On cosmological isotropy, quantum cosmology

arXiv:gr-qc/9212006v1 10 Dec 1992On cosmological isotropy, quantum cosmologyand the Weyl curvature hypothesisSean A. HaywardMax Planck Institut f¨ur AstrophysikKarl Schwarzschild Straße 18046 Garching bei M¨unchenGermany24th June 1992Abstract. The increasing entropy, large-scale isotropy and approximate flatness ofthe universe are considered in the context of signature change, which is a classical modelof quantum tunnelling in quantum cosmology.

The signature change hypothesis implies aninitial inflationary epoch, the magnetic half of the Weyl curvature hypothesis, and a closeanalogue of the conformal singularity hypothesis. Adding the electric half of the Weyl cur-vature hypothesis yields, for a perfect fluid, only homogeneous and isotropic cosmologies.In the cosmological-constant case, the unique solution is the Vilenkin tunnelling solution,which gives a de Sitter cosmology.Explaining the large-scale structure of the universe is the ultimate goal of cosmology.Although the dynamical evolution of the universe is usually accepted as being determinedby general relativity, various particular properties of the universe can only be explainedin terms of particular initial conditions.

The most striking cosmological observations ofthis type are the thermodynamic arrow of time, the high isotropy of the universe at largescales, and the approximate flatness of the universe. Inflation [1–3] is widely regarded asproviding a plausible explanation of the last two observations, albeit at a phenomenologicalrather than fundamental level.It seems likely that only a quantum theory of gravity can adequately explain theparticular initial conditions of the universe.

Nevertheless, it is possible to make conjecturesin classical terms by appealing to the expected properties of a correct quantum theory ofgravity.In particular, there is the Weyl curvature hypothesis of Penrose [4–6], whichasserts that the Weyl tensor vanishes at the initial singularity, and is intended to expressa low initial entropy. The initial cosmological singularity is predicted by general relativityprovided that positive-energy conditions are satisfied, but creates various philosophicaland mathematical difficulties, which may be resolved either by abolishing the singularityor by accomodating it in some way.

A suggestion in the latter direction, due to Goode& Wainwright [7–10] and in a simpler form to Tod [11–13], is that the singularity beconformal or isotropic. On this assumption, together with the Weyl curvature hypothesis,R.

Newman has shown that for a radiative perfect fluid, the resulting universe must behomogeneous and isotropic [14]. This remarkable result raises the question of whether theassumptions can indeed be justified by quantum gravity.A promising candidate for a correct quantum theory of gravity is quantum cosmologyaccording to the Hartle-Hawking programme [15–18].

Unfortunately, this involves variousmathematical and interpretational problems, and obtaining reliable predictions is rather1

difficult. One way to obtain simple predictions is to take a classical limit of the theory.

Itmight seem that this would merely yield classical cosmology according to general relativity,but an alternative suggestion is that the natural classical model of the Hartle-Hawkingproposal is a universe whose signature is initially Riemannian but subsequently becomesLorentzian. In quantum cosmology, these are known as tunnelling solutions [19–22].

Thechange of signature can only occur at the Planck epoch, since it is classically unstable [23].Under the signature change hypothesis, the initial singularity is abolished in favour of acompact non-singular Riemannian region. To make sense of the Weyl curvature hypothesisin this context, it is natural to consider a vanishing Weyl tensor at the signature-changejunction, since this will be observationally indistinguishable from the original condition.Then the following remarkable result is found: the junction conditions for signature changeimply the magnetic half of the Weyl curvature hypothesis and an analogue of the conformalsingularity hypothesis.

Moreover, an initially inflationary universe is also predicted.The derivation uses the standard ‘3+1’ decomposition [24–25] with unit lapse and zeroshift, where the metric gab is decomposed into a spatial 3-metric hab and a normal vectorna by gab = hab −nanb, ncnc = −1. The second fundamental form may be divided intoan expansion θ = 12hcdLnhcd and a traceless shear σab = 12(Lnhab −13habhcdLnhcd), whereLn is the Lie derivative along na.

The energy tensor Tab of the matter may be dividedinto an energy density ρ = Tcdncnd, a momentum ja = hcaTcdnd, a pressure p = 13hcdTcdand a traceless stress ςab = hcahdbTcd −13habhcdTcd. The Einstein equations are then0 = 3σcdσcd −2θ2 −3R + 6ρ,(1)0 = 3Dcσac −2Daθ −3ja,(2)Lnhab = 2(σab + 13θhab),(3)Lnθ = −14(2θ2 + 3σcdσcd + R + 6p),(4)Lnσab = 2σcaσbc −13θσab −Gab −16Rhab + ςab,(5)where R is the Ricci scalar, Gab the Einstein tensor and Da the covariant derivative of hab.The electric and magnetic parts of the Weyl tensor areEab = 12(Gab + 16Rhab) −12(Lnσab −13habhcdLnσcd) + 16θσab,(6)Bab = εcda(2Dcσbd −hbcDeσde + 23hbcDdθ),(7)respectively, where εabc is the alternating form of hab.

