Large classical universes emerging from quantum cosmology

Nelson Pinto-Neto

ICRA—Centro Brasileiro de Pesquisas Fısicas—CBPF, rua Xavier Sigaud, 150, Urca, CEP22290-180, Rio de Janeiro, Brazil(Received 18 November 2008; published 16 April 2009)

It is generally believed that one cannot obtain a large universe from quantum cosmological models

without an inflationary phase in the classical expanding era because the typical size of the universe after

leaving the quantum regime should be around the Planck length, and the standard decelerated classical

expansion after that is not sufficient to enlarge the universe in the time available. For instance, in many

quantum minisuperspace bouncing models studied in the literature, solutions where the universe leaves

the quantum regime in the expanding phase with appropriate size have negligible probability amplitude

with respect to solutions leaving this regime around the Planck length. In this paper, I present a general

class of moving Gaussian solutions of the Wheeler-DeWitt equation where the velocity of the wave in

minisuperspace along the scale factor axis, which is the new large parameter introduced in order to

circumvent the above-mentioned problem, induces a large acceleration around the quantum bounce,

forcing the universe to leave the quantum regime sufficiently big to increase afterwards to the present size,

without needing any classical inflationary phase in between, and with reasonable relative probability

amplitudes with respect to models leaving the quantum regime around the Planck scale. Furthermore,

linear perturbations around this background model are free of any trans-Planckian problem.

DOI: 10.1103/PhysRevD.79.083514 PACS numbers: 98.80.Cq, 04.60.Ds

I. INTRODUCTION

The existence of an initial singularity [1] is one of themajor drawbacks of classical cosmology. In spite of thefact that the standard cosmological model, based in clas-sical general relativity sourced by ordinary matter, hasbeen successfully tested until the nucleosynthesis era, theextrapolation of this model to higher energies leads to abreakdown of the geometry in a finite cosmic time. Itindicates the failure of this conventional approach at highenergies, which should be complemented through the in-tervention of some new physics (presence of exotic matter,modifications of general relativity through nonminimalcouplings, nonlinear curvature terms in the Lagrangian,quantum effects of the gravitational field, etc.), leading toa complete regular cosmological model.

In the framework of quantum cosmology in minisuper-space models, nonsingular bouncing models have beenobtained,1 where the bounce occurs due to quantum effectsin the background [6–8]. Some approaches have used anontological interpretation of quantum mechanics, theBohm-de Broglie [9] one, to interpret the results [7,8]because, contrary to the standard Copenhagen interpreta-tion, this ontological interpretation does not need a classi-

cal domain outside the quantized system to generate thephysical facts out of potentialities (the facts are thereab initio), and hence it can be applied to the universe asa whole. Of course there are other alternative interpreta-tions which can be used in quantum cosmology, like themany worlds interpretation of quantum mechanics [10],but I will not use them in this paper.In the Bohm-de Broglie interpretation, quantum

Bohmian trajectories, the quantum evolution of the scalefactor aqðtÞ, can be defined through the relation _a /@S=@a, where S is the phase of an exact wave solution�ða; tÞ of the Wheeler-DeWitt equation. It satisfies amodified Hamilton-Jacobi equation, augmented with aquantum potential term derived from �ða; tÞ, and henceaqðtÞ is not the classical trajectory: in the regions where thequantum effects cannot be neglected, the quantum trajec-tory aqðtÞ performs a bounce which connects two asymp-

totic classical regions where the quantum effects arenegligible. One then has in hand a definite function oftime for the homogeneous and isotropic background partof the universe, even at the quantum level, which realizes asoft transition from the contracting phase to the expandingone.When studying the evolution of quantum cosmological

perturbations on these backgrounds, which was done in theseries of papers [11–14] for the case of one perfect fluidwith equation of state p ¼ w�, one arrives at the resultthat, in order to obtain wavelength spectra and amplitudescompatible with CMB data, one must have [14] jwj � 1

and w1=4L0 � 102lpl, where L0 is the curvature scale at thebounce and lpl is the Planck length. Hence this analysis

shows that the model is self-consistent because observa-tional constraints impose that the curvature scale at thebounce must be at least a few orders of magnitude greater

1There are many other frameworks where bounces connectingthe present expanding phase with a preceding contracting onemay occur [2,3]. In this case, the universe is eternal, there is nobeginning of time, nor horizons. The new features of thesemodels introduce a new picture, where the usual problems ofinitial conditions [4] (as, for instance, the almost homogeneousbeginning of the expanding phase might be explained throughthe dissipation of nonlinear inhomogeneities when the universewas very large and rarefied in the asymptotic far past of thecontracting phase) and the evolution of cosmological perturba-tions [5] are viewed from a very different perspective.

