Horava-Lifshitz quantum cosmology

Orfeu Bertolami*

Departamento de Fısica e Astronomia, Faculdade de Ciencias da Universidade do Porto,Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Carlos A.D. Zarro†

Centro de Fısica do Porto, Departamento de Fısica e Astronomia, Faculdade de Ciencias da Universidade do Porto,Rua do Campo Alegre 687, 4169-007 Porto, Portugal(Received 3 June 2011; published 18 August 2011)

In this work, a minisuperspace model for the projectable Horava-Lifshitz gravity without the detailed-

balance condition is investigated. The Wheeler-DeWitt equation is derived and its solutions are studied

and discussed for some particular cases where, due to Horava-Lifshitz gravity, there is a ‘‘potential

barrier’’ nearby a ¼ 0. For a vanishing cosmological constant, a normalizable wave function of the

Universe is found. When the cosmological constant is nonvanishing, the WKB method is used to obtain

solutions for the wave function of the Universe. Using the Hamilton-Jacobi equation, one discusses how

the transition from quantum to classical regime occurs and, for the case of a positive cosmological

constant, the scale factor is shown to grow exponentially, hence recovering the general relativity behavior

for the late Universe.

DOI: 10.1103/PhysRevD.84.044042 PACS numbers: 04.60.Ds

I. INTRODUCTION

Horava-Lifshitz (HL) gravity is a quite original proposalfor an ultraviolet (UV) completion of general relativity(GR) [1], in which gravity turns out to be power-countablerenormalizable at the UV fixed point. GR is supposed to berecovered at the infrared (IR) fixed point, as the theorygoes from high-energy scales to low-energy scales. Inorder to obtain a renormalizable gravity theory one aban-dons Lorentz symmetry at high energies [1,2]. Even thoughthe idea that the Lorentz symmetry is a low-energy sym-metry has been previously considered [3], the novelty ofthe HL proposal is that the breaking of Lorentz symmetryoccurs the very way as in some condensed matter models(cf. Ref. [1] and references therein), that is through ananisotropic scaling between space and time, namely,~r ! b~r and t ! bzt, with b a scale parameter. The dynami-cal critical exponent z is chosen in order to ensure thatthe gravitational coupling constant is dimensionless,which makes possible a renormalizable interaction. Asthe Lorentz symmetry is recovered at the IR fixed point,z flows to z ¼ 1 in this limit.

The anisotropy between space and time leads rathernaturally to the well-known 3þ 1 Arnowitt-Deser-Misner (ADM) splitting [4], originally devised to expressGR in a Hamiltonian formulation. Following Ref. [1], afoliation, parametrized by a global time t, is introduced.Since the global diffeomorphism is not valid anymore, oneimposes a weaker form of this symmetry, the so-called

foliation-preserving diffeomorphism. Choosing this ap-proach, the lapse ADM function, N, is constrained to bea function only of the time coordinate, i.e. N ¼ NðtÞ. Thisassumption satisfies the projectability condition [1]. Inorder to match GR, one could also choose N ¼ Nð~r; tÞ, amodel dubbed nonprojectable and which has been inves-tigated in Refs. [5,6]. The next step involves getting agravitational Lagrangian into this anisotropic scenario.For this purpose, the effective field theory formalism isused: Every term that is marginal or relevant at the UVfixed point (z � 1) is included and, at the IR fixed point,only the z ¼ 1 terms survive. GR is then presumablyrecovered. The number of terms that must be includedsplits HL gravity into two classes, depending on whetherone adopts the detailed-balance condition or not. It isargued in Ref. [1] that if one allows every relevant termto be included into the Lagrangian, the number of couplingconstants would be so large that any analysis wouldbecome impracticable. The detailed-balance condition isinspired by nonequilibrium thermodynamics [7] and,loosely speaking, it states that the potential terms of aD-dimensional action are obtained using a (D� 1)-dimensional function, the superpotential. It is argued thatalthough detailed balance is a simplifying assumption, it isby no means a necessary one [8,9]. It is shown that the listof allowed terms is not so large after all and the detailed-balance Lagrangian is obtained after the proper choice ofcoefficients.A common problem plaguing all HL versions is the

presence of a scalar field mode, which has a trans-

Bogoliubov dispersion relation with ~k6 term [10,11]. Thisscalar mode appears, likewise a Goldstone boson, afterbreaking a continuous symmetry. To avoid this mode,

*Also at Instituto de Plasmas e Fusao Nuclear, InstitutoSuperior Tecnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal.orfeu.bertolami@fc.up.pt

†carlos.zarro@fc.up.pt

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one has to introduce more symmetries: Besides thefoliation-preserving diffeomorphism, a local Uð1Þ symme-try can be introduced [12] and it is shown that the scalarmode is then gauged away. This version of HL gravity isreferred to as a general covariant, given that the number ofdegrees of freedom matches that of GR. In Ref. [12] it wasargued that this extra symmetry is sufficient to ensure that� ¼ 1; however, it was shown in Ref. [13] that this is not soand that despite this symmetry � � 1 values are possible.For the general covariant version, the detailed-balancecondition was investigated in Ref. [14]. For reviews onthese versions of HL gravity, the reader is referred toRefs. [7,10,11].

