An Early Universe Model with Stiff Matter and a Cosmological Constant
G. Oliveira-Neto, G. A. Monerat, E. V. Corrêa Silva, C. Neves, L. G. Ferreira
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AN EARLY UNIVERSE MODEL WITH STIFF MATTER AND ACOSMOLOGICAL CONSTANT
G. OLIVEIRA-NETO ∗ Departamento de F´ısica,Instituto de Ciˆencias Exatas,Universidade Federal de Juiz de Fora,CEP 36036-330 - Juiz de Fora, MG, Brazilgilneto@fisica.ufjf.br
G. A. MONERAT, E. V. CORRˆEA SILVA, C. NEVES, L. G. FERREIRA FILHO † Departamento de Matem´atica, F´ısica e Computa¸c˜ao,Faculdade de Tecnologia,Universidade do Estado do Rio de Janeiro,Rodovia Presidente Dutra, Km 298, P´olo Industrial,CEP 27537-000, Resende-RJ, [email protected], [email protected], cliff[email protected], [email protected]
Received October 14, 2018Revised Day Month YearIn the present work, we study the quantum cosmology description of a Friedmann-Robertson-Walker model in the presence of a stiff matter perfect fluid and a negativecosmological constant. We work in the Schutz’s variational formalism and the spatialsections have constant negative curvature. We quantize the model and obtain the appro-priate Wheeler-DeWitt equation. In this model the states are bounded therefore we com-pute the discrete energy spectrum and the corresponding eigenfunctions. In the presentwork, we consider only the negative eigenvalues and their corresponding eigenfunctions.This choice implies that the energy density of the perfect fluid is negative. A stiff mat-ter perfect fluid with this property produces a model with a bouncing solution, at theclassical level, free from an initial singularity. After that, we use the eigenfunctions inorder to construct wave packets and evaluate the time-dependent expectation value ofthe scale factor. We find that it oscillates between maximum and minimum values. Sincethe expectation value of the scale factor never vanishes, we confirm that this model isfree from an initial singularity, also, at the quantum level.
Keywords : Stiff matter; Wheeler-DeWitt equation; negative cosmological constant.PACS numbers: 04.40.Nr,04.60.Ds,98.80.Qc ∗ permanent address † permanent address of all four authors 1 ctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected G. Oliveira-Neto et al.
1. Introduction
The great importance of cosmological models where the matter content is rep-resented by a stiff matter perfect fluid was recognized since its introduction byZeldovich.1 , p = αw , with α = 1, where w and p are, respectively, the fluid energy density and pressure. Itcan also be described by a massless free scalar field. In order to understand betterthe importance of this perfect fluid for cosmology, one has to compute its energydensity. In the temporal gauge ( N ( t ) = 1), this quantity is proportional to 1 /a ( t ) ,where N ( t ) is the lapse function and a ( t ) is the scale factor. On the other hand,in the same gauge, the energy density of a radiative perfect fluid is proportional to1 /a ( t ) . This result indicates that there may have existed a phase earlier than thatof radiation, in our Universe, which was dominated by stiff matter. Due to that im-portance, many physicists have started to consider the implications of the presenceof a stiff matter perfect fluid in FRW cosmological models. The first important im-plication of the presence of stiff matter in FRW cosmological models is in the relicabundance of particle species produced after the ‘Big Bang’ due to the expansionand cooling of our Universe. 3 , , , , , ,
10 It may also play an important role in the spectrum of relic gravitywaves created during inflation.11 Since there may have existed a phase earlier thanthat of radiation which was dominated by stiff matter some physicists consideredquantum cosmological models with this kind of matter.12 , , ,
16 Wequantize the model and obtain the appropriate Wheeler-DeWitt equation. In thismodel the states are bounded therefore we compute the discrete energy spectrumand the corresponding eigenfunctions. In the present work, we consider only the neg-ative eigenvalues and their corresponding eigenfunctions. This choice implies thatthe energy density of the perfect fluid is negative. A stiff matter perfect fluid withthis property has already been considered in the literature.17 , ,
