CHAPTER FOUR
4.1 The Geometry of the Universe
One of the fundamental characteristics of the universe is its geometry. Geometry is the study of the relations between points, lines, angles, surfaces, etc. Let’s go back to thinking about space. We have noted that it can expand at different rates. It is also malleable and can be warped or curved, as we saw with gravitational lensing by the Bullet Cluster. For lensing, it was not too much of a stretch to think of the space near a massive object as being curved. But now we want to think about our whole three-dimensional space as being curved into a fourth spatial dimension. This is a little more challenging.
In the mid-1800s, Georg Friedrich Bernhard Reimann showed that even without going to the next higher dimension we can tell if we live in a curved space. To understand his insight we will work with two-dimensional surfaces curved into our familiar three-dimensional space, as shown in figure 4.1. Imagine that you are an ant walking around on the two-dimensional surface between the three vertices of a triangle. Think of the surface as being very large, the ant as very small with negligible height, and assume that all motion is confined to the surface. If you walk the perimeter of any triangle on a flat piece of paper and sum up the interior angles, you will get 180°. The piece of paper is said to have a “flat” geometry and the two-dimensional space is infinite so that there are no edges.1 By convention, we use the word “flat” even if we make the triangles in three-dimensional space with all possible orientations as opposed to on a flat sheet of paper.
FIGURE 4.1. Examples of possible geometries of two-dimensional space. On the left is the saddle-surface-like open geometry. Think of it going on forever. The thick dark lines show a triangle whose interior angles sum to less than 180°. The middle shows a flat sheet-of-paper-like geometry. Think of it too as going on forever. Here the thick dark lines show a triangle whose interior angles sum up to 180°. On the right is a spherical-surface-like geometry. This one is finite. Here the thick dark lines show a triangle whose interior angles sum up to greater than 180°.
Let’s consider instead a spherical shell. This is an example of a finite and closed, positively curved space. The ant could walk a triangular path from the North Pole to the equator, around the equator by a quarter of the circumference, and then back to the North Pole. Now the ant would find the sum of the interior angles to be greater than 180°, and for this particular path the sum would be 270°. The larger the triangle, the larger the sum of interior angles. If the ant shot a laser beam out in this space, it would come back and hit his back side because in two dimensions the laser beam is confined to follow the surface of the sphere.
A horse saddle is an example of an open, negatively curved space. Unlike the case for the spherical shell, the saddle surface, like the flat piece of paper, goes on forever. If the ant were to walk the perimeter of a triangle on the saddle surface and sum up the interior angles, he would find that they were less than 180°. If we think of the leather as space and if we tried to flatten the saddle onto a flat surface, we would have a bunch of folds left over. An open, negatively curved space is one in which there is more available space, more leather, the farther away we go.
This same process of determining the overall geometry by measuring the sum of the interior angles of a triangle on a two-dimensional surface also works for a three-dimensional space that is curved into a fourth spatial dimension. All we have to do to figure out the geometry of our three-dimensional space is to traverse a large triangular path and sum up the angles. Locally we might do this by making a triangle out of the Sun, Moon, and Earth and measuring the interior angles, but the best test is to make a very large triangle. The CMB gives us a way to do this, but on a cosmic scale.
You may recall from high school geometry that to determine all the angles in a triangle, you need three pieces of information. They could be, say, the lengths of two sides and one of the angles. The CMB hot or cold spots form one side of the triangle. In effect we average all the spot sizes together. As we discussed in the previous section, we can compute to high accuracy the average physical size of a hot or cold spot in, say, light-years. Using our maps we can measure the average angular size to high accuracy as shown in figure 3.3 and plates 8a and 8b. We need to know one more bit of information to completely specify the triangle that has, say, a hot spot as the far side. Although it is not obvious, that piece of information is the Hubble constant because it links the physical size of the hot spot to the distance to the hot spot. The result of the accounting is that the sum of the interior angles is 180°. To the limit of measurement, the geometry of the universe is “flat.”
A simple calculation can help us get a sense for how the pieces fit together. Earlier we found that the computed size of a hot spot is about 900,000 light-years across at decoupling. The universe has expanded almost a factor of 1100 since then, so today this is 990 million light-years across. Its angular size is measured to be about 1.2° across by the Planck and WMAP satellites. From this we compute the distance to the decoupling surface to be about 46 billion light-years, which we recognize as the radius of the observable universe.2 If the geometry of the universe had been closed, we would have measured a larger angular size for the spots; if it had been open, we would have measured them to be smaller. It all hangs together!
