Science in Christian Perspective

MODERN SCIENTIFIC
COSMOGONIES
THOMAS H. LEITH

From: JASA 16 (September 1964): 73-81.

This article is an analysis of the major ideas as to the origin of the universe found among 'scientific' cosmologists of this century. It discusses first the early applications of general relativity to the construction of models of the universe and the related questions each raised as to the origin and past history of the physical world. The remainder of the paper analyses later experimental and theoretical work either based upon these models or upon a conscious rejection of them, emphasis being placed upon the cosmogenic aspects of these labors. The views of Lemaitre, Gamow and his co-workers, the matter-antimatter theorists, and the Gi;del-Heckmann-Omer school are discussed in some detail as examples of modern development of the earlier models. In contradistinction Milne's kinematic ideas, Eddington's fundamental theory, the views of Dirac and Jordan, and the positions of the steady-state cosmologists are explored. In all cases, fundamental assumptions and experimental difficulties are brought out fairly sharply.

It is our purpose in this paper to survey fairly comprehensively, but as briefly as possible, the major ideas as to the origin of the 'universe' to be found in recent cosmological speculation. To this end we will restrict our discussion to origins as they are conceived within schemes which one might call scientific rather than as they are seen in more philosophical systems. The distinction here is not as sharp as might be supposed but it cannot be explored in this paper; hence it will be retained for discussion only in a second comparison paper to follow. For the present, we shall base the difference upon whether scientists or philosophers, and their respective professional journals, have paid most attention to any given scheme. In our second paper we will also have something more to say about the somewhat vague term 'universe' mentioned above, because the writer feels sure that the vagaries of usage of this word will become apparent to anyone who thinks at all upon our remarks which follow below.

May we begin with some historical matter? After the development of the general theory of relativity, wherein the geometry of a spatial region was made dependent upon its relation to matter (or perhaps the very distinction of space and matter was reduced to the change in intensity of an energy field from place to place and thus to variation in space-time curvature), Einstein noticed that, on a very large scale and assuming the random distribution of galaxies and intergalactic matter and radiation, one could sensibly speak of the average curvature of an immense region.¹

The eventual result of his thought here was the postulation of a model² in which matter and radiation were seen as smoothed out on a very large scale and in which there was no large-scale relative motion. The metric appropriate to this he took to be a Riemannian

Dr. Thomas H. Leith is Associate Professor of Natural Science, York University, Toronto, Ontario.

space of constant positive curvature, i.e. a closed space in which the curvature was the residue remaining after the smoothing out mentioned above. The radius of such a model depends upon the total matter-energy content assumed; the value for this latter figure being suggested by astrophysical analysis and observation.

To achieve the stability of his model Einstein believed that it was necessary to add a tensor containing the famous 'cosmic constant' X to the tensor field equation of general relativity which was, of course, the basic theory behind the model.³ This constant was then really descriptive of a sort of repulsion superimposed upon gravitational attraction so as to cancel it out on the large scale. It also served, he believed, to prevent there being any solutions for the field equations of the universe contained no matter. But therein lay at least one fatal flaw to the model - the field equations turn out to have a solution in the absence of mass even with the k term.

This was shown in the same year by de Sitter.⁴ Thus in turn, he suggested an alternative model, an empty one in which, were matter particles to be introduced, they would recede with increasing velocity with distance from any observer. Such motion is, of course, the consequence of retaining the cosmic repulsion without the presence of a balancing gravitational attraction. This and Einstein's model interest us here only because both are essentially static,⁵ Einstein's in a true sense and de Sitter's because when it is empty repulsive motion is not apparent, and thus they suggest no beginning or end to the universe which they picture.

In the years following 1917, attention turned to increasing evidence that there was a recessional galactic motion not permitted by Einstein's model, to an increasing awareness that the average density of the visible universe, while quite small, was hardly de Sitterian, and to the X term which, whatever the logical or esthetic grounds for introducing it as positive, might empirically turn out to be zero or even negative in magnitude. This combination of new empirical observation and varied mathematical possibilities led to the major work of Friedmann,⁶ Robertson,⁷ Lemaitre,⁸ and others.⁹ We will make no attempt to sort out the differences of analysis here (for example, Robertson used a kinematic approach which ignores the dynamical field equations of general relativity) nor the contribution of each, with 1he exception of Lemaitre who requires further mention later. Instead we will try to summarize briefly the net mathematical product of this work.

When we consider the case in which the cosmic constant is taken as zero in value, it turns out that we have three types of model dependent on whether the curvature is taken as positive (a closed model in analogy to a sphere or ellipse), zero, or negative (the model is open in analogy to a hyperboloid). The first of these has no unique beginning nor end, it pulsates endlessly from zero (or some finite) radius to some larger size and back again. The second has a singularity in time and size: it either contracts toward this from an infinite past or expands from it without end. The last model is not unlike the second, save that it contracts from or expands toward a de Sitter state far in the past or future. Consequently, the first can, at best, be given an age only as far as the time since the present pulse began is concerned; the second and third (if we ignore the contracting options in each, the universe to the best of our knowledge showing expansion) suggest a definite age since the model began expansion from near-zero radius and infinite density.

But there still remain the options wherein the cosmic constant is negative or positive. The negative cases, for any of the three possible curvatures, turns out to be fairly simple: all of them are pulsing models with a minimum radius of some size lying between zero and the radius of the Einstein universe. None then has a real beginning - the only meaningful age is given by the possibility of dating the beginning of the last pulsation, in which the regions of the universe which we see appear to be still on the expanding phase. The positive value for the cosmic constant eventuates, unlike the negative case, in quite a complex set of models. When the curvature is negative or zero the resulting schemes all involve a single expansion from zero radius to infinity with the passing of time, the models then ultimately becoming empty de Sitter universes.

