Observational and dynamical properties of small universes

Volume 115, number 3 PHYSICS LETTERS A 31 March 1986

O B S E R V A T I O N A L A N D D Y N A M I C A L P R O P E R T I E S O F S M A L L U N I V E R S E S

G.F .R . E L L I S 1 and G. S C H R E I B E R

Department of Applied Mathematics, University of Cape Town, Rondebosch 7700, Cape Town, South Africa

Received 9 December 1985; revised manuscript received 28 January 1986; accepted for publication 28 January 1986

It is possible that we live in a "small universe", that is, we might have seen round the universe many times since decoupling so that there are many images of each galaxy. We estimate the number of images per galaxy in different small universes, and review how apparent isotropy of galactic observations and background radiation measurements can naturally occur in such universes. Furthermore interesting physical effects may result; in particular, in some cases the dynamical behaviour of such space-times is necessarily like that of a Friedmann universe, so that inhomogeneous small universes may both look like and evolve like the standard isotropic and spatially homogeneous world models.

Introduction. A currently popular way of explaining the apparent isotropy o f the universe on large angular scales is to adopt the standard interpretat ion that this reflects spatial homogenei ty [1 ] , in turn explain- ed as due to an " inf la t ionary" period in the early universe [2,3]. However there is at present no direct ex- perimental or observational evidence supporting the inflationary scenario; and apart from the question of galaxy formation, there are various problems with inflation related bo th to the grand unified theory supposed to underly it, and the application of this theory to cosmology (see e.g. refs. [4,5 ] ).

One alternative way of explaining apparent homogeneity is the possibility that the universe might be a "small universe", that is, might have compact spatial sections with non-trivial topology [6,7], small enough that we have had time to see round the universe many times since decoupling, so that there are many images of each galaxy we observe [8 -11 ] . If this were true it would imply various attractive features, apart from explaining apparent homogenei ty [12] ; in particular, (a) the machian conditions resulting from having compact spatial sections, whose desirability has been em- phasized particularly by Einstein [13] and Wheeler [14,15] (see also ref. [16]) would be satisfied; (b)

l Address from 25 November until 25 December 1985: Max- Planck-Institute for Physics and Astrophysics, Karl Schwarzschildstrasse 1, D-8046 Garching, FRG.

0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

only in such universes do we have enough information to determine the structure of the entire universe, and to predict with certainty events in the future such as solar eclipses and the return of Halley's comet (in the standard universe models, we have observational access to only a very small part of the universe, and uncer- tainty arises because of the possible intrusion o f gravi- tat ional waves or other information travelling in from large distances at the speed of light from sources about which we have no information whatever [9,17] ; (c) even in the case of a low-density universe, only a ffmite number of galaxies would exist, avoiding problems related to the creation o f an inifinite amount of matter at one instant and the uneasthetic feature of infinite genetic duplication of each person [ 18 ] .

In this note we present some results on the dy- namics and observational properties of small universes which confirm the potential utility of the concept in explaining major features of our observations o f the universe.

The concept. The essential feature of a "small universe" is that it has compact spatial sections S(t) which are small enough that at the present time t o we can have seen at least once round the universe in every direction since the time of decoupling t d. More pre- cisely, we assume space - t ime is the direct product R X S O where

(a) S O is a compact three-manifold,

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(b) the point P corresponding to our present spacetime position in the universe ("here and now") lies in S(t0), and S(td) is the surface of decoupling of matter and radiation;

(c) S(t) are complete spacelike surfaces for t d < t < t o ; and,

(d) S(td) lies in the causal past J - (P ) of P. The implication is that the world-line of every fun-

damental observer passes through S(td) and intersects the past light cone C-(P) of P, and so all the matter present in the universe between t d and t o can have been seen at least once by an observer present at the event P (we neglect here the obscuring effect of inter- vening matter). For the resulting effects to be interesting, the scale of the surfaces S(t) would be such that we have seen many times round the universe since decoupling so that each massive luminous object is visible to us through many images [8-11 ].

To pursue the topic further, we need to distinguish two distinct cases: those of spatially homogeneous and spatially inhomogeneous universes.