In order to verify the consistency of these versions, onehas to check the issues of stability, the presence of ghostmodes, and the strong coupling. The projectable versionwithout detailed balance has either ghosts or instabilitiesfor the extra scalar mode [9,15]. The Minkowski space-time is shown to be unstable [15] and in Ref. [16], thestability of the de Sitter space-time was obtained. InRefs. [17,18], it was shown that there exists a strongcoupling problem. Abandoning the projectable versiondoes not eliminate the scalar mode, but leads to an im-provement of behavior given that for some ranges ofparameters the resulting version is ghost free and stable[5,6]. Nonetheless, despite this somewhat better behavior,the strong coupling problem persists [19]. A possible so-lution to this problem consists of introducing a new energyscale M�, above which the higher order terms are sup-pressed so that M� <Msc, where Msc is the energy scalewhere the coupling gets strong [20]. This method was usedin Ref. [18] to solve the strong coupling problem of theprojectable version and some phenomenological bounds toM� were found [10]. For the generalized covariant versionproposed in Ref. [13], all the mentioned problems aresolved [21], and in Ref. [22] it is shown that by introducinga new energy scale M�, as discussed above, the strongcoupling problem is resolved. Interestingly, the methodof Ref. [20] is not the unique way to circumvent the issueof the strong coupling. It can be shown that the strongcoupling arises when considering a theory with � � 1 andthen taking the GR limit � ! 1 (see, e.g., Refs. [10,23]).This problem is analogous to the one encountered in mas-sive gravity theory, investigated in Ref. [24], where aperturbative approach about zero graviton mass is impos-sible; however one can find a static spherically symmetricsolution that is continuous in the limitm ! 0. This methodwas applied in Ref. [23], which shows the so-called non-perturbative continuity of a static spherically symmetricspace-time in the limit � ! 1.

Cosmological considerations have been extensivelystudied in the context of HL gravity (for reviews, seeRefs. [23,25]). One subtle point that arises in the project-able version concerns the lapse function which, being just afunction of time, implies that the classical Hamiltonian

constraint of GR is no longer local and must be integratedover spatial coordinates. It is shown in Ref. [26] that thisyields an additional term that mimics dust into theFriedmann equations. However, as noted in Ref. [9], theRobertson-Walker metric is homogeneous, so this spatialintegral is simply the spatial volume of the space and hencethis ‘‘dark dust’’ constant must vanish [27]. The presenceof higher spatial curvature terms in HL gravity gives rise toa plethora of new cosmologies that exhibit, for instance,some bouncing and oscillating solutions [28–30]. Aclassification of Friedmann-Lemaıtre-Robertson-Walker(FLRW) cosmologies for HL gravity has been performedin Refs. [27,31]. One should notice that the analysis carriedout in these references is entirely classical and basedon the resulting Friedmann-like and Rauchaudhury-likeequations.Quantum cosmology (QC) is an interesting step toward

the understanding of quantum gravity and the initial con-ditions of the Universe. Its setup consists of splitting space-time, using the ADM formalism and applying well-knownquantum mechanical considerations for constrained sys-tems. The cosmological principle is evoked so that space-time is foliated in leaves with a constant global time. Toimplement the quantum scenario, one promotes theHamiltonian constraint H ¼ 0 to an operator equation,

the Wheeler-DeWitt (WDW) equation, Hc ¼ 0, wherec is the wave function of the Universe [32]. The WDWequation is a hyperbolic equation on the space of all3-metrics, the so-called superspace. Its complexity makesthe task of obtaining solutions a formidable one. To dealwith this equation, one often considers simpler spaces,such as for instance the FRLW space-times, which leadsto a minisuperspace model, where the number of degreesof freedom is considerably reduced from infinite (any3-metrics) to 1 (the scale factor) [33]. Despite their relativesimplicity the minisuperspace models are not completelyfree from problems. Indeed, one can point out, for instance,the fact that the wave function of the Universe is not inmany cases normalizable, which implies that the usualinterpretation of quantum mechanics cannot be used.However, in the context of some particular models normal-izable wave functions have been found and discussed[34–36]. For, comprehensive reviews, see e.g. Refs. [37,38].We argue that quantum cosmology allows for a valuable

insight of HL gravity in the quantum context. In bothapproaches, one foliates the space-time in constant globaltime leaves, a procedure that automatically satisfies theprojectability condition. But when adopting the QC for-malism in HL gravity, one faces the problem of turning theHamiltonian constraint into the WDW equation, since theHamiltonian constraint in the HL gravity is not local.Nevertheless, choosing a FLRW metric minisuperspacemodel or, more generally, a spatially homogeneous cos-mological metric, one can argue that the spatial integrationyields a local constraint. Notice that another suitable

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feature of HL gravity is that it does not introduce higherthan first order time derivatives of the scale factor on theaction, making the quantization procedure straightforwardas a mixture between time and spatial derivatives andpowers of momentum are not found. Indeed, the kineticpart of the Hamiltonian has the same structure as the one inthe usual QC, and HL gravity introduces only higherspatial derivative terms, which dominate on the very smallscales. Notice that the problem of high order time deriva-tives imposes severe obstacles for applying QC on highorder derivative gravities and string theory, but not for HLgravity. One concludes then that the minisuperspace modelcan be naturally implemented in the HL gravity proposal.In the minisuperspace model, the WDW equation for theHL gravity was obtained in Ref. [39]; however in there theinterest was on the cosmological constant problem andthe HLWDWequation was neither discussed nor solutionspresented.

In this work, one investigates the projectable HL gravitywithout detailed balance in the context of the minisuper-space model of quantum cosmology for a FLRW universewithout matter. This particular choice, despite being muchsimpler than the nonprojectable version [5,6] and the gen-eral covariant approach [12], exhibits the main features ofthe HL gravity and contains the detailed balance as alimiting case. A matter sector is not introduced, giventhat the main interest in the very early Universe, wherethe HL new terms dominate and for the late Universe, anepoch dominated by the cosmological constant. Moreover,the inclusion of the matter sector and how it is coupled toHL gravity remains an open question [10].

This paper is organized as follows: In Sec. II, the min-isuperspace HL is presented and the WDW equation isobtained. In Sec. III, the solutions of the WDW are foundand discussed. In Sec. IV, the wave function of a HL isinterpreted and an analysis of the Hamilton-Jacobiequation is performed. Finally, concluding remarks arepresented in Sec. V.

