Chapter 3

How Clusters Form and Grow—Meatballs in Space

I met Jim Gunn when I was a first-year graduate student at Princeton. I already knew of him by reputation. He had been a star graduate student at Caltech and had come to Princeton as a new assistant professor. If you want to visualize him, think of a young George Lucas at the time he was directing his first Star Wars movie. Like me, Jim had been an amateur astronomer as a teenager, but he had built his own telescope, which had been covered in Sky and Telescope magazine. At Caltech he had codiscovered what came to be known as the Gunn-Peterson effect: a method to detect intergalactic hydrogen by observing a broad absorption band in the spectra of highly redshifted quasars.

Princeton’s astrophysics department has a system where graduate students work with different professors on three different research projects in their first 2 years. I was assigned to work with Jim Gunn, and I was quite excited to see what project he would have in mind. He wanted to study the formation and growth of clusters of galaxies. It would be my job to work out the formulas. Jim felt that after clusters had formed, further material would fall in and this process could be used to measure and test for the presence of intergalactic gas.

Jim and I were going to be working in the context of the standard hot Big Bang model founded by George Gamow and his students Alpher and Herman. They had predicted that if the universe started in a Big Bang, at early times it should have been hot—filled with hot thermal radiation. Compressing gas as one traced backward in time toward the beginning would make it hotter and hotter. At about 1 second after the beginning, the temperature would be about 10 billion degrees. As the universe expanded, the thermal radiation would redshift to longer wavelengths due to the stretching of space and therefore cool. At about 3 minutes after the Big Bang, the temperature would drop enough for protons and neutrons to start fusing into nuclei. This nuclear burning could create the deuterium, helium, and lithium that we observe in the universe today. More helium and heavier elements could be built later in stars.

As the universe expanded by a factor of 10, the radiation would become 10 times longer in wavelength and ¹/₁₀ the temperature. Eventually, at about 380,000 years after the Big Bang, the temperature of this thermal radiation permeating the universe would drop to about 3,000 degrees on the Kelvin scale (3,000 K)—cool enough for electrons to combine with protons to make neutral hydrogen. Before that epoch, the electrons, which carry electric charges and are independently moving around, would have been coupled closely to the thermal photons and the electrons would drag the protons with them. Thermal radiation has a large radiation pressure and is stiff—it resists being squeezed, thereby preventing any fluctuations in the density of ordinary matter from growing. But after the universe has become neutral—after the negatively charged electrons have combined with the positively charged protons to create neutral hydrogen (with zero total charge)—the neutral atoms are free to move relative to the thermal radiation. Astronomers call this epoch recombination, even though the electrons are really combining with the protons for the first time. After this epoch, the universe becomes transparent, and photons from that epoch can fly freely to us today. We should be able to see them.

After recombination, the universe expands a further factor of 1,090 to reach the present epoch. The thermal radiation cools off to a temperature of 2.725 K, just slightly above absolute zero on the Kelvin scale. (This is quite cold—100 times colder than the temperature of ice cubes melting in your drink.) The wavelengths of the photons in this thermal radiation have by the present epoch redshifted (stretched) until they are in the microwave region of the electromagnetic spectrum. This radiation fills the universe today and comes at us from all directions. In 1948, Herman and Alpher had predicted its current temperature to be 5 K, based on calculations of the formation of light elements in the early universe and comparison with the observed amounts of these elements seen today. Penzias and Wilson discovered this radiation in 1965. Its temperature (2.725 K) is very close to Herman and Alpher’s original estimate. We call this radiation the cosmic microwave background radiation. When we see this radiation, which comes to us from all over the sky, we are looking out in space 13.8 billion light-years and back in time 13.8 billion years, to just 380,000 years after the Big Bang.¹ That is the last time the photons we are seeing today had been scattered by electrons before coming directly to us during the universe’s transparent era.

For our research project, Jim Gunn and I started with a uniformly expanding universe at recombination, containing a spherical region with a slight density enhancement built in. This would be the seed from which a cluster of galaxies would grow. We had a very simple model: inside a spherical volume in the early universe the density was above average, while beyond that sphere the universe was of exactly average density. It was a lone cluster trying to form in an otherwise average-density universe. The sphere would start off with an expansion rate determined by Hubble’s law, just as for the rest of the universe. But the extra mass it had inside would cause it to decelerate more than the universe as a whole, and so, after a while, the radius of the sphere would have expanded by a factor that was less than the factor by which the universe as a whole had expanded. Let ρ be the average density in the universe and δρ be the excess density (above the average) within the sphere. Because the sphere is not expanding as fast as the universe due to the action of gravity, the fractional density enhancement inside the sphere relative to the rest of the universe, δρ/ρ, will become larger than it was originally. Eventually, the extra gravity inside the sphere would stop the expansion of the sphere and it would reach a maximum radius, R_max, as it stopped expanding. Once the sphere stops expanding, it begins to collapse under the influence of gravity.

