Chapter 6
A Cosmic Sponge
Did the visible universe’s structural geometry most resemble meatballs or Swiss cheese? These metaphors seemed to summarize the only two possibilities. If the low-density regions of the universe were in one connected piece, then the high-density regions must be in separate pieces, like meatballs in a cosmic soup. If, on the other hand, the high-density regions were in one connected piece, the low-density voids must be in separate pieces, like the holes in Swiss cheese. Apparently it had to be one way or the other. But that was two-dimensional thinking. If one thought in 3D, another possibility emerged, one that hadn’t been considered before. I knew that a third possibility existed because of a science project I had done in high school. I had discovered some peculiar arrangements of polygons that had a structural geometry similar to that found in a marine sponge. I knew some of my spongelike polyhedrons divided space into two equivalent and interlocking parts. A sponge is something whose insides and outsides can be the same. A marine sponge is all in one piece but has water passages percolating through it. This brings nutrients to all parts of the sponge. If I were to pour concrete into the water, let it harden, and then dissolve away the unfortunate sponge with acid, I would be left with a concrete sponge. The concrete sponge would be all in one piece, with air passages percolating through it. It would be essentially the 3D inverse of the living sponge.
In the theory of inflation, the density fluctuations in the early universe were produced by random quantum fluctuations. Random quantum fluctuations could either be positive or negative. A positive fluctuation would cause the density in that region to be slightly above average; a negative fluctuation would cause the density to be slightly below average. Over the course of 13.8 billion years, gravity would cause these fluctuations to grow. A region that was of higher-than-average density would expand more slowly than the rest of the universe, because the gravitational attraction of its different parts for each other would be greater than average. As a high-density region began to expand more slowly than the rest of the universe, it would grow denser still relative to the rest of the universe. A region that was less dense than average would in turn have less gravitational self-attraction to slow down its expansion. It would end up expanding faster than the rest of the universe, becoming even less dense relative to the rest of the universe as it expanded. Over time, the density fluctuations would grow in magnitude, as we have discussed. At some point, the overdense regions would grow to have relative densities of 1.01, while comparable underdense regions would have relative densities of 0.99. Still later, the overdense regions would grow to have relative densities of 1.1, while comparable underdense regions would have relative densities of 0.9. Positive and negative fractional density fluctuations (δρ/ρ) would grow by equal factors. This simple process is called growth by gravitational instability in the linear regime. All fluctuations grow by the same linear factor. As long as we look on large scales where the density fluctuations are of order less than 1 today, we should see a pattern of overdense and underdense regions resembling the initial conditions—but just enhanced in contrast. But what should the initial conditions produced by inflation look like?
If inflation produces density fluctuations from random quantum fluctuations, then the geometry of the high- and low-density regions must be identical. That’s because random fluctuations are equally likely to be positive or negative. These inflationary fluctuations start out small—think of sinusoidal waves like ocean waves crossing space in all different random directions. These waves can be generated by a random-number generator when making a computer simulation of the universe. Make such a set of waves, going in all different random directions. The phases of the waves, the positions of their crests and troughs along the direction of motion, are also random. The amplitude of the waves will be drawn from a Gaussian, or bell-shaped, distribution. This is, therefore, called a Gaussian random-phase distribution. It is what inflation predicts. Some regions will be above average in density and some will be below average in density when all the waves are added up. That is a perfectly good set of initial conditions. But now multiply all those random numbers by −1. That will turn all the wave crests into troughs and the troughs into crests. It will turn all the high-density regions into low-density regions, and vice versa. That is just as good a set of initial conditions as the first one!
Suppose we had initial conditions that had a meatball topology. The high-density regions would be isolated meatballs and the low-density regions would be one connected soup. If we multiplied all our initial fluctuations by −1, it would turn the high-density meatballs into lowdensity voids (holes) and the soup into one connected, high-density piece of cheese. If random fluctuations always made a meatball topology, then multiplying these fluctuations by −1 would turn them into equally random fluctuations but suddenly with a Swiss cheese topology—that’s inconsistent. Thus, random fluctuations cannot produce an initial meatball topology. Random fluctuations cannot produce an initial Swiss cheese topology either—multiplying those fluctuations by −1 would turn them into a meatball topology. From this reasoning, neither meatballs (the American school of cosmology) nor Swiss cheese (the Russian school) could be produced by the random initial conditions predicted by inflation! Both led to a logical contradiction. By contrast, a sponge can have insides and outsides that are the same. Imagine our marine sponge example again, with water percolating through it. Multiply the fluctuations by −1, and geometry of the new high-density regions (concrete) is just as spongelike as before. Random initial conditions can produce this. Thus, a spongelike geometry with insides and outsides that look alike is what is required for Gaussian random-phase initial conditions. I got this idea from my high school science project.
A High School Science Project
I was always drawn to geometrical problems. After eighth grade, I attended a summer program in mathematics at Rollins College in Florida sponsored by the National Science Foundation and run by Professor Bruce Wavell. This program had wonderful courses on mathematical logic and a course on special relativity based on Max Born’s book. I frequented the library and found exciting books on four-dimensional geometry. At this summer program I began a project on the geometrical arrangement of atoms in metallic crystals.