(a) Spatially homogeneous small universes are necessarily locally identical to a Fr iedman-Lemait re- Robertson-Walker (FLRW) universe if isotropic, and either a Bianchi or Kantowski-Sachs model if aniso- tropic. These small universes can be regarded as obtained from those models by suitable identifications under a discrete group of isometrics; the original spacetime is then regained from the small universe by going to its universal covering space [1 ].

In these cases it is natural to choose the surfaces S(t) as the surfaces of homogeneity in the universe. These will be spacelike everywhere, and so will be Cauchy surfaces. This follows immediately in the FLRW and Kantowski-Sachs cases, and in all Bianchi cases [19] except the class B models where horizon- crossing and whimper singularities occur. One cannot make identifications to obtain a small universe in these exceptional class B universes, because then on the horizon a null Killing vector field is parallel to geodesic null rays that necessarily converge [20]. One can obtain small universes for each sign of the spatial curvature in the FLRW universes [8,21,22], so in particular they are possible in the low-density (k < O, q0 < 1/2) ever-expanding case. The surfaces S(t) will not be simply connected, and indeed many distinct topologies are possible for them; the simplest are the torus topology in the FLRW case k = O, and the elliptic topology

in the case k = + 1. In one exceptional case, no identifications need be made: if k = + 1 in a FLRW universe and there is a large positive cosmological constant, one will have time to see right round the compact (S 3) space-sections since decoupling. However we will not consider this case further here for two reasons: it has already been analysed extensively in the literature (e.g. refs. [23-26] ), and we are primarily concerned with the more standard situation where the cosmological constant vanishes.

Observational properties of these small universes are easily found by performing the relevant calculation in the associated FLRW, Kantowski-Sachs, or Bianchi model, and then making the appropriate identifications to obtain the small universe (fig. 1). Provided the identification scale is small enough the past light cone in a small universe will intersect each fundamental world line many times. Strictly speaking, these universes will be observationally indistinguishable from the corresponding standard models, for in a spatially homogeneous universe there are no identifiable features to use as markers. However assuming the existence of individually identifiable objects, one will see each galaxy (including our own) many times over, and so could hope to observationally determine that one did indeed live in a small universe from the pattern of repeated images of particular objects.

Two points are clear from these examples. Firstly, the spatial closure in these small universes is essential- ly due to their global connectivity rather than being a consequence of large spatial curvature [27,28]. This kind of closure is possible because the Einstein field equations are differential equations that do not specify the topology of space-time. Secondly, these models are compatible with the physical processes leading to inflation; so it is perfectly possible to conceive of an inflationary small universe model [29].

(b) A general small universe will be inhomogenous ("lumpy"). Again the surfaces S(t) will not be simply connected; and in general they will not be spacelike at very early or very late times. To explore the properties of such a spacetime, it is convenient to go to its universal covering space; this will be invariant under a discrete group of isometrics, and the small universe is recovered from it by identification under this group of isometrics. One has then effectively constructed a simply connected space-time by exact repetition of a basic inhomogeneous cell;the small universe is re-ob-

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.q..._ L-2 __ . .~

. .J . • • * O • P • •

• " - . ~ . ~ . _ . L J " ~ • .

in

Z ~

4 - - - L2 ---~

t 1

A

Fig. 1. (of. ref. [ 17 ] ): (right) A spatial section S(t) of a small universe, with the opposite sides A, B of the basic cell identified to give a torus topology; and (left) part of the representation S'(t) of the spatial sections of that small universe by indefinite identical repetition of a basic cell (giving its "universal covering space"). L m is the minimum scale on which repetition can be caused by the small universe structure (i.e. is the distance from any galaxy to its nearest identical image), while L M is the scale beyond which repetition of structure is enforced (it is the distance from any galaxy to the furthest of its neighbouring images). The observer, represented as at O, will see the same galaxies (including his own) many times over, as if he lived in the universe S'(t). The circle with present radius D represents images observed at the redshift z*; images beyond will be omitted from a catalogue with cut-off redshift z*. When D > R2 = LM/2, the observer at O' will see all the matter in the universe at a redshift of z* or less. It can be seen that different segments of the sky subtending an angle a will appear different to the observer at O: the spatially repeating pattern will not necessarily be obvious to him.

tained when all these identical cells are identified as the same cell (fig. 1).

