AAn alternative interpretation of the cosmological vacuum
Thomas Sch¨urmann ∗ D¨usseldorf, Germany
The present contribution aims at obtaining the energy/dark energy fraction of the universe bystarting from the Sitter vacuum only and without involving any additional source of energy. Todo so, we consider two different standard solutions of the Einstein vacuum equations with positivecosmological constant. In accordance with [9][10], we initially derive an uncertainty principle for theassociated spherical and hyperbolical time-slices to highlight the connection between the classicalnotion of spatial curvature and the quantum mechanical uncertainty of position and momentumin de Sitter space (Theorem). Based on the positive and negative curvatures of these foliations,an alternative notion of (time-dependent) energy and dark energy of the vacuum is established.This opens the possibility of a formal derivation of Einstein’s gravitational constant κ by matchingthe dark energy contribution at the Planck scale at one Planck time after the initial singularity.Moreover, for the fraction of 70% dark energy, the age of the universe is estimated to be about13.7 billion years. Finally, we verify that the metric corresponding to a suitable junction of thehyperbolic and the spherical foliation is a solution of Einstein’s field equations. This suggests acosmology given by a de Sitter space embedded into an asymptotic Schwarzschild-Anti-de Sitterbackground. PACS numbers: 04.60.-m, 04.60.Bc, 02.40.Ky
I. INTRODUCTION
At present, one of the deepest problems in theoreticalphysics is harmonizing the theory of general relativity(GR), which describes gravitation, and applications tolarge-scale structures (stars, planets and galaxies), withquantum mechanics, which describes the other three fun-damental forces acting on the atomic scale.General relativity models gravity as curvature of space-time. On the other hand, quantum field theory (QFT) istypically formulated in the flat spacetime used in specialrelativity. No theory has yet proven successful in describ-ing the general situation where the dynamics of matter,modelled with quantum mechanics, affect the curvatureof spacetime. Even in the simpler case where the curva-ture of spacetime is fixed a priori, developing QFT be-comes more mathematically challenging, and many ideasphysicists use in QFT on flat spacetime are no longer ap-plicable [1]. A conceptual difficulty in combining quan-tum mechanics with GR also arises from the contrastingrole of time within these two frameworks. In quantumtheories time acts as an independent background throughwhich states evolve, with the Hamiltonian operator act-ing as the generator of infinitesimal translations of quan-tum states through time. In contrast, GR treats time asa dynamical variable which interacts directly with mat-ter and moreover requires the Hamiltonian constraint tovanish [2], removing any possibility of employing a notionof time similar to that in quantum theory.One of the biggest confrontations between both theo-ries is the Cosmological Constant Problem [3]. Quantumfield theory predicts a huge vacuum energy density fromvarious sources. On the other hand, GR requires thatevery form of energy gravitates in the same way. When combining these concepts together, it is widely supposedthat the vacuum energy gravitates as a cosmologicalconstant. However, depending on the Planck energycutoff and other factors, the discrepancy between theobserved cosmological constant and the prediction ofQFT is as high as 50-120 orders of magnitude.In the ΛCDM approach, the universe is approximatedat late times by two fluids: pressureless matter and acosmological constant Λ. Both baryonic matter and colddark matter are unable to push the universe to accelerate[4]. Thus, besides dust-like fluids, one needs to includeΛ to account for the observed speedup. However, themagnitude of Λ predicted by quantum fluctuations offlat spacetimes leads to a severe fine-tuning problemwith the observed value of Λ. Even considering a curvedspacetime one cannot remove the problem [3]. Further,both matter and Λ magnitudes are extremely closetoday, leading to the well-known coincidence problem[5][6][7]. Under these aspects the ΛCDM model