Positive values for the curvature give, however, quite distinctive results depending upon the precise value chosen for k. If we pick a value which is larger than that which Einstein chose for his static model the consequence is a scheme either contracting from a de Sitter state toward a small or zero radius or a scheme going the opposite way. The plot of expansion with time shows very rapid change in radius when the model is small; when it approaches the size of the Einstein model it slows down considerably; and then it speeds up again as it moves to larger size and closer approximation to the de Sitter empty state. If, however, we choose the value of X so that it is the same as Einstein's value, we get either his scheme which maintains constant volume with time or two non-static models: one expanding from a very small radius toward the Einstein size asymptotically with time and another beginning asymptotic to the Einstein state infinitely long ago and expanding thereafter toward the empty de Sitter case. Finally, if we choose X greater than zero, but less than the Einstein value, we get either models contracting toward the Einstein radius which slowly reverse direction and expand or models expanding and contracting from and to a singularity, with size somewhere between zero radius and the Einstein dimensions.¹⁰

One's first reaction might be to smile indulgently at the manifest fertility of the mathematical imagination as universe after universe falls from page after page of equations. Yet science often advances by putting to test just such suggestive ideas, and in the third and fourth decades of our century these cosmological models were behind a great deal of astronomical research- Certainly the general relativity theory on which they were usually based, the assumptions of homogeneity and isotropy of the universe on the large scale, and the more detailed physical theses of most of the models seemed to be, if nothing else, at least not improbable. On such a basis workers might well examine the theoretical and empirical consequences of the specific model types individually - and this they did. Three things only interest us here: which, if any, of these cosmological schemes have stood the test of time, whether the models in which a mathematical singularity occurs can explain what goes on around this point within the model, and whether pulsing models or those with an infinite past logically provide an escape from 'creation' ideas.

Let us turn first to the ideas of origin in models which, in the above outline, involve a single expansion, i.e. they do not pulse. In 1930 Lemaitre suggested a scrutiny of the case wherein the universe began close to the Einsteinian form but which has thereafter expanded, due to instabilities within this supposedly static model, toward a de Sitter state. As a result of this suggestion he incorporated a preEinsteinian stage, one in which matter was originally highly compressed in a very small space - a veritable primordial atom. This scheme he has fleshed out over the years and we may present his recent thought as indicative of the latest thinking on this sort of picture of how our universe began.¹¹

The initial state of the universe he takes to be a ,quantum' or atom which explodes the instant it appears. This odd entity is postulated because he believes that entropy considerations teach us that the universe is moving toward maximizing the degradation of energy so that it must move from some state of minimum entropy, a state which can exist only for a moment but which is the simplest we know how to describe physically - a single undivided 'bit' of energy. Lemaitre suggests an analogy here to a lesson of quantum theory as he sees it: that a physical system may be described as an assemblage of potential states, the most probable distribution for which is to have all states occupied (maximizing entropy) and the least probable is that which has only one filled (thus minimizing entropy).

There is an ancillary idea involved in assuming this primeval 'quantum' which is of interest and may be noted briefly. This is that the properties of such an entity will always be only crudely describable; we can neither physically explain its origins nor can we predict deductively from its character the precise consequences of its disintegration. The first alternant implies that any philosophical view of its cause is permissible.as far as science takes us, but that such ideas as the thesis that it must result from a prior contraction of the universe are impossible to prove - indeed one would expect on such a ground that the end of a contracting phase would be a state of maximum rather than minimum entropy. The second alternant requires that there be no way to predict just how it will split up or just what precise consequences will eventuate. Only when division has proceeded far enough to give a very large number of smaller 'bits' will laws based on statistical predictability enable us to describe the likely consequences of disintegration with some precision.

Lemaitre proceeds to this general statistical description. After the primeval disintegration the 'zero' space of the 'quantum' expands and fills with the pieces which lose kinetic energy in proportion to this expansion. As time passes these fragments disintegrate into protons, electrons, and gamma radiation though some still remain as uranium and thorium atoms, atoms of long half-life. After a few billion years of expansion most of the naturally - occurring atoms which we know had been formed and had settled down into a near statistical equilibrium in the form of a gas. This was a close approximation to an Einstein universe, a space-time equilibrium with space full of matter and radiation. After several billion years the gas condensed locally into proto-galaxies, and under the resultant repulsion, space resumed its expansion in these regions. Some regions, where the gas is not yet condensed, remain today as cosmic dust clouds and nebular clusters. At present, Lemaitre believes that the universe has expanded for some four billion years from the Einstein stage so that it is some one hundred times the Einstein volume and some 10¹⁷ times as big as the original atom.

We should not, however, leave this model without a brief critique, for it has several weaknesses. one such flaw lies in the highly speculative character of the primeval atom. It is argued that its nature may be generally described in analogy to radioactivity and quantum theory but to say the least this extrapolation is fraught with hazard, especially when his interpretation of quantum theory itself is rather debatable. It is also argued that it cannot have a prior physical history both because it explodes as soon as it appears and thus cannot have a stable history, and because a contracting universe prior to it should result in maximum rather than minimum entropy. However, Lemaitre seems to forget that if time direction is in some way connected with expansion and entropy increase in our present universe, a prior phase which contracted might well define a time of opposite sense leading to a minimizing in entropy. Whatever other difficulties this argument may have, it at least reveals the possibility that the primeval atom of minimum entropy, if there ever was such a thing, could logically have had an indefinitely long prior history.

