Canonical quantization of anisotropic Bianchi I cosmology from scalar vector tensor Brans Dicke gravity
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Canonical quantization of anisotropic Bianchi Icosmology from scalar vector tensor Brans Dickegravity
Hossein Ghaffarnejad
Faculty of Physics, Semnan University, Semnan, IRAN, 35131-19111E-mail: [email protected]
Abstract.
We applied a generalized scalar-vector-tensor Brans Dicke gravity model to studycanonical quantization of an anisotropic Bianchi I cosmological model. Regarding an anisotropicHarmonic Oscillator potential we show that the corresponding Wheeler de Witt wave functionalof the system is described by Hermit polynomials. We obtained a quantization condition onthe ADM mass of the cosmological system which raises versus the quantum numbers of theHermit polynomials. Our calculations show that the inflationary expansion of the universe canbe originate from the big bang with no naked singularity due to the uncertainty principle.
1. Introduction
The standard cosmology has a great success in explaining the observations of the cosmicmicrowave background radiation (CMBR) temperature [1, 2, 3, 4]. This model is based onthe validity of the cosmological principle (the spatial homogeneity and isotropy but not inthe time direction) and the Einstein‘s general relativity explain most large-scale observationswith unprecedented accuracy. However, several directional anomalies have been reported invarious large-scale observations (see [5] and references therein). They have no place in standardcosmology and are not being studied. In fact origin of these anomalies do not understoodand so there are two different proposals to understand them as follows: a) Perhaps they areoriginated from cosmological effects which should be described via alternative gravity theoriesinstead of the Einstein‘s general theory of relativity. b) Other possibility which arises thesedirectional anomalies can be systematic errors or contaminations of measuring instruments andetc., which should be exclude from the future data analysis. In the latter case one usually acceptvalidity of the standard cosmological Λ
CDM model while in the former proposal one use analternative gravity model instead of the Einstein‘s general theory of relativity. Zhao and Santos,provided full review about these proposals in ref. [6] where the directional anomalies predict apreferred axis ”
Axis of Evil ” in large scale of the Universe. However, the general covarianceprincipal leads us to believe that these anomalies have cosmological origin and it can be describedby some alternative gravity models where the cosmological principals should be violated (see([5]) and reference therein). To describe the above mentioned anomalies the anisotropic Bianchicosmological models are applicable [7, 8, 9] for anisotropic cosmological constant and dark energy.As an alternative gravity model we consider scalar-vector-tensor gravity model [10, 11] whichis made from generalization of the well known Jordan-Brans-Dicke scalar tensor gravity [12] byransforming the background metric as g µν → g µν + 2 N µ N ν . N µ is dynamical four vector fieldwhich can be called as four velocity of a preferred reference frame. Several classical and quantumapplications of this model are studied previously for FLRW cosmology which are addressed inreferences of the work [13]. As an application of the gravity model for anisotropic cosmologicalmodels we applied it to study Bianchi I classical cosmology in ref. [5] where the correspondingFreedmann equations read to anisotropic inflationary expanding universe and they satisfiedobservational data successfully. In the present work we want to resolve the naked singularity ofthe anisotropic Bianchi I cosmological model by applying the canonical quantization approach.To do so we use self-interaction anisotropic three dimensional Harmonic Oscillator potential tocalculate Hamiltonian operator of the system. Then we obtain Wheeler de Witt wave functionalof the anisotropic Bianchi I background metric versus the Hermit polynomials. Outlook of thiswork is to present a quantization condition on the energy density of the system with no nakedsingularity where the system is stable at high energy quantum level. Organization of the paperis as follows.In section 2, we introduce the scalar-vector-tensor gravity model [10, 11] under consideration.In section 3, we use the Bianchi I type of the background metric to obtain exact form ofHamiltonian density. In section 4 we solve Wheeler de Witt wave equation of the system andobtain a quantization condition on the energy density of the Bianchi I quantum cosmologicalsystem. The obtained Wheeler de Witt wave solution is described versus the quantum anisotropicharmonic Oscillator (Hermit polynomials) for three dimensional anisotropic Oscillator potential.In section 5 we discuss about the quantization of the ADM energy of the system. In section 6we discuss about the big bang naked singularity of the expanding universe which how can it isremoved in the quantum perspective of the system. Section 7 denotes to concluding remark.