seemsto be incomplete, whereas from the observational pointof view it adapts well to data.In the following, our intention is to obtain the energyand dark energy fraction of the universe by starting onlyfrom the de Sitter vacuum and without involving anyother additional source of energy. We consider two dif-ferent standard solutions of the Einstein vacuum equa-tions with positive cosmological constant. In Sect. II,we initially derive an uncertainty principle for the as-sociated spherical and hyperbolical time-slices. Based onthe positive and negative curvatures of these foliations,in Sect. III, an alternative notion of (time-dependent) en-ergy and dark energy of the vacuum is established anddiscussed. In Sect. IV, we introduce the corresponding a r X i v : . [ phy s i c s . g e n - ph ] J u l cosmology given by a de Sitter space embedded into anasymptotic Schwarzschild-Anti-de Sitter background. Asummary with outlook is given in Sec. V. II. THE UNCERTAINTY PRINCIPLE IN DESITTER SPACE
As known from the history of Riemannian geometryand general relativity, the property of diffeomorphisminvariance is one of the most important features for thegeneralization of physical laws to curved spaces.For uncertainty principles given in 3-dimensional spacethis means that the applied measures of uncertaintyshould be chosen with caution. When the standard devi-ation of the momentum is based on the Laplace-Beltramioperator, then one can be sure that invariance underchange of coordinates is satisfied. On the other hand,a proper choice for the measure of position uncertaintyis hard to obtain if one is only concerned with applyingthe concept of standard deviation. As recently shown[8][9][10][11], fortunately the choice of a standard devi-ation in position space is not really necessary or evenappropriate. Especially from the concept of projection-valued measures it becomes clear, alternatively, to con-sider suitable spatial domains for the representation ofposition uncertainty. Moreover, from the theory of spec-tral analysis on manifolds, we know that geodesic ballsplay an important role because these are the distin-guished domains in many variational approaches. Sincegeodesic balls are uniquely classified by their geodesicradius (or diameter) it becomes self-evident that thegeodesic radius is the appropriate measure for the rep-resentation of position uncertainty in curved spaces. Forthat reason it becomes clear why the requirement of co-ordinate invariance is hard to obtain by the known GUPand EUP in literature.More precisely, in order to measure the momentum oneneeds to consider a measure of position uncertainty. Thisis given by a domain D (typically the geodesic ball B r )with boundary ∂D characterized by its geodesic radius r or diameter d and Dirichlet boundary conditions suchthat the wave function of the particle is confined in D .The method then reduces to the solution of an eigenvalueproblem for the wave function ψ ∆ ψ + λψ = 0 (1)inside D with the requirement that ψ = 0 on the bound-ary ∂D , while λ denotes the eigenvalue and ∆ is theLaplace-Beltrami operator of the corresponding mani-fold. Then, one can write the following general inequality[8] σ p ≥ (cid:126) (cid:112) λ , (2)where λ denotes the first Dirichlet eigenvalue of theproblem. For the general class of 3-dimensional Rieman- nian manifolds of constant curvature k , there is a closedform solution and it was found that [8] σ p r ≥ π (cid:126) (cid:114) − kπ r , (3)where the corresponding position uncertainty of the par-ticle is represented by the radius r of the associatedgeodesic ball. The underlying metric in spherical andhyperbolic coordinates can be written as ds = dr + sin( √ k r ) k d Ω , (4)with the 2-dimensional measure d Ω = dθ + sin( θ ) dϕ , (5)for 0 ≤ θ ≤ π and 0 ≤ ϕ ≤ π . Note that we usethe formal identity sin( ix ) ≡ i sinh( x ), such that thisrepresentation can be used for positive and negative k .For k ≥
0, we have the domain 0 ≤ r < π/ √ k . For k < r >