PHYSICAL REVIEW D 79, 083514 (2009)

1550-7998=2009=79(8)=083514(8) 083514-1 � 2009 The American Physical Society

than the Planck length, a region where one can trust theWheeler-DeWitt equation without having been spoiled byhigh order quantum gravity effects. Of course this modelshould be extended to include radiation. In Ref. [4] it isshown that the requirement jwj � 1 is important only atthe moment when the perturbation wavelength becomesgreater than the curvature scale: the fluid which dominatesat the bounce is irrelevant for the spectral index (but isimportant for the amplitude, as we will see in futurepublications).

However, this scenario has a problem on the quantumbackground solution itself: in order for the model to de-scribe the big universe we live in, the scale factor at thebounce a0 must be somewhat large, and the probability onecan obtain from the wave function of the model for theoccurrence of this value is incredibly small (in some casesexpð�1089Þ, as we will see later on). Hence, either there isan inflationary phase after the bounce in order to enlargethe universe from a small a0 (which may lead to trans-Planckian problems [15] and nonlinear inhomogeneities atthe bounce because of the growth of linear perturbations inthe contracting phase if a0 is small), or one should rely verystrongly on some anthropic principle in a situation muchworst than in the landscape scenario.

The aim of this paper is to overcome this difficulty byproposing some more general wave solutions of theWheeler-DeWitt equation which lead to realistic bouncingscenarios with parameters with reasonable probabilitiesand without any trans-Planckian problem. In Ref. [14],the wave function at the bounce was chosen to be a staticGaussian of the scale factor centered at a ¼ 0. As we willsee, the ratio between the scale factor at the bounce a0 andthe width of the Gaussian must be very large in order toyield the big universe we live in, yielding the very smallprobability of occurrence of these parameters I mentionedabove. However, if one generalizes the wave function to bea moving Gaussian on the a-axis with velocity u, there is aminimum value of this parameter from where one canobtain a large universe, with reasonable probability ofoccurrence, and without any trans-Planckian problems.The parameter u induces a very large acceleration aroundthe bounce, leading to a sufficiently large scale factor whenthe quantum regime is over.

This paper is organized as follows: in Sec. II I describein detail the problem I want to solve. In Sec. III I presentthe generalized wave solutions from which this problemcan be circumvented. I conclude in Sec. IV with a discus-sion of our results, their physical meanings, and prospectsfor future work.

II. QUANTUM BOUNCE SOLUTIONS FROMSTATIC INITIAL GAUSSIANS AND THEIR

PROBLEMS

The Hamiltonian constraint describing a cosmologicalmodel with flat homogeneous and isotropic closed space-

like hypersurfaces with comoving volume V ¼ 1, and aperfect fluid satisfying p ¼ !�, where � is the perfect fluidenergy density, p is the pressure, and! is a constant, reads[11]

H0 � PT

a3!� P2

a

4a; (1)

where a is the scale factor, Pa its canonical momentum,and the conserved quantity PT , the momentum canonicallyconjugated to the degree of freedom of the fluid T [16], isassociated with the constant appearing in the energy den-

sity of the fluid through the relation � ¼ PT=a3ð1þwÞ. All

quantities are in Planck unities. One can verify that theHamiltonian H ¼ NH0 generates the usual Friedmannequations of the model.The wave function�ða; TÞ satisfies the Wheeler-DeWitt

equation H0� ¼ 0,

i@

@T�ða; TÞ ¼ að3!�1Þ=2

4

@

@a

�að3!�1Þ=2 @

@a

��ða; TÞ; (2)

where I have chosen the factor ordering in a in order toyield a covariant Schrodinger equation under field redefi-nitions. The fluid selects a preferred time variable.I change variables to

� ¼ 23ð1�!Þ�1a3ð1�!Þ=2;

obtaining the simple equation

i@�ð�; TÞ

@T¼ 1

4

@2�ð�; TÞ@�2

: (3)

This is just the time reversed Schrodinger equation for aone-dimensional free particle constrained to the positiveaxis. As a and � are positive, solutions which have unitaryevolution must satisfy the condition

��? @�

@���

@�?

@�

����������¼0¼ 0 (4)

(see Ref. [8] for details). I can choose the initial normalizedwave function

�ðinitÞð�Þ ¼�

8

T0�

�1=4

exp

���2

T0

�; (5)

where T0 is an arbitrary constant. The Gaussian �ðinitÞsatisfies condition (4), and it gives the probability densityfor the value of � at T ¼ 0 with minimum uncertainty.Using the propagator procedure of Refs. [8], we obtain

the wave solution for all times in terms of a:

NELSON PINTO-NETO PHYSICAL REVIEW D 79, 083514 (2009)

083514-2

�ða; TÞ ¼�

8T0

�ðT2 þ T20Þ�1=4

exp

� �4T0a3ð1�!Þ

9ðT2 þ T20Þð1�!Þ2

�

� exp

��i

�4Ta3ð1�!Þ

9ðT2 þ T20Þð1�!Þ2

þ 1

2arctan

�T0

T

�� �

4

��: (6)

