2. Entropy and the Classical Cyclic Universe

From an early date the second law of thermodynamics was applied to the universe at large; witness that Rudolf Clausius’ canonical formulation of the principle, dating from 1867, reads as follows: “The entropy of the universe tends to a maximum” (Clausius Reference Clausius and Hirst1867, 365). The law led to the prediction that in the far future the universe would suffer a “heat death” corresponding to the irreversible state of maximum entropy where all temperature differences had vanished. It was further argued that since entropy ever increases, and since we manifestly do not live in a high-entropy world, the universe cannot have existed for an eternity of time. It must have had a beginning, presumably in a creative act which was commonly believed to be divine – what else could it have been?Footnote ³ From this kind of reasoning, allegedly based on the authoritative laws of thermodynamics, it followed that cyclic and other eternal conceptions of the universe were ruled out. As early as 1868, three years after he had introduced the notion of entropy, Clausius used the second law to conclude that a cyclic conception of the universe was scientifically impossible (Clausius Reference Clausius1868, 417). Of course, advocates of a cyclic world view – many of them materialists, positivists, or atheists – vehemently denied that this was the case.

It is important to realize that the nineteenth-century concept of the cyclic universe or the end of the world was in a sense very different from the concept adopted by later generations of physicists. They argued that even though parts of the infinite universe may be running down and for some period end in a state of equilibrium, there will always be other parts in which constructive entropy-consuming processes dominate the cosmic scene. High-entropy areas were not eternally dead, for they could be awakened and would then turn into active states of low entropy. Although people at the time referred to this conception as cyclic, it may be more appropriate to speak of a regenerative or reprocessing universe. The general conception of the universe in the late nineteenth century was that it consisted of a multitude – possibly an infinity – of stars floating around in space and that the stars and nebulae had been formed from a primeval mist. This conception might include a notion of the size of the sidereal universe, but not a notion of the size of cosmic space and certainly not a notion of cosmic space varying in time.

The spatial or geometric sense that we associate with the notion of a cyclic universe was foreign to the Victorian mind. Within the framework of the standard cosmological models of general relativity theory, this notion presupposes the notion of a closed space with a curvature that varies periodically in time. Although the concept of a closed space had been familiar to mathematicians since the middle of the nineteenth century, it only became an effective part of physics with Einstein's general theory of relativity. However, it is worth noting that several decades before this happened the controversial German astrophysicist Karl Friedrich Zöllner suggested that our world is not spatially flat (Euclidean) but positively curved. He believed that the universe contains only a finite amount of matter, and this hypothesis led him to “ascribe to the constant curvature of space not the value zero but a positive value, however small.” Arguing that neither cosmic time nor space can be limited, Zöllner suggested that in a positively curved space all physical processes would occur cyclically, indeed that the universe itself would be cyclic and that time would thus go on forever:

That is, Zöllner's universe was cyclic in the sense that all physical processes occurred periodically, not in the sense that the size of the universe varied cyclically in time.

The latter possibility was independently entertained by a few of Zöllner's contemporaries, although within the framework of ordinary Euclidean space. They thought of the material universe in terms that have a certain analogy to later views of the cyclic universe. According to Rudolf Falb, an Austrian amateur scientist, space could not exist without matter, and he shared the belief of many astronomers that there were only a limited number of stars and nebulae. A sidereal universe consisting of a huge number of stars would eventually contract gravitationally to a giant sun which, so he postulated, would immediately evaporate and form an expanding gas, a nebulous body of the same kind that had originally given rise to the celestial bodies. Falb contended that the process would continue indefinitely: “The life of the world is to be conceived as a recurrence of expansion and contraction, like the breaths of a monstrous colossus. In this way the eternity of processes becomes understandable, that is, the infinite duration of the universe. . . . The end of the world is at the same time the beginning of the world” (Falb Reference Falb1875, 202).

A somewhat similar picture of a pulsating material universe, eternally varying between a highly compressed state and a maximum size, was developed by Ludwig Zehnder, a Swiss-German professor of physics at the Berlin Technical University (Zehnder Reference Zehnder1897). According to Zehnder, space was filled throughout with ether atoms which he assumed were responsible for electrical phenomena. As a result of the gravitational collapse, electrical repulsion between atoms of matter would increase and lead to an expansion of the material universe. The competition between gravitational attraction and electrical repulsion was the motor that secured Zehnder's eternally oscillating universe.

3. From Friedmann to Tolman and Beyond

A completely new chapter in the history of cosmology had its start in 1917, when Einstein first applied his general theory of relativity to the universe at large. Although not recognized at the time, this conceptual innovation was to form the basis of a new paradigm of cosmological thought, a radical disjuncture that comes close to being a Kuhnian revolution. Einstein's original model was closed and static, that is, with a curvature radius that did not depend on time, a result he secured by introducing his famous cosmological constant (symbol Λ). Only in the 1920s did Alexander Friedmann in Russia and Georges Lemaître in Belgium demonstrate that the cosmological field equations include dynamical models in which the curvature of space varies in time. This important insight was crucial to the later development of cosmology, but for several years it was ignored by contemporary astronomers and physicists.

In his important but ill-fated work of 1922, Alexander Friedmann [Aleksandr Fridman] analyzed from a mathematical perspective the various solutions to Einstein's field equations for an isotropic and homogeneous universe. These solutions had the form R = R(t), where R is the scale factor (relating to the radius of curvature) and t the cosmic time. For a particular case, corresponding to a closed space with Λ ≤ 0, he showed that R(t) became a periodic function: “The radius of curvature varies between 0 and x ₀. We shall call this universe the periodic world” (Friedmann Reference Friedmann1922, 385; idem Reference Friedmann1999). As an illustration he calculated that for Λ = 0 and a world mass of M = 5 × 10²¹ solar masses, the world period became of the order of 10 billion years. Friedmann did not describe the universe as oscillating in the sense that cycle followed after cycle, but only referred to a single cycle from bang to crunch. However, in a semipopular book published in Russian in 1923, The World as Space and Time, he connected his model with the old idea of a perpetually oscillating universe: “Cases are also possible when the radius of curvature changes periodically. The universe contracts into a point (into nothing) and then increases its radius from the point up to a certain value, then again diminishes its radius of curvature, transforms itself into a point, etc.” (Friedmann Reference Friedmann and Singer2000, 109).Footnote ⁴

Einstein, who back in 1922 had rejected Friedmann's theory of a dynamical universe, came to a different conclusion after the redshift measurements of Melvin Slipher and Edwin Hubble had made the expanding universe a preferred alternative to the static model. In a brief paper in the Sitzungsberichte of the Prussian Academy of Science from the spring of 1931 he belatedly recognized Friedmann's pioneering work and decided that there was no longer any need for the cosmological constant (Einstein Reference Einstein1931). He discussed a Friedmann single-cyclic model with Λ = 0, but without extending it to possible previous or later cycles. That is, neither Friedmann's model of 1922 nor Einstein's of 1931 was oscillatory in the strict sense.

