Applications of canonical quantum gravity to cosmology
aa r X i v : . [ g r- q c ] A ug APPLICATIONS OF CANONICAL QUANTUM GRAVITYTO COSMOLOGY
CLAUS GERHARDT
Abstract.
We apply quantum gravitational results to spatially un-bounded Friedmann universes and try to answer some questions relatedto dark energy, dark matter, inflation and the missing antimatter.
Contents
1. Introduction 12. The dark energy density 63. The inflationary period 94. The dark matter 105. The missing antimatter 156. Spherically symmetric eigenfunctions in hyperbolic space 16References 171.
Introduction
The quantization of gravity is one of the most challenging open problemsin physics. The Einstein equations are the Euler-Lagrange equations of theEinstein-Hilbert functional and quantization of a Lagrangian theory requiresto switch from a Lagrangian view to a Hamiltonian view. In a ground break-ing paper, Arnowitt, Deser and Misner [3] expressed the Einstein-HilbertLagrangian in a form which allowed to derive a corresponding Hamilton func-tion by applying the Legendre transformation. However, since the Einstein-Hilbert Lagrangian is singular, the Hamiltonian description of gravity is onlycorrect if two additional constraints are satisfied, namely, the Hamilton con-straint and the diffeomorphism constraint. Dirac [7] proved how to quantizea constrained Hamiltonian system—at least in principle—and his method hasbeen applied to the Hamiltonian setting of gravity, cf. the paper by DeWitt[6] and the monographs by Kiefer [16] and Thiemann [18]. In the general
Date : August 7, 2019.2000
Mathematics Subject Classification.
Key words and phrases. quantization of gravity, quantum gravity, Friedmann universe,dark energy density, dark matter, inflation, missing antimatter, cosmology, negative cos-mological constant. case, when arbitrary globally hyperbolic spacetime metrics are allowed, theproblem turned out to be extremely difficult and solutions could only befound by assuming a high degree of symmetry, cf., e.g., [8].However, in the papers [9, 10] we developed a model for the quantization ofgravity for general hyperbolic spacetimes. In these papers we eliminated thediffeomorphism constraint by reducing the number of variables and provingthat the Euler-Lagrange equations for this special class of metrics were stillthe full Einstein equations. The Hamiltonian description of the Einstein-Hilbert functional then allowed a canonical quantization.We quantized the action by looking at the Wheeler-DeWitt equation ina fiber bundle E , where the base space is a Cauchy hypersurface of thespacetime which has been quantized and the elements of the fibers are Rie-mannian metrics. The fibers of E are equipped with a Lorentzian metric suchthat they are globally hyperbolic and the transformed Hamiltonian, whichis now a hyperbolic operator, ˆ H , is a normally hyperbolic operator actingonly in the fibers. The Wheeler-DeWitt equation has the form ˆ Hu = 0with u ∈ C ∞ ( E, C ) and we defined with the help of the Green’s operator asymplectic vector space and a corresponding Weyl system.The Wheeler-DeWitt equation seems to be the obvious quantization ofthe Hamilton condition. However, ˆ H acts only in the fibers and not in thebase space which is due to the fact that the derivatives are only ordinarycovariant derivatives and not functional derivatives, though they are supposedto be functional derivatives, but this property is not really invoked when afunctional derivative is applied to u , since the result is the same as applyinga partial derivative.Therefore, we discarded the Wheeler-DeWitt equation in the paper [14]and also in the monograph [15], and expressed the Hamilton condition dif-ferently by looking at the evolution equation of the mean curvature of thefoliation hypersurfaces M ( t ) and implementing the Hamilton condition onthe right-hand side of this evolution equation. The left-hand side, a