Because of the chosen factor ordering, the probabilitydensity �ða; TÞ has a nontrivial measure and it is given by

�ða; TÞ ¼ að1�3!Þ=2R2, where R2 ¼ j�ða; TÞj2. Its con-tinuity equation, one of the equations coming fromEq. (2) after substitution of � ¼ ReiS in it, reads

@�

@T� @

@a

�að3!�1Þ

2

@S

@a�

�¼ 0; (7)

which implies, in the Bohm interpretation [9], the defini-tion of a velocity field

_a ¼ �að3!�1Þ

2

@S

@a; (8)

in accordance with the classical relations _a ¼ fa;Hg ¼�að3!�1ÞPa=2 and Pa ¼ @S=@a.

Note that S satisfies the other equation coming from (2),

@S

@T� að3!�1Þ

4

�@S

@a

�2 þ að3!�1Þ=2

4R

@

@a

�að3!�1Þ=2 @R

@a

�¼ 0;

(9)

which is a Hamilton-Jacobi-like equation with an extraquantum term, called the quantum potential, given by

Q � � að3!�1Þ=2

4R

@

@a

�að3!�1Þ=2 @R

@a

�: (10)

Hence, the trajectory (8) will not coincide with the classi-cal trajectory whenever Q is comparable with the otherterms present in Eq. (9) because Swill be different from theclassical Hamilton-Jacobi function.

Inserting the phase of (6) into Eq. (8), I obtain theBohmian quantum trajectory for the scale factor:

aðTÞ ¼ a0

�1þ

�T

T0

�2�ð1=3ð1�!ÞÞ

; (11)

or, in terms of �ðTÞ,

�ðTÞ ¼ �0

�1þ

�T

T0

�2�1=2

: (12)

Note that �0 is the value of � at T ¼ 0, the moment of thebounce, �0 ¼ �bounce, and the scale factor at the bounce a0is connected to �0 through

a0 ¼�3

2ð1�!Þ�0

�2=½3ð1�!Þ�

: (13)

Solution (11) has no singularities and tends to the clas-sical solution when T ! �1. Remember that I am in the

gaugeN ¼ a3!, and T is related to conformal time through

NdT ¼ ad� ) d� ¼ ½aðTÞ�3!�1dT: (14)

The solution (11) can be obtained from other initial wavefunctions (see Ref. [8]).However, the above solution suffers from the following

drawback: the curvature scale at the bounce readsLbounce � T0a

3w0 , and the quantity PT associated in the

classical limit jTj ! 1 with the constant appearing in

the energy density of the fluid through the relation � ¼PT=a

3ð1þwÞ, can be obtained in the Bohmian approach fromthe wave function through the relation PT ¼ @S=@T. Itreads

PT ¼ @S

@T¼ T0

2ðT2 þ T20Þ

� �ðTÞ2ðT20 � T2Þ

ðT2 þ T20Þ2

: (15)

Inserting the solution (12) in Eq. (15) and taking theclassical limit jTj ! 1, one obtains

PT ¼ �20

T20

: (16)

In the case of dust, PT is the total dust mass of theuniverse, yielding PT � 1060. If one takes the curvaturescale at the bounce some few orders of magnitude largerthan the Planck length, say 103, in order to not spoil theWheeler-DeWitt approach used above due to strong quan-tum gravitational effects, one has to have �0 � 1033, withprobability less than expð�1063Þ to occur (see Eq. (5)).The situation is similar with radiation, where now PT ¼

�20=T

20 � 10116. Note that in this case �0 ¼ a0, T ¼ �, and

the curvature scale at the bounce reads Lbounce � T0a0.Combining the constraints a0=T0 � 1058 and a0T0 �103, one arrives at the very low probability expð�1089Þfor these parameters to occur.The source of the problem is the fact that the constant �0

appearing in Eq. (16) is also the value of � at T ¼ 0, the �at the bounce, and the fact that PT must be large, induces alarge �2=T0 in the Gaussian (5). One possibility to scapefrom this drawback is to find a different wave solution toEq. (2) which either modifies Eq. (16) or yields Bohmiantrajectories where �0 is not anymore the value of � at T ¼0, allowing the possibility of having a small initial �, hencea small �2=T0 in (5), and a huge �0, perhaps through thepresence of an inflationary phase between the bounce andthe standard decelerated expansion. I will show in the nextsection that it is indeed possible to obtain a more generalclass of wave solutions of Eq. (2) where the above-mentioned problem is circumvented.