But the major flaw in the model lies in discussing anything prior to or during the approximately Einsteinian stage. We noted earlier that such a state is presumably static; consequently, if our universe is now expanding, it must have only approximated such a condition, the closeness of the approximation determining how long it took for sufficient instability to develop so that we have had expansion ever since. The problem of stability has been studied extensive ly,¹² but it is safe to say that the closer our universe might have been at one time to an Einstein state, the longer it would have been close to stability, and thus the more unlikely is the value of any extrapolation from it to yet earlier processes. Whatever processes are suggested, and they are usually very ad hoc, for giving rise to instability and the processes and character of the presently observed universe, they will have occurred so slowly that it is simply fiction to presume that one can credibly explain these tiny changes in terms of physical entities and reactions all the way back to some primitive 'quantum.' It would seem far better either to presume our universe began in something like an Einstein state or to presume it has expanded continuously from some super-dense initial state along the lines of Lemaitre's 'quantum.'

The former option has received its most elaborate, and one might add mystifying, treatment in the hands of Arthur Eddington. Unfortunately the analysis of his ideas requires technicalities and criticism far beyond the possibilities of this paper, and the reader is therefore referred to other sources.¹³ We may salve our conscience, however, because Eddington's a priori views as to the mass and size of the Einstein state are rather more suggestive to philosophy than they are as yet useful to science. If we can ignore Whitehead in this paper, we can presumably do much the same for this other great theorist and for the same reasons, but we will nonetheless have a few more words to say on Sir Arthur later.

The option for continual expansion from a small initial state has been given rather-better-known analysis. Here Gamow and his co-workers¹⁴ have suggested a very condensed early universe in which neutrons are the stable constituent. As this expands and pressure is relieved, some neutrons decay to protons which, in the enormous flux of remaining neutrons, capture these successively and along with electron decay form the nuclei of the elements which we know. The entire process involves, presumably, only minutes. Later, in the highly supersonic turbulence of this expanding hot gas, denser areas are established. Stabilized by internal gravitation it is these which form the origins of the galaxies we know.

The thesis is based largely upon a theory of element formation now held suspect in most quarters. Gamow and his fellows had argued that the amazingly close relation between their figures for the abundance of any given elements in the universe and the target cross-sections of the various nuclei for neutron capture (when neutrons, with presumably the same energy distribution as these suggested for the early universe, were fired at them in the laboratory) could hardly be fortuitous. The cross-section determines the rate at which nuclei capture these neutrons; thus nuclei of small cross section would capture neutrons slowly and increase in relative abundance since they don't change quickly to the nuclei of the next highest atomic weight. The result would be the inverse relation between cross-sections and relative abundances of each element actually observed.

But there are difficulties. In the first case, it is very hard to feel confident in any available figures for relative abundance of the elements in the universe and this entire scheme depends on assuming we have good values. Indeed the iron group fits poorly even with the date available now. More important, however, are two theoretical problems. One of these lies in the fact that some nuclei which do exist should never exist at all on this picture since their stable isobars terminate the electron decay before the nuclei in question could be produced. The other difficulty is that there are no known stable atoms of mass 5 or 8, thus we cannot get past these to build more massive atoms. Finally, there is the practical question of how, since expansion decreases density and temperature in this model, the synthesis of elements with increasing charge and increasing electrical barriers, which require just the opposite situation, can occur as times goes on.

Most theoreticians have, as a consequence, felt that both the origin of the elements and the formation of stars and galaxies are better explained by alternative schemes. Stellar and galactic formation seems to be going on continually, as far as astrophysicists can tell, and do 'not appear to show the wide agreement in ages which one would expect on the Gamow model.¹⁵ Similarly, the origin of the elements is now most commonly related to stellar interiors at different stages of development. But it should be noted, as Gamow has himself pointed out recently, that some sort of combination of these differing positions might be possible: perhaps, beyond those atoms with atomic number four, synthesis might occur within stars. The reasons for the apparently greatly varying ages of stars and galaxies is, however, still a mystery on the basis of Gamow's type of model.

When Gamow is mentioned most people do not think so much of our above remarks as of his thesis that the universe has, contrary to Lemaitre's position, a physical history prior to the present expansion. The Irish astronomer Opik has argued in a similar manner, ¹⁷ and we might comment briefly on this idea. In essence, both theories are attempts to avoid any real temporal beginning to the universe, any 'age' we give the universe being that applicable to the present expanding phase only. To be sure Gamow and 6pik differ in detail: Gamow and his followers suggest either prior oscillations of increasing amplitude or a single contraction from extremely large size, with the present expansion probably having such velocity that it will never reverse, while Opik postulates an endless series of pulsations of roughly equal amplitude.

One may say, however, that while such ideas are logically meaningful, they are in no way scientifically testable in the light of our best knowledge today. Indeed, it seems that such should remain the case, for if we may follow Lemaitre's suggestion that our knowledge of some hyperdense phase is hardly likely to be much use in predicting the future of the universe, may we not equally well say it is even less useful for retrodiction into some earlier phase?¹⁸ If we really don't know what the properties of matter might be like in such a dense state, how can we hope to extrapolate with confidence to some prior phase or phases and above all how are we to test our beliefs? To be sure one might find evidence that our present universe is best described as a pulse, but there is really no basis other than mathematical symmetry, esthetic considerations, or some desire to avoid a beginning to the matter-energy of the universe for presuming that one pulse demands others before it. On the other hand, if Gamow is correct, and this present phase is unique in likely being open in topology and thus subject to endless expansion, might one not justifiably argue that economy of thought might suggest its equal uniqueness in origin?