II. THE WHEELER-DEWITT EQUATION

A. Metric

One considers the Robertson-Walker (RW) metric withR� S3 topology

ds2 ¼ �2ð�NðtÞ2dt2 þ a2�ijdxidxjÞ; (1)

where i, j ¼ 1, 2, 3, �2 is a normalization constant, NðtÞis the lapse function, and �ij is the metric of the unit

3-sphere. Its metric is given by

�ij ¼ diag

�1

1� r2; r2; r2sin2�

�:

The extrinsic curvature takes the form:

Kij ¼ 1

2�N

��@gij

@tþriNj þrjNi

�; (2)

where Ni is the ADM shift vector and ri denotes the3-dimensional covariant derivative. As Ni ¼ 0 forRW-like spaces in study,

Kij ¼ � 1

�N

_a

agij: (3)

Taking the trace one gets

K ¼ Kijgij ¼ � 3

�N

_a

a: (4)

The Ricci components of the 3-metric can also be obtainedas the foliation is a surface of maximum symmetry

Rij ¼ 2

�2a2gij; (5)

R ¼ 6

�2a2: (6)

B. Horava-Lifshitz action

The action for the projectable HL gravity without de-tailed balance is given by [8,9]:

SHL ¼ M2Pl

2

Zd3xdtN

ffiffiffig

p fKijKij � �K2 � g0M

2Pl � g1R

� g2M�2Pl R

2 � g3M�2Pl RijR

ij � g4M�4Pl R

3

� g5M�4Pl RðRi

jRjiÞ � g6M

�4Pl R

ijR

jkR

ki

� g7M�4Pl Rr2R� g8M

�4Pl riRjkriRjkg; (7)

where gi are coupling constants, MPl is the Planck mass,and ri denote covariant derivatives. The time coordinatecan be rescaled in order to set g1 ¼ �1, recovering the GRvalue. One also defines the cosmological constant � as2� ¼ g0M

2Pl. An important feature of the IR limit is the

presence of the constant � on the kinetic part of the HLaction. GR is recovered provided � ! 1 (corresponding tothe full diffeomorphism invariance); however � must be arunning constant, so there is no reason or symmetry thata priori yields � ¼ 1 GR value. Phenomenological boundssuggest however that the value of � is quite close to the GRvalue [10].Performing these redefinitions the HL action reads

SHL ¼ M2Pl

2

Zd3xdtN

ffiffiffig

p fKijKij � �K2 þ R� 2�

� g2M�2Pl R

2 � g3M�2Pl RijR

ij � g4M�4Pl R

3

� g5M�4Pl RðRi

jRjiÞ � g6M

�4Pl R

ijR

jkR

ki

� g7M�4Pl Rr2R� g8M

�4Pl riRjkriRjkg: (8)

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C. Horava-Lifshitz minisuperspace action

In order to consistently reduce the number of degrees offreedom when restricting the 3-metrics of the superspace tobe isotropic and homogeneous, one can consider that therestriction is performed directly into the equations of mo-tion, or through the substitution of the RW metric into theLagrangian density and then obtain the equations of mo-tion for the remaining degrees of freedom. In general, thephysical content of these two ways is different, showingthat the restriction cannot be done over the Lagrangianunless one properly solves the arising constraints. For theRW metric, without matter fields, these procedures areshown to lead to the same results [40]. Since the HLproposal introduces only an anisotropy between spaceand time, it does not alter the homogeneity of the RWmetric and hence the metric (1) can be substituted intoEq. (8) yielding the HL minisuperspace action

SHL¼M2Pl�2�2�3ð3��1Þ�2

2

�ZdtN

�� _a2a

N2þ 6a

3ð3��1Þ�2��2a3

3ð3��1Þ�M�2

Pl � 12

3ð3��1Þ�2a�ð3g2þg3Þ

�M�4Pl � 24

3ð3��1Þ�4a3�ð9g4þ3g5þg6Þ

�; (9)

where the spatial integralRd3x

ffiffiffiffi�

p ¼ 2�2 has been per-

formed. A further simplification is obtained after choosingunits so to satisfy�2 � 6�2 � ð3�� 1ÞM2

Pl ¼ 1. The min-

isuperspace action then reads

SHL¼1

2

ZdtN

�� _a2a

N2þ 2a

ð3��1Þ��M�2

Pl a3

18�2ð3��1Þ2

�24�2ð3g2þg3Þa

�288�4ð3��1Þð9g4þ3g5þg6Þa3

�:

(10)

Following Ref. [27] the dimensionless coupling con-stants are redefined as

gC ¼ 2

3�� 1; g� ¼ �M�2

Pl

18�2ð3�� 1Þ2 ;

gr ¼ 24�2ð3g2 þ g3Þ;gs ¼ 288�4ð3�� 1Þð9g4 þ 3g5 þ g6Þ:

(11)

Notice that gC > 0, which stands for the curvature cou-pling constant. The sign of g� follows the sign of thecosmological constant. These two terms are alreadypresent in the minisuperspace GR model, but now theydepend on �. The coupling constants gr and gs can beeither positive or negative as their signal does not alterthe stability of the HL gravity (cf. Ref. [27]). As discussedin Ref. [9], physically, gr corresponds to the coupling

constant for the term behaving as a radiaton and gs standsfor the term behaving as ‘‘stiff’’ matter (p ¼ � equation ofstate). The minisuperspace action is finally written as [9]

SHL ¼ 1

2

Zdt

�N

a

����a

N_a

�2 þ gCa

2 � g�a4 � gr � gs

a2

�:

(12)

D. Horava-Lifshitz minisuperspace Hamiltonianand Wheeler-DeWitt equation

The canonical conjugate momentum associated with a isobtained using Eq. (12)

�a ¼ @L@ _a

¼ � a

N_a: (13)

The Horava-Lifshitz minisuperspace Hamiltonian densityis performed using Eqs. (12) and (13)

H ¼ �a _a�L

¼ 1

2

N

a

���2

a � gCa2 þ g�a

4 þ gr þ gsa2

�: (14)