In a metal there is a periodic array of atoms, but since all the atoms are identical, one might expect the arrangement of atoms around any given atom to be the same. I thought such periodic structures could be related to volume-filling packings of identical polyhedrons. As we have discussed, we can stack cubical boxes to fill a warehouse leaving no empty spaces. Imagine an atom fitting inside each box. The nucleus of each atom would be at the center of each box. The points within a given box represent the set of points in space closer to the center of that atom than to the center of any other atom. A warehouse filled with such cubical boxes with one atom in each would produce a cubic crystal structure. The atoms would be located at the vertices of a cubical jungle gym. But what other kinds of identical cells could equally well fill an infinite warehouse, leaving no gaps? I found a thick book showing all kinds of polyhedrons and set out to test each one. Using spherical trigonometry, I wrote a computer program to figure out which polyhedrons would fit together to fill space. I ran this program on an early IBM 1620 computer, which used punched cards and was about the size of a washer and dryer. An iPhone would exceed its computing power by a large factor today, but it was the state-of-the-art computer available for general use at that time.
I found a number of different polyhedrons that could stack together to fill space. For example, triangular prisms would fit together in this way: a triangular prism has a triangular top and bottom with three square sides connecting them. Lay out equilateral triangles to tile a plane; identical triangular prism boxes constructed on top of each triangle make a layer that covers the entire floor of the warehouse; then put a second layer on top of that and continue until the warehouse is filled. This makes a pattern like the arrangement of carbon atoms in graphite.
One particularly interesting solution used truncated octahedrons to fill space. A truncated octahedron is an octahedron with its corners cut off—it has six square faces and eight hexagonal faces.
If I have a warehouse filled with truncated octahedral boxes, as depicted in Figure 6.1, the atoms centered in each box will form a body-centered cubic array. It’s as if you had a jungle gym with atoms at all the intersection points between rods and also an atom suspended in the middle of each cubic open space in the grid. This is the structure of metallic sodium. I noted, in addition, that the shape of the occupied electron orbitals in metallic sodium actually looks approximately like a truncated octahedron.
Figure 6.1. How truncated octahedrons can be stacked to fill space in a body-centered cubic pattern. Eight truncated octahedrons form a cube—with another one filling the center of the cube. One more truncated octahedron sits on top of that central one, thereby starting the next layer. This pattern can be extended indefinitely to fill space. Note that the vertex on the left, between the two frontward-facing squares on the left, is surrounded by four hexagons in a butterfly pattern. These four hexagons form a saddle-shaped surface around the vertex. This will prove to be important later. (Credit: A.J.S. Hamilton, J. Richard Gott, and D. Weinberg, Astrophysical Journal, 309: 1, 1986)
This project won first place in Physical Sciences at the Kentucky State Science Fair and that got me a place in the National Science Fair International, which had entrants from all over the United States and from a number of other countries. There, my crystal project won a first prize from the American Metallurgical Society and a runner-up award for a trip to Japan from the U.S. Navy to exhibit at the Japanese Science Fair. Over the summer I improved my project by doing X-ray diffraction studies of metallic sodium to prove that it indeed had a body-centered cubic pattern and submitted it to the Westinghouse Science Talent Search.
The Westinghouse Science Talent Search (now the Intel Science Talent Search) has been nicknamed the “Nobel Prize” of high school science competitions. From 300 semifinalists, 40 winners are brought to Washington, D.C., where they are interviewed over several days by a panel of judges. I was selected as one of the 40 winners. One of the other winners my year was Ray Kurzweil—well known today for his invention of the Kurzweil reader for the blind and for his ideas on artificial intelligence and the implications of increasingly powerful computer technology for the future.
One of our judges was Glenn Seaborg (who discovered plutonium and other transuranic elements). He seemed to like my project and asked me questions about sodium, which I was able to answer. Finally came the awards banquet, a black-tie affair held in a fancy hotel ballroom. Ten scholarship winners were read off one by one, starting with tenth place. In the end, I came in second. I was thrilled—with a few names called in reverse order, you are happy to hear your name come up at all!1
The scholarship from the contest paid for about half of my Harvard undergraduate education. But, in a way, the most important thing occurred before I got to Washington, when I was rushing to prepare my exhibit for the judges. I had made a plastic model of transparent truncated octahedrons stacked to fill space (like Figure 6.1). Inside each was a Styrofoam ball representing a sodium atom. The truncated octahedrons themselves were made from Plexiglas® sheets that I had cut into hexagons and squares. They had beveled edges that I had cut on a bench saw at the proper angle so that they would glue together to form the truncated octahedrons. I had assembled a set of hexagons glued together in pairs, and I noticed that if I put two pairs of beveled hexagons together, the four hexagons formed a saddle-shaped surface. I knew that a saddle shape corresponded to a negatively curved surface. (A sphere has positive curvature, a plane has zero curvature, and a Western saddle has a negative curvature.) This fact would later have an important application for large-scale structure because sponges turn out to have negatively curved surfaces.
A triangle drawn on a Western saddle will have a sum of angles that is less than 180°. The circumference of a circle drawn on a Western saddle will be larger than 2π times its radius. That’s because the saddle goes up and down as you circle the center, making that circumference longer than it would be on a flat plane.
Mathematician Carl Friedrich Gauss proved that the intrinsic curvature of a 2D surface is equal to 1/r1r2, where r1 and r2 are the principal radii of curvature of the surface.
For example, a flat plane has infinite radii of curvature, so 1/r1r2 = 0, giving a flat plane zero curvature. A cylinder also has a curvature given by 1/r1r2, but in this case, r1 is the radius of the cylinder, while r2, the radius of curvature along its length, is infinite, because the cylinder is straight along the direction of its length. No matter the value of r1, r1 times infinity is infinity. And 1 over infinity is 0, so the Gaussian curvature of a cylinder is also 0. You can make a cylinder out of a flat piece of paper without distortion.
On a sphere, the two principle radii of curvature, r1 and r2, are both equal to the radius of the sphere, and both point in the same direction. If you sit on top of a sphere, your two legs can hang down on opposite sides, while the sphere also curves down in the front and back direction as well. It curves downward in both perpendicular directions. The curvature of a spherical surface is thus positive, equal to 1/r2, where r is the radius of the sphere.