III. NEW BOUNCING SOLUTIONS

I will generalize the initial wave function by inserting avelocity term in Eq. (5), which of course must satisfy theboundary condition (4), and now reads

LARGE CLASSICAL UNIVERSES EMERGING FROM . . . PHYSICAL REVIEW D 79, 083514 (2009)

083514-3

�ðinitÞð�Þ ¼�

2

T0�

�1=4½1� expð�u2T0=8Þ��1=2

�½expðiu�=2Þ � expð�iu�=2Þ� exp���2

T0

�:

(17)

This initial wave function represents two Gaussians trav-elling from the origin in opposite directions (keeping inmind that only the tail of the Gaussian traveling in thenegative direction with support on the positive a axis hasphysical meaning). The solution for all times reads

�ð�; TÞ ¼�

2T0

�ðT2 þ T20Þ�1=4ð1� expð�u2T0=8Þ�1=2

��exp

��T0ð�� uTÞ2

ðT2 þ T20Þ

�exp

��i

�Tð�� uTÞ2ðT2 þ T2

0Þþ 2uð�� uT=2Þ þ 1

2arctan

�T0

T

�� �

4

��

� exp

��T0ð�þ uTÞ2

ðT2 þ T20Þ

�exp

��i

�Tð�þ uTÞ2ðT2 þ T2

0Þ� 2uð�þ uT=2Þ þ 1

2arctan

�T0

T

�� �

4

���;

(18)

which I write as

� ¼ AðR�eiS� � RþeiSþÞ;

where

R� � exp

��T0ð�� uTÞ2

ðT2 þ T20Þ

�;

S� ���Tð�� uTÞ2

ðT2 þ T20Þ

� 2uð�� uT=2Þ � 1

2arctan

�T0

T

�

þ �

4

�;

A ��

2T0

�ðT2 þ T20Þ�1=4ð1� expð�u2T0=8Þ�1=2:

From these equations one obtains the total amplitude andphase as

R ¼ AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2þ þ R2� � 2RþR� cosðSþ � S�Þ

q;

S ¼ arctan

�Rþ sinðSþÞ � R� sinðS�ÞRþ cosðSþÞ � R� cosðS�Þ

�:

The derivative of S with respect to some variable x reads

@S

@x¼ R2þ

@Sþ@x þ R2�

@S�@x � ð@Sþ@x þ @S�

@x ÞRþR� cosðSþ � S�Þ � ðR�@Rþ@x � Rþ

@R�@x Þ sinðSþ � S�Þ

R2þ þ R2� � 2RþR� cosðSþ � S�Þ:

The guidance relation (8) leads to the exact differentialequation

_�ðTÞ ¼ T�ðTÞT2 þ T2

0

þ uT20

T2 þ T20

�sinhð� TT0Þ � T

T0sin�

coshð� TT0Þ � cos�

�;

(19)

where

� � 4uT20�ðTÞ

T2 þ T20

: (20)

From the solution (18), the new PT is now given by

PT ¼ @S

@T

¼ T0

2ðT2þT20Þþ ½u2T2

0 ��2ðTÞ�ðT20 �T2Þ

ðT2þT20Þ2

þ½4uT20T�ðTÞ sinhð� T

T0Þ� 2T0u�ðTÞðT2�T2

0Þ sin��ðT2þT2

0Þ2½coshð� TT0Þ� cos�� :

(21)

From now on I will work with the plus sign solutiongiven in Eq. (18). The minus sign solution yields the samequalitative results.

A. Quantum solutions for small u

Taking u � 1, one has

_�ðTÞ ¼ T�ðTÞT2 þ T2

0

þ 4�ðTÞu2TT30

ðT2 þ T20Þ2

; (22)

with solution

�ðtÞ ¼ �0

T0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 þ T2

0

qexp

��2u2T30

T2 þ T20

�

¼ �0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ 1

pexp

��2u2T0

x2 þ 1

�; (23)

where in the last step I wrote the solution in terms of x ¼T=T0.Solution (23) has very nice properties. First of all, one

can see that the values of � and the curvature scale at thebounce are now given by

�bounce ¼ �0 expð�2u2T0Þ; (24)

and

NELSON PINTO-NETO PHYSICAL REVIEW D 79, 083514 (2009)

083514-4

Lbounce ¼ exp

�� 4wu2T0

ð1� wÞ�

T0a3w0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 4u2T0

p : (25)

Inserting solution (23) into Eq. (21) in the limit jTj ! 1yields

PT ¼ �20

T20

: (26)

Note that now �0 � �bounce, the value of � at T ¼ 0. Infact, from Eq. (24), one may have �0 �bounce, dependingon the value of T0, because of the huge acceleration onemay obtain near after the bounce as compared with the casewhere u ¼ 0: it is a bounce followed by inflation. Hence,one may have reasonable probability amplitudes for thefree parameters of the theory which are compatible with ahuge PT .

