18) initialized to the Bunch-Davies vacuum the energy-momentum of a quantum fieldhas a thermal character when in the density matrix oneincludes only the observable states inside the horizon.The energy density inside the horizon is maintained at aconstant temperature by a continuous flux of radiationincoming from the horizon that precisely cancels the dilu-tion from expansion. This results in ρ m + p m = 0, whichis independent of the details of renormalization and theconformal anomaly due to the symmetries of the counterterms in de Sitter space. Thus even at the limit whenthe distance to the horizon is very large and the resultis isotropic and homogeneous the sum of the energy andpressure densities does not cancel and because of thisthe dynamical Friedmann equation (9) then implies thata strictly constant Hubble rate H is not a consistent so-lution. This is also visible in the result given in the staticcoordinates (101), which does not solve Einstein’s equa-tion if the background is assumed to be strictly de Sitter. VI. SELF-CONSISTENT BACK REACTION
If, as the arguments of the previous section imply, deSitter space is affected by back reaction in the prescrip-tion we have chosen, this naturally leads one to investi-gate how precisely is the strict de Sitter solution modi-fied. Ultimately, this is determined by the semi-classicalEinstein equation.Taking the limit of begin close to the center of the Hub-ble sphere, which is the same as assuming that the hori-zon is far away, the equations (102 – 104) correspond to ahomogeneous and an isotropic solution. One would then expect that at this limit also the back reaction is homoge-neous and isotropic parametrizable with a FLRW line el-ement. In fact strictly speaking, we can only consistentlystudy back reaction if the homogeneous and isotropic ap-proximation holds, since the calculation of the previoussection was made by assuming the FLRW line element.In this case the quantum corrected Hubble rate can beself-consistently solved by using ρ m = H π ; p m = 13 H π , (106)where in the above for simplicity we have absorbed the ± H / (960 π ) contributions in (102) and (103) into the(re)definition of the cosmological constant, making thedistinction between ρ Sm and ρ m irrelevant and left the O ( H ¯ x ) notation as implicit. However, before proceed-ing we must address a crucially important implication ofhaving a constant energy and pressure density in a space-time described by a FLRW line element: (106) are notconsistent with covariant conservation˙ ρ + 3 H ( ρ + p ) = 0 , (107)where ρ = ρ m + ρ Λ and similarly for p . This is expected,since we have not included the effect of the flux (104),which continuously injects the system with more energy.As argued in [37] for the analogous black hole case aflux can be seen to imply a change in the size of horizon:a positive flux coming from the horizon is equivalent to anegative flux going into the horizon. Note that when thecosmological horizon or ρ Λ absorbs negative energy thehorizon radius will grow, where as precisely the oppositerelation holds for the horizon and mass of a black hole.If we make the assumption that the rate of changeof the vacuum energy equals the energy injected by theflux (104) we can satisfy (107) while matching the right-hand side of the dynamical Friedmann equation (9) with(105), by re-writing the Friedmann equations (7) withdynamical vacuum energy, ρ Λ −→ ˜ ρ Λ , for which˜ ρ Λ = − ˜ p Λ = ρ Λ − tH ( ρ m + p m ) = Λ M − t H π ⇒ ˙˜ ρ Λ = − H ( ρ m + p m ) , (108)where small terms of O ( ˙ H ) are beyond our approxima-tion and are neglected. The dynamical Friedmann equa-tion (9) then allows us to solve for the evolution of H − HM = H π ⇒ H = H (cid:16) H π M t + 1 (cid:17) / . (109)The crucial observation is that it is impossible for the deSitter approximation to hold for an arbitrarily long time:after the time scale t / = 1680 π (cid:18) M pl H (cid:19) H − , (110)5which is defined as the half-life of the Hubble rate, thesystem seizes to be de Sitter and we can conclude thatthe cumulative effect of the quantum back reaction hasbecome significant enough to dominate