To summarize, the geometry of the universe is like the geometry many of us learned in high school. It is the simplest one we can think of. It is what you would have expected if you had never heard of Einstein or Reimann. More important, the geometry has been determined by measurement and can be checked with different types of measurements of the universe.
4.2 The Seeds of Structure Formation
The very very earliest instants of the universe are still not well understood. The reason is that as yet there is no fundamental theory that combines gravity with the standard model of particle physics. In place of this we have “effective theories” and paradigms that are deeply rooted in the physics we know and that can explain the observations. The best known of these is “inflation.” We touch on aspects of it now, but keep in mind that this is still a very active area of theoretical research.
One of the great mysteries of the cosmos before the inflation model was invented was: “why are the properties of the universe in two opposite directions so similar?” To be concrete, let’s take the north celestial pole and south celestial pole as our opposite directions and consider how the CMB can have the same temperature in the two directions. According to the picture we have been developing, the light from both sides of the observable universe is just reaching us now. No information can travel faster than light, so there is no way that radiation from the north celestial polar direction could have passed us and gone on to affect what we see in the south celestial polar direction, and vice versa. Yet, they are nearly the same temperature, 2.725 K, and have the same properties.
In the inflation model, the space in the very early universe, before there were any particles, had an enormous energy density. Associated with this energy density was a pressure that made space, or caused an expansion, at an unimaginably fast, exponential pace. At the start of the process imagine you have two regions, call them Alice and Bob, that are right next to each other and that share information. In inflation, space between Alice and Bob is made so rapidly that they can no longer communicate with each other. Their apparent speed of separation is faster than the speed of light. They are separated beyond, perhaps many many times beyond, the distance over which they can subsequently affect each other.
Inflation takes place over an extremely short period of time, very roughly a billionth of a billionth of a billionth of a billionth of a second. After inflation ends, the universe settles into a calmer pace of expansion. As the universe ages, the observable universe gets larger and larger because we can look farther and farther away. At some point, Alice and Bob come into view with Alice, say, in the north polar direction and Bob in the south. Now we have a mechanism for saying why opposite sides of the universe might look the same. They communicated with each other very very early on, became hugely separated during the inflation epoch, and are just now coming into our observable universe. We also need a mechanism to explain why they would not separately evolve when they are out of our sight, but that too is part of the model.
There are many variants of inflation, but the simplest model has two other features that are relevant to the CMB. The first is that the universe is geometrically flat to at least a part in 10,000. This corresponds to trying to determine if a meter stick is flat or bowed up at one end by 100 microns. As the inflation model was proposed prior to the observations, it gained a lot of credibility when the data showed the universe had a flat geometry. The idea is that even if the earliest geometry were, say, positively curved, inflation would have expanded it so much that it would effectively be flat. It is not difficult to imagine this in two dimensions. If you are on the surface of a sphere, such as the Earth, you can tell the surface is curved. But, if the radius were a billion billion times larger, it would be difficult to tell you were on a sphere. To the limits of measurement, our geometry is flat but we cannot rule out the possibility that it is just ever so slightly positively or negatively curved.
The second feature is that inflation incorporates a mechanism for generating the seeds for the formation of cosmic structure. The seeds are quantum fluctuations in the primordial energy density. What do we mean by quantum fluctuations? Think of them as tiny localized fluctuations in energy on the subatomic scale. Quantitatively they are understood using the Heisenberg uncertainty principle. Let’s say that in the lab you created the best possible vacuum with the strongest possible pumps and removed every atom from some volume inside a container. It’s not physically possible, but we can imagine doing it. Even then, inside your container at the subatomic level so-called virtual particles are continuously coming into existence and then vanishing on a timescale inversely proportional to their energy. The vacuum is roiling with activity. While this may seem a bit out there, the effects of the roiling vacuum on atoms, after having let some back into your container, can be computed and measured to high accuracy. Quantum fluctuations are a well-established phenomena in laboratory experiments.
The model is that quantum fluctuations in the primordial energy density were stretched out to cosmic scales through the inflation of space. The fluctuations in the primordial field are now seen as the gravitational landscape that produced the hot and cold spots in the CMB. This means that when we look at the CMB we are looking directly at a manifestation of quantum processes. The random distribution in space of the hot and cold patches is a result of our quantum origins. We usually associate quantum processes with taking place on an atomic or subatomic scale. This is still true; it is just that inflation expands space so much that the quantum scale becomes the cosmic scale, a mind-blowing concept.