time de-rivative, we replaced by the corresponding Poisson brackets. After canonicalquantization the modified evolution equation was transformed to an equationsatisfied by operators which acted on functions u ∈ C ∞ ( E, C ).Since the Poisson brackets became a commutator we could now employ thefact that the derivatives are functional derivatives, since we had to differen-tiate the scalar curvature of a metric when we applied the operator equationto a smooth function and tried to simplify the resulting equation. As a resultof the simplification of the commutator action we obtained an elliptic differ-ential operator in the base space, the main part of which was the Laplacianwith respect to a fiber element. Here, we considered functions u dependingon the variables ( x, g ij ), where x is a point in the base space S , x ∈ S , and g ij is an element of the fibers. The fiber metrics have the form(1.1) g ij = t n σ ij , PPLICATIONS OF CANONICAL QUANTUM GRAVITY TO COSMOLOGY 3 where 0 < t < ∞ is a time-like fiber variable, which is referred to as time, n ≥
3, is the dimension of S and σ ij is a Riemannian metric, depending onlyon x , subject to the requirement(1.2) det σ ij = det χ ij , cf. [15, equs. (1.4.103) & (1.4.104), p. 29] and also [15, Remark 1.6.8]. Thearbitrary, but fixed, metric χ ij in S had been introduced to transform thedensities det g ij to functions.On the right-hand side of the evolution equation the interesting term was H , the square of the mean curvature. It transformed to a second timederivative, the only remaining derivative with respect to a fiber variable,since the differentiations with respect to the other variables canceled eachother. The resulting quantized equation is then a wave equation in a globallyhyperbolic spacetime(1.3) Q = (0 , ∞ ) × S , of the form(1.4) 132 n n − u − ( n − t − n ∆u − n t − n Ru + nt Λu = 0 , where S is a Cauchy hypersurface of the original spacetime and the Laplacianand the scalar curvature R are formed with respect to a metric σ ij satisfying(1.2) and Λ is a cosmological constant. The function u depends on ( x, t, σ ij ).Since the metric χ ij is also a fiber metric we may choose σ ij = χ ij andbecause it is also arbitrary we may set χ ij to be the original metric of theCauchy hypersurface S , cf. [15, Remark 1.6.8 on page 49]. The function u then only depends on ( t, x ), u = u ( t, x ). For a detailed derivation of equation(1.4) we refer to [15, Chapter 1.6] or [14, Section 6].When S is a space of constant curvature then the wave equation, consid-ered only for functions u which do not depend on x , is identical to the equa-tion obtained by quantizing the Hamilton constraint in a Friedmann universewithout matter but including a cosmological constant, cf. [15, Remark 1.6.11on page 50] or [14, Remark 6.11].There exist temporal and spatial self-adjoint operators H resp. H suchthat the hyperbolic equation is equivalent to(1.5) H u − H u = 0 , where u = u ( t, x ). The operator H is defined by(1.6) H w = ϕ − {− n n − w − nt Λw } , where w = w ( t ), w ∈ C ∞ c ( R + , C ), and ϕ = t − n , while the definition of H is given by(1.7) H v = − ( n − ∆v − n Rv,
CLAUS GERHARDT where v = v ( x ), v ∈ C ∞ c ( S , C ). More precisely, the operators H i , i = 0 , Λ < H has a pure point spectrum withpositive eigenvalues λ i , cf. [15, Chapter 6.2], especially [15, Theorem 6.2.5 onpage 144], while, for H , it is possible to find corresponding eigendistributionsfor each of the eigenvalues λ i , if S is asymptotically Euclidean or if thequantized spacetime is a black hole with a negative cosmological constant,cf. [12, 11, 13] or [15, Chapters 3–5], and also if S is the hyperbolic space S = H n , n ≥
