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Anisotropic and inhomogeneous cosmologies

arXiv:gr-qc/9212014v1 21 Dec 1992Anisotropic and inhomogeneous 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.uk1INTRODUCTIONMy first impressions of Dennis Sciama came from a short introductory astrophysicscourse he gave to undergraduates in 1964.

Then in 1966-7 I took his Cambridge PartIII course in relativity, in which he charitably ignored my inadvertent use of Euclideansignature in the examination (an error I spotted just at the very end of the allowedtime) and gave me a good mark. In both these courses he showed the qualities ofenthusiasm and encouragement of students with which I was to become more familiarlater in 1967 when I began as a research student.A project on stellar structurehad taught me that I did not want to work on that, and I began under Denniswith the idea of looking at galaxy formation.

However, by sharing an office withJohn Stewart I came to read John’s paper with George Ellis [Stewart and Ellis, 1968]and its antecedent [Ellis, 1967] and developed an interest in relativistic cosmologicalmodels, which led to George becoming my second supervisor.I was still in Sciama’s group, and I learnt a lot from the tea-table conversations,which seemed to cover all of general relativity and astrophysics. Dennis taught us byexample that the field should not be sub-divided into mathematics and physics, orcosmological and galactic and stellar, but that one needed to know about all thosethings to do really good work.

Of course he was not uncritically enthusiastic: hisown opinions were strongly enough held that we used to joke that if we wanted tostop research all we need do was say loudly in the tea-room that we did not believein Mach’s principle. But it was a very supportive and stimulating atmosphere forwhich I will always be grateful.This is a review of what we have learnt from the study of non-standard cos-

mologies in which I got involved 25 years ago. Only exact solutions will be consid-ered: the perturbation theory will be left for others to discuss.

In earlier reviews[MacCallum, 1979, MacCallum, 1984] I started from the mathematical classificationof the solutions but here I want to take a different route and consider the applicationareas. So let me just quickly remind readers of the general groups of models to whichI will later refer.

They are:[1] Spatially-homogeneous and isotropic models. In relativity these give the Friedman-Lemaˆıtre-Robertson-Walker cosmologies, and the “standard model”.

[2] Spatially-homogeneous but anisotropic models. These are the Bianchi models,in general, the exceptions being the Kantowski-Sachs models with an S2 × R2topology.

[3] Isotropic but inhomogeneous models. These are spherically symmetric models,whose dust subcases, having been first discussed by Lemaˆıtre (1933), are calledTolman-Bondi models.

[4] Models with two ignorable coordinates, usually with a pair of commuting Killingvectors. These may be plane or cylindrically symmetric.

[5] Models with less symmetry than those above.Only a few special cases areknown exactly.In giving this review I only had time to mention and discuss some selected papersand issues, not survey the whole vast field. Thus the bibliography is at best a rep-resentative selection from many worthy and interesting papers, and authors whosework is unkindly omitted may quite reasonably feel it is unrepresentative.2OBSERVATIONS, THE STANDARD MODELAND ALTERNATIVESWhat is it that a cosmological model should explain?

There are the following mainfeatures:[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.The standard big-bang model at the time I started as a student was:[1] Isotropic at all points and thus necessarily. .

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

[3] Satisfied 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 decadeadded 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, thesource 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 strong1Why so anatomical?

for 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 the-ory has shown that inflation does not uniquely predict Ω= 1 and partly becauseobservations give somewhat variant values. Some authors have pointed out that ourknowledge of the physics valid at nucleogenesis and before is still somewhat uncertain,and we should retain some agnosticism towards our account of those early times.Finally, we should recognize that our belief in homogeneity has very poor obser-vational support.

We have data from our past light cone (and those of earlier humanastronomers) and from geological records [Hoyle, 1962]. Studying homogeneity re-quires us to know about conditions at great distances at the present time, whereaswhat we can observe at great distances is what happened a long time ago, so to testhomogeneity we have to understand the evolution both of the universe’s geometryand of its matter content3.