The next step in implementing the quantum cosmologyprogram involves promoting the classical minisuperspaceHamiltonian into an operator on which the so-called wavefunction of the Universe is applied [32,33].This is a subtle point in HL gravity since there is no

global diffeomorphism, just a foliation-preserving diffeo-morphism [1]. This can also be seen as the lapse functionno longer depends on the space-time variables, as in GR,but now it depends only on the global time N ¼ NðtÞ, asdiscussed in Sec. I. This implies that the Hamiltonianconstraint is not local; however this problem can be cir-cumvented for an homogeneous metric, such as Eq. (1), asthe integration over space can be performed as seen above.The canonical quantization is obtained by promoting thecanonical conjugate momentum into an operator, i.e.�a � �i d

da . Because of ambiguities in the operator order-

ing, one chooses �2a ¼ � 1

apdda ðap d

daÞ [33]. The resulting

WDW equation is then obtained�1

apd

da

�ap

d

da

�� gCa

2 þ g�a4 þ gr þ gs

a2

��ðaÞ ¼ 0:

(15)

The choice of p does not modify the semiclassicalanalysis [41]; hence one chooses p ¼ 0, and the WDWequation is written as

�d2

da2� gCa

2 þ g�a4 þ gr þ gs

a2

��ðaÞ ¼ 0: (16)

This equation is similar to the one-dimensionalSchrodinger equation for ℏ ¼ 1 and a particle withm ¼ 1=2 with E ¼ 0 and potential

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VðaÞ ¼ gCa2 � g�a

4 � gr � gsa2

: (17)

E. Horava-Lifshitz minisuperspace potentials

TheWDWequation derived in the last section resemblesan unidimensional Schrodinger equation with potentialgiven by Eq. (17). Classically the allowed regions aresuch that VðaÞ � 0 since E ¼ 0. A complete analysis ofthe phase structure of HL FLRW cosmologies was per-formed in Ref. [27].

Notice that the first two terms of Eq. (17) are the usualGR terms of the quantum cosmology analysis [33]. HLgravity introduces the last two terms which dominate fora � 1, i.e. are relevant at short distances, presumably atthe very early Universe, where the GR description must bereplaced by the quantum gravity one. At the very earlyUniverse, this potential is dominated by the term �gs=a

2,implying that for gs < 0 this potential exhibits a ‘‘barrier’’that might prevent space-time from being singular. Thecase gs > 0 is not examined as it leads to a cosmology thatcannot be suitably investigated using QC techniques.Notice that the detailed-balance condition yields gs ¼ 0.

The choice gs < 0 splits the discussion into three dis-tinct scenarios, for positive, negative, and vanishing cos-mological constants. In what follows one studies the casesensued by these choices for the coefficients of Eqs. (11).

1. The � ¼ 0 case

In this case the curvature term dominates at large dis-tances. The Universe oscillates between a1 and a2. Thepotential, depicted in Fig. 1, is written as

V�¼0ðaÞ ¼ gCa2 � gr � gs

a2: (18)

2. The � � 0 case

For large a, the potential is dominated by the cosmo-logical term, and given by

V��0ðaÞ ¼ gCa2 � g�a

4 � gr � gsa2

: (19)

The sign of g� follows the sign of the cosmological con-stant�. For positive� the potential is depicted in Fig. 2(a)and the negative � case in Fig. 2(b). For a positive cos-mological constant, one considers a potential that has threepositive roots (a1, a2, and a3); hence there are two classi-cally allowed regions for a1 < a< a2 and a3 < a and aforbidden region where a2 < a< a3. There is anotherpossibility, discussed in Refs. [23,27], in which the poten-tial has only one real positive root, namely, a1. The ex-pression for the roots of Eq. (19) can be found in Ref. [42]and will not be presented here as their expressions willnot play any role in what follows. The potential can befactorized as

V�>0ðaÞ ¼ �g�a2

ða2 � a21Þða2 � a22Þða2 � a23Þ: (20)

By the same token, for a negative cosmological constant,one finds a similar behavior already present in the � ¼ 0case: Classically, the Universe oscillates between a1 anda2. This implies that the potential Eq. (19) reads

V�<0ðaÞ ¼ �g�a2

ða2 � a21Þða2 � a22Þða2 þ a2i Þ; (21)

where a ¼ �iai are the imaginary roots of this potentialand ai is real.

III. SOLUTIONS OF THE WHEELER-DEWITTEQUATION

Having described the three types of potentials one en-counters, the task is to solve the WDW equation (16).If � � 0, the cosmological constant term is quartic soEq. (16) cannot be solved in a closed form, and theWKB approximation will be employed.

a1 a2a

V 0 a

FIG. 1 (color online). Potential for � ¼ 0.

a1 a2 a3a

V 0 a

a1 a2a

V 0 a

FIG. 2 (color online). Potentials for nonvanishing cosmological constant.

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A. Boundary conditions

To find suitable boundary conditions for the WDWequation one has to rely on physical assumptions. Themost discussed choices are the DeWitt boundary condition[32], the ‘‘no-boundary’’ proposal [33,37], and the tunnel-ing boundary condition [43].

The DeWitt boundary condition [32] is the one in whichthe wave function of the Universe is required to vanishwherever there is a classical singularity. It is inspired onquantum mechanics, and it is suitable for the cases understudy here, as there is a potential barrier (bounce) fora � 1 yielding that the singularity is inside a classicallyforbidden region (cf. Ref. [34]). The DeWitt boundarycondition is expressed, for FLRW models as

c dWða ¼ 0Þ ¼ 0: (22)

The no-boundary condition, of Hartle and Hawking [33],arises from using the Euclidean path integral formalism. Inthat formalism, the ground state for the wave function ofthe Universe is written as (cf. [44])

c ðaÞ ¼ZC½da� expð�IÞ; (23)

where C denotes the integral taken over compact mani-folds, and I is the Euclidean version of the action definedin Sec. II such that the corresponding Euclidean actionI ¼ �iSHL can be obtained from Eq. (12) using d� ¼ iNdtand the N ¼ 1 gauge:

I ¼ �iSHL

¼ 1

2

Zd�

��a

�da

d�

�2 � gCaþ g�a

3 þ graþ gs

a3

�; (24)

where � is the Euclidean time. It is possible to evaluatec ðaÞ nearby a ¼ 0 [44,45]. In this case, it can be proventhat for � � 1 (close to a ¼ 0), one has da

d� ¼ 1 [37].