3, cf. Section 6 on page 16.Let w i , i ∈ N , be an orthonormal basis for the temporal eigenvalue prob-lems(1.8) H w i = λ i w i and v i be corresponding smooth eigendistributions for the spatial eigenvalueproblems(1.9) H v i = λ i v i , then(1.10) u i = w i v i are special solutions of the wave equation (1.4).The temporal eigenvalues λ i all have multiplicity 1, the spatial eigenval-ues are the same eigenvalues, but they may have higher multiplicities. Incase of black holes this is caused by very compelling intrinsic mathematicalreasons, cf. [15, Chapter 6.4], but unless there are either convincing intrin-sic or extrinsic reasons, like data, we choose the spatial eigenspaces to beone-dimensional, because the spatial eigenvalues belong in general to thecontinuous spectrum of the spatial Hamiltonian H . If S is the Cauchy hy-persurface of a Friedmann universe we only considered smooth sphericallysymmetric spatial eigenfunctions, which also leads to one-dimensional spa-tial eigenspaces, cf. [15, Chapter 6.6] for the Euclidean case and Section 6 onpage 16 for the hyperbolic case.One can then define an abstract Hilbert space H spanned by the u i and aself-adjoint operator H , unitarily equivalent to H , such that(1.11) Hu i = λ i u i .e − βH is then of trace class in H for all β > H to the corresponding symmetric Fock space F , which is still called H ,shares this property. Hence, we can define the partition function Z ,(1.12) Z = tr e − βH , the operator density(1.13) ˆ ρ = Z − e − βH , ∀ β > , PPLICATIONS OF CANONICAL QUANTUM GRAVITY TO COSMOLOGY 5 the average energy and the von Neumann entropy in F . The eigenvectors u i can also be viewed as to be elements of F and they are then also eigenvectorsof ˆ ρ .In the present paper we want to apply these quantum gravitational resultsto cosmology by looking at a Friedmann universe(1.14) N = I × S , where S is a n -dimensional simply connected space of constant curvature ˜ κ ,(1.15) ˜ κ ∈ { , − } , i.e., S is either R n or the hyperbolic space H n , n ≥
3. We tried to an-swer some questions related to dark energy, dark matter, inflation and themissing antimatter. In doing so we shall also show that assuming a negativecosmological constant is not a contradiction to the observational result of anexpanding universe. Usually a positive cosmological constant is supposed tobe responsible for the dark energy and dark matter is sometimes explainedby assuming so-called extended theories of gravity, confer, e.g., the papers[5] and [4]. In this paper we rely on general relativity combined with somequantum gravitational ingredients.Let us summarize the main result as a theorem, where ρ dm resp. ρ de refer tothe dark matter resp. dark energy densities, which we defined as eigenvaluesof the operator density ˆ ρ in F , and ρ is a conventional density. Z is thepartition function, T > λ > H . Theorem 1.1.
Let the cosmological constant Λ , (1.16) − < Λ < , be given and consider the perfect fluid defined by the density (1.17) ρ = ρ dm + ρ de + ρ satisfying the assumptions (4.23) , (4.24) , (4.36) and (4.37) . Moreover, wesuppose that β = T − and the scale factor a are functions depending on t .The initial value problems (1.18) ¨ aa = − κ n ( n − { ( n − ρ + np } + 2 n ( n − Λ and (1.19) ˙ β = − n ρ dm ∂∂β ( ρ dm + ρ de ) a − ˙ a. with initial values ( β , a , ˙ a ) are then solvable in I = [ t , ∞ ) provided β > is so large that (2.12) on page as well as (1.20) 2 κ n ( n − Z − { −
12 ( n − α e − βλ } + 2 n ( n − Λ > CLAUS GERHARDT are valid at β = β and a > has to be chosen such that after adding (1.21) − κ n ( n −
1) ( n (1 + ω ) − γ a − n (1+ ω )0 to the left-hand side of (1.20) the inequality still remains valid at β = β . Theinitial value ˙ a is supposed to be positive. The solution ( β, a ) then satisfies (1.22) ˙ β > , (1.23) ˙ a > , (1.24) ¨ a > and (1.25) 2 n ( n − κ ρ + 2 n ( n − Λ − ˜ κa − > . In order that ( β, a ) also satisfies the first Friedmann equation ˙ a has to bechosen appropriately, namely, such that the first Friedmann equation is validfor t = t , which is possible, in view of (1.25) . Remark 1.2.
