Can singularities be avoided in quantum cosmology?

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<ul><li><p>Ann. Phys. (Berlin) 19, No. 3 5, 211 218 (2010) / DOI 10.1002/andp.201010417</p><p>Can singularities be avoided in quantum cosmology?</p><p>Claus Kiefer</p><p>Institut fur Theoretische Physik, Universitat zu Koln, Zulpicher Strasse 77, 50937 Koln, Germany</p><p>Received 15 November 2009, accepted 4 January 2010Published online 23 February 2010</p><p>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.</p><p>c 2010 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim</p><p>1 Introduction</p><p>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.</p><p>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.</p><p>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.</p><p>2 Classical singularities in cosmology</p><p>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]:</p><p>Theorem 2.1. A spacetime M cannot satisfy causal geodesic completeness if, together with Einsteinsequations, the following four conditions hold:</p><p> E-mail:, Phone: +49 221 470 4301, Fax: +49 221 470 2189</p><p>c 2010 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim</p></li><li><p>212 C. Kiefer: Quantum cosmology</p><p>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</p><p>through P changes sign somewhere to the past of P , or a compact spacelike hypersurface</p><p>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</p><p> +</p><p>i</p><p>pi 0, + pi 0, i = 1, 2, 3 . (1)</p><p>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.</p><p>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.</p><p>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 &gt; 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.</p><p>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</p><p>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</p><p>phantom dark energy, that is, matter with an equation of state p = w, w &lt; 1. Phantoms notonly violate the strong energy condition (cf. the theorem quoted above), but also all the other energyconditions.</p><p>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.</p><p>c 2010 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim</p></li><li><p>Ann. Phys. (Berlin) 19, No. 3 5 (2010) 213</p><p>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.</p><p>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.</p><p>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).</p><p>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.</p><p>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.</p><p>3 Quantum geometrodynamics</p><p>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.</p><p>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.</p><p>The HamiltonJacobi equation for general relativity was formulated by Peres [13]. In the vacuum case,it reads (we set c = 1 in the following)</p><p>16GGabcdS</p><p>hab</p><p>S</p><p>hcd</p><p>h</p><p>16G( (3)R 2) = 0 . (2)</p><p>The functional S[hab] depends on the three-dimensional metric, hab. It also obeys the equation</p><p>DaS</p><p>hab= 0 , (3)</p><p>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)</p><p>in the semiclassical limit (WKB approximation) where the wave functional reads</p><p>[hab] = C[hab] exp(</p><p>iS[hab]</p><p>)(4)</p><p> c 2010 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim</p></li><li><p>214 C. Kiefer: Quantum cosmology</p><p>with a rapidly varying phase and a slowly varying amplitude. In the vacuum case, one then nds from (2)and (3) the equations</p><p>H (16G2Gabcd </p><p>2</p><p>habhcd (16G)1</p><p>h(</p><p>(3)R 2))</p><p> = 0 , (5)</p><p>and</p><p>Da 2b ihab</p><p>= 0 . (6)</p><p>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.</p><p>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).</p><p>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]</p><p>|[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).</p><p>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</p><p>hab = NGabcdS</p><p>hcd+ 2D(aNb) , (8)</p><p>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,</p><p>t|(t) =</p><p>d3x hab(x, t)</p><p>hab(x)|[hab] ,</p><p>one obtains a functional Schrodinger equation for quantized matter elds in the chosen external classicalgravitational eld:</p><p>i</p><p>t|(t) = Hm|(t) ,</p><p>Hm </p><p>d3x{N(x)Hm(x) + Na(x)Hma (x)</p><p>}. (9)</p><p>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.</p><p>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</p><p>c 2010 WILEY-VCH Verlag GmbH &amp; Co. KGaA, Weinheim</p></li><li><p>Ann. Phys. (Berlin) 19, No. 3 5 (2010) 215</p><p>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.</p><p>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.</p><p>4 Quantum avoidance of singularities</p><p>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...</p></li></ul>