0. It should also be mentioned that (3)is independent of the coordinate system (diffeomorphisminvariance) and not of the same kind as the ordinary EUPor GUP in literature because it features the characteristiclength of the confinement corresponding to r . Thus, r should be interpreted rather as uncertainty and does notdescribe the standard deviation of position [8][11].Now, let us turn to the Einstein equations. Every n -dimensional space of constant curvature K is also an Ein-stein space defined by the standard condition R ij = ( n − Kg ij (6)with Ricci tensor R ij and metric g ij , where i, j =1 , , ..., n . This is also the case for de Sitter spacetimeswhich are defined as the solution of Einstein’s vacuumfield equations with cosmological constant Λ, R µν = Λ g µν , (7)for µ, ν = 0 , , ,
3. In four dimensions we have Λ = 3 K .It has been shown in [8] that inequality (3) only holds inspaces of dimension three. That is why k in (3) cannotsimply be replaced by K .Since quantum fluctuations are expected to be rele-vant only on very small time scales compared to thecosmological circumstances, it is obvious to consider 3-dimensional foliations at a temporal vicinity of a fixedinstance of cosmological time. Now, let us proceed un-der the assumption that the universe is homogeneous andisotropic. Then, there exists a one-parameter family ofspacelike hypersurfaces Σ τ , foliating the spacetime intopieces labelled by the proper time, τ , of a clock carriedby any isotropic observer. In these coordinates the space-time metrics can be written as ds = − c dτ + a i ( τ ) dχ + sin χ d Ω , ≤ χ ≤ πdχ + χ d Ω , χ ≥ dχ + sinh χ d Ω , χ ≥ τ are 3-dimensional subspaces of constant curvature. The gen-eral form of (8) is called the Robertson-Walker cosmolog-ical model. In the case of the vacuum field equations, itis sufficient to consider only one differential equation cor-responding to the first component of the Einstein tensor, G νµ = − Kδ νµ , which is explicitly given by˙ a ( τ ) c = Ka i ( τ ) − (cid:15), (9)with the scale function a i ( τ ) of (8), and (cid:15) = 0 , ± a i ( τ ) of this equation are asfollows [12][13]: K > a ( τ ) = 1 √ K cosh( √ K cτ ) (cid:15) = +1 (10) a ( τ ) = 1 √ K sinh( √ K cτ ) (cid:15) = − a ( τ ) = 12 √ K e √ K cτ (cid:15) = 0 (12)
K < a ( τ ) = 1 (cid:112) | K | cos( (cid:112) | K | cτ ) (cid:15) = − a ( τ ) = 1 (cid:112) | K | sin( (cid:112) | K | cτ ) (cid:15) = − a i ( τ ), i = 1 , ...,
5, cannot arbitrarilybe chosen, but are uniquely determined. For this reason,the corresponding curvature is well defined.The periodic solutions corresponding to the case of
K <
Theorem.
The uncertainty principle of positionand momentum, corresponding to the metric (8) withconformal factors (10)-(14) at fixed time τ , is given bythe inequality σ p r ≥ π (cid:126) (cid:114) − k i ( τ ) π r , (15)where, for K (cid:54) = 0, the spatial curvature k i ( τ ), i = 1 ,
2, isgiven by k ( τ ) = K cosh ( √ K cτ ) ≡ a (16) k ( τ ) = − K sinh ( √ K cτ ) ≡ − a . (17)For K >
0, there is a spatially flat case k ( τ ) = 0 . (18) Remark:
As will be shown in the proof, the remainingcases k and k , corresponding to (13) and (14), arecontained in k and k and will be obtained by analyticcontinuation. Proof.
See appendix.The time-dependence of k and k for some values K is shown in Fig. 1 and Fig. 2. For K >
0, there is anasymptotic behaviour given by | k i | ∼ Ke − √ K cτ , (19)for τ → ∞ , i = 1 ,
2. That is, in the long run therelation (15) is simplified to the case of σ p r ≥ π (cid:126) . Theturning point of k , for K >
0, is given by the solutionof cosh(2 √ Kcτ ) = 2. This condition can be solvednumerically and gives τ ∗ = 0 . / √ Kc .Moreover, the time behaviour of k remains finitefor τ → k ( τ ) = K + O ( τ ) . (20)On the other hand, the leading term of the asymptoticexpansion of k , for τ →
0, is independent of K and givenby k ( τ ) = − c τ + O (1) . (21)Actually, this independence will be an important pointin the resolution of the cosmological constant problemdiscussed below. Although k and k have a verydifferent time-dependence near the beginning of time,we have the following decomposition of K : Corollary.
For K (cid:54) = 0, there is the identity K = k k k + k . (22) FIG. 1: Spatial curvature k , for K = 10 (blue) and K = − c . For positive K , there is a turning point at τ ∗ ≈ .
2. For negative K ,there is a singularity at π/ k , for K = 1 (blue) and K = − c . For negative K , there is an additional singularity π (see text). Proof.