over the classicalsolution. The time scale for the destabilization of de Sit-ter t ∼ M /H was similarly obtained for a fully quan-tized model in [35], where it is also argued that after thistime the full quantum evolution must depart completelyfrom the classical one, in agreement with our analysis .Furthermore, it also follows from the results of section10.4 of [13].The time scale (110) is quite large, at least in the ob-vious cosmological applications: for inflation with themaximum scale allowed by the non-observation of ten-sor modes H ∼ GeV [99] gives t / ∼ H − ,which corresponds to 10 e -folds of inflation. The break-down of the Dark Energy dominated late time de Sitterphase can be estimated by using the current Hubble rate H ∼ − GeV [100] giving t / ∼ H − , where H − corresponds to the age of the Universe.The observation that in an expanding, homogeneousand isotropic space a non-diluting particle density neces-sitates a decaying vacuum energy was already made in[101, 102] and has since been studied in [31, 103–108].The use of the results of the previous section, whichwere calculated on a fixed background, is a good ap-proximation only when the modification from back re-action is very small. Since without back reaction theresult is strict de Sitter space this translates as de-manding validity of the adiabatic limit i.e. H shouldchange very gradually. From (109) we see this to betrue, − ˙ H/H ∼ H /M ≪ H are neglected. One may fur-thermore check the robustness of the cosmological eventhorizon and our coarse graining prescription under backreaction: from (109) one gets the scale factor a ( t ) ∝ exp (cid:26) π M H (cid:18) H π M t + 1 (cid:19) / (cid:27) , (111)with which the cosmological event horizon (20) in thepresence of back reaction reads a ( t ) Z ∞ t dt ′ a ( t ′ ) = H − h O (cid:0) H/M pl (cid:1) i , (112)so to a very good approximation the event horizon tracks1 /H , as required.The problem with vacuum energy changing its value isthat classically it is proportional to a constant parameterΛ in the gravitational Lagrangian (3) and it is not ob-vious how a parameter characterising different de Sitter We thank the authors of [35] for clarifying this issue. configurations can change dynamically. In a quantizedtheory however, the situation changes completely sincethe zero-point energy and pressure, which all particlespossess, satisfy the same equation of state as the contri-bution resulting from the cosmological constant Λ. Forexample, for a massive scalar field when dimensionallyregularizing the sum of the zero-point energy and pres-sure contributions one has Z d n − k (2 π ) n − √ k + m Z d n − k (2 π ) n − k n − √ k + m = 0 . (113)Unlike in flat space, the zero-point terms are gravitation-ally significant which is the key issue behind the cosmo-logical constant problem and was first discussed in [109].Taking this idea further, if all quantum fields can con-tribute to the vacuum energy which in turn couples togravity, it seems natural to assume that in curved spacethe amount vacuum energy a particular field is responsi-ble for is not fixed, but a dynamical quantity much likethe field itself. With this in mind we propose the follow-ing physical picture of the continuous particle creationprocess required by (106): when a particle pair is createdit leaves behind a hole of negative vacuum energy in or-der not to violate covariant conservation. This causes theoverall vacuum energy density to decrease, analogously tothe interpretation of black hole evaporation where holesof negative energy fall into the black hole causing it tolose mass [37].As we discuss in the next section, the proposal thatvacuum energy is dynamical also has a deep connec-tion with the thermodynamic interpretation of de Sitterspace, which gives it a more solid footing. VII. DERIVATION FROM HORIZONTHERMODYNAMICS