Leaving this discussion we may, before turning to different considerations, remark on the application of anti-matter ideas to models which involve some very small volume at some time in the past, as do the Lemaitre-Gamow schemes. The literature here is fairly interesting but highly speculative.¹⁹ It has been argued that the production of elementary particles should result in quantitative symmetry of matter and anti-matter particles but the question hen arises as o where all the anti-matter is. We see no firm recognizable evidence of their frequent violent union in observational astronomy, and there would be major difficulties in the entire general relativity theory were antimatter present in anything like the amounts we see of matter, even if we assumed they repel one another. Perhaps parity considerations provide a segregation mechanism, but, if so, we then must have the apparently quite untestable thesis that there are really two types of cosmos of opposite handedness.

Kevane has suggested a model which rejects general relativity and utilizes anti-gravity possibilities. It originates as a plasma of both types of particles which, being unstable, expands continuously at an accelerating rate. The idea has some interest but is hardly testable at present. Kapp suggests relating anti-particles to the forming of space and particles to its disappearance. We will mention this theory later. Finally, Reiser suggests a model in which matter and anti-matter represent opposing vortices in a hyperdimensional and pantheistic cosmic field. This is pure speculation and presently quite uncorroborable. In sum, though we may expect to hear more from matterantimatter considerations, it appears to this writer that, in direct proportion, we may expect an increase in the metaphysical or ad hoc content of cosmological theorizing, if present discussion is any indication, and a consequent decrease in scientific testability of much of the argument.

This exhausts our rapid over-view of the usual models growing from Einstein's work directly. Before turning for a brief look at rather different schemes reference should be made, however, to three rather far-out schemes which deny the homogeneity and/or isotropy of all the models previously considered. These are the work of Kurt G6del, Omer, and Heckmann.²⁰ The result of their discussion is the avoidance either of any 'cosmic time' applicable to the entire universe, and thus of any unique beginning, or the avoidance of an extremely dense origin, approximating zero size, for the present expansion we observe. The latter eventuates instead in simply bringing the galactic material closely together at some early date, as de Sitter originally suggested in 1933.²¹

These models variously require such odd things as the idea of absolute rotation of the entire content of the universe (a thesis violating general relativity, as usually understood, and of the so-called Machian principle of inertia), and large scale anisotropic expansion. At present none seems testable but they are perhaps potentially corroborable or falsifiable by observation and thus of some little importance. In our present discussion their interest lies in their possibly providing in future some way out of the difficulties of getting back beyond some early compressed state of our universe to the description of some earlier phase. For if matter were never sufficiently condensed to lose the character it may have had earlier, an understanding of the laws of change in this state might enable us to extrapolate to this character. However, all this is presently only a hope and nothing more.

Not all cosmologists have felt that Einstein got cosmic model-making off on the right track. One of these was Milne whose kinematic relativity model had for a time a considerable VogUe.²² The approach here is unlike most of the method used in our earlier models: Milne believes that a correct understanding of the rational foundations of cosmology provides a means of deducing the necessity of the physical laws which careful experiment can only show to be probable. Milne begins by assuming, unlike general relativity, that not all frames of reference are equivalent for describing the universe but only the centers of the galaxies which we see receding from us. Consonant with this is the belief that the laws of nature are describable in the same way from all such centers. On these foundations, Milne proceeds to attempt to discover what the laws of nature would be in such a universe. To do this he assumes also that the galactic centers separate with uniform velocity, that a hypothetical observer at each center would be aware of the passage of time, that such observers could send light signals to one another, and that the space in which they move is Euclidean. The result is a model with an expanding spherical swarm of centers filling its space, its kinematics being that of special relativity. The background of centers provides what he calls a 'substratum', the necessary frame of reference for the subsequent construction of various theorems of dynamics. And on this dynamics he builds a gravitational theory and electrodynamic laws.

All of this is highly technical and approached variously in his different works, but certain aspects of it interest us here. One such facet is Milne's claim that, if the 'substratum' of the universe were not as he describes it, the structure of the world could never be known. This implies the deliberate creation of the universe, an event which he places at zero time in Ahe kinematic equations when all the galactic centers now in uniform relative motion were in coincidence. Milne also believes that an infinity of such centers was involved at the creation in order that no preferential velocity frame be present immediately after the beginning. He believes too that such centers must have had initial velocities ranging from zero to infinity in a non-Maxwellian distribution. The detailed reasons for the last two beliefs are, I am afraid, too complex for a brief survey such as this.

A second facet arises from the fact that galactic centers form only a reference frame, a background for phenomena. Milne thus adds what he calls 'free' particles into this background and on this basis constructs his dynamics, An odd consequence arises: the motion of the 'free' particles is accelerated with respect to the 'substratum', but if observers appropriately regraduate the clocks they have set by signalling one another as mentioned above, this acceleration can be reduced to a uniform motion as in Newton's first law. When this is carried out, the two -time scales differ in having t==O on the kinematic scale equivalent to T~ - 00 on the dynamic scale. For kinematics there is a creation, but on the basis of the usual dynamic considerations of physics the same event disappears infinitely into the past!