In a Western saddle, although the saddle curves downward from side to side where your legs go, it curves upward from front to back to fit on the horse’s back. Thus the two radii of curvature in a Western saddle point in opposite directions and their product is negative, giving the Western saddle a negative curvature.
So when I noticed that those four hexagons glued together (where two of the truncated octahedrons were joined together) formed a saddleshaped surface, I immediately related it to a negatively curved surface (see Figure 6.1).
Positively curved surfaces like the sphere can be approximated by regular polyhedrons. The cube is a very rough approximation to a sphere. A cube is made up of squares meeting 3 at a point. In a cubical room, the ceiling meets 2 side walls at a vertex—look up at the corner of the room you are sitting in to verify this. Each square face of a cube has a 90° angle at its corner. So at the corner of a cube, three square faces meet at a vertex, and each has a 90° angle at its corner, making the sum of the angles around a point at the corner 3 × 90°, or 270°—that is 90° less than one would get in a plane. A plane can be tiled by squares 4 around a point to make a checkerboard pattern, where the sum of angles around each point is therefore 4 × 90°, or 360°—just what one would expect. On a polyhedron that approximates a positively curved surface, the sum of angles around a vertex is less than 360°. The cube has an angle deficit of 90° at each vertex. What about my 4 hexagons? Hexagons have 120° angles at their vertices, which makes the sum of angles around a vertex in my saddle-shaped surface 4 times 120°, or 480°! That is more than 360°. There is an angle excess of 120° at the vertex. It all made sense. If one has polygons approximating positively curved surfaces (as occurs in the 5 regular polyhedrons), they will have less than 360° around a point; if they approximate a plane surface (as in a checkerboard), they will have exactly 360° around a point; and if they approximate a negatively curved surface, they will have more than 360° around a point. (In fact, angle deficits and angle excesses would later enable us to write a computer program to measure the curvature of the complex surfaces we encounter in studying large-scale structure in the universe. Unknowingly, I was learning in high school the tools I would need later to study the topology of large-scale structure.)
Could this pattern of four hexagons around a point be extended to continue the negatively curved surface? I noticed that it could! In fact, if you just took my plastic model of truncated octahedrons filling space and deleted all the squares, you would be left with a single convoluted surface—composed only of hexagons meeting four at a point in every case. The hexagons formed a spongelike surface that divided space into two equal parts that were identical to each other but interlocked. In high school I already realized that this surface made of hexagons had numerous holes, like donut holes. I had found a new regular polyhedron—an infinite spongelike one—composed of hexagons meeting four at a point. The astronomical connection is this: it is such a spongelike surface that inflation would require for the geometry of the median-density contour surface separating the high- and low-density regions.
Regular polyhedrons are defined as having regular polygons (i.e., closed planar figures having equal sides meeting at equal angles) as faces, with the same configuration of faces around each vertex. Finding a new one was exciting because only five regular polyhedrons had been known from ancient Greek times (Figure 6.2). Each has vertex angles totaling less than 360°. These were the only possibilities that added up to less than 360° to make finite closed figures roughly approximating positively curved spheres.
But, in addition, there were three regular planar networks (shown in Figure 6.3). Johannes Kepler recognized that these long-known planar networks were like regular polyhedrons but with an infinite number of faces.
Figure 6.2. Euclidean geometry identifies five regular polyhedrons: (a) tetrahedron, (b) octahedron, (c) icosahedron, (d) cube, and (e) dodecahedron. These are polygon networks with (a) triangles, 3 around a point; (b) triangles, 4 around a point; (c) triangles, 5 around a point; (d) squares, 3 around a point; and (e) pentagons, 3 around a point, respectively. (Credit: Robert Webb’s Stella software is the creator of these images: http://www.software3d.com/Stella.php [here adapted as black and white])
Figure 6.3. The regular planar networks: (a) triangles, 6 around a point; (b) squares, 4 around a point; and (c) hexagons, 3 around a point. (Adapted from: R. A. Nonenmacher)
Now I had found a regular polygon network (hexagons, 4 around a point) that was negatively curved (having a sum of angles around a point of 480°, which is greater than 360°) and was also spongelike, with an infinite number of faces and an infinite number of holes. I quickly found six other spongelike networks. I called these pseudopolyhedrons (Figure 6.4), after the pseudosphere which is a surface of constant negative curvature encountered in the non-Euclidean geometry of Nikolai Lobachevsky and Janos Bolyai.
These were seven regular polygon networks that were every bit as legitimate as the ones known from ancient times! This was a much better science project than the one I was going to take to Washington for the Westinghouse contest. I was just turning eighteen at the time. I wondered if I should substitute this project for the exhibit I was planning to take. I decided there was not enough time to make it. But I did think that I might turn this into an exhibit for the science fair later in the spring. This meant constructing cardboard models of all these pseudopolyhedrons. I made the convoluted surfaces white on one side and red on the other. (In Figure 6.4 these surfaces are colored white on one side and black on the other.) A number of these pseudopolyhedrons divided space into two equal parts—these were sponges whose insides and outsides were identical: squares, 6 around a point; hexagons, 4 around a point; hexagons, 6 around a point; pentagons, 5 around a point; and triangles, 10 around a point.
In the end, my pseudopolyhedron project won first place in mathematics in the Louisville Science Fair and the top prize over all, so I got to go to the National Science Fair International held in St. Louis. My wonderful math teacher Mrs. Ruth Pardon got to go along as well. There, it won first place in mathematics, and one of three grand prizes, a trip to Japan sponsored by the U.S. Navy. I ended up traveling to Japan after all! I took the trouble to translate my project’s labels into Japanese (Figure 6.5).
