CHAPTER TWO
There are three major components to the universe: radiation, matter, and dark energy. We think of each as a density—that is, as an energy or mass per volume. As mentioned earlier, by using E = mc2, we can convert a mass to an energy, or vice versa, and treat the three components on the same footing. We can then say that the energy density of the cosmos, averaged over a large volume, is made up of x% radiation, y% matter, and z% dark energy. Let’s briefly review these three terms before getting into the details of the x, y, and z.
The universe is filled with radiation in the form of thermal energy. This is the CMB. As we shall see, the CMB really is a fossil of the infant universe, but it is a fossil in primordial light as opposed to something more tangible. Like a dinosaur footprint, it is not so important for understanding the current state of the universe but it is essential for telling us how we got to where we are.
The matter component is divided into two subcomponents, atoms and dark matter. When we peer deep into the night sky, with say the Hubble Space Telescope, we see galaxies because the atoms in them emit light. Not only are the atoms we see a small fraction of all the atoms, but the atoms taken together are just 17% of all the mass, and furthermore all the mass accounts for just 30% of the total energy density. When we observe an image of galaxies, it is as though we are flying over land at night and trying to figure out what’s beneath us—mountains, forests, deserts, lakes—by looking at the distribution of house lights. The house lights are like galaxies and Earth’s surface is like the universe. In areas near cities you can tell what’s below you, but for most of the flight you need more than just a snapshot of the lights. By observing the universe in different ways, we can get that additional information to determine the cosmic composition.
The third major component is the dark energy. In contrast to the CMB, it is important for understanding the current state of the universe and its future expansion, but was insignificant in the early universe. It is the component we understand least. We’ve only known of its existence since the late1990s and are still trying to connect it to the rest of physics.
At different epochs in our cosmic history, one of these three forms of energy density dominates over the others. For the first roughly 50,000 years of cosmic history, radiation in the form of the CMB was the dominant form of energy. Then, for the next ten billion years, matter dominated; or, to put it in the same terms as the CMB, its equivalent energy dominated. And, most recently, for the past 3.8 billion years, dark energy has come to dominate. We now examine these three components in more detail to see how they interact over time to produce cosmic structure.
2.1 The Cosmic Microwave Background
The primary characteristic of the CMB is its temperature, 2.725 K. In this section we interpret what this means. Another aspect of the CMB is its small temperature differences from place to place in the universe, or, as viewed by us, from position to position on the night sky. A third aspect is its polarization. We will address temperature differences and polarization in later sections.
The fact that we can characterize the CMB as a temperature is a profound statement on its own. The CMB is thermal radiation, or radiant energy, of a very particular form called “blackbody radiation.” Things that emit blackbody radiation are called blackbodies.
To get a sense for thermal emission, let’s consider a simple comparison. A black piece of paper left in the Sun gets hotter than a white one, which gets hotter still than a perfect mirror. The black piece of paper absorbs the radiation that lands on it, the white piece of paper absorbs some of the radiation but scatters most of it away, and the perfect mirror reflects all the radiation that lands on it and doesn’t absorb any.1 From the laws of thermodynamics, we can deduce that a good absorber of radiation is also a good emitter. So, if you put your hand over, not on, the black piece of paper that has been exposed to the Sun, you will feel that it radiates more energy than the white one or a mirror. Even better examples of blackbody radiators are the Sun or a pottery kiln.
Objects emit their thermal energy over a range, or spectrum, of wavelengths. However, even for blackbodies, most of that energy comes out over a limited portion of the spectrum; in other words, blackbodies emit predominantly over a relatively small span of wavelengths. For the Sun, almost half of the energy comes out at wavelengths between 0.4 and 0.8 microns. It is no coincidence that this is the visible spectrum we detect with our eyes, which likely evolved to take advantage of the Sun’s spectrum. We know the Sun also emits UV radiation. For example, “UVB,” the primary source of sunburn, is at 0.3 microns, but we cannot see that radiation. The Sun also emits in the “near-infrared” region, but we can’t see that light either.
The lower the temperature of an object, the longer the predominant wavelength of emission. This is known as the Wien displacement law, which says that the dominant wavelength of emission of a blackbody in microns is roughly 3000 divided by its temperature in kelvin. For example, the Sun with its temperature of about 6000 K (see note 1 in the preface) emits predominantly at a wavelength of 3000/6000 or 0.5 microns. The Milky Way is about two hundred times colder than the Sun, or 30 K, so it emits predominantly at a wavelength that is two hundred times longer. Using our simple law, we find the wavelength is 3000/30 or 100 microns. This is the wavelength for the far-infrared radiation that DIRBE measured, as shown in plate 3. Although the Wien law applies to a single wavelength, we should really think of most of the radiation as coming from a range of wavelengths around the predominant one. With this simple relationship, we can link the temperature of an object to the wavelength of light it emits.