All sorts of unusual consequences follow from reverting from the dynamic to the kinematic scale.²³ Constant length becomes steadily increasing length, relative rest becomes uniform translation, a stationary universe becomes an expanding one, decreasing frequencies of light emission with the past become redshifts interpreted as due to recession, and a hyperbolic space of constant density becomes a Euclidian space of decreasing density. I have discussed these at length elsewhere,²⁴ but it may suffice here if we say that there are a number of serious technical difficulties in various aspects of the thesis. Though the model is perhaps the most elaborate modern attempt at an argument to the existence of a creator from the design of nature, I am afraid these difficulties are in many ways sufficient to make the rationale as a whole quite suspect. This is not to say that the kinematic method of analysis in cosmology is to be ignored, for several notable figures at present use it widely, but it is to say that Milne's detailed attempt to show that a crea ion could be demanded on its basis and to describe the world at this time, together with the later consequences of its nature, must be considered unconvincing.

Our last models for discussion are quite different from anything considered heretofore: they all imply that the creation of the universe is a continuing affair. Possibly the earliest of the modern advocates of this position were Macmillan, Millikan, and Jeans,²⁵ but their views received only casual interest until the stimulating suggestions of the last fifteen years by the British workers Bondi, Gold, and Hoyle. However, before turning to them we might note several rather different schemes with a similar theme which are, perhaps, not as widely considered. One such model is that of Dirac.²⁶

Working along similar lines to parts of Eddington's 'fundamental theory', wherein certain constants of nature (the 'fine structure constant', the ratio of proton to electron mass, the ratio of the gravitational to electrical attraction between an electron and proton, and so on) are related to one another, Dirac suggested that the age of the universe could be expressed, in terms of units given by atomic constants, as the large number 1039. Noting that this was also very nearly the value, or the square root of the value, of certain of Eddington's constants, he theorized that such constants were really not constant at all but were related to the age of the universe in atomic units. Detailed differences he presumed would be resolved later when we attained a sufficiently comprehensive theory of cosmology and atomicity.

The consequence of this, of interest here, is that the number of protons and neutrons must increase in proportion to t², that is new matter must enter space, where t is the present age of the universe in the atomic units noted above. Of course, evidence of this increase directly would be unlikely since it is minute in any short time and likely takes place within stars in any case. But a check might be made based upon another consequence of his model: the gravitational constant should be proportional to 1/t. He also predicted that, since the average mass of stars and galaxies cluster around t^1.5 and t^1.75 respectively, their masses should increase in that proportion and thus one might determine the resultant rate of increase of the average number of stars in a galaxy. A final consequence, too complex to outline here, is that space must have zero spatial curvature - a presumably testable deduction.

There are certain internal and empirical difficulties with this interesting thesis, but its importance lies in pointing out the odd relationship among the large physical constants which Eddington had also noted. It is safe to say that, whatever becomes of Dirac's peculiar model, this relationship will remain a challenge to future theorizing. One cosmologist who feels this way is Pascual Jordan.

In his scheme,²⁸ Jordan builds upon what he considers the six fundamental physical things in the universe: the velocity of light, the gravitational constant, the age of the oldest bodies (A), the mean density of mass in the universe, the Hubble constant, and the radius of the universe. Certain ratios between these turn out to be very near unity. Working on this base, and assuming that the negative potential energy of gravitation for the entire universe just cancels the rest energies of the masses of all the stars so that the universe has zero total energy, Jordan believes that the number of elementary particles is of the order of A2.

This implies the continuous appearance of matter into our universe, a mass for stars proportional to the three-halves power of the age of the universe at the time they are formed, and a gravitational constant which varies with the age of the universe. The first of these requires that matter enter the universe with mass such that their negative gravitational energy just balances the energy of the matter within them so that the total energy of the universe is unaffected. Today this requires the mass be of the order of a star! New stars, appearing as a unit, are the signs of such new matter arriving on the scene.

Jordan suggests that the universe may be pictured as a four-dimensional manifold with a cone shape but with many subsidiary apices. At any given time, a cut across a cone may leave a large three-dimensional part and perhaps several smaller isolated parts, each unfolding in time. Eventually any smaller part coalesces with the larger universe, and we have ourselves a new star which Jordan believes is the source of the Type I Supernovae of observational astronomy.

Several features of this model, as with Dirac's, may be questioned, however. It implies that stars are not the product of condensation from diffuse matter, a thesis which disagrees with most observational theorizing today. It also seems to require a rate of supernova appearance several hundred times that observed. Finally, it and Dirac's model before it, require serious rethinking in the light of new estimates for the age of many old objects seen in astronomy which necessitate increasing the ages Jordan and Dirac use (and use fundamentally, it should be noted) by a sizeable factor.

As our last model before turning in conclusion to Bondi, Gold, and Hoyle, let us note that of Reginald Kapp.²⁹ Based upon the rejection of a single creation because this makes two moments of time unique - the moment creation began and the moment it ended - and on the rejection of an infinite age for matter, since entropy and other considerations empirically seem to deny it, and yet upon the thesis that past time is endless, Kapp constructs a scheme which is to say the least, different. Assuming an infinite time and the continual appearance of new matter, and because such a universe under all prior theories should have infinite density which it obviously has not, Kapp concludes that matter continually disappears as well!

In this model, matter arises at random in space and disappears in the same manner. But because, in the interval, much of it is attracted into neighboring existing stars, the disappearance occurs largely within them - stars therefore lose mass in proportion to their size. However, though stars, and the galaxies in which they are situated, gain and lose mass continually, there comes a time when a rough equilibrium is established so that stars and galaxies should eventually all attain a more or less uniform miximum size. Unfortunately, apart from numerous other problems in Kapp's type of theorizing, this point seems to falsify the model, for, if we calculate the rate at which his thesis requires that matter be created and if we assume it is lost in large stars and galaxies at roughly the same rate, the estimate turns out impossibly large in the face of observation. We need delay no longer then in moving to the famous 'steady-state' theories to follow.