19 It produces amodel with a bouncing solution, at the classical level, free from an initial singular-ity. After that, we use the eigenfunctions in order to construct wave packets andevaluate the time-dependent expectation value of the scale factor. We find that itoscillates between maximum and minimum values. Since the expectation value ofthe scale factor never vanishes, we confirm that this model is free from an initialsingularity, also, at the quantum level.The presence of a negative cosmological constant in the present model impliesthat the universe described by it has a maximum size, in other words it is bounded.Taking into account the current cosmological observations, a negative cosmologicalctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected
An Early Universe Model with Stiff Matter and a Cosmological Constant constant will not be able to describe the present accelerated expansion of our Uni-verse. It is not our intention to describe the present state of our Universe with thismodel. On the other hand, it is our intention to describe a ‘possible’ state of ourprimordial Universe. One important theory which is a strong candidate to describethe unification of all known physical interactions is superstring theory.20 ,
21 Dueto that, many physicists believe that superstring theory will correctly describe thequantum gravity effects that took place at the beginning of our Universe. There isan important conjecture which tells that Type IIB string theory on (
AdS × S ) N plus some appropriate boundary conditions is dual to N = 4 d = 3 + 1 U ( N ) super-Yang-Mills.22 It means that, possibly, for an appropriate description of the knownphysical interactions through superstring theories, the strings have to exist in anAnti-DeSitter spacetime. The Anti-DeSitter spacetime has a negative cosmologicalconstant, therefore it seems worthwhile to study spacetimes with a negative cosmo-logical constant if one wants to understand more about a ‘possible’ initial state ofour primordial Universe. Of course, after that initial state the Universe would haveto undergo a transition where the cosmological constant would change sign. Besidesthat, several important theoretical results and predictions in quantum cosmologyhave been obtained with a negative cosmological constant.23 , , ,
26 Consideringa subset of all four-dimensional spacetimes with constant negative curvature andcompact space-like hypersurfaces, S. Carlip et al showed how to compute the sumover topologies leading to the no-boundary wave-function.23 ,
24 These spacetimesare curved only due to the presence of a negative cosmological constant. In Ref. 23it was shown how to obtain a vanishing cosmological constant as a prediction fromthe no-boundary wave-function and in Ref. 24 it was shown how to obtain predic-tions about the topology of the Universe from the no-boundary wave-function. Wemay also mention the result in Ref. 25, where the WKB no-boundary wave-functionof a homogeneous and isotropic Universe with a negative cosmological constant wascomputed. Due to the regularity condition imposed upon the spacetimes contribut-ing to the no-boundary wave-function, it was shown that only a well defined, discretespectrum for the cosmological constant is possible. It was also found that amongthe spacetimes contributing to the wave function, there were two complex conjugateones that showed a new type of signature change.The present paper is organized as follows. In Sec. 2, we introduce the classicalmodel and obtain the appropriate Friedmann equation. With the aid of the poten-tial curve coming from this equation, we comment on the general behavior of theclassical solutions. In Sec. 3, we quantize the model by solving the correspondingWheeler-DeWitt equation. The wave-function depends on the scale factor a and onthe canonical variable associated to the fluid, which in the Schutz variational for-malism plays the role of time ( T ). We separate the wave-function in two parts, onedepending solely on the scale factor and the other depending only on the time. Thesolution in the time sector of the Wheeler-DeWitt equation is trivial, leading to animaginary exponential of the type e − iEτ , where E is the system energy and τ = − T .The scale factor sector of the Wheeler-DeWitt equation gives rise to an eigenvaluectober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected G. Oliveira-Neto et al. equation. We find approximate solutions. In Sec. 4, we construct wave packets fromthe eigenfunctions and compute the time-dependent, expectation value of the scalefactor. We find that the expectation value of the scale factor shows bounded oscilla-tions. Since the expectation value of the scale factor never vanishes, we confirm thatthis model is free from a big bang singularity, also, at the quantum level. Finally, inSec. 5, we summarize the main points and results of our paper.
