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Inhomogeneous and anisotropic cosmologies

arXiv:gr-qc/9212013v1 21 Dec 1992Inhomogeneous and anisotropic cosmologiesM.A.H. MacCallumSchool of Mathematical SciencesQueen Mary and Westfield College, University of LondonMile End Road, London E1 4NS, U.K.E-mail: M.A.H.MacCallum@qmw.ac.uk1IntroductionAlthough the purpose of this workshop is to discuss the structures in the universe,which are inhomogeneous, homogeneous models have been used in considering manyof the cosmological issues raised in that discussion, so I have also included in this sur-vey the anisotropic homogeneous models and their implications.

Only exact solutionswill be covered: other speakers at Pont d’Oye (e.g. Bardeen, Brandenberger, Dunsbyand Ellis) gave very full discussions of perturbation theory.

Here I continue my pre-vious practice (MacCallum 1979, 1984), by using the mathematical classification ofthe solutions as an overall scheme of organization; the earlier reviews give additionaldetails and references. (A survey organized by the nature of the applications is toappear in the proceedings of Dennis Sciama’s 65th birthday meeting.) Other usefulreviews are: Ryan and Shepley (1975) on homogeneous anisotropic models; Krasinski(1990) on inhomogeneous models; and Verdaguer (1985, 1992) on models of solitoniccharacter.In section 2 I will consider the spatially-homogeneous but anisotropic models.These are the Bianchi models, in general, the exceptions being the Kantowski-Sachsmodels with an S2 × R2 topology.

Such models could be significant in understandingthe background in which structure is formed, but they do not themselves model thatstructure. However, I will include here some remarks about inhomogeneous modelswhich are closely related to calculations done with Bianchi models.

Then in section3 I will consider the inhomogeneous models, which fall into several classes. Theycan be used both as local models of structure and as possible global models of thebackground in which structure forms (and are in some cases used for both purposessimultaneously).

A final section attempts a synthesis and makes some summarizingremarks.What is it that a cosmological model should explain? There are the following

main features:[1] Lumpiness, or the clumping of matter. The evidence for this is obvious.

[2] Expansion, shown by the Hubble law. [3] Evolution, shown by the radio source counts and more recently by galaxycounts.

[4] A hot dense phase, to account for the cosmic microwave background radiation(CMWBR) and the abundances of the chemical elements. [5] Isotropy, shown to a high degree of approximation in various cosmological ob-servations, but especially in the CMWBR.

[6] Possibly, homogeneity. (The doubt indicated here will be explained later.

)[7] The numerical values of parameters of the universe and its laws, such as thebaryon number density, the total density parameter Ω, the entropy per baryon,and the coupling constants[8] (Perhaps) such features as the presence of life.Originally, the standard big-bang models were the Friedman-Lemaˆıtre-Robertson-Walker (FLRW) models characterized as:[1] Isotropic at all points and thus necessarily. .

. [2] Spatially-homogeneous, implying Robertson-Walker geometry.

[3] Satisfying Einstein’s field equations[4] At recent times (for about the last 1010 years) pressureless and thus governedby the Friedman-Lemaˆıtre dynamics. [5] At early times, radiation-dominated, giving the Tolman dynamics and a ther-mal history including the usual account of nucleogenesis and the microwavebackground.To this picture, which was the orthodox view from about 1965-80, the last decadehas added the following extra orthodoxies:[6] Ω= 1.

Thus there is dark matter, for which the Cold Dark Matter model waspreferred. [7] Inflation – a period in the early universe where some field effectively mimicsa large cosmological constant and so causes a period of rapid expansion longenough to multiply the initial length scale many times.

[8] Non-linear clustering on galaxy cluster scales, modelled by the N-body simula-tions which fit correlation functions based on observations.and also added, as alternatives, such concepts as cosmic strings, GUTs or TOEs1 andso on.The standard model has some clear successes: it certainly fits the Hubble law, the1Why so anatomical?

source count evolutions (in principle if not in detail), the cosmic microwave spectrum,the chemical abundances, the measured isotropies, and the assumption of homogene-ity.Perhaps its greatest success was the prediction that the number of neutrinospecies should be 3 and could not be more than 4, a prediction now fully borne outby the LEP data.However, the model still has weaknesses [MacCallum, 1987]. For example, the trueclumping of matter on large scales, as shown by the QDOT data [Saunders et al., 1991]and the angular correlation functions of galaxies [Maddox et al., 1990], is too strongfor the standard cold dark matter account2.

The uniformity of the Hubble flow isunder question from the work of the “Seven Samurai” [Lynden-Bell et al., 1988] andothers. The question of the true value of Ωhas been re-opened, partly because theoryhas shown that inflation does not uniquely predict Ω= 1 (cf.

Ellis’ talk at Pontd’Oye) and partly because observations give somewhat variant values. Some authorshave pointed out that our knowledge of the physics valid at nucleogenesis and beforeis still somewhat uncertain, and that we should thus retain some agnosticism towardsour account of those early times.Finally, we should recognize that our belief in homogeneity on a large scale hasvery poor observational support.

We have data from our past light cone (and thoseof earlier human astronomers) and from geological records [Hoyle, 1962]. Studyingspatial homogeneity requires us to know about conditions at great distances at thepresent time, whereas what we can observe at great distances is what happened along time ago, so to test homogeneity we have to understand the evolution both ofthe universe’s geometry and of its matter content3.