Clearly many lumpy small universes may be obtained by constructing a smooth small universe model from a FLRW universe (as in (a) above) and then per- turbing this small universe; the universal covering space will then be like a FLRW universe with an exactly repeated perturbation superimposed. It is not known how like such a regularly perturbed FLRW universe the universal covering space of an arbitrary small universe is, i.e. whether most small universe models can be regarded as lumpy versions of some identified FLRW universe; this is an interesting geometrical question. It is possible there are families of small universes which cannot be regarded in this way.

Considering the universal covering space it becomes clear that in the case of these lumpy small universes also one will see many images of each galaxy since de-

coupling. An observer in the small universe effectively "sees" the universal covering space - his observations will be exactly the same as those of an observer in such a simply connected space-t ime with an exactly repeating set of galaxies (fig. 1). The observational problem is to distinguish this situation from that expected to hold in a perturbed FLRW universe model, where one would be observing statistical (rather than exact) repetition of a basic cell. If there were substantial variations between similar cells in the FLRW model, this would belie the underlying supposed spatial homogeneity of that universe. Taking observational problems

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and selection effects into account, the difficulty of observational distinguishing a small universe from a lumpy FLRW universe model starts to become clear.

To clarify the observational problem, it is convenient to define minimum and maximum length scales Lm(t), LM(t ) for the surfaces S(t). Here L m is the smallest distance between two copies of the same galaxy in the universal covering space S'(t) of S(t), and LM(t ) is the largest distance between copies of the same galaxy in neighbouring copies of the basic cell. Thus no identifications (or repetitions of structure due to these identifications) occur in the coveting space for lengths less than Lm, but there are always identifications (and resulting repetitions of structure) for lengths greater than L M ; as expressed by Sokolov and Schvartsmann [10] and Gott [11 ] , there is not a single ghost image seen for distances less than R 1 = Lm/2 , but there is not a single original for distances greater than R 2 = LM/2. It is convenient also to define an average length scale R(t) by the condition: the volume V(t) of the unit cell scales as R 3(0. This length scale then corresponds to the FLRW"radius function"R(t), in representing the volume behaviour of the small universe; as it will lie between L m and LM, it can also be taken as a typical spatial repetition length scale.

These lengths depend both on the shape and dimensions of the basic cell. For example, in a k = 0 FLRW universe the spatial sections are fiat; suppose the basic cell is a rectangle with sides L1, L2, L 3. 2 ThenL m = rnin(L1,L2,L3) andL M = [(L1) 2 + (L2) + (/,3)2] 1/2 (see fig. 1), while R = (L1L2L3) 1/3.

The importance of these scales is that they define the maximum size of possible distinguishable structure in a small universe: scale larger than L m will in general show some exact repetition of structure, while such repetition must occur on all scales larger than L M . Indeed on very large scales, what would be observed is a multiple repetition of structures with scales between Lrn and L M (typically, at length scales R). Conversely, if one can observationally distinguish structures in the universe up to a scale D, then a small universe situation is only possible i fL m > D. On what scales can one in fact distinguish different regions in universe? Various proposals have been made for limits on L m and LM, for example Sokolov and Schvartsmann [10] and Gott [11] conclude R 1 > 15 h61 Mpc,

R 2 > 600 Mpc ho f and R 1 > 60 h0 -1 ,R 2 >-400 Mpc h~ 1 respectively, where h 0 = (H0/50) km s -1 Mpc.

Fang and Sato suggest R ~ 600 Mpc [30], while Fairall [31 ] suggests that we can already distinguish distinct structures up to a scale of 500 Mpc. However Tyson suggests we cannot at present distinguish distinct structures on scales larger than 200 Mpc [32]. It seems clear that we can safely claim that an identification scale > 600 Mpc is uncontroverted by present observations (as no present observations clearly distinguish different regions on this scale) and might be able to get away with a value as small as 200 Mpc.