We can also think of thermal emitters such as the Sun in terms of atomic processes. The hotter an object is, the more the atomic constituents jostle around and emit light. The more they jostle, the more energy they emit; the more energy they emit, the shorter the predominant wavelength. The difference between a blackbody emitter and an ensemble of energetic atoms is that to get blackbody radiation you need a lot of atoms each absorbing the radiation of its neighbors and then re-radiating it. We might imagine the radiation as beach balls and the atoms as a special group of beachgoers who always play with beach balls. On a hot day there would be so many beachgoers that you’d need smaller balls just to play. From a distance you’d see a swarm of small beach balls energetically passed around and in the air. This corresponds to short-wavelength high-temperature radiation. On a cold day, there would be fewer people out so they could use larger balls, and we can imagine folks being less excited about playing the game, so there would be fewer balls in the air. This corresponds to long-wavelength low-temperature radiation.
It is a deep aspect of blackbody radiation that all you have to do is specify its temperature and you know how much energy is radiated at all wavelengths. That is, the temperature describes the entire spectrum, not just the predominant wavelength. By definition, objects that emit blackbody radiation are in thermal equilibrium with that radiation. In other words, the temperature of the radiation corresponds to the temperature of the object. Say you embedded a thermometer in the walls of a kiln well away from the radiation. The temperature that you would ascribe to the radiation in the kiln by measuring the amount of energy at each wavelength would be the same as the kiln’s wall temperature as read by your thermometer. Using our earlier analogy, you can tell how many beachgoers there are, and the beachgoers’ temperature, simply by looking at the beach balls in the air.
In 1900, Max Planck derived the celebrated formula that describes blackbody radiation. The CMB has by now been measured at many wavelengths, and to the limits of measurement it follows the Planck formula. So, we know that it came from an era when the matter in the universe was in thermal equilibrium with the radiation. Figure 2.1 shows the measurement of the CMB spectrum from the COBE satellite and other instruments. Many have tried to explain the spectrum with different sources of radiation in order to find alternative explanations to the Big Bang. One proposal was that the CMB was emission by distant clouds of cool dust. Such attempts have not succeeded because the predicted spectrum from alternative sources of radiation does not match the observations. Nevertheless, searching for departures from a Planck spectrum is important. A departure can tell us, for example, if there was an injection of energy into the universe, say from a decaying particle, from some earlier epoch.
FIGURE 2.1. The spectrum of the CMB. The x-axis shows the wavelength and the y-axis shows the emitted power. The thin black line shows Planck’s celebrated formula for a 2.725 K blackbody. The continuous gray line shows the measurement from the FIRAS instrument on the COBE satellite. The error bars are smaller than the thickness of the line. Some selected measurements at wavelengths longer than those from FIRAS are shown in gray as well. The agreement between the observations and the blackbody formula is clear.
In an historic step for physics marking the birth of quantum mechanics, Planck hypothesized that electromagnetic radiation was quantized to derive his formula. This means that radiation can be described as discrete packets or quanta of energy. These quanta are called “photons” or “particles of light.” Part of the foundation of quantum physics is that the interaction of radiation and matter may be considered either as involving waves and matter or as involving photons and matter. At times one formulation is easier to work with than the other. Our beachgoers playing with beach balls are analogous to atoms absorbing and emitting photons. For the CMB, there are currently 400 photons in every cubic centimeter of the universe. Once we know the radiation is a blackbody, specifying the photon density is equivalent to specifying the temperature.
We do not know a priori that the universe started off in an incredibly hot state. Leaving the CMB aside, the whole picture we have been developing of the expanding universe could in principle work with a relatively cool early universe. However, because the CMB exists, we know the early universe was hot and in thermal equilibrium. Here is how it works.
When the universe expands, the wavelengths of light are stretched in proportion to the expansion. Imagine you had a slinky. Think of each full turn of your slinky as corresponding to a wavelength of light. Let’s say that the slinky is initially 10 cm long. Now stretch it to 20 cm. The total number of turns is the same but the space occupied by each turn has increased. This is analogous to the stretching of the wavelength of light as the universe expands by a factor of two.
Not only do the wavelengths of the CMB get stretched on their way to us, but the wavelengths of all light from all distant objects get stretched. We see distant objects not only as they were when they were younger, we also see them through stretched wavelengths.
There is another way to think about the stretching wavelengths. A state trooper patrolling the highway might point a Doppler radar gun at your car to see how fast you’re driving. When the radar beam bounces off your car and back to the trooper, its wavelength is slightly shifted, in fact shortened if you are moving toward each other. This is known as the Doppler effect. It occurs because through the reflection, your car in effect becomes a moving radar source; and a moving source emits a different wavelength than one at rest. The trooper can tell how fast you are moving from the difference in the transmitted and received wavelengths. The shift is small but may be computed accurately with the Doppler equation. If instead the trooper receives a signal from a source moving away, its wavelength is stretched. Because red is at the long wavelength end of the visible spectrum, we say the light is redshifted.
Hubble and Lemaître used the redshifted light from recognizable atoms in distant galaxies to determine their speeds. There is an interesting subtlety, though, related to our earlier discussion of how to think about the expanding universe. For very distant objects, much farther away than those known to Hubble and Lemaître, that are moving away at an appreciable fraction of the speed of light, the recessional speed is not described by the Doppler equation or even its relativistic extension. The apparent speed of a distant object, the one that enters in the Hubble-Lemaître law, is from the expansion of space. This phenomenon is called the cosmological redshift. Let’s now apply these concepts to the most distant light we observe, the CMB.