After I got to Harvard, math instructor Tom Banchoff2 encouraged me to submit my paper on pseudopolyhedrons to the American Mathematical Monthly, which I did. The referee’s report was quite positive but noted that three of my polygon networks had been discovered before! The reference was to a paper, which I had never heard of, by H.S.M. Coxeter in 1937. That paper described how the first of these figures to be discovered—squares, 6 around a point—was found in 1926 by John Petrie, who also discovered hexagons, 4 around a point (the one I found first). Therefore, Petrie gets credit for discovering this entire class of figures. Coxeter himself discovered hexagons, 6 around a point. They did this work in 1926 when they were both 19. In addition to demanding that the configuration of polygons around each vertex be identical (as I did), their criteria for regularity also demanded that the angles between all adjacent pairs of faces be equal. With those conditions they were able to prove that the three examples they found were the only regular figures of this type. They called them regular skew polyhedrons. I was happy to add the Coxeter-Petrie reference. My paper was still publishable, the referee said, because I had discovered four new pseudopolyhedrons. I still required the configuration of polygons around each vertex to be identical but allowed the angles between adjacent faces to vary. I rediscovered all three structures discovered by Petrie and Coxeter as well as finding four new ones allowed by my more lenient rules. My paper appeared in print in 1967. It was my first published scientific paper. When Siobhan Roberts wrote her definitive biography of Coxeter, King of Infinite Space, in 2006, I was happy to contribute my story of the astronomical applications these figures later had in understanding the distribution of galaxies in space.
Figure 6.4. Regular pseudopolyhedrons expand the class of regular polyhedrons. (Credit: J. Richard Gott, American Mathematical Monthly, 74: 497, 1967)
Figure 6.5. The author exhibiting his high school science project on pseudopolyhedrons at the 1965 japanese Science Fair. (Credit: j. Richard Gott, personal collection)
Additional regular pseudopolyhedrons have been discovered by the noted crystallographer A. F. Wells: including triangles, 7 around a point; triangles, 9 around a point; and triangles, 12 around a point. These are illustrated in Wells’s 1969 paper and his 1977 book, Three Dimensional Nets and Polyhedra. He, like I, did not demand equal angles between adjacent faces. All are spongelike with an infinite number of faces and an infinite number of holes.
Melinda Green rediscovered my pentagons, five around a point, and has illustrated many pseudopolyhedrons on her geometry Web page. Her illustration of pentagons, 5 around a point, appears in Color Plate 1.
Avraham Wachmann, Michael Burt, and Menachem Kleinman have discovered many semiregular spongelike polyhedrons, composed of polygons of more than one kind, for example, two squares and two hexagons around each point. These are illustrated in their 1974 book Infinite Polyhedrons.
In mathematics, the Schwarz P surface is a negatively curved surface that approximates the geometry of the pseudopolyhedron having hexagons, four around a point; it divides space into two equal spongelike parts. In biology, the Schwarz P surface has been observed in the biological membranes of various cells by transmission electron microscopy. Thus, microscopic spongelike surfaces have been observed in nature.
Would the universe at large scales also be spongelike? That is what inflation seemed to predict. If random quantum fluctuations produced the small fluctuations present in the initial conditions, then the geometry of those regions above average and below average in density had to be identical. The geometry of a sponge could achieve this, as illustrated by the pseudopolyhedron squares, six around a point (Figure 6.6). The high-density parts would constitute one connected sponge with many holes, and the low-density regions would form a complementary sponge of low-density voids connected by low-density tunnels.
I set out to test this. I called up Adrian Melott at the University of Kansas, who had done some of the world’s largest N-body simulations of growth of structure in the universe. I knew from his papers that he had programs for drawing density contour surfaces in 3D and that these could be applied both to his initial conditions and to the outcome today.
When measuring the topology of large-scale structure today, one has the positions of individual galaxies to work with—a set of points in 3D space. We are interested in how these points are clustered: where are the regions where there are more points than average and where are there fewer than average? So we must take averages. We do this by smoothing, or blurring, the data to produce a smoothly varying average density field. It is important to smooth the data over a length scale that is at least as large as the average separation between galaxies and on a scale where High density Low density the average density fluctuations from place to place are small compared with the average density for the universe as a whole. In practice, this means smoothing, or blurring, the data over a scale of at least 24 million light-years. It’s like producing a population density map of the United States by averaging over counties rather than showing the points representing each of the people.
Figure 6.6. This pseudopolyhedron (squares, 6 around a point) divides space into equivalent, spongelike high- and low-density halves. (Credit: J. Richard Gott, A. L. Melott, and M. Dickinson, Astrophysical Journal, 306: 341, 1986)
First, we smoothed the random initial conditions of Melott’s simulation based on inflation and plotted the median density contour. Half the volume was on the high-density side of this contour and half was on the low-density side. This contour surface was spongelike, just as I had predicted it might be! The high-density side was multiply connected with many holes, and the low-density side was also multiply connected and interlocking with the high-density side. Space was divided into two complementary sponges—like the marine sponge and the water. This was, of course, a random sponge; my pseudopolyhedrons were periodic sponges. But both were sponges that divided space into two complementary parts.