Galaxy observations. Each galaxy (including our own) will be seen many times in a small universe, because of the effective repetition of an identical cell until observational cut-off (say at redshift z*). But each time the matter in this ceil (and so each galaxy) is observed, it is seen at different times in its history and so with different evolution; the redshifts, area distances, and absorbtion, and so the selection effects, are different for each image; in each case the galaxy is seen effectively from a different direction, and also effectively at a different position in the cell if proper motions are significant; and there will be different distortion and lensing effects for each image. Therefore the problem of identification of different images of the same galaxy is non-trivial (basically this is because the image identification that must be made is non-local, unlike the case of lensing of QSOs where the imaging is local and the observed redshifts, area distances and viewing angle are almost identical for each image). Some of the problems have been discussed for the case o f k = + 1 FLRW models with large A terms [23-26] , but then there are only two images for each galaxy and they should occur antipodally in the sky. Much more complex situations are possible in a general small universe; the problem is to identify the repeated cell, and the pattern of repetition (i.e. the group G which controls the small cell topology).

In order to explore this, firstly we have estimated the number of images per galaxy in the following way. Consider a smooth small universe obtained by identification from a FLRW universe with Hubble constant H 0 and deceleration parameter q0. The proper distance d from our past world line to the light cone at a redshift z* is easily found in the FLRW universe, and then converted to a proper distance D = (1 + z*) d representing the present distance to the furthermost galaxies visible if the cut-off redshift is z*. The ratio

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Table 1 Present distances and volumes corresponding to a cut-off rodshift z* in a FLRW universe, expressed in terms of ratios relative to the scale and volume of a small universe basic cell (see text). The volume ratio is an estimate of the number of images observed for a single galaxy in a small universe with parameters as indicated.

Scale Redshift Hubble Qo Distance Length Volume (Mpc) z* constant (Mpc) ratio ratio

600.00 0.200 100 1.00 502.34 0.84 2.44 0.50 522.77 0.87 2.77 0.02 545.91 0.91 3.18

75 1.00 669.79 1.12 5.79 0.50 697.03 1.16 6.57 0.02 727.88 1.21 7.53

50 1.00 1004.69 1.67 19.56 0.50 1045.55 1.74 22.17 0.02 1091.82 1.82 25.40

0.400 100 1.00 869.25 1.45 12.53 0.50 929.07 1.55 15.55 0.02 1005.63 1.68 20.15

75 1.00 1159.01 1.93 29.69 0.50 1238.77 2.06 36.86 0.02 1340.85 2.23 47.77

50 1.00 1738.51 2.90 100.20 0.50 1858.15 3.10 124.42 0.02 2011.27 3.35 161.22

1.000 100 1.00 1570.80 2.62 71.15 0.50 1757.36 2.93 105.25 0.02 2061.37 3.44 185.94

75 1.00 2094.39 3.49 168.64 0.50 2343.15 3.91 249.48 0.02 2748.49 4.58 440.76

50 1.00 3141.59 5.24 569.17 0.50 3514.72 5.86 841.99 0.02 4122.74 6.87 1487.55

200.00 0.200 100 1.00 502.34 2.51 66.00 0.50 522.77 2.61 74.81 0.02 545.91 2.73 85.73

75 1.00 669.79 3.35 156.45 0.50 697.03 3.49 177.32 0.02 727.88 3.64 203.20

50 1.00 1004.69 5.02 528.03 0.50 1045.55 5.23 598.46, 0.02 1091.82 5.46 685.81

0.400 100 1.00 869.25 4.35 338.18 0.50 929.07 4.65 419.90 0.02 1005.63 5.03 544.11

75 1.00 1159.01 5.80 801.60 0.50 1238.77 6.19 995.33 0.02 1340.85 6.70 1289.73

50 1.00 1738.51 8.69 2705.42 0.50 1858.15 9.29 3359.23 0.02 2011.27 10.06 4352.85

1.000 100 1.00 1570.80 7.85 1920.95 0.50 1757.36 8.79 2841.71 0.02 2061.37 10.31 5020.49

75 1.00 2094.39 10.47 4553.35 0.50 2343.15 11.72 6735.91 0.02 2748.49 13.74 11900.43