2. The Classical Model
The present Friedmann-Robertson-Walker cosmological model is characterized bythe scale factor a ( t ) and has the following line element, ds = − N ( t ) dt + a ( t ) (cid:18) dr r + r d Ω (cid:19) , (1)where d Ω is the line element of the two-dimensional sphere with unitary radius, N ( t ) is the lapse function and we are using the natural unit system, where ~ = c =8 πG = 1. In this model the spatial sections are some closed three-dimensional solidwith negative constant curvature, locally isometric to H .27 The matter contentof the model is represented by a perfect fluid with four-velocity U µ = δ µ in thecomoving coordinate system used, plus a negative cosmological constant (Λ). Thetotal energy-momentum tensor is given by, T µν = ( w + p ) U µ U ν − pg µν − Λ g µν , (2)As mentioned above, here, we assume that p = w , which is the equation of state forstiff matter.Einstein’s equations for the metric (1) and the energy momentum tensor (2) areequivalent to the Hamilton’s equations generated by the total Hamiltonian N ( t ) H ,where, H = − p a a + 3 a + Λ a + p T a . (3)The variables p a and p T are the momenta canonically conjugated to the variables a and T , respectively.The classical dynamics is governed by the Hamilton’s equations, derived fromthe total Hamiltonian N ( t ) H Eq. (3). In the gauge N ( t ) = a ( t ), they are, ˙ a = ∂N H ∂p a = − p a , ˙ p a = − ∂N H ∂a = − a − a + 2 p T a , ˙ T = ∂N H ∂p T = 1 a , ˙ p T = − ∂N H ∂T = 0 . (4)ctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected An Early Universe Model with Stiff Matter and a Cosmological Constant Where the dot means derivative with respect to the conformal time τ ≡ N t , whichin the present gauge is equal to at .We also have the constraint equation H = 0. It gives rise to the Friedmannequation, ˙ a + V c ( a ) = 0 , (5)where the potential V c ( a ) is equal to, V c ( a ) = − a − Λ a − p T a . (6)For the present situation where Λ <
0, we have bounded solutions. The classicalsolutions are bouncing ones, free from an initial singularity, because we are con-sidering a stiff matter perfect fluid with negative energy density. Those results canbe directly seen from the potential expression. A particular example of V c ( a ), forΛ = − . p T = −
3. The Quantum Model
We wish to quantize the model following the Dirac formalism for quantizing con-strained systems.28 , , ,
31 First we introduce a wave-function which is a functionof the canonical variables ˆ a and ˆ T ,Ψ = Ψ(ˆ a, ˆ T ) . (7)Then, we impose the appropriate commutators between the operators ˆ a and ˆ T andtheir conjugate momenta ˆ p a and ˆ p T . Working in the Schr¨odinger picture, the oper-ators ˆ a and ˆ T are simply multiplication operators, while their conjugate momenta Fig. 1. V c ( a ) for Λ = − . p T = − ctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected G. Oliveira-Neto et al. are represented by the differential operatorsˆ p a → − i ∂∂a , ˆ p T → − i ∂∂T . (8)Finally, we demand that ˆ H , the superhamiltonian operator corresponding to (3),annihilate the wave-function Ψ Eq. (7), which leads to Wheeler-DeWitt equation (cid:18) − ∂ ∂a − (3 a + Λ a ) (cid:19) Ψ( a, τ ) = i a ∂∂τ Ψ( a, τ ) , (9)where the new variable τ = − T has been introduced.The operator ˆ H is self-adjoint 32 with respect to the internal product,(Ψ , Φ) = Z ∞ da a Ψ( a, τ ) ∗ Φ( a, τ ) , (10)if the wave-functions are restricted to the set of those satisfying either Ψ(0 , τ ) = 0or Ψ ′ (0 , τ ) = 0, where the prime ′ means the partial derivative with respect to a .The Wheeler-DeWitt equation (9) may be solved by writing Ψ( a, τ ) as,Ψ( a, τ ) = e − iEτ η ( a ) (11)where η ( a ) depends solely on a . Then η ( a ) satisfies the eigenvalue equation − d η ( a ) da + V ( a ) η ( a ) = 12 E a η ( a ) , (12)where the potential V ( a ) is given by V ( a ) = − a − a . (13)In the same way as in the classical regime, the potential V ( a ) Eq. (13) givesrise to bound states. Therefore, the energies E , Eq. (12), of those states will form adiscrete set of eigenvalues E n , where n = 1 , , , ... . For each eigenvalue E n there willbe a corresponding eigenvector η n ( a ). The general solution to the Wheeler-DeWittequation (9) is a linear combination of all those eigenvectors,Ψ( a, τ ) = ∞ X n =1 C n η n ( a ) e − iE n τ , (14)where C n are free coefficients to be specified.We are going to use the Galerkin or spectral method (SM), in order to solve theeigenvalue equation (12). This method is well presented in Ref. 33 and it has alreadybeen used in quantum cosmology.34 ,