If we go back in time, the wavelengths that comprise the CMB decrease because space is more compact. At its current temperature of 2.725 K, the most prominent wavelength of emission is 3000/2.725 or approximately 1000 microns (0.1 cm as in figure 2.1) as determined by the Wien displacement law. You can show that the Planck formula keeps its same form if the universe is more compact, but to do so the temperature has to increase inversely with its size. Going back in time to when the universe was twice as compact, as it was 8 billion years ago, the temperature of the CMB was double the current temperature, or about 5.2 K: its predominant wavelength was half of what it is today, or 500 microns; there were 3200 photons per cubic centimeter; and the spectrum was still that of a blackbody.
We can keep going back to when the universe was much more compact. At 400,000 years after the Big Bang when the universe was 1,000 times more compact, the CMB was 2725 K, about half as hot as the Sun. It was just about to be energetic enough to rip electrons away from the proton nuclei in hydrogen atoms.
Going back even more, to about three minutes after the Big Bang, when the universe was about a third of a billion times as compact as today and thus at a billion kelvin, the radiation was so intense that the nuclei of helium could just barely hold together. The energy that binds the neutrons and protons together in a helium nucleus is about a million times greater than the energy that binds the electrons to the protons, so the radiation has to be about a million times hotter, with wavelengths a million times shorter, to rip the nuclei apart.
At yet another factor of 3,000 times more compact and hotter, about 25 millionths of a second after the Big Bang, neutrons and protons did not independently exist and the universe was a “quark-gluon plasma.” (Quarks are the elementary particles that make up protons and neutrons.) This state of matter has been reproduced on Earth in the Relativistic Heavy Ion Collider (RHIC) on Long Island, New York. If we go back to a state where the universe was about 50 million billion times as compact as today, about a hundred thousandth of a billionth of a second after the Big Bang, the energy in the photons was roughly the energy of a collision between protons in the Large Hadron Collider in Geneva, Switzerland. These are the highest energy elementary particles produced by humankind so far. Yet, using the universe we can explore even greater energies.
Let us now tie in the idea of a hotter younger universe with the spatial picture we have been developing. If we could travel anywhere in the universe instantaneously right now, the temperature would be 2.725 K everywhere. We can call this the current temperature of the universe. If instead you could have traveled instantaneously around the universe when it was twice as compact, you would have measured the temperature everywhere in space to be 5.2 K. A galaxy at this time would be 8 billion years younger. If we observed this same galaxy today from Earth, we would see it as much younger and the wavelengths would all be a factor of two longer because of the expansion of the universe since the time the galaxy emitted its light.
When we measure the CMB, where does the light come from? Let’s go back to plate 5. The CMB light that lands on our detectors has been traveling to us since just after the Big Bang. It started on its path toward us before there were stars or galaxies and of course before the Earth existed. Back then, it was more energetic and was still described by the Planck function, but with a much, much higher temperature. On its way to us, the universe expanded, the wavelengths stretched, and the radiation cooled. We now see the remnant glow of the Big Bang 13.8 billion years ago in a conceptually similar way to how we see the supernovae light from stars that no longer exist. Unlike the supernovae, the CMB comes to us from all directions.
To summarize, early in our cosmic history, when the CMB was incredibly hot, it was the dominant form of energy density. Being in the universe then was like being inside an unimaginably hot and large pottery kiln. Now, because of the expansion, the CMB is but a cool, dim afterglow with a nearly negligible effect on the current universe. As the universe expanded and the CMB dimmed, matter became the dominant form of energy density, leading to a new set of phenomena. Most important, it allowed structure to form. Before fitting the pieces together, we need to explain a little more about matter.
2.2 Matter and Dark Matter
All matter with which we have direct experience is made up of protons, neutrons, and electrons. These are the building blocks of atoms. The atoms interact with each other both gravitationally and through the exchange of photons, that is through various wavelengths of light. To be sure, there are other fundamental particles and other forces, but mostly they are not part of our daily lives.
Before addressing dark matter, let’s review the known particles and atoms most relevant to cosmology. The simplest atom is hydrogen. It consists of one positively charged proton in the nucleus orbited by a negatively charged electron roughly one ten-thousandth of a micron away. The proton is 2,000 times heavier than the electron. If the hydrogen is in a high-temperature environment, the electron can be ripped off by the energetic photons so that both the proton and electron are free. The hydrogen is then said to be ionized. The next simplest atom is deuterium, which still has only one electron but has a proton and neutron in the nucleus. A neutron has roughly the same mass as a proton but is neutral. Because deuterium has the same number of protons, it is called an isotope of hydrogen, and is often referred to as “heavy hydrogen.”2 Going up in mass, the next familiar atom, and the next one in the periodic table, is helium. It has two protons and two neutrons in the nucleus orbited by two electrons.