50 1.00 3141.59 15.71 15367.56 0.50 3514.72 17.57 22733.71 0.02 4122.74 20.61 40163.95


r 1 = D / L of this distance to the present small universe scale size is an estimate of the number o f times one would have seen around that small universe up to a redshift o f z*. These numbers are given in table 1. The present volume V encompassed in the FLRW model up to the distance D is approximately 41riD 3 ;

its value allowing for the spatial curvature can be found straightforwardly, and the volume ratio r 2 = I1/

L 3 calculated (for simplicity, we here estimate the volume o f the basic cell by approximating it as a cube of side L in fiat space). This gives an estimate o f the number of images to be expected per galaxy in the small universe. These numbers are also given in table 1.

As expected,the volume ratio r 2 varies as the inverse cube of L and H 0 (for given z*, the distance limit observed scales as Ho), but as the cube of the catalogue cut-off redshift z* (for small z*). It also varies inver- sely with q0. For reasonable values of these parameters, r 2 varies over a very large range, reaching the astonish- ing figure o f 40163 i fL = 200 Mpc, z* = 1 ,H 0 = 50 km s -1 Mpc, and q0 = 0.5. This corresponds to seeing approximately 20 times around the universe.

Secondly, we have examined the images produced in a smooth small universe constructed by making identifications in a k = 0 (Einstein-de Sitter) FLRW universe. The null geodesic equations in these universes may easily be integrated to obtain the observed images of a single galaxy in the basic cell, given (a) the cell topology (when k = 0 there are six distinct orientable topologies, which we label T, T = 1 to 6, see table 2; for T = 1 to 4 the cell is conveniently chosen rectangular but for T = 5, 6 it is chosen hexagonal); (b) the cell dimensions L u; (c) the cut-off redshift z* (we assume

Table 2 The 6 orientable topologies in the FLRW k = 0 case (flat spatial three-spaces) can be obtained by identifying opposte sides of a basic unit cell as shown in this table.

'Topology B a s i c Identifications type T cell of opposite faces

1 rectangular non rotated 2 one pair rotated 180 ° 3 one pair rotated 90 ° 4 all 3 pairs rotated 180 ~

5 hexagonal 6

top rotated 120 ° relative to bottom top rotated 60 ° relative to bottom

that all images up to this redshift are observed, and none for greater values o f redshift); (d) the Hubble constant H 0 ,1 ; (e) the position o f the galaxy in the cell (for convenience taking our own galaxy at the centre o f the cell). The resulting images have been plotted using the Russell-Aitoff equal area projec- tion [33], either as single plots of all the images, or as series o f plots showing the apparent distribution of the observed galaxies in depth. Since it is difficult to reproduce and view the latter, they have also been converted to stereo pairs that can be viewed using a standard stereo viewer (or without it, by viewing from an appropriate distance and suitably crossing your eyes!) The computer programme used can plot images o f a number o f galaxies (either at specified positions or randomly distributed) in the basic cell in any of these ways, and displays the distribution of redshifts, area distances, or observed orientations as desired.

Some of the image distributions are shown in figs. 2--4. There are in general clear patterns in the images when Nis small,in particular the identification pattern shows up in "planes of avoidance" in the sky (very few images are observed in directions close to the pro- jected axes, cf. ref. [7] ); however obvious patterns disappear when there are a reasonable number o f galaxies in the basic cell (i.e.N>> 1), and one obtains a general impression of approximate isotropy. I fa single spiral galaxy is considered, it will be effectively seen from different viewpoints in its different images in the sky; again a rather clear pattern will exist in the orientation o f images o f a single galaxy, but the pattern will be difficult to discern when there are numer- ous galaxies. The number of images up to the cut-off redshift z* has been determined directly from the plots we have obtained. The cut-off of images for a galaxy depends on its position in the cell (which determines the observed redshift for each image), so with randomly placed galaxies we can get different numbers of images even when all the other parameters are the same. We find results consistent with the results of table 1 ; and find that the number of images is insen- sitive to the topology.

Even in the case of a single galaxy, histograms of

,1 In a general FLRW tmiverse, there will also be a depen- dence on the deceleration parameter q0 ; in the case considered, q0 = 1/2.