When considering signature change,the Einstein equations in the Riemannian region take a similar form with some signchanges, and the Einstein equations at the junction itself take the form of junction condi-tions [23].An na-orthogonal 3-surface S is isotropic, both intrinsically and extrinsically, only ifσab|S = (Gab + 16Rhab)|S = ςab|S = 0,ja|S = 0,(8)since these quantities, if non-zero, yield preferred directions. It follows from equations(1–8), using the Bianchi identities2DcEac = Da(p −ρ) −2jc(σac + 13θhac) −Dcςac,DcGac = 0,2

thatEab|S = Bab|S = 0,Daθ|S = DaR|S = Daρ|S = Dap|S = 0,so that the solution on S is homogeneous if it is isotropic.To deduce that the entirespacetime is homogeneous and isotropic given the isotropy conditions (8) requires an as-sumption on the matter, since a matter field with internal degrees of freedom can break thesymmetry of the gravitational field. However, the standard Cauchy uniqueness theoremshows that a homogeneous and isotropic cosmology arises from (8) for a perfect fluid, givenbyTab = (ε + π)vavb + πgab,(9)where ε is the energy density, π(ε) the pressure and va the flow direction: vcvc = −1.

Noenergy conditions are assumed here, so that, for instance, a cosmological constant Λ > 0can be included as the case ε = −π = Λ.Before discussing the signature change model, consider for comparison the conformalsingularity model [7–14], where a conformal transformation gab = Ω2ˆgab is assumed, suchthat the initial singularity at Ω= 0 is a regular surface ˆS in the conformal manifold. TheWeyl tensor is conformally invariant, ˆEab = Eab, ˆBab = Bab.

Tod [12–13] shows that theconformal second fundamental form vanishes at the singularity:ˆσab| ˆS = 0,ˆθ| ˆS = 0. (10)It follows that the magnetic half of the Weyl tensor vanishes initially, ˆBab| ˆS = 0 [7–14].Adding the electric half ˆEab| ˆS = 0 of the Weyl curvature hypothesis, and assuming aradiative perfect fluid π = 13ε, Newman [14] shows that only homogeneous and isotropicspacetimes result.

This is an intriguing result, though the status of the conformal singu-larity hypothesis as a fundamental principle is questionable, and like the Weyl curvaturehypothesis it presumably requires an appeal to the conjectured properties of a correct quan-tum theory of gravity. Happily, a very similar condition does arise in quantum cosmology,or at least the signature-change model thereof, as follows.The junction conditions for signature change are that the second fundamental formvanishes at the junction S,σab|S = 0,θ|S = 0,(11)with possible further conditions on the matter fields [20–23].

Such an initially stationaryuniverse must have been strongly inflationary during early epochs in order to expand to asize consistent with observation. In particular, sinceLnθ = −13θ2 −σcdσcd −12(ρ + 3p),initial inflation Lnθ|S > 0 occurs if and only if (ρ+3p)|S < 0.

Conversely, if ρ+3p > 0, theuniverse would have rapidly collapsed to a caustic. Thus there is a prediction of inflationand its consequences for isotropy and flatness.The junction conditions (11) are analogues of the conformal singularity conditions(10), and various analogous results follow directly from the Einstein equations (1–5) and3

Weyl identities (6–7), without the technical difficulties caused by the singular nature ofthe conformal field equations. In particular, the magnetic half of the Weyl tensor vanishesinitially, Bab|S = 0, as does the momentum density, ja|S = 0.

For the particular case of aperfect fluid (9), this means that the flow is orthogonal to S, so that the traceless stress alsovanishes, ςab|S = 0. Adding the electric half Eab|S = 0 of the Weyl curvature hypothesisthen yields the isotropy conditions (8), and hence only homogeneous and isotropic cos-mologies.

For instance, in the cosmological-constant case, the Vilenkin tunnelling solution[19] is the unique solution [23]. The Lorentzian part of this is de Sitter spacetime, whichis inflationary and has a density parameter which approaches one asymptotically.

If this istaken as a model for the early evolution of the universe, the observed density fluctuationscould perhaps be explained by initial quantum fluctuations, since the uncertainty principleforbids a precisely zero Weyl tensor.Given that the entropic motivation for the Weyl curvature hypothesis appears to besatisfied by the magnetic half alone, it is tempting to dispense with the electric half. Also,the original formulation of the Weyl curvature hypothesis [4] required the Weyl tensor tobe initially only finite, which is certainly the case in the context of signature change.

ThusI suggest simply using the signature change model as it stands, since it can be justified asthe classical limit of Hartle-Hawking quantum cosmology. Determining the possible initialconditions for the universe is reduced to a problem in Riemannian geometry: classifycompact, simply-connected 4-manifolds with boundary, with a non-singular Riemannianmetric satisfying the Einstein equations coupled to some well-defined matter source, suchthat the boundary is totally geodesic [20–23].

(Compare with compact instantons [26–27]. )The Lorentzian spacetimes determined by the initial data on the junction then constitutethe set of predicted cosmologies, and the spacetimes predicted by full quantum cosmologywill presumably be close to these.

If the Riemannian region is topologically a 4-hemisphere,the only known solution is the Vilenkin solution, and it may be conjectured to be unique,given the matter field. This solution is at least known to be isolated [28].In conclusion, the signature change hypothesis reveals various intriguing connectionsbetween quantum cosmology, inflation, the conformal singularity hypothesis and the Weylcurvature hypothesis, and offers hope of a full explanation of the cosmological observationsof increasing entropy, large-scale isotropy and approximate flatness.4

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