All the other elements are formed from these basic elements, as we will describe later. When we peer out into the night sky with telescopes, we see that the light from distant galaxies comes from some of the same atoms we find on Earth; not just the simple ones mentioned in the previous paragraph, but a host of more complex atoms and molecules as well. These distant galaxies are made of the same stuff we are. This simple observation suggests a common origin.
There is one more particularly relevant fundamental particle in cosmology—the neutrino. As its name implies, like the neutron, it is neutral. Neutrinos barely interact with anything. They are products of nuclear decays and nuclear interactions. For example, a free neutron will decay, on average, in just over ten minutes to a proton, an electron, and a neutrino.3 As another example, the fusion reactions that power the Sun (and sustain life on Earth) generate about 100 billion neutrinos that go through each of our fingernails each second. We are sieves to these particles. They go right through the Earth as well.
Because they interact so little, neutrinos are especially difficult to study. We know there are three types, but we don’t know the masses of any of them. All we know is the mass difference between different types. Multiple experiments are being conducted around the world to determine their properties. From the nuclear interactions in the early universe, there should be 300 neutrinos per cubic centimeter throughout the universe today traveling at a few percent the speed of light. This means roughly the same number of neutrinos from the early universe go through each of your fingernails in one second, as do neutrinos from the Sun. Even though there are almost as many primordial neutrinos as CMB photons, they have yet to be detected.
The early universe is simple. From about the time that protons and neutrons emerged from the quark-gluon plasma until the first stars formed some 200 million years after the Big Bang, the most important forms of known matter are the proton, neutron, electron, and neutrino along with their antiparticles. With regard to the matter in the universe, cosmic evolution is determined by how these four particles interact with each other, the CMB photons, and the gravitational attraction of the dark matter in an ever-cooling universe.
Dark Matter
If you looked up in the night sky and saw that, over a period of time, a distant star was following a circular path of, say, two full moons (a degree) in diameter, you would immediately conclude that it was in orbit around another object. For an object to move in a circle there must be a force acting on it.4 In the cosmos, that force is gravity. You might then train your telescope to look for the companion. You know that something has to be applying a gravitational force on the star. There has to be some “missing matter.” It might be a black hole or a dim star that you had not noticed at first.
For many decades, astronomers have been making observations similar in spirit to the one above (although in much more clever ways) with different systems and with less obvious geometries. In the cosmological context, the existence of missing matter was first proposed by Fritz Zwicky in 1933 based on observations of the Coma cluster of galaxies. Others extended his findings. Of particular note were observations of the orbital velocities of stars and star-forming regions as well as of diffuse hydrogen gas in the Andromeda galaxy, an excellent laboratory because, as shown in figure 1.2, it is nearby and looms large. In 1970, Vera Rubin and Kent Ford showed clearly that the velocities of the stars they observed agreed with earlier measurements of the diffuse hydrogen gas velocity. Subsequent models of the orbits of the observed stars and gas in Andromeda showed that there had to be additional matter that was not in either luminous stars or diffuse gas in order to explain the measured velocity profiles. More generally, regardless of the size of the system, from nearby in our galaxy to distant galaxies and groups of galaxies, astronomers found that there is not enough observable matter to account for the motions of stars and galaxies.
The amount of missing matter is not small, nor is its effect subtle. By observation, there is more than about five times as much missing matter as observable matter. The best accounting of it comes from measurements of the spatial variations in the CMB, which we discuss in chapter 3. Here, though, we focus on the characteristics of the missing matter that are independent of the CMB.
The path from not being able to find the missing matter to concluding that there must be a new form of invisible matter, or dark matter, has involved thousands of scientists and multiple lines of evidence. In part the missing matter has been characterized by process of elimination. We know what it is not. We know it cannot be an assemblage of planets, say “Jupiters,” that are just too dim to see. We know it cannot be atomic, such as hydrogen gas, or be the same as the stuff of which we are made. We know that it cannot be black holes of the type that have been observed so far. We know that it cannot be one of the three types of neutrinos, even though there are almost as many neutrinos in the universe as CMB photons.
One assumption is that the missing matter is a new type of elementary particle, but it may just as well be a new family of particles, multiple families, or a combination of different types of particles. Generically we call these possibilities “dark matter.” If dark matter is a particle, we do not know how it interacts with other particles or even, if two dark matter particles collide, with itself. We know that it does not interact significantly with photons, which is why it is called “dark.” Observationally, all we know is that the dark matter interacts gravitationally. Its character is a grand mystery, it is unambiguous that dark matter exists in vast quantities and that it is not a form of matter we have encountered in our laboratories.
One of the clearest astronomical observations that shows the need for dark matter is of the Bullet Cluster, as seen in plate 6. The image actually shows two clusters of galaxies that have collided and passed through each other. The one on the right, as suggested by the pink shape, is the “bullet.” Before they collided, the clusters were full of a moderately uniform mixture of normal matter in the form of a diffuse hot gas and stars in individual galaxies, and dark matter. In both, the amount of mass in the hot gas was much greater than in all the stars that made up the galaxies, and the mass in dark matter was much greater again than the mass in the hot gas in this system. When the clusters collided, the galaxies and dark matter passed through each other almost unscathed, but the gas interacted. Think of it this way. If you filled both hands with pebbles and threw them toward each other with their trajectories crossing a short distance in front of you, most of the pebbles would not collide. The pebbles are like the galaxies and dark matter. If instead you aimed two garden hoses at the same meeting point, the water from each hose would collide and interact. The water is more analogous to the hot gas.