2. The gravity model
Let us start with the Brans-Dicke scalar-vector tensor gravity [10, 11] I = 116 π Z d x √− g (cid:26) φR − ωφ g µν ∇ µ φ ∇ ν φ (cid:27) + 116 π Z d x √− g { ζ ( x ν )( g µν N µ N ν + 1) + 2 φF µν F µν + U ( φ, N µ ) − φN µ N ν (2 F µλ Ω νλ + F µλ F νλ + Ω µλ Ω νλ − R µν + 2 ωφ ∇ µ φ ∇ ν φ ) } , (1)where g is absolute value of determinant of the metric tensor g µν with signature as (-,+,+,+), φ is the Brans-Dicke scalar field and ω is the Brans-Dicke adjustable coupling constant. Thetensor fields F µν and Ω µν are defined versus the time like vector field N µ as follows. F µν = 2( ∇ µ N ν − ∇ ν N µ ) , Ω µν = 2( ∇ µ N ν + ∇ ν N µ ) (2)The above action is written in units c = G = ¯ h = 1 and the undetermined Lagrange multiplier ζ ( x ν ) controls N µ to be an unit time-like vector field. φ describes inverse of variable Newton’sgravitational coupling parameter and its dimension is ( lenght ) − in units c = G = ¯ h = 1. Presentlimits of dimensionless BD parameter ω based on time-delay experiments [14, 15, 16, 17] requires ω ≥ × . Varying the above action functional with respect to ζ we obtain time-like conditionon the vector field N µ as follows. g µν N µ N ν = − . (3)In the next section we apply the above mentioned action functional to study the anisotropicBianchi I cosmological model. . Bianchi I quantum cosmology Spatially homogenous but anisotropic dynamical flat universe is defined by the Bianchi I metricwhich from point of view of free falling (comoving) observer is defined by the following lineelement [18]. ds = − dt + e a ( t ) { e − b ( t ) dx + e b ( t ) ( dy + dz ) } (4)where x, y, z are cartesian coordinates of the comoving observer and t is cosmic time. In theabove metric equation we assume that the spatial part has cylindrical symmetry for which e a ( t ) is an global isotropic scale factor and b ( t ) represents a deviation from the isotropy. Substituting(4) into the equation (3) we obtain N µ ( t ) = N t N x N y N z = cosh αe a − b sinh α cos βe a + b sinh α sin β cos γe a + b sinh α sin β sin γ (5)where ( α, β, γ ) are angular constant parameters of the vector field N µ which makes as fixed itsdirection at the 4D anisotropic space time (4). Substituting (4) and (5) into the action functional(1) with some simple calculations, one can show that the action functional read I = 116 π Z dxdydz Z dte a (cid:20) − ω ˙ φ φ − φ ˙ a + 6 φ ˙ b − U ( a, b, φ ) (cid:21) (6)where dot ˙ denotes to ‘cosmic‘ time derivative ddt and we used the ansatz α = 0 to eliminatefrictional terms ¨ a and ¨ b in the action functional (6) (see ref.[5] for more detail). In general, ifan action functional contains time derivative of velocity of the dynamical fields then there willbe some frictional forces which cause that the extremum point of the action functional doesnot fixed. The latter kind of dynamical systems are not closed and so stable. They behaveusually as chaotic dynamical systems. Hence we should eliminate these acceleration terms ofthe dynamical fields to fix extremum points of the system. Usually one obtained an effectiveaction functional by integrating by part and removing divergence-less counterpart of the actionfunctional. Here we can eliminate the frictional terms ¨ a and ¨ b without to use an effective actionfunctional instead of the action functional (6) just by setting α = 0. This restrict us to choosea particular direction for the time-like dynamical vector field (5) where the lagrangian of thesystem has not friction terms ¨ a and ¨ b .When we study canonical quantization of a mini-super-space quantum cosmology, then therethe local coordinates do not have an important role but the dynamical fields themselves play animportant role. Hence it will be useful to apply a conformal frame with the following conformaltime τ ( t ) in what follows. dt = F ( t ) dτ, F = e a φ π (7)where F is called as lapse (red shift) function in the ADM formalism of the decomposition ofthe background metric. Substituting (7) and φ = e Z ( τ ) G (8)into the action functional (6) we obtain I = R dxdydz R L dτ where G is Newton‘s couplingconstant and L is the Lagrangian density of the system from point of view of the conformalframe. L = − ωZ ′ − a ′ + 6 b ′ − V ( a, b, Z ) (9)here ′ denotes to derivative with respect to the conformal time τ and V = F Uφ (10)is a suitable super-potential. Calculating canonical momenta of the fields ( a, b, Z ) asΠ a = ∂ L ∂a ′ , Π b = ∂ L ∂b ′ , Π Z = ∂ L ∂Z ′ (11)and applying definition of the Hamiltonian density H = a ′ Π a + b ′ Π b + Z ′ Π Z − L (12)one can infer H = − Π a
24 + Π b − Π Z ω + V ( a, b, Z ) (13)where we see that signature of the de Witt superspace metric is Lorentzian form ( − , + , − ) . Inthe next section we see that the canonical momentum operator of the anisotropic counterpart ofthe metric field b behaves as a time-evolution parameter of the system in the canonical quantumcosmology of the minisuperspace de Witt metric. However we obtain now Wheeler de Wittprobability wave solution of the cosmological system under a quantization condition on theADM energy of the system.