In 1948 Bondi and Gold, and independently Hoyle, presented papers involving schemes of considerable current interest.³⁰ They, and others, have since refined these first suggestions and, in what we will say, we shall look at the composite result. Bondi and Gold built their thesis upon a rejection of all the models based on Einstein's early work for they believe that, if the universe changes in size with time it must also change in density, leaving the laws of physics altered in the past in an unknownable way. The result would be that observations of very distant galaxies, and thus of the universe as it was long ago, will not be interpretable in terms of the laws of nature as we know them today on earth and we must therefore be agnostic as to their real character.

To avoid this, which must remove cosmology from any serious scientific interest, they postulate a 'perfect cosmological principle' in which the universe and its laws remain unchanged on the large scale with time and place. At the same time they reject the theory of general relativity and utilize instead certain aspects of kinematic argument. As a result, since observation shows the recession of distant galaxies and since it is essential that the average density of any large region of the universe remain approximately constant, they postulate the continual 'creation' of new matter between the mutually separating galaxies.

From this matter, presumed to be hydrogen atoms or (in Bondi and Lyttleton's 1959 paper) protons and electrons with the former either in slight excess or with a slight excess of charge, new galaxies eventually condense. The result is that any large volume will show galaxies of greatly varying age, but with an average unchanging with any place or time one makes observation. Also, in time, galaxies will tend to cluster due to gravity, and this clustering is actually observed. But the point of greatest interest here is that the age of our galaxy may well be much greater than the age given by the Hubble constant in working back to the time when all the galaxies we observe might have had their world-lines (that is their extrapolated course back through space and time) coalesce. In other words, unlike most prior models the Hubble 'age' is of no real significance and certainly is not an age for the universe as a whole.

Hoyle's model is similar yet distinctive. It is constructed on the assumption that general relativity is valid and yet that the average density of matter in the universe remain steady. This requires certain revisions in Einstein's field equations wherein a Cvector appears in the place of the 'cosmic constant' mentioned earlier in the paper. The technical details are much too sophisticated to outline here, but they have resulted in some esoteric discourse between Hoyle and his friends, Bondi and Gold, Which is noted in our references. We need only note that the net effect is quite similar to the Bondi-Gold scheme, with matter appearing continually in the model. The major differences lies in the fact that the 'perfect cosmological principle' here arises only as a consequence of Hoyle's axioms and is not fundamental as it is with the other men.

There can be little doubt that in assessing the consequences of these schemes, these three theoreticians and their followers have perhaps done more for theoretical astrophysics than has all the work predicated on the more Einsteinian constructs. The lite-rapture is permeated with the fruits of this beginning. However, this does not necessarily demonstrate the truth of their general thesis, since indeed it has a number of serious problems observationally and otherwise. Perhaps then, we should complete our survey by remarking rapidly upon the general status of the plethora of models we've discussed, for after all, any cosmological model must be judged on whether it is testable, whether it survives testing and whether among those which survive test it is most suggestive of future study.

Yet here we find a phenomenon not unknown to the history of science: the theory has outrun available test data, or even any testing potentially available in the near future. While I have given this problem in currently cosmology several hundred pages of discussion in my work noted in the references we must make a few comments of interest to our present theme.

Obviously one way to falsify any specific interpretation of a cosmogony is to disconfirm the model from which it is deduced. At times this may be done by spotting certain incoherences within the construct, as we have noted on occasion earlier, but usually it is a question of observational data providing the test as should be the case in science. Several such potential tests are the famous red shift-distance law and the density counts of galaxies at increasing distance from us. We can only report certain tentative conclusions here, because the data is not without question. The result is that presently the steady-state models of Bondi, Gold, and Hoyle are in some difficulty and the specific Einstein-de Sitter, Lemaitre, Dirac, and Milne models likewise seem rather unten able. However, we as yet cannot settle the value of the 'cosmic constant' nor the curvature permitted to the models, except that there is some possible preference for small negative values for both k and the curvature. This leaves a number of our models permissible on these grounds.

Another test is the ability of any model to explain stellar and galactic theories of age and evolution which seem to fit observation rather well. One problem with some of the schemes such as Eddington's, Jordan's, Dirac's, and the earlier expanding and pulsing models, has been that they inadequately handled such theories of stellar and galactic origin and also that they gave ages for the present expanding universe which were too small to fit the ages of the things they contain. However, while this is quite serious in the case of Eddington, Jordan, Dirac, Lemaitre, and Milne it is not so important, perhaps, in many of these other models. The reason is that it is not at all beyond possibility that, while all matter might have a finite age (that is, the universe had a beginning), much of the matter may have appeared subsequent to this origin as suggested by the various continuous 'creation' theses. This is not to say that Jordan's, Kapp's, or the Bondi-Lyttleton suggestions are therefore acceptable, for all have other major problems, but it is to say that a combination of certain facets of the Bondi-Gold-Hoyle cosmogony with some of the earlier general relativity models is a tenable option at present.

Of course future work is going to have to consider the suggestions of the G6del-Heckmann-Omer theses as well as the odd large dimension-less constants of Eddington, Dirac, and Jordan and the results may well be exciting. Nonetheless, we presently face the interesting tenability of a fairly broad spectrum of models, however we may refine these, and it looks as if this leaves us with even the maximum age and origin of our universe rather undecidable at the moment. In such a case, some hard thinking is going to have to be put into which models of this spectrum may be most quickly put to new tests (for only in this way can cosmology advance as a science) and which best fit broader esthetic, philosophical, and logical requirements. In our paper to follow, we shall assay the last of these points, insofar as it directly involves the question of origin and age. We have surveyed the landscape; it may be well now to do something which we might call a geological analysis of some of the structure beneath it.