Let us also mention that we use (modified) Planck units inthis paper, i.e.,(1.26) c = κ = ~ = K B = 1 , where κ is the coupling constant connecting the Einstein tensor with thestress-energy tensor(1.27) G αβ + Λ ¯ g αβ = κ T αβ . The dark energy density
In [15, Remark 6.5.5] we proposed to use the eigenvalue of the densityoperator ˆ ρ with respect to the vacuum vector η , which is Z − ,(2.1) ˆ ρη = Z − η, as the source of dark energy density, and though this eigenvalue is the vac-uum, or zero-point, energy and many authors have proposed the vacuumenergy to be responsible for the dark energy, these proposals all assumed thecosmological constant to be positive, while we assume Λ <
Λ < Z − dominates Λ which will only be the case if(2.2) T < T = T ( | Λ | ) . Note that Z depends on the eigenvalues λ i and on(2.3) β = T − . First, we emphasize that we shall treat(2.4) ρ de = Z − as a constant, i.e., we shall define the perfect fluid stress-energy tensor by(2.5) T αβ = − ρ de ¯ g αβ . Let λ i > i ∈ N , be the eigenvalues of the temporal Hamiltonian H fora given Λ < λ i be the eigenvalues for(2.6) Λ = − , then(2.7) λ i = ¯ λ i | Λ | n − n , cf. [15, Lemma 6.4.9, p. 172], and define the parameter τ by(2.8) τ = | Λ | n − n , where we now assume(2.9) | Λ | < , throughout the rest of the paper. We proved in [15, Theorem 6.5.6, p. 180]that(2.10) lim τ → Z = ∞ , or equivalently, that(2.11) lim τ → ρ de = 0 . However, we shall now derive a more precise estimate of ρ de = Z − involving β and Λ . Lemma 2.1.
For any Λ satisfying − < Λ < , there exists exactly one T > such that (2.12) Z − ( β ) > | Λ | ∀ β > β = T − , where we recall that (2.13) β = T − . Proof.
In view of (2.7) we deduce that(2.14) Z ( β ) ≡ Z ( β, λ i ) = ¯ Z ( γ, ¯ λ i ) ≡ ¯ Z ( γ ) , where(2.15) γ = β | Λ | n − n . From the relations(2.16) 0 < E = − ∂ log Z∂β = ∂ log Z − ∂β , cf. [15, equations (6.5.30) and (6.5.32), p. 176],(2.17) lim β →∞ Z ( β ) = 1 , CLAUS GERHARDT and(2.18) lim β → Z ( β ) = ∞ , cf. [15, Theorem 6.5.8, p. 181], we then conclude that there exists exactlyone γ such that(2.19) ¯ Z − ( γ ) = | Λ | and, furthermore, that(2.20) ¯ Z − ( γ ) > ¯ Z − ( γ ) ∀ γ > γ , completing the proof of the lemma. (cid:3) Thus, defining the dark energy density by (2.4) and (2.5), we immediatelydeduce:
Theorem 2.2.
Let T be the temperature defined in Lemma and as-sume that the temperature T satisfies T < T , then the dark energy densityguarantees that the Friedmann universe with negative cosmological constant Λ , (2.21) − < Λ < , is expanding such that (2.22) ˙ a > as well as (2.23) ¨ a > . Proof.
The Friedmann equations for a perfect fluid with energy ρ and pressure p are(2.24) ˙ a a = 2 n ( n − κ ρ + 2 n ( n − Λ − ˜ κa − and(2.25) ¨ aa = − κ n ( n − { ( n − ρ + np } + 2 n ( n − Λ. Choosing ρ = ρ de we also specified(2.26) p = − ρ de yielding(2.27) ¨ aa = 2 κ n ( n − ρ de + 2 n ( n − Λ. Moreover, in our units,(2.28) κ = 1and we also only consider space forms satisfying(2.29) ˜ κ ≤ , PPLICATIONS OF CANONICAL QUANTUM GRAVITY TO COSMOLOGY 9 hence the theorem is proved in view of Lemma 2.1. (cid:3) The inflationary period
Immediately after the big bang the development of the universe will have tobe governed by quantum gravitational forces, i.e., by the eigenfunctions resp.eigendistributions of the corresponding temporal and spatial Hamiltonians,which we have combined to a single Hamiltonian H acting in an abstractseparable Hilbert space H spanned by the eigenvectors u i (3.1) Hu i = λ i u i , where the eigenvalues all have multiplicity 1, are ordered(3.2) 0 < λ < λ < · · · and converge to infinity(3.3) lim i →∞ λ i = ∞ . The dominant energies near the big bang will therefore be the eigenvalues(3.4) λ i = h Hu i , u i i for large