Can singularities be avoided in quantum cosmology?

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  • Ann. Phys. (Berlin) 19, No. 3 5, 211 218 (2010) / DOI 10.1002/andp.201010417

    Can singularities be avoided in quantum cosmology?

    Claus Kiefer

    Institut fur Theoretische Physik, Universitat zu Koln, Zulpicher Strasse 77, 50937 Koln, Germany

    Received 15 November 2009, accepted 4 January 2010Published online 23 February 2010

    Key words Quantum cosmology, WheelerDeWitt equation, cosmological singularities.Many cosmological models based on general relativity contain singularities. In this contribution I address thequestion whether consistent models without singularities can exist in quantum cosmology. The discussion isbased on the WheelerDeWitt equation of quantum geometrodynamics. The models under consideration aremotivated by recent discussions of dark energy. Employing some natural criteria of singularity avoidance inthe quantum theory, I show that this can indeed happen in these models.

    c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

    1 Introduction

    The search for a quantum theory of gravity belongs to the most important tasks in fundamental physics [1].While such a theory is not yet available in nal form, it is more or less obvious which expectations itshould fulll. Among them is the fate of the classical singularities in quantum gravity: a reasonable theoryof quantum gravity should predict their avoidance.

    It is clear that general statements cannot be made, given the incomplete understanding of quantum grav-ity. However, the issue of singularity avoidance may be addressed within the context of simple models.This is the topic of my contribution. I shall restrict myself to singularities for homogeneous and isotropiccosmological models. For such models, the quantization can be performed and the question can be inves-tigated whether singularity-free solutions exist and whether they are natural.

    My paper is organized as follows. In Sect. 2, I shall quote one of the famous singularity theorems andgive an overview of cosmological singularities in Friedmann models; here, one does not only have thestandard big-bang and big-crunch singularities, but also more exotic ones such as a big rip or a big brake.Section 3 presents the general framework of my discussion: quantum geometrodynamics and the WheelerDeWitt equation. In Sect. 4, then, I review the quantum cosmology of some of these models and show thatthere, indeed, singularities can be avoided in a natural way. One could imagine that singularity avoidanceis similarly implementable in full quantum gravity. I end with some brief conclusions.

    2 Classical singularities in cosmology

    We call a spacetime singular if it is incomplete with respect to a timelike or null geodesic and if it cannotbe embedded in a bigger spacetime. There exists a variety of theorems which postulate the existence ofsingularities in general relativity [2]. Typically, they involve an energy condition, a condition on the globalstructure, and the condition that gravitation be strong enough to lead to the existence of a closed trappedsurface. Let me quote one of the famous theorems proved by Hawking and Penrose in 1970 [3]:

    Theorem 2.1. A spacetime M cannot satisfy causal geodesic completeness if, together with Einsteinsequations, the following four conditions hold:

    E-mail: kiefer@thp.uni-koeln.de, Phone: +49 221 470 4301, Fax: +49 221 470 2189

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  • 212 C. Kiefer: Quantum cosmology

    1. M contains no closed timelike curves.2. The strong energy condition is satisfied at every point.3. The generality condition (. . . ) is satisfied for every causal geodesic.4. M contains either a trapped surface, or a point P for which the convergence of all the null geodesics

    through P changes sign somewhere to the past of P , or a compact spacelike hypersurface

    If we take an ideal uid as a model for matter, with as the energy density and pi (i = 1, 2, 3) as theprincipal pressures, the strong energy condition used in this theorem reads as

    +

    i

    pi 0, + pi 0, i = 1, 2, 3 . (1)

    In fact, as was already shown in [3] and discussed in [2], the observation of the Cosmic Microwave Back-ground Radiation indicates that there is enough matter on the past light-cone of our present location P toimply that the divergence of this cone changes somewhere to the past of P . Thus, if in addition the strongenergy condition is fullled, the conditions stated in the above theorem will apply and the origin of ourUniverse cannot be described within general relativity.