Then we watched the structure grow in the simulation and waited until the simulation reached the present day. The initial fluctuations were so small as to be barely visible, but gravity worked on them to make them bigger, with the high-density parts getting ever denser relative to the average density, while the low-density parts emptied out. Eventually the very densest parts in the initial conditions stopped expanding with the universe and collapsed to form clusters of galaxies, while low-density voids also appeared. In the end, the clusters were connected by filaments of galaxies to make a spongelike pattern—what we now call the cosmic web. The low-density region cleared out to make a complementary low-density sponge—empty voids connected by low-density tunnels. We smoothed the map showing the density at the present epoch by blurring it on a scale of 94 million light-years. Then we had the computer again draw the median density contour. This contour contained half the volume on the high-density side and half the volume on the low-density side, just like the one we computed in the initial conditions. It was also a sponge. Furthermore, it was almost exactly the same sponge we started with—it had the same holes in the same places (Figure 6.7). They had moved a little and their shapes had changed a little, but their topology looked the same. One could recognize the same pattern of holes, implying that the universe “remembered” what its initial conditions looked like (see Figure 6.7). The important point here is that the topology on large scales is preserved, despite the strong evolution of clustering on small scales. Smoothing more or less recovers the initial conditions. In the computer simulation, those initial conditions were the random fluctuations predicted by inflation. With this topology tool, we could now look at the universe today to deduce what it looked like at the beginning—to check whether it had the spongelike initial conditions expected from inflation.
Figure 6.7. Initial and final conditions for the Gott-Melott-Dickinson computer simulation are displayed for both high- and low-density halves. The topology in the initial conditions is spongelike, with interlocking (a) high-density and (b) low-density regions, as expected for Gaussian random-phase initial conditions. The final conditions, (c) and (d), are also spongelike, showing holes in the same places as in the initial conditions with only slight differences. On these large scales, gravity causes the structures to grow in place, increasing in contrast with time. This simulation featured heavy neutrinos (favored by Zeldovich). (Credit: J. Richard Gott, A. L. Melott, and M. Dickinson, Astrophysical Journal, 306: 341, 1986)
The next step was for me to enlist my then-undergraduate student Mark Dickinson (now at Kitt Peak National Observatory in Arizona) to look at the observational data and apply the same test. Using data from the Center for Astrophysics (CfA) survey, we constructed a uniform cubic sample stretching beyond the local Virgo Supercluster of galaxies. including all galaxies above a certain luminosity all the way out to the sample’s outer edge. We thus had a complete sample of bright galaxies inside the cubical region we were exploring, with our Milky Way Galaxy located at the bottom-front corner of the cube. We could get a clear view looking out of the plane of our galaxy into a cubic survey region beyond. Mark then smoothed the data on a scale of 47 million lightyears and had the computer draw the median density contour (see Figure 6.8). It was spongelike—just like the simulations.3 The high-density parts, which included the Milky Way and the Virgo Supercluster, were on the high-density side of the spongelike contour, and the low-density parts were on the other side. It was a very small sample, but we felt it was a eureka moment nevertheless, and we knew that it was only a matter of time before much-larger observational samples would become available, allowing us to test our theory in great detail. Our paper (Gott, Melott, and Dickinson 1986) was titled “The Sponge-like Topology of Large-scale Structure in the Universe” and was published in the Astrophysical Journal.
We wanted to quantify the topology of these structures. Topology is the field of mathematics studying those properties of geometric figures that remain unchanged under distortion. Topologists talk about genus. A donut has a genus of 1. It has 1 hole. A coffee cup with one handle also has a genus of 1. To a topologist, a donut and a coffee cup look just alike! One can be distorted into the other without breaking. If you had a donut made out of clay, you could continue molding it without ever breaking it until you had made a coffee cup with a handle. But if you cut through the coffee cup handle, its genus becomes 0, because the cup now has no holes. A sphere has a genus of 0. You can take a sphere of clay and mold it into a coffee cup (without a handle). The cup has a depression where the coffee goes, but that is not a hole—it doesn’t go all the way through. Eyeglass frames have 2 holes: they have a genus of 2. A trophy with two handles has a genus of 2 as well. If you cut through one of its handles, you destroy that hole and reduce the genus by 1, leaving it with a genus of 1, like a coffee cup. So genus corresponds to the number of holes in an isolated figure; alternatively, one can say it is equal to the maximum number of complete cuts you can possibly make in the figure without it falling apart into two pieces.
Figure 6.8. Observations (from the CfA sample) are shown divided into (a) high-density and (b) low-density halves. We studied a cubical observational sample, 140 million lightyears on a side, with an effective smoothing scale of 47 million light-years. Earth is at the bottom front corner. The topology is spongelike. (Credit: J. Richard Gott, A. L. Melott, and M. Dickinson, Astrophysical Journal, 306: 341, 1986)
For cosmology we needed a new definition of genus—one that could be applied to many structures, not just one. We redefined the genus of a density contour surface as
genus = number of donut holes − number of isolated regions.
By this new definition, a donut has a genus of 0, because it has one donut hole and is itself one isolated region (1 − 1 = 0). A sphere (e.g., a spherical density contour surrounding an isolated cluster of galaxies) would have a genus of −1 under this new definition (0 holes − 1 isolated region = −1). Two spheres would have a genus of −2 because they have no donut holes and would enclose two isolated regions (0 − 2 = −2). If you cut a sphere in half, you would lower its genus by 1: as a single sphere, it was 1 isolated region with no holes and a genus of −1; after it was cut, it would be two isolated hemispheres (0 holes − 2 isolated regions), giving a genus of −2. Every time you cut, you lower the genus by 1, just as before. With this new definition of genus, we can keep track of how many holes are found in the density contour surface as well as how many isolated regions it is broken up into. If we had a meatball topology, for example, the median density contour surface would show isolated clusters and we would have many isolated regions and a negative genus. If we had a Swiss cheese topology, the median density contour would likewise have many isolated voids and a negative genus. We could distinguish these two topologies by noting whether the isolated regions contained the high- or low-density parts. But if we had a spongelike topology, the median density surface would be one convoluted piece, multiply connected, with many holes. It would have no isolated regions but many holes, making its genus positive. We were able to prove (using a famous Gauss-Bonnet theorem from mathematics) that the newly defined genus could be calculated by integrating the curvature over the contour surface and dividing by −4π. To understand this better, let’s consider some properties of curvature.