Why did Einstein choose the cyclic model rather than some of the other Λ = 0 solutions to the Friedmann equations? It is well known that Einstein nourished a deep admiration for Spinoza's philosophy, and it has been suggested that this is what motivated him to propose a cyclic model in 1931 (Steinhardt and Turok Reference Steinhardt and Turok2007, 177).Footnote ⁵ According to Spinoza, God would not have made a world which was empty and limited in time, and so (the argument goes) Einstein might have found it natural to think of the expanding universe as a stage in an eternal succession of cycles. However, the speculation is far from convincing. For one thing, if Einstein had been committed to a universe with no absolute beginning in time, why did he so readily abandon it for the finite-age Einstein-de Sitter model of 1932?Footnote ⁶ His admiration for Spinoza persisted, and yet he never returned to the oscillating universe. Moreover, given that his 1931 model was not in fact perpetually oscillating, it did not live up to Spinoza's claim of an eternal universe. Einstein's model included such un-spinozist features as a beginning and an end of time, and neither then nor later did he suggest extending the model to an infinity of cycles.

Einstein's 1931 model of a pulsating universe was well known and for a while it attracted considerable attention. However, as Einstein was well aware, it was plagued by at least two major problems. First, it resulted in an age of the universe that was much too short, amounting to at most 1.5 billion years. Second, it contained two singularities (R = 0) where the density of matter would become infinite. In both cases Einstein suggested that the problems were caused by the approximations of the model, and that they might disappear if assumptions of homogeneity and symmetry were abandoned in a more realistic version. Although the Einstein model described a single cycle only, it was often seen as a multicycle model and inspired new interest in the idea of endless oscillations. Lemaître did not believe in such models, but he found them to be fascinating “from a purely aesthetic point of view.” As he wrote in a paper of 1933: “Those solutions where the universe expands and contracts successively while periodically reducing itself to an atomic mass of the dimensions of the solar system, have an indisputable charm and make one think of the Phoenix of legend” (Lemaître Reference Lemaître1997, 679, a translation of Lemaître Reference Lemaître1933).

A thorough investigation of cyclic models was first undertaken in a series of works from 1931 to 1934 by Richard Chase Tolman, an eminent American physical chemist and mathematical physicist at the California Institute of Technology, where he stayed from 1922 to his death in 1948. Tolman was not only a specialist in relativity and cosmology, but also in thermodynamics, and he decided to investigate cosmological models from the point of view of a relativistic generalization of the laws of thermodynamics. A major result of his investigations was a drastic revision of the results based on the classical thermodynamical reasoning of the late nineteenth century. According to Tolman, there was no justification for either entropic creation or the heat death. As he formulated it in his important textbook of 1934, Relativity, Thermodynamics and Cosmology: “It would seem wisest, if we no longer dogmatically assert that the principles of thermodynamics necessarily require a universe which was created at a finite time in the past and which is fated for stagnation and death in the future” (Tolman Reference Tolman1934, 444).

In the fall of 1931 Tolman concluded that a model universe such as Einstein's could expand and contract reversibly without increase in entropy (Tolman Reference Tolman1931), and the following year he extended the investigation to systems with irreversible processes. It would thus be possible for a closed universe with Λ ≤ 0 to undergo a continual series of cycles. In general, “relativistic thermodynamics could not impose restrictions which would prevent such a series of expansions and contractions” (Tolman Reference Tolman1932a, 331). However, the cycles would not be identical, for the maximum value of R would increase with the number of cycles and so would the period; at the same time the rate of change of R at a given value would also get greater. The entropy would indeed increase from one cycle to the next, but without ever approaching a limit. Tolman was pleased to have found, for the first time, a model which showed a continual succession of expansions and contractions “without ever coming to that dreadful final state of quiescence predicted by the classical thermodynamics” (Tolman Reference Tolman1932b, 372).

Strictly speaking, Tolman's mathematical analysis of a pulsating universe filled with matter and radiation was valid only for a single cycle, but he argued that an extension to an endless series of successive expansions and contractions was reasonable from a physical point of view. In an important paper coauthored with his student Morgan Ward he spoke in favor of the “possibility that the actual universe or parts thereof might also exhibit such a continued succession of expansions and contractions” (Tolman and Ward Reference Tolman and Ward1932, 843). Although he was unable to provide any physical mechanism for the passage from one cycle to the next, in his book of 1934 he boldly suggested that such passages were “physically inevitably necessary.” More generally he found it “difficult to escape the feeling that the time span for the phenomena of the universe might be most appropriately taken as extending from minus infinity in the past to plus infinity in the future” (Tolman Reference Tolman1934, 486).

In spite of his great interest in oscillating models, Tolman was not emotionally committed to a cosmology based on periodic expansions and contractions. At the end of his book, he not only warned against interpreting Lemaître's new big-bang model realistically, he also extended his warning to cover cyclic models. Such models, he pointed out, “must not be confused with a proof that the actual universe will always provide a stage for the future role of man” (Tolman Reference Tolman1934, 488, and similarly in Tolman Reference Tolman1932b, 373). Tolman's lack of strong commitment to an ever-oscillating universe is shown in his later publications in cosmology. From 1934 to his death in 1948 he continued doing important work in this area of research, but without paying attention to oscillating models of the kind he had analyzed in his earlier works.