i and we shall assume, when considering the development of a Fried-mann universe, that this development is driven by a perfect fluid(3.5) T αβ = − ρ i ¯ g αβ , where(3.6) ρ i = λ i . Looking at the Friedmann equations(3.7) ˙ a a = 2 κ n ( n − ρ i + 2 n ( n − Λ − ˜ κa − and(3.8) ¨ aa = 2 κ n ( n − ρ i + 2 n ( n − Λ we conclude that the universe is expanding rapidly depending on the eigen-value ρ i = λ i . The corresponding eigenvector, or particle, u i will decay aftersome time and produce lower order eigenvectors or maybe particles that canbe looked at as matter or radiation satisfying the corresponding equations ofstate.After some time the inflationary period will have ended and only the stableground state u ,(3.9) Hu = λ u , together with conventional matter and radiation will be responsible for thefurther development of the Friedmann universe.The eigenvalue λ is of the order | Λ | n − n in view of (2.7) on page 7, henceit will dominate Λ for small values of | Λ | . The dark matter
Let ˆ ρ be the density operator acting in the Fock space F ,(4.1) ˆ ρ = Z − e − βH , where we use the same symbol H to denote the self-adjoint operator H in theseparable Hilbert space H as well its canonical extension to the correspondingsymmetric Fock space F + ( H ) ≡ F . In Section 2 we defined the dark energydensity ρ de by(4.2) ρ de = h ˆ ρη, η i = Z − and we propose to define the dark matter density by(4.3) ρ dm = α h ˆ ρu , u i = α e − βλ Z − , where u is a unit eigenvector of H satisfying(4.4) Hu = λ u and(4.5) α > β > ∂∂β ( ρ dm + ρ de ) < ∀ β ≥ β , as we shall now prove: Lemma 4.1.
Let α satisfy (4.5) and Λ (4.7) − < Λ ≤ Λ < , then there exists β = β ( α , | Λ | ) such that the inequality (4.6) is valid.Proof. In view (2.16) on page 7 we have(4.8) ∂∂β ( ρ dm + ρ de ) = − α λ e − βλ Z − + α e − βλ Z − E + Z − E, where(4.9) E = ∞ X i =0 λ i e βλ i − λ e βλ − ∞ X i =1 λ i e βλ i − , cf. [15, equ. (6.5.32), p. 176] or simply differentiate. Hence, we obtain(4.10) Ee βλ = λ e βλ e βλ − ∞ X i =1 λ i e β ( λ i − λ ) − e − βλ ≤ λ e βλ e βλ − ∞ X i =1 λ i e β ( λ i − λ ) − PPLICATIONS OF CANONICAL QUANTUM GRAVITY TO COSMOLOGY 11 and we conclude(4.11) lim β →∞ Ee βλ = λ , since(4.12) ∞ X i =1 λ i e β ( λ i − λ ) − ∞ X i =1 λ i − λ e β ( λ i − λ ) − ∞ X i =1 λ e β ( λ i − λ ) − ≤ ∞ X i =1 µ i e βµ i − λ ( λ − λ ) − ∞ X i =1 µ i e βµ i − , where µ i is defined by(4.13) µ i = λ i − λ ≥ λ − λ > ∀ i ≥ . Thus the right-hand side of (4.12) is estimated from above by(4.14) (1 + λ ( λ − λ ) − ) E ( β, µ i )and(4.15) lim β →∞ E ( β, µ i ) = 0 , cf. [15, equ. (6.5.71), p. 181]. Furthermore, we know(4.16) λ = ¯ λ | Λ | n − n , cf. (2.7). Combining these estimates we conclude that there exists(4.17) β = β ( α , | Λ | )such that(4.18) ∂∂β ( ρ dm + ρ de ) ≤ − α − λ e − βλ Z − ∀ β ≥ β . The limits in (4.11) and (4.15) are also uniform in | Λ | because of (4.7). (cid:3) Dark matter is supposed to be dust, i.e., its pressure vanishes, and hence, ρ dm cannot be constant which is tantamount to(4.19) β const , since we assume that Λ is constant. Thus, ρ de is also not constant, thoughwe still assume that its stress-energy tensor is defined by(4.20) T αβ = − ρ de ¯ g αβ . Therefore, we can only establish the continuity equation for(4.21) ρ dm + ρ de and not for each density separately. Let a dot or a prime indicate differenti-ation with respect to time t , then the continuity equation has the form(4.22) ( ρ dm + ρ de ) ′ = − nρ dm a − ˙ a, because(4.23) p dm = 0 and(4.24) p de = − ρ de . The left-hand side of (4.22) is equal to(4.25) ∂∂β ( ρ dm + ρ de ) ˙ β and we see that the continuity equation can only be satisfied if(4.26) ˙ β = − n ρ dm ∂∂β ( ρ dm + ρ de ) a − ˙ a. From Lemma 4.1 we immediately derive
Lemma 4.2.