The gas in the clusters is roughly ten million kelvin. It is so hot that it emits X-rays which are observed by NASA’s Chandra X-ray Observatory. In plate 6 the hot gas is pink. In other words, the pink shows us the location of most of the normal matter. The blue shows us the location of primarily the dark matter, but also the galaxies. To understand how we know the matter is there we need to take a small detour to discuss the bending of light.
FIGURE 2.2. An example of bending light in a curved two-dimensional space. The Sun is represented as the ball in the center of the image. It warps space as would a bowling ball on a large rubber sheet. Light’s path by the Sun is analogous to a quickly rolled small marble going past the bowling ball. The marble follows the contour of the two-dimensional rubber sheet and its trajectory is bent toward the bowling ball, away from a straight path. It rolls along the easiest path. Similarly, in three dimensions a light ray going by the Sun follows the easiest path. Its trajectory is bent by the curvature of three-dimensional space or, equivalently, by the force of gravity. In the figure the deflection is greatly exaggerated.
Let’s go back to thinking about space. Not only can it expand but it can also be warped or curved. When light from a distant star travels to us on a path that goes close to the Sun, it is deflected ever so slightly. This can be thought of as the Sun’s gravitational pull on the light. A better way to think about it is that the space around the Sun is curved and that the light from the distant star follows the easiest path on its way to us.5 Figure 2.2 shows one way to visualize this. The bending of light rays by the gravitational field of a large mass is directly analogous to the bending of light by a camera lens. The phenomenon is called gravitational lensing. You can determine the amount of mass from the amount of bending.
We can now understand the blue areas in plate 6. Distant galaxies far behind the Bullet Cluster were observed through the Bullet Cluster. From the distortion of the distant galaxies, the effective lensing and mass distribution in the Bullet Cluster were determined. The image shows that most of the mass is in two distinct regions. The essential feature of the image is that the dark matter is clearly separated from the normal matter. The hot diffuse gas interacted during the collision and got left behind.
The search for dark matter is a very active area of physics. Multiple experiments are trying to detect it directly. Some are trying to identify a direct hit of a dark matter particle with a target atom of germanium, argon, or xenon. These experiments are often constructed deep underground to shield the target atoms from other known particles that can’t easily penetrate the Earth. Other experiments take much different approaches to search for different types of interactions and different forms of dark matter. There have been hints of possible detections and reported detections that have not withstood further scrutiny. As of 2019, no iron-clad direct detections have been made. We hope that dark matter particles will be detected in the Large Hadron Collider.
The discovery of new elementary particles has mostly taken place in particle accelerators that were precursors to the Large Hadron Collider. There is an enormously successful “standard model of particle physics” that has 17 different fundamental elementary particles, including the quarks that make up the protons and neutrons, the electrons and neutrinos, and most recently the Higgs boson. Although comprehensive, predictive, and well tested, we know the standard model of particle physics is not complete because there are measurements of elementary particles that it cannot explain, such as the mass of the neutrino. We hope that the detection and characterization of the dark matter in the lab will show us how to advance our model of particle physics.
Is it possible that there are no dark matter particles and that our laws of physics are incomplete? A lot of research has gone into investigating how general relativity might be wrong on large scales and so, in fact, there is no missing matter and instead a new force accounts for the observations. These new theories generally go under the name MOdified Newtonian Dynamics, or MOND. Fortunately, MOND makes predictions that can be tested, and some of the predictions do not agree with observations. In contrast, there has yet to be an observation in disagreement with general relativity. Therefore the large majority of cosmologists do not agree with MOND. Of course it is quite possible that there are other forces or laws of physics we simply have not yet discovered.
2.3 The Cosmological Constant
Earlier, we gave an approximate value for the current expansion rate of the universe as 15 miles per second for every million light-years’ distance. Put another way, a galaxy 10 million light-years distant appears to be moving away from us at a speed of 150 miles per second. In the late 1990s it was discovered that the expansion rate is increasing. In other words, the expansion is accelerating. In one billion years that same galaxy will move away at 156 miles per second; one billion years ago it was moving away at 144 miles per second.6
This remarkable observation was made by two independent groups, the Supernovae Cosmology Project and the High-Z Supernovae Search Team, and has been confirmed by others. As their names suggest, they used supernovae to look back to when the universe was just a few billion years old. The trick was to find objects for which they could determine accurate distances and speeds and compare the expansion rate then to the current expansion rate.
One way to think about this observation is that space is being made at an accelerated pace. Not only is the concept of “expanding space” everywhere a convenient way of thinking about the expansion of the universe, but we are now almost forced to think about it this way. In a static space, we can imagine that two galaxies could be moving apart at a nearly constant speed, slightly slowing down due to their gravitational attraction, but we can’t come up with a way for them to accelerate away from each other. Acceleration requires a force, and in a static space the only force available is gravity which, if anything, would tend to decelerate the expansion.