Much like for a black hole, it is expected that the firstlaw of thermodynamics dU = T dS − P dV , (114)can be expressed as a relation connecting internal energy U , the horizon and the pressure P of de Sitter space[41, 44]. However, if the de Sitter solution is assumedto be determined only by the cosmological constant termΛ, a parameter of the Einstein-Hilbert Lagrangian, thechange in internal energy dU requires the problematicconcept of varying Λ [13, 111]. If however, we adopt theproposal that semi-classical back reaction is sourced bythe coarse grained energy-momentum tensor as discussedin V A, the derivation of the two previous sections implythat the vacuum energy becomes a dynamical quantitydue to the inevitable contribution of quantum fields andthis issue is evaded. In fact quite remarkably, allowingthe vacuum energy to vary and by using the standard6concepts of horizon thermodynamics we can derive theresults (109) and (108) in a mere few lines.In general a spacetime horizon contains entropy pro-portional to its area S = A G , (115)which in de Sitter space is given by the de Sitter horizon A = 4 πH − [41]. We can then define the internal energycontained inside the Hubble sphere as ρ Λ V = ρ Λ 4 π H − ,where ρ Λ is now the dynamical quantity we denoted with˜ ρ Λ in the previous section. In the case of a cosmologi-cal horizon there is however an important subtlety givingrise to a few additional minus signs: the change in vol-ume and internal energy in (114) refer to the region thatis hidden from the observer, the space beyond the cosmo-logical horizon. Energy lost from the Hubble sphere willin fact be gained by the degrees of freedom beyond thehorizon and similarly when the horizon radius increasesthe volume of the hidden region decreases: dU = − d (cid:18) ρ Λ π H (cid:19) ; P dV = − p Λ d (cid:18) π H (cid:19) . (116)When the temperature is given by the Gibbons-Hawking relation T = H/ (2 π ) and by using the de Sitterequation of state ρ Λ + p Λ = 0, from the first law (114)with (115) and (116) we straightforwardly get a relationbetween the Hubble rate and the change in vacuum en-ergy, − d (cid:18) ρ Λ π H (cid:19) = H πG d (cid:18) πH (cid:19) + p Λ d (cid:18) π H (cid:19) ⇔ HM = ˙ ρ Λ H . (117)If ρ Λ was given by the potential of a scalar field one mayrecognize (117) as one of the slow-roll equations usedin the inflationary framework [110], which we have herederived by making no reference to Einstein’s equation.Then we assume that in addition to vacuum energy thetheory contains a massless scalar field with the energydensity ρ m . If the de Sitter horizon is a thermodynamicobject with the temperature T = H/ (2 π ), when givenenough time we can expect the cavity enclosed by thehorizon to contain a thermal distribution of particles. Inthe static coordinates (22) this gives the ”hot tin can”description of de Sitter space [60, 112, 113]. However, aswe showed in sections V and VI the static line element isnot a solution of the Einstein equation when the effect ofthe horizon is included in the quantum averaging leadingto a thermal energy-momentum tensor. Hence we mustuse coordinates that can accomodate also non-de Sittersolutions such as the cosmologically relevant expandingde Sitter patch as described by the FLRW coordinates(18). In these coordinates the tin can picture must be generalized to account for continuous energy loss due tothe expansion of space. Simply put, in an expandingspace the tin leaks. More concretely, the expansion ofspace will lead the energy density of the massless particlesto dilute as ∝ a − , which is a purely geometric statement.We will denote the loss of energy density per unit timefrom dilution with ∆, which here has the expression∆ = − Hρ m . (118)If the cosmological horizon maintains thermal equilib-rium with an otherwise constantly diluting and thus cool-ing energy density, an equal amount of heat must flowfrom the horizon ”in” to the bulk that is lost ”out” bydilution ˙ ρ Λ = ∆ , (119)where we have neglected small terms of O ( ˙ H ). When theabove is inserted in (117) and with (118) one gets2 ˙ HM = − ρ m . (120)For a thermal ρ m with the temperature T = H/ (2 π )equation (120) precisely coincides with (109) and fur-thermore the change in vacuum energy (119) with (118)agrees with (108).Assuming that the thermodynamic features persisteven when spacetime has evolved away from de Sitter, aslong as the horizon has heat it will continue to radiate andlose energy by dilution and thus to grow without bound.In this case the ultimate fate of the Universe would notbe an eternal de Sitter space with finite entropy, but anasymptotically flat spacetime with no temperature, aninfinitely large horizon and hence infinite entropy. VIII. SUMMARY AND CONCLUSIONS