By this time, galaxies have already formed (a process I will discuss later). If the cluster were perfectly uniform, it would collapse to a point within a time equal to the time it took to expand out to its maximum radius. But it is not perfectly uniform, and, therefore, the galaxies will miss each other as they fall through the center. We called the time from the Big Bang to the completion of the collapse of the cluster the collapse time, T_c. The cluster starts off expanding like the rest of the universe but slows its expansion due to the extra gravitational attraction of the excess mass inside and eventually stops expanding at a radius R_max at a time after the Big Bang of ½T_c. It takes an equal time ½T_c to collapse into the center. After the point of collapse, the galaxies pass through the center in a disordered way and expand back out; the distribution sloshes around for a while and quickly reaches equilibrium. This occurs at a time ³/₂T_c after the Big Bang. At this point, the random velocities of the galaxies are resisting the gravitational attraction of the mass of the cluster, as the galaxies orbit within the cluster. The equilibrium radius of the cluster is about ½R_max. Because the volume of the cluster is 8 times less than before, the equilibrium cluster is about 8 times as dense as it was when it reached maximum expansion with a radius of R_max. See Figure 3.1.

Meanwhile the universe has been continuing to expand since the moment when the cluster reached a radius of R_max. In a high-density universe that has just the critical density, so that Ω₀ = Ω_m = 1, we found the cluster was 5.5 times denser than the universe as a whole when the cluster turned around; by t = ³/₂T_c the cluster has grown 8 times denser, while the universe has thinned out by a factor of 9 due to its continuing expansion. That makes the just-formed equilibrium cluster 5.5 × 8 × 9 = 396 times as dense as the universe as a whole. After that, the cluster stays the same size and the same density (it’s in equilibrium), while the universe continues to expand, its density continuing to go down with time. Thus, if we see a relaxed cluster in equilibrium, we expect it to be at least 396 times as dense as the universe as a whole.

This is the case for a high-density universe with Ω_m = 1. If the universe has a lower density than this (if Ω_m < 1), the density enhancement of the cluster over the background is larger than the background by a factor of at least 396/Ω_m. This is how clusters formed after the Big Bang. Only a small positive density fluctuation was needed at recombination. The operation of gravity inexorably turns it into a large density enhancement that is easy to see later. We found that the collapse time of the cluster T_c was approximately inversely proportional to the initial fractional density enhancement δρ/ρ raised to the ³/₂ power (written as T_c proportional to [δρ/ρ]^−³/₂). A bigger initial fractional density excess will cause a shorter collapse time T_c.

Figure 3.1. Cluster collapse and infall. Time is vertical. The radius of the cluster proper is shown expanding with the universe at first (1), then collapsing and relaxing (this is called virialization) to constant size. Shells of matter beyond the cluster proper are shown infalling later (2) to add to the cluster’s mass. Shells of matter further out (3) are still infalling today (at time t₀). The overall background expansion of the universe is also plotted for two cases: a Ω = 1 Friedmann universe of just-critical matter density, which barely expands forever; and a low-density Ω = 0.1 universe, which easily expands forever. (Credit: J. Richard Gott and M. J. Rees, Astronomy & Astrophysics, 45: 365, 1975)

How do galaxies form? Basically by the same mechanism. The distribution of mass inside the cluster initially is not perfectly uniform. The little spherical region that will form the galaxy will have an even higher value of δρ/ρ than the cluster and will have a collapse time that is shorter than that of the cluster. Galaxies would form first, followed by clusters. Galaxies might have typical collapse times of about a billion years, whereas clusters of galaxies would have collapse times of several billion years.

Gunn and I were interested in what happens after the cluster formed. Consider a suburban galaxy, that is, one forming outside the spherical cluster, but close to it. The excess density inside the cluster proper also creates extra mass, pulling the suburban galaxy toward the center. The suburban galaxy will be decelerated by the gravity of the cluster but not by as much deceleration as experienced within the sphere itself. The cluster reaches its point of maximum expansion at a radius R_max. At that time the suburban galaxy is still moving outward, but later it will also stop (at a radius larger than R_max), turn around, and fall into the center of the cluster (after the time T_c). It will slosh through the center and find an equilibrium orbit, but one larger in size than the orbits of the galaxies in the original cluster. This infall will add mass to the cluster as the cluster acts like a gravitational vacuum cleaner, pulling in stuff from outside. We found that the mass of the cluster after the time T_c grew approximately like the ⅔ power of the time. This infall material created an extended envelope for the cluster that was less tightly bound and extended to larger distances. The density in the envelope fell off approximately like 1/r^2.25. This was similar to the envelopes seen in great clusters.

Gunn and I applied our results to the Coma Cluster of galaxies. Zwicky had already estimated its total mass at about 10¹⁵ solar masses. Galaxies in this cluster were orbiting with typical velocities of about 1,700 kilometers/second. It had an envelope typical of what we might have expected from infall. In the center, the core of the cluster was dominated by two giant elliptical galaxies. Gunn and I were able to estimate their masses from their observed separation and their relative velocities. They weighed about 10¹³ solar masses, an order of magnitude more massive than our own galaxy. We then considered what might happen to the infall material. At that time people wondered if much of the mass of the universe could be hidden in the form of intergalactic gas. If that were the case, gas in the region beyond the cluster proper would infall as well as galaxies. When this gas slammed into the center of the cluster, it could not just pass through. The intergalactic gas is spread out, and its particles would tend to collide with other gas particles headed into the cluster from the opposite direction. This would heat the gas until the gas particles were also moving about the cluster with velocities of order 1,700 kilometers/second, reaching a temperature of about 10 million Kelvin, hot enough to glow in the X-rays. Any gas present in the cluster or in the surrounding region that fell in should be heated so that it would glow in the X-rays. When we started our research, no X-rays had been detected from clusters of galaxies. We thus had a prediction that could be tested. But before we could publish, such X-ray emissions were actually detected from clusters of galaxies. Given that, we had an explanation for this discovery already in hand, and we were eager to publish without delay.