The expansion in inflation is similar in character to that for the cosmological constant discussed earlier, but in inflation the pressures are much much greater. Perhaps the origin of the processes is related. We don’t know. Also, it may be that inflation is not the correct paradigm. It is possible that the universe goes through cycles of expansion and that we are in just one of the cycles. Even in this case, though, the origin of the CMB anisotropy can be traced to quantum fluctuations.
Let’s revisit the maps of the CMB anisotropy in plate 7. We can now look at them with a new perspective. These maps show quantum processes now writ large across the sky. It is as though the evolution of the universe acts like a microscope to show us our quantum origins.
4.3 Pulling It All Together
While explaining the CMB has guided our path to an understanding of the universe, there are many other ways to study the universe. Cosmology is a broad field. The physics brought to bear includes everything from general relativity, to thermodynamics, to elementary particle theory. Observations are made in nearly every wavelength regime accessible to measurement and with state-of-the-art particle detectors. The observations come from nearby and from the farthest reaches of space. All of this evidence and theory is encompassed in the surprisingly simple standard model. Before summarizing the model, we touch on two major frontiers we have not discussed in much detail.
The most time-honored approach to cosmology is through observations of galaxies. As we saw, this was how Hubble and Lemaître pointed out that the universe was expanding. In addition to telescopes like the Hubble Space Telescope that can peer deeply and with high resolution in a given direction, there are others that measure the properties of millions of galaxies over more than a third of the sky. Probably the best known is the Sloan Digital Sky Survey. From this and related efforts we now have maps of the three-dimensional distribution of galaxies throughout much of the observable universe. We see in detail how galaxies clump. We can see how light from distant galaxies is bent on its way to us by the curved space near the intervening galaxies. By averaging over large volumes we can even see that there is a characteristic size for the clumping of galaxies that corresponds to the average CMB hot spot and cold spot sizes in plate 8a. For galaxies, the special spot size is called the “baryon-acoustic oscillation scale.” To emphasize, the signature of the physical processes that produced the hot and cold spots in the CMB is also detected in the distribution of galaxies.
Quite independently of the galaxy and CMB observations, cosmologists have worked out the nuclear physics of the first three minutes of the universe in a study called Big Bang Nucleosynthesis. The inputs to the calculation are the CMB temperature and nuclear interaction rates as measured in laboratories. The outputs are the abundances of the lightest elements: hydrogen, deuterium, helium, lithium, and beryllium. The first atomic nucleus to form was the deuteron, the nucleus of deuterium. It consists simply of one proton and one neutron. Before about 100 seconds, whenever a deuteron tried to form, it would be split apart by energetic photons corresponding to more than a billion kelvin. By 100 seconds, the universe had expanded and cooled enough for the forces that bind the proton and neutron to overcome the collisions with the photons that were trying to tear them apart. Thus, the deuteron could survive intact. In roughly the next 100 seconds, the deuterium was converted to helium through a series of nuclear interactions. By 1000 seconds, the other light nuclei formed. The process was a competition between the force that binds the neutrons and protons, the photons losing their energy because of the expansion of the universe, and the 10-minute decay time of the neutron.
The main predictions of the nucleosynthesis calculations are the overall cosmic fractions of atoms. As you might imagine from the above, there is an intimate relation between the energetics of the photons and the number of nuclei that are made. For the predicted nuclear abundance to match observations, there must have been about two billion photons per proton. These photons are of course the CMB.
The calculations predict that the atoms in the universe are primarily hydrogen (75% by mass) and helium (25% by mass) with only trace amounts of the other elements. Not only is this observed on the cosmic scale, but the Sun is 75% hydrogen and 25% helium. The calculations show that elements heavier than beryllium could not have been formed in the early universe. In general, the measurements of cosmic abundances of the light elements are in agreement with what one would infer from the CMB with one exception—lithium. Less is found than predicted. It is likely that the early stars eat the lithium but the mismatch between expectations and measurement may be telling us that there is an element of the calculation or model we are missing.
We now summarize the six parameters3 of the cosmological model. The particular values we give result from fitting the gray curve in figure 3.3 to the CMB data. The values don’t shift much, and the uncertainties improve, when additional datasets, such as the distribution of galaxies, are combined with the CMB. We give the specific symbols for the parameters as they are often encountered in the scientific literature.