Substituting these conditions into the Euclidean versionof action (24), and integrating from 0 to ��, one finds

I ¼ 1

2

Zd�

��ð1þ gCÞ�þ g��

3 þ gr�þ gs

�3

�; (25)

which is I ! �1, for gr � 0 and gs � 0 yielding a di-vergent wave function. This shows that the no-boundarycondition is not suitable for the problem under study.

It is important to notice that the boundary conditionc ð0Þ ¼ 0 does not mean that there is a quantum avoidanceof the classical singularity given that it is a sufficient butnot a necessary condition [34–36]. In the above references,some examples are given where c ! 0, but

Rdajc ðaÞj2

diverges, and conversely cases whereRdajc ðaÞj2 ! 0,

but c diverges. The conditions under which the classicalsingularity is removed or avoided by quantum mechanicsare understood only in specific cases (cf. [34,36] andreferences therein).

B. WDW equation solution for a � 1

This region corresponds to the very early Universe,where the HL terms dominate. This HL epoch is expectedsince any theory of quantum gravity is supposed to alter theGR description of the structure of the space-time at smalldistances. For this case, the WDW Eq. (16) reads�

d2

da2þ gs

a2

�c ðaÞ ¼ 0; (26)

which is an Euler equation whose solution is c ðaÞ ¼ a�

and � ¼ 12 � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4gs

p. As the solution must be real

and c ð0Þ ¼ 0, one finds that the wave function fora � 1 goes as

c ðaÞ � a1=2þ1=2ffiffiffiffiffiffiffiffiffiffi1�4gs

p; (27)

which yields gs � 1=4, a ‘‘quantum’’ bound for gs. Noticethat this coefficient is unconstrained by classical consid-eration [27]. Although the potential for gs > 0 correspondsto an infinite well, this quantum bound gives rise to a mildsingularity, which admits a well-defined mathematicaltreatment (cf. Ref. [46]).

C. WDW equation solutions for a 1

This limit corresponds to the very late Universe, which isdominated by the curvature and cosmological constantterms. These terms are already present in the usual GRquantum cosmology setup [33,37], reflecting the fact thatGR behavior is recovered at large distances. For � ¼ 0,Eq. (16) for a 1 is given by

�d2

da2� gCa

2

�c ðaÞ ¼ 0; (28)

which has the following asymptotic solution:

c ðaÞ � e�ð ffiffiffiffigC

p=2Þa2 : (29)

Thus, as expected, the wave function has an exponentialbehavior, since a 1 corresponds to a classically forbid-den region for the potential Eq. (18).For the positive cosmological constant case (g� > 0),

the WDW Eq. (16) reads�d2

da2þ g�a

4

�c ðaÞ ¼ 0; (30)

whose asymptotic solution is given by a combinationof Bessel and Neumann functions with ¼ 1=6[cf. Eq. 8.491.7) of Ref. [47]]. Since the limit a 1 isbeing considered and the Neumann functions only divergeat a ¼ 0, these two functions are admissible:

c ðaÞ ¼ �C1

ffiffiffia

pJ1=6

� ffiffiffiffiffiffig�

p3

a3�þ �C2

ffiffiffia

pN1=6

� ffiffiffiffiffiffig�

p3

a3�;

(31)

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where �C1 and �C2 are constants. A further analysis showsthat the asymptotic expansions for Bessel and Neumannfunctions of any order () and large arguments (jzj ! 1)are given by [cf. Eqs. (9.2.1) and (9.2.2) of Ref. [42]]

JðzÞ ffiffiffiffiffiffi2

�z

scos

�z� �

2� �

4

�;

NðzÞ ffiffiffiffiffiffi2

�z

ssin

�z� �

2� �

4

�:

(32)

Substituting these asymptotic expressions into Eq. (31)one finds the asymptotic behavior of the wave function forlarge a:

c ðaÞ ¼ C1

acos

� ffiffiffiffiffiffig�

p3

a3 � �

12� �

4

�

þ C2

asin

� ffiffiffiffiffiffig�

p3

a3 � �

12� �

4

�; (33)

where Ci ¼ �Ci

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6=�

ffiffiffiffiffiffig�

pq, for i ¼ 1, 2. Notice that this

wave function is oscillatory, denoting that this is a classi-cally allowed region and damped as jc ðaÞj2 � a�2. Notsurprisingly, the same behavior is found in Ref. [33] whenthe cosmological constant dominates the evolution ofthe Universe, and the GR regime is recovered. This issuewill be discussed in Sec. IV. Interestingly, the WKBmethod yields the same asymptotic expression given byEq. (31) [36,46].

For �< 0 ) g� < 0, Eq. (23) is written as�d2

da2� ð�g�Þa4

�c ðaÞ ¼ 0; (34)

whose asymptotic solution is a combination of the modi-fied Bessel functions, IðzÞ and KðzÞ [cf. Eq. (8.406) ofRef. [47]] of order ¼ 1=6. However, IðzÞ grows expo-nentially as z ! 1 [cf. Eq. (36)], hence only KðzÞ repre-sents an acceptable solution for large a. The wave functionin that limit is given by

c ðaÞ � ffiffiffia

pK1=6

� ffiffiffiffiffiffiffiffiffiffi�g�p

3a3�: (35)

As above, asymptotic expansions for modified Bessel func-tions of any order () and large arguments (jzj ! 1) areobtained using Eqs. (9.7.1) and (9.7.2) of Ref. [42]:

IðzÞ ezffiffiffiffiffiffiffiffiffi2�z

p ; KðzÞ ffiffiffiffiffi�

2z

re�z: (36)

One then gets the wave function for the very late Universe

c ðaÞ � 1

aeð�

ffiffiffiffiffiffiffiffi�g�p

=3Þa3 : (37)

Thus, one concludes that for a 1 the wave function is, asexpected, strongly suppressed in this limit, given that thisregion is classically forbidden.