    Modern cosmology entertains the idea that a period of quasi-exponential acceleration (ination) tookplace in the very early Universe. Since such an inationary phase may violate the strong energy conditions,it has been speculated that the singularity can thereby be avoided. Is this true? There are two facets of this.On the one hand, it was shown by Borde et al. in [4] that singularities are not avoided by an inationaryphase if the universe has open spatial sections or if the Hubble expansion rate is bounded away from zero inthe past. On the other hand, Ellis et al. have presented an example of a singularity-free inationary modelwith closed spatial sections [5]. Thus, independent of what describes more precisely the real conditions inthe early Universe, one should at least envisage the possibility that the classical solution describing it issingular.

    Cosmological singularities cannot only happen in the past. There is the option (presently not supportedby observations) that the Universe will recollapse in the future and encounter a big-crunch singularity.But even if the Universe continues to expand, it can encounter singularities in the future for non-vanishingscale factor. This can happen for various equations of state which can describe a form of dark energy. (Suchsingularities may also occur at a nite value a > 0 in the past). In fact, present observations still allow thepossibility of an equation of state leading to such exotic singularities, cf. [6] and the references therein.In the light of the above-quoted singularity theorem it is of interest to remark that these exotic singularitiesmay or may not violate energy conditions, so this theorem is not directly applicable and one has to exploreadditional options for relevant energy conditions.

    One can classify cosmological singularities for homogeneous and isotropic spacetimes following theschemes proposed in [6,8]. Different singularities can be distinguished according to the behaviour of scalefactor a, energy density , and pressure p. The classication is

    Big Bang/Crunch: a = 0 at nite proper time, but diverges.Type I (Big Rip): a diverges in nite proper time, and both and p diverge. A big rip is obtained from

    phantom dark energy, that is, matter with an equation of state p = w, w < 1. Phantoms notonly violate the strong energy condition (cf. the theorem quoted above), but also all the other energyconditions.

    Type II (Sudden singularity): a and remain nite, p diverges, but the Hubble parameter stays nite(H diverges). Sudden singularities only violate the dominant energy condition. Special cases are thebig-brake singularity in the future and the big-demarrage singularity in the past; these are character-ized as sudden singularities arising from a generalized Chaplygin equation of state, p = A/ . Forsudden singularities, geodesics can be continued through these singularities, but tidal forces diverge.

    c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org

  • Ann. Phys. (Berlin) 19, No. 3 5 (2010) 213

    Sudden singularities can appear as close as 8.7 million years in the future [6]. There exist also gener-alized sudden singularities, which are singularities of pressure derivatives and which fulll all energyconditions.

    Type III (Finite scale factor singularity): a remains nite, but both and p diverge (as well as H andH); particular examples are the big-freeze singularities (both in the past and the future), which arecharacterized by a generalized Chaplygin equation of state.

    Type IV (Big Separation): a, , and p remain nite, but the rth derivative of a and the (r1)th deriva-tive of H diverge (r 3).

    w-singularity: The only singularity here is in the barotropic index w [7].In extension of the Type I to IV classication, one could call big bang/big crunch Type 0 and the w-singularity Type V.

    It is now of interest to see whether some or all of these singularities can be avoided in quantum gravity.We have investigated the quantum theory for some of these models [911], and some of the results will bediscussed below. Before that, however, I shall briey introduce the framework on which the investigationis based.

    3 Quantum geometrodynamics

    A full quantum theory of gravity remains elusive [1]. Can one nevertheless say something reliable aboutquantum gravity without knowing the exact theory? In [12] I have made the point that this is indeedpossible. The situation is analogous to the role of the quantum mechanical Schrodinger equation. Althoughthis equation is not fundamental (it is non-relativistic, it is not eld-theoretic, etc.), important insights canbe drawn from it. For example, in the case of the hydrogen atom, one has to impose boundary conditionsfor the wave function at the origin r 0, that is, at the centre of the atom. This is certainly not a regionwhere one would expect non-relativistic quantum mechanics to be exactly valid, but its consequences, inparticular the resulting spectrum, are empirically correct to an excellent approximation.