A normal spiral galaxy like our own forms when a dense core collapses. The density in this core is relatively high when it reaches R_max and begins to recollapse. The star-formation rate typically depends on the square of the gas density, while the collapse rate depends on the square root of the density at turnaround at R_max. Thus, in this high-density central region, stars can form by the time the central region completes its collapse. The stars, being tiny, miss each other as they pass through the center and relax to form an elliptical bulge, where they orbit but are not drawn further in. Gas outside this region falls in later; it is at a lower density as it turns around and falls in and does not turn into stars before its collapse is completed. The gas heats up and then rapidly cools off, dissipating energy in the form of radiation. Because the gas cannot get rid of its rotational angular momentum, a rotating disk of gas is left. This gas disk can later slowly turn itself into stars. This produces a stellar disk (surrounding the central elliptical bulge), like the one in our own galaxy, which also contains some leftover gas, still forming new stars today.

In our own galaxy we see star formation still occurring. New highmass stars are forming today—bright blue stars that light up the galaxy. As gas in the disk circles the center, gravitational traffic jams occur, creating spiral density waves. The gas is compressed as it enters the spiral density wave, and bright, short-lived, massive blue stars are formed. Being very bright, they use up energy at a prodigious rate and deplete their hydrogen fuel rapidly, dying quickly. The bright blue spiral arms are outlined by these short-lived stars. Lower-mass, lower-luminosity stars like the Sun and stars of even lower mass last longer and stay in the stellar disk. Thus, the disks of spiral galaxies are typically blue in color, dominated by the light of the bright blue stars. The bulge in the center of a spiral galaxy is redder. It is filled only with old low-mass stars, called Population II stars, as opposed to the Population I stars that are seen in the disk.

In the center of the Coma Cluster we see only elliptical galaxies and S0 galaxies—there are no spiral galaxies at all. S0 galaxies have elliptical bulges at their centers and disks of stars, but they have no gas currently forming stars. Gunn and I proposed that these galaxies used to be spirals that had fallen into the cluster and had had their gas stripped out as they plowed through the hot cluster gas. Their bright blue stars simply burnt out, and with the gas gone, no new stars could form to replace them. The hot gas in the cluster strips the gas from any spiral that falls in, turning it into an S0 galaxy. The burning out of the remaining blue stars occurs so fast (millions of years instead of billions) that it is hard to catch a dying spiral in the short interval before it turns into an S0. If two galaxies collided within the cluster, this could create a gravitational pileup, a jumble of stars that would later settle down to form an elliptical galaxy.

Jim Gunn and I emphasized that in the case of great clusters, the hot cluster gas was not going to cool in the age of the universe up to now, and so it would be left there within the cluster as hot gas, able to strip any spirals falling in. The cluster would gravitationally collect material from the neighborhood outside the cluster, and any intergalactic gas out there would be heated and show up in the X-rays. Intergalactic gas in the universe generally might be cool and invisible to us, but if it were out there, it would fall into clusters and heat up so we could see it in X-rays. This could give us a way of detecting and taking a census of intergalactic gas. Jim and I concluded that, given the X-ray observations, the universe did not have enough intergalactic gas ever to halt and reverse its expansion. The universe would expand forever. We called our paper, “On the infall of matter into clusters of galaxies and some effects on their evolution” (Gunn and Gott 1972). It is my most cited and Gunn’s fourth-most cited paper, the first three being associated with data from the Sloan Digital Sky Survey (discussed in Chapter 9). It was a great privilege for me to work with Jim Gunn on this paper and others that we would do later. Jim went on to win numerous awards, including the MacArthur “genius” award, the Crafoord Prize, and the National Medal of Science. Jim may be said to be the person who really brought CCD digital cameras to astronomy; these cameras are about 100 times as sensitive as film and have revolutionized astronomy. He designed a CCD camera for the 200-inch telescope at Palomar Mountain, the Wide Field and Planetary camera on the Hubble Space Telescope, and the camera for the Sloan Digital Sky Survey.

Applications to Galaxy Formation

Jim Peebles at Princeton would subsequently show how galaxies got their angular momentum (rotation). As galaxies collapsed, they were not perfectly spherical. They would be subject to tidal forces from neighboring galaxies. If a galaxy had a barlike shape that was tipped at 45° relative to the line of sight to a neighboring galaxy, the end of the bar closest to the neighboring galaxy would be pulled more strongly toward it, while the end of the bar further away would be pulled less strongly. This would start the bar slowly spinning as it began to collapse. James Binney at Oxford then proved that elliptical galaxies forming from gravitational collapse would usually end up as triaxial ellipsoids—having elliptical shapes but with different diameters in three perpendicular directions.