Measuring Curvature—David Weinberg’s CONTOUR3D Program
A sphere has a uniform curvature of 1/r2 over its entire surface. The area of the sphere is 4πr2, so if we integrate, or add up, the curvature over the entire area of the sphere, we get 1/r2 times 4πr2, or 4π. This is independent of the radius of the sphere. If the sphere is twice as big, its curvature is ¼ as large but its area is 4 times as big, so the answer is 4π, just as before. If we divide 4π by −4π, we will get a genus of −1 for the sphere (one isolated region with no holes). It doesn’t depend on how big the sphere is.
For a sphere the curvature is uniformly spread over the entire surface. But for a polyhedron like a cube, all the Gaussian curvature is concentrated in the vertices. The faces of the polyhedron are flat, and the edges are like bent cylinders, which also have zero curvature as defined by Gauss. If you have ever tried to flatten some cardboard boxes for recycling, you will realize that the problem occurs not at the edges but at the vertices—you have to do some tearing to get the job done there. In a polyhedron, the integral of the curvature over a vertex is equal to the angle deficit at that vertex. Mathematicians measure angles in radians, where 2π radians are taken to be equal to 360°. The angle deficit at each vertex of the cube is 90°, or π/2 radians. The cube has 8 vertices, 4 on the top and 4 on the bottom, so the total angle deficit for the cube is 8 times π/2 radians, or 4π radians. Thus the integral of the curvature over the entire cube is 4π, just as it was for the sphere. The cube is one isolated region with no holes, so by our definition it should have a genus of −1, just like the sphere. To a topologist, a sphere and a cube look the same.
My spongelike pseudopolyhedrons, by contrast, have angle excesses (i.e., more than 360°) around each vertex. An angle excess is the opposite of an angle deficit—in other words, a negative angle deficit. And this corresponds to a negative curvature. When we integrate the curvature over our spongelike pseudopolyhedrons, we will get a negative number that will tell us how many holes we are looking at. In our paper, we explained how to write a computer program that would do this calculation automatically. The volume of the computer simulation is divided into little cubic voxels, or 3D volume elements, like pixels but in 3D. The boundary between the high-density and low-density voxels will be a corrugated surface of square faces. Our program would then look at each vertex in this corrugated surface: if three squares came together at a point, it would have a positive curvature like the cube; if 4 squares came together at a point, it would have zero curvature like a plane; and if 5 or 6 squares came together at a point (as they do in the pseudopolyhedrons squares, 5 around a point, or squares, 6 around a point), then the curvature would be negative. Polygon networks (as in my high school science project) were now going to be used to measure the curvature. The program just adds up the contributions from each vertex to determine the answer. Dividing the answer by −4π would tell us the number of holes minus the number of isolated regions the density contour surface had. My graduate student at the time, David Weinberg (now a professor at Ohio State University), wrote this program, which we called CONTOUR3D. It allowed us to calculate the genus of any contour surface.
The Genus Predicted by Inflation
I next enlisted Andrew Hamilton at Princeton (now at the University of Colorado) to help with calculating what the genus predicted by inflation would be. I designed the voxels for this calculation—in this case, truncated octahedrons (as it happens, just like the cells for sodium atoms in my high school Westinghouse project)—and he did the calculation. Given the probabilities of different voxels being above or below the contour threshold, one could calculate the probability of a particular curvature occurring at each vertex and, therefore, the genus. Inflation predicted Gaussian random-phase fluctuations in the initial conditions because these were produced by random quantum effects. Think again of random waves in an ocean. Inflation also told you how much power to expect at different wavelengths. The power is proportional to the square of the height of a typical wave of a certain wavelength. Hamilton’s calculations gave a formula for the genus. If you want to see what it looks like, here it is: genus = A(1 − ν2)exp(−ν2/2). It is a simple formula. A is a positive constant that depends on the amount of power at different wavelengths, ν is the number of standard deviations above the mean for the contour surface, and exp(x) (the number e = 2.71828 … raised to the power x) is the exponential function, which is always positive. The function exp(−ν2/2) is a bell-shaped or Gaussian distribution, which is always positive. Suffice it to say, at the median density contour surface (ν = 0), which divides space into two equal parts, the genus was always positive—indicating a spongelike topology. If, however, we look at the highest-density regions, comprising only 7% of the volume (ν = 1.45), we will see a negative genus—only isolated clusters. If we look at the lowest-density regions, comprising only 7% of the volume (ν = −1.45), we will find isolated voids—also a negative genus. This gives a symmetric genus curve, as we go from low-density contours toward high-density contours: negative genus (isolated voids) at first, positive genus (in the middle-density ranges), and then negative genus again (isolated clusters)—creating a W-shaped curve. Reverse the sign of initial conditions, replacing high-density regions with low-density regions, and the curve stays the same—as must be true for random quantum fluctuations. Now we had a whole theoretical curve to test against both our simulations and against actual observational data sets.