Thus we cannot test homogeneity, only check that it isconsistent with the data and our understanding of the theory. The general beliefin homogeneity is indeed like the zeal of the convert, since until the 1950s, whenBaade revised the distance scale, the accepted distances and sizes of galaxies werenot consistent with homogeneity.These comments, however, are not enough to justify examination of other models.Why do we do that?

I think there are several reasons. Alternative models providefully non-linear modelling of local processes.

They may show whether characteristicsthought to be peculiar to the standard model, and thus a test of it, can occur else-where. They may be used in attempted proofs that no universe could be anisotropicor inhomogeneous, by proving that any strong departures from the standard modeldecay away during evolution.

They can be comparators in data analysis, to showthat only standard models fit. Finally, they may even be advanced as replacementsof the standard model.Before starting to examine how the alternatives fare in these various rˆoles, Imust point out two major defects in work up to now.

One is that the matter con-tent is almost always assumed to be a perfect fluid. Yet even in the simplest non-standard models, the Bianchi models, as soon as matter is in motion relative to thehomogeneous surfaces (i.e.

becomes ‘tilted’)4 it experiences density gradients which2These 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.4Such models have recently been used to fit the observed dipole anisotropy in the CMWBR[Turner, 1992], though other explanations seem to me more credible.

should lead to heat fluxes [Bradley and Sviestins, 1984]: similar remarks apply toother simple models. Attempting to remedy this with some other mathematicallyconvenient equation of state is not an adequate response; one must try to base thedescription of matter on a realistic model of microscopic physics or thermodynam-ics, and few have considered such questions [Salvati et al., 1987, Bona and Coll, 1988,Romano and Pavon, 1992].

The other objection is that we can only explore the math-ematically tractable subsets of models, which may be far from representative of allmodels.3MODELLING LOCAL NON-LINEARITIESCosmic strings have been modelled by cylindrically symmetric models, starting withthe work of Gott , Hiscock and Linet in 1985. These studies have usually been donewith static strings5, and have considered such questions as the effects on classical andquantum fields in the neighbourhood of the string.Similarly, exact solutions for domain walls, using plane symmetric models, usuallystatic, have been considered [Vilenkin, 1983, Ipser and Sikivie, 1984, Goetz, 1990,Wang, 1991]6.Galactic scale inhomogeneities have frequently been modelled by spherically sym-metric models, usually Tolman-Bondi.

They have been used to study galaxy forma-tion (e.g. Tolman (1934), Carr and Yahil (1990), and Meszaros (1991)), to estimatedepartures from the simple theory of the magnitude-redshift relations based on asmoothed out model7 (e.g.

Dyer (1976), Kantowski (1969b) and Newman (1979):note that these works show that the corrections depend on the choice of modelling),and as the simplest models of gravitational lenses8.Spherically symmetric inho-mogeneities have also been used to model the formation of primordial black holes[Carr and Hawking, 1974].On a larger scale still inhomogeneous spacetimes have been used to model clusters5There is some controversy about whether these can correctly represent strings embedded in anexpanding universe [Clarke et al., 1990].6Note 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.7The 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.8The very detailed modern work interpreting real lenses to study various properties of individualsources and the cosmos mostly uses linearized approximations.

of galaxies [Kantowski, 1969a], variations in the Hubble flow due to the superclus-ter [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].The references just cited are only the tip of the iceberg. For his mammoth surveyof all inhomogeneous cosmological models which contain, as a limiting case, Friedman-Robertson-Walker models, Krasinski now has read about 1900 papers (as reportedat the GR13 conference in 19929).As well as the issues mentioned above, thesepapers discuss many others including models for interactions between different formsof matter, generation of gravitational radiation, and the nature of cosmic singularities.There is not enough space here to do all these arguments justice, and anywayit would be unfair to pre-empt Krasinski’s conclusions.