35 In the SM, one must choose orthonormalbasis functions in order to expand the solution to the eigenvalue equation. Thesolutions to the present eigenvalue equation (12) must fall sufficiently fast for largescale factor values ( a ). It means that we must restrict the initial infinity domain ofour variable a , to a finite domain. Say, 0 < a < L , where L is a finite number thathas to be fixed. Here, we shall consider solutions satisfying the condition Ψ(0 , τ ) = 0.Putting together all the above properties of the solutions to Eq. (12), it is convenientctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected An Early Universe Model with Stiff Matter and a Cosmological Constant (but not mandatory) to choose our basis functions to be sine functions. Therefore,we may write η n ( a ) in Eq. (12) as, η n ( a ) = ∞ X n =1 A n r L sin (cid:16) nπaL (cid:17) , (15)where the A n ’s will be determined by the SM. In the same a domain, we may alsoexpand, in the same basis, the other two important functions of a appearing in Eq.(12), V ( a ) η n ( a ) = ∞ X n =1 B n r L sin (cid:16) nπaL (cid:17) , (16) (cid:18) a (cid:19) η n ( a ) = ∞ X n =1 B ′ n r L sin (cid:16) nπaL (cid:17) , (17)Where V ( a ) is given in Eq. (13) and the coefficients B n and B ′ n can be easilydetermined. They are determined with the aid of Eq. (15) and the fact that thebasis functions are orthonormal. The coefficients B n and B ′ n are given by, B n = ∞ X m =1 C m,n A m , (18) B ′ n = ∞ X m =1 C ′ m,n A m , (19)where, C m,n = 2 L Z L sin (cid:16) mπaL (cid:17) V ( a ) sin (cid:16) nπaL (cid:17) da, (20) C ′ m,n = 2 L Z L sin (cid:16) mπaL (cid:17) (cid:18) a (cid:19) sin (cid:16) nπaL (cid:17) da. (21)Introducing the results Eqs. (15)-(21) in the eigenvalue equation (12) and using thefact that the basis functions are orthonormal, we obtain, (cid:16) nπL (cid:17) A n + ∞ X m =1 C m,n A m = E ∞ X m =1 C ′ m,n A m . (22)In order to derive some numerical results we must fix a finite number of basisfunctions, in other words, a finite number for the maximum value of the summationindices. Let us call this number N . The greater the value of N , the closer our resultswill be to the exact ones. We shall be restricted by our computational resources.Equation (22), may be written in a compact notation as, D ′− D A = E A , (23)ctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected G. Oliveira-Neto et al. where D and D ′ are N × N square matrices and their elements are obtained from Eq.(22). The solution to Eq. (23) gives the eigenvalues and corresponding eigenfunctionsto the bound states of our quantum cosmology model.It is important to mention that the most correct form of η ( a ) in the limit when a → η ( a ), onehas to introduce the ansatz η ( a ) = Ca α (where C is a constant and α is a numberto be determined) in Eq. (12). After that, one has to discard the terms whichhave as coefficients the cosmological constant (Λ) and the curvature of the spatialsections ( k = − a α +4 and a α +2 and should be less important, in the limit a → α . Finally, one solvesthe resulting equation imposing that the coefficient of the only remaining term,proportional to a α − , vanishes. This gives rise to a second order algebraic equationfor α , which solution satisfying the boundary condition η (0) = 0 is given by: α =0 . p .
25 + 12 | E | . Therefore, the most correct form for η ( a ) in the limit when a → η ( a ) = Ca . √ . | E | . (24)We notice that it cannot represent the correct expression for η ( a ) in the limit a → ∞ because it diverges in that limit. That solution is already known in the literature.36
4. Energy Spectrum, Wave Packet and Mean Value.
In this section we will solve Eq. (12) using the SM. In order to choose the numberof basis functions N and the values of Λ, we performed the following numericalprocedures. First of all, in order to choose the values of Λ, we solved numerically,the eigenvalue equation (23) for several different values of Λ and fixed values of N and L . We noticed that, although, we are free to choose any value of Λ, theresults accuracy for small absolute values of Λ is better than for large absolutevalues of Λ, for a given number of basis functions N . This means that, if we usea large absolute value of Λ, we need to increase N to obtain the same accuracyof the case with a small absolute value of Λ. This, of course, would increase thecomputation time. Therefore, we have decided to use small absolute values of Λ.Taking in account this considerations, we choose Λ = − . N , we solved numerically, the eigenvalue equation(23) for several different values of N and fixed values of Λ and L . Then, we comparedthe eigenvalues coming from the different choices of N . We noticed that only someof the first eigenvalues, for each different choice of N , remained the same up to asatisfactory accuracy. Therefore, we have decided to use N = 100 and take onlythe first eighteen eigenvalues to construct the wave packet. The accuracy of theeigenvalues, in this case, is in the tenth digit after the dot. Finally, it is importantto mention that based on comparisons with results of other models studied with theSM,35 we chose L = 6, in the present case.ctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected An Early Universe Model with Stiff Matter and a Cosmological Constant Now, using all these values: N = 100, Λ = − . L = 6, in the determinantconstructed from Eq. (23), we solve it and obtain the first 100 energy eigenvaluesfor the present case. From those, we take the first 18 energy levels and list them inTable 1. Table 1. The eighteen lowest energy levels for a FRW model with k = −
1, Λ = − .