In 1977, Jerry Ostriker (at Princeton) and Martin Rees (at Cambridge University in England) showed that for collapsing objects (like galaxies) smaller than 250,000 light-years across, the remaining gas would be sufficiently dense to cool and form a cold gas disk, while for collapsing objects larger than 250,000 light-years across (like the Coma Cluster), remaining gas would stay hot and not have time to cool off. This marked the difference between galaxies and clusters of galaxies. Gas falling into galaxies still in formation could cool and form a gas disk, which could then form stars and make a spiral galaxy. An especially dense galaxy could complete its star formation early and form an elliptical galaxy. Thus, galaxies form a distinct class of objects and then aggregate to form groups and clusters.

Gunn would apply our model of cluster expansion and collapse to the formation of our Local Group of galaxies, which is dominated by our own Milky Way and the Andromeda Galaxy. Originally, the Milky Way and Andromeda each formed from small fluctuations in the density distribution of the universe at recombination. Because of the excess density within each protogalaxy, each protogalaxy slowed its expansion relative to the rest of the universe and stopped expanding. The Milky Way and the Andromeda Galaxy then each collapsed and formed approximately a billion years after the Big Bang. Meanwhile the two galaxies continued expanding away from each other at a rate that was decelerated relative to the expansion of the universe as a whole. They eventually stopped moving away from each other and began to fall toward each other under their mutual gravitational attraction. The Andromeda Galaxy is now approaching us (one of the few galaxies to show a blueshift in its spectral lines, indicating a velocity toward us rather than away). It should collide with us about 4 billion years from now. After the stars of both galaxies have passed through each other and formed a jumble and the gas has collided, we should settle down ultimately to make one galaxy. Thus, as a cluster, our Local Group has a collapse time T_c, which is about 18 billion years—longer than the current age of the universe (13.8 billion years). Although the Local Group has not yet completed its collapse, it is still an appreciable density enhancement relative to the universe as a whole. Many loose clusters are in a similar state: loose associations of galaxies that are gravitationally bound but have not yet completed their collapse and have not reached a centrally condensed, relaxed state. These loose clusters do not have hot gas (glowing in X-rays) and are rich in spiral galaxies.

Jim Peebles believed that galaxies and clusters were formed “from the ground up”—smaller structures forming first. He knew that globular clusters (like those Shapley studied in the halo of our galaxy) had some of the oldest stars known, and he thought these globular clusters might be the first objects in the universe to collapse and form stars. He calculated that the first mass scale to be unstable to gravitational collapse after recombination would be the mass scale associated with globular clusters, about a million solar masses. These globular clusters would then cluster themselves to form the building blocks of galaxies. The galaxies would then cluster to form clusters of galaxies. Clusters of galaxies would draw together against the general expansion to form superclusters.

Bill Press and Paul Schechter at Caltech investigated this scenario via computer simulation. They started with seed masses of 3 × 10⁷ solar masses distributed at random locations at recombination. These were just a bit larger than globular clusters. This random (Poisson) distribution of masses leads to random statistical fluctuations. For example, consider a region that should contain on average 100 masses. If you examine many such regions, you find that they typically contain 100 ± 10 masses. (We call the 10 masses the standard deviation from the mean.) Sometimes they will contain as few as 90 masses and sometimes they will contain as many as 110 masses. When one contains 110 masses, it will constitute a fractional density enhancement of 10%, and will, according to the Gunn and Gott paper, collapse within a certain collapse time. If it has 90 masses, it will constitute an underdensity and will thin out. Eventually the collapsing centers will draw in this extra material by infall and add it to their own mass.

In a simulation starting with a random distribution of masses, a spherical region of sufficient size to contain on average N masses, will have typical fluctuations in this count of order . That gives typical fractional density excesses (or decrements) of order . If a galaxy is defined as any object that collapses within 10⁹ years, then (according to the Gunn and Gott formulas) this must start with the mass fluctuating upward by about 1% at recombination. Such fluctuations occur regularly in regions large enough to contain 10,000 of the seed masses on average. That’s because , and we expect these regions to typically contain 10,000 ± 100 masses, giving fractional density fluctuations of ±100/10,000, or ±1%. Thus, we would expect typical galaxy masses to be of order 10,000 times the original seed mass (i.e., 10,000 × 3 × 10⁷ solar masses), or 3 × 10¹¹ solar masses, which is about right—similar to the Milky Way (which is about 10¹² solar masses). Bill Press and Paul Schechter studied the distribution of masses for the collapsed galaxies that would result from such a simulation. In essence, they went searching in the initial conditions at recombination for the largest isolated spherical regions that had an excess density of 1% over the average. These would collapse in a billion years to form galaxies. They could figure out the distribution of galaxy masses that would result and write a formula for it. Schechter was able to fit the observed luminosity distribution of galaxies in large clusters well with a formula of this form. The Press and Schechter formula showed that random fluctuations in mass density could potentially explain the distribution of luminosities of galaxies that were observed, provided that the mass-to-light ratios of different galaxies were roughly equivalent, which seemed plausible. These mass perturbations were isothermal fluctuations—the temperature and density of the microwave background photons remained constant throughout, while the density of matter varied from place to place. It was a random distribution with fluctuations on a mass scale M at recombination of δρ/ρ = ±1% (M*/M)^½, where M* is the mass of a typical bright galaxy. On a mass scale of 16M*, the fluctuations would typically be about ±0.25%, for example. If these happened to be positive in sign (i.e., above average in density by 0.25%), they would be able to collapse in 8 billion years and form a cluster of 16 bright galaxies.