By substitution of (16) and (17).This decomposition of K will be the key identityfor the description of the Schwarzschild-Anti-de Sittercosmology in Section IV. III. ALTERNATIVE VACUUM ENERGY IN DESITTER SPACE
In quantum field theories the notion of empty spacehas been replaced with that of a vacuum state, definedto be the ground state (lowest possible energy density)of a collection of quantum fields. A quantum mechanicalfeature of quantum fields is that they exhibit zero-pointfluctuations everywhere in space, even in regions whichare otherwise empty. These zero-point fluctuations giverise to a vacuum energy density u QFT . This vacuum en-ergy density is believed to act as a contribution to thecosmological constant [14].On the other hand, when κ = 8 πG/c is Einstein’sgravitational constant, the vacuum energy and the cos-mological constant have identical behaviour in generalrelativity, as long as the vacuum energy density is iden-tified with u Λ = Λ κ ≈ . × − Jm , (23)where Λ = 1 . × − m − is the empirical esti-mate of the cosmological constant. Since the cosmo-logical constant corresponds to the curvature K of the4-dimensional spacetime, an experimental setup to mea-sure it needs to include the dimension of time for itsdetermination. Measurements of Λ in astrophysics aretypically performed by (indirectly) comparing huge spa-tial distances or velocities of objects corresponding todifferent instances of cosmological time associated withsignals coming from very far away.However, the outstanding problem is that most quan-tum field theories predict a huge value of u QFT for thequantum vacuum. A (simplified) standard argumentfor the determination of vacuum energy densities corre-sponding to vacuum fluctuations is given by summationof zero-point energies according to u QFT = 1(2 π (cid:126) ) (cid:90) | p |≤ Γ d p E p , (24)with cutoff parameter Γ >
0, and the energy spectrum E p = ( p c ) + ( mc ) . In contrast to the measurement ofΛ from the astrophysical point of view, in the quantumfield theoretic approach the measurement of vacuum en-ergy is restricted to 3-dimensional (spatial) domains cor-responding to the vicinity of only one specific instant ofworld time. No observed data at very different cosmolog-ical times are taken into account. Therefore, we ask for adecomposition of Λ into components which are associatedwith spatial foliations of spacetime and which are corre-sponding to the measurements of quantum field theoryin the vicinity of a given world time. Here, we propose toconsider the energy densities corresponding to k and k .According to (23), it is obvious to introduce the notationΛ i = 3 k i , (25)for i = 1 ,
2. As a consequence from the corollary, weobtain the following decomposition of the cosmologicalconstant Λ = Λ Λ Λ + Λ . (26)The left-hand side is proportional to the curvature in 4-dimensional spacetime. The right-hand side is composedby components of curvatures in 3-dimensional space at agiven instant of world time. Obviously, we can identifythe corresponding spatial vacuum energies u i , i = 1 , u i = Λ i κ . (27)After a few algebraic manipulations, the energy densitycorresponding to Λ is given in terms of the compositionlaw u Λ = u u u + u . (28)For Λ >
0, it follows that u >
0, and u <
0, for all τ ≥
0. Thus, it is obvious to consider u to be the positiveenergy density, and u to be the contribution of the darkenergy density. Now, it is interesting to consider the timeevolution of the relative fractions with respect to the totalamount of energy density by q ( τ ) = u u + | u | = 12 (cid:104) − sech (cid:16) c √ Kτ (cid:17)(cid:105) , (29) q ( τ ) = | u | u + | u | = 12 (cid:104) (cid:16) c √ Kτ (cid:17)(cid:105) , (30)where sech( x ) = 1 / cosh( x ) is the hyperbolic secant of x . According to these fractions, there is a high relativedensity of dark energy at τ = 0, followed by a continuousdecrease to approach 50 % in the long run for τ → ∞ . Af-ter 7 .
68 billion years there is a turning point. The presentfraction of about 70 % dark energy density is reached atan age of τ = 13 .
64 billion years . (31)This fits well to the best known estimate of 13 .
77 billionyears obtained by the Planck Collaboration in [15]. Weexpressed the time evolution in Fig. 3.Since we have obtained a time evolution of vacuumenergy densities, let us also verify the situation at thePlanck scale. For the moment at one Planck time τ P = (cid:112) (cid:126) G/c after the initial state, we can restrict our con-siderations to the dark energy density u , because therelative fraction q of positive energy density gives nearlyzero at this time (see data below). According to (25) and(27), the absolute value of the dark energy density at thePlanck time τ P after the initial state is given by | u ( τ P ) | = 5 . × Jm . (32) FIG. 3: Relative fraction of energy density (orange) anddark energy density (blue) over the age τ of the universe.The fraction of 70 % dark energy (black dot) is reachedat the age of 13.64 billion years. There are also turningpoints at 7 .