Let the assumptions of Lemma be satisfied and supposethat ˙ a > , then, for any solution β = β ( t ) of (4.26) in the interval (4.27) I = [ t , b ) , t < b ≤ ∞ , with initial value (4.28) β ( t ) ≥ β the inequality (4.29) ˙ β > is valid and hence (4.30) β ( t ) ≥ β ∀ t ∈ I. Furthermore, ˙ β can be expressed in the form (4.31) ˙ β = nδ ( α − − α a − ˙ a, where δ = δ ( t, β ) satisfies (4.32) 1 ≤ δ ≤ and (4.33) lim β →∞ δ = 1 , i.e., (4.34) β ( t ) − β ( t ) ≈ nδα ( α − − (log a ( t ) − log a ( t )) . Proof. ”(4.29)“ Follows from (4.6) and (4.26).”(4.31)“ To prove the claim we combine (4.8), (4.26) and (4.11).”(4.32)“ and ”(4.33)“ Same argument as before.”(4.34)“ Obvious in view of (4.31) and (4.33). (cid:3)
PPLICATIONS OF CANONICAL QUANTUM GRAVITY TO COSMOLOGY 13
Now, we are prepared to solve the Friedmann equations (2.24) and (2.25)on page 8 for(4.35) ρ = ρ dm + ρ de + ρ , where ρ is a conventional density satisfying the equation of state(4.36) p = ω ρ assuming(4.37) ω > − .ρ is only added for good measure and we are allowed to assume(4.38) ρ = 0 , since its presence is not essential.We also emphasize that we have to solve an additional third equation,namely, equation (4.26). We shall solve the Friedmann equations and (4.26)in the interval(4.39) I = [ t , ∞ ) , t > , for the unknown functions ( a, β ) with prescribed positive initial values( a , ˙ a , β ). β can be arbitrary but large enough such that the assump-tions in Lemma 4.1 and Lemma 4.2 are satisfied. If ρ vanishes then a > a > t = t .If these assumptions are satisfied then we shall solve the equations (2.25)on page 8 and (4.26). The first Friedmann equation will then be valid auto-matically. For simplicity we shall only consider the case(4.40) ρ > ρ = γ a − n (1+ ω ) , where γ > Theorem 4.3.
Let the cosmological constant Λ , (4.42) − < Λ < , be given and consider the perfect fluid defined by the density (4.43) ρ = ρ dm + ρ de + ρ satisfying the assumptions (4.23) , (4.24) , (4.36) and (4.37) . Moreover, wesuppose that β = T − and the scale factor a are functions depending on t .The initial value problems (4.44) ¨ aa = − κ n ( n − { ( n − ρ + np } + 2 n ( n − Λ and (4.45) ˙ β = − n ρ dm ∂∂β ( ρ dm + ρ de ) a − ˙ a. with initial values ( β , a , ˙ a ) are then solvable in I = [ t , ∞ ) provided β > is so large that (2.12) on page as well as (4.46) 2 κ n ( n − Z − { −
12 ( n − α e − βλ } + 2 n ( n − Λ > are valid at β = β and a > has to be chosen such that after adding (4.47) − κ n ( n −
1) ( n (1 + ω ) − γ a − n (1+ ω )0 to the left hand side of (4.46) the inequality still remains valid at β = β . Theinitial value ˙ a is supposed to be positive. The solutions ( β, a ) then satisfy (4.48) ˙ β > , (4.49) ˙ a > , (4.50) ¨ a > and (4.51) 2 n ( n − κ ρ + 2 n ( n − Λ − ˜ κa − > . In order that ( β, a ) also satisfy the first Friedmann equation ˙ a has to bechosen appropriately, namely, such that the first Friedmann equation is validfor t = t , which is possible, in view of (4.51) .Proof. By introducing a new variable(4.52) ϕ = ˙ a we may consider a flow equation for ( β, a, ϕ ), where ˙ ϕ replaces ¨ a and(4.53) ˙ a = ϕ is an additional equation.Choosing then β , a as above and ϕ > I = [ t , t ) , t > t , because of Lemma 4.1 and Lemma 4.2. It is also obvious that the relations(4.48)–(4.51) are valid, in view of these lemmata.Furthermore, if the interval I was bounded, then the flow would have asingularity at t = t which is not possible, in view of the relation (4.34),which would imply that β , ˙ β as well as a and ˙ a would tend to infinity byapproaching t which, however, contradicts the second Friedmann equation(4.44) from which we then would infer(4.55) 0 < ¨ a ≤ ca ∀ t ∈ I PPLICATIONS OF CANONICAL QUANTUM GRAVITY TO COSMOLOGY 15 an apparent contradiction. Hence we deduce(4.56) I = [ t , ∞ ) . It remains to prove that the first Friedmann equation is satisfied if ˙ a ischosen appropriately, Define(4.57) Φ = ˙ a − { n ( n − κ ρ + 2 n ( n − Λ } a + ˜ κ, then we obtain(4.58) ˙ Φ = 0 , in view of the continuity equations and (4.44), yielding(4.59) Φ ( t ) = Φ ( t ) = 0 ∀ t ∈ I. (cid:3) The missing antimatter