In addition to the problem of cosmic singularities, the oscillating model faced two related empirical problems, although neither of them definitely ruled it out. The time scale problem was serious, but it was a problem shared with several other cosmological models, including the Einstein-de Sitter critical model. The other problem was the high density of matter required for the universe to reverse its motion in the future. On the other hand, it was generally recognized that the most favored observational estimate of 10⁻³⁰ g/cm³ was uncertain and that a considerably greater value remained a possibility. In The Realm of the Nebulae of 1936, Hubble stated 10⁻³⁰ g/cm³ as a lower limit and 10⁻²⁸ g/cm³ as an upper limit (Hubble Reference Hubble1936, 189).

The revision of the Hubble constant in the 1950s, when it was reduced by a factor 3–10, made life a little easier for closed models because it implied that the mean density of matter needed to close the universe became smaller and therefore less difficult to justify observationally. In spite of this change, cyclic models of the universe were not highly regarded by the majority of cosmologists. The general skepticism with regard to oscillating models may be illustrated by the French astronomer Paul Couderc, who had no confidence in closed models with Λ ≤ 0. They have, he wrote, “an ephemeral existence, bounded at either end by a catastrophic condition (nil radius); their life, equal at most to 10⁹ years, could not provide for our Archaean rocks” (Couderc Reference Couderc and Sidgwick1952, 195). Still, it was possible to appeal to large amounts of invisible matter in the universe, and it was in any case realized that observational data were highly uncertain. The oscillating universe remained alive.

In fact, some observations seemed to favor a closed universe and therefore reopened the possibility of an endless chain of cycles. The observational cosmologist Allan Sandage and his collaborators at Mount Palomar Observatory found in 1956 from new redshift-magnitude data that the universe is probably closed, a result that he welcomed. Not only did it rule out the steady-state theory (which Sandage disliked), it also suggested that the universe might evolve cyclically, a possibility he found appealing. In a popular exposition of his work he wrote: “If the expansion of the universe is decelerating at the rate the photographic data suggest, the expansion will eventually stop and contraction will begin. If it were to return to a superdense state and explode again, then in the next cycle of oscillation, some 15 billion years hence, we may find ourselves again pursuing our present tasks” (Sandage Reference Sandage and Piel1957, 97–98).Footnote ⁷

4. The Cyclic Universe Bounces Back

Among those who defended oscillating models in the 1950s, two astronomers in particular merit attention. In 1957, the distinguished Dutch astrophysicist Herman Zanstra, professor of astronomy at the University of Amsterdam, published a careful study of bouncing and pulsating universes which in some respects went beyond Tolman's early works (Zanstra Reference Zanstra1957; see also Zanstra Reference Zanstra1964). For example, he considered the role of stellar nuclear reactions which in the present state of the universe convert hydrogen into helium and radiation energy. He suggested that in the contracting phase a complete regeneration of energy would occur, so at the bounce the universe would have returned to a pure state of hydrogen. Then, following the bounce, a new cycle would start afresh. Whereas Zanstra confirmed Tolman's result that new cycles would grow increasingly bigger, with greater values of the maximum curvature radius and the period, he also found from thermodynamic arguments that “the universe if of the pulsating type could not have existed prior to a certain time” (Zanstra Reference Zanstra1957, 300). In other words, there could have been only a finite number of previous cycles.

Zanstra realized that the known laws of physics would lead the contraction to end in a singularity, but in agreement with Einstein and Tolman he felt free to assume that these laws would break down near R = 0 and thus leave open the possibility of a continual succession of expansions and contractions. Although he admitted that the oscillating universe was not supported by observations, he thought it was just observationally possible and at any rate a model to be preferred from a philosophical and aesthetic point of view (see section 5).

In the 1950s William Bonnor, a British physicist and expert in general relativity theory, investigated theories of galaxy formation. In this connection he argued for an oscillating universe in which the inhomogeneities of the early universe, out of which seeds galaxies were formed, were fossils from the preceding contraction. He suggested that under certain circumstances a contracting Friedmann model could change into an expanding one without passing through a singular state (Bonnor Reference Bonnor1954; idem 1957). This would be possible, he thought, if radiation were converted into matter during the compression, and a negative pressure was in this way built up near the minimum.

Bonnor's favored candidate was thus a cyclic model in which the universe oscillated smoothly, bounced between states of high but not infinitely high density. What mattered to Bonnor was not really the oscillatory behavior of the universe, but to have a dynamical and singularity-free model with an unlimited past and future. As he admitted, this was an epistemic desideratum, for he thought it would be contrary to the spirit of science to restrict the scope of its hypotheses (Bonnor Reference Bonnor1957, 162). If it were accepted that the universe was limited in time, questions about its state before t = 0 and after t = t _max would be forbidden, whereas this restriction was not imposed by the model of infinite oscillations.

The hypothesis of an ideal fluid with negative pressure came to play an important role in cosmology after about 1960, both in big-bang models, oscillating models, and revised steady-state models. Introduced by the Irish-born physicist William McCrea in 1951 in the context of the steady-state theory of the universe, it resulted in several new ideas of universes oscillating gently without bangs or crunches (McCrea Reference McCrea1951; Harrison Reference Harrison1967). Among the first of these models was a proposal by the Polish physicist Jaroslav Pachner that “our universe is oscillating and no singularity with an infinite density of matter occurs during its periodical pulsations between two extreme values” (Pachner Reference Pachner1965, 173). To obtain the bounce he made use of “a natural extension of McCrea's ideas,” and arrived in this way at a negative pressure varying with the curvature of space. The extreme state of maximum contraction was assumed to result in a density of at least the order of nuclear density, that is, ρ ≥ 10¹⁶ g/cm³.

Even with the assumption of a negative pressure as a way to avoid the singularity at the bounce, oscillating models were not highly considered by most cosmologists during the last third of the twentieth century. At a time when the standard hot big-bang theory, with or without inflation, dominated cosmology there seemed little reason to reconsider cyclic models. After all, models with a finite past time seemed to agree with observations, and the hypothetical series of endless cycles could be and often was dismissed as speculative.