As a foundation, the model stipulates that the universe is geometrically flat. We showed earlier that we could use measurements of the CMB and Hubble’s constant to demonstrate that the geometry was flat. Indeed, when we do the calculation the result is consistent with flatness. However, instead of making the geometry part of the fit, we assume geometrical flatness and deduce Hubble’s constant. This gives us the opportunity to compare the Hubble constant derived from the CMB, that is from the early universe, with the Hubble constant directly measured from the recession of galaxies versus distance. While the agreement is very good, it is not perfect. This too may be telling us that there is an element of the model we are missing—an exciting prospect—or that, perhaps, there are systematic errors in the measurements. The jury is out. Fortunately there are other ways to check the cosmic geometry, and they also tell us it is flat to the limits of measurement.
The first three parameters tell us about the contents of the universe. They are specified as fractions of the total, like the components in a typical pie chart as we discussed earlier.
1. Atoms account for about 5% of the universe. In the CMB anisotropy spectrum, figure 3.3, the ratio of the height of the first to the second peak gives a measure of the density of atomic nuclei in the early universe. It is not obvious that this should be the case and was only understood after we knew how to compute the curve in the figure. The value from the CMB anisotropy agrees with the value from Big Bang Nucleosynthesis. The fact that the stuff of which we are made accounts for just 5% of the net cosmic energy density gives a new perspective on our place in the universe. We specify this fraction with the Greek letter omega and say Ωatoms = 0.05.
2. Dark matter accounts for 25% of the universe. In the CMB anisotropy spectrum, the ratio of the height of the first to the third peak gives a measure of the dark matter density. This too is not obvious and was only understood after we knew how to compute the gray curve in figure 3.3. What is remarkable is that the amount of dark matter derived from the CMB anisotropy agrees with the value deduced from observations of the motions of stars and galaxies discussed in section 2.2, but the value from the CMB is much more precise. In addition, because the CMB comes to us from the decoupling era, the third peak tells us that dark matter existed in the early universe.4 There must be new fundamental particles in Nature from the Big Bang that have never been detected in the lab. To specify the fraction of the universe that is dark matter, we write ΩDM = 0.25. What’s more, we see that the stuff of which we are made accounts for just one sixth the total mass in the universe.
3. The cosmological constant accounts for 70% of the universe. We don’t know what it is, but we have directly measured its presence through the cosmic acceleration. In the CMB, we determine it from the position of the first peak in figure 3.3. The value from the supernovae observations agrees with the value from the CMB. We write ΩΛ = 0.70.
There are of course other components such as the CMB radiation itself and the mass fraction of neutrinos. We know they are there but they are not significant enough that, with the current level of precision, they need to be included in the overall budget.
The next parameter is the most astrophysical one. It captures our rather scant knowledge of the entire complex process of the formation and subsequent explosion of the first stars, and the formation of the first galaxies. The intense light from these early stars and galaxies broke apart the hydrogen into its constituent protons and electrons, reionizing the universe. With the current level of precision, we need just one parameter to account for what no doubt we will one day find to be a rich process.
4. In the process of reionization, about 5–8% of the CMB photons were rescattered. In the analogy used to describe decoupling, it is as though a bit of fog rolled in. Not too much—you could still see a distant shore—but the visibility would not be perfect. The symbol that describes the scattering is τ and is called the “optical depth.” We measure τ = 0.05 − 0.08. But τ cannot be determined with the temperature anisotropy alone. It takes a measurement of the polarization of the CMB, a topic we have not discussed. Polarization, along with intensity and wavelength, is one of the three characteristics of a light wave. The polarization specifies the direction in which the light wave is oscillating. For example, light reflected off the hood of your car is horizontally polarized. That is, the light wave oscillates back and forth horizontally. Polarized sunglasses block this oscillation direction and its associated reflected glare. Similarly, the electrons freed up in reionization scatter and polarize the CMB. If you could look at the CMB with polarized “sunglasses” it would look slightly different. In figure 3.3 reionization causes an overall suppression of the spectrum, with just slightly more suppression at the largest angular scales. The optical depth is the least well known of the cosmological parameters.
The next two parameters characterize the seeds of the fluctuations that gave rise to all the structure in the universe. The concepts underlying these are beyond the scope of this book, but we include them for completeness. These seeds led to the CMB anisotropy spectrum and to the fluctuations in the total matter in spheres of 25 million light-years in diameter that were discussed in section 1.1. These primordial fluctuations are described by the “primordial power spectrum.” This is similar in character to the CMB anisotropy power spectrum (figure 3.3), but instead of describing the decoupling surface, it describes fluctuations in density in three-dimensional space. As we look around the cosmos today, the three-dimensional fluctuations in density are large. In some places there are galaxies, in others clusters, and in others almost nothing. Before there were identifiable objects, the density fluctuations were much smaller. As we discussed, at decoupling the contrast was a part in 100,000. We use the primordial power spectrum to quantify the density fluctuations back at the beginning of the cosmic expansion.
