Historical and philosophical reflections on the Einstein-de Sitter model
11 Reflections on the Einstein-de Sitter model
Cormac O’Raifeartaigh a , Michael O’ Keeffe a and Simon Mitton b a School of Science and Computing, Waterford Institute of Technology, Cork Road, Waterford, Ireland. b St Edmund’s College, University of Cambridge, Cambridge CB3 0BN, United Kingdom.
Author for correspondence: [email protected] Abstract We present some historical and philosophical reflections on the paper “
On the Relation Between the Expansion and the Mean Density of the Universe ”, published by Albert Einstein and Willem de Sitter in 1932. In this famous work, Einstein and de Sitter considered a relativistic model of the expanding universe with both the cosmological constant and the curvature of space set to zero. Although the paper served as a standard model in ‘big bang’ cosmology for many years, we note that it is in fact couched in the framework of a cosmos that expands outward from an initial cosmic radius of several billion lightyears, in a manner similar to Georges Lemaître’s cosmic model of 1927. We consider claims that the paper was neither original nor important; we find that, by providing the first specific analysis of the case of a cosmology without a cosmological constant or spatial curvature, the authors delivered a unique, simple model with a straightforward relation between cosmic expansion and the mean density of matter that set an important benchmark for both theorists and observers. We consider some philosophical aspects of the model and provide a brief review of its use as a standard ‘big bang’ model over much of the 20 th century. Introduction
In 1932, Albert Einstein and Willem de Sitter jointly proposed a relativistic model of the expanding universe in which the cosmological constant and the curvature of space were set to zero (Einstein and de Sitter 1932). This model, soon known as the Einstein-de Sitter universe, went on to become a standard model of modern ‘big bang’ cosmology. One reason was the model’s great simplicity; by removing two major unknowns, the authors provided a cosmology with a simple relation between two observables, the rate of spatial expansion and the mean density of matter, that could be tested against astronomical observation. Another reason was that the critical density of matter predicted by the model provided a simple benchmark for the classification of cosmic models by theorists. In any event, the theory became the prototype ‘big bang’ model for much of the 20 th century (Nussbaumer and Bieri 2009 p. 152; Realdi 2019 p. 113) . For example, in the long debate between steady-state and ‘big bang’ cosmologies in the 1950s and 60s, the Einstein-de Sitter model became the default example of the latter (Kragh 2007, pp. 215-216; Kragh 2014). In the 1970s and 80s, the long search by astronomers to establish a reliable estimate of the mean density of matter was conducted with reference to the Einstein-de Sitter model, although it became increasingly clear that the density lay far below the critical value required for flatness (Longair 2006 pp. 340-361; Peebles 2020 pp. 105-111). With the hypothesis of cosmic inflation in the 1980s, the model returned to the fore, at least for many theoreticians, and by the early 2000s, precision measurements of the cosmic microwave background had provided the first observational evidence of a universe of Euclidean geometry. However, by this time, evidence had emerged of a dark energy component for the energy density of the universe, giving rise to today’s Lambda-Cold-Dark-Matter or ɅCDM model (Martínez and Trimble 2009; Calder and Lahav 2010). It is therefore quite surprising to note that the cosmic model proposed by Einstein and de Sitter in 1932 is not in fact a model of the ‘big bang’ type. Although not explicitly stated in the text, a close study of the paper suggests a cosmology that expands outwards from an initial radius of about two billion lightyears, in a manner similar to Georges Lemaître’s cosmic model of 1927 (Lemaître 1927, 1931a). On reflection, this should not come as too great a surprise. While the hypothesis of a universe expanding outwards from an initial singularity was first considered by Alexander Friedman in 1922 (Friedman 1922), this work did not receive much attention until the 1930s (Nussbaumer and Bieri 2009 p. 92,110; Kragh 2014). On the other hand, the notion of an expanding cosmos with a physical ‘fireworks’ origin was first mooted by Lemaître in 1931 (Lemaître 1931b, 1931c); however, the hypothesis was considered highly speculative by many researchers at first (Kragh 2008). It is thus interesting to the historian of science that a paper that later served as an archetypal ‘big bang’ model does not, technically speaking, belong to this class. Instead, the paper can be seen as a special case of the early ‘emergent’ models of the expanding universe that were proposed by theoreticians such as Georges Lemaître, Willem de Sitter and Arthur Stanley Eddington (Lemaître 1927; de Sitter 1930a; Eddington 1930), as discussed below. With this is mind, we thought it useful to provide a guided tour and analysis of the 1932 paper by Einstein and de Sitter. The astute reader will soon notice that the model is framed in terms of R A , the radius of Einstein’s static cosmology of 1917 (Einstein 1917), and R B , a characteristic length from de Sitter’s empty cosmology of 1917 (de Sitter 1917). Indeed, in some ways the paper can be read as a closure of their decade-long debate concerning the relative merits of the Einstein and de Sitter models of 1917. After the tour, we consider claims that the paper was neither original nor important in the history of cosmology. In section 5, we consider the philosophical implications of the paper and in section 6, we provide a brief history of the paper as a standard ‘big bang’ model over much of the 20 th century. Historical context
With the publication of Edwin Hubble’s observation of an approximately linear relation between the redshifts of the spiral nebulae and their distance (Hubble 1929), many theorists began to consider the possibility of a universe of expanding radius. Thus, the Einstein-de Sitter paper should be considered in the context of a number of works on relativistic cosmology by theorists such as Georges Lemaître Arthur Stanley Eddington, Willem de Sitter, Albert Einstein, Howard Percy Robertson and Richard Tolman. The earliest of these cosmologies followed Lemaître’s lead in proposing an ‘emergent’ universe that expanded from a pre-existing Einstein radius (Lemaître 1927, 1931a; de Sitter 1930a; Eddington 1930, 1931). Indeed, Eddington’s consideration of the expansion of the universe as a result of instabilities in the static Einstein universe led to such models becoming known as Eddington-Lemaître models (Bondi 1952 pp. 84, 118; North 1965 pp. 122-125). However, as the cosmology of Alexander Friedman (Friedman 1922) became better known, attention also turned to slightly more mathematical models that expanded from a singularity (de Sitter 1930b, 1930c, 1931a; Despite hundreds of references to the paper, we are aware of only one article that notes that the Einstein-de Sitter paper is not a ‘big bang’ model (Kragh 1997). See (Kerzberg 1989; Realdi and Peruzzi 2009) for a discussion of this debate.