68 billion years after the initial state (graydots). In the long run, both energy densities approachan equal fraction (see text).
On the other hand, the textbook expression of the Planckenergy density is given by E P l P = c (cid:126) G = 4 . × Jm , (33)which is only about one order of magnitude larger than(32). Nevertheless, we would like to refine this compar-ison as follows: First, we replace the cubic Planck vol-ume of edge length l P by the spherical domain of volume V P = 4 πl P /
3, of Planck radius l P and define the followingadjusted Planck energy density u P ≡ E P V P = 34 π E P l P , (34)which is slightly different to (33) by the factor 3 / π . Fur-thermore, we have to complete the Planck energy by re-garding the factor 1/2 corresponding to the zero-pointmodes of the vacuum energy (24). Thus we have toequate | u ( τ P ) | = u P . (35)Since the Planck time is very small, we can properly ap-ply the asymptotic representation u = − κ c τ + Λ3 κ + O ( τ ) , (36)which is independent of the cosmological constant in theleading term. The second term is about 1 . × − and therefore negligible right now. After substitutionof the leading term into (35), we obtain the equivalentexpression κ = 8 πGc , (37)which is identically satisfied by the definition of Einstein’sconstant κ . Thus, we see that the vacuum energy density u ( τ ), at Planck time τ P , is identical to the Planck energydensity. This is a remarkable result, because it can beconsidered as a calibration of general relativity ( κ ) tothe Planck scale of QFT.Up to this point there is no indication in our approachthat gravity breaks down even for τ →
0. Thus, onecould suppose that something in quantum field theoryhas to be completed. One known way to do that is theintroduction of a cutoff Γ to fix the ultra-high energydensity scale to the amount of energy density given bythe theory of gravity. Therefore, we consider the zero-point energy expression (24). For zero mass, its exactvalue is given by the closed form expression u QFT = c Γ π (cid:126) . (38)The corresponding cutoff Γ can be fixed by equating itat one Planck time after the initial state according to | u ( τ P ) | = u QFT , (39)and is given by Γ = (6 π ) m P c, (40)with Planck momentum m P c = (cid:126) /l P . Although the cut-off is obtained purely from the vacuum field equations ofgeneral relativity, it is as far as possible independent ofthe cosmological constant itself. This seems to be neces-sary to overcome so many orders of magnitude to reachthe Planck scale at all. Actually, such results are whatone might require to overcome the cosmological constantproblem.There have been assumptions in literature that alsonon-zero masses might be necessary to fit the Planckscale. This is not confirmed in our approach becausea suitable cutoff can already be reached for zero mass.However, we cannot exclude that there are also masscontributions in the cutoff, but we think that they areof minor effect in the very early Planck stage after thebeginning of time.Let us now consider some observable quantities at laterages of the universe. In Table I, we have summarizedsome steps in the time evolution of the cosmological vac-uum energy densities. At the initial state the positiveenergy contribution is finite and given by u Λ . The lat-ter is our main interpretation of the quantity u Λ . Itsdark counterpart is characterized by an infinite densityof negative sign. At one Planck time τ P right after the initial state, thepositive energy density is nearly unchanged, but the neg-ative contribution has become finite and is equal to thehuge energy density corresponding to the Planck scale(34).After about 7 billion years later, there is a turningpoint when the fractions ( q and q ) of both energydensities stop decreasing progressively in time, but beginto relax exponentially slower (Fig. 3). Subsequently,when the fraction of dark energy density has decreasedto 70 %, then the present age has been reached. Lateron, both energy densities asymptotically tend to zero.On the other hand, there is an exponential growth rateof space which is asymptotically preserved because thenegative fraction of energy always dominates its positivecounterpart. This phenomenon is one of the mainadvantages in our approach because the horizon problem does not come into the picture.Let us also compare our results with the mass den-sity given by the Wilkinson Microwave Anisotropy Probe(WMAP). The WMAP determined that the universe isflat, from which it follows that the mean energy density inthe universe is equal to the critical density. This is equiv-alent to a mass density of 9 . × − g/cm . Of this totaldensity, we know (as of January 2013) the breakdown tobe 71.4 % dark energy and 28.6 % of atoms and cold darkmatter. For this fraction of dark energy, our estimatedage of the universe is about 13 billion years. Accordingto our approach, the corresponding mass densities at thisage are given in the first line of Table II.Even though our values are 25 % above the estima-tions of the WMAP, they are still within the bounds ofpossibility. This is remarkable because the same formulaalso fits to the Planck scale shortly after the initial state.Since there is only one single parameter (Λ) in our ap-proach, this description of the large scale in space andtime encourages us to unify both a and a in a closedform static solution of the field equations. Cosmological event τ u /u Λ u /u Λ Initial singularity 0 1 −∞ Planck time τ P ≈ − . × Turning point 7 .