    Erwin Schrodinger has found his equation by guessing a wave equation from which the HamiltonJacobi equation of classical mechanics can be recovered in the limit of small wavelengths, in analogy tothe limit of geometric optics from wave optics. The same approach can be applied to general relativity.One can start from the HamiltonJacobi version of Einsteins equations and guess a wave equation fromwhich it can be recovered in the classical limit. The only assumption that is required is the universal validityof quantum theory, that is, its linear structure. It is not yet needed for this step to impose a Hilbert-spacestructure. Such a structure is employed in quantum mechanics because of the probability interpretationfor which one needs a scalar product and its conservation in time (unitarity). Its status in quantum gravityremains open.

    The HamiltonJacobi equation for general relativity was formulated by Peres [13]. In the vacuum case,it reads (we set c = 1 in the following)

    16GGabcdS

    hab

    S

    hcd

    h

    16G( (3)R 2) = 0 . (2)

    The functional S[hab] depends on the three-dimensional metric, hab. It also obeys the equation

    DaS

    hab= 0 , (3)

    which guarantees that S[hab] is invariant under three-dimensional coordinate transformations.The task is now to nd a wave equation which yields the HamiltonJacobi equation (2) as well as Eq. (3)

    in the semiclassical limit (WKB approximation) where the wave functional reads

    [hab] = C[hab] exp(

    iS[hab]

    )(4)

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  • 214 C. Kiefer: Quantum cosmology

    with a rapidly varying phase and a slowly varying amplitude. In the vacuum case, one then nds from (2)and (3) the equations

    H (16G2Gabcd

    2

    habhcd (16G)1

    h(

    (3)R 2))

    = 0 , (5)

    and

    Da 2b ihab

    = 0 . (6)

    The rst equation is called the WheelerDeWitt equation, the second one which guarantees the invari-ance of the wave functional under three-dimensional coordinate transformations is called the quantumdiffeomorphism (momentum) constraint. Whether these equations hold at the most fundamental level ornot, they should approximately be valid away from the Planck scale, provided that quantum theory isuniversally valid.

    We recognize from (5) und (6) that no external time is present anymore spacetime has disappearedand only space remains [1]. Nevertheless, a local intrinsic time can be dened through the local hyperbolicstructure of the WheelerDeWitt equation. This intrinsic time is solely constructed from the three-geometry(and matter degrees of freedom, if present).

    If non-gravitational degrees of freedom are present, the Eqs. (5) and (6) have to be augmented by thecorresponding terms. For the semiclassical approximation one then makes the following BornOppen-heimer-type of ansatz [1]

    |[hab] = C[hab]eim2PS[hab]|[hab] , (7)where the bra and ket notation refers to non-gravitational elds, for which we can assume that a standardHilbert-space is at our disposal (mP denotes the Planck length).

    One then evaluates the wave functional |[hab] along a solution of the classical Einstein equations,hab(x, t), corresponding to a solution, S[hab], of Eqs. (2) and (3); this solution is obtained from

    hab = NGabcdS

    hcd+ 2D(aNb) , (8)

    where N and Nb denote the lapse function and the shift vector, respectively [1]. Employing in the semi-classical limit the following denition of an approximate time parameter t,

    t|(t) =

    d3x hab(x, t)

    hab(x)|[hab] ,

    one obtains a functional Schrodinger equation for quantized matter elds in the chosen external classicalgravitational eld:

    i

    t|(t) = Hm|(t) ,

    Hm

    d3x{N(x)Hm(x) + Na(x)Hma (x)

    }. (9)

    Here, Hm denotes the matter-eld Hamiltonian in the Schrodinger picture, parametrically depending onthe (generally non-static) metric coefcients of the curved spacetime background; the WKB time tcontrols the dynamics in this approximation.

    The special ansatz (7) can be justied in retrospect through the process of decoherence [14]. Unobserveddegrees of freedom such as tiny gravitational waves interact with the relevant degrees of freedom (e.g. the

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  • Ann. Phys. (Berlin) 19, No. 3 5 (2010) 215

    volume of the three-dimensional space) in such a way that, for example, the superposition of (7) with itscomplex conjugate cannot be distinguished from a corresponding mixture; the state (7) can thus be treatedseparately.