4. Canonical quantization
To study quantum stability of the Bianchi I cosmology we should first fix the potential V ( a, b, Z )where we choose anisotropic harmonic Oscillator potential defined on the mini-super-space deWitt metric as follows. V ( a, b, Z ) = 12 K a a − K b b + 12 K Z Z . (14)Substituting the Dirac‘s canonical quantization condition for the momentum operators asˆΠ a = − i δδa , ˆΠ b = − i δδb , ˆΠ Z = − i δδZ , (15)and the potential form (14) into the Hamiltonian density (13) we obtain the correspondingWheeler de Witt wave equation ˆ H Ψ( a, b, X ) = 0 as follows. (cid:26) δ δa − δ δb + 14 ω δ δZ − K a a + K b b − K Z Z (cid:27) Ψ = 0 . (16)Negativity sign of the differential operator δ δb in the above equation shows that the anisotropyfield Y can behave as the time evolution parameter in the three dimensional de Witt superspacemetric. In other word the Kinetic terms in the above differential equation is similar to aKlein Gordon operator defined on the mini-super-space de Witt metric. To solve the equation(16) one can apply the standard method of separation of variables as follows. We assume theWheeler de Witt wave solution to be separable as Ψ( a, b, Z ) = A ( a ) B ( b ) C ( Z ) and substituteit into the equation (16), then we can obtain the following differential equations for the fields A ( a ) , B ( b ) , C ( Z ) respectively as follows. (cid:20) δ δa + ǫ a − K a a (cid:21) A ( a ) = 0 (17) δ δb + ǫ b − K b b (cid:21) B ( b ) = 0 (18) (cid:20) ω δ δZ + ǫ Z − K Z Z (cid:21) C ( Z ) = 0 (19)where the constants of separation of variables ǫ a,b,Z satisfy the following relation. ǫ a + ǫ Z = ǫ b . (20)One can show that the equations (17), (18) and (19) can be described by the well known Hermitpolynomials if we set the following quantization conditions on the parameters ǫ X,Y,Z .ǫ a = (cid:18) N a + 12 (cid:19)s K a , ǫ b = (cid:18) N b + 12 (cid:19)s K b , ǫ Z = (cid:18) N Z + 12 (cid:19)s K Z ω (21)in which N { a,b,Z } can take quantized values 0 , , , , · · · . In the latter case we will obtainnormalized form of the solutions A ( a ) , B ( b ) and C ( Z ) as follows. A N a ( a ) = H N a ( λ a a ) e − λ a a / q N a N a ! √ π , λ a = (12 K X ) (22) B N b ( b ) = H N b ( λ b b ) e − λ b b / q N b N b ! √ π , λ b = (12 K b ) (23) C N Z ( Z ) = H N Z ( λ Z Z ) e − λ Z Z / q N Z N Z ! √ π , λ Z = (2 ωK Z ) (24)in which H N a , H N b , H N Z are Hermit polynomials. Multiplying the above solutions we obtainΨ N a N b N Z = A N a ( a ) B N b ( b ) C N Z ( Z ) which in fact describes quantum fluctuations of the metricfield of the Bianchi I cosmology defined by (4) when the cosmological system is in quantum state( N a , N b , N Z ) with corresponding eigne energy ( ǫ N a , ǫ N b , ǫ N Z ). Now we can show that generalsolution of the Wheeler de Witt wave equation (16) by expanding it versus the eigne functionalssuch that Ψ( X, Y, X ) = Σ ∞ N a =0 Σ ∞ N b =0 Σ ∞ N Z =0 D N a N b N Z Ψ N a N b N Z ( a, b, Z ) (25)where the coefficient D N X N Y N Z describes probability of the quantum Bianchi I cosmology whichtakes the eigne state Ψ N X N Y N Z . This can be determined by regarding the initial condition of thesystem and orthogonal condition on the Hermit polynomials. However no one does not knowabout initial condition of ”physical cosmology” but we can predicts some physical statementsabout our obtained solutions in what follows. Substituting (21) into the condition (20) we canobtain allowable eigne states of the system as follows. (cid:18) N a + 12 (cid:19)s K a K b + (cid:18) N Z + 12 (cid:19)r ω s K Z K b = (cid:18) N b + 12 (cid:19) . (26) . ADM energy and mass In the previous section we assume the