However, one must also check whether the Bohmiantrajectory (23), with such appropriate choice of parameters,reaches classical evolution (x 1) before the nucleosyn-thesis epoch. Let us concentrate on the case of radiation(w ¼ 1=3), as it is the most interesting physical situation(one expects the quantum effects and the bounce to occurin a very hot radiation dominated universe). Suppose theclassical limit is already valid at a conformal time wherethe energy density of radiation is minimally greater thanthe energy density before nucleosynthesis, say, the energydensity around the freeze-out of neutrons, �f � 10�88.

Then, from Eqs. (23) and (26), and from �f ¼ PT=a4f,

with af ¼ a0�f=T0, one obtains xf � �f=T0 �1022=ðT0P

1=4T Þ. Using that PT � 10116, one gets that xf

10�7=T0. Hence, xf 1 if and only if T0 � 10�7.

However, as u � 1, then u2T0 � 1, and the exponential in(23) would be irrelevant, turning solution (23) very close tosolution (11) for w ¼ 1=3, taking us back to the previousproblem. Concluding, the only way to obtain a huge PT

with parameters with reasonable probability amplitudes inthis framework is through choices which will change theusual scale factor evolution during nucleosynthesis, spoil-ing its observed predictions.

B. Quantum solutions for large u

If u 1, for jTj not very small, and noting that theunique possible asymptotic behavior of a solution �ðTÞ ofEq. (19) is �ðTÞ / T, then the hyperbolic functions in (19)are very large and much greater than the terms with trigo-nometric functions, yielding

_�ðTÞ ¼ T�ðTÞT2 þ T2

0

� uT20

T2 þ T20

; (27)

with solution

�ðTÞ ¼ �0

T0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2 þ T2

0

q� uT; (28)

where the � sign corresponds to positive and negative

values of T, respectively. For jTj � 0, one has to rely onnumerical calculations. However, as shown in Fig. 1 below,for large u this difference is quite unimportant. Hence,again, �0 is very close to the value of � at the bounce.Inserting solution (28) into Eq. (21) in the limit jTj !

1, we obtain

PT ¼��0

T0

þ u

�2: (29)

Hence, the huge values of PT can be obtained from largevalues of u, without any imposition on the parameters �0

and T0.Let us calculate the constraints on the parameter space in

order to obtain a sensible model. As discussed in Sec. II,one should have a20=T0 1 in order to have a reasonable

probability amplitude for a0, and the curvature scale at thebounce should not be very close to the Planck length inorder to avoid strong quantum gravitational effects. I willalso impose that a0 > 1 in order to avoid trans-Planckianproblems (see below), which implies that a0=T0 1.The curvature scale at the bounce reads

Lbounce ¼ a0T0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0½1þ cosð4ua0Þ�

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0½1þ cosð4ua0Þ� þ uT0½4ua0 þ sinð4ua0Þ�

p

� a0ffiffiffiffiffiT0

p2u

; (30)

where in the last approximation I used that uT0=a0 1,which follows from u 1, a0=T0 1, and I assumed that4ua0 � ð2nþ 1Þ�, in order to avoid Lbounce � 1. Hence,

as a0 T1=20 , then

1.2

1.4

1.6

1.8

2

2.2

–0.1 –0.05 0 0.05 0.1

t

FIG. 1. Figure showing �ðtÞ against t, comparing solution (28)(continuous line) for u ¼ 12, �0 ¼ 1, T0 ¼ 1 with the numericalsolution of Eq. (19) for u ¼ 12, �ð10Þ ¼ 130 (dotted line). Thealmost parallel line corresponds to the bouncing solution for u ¼0, �0 ¼ 1:155.

LARGE CLASSICAL UNIVERSES EMERGING FROM . . . PHYSICAL REVIEW D 79, 083514 (2009)

083514-5

Lbounce T0

2u: (31)

Demanding that Lbounce � 103, then

T0 > u103 1: (32)

One must check again whether one recovers the classicalradiation dominated evolution before nucleosynthesis. Asbefore, I concentrate on the case w ¼ 1=3, which impliesthat � ¼ a and T ¼ �. I will do this in two steps: first Icheck whether solution (28) is valid before nucleosynthe-sis, and then whether quantum effects are negligible there.