67 0 . − . q = 0 .
7) 13 .
64 0 . − . ∞ − u Λ ( (cid:54) = u + | u | ). The time ismeasured in billions of years from the initial state. For thenumerical value of the Planck energy density see (35). ρ [g/cm ] ρ [g/cm ] ρ [g/cm ]From u , u of (27) 12 . · − . · − . · − WMAP (2013) 9 . · − . · − . · − TABLE II: Total mass density ρ and its fractions ρ , ρ cor-responding to 26.8 % of atoms/dark matter and 71.4 % of darkenergy (WMAP). The densities of the first line are our esti-mations for the present age and ρ i ( τ ) = q i ( τ ) ( u + | u | ) /c ,for i = 1 , IV. SCHWARZSCHILD-ANTI-DE SITTERCOSMOLOGY
The results of the previous sections can now be ap-plied to consider the corresponding Schwarzschild-Anti-de Sitter (SAdS) solution. But before we proceed, letus briefly describe the state-of-the-art situation of theordinary Schwarzschild-de Sitter (SdS) approach. Thismetric describes a static spherically symmetric vacuumsolution of the Einstein equations G µν + Λ g µν = 0 and isgiven by ds = − f ( r ) d ( ct ) + dr f ( r ) + r d Ω , (41)with f ( r ) = 1 − r s r − Λ3 r (42)and Schwarzschild radius r s = 2 GMc . (43)It describes one part of the maximal extension of the SdSspacetime. Cosmologically, this is a good model of anisolated region in an asymptotic background of constantpositive curvature Λ /
3. Negative values of Λ in (42) arealso possible, but its order of magnitude would be muchtoo small to solve the cosmological constant problem.As opposed to this, our approach is leading to a staticcosmology composed by a 3-sphere of constant curva-ture and energy u in an asymptotic background of neg-ative curvature corresponding to the energy u , but stillrecognizing Λ >
0. This picture fits well into a staticSAdS solution of the field equation, where we have toconsider both internal and external parts, which are con-tinuously matched together at a suitable hypersurface ofboth spacetime. More formally, we know that for a fixedtime slice τ , the curvatures of both regions are given by k and k of the theorem. For reasons of compatibilitywe apply the notation Λ and Λ of (25). Then, for theinterior static solution, we consider the following fieldequation G µν + Λ g µν = 0 , (44) whereas for the exterior metric we have to solve G µν + Λ g µν = 0 . (45)We denote the associated solutions by g (1) µν and g (2) µν . Theyhave to match continuously at the boundary betweentheir domains. The Birkhoff theorem states that the vac-uum metric of a spherical symmetric distribution of mass(energy) is static and identical to the Schwarzschild met-ric of the enclosed total gravitational mass, while thevacuum region can either be outside all mass, or interiorto some or all mass [16]. Therefore, the interior regionis not affected by the homogeneous (dark) backgroundof the exterior region and the solution of (44) can beconsidered as a special case of the interior Schwarzschildmetric, when the domain inside is completely filled bythe vacuum fluid u . In this case, we obtain ds = − g ( r ) d ( ct ) + dr g ( r ) + r d Ω , (46)with g ( r ) = 1 − Λ r . (47)Actually, this solution also solves (44) when there is atime-dependent factor in front of the first term of (46).For the original interior Schwarzschild solution this fac-tor is a constant and given by 1 /
4. At this point, thecommon practice is to redefine the time coordinate toabsorb the factor, so that g ≡ − g − . But such a choicemakes g discontinuous across the boundary and canlead to observational effects such as light deflection ordelay. However, we will match the interior and exteriorsolution without any change of the time coordinate toserve the continuity of time across the boundary.For the exterior solution of the field equation (45), weobtain the static Schwarzschild-Anti-de Sitter metric ds = − g ( r ) d ( ct ) + dr g ( r ) + r d Ω , (48)where g ( r ) = 1 − r r − Λ r . (49)It should be mentioned here that this metric is of typeSAdS because according to our theorem we always haveΛ <
0. Moreover, this metric solves (45) for every con-stant value of r . This is an important degree of freedomto get a proper matching of both solutions. The trivialcase r = 0 is not appropriate to continuously fit the in-terior solution. We require connecting both solutions atthe curvature radius a of the interior solution, which isgiven in (16) of our theorem. Accordingly, we have toconsider the equation g ( a ) = g ( a ). From this condi-tion it uniquely follows r = a (cid:18) − Λ Λ (cid:19) . (50)We have chosen a mixed notation to emphasise the mean-ing of the pre-factor as the curvature radius of the interiordomain. This matching implies that the only singular-ity of the exterior solution is given at r = a , and wehave g ( r ) ≥