Einstein 1931; Robertson 1932; Tolman 1930, 1932). In all of these early models of the expanding universe, there was little consideration of an origin for the cosmos in the physical sense; the emphasis was on whether relativity could account for present observations, i.e., the redshifts of the nebulae (North 1965 pp. 125-126). We note also that almost all of these models assumed closed spatial curvature, in the same manner as the static cosmologies of Einstein and de Sitter (Einstein 1917; de Sitter 1917) and the early dynamic cosmologies of Friedman and Lemaître (Friedman 1922; Lemaître 1927). In particular, Einstein lost little time in investigating whether relativity could account for a non-static cosmos without the use of the cosmological constant term in the field equations. Adopting Alexander Friedman’s analysis of a relativistic universe of positive spatial curvature expanding outwards from a singularity (Friedman 1922), Einstein set the cosmological constant to zero and arrived at a model of a universe that first expands and then contracts, a cosmology that is sometimes known as the Friedman-Einstein universe (Einstein 1931). Meanwhile, Willem de Sitter pursued a much more general investigation of expanding models (de Sitter 1930b, 1930c, 1931a), although he retained a preference for emergent models (de Sitter 1931b, 1933). It is known that the Einstein-de Sitter paper was written over the course of a few days in early January 1932, while Einstein and de Sitter were both visiting Caltech in Pasadena (figure 1). Indeed, the two great physicists were both housed at the luxurious Athenaeum at Caltech and worked intensely together for a few days (Eisinger 2011 p. 141; Guichelaar 2018 pp. 257-259). Given their decade-long debate on the relative merits of the Einstein and de Sitter models of the cosmos, and the groundbreaking observations of astronomers at the nearby Mt Wilson Observatory, it is no surprise that their conversation turned to models of the expanding universe. Indeed, it is known that de Sitter also had many discussions with Edwin Hubble and his assistant Milton Humason during this period, concerning their ongoing observations of the redshift/distance relation of the spiral nebulae (Guichelaar 2018 pp. 257-259). During this time, de Sitter also gave a series of lectures at Caltech on relativity, cosmology and the expanding universe. Einstein attended at least one of these lectures and commented most favourably on it (Guichelaar 2018 pp. 257-259). Of course, Einstein himself was no stranger to cosmological inspiration in sunny Pasadena; his sojourn at Caltech the previous winter had inspired his cosmic model of 1931 (Eisinger 2011 pp. 109-115). We shall probably never know whether it was Einstein or de Sitter who drew the other’s attention to Otto Heckmann’s cosmological paper of 1931. In this article, Heckmann noted that a non-static cosmos could exhibit positive spatial curvature, negative curvature or no curvature at all (Heckmann 1931). Up to this point, almost every cosmic model had assumed positive spatial curvature. Einstein and de Sitter immediately recognized that a dynamic, matter-filled universe of zero spatial curvature represented an intriguing class of cosmic models; with the cosmological constant and the pressure of matter set to zero, such a universe would expand indefinitely at an ever-slower rate of expansion. Most importantly, the model predicted a simple relation between two observables, the rate of expansion and the mean density of matter, that could be tested against observation. As regards the style of the paper, there is little question that the model resembles a special case of de Sitter’s first paper on expanding cosmologies (de Sitter 1930a), which was in turn based on Lemaître’s emergent model of 1927 (Lemaître 1927, 1931a). A guided tour of the paper
The authors begin the paper by noting an observation by the German theorist Otto Heckmann to the effect that, in a matter-filled universe of dynamic radius, positive spatial curvature is not a given:
In a recent note in the Gottinger Nachrichten, Dr. 0. Heckmann has pointed out that the non-static solutions of the field equations of the general theory of relativity with constant density do not necessarily imply a positive curvature of three-dimensional space, but that this curvature may also be negative or zero.
Although an exact reference is not given, there is little question that the paper the authors are referring to Heckmann’s 1931 paper ‘Über die Metrik des sich ausdehnenden Universums’ (Heckmann 1931) or ‘
On the Metric of the Expanding Universe’ . In this paper, Heckmann points out that an expanding universe may be of positive, negative or zero spatial curvature. However, he does not explicitly explore the case of flat geometry, as discussed in section 4. In the second paragraph of their paper, Einstein and de Sitter point out that neither the sign nor the magnitude of spatial curvature can be derived from observation. An interesting question arises, namely, whether a cosmic model entirely devoid of spatial curvature can account for observations such as the rate of expansion and the density of matter:
There is no direct observational evidence for the curvature, the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor the value of the curvature, and the question arises whether it is possible to represent the observed facts without This point is sometimes disputed and will be discussed further in section 4 . introducing a curvature at all. In the third paragraph, the authors recall that the cosmological constant term was introduced to the field equations in order to account for a finite density of matter in a universe that was assumed to be static. For the case of a non-static universe, this term may not be necessary:
Historically the term containing the "cosmological constant" λ was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ.