What is more, the old entropy problem reappeared in a new way, now related to the finite value of the entropy per baryon (“baryon” is a generic name for subatomic particles such as protons and neutrons). In each cycle the entropy increases by a finite amount, from which follows that the universe cannot have lived through an infinite number of cycles. As mentioned, a similar result had been obtained by Zanstra in 1957. The Russian physicists Igor Novikov and Yakov Zel'dovich found the cyclic universe to be an “appealing picture,” but they also noted that “The multicycle model has an infinite future, but only a finite past,” which they considered much less appealing (Novikov and Zel'dovich Reference Novikov and Zel'dovich1973, 402). Robert Dicke and Jim Peebles, two of the key figures in the revival of the big-bang theory in the 1960s, similarly pointed out that “the Phoenix could reproduce only a very limited number of times” (Dicke and Peebles Reference Dicke, Peebles, Hawking and Israel1979, 512; see also Rees Reference Rees1969, 197). Obviously, an oscillating universe that can be traced back to a first cycle loses much of its philosophical appeal, since the question of an absolute origin then reappears.

Apart from theoretical problems such as the entropy problem and overproduction of black holes in the contracting phase (Saslaw Reference Saslaw1991), there were observational problems. A classical Friedmann-type cyclic universe assumes space to be closed, and according to observations in the 1970s this assumption was contradicted by observational evidence. From a careful weighing of the observational evidence a group of American astronomers concluded that the expansion of the universe cannot be halted or reversed. They described the objections to closed models as “formidable,” although cautiously adding that they might not be fatal (Gott et al. Reference Gott, Gunn, Schramm and Tinsley1974, 552; see also Gott et al. Reference Gott, Gunn, Schramm and Tinsley1976).

All this was not enough to kill the oscillating universe model, which continued to be explored by a minority of physicists and astronomers. One of the models, fairly well known in the cosmology community, was due to the Israeli physicists Mark Israelit and Nathan Rosen (Israelit and Rosen Reference Israelit and Rosen1989; idem Reference Israelit and Rosen1991). Rosen, by then 80 years old, was a former collaborator of Einstein and coauthor of the famous EPR (Einstein-Podolsky-Rosen) paper of 1935 (Einstein, Podolsky, and Rosen Reference Einstein, Podolsky and Rosen1935). He also worked with Einstein on gravitational waves and formulated the basic concepts of the so-called “Einstein-Rosen bridge,” a precursor of what later became known as wormholes in the general theory of relativity (Kennefick Reference Kennefick2007, 79–104; Israelit Reference Israelit1996). According to the Rosen-Israelit oscillating model the expansion of the present universe began with a “cosmic egg” assumed to have a density of the order of the Planck density ρ_P = c ⁵/hG ² ≅ 10⁹⁴ g/cm³, where h denotes Planck's constant, c is the velocity of light, and G Newton's constant of gravitation. This “prematter,” as they called it, was assumed to have an enormous negative pressure and would therefore blow up at a gigantic rate. Whereas the initial radius of the cosmic egg was about 10⁻³⁴ cm, after a time of 10⁻⁴² s it would have increased to about 10⁻⁵ cm and then the expansion would decelerate, the universe developing much in the same way as in the conventional big-bang theory. The period of their oscillating model was stated to be 1.2 × 10¹² years, the amplitude R _max ≅ 10²⁹ cm, and the present age 1.4 × 10¹⁰ years.

This picture of the early universe has some similarity to the inflation scenario, which at the time had become accepted by a majority of cosmologists, and Israelit and Rosen indeed offered their theory as an alternative to the “complicated” and “highly speculative” inflation theory. They showed that it was able to take care of the flatness problem and the horizon problem, the two problems that had motivated the inflation theory in particular. At about the same time Ruth Durrer and Joachim Laukenmann, two Swiss physicists, suggested that the oscillating universe was a viable alternative to the inflation “cosmological dogma” (Durrer and Laukenmann Reference Durrer and Laukenmann1996). Not only did they show that the flatness and horizon problems can be solved in this way, they also argued that no entropy problem needed to occur. Several more works were published on oscillating models in the late twentieth century, but there is no need to mention them all. Instead I shall take a look at some of the extrascientific considerations that were associated with this kind of cosmological models during the period from about 1930 to 1990.

5. Aesthetic and Emotional Responses

Extrascientific arguments are common in all sciences at a fundamental level and especially in cases where empirical data are insufficient to choose between theories. There have always been more cosmological theories and models than relevant observational data, which makes it understandable that extrascientific considerations have played and still play a significant role in this science. The lack of reliable cosmological facts was a major reason why extrascientific arguments were very important in the early evaluations of the big-bang theory and, not least, in the heated controversy between relativistic evolution theories and the steady-state theory. The choice between cosmological models was often conceived to be a choice between different world views, something that could not be made on a purely scientific or rational basis. To a large extent it had to rely on the scientist's personal “taste” (Kragh Reference Kragh1996, 219–227).

One may distinguish between two classes of extra- or transscientific considerations. One class is philosophical in the strict sense of including methodological and epistemic criteria that a good theory is supposed to live up to. If two competing theories A and B fare equally well in empirical tests, and A is clearly superior when it comes to those philosophical criteria, theory A will usually be preferred over theory B. The kind of criteria included in this class is well known from textbooks in philosophy of science and often used, more or less spontaneously, by research scientists. Thus, a good theory is expected to be simple, have a high degree of testability, contain no ad hoc elements, be complete, have great explanatory and predictive power, be systemically connected to established knowledge, etc.

Such criteria routinely enter theory evaluation, whereas the other class of extrascientific considerations rarely appear explicitly. These considerations too may be called philosophical, but now in a much looser sense. They are aesthetic and emotional criteria, and for this reason they are basically subjective. A theory may be judged to be “appealing” or “beautiful,” or an alternative theory may be criticized for being “unnatural” or “implausible.” In some sciences, including cosmology and branches of biology, human values and desires – such as the possibility of endless life – may play a role. Religiously motivated arguments may enter this category as well. The problem with such considerations is that they only make sense or carry force to the believers. What one scientist finds appealing, another one may find unappealing. And yet aesthetic factors of this kind often play a role in the sciences in addition to the epistemic and methodological factors. Whether this role is legitimate or not is a matter of debate. Most scientists probably hold that they should only be provisionally accepted until they can be replaced by rational and empirical criteria. Extra- or transscientific notions come into play in particular in the “context of discovery” (or context of pursuit), whereas they are more rarely found in the public “context of justification.”