So the question is, why is space being made at an accelerated rate? We do not know. What it means is that space, the vacuum, appears to have an energy density associated with it. This energy density acts like a pressure that expands the universe or, more prosaically, “expands space.” The energy density is quantified as a cosmological constant denoted by the Greek letter Lambda, Λ. This is a new constant of Nature that actually may not even be constant.
Einstein introduced Λ in 1917, before Hubble’s observations. He thought the universe was static—that is, not expanding as Hubble’s observations showed. To understand his motivation, imagine two isolated galaxies in the universe. They are attracted to each other by gravity and would fall toward each other. The cosmological constant provides a new kind of pressure that balances out the gravitational attraction between them and holds them in place. By extension, this would apply to a universe full of galaxies. However, after Hubble’s observation, Einstein abandoned Λ. We now know this pressure exists at an even larger value than Einstein thought necessary.
There are other explanations for the accelerated expansion then the cosmological constant. In general they posit some form of “dark energy” that is not constant. These alternatives make predictions for the acceleration versus the age of the universe. Measurements are in progress to test these predictions. We do not know if the dark energy is, say, a substance, or if it is constant throughout space. Perhaps we are missing some fundamental element of one of our theories. At the moment, the most straightforward explanation that agrees with all the data is that space is described by a cosmological constant that is constant in space and over time. So, let’s adopt this point of view.
The mere existence of the cosmological constant is deep. It is not part of any fundamental theory in physics. It has no bearing on life or physics on Earth. No laboratory experiments have been developed that can measure it. It is a constant that allows us to quantify the cosmic acceleration and in so doing tells us that there is an energy density or pressure associated with space.
What does this mean for the future? We will set aside the cautionary words in the introduction and extrapolate. If you are in your car on the highway and accelerate at a constant rate, you of course go faster and faster. The case is similar for the universe, but more extreme than a constant highway acceleration. In the universe, the space between galaxies is growing exponentially. Galaxies that are widely separated now will soon apparently be moving apart faster than the speed of light. Here is an example where some of the analogies we sometimes use to describe the expanding universe break down. It is not physically possible for a rubber sheet to expand in this way. There are simply no material objects that can act like the expanding universe.
There is no contradiction between the accelerating expansion and the special theory of relativity, which requires only that information and massive particles cannot be transmitted from one place to another faster than the speed of light. For the galaxies, the space in their cosmic neighborhoods is just expanding at an exponentially increasing rate. No information is being transmitted faster than light. From the point of view of someone in the Milky Way, distant galaxies that we can currently observe will simply fade away in the future. We do not know how long the exponential expansion will last.
Let’s take account of where we are. We now know what the universe is made of. That’s a big step. We still have to explain how we know the cosmic constituents so precisely but we will get to that in chapter 4. For now we can use a pie chart to summarize what we know. Today, the slice for atoms is 5%, dark matter, 25%, and the cosmological constant, 70%. The slice for radiation is less than 0.01% and not that significant. The slices of the pie chart change as the universe evolves. Early on, radiation dominated and the other components were insignificant. Next, matter dominated. Now, in the current epoch, the cosmological constant dominates. In the future, the cosmological constant will increasingly dominate and the atoms plus dark matter will be comparatively insignificant.
We can relate these fractions to the average energy density of the universe. The whole pie corresponds to an effective mass density of about five and a half protons (or the equivalent energy computed with E = mc2) per cubic meter of space. We can think of one and a half of those protons as representing all the matter (the dark matter plus the familiar matter) and the other four representing the cosmological constant. Of course there is no such thing as half a proton, but for us it just represents an amount of mass. Let’s put imaginary walls around that cubic meter and let the universe expand by a factor two. The volume inside our walls increases by a factor of eight. The mass inside the walls remains the same, so its average density drops to about a fifth of a proton per cubic meter. What happens to the effective mass density that represents the cosmological constant? It stays the same at four effective protons per cubic meter! It really is like the vacuum has an energy density associated with it, which is why we count it as a component in the pie chart. We can also see why over time the cosmic pie chart becomes dominated by the cosmological constant. With a factor of two expansion over the present, the pie will have 5% for all matter and 95% for the cosmological constant.
Once we know the contents of the universe and the Hubble constant today we can determine the compactness (and temperature) of the universe throughout cosmic history. The result comes from a solution to the “Friedmann equation” which Alexander Friedmann derived from the general theory of relativity in 1922. The inputs to the equation are the matter density, radiation density, and the energy density associated with the cosmological constant; from these the equation gives the Hubble parameter versus cosmic time with today’s value as a reference point. Figure 1.5 shows the solution to the Friedmann equation for a galaxy that is 110 million light-years away today. You can see that the recessional speed of the galaxy was faster 2 billion years after the Big Bang, then slowed down nearly 6 billion years due to the gravitational attraction to other matter, and is now speeding up due to the effect of the cosmological constant.
