Hubble's finding that nebular red shifts are linearly proportional to distance was interpreted by researchers working in general relativity as conclusive evidence that the universe is expanding. The red shifts, which were, for the sake of convention, nominally listed by Hubble as radial velocities, were regarded as true velocities resulting from the motions of galaxies relative to the solar system. Theoreticians also accepted the cosmological principle, which stipulated that there is no special or privileged vantage point in the universe. What we observe from one place in the universe is on a large scale the same as what is observed from any other place in the universe. Hence it follows from Hubble's law that every two galaxies in the universe are moving away from each other with a velocity that is proportional to the distance between them. An analogy to help understand the concept of the expanding universe is a balloon with dots distributed over its surface. The surface is the universe, and the dots are the galaxies. As the balloon inflates, the surface expands, and each dot moves away from every other dot with a speed proportional to the distance along the surface separating them.

The original static Einstein solution and the de Sitter solution were not able to describe an expanding universe. The de Sitter solution predicts a general reddening at large distances, but this model contains no matter, and the spectral shifts do not represent real velocities. It was apparent by the late 1920s that the density of matter in the universe was nowhere near small enough for de Sitter's solution to be valid. However, soon after the publication of Hubble's law, researchers in general relativity such as Eddington and de Sitter himself called attention to Lemaitre's 1927 paper and his dynamical solution, which did describe an actual expanding universe. This paper was translated into English and published by the Royal Society in 1931, thereby ensuring the wide dissemination of Lemaitre's results. Friedmann's and Robertson's work also became the subject of renewed interest among relativists. Cosmological research within the theory of relativity during the 1920s was seen by scientists as leading up to Hubble's discovery, with Lemaitre's contribution being regarded historically as the most significant. It was soon recognized that a range of relativistic solutions were available to describe the evolution of an expanding universe. It is customary to describe these solutions in terms of what is known as an R-t diagram, giving the scale function R(t) graphed as a function of t. Three important models during this period were advanced by Lemaitre, by Eddington, and by Einstein and de Sitter.

The Lemaitre universe was proposed by the Belgian physicist in 1932 and differed somewhat from the solution given in his 1927 paper. One supposes that the world begins from a single point and expands outward, with the rate of expansion decreasing with time. At a certain point the rate of expansion begins to increase with time and continues in this way forever (figure 8.2). An essential Figure 8.2: Lemaitre's universe (1932). feature of this universe is the presence of a

positive cosmological constant in the gravitational field equations. Lemaitre's model was not a very popular one and was hardly considered at all in the literature of the second half of the twentieth century. However, since 1998 and the discovery of universal acceleration, there has been a marked revival of interest in

Eddington's universe was very similar to the one presented in Lemaitre's 1927 paper and was described in his popular 1933 book The Expanding Univer.se. Eddington made essential use of the cosmological constant, a fact that was in keeping with the curious emphasis he placed on the numerical properties of the constants of nature. In this conception the universe begins in a static Einstein state. At a certain instant, there is instability or a disturbance, and the universe begins and continues henceforth to expand outward (figure 8.3). What exactly triggers the expansion is not identified, but the initial static state may be regarded as a time when the stars and galaxies were formed. The cosmic repulsion corresponding to the cosmological constant is responsible for the continued expansion of the universe. Despite the enthusiasm of its founder, the Eddington universe has never enjoyed popularity among cosmologists.

Einstein and de Sitter devised a solution which did away with the cosmological constant. As Einstein saw it, it was no longer necessary to include the constant in the field equations because the universe is no longer assumed to be static. In the Einstein-de Sitter universe the world expands from a point, and the rate of expansion continually decreases with time. This decrease is caused by the braking force of gravity (figure 8.4). In the second half of the twentieth century this solution, or something like it, was the one preferred by cosmolo-gists, although since 1998, its validity has been called into question. From its initial formulation, an objection to the Einstein-de Sitter world concerned the very low age for the universe implied by it. If we assume that the expansion of the universe is slowing down, then an upper bound on the age of the universe is given by 1/H. This quantity is obtained by running the expansion at its current rate backward and calculating the time from the present to the birth of the universe at a point. Using Hubble's value for H of 550, this procedure leads to an age of less than two billion years, a value that was very difficult to square with the time scales required in geology, much less those required for the universe as a whole.

Figure 8.4: Einstein-de Sitter universe (1932).


Figure 8.3: Eddington's universe (1933).

Figure 8.4: Einstein-de Sitter universe (1932).

The so-called age problem would be a recurrent theme in modern cosmology up to the very present.

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