A review of the literature shows, perhaps contrary to expectations, that extrascientific considerations only turned up in relatively few cases in the scientific works that related to cyclic models of the universe during the period from about 1920 to 1990. None of the pioneers of this kind of model – Friedmann, Einstein, and Tolman – assigned great philosophical significance to it or was emotionally attached to it. The philosophical appeal of the oscillating universe is closely associated to the model being multicyclic, a possibility which was not seriously considered by Friedmann and Einstein. To the extent that emotions and human desires were involved in the early phase of oscillating universe cosmology, they appeared in non-scientific contexts and were more often negative rather than positive evaluations.

No physicist or astronomer in the early period expressed strong commitment to the idea, whereas a few reacted emotionally against it. As early as 1928, Arthur Eddington admitted that he was “no Phoenix worshipper.” He explained his aversion as follows:

Eddington did not at the time know of Friedmann's work and his view thus referred to the older, prerelativistic conception of a cyclic universe. But his opposition was of a general kind and he repeated it after having become acquainted with the relativistic models of an oscillating universe. His dismissal of the cyclic universe was not scientific, but rather based in his religious and moral sentiments, which he made clear in 1935: “From a moral standpoint the conception of a cyclic universe, continually running down and continually rejuvenating itself, seems to me wholly retrograde” (Eddington Reference Eddington1935, 59).

Eddington was not the only one to feel this way. The Dutch astronomer Willem de Sitter, another of the pioneers of relativistic cosmology, felt the oscillating model to be positively unappealing: “Personally I have, like Eddington, a strong dislike to a periodic universe, but that is a purely personal idiosyncracy, not based on any physical or astronomical data” (de Sitter Reference De Sitter1933, 630). In a lecture of 1940, the distinguished American astrophysicist Henry Norris Russell expressed his surprise at “the wide-spread desire to believe in some cyclical restoration of however great intervals” (Russell Reference Russell1940, 27). With regard to this question, which he considered to be aesthetically rather than religiously based, “for religion has never concerned itself much about the fate of material things,” he sided with Eddington and de Sitter.

To jump ahead in time, the Russian theoretical physicist and famous political dissident Andrei [Andrey] Sakharov found oscillating models to be most appealing and of more than just scientific interest. At the end of his Nobel Peace Prize lecture of 1975 he expressed support of “the cosmological hypothesis that states that the development of the universe is repeated in its basic characteristics an infinite number of times.” Characteristically, he did not mention the scientific merits of this hypothesis but its relation to the values of life. “This weltanschauung,” he said, “cannot in the least devalue our sacred inspirations in this world, into which, like a gleam in the darkness, we have appeared in the darkness for an instant from the black nothingness of the ever-conscious matter, in order to make good the demands of reason and create a life worthy of ourselves and of the goal we only dimly perceive” (Drell and Okun Reference Drell and Okun1990, 32).Footnote ⁹

Sakharov investigated in several works models of the oscillating universe or what he preferred to call “multisheet models.” When he realized that an oscillating universe can contain only a limited number of cycles, he was disappointed because it implied a first cycle beyond the limits of scientific enquiry. His disappointment was epistemically rooted: “An approach that places no limit on the scientific investigation of the material world and space-time in my opinion is better and more fruitful, even though it leaves no room for an act of Creation: the basic religious concept of divine meaning of existence does not concern science and lies beyond its limits” (Sakharov Reference Sakharov1990, 543).

What are perhaps the clearest expressions of an emotional sympathy for the oscillating universe in the period here considered appeared in two works of the 1950s. In an essay on the age of the universe, the eminent Estonian-Irish astronomer Ernst Julius Öpik noted that Lemaître's primeval atom hypothesis might as well be interpreted as the outcome of a previous, indefinitely oscillating universe. The choice between the two conceptions, he said, “will remain a matter for aesthetic judgment, not for positive science as theory verified by observation” (Öpik Reference Öpik1954b, 91).Footnote ¹⁰ Although not belonging to science proper, Öpik believed that extrascientific factors of an aesthetic kind were not only valuable but indispensable: “There is no proof in purely aesthetic matters. This does not mean that aesthetic methods of approach to scientific problems are worthless. On the contrary, scientific theories are created by intuition, or by an essentially aesthetic process. However, without the flesh and bone of experiment such theories remain mere shadows of possibilities” (ibid.).

Öpik's preference was clearly for an eternal and cyclic universe, or what he called the “Great Repetition” in which the universe would return to an initial state of nuclear fluid an infinity of times. He stressed however that this was not the same as an eternal recurrence in the sense of Nietzsche, for each cycle would be new and offer its own unlimited variety of events and prospects of evolution. Although he maintained that the idea of an oscillating universe did not really belong to the realm of science, nor did he think that this was the case with Lemaître's idea of a universe with an absolute age, Öpik took it very seriously. In 1960 he published a monograph on the subject in which he described how the old universe gave birth to a new one:

Zanstra was no less attracted to cyclic models of the universe than Öpik, and no less explicit with regard to the extrascientific roots of the attraction. As he saw it, “philosophical desires” were both unavoidable and legitimate in cosmological research. Thus, “If one has to choose between a universe which runs down to a final state of thermal equilibrium and a universe where regeneration of energy takes place either continually or periodically, one certainly would choose the latter” (Zanstra Reference Zanstra1957, 302). The similarity to the rhetoric of the nineteenth-century discussion over the heat death is striking. Of course Zanstra did not claim that scientific truth was dictated by philosophical desires, but he thought they should result in working hypotheses to be further investigated. Observations would then decide whether or not the hypothesis could be maintained.