Being able to get this solution is already quite an accomplishment. However, our understanding of the universe is much more comprehensive and the cosmological model is more far-reaching than simply knowing the compactness for all times. For example, we can also understand why the universe looks the way it does, as we hope to show in the following.
2.4 Structure Formation and the Cosmic Time Line
We now combine our framework of an expanding universe from chapter 1 with our knowledge of the major components. Our goal in this section is to develop a picture for how “structure” forms. By structure we mean objects that are held together by gravity. There is a magnificent and varied array of different types of objects that range in size from stars, to galaxies, to clusters of galaxies. As can be inferred from figure 1.2 and plate 4, the space between the objects is vast and cold. In contrast, the early universe is a near uniform primordial soup of hot thermal radiation (CMB photons), electrons, protons, neutrons, neutrinos, and dark matter. How did the universe get from one state to the other? In other words, how did structure form and grow? Although we do not know the details of, say, how a galaxy forms, cosmologists have a framework that explains how there can be such a wide array of structure over such a large range of mass and how the formation process got started. Keeping in mind that this is an active area of research, we just touch on the well-established key elements of the process as it pertains to the CMB and the standard model of cosmology.
We pick up the story five minutes after the Big Bang. At this time the temperature of the universe was a little under a billion kelvin, the expansion rate was 3 million times what it is today, and the cosmological constant was irrelevant because its energy density was so much less than that of everything else. The properties of matter at this temperature, roughly 70 times hotter than the center of the Sun, are well understood. The atomic nuclear composition of the universe was roughly 75% hydrogen and 25% helium by mass. These percentages are nearly the same as they are today. This ratio was already set by nuclear reaction rates between the protons, neutrons, and neutrinos during the first three minutes, a topic to which we return later.
Initially, these primordial nuclei were in a gas with the electrons and photons, but the temperature was too hot for neutral atoms to form. Such a gas is called a “plasma,” sometimes termed the fourth state of matter after solid, liquid, and gas. In the cosmological plasma, for every electron there were almost 2 billion CMB photons, and for every proton there was a little more than five times the mass in dark matter particles. Other than their participation in nuclear reactions, the neutrinos were not directly part of the formation of structure at this epoch because they barely interact with the other matter and are moving so quickly.
With the ingredients and the state of the universe in hand we now turn to the physical process. Let’s put aside for a moment our concept of an expanding universe. Imagine an infinitely long one-dimensional string of equally spaced and stationary objects all of the same mass. These masses are attracted to each other gravitationally. Let’s assume that gravity is the only force on them. This configuration is not stable because gravity is only attractive. Pick any mass and displace it ever so slightly to the right. Now, it is closer to its right-hand neighbor than its left-hand neighbor. Because the gravitational force is inversely proportional to the separation squared, the attraction to the right is even stronger than the initial attraction to the left: the mass and its right-hand neighbor fall toward each other. Once the spacing is changed anywhere, the whole string becomes unstable and the masses start to clump together.
The physical process behind the formation of cosmic structure is gravitational instability. We need something to get the process going, a “seed,” but once it gets going, a once-uniform gas of dark matter and plasma can form structure. Of course, the one-dimensional string of masses is overly simplistic. In the full picture we have to consider all the constituents in a rapidly expanding universe. Now let’s go through the process. We will put off discussing the source of the seeds until section 4.2.
There are multiple processes that take place at the same time. In one process, the seeds of structure formation initiate the clumping of the dark matter, but for the first 50,000 years the universe is expanding too rapidly for structure to form. In our one-dimensional analogy the masses start to fall together but the universe is expanding too quickly for them to clump. While this is happening, in a different process, the electrons and the CMB photons are strongly interacting, constantly scattering off each other. This is somewhat akin to being in a thick fog in which the light scatters off the water vapor so that every direction looks the same and you can’t see very far. And, in a third process, the negatively charged electrons attract the positively charged protons (the hydrogen nuclei and in the helium nuclei) simply because opposite charges attract. The picture to have in mind is that the CMB acts most effectively with the electrons, because they are so much less massive than the protons, and the electrons pull on the protons but they can never combine to form atoms because it is too hot: the atoms would be instantly ionized. The combined interactions in the second two processes are much stronger than the force of gravity and so, even if the universe were not expanding so rapidly, the plasma would not clump. The electrons and thus protons are kept from clumping by the intense interaction with the radiation. Again, all three processes—clumping, scattering, and electronic attraction—are taking place at the same time.
As the universe expands, the radiation cools and the rate of expansion decreases. Soon after 50,000 years, when the matter comes to dominate the energy density, the expansion rate is slow enough for the dark matter to start clumping but it is still too hot for the plasma to clump. The interactions between the CMB and the electrons overwhelm the gravitational forces.
After 400,000 years the universe cools to the point where hydrogen atoms can form. In a relatively short time, the electrons bind to protons. While an electron is free, it can interact with radiation of all wavelengths, but once it is bound its interactions are restricted because it has to obey the rules of atomic physics. Shortly after the binding occurs, the electrons no longer scatter the CMB; the second and third processes mentioned above cease. Without the photon scattering, the hydrogen can begin to clump. (The helium undergoes a similar process, but slightly earlier.) Agglomerations of mass already exist because clumps of dark matter have been forming since the universe was 50,000 years old. The atoms fall into the dark matter structure.
