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Fig. 2.1. A sketch of Willem de Sitter on the occasion of his explanation of the idea of an expanding universe in a Dutch newspaper in 1930. His body is sketched as the Greek symbol lambda, or A, which represents Einstein's cosmological constant. As will be discussed, this constant was taken seriously then and came back into fashion.

other galaxy would see the same motion of general recession in all directions. The key point illustrated here is that this model universe is expanding but has no center of expansion: expansion is happening everywhere in the two-dimensional space. In the cosmology of our universe an observer in any galaxy in our three-dimensional space sees the same effect: the other galaxies are moving away.

A little thought about this expanding balloon model may convince you that an observer at rest in a galaxy sees that galaxies at greater distance r from the observer - measured along the balloon surface - are moving away at greater speed v. The recession velocity is proportional to the distance, following the linear relation v = Hor. (2.1)

The same argument, and this linear relation, applies to the expansion of the three-dimensional space of our universe.

Equation (2.1) is called Hubble's law, after Edwin Hubble (1929), who was the first to find reasonably convincing evidence of this relation. The multiplying factor, H0, is called Hubble's constant.3

The speed v of recession of a galaxy is inferred from the Doppler effect. Motion of a source of light toward an observer squeezes wavelengths, shifting features in the spectrum of the source toward shorter - bluer - wavelengths, while motion away shifts the spectrum to the red, to longer wavelengths. The spectra of distant galaxies are observed to be shifted to the red, as if the light from the galaxies were Doppler shifted by the motion of the galaxies away from us. This is the cosmological redshift.

You will recall from the balloon model that in this expanding universe an observer in any galaxy would see the same pattern of redshifts, and hence also observe Hubble's relation v = H0r. It is of course a long step from the observation that the light from distant galaxies is shifted to the red to the demonstration that all observers in our universe actually see the same general expansion. But the proposition can be tested; that is one of our themes.

A numerical measure of the redshift is the ratio of the observed wavelength Aobs of a spectral feature in the light from a galaxy to the wavelength Aem of emission at the galaxy. In an expanding universe the ratio Aobs/Aem of observed and emitted wavelengths is greater than one. Astronomers subtract unity from this ratio, defining the cosmological redshift z as

3 In equation (2.1) Ho often is called the "constant of proportionality." That can be confusing, because in the standard cosmology this factor of proportionality changes with time.

Thus when the redshift vanishes, z = 0, the wavelength is unchanged.

The redshift z does not depend on the wavelength of the spectral feature used to measure it. That means we can define a single measure of the wavelength shift by the equation

The radiation was emitted from the galaxy at time tem and received by the observer at the later time tobs. The parameter a(t) defined in this equation depends on time, but it does not depend on the wavelength, because we have observed that Aobs/Aem does not depend on the wavelength. The parameter a(t) serves as a measure of how the wavelength of radiation moving from one galaxy to another is changing now and has changed in the past.

Now let us consider how distances between galaxies change with time. As the universe expands the distance d between a well-separated pair of galaxies increases. Very conveniently, the theory says that the distance is stretched in the same way as the stretching of the wavelength of light moving from one galaxy to the other. That means the distance d(t) between two galaxies -any pair of well-separated galaxies - is increasing as d(t) rc a(t). Thus we call a(t) the expansion parameter.4 When its value has doubled the mean distance between galaxies also has doubled. It follows that the mean number density of galaxies decreases as the universe expands, as n(t) rc a(t)-3, (2.4)

as long as galaxies are not created or destroyed.

In short, if we knew a(t) we would have a measure of the history of the expansion of the universe. It is an interesting exercise for the student to calculate the rate of change of the distance d(t) between a pair of galaxies in terms of a(t); check that the result agrees with Hubble's law in equation (2.1); and find Hubble's constant H0 in terms of the present values of a(t) and its first time derivative. The rest of us may move on.

4 To reduce confusion we urge the reader to bear in mind that our standard of length — be it a meter or a megaparsec — is fixed. Large-scale distances measured in terms of this standard are increasing. On the other hand, objects like ourselves or meter sticks are not expanding. A galaxy that is not accreting or losing matter is not expanding either. Its size is fixed by the gravity that is holding it together. The same is true of a gravitationally bound cluster of galaxies. The expansion parameter a(t) describes the increasing distances between galaxies which are well-enough separated that we can ignore the local clumping of mass in galaxies and clusters of galaxies.

The standard cosmology of the early 21st century is based on Einstein's general relativity theory, the commonly accepted and successful theory of gravity. The use of this theory in the early days of cosmology was speculative, because there were few significant observational tests. But general relativity strongly influenced thinking, as follows.

In general relativity the rate of expansion of the universe changes as the universe expands. The gravitational attraction of the mass of the universe tends to slow its expansion. If the cosmological constant term mentioned in the caption in Figure 2.1 is present, and positive, then it tends to speed the expansion. The resulting acceleration - the second time derivative - of the expansion parameter a(t) in equation (2.3) is represented by the equation d2a ^ 1. rN

Newton's constant of gravity is G and the mean mass density, averaged over local irregularities, is p. The minus sign in front of this mass density term signifies the gravitational effect of the mass: it tends to slow the rate of expansion of the universe. Einstein's cosmological constant appears in the last term. The style has changed here: people nowadays write it as an upper case Greek lambda, A, reserving the symbol A for wavelength. (Note also that in Figure 2.1 the artist drew A backward from the current convention, but in a style similar to Einstein's way of writing it.) If A is positive it opposes the effect of gravity. If A is positive and large enough it causes the rate of expansion to increase, or accelerate. The evidence reviewed in the last chapter of this book is that this is the situation in the universe now.