D. WDW equation solution for � ¼ 0

If the cosmological constant vanishes theWDWEq. (16)reads1 �

d2

da2� gCa

2 þ gr þ gsa2

��ðaÞ ¼ 0: (38)

After a change of variables, x ¼ g1=4C a, Eq. (38) reads�d2

dx2� x2 þ gr

g1=2C

þ gsx2

��ðxÞ ¼ 0: (39)

This equation can be exactly solved in terms of theassociate Laguerre functions. Indeed, the following differ-ential equation [cf. Eq. (22.6.18) of Ref. [42]]:

�d2

dx2þ 4nþ 2þ 2� x2 þ 1� 42

4x2

�yðxÞ ¼ 0; (40)

has the solution of yðxÞ ¼ e�x2=2xþ1=2LðÞn ðx2Þ. Here n is a

positive integer, and LðÞn are associate Laguerre functions.

Comparing Eqs. (39) and (40), one finds that

¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4gs

p;

grffiffiffiffiffiffigC

p ¼ 4nþ 2þ 2;

c ðaÞ ¼ Ne�ð ffiffiffiffigC

pa2=2Þðg1=4C aÞþ1=2LðÞ

n ð ffiffiffiffiffiffigC

pa2Þ;

(41)

where N is a normalization constant to be obtained below.As gs < 0, > 0, the wave function c ð0Þ is regular.Comparing with Eq. (27), one also finds that the ratiogr=

ffiffiffiffiffiffigC

pmust be quantized. This is not surprising given

that one is solving the Schrodinger equation with E ¼ 0,which for a bounded potential has a discrete spectrum.Thus, E ¼ 0 is an eigenvalue only for specific values ofthe coefficients and these values must be quantized.Another interesting feature of this solution is that it isnormalizable. Indeed, using Eq. (41) and Eq. (8.980) ofRef. [47], one obtains for the normalization condition that

N ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2n!g1=4C

�ðnþ þ 1Þ

vuut:

The complete solution, for the � ¼ 0 WDW equationsatisfying the c ð0Þ ¼ 0 boundary condition, is given by

c ðaÞ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2n!g1=4C

�ðnþþ1Þ

vuute�ð ffiffiffiffi

gCp

a2=2Þðg1=4C aÞþ1=2LðÞn ð ffiffiffiffiffiffi

gCp

a2Þ:(42)

This wave function behaves as Eq. (27) for a � 1 and asEq. (29) for a 1. The Laguerre associated function L

n is

1In a quantum mechanical context, this equation was solved inproblem 4 of Sec. 36 of Ref. [48].

HORAVA-LIFSHITZ QUANTUM COSMOLOGY PHYSICAL REVIEW D 84, 044042 (2011)

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an nth order polynomial which yields that the wave func-tion has n nodes. It is not difficult to verify that for a fixedgs, any gr obtained through the quantization conditionEq. (41) implies that the potential Eq. (18) has twopositive real roots and so the Universe is oscillating what-ever value of n � 0 is chosen. For large gr values, whichimplies that n is large, the very structure of the wavefunction shows that the Universe is almost classical [thiscan be also seen by inspection of the probability densityplots for large n as shown in Fig. 3(d)]. This result can beunderstood in terms of Bohr’s correspondence principle,according to which the classical behavior is obtained fromthe quantum one in the limit of large quantum numbers.The probability density distribution for the wave functionof the Universe, jc ðaÞj2, for some n values are plotted inFig. 3. One clearly sees that the solution is highly sup-pressed in the classically forbidden region, and it is oscil-lating with n nodes in the classically allowed region.Finally, it is straightforward to show that the singularityis avoided in this model given that the probability to findthe Universe at a ¼ 0 vanishes due to the HL gravityterms.

E. WDW solution for � > 0

If g� � 0, Eq. (16) cannot be analitically solved. Thebehavior of the wave function for large a and nearby thesingularity a ¼ 0 were already discussed. For the inter-mediate regions where the curvature term starts to becomerelevant, after the HL epoch (very early Universe), one hasto rely on the WKB approximation, which for the classi-cally allowed region is given by [48]

c ðaÞ 1

jVðaÞj1=4 exp

��i

Z a

a1

ffiffiffiffiffiffiffiffiffiffiffiffiffijVðaÞj

qda

�; (43)

where the � symbol denotes that one must consider acombination of these two exponentials, VðaÞ is the poten-tial Eq. (20) and a1 is the classical turning point, i.e.Vða1Þ ¼ 0. The following integral must be solved:

Z a

a1

ffiffiffiffiffiffiffiffiffiffiffiffiffijVðaÞj

qda¼ ffiffiffiffiffiffi

g�p Z a

a1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2�a21Þða22�a2Þða23�a2Þ

qa

da:

(44)

One uses that VðaÞ � 0 for the classically allowed regionand, hence jVðaÞj ¼ �VðaÞ. This integral is valid fora1 < a< a2 < a3. Changing the variables to t ¼ a2 andrationalizing the square root, one finds

Z a

a1

ffiffiffiffiffiffiffiffiffiffiffiffiffijVðaÞj

qda ¼

ffiffiffiffiffiffig�

p2

Z a2

a21

ðt� a21Þða22 � tÞða23 � tÞt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� a21Þða22 � tÞða23 � tÞ

q dt:

(45)

This integral can be written as a sum of elliptic integrals[47,49]. Using the formulas (3.131.3), (3.132.2), and(3.137.3) of Ref. [47] and the reduction formula (230.01)of Ref. [49] one finds after a rather long, although straight-forward computation

a

n 0

a 2

an 1

a 2

a

n 2

a 2

a

n 10

a 2

FIG. 3 (color online). Probability density for the wave function of the Universe for diferent values of n. The dashed line plotsrepresent the potential Eq. (18).