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. . : . - , . . . : - ~ . . . - . ~ ' : , ; : - ; ' : . . . , . . ' ; ~ . . . . r ~ ' % ' , , . ' ~ " , , " . . ~.° o . . * . . . * • " . ' , ~ * . * * , ~ .~ * * * * : , * , ~ . . ~ ' . * • . .

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Fig. 2. Stereo pair showing distribution in the sky of 476 images of a single galaxy in a small Einstein-de Sitter universe with a Hubble constant Ho equal to 75 km/s Mpe, identification scales Lu of 600 Mpc, and cut-off redshift z* of 1. The topology is that of a three<limemional torus (type 1). The apparent distribution in depth of the images reflects the area distances that would be measured for them.

• • • %

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Fig. 3. As for fig. 2, but for 30 galaxies in a small universe with cut-off redshift z* = 0.5 and identification scales L~ = 800 Mpc. There are 683 images of these galaxies. In this case the distribution in depth represented is the redshift distribution.

• -e.." ~ . . ~ . , ~ . - .

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• ",,.- _ ~ L , ~ - .'~'~..~"91~" •

Fig. 4. As for fig. 3, but with topology 4. There are 679 images of the 30 galaxies.

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observed area distance or redshift in general show no remarkable features; rather in general one simply sees more and more images up to the cut-off redshift (as predicted by standard number.count calculations). Thus we do not find a marked effect in such histograms, such as that claimed to have been observed in QSO observations and interpreted by Fang and Sato [30] as indicating a small universe (but interpreted by Box and Roeder [34] as due to selection effects). Similarly we do not expect significant differences between the observed angular correlation functions in a small universe and in an ordinary FLRW model, provided the clustering in the basic cell accords with that indicated by the correlation function observations for distances less than L m ; for although there will be an exact correlation in the spatial position of images on length scales larger than L M at any time, one cannot observe these positions at a fixed time, and the images from the effectively repeated cells will be superimposed on each other (see fig. 1 and ref. [11 ] ) at different orientations and with different apparent sizes, lumi- nosities, evolution, and selection effects. Towards the limiting redshift z* one would see the same cellular structure repeated at angles corresponding to lengths between L m and LM, but again observed from effectively different directions. Because of all these factors which tend to hide the spatial correlations that exist, the angular correlation function in a smooth isotropic small universe may be expected to be broadly the same as that predicted in FLRW universe models, with possibly some features showing up at the angular scales corresponding to the minimum angular size of the basic cell at the cutoff redshift z*. Our exploratory numerical experiments confirm this view, as do those of Gott [11 ]. These calculations apply directly to smooth (k = O) small universes (which would include the case of an inflationary small universe); however we may expect similar results at least in the other (k = + 1 and k = - 1) smooth small universe models, and those lumpy small universe models obtained by perturbations of the smooth ones.

The best hope of confirming a small universe would be by observing some major feature, such as the fila- ments and voids now observed on large scales, with repetitive patterns which can be interpreted as due to a small universe structure with some appropriate unit cell. We have not carried out detailed simulations to explore this. On the other hand one can disprove the

existence of small universe structure with L m below some value D by observing clearly non-repeating structures up to a scale D; as discussed above, this puts a definite lower limit on L of about 200 Mpc and possible limit of about 500 to 600 Mpc. As better and better observations accumulate, either a possible repetitive pattern will be discerned, indicating a small universe structure; or the lower limit on L m will be increased to larger and larger values until the hypothesis loses much of its attractiveness by losing much of its predictive power. However it is clear that if we could discern such patterns for small values of Lm, the model indeed has great predictive power: e.g. many of the models we have examined have up to 6690 images per galaxy, giving extraordinary economy of explanation of a very large number of images.

Microwave background radiation observations. Let the small universe considered be characterised by a maximum identification scale L M at the present time t o . At the time of decoupling td, the corresponding scale was L M / 1000 (because co-moving lengths have then been scaled by factor 1000). From the area distance relation for FLRW universe models we can find the corresponding angle ct in a FLRW universe, which is the presently observed angle in the sky corresponding to the size of a present length scale L M at the time of decoupling. The angle tx depends on L M, q0, and H 0 ; representative values are given in table 3. In a smooth isotropic small universe, comparison of observations at angles larger than tx would correspond to repeated observation of the same physical events in the single basic cell at the time of decoupling. In a lumpy small universe we might expect approximately the same an-

Table 3 The angular scale on which the microwave background radiation would be smoothed out because of the small universe structure.