0, for all r ≥ a . Moreover, there is anabsolute minimum of r when Λ ≈ .
44 Λ, so that r remains strictly positive for every instant of cosmologicaltime. However, the value of r approaches to infinity ifΛ is chosen to be one of its extreme values 0 or Λ. Theadvantage of our solution (49) against the approach in(42) is threefold. On the one hand our solution is of typeSAdS although Λ is positive. Furthermore, the value ofΛ can increase (negatively) beyond all bounds depend-ing on the cosmological time which is considered. Thisproperty opens the possibility to reach even the Planckscale in the vicinity of the initial state. Finally, there is anatural relation between both terms r and Λ , such thatthey are not different kinds of objects but are fundamen-tally related. This property is missing in the commonsolution (42). However, black hole’s inside the interiorregion can be expressed by applying Λ in (42), insteadof Λ.Since the interior region is corresponding to a 3-sphere,we can compute the maximum possible (physical) dis-tance of two points in this region. At the present age t = τ , we obtain a value of d ( τ ) = πa ≈ . × m.For the diameter just at the initial state we obtain thefinite value of d (0) = π (cid:112) / Λ ≈ . × m, which isnot very far apart from the present value. On the otherhand, the physical distance d obs of two photons sent inopposite directions is twice the observable radius (or theparticle horizon) and given by d obs ( t ) = 2 a ( t ) (cid:90) t c dτa ( τ )= 2 a ( t ) gd( √ Kc t ) , (51)where gd( · ) is the Gudermann function [17] defined bygd( x ) = (cid:82) x sech ξ dξ . For the present age we obtain avalue of d obs ≈ . × m, which might be comparedwith the diameter of the observable universe in literatureof about 8 . × m . For the exterior region no horizonis present.All in all, at the initial state, the interior solution cor-responds to a 3-sphere (closed universe) with finite cur-vature Λ / > k → −∞ ). Lateron ( τ > ∝ e √ Λ / cτ , which is consis-tent with the necessity of inflation. Therefore, the phe- nomenon of inflation is intrinsically contained in this ap-proach, such that the horizon problem does not appear. V. SUMMARY AND OUTLOOK
We have proposed a reinterpretation of two non-standard de Sitter solutions of Einstein’s vacuum fieldequations with Λ >
0. This approach is new insofar thatit takes into account that the spatial curvatures k and k of the associated hypersurfaces are non-zero and there-fore explicitly dependent on the given time-slice. Thevacuum energy densities corresponding to these curva-tures have been discussed and properly matched at thePlanck scale. For every instant of cosmological time,we also introduced the associated field equation andmatched the corresponding interior and exterior solu-tion. This spacetimes provide the possibility to imple-ment quantum field theories at any instant of cosmic timeand even near the initial state.In the present day experiments we measure the shapeof the radiation spectrum in the universe while the en-ergy of the radiation is proportional to T . The deriva-tion of this law uses the relation between radiation pres-sure and the internal energy density of a black body. Inthe cosmology described above, the history of the uni-verse is mainly dominated by the dark energy contri-bution, which homogeneously surrounds the interior 3-sphere of positive vacuum energy. Photons are their ownanti-particles. Therefore, the corresponding temperaturedistribution of the radiation given by our dark surround-ing might be obtained by a generalization of Planck’s lawto the case of negative energy densities. Then, the rela-tion between energy density and temperature depends onthe cosmological time and might be given byΛ + Λ ≈ − κ σ T , (52)where σ is the radiation constant [18]. We alreadyknow from the previous sections that our dark energyapproach fits well into the large scale and one can expectthat the time-dependence of temperature will also beappropriate, especially at the Planck scale or at theearly universe. When time goes to infinity the left-handside approaches zero and we obtain T →
0, whichseems to be consistent too. However, at the presentcosmological age this law cannot carry up because alsoother kinds of energy sources become more relevant andit doesn’t fit into the generalized Stefan-Boltzmann law.A rough estimation of today’s temperature accordingto (52) gives 27 K, which is one order of magnitudeabove 2 .