Thus we cannot test homogeneity,only check that it is consistent with the data and our understanding of the theory.The general belief in homogeneity is indeed like the zeal of the convert, since untilthe 1950s, when Baade revised the distance scale, the accepted distances and sizes ofgalaxies were not consistent with homogeneity.These comments, however, are not enough to justify examination of other models.Why do we do that? The basic reason is to study situations where the FLRW models,even with linearized perturbations, may not be adequate.

Three types of situationcome to mind: the fully non-linear modelling of local processes; exploration of theuniqueness of features of the FLRW models; and tests of the viability of non-FLRWmodels. The uniqueness referred to here may lie in characteristics thought to bepeculiar to the standard model; in attempted proofs that no model universe could2These discoveries made it possible for disagreement with the 1980s dogmatism on such mattersto at last be listened to.3Local measures of homogeneity merely tell us that the spatial gradients of cosmic quantities arenot too strong near us.

be anisotropic or inhomogeneous, by proving that any strong departures from thestandard model decay away during evolution; or in comparisons with observation, toshow that only the standard models fit.Some defects of the present survey should be noted. One is that the matter contentis generally assumed to be a perfect fluid, although this is strictly incompatible withthe other assumed physical properties.

Attempting to remedy this with some othermathematically convenient equation of state is not an adequate response; one must tryto base the description of matter on a realistic model of microscopic physics or ther-modynamics, and few have considered such questions [Bradley and Sviestins, 1984,Salvati et al., 1987, Bona and Coll, 1988, Romano and Pavon, 1992].A second limitation is that we can only explore the mathematically tractablesubsets of models4, which may be far from representative of all models. To avoidthis restriction, we will ultimately have to turn to numerical simulations, includingfully three-dimensional variations in the initial data.

Some excellent pioneering workhas of course been done, e.g. Anninos et al.

(1991b), but capabilities are still limited(for example Matzner (1991) could only use a space grid of 313 points and 256 timesteps). Moreover, before one can rely on numerical simulations one needs to provesome structural stability results to guarantee that the numerical and exact answerswill correspond.As a final limitation, in giving this review I only had time to mention and discusssome selected papers and issues, not survey the whole vast field.

For his mammothsurvey of all inhomogeneous cosmological models which contain, as a limiting case, theFLRW models, Krasinski now has read about 1900 papers (as reported at the GR13conference in 1992)5. Thus the bibliography is at best a representative selection frommany worthy and interesting papers, and authors whose work is unkindly omittedmay quite reasonably feel it is unrepresentative.

In particular, I have not attemptedto cover the higher-dimensional models discussed by Demaret and others.2Spatially-homogeneous anisotropic models4Kramer et al. (1980) provides a detailed survey of those classes of relativistic spacetimes wherethe Einstein field equations are sufficiently tractable to be exactly solved.5The survey is not yet complete and remains to be published, but interim reports have appearedin some places, e.g.

Krasinski (1990).

2.1Metrics and field equationsAs already mentioned, this class consists of the Bianchi and Kantowski-Sachs models.They have the advantage that the Einstein equations reduce to a system of ordinarydifferential equations, enabling the use of techniques from dynamical systems theory,and there is thus again a vast literature, too big to fully survey here.The Bianchi models can be defined as spacetimes with metricsds2 = −dt2 + gαβ(t)(eαµdxµ)(eβνdxν)where the corresponding basis vectors {eα} obey[eα, eβ] = Cγαβeγin which the C’s are the structure constants of the relevant symmetry group. Thedifferent Bianchi-Behr types I-IX are then defined (see e.g.

Kramer et al. (1980)) byalgebraic classification of these sets of structure constants.The Kantowski-Sachs metric isds2 = −dt2 + a2(t) dx2 + b2(t)(dθ2 + sin2 θ dφ2).

(The other metric given in the original paper of Kantowski and Sachs was in fact aBianchi metric, as pointed out by Ellis. )The adoption of methods from the theory of dynamical systems has considerablyadvanced the studies of the behaviour of Bianchi models, beginning in the early 70swith the discussion of phase portraits for special cases [Collins, 1971].

Subsequently,more general cases were discussed using a compactified phase space. In the last decadethese methods have been coupled with the parametrization of the Bianchi mod-els using automorphism group variables [Collins and Hawking, 1973, Harvey, 1979,Jantzen, 1979, Siklos, 1980, Roque and Ellis, 1985, Jaklitsch, 1987].The automorphism group can be briefly described as follows.

Take a transforma-tionˆeα = Mαβeβ.This is an automorphism of the symmetry group if the {ˆeα} obey the same commu-tation relations as the {eα}. The matrices M are time-dependent and are chosenso that the new metric coefficients ˆgαβ take some convenient form, for example,become diagonal.The real dynamics is in these metric coefficients.This idea is

present in earlier treatments which grew from Misner’s methods for the Mixmastercase [Ryan and Shepley, 1975] but unfortunately the type IX case was highly mis-leading in that for Bianchi IX (and no others except Bianchi I) the rotation group isan automorphism group.The compactification of phase space, introduced for general cases by S.P. Novikovand Bogoyavlenskii (see Bogoyavlenskii (1985)) entailed the normalization of config-uration variables to lie within some bounded region, which was then exploited by(a) finding Lyapunov functions, driving the system near the boundaries of the phasespace and (b) using analyticity, together with the behaviour of critical points andseparatrices, to derive the asymptotic behaviour.Three main groups have developed these treatments: Bogoyavlenskii and his col-leagues (op.

cit); Jantzen, Rosquist and collaborators (e.g. Jantzen (1984), Rosquistet al.