In [15, Theorem 4.3.1, p. 110] we proved that a temporal eigenfunction w = w ( t ) defined in R + can be naturally extended past the big bang singu-larity { t = 0 } by defining(5.1) w ( − t ) = − w ( t ) , ∀ t > . The extended function is then of class C ,α ,(5.2) w ∈ C ,α ( R ) , for some 0 < α < { t < } is also a solution of thevariational eigenvalue problem. Hence we have two quantum spacetimes(5.3) Q − = R − × S and(5.4) Q + = R + × S and a C ,α transition between them. If we assume that the common timefunction t is future directed in both quantum spacetimes, then the singularityin { t = 0 } would be a big crunch for Q − and a big bang for Q + and simi-larly for the corresponding Friedmann universes N ∓ governed by the Einsteinequations. No further singularities will be present, i.e., the spacetime N − willhave no beginning but will end in in a big crunch and will be recreated witha big bang as the spacetime N + .This scenario would be acceptable if it would describe a cyclical universe.However, there are no further cycles, there would only be one transition froma big crunch to a big bang. Therefore, the mathematical alternative, namely,that at the big bang two universes with opposite light cones will be created, ismore convincing, especially, if the CPT theorem is taken into account whichwould require that the matter content in the universe with opposite timedirection would be antimatter. This second scenario would explain whathappened to the missing antimatter. Spherically symmetric eigenfunctions in hyperbolic space
The spatial Hamiltonian H is a linear elliptic operator(6.1) H v = − ( n − ∆v − n Rv, where the Laplacian is the Laplacian in S and R the corresponding scalarcurvature. We are then looking for eigenfunctions or, more precisely, eigendis-tributions v ,(6.2) H v = λv, such that, for each temporal eigenfunction ( λ i , w i ) there exists a matchingspatial pair ( λ i , v i ). The product(6.3) u i = w i v i would then be a solution of the wave equation (1.4) on page 3.If S is the hyperbolic space H n , n ≥
3, we have(6.4) R = − n ( n − λ i , we would have to find functions v i satisfying(6.5) − ( n − ∆v i = ( λ i − n n − v i . We are also looking for spherically symmetric eigenfunctions v i . In hyperbolicspace the radial eigenfunctions, known as spherical functions, are well-known:For each µ ∈ C there exists exactly one radial eigenfunctions ϕ µ of theLaplacian satisfying(6.6) − ∆ϕ µ = ( µ + ρ ) ϕ µ and(6.7) ϕ µ (0) = 1 , where(6.8) ρ = n − , see e.g., [1, Section 2] and the references therein. Here, we introduced geodesicpolar coordinates ( r, ξ ) in H n and the ϕ µ only depend on r . The ϕ µ have theintegral representation(6.9) ϕ µ ( r ) = c n (sinh r ) − n Z r − r (cosh r − cosh t ) n − e − iµt dt, cf. [2, equation (6), p. 4].Since the ϕ µ are distributions they are smooth in H n , cf. [17, Theorem3.2, p. 125]. Furthermore, for each i ∈ N we can choose µ i ∈ C such that(6.10) ( n − µ i + ρ ) = λ i − n n − . PPLICATIONS OF CANONICAL QUANTUM GRAVITY TO COSMOLOGY 17
Obviously, there are two solutions µ i and − µ i , but the corresponding eigen-functions are identical as can be easily checked. References [1] Jean-Philippe Anker and Vittoria Pierfelice,
Wave and Klein-Gordon equations onhyperbolic spaces , Anal. PDE (2014), 953–995, arXiv:1104.0177.[2] Jean-Philippe Anker, Vittoria Pierfelice, and Maria Vallarino, The wave equation onhyperbolic spaces , (2010), arXiv:1010.2372.[3] R. Arnowitt, S. Deser, and C. W. Misner,
The dynamics of general relativity , Grav-itation: an introduction to current research (Louis Witten, ed.), John Wiley, NewYork, 1962, pp. 227–265.[4] Salvatore Capozziello, Francisco S. N. Lobo, and Jos´e P. Mimoso,
Generalized energyconditions in Extended Theories of Gravity , Phys. Rev.