After completing this calculation, Hamilton found that Doroshkevich in 1970 had derived essentially the same formula by an independent method. Doroshkevich even remarked in passing that the contour surface for ν2< 1 would be complex and multiply connected, but he was just interested in galaxy formation and in using the formula to count peaks for ν > 1. Bardeen, Bond, Kaiser, and Szalay in 1986 also used it to count high-density peaks, but no one had applied it to examine the median density contour in galaxy clustering. It turns out that mathematician R. J. Adler had calculated these topology measures in N dimensions in 1976 and discussed them in his 1981 book, The Geometry of Random Fields. They also go by the name of Minkowski functionals. Hikage and colleagues (2003, 2006), who have measured topology following our work, have used this nomenclature for the genus, as did Mecke and colleagues (1994). In 3D, the genus, as we are defining it, for a density contour surface is related to the Euler characteristic (V − E + F). A 3D density contour surface, if pixelated to make a network of polygons, will have a number of faces (F), edges (E), and vertices (V) satisfying the relation:
genus = −(V − E + F)/2 = −Euler characteristic/2,
where the 3D genus is as we have defined it (number of holes − number of isolated regions).
That the value of (V − E + F) was related to topology was proven by the great mathematician Leonhard Euler (1707–83). For example, a cube has 8 vertices, 12 edges, and 6 faces, giving it an Euler characteristic of 2 and a genus of −1 for one isolated region, by our definition. In my high school science project, using ratios of vertices, edges, and faces, I had already realized that the Euler characteristic of my infinite spongelike polyhedrons was negative and infinite, allowing me to show that they had an infinite number of holes.4 Euler’s formula is a piece of genius.
Thus, we have brought some famous old mathematics to bear on testing whether the universe started with Gaussian random-phase initial conditions as predicted by inflation.
Measuring the Genus Topology
We calculated the genus at various density contours in the initial conditions of our computer simulation and found excellent agreement with the formula. Since the initial conditions were calculated according to the initial conditions for inflation, that was no surprise. But the results also matched the formula with little difference at the present epoch. If an appropriately large smoothing length was applied, the density contours of cold dark matter in terms of volume fraction at present still looked quite like the similarly smoothed density contour surfaces in the initial conditions. The contrast in the picture at the present epoch was larger, but the topology of the map was much the same as in the initial conditions (as shown in Figure 6.7). It’s as if the mountains got higher and the valleys got lower while staying in the same places. Gravity grows the fluctuations in place. In the cold dark matter model, we expect galaxies to track the cold dark matter, but we expect them to be biased toward forming in the highest density dark matter regions. As long as the probability of making a galaxy was a monotonic (ever-increasing) function of the density of the cold dark matter, the topology of the biased (galaxy) map and the cold dark matter map should be similar. The voxels of cold dark matter and galaxies should be ranked similarly, and the median density contour by volume should be similar. Thus, great clusters today are located where tiny density enhancements in the cold dark matter occurred in the initial conditions, and great voids today are located where tiny density decrements in cold dark matter occurred in the initial conditions. Figure 6.9 shows the results we obtained studying a computer simulation that assumed cold dark matter. The spongelike median density contour (50% low—50% high) that divides space into two equal volumes hardly moves at all between the initial conditions and between the final and final biased distributions. The holes are all in the same places. Thus we can measure the topology in the smoothed galaxy distribution today, and deduce the topology of the initial conditions, and compare it to the predictions of inflation. At the bottom, we show our small cubical CfA observational sample at the same scale and with the same smoothing length. It displays a spongelike median density contour (the 50% low—50% high pictures) and isolated voids at the 7% low contour and isolated clusters at the 7% high contour. It also shows a similar number of structures per unit volume as the Cold Dark Matter simulations.
Figure 6.9. Cold dark matter simulations. Initial refers to the topology in the low-contrast initial conditions at recombination. Final is the topology of the clustering of the cold dark matter particles in the final conditions, at the present epoch. Biased refers to the topology expected for the galaxy distribution if biased galaxy formation causes galaxies to form preferentially in regions of higher cold dark matter density. These are compared with the CfA observational survey sample (to scale). At the median (50%) density contour, the topology is always spongelike. The 50%-high-density and 50%-low-density regions add to make the entire cube. If we look at the highest density 7% of the volume, we find isolated clusters. Symmetrically, if we look at the lowest density 7% of the volume, we find isolated voids. (Credit: J. Richard Gott, D. H. Weinberg, and A. L. Melott, Astrophysical Journal, 319: 1, 1987)
Figure 6.10 (top graph) shows the genus curve for the cold dark matter initial conditions, the cold dark matter final conditions, and the final biased conditions—tracing where we expect to find the galaxies. As expected from Figure 6.9, at the 7% low-density contour, we find isolated voids, at the median (50%) contour surface, we find a spongelike topology, and at the 7% high-density contour, we find isolated clusters. The genus in Figure 6.10 (top) starts out negative (isolated voids), becomes positive (spongelike genus) in the middle, and becomes negative again (isolated clusters) at the end. For comparison, the theoretical curve is shown. The fit is very good, within the errors.
Then we measured again our small CfA observational sample, which included the Virgo Supercluster. Its median density contour had four holes. It was spongelike. It had two isolated high-density clusters and two isolated low-density voids. With twice as many holes as clusters or voids, its observed genus curve couldn’t have possibly looked more like the theoretical curve, given the small number of structures we were measuring (see Figure 6.10, bottom graph).
Simulations of Swiss Cheese Universes
We could also calculate genus curves for the Zeldovich model. The genus curves for a Voronoi honeycomb look quite different. Here we placed 8 points at random in our simulation box and constructed a Voronoi honeycomb with these as the 8 cell centers. We placed all the galaxies on the cell walls. This was the Zeldovich model. Then we smoothed the data and constructed density contour surfaces (Figure 6.11).