Deceleration Hubble Present Apparent parameter constant size angular qo H o L (Mpc) size

0.5 50 200 59' 13"

100 600 5 ° 55' 21"

0.02 50 200 3' 7" 100 200 6' 15"

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gular relations to hold; hence, irrespective of what material inhomogeneity might occur in the basic cell, identical conditions leading to the emission of the black body background radiation would be repeated- ly observed on angular scales larger than a. Thus only very small angular variation should occur in the cosmic microwave background radiation (cmwbr) on angular scales larger than a (these variations would result from variation in (1) the observed redshift, and/or (2) the effective orientation of the basic cell relative to the past light cone of the observer, resulting in the repetition length scale varying between L m and LM).

While the small universe hypothesis immediately gives a reason why only small variations in the cmwbr could occur on angular scales larger than a it does not itself explain why the small cell should be homogeneous, and hence why we observe near microwave isotropy on angular scales smaller than a. To investigate this, it is useful to examine the horizon size at the time of decoupling. Using a k = 0 FLRW model, matter dominated after decoupling and radiation dominated before decoupling, leads to the relation

dph = 3vq-6/ 0

for the metric size of the horizon at the time of decoupling. ThusH 0 = 50 km s -1 Mpc shows the present scale corresponding to the horizon size at the time of decoupling is Dph = 1000dph = 190 Mpc. This is just on the order of the smallest scale size we are able to consider for a small universe. Thus if we take L = 190 Mpc in a small universe, efficient physical processes can in principle explain cmwbr isotropy up to angles corresponding to this size, and the repetition of identical units can explain it on larger angular scales.

The calculation summarised above ignores the possibility of an inflationary era. However if we have inflation in a small universe, we can easily guarantee homogeneity, and hence cmwbr isotropy, for then the horizon time at decoupling would be many times the identification scale (in this case, the universe was very small at the pre-inflation era).

Dynamical properties. We have seen that a small universe may be expected to look like a FLRW universe in terms both of number counts of galaxies and of the microwave background radiation. A further major issue that needs consideration is its likely dynamical evolution, determining the magnitude redshift relation

and the synthesis of helium and other light elements in the early universe.

If the small universe is a smooth model, obtained by identification from a FLRW, Kantowski-Sachs or Bianchi universe, it is locally identical to that universe model and so its dynamical behaviour will be exactly similar; in particular, an isotropic smooth model will evolve just as a FLRW model does.

If the small universe is lumpy, and particularly if it is not just a perturbation of a FLRW model, the issue is: can we determine its average volume behaviour? A remarkable paper by Carfora and Marzuoli [35], written with a different context in mind, is very encouraging in this regard. Consider an initial surface S(t0) in small lumpy universe, with metric h~x. If the spatial curvature is positive definite in the sense that 3Ricci(h) > 0, it follows from their results that one can define a deformation procedure h~x ~ kKx(/3 ) such that

(a) kKx(0 ) = hKx , k~x(1 ) = (FLRW), i.e. the lumpy universe is continually deformed into a smooth universe;

(b) for every value of/~, the Einstein initial value equations are satisfied for suitably defined average values of the matter tensor in the small universe;

(c) the space-average of the second fundamental form trace stays constant under the deformation, i.e. the volume behaviour is the same for the lumpy and the smooth universes;

(d) symmetries are preserved under the deformation, so a smooth small universe is left invariant by the process.

Thus in this case, the deformation procedure shows that on average, the initial data of the lumpy model can be smoothed into that for a FLRW model, and so the universe necessarily evolves like a (identified) FLRW model, as well as looking like one. The issue that is presently unresolved is if this result could be extended to all small universes; this initial result is certainly very encouraging.