The approximation leading to solution (28) requires thatthe argument of the hyperbolic functions in Eq. (19) be

large, �T=T0 1, which implies that x ¼ �=T0 ð4u2T0 � 1Þ�1=2 � 1=ð2uT1=2

0 Þ. Hence, the values of the

conformal time for which solution (28) is reliable are � T1=20 =ð2uÞ.Let us now verify whether solution (28) is valid around

the freeze-out of neutrons, before nucleosynthesis. This

will be true if �f T1=20 =ð2uÞ. However, �f �

1022=P1=4T ¼ 1022=u1=2 which, when combined with �f

T1=20 =ð2uÞ, implies that 1044 T0=ð4uÞ � Lbounce, where I

used Eq. (31). As the curvature scale around freeze-out ofneutrons, Lf, satisfies Lf � 1044, this condition is just the

reasonable constraint that

Lbounce � Lf � 1044: (33)

Hence, if the curvature scale at the bounce is much smallerthan the curvature scale around the freeze-out of neutrons,then solution (28) must be valid at nucleosynthesis period.Finally, one must check that in the regime where solu-

tion (28) is valid we are already in the classical limit, even

though the above condition x 1=ð2uT1=20 Þ may still con-

tain a region were x � 1 because u and T0 are large. To

prove this, note first that at� T1=20 =ð2uÞ, the term u�

T1=20 =2 dominates over a0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ 1

pin Eq. (28), either for

x � 1, because a0 T1=20 , as for x 1, because

a0=T0 � u. Hence, the quantum potential given by

Q :¼ � @2R

4R@�2¼: Q1 þQ2

2; (34)

where

Q1 ¼ �T0f½4T0ð�2 þ u2T2Þ � ðT2 þ T20Þ� coshð�T=T0Þ

� 8uT0T� sinhð�T=T0Þ þ ½4T0ð�2 � u2T20Þ

� ðT2 � T20Þ� cos�þ 8T2

0�u sin�gf2ðT2 þ T20Þ2

� ½coshð�T=T0Þ þ cos��g�1; (35)

and

Q2 ¼ T0

X½coshð�T=T0Þ þ cos�� � u½T sinhð�T=T0Þ � T0 sin��ðT2 þ T2

0Þ½coshð�T=T0Þ þ cos�� ; (36)

reads, around the Bohmian trajectory a � u�,

Q � 1

coshð2�T=T0Þ � 1; (37)

while the kinetic term of the Hamilton-Jacobi-like equa-tion (9) is given by�

@S

2@a

�2 � u2 1: (38)

Note that near the bounce at � � 0, the kinetic term isalmost null, while the quantum potential is finite: quantumeffects are dominant only very near the bounce. The tran-sition from the quantum to classical regime should bearound �T=T0 � 1, where Q � 1 and solution (28) is not

reliable. A little bit later, when x ¼ �=T0 > ð4u2T0 �1Þ�1=2 � 1=ð2uT1=2

0 Þ, and knowing from Eq. (32) that T0 >1061 because u > 1058, we obtain that the scale factor at thebeginning of the classical regime is a > 1031 which is theminimum value required for the model to reach the size ofthe observed universe without needing any classical infla-tionary phase afterwards. Note from Fig. 1 that the pres-ence of the u term in Eq. (28) induces a much bigger

acceleration at the bounce in comparison with the solutionwith the same a0 but without the u term. It is this termwhich is responsible for the big value a > 1031 when themodel enters the classical regime.Concluding, solution (28) reaches the standard cosmo-

logical model before nucleosynthesis and can indeed de-scribe the observed universe in the radiation dominatedphase with parameters with reasonable relativeprobabilities.In order to avoid the trans-Planckian problem for the

scales of physical interest today, 1054 < �physicaltoday < 1060,

one should have �physicalbounce corresponding to these scales not

smaller than, say, 103. As

�physicalbounce ¼ a0

atoday�physicaltoday ; (39)

this problem can be avoided if

a0atoday

> 10�51; (40)

which implies that a0 > 109.

NELSON PINTO-NETO PHYSICAL REVIEW D 79, 083514 (2009)

083514-6

Note that there is an upper limit for a0 coming from

Lbounce � a0T1=20

2u� 1064

�a0

atoday

�2; (41)

where I used that a20=T0 1, PT � u2 � a4today10�128. The

constraint Lbounce � 1044 (see Eq. (33)) then implies that

a0atoday

� 10�10: (42)

Hence, there is a large domain of values of a0 where thetrans-Planckian problem can be avoided (see Eqs. (40) and(42)).

IV. CONCLUSION

I have shown in this paper how a sufficiently big uni-verse can emerge from a quantum cosmological bounce,without needing any classical inflationary phase afterwardsto make it grow to its present size. This is caused by a hugeacceleration during the quantum bounce, which may beviewed as a quantum inflation. These results were obtainedfrom a moving Gaussian function of the scale factor, whichis a solution of the Wheeler-DeWitt equation coming fromthe canonical quantization of general relativity sourced byrelativistic particles. The solution is exact, there is noWKB approximation involved here. Its value at T ¼ 0yields reasonable relative probability amplitudes of havingthe scale factor at the bounce with the value a0 required toavoid any trans-Planckian problem, and to allow that thecurvature scale at the bounce be some few orders ofmagnitude greater than the Planck length, a region whereone can rely on this simple quantization scheme. In fact,since the maximum value the curvature scale can have is atthe bounce itself, one never reaches energy scales wheremore involved quantum gravity theories, like string theoryand loop quantum gravity (see Ref. [17] about issuesconcerning this approach), must be invoked: the model isself-contained.