Moreover, I believe the is-sues for which I have given a few detailed references are (together with some ap-pearing later in this survey) the most important astrophysically.So I will justmention two more points which have arisen.One is that some exact non-linearsolutions obey exactly the linearized perturbation equations for the FLRW models[Goode and Wainwright, 1982, Carmeli et al., 1983]. The other is a jeu d’esprit inwhich 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].4WHICH FLRW PROPERTIES ARE SPECIAL?The 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.A later similar investigation was to see if the helium abundance, as known inthe 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 is9The survey is not yet complete and remains to be published, but interim reports have appearedin some places, e.g.

[Krasinski, 1990].

that anisotropy speeds up the evolution between the time when deuterium can firstform, because it is no longer dissociated by the photons, and the time when neutronsand protons are sufficiently sparse that they no longer find each other to combine.Hawking and Tayler (1966) were pioneers in this effort, which continued into the1980s but suffered some 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 models, not obeyingthe limits deduced from perturbed FLRW models, can also produce correct elementabundances, though they may violate other constraints [Matravers and Madsen, 1985,Matravers et al., 1985].The above properties turn out not to be special to FLRW geometry. One thatmight be thought to be is the exact isotropy of the CMWBR.

To test this, many peoplein the 1960s and 70s computed the angular distribution of the CMWBR temperaturein Bianchi models (e.g. Thorne (1967), Novikov (1968), Collins and Hawking (1972),and Barrow et al.

(1985)). These calculations allow limits to be put on small devia-tions from isotropy from observation, and also enabled, for example, the predictionof ‘hot spots’ in the CMWBR in certain Bianchi models, which could in principle besearched for, if there were a quadrupole component10, to see if the quadrupole verifiesone 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].One property, the nature of the big-bang singularity, as distinct from its existence,has been so extensively discussed as to demand a section of its own.10Which, since the meeting this survey was given at, has been shown to exist in the COBE data.

5THE ASYMPTOTIC BEHAVIOUR OF CLAS-SICAL COSMOLOGIESOne can argue that classical cosmologies are irrelevant before the Planck time, butuntil a theory of quantum gravity is established and experimentally verified (if indeedthat will ever be possible) there will be room for discussions of the behaviour ofclassical 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 [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 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 smallerliterature.

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 (vacuumBianchi) cosmology have been more rigorously characterized and the results justified[Eardley et al., 1971, Holmes et al., 1990].

Sadly this does not settle the more gen-eral question, and attempts to handle the whole argument on a completely rigorousfooting11 have so far failed.11One of them made by Smallwood and myself.

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].Studies of the behaviour of Bianchi models have been much advanced by theadoption of methods from the theory of dynamical systems.

In the early 70s thisbegan with the discussion of phase portraits for special cases [Collins, 1971] and wasextended in work in which (a) the phase space was compactified, (b) Lyapunov func-tions, driving the system near the boundaries of the phase space, were found and(c) analyticity together with the behaviour of critical points and separatrices wasused to derive the asymptotic behaviour [Bogoyavlenskii, 1985]. 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.

Writing the Bianchimetrics asds2 = −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, one usesa transformationˆeα = Mαβeβchosen so that the {ˆeα} obey the same commutation relations as the {eα}.Thematrices M are time-dependent and can be chosen so that the new metric coefficientsˆgαβ take some convenient form. The real dynamics is in these metric coefficients.The idea is present in earlier treatments which grew from Misner’s methods for theMixmaster case [Ryan and Shepley, 1975] but unfortunately the type IX case was

highly misleading in that for Bianchi IX (and no others except Bianchi I) the rotationgroup is an automorphism group.A long series of papers by Jantzen, Rosquist and collaborators [Jantzen, 1984,Rosquist et al., 1990] have coupled these ideas with Hamiltonian treatments in apowerful formalism. Using a different, and in some respects simpler, set of variables,Wainwright has also attacked the asymptotics problem [Wainwright and Hsu, 1989]:his variables are well-suited for those questions because their limiting cases are phys-ical evolutions of simpler models rather than singular behaviours.The conclusions of these studies have justified the work of Belinskii et al.