1, astiff matter perfect fluid ( p = ρ ), N = 100 and L = 6. E = -380.2201284331828 E = -342.1147751350869 E = -305.9147225014253 E = -271.6319016521779 E = -239.2791871064332 E = -208.8705235210961 E = -180.4210774184946 E = -153.9474207233139 E = -129.4677552918257 E = -107.0021912238143 E = -86.57309703051190 E = -68.20554802882278 E = -51.92791264850499 E = -37.77263895532904 E = -25.77734426711615 E = -15.98638945709977 E = -8.453288958554614 E = -3.244733126937446 In order to give an idea how the energy spectrum depends on Λ, we have con-structed the curve of the fundamental energy level E versus Λ. It is given in Figure2. We notice that E decreases when one increases Λ.It is important to mention that even though the expression of η ( a ) given byEq. (15) is not the most correct one, in the limit a →
0, we were able to determinenumerically that it converges rapidly to zero in that limit. In particular, for the firstenergy level the corresponding eigenfunction oscillations about the zero value, in aregion very close to it (0 ≤ a ≤ . − . For the eighteenthenergy level the corresponding eigenfunction oscillations about the zero value, ina region very close to it (0 ≤ a ≤ . − . For the othereigenfunctions the oscillations about zero, in regions very close to zero, have valuesbetween the ones given above.Next, we construct the wave packet Ψ( a, τ ) Eq. (14), with the aid of η n ( a ) Eq. Fig. 2. Dependence of E with Λ, for N = 100 and L = 6. ctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected G. Oliveira-Neto et al. (15) and the energy levels in Table 1. Our numerical study showed that, the energyeigenfunctions in the linear combination Eq. (14) are orthonormal and Ψ( a, τ ) Eq.(14), has a constant norm. In the linear combination Eq. (14), we set C n equal toone, for the first eighteen values of n and C n equal to zero for the other valuesof n . In Figure 3, we show, as an example, the modulus squared of a wave packetconstructed with the first eighteen energy levels, given in Table 1, for τ = 1000, L = 6 and Λ = − . a, τ ) we compute the mean value for the scalefactor a , according to the following expression, h a i ( τ ) = R ∞ a − | Ψ( a, τ ) | da R ∞ a − | Ψ( a, τ ) | da . (25)We computed this quantity for many different time intervals. For all this differentvalues, we observed that h a i performs bounded oscillations and never assume thezero value. Therefore, we confirm that these models are free from singularities, also,at the quantum level. As an example, we show in Fig. 4 the mean value computedwith the wave packet constructed with the first eighteen energy levels, given inTable 1, for the interval from τ = 0 until τ = 1000, L = 6 and Λ = − .
5. Conclusions.
In the present paper, we quantized a Friedmann-Robertson-Walker model in thepresence of a stiff matter perfect fluid and a negative cosmological constant. We usedthe variational formalism of Schutz. The model has spatial sections with negativeconstant curvature. The quantization of the model gave rise to a Wheeler-DeWittequation, for the scale factor. We found the approximate eigenvalues and eigenfunc-tions of that equation by using the Galerkin or spectral method. In the present
Fig. 3. Modulus squared of a wave packet constructed with the first eighteen energy levels for τ = 1000, L = 6 and Λ = -0.1. ctober 14, 2018 17:17 WSPC/INSTRUCTION FILEG-Oliveira-Neto˙corrected An Early Universe Model with Stiff Matter and a Cosmological Constant work, we considered only the negative energy eigenvalues and their correspondingeigenfunctions. This choice implies that the energy density of the perfect fluid isnegative. A stiff matter perfect fluid with this property produces a model with abouncing solution, at the classical level, free from an initial singularity. After that,we used the eigenfunctions in order to construct wave packets and evaluate the timedependent, expected value of the scale factor. We found that the expected value ofthe scale factor evolve with bounded oscillations. Since the expectation value of thescale factor never vanish, we confirm that this model is free singularities, also, atthe quantum level. Acknowledgments
E. V. Corrˆea Silva (Researcher of CNPq, Brazil), G. A. Monerat, G. Oliveira-Neto,C. Neves and L. G. Ferreira Filho thank CNPq and FAPERJ for partial financialsupport. G. Oliveira-Neto thanks FAPEMIG for partial financial support.
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