ORFEU BERTOLAMI AND CARLOS A.D. ZARRO PHYSICAL REVIEW D 84, 044042 (2011)

044042-8

Z a

a1

ffiffiffiffiffiffiffiffiffiffiffiffiffijVðaÞj

qda ¼

ffiffiffiffiffiffig�

p3

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 � a21Þða22 � a2Þða23 � a2Þ

q

� ða21 þ a22 þ a23Þ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � a21

q� Eð�; qÞ

þ�2a21a

22 þ a21a

23 þ a22a

23 � a43ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a23 � a21

q�Fð�; qÞ

� 3a22a23ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a23 � a21

q �

��;

a21 � a22a21

; q

��; (46)

where

� ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � a21a22 � a21

s; q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22 � a21a23 � a21

s:

Fð’; kÞ is an elliptic integral of the first kind, Eð’; kÞ is anelliptic integral of the second kind, and �ð’; n; kÞ is theelliptic integral of the third kind (cf. [47]). SubstitutingEqs. (20) and (46) into Eq. (43) one finds the WKB wavefunction of the Universe. This approximation is valid only

if dVðaÞda � jVðaÞj3=2 [48].For a positive cosmological constant, the classically

forbidden region, a1 < a2 < a< a3, has the followingWKB wave function:

c ðaÞ ¼ C1

jVðaÞj1=4 exp

�Z a

a2

ffiffiffiffiffiffiffiffiffiffiffiffiffijVðaÞj

qda

�

þ C2

jVðaÞj1=4 exp

��Z a

a2

ffiffiffiffiffiffiffiffiffiffiffiffiffijVðaÞj

qda

�; (47)

and one has to solve

Z a

a2

ffiffiffiffiffiffiffiffiffiffiVðaÞp

da¼ ffiffiffiffiffiffig�

p Z a

a2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2�a21Þða2�a22Þða23�a2Þ

qa

da:

(48)

Following the same steps as before, this integral can bewritten as

Z a

a2

ffiffiffiffiffiffiffiffiffiffiVðaÞ

pda ¼

ffiffiffiffiffiffig�

p2

Z a2

a22

ðt� a21Þðt� a22Þða23 � tÞt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� a21Þðt� a22Þða23 � tÞ

q dt;

(49)

which can be solved using the formulas (3.131.5),(3.132.4), and (3.137.5) of [47] and the reduction formula(230.01) of [49]. One gets

Z a

a2

ffiffiffiffiffiffiffiffiffiffiVðaÞ

pda ¼

ffiffiffiffiffiffig�

p3

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 � a21Þða2 � a22Þða23 � a2Þ

q

� ða21 þ a22 þ a23Þða22 � a21Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � a21

q �ð�; p2; pÞ

þ�a41 � a21a

22 � a21a

23 þ a22a

23ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a23 � a21

q�Fð�; pÞ

þ 3a23ða22 � a21Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � a21

q �

��;

p2a21a22

; p

��; (50)

where

� ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða23 � a21Þða2 � a22Þða23 � a22Þða2 � a21Þ

s

and

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � a22a23 � a21

s:

Substituting Eqs. (20) and (50) into Eq. (47), one finds theWKB wave function for a1 < a2 < a< a3.

F. WDW solution for � < 0

One is interested in the WKB wave function for theclassically allowed region a1 < a< a2. The WKB wavefunction is given by Eq. (43). The steps are the very onesof the above procedure, however, following the discussionof Sec. II, one must consider that the smaller root (a2) isnegative and real [cf. Eq. (21)]. Using the formulas(3.131.5), (3.132.4), and (3.137.5) of Ref. [47] and thereduction formula (230.01) of Ref. [49], one gets

Z a

a1

ffiffiffiffiffiffiffiffiffiffiffiffiffijVðaÞj

qda¼

ffiffiffiffiffiffiffiffiffiffi�g�p

3

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða22�a2Þða2�a21Þða2þa2i Þ

q

�ða21þa22�a2i Þða21þa2i Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22�a2i

q �ð�;p2;pÞ

þ�a4i þa2i a

22þa2i a

22�5a22a

21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a22þa2i

q�Fð�;pÞ

�3a22ða21þa2i Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia22þa2i

q �

��;�p2a2ia21

;p

��; (51)

where

� ¼ arcsin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða23 � a21Þða2 � a22Þða23 � a22Þða2 � a21Þ

s

and

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia23 � a22a23 � a21

s:

HORAVA-LIFSHITZ QUANTUM COSMOLOGY PHYSICAL REVIEW D 84, 044042 (2011)

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The WKB wave equation is obtained after insertingEqs. (21) and (51) into Eq. (43). This completes the WDWsolutions for the potentials given by Eqs. (18) and (19). Anyanalysis of theWKBwave function in this regime, due to itscomplex expressions, is somewhat difficult.

IV. INTEPRETATION OF THE WAVE FUNCTION

To analyze the behavior of the wave function, one mustcompute K2 ¼ KijK

ij, the trace of the square of the

extrinsic curvature Eq. (2) [44,45]. If a wave function isoscillatory (exponential), K2 has positive (negative) ei-genvalues. Using Eqs. (3) and (13), one gets

K 2 ¼ � 9

�2a4d2

da2: (52)

Defining the auxiliary quantity W :¼ K2c ðaÞc ðaÞ and the

asymptotic expression Eq. (27), for a � 1, one obtainsthat W < 0. When the HL gravity terms dominate theUniverse, the wave function is exponential, correspondingto a Euclidean geometry.