for thehomogeneous case (but do not affect the arguments about the inhomogeneous cases)and have enabled new exact solutions to be found and some general statements aboutthe occurrence of these solutions (which in general have self-similarity in time) to bemade [Wainwright and Hsu, 1989], in particular showing their rˆoles as attractors ofthe dynamical systems.The other class of models where techniques have improved considerably are themodels with two commuting Killing vectors, even when these vectors are not hypersurface-orthogonal. Some studies have focussed on the mathematics, showing how known vac-uum solutions can be related by solution-generating techniques [Kitchingham, 1984],while others have concentrated on the physics of the evolution of fluid models (notobtainable by generating techniques, except in the case of ‘stiff’ fluid, p = ρ) and in-terpretative issues [Wainwright and Anderson, 1984, Hewitt et al., 1991].

It emergesthat the models studied are typically Kasner-like near the singularity (agreeing withthe LK arguments), and settle down to self-similar or spatially homogeneous modelswith superposed high-frequency gravitational waves at late times. However, somecases have asymptotic behaviour near the singularity like plane waves, and othersare non-singular [Chinea et al., 1991].

The Penrose conjecture has been particularlydeveloped, using exact solutions as examples, by Wainwright and Goode, who havegiven a precise definition to the notion of an ‘isotropic singularity’ [Goode et al., 1992,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 behaviour buttypically generate shears of the order of 25% of their expansion rates.

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 withColey, Ellis, Farina, Van den Bergh and others [Coley and Tavakol, 1992].6DO NON-FLRW MODELS BECOME SMOOTH?Attempts to smoothe 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. This was one of the arguments which rebutted Misner’singenious suggestion that viscosity in the early universe could explain the presentlevel of isotropy and homogeneity regardless of the initial conditions.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].Objections to some specific calculations have been given [Rothman and Ellis, 1986].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-bility12 [Ellis, 1991].However, if one accepts there is a flatness problem, then there is also an isotropy12One 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.

problem, 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.Incidentally, one may note that inflation does not solve the original form of the‘horizon problem’, which was to account completely for the similarity of points onthe last scattering surface governed by different subsets of the inital data surface.Inflation leads to a large overlap between these initial data subsets, but not to theirexact coincidence.

Thus one still has to assume that the non-overlap regions are nottoo different. While this may give a more plausible model, it does not remove theneed for assumptions on the initial data.A further interesting application of non-standard models has come in a recentattempt to answer the question posed by Ellis and Rothman (unpublished) of howthe universe can choose a uniform reference frame at the exit from inflation when atruly de Sitter model has no preferred time axis.

Anninos et al. (1991a) have shownby taking an inflating Bianchi V model that the answer is that the memory is retainedand the universe is never really de Sitter.Finally, one may comment that if inflation works well at early times, then inflationactually enhances the chance of an anisotropic model fitting the data, and that sincethe property of anisotropy cannot be totally destroyed in general (because it is codedinto geometric invariants which cannot become zero by any classical evolution) theanisotropy could reassert itself in the future!7ASTROPHYSICAL AND OBSERVABLECONSEQUENCES OF NON-STANDARDMODELSGalaxy formation in anisotropic models has been studied to see if they could overcomethe well-known difficulties of FLRW models (without inflation), but with negativeresults [Perko et al., 1972].As mentioned above solutions with two commuting Killing vectors provide models

for universes with gravitational waves. Aspects of these models have been consideredby several authors, e.g.