Zanstra preferred an eternally recycling universe, which alone “give[s] a permanence and constancy of properties to the universe denied to human beings.” He even claimed that “it goes against one's pre-conceived ideas to have a perishable universe” (ibid.). What he meant of course was that it went against his own preconceived ideas. The Dutch astronomer realized that his desired eternal universe was contradicted by his discovery that an oscillating universe cannot have existed for an infinity of time, and for this reason he did not find oscillating models to be fully satisfactory. On the other hand, he concluded that they were much more appealing than ever-expanding models, and he consequently found it imperative to study the cyclic universe from the point of view of physics as well as metaphysics (Zanstra Reference Zanstra1964).

A few scientists found the oscillating world picture appealing either because it allowed in principle eternal life or because it avoided the thorny question of an absolute beginning. The first argument belongs to the aesthetic-emotional category, while the other is epistemic. A cyclic model constructed by the Russian astrophysicist Nikolay Kardashev was announced as an “optimistic cosmological model,” the reason being that “it does not lead to the extermination of life as a result of the unlimited expansion of the Universe and of a density decrease or collapse to a singularity” (Kardashev Reference Kardashev1990, 255). William Saslaw, an American astronomer, proposed at about the same time an oscillating model of the universe in which “the formation of structure, as well as the existence of life, always gets another chance” (Saslaw Reference Saslaw1991, 45).

One of the fathers of modern big-bang cosmology, Robert Dicke of Princeton University had a predeliction for the ever-oscillating closed universe which he believed avoided the problem of explaining the original creation of matter and made it possible to understand how the heavy elements could be transformed back into hydrogen. In an important paper in the Astrophysical Journal of 1965, in which the recently discovered cosmic microwave background radiation was explained as a relic from the early universe, Dicke and his collaborators referred positively to the hypothesis of an oscillating universe. The hypothesis was said to be attractive because it “relieves us of the necessity of understanding the origin of matter at any finite time in the past” (Dicke et al. Reference Dicke, Peebles, Roll and Wilkinson1965, 415).

Among the extrascientific arguments belong considerations of a religious nature, but these played almost no role in connection with twentieth-century cyclic models and I have not found them explicitly mentioned in any of the research publications. In his popular book of Reference Bonnor1964, Bonnor (who was an atheist) associated the big-bang theory with a theological – hence unscientific – explanation of the original creation (Bonnor Reference Bonnor1964, 117–120), but in his scientific articles references to God were replaced by epistemic arguments. For example:

In Bonnor's view, a singularity-free eternally oscillating universe was the only model that avoided a divine miracle and at the same time kept on the safe ground of general relativity theory.

6. The Recent Scene: Inflation or Eternal Cycles?

In the final years of the twentieth century cosmology experienced a major transition with the observationally based recognition that the universe is in a state of acceleration and will presumably expand forever. The acceleration was explained in terms of a new repulsive “dark energy,” generally considered to be an expression of a positive cosmological constant. The belief that the universe consists essentially of dark energy and cold dark matter (the “ΛCDM paradigm”) became accepted by a majority of cosmologists. Moreover, the new picture of the universe could be brought in agreement with the inflation theory of Alan Guth and others, generally seen as the authoritative theory of the early universe.Footnote ¹¹

However, not all physicists and astronomers were comfortable with the inflation scenario in whatever of its many versions, and there was no shortage of cosmological models based on alternative ideas.Footnote ¹² As mentioned above, the cyclic model of Israelit and Rosen was one such alternative and there were several others. However, relying as it did on closed cycles the Rosen-Israelit model did not survive the observations of 1997–99. In general the flat accelerating universe seemed to herald the death of all realistic models of an oscillating universe.

Paul Steinhardt of Princeton University, one of the architects of the so-called new inflation theory of 1982, disagreed. In spite of his vested interest in this highly successful theory he suggested after twenty years, together with Neil Turok of Cambridge University, a radical alternative to it in the form of a new cyclic model of the universe (Steinhardt and Turok Reference Steinhardt and Turok2002a; idem Reference Steinhardt and Turok2002b). Turok, who used to be skeptical of cyclic models, was convinced that the new theory was not only mathematically consistent but also aesthetically pleasing. “Time is infinite, space is infinite, and they have always been here,” he said to Science, emphasizing the feature of eternity rather than the cyclic feature. Interestingly, he compared the new theory with the long discarded steady-state model: “It's exactly what the steady-state people wanted. Our model really realizes their goal” (Seife Reference Seife2002). Indeed, the new cyclic cosmos can be viewed as “a remarkable reincarnation of Fred Hoyle's steady-state model of the universe” (Steinhardt and Turok Reference Steinhardt and Turok2002b, 18).Footnote ¹³

Contrary to other alternatives, the Steinhardt-Turok theory has received much attention both within and without the scientific community.Footnote ¹⁴ The authors deliberately promote the theory as a “new paradigm” (Steinhardt and Turok Reference Steinhardt and Turok2002c, 1), although what they mean is presumably a candidate for a new paradigm. It is, they say, “a promising and provocative alternative to the standard big bang inflationary picture” (Steinhardt Reference Steinhardt2004, 469). Claimed to be the alternative to the inflationary big bang theory, cosmologists are faced with a choice between two widely different pictures of the world. At least, so we are told: “We now have two disparate possibilities: a universe with a definite beginning and a universe that is made and remade forever” (Steinhardt and Turok Reference Steinhardt and Turok2002a, 1439). This sounds much like the situation in the 1950s, when the choice was between big bang and steady state, but it refers to the early twenty-first century.