In this work we have studied the stability of de Sit-ter space in the semi-classical approach for a model witha non-interacting conformally coupled scalar field and acosmological constant i.e. vacuum energy. Back reac-tion was derived in a prescription where the expectationvalues sourcing the semi-classical Einstein equation werecalculated via a coarse grained density matrix contain-ing only states that are observable to a local observer.For the chosen initial condition of the Bunch-Davies vac-uum this prescription translates as neglecting all degreesof freedom located beyond the cosmological event hori-zon. As we have shown via a detailed argument, in ourapproach de Sitter space is not stable and in agreementwith [13] (section 10.4) but in disagreement with [41].Coarse graining over unobservable states in the den-sity matrix is made frequent use in various contexts suchas the decoherence program and black hole informationparadox, but rarely considered in cosmological applica-tions, in particular semi-classical backreaction via the7Friedmann equations as done in this work. Our studyindicates that loss of information from coarse grainingstates beyond the horizon leads from the initial Bunch-Davies vacuum, a pure state, to a thermal density matrixand manifestly breaks de Sitter invariance.Our result also shows that a local observer who is onlycausally connected to states inside the horizon will inthe cosmologically relevant expanding FLRW coordinatesview de Sitter space as filled with a thermal energy den-sity with a constant temperature given by the Gibbons-Hawking relation T = H/ (2 π ) that is maintained by acontinuous incoming flux of energy radiated by the hori-zon. Without such a flux the expansion of space woulddilute and cool the system quickly leading to an emptyspace.From the semi-classical Friedmann equations we madethe simple but nonetheless important observation thatspace filled with thermal gas, which in our prescriptionfollows from de Sitter space possessing the cosmologicalevent horizon, is not a solution consistent with havinga constant H . This follows trivially from the fact thatthermal particles do not have the equation of state ofvacuum energy and is the key mechanism behind the ob-tained non-trivial back reaction. This can also be seen inthe static coordinates, which are not a solution of Ein-stein’s equation when the energy-momentum tensor de-scribes thermal gas.By modifying the Friedmann equations to containgradually decaying vacuum energy we were able to pro-vide a self-consistent solution for the evolution of theHubble rate. The solution had the behaviour where H remained roughly a constant for a very long time, buteventually after a time scale ∼ M /H the system nolonger resembled de Sitter space. As a physical pictureof the process we proposed that a quantum field in curvedspace may exchange energy with the vacuum making vac-uum energy a dynamical quantity instead of a constantparameter fixed by the Lagrangian. In this interpretationparticle creation occurs at the expense of creating a neg-ative vacuum energy contribution. This provides a mech-anism allowing the overall vacuum energy to decrease, avery closely analogous picture to black hole evaporationwhere a black hole loses mass due to a negative energyflux into the horizon.Finally, we presented an alternate derivation of themain result by using the techniques of horizon thermo-dynamics. In the thermodynamic derivation the conceptof dynamical vacuum energy proved a crucial ingredient,as it gives a well-defined meaning to the differential of in-ternal energy in the cosmological setting allowing a clearinterpretation of the first law of thermodynamics for deSitter space. The derivation via horizon thermodynam-ics turned out to be remarkably simple providing insightsalso to spacetimes