(1990)) who have coupled the automorphism variables with Hamiltonian treat-ments in a powerful formalism; and Wainwright and colleagues (e.g. Wainwright andHsu (1989)) who have used a different, and in some respects simpler, set of automor-phism variables, which are well-suited for studying asymptotic behaviour becausetheir limiting cases are physical evolutions of simpler models rather than singularbehaviours.

Similar ideas can be used for the Kantowski-Sachs models too. As wellas qualitative results, some of them described below, these methods have enablednew exact solutions to be found, and some general statements about the occurrenceof these solutions to be made.Many of the geometrical properties of Bianchi cosmologies can be carried over tocases where the 3-dimensional symmetry group (which is still classifiable by Bianchitype) acts on timelike surfaces.

A number of authors have considered such metrics,for example Harness (1982) and myself and Siklos (1992). Although of less interest,since they do not evolve in time, than the usual Bianchi models, some of these modelsreappear as (spatially) inhomogeneous static or stationary models below.Since the present-day universe is not so anisotropic that we can readily detect itsshear and vorticity, the Bianchi models can be relevant to cosmology only as modelsof asymptotic behaviour, in the early or late universe, or over long time scales, suchas the time since the “last scattering”.

They have also been used, in these contexts,as approximations in genuinely inhomogeneous universes, and one has to be carefulto distinguish the approximate and exact uses.

2.2Asymptotic behaviour: the far past and futureThe earliest use of anisotropic cosmological models to study a real cosmological prob-lem was the investigation by Lemaˆıtre (1933) of the occurrence of singularities inBianchi type I models. The objective was to explore whether the big-bang whicharose in FLRW models was simply a consequence of the assumed symmetry: it wasof course found not to be.One can argue that classical cosmologies are irrelevant before the Planck time,but until a theory of quantum gravity is established and experimentally verified (ifindeed that will ever be possible) there will be room for discussions of the behaviourof classical models near their singularities.In the late 1950s and early 60s Lifshitz and Khalatnikov and their collaboratorsshowed (a) that singularities in synchronous coordinates in inhomogeneous cosmolo-gies were in general ‘fictitious’ and (b) that a special subclass gave real curvaturesingularities, with an asymptotic behaviour like that of the Kasner (vacuum BianchiI) cosmology [Lifshitz and Khalatnikov, 1963].

From these facts they (wrongly) in-ferred that general solutions did not have singularities. This contradicted the latersingularity theorems (for which see Hawking and Ellis (1973)), a disagreement whichled to the belief that there were errors in LK’s arguments.

They themselves, in col-laboration with Belinskii, and independently Misner, showed that Bianchi IX modelsgave a more complicated, oscillatory, behaviour than had been discussed in the ear-lier work, and Misner christened this the ‘Mixmaster’ universe after a brand of foodmixer. The broad picture of the rˆoles of the Kasner-like and oscillatory behaviourshas been borne out by the more rigorous studies by the methods described in theprevious section.

There is also an interesting and as yet incompletely explored resultthat after the oscillatory phase many models approximate one of a few particularpower-law (self-similar) solutions [Bogoyavlenskii, 1985].The detailed behaviour of the Mixmaster model has been the subject of still-continuing investigations: some authors argue that the evolution shows ergodic andchaotic properties, while others have pointed out that the conclusions depend cru-cially on the choice of time variable [Barrow, 1982, Burd et al., 1990, Berger, 1991].Numerical investigations are tricky because of the required dynamic range if one isto study an adequately large time-interval, and the difficulties of integrating chaoticsystems.The extension of these ideas to the inhomogeneous case, by Belinskii, Lifshitzand Khalatnikov, has been even more controversial, though prompting a smaller

literature. It was strongly attacked by Barrow and Tipler (1979) on a number oftechnical grounds, but one can take the view that these were not as damaging to thecase as Barrow and Tipler suggested [Belinskii et al., 1980, MacCallum, 1982].

In-deed the ‘velocity-dominated’ class whose singularities are like the Kasner cosmologyhave been more rigorously characterized and the results justified [Eardley et al., 1971,Holmes et al., 1990]. Sadly this does not settle the more general question, and at-tempts to handle the whole argument on a completely rigorous footing6 have so farfailed.General results about singularity types have been proved.