The authors then give an extremely short passage of relativistic analysis. Assuming a time-dependent line element with no spatial curvature, and setting both the cosmological constant and pressure of matter to zero, a differential equation can be derived from the field equations that relates the fractional expansion of cosmic radius with the density of matter ρ : If we suppose the curvature to be zero, the line-element is 𝑑𝑠 = −𝑅 (𝑑𝑥 𝑑𝑦 + 𝑑𝑧 ) + 𝑐 𝑑𝑡 (1) where R is a function of t only, and c is the velocity of light. If, for the sake of simplicity, we neglect the pressure p , the field equations without λ lead to two differential equations, of which we need only one, which in the case of zero curvature reduces to: ( 𝑑𝑅𝑐𝑑𝑡) = 13 𝜅𝜌 (2) In the fifth paragraph, the authors note that the fractional rate of expansion and the density of matter can be derived from observation:
The observations give the coefficient of expansion and the mean density:
1𝑅 𝑑𝑅𝑐𝑑𝑡 = ℎ = 1𝑅 𝐵 ; 𝜌 = 2𝜅𝑅 𝐴2 (2 ′ ) In the first expression, the authors are preparing to use Hubble’s measurements of the redshift/distance relation of the nebulae as an empirical estimate of the fractional rate of expansion; as defined, the quantity h is simply the Hubble constant divided by the speed of light. The quantity R B is not explicitly described in the paper; from equation (2’) and from previous papers by de Sitter (de Sitter 1930a, 1930b, 1930c), it can be taken as the radius of Equation (2) is a special case of the so-called Friedman equation, although the authors don’t state this. As this equation is not numbered in the original paper, we use the label (2’) for reasons of clarity . the de Sitter universe (this model was known to the authors for many years as solution B ). Similarly, in the second relation of equation (2’), the density of matter is related to a quantity R A that is not specifically defined in the text; from the equation, it corresponds to the radius of the static Einstein universe, first proposed in 1917 and known to the authors as solution A . Einstein and de Sitter then proceed to put theory together with experiment. Taking a value of 500 km s -1 Mpc -1 for the Hubble constant, they first compute a value of R B using equation (2’). Since equations (2) and (2’) imply the relation R A / R B = 2/3, they then calculate a value for R A and from this the matter density: Therefore we have, from (2), the theoretical relation ℎ = 13 𝜅𝜌 (3) or 𝑅 𝐴2 𝑅 𝐵2 = 23 (3 ′ ) Taking for the coefficient of expansion h = 500 km./sec. per 10 parsecs, (4) or R B = 2 x 10 cm., we find R A = 1.63 x 10 cm., or ρ = 4 x 10 -28 gr. cm .-3 , (5) which happens to coincide exactly with the upper limit for the density adopted by one of us. We note first a slight inconsistency in notation and units. The quantity h used in the central equation h = κρ /3 (equation (3)) is defined in equation (2’) as h = ( and has the dimensions of inverse length. On the other hand, the observational parameter h = 500 km s -1 Mpc -1 in equation (4) has the units of inverse time; this latter quantity is usually denoted as H and really corresponds to hc. We also note that the authors could have calculated the density of matter directly from equation (3); taking the Einstein constant κ as 1.866 x 10 -26 m/kg, we obtain ρ = 3 h /κ ~ 4 x 10 -25 kg/m . Instead they calculate the matter density via the radius of the Einstein and de Sitter universes, a clear indication that the model is couched in terms of a universe that expands outwards from an initial radius R A of 1.63x10 cm or 2x10 lightyears. The result is of course the same, and the authors are pleased to note that their estimate for the density of matter is not inconsistent with values estimated by de Sitter from astronomical observations (de Sitter 1931a). The underlying reason for this is that the equation relating the Doppler shifts of the nebulae to cosmic expansion is given by
R’/R = v/cr where r is the distance of the source (Lemaître 1927). In the last section of the paper, the authors consider the uncertainty in observational estimates of the rate of cosmic expansion and of the matter density. They note that the main source of error in determining each of these parameters lies in the significant uncertainty associated with the distances of the nebulae. Errors in observational estimates of the matter density may also arise due to the assumption that all of the material mass of the universe resides in the nebulae, although the authors doubt this assumption will introduce any appreciable error:
The determination of the coefficient of expansion h depends on the measured red-shifts, which do not introduce any appreciable uncertainty, and the distances of the extra-galactic nebulae, which are still very uncertain. The density depends on the assumed masses of these nebulae and on the scale of distance, and involves, moreover, the assumption that all the material mass in the universe is concentrated in the nebulae. It does not seem probable that this latter assumption will introduce any appreciable factor of uncertainty. The authors then consider the ratio of the observables h and ρ . Assuming a nebula occupying a spatial cube of side 1x10 lightyears, they note that their derived density of 4x10 -28 g/cm corresponds to a mass of 2x10 solar masses. This is a second check on their estimate of matter density, as the latter mass is not inconsistent with estimates of the mass of the Milky Way from astronomy: Admitting it, the ratio h / ρ , or R A2 /R B2 , as derived from observations, becomes proportional to Δ/M, Δ being the side of a cube containing on the average one nebula, and M the average mass of the nebulae. The values adopted above would correspond to Δ = 10 light years, M = 2x10 ʘ , which is about Dr. Oort's estimate of the mass of our own galactic system . Thus, the authors conclude that a cosmic model that assumes no spatial curvature gives an estimate for the density of matter that is not inconsistent with observation:
Although, therefore, the density (5) corresponding to the assumption of zero curvature and to the coefficient of expansion (4) may perhaps be on the high side, it certainly is of the correct order of magnitude, and we must conclude that at the present time it is possible to represent the facts without assuming a curvature of three dimensional space.