D91 (2015), no. 12, 124019,1407.7293, doi:10.1103/PhysRevD.91.124019.[5] Sayantan Choudhury, Manibrata Sen, and Soumya Sadhukhan,
Can dark matter bean artifact of extended theories of gravity? , The European Physical Journal C (2016), no. 9, 494, doi:10.1140/epjc/s10052-016-4323-2.[6] Bryce S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory , Phys. Rev. (1967), 1113–1148, doi:10.1103/PhysRev.160.1113.[7] Paul A. M. Dirac,
Lectures on quantum mechanics , Belfer Graduate School of ScienceMonographs Series, vol. 2, Belfer Graduate School of Science, New York, 1967, Secondprinting of the 1964 original.[8] Claus Gerhardt,
Quantum cosmological Friedman models with an initial sin-gularity , Class. Quantum Grav. (2009), no. 1, 015001, arXiv:0806.1769,doi:10.1088/0264-9381/26/1/015001.[9] , The quantization of gravity in globally hyperbolic spacetimes ,Adv. Theor. Math. Phys. (2013), no. 6, 1357–1391, arXiv:1205.1427,doi:10.4310/ATMP.2013.v17.n6.a5.[10] , A unified quantum theory I: gravity interacting with a Yang-Millsfield , Adv. Theor. Math. Phys. (2014), no. 5, 1043–1062, arXiv:1207.0491,doi:10.4310/ATMP.2014.v18.n5.a2.[11] , The quantization of a black hole , (2016), arXiv:1608.08209.[12] ,
The quantum development of an asymptotically Euclidean Cauchy hypersur-face , (2016), arXiv:1612.03469.[13] ,
The quantization of a Kerr-AdS black hole , Advances in Mathemati-cal Physics vol. 2018 (2018), Article ID 4328312, 10 pages, arXiv:1708.04611,doi:10.1155/2018/4328312.[14] ,
The quantization of gravity , Adv. Theor. Math. Phys. (2018), no. 3, 709–757, arXiv:1501.01205, doi:10.4310/ATMP.2018.v22.n3.a4.[15] , The Quantization of Gravity , 1st ed., Fundamental Theories of Physics, vol.194, Springer, Cham, 2018, doi:10.1007/978-3-319-77371-1.[16] Claus Kiefer,
Quantum Gravity , 2nd ed., International Series of Monographs onPhysics, Oxford University Press, 2007.[17] J.-L. Lions and E. Magenes,
Non-homogeneous boundary value problems and appli-cations. Vol. I , Springer-Verlag, New York, 1972, Translated from the French by P.Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.[18] Thomas Thiemann,
Modern canonical quantum general relativity , Cambridge Mono-graphs on Mathematical Physics, Cambridge University Press, Cambridge, 2007, Witha foreword by Chris Isham.
Ruprecht-Karls-Universit¨at, Institut f¨ur Angewandte Mathematik, Im Neuen-heimer Feld 205, 69120 Heidelberg, Germany
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