The ‘locally extendible’singularities, in which the region around any geodesic encountering the singularitycan be extended beyond the singular point, can only exist under strong restrictions[Clarke, 1976], while the ‘whimper’ singularities [King and Ellis, 1973], in which cur-vature invariants remain bounded while curvature components in some frames blowup, have been shown to be non-generic and unstable [Siklos, 1978].Examples ofthese special cases were found among Bianchi models, and both homogeneous andinhomogeneous cosmologies have been used as examples or counter-examples in thedebate.A further stimulus to the study of singularities was provided by Penrose’s con-jecture that gravitational entropy should be low at the start of the universe and thiswould correspond to a state of small or zero Weyl tensor [Penrose, 1979, Tod, 1992].Many authors have also considered the far future evolution (or, in closed models,the question of recollapse, whose necessity in Bianchi IX models lacked a rigorousproof until recently [Lin and Wald, 1991]). From various works [MacCallum, 1971,Collins and Hawking, 1973, Barrow and Tipler, 1978] one finds that the homogeneousbut anisotropic models do not in general settle down to an FLRW-like behaviourbut typically generate shears of the order of 25% of their expansion rates; see also[Uggla et al., 1991].

From the dynamical systems treatments, it is found that cer-tain exact solutions (which in general have self-similarity in time) act as attractorsof the dynamical systems in the future [Wainwright and Hsu, 1989]. (All such ex-act solutions are known: see Hsu and Wainwright (1986) and Jantzen and Rosquist(1986).

)This last touches on an interesting question about our account of the evolutionof the universe: is it structurally stable, or would small changes in the theory ofthe model parameters change the behaviour grossly? Several instances of the latterphenomenon, ‘fragility’, have recently been explored by Tavakol, in collaboration with6One of them made by Smallwood and myself.

Coley, Ellis, Farina, Van den Bergh and others [Coley and Tavakol, 1992].2.3Long time effects: the cosmic microwave backgroundTo test the significance of the observed isotropy of the CMWBR, many people inthe 1960s and 70s computed the angular distribution of the CMWBR temperature inBianchi models (e.g. Thorne (1967), Novikov (1968), Collins and Hawking (1972), andBarrow et al.

(1983)). These calculations allow limits to be put on small deviationsfrom isotropy from observation, and also enabled, for example, the prediction of ‘hotspots’ in the CMWBR in certain Bianchi models, which could in principle be searchedfor, if there were a quadrupole component, as there is in the COBE data (thoughperhaps not for this reason), to see if the quadrupole verifies one of those models.Similar calculations, by fewer people, considered the polarization [Rees, 1968,Anile, 1974, Tolman and Matzner, 1984] and spectrum [Rees, 1968, Rasband, 1971].More recently still, work has been carried out on the microwave background in someinhomogeneous models [Saez and Arnau, 1990].

It has been shown that pure rotation(without shear) is not ruled out by the CMWBR [Obukhov, 1992], but this result maybe irrelevant to the real universe where shear is essential to non-trivial perturbations[Goode, 1983, Dunsby, 1992]; in any case shearfree models in general relativity are avery restricted class [Ellis, 1967].An example of the problem with assuming a perfect fluid is that in Bianchimodels, as soon as matter is in motion relative to the homogeneous surfaces (i.e.becomes ‘tilted’) it experiences density gradients which should lead to heat fluxes[Bradley and Sviestins, 1984]: similar remarks apply to other simple models. Suchmodels have recently been used to fit the observed dipole anisotropy in the CMWBR[Turner, 1992], though other explanations seem to me more credible.2.4Early universe effectsGalaxy formation in anisotropic models has been studied to see if by this means onecould overcome the well-known difficulties in FLRW models (without inflation), butwith negative results [Perko et al., 1972].A similar investigation was to see if the helium abundance, as known in the 1960s,could be fitted better by anisotropic cosmologies than by FLRW models, which at the

time appeared to give discrepancies. The reason this might happen is that anisotropyspeeds up the evolution between the time when deuterium can first form, becauseit is no longer dissociated by the photons, and the time when neutrons and protonsare sufficiently sparse that they no longer find each other to combine.

Hawking andTayler (1966) were pioneers in this effort, which continued into the 1980s but sufferedsome mutations in its intention.First the argument was reversed, and the good agreement of FLRW predictionswith data was used to limit the anisotropy during the nucleogenesis period (see e.g.Barrow (1976), Olson (1977)). Later still these limits were relaxed as a result of con-sidering the effects of anisotropic neutrino distribution functions [Rothman and Matzner, 1982]and other effects on reaction rates [Juszkiewicz et al., 1983].

It has even been shown[Matravers et al., 1984, Barrow, 1984] that strongly anisotropic Bianchi models, notobeying the limits deduced from perturbed FLRW models, can produce correct ele-ment abundances, though they may violate other constraints [Matravers and Madsen, 1985,Matravers et al., 1985].3Inhomogeneous cosmologies3.1Self-similar modelsSome of the self-similar models, especially those relevant to modelling structure for-mation, are reviewed in much greater detail in a complementary talk by Carr, so Iwill give here only a few details of other cases.The geometry of the self-similar models first considered in cosmology is somewhatlike that of the Bianchi models, except that one of the isometries is replaced by ahomothety, that is to say by a vector field satisfyingξ(a;b) = 2kgabwhere k is a constant. This class, where the homothety and two independent sym-metries act, was considered by a number of authors [Eardley, 1974, Luminet, 1978,Wu, 1981, Hanquin and Demaret, 1984]7, and many details, parallel in nature tothose covered by the detailed studies of Bianchi models, can be found in those works.7Due to Western confusion over Chinese name order, Wu Zhong-Chao is sometimes incorrectlyreferred to as W.Z.

Chao rather than Wu, Z-C.