Hierarchical Clustering

Jim Peebles predicted how this clustering should go. A galaxy would form, and then it would tend to bond gravitationally with the galaxy nearest to it in the initial conditions. These would form a density enhancement, which would collapse to form a binary galaxy. The binary galaxy would bond with another binary galaxy to form a quadruple. Two quadruples would be pulled together to form an octuple, and then two of the octuples would form a cluster of 16 galaxies with a collapse time of about 8 billion years, according to Press and Schechter’s simulation. There would be a hierarchy of clustering. Peebles and his graduate student Ray Soniera produced a hierarchical clustering model in just this way. They found that a galaxy today was likely to be in a tight binary, inside a quadruple galaxy system, inside an octuple—that is, hierarchically clustered. They placed the galaxies at random angles but at the proper distances to form the correctly bound hierarchy. If galaxies are clustered today, they should have more nearby neighbors than would be expected if galaxies were laid down at random today.

Peebles developed a very useful quantitative measure of this excess “neighborliness” among galaxies. If you sat on a random galaxy, the covariance function was defined as the average excess probability of finding a galaxy in a narrow shell of radius r away from you above and beyond the average density of galaxies in the universe. The covariance function was thus an elegant measure of the galaxy clustering. If the galaxies were initially in a Poisson distribution (distributed at random), as those point masses were in the initial conditions of the Press and Schechter model, the covariance function would be zero at all radii—no clustering on average initially. But in Poisson initial conditions, there are nevertheless random regions of excess density and underdensity with δρ/ρ = ±1% (M*/M)^½. The galaxies in the random density enhancements would be drawn together by gravity and eventually collapse to form clusters. Peebles then did a little fancy algebra,² and he concluded that if you are sitting on a typical galaxy living in this hierarchy of binaries-withingroups-within-clusters and you look out, the number density of galaxies you see around you should fall off like r^−1.8. Since the clusters are largedensity enhancements, the excess density is nearly equal to the total density (the mean density of the universe being small by comparison), so the covariance function should be proportional to r^−1.8.

Peebles then went ahead and measured the covariance function of galaxies in the real sky (using the Zwicky and Shane-Wirtanen catalogs of galaxies). By mathematical inference he could deduce from the projected positions of the galaxies on the sky their covariance function in 3D. This is possible because the projected separation between two galaxies on the sky is statistically related to their 3D separation. Averaging over all galaxies, Peebles found that the covariance function—denoted as ξ(r)—was ξ(r) ≈ (r/24 million light-years)^−1.77. Rounding to two significant figures, it was proportional to r^−1.8, matching the theoretical calculation of what a hierarchy of clusters formed out of Poisson initial conditions should produce! This agreement reinforced the idea that the (actual) initial conditions were a random (Poisson) distribution of seed masses.

The covariance function was equal to 1 at a radius of 24 million light-years. This meant that if you sat on any galaxy and looked out to a distance of 24 million light-years away from you in any direction, you would find on average about 1 + 1, or twice the average number density of galaxies found in the universe. You would find in a thin radial shell at that radius about twice as many galaxies as you expected. (By the way, the mean separation between galaxies in the universe is also about 24 million light-years—an interesting coincidence.) One has to add up the result over many galaxies and take the average. Peebles’ theoretical calculation was based on an Ω_m = 1 Friedmann universe, where the matter density was equal to the critical density, one that had barely enough kinetic energy in its expansion to overcome the gravitational attraction of the matter and continue to expand forever. Peebles thought the fact that this model produced a covariance function matching what was observed supported the idea of a universe with a critical mass density—pretty simple and elegant, all the way around.

Jim Peebles came to Caltech to give a colloquium. I was there as a postdoc. Peebles had done the world’s largest N-body computer simulation. He had placed 1,000 point masses (N = 1,000), each representing a galaxy, initially sprinkled at random (a Poisson distribution) inside a spherical volume. Then he started the sphere off with a Hubble expansion, adjusted so that the universe met the conditions of the critical density case. The radius of the sphere would then grow as the ⅔ power of the time. This means that the sphere grew in size by a factor of 4 as the time since the Big Bang increased by a factor of 8 (because 4 = 8^⅔). The expansion gradually slowed down because of the mutual gravitational attraction of the galaxies. As the sphere expanded, the galaxies began to cluster just as Peebles had claimed. He made a movie whose scale expanded with time in synch with the expansion of the sphere. Thus, on the screen, the sphere of galaxies remained the same size. But you could watch with time as the galaxies went from their initial Poisson random distribution to the clustered distribution we see today. It was like watching the history of the universe! First, nearby galaxies fell together and made tight binaries. Soon, small groups could be seen forming, and then groups fell together to form clusters. All the while, Jim was standing in front of the screen, dramatically pointing to the places where he knew a cluster was about to form, giving a play-by-play like a sportscaster. Then, Richard Feynman shouted out, “Get away from the screen, let us see it!”—gesturing for Peebles to move aside so he could get a better view. He was clearly enjoying the show and was smart enough to figure it out for himself without commentary! By the end, there were even great clusters like the Coma cluster. It was quite a success.