The new cyclic model was inspired by many-dimensional string theory and the notion of brane-worlds, especially the so-called ekpyrotic model suggested in 2001. In spite of this heritage, it does not depend crucially on concepts from string theory, meaning that it can be formulated in the language of more conventional physics. Although Steinhardt and Turok were and are much concerned with comparing their cyclic model with the popular inflation theory, in the present context it is of greater interest to compare it with the classical models of the oscillating universe. The new cyclic picture of the universe shows some similarity to models of the Friedmann-Tolman type, but it also differs from them in important respects. For instance, the cycles are not completely independent. “The events that occurred before the last bang shaped the large-scale structure of the universe observed today, and the events that are occurring today will determine the structure of the universe in the cycle to come” (Steinhardt and Turok Reference Steinhardt and Turok2007, xiv). In an address to the American Philosophical Society, Steinhardt summarized in non-technical language the essential features of the new model as follows:

First of all, and most remarkably, the Steinhardt-Turok theory leads to an eternal sequence of cycles of expansion and contraction without relying on a closed universe; on the contrary, its universe is infinite and flat, in agreement with observations. Another remarkable feature is that it deals directly with the repeated cosmic events where the universe bounces from a crunch to a bang. These events are “tamed,” with temperature and density remaining finite. There is no big bang in the sense of the traditional hot big-bang model, and also there is no superluminal inflationary phase. Moreover, contrary to the Tolman model and most later models of the oscillating universe the cycles are nearly identical, namely in the sense that a full cycle ago the physical conditions were the very same as they are today. (Of course, the cycles are not identical in the stronger sense of the old idea of eternal recurrence favored by Nietzsche and others.) Whereas in the Tolman models the lengths of the cycles increase continuously, according to the new cyclic model all the cycles are of the same duration.

This result may be seen as comforting as it allows an infinite number of cycles and thus avoids the unpleasant problem of a first cycle that plagued earlier models.Footnote ¹⁵ It remains the case that the entropy increases from cycle to cycle, but what matters is not the entropy itself but the entropy density. This density increases in the model Tolman considered, which led to the longer periods of the cycles. In the new cyclic theory the entropy density does not build up from cycle to cycle. Of course, Tolman was unaware of dark energy and cosmic acceleration, elements which are crucially important in the recent cyclic theory. In contrast to the inflation scenario there is only one kind of acceleration, the one that has been found observationally, which may be taken as a sign of the new model being more economical. The acceleration is supposed to last an exceedingly long time (“trillions of years”) until the universe begins to contract, a result caused by the decay of dark energy. Finally, the density fluctuations in the microwave background are produced in the early contracting phase, not in any inflationary phase.

Although the model of Steinhardt and Turok is by far the most discussed cyclic alternative to the standard theory, it is not the only game in town of its kind. In part motivated by what they call aesthetic objections to the ever-expanding standard theory, Lauris Baum and Paul Frampton of the University of North Carolina have developed a model based on “phantom energy,” a hypothetical representative of dark energy which increases to infinity within a finite span of time (Baum and Frampton Reference Baum and Frampton2007). According to this oscillating model, the present expansion continues almost to the point where the catastrophic “big rip” would happen, but briefly before it will undergo a dramatic deflation. The universe will be left very small, very empty, and nearly without entropy. Immediately after the bounce, an inflationary phase will set in, as in ordinary inflation theory. At the turnaround the universe will have fragmented in a huge number of causally separate subuniverses, each of which will deflate. Thus, the Baum-Frampton model is a multiverse scenario as well as an eternal-cyclic scenario. The link to eternal inflation distinguishes this model from the Steinhardt-Turok model.

Another recent proposal of an oscillating universe based on phantom energy operates with modified Friedmann equations. In this way, and by using ideas from string theory, the authors of the theory claim to produce an oscillating scenario that does not conflict with observations (Brown, Freese, and Kinney Reference Brown, Freese and Kinney2008). In addition to these models, there is a class of models based on so-called loop quantum gravity, an alternative to string theory. The universe of loop quantum cosmology is bouncing, with no singularity at the big bang (Bojowald Reference Bojowald2008). Meanwhile studies of more classical versions of oscillating universes, with no connection to string theory, continues. For example, John Barrow and T. Clifton have examined a wide class of models, including a flat universe containing radiation and vacuum energy where the initial oscillations of the scale factor turns into a period of endless expansion (Clifton and Barrow Reference Clifton and Barrow2007). What matters is that in the early years of the twenty-first century oscillating models of the universe are alive and well. Indeed, they seem to proliferate.

7. Concluding Remarks

In this review I have surveyed the development of cyclic or oscillating models of the universe, with an emphasis on the twentieth century. I have argued that although there are some similarities to the earlier, prerelativity conceptions, the dissimilarity is probably more striking. What was “missing” in the nineteenth century was the notion of a closed space with a curvature varying in time. It is only in an allegorical and philosophical sense that relativistic models of the oscillating universe can be seen as descendants of the earlier tradition.

Modern cosmologists may believe that nothing interesting took place in the study of oscillating universes between Tolman in the early 1930s and the new cyclic model of Steinhardt and Turok in the early years of the twenty-first century. This is the message of Endless Universe, where Steinhardt and Turok refer to Tolman's proof that the period of the cycles increase and the number of past cycles is therefore finite: “For the next seventy years, this argument dissuaded most cosmologists from pursuing the cyclic idea” (Steinhardt and Turok Reference Steinhardt and Turok2007, 182). This is however misleading, for, as we have seen, interest in some kind of oscillating universe never faded and the idea was investigated by a fairly large number of cosmologists. Moreover, Tolman did not actually conclude that there had been only a finite number of earlier cycles. Much of the appeal of the oscillating universe is related to the possibility of an endless series of cycles, and it may be tempting to infer that this is the very raison d'être of such models. In his best-selling The First Three Minutes, the physicist and Nobel laureate Steven Weinberg indicated as much: “Some cosmologists are philosophically attracted to the oscillating model, especially because, like the steady-state model, it nicely avoids the problem of Genesis” (Weinberg Reference Weinberg1977, 154).

But, as noted in Section 1, this is to overrate the importance of the argument, which ignores that it has been generally recognized since the 1960s that Friedmann-type models can include only a finite number of cycles, for which reason they do not really avoid the problem of creation. Moreover, the claim that the interest in oscillating models relates to their being eternal, hence avoiding absolute creation, is hardly satisfactory. Why should the alternative to a universe with a closed past be cyclic? Why not a universe of the steady-state type? There are many other solutions to the cosmological field equations which represent models without a beginning in time. One picture, which from this point of view may seem preferable, is an open universe which contracts in the past (−∞ < t < t ₀) and then turns into an expanding state (t > t ₀), where it remains for ever (Zel'dovich and Novikov Reference Zel'dovich and Novikov1983, 660).