5. The amplitude of the primordial power spectrum is encoded in the formidable symbol 
. If we had a complete model of the universe that began with the quantum fluctuations and predicted, say, the fluctuations in matter in spheres of 25 million light-years diameter, we could relate to the rest of physics and its value would be known. Unfortunately, while we have a very successful framework, we do not yet know all the connections and so require it as a parameter.
6. The final parameter, called the “scalar spectral index” or ns, is the most difficult to understand, but is also our best window into the birth of the universe. Like , it tells us about the primordial fluctuations. In contrast to specifying the overall amplitude, it tells us how the primordial fluctuations depend on angular scale. To better grasp this, let’s go back to the musical analogy that we used to describe figure 3.3. For a moment, let’s put aside those peaks and troughs in the spectrum, from which we learn so much, and imagine the plot representing “white noise.” In this case, the data points would lie along a flat horizontal line. All frequencies (all angular scales) would have the same loudness (or variance as measured on the y-axis). The parameter ns allows us to distinguish between “white noise” and, say, “pink noise,” in which the bass notes have a somewhat greater loudness than the treble notes.5 Using the CMB, we find that the primordial fluctuations, the “seeds,” were ever so slightly larger in amplitude at large angular scales than they were at smaller ones. That is, the primordial cosmic noise is slightly pink.
When the process of cosmic structure formation was originally being studied, the scalar spectral index was argued to be unity, or ns = 1, on general grounds. It was called the Harrison-Peebles-Zel’dovich spectrum after its authors. This value corresponds to white noise. Then, in the early 1980s, it was realized by Viatcheslav Mukhanov and Gennady Chibisov that this quantity could be computed from quantum principles operating as the universe was being born. We now know this index differs from unity by about 5%, that is ns = 0.95, corresponding to just slightly “pink.” This is the evidence that all the structure in the universe arose from quantum processes operating at a time when the universe was so compact and energetic that no known particles yet existed.
With these six parameters we can compute the properties and spectrum (the gray line in figure 3.3) not only for the CMB but for any cosmological measurement. We can compute the age of the universe. The single most constraining observation is the CMB anisotropy, but the model is consistent with all measurements. In short, no matter how we look at the cosmos—with galaxy surveys, through exploding stars, through the abundance of the light elements, through the speeds of galaxies, or through the CMB—we need only the six parameters given above and the physical processes we explained in the preceding sections to describe what we observe.
In 1970, Allan Sandage wrote an article for Physics Today entitled “Cosmology: A search for two numbers.” We now know it takes six, but with them we can account for more than Sandage thought possible. What does it mean to be able to describe something so simply and quantitatively? It means we understand how the pieces fit together, all the ones discussed in chapters 1–3 and more, to form a whole. We understand some deep connections in Nature. It means we can be proved wrong, not by different arguments but by a better quantitative model that describes more aspects of Nature. There are few systems studied by scientists that can be described so simply, completely, and with such high accuracy. We are fortunate that the observable universe is one of them.
1 A Möbius strip is an example of a “flat” geometry that is not infinite. It is said to have a non-trivial topology. Topology describes the way in which space is connected. For example, a doughnut has a different topology than a sphere because you can’t deform one into the other. In this book we assume that the universe is characterized by its geometry and not its topology. Predictions from cosmologies with non-trivial topologies can be tested with CMB maps. So far, there is no strong evidence for a non-trivial topology.
2 For a flat geometry, the ratio of an object’s angular size to the 360° in a circle is the same as the ratio of the object’s physical size in, say, light-years to the circumference of the circle also in light-years. For a hot spot, this translates to 1.2°/360° being the same as (hot spot size)/(2π distance). After plugging in numbers we find a distance of 47.2 billion light-years. Had we used less approximate numbers, the agreement with 46 billion light-years would be even better.
3 Some advocate that the temperature of the CMB, 2.725 K, should be included in the list of parameters for a total of seven.
4 Although the matter fraction of the total cosmic density changes with time, the ratio of atomic matter to dark matter was fixed well before decoupling.
5 While our analogy is a reasonable representation, in practice ns applies to the full three-dimensional primordial power spectrum and not the two-dimensional version of it that characterizes the CMB anisotropy. Also, there is a subtle convention, beyond our scope, that specifies when the loudness of a given pitch is defined. Last, for the experts, this use of the term “white noise” refers to the CMB power spectrum as plotted, as opposed to a constant Cℓ.