Finally, the authors stress that the spatial curvature may not in fact be zero, and suggest that an increase in the precision of observational data will allow for the determination of its sign and value:
The curvature is, however, essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and to determine its value. Discussion
Considering the cosmological constant first, we have noted above that Einstein had already removed this term in his 1931 model of the expanding cosmos (Einstein 1931). Although this paper is far less well-known than the 1932 paper of Einstein and de Sitter, it offers many insights into Einstein’s cosmology. We note here that Einstein’s justification for the removal of the cosmological constant term in his 1931 model is identical to that of the 1932 paper, namely that one could account for a finite density of matter in an expanding universe without it (section 3). We also note that in the 1931 model, Einstein derived a relation between the rate of expansion and the mean density of matter that is mathematically very similar to that of the present paper. However, an important difference is that the 1931 derivation necessitated several assumptions and approximations concerning the current phase of the cosmos in its timeline of evolution that are obviated in the 1932 paper by setting the spatial curvature to zero (O’Raifeartaigh and McCann 2014). It is also true that the 1932 paper by Einstein and de Sitter was not the first in which it was noted that non-static cosmologies allow the possibility of a universe with no spatial curvature. Many commentators have suggested that, quite apart from Otto Heckmann, , the possibility of Euclidean geometry for the cosmos had been explored by Alexander Friedman, Georges Lemaître and Howard Percy Robertson. On this basis, it has often been suggested that the Einstein-de Sitter paper of 1932 was neither original nor significant (Kragh 1996 p. 35; Kragh 2007 p. 156; Nussbaumer 2014) and it has even been suggested that the paper would hardly have been published had it been submitted by less illustrious authors (Nussbaumer and Bieri 2009 p. 150; Barrow 2011 p. 75). We do not agree with this view. Considering the case of Friedman first, there is little question that he delivered a comprehensive analysis of static and non-static cosmologies of positive spatial curvature in 1922 (Friedman 1922) and an analysis of static and non-static cosmologies of negative curvature in 1924 (Friedman 1924). However, we find no evidence that Friedman explored the specific case of a universe of flat geometry in any of his major publications. Turning to the case of Lemaître, it is certainly true that, in his 1925 analysis of We have given an analysis and first English translation of the paper in (O’Raifeartaigh and McCann 2014). This impression is strengthened by a well-known anecdote from Eddington that suggests that the authors themselves did not attach too much significance to the work (Eddington 1940, p128; Nussbaumer and Bieri 2009, p152). the de Sitter model, Lemaître was led to the case of a time-varying universe of Euclidean geometry (Lemaître 1925). However, Lemaître did not analyse this cosmology, but dismissed it outright on the grounds that “ we are led... to the impossibility of filling up an infinite space with matter which cannot but be finite” (Lemaître 1925). As for Robertson, it could be said that his 1929 exploration of a general (static or non-static) line-element for relativistic cosmology, based on general assumptions of homogeneity and isotropy, implicitly included the case of a universe of Euclidean geometry (Robertson 1929). However, this possibility is not explicitly explored and the physics of the paper is in any case firmly rooted in the context of a cosmos that is assumed to be static. Thus, we find that Einstein and de Sitter were correct to cite Heckmann as the first to consider the specific case of a time-varying universe of flat geometry, Even here, Heckmann touched on the case as one theoretical possibility amongst others and made no attempt to derive an expression for cosmic parameters that could be tested by observation (Heckmann 1931). By contrast, Einstein and de Sitter constructed a specific cosmic model with both spatial curvature and the cosmological constant set to zero with the express purpose of establishing a simple relation between the rate of expansion and the mean density of matter that could be compared with observation. . As noted above, setting both spatial curvature and the cosmological constant to zero in their model enabled the authors to derive a simple relation between the rate of expansion and the mean density of matter that could be tested against observation. Thus, taking Hubble’s redshift/distance value of 500 km s -1 Mpc -1 for the fractional rate of expansion, they derived a value of 4x10 -28 g/cm for the density of matter. (We have already noted that the authors do not obtain this estimate directly from the relation ρ = 3 h /κ (equation 3), but from estimates of the radii of the Einstein and de Sitter universes, a clear indication that the model is couched in terms of a universe that expands outwards from an initial radius R A ). They note that the resulting estimate for the mean density of matter lay at the upper bound of a range of values estimated by de Sitter from astronomical observations (de Sitter 1931a). The authors provide a second check on their estimate of matter density by means of a simple order-of-magnitude calculation. Assuming the material mass of the universe is contained within the nebulae and assuming that a single nebula occupied a cubic volume of side 1x10 lightyears, simple calculation suggested that the authors’ estimate of the density This was a common assumption at the time (de Sitter 1930). of matter corresponded to a nebular mass of 2x10 solar masses. As they note, this estimate was consistent with estimates of the mass of the Milky Way provided by Jan Oort, de Sitter’s colleague at the Leiden Observatory. Although a reference is not given, Oort’s estimate was based on determinations of the local mass density of the Milky Way from stellar velocity dispersions and distributions that included a contribution from dark matter, in the tradition of earlier estimates by James Jeans and Johannes Kapteyn (Oort 1932; Trimble 1990). Thus, it could be argued that the hypothesis of dark matter is implicit in the Einstein-de Sitter model (Longair 2004; Longair 2006 p. 116). One of the most notable aspects of the Einstein-de Sitter paper is the lack of consideration of the timespan of expansion. Whereas relativistic cosmology leads naturally to the derivation of two independent differential equations from the field equations, the authors concentrated on only one, equation (2) above. At first sight, this omission was unfortunate, as, in common with many of the models at the time, the timespan of expansion implicit in the model was in fact puzzling. It is easily shown that the Einstein-de Sitter model expands as R ( t ) = ( t/t ) , suggesting a time of expansion of t = 2/(3 H ); with H = 500 km s -1 Mpc -1 , this corresponded to a time of expansion of about 1.2 billion years, well below estimates of the age of the stars and of the earth (Longair 2006 p. 116); Kragh 2007 pp. 159-161). One reason for this omission is made clear in the title of the paper; only one differential equation is necessary to establish the main goal of the paper, a simple relation between the rate of spatial expansion and the density of