There are two internal parameters of the wave functionwhich must be big in order to obtain a large classicaluniverse from a quantum bounce. The first one is theparameter T0, the square root of the width of theGaussian at the moment of the bounce (see Eq. (17)),which must satisfy T0 > 1061 (in Planck unities). Thisvalue yields the sufficient large value a > 1031 for the scalefactor in the beginning of the classical regime and guaran-tees that the curvature scale at the bounce be some mini-mum orders of magnitude greater than the Planck length inorder for the Wheeler-DeWitt equation I used to be reli-able. The other one is the velocity u of the Gaussian alongthe scale factor axis which must satisfy u > 1058 in order toyield the amount of radiation we observe in the universe

today, without appealing to some huge production of pho-tons during the bounce.From these considerations, one can see that the parame-

ters emerging from the quantum era of the universe are notnecessarily Planckian;: they depend also on the quantumstate of the system, on the internal parameters of the wavefunction of the universe. Hence, it is not surprising that onemay have quantum gravity effects in large (when comparedwith the Planck length) universes [18], which could bedramatically seen in a big rip [19].However, one may ask why the internal parameters of

the wave function we obtained are so large. Note first thatthese are not coupling constants, but parameters in thequantum state of the universe. Hence, to answer this ques-tion, one should rely on some deep understanding ofquantum cosmology and/or new principles which are notavailable today. Note, however, that the big value of T0

leads to a widely spread Gaussian, and hence almost allscales at the bounce are equally probable. This is a rea-sonable assumption about the wave function of the uni-verse one can make: it should not intrinsically select anypreferable scale at the bounce without any special reason.Concerning the u variable, its large value implies that thepeak of the initial wave packet moves very fast towardslarge scale factors, which induces a large universe. Perhapssome version of the anthropic principle could justify thepreference for large classical universes, and as a conse-quence for a large u, but I think the important message hereis the possibility of obtaining a large universe from a hugeacceleration of the scale factor in the far past, whose origindiffers fundamentally from those considered in usual infla-tionary scenarios.In future publications, we will calculate the evolution of

linear quantum perturbations and particle production onthese quantum backgrounds, as in Ref. [14], and comparethe results with observations.As a final remark, I would like to repeat a comment we

made elsewhere [4]: in contradistinction with models inwhich time begins, there is no point on asking what is theprobability of appearance of some particular eternal modelout of nothing. Contrary to usual perspectives, one can aswell assume existence to be conceptually prior to nonexis-tence, i.e. existence itself may not be deserving explana-tion. This is the idea underlying our category of models:the universe always existed and its ‘‘appearance’’ is thus anonquestion.

ACKNOWLEDGMENTS

I would like to thank CNPq of Brazil for financialsupport. I very gratefully acknowledge various enlighten-ing conversations with Felipe Tovar Falciano, PatrickPeter, Emanuel Pinho, and specially Andrei Linde, whosecriticisms inspired this work. I also would like to thankCAPES (Brazil) and COFECUB (France) for partial finan-cial support.

LARGE CLASSICAL UNIVERSES EMERGING FROM . . . PHYSICAL REVIEW D 79, 083514 (2009)

083514-7

[1] S.W. Hawking and G. F. R. Ellis, The large ScaleStructure of Space-time (Cambridge University Press,Cambridge, England, 1973); R.M. Wald, GeneralRelativity (University of Chicago, Chicago, 1984); A.Borde and A. Vilenkin, Phys. Rev. D 56, 717 (1997).

[2] M. Novello and S. E. Perez Bergliaffa, Phys. Rep. 463,127 (2008).

[3] R. C. Tolman, Phys. Rev. 38, 1758 (1931); G. Murphy,Phys. Rev. D 8, 4231 (1973); M. Novello and J.M. Salim,Phys. Rev. D 20, 377 (1979); V. Melnikov and S. Orlov,Phys. Lett. A 70, 263 (1979); E. Elbaz, M. Novello, J.M.Salim, and L.A. R. Oliveira, Int. J. Mod. Phys. D 1, 641(1992); J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N.Turok, Phys. Rev. D 64, 123522 (2001); V. A. De Lorenci,R. Klippert, M. Novello, and J.M. Salim, Phys. Rev. D 65,063501 (2002); J. C. Fabris, R. G. Furtado, P. Peter, and N.Pinto-Neto, Phys. Rev. D 67, 124003 (2003).

[4] P. Peter and N. Pinto-Neto, Phys. Rev. D 78, 063506(2008).