    The semiclassical approximation and the ensuing time parameter exist, of course, only under specialconditions. Only then can the notions of spacetime and geodesics, needed for the application of the clas-sical singularity theorems, be employed. In the quantum case we have to think rst about the meaning ofsingularities and singularity avoidance.

    4 Quantum avoidance of singularities

    There does not yet exist any general agreement on the necessary criteria for quantum avoidance of sin-gularities. I shall not attempt here to provide such necessary criteria either, but shall focus instead on twosufcient criteria which have turned out to be useful in the discussion of the models presented below. Therst criterium dates back to the pioneering work of Bryce DeWitt on canonical quantum gravity [15]. Hehas suggested that the wave function should vanish at the point of the classical singularity. We shall thusinterpret the vanishing of the wave function in the region of the classical singularity as singularity avoid-ance. As for the second criterium, we shall adopt the spreading of wave packets when approaching theregion of the classical singularity: the semiclassical approximation discussed above then breaks down andno classical time parameter is available. The classical singularity theorems can then no longer be applied.

    It must be emphasized that 0 is really only a sufcient, but by no means a necessary criteriumfor singularity avoidance. Consider, for example, the solution of the Dirac equation for the ground state ofhydrogen-like atoms (in standard notation). This solution diverges at the origin,

    0(r) (2mZr)

    1Z221emZr r0 ,but

    dr r2|0|2 remains nite. So the important quantity is, in fact, the inner product. As already men-

    tioned, there is no general consensus about the role of the inner product in quantum cosmology, but thesituation could be analogous to the hydrogen atom. For the WheelerDeWitt equation for a Friedmannuniverse with a massless scalar eld, for example, the simplest solution diverges for scale factor a 0,

    K0(a2/2) a0 c ln a ,

    but nevertheless the integral

    dad|G||(a, )|2, where G here denotes the determinant of the DeWitt

    metric, may be nite.For a closed FriedmannLematre universe with scale factor a, containing a homogeneous scalar eld

    with potential V (), that is, a two-dimensional minisuperspace, the classical line element is

    ds2 = N2(t)dt2 + a2(t)d23 .The WheelerDeWitt equation reads (with units 2G/3 = 1)

    12

    (2

    a2

    a

    (a

    a

    )

    2

    a32

    2 a + a

    3

    3+ 2a3V ()

    )(a, ) = 0 ,

    where the factor ordering has been chosen in order to achieve covariance in minisuperspace; this is theLaplaceBeltrami factor ordering. (For a general overview of quantum cosmology, see, for example, [1,16]and the references therein.)

    Let us consider now various models with classical singularities. The rst one is a model with phan-toms. These are elds with negative kinetic energy, which are certainly very exotic but which cannot yetbe excluded on the basis of supernova data. Classically, the ensuing dynamics develops a big-rip singular-ity, that is, and p diverge as a goes to innity at a finite time (see above). The corresponding quantization

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  • 216 C. Kiefer: Quantum cosmology

    was discussed in [9]. It was found that wave-packet solutions of the WheelerDeWitt equation necessarilydisperse in the region of the classical big-rip singularity; time and the classical evolution thus come toan end and only a stationary quantum state is left. Quantum effects thus become important for a big uni-verse, not only for a small Planck-size universe. Interestingly, it was shown that an alternative quantization,employing the Bohm interpretation of quantum theory, cannot avoid the big-rip singularity [17].

    Another model is cosmology with a big brake [10]. We take an equation of state of the form p = A/,A > 0, that is, an Anti-Chaplygin gas. For a Friedmann universe with scale factor a(t) and a scalar eld(t), this equation of state can be realized by the potential

    V () = V0

    sinh(32||) 1

    sinh(

    32||) ; V0 = A/4 ,

    where 2 = 8G. The classical dynamics develops a pressure singularity (only a(t) becomes singular)and comes to an abrupt halt in the future.