The contrast between the collections of dark matter in different regions is very small. One region may be more massive than another by a few parts in 100,000 which is like the tip of your little finger compared to your entire mass. That’s all it takes to start the clumping of atoms.
The time at which hydrogen atoms form is called “decoupling” because the CMB photons decouple from, or stop interacting with, the electrons which have become bound up in atoms. The photons are then free to roam the universe. It is as though the fog has lifted and light from a distant shore can now reach you. To a reasonable approximation, the photons that land on our detectors were last scattered in this process and have since traversed the radius of the observable universe to get to us. Thus they bring to us a picture of the universe from 13.8 billion years (minus 400,000 years) ago, very much like the light from a distant galaxy brings us an image of the galaxy in its youth. The main difference is that the CMB comes to us from a time before there were stars and galaxies, from a time when matter was just beginning to form structure. This is why the CMB is sometimes called the “baby picture of the universe.”
After decoupling the universe is neutral and enters a period somewhat playfully called the “dark ages” (see plate 5) because there were no stars to shine and the CMB had been cooled enough by the expansion that it did not emit visible radiation. During this time, the atoms continued their fall into concentrations of dark matter. The clumping triggered by the gravitational instability took place on all scales, from stellar sizes to huge filaments containing countless protogalaxies. The first objects to form, though, were stars. They lit up the universe, ending the dark ages.
Star formation took place about 200 million years after the Big Bang. The first generation was made of hydrogen and helium, but in their cores they produced heavier elements such as carbon, nitrogen, and oxygen through the process of nuclear fusion. These stars aged and exploded in supernovae, spewing the heavy elements throughout the universe. We are made of these heavier elements. There are ongoing searches to identify remnants and signatures of these stars; some even may have become black holes. Nevertheless, we know they have to exist because we see their ashes. More recently formed stars, such as the Sun,7 contain elements in their surfaces heavier than helium, and these elements could not have been formed prior to the first generation of stars in the quantities we now observe. As Joni Mitchell sang in 1969, “We are stardust, billion-year-old carbon.” Our knowledge of the universe has increased enormously since these lyrics were written, but “We are stardust, 13.6 billion-year-old carbon” doesn’t have quite the same ring.
We also know that the first stars produced enough energy to tear the electrons from the hydrogen nuclei (the protons) by bombarding them with energetic photons. Thus, the universe began as an ionized plasma with no structure, became a neutral gas of hydrogen and helium after decoupling, and was then reionized by the first stars predominantly at an age between 500 million and one billion years. By this point though, the universe had expanded enough, and the CMB was cool enough, that structure could continue to form. Nevertheless, the reionization left its mark, the newly freed electrons scattered roughly 5–8% of the CMB photons, an effect that is observed in the CMB. As with the formation of the first stars, the process of reionization is complicated and not yet well understood. It’s an active area of investigation. Regardless, we know the process took place because we observe that intergalactic space is still ionized today.
As the universe ages, new stars come on the scene, galaxies begin to form, and clusters of galaxies begin to grow. The largest structures are still forming today. Although we have laid out the process sequentially, structure formation on all scales takes place to varying degrees at the same time.
The scenario of structure formation and the cosmic time line might, at first glance, seem a bit contrived. It is awfully detailed. However, the physics is straightforward and well tested. The model is predictive and there are multiple ongoing efforts to check those predictions. The picture we painted is rooted in measurements. Telescopes of all kinds are mapping out the process by looking at different structures from different epochs. If gravity acts differently than we think, if the cosmological constant isn’t constant, if we don’t have the correct ratio between protons and dark matter, if a new process or particle comes into play, or if neutrinos play a large role in structure formation, we can see the effects through ongoing and detailed measurements of how cosmic structure grows over time. One of the reasons for confidence in the scenario is that we know how the process started. That’s one of the things we learn from the CMB anisotropy, our next topic.
1 It is difficult to make a perfect mirror. Aluminum is an obvious candidate, but it is a good absorber of ultraviolet radiation and so gets hot in the Sun. If our eyes could see ultraviolet radiation, which they cannot, an aluminum mirror would look dark.
2 When the hydrogen in water is replaced by deuterium, you get “heavy water,” which is used in some nuclear power plants.
3 More specifically, the neutron decays to a proton, electron, and an electron antineutrino. To the limits of measurement, free protons do not decay.
4 This follows from Newton’s second law.
5 In general relativity, the description of the space around a massive object is mathematically different from the description of the geometry of the universe. For light moving by an object, the effect of gravity on the passage of time is also a significant effect.
6 An accelerating universe corresponds to the Hubble parameter approaching a constant value. This is because the Hubble parameter is the expansion rate divided by the scale factor (as given in appendix A.3).
7 The Sun is 4.6 billion years old and is expected to stay in its current form for roughly another 5 billion years. It is not massive enough to produce a supernovae.