1. See Einstein's 1917 paper, "Cosmological Considerations on on the General Theory of Relativity" in A. Einstein et at. The Ptinciple of Relativity (New York: Dover Publications, 1923), 175488.

2. We will take this term here and subsequently to mean a construct postulated as coherently describing the major features of the physical universe. Behind it will lie a theory considered to explain the processes of the model. The conjunction of model and theory Is a hypothesis about the laws governing astrophysical processes as a whole and the character of future observation which should be possible If the laws are true.

3. See Einstein's paper in Milton K. Munitz (ed.), Theories of the Universe (Glencoe: Free Press, 1957), pp. 275-279.

4. W. de Sitter's paper appeared in the Mon. Not. Row Astron. Soc., 78 (1917). 10 ff. See also Wm. Wilson. "De Sitter and the Expanding Universe," Science Progress, 48 (1960), 43-47. See also the excerpt from his Kosmos in M. K. Munitz, Op. Cit. pp. 302-319.

5. This static character was not a serious limitation to the models at the time since the observation of red-shift in the radiation from distant galaxies, widely taken today to indldate high velocities of recession relative to us, was as yet inconclusive and fragmentary.

6. A. Friedmann, "Uber die Krummung des Raumes," Zeit. fur Physik, 10 (1922), 377 ff. Also Op. Cit., 21 (1924), pp. 326 ff.

7. H. P. Robertson, "On the Foundations of Relativistic Cosmology," Proc, Nat. Acad. Sci., 15 (1929), 822-289.

8. Translations of his work appeared In the Mbn. Not. Roy Astron. Soc., 91 (1931), 483-501 and 93 (1933), pp. 628ff. Five of his most important papers from 1929 to 1945 appear in George Lemaitre, The Primeval Atom, trans. B.H. & S.A. Xorff (New York: Van Nostrand, 1950).

11. See his "The Primeval Atom Hypothesis and the Problem of the Clusters of Galaxies" in R. Stoops (ed.), La structure et l1evolution de Punivers (Brussels: Inst. Internl de Physique Solvay, 1958), pp. 1-31. Also his paper and Einstein's reply In P.A. SchlIpp (ed.), Albert Einstein: Philosopher-Scientist, Vol. 11 (New York: Harper, 1959), pp. 439-456, 684-685.

12. A. S. Eddington, "On the Instability of Einstein's Spherical World," Mon. Not. Roy Astron. Soc., 90 (1930), 668-6781 See also papers by W. H. McCrea and G. C. MeVittie, Op. Cit., 91 G. C. McVIttie, Op. Cit., 92 (1932)$ 500-518 and 93 (1933), 325339. Also H. Dingle, Op. Cit., 94 (1934)3 134-158. On the Einstein-de Sitter model In general see de Sitter in Proc. Astron. Soc. Pacific, 44 (1932) 89-104.

13. See J. Witt-Hansen, Exposition and Critique of the Conceptions of Eddington Concerning the Philosophy of Physical Science (Copenhagen: G.E.C. Gads, 1958); John Yolton, The Philosophy of Science of A. S. Eddington (The Hague: M. Nijhoff, 1960); and Kfimister and Tupper Eddington's Statistical Theory (Cambridge: Cambridge Univ. Press, 1962). Primary sources are the popular The Philosophy of Physical Science (Cambridge: Cambridge Univ. Press, (1939) 1949); the papers In the Mon. Not. Roy. Astron. Soc., 92 (1931),3-7; 45 (1935), 636-638; and 104 (1944), 200ff; and the very technical posthumously published Fundamental Theory. Here see Noel Slater, Eddington's Fundamental Theory (Cambridge: OUP, 1957).

14. George Gamow, Creation of the Universe (New York: VikIng, 1956); "Origin of the Protogalaxies," Astron, Jour., 58 (1953), 39ff; and D. Layzer's paper Op. Cit., 59 (1954), 170-173. See also T. Kohman, "Chronology of Nucleosynthesis and Extinct Radioactivity," Jour. Chem. Educ., 38 (Feb., 1961), 73-82 and R. Alpher et al, "Physical Conditions in the Initial Stages of the Expanding Universe," Phy. Rev., 92 (1953), 1347 ff.

15. Compare F. D. Kahn in Times Science Review, 36 (1960), 3-5; V.C.Reddish in Set. Prog., 48 (1960), 241-251; Jan Oort in Sci. Amer, 195 (Sept., 1956); W. J. Luyten In Amer. Scientist, 48 (1960), 30-39; and M. & G. Burbidge in Sci. Amer., 199 (Nov., 1958), 44-50.

16. Compare W. Fowler In Sci. Mon., 84 (1957), 84-100; Charles Coryell in Jour. of Chem. Educ., 38 (Feb., 1961 ' 72 ff; M. & G. Burbidge in Science, 128 (1958), 387-399; J. Greenstein in Amer. Scientist, 49 (1961), 449-473; and W. Fowler et at, Geophy. Jour., 6 (1962), 148-220.

17. E. J. Opik, The Oscillating Universe (New York: New Amer. Library, 1960).

18. Compare Gamow, Op. Cit., p. 29; W. B. Bonnor In H. Bondi et at, Rival Theories of Cosmology (Oxford: OUP, 1960), pp. 8-9; G. C. MeVittle, Fact and Theory in Cosmology (London: Eyre & Spottiswoods, 1961), p. 152; and K. Harrison et at in R. Stoops (ed.), Op. Cit., pp. 124-145.