[5] R. Brandenberger and F. Finelli, J. High Energy Phys. 11(2001) 056; P. Peter and N. Pinto-Neto, Phys. Rev. D 66,063509 (2002); F. Finelli, J. Cosmol. Astropart. Phys. 10(2003) 011; V. Bozza and G. Veneziano, Phys. Lett. B 625,177 (2005); J. Cosmol. Astropart. Phys. 09 (2005) 007; F.Finelli and R. Brandenberger, Phys. Rev. D 65, 103522(2002); S. Tsujikawa, R. Brandenberger, and F. Finelli,Phys. Rev. D 66, 083513 (2002); P. Peter, N. Pinto-Neto,and D.A. Gonzalez, J. Cosmol. Astropart. Phys. 12 (2003)003; J. Martin and P. Peter, Phys. Rev. Lett. 92, 061301(2004); J. Martin, P. Peter, N. Pinto-Neto, and D. J.Schwarz, Phys. Rev. D 65, 123513 (2002); 67, 028301(2003); C. Cartier, R. Durrer, and E. J. Copeland, Phys.Rev. D 67, 103517 (2003); L. E. Allen and D. Wands,Phys. Rev. D 70, 063515 (2004); T. J. Battefeld and G.Geshnizjani, Phys. Rev. D 73, 064013 (2006); A. Cardosoand D. Wands, Phys. Rev. D 77, 123538 (2008); Y. Cai, T.Qiu, R. Brandenberger, Y. Piao, and X. Zhang, J. Cosmol.Astropart. Phys. 03 (2008) 013.

[6] M. Bojowald, Phys. Rev. Lett. 86, 5227 (2001); Phys. Rev.D 64, 084018 (2001); A. Ashtekar, M. Bojowald, and J.Lewandowski, Adv. Theor. Math. Phys. 7, 233 (2003).

[7] R. Colistete, Jr., J. C. Fabris, and N. Pinto-Neto, Phys. Rev.D 62, 083507 (2000); N. Pinto-Neto, E. Sergio Santini,and Felipe T. Falciano, Phys. Lett. A 344, 131 (2005).

[8] F. G. Alvarenga, J. C. Fabris, N. A. Lemos, and G.A.Monerat, Gen. Relativ. Gravit. 34, 651 (2002); J. Acaciode Barros, N. Pinto-Neto, and M.A. Sagioro-Leal, Phys.Lett. A 241, 229 (1998).

[9] D. Bohm, Phys. Rev. 85, 166 (1952); D. Bohm, B. J. Hiley,and P.N. Kaloyerou, Phys. Rep. 144, 321 (1987); P. R.Holland, The Quantum Theory of Motion: An Account ofthe de Broglie-Bohm Causal Interpretation of QuantumMechanichs (Cambridge University Press, Cambridge,England, 1993).

[10] The Many-Worlds Interpretation of Quantum Mechanics,edited by B. S. DeWitt and N. Graham (PrincetonUniversity, Princeton, NJ, 1973).

[11] P. Peter, E. Pinho, and N. Pinto-Neto, J. Cosmol.Astropart. Phys. 07 (2005) 014.

[12] P. Peter, E. J. C. Pinho, and N. Pinto-Neto, Phys. Rev. D73, 104017 (2006).

[13] E. J. C. Pinho and N. Pinto-Neto, Phys. Rev. D 76, 023506(2007).

[14] P. Peter, E. J. C. Pinho, and N. Pinto-Neto, Phys. Rev. D75, 023516 (2007).

[15] J. Martin and R.H. Brandenberger, Phys. Rev. D 63,123501 (2001); R. H. Brandenberger and J. Martin,Mod. Phys. Lett. A 16, 999 (2001); J. C. Niemeyer,Phys. Rev. D 63, 123502 (2001); M. Lemoine, M. Lubo,J. Martin, and J. P. Uzan, Phys. Rev. D 65, 023510 (2001).

[16] V. G. Lapshinskii and V.A. Rubakov, Theor. Math. Phys.33, 1076 (1977); F. J. Tipler, Phys. Rep. 137, 231 (1986).

[17] H. Nicolai, K. Peeters, and M. Zamaklar, ClassicalQuantum Gravity 22, R193 (2005); T. Thiemann, Lect.Notes Phys. 721, 185 (2007).

[18] A. Linde, J. Cosmol. Astropart. Phys. 10 (2004) 004; J. J.Halliwell and J. B. Hartle, Phys. Rev. D 41, 1815 (1990);N. Pinto-Neto and E. S. Santini, Phys. Lett. A 315, 36(2003).

[19] E.M. Barboza, Jr. and N.A. Lemos, Gen. Relativ. Gravit.38, 1609 (2006); M. P. Dabrowski, C. Kiefer, and B.Sandhofer, Phys. Rev. D 74, 044022 (2006).

NELSON PINTO-NETO PHYSICAL REVIEW D 79, 083514 (2009)

083514-8