More recently Wainwright, Hewitt and colleagues [Hewitt et al., 1988, Hewitt and Wainwright, 1Hewitt et al., 1991] have considered cases where the homothety has a timelike ratherthan spacelike generator. Like the former class, these solutions are in fact special casesof “G2 solutions” (discussed below) with perfect fluid matter content.

It is found thatthe spatial variations can be periodic or monotone; the asymptotic behaviour maybe a vacuum or spatially homogeneous model; the periodic cases are unstable toincreases in the anisotropy; and the singularities can be acceleration-dominated.3.2Spherically symmetric modelsThese have a metricds2 = −e2νdt2 + e2λdr2 + R2(dθ2 + sin2 θ dφ2)where ν, λ and R are functions of r and t. The precise functional forms in the metricdepend on the choice of coordinates and the additional restrictions assumed. It shouldbe noted that there are so few undetermined functions that a sufficiently-complicatedenergy-momentum will fit a totally arbitrary choice of the remaining functions: in myview this should not be regarded as a solution, since no equation is actually solved!Some important subcases have been studied, notably:[1] The dust (pressureless perfect fluid) cases, originally studied by Lemaˆıtre, butusually named after Tolman and Bondi;[2] McVittie’s 1933 solution representing a black hole in an FLRW universe;[3] The “Swiss cheese” model constructed by matching a Schwarzschild vacuumsolution inside some sphere to an exterior FLRW universe;[4] Shearfree fluid solutions [Wyman, 1946, Kustaanheimo and Qvist, 1948, Stephani, 1983,McVittie, 1984];[5] Self-similar solutions, discussed in Carr’s contribution at Pont d’Oye.Spherically symmetric models, especially Tolman-Bondi, have often been used tomodel galactic scale inhomogeneities, in various contexts.

Galaxy formation has beenstudied (e.g. Tolman (1934), Carr and Yahil (1990)): Meszaros (1991) developed avariation on the usual approach by considering the shell-crossings, with the aim ofproducing “Great Wall” like structures, rather than the collapse to the centre pro-ducing a spherical cluster or galaxy.

Some authors have used spherically symmetriclumps to estimate departures from the simple theory of the magnitude-redshift re-lations based on a smoothed out model8 (e.g. Dyer (1976), Kantowski (1969a) and8The point is that the beams of light we observe are focussed only by the matter actually insidethe beam, not the matter that would be there in a completely uniform model.

Newman (1979)): note that these works show that the corrections depend on thechoice of modelling, since Newman’s results from a McVittie model differ from theones based on Swiss cheese models. The metrics also give the simplest models ofgravitational lenses9 and have also been used to model the formation of primordialblack holes [Carr and Hawking, 1974].On a larger scale, inhomogeneous spherical spacetimes have been used to modelclusters of galaxies [Kantowski, 1969b], variations in the Hubble flow due to the super-cluster [Mavrides, 1977], the evolution of cosmic voids [Sato, 1984, Hausman et al., 1983,Bonnor and Chamorro, 1990 and 1991], the observed distribution of galaxies and sim-ple hierarchical models of the universe [Bonnor, 1972, Wesson, 1978, Wesson, 1979,Ribeiro, 1992a].

Most of this work used Tolman-Bondi models, sometimes with dis-continuous density distributions.Recent work by Ribeiro (1992b), in the course of an attempt to make simplemodels of fractal cosmologies using Tolman-Bondi metrics, has reminded us of theneed to compare data with relativistic models not Newtonian approximations. Takingthe Einstein-de Sitter model, and integrating down the geodesics, he plotted thenumber counts against luminosity distances.At small distances, where a simpleinterpretation would say the result looks like a uniform density, the graph is irrelevantbecause the distances are inside the region where the QDOT survey shows things arelumpy [Saunders et al., 1991], while at greater redshifts the universe ceases to have asimple power-law relation of density and distance.

Thus even Einstein-de Sitter doesnot look homogeneous!One must therefore ask in general “do homogeneous models look homogeneous?”.Of course, they will if the data is handled with appropriate relativistic corrections, butto achieve such comparisons in general requires the integration of the null geodesicequations in each cosmological model considered, and, as those who have tried itknow, even when solving the field equations is simple, solving the geodesic equationsmay not be.Many other papers have considered spherically symmetric models, but there isnot enough space here to review them all, so I will end by mentioning a jeu d’espritin which it was shown that in a “Swiss cheese” model, made by joining two FLRWexteriors at the two sides of a Kruskal diagram for the Schwarzschild solution, onecan have two universes each of which can receive (but not answer) a signal from theother [Sussman, 1985].9The very detailed modern work interpreting real lenses to study various properties of individualsources and the cosmos mostly uses linearized approximations.

3.3Cylindrically symmetric and plane symmetric (static)modelsThese have been used to model cosmic strings and domain walls. One should notethat locally the metrics may be the same for these two cases, the difference lying inwhether there is or is not a Killing vector whose integral curves have spatial topologyS1.