Physical properties. There are a series of very in- triguing further physical consequences of the small universe hypothesis that should be investigated further. These really all flow from the boundary condition of having no boundary, implying at least the following:

(1) The overall electric charge is related to the topology: for example it must necessarily be zero if the topology is S 3 or T 3 (by Gauss' theorem applied

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to the spatial sections), thus explaining one feature of the observed universe that is otherwise an arbitrary initial condition.

(2) In a similar vein, the overall gravitational mass of the matter, as measured by Gauss' theorem applied to the spatial sections, is related to the topology and may even be undefinable [36]. The implications of this are unclear; presumably they relate to the gravitational behaviour of bounded systems. It is therefore interesting to note that various of the numerical simulations of galaxy formation that have recently been carried out, and have reasonably successfully simulated the observed distribution of matter, have assumed periodic spatial boundary conditions [37], i.e. have been carried out in a small universe setting. This has been treated simply as a calculational convenience. However what has actually been proved through these numerical calculations is that galaxy formation processes work successfully in a small universe context.

(3) The closure of a small universe puts constraints on the possible creation of strings, walls, and mono- poles through physical processes in the early universe. The nature of the allowed such features will depend on the topology of the small universe spatial sections; it should be possible to investigate this fairly easily in the case o f k = 0 FLRW models, but the k = - 1 case will be extremely complex.

(4) Because of the spatial closure, there will be effective periodic boundary conditons with length-scales between L m and LM, and so discrete spectra of eigen- functions of all fields and a maximum possible wave- length for any wave whatever in a small universe. Con- sequently there will be a minimum energy possible for each wave mode. This could be significant in af- fecting various physical processes, for example particle creation in the early universe [38] ; the effect could be very dramatic in the case of an inflationary small universe where the scale of spatial repetition enforced will be extraordinarily small in the pre-inflation period.

Conclusion. The "small-universe" concept is an in- triguing one, for it opens up a natural explanation of apparent observed spatial homogeneity and microwave background radiation isotropy independent of other physical mechanisms at work. A general small universe looks like a FLRW universe, even if it is quite inhomogeneous. Because of this similarity, observational proof or disproof of the small-universe possibility is

quite non-trivial provided the scale of the spatial sections (which is the scale of observed effectibe repetition of physical features) is suitably chosen.

A remarkable feature is that the volume behaviour of a small universe will also be like that of a FLRW universe, at least in the case where its spatial Ricci curvature is positive. Additionally there is the possibility of various interesting connections between local physical behaviour and the small universe topology and scale size. These are certainly worth pursuing, because of the way they relate large-scale and small-scale properties of the universe.

The problem highlighted is the arbitrariness of the topology of solutions allowed by Einstein's equations. The disadvantage of these models is that no explanation is currently given for either the topology or the scale of the small spatial sections. Three comments are in order here. Firstly, the "standard" model also does not explain its topology, which is simply assumed (cf. ref. [39] ). Secondly, it has been claimed that compactness of space is a necessary condition for the spontaneous birth of the universe from nothing [40] ; and preliminary calculations by Madsen [41] suggest that k = 0 toroidal small universes are as likely to be created by the Hartle-Hawking creation process [42] as are universe models with the "standard" topology. Thirdly, the relation of small universes to observational evidence is so different from that in usual universe models as to represent a quite different relation of man to the astronomical universe. For example, in a very clear review article Davis states the CfA galaxy survey sample "is perhaps large enough to represent a fair sample volume of the universe" [43]. In k = 0 or k = - 1 FLRW universe models, this statement cannot be correct - no observable volume is an appreciable fraction of all the matter in the universe in these cases (for particle horizons limit the number of galaxies we can observe, but there is then an infinte amount of matter present). Assuming the normal topologies, only i fk = + 1 could this statement be true; and the observational evidence does not support this high-density case [43]. However if and only if the small universe sup- positition is true, we have seen not merely a fair sample - we have seen all the matter in the universe. Thus only these universes can truly be mapped by astronomical observation [9,12,17].

The work presented here is a continuation of work

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completed by Miss J. Biesheuwel as an Honours pro- ject at the University of Cape Town, which provided a good foundation for the present investigation. We thank A ~ . FairaU and T. Tyson for helpful discussions.

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Observational and dynamical properties of small universes