that are not to a good approximationde Sitter. The thermal argumentation implied that thefate of the Universe is in fact an asymptotically flat spaceinstead of eternal de Sitter expansion.The possible decay or evaporation of the de Sitter hori- zon seems like a prime candidate for explaining the un-naturally small amount of vacuum energy that is consis-tent with observations. Importantly, in our prescriptionfor semi-classical gravity a gradual decrease of H is re-covered. Unfortunately, the predicted change is quiteslow. For the Early Universe and in particular inflationthe gradual decrease of H from back reaction is muchsmaller than the slow-roll behaviour usually encounteredin inflationary cosmology. Of course due to the multitudeof various models of inflation, an evaporation mechanismcould potentially provide a novel block for model buildingin at least some cases.Perhaps the most profound implication of this workis that it suggests that potentially the eventual de Sit-ter evolution of the Universe as predicted by the currentstandard model of cosmology, the ΛCDM model, is noteternal. This indicates that at least some of the prob-lems associated with the finite temperature and entropyof eternal de Sitter space and in particular the issues withBoltzmann Brains [114] could be ameliorated.Of course all of the perhaps rather significant predic-tions from this work rest on the coarse graining prescrip-tion we have introduced in the semi-classical approachto gravity. Quite unavoidably, it results in an inherentlyobserver-dependent approach due to the observer depen-dence of the de Sitter horizon. This is in accord with thestatements of [41], but from a fundamental point of viewappears to result in rather profound conclusions such asEverett - Wheeler or many-worlds interpretation of quan-tum mechanics, as discussed in [41]. In a semi-classicalapproximation however no obvious inconsistencies seemto arise when one simply includes the additional step ofcoarse graining the quantum state with respect to theperceptions of a particular observer, although more workin this regard is required.Coarse graining over unobservable information givesrise to several natural features: it allows for the gener-ation of entropy, the quantum-to-classical transition viadecoherence and by definition leads to a result contain-ing only the information an observer may interact with.When tracing over information beyond the event horizonof de Sitter space it also leads to an energy-momentumwith a divergence on the horizon, which may signal abreakdown of the semi-classical approach but more in-vestigation is needed. We end by emphasizing that inthis work we have not presented a complete analysis ofall physical implications of the prescription, which needsto be done in order to ultimately determine its viability. ACKNOWLEDGMENTS
We thank Paul Anderson and Emil Mottola for illumi-nating discussions and Thanu Padmanabhan for valuablecomments on the draft. This research has received fund-ing from the European Research Council under the Euro-pean Union’s Horizon 2020 program (ERC Grant Agree-ment no. 648680) and the STFC grant ST/P000762/1.8
Appendix A: Details on mode normalization
Here we provide the details for the normalization insection V C.Starting with the mode defined in the unobservablepatch of the expanding half of the de Sitter manifold (94)we first introduce an infinitesimal convenience factor inthe arguments of the hypergeometric function k −→ k − i ǫH , (A1)which gives f B ℓk (¯ r B ) = D B ℓk ( H ¯ r B ) ℓ (cid:2) ( H ¯ r B ) − (cid:3) − ik H × F (cid:20) ℓ − ik H + 12 − ǫ, ℓ − ik H + 1 − ǫ ; ℓ + 32 ; ( H ¯ r B ) (cid:21) . (A2)We can then use the standard relation F (cid:2) a, b ; c ; z (cid:3) = Γ[ c ]Γ[ c − a − b ]Γ[ c − a ]Γ[ c − b ] × F (cid:2) a, b ; a + b + 1 − c ; 1 − z (cid:3) + Γ[ c ]Γ[ a + b − c ]Γ[ a ]Γ[ b ] × (1 − z ) c − a − b F (cid:2) c − a, c − b ; 1 + c − a − b ; 1 − z (cid:3) , (A3)to write two important results F (cid:20) ℓ − ik H + 12 − ǫ, ℓ − ik H + 1 − ǫ ; ℓ + 32 ; ( H ¯ r B ) (cid:21) H ¯ r B → = Γ (cid:2) ℓ + (cid:3) Γ (cid:2) ikH (cid:3) Γ (cid:2) (cid:0) ℓ + ikH + 1 (cid:1)(cid:3) Γ (cid:2) (cid:0) ℓ + ikH + 2 (cid:1)(cid:3) + O ( ǫ ) , (A4)and (cid:2) ( H ¯ r B ) − (cid:3) ∂∂r F (cid:20) ℓ − ik H + 12 − ǫ, ℓ − ik H + 1 − ǫ ; ℓ + 32 ; ( H ¯ r B ) (cid:21) H ¯ r B → = 0 . (A5)The validity of (A4) and (A5) is only strictly true withthe infinitesimal shift (A1) introducing the factor ∝ (cid:2) ( H ¯ r B ) − (cid:3) ǫ , (A6)which suppresses the second term coming from (A3).With (A4) and (A5) one may verify the correct normal-ization of (A2) by evaluating the inner product (93) atthe time instant H ¯ r B →