Carr and Verdaguer (1983), Ibanez and Verdaguer (1983),Feinstein (1988). There are in fact several mathematically related but physically dis-tinct classes of solutions of the Einstein equations accessible by generating techniques:stationary axisymmetric spacetimes, colliding wave solutions (nicely summarized inGriffiths (1991) and Ferrari (1990)), and cosmological solutions, the differences aris-ing from the timelike or spacelike nature of the surfaces of symmetry and the natureof the gradient of the determinant of the metric in those surfaces.The generating techniques essentially work for forms of matter with characteristicpropagation speed equal to the velocity of light, and use one or more of a batteryof related techniques: B¨acklund transformation, inverse scattering, soliton solutionsand so on.

One interesting question that has arisen from recent work is whethersolitons in relativity do or do not exhibit non-linear interactions: Boyd et al. (1991),in investigations of solitons in a Bianchi I background, found no non-linearity, whileBelinskii (1991) has claimed there is a non-linear effect.Work on the observable consequences of non-standard models has been done bymany, as mentioned above.

One intriguing possibility raised by Ellis et al. (1978)is that the observed sphere on the last scattering surface could lie on a timelike(hyper)cylinder of homogeneity in a static spherically symmetric model.

This makesthe CMWBR isotropic at all points not only at the centre, and although it cannot fitall the other data, the model shows how careful one must be, in drawing conclusionsabout the geometry of the universe from observations, not to assume the result onewishes to prove.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 geodesic

equations 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.Ultimately we will have to refine our understanding with the help of numericalsimulations which can include fully three-dimensional variations in the initial data,and some excellent pioneering work has of course been done, e.g. Anninos et al.

(1991b), but capabilities are still limited (for example Matzner (1991) could only usea space grid of 313 points and 256 time steps). Moreover before one can rely onnumerical simulations one needs to prove some structural stability results.8IS THE STANDARD MODEL RIGHT?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 in section 5, even without con-sidering quantum gravity.

Similarly, as also mentioned in section 5, the universe maynot be isotropic in the far future. Moreover, there is the question of on what scale,if any, the FLRW model is valid.

Its use implies some averaging, and is certainlynot correct on small scales. Is it true on any scale?

If so, on what scales? Theremay be an upper as well as a lower bound, since we have no knowledge of conditionsoutside our past null cone, where some inflationary scenarios would predict bubblesof differing FLRW universes, and perhaps domain walls and so on.If the universe were FLRW, or very close to that, this means it is in a region, inthe space of all possible models, which almost any reasonable measure is likely to sayhas very low probability (though note the remarks on assignments of probabilities insection 6).

One can only evaluate, and perhaps explain, this feature by consideringnon-FLRW models. It is noteworthy that many of the “problems” inflation claimsto tackle are not problems if the universe simply is always FLRW.

Hence, as alreadyargued above, one has a deep problem in explaining why the universe is in the unlikelyFLRW state if one accepts the arguments about probabilities current in work oninflation.

Moreover, suppose we speculated that the real universe is significantly inhomo-geneous at the present epoch (at a level beyond that arising from perturbations inFLRW). What would the objections be?

There are only two relevant pieces of data,as far as I can see.One is the deep galaxy counts made by the automatic platemeasuring machines, which are claimed to restrict variations to a few percent, andthe other is the isotropy of the CMWBR. Although the latter is a good test for largelumps in a basically FLRW universe, one has to question (recalling the results of Elliset al.) whether it really implies homogeneity.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 thatan approximately isotropic gas distribution, at all points, would imply an approxi-mately Robertson-Walker metric. (It is this assumption which underlies some of thearguments used, for example, by Barrow in his talk at this meeting.

)Whether the standard model is correct or not, I feel confident in concluding thatone of the more outstanding inhomogeneities is the dedicatee of this piece, DennisSciama, and I hope some small part of his talents has been shown here to have beenpassed on to me. To show how it has influenced the subject, I have marked authorscited in the bibliography below who also appear in the Sciama family tree by anasterisk.I would like to thank G.F.R.

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