19. See C. J. Kevane, "On Antimatter and Cosmology," Science, 133 (1961), 580-581; 0. L. Reiser, "Matter, Anti-matter, and Cosmic Symmetry," Phil. Sci., 24 (1957), 271-274; Goldhaber, "Speculations on Cosmogony," Science, 124 (1956), 218; R. A. Alpher & R. Herman, "On Nuclean-Antinucleon Symmetry in Cosmology," "Science, 128 (1958), 904; T. Gold in R. Stoops (ed.), Op. Cit., pp. 90-91; R. 0. Kapp, Towards a Unified Cosmology (London: Hutchinson, 1960), pp. 277-282; G. Burbridge & F. Hoyle, "Antimatter," Sci. Amer., 198 (Apr., 1958), 34,39; and Alfven & Klein In Arkiv. Fysik, 23 (1963) 187-194.

20. See Godel's papers in Rev. Mod. Phys., 21 (1949), 447 ff., In P. A. Schilpp (ed.), Op. Cit., Vol. II, pp 557-562; See Heckmann's Theorien der Kosmologie (Berlin: Springer, 1942) and the reports of his London lectures in 1959 In The Observatory, 79 (1959), 130-131; and see general discussion of these ideas in J. L. Synge, Relativity: The General Theory (Amsterdam: North Holland, 1960), pp. 331-338; G. J. Whitrow, Natural Philosophy of Time (London: Thomas Nelson, 1961), pp. 257-261; and E. Schucking and 0. Heckmann In R. Stoops (ed.), Op. Cit., pp. 149-159.

22. See E. A. Milne in Proc. Roy. Soc., A, 156 (1936), 62 ff; Nature, 139 (1937), 409, 997-999; Mon. Not. Roy Astron. Soc., 93 (1933), 519-529, 668-680 and 94, (1934), 3-14; and Zeit fur Astrophysik, 6 (1933), 1-95. Later, and somewhat revised presentations, occur in Mon. Not. Roy Astron. Soc., 104 (1944), 120-136; Bull. de IAcademie Internationale de Phil. des Sciences, A, 3 (1947), 7-51 and discussion; Philosophy, 16 (1941), 363; Kinematic Relativity (Oxford: Clarendon, 1948); and Modern Cosmology and the Christian Idea of God (Oxford: Clarendon, 1952).

23. Discussion of these and other matters may be found In Otto Bluh's articles in the Jour. Roy Astron. Soc. Canada, 43 (1949), 169-180; in MeVittle and Wyatt's article In the As~ trophy. Jour, 130 (1959), 1-11; in Dudley Shapere's article in Phil. Rev., 69 (1960), 376-385; in J. Singh, Great Ideas and Theories of Modern Cosmology (New York: Dover Pubns., 1961), pp. 113-133; in G. C. McVittie, Cosmological Theory (London: Methuen, (1949) 1952), pp 70-100; and H. Bondi, Cosmology (Cambridge: CUP, 1952) pp. 123-139.

24. T. H. Leith, Popper's View of Theory Formation Compared with the Developmen tof Post Relativistic Cosmological Models (Boston Univ. PhD dissertation), pp. 231-242, 301-302, 323, 486491. Many other points in the present paper are to be found in this source.

25. See W. D. Macmillan, "On Stellar Evolution," Astrophy. Jour., 48 (1918), 37-49 and his note In Nature, 129 (1932), 93; R. Schlegel, "Steady-State and Theory at Chicago," Amer. Jour. Phy., 26 (1958), 601-604; R. A. Millikan, Science and the New Civilization (New York: Scribner's, 1931), pp. 106-109, 1963; and Sir James Jeans, Astronomy and Cosmogony (Cambridge: CUP, 1928), p. 352.

26.See his notes in Nature, 139 (1937), pp. 323, 1001-1002 and his paper in Proc. Roy. Soc., A, 165 (1938), pp. 199ff. Also S. Chandrasekhar in Nature, 139 (1937), 757-758. Later notes may be found in Nature, 189 (1962), pp. 664, 766 and 190 (1962), p. 587. See also Dicke and others in Jour. Geophy. Research, 67 (1962), 4063-4047; in Science, 138 (1962), 53-664; and in Space Research 11 (Amsterdam: North-Holland, 1961), pp 287291.

28. See his "Formation of the Stars and Development of the Universe," Nature, 164 (1949), 637-640 and his Schwerkraft und Weltall (Braun-Schweig: F. Viewag, 1955). See also H. L. Dorrie, Genesis (Munich: C. H. Beek, 1959); P. Douderc, The Expansion of the Universe (London: Faber and Faber, 1952), pp. 222-224; P. Jordan and others in Ann. der Physik, 1(1947), 219 ff. and 2 (1948), 76ff; and Jordan in Rev. Mod. Phy, 34 (1962), 596-600.

29. See his work noted in reference nineteen and also his note in The Observatory, 73 (1953), 113-116 and in Nature, 165 (1950), 68-69.

30. See H. Bondi, "Review of Cosmology," Mon. Not. Roy. Astron. Soc., 108 (1948), 104-120; H. Bondi and T. Gold, "The Steady-state Theory of the Expanding Universe," as above, pp. 252-270; and H. Bondi and R. A. Lyttleton in Proc. Roy Soc., A, 252 (1959), 313-333. Also F. Hoyle, "A New Model for the Expand.Sng Universe," Mon. Not. Roy Astron. Soc., 108 (1948), 372-382; and his paper, Op. Cit., 109 (1949), 365-371. The various books by Bondi and by Hoyle need not be noted. They will be in any adequate library. See also the extended discussion I give these models in my work noted earlier.