1. Equation (A4) also allowsone to easily show that close to the horizon with (A2)the mode becomes a simple plane wave.Introducing a similar infinitesimal factor in the hyper-geometric function of the mode in the observable patchof the expanding half of the de Sitter manifold (86) with k → k + i ǫH and using (A4) we can verify that thehorizon limit is again a plane wave with a prefactor coin-ciding with the horizon limit of (A2), which shows thatit is also correctly normalized.Strictly speaking, by introducing the infinitesimal ǫ factor we are effectively solving a different equation ofmotion than when starting with ǫ = 0. However, at theend of a calculation once a physical observable has beenderived we set ǫ → ǫ is useful as it simplifies some of the inter-mediate steps of the derivation.We acknowledge fruitful discussions with the authorsof [115]. Appendix B: Energy density far from the horizon
In this appendix we calculate the energy momentumtensor in the static coordinates (21) with the line element(22) in a region close to the center of the Hubble sphere.In order to calculate the expectation value of the en-ergy density (again dropping the A labels) h ˆ T ¯0¯0 i = h ∇ ρ ˆ φ ∇ ρ ˆ φ + ∇ ¯0 ˆ φ ∇ ¯0 ˆ φ + 16 (cid:2) G ¯0¯0 − ∇ ¯0 ∇ ¯0 − (cid:3) (cid:3) ˆ φ i , (B1)it proves convenient to make use the equation of motion − (cid:3) ˆ φ = − H ˆ φ and the relation ∇ ¯0 ∇ ¯0 h ˆ φ i = O ( H ¯ x ) , (B2)to arrive at the expression h ˆ T ¯0¯0 i = h h g ¯0¯0 (cid:0) ∂ ¯0 ˆ φ (cid:1) + g ¯ r ¯ r (cid:0) ∂ ¯ r ˆ φ (cid:1) + g ¯ θ ¯ θ (cid:0) ∂ ¯ θ ˆ φ (cid:1) + g ¯ ϕ ¯ ϕ (cid:0) ∂ ¯ ϕ ˆ φ (cid:1) − H ˆ φ i + (cid:0) ∂ ¯0 ˆ φ (cid:1) i + O ( H ¯ x ) . (B3)The simplifications (B2) and (B3) follow from the factthat to leading order in O ( H ¯ x ) the radial contribution tosolutions (86) simplifies significantly and that the staticline element coincides with Minkowski space. To thisaccuracy the only relevant modes are ψ k = D k Y e − ik ¯ t + O ( H ¯ x ) (B4) ψ m k = D k H ¯ rY m e − ik ¯ t + O ( H ¯ x ) ; m ∈ {− , , } , (B5)where from (91) we have | D k | = kπ , | D k | = H k + k H π , (B6)9and the spherical harmonics read Y = 12 r π , (B7) Y − = 12 r π sin θ e − iϕ , (B8) Y = 12 r π cos θ , (B9) Y = − r π sin θ e iϕ . (B10)With the above, we can now evaluate the expression (B3)piece-by-piece. The first is h g ¯0¯0 (cid:0) ∂ ¯0 ˆ φ (cid:1) i = −h (cid:0) ∂ ¯0 ˆ φ (cid:1) i = − D(cid:26) Z ∞ dk h − ikψ k ˆ a k + H . C . i(cid:27) E = − Z ∞ dk k π h h ˆ n k i i = − Z d k (2 π ) (cid:20) k ke πk/H − (cid:21) , (B11)where we have left the accuracy O ( H ¯ x ) implicit andused h ˆ a † ℓmk ˆ a m ′ ℓ ′ k ′ i = δ mm ′ δ ℓℓ ′ δ ( k − k ′ ) h ˆ n ℓmk i = δ mm ′ δ ℓℓ ′ δ ( k − k ′ ) 1 e πk/H − , (B12) valid for the thermal density matrix (100). We can con-tinue in a similar fashion h g ¯ r ¯ r (cid:0) ∂ ¯ r ˆ φ (cid:1) i = D(cid:26) ∂ ¯ r X m = − Z ∞ dk h ψ m k ˆ a m k + H . C . i(cid:27) E = Z ∞ dk H k + k π X m = − | Y m | × h h ˆ n m k i i = 13 Z dk H k + k π h e πk/H − i . (B13)Repeating these steps for the remaining contributions in(B3) gives up to O ( H ¯ x ) h g ¯ r ¯ r (cid:0) ∂ ¯ r ˆ φ (cid:1) i = h g ¯ θ ¯ θ (cid:0) ∂ ¯ θ ˆ φ (cid:1) i = h g ¯ ϕ ¯ ϕ (cid:0) ∂ ¯ ϕ ˆ φ (cid:1) i = 13 h (cid:0) ∂ ¯0 ˆ φ (cid:1) i + H h ˆ φ i , (B14)so the terms in the square brackets of (B3) cancel so that(B3) and (B11) finally give h ˆ T ¯0¯0 i = h (cid:0) ∂ ¯0 ˆ φ (cid:1) i = Z d k (2 π ) (cid:20) k ke πk/H − (cid:21) + O ( H ¯ x ) , (B15)as written in (101). [1] E. Mottola, Particle Creation in de Sitter Space,
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