    The WheelerDeWitt equation for this model reads

    2

    2

    (2

    62

    2

    2

    2

    )(, )+V0e6

    sinh(32||) 1

    sinh(

    32||)(, ) = 0 ,

    (10)where = ln a, and LaplaceBeltrami factor ordering has again been used. The vicinity of the big-brakesingularity is the region of small ; we can therefore use the approximation

    2

    2

    (2

    62

    2

    2

    2

    )(, ) V0|| e

    6(, ) = 0 ,

    where V0 = V0/32.It was shown in [10] that all normalizable solutions read

    (, ) =

    k=1

    A(k)k3/2K0

    (16

    V2k

    )(

    2Vk||)

    eVk|| L1k1

    (2Vk||)

    , (11)

    where K0 is a Bessel function, L1k1 are Laguerre polynoms, and V V0e6. They all vanish at theclassical singularity. This model therefore implements our other criterium above: the vanishing of the wavefunction in the spirit of DeWitt. A similar result holds for the corresponding loop quantum cosmology.Interestingly, the solutions also implement the avoidance of the big-bang singularity, 0 for .

    The big-brake singularity is an example of type-II singularities. A more general class of type-II andtype-III singularities was discussed in [11]. The equation of state in this class is chosen to be that of ageneralized Chaplygin gas:

    p = A

    with general real parameters A and . For example, in the big-freeze (type III) singularity, both H and Hblow up in the past at a nite value of the scale factor. The big-freeze singularity occurs in the past at aminimal scale factor amin > 0; there are thus no classical solutions in the limit . For this reasonwe have to demand that the wave function go to zero in the classically forbidden region, 0,because otherwise one would not obtain the correct classical limit.

    c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org

  • Ann. Phys. (Berlin) 19, No. 3 5 (2010) 217

    The class of solutions then reads

    k(, ) ||J(k||)

    [b1ei

    6k + b2ei

    6k

    ],

    with as a function of . These solutions obey, in fact, DeWitts boundary condition at the singularity,k(0, 0) = 0. The same also holds for the other cases discussed in [11].

    The vanishing of the wave function at the classical singularity is also a feature in other approaches. Anespecially interesting example are the supersymmetric quantum cosmological billiards discussed in [18].In D = 11 supergravity, one can employ near a spacelike singularity a cosmological billiard descriptionbased on the KacMoody group E10 and address the corresponding WheelerDeWitt equation. It wasfound there, too, that 0 near the singularity. DeWitts criterium of singularity avoidance may thus bea viable criterium which is implementable in a wide range of quantum cosmological models.

    5 Conclusion

    I have shown that the classical cosmological singularities of various models can be avoided in quantumcosmology. Two sufcient (but by no means necessary) criteria have been employed for this purpose:unavoidable spreading of the wave function when approaching the singular region and vanishing of thewave function at the singularity itself.

    The models discussed here can have (in the classical theory) singularities at large values of the scalefactor, that is, far away from the Planck length. The corresponding quantum avoidance thus means thatquantum gravitational effects can occur for large universes an intriguing thought. Such macroscopicquantum effects have hitherto only been envisaged near the turning point of a classically recollapsinguniverse [19,20]. It is clear that these scenarios have also consequences for the arrow of time in the universe[19, 21].

    Singularity avoidance also occurs in the framework of loop quantum cosmology, although in a somewhatdifferent way, see [22]. The big-bang singularity can be avoided by solutions of the difference equationwhich there replaces the WheelerDeWitt equation. The avoidance can also be achieved by the occurrenceof a bounce in the effective Friedmann equations. For the singularities at large scale factors of the big-rip orbig-brake type, the resulting singularity avoidance is similar to the one discussed above for the WheelerDeWitt equation, but there are other sudden singularities that do not seem to be avoided [23].

    Acknowledgements I would like to thank Mariusz Dabrowski for inviting me to an interesting and pleasant con-ference. I also want to acknowledge his useful comments on this manuscript. I am grateful to my co-authors MariamBouhmadi-Lopez, Mariusz Dabrowski, Alexander Kamenshchik, Barbara Sandhofer, and Paulo Vargas Moniz for ourcollaboration on this fascinating subject.

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  • 218 C. Kiefer: Quantum cosmology

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    c 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org