Einstein (1917) found that his original form of general relativity theory, without the A term, cannot apply to a universe that is homogeneous and, as he supposed, unchanging. You can see that from equation (2.5): if the universe were momentarily at rest then in the absence of the A term the attraction of gravity would cause the universe to start collapsing. That led Einstein to adjust the theory by adding the cosmological constant term, which he could choose so that the attraction of gravity and the effect of a positive A just balance: the right-hand side of equation (2.3) vanishes. That allows the static universe that made sense to him (since he was writing before Hubble's discovery). It takes nothing away from Einstein's genius to notice that he overlooked the instability of his model universe: a slight disturbance would reduce or increase the mass density p, and that would cause the universe to start expanding or contracting. (More generally, a local departure from exact homogeneity would grow and eventually make the universe much more clumpy than is observed.)

Aleksandr Friedmann (1922), in Russia, was the first to show that general relativity theory allows Einstein's homogeneous universe to expand or contract. He had the misfortune to do it a few years before there was a hint from astronomical observations that the universe is in fact expanding. Georges Lemaitre (1927), in Belgium, rediscovered Friedmann's result and recognized that it meant Einstein's static universe is unstable. Lemaitre also saw that the expansion of the universe might account for the astronomers' discovery that the spectra of galaxies are shifted toward the red, perhaps by the Doppler effect. Figure 2.1 shows de Sitter's explanation of Lemaitre's idea. De Sitter is quoted as saying, "what causes the balloon to expand? That is done by the lambda. Another answer cannot be given." De Sitter is explaining Lemaitre's idea that the universe was in Einstein's static condition, and that some disturbance had allowed the A term to push the universe into expansion.

Lemaitre (1931) soon saw that the expansion could instead trace back to an exceedingly dense early state that he termed the "primeval atom." The evidence is that the universe did expand from a state that was dense, as Lemaitre proposed, and hot. We will use the more familiar term for it, the hot big bang.5

It was soon recognized that the expansion of the universe does not require the cosmological constant, provided one is willing to live in a universe that expanded from a big bang. Einstein accordingly proposed that we do away with the A term. The physicist George Gamow (1970) quotes Einstein as saying that his introduction of A was his biggest "blunder." We might suppose Einstein meant that if he had stayed with his original theory, and kept to the idea that the universe is homogeneous, he could have predicted that the universe is evolving, either expanding or contracting. It is a curious historical development that Einstein's cosmological constant is back in style, for the reasons indicated in Chapter 5. The reasons are different from Einstein's original argument, but we imagine Einstein might not have been too disturbed by that. The cosmological constant was his invention, after all.

The names "primeval atom" and "big bang" are meant to indicate that, if the A term does not prevent it, general relativity theory predicts that there

5 The evidence Mitton (2005) assembles is that Hoyle coined the term "big bang" in a lecture on BBC radio in March 1949. Mitton quotes Hoyle: "We come now to the question of applying the observational tests to earlier theories. These theories were based on the hypothesis that all matter in the universe was created in one big bang at a particular moment in the remote past." The connotation of a localized explosion is unfortunate — the theory deals with evolution of the near-uniform observable universe from a dense early state — but its usage is firmly established.

was a time in the past when the expansion parameter a(t) in equation (2.3) vanished. The effect may be easier to see qualitatively by imagining the expansion of the universe running backward in time. The distances between galaxies are smaller in the past, and approach zero as a(t) approaches zero going back in time. This means there was a time when the density of matter was arbitrarily large. If the effects of gravity and A are ignored then the recession speed v of a galaxy does not change, and in this case one sees from equation (2.1) (remembering that distance traveled is speed times time) that the distances between galaxies vanished at time H—1 in the past, or about 10 billion years ago.6 This marks the moment of formally infinite density. It is conventional to speak of this moment as the beginning of the history of the universe as we know it, when a = 0. We include ourselves among the many who suspect that better physics to be discovered, perhaps within the concept of cosmological inflation, will remove this singularity, and teach us what happened "before the big bang," or "at the big bang," or whatever is the suitable term.

In the early 1960s another world view was under discussion. In the steady state cosmology proposed by Bondi and Gold (1948) and Hoyle (1948) matter is continually created - at a rate that would be unobservably small in the laboratory - and collects to form young galaxies which fill the spaces that are opening up as older galaxies move apart. The mean distance between galaxies - about 10 million light years (or about 3 Mpc) for relatively large ones such as the Milky Way - thus would stay constant. The universe on the whole would not be changing: there would be no singular start to the expansion and no end of the world as we know it. Einstein's (1917) original world model, taken literally, has no beginning or end of time either. But, if energy were conserved, all the stars would eventually exhaust their supplies of fuel and die, or if energy were not conserved and stars shone forever, space would become filled with starlight. The steady state cosmology offers an elegant solution: the expansion of the universe dilutes away the starlight and the dead stars, and continual creation supplies matter for unlimited generations of new stars. But this is not the way our universe operates. Part of the story of how that was established commences on page 51, where we consider the state of research in cosmology in the early 1960s. Chapters 4 and 5 describe what happened after that, and the role of the thermal radiation that fills space in teaching about the evolution of the universe. Let us consider now some properties of this radiation.

6 For this reason H—1 is called the Hubble time, or the Hubble length measured in light travel time.

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