Two astronomers at the Lick Observatory, Donald Shane and Carl Wirtanen, had carefully counted galaxies on photographic plates of the sky for years. (I’ve already mentioned how Peebles made use of their catalog to calculate the covariance function in the universe.) The Lick survey covered about one quarter of the entire sky centered on the North Galactic Pole. They had counts of galaxies in small bins covering the whole survey region. In all, the two had laboriously counted a million galaxies. They would patiently count galaxies on these glass plates, all day long, day after day. In the end, they were so exhausted that, except for the counts they had obtained, they had little time to study their own catalog! Jim Peebles and his colleagues entered the data into a computer so that a digital image could be constructed. On the largest scales the distribution was uniform (as Hubble had noted), with approximately as many galaxies on the left side of the picture as on the right side. But on smaller scales the clumping of galaxies was obvious. Most prominent in the picture, near the center, was the dense kernel of the Coma Cluster. Peebles could now compare this picture with the hierarchical clustering model he and Soniera had produced, which also had a million galaxies. The agreement was quite impressive. The hierarchical model reigned supreme. The universe seemed to be described best as clusters within clusters within clusters—or “meatballs” (as we would nickname them) within meatballs within meatballs—all floating in a low-density sea. Clusters formed from positive-density fluctuations in the initial conditions, and they grew by drawing in additional material from adjoining low-density regions. The meatballs got more and more dense relative to the background density. The message was: “Keep your eye on the clusters—that’s where the action is.” This was the view of the American school of cosmology. Before long, all the American astronomical community had seen Peebles’ movie and his results on the covariance function, and they were impressed.

The hierarchical model would ultimately be topped off by the 1983 paper of Neta Bahcall and Ray Soniera, who showed that great clusters were themselves clustered; these great clusters had a covariance function with other great clusters that was a power law with the same slope as the galaxy covariance function: ξ(r) = (r/120 million light-years)^−1.8. This result is exactly what one would have expected from the hierarchical model of Peebles and Soniera. There were clusters of clusters—superclusters of galaxies. As early as 1953, Gerard de Vaucouleurs had noticed that the Virgo cluster of galaxies had extra galaxies in its neighborhood with the number density of galaxies falling off like r^−1.7 as a function of radius measured from the Virgo Cluster center. The Virgo Cluster was accompanied by surrounding galaxies. De Vaucouleurs ultimately called this our Local Supercluster. Today it is known as the Virgo Supercluster. Its diameter is about 100 million light-years, and our own Local Group of galaxies is in its outskirts. Our address in space is: Earth, Solar System, Milky Way, Local Group, Virgo Supercluster: clusters within superclusters, meatballs within meatballs.

A Year at Trinity College, Cambridge University

After Caltech, I went to Cambridge University and worked with Martin Rees, who was head of the Institute of Astronomy at that time. He found me a place at Trinity College, one of the many colleges that make up Cambridge University. Trinity College was Isaac Newton’s old college. It was quite a privilege to be there. As a visiting Fellow Commoner, I ate at the high table every night with many distinguished scientists. I got to know Alan Baker, a number theorist who had won the Fields Medal, Brian Josephson, who invented the Josephson junction, and Douglas Heggie, a Scottish astronomer with an interest in Paleolithic astronomy. The senior fellows sat at the head of the table and escorted us upstairs to port after dinner every night. Some of the fellows were in their nineties but still had plenty of energy to share reminiscences. The most senior fellow was Lord Adrian, who had discovered the reflex reaction that gets you to lift your finger off a hot stove before the nerve signal gets to your brain. Next to him sat Littlewood, the famous mathematician, who would tell stories of his legendary colleagues Ramanujan and Hardy. Next to them was Mr. Nicholas, the healthiest 87-year-old I had ever seen. He had been a geologist and had a ruddy complexion and perfectly white hair; except for the white hair, he would have looked 65. As a junior fellow, he had lived in the college until he married at age 30. He and his wife celebrated their fiftieth anniversary just before she died, when he was 80. He showed me her picture. He had moved back into the college after she was gone and had been living there again for 7 years when I met him. Once I asked him if he had seen Halley’s Comet when it appeared in 1910. Yes, he had. But as a young man he also remembered talking to an old fellow of the college who had seen it on its previous visit, in 1835! Such were the conversations one could have at Trinity College. Ultimately, Mr. Nicholas lived to be 101, still entertaining the fellows with stories on his 100th birthday.

At times during my year at Trinity, I felt as though I was living in a starship that had been traveling toward its goal for centuries. I could visit Newton’s rooms, where he had written the Principia, outlining his theory of gravity. Newton’s own copy of the book, with his marginal notes for the second edition, was on display in the college library. In the antechapel stood a statue of Newton, honored by Wordsworth with these lines from his poem The Prelude (III.58–63):

And from my pillow, looking forth by light

Of moon or favouring stars, I could behold

The antechapel where the statue stood

Of Newton with his prism and silent face,

The marble index of a mind for ever

Voyaging through strange seas of Thought, alone.

The Great Court of the college looked much as it had in Newton’s day. In the center was the fountain where, legend has it, young Lord Byron used to tether his pet bear!