One can see at a glance that this produces a Swiss cheese topology. Just look at the 50% high picture, showing the parts that are higher than the median density—it looks exactly like a block of Swiss cheese.5 The 50% low picture shows the low-density voids—the low-density Voronoi cells. As we go to still-lower density contours, the 8 low-density voids are still seen intact in the 24% low, 16% low, and 7% low pictures. If we continue to higher-and-higher-density contours (above the median), we see high-density filaments where the cell walls meet, in the 24% and 16% high pictures, just as in the Zeldovich model, and, finally, at the highest density, in the 7% high picture, we see isolated clusters where the filaments meet. The genus curve corresponding to these pictures (Figure 6.12) is quite different from the symmetric curve seen for a sponge-like topology.
Figure 6.10. Top graph: Genus curves for the Cold Dark Matter simulations (initial, final, and biased; see Figure 6.9) are shown in the top graph. Mean values with error bars are shown. The smooth solid curve is the genus curve predicted for Gaussian random-phase initial conditions expected from inflation; f is the fraction of the volume on the high-density side of the contour surface being measured. Bottom graph: Genus curve for the observational CfA sample. It is only a small sample but still has approximately the same shape as the genus curve for Gaussian random-phase initial conditions shown in the simulations above. (Credit: J. Richard Gott, D. H. Weinberg, and A. L. Melott, Astrophysical Journal, 319: 1, 1987)
Figure 6.11. A Voronoi honeycomb with 8 cells (or bubbles) was created to simulate a Swiss cheese universe. The topology of the simulation, with appropriate smoothing, at different density thresholds is shown. The 50%-high-density picture looks very much like a block of Swiss cheese. (Credit: J. Richard Gott, D. H. Weinberg, and A. L. Melott, Astrophysical Journal, 319: 1, 1987)
Figure 6.12. Genus curve for the Voronoi honeycomb shows a dramatic shift to the left—a Swiss cheese shift. The genus value is −8 and constant for all low-density contours (on the right) showing 8 isolated voids. The median density contour ν = 0 is negative, due to isolated voids. This does not look like the genus curve for the observations (Figure 6.10, bottom graph). (Credit: J. Richard Gott, D. H. Weinberg, and A. L. Melott, Astrophysical Journal, 319: 1, 1987)
The left hand of the genus curve is flat at the value of genus = −8, as it identifies the 8 low-density voids created by the Voronoi Honeycomb. The genus at the median contour (in the center of the graph) is negative, showing a Swiss cheese topology at the median density contour. At higher-density contours, the genus becomes positive and the topology becomes spongelike (Zeldovich’s network of filaments at the edges of the cells) before it finally breaks apart into isolated clusters at very high density (at the corners of the cells). This shifts the genus curve radically to the left—what we call a Swiss cheese shift. The genus curve for the observations (Figure 6.10, bottom) does not look like the Swiss cheese model at all: the observations show a symmetrical genus curve, which is spongelike at the median density contour in the center.
Figure 6.13. Stereo pairs: cold dark matter–50%-high final simulation (top pair), and the 50%-high CfA observational sample (bottom pair). To observe in 3D, touch your nose to the page. The left-eye image of cold dark matter–50%-high pair will be directly in front of your left eye, and the right-eye image will be directly in front of your right eye. They will fuse into one blurry image. Slowly move your head back, keeping the image fused. As you move your head back, the fused image in the center will come into focus—in 3D! It takes some practice to do this—give it a try. Move down and repeat the procedure to see the 3D image for the CfA data. (Credit: Adapted from J. Richard Gott, David H. Weinberg, and A. L. Melott, Astrophysical Journal, 319: 1, 1987)
Figure 6.14. David Weinberg (far left) and the author view slides of the topology in 3D using red-blue glasses and discuss the topology results (right) in 1987. (Credit: Photos courtesy of Princeton University, Bob Matthews)
Finally, we show two stereo pairs (Figure 6.13): the cold dark matter (CDM) simulation, final conditions; and the original cubical CfA observational dataset (adapted from Gott, Weinberg, and Melott 1987). In both cases we show the high-density halves. Both are sponges, where the high-density half is in one multiply connected piece, full of holes, allowing the low-density half to percolate through it.
We could view these 3D images by projecting a red left-eye image and a blue right-eye image on the screen and viewing them with red-blue 3D glasses, as used in 1950s’ sci-fi movies (see Figure 6.14).
As he relates in his 2014 book Dark Matter and the Cosmic Web Story, when Einasto saw our Gott, Melott, and Dickinson (1985) preprint on the spongelike topology of the universe during a visit to the European Space Agency, he realized that he had made an error in not writing a paper attempting to characterize the topology of the distribution. He set out immediately to do so. He used a code to measure the maximum size of connected systems. He set density thresholds for the voxels in a computer simulation. He found that at medium-density thresholds, the low-density voxels formed a connected system whose size was equal to the simulation box size, as did the high-density voxels. That supported the spongelike topology we were claiming. In the observations, since he was not smoothing the data, he could not study the connectedness of voids—empty voxels always filled the volume, but he did find that the high-density voxels (containing galaxies) produced connected structures that extended completely across the observational box as one lowered the density threshold to medium densities. He rushed out a preprint on this. It supported our spongelike topology and reflected the view he expressed at the 1983 Crete Symposium (which I attended), where he emphasized filaments connecting clusters. I was quite happy to see this preprint; it provided some unexpected support from the Soviet school. We were able to add this preprint to our references in the final version of our paper, which appeared in the Astrophysical Journal in 1986.
On November 9, 1986, the New York Times covered our paper in a page 1 story titled “Rethinking Clumps and Voids in the Universe,” written by the eminent science writer James Gleick. This article included a kind quote from Jerry Ostriker: “It’s a clever new approach. It looks to be a powerful tool for discriminating between different physical models for how the universe got its structure, and that’s the really exciting thing.”
We were now eager to test our theory on larger data sets.