ADM mass of the cosmological system has a zero value.If it is not permissible then we should solve the extended Wheeler de Witt wave equation asˆ H Ψ = M Ψ in which M is called to be the ADM mass of the system. In the latter case theequation (20) should be extended to the following form. M = ǫ b − ( ǫ a + ǫ Z ) (27)which by substituting the quantization conditions (21) we obtain a quantization condition onthe ADM mass of the Bianchi I cosmological model such that M N a N b N Z = (cid:18) N b + 12 (cid:19)s K b − (cid:18) N a + 12 (cid:19)s K a − (cid:18) N Z + 12 (cid:19)s K Z ω (28)In fact ADM energy is a special way to define the energy in general relativity, which is onlyapplicable to some special geometries of spacetime that asymptotically approach a well-definedmetric tensor at infinity. If the background metric approaches to Minkowski space asymptoticallythen the Neother‘s theorem implies that the ADM energy or mass should be invariant because oftime independence of the Minkowski metric. According to general relativity, the conservation lawfor the total energy does not hold in more general. For instance for time-dependent backgroundmetrics it will be violated. For example, it is completely violated in physical cosmology. Infact cosmic inflation in particular is able to produce energy or mass from ”nothing” because thevacuum energy density is roughly constant, but the volume of the Universe grows exponentially.Here we can modeled the violation of the ADM energy in the anisotropic Bianchi I inflationarycosmology by quantization approach. Here we show that the raising the ADM energy in theBianchi I cosmology can be described by quantum fluctuations of the mass parameters of thefields ( a, b, Z ) which we called with K a , K b , K Z respectively.
6. Big Bang naked singularity
The background metric (4) shows that the naked singularity lies on the particular hypersurface( a, b ) → ( −∞ , −∞ ) at the classical cosmological feature where the big bang originated. Applying( a, b ) → ( −∞ , −∞ ) one can infer that for naked singularity we have A ( −∞ ) = 0 = B ( −∞ ) . This reads lim ( a,b ) → ( −∞ , −∞ ) Ψ( a, b, Z ) = 0 . (29)Physical interpretation of the above result can be described as follows: There is a zero probabilitywhere the big bang originates from a naked singularity. In other words the big bang originatesfrom an anisotropic quantum Oscillation of the quantum matter fields ( a, b, Z ) at small scalesof the space time. This happened because of the uncertainty relations between the fields andthe corresponding canonical momenta as ∆ X i ∆Π X i ≡ X i = ( a, b, Z ) .
7. Concluding remarks
In this paper we choose scalar vector tensor Brans Dicke gravity to study anisotropic Bianchi Icosmology in the canonical quantization approach. We solved Wheeler de Witt wave equationand obtained its solutions versus the Hermit polynomials for an anisotropic harmonic Oscillatorpotential. We obtained a quantization condition on the ADM boundary energy of the system.Mathematical calculations predict that the inflation of the universe can be originated from a bigbang state without a naked singularity. Furthermore anisotropic counterpart of the metric fieldcan play as a time evolution parameter on the mini-super-space because of Lorentzian signatureof the de Witt metric. As a future work one can study backreaction of the Hawking thermaladiation of the quantum matter (the vector and the Brans Dicke scalar) fields on the solutionsof the Wheeler de Witt equations and so the corresponding eigne energies (quantized ADMenergy) of the system which does not considered here.
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