Martin Rees lived down the road in Kings College (renowned for its magnificent chapel and choir), in rooms formerly occupied by the worldfamous economist John Maynard Keynes. In 1975 Martin was a young, but already distinguished, professor. (He would later become Astronomer Royal of England and Master of Trinity College, be knighted, and become Lord Rees and then Baron Rees of Ludlow.) Our 1975 research concerned large-scale structure. We were trying to find out where those very early isothermal fluctuations, needed by Peebles and by Press and Schechter, really came from. Martin was familiar with the work of Yakov Zeldovich and the Russian school of cosmology. Zeldovich was working on large-scale structure from a completely different angle. He realized that in the standard Big Bang model, we continue to see larger and larger regions of the universe as time goes on. When the universe is 1 second old, we can see out to a distance of 1 light-second—the distance light can travel in 1 second. When the universe is 1 year old, we can see out 1 light-year. Now, when the universe is 13.8 billion years old, we can see out to a distance of 13.8 billion light-years. The universe we see on the largest scales is nearly, but not perfectly, uniform. To first order, the universe on large scales is approximately uniform with similar counts of galaxies in different directions, but superimposed on this uniformity are small fluctuations. Without fluctuations, the universe would remain perfectly uniform and no galaxies or stars would ever form. We would not be here. We needed a uniform universe plus small fluctuations.

Zeldovich postulated that the ratio δρ/ρ (density fluctuations/average density) on the largest scales visible should remain constant as the universe expanded, as we saw larger and larger regions with time—that is, as light had more time to travel to us from distant realms. It was a bold hypothesis. It was simple. Martin and I applied this to isothermal fluctuations (fluctuations in matter density alone while the radiation temperature and density remained constant—like Press and Schechter had postulated). We realized that as the universe expands, the matter density goes down as the cube of the expansion factor. If the universe expands by a factor of 2, the density of matter goes down by a factor of 8 as the matter particles fan out into a volume 2 × 2 × 2 = 8 times larger. The energy density of the cosmic microwave background radiation goes down like the fourth power of the expansion factor. As the universe expands by a factor of 2, the number density of photons in the microwave background decreases by a factor of 8. But the wavelengths of these photons are also stretched by a factor of 2 because of the stretching of space. The energy of a photon is, by Einstein’s formula, proportional to 1 over its wavelength. Therefore, when the universe expands by a factor of 2, the energy of each photon drops by a factor of 2 as well. If the number density of photons drops by a factor of 8 and the energy of each photon also drops by a factor of 2, then the overall energy density of the cosmic microwave background radiation must drop by a factor of 16 (or 2⁴). This means that the energy density in the thermal radiation falls faster than the matter density as the universe expands.

Conversely, if we trace backward in time to the moment of the Big Bang, we will find the radiation density increasing faster than the matter density. As we get close enough to the Big Bang, the radiation must become dominant. George Gamow understood this. When the mass scales associated with galaxies and clusters of galaxies come into view after the Big Bang, we are still at early times when radiation is dominant. The mass in matter we can see goes up as a function of time as we get further and further away from the Big Bang and light has had more time to travel. The ratio of matter density to radiation density in the universe is also increasing with time. If Zeldovich was right, Martin and I reasoned, total δρ/ρ on the largest visible scales would stay constant with time. If the perturbations were isothermal, the radiation would be the same temperature everywhere at a given epoch and would have no intrinsic fluctuations itself. The δρ/ρ in total density would have to be produced by fluctuations in the matter alone. At earlier times when the matter was a smaller part of the total density, the fluctuations in the matter would have to be larger in order to keep the total density fluctuations at the same Zeldovich amplitude. At earlier times, one could see out only a short distance because of the finite velocity of light, so one could see only a small amount (mass) of matter. This would mean that the fluctuations in the matter on small scales (visible at earlier times) had to be larger than the fluctuations in the matter on large mass scales (which became visible later). Martin and I calculated that (δρ/ρ)_matter should be proportional to M^−⅓, where M is the mass scale of the fluctuation. This was close to the (δρ/ρ)_matter proportional to M^−½ in the initial conditions that Peebles got for his Poisson fluctuations. These matter fluctuations would remain frozen with the cosmic microwave background as the universe expanded until recombination. Matter is coupled tightly to the photons because the matter is ionized, so there are negatively charged electrons plus protons, primarily. By the time recombination occurs, the temperature of the radiation has cooled enough for the electrons and protons to combine to make neutral hydrogen gas, which is free to move relative to the radiation. Gravitational clustering takes over, just as in Peebles’s computer simulation.

The small difference between the M^−⅓ relation that Rees and I found and the M^−½ relation Peebles used was actually helpful. The catalog of groups of galaxies compiled by Ed Turner and myself at Caltech had detected many binary galaxies and small groups while also finding the famous Virgo cluster of galaxies. The observed distribution of groups and clusters of galaxies was broader in mass than the Press and Schechter formulas would have suggested if (δρ/ρ)_matter in the initial conditions was proportional to M^−½, but the data were fit well by the M ^−⅓ relation. Also, Peebles’s calculations for the covariance function had been calculated on the basis of a cosmology having critical density. But observations suggested that the universe actually had a matter density that was less than the critical density. Ostriker, Peebles, and Yahil (1974), for example, had estimated it as one-fifth the critical density. In such a low-density universe, growth of structure at early epochs was similar to that in a critical-density model, but structure grew less at late epochs when the matter density began to depart significantly from the critical density as the matter thinned out. Thus, in such low-density universes, there was less growth of structure at large scales, relative to small scales. If the initial conditions were Poisson with the M^−½ distribution, this would make the covariance function too steep in a low-density universe. But if one started with an M^−⅓ distribution, this would compensate for less growth at large scales in a low-density universe. Martin and I concluded that the covariance function in a low-density universe would be okay if one started with fluctuations at recombination in the matter, (δρ/ρ)_matter, proportional to M^−⅓, (see Figure 3.2).