Plane symmetric metrics should have a rotational symmetry in the plane, butto add to the possible confusions some authors use the term “plane” for solutionswithout such a rotation: the term “planar” would be a useful alternative.The usual (though not the only) form for the cylindrically symmetric metrics isds2 = −T 2dt2 + R2dr2 + Z2dz2 + 2W dz dφ + Φ2dφ2where T, R, Z, W and Φ depend on r (and, in the non-stationary case, t) and φ isperiodic, and, for the plane symmetric case,ds2 = −T 2dt2 + R2dr2 + X2(dx2 + x2dφ2)where T, R and X are functions of r (and perhaps t). The static cases all belong inHarness’s (1982) general class.Plane symmetric models, usually static, solutions have been used to model domainwalls [Vilenkin, 1983, Ipser and Sikivie, 1984, Goetz, 1990, Wang, 1991b]10.

The cylin-drically symmetric models have been used for cosmic strings, starting with the workof Gott, Hiscock and Linet in 1985. These studies have usually been done with staticstrings11, and have considered such questions as the effects on classical and quantumfields in the neighbourhood of the string.3.4G2 cosmologiesI use the above title as a general name for all cosmological metrics with two spacelikeKilling vectors (and hence two essential variables).

The cylindrical and plane metrics,and many of the Bianchi metrics, are special cases of G2 cosmologies.G2 cosmologies admit a number of specializations, such as:10Note that since the sources usually have a boost symmetry in the timelike surface giving the wall,corresponding solutions have timelike surfaces admitting the (2+1)-dimensional de Sitter group.11There is some controversy about whether these can correctly represent strings embedded in anexpanding universe [Clarke et al., 1990].

[1] the Killing vectors commute;[2] the orbits of the G2 are orthogonal to another set of 2-dimensional surfaces V2;[3] the Killing vectors individually are hypersurface-orthogonal;[4] the matter content satisfies conditions allowing generating techniques.Among the classes of metrics covered here are colliding wave models, cosmologies withsuperposed solitonic waves, and what I call “corrugated” cosmologies with spatialirregularities dependent on only one variable.The metrics where the Killing vectors do not commute have been very little stud-ied: it is known they cannot admit orthogonal V2 if the fluid flows orthogonal to thegroup surfaces (unless they have an extra symmetry) and that if the fluid is thusorthogonal it is non-rotating [Bugalho, 1987, van den Bergh, 1988]. So we now takeonly cases where the Killing vectors commute.The case without orthogonal V2 has also been comparatively little studied, butrecently some exact solutions which have one hypersurface-orthogonal Killing vectorand in which the metric coefficients are separable, have been derived and studied[van den Bergh et al., 1991, van den Bergh, 1991].

One class consists of metrics ofthe formds2 = e2(K+k)(−dt2 + dx2) + e2(S+s)[(eF +fdy)2 + (e−(F +f)θ)2]where: K, S and F depend on t; k, s and f depend on x; θ = dz + 2ω dx; and ωdepends on t and x. Some perfect fluid solutions are known explicitly but usuallyturned out to be self-similar, with big-bang singularities of the usual types.

The “stifffluid” (γ = 2) is a special case, discussed in detail by van den Bergh (1991). Mostof the solutions have singularities at finite spatial distances or can be regarded asinhomogeneous perturbations of the Bianchi V I−1 models.The cases with orthogonal V2 were classified by Wainwright (1979;1981), and anumber of specific examples are known (e.g.

Wainwright and Goode (1980); Kramer(1984)). A recent solution found by Senovilla (1990) attracted much attention, be-cause it is non-singular [Chinea et al., 1991], evading the focussing conditions in thesingularity theorems by containing matter that is too diffuse: it is closely relatedto an earlier solution of Feinstein and Senovilla (1989)12.

The metrics investigatedin this class generally have Kasner-like behaviour near the singularity (though somehave a plane-wave asymptotic behaviour [Wainwright, 1983]) and become self-similaror spatially homogeneous in the far future.Finally we come to the most-studied class, those where the generating techniques12Some recent work has given generalizations of these solutions [Ruiz and Senovilla, 1992,van den Bergh and Skea, 1992]; also S.W. Goode at GR13 (unpublished).

are applicable. The matter content must have characteristic propagation speeds equalto the speed of light, so attention is restricted to vacuum, electromagnetic, neu-trino and “stifffluid” (or equivalently, massless scalar field with a timelike gradient)cases.

However, FLRW fluid solutions can be obtained by using the same methods inhigher-dimensions and using dimensional reduction. There are useful reviews cover-ing the cosmological, cylindrical, and colliding wave sub-classes [Carmeli et al., 1981,Verdaguer, 1985, Verdaguer, 1992, Ferrari, 1990, Griffiths, 1991].

The metrics can bewritten in a form covering also the related stationary axisymmetric metrics asds2 = ǫfABdxAdxB + δe2γ((dx4)2 −ǫ(dx3)2)/fwhere A, B take values 1, 2 and the values of fAB can be written as a matrix f−fω−fωfω2 + ǫ(x3)2/f!The case δ = −ǫ = 1 gives the stationary axisymmetric metrics, the case δ = ǫ = 1the cylindrical cases and ǫ = −δ = 1 the cosmological cases. Physically these classesdiffer in the timelike or spacelike nature of the surfaces of symmetry and the natureof the gradient of the determinant of the metric in those surfaces.Some studies have focussed on the mathematics, showing how known vacuum so-lutions can be related by solution-generating techniques [Kitchingham, 1984], whileothers have concentrated on the physics of the evolution and interpretative issues.The generating techniques use one or more of a battery of related methods: B¨acklundtransformation, inverse scattering, soliton solutions and so on.