The idea of an oscillating universe may carry with it philosophical and emotional aspects, such as may be the case with the idea of an absolutely created universe. Yet, as we have seen, only relatively few astronomers and physicists in the twentieth century expressed strong opinions about cyclic models. The issues of creation and eternity were sometimes mentioned in the early literature, but they played no great role and were not the main motivation for scientists to deal with oscillating world models. From Friedmann's pioneering work in 1922 to the Steinhardt-Turok model of the twenty-first century, motivations of an intrinsic scientific nature have been much more important. The main motivation for Steinhardt and Turok to develop their cyclic model was not any philosophical desire for an endless universe, but to challenge the standard inflation big-bang theory.

There is in relativistic cosmology a long and recognized tradition for examining solutions of the Einstein field equations with interesting properties, whether or not these models, “toy universes” or not, have anything in common with the observed universe. Since the days of Friedmann, this tradition has included examinations of oscillating solutions. Only a minority of those scientists who have investigated cyclic models have been committed to them in the realistic sense of advocating them as models that correspond to the actual universe. To this minority belong the modern theories proposed by Steinhardt and Turok and by Baum and Frampton.

From about 1970, when cosmology developed into a mature science, works on cyclic models became increasingly technical and oriented towards theoretical issues. Works in this class of world models did not differ significantly from works in other areas of theoretical cosmology. They were mostly motivated by internal problems and new developments in cosmology and particle physics. Could the overproduction of black holes in the contracting phase be avoided? How would the inclusion of negative pressure affect the cycles? Did concepts from string theory provide new theoretical resources? Could a cyclic model be developed as an alternative to the standard inflation picture? It was such kinds of problems that motivated the new wave of interest in cyclic models, not so much the old philosophical dream of an eternal yet dynamical universe. Extrascientific considerations have not disappeared, and it is unlikely that they ever will, but they are secondary and do not drive the research. We could also say that cosmology from about 1970 developed into a normal science in the sense of Thomas Kuhn, occupied with technical rather than foundational problems.Footnote ¹⁶

It is common to include brief references to the aesthetic and epistemic merits of cyclic models, or to compare them philosophically with the standard model, but I am not aware of more thoughtful and elaborate discussions of the kind that existed in the 1950s among steady-state cosmologists in particular. A few examples from the more recent literature will illustrate that extrascientific considerations, such as those mentioned in section 5, are still to be found. One cosmologist finds it “satisfactory at least from a philosophical point of view” that cyclic non-singular models can be constructed, but characteristically he leaves it at that (Dabrowski Reference Dabrowski1995, 3). To Baum and Frampton it is a valid aesthetic objection to the standard model that “the predicted fate of the Universe is an infinitely long expansion” (Baum and Frampton Reference Baum and Frampton2007, 4). In a popular exposition Frampton similarly claims that an oscillating universe with no limits in time is “more aesthetically acceptable” than cosmologies of the big-bang type (Frampton Reference Frampton2007, 54). He does not explain what this is supposed to mean.

Steinhardt and Turok, as well as some other advocates of oscillating models, have emphasized the danger of dogmatism associated with a paradigm or standard theory. “Seeking an alternative is just plain good science,” they comment, in this way justifying their new cyclic model (Steinhardt and Turok Reference Steinhardt and Turok2004, 1). They further suggest that the paradigmatic status of the inflation big-bang theory is historically contingent: had the historical circumstances been different, the new cyclic picture might conceivably have been the accepted picture of the universe today. Among the virtues of their own alternative they mention simplicity, parsimony, completeness, and explanatory and predictive power. In a recent article they compare their theory with the eternal-inflation picture, claiming that “a theory that predicts that life can exist almost everywhere is overwhelmingly preferred by Bayesian analysis (or common sense) over a theory that predicts it can exist almost nowhere” (Steinhardt and Turok Reference Steinhardt and Turok2006, 1182).Footnote ¹⁷ This is of course a matter of opinion, not a scientific argument.

Although cosmologists, in so far they are concerned with the real universe (and not with toy models), acknowledge the role of extrascientific criteria in theory choice; however, they all agree that ultimately the choice has to be made by comparing theories with observational data. This is a faith shared by all physicists, a defining feature of their trade. There may be a rhetorical element in their allegiance to this standard empiricist criterion, but not all of it is rhetoric. In fact, inflation cosmology and the Steinhardt-Turok cyclic model differ in their predictions, although the differences are so small that they escape detection by today's instruments. Inflation theory predicts that space is filled with gravitational waves, and these are expected to leave their imprint on the polarization of the cosmic microwave background. According to the cyclic scenario the signature produced by gravitational waves should not exist. Thus, the new cyclic model is falsifiable as it predicts certain details in the microwave background that can be measured in very sensitive experiments and which differ from those predicted by the rival inflation theory. If these details are not found, “this would support the inflationary picture and definitively rule out the cyclic model” (Steinhardt Reference Steinhardt2004, 469). Karl Popper would have been pleased.Footnote ¹⁸

Seen in a longer frame of time, the notion of a cyclic universe may be considered a “themata” in the sense of Gerald Holton: a theme which can be followed nearly invariantly through the history of science. Indeed, in 1988 Holton pointed out that although big-bang creation cosmology appeared to have finally triumphed over the rival view of eternal cycles, then “we can be sure that this thema will come in again through the back door” (Holton Reference Holton1988, 46). Themata are ideas that are too basic and too general to be simply proved or disproved. A particular model of the oscillating universe, such as Einstein's model of 1931 or the modern Steinhardt-Turok model, may be shot down observationally. It is a different matter with the more general idea made up of all classes of oscillating models, whether based on standard general relativity or something else.

As I have suggested, although oscillating models are currently discussed in a normal-science framework, there is a kernel in such works that reflects the deep extrascientific attraction that this kind of universe holds on some researchers. The general idea of an eternally changing universe seems so appealing and, not least, so flexible that it cannot be put into the grave just by theoretical arguments or because it disagrees with some ugly observational facts. It would be wise to take the falsificationist rhetoric of modern cosmologists with more than a grain of salt.