The other kind of density fluctuations people often discussed were adiabatic fluctuations—fluctuations in the density of the thermal radiation in the hot Big Bang accompanied by equal fluctuations in the matter density. A region that had more photons also had more protons, neutrons, and electrons as well. Thermal radiation has an energy density, and according to Einstein’s equation E = mc², this corresponds to a certain mass density. Fluctuations in the temperature of the radiation cause fluctuations in the energy density of the radiation. Unfortunately, these adiabatic fluctuations got erased at mass scales smaller than the Silk Mass (10¹² to 10¹⁴ solar masses) as we shall discuss in the next chapter. In 1974, Doroshkevich, Sunyaev, and Zeldovich had shown that adiabatic fluctuations (obeying Zeldovich’s hypothesis of constant amplitude on the largest scales becoming visible as a function of time) would also produce matter fluctuations proportional to M^−⅓ on mass scales larger than the Silk Mass (see Figure 3.2). Thus, if one had only adiabatic fluctuations in the matter plus radiation occurring equally and in synch, then on very large scales the fluctuations should also scale as M^−⅓. General density fluctuations (in the matter and in the radiation) could always be decomposed into isothermal and adiabatic components, so it was promising that both components suggested matter fluctuations proportional to M^−⅓ on the largest mass scales. But Martin Rees and I wanted to focus on isothermal fluctuations because these could extend down to small scales and allow us to make galaxies first, as Peebles had argued. We wanted to make galaxies first and then have them cluster.

Figure 3.2. Fractional fluctuations(δρ/ρ) in normal matter (baryons) expected at recombination from different models (vertical axis) as a function of mass scale in solar masses (horizontal axis) in low-density (Ω = 0.1, top panel) and high-density (Ω = 1, bottom panel) universes. These calculations (circa 1979) follow Zeldovich’s hypothesis that fluctuations are of constant amplitude (δρ/ρ = 10^–4) as they first become visible. GR—(Gott–Rees) isothermal fluctuations; DSZ—(Doroshkevich–Sunyaev–Zeldovich) adiabatic fluctuations with instantaneous recombination; and PV—(Press and Vishniac) adiabatic fluctuations with a realistic finite timescale for recombination. M_Jeans is the smallest mass unstable to collapse after recombination. Adiabatic fluctuations on scales smaller than the Silk mass (from 10¹² to 10¹⁴ solar masses) are damped out (erased), and can’t be used to make galaxies (10¹² solar masses) directly. (Credit: J. Richard Gott, Lecture Notes of Les Houches summer school, 1979)

I was able to check some of these results by doing large N-body computer simulations at Cambridge with Sverre Aarseth and Ed Turner. Besides being one of the world’s top experts on large N-body simulations, Aarseth was an accomplished mountaineer, having summited a number of high Himalayan peaks, and he was one of the 10 best postal chess players in the world—quite an interesting fellow! Aarseth had developed a very sophisticated N-body code. Usually, when a tight binary formed, one had to slow down the whole calculation as one took tiny time steps to follow this rapidly orbiting pair using Newton’s laws. Aarseth simply computed the orbital elements for the Keplerian elliptical orbit that the binary would have, allowing him to return to it many orbits later when the big simulation had taken another normal time step. We were thus able to follow the evolution of 4,000 particles in an expanding cosmology. This was 4 times larger than the simulation Peebles had shown at Caltech. We found that a low-density (Ω_m = 0.097) universe with initial (δρ/ρ)_matter proportional to M^–½ was able to produce a covariance function proportional to r^−1.9, close to the observed r^−1.8 relation. A high-density (Ω_m = 1) universe with initial (δρ/ρ)_matter proportional to M^–½ gave nearly identical results. See Figure 3.3.

Figure 3.3. N-body computer simulations of galaxy clustering by Aarseth, Gott, and Turner produced these covariance functions of galaxies as a function of radius (ξ(r)). The radius of the simulation volume is r_sp. Results at the present epoch from two simulations are shown: a high-density (Ω = 1, open circles) universe with initial conditions having (δρ/ρ)_matter proportional to M^−½, as Peebles proposed; and a low-density (Ω = 0.097, filled circles) universe with initial conditions having (δρ/ρ)_matter proportional to M^−⅓, as Martin Rees and I proposed. The two results are essentially identical, and close to the observations as measured by Peebles. (Credit: S. J. Aarseth, E. L. Turner, and J. Richard Gott, Astrophysical Journal, 228: 664, 1979)

These were the models and lines of evidence the Americans were considering. We were following the lead established by Jim Peebles: a “meat-ball” universe—isolated clusters forming and growing by gravitational attraction in a low-density soup: meatball soup. Later, Jim Peebles, Jim Gunn, and Martin Rees would share the Crafoord Prize awarded by the Swedish Academy for their contributions to understanding large-scale structure. But in the Soviet Union, Zeldovich and his colleagues were cooking up a different story.