matter that could be tested against observation. Another reason may be that the authors did not feel that considerations of the timespan of expansion were relevant to the model in question. We have already noted that the Einstein-de Sitter paper is framed in the context of a cosmos that expands outwards from a universe of radius 2x10 lightyears, in the manner of Lemaître’s model of 1927. Despite their considerations of other cosmologies, this type of model remained the favoured model for several theorists such as de Sitter, Eddington and Richard Tolman (de Sitter 1933; Eddington 1933 pp. 55-56; Tolman 1934 pp. 485-486). As Eddington put it: “ ... it has seemed to me that the most satisfactory theory would be one which made the beginning not too un-aesthetically abrupt ... Philosophically the notion of a beginning of the present order of the Universe is repugnant to me.” (Eddington 1933 pp. 55-56). By contrast, Einstein’s 1931 model is framed in the context of a cosmos that expands from a singularity, in the manner of Friedman’s cosmology of 1922. In his 1931 paper, Einstein derived a simple expression for the timespan of expansion and noted that it implied an age for the universe that was problematic in comparison with estimates of the age of the earth. It seem likely that such considerations were not considered relevant to a cosmos that expands from a pre-existing universe of radius 4x10 cm, just as they weren’t for George Lemaître in 1927 (Lemaître 1927). As de Sitter put it (de Sitter 1933):
Nevertheless the “age of the universe,” i.e., the time passed since y passed through its minimum, has very generally been assumed to be also the age of the stars, and consequently the opinion has generally been held the long time-scale of 10 or 10 years, demanded by modern theories for the evolution of the stars, would have to be given up, and the theories of evolution would have to be modified so as to give a shorter timescale. I think this identification of the time of minimum of y with the “beginning of the world” is entirely gratuitous. In the case of the expanding universes of the second kind it is at once evident that the minimum is not a very remarkable point on the curve at all. There is no singularity at that point and no discontinuity in the motion, no more than at the perihelion of a planetary or cometary orbit. A similar attitude is summarized by Richard Tolman in his book
Relativity, Cosmology and Thermodynamics (Tolman 1934 p. 486):
It is to be emphasized, as has been done particularly by de Sitter, that there is no necessity for regarding the beginning of the expansion as in any sense the beginning of the universe, and no reason for expecting an identity between the time scales for stellar evolution and nebulae expansion… the difference between the timescales for stellar evolution and nebular expansion suggests that no definiteness could now be attached to any idea as to the beginning of the physical universe. Indeed, it is 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 . As for Einstein, we note that, in his later reviews of cosmology, he considered only models that expanded outwards from a singularity, attributing the timespan puzzle to shortcomings of theory in describing the high-density conditions of the early universe (Einstein 1945 p. 128). His explanation for this paradox was to question the assumption of homogeneity in the model (Einstein 1931; O’Raifeartaigh and McCann 2014). It appears from handwritten drafts of Lemaître’s 1927 paper that he may have chosen his ‘emergent’ model specifically to avoid the timespan problem (Luminet 2013). In this paper, “expanding universes of the second kind” refers to cosmologies with a non-singular origin. From a philosophical point of view, the innovative aspect of the Einstein-de Sitter paper was the proposal of a cosmic model of open spatial geometry. It is often forgotten that, following in the footsteps of Friedman and Lemaître, almost all of the dynamic cosmic models that were proposed in years 1929-1932 in the wake of Hubble’s observations were framed in terms of a cosmology of positive spatial curvature. The hypothesis of positive spatial curvature can be traced back to the very first relativistic model of the cosmos, Einstein’s model of 1917 (Einstein 1917). Having struggled with the concept of a finite density of matter in an unbounded static cosmos, Einstein proposed a universe of closed, spherical curvature in order to satisfy his view of Mach’s principle and the relativity of inertia (Einstein 1918; Barbour 1990; O’Raifeartaigh et al. 2017). Einstein’s expanding model of 1931 was of positive curvature and not in obvious conflict with this belief; however, the same could hardly be said of the Einstein-de Sitter model of 1932. That Einstein proposed such a model is perhaps another indication of a gradual change in his attitude to Mach’s principle in these years (Einstein 1949 p. 29;
Tian Yu Cao 1997 pp. 93-94;
Gutfreund and Renn 2017 p. 40). We have noted earlier that Lemaître was led in his 1925 analysis of the de Sitter model to consider the case of a non-static cosmos of Euclidean geometry, but dismissed the possibility on the basis that infinite space could not be filled by a finite amount of matter. Here, it appears that Lemaître conflated the requirement of a finite mean density of matter in an expanding universe with a requirement of a finite quantity of matter. The most likely explanation for this uncharacteristic error is that, although Lemaître was well aware of the non-static character of the de Sitter metric in a mathematical sense, he was not truly thinking in 1925 in terms of a universe that is physically expanding. Thus, it could be argued that the Einstein-de Sitter paper of 1932 was an important advance in philosophical terms. The authors shook off the heritage of positive spatial curvature, a legacy from static models of the cosmos that had dogged the first tranche of expanding cosmologies. For the first time, physicists took seriously the prospect of an unbounded universe in which matter played a subsidiary role. More pragmatically, by setting the curvature to zero, along with the cosmological constant, the authors delivered a cosmic model with a simple Friedman’s paper of 1924 was not well-known. We note that de Sitter’s empty model of 1917 was also of positive spatial curvature (de Sitter 1917). A similar observation can be made about the cosmologies of Hermann Weyl, Cornelius Lanczos and Howard P. Roberston (Nussbaumer and Bieri 2009 pp. 78-82). relation between cosmic expansion and the density of matter that could be tested against observation. It is interesting that, in contrast with Einstein’s deliberations of 1917, few physicists appeared to have been concerned with the philosophical implications of a cosmos of open geometry. Yet such geometries were not without concern as, unlike models of closed curvature, they did not avoid the concept of an ‘actual infinite’ (North 1965 p. 135). As the theoretician Leopold Infeld later remarked (Infeld 1949 pp. 495-496): Yet every mathematician - if given the choice - would rather see our universe close than open. There is mathematical beauty in such a universe which reveals itself when we consider any mathematical problem on such a cosmological background. In such a closed universe we have simple boundary conditions and we do not need to worry about infinities in time and space. Compared with the closed universe the open one of Einstein-de Sitter appears to be dulled and uninspired.