One interesting ques-tion that has arisen from recent work is whether solitons in relativity do or do notexhibit non-linear interactions: Boyd et al. (1991), in investigations of solitons ina Bianchi I background, found no non-linearity, while Belinskii (1991) has claimedthere is a non-linear effect (see also Verdaguer (1992)).The applications in cosmology, which have generated far too many papers to listthem all here, have been pursued by a number of groups, notably by Carmeli, Charachand Feinstein, by Verdaguer and colleagues, by Gleiser, Pullin and colleagues, andBelinski, Curir and Francaviglia, with important contributions by Ibanez, Kitching-ham, Yurtsever, Ferrari, Chandrasekhar and Xanthopoulos, Letelier, Tsoubelis andWang and many others.One use of these metrics is to provide models for universes with gravitationalwaves.

It emerges that the models studied are typically Kasner-like near the sin-gularity (agreeing with the LK arguments), and settle down to self-similar or spa-tially homogeneous models with superposed high-frequency gravitational waves at

late times [Adams et al., 1982, Carmeli and Feinstein, 1984, Feinstein, 1988].An-other use is to model straight cosmic strings in interaction with gravitational orother waves (e.g. Economou and Tsoubelis (1988), Verdaguer (1992)).One canalso examine the gravitational analogue of Faraday rotation [Piran and Safier, 1985,Tomimatsu, 1989, Wang, 1991a] and there are even solutions whose exact behaviouragrees precisely with the linearized perturbation calculations for FLRW universes[Carmeli et al., 1983].3.5Other modelsSolutions with less symmetry than those above have been little explored.

FollowingKrasinski one can divide the cases considered into a number of classes (in which Ionly mention a few important special subcases).1. The Szekeres-Szafron family (also independently found by Tomimura).

Thesehave in general no symmetries, and contain an irrotational non-acceleratingfluid. Tolman-Bondi universes are included in this class, as are the Kantowski-Sachs metrics; some generalizations are known, such as the rotating inhomoge-neous model due to Stephani (1987).

Like the G2 solutions mentioned earlier,some Szekeres models obey exactly the linearized perturbation equations forthe FLRW models [Goode and Wainwright, 1982].2. Shearfree irrotational metrics [Barnes, 1973] which include the conformally flatfluids (Stephani 1967a, 1967b) and McVittie’s spherically symmetric metric.Bona and Coll (1988) have recently argued that the Stephani cases can onlyhave acceptable thermodynamics if the metrics admit three Killing vectors.3.

The Vaidya-Patel-Koppar family, which represent an FLRW model contaning a“Kerr” solution using null radiation and an electromagnetic field. The physicalsignificance of these metrics is dubious.4.

Some other special cases such as Oleson’s Petrov type N fluid solutions.4Syntheses and conclusions: what have we learnt?Here I collect up the outcome of the work surveyed above, without repeating alldetails, and review some relevant extra references, but many interesting aspects are

still omitted. For example, the literature covers such issues as models for interactionsbetween different forms of matter, and generation of gravitational radiation.4.1The classical singularityThe occurrence of a “big-bang” in FLRW models is not just a consequence of thehigh symmetry.

Its nature in general models is probably a curvature singularity, andthe best guess so far is that the asymptotic behaviour would be oscillatory but otherpossibilities exist. The Penrose conjecture, which would be a selection principle onmodels, has been particularly developed, using exact solutions as examples, by Wain-wright and Goode, who have given a precise definition of the notion of an ‘isotropicsingularity’ [Goode et al., 1992, Tod, 1992].4.2Occurrence of inflationIn “old” inflation in Bianchi I models, inflation need not occur [Barrow and Turner, 1981],but in “new” inflation it was predicted [Steigman and Turner, 1983].

In a large classof chaotic inflation models it is also expected [Moss and Sahni, 1986]. Further pa-pers by a number of authors have suggested that inflation need not always occur (seeRothman and Ellis (1986) for some criticisms of earlier papers).4.3Removal of anisotropy and inhomogeneityThree means of smoothing the universe have been explored over the years: the use ofviscosity in the early universe; the removal of horizons in the Mixmaster universes;and removal during inflation.

The first two of these ingenious suggestions are due toMisner.Attempts to smooth out anisotropies or inhomogeneities by any process obeyingdeterministic sets of differential equations satisfying Lipschitz-type conditions aredoomed to fail, as was first pointed out by Collins and Stewart (1971) in the contextof viscous mechanisms. The argument is simply that one can impose any desiredamount of anisotropy or inhomogeneity now and evolve the system backwards intime to reach initial conditions at some earlier time whose evolution produces thechosen present-day values.

The same argument also holds for inflationary models. Inflation in itself, withoutthe use of singular equations or otherwise indeterminate evolutions, cannot wholly ex-plain present isotropy or homogeneity, although it may reduce deviations by large fac-tors [Sirousse-Zia, 1982, Wald, 1983, Moss and Sahni, 1986, Futamase et al., 1989].Although one can argue that anisotropy tends to prolong inflation, this does notremove the difficulty.Since 1981 I have been arguing a heretical view about one of the grounds forinflation, namely the ‘flatness problem’, on the grounds that the formulation of thisproblem makes an implicit and unjustified assumption that the a priori probabilitiesof values of Ωis spread over some range sufficient to make the observed closeness to1 implausible.