For the very late Universe, the behavior of the wavefunction is very different depending on the value of thecosmological constant. If � ¼ 0, the wave function isgiven by Eq. (29) and it is easy to show that W < 0,showing that the behavior is exponential. If �> 0, thewave function is given by a combination of oscillatoryfunctions Eq. (33), givingW > 0, meaning that the geome-try is Lorentzian or classical. Finally, for negative values ofthe cosmological constant, Eq. (37) yields W < 0. In thecase studied here, the computations are quite simple, and itis not surprising to find this result since the nature of thewave function given by Eqs. (27), (29), (33), and (37) canbe obtained directly by inspection.

The semiclassical approximation implies that the con-figurations will oscillate about the classical solution [38].In order to verify whether GR can be recovered for thelow-energy limit in this approximation, one obtains theHamilton-Jacobi equation from theWDWequation throughtheWKBmethod. TheWDWequation [cf. Eq. (16)] can bewritten as

�d2

da2� VðaÞ

��ðaÞ ¼ 0: (53)

Substituting a wave function of the form c ¼ Re½CeiS�,where C is a slowly varying amplitude and S is the phase,one obtains the Hamilton-Jacobi equation�

dS

da

�2 þ VðaÞ ¼ 0: (54)

Notice that S is real provided VðaÞ � 0 (classically allowedregion), denoting that the wave function is oscillatory. IfVðaÞ> 0 (classically forbidden region), S is imaginaryand the wave function has an exponential behavior. Using

Eq. (54), one finds that S ¼ R ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�VðaÞpda is the phase for

the classically allowed region. This integral was discussedin Sec. III. Applying �a to the wave function of theUniverse, c ¼ Re½CeiS�, one gets

�ac ¼ �idc

da¼ �i

�dC

daþ iC

dS

da

�eiS:

The WKB assumption, j dCda j � j dSda j, yields that �a ¼ dSda .

Using Eqs. (13) and (54) one is led to

t ¼Z aðtÞ

að0Þda

affiffiffiffiffiffiffiffiffiffiffiffiffiffi�VðaÞp ; (55)

where this equation relates the global time and the scalefactor. One intends to investigate the regions where the posi-tive cosmological constant dominates, so VðaÞ � �g�a

4;substituting this asymptotic potential into Eq. (55), one getsthat the time evolution for the scale factor when a 1 isgiven by

aðtÞ � effiffiffiffiffig�

pt: (56)

This corresponds to a de Sitter expansion phase, a behaviorexpected for the GR regime, which is recovered when� ¼ 1. Of course, this does not prove that GR is recoveredas an IR fixed point of the HL gravity, but shows that a HLFLRW cosmology yields for a 1 a semiclassical solutionthat corresponds to theGRone.Apossible approach to tacklethe problem would involve considering a scaling � ¼ �ðaÞ,and expect that � ! 1 for a 1. Considerations of thisnature were developed for cosmological models withscale-dependent Newtonian gravitational coupling(cf. Refs. [50,51]).

V. CONCLUSIONS

In this work, the quantum cosmology for the Horava-Lifshitz gravity without matter is investigated for theclosed Universe. In the minisuperspace model, the WDWequation is derived, and it is shown that the HL gravityintroduces terms that are dominant for short distances,modifying the behavior of GR on these scales. One choosesthe configurations for which the HL gravity new terms actas a ‘‘potential barrier’’ close to the singularity, a ¼ 0.The solutions for the WDW equation are obtained con-

sidering the DeWitt boundary condition, Eq. (22), whichstates that the wave function vanishes at the singularity. Fora � 1, corresponding to the very early Universe when theHL gravity terms dominate, the wave function is an ex-ponential that is typical of the classically forbidden region.A quantum bound for the coefficient gs is found.For the very late Universe, when a 1, the curvature

and the cosmological constant terms dominate and, onefinds that, for � ¼ 0 or �< 0, the wave function is ex-ponentially suppressed, denoting as before, that this region

ORFEU BERTOLAMI AND CARLOS A.D. ZARRO PHYSICAL REVIEW D 84, 044042 (2011)

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is classically forbidden. For a positive cosmologicalconstant case, one finds a damped oscillatory behavior,already found in the usual QC for GR.

For the vanishing cosmological constant, an exact solu-tion can be obtained. In this case, one finds that the singu-larity is avoided due to quantum effects as the probability toreach the singularity a ¼ 0 vanishes and that gr is quan-tized. Fixing the value of gs, for large values of n (large grvalues), one can obtain a classical Universe according to theanalog of the correspondence principle of old quantummechanics. The complete exact solution for � � 0 cannotbe obtained, although wave functions in the WKB approxi-mation can be obtained for the intermediate regions.

Finally, a discussion of how the classical solutionemerges from the semiclassical analysis is performed solv-ing the Hamilton-Jacobi equation: One encounters a semi-classical solution oscillating nearby the classical solution.For �> 0 and a 1, this leads to a de Sitter space-time,as expected from GR.

One then concludes that quantum cosmology appliedto HL gravity suggests that this proposal matches theexpectations of a quantum gravity model for the veryearly Universe, as it provides, for instance, a hint forthe singularity problem for the � ¼ 0 case. In whatconcerns specific solutions, the model suggests that GRbehavior is recovered at the semiclassical limit.

ACKNOWLEDGMENTS

C. Z. would like to thank the Fundacao para a Ciencia e aTecnologia (FCT) for financial support under ProjectsNo. SFRH/BPD/75046/2010 and No. CERN/FP/116358/2010, the Centro de Fısica do Porto for hospitality duringthe period that this paper was prepared, and Miguel Costafor stimulating discussions about some issues related tothis work. The work of O. B. is partially supported by FCTProjects No. PTDC/FIS/111362/2009 and No. CERN/FP/116358/2010.

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