Some further considerations can be found by Lemaître in his 1929 work
La Grandeur de l'Espace (Lemaître 1929; Lemaître 1950 pp. 22-56). However once again, he appears to conflate the issue of the counting of an infinite number of objects (stars and galaxies) with the postulate of a finite mean density of matter in infinite space. Eddington continued to argue strongly for a positive curvature of space (Eddington 1933 pp. 29-65). However, his argument was based on considerations of the dimensions of elementary particles and did not attract much support (Milne 1933 pp. 28-29; North 1965 pp. 281-282). In general, few theorists and astronomers seemed perturbed by the proposal of open geometry for the cosmos, perhaps an indication that they viewed the Einstein-de Sitter model as a useful hypothetical tool rather than a literal description of the universe. After all, the authors themselves suggest in the concluding section of the paper that spatial curvature of the cosmos could one day be observed: “
The curvature is, however, essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and to determine its value” (Einstein and de Sitter 1932). It seems likely that most scholars saw little advantage in perusing the philosophical implications of the model until better estimates of cosmic parameters such as spatial curvature, material pressure and the cosmological constant were forthcoming from observation. One could argue that this phase of cosmology resembled the ‘shut up and calculate’ phase of quantum field theory (Kaiser 2011 pp. 1-25). The Einstein-de Sitter model became very well-known and went on to play a significant role in 20 th century cosmology. For theorists, it marked an important hypothetical case in which the expansion of the universe was precisely balanced by a critical density of matter, given by equation (3) as ρ c ( t ) = /κ = (t)/8πG . This allowed for a useful classification of cosmic models. Assuming a vanishing cosmological constant, a cosmos of mass density higher than the critical value would be of closed spatial geometry and eventually collapse, while a cosmos of mass density less than the critical value would be of open spatial geometry and expand at an ever increasing rate; in between lay the critical case of a cosmos with Euclidean geometry that would expand at an ever decreasing rate. Indeed, this classification of expanding models became a staple of cosmology textbooks (Harrison 1981 p. 298; Liddle 1999 p. 37). The geometry of such models is usefully described in terms of the dimensionless density parameter Ω , defined as the ratio of the actual matter density of the universe ρ to the critical density 𝜌 𝑐 required for spatial closure, i.e., Ω = ρ / ρ c . This simple classification scheme could be generalized to models with a cosmological constant and radiation pressure by defining the energy density parameter as Ω = Ω M + Ω λ + Ω R , where Ω M , Ω λ and Ω R represented the energy density contributions due to matter, the cosmological constant and radiation respectively. In this scheme, the Einstein-de Sitter universe is neatly specified as (Ω=1: Ω M =1, Ω λ =0, Ω R =0) with Ω M = ( / ) ρ M . One immediately sees that the Einstein-de Sitter model represents a very special case as Ω = Ω M = 1 for all time (Liddle 1999 pp. 49-53). The Einstein-de Sitter model also marked an important benchmark case for observers; in the absence of empirical evidence for spatial curvature or a cosmological constant, it seemed the cosmos could be described in terms of just two parameters, each of which could be determined by astronomy. In addition, it was soon realised that, in addition to astronomical methods such as galaxy counts and stellar dynamics, the mean density of matter could be estimated by measuring the rate of expansion at different epochs. Defining the expected slowing of expansion over time in terms of a deacceleration parameter q , it was easily shown that for models without a cosmological constant q = Ω M /2. One could therefore expect a value of q > 1/2 for a cosmos of closed spatial curvature, q < 1/2 for a cosmos of open geometry and q = 1/2 for a universe of Euclidean geometry (Liddle 1999 pp. 50-53). Thus, the Einstein-de Sitter model soon became a significant benchmark model for astronomers (North 1965, p. 134; Kragh 1996, p. 35; Nussbaumer and Bieri 2009, p. 152). The opening of the 200-inch Hale telescope at the Palomar Observatory in California in 1949 heralded a new era of observational cosmology. In particular, the hypothesis of steady-state cosmology as an alternative to evolving models of the cosmos spurred new efforts to determine key cosmological parameters (Sandage 1961; Longair 2004). In this work, attention focused on the Einstein-de Sitter model and the determination of two parameters, the current rate of cosmic expansion H and the deacceleration parameter q . Indeed, the challenge to establish observational values for these parameters was later dubbed “the search for two numbers” (Sandage 1970). By the mid-1960s, estimates of the rate of cosmic expansion had decreased by almost a factor of ten, temporarily easing the age problem associated with evolving models (Longair 2006 pp. 340-361). By contrast, it became more and more apparent during the 1960s and 70s that estimates of the average density of matter from astronomy fell a long way below the critical value of the Einstein-de Sitter model. Even accounting for the existence of dark matter, estimates of the matter density from galaxy counts and rotational dynamics remained below 30% of that required for flatness. Similarly, measurements of the deacceleration parameter q also suggested a very low value for the density of matter (Sandage 1971; Longair 2006 pp. 340-361). An intriguing mathematical puzzle associated with the spatial curvature of the cosmos emerged in the 1970s. In a detailed consideration of the evolution of the geometry of the universe over time, the American theorist Robert Dicke noted that the smallest deviation from flat geometry in the early universe would quickly have resulted