73 K. At this point it might be appropriate alsoto include additional sources (e.g. cosmological dust)to complete the picture. This task is left for a futureconsideration.
Acknowledgement.
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Proof of the Theorem.
For
K >
0, we express thespatial part of the first solution corresponding to (10) by ds τ = cosh ( √ Kc τ ) (cid:104) d ˜ r + sin ( √ K ˜ r ) K d Ω (cid:105) , (53)where we have introduced the notation ds τ to indi-cate the restriction to the 3-dimensional space consid-ered at fixed τ . Then, we apply the transformation r = ˜ r cosh( √ Kc τ ), to get ds τ = dr + sin ( √ k r ) k d Ω , (54)with k defined by (16) and the domain0 ≤ r ≤ π √ k . (55)After applying (3) and (4), we obtain the first statement(16) of the theorem. Next, we consider the (spatially)hyperbolic representation corresponding to (11), by ds τ = sinh ( √ Kcτ ) (cid:104) d ˜ r + sinh ( √ K ˜ r ) K d Ω (cid:105) , (56)and apply the transformation, r = ˜ r sinh( √ Kc τ ). Then,we obtain the representation ds τ = dr + sinh ( (cid:113) | ˜ k | r ) | ˜ k | d Ω , (57)where the curvature ˜ k is identified by˜ k ( τ ) = − K sinh ( √ K cτ ) ≡ k . (58)The remaining two cases (13) and (14) correspond to K <
0. Both of them have only hyperbolic representations inthe spatial domain. For the first of them, we write ds τ = cos ( (cid:112) | K | cτ ) (cid:104) d ˜ r + sinh ( (cid:112) | K | ˜ r ) | K | d Ω (cid:105) . (59)This can be transformed by r = ˜ r cos( (cid:112) | K | c τ ), and weget the representation ds τ = dr + sinh ( (cid:113) | ˜ k | r ) | ˜ k | d Ω , (60)with curvature˜ k ( τ ) = −| K | cos ( (cid:112) | K | cτ ) (61)= K cosh ( √ K cτ ) ≡ k . (62)0The analytical continuation of the last line is performedaccording to the identity cos( ix ) = cosh( x ).The metric of the remaining hyperbolic case (14)is ds τ = sin ( (cid:112) | K | cτ ) (cid:104) d ˜ r + sinh ( (cid:112) | K | ˜ r ) | K | d Ω (cid:105) , (63)which is expressed by ds τ = dr + sinh ( (cid:113) | ˜ k | r ) | ˜ k | d Ω , (64)with curvature˜ k ( τ ) = −| K | sin ( (cid:112) | K | cτ ) (65)= − K sinh ( √ K cτ ) ≡ k . (66) Here, we have applied the identity sin( ix ) = sinh( x ).Finally, we consider the spatially flat solution cor-responding to (12), which is easily expressed by ds τ = e √ K cτ K (cid:104) d ˜ r + ˜ r d Ω (cid:105) . (67)The spatial flatness of the corresponding subspace isequivalent to statement (18).(67)The spatial flatness of the corresponding subspace isequivalent to statement (18).