Unless one can justify the a priori distribution, there is no implausi-bility13 [Ellis, 1991].However, if one accepts there is a flatness problem, then there is also an isotropyproblem, since at least for some probability distributions on the inhomogeneity andanisotropy the models would not match observation. Protagonists of inflation cannothave it both ways.

Perhaps, if one does not want to just say “well, that’s how theuniverse was born”, one has to explain the observed smoothness by appeal to the‘speculative era’, as Salam (1990) called it, i.e. by appeal to one’s favourite theory ofquantum gravity.If inflation works well at early times, then inflation actually enhances the chanceof an anisotropic model fitting the data, and since the property of anisotropy cannotbe totally destroyed in general (because it can be coded into geometric invariantswhich cannot become zero by any classical evolution) the anisotropy could reassertitself in the future!

(This of course will not happen if a non-zero Λ term persists, asthe “cosmic no-hair” theorems show [Wald, 1983, Morrow-Jones and Witt, 1988]. )The Mixmaster horizon removal suggestion was shown to fail when more detailedcomputations than Misner’s were made [MacCallum, 1971, Doroshkevich et al., 1971,Chitre, 1972].

Incidentally, one may note that inflation does not solve the originalform of the ‘horizon problem’, which was to account completely for the similarity ofpoints on the last scattering surface governed by different subsets of the inital datasurface. Inflation leads to a large overlap between these initial data subsets, but notto their exact coincidence.

Thus one still has to assume that the non-overlap regionsare not too different. While this may give a more plausible model, it does not remove13One can however argue that only Ω=1 is plausible, on the grounds that otherwise thequantum theory before the Planck time would have to fix a length-scale parameter much largerthan any quantum scale, only the Ω= 1 case being scale-free.

I am indebted to Gary Gibbons forthis remark.

the need for assumptions on the initial data.4.4The exit from inflationA further interesting application of non-standard models has come in a recent attemptto answer the question posed by Ellis and Rothman (unpublished) of how the universecan choose a uniform reference frame at the exit from inflation when a truly de Sittermodel has no preferred time axis. Anninos et al.

(1991a) have shown by taking aninflating Bianchi V model that the answer is that the memory is retained and theuniverse is never really de Sitter.4.5The helium abundanceThis is still used to set limits on anisotropy during the nucleosynthesis phase.4.6The cosmic microwave backgroundObservations limit the integrated effect since “last scattering”: note this can in princi-ple permit large but compensating excursions from FLRW. One intriguing possibilityraised by Ellis et al.

(1978) is that the observed sphere on the last scattering surfacecould lie on a timelike (hyper)cylinder of homogeneity in a static spherically sym-metric model. This makes the CMWBR isotropic at all points not only at the centre,and although it cannot fit all the other data, the model shows how careful one mustbe, in drawing conclusions about the geometry of the universe from observations, notto assume the result one wishes to prove.There is a theorem by Ehlers, Geren and Sachs (1968) showing that if a congruenceof geodesically-moving observers all observe an isotropic distribution of collisionlessgas the metric must be Robertson-Walker.

Treciokas and Ellis (1971) have inves-tigated the related problem with collisions. Recently Ferrando et al.

(1992) haveinvestigated inhomogeneous models where an isotropic gas distribution is possible.These studies throw into focus a conjecture which is usually assumed, namely that anapproximately isotropic gas distribution, at all points, would imply an approximatelyRobertson-Walker metric. (It is this assumption which underlies the arguments nor-mally used in analysing data like that from COBE to get detailed information on

allowed FLRW perturbations. )4.7The far futureAnisotropy will in general become apparent, if it is present and if the cosmologicalconstant Λ is zero: isotropy is not stable.

Inhomogeneities may become significanteven faster.4.8The origin of structureNone of the work discussed above accounts for the origin of structure, although itoffers suitable descriptions for the evolution, or the background spacetime in whichthe evolution takes place. I feel it does, however, indicate strongly that the true originlies in the perhaps unknowable situation in the Speculative Era, and the resultinginitial conditions for the later evolution.4.9A genuinely anisotropic and inhomogeneous universe?While I do not think one can give a definitive answer to this question, I wouldpersonally be very surprised if anisotropic but homogeneous models turned out tobe anything more than useful examples.

However, the status of fully inhomogeneousmodels is less clear.One argument is that while the standard models may be good approximationsat present, they are unstable to perturbations both in the past and the future. Thepossible alternative pasts are quite varied, as shown above, even without consideringquantum gravity.Similarly, the universe may not be isotropic in the far future.Moreover, we have no knowledge of conditions outside our past null cone, wheresome inflationary scenarios would predict bubbles of differing FLRW universes, andperhaps domain walls and so on.If the universe were FLRW, or very close to that, this means it is in a region, in thespace of all possible models, which almost any reasonable measure is likely to say hasvery low probability (though note the earlier remarks on assignments of probabilities).One can only evaluate, and perhaps explain, this feature by considering non-FLRW

models. It is noteworthy that many of the “problems” inflation claims to tackle arenot problems if the universe simply is always FLRW.

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