in a runaway open or runaway closed universe (Dicke 1970 p. 62; Dicke and Peebles 1979). Assuming a cosmology without a cosmological constant, it could be shown that an infant universe with a matter density greater than the critical density ρ c would soon become so dense it would soon cease expanding and collapse; on the other hand, an infant universe with a matter density less than the critical density would soon become essentially empty. In either case, the universe would not evolve to contain complex structures such as galaxies and stars. Thus, Dicke’s analysis implied a cosmos that must have been extremely close to the special case of flat geometry in the first moments. Many theorists found this apparent fine-tuning of the early universe puzzling, a conundrum that became known as the flatness problem . More pragmatically, the observation argued for a universe of Euclidean geometry in the present epoch, a resulted that appeared to be in conflict with estimates of the density of matter from astronomical observations. In the early 1980s, the theory of cosmic inflation was proposed in order to address numerous puzzles associated with evolving models of the universe (Guth 1981; Smeenk 2005). Note for example that steady-state models predicted a value of q = -1. Strong observational evidence for the existence of dark matter emerged in the 1960s (Trimble 1990).7
The curvature is, however, essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and to determine its value” (Einstein and de Sitter 1932). It seems likely that most scholars saw little advantage in perusing the philosophical implications of the model until better estimates of cosmic parameters such as spatial curvature, material pressure and the cosmological constant were forthcoming from observation. One could argue that this phase of cosmology resembled the ‘shut up and calculate’ phase of quantum field theory (Kaiser 2011 pp. 1-25). The Einstein-de Sitter model became very well-known and went on to play a significant role in 20 th century cosmology. For theorists, it marked an important hypothetical case in which the expansion of the universe was precisely balanced by a critical density of matter, given by equation (3) as ρ c ( t ) = /κ = (t)/8πG . This allowed for a useful classification of cosmic models. Assuming a vanishing cosmological constant, a cosmos of mass density higher than the critical value would be of closed spatial geometry and eventually collapse, while a cosmos of mass density less than the critical value would be of open spatial geometry and expand at an ever increasing rate; in between lay the critical case of a cosmos with Euclidean geometry that would expand at an ever decreasing rate. Indeed, this classification of expanding models became a staple of cosmology textbooks (Harrison 1981 p. 298; Liddle 1999 p. 37). The geometry of such models is usefully described in terms of the dimensionless density parameter Ω , defined as the ratio of the actual matter density of the universe ρ to the critical density 𝜌 𝑐 required for spatial closure, i.e., Ω = ρ / ρ c . This simple classification scheme could be generalized to models with a cosmological constant and radiation pressure by defining the energy density parameter as Ω = Ω M + Ω λ + Ω R , where Ω M , Ω λ and Ω R represented the energy density contributions due to matter, the cosmological constant and radiation respectively. In this scheme, the Einstein-de Sitter universe is neatly specified as (Ω=1: Ω M =1, Ω λ =0, Ω R =0) with Ω M = ( / ) ρ M . One immediately sees that the Einstein-de Sitter model represents a very special case as Ω = Ω M = 1 for all time (Liddle 1999 pp. 49-53). The Einstein-de Sitter model also marked an important benchmark case for observers; in the absence of empirical evidence for spatial curvature or a cosmological constant, it seemed the cosmos could be described in terms of just two parameters, each of which could be determined by astronomy. In addition, it was soon realised that, in addition to astronomical methods such as galaxy counts and stellar dynamics, the mean density of matter could be estimated by measuring the rate of expansion at different epochs. Defining the expected slowing of expansion over time in terms of a deacceleration parameter q , it was easily shown that for models without a cosmological constant q = Ω M /2. One could therefore expect a value of q > 1/2 for a cosmos of closed spatial curvature, q < 1/2 for a cosmos of open geometry and q = 1/2 for a universe of Euclidean geometry (Liddle 1999 pp. 50-53). Thus, the Einstein-de Sitter model soon became a significant benchmark model for astronomers (North 1965, p. 134; Kragh 1996, p. 35; Nussbaumer and Bieri 2009, p. 152). The opening of the 200-inch Hale telescope at the Palomar Observatory in California in 1949 heralded a new era of observational cosmology. In particular, the hypothesis of steady-state cosmology as an alternative to evolving models of the cosmos spurred new efforts to determine key cosmological parameters (Sandage 1961; Longair 2004). In this work, attention focused on the Einstein-de Sitter model and the determination of two parameters, the current rate of cosmic expansion H and the deacceleration parameter q . Indeed, the challenge to establish observational values for these parameters was later dubbed “the search for two numbers” (Sandage 1970). By the mid-1960s, estimates of the rate of cosmic expansion had decreased by almost a factor of ten, temporarily easing the age problem associated with evolving models (Longair 2006 pp. 340-361). By contrast, it became more and more apparent during the 1960s and 70s that estimates of the average density of matter from astronomy fell a long way below the critical value of the Einstein-de Sitter model. Even accounting for the existence of dark matter, estimates of the matter density from galaxy counts and rotational dynamics remained below 30% of that required for flatness. Similarly, measurements of the deacceleration parameter q also suggested a very low value for the density of matter (Sandage 1971; Longair 2006 pp. 340-361). An intriguing mathematical puzzle associated with the spatial curvature of the cosmos emerged in the 1970s. In a detailed consideration of the evolution of the geometry of the universe over time, the American theorist Robert Dicke noted that the smallest deviation from flat geometry in the early universe would quickly have resulted in a runaway open or runaway closed universe (Dicke 1970 p. 62; Dicke and Peebles 1979). Assuming a cosmology without a cosmological constant, it could be shown that an infant universe with a matter density greater than the critical density ρ c would soon become so dense it would soon cease expanding and collapse; on the other hand, an infant universe with a matter density less than the critical density would soon become essentially empty. In either case, the universe would not evolve to contain complex structures such as galaxies and stars. Thus, Dicke’s analysis implied a cosmos that must have been extremely close to the special case of flat geometry in the first moments. Many theorists found this apparent fine-tuning of the early universe puzzling, a conundrum that became known as the flatness problem . More pragmatically, the observation argued for a universe of Euclidean geometry in the present epoch, a resulted that appeared to be in conflict with estimates of the density of matter from astronomical observations. In the early 1980s, the theory of cosmic inflation was proposed in order to address numerous puzzles associated with evolving models of the universe (Guth 1981; Smeenk 2005). Note for example that steady-state models predicted a value of q = -1. Strong observational evidence for the existence of dark matter emerged in the 1960s (Trimble 1990).7 Inflation certainly gave new life to the prediction of a cosmos of Euclidean geometry, at least amongst many theorists, although others pointed out that estimates of the mean density of matter from astronomy remained far below the critical value (Coles and Ellis 1994; Peebles 2020 pp. 82-105). In the 1990s, new data from the Hubble Space Telescope (HST) and from ground-based telescopes suggested an observational value of 87 ± 7 km s -1 Mpc -1 for the present rate of expansion (Pierce et al. 1994). Put together with measurements of the cosmic microwave background, models of structure formation and constraints on the matter content of the universe set by primordial nucleosynthesis, a number of theorists began to argue forcefully for a cosmic model with a positive cosmological constant (Krauss and Turner 1995; Ostriker and Steinhardt 1995). This proposal received an enormous boost when a new generation of observational programmes to measure the deacceleration parameter q using supernovae as standard candles gave the first evidence of a negative deacceleration, i.e., of an acceleration in expansion. In the early years of the 21 st century, new precision measurements of the cosmic microwave background provided the first observational evidence that we do indeed inhabit a universe with spatial curvature close to zero. Put together, these measurements resulted in today’s model of a universe of Euclidean geometry with energy contributions of 𝛺 𝑀 ~ 0.3 and 𝛺 𝛬 ~ 0.7 from matter and from dark energy respectively (Martínez and Trimble 2009; Calder and Lahav 2010). We note finally that, while the dark energy component is represented mathematically by a positive cosmological constant in today’s Ʌ-CDM model, the physical nature of dark energy remains elusive (Brax 2018). As a result, some theorists propose alternate cosmologies in which the accelerated expansion of the cosmos is represented without the use of a cosmological constant. It is interesting to note that such alternate cosmologies are usually couched in the framework of the Einstein-de Sitter model (Bull et al. 2016). We note also that some theorists have become interested once again in the hypothesis of a universe that expands from a pre-existing radius after an indefinite period of time, this time in the context of certain models of inflation. It is thought that this scenario, known as ‘the emergent universe’, might avoid major difficulties in theoretical cosmology such as the horizon problem, the quantum gravity era and the initial singularity (Barrow et al. 2003; Ellis and Maartens 2004). Whether the emergent universe will offer a plausible, consistent description of the evolution of our universe is not yet known, but we note, as so often, the relevance of past models of the universe for today’s research. Conclusions It is intriguing for the historian to note that, although the Einstein-de Sitter model of 1932 served as the prototype ‘big bang’ model for much of the 20 th century, the paper is in fact couched in the framework of a cosmos that expands outward from an initial cosmic radius of about two billion lightyears. Thus, the model owes more to Lemaître’s cosmology of 1927 than to Friedman’s model of 1922. That said, by providing the first detailed analysis of the specific case of a cosmology without a cosmological constant or spatial curvature, the authors delivered a unique, simple model with a straightforward relation between cosmic expansion and the mean density of matter that became an important benchmark for both theorists and observers. The model also represented a significant philosophical advance, as it was the first well-known cosmology that was not of closed spatial curvature. While today’s Ʌ-CDM model of a cosmos of Euclidean geometry with a positive cosmological constant is a much better fit to current observational data, the physical meaning of dark energy remains elusive. Acknowledgements
Cormac O’Raifeartaigh thanks Professor Jim Peebles and Professor G. F. R. Ellis for helpful discussions. Simon Mitton thanks St Edmund’s College, University of Cambridge for the support of his research in the history of science. References
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A famous photo of Einstein and de Sitter at work together at Caltech, Pasadena in 1932. ©Associated Press.7