CHAPTER THREE
We can give the detailed accounting we have—the cosmic energy densities versus time, the ratio of hydrogen to helium, the epochs for different processes—because these quantities affect the CMB in characteristic and measurable ways. To understand how we can learn so much, we now focus on the small temperature differences from position to position on the night sky. The variation of temperature with position is called the temperature anisotropy. The word “isotropic” means “having a physical property that has the same value when measured in different directions.” Anisotropic means not isotropic. The CMB is not isotropic, but the difference in temperature for different directions in the sky is tiny, typically one ten-thousandth of a kelvin or 0.003%.
The CMB anisotropy has been measured with exquisite precision over the entire sky by the WMAP and Planck satellites. The maps are usually shown in a Mollweide projection, which simply specifies the manner in which you represent something that is intrinsically a spherical shell, like the Earth’s surface, on a flat piece of paper. Figure 3.1 shows the Earth in a Mollweide projection. The equator runs horizontally along the middle of the map, the North Pole is on top, and the South Pole is on the bottom.
FIGURE 3.1. Left: A Mollweide projection map of the Earth’s surface. Credit: Daniel R. Strebe August, 15, 2011. Right: A Mollweide projection map of the CMB dipole as measured by COBE/DMR at a wavelength of 0.6 cm. Relative to our cosmic reference frame, the solar system is traveling away from the lower-left dark region and toward the upper-right lighter region at 0.1% the speed of light. Parts of the Milky Way’s galactic plane are just visible at this temperature range. For example, the circular feature in the center left is the Cygnus region seen in plate 3. While both maps use the Mollweide projection, the relative orientation on the sky of the Earth’s equator is tilted about 50° relative to the galactic plane, as can be seen in figure 1.1. Credit: NASA/COBE Science Team.
Plate 7 shows maps of the CMB anisotropy from both WMAP and Planck. Whereas the left side of figure 3.1 is made looking down on Earth from space, the images in plate 7 are made looking up into the sky. These maps are oriented so that their equators are aligned with the Milky Way.1 The center of the map corresponds to the center of the galaxy; the top is the “north galactic pole” and the bottom is the “south galactic pole.” The map from WMAP was made at a wavelength of 0.5 cm (5000 microns). The one from Planck was made at a wavelength of 0.2 cm. Both satellites made maps at multiple wavelengths; these are just representative images. In general, the Planck maps have higher precision than WMAPs but the similarity in the maps once you move away from the galactic equator is striking.
Neither WMAP nor Planck measures the absolute temperature of the CMB. If they did, the maps would be one solid color corresponding to the temperature of the CMB. Instead, they measure only deviations from the average temperature of 2.725 K. The largest spatial variation is called the CMB dipole, as shown in figure 3.1, and it too is subtracted before producing the images in plate 7. The amplitude of the dipole is 3350 millionths of a kelvin so, if shown, it would saturate the color scale. The dipole arrises because the satellites have a net velocity relative to the CMB. As you might imagine from the Doppler effect, if you are moving toward a blackbody, it appears slightly hotter, and if you are moving away, it appears slightly colder. Thus, by looking around, you can tell if you are stationary relative to the blackbody.
The existence of the dipole gives us another insight into the universe. It means that there is a universal cosmic reference frame. This does not violate any laws of physics because we are simply defining a reference frame. It is the same frame relative to which galaxies are on average at rest after subtracting their motion from the cosmic expansion. Most galaxies have some peculiar velocity relative to this frame, and the Milky Way is no exception. Our net velocity relative to the cosmic reference frame is about 0.1% the speed of light. We are moving quite fast with respect to the rest of the universe. The dominant components that make up this velocity are the motion of the Earth around the Sun (0.01% the speed of light), the motion of the Sun around the center of the Milky Way (0.08% the speed of light), the motion of the Milky Way in the Local Group, and the motion of the Local Group relative to the rest of the galaxies. Their velocities are in different directions so you have to be careful when combining them.
The component from the Earth orbiting the Sun is especially important for CMB measurements. It’s called the orbital dipole and allows us to calibrate the instruments to high accuracy. Because the Earth’s velocity can be measured accurately independent of the CMB, the amplitude of the orbital dipole can be predicted to similar accuracy—about 270 millionths of a kelvin. Over the course of a year, this component of the overall dipole is measured by the satellites. It is satisfying to think of calibrating variations in light from the edge of the universe with the motion of the Earth around the Sun.
In plate 7 the color bar shows the magnitude of the deviations from the average. Let’s consider the regions above and below the dashed lines in the top image. Some parts are hotter than the average—the reddest places are about 2.7253 or more above absolute zero—and some are colder—the bluest places are about 2.7247 or less above absolute zero. We say “or more or less” because those colors are at the ends of the color bar. The broad hot stripe down the equator of each map is the emission from the Milky Way at these wavelengths. The map in plate 1 has had a model of this “galactic emission” subtracted. We show the full picture here so that the next time you look at the Milky Way, you can think about it at longer wavelengths and picture its relation to the CMB anisotropy. For understanding cosmology, though, the emission from the Milky Way and other galaxies is a contaminant.
3.1 Measuring the CMB
Before digging into the details of how we make sense of the maps in plate 7, let’s go over how the measurements are actually made. The CMB was discovered by Arno Penzias and Robert Wilson in 1965. They were working at Bell Lab’s Crawford Hill Laboratory, in Holmdel, New Jersey, on a telescope designed to receive signals from a communications satellite. With their state-of-the-art and well-calibrated receiver they detected an unexpected signal: that the whole sky was glowing at roughly 3.5 K. The primary reason for believing that the signal was cosmic was that it was the same in all directions.2 Since then, the CMB has been measured with many different methods. The promise of what we might learn has driven the development of multiple new technologies and impressive instruments for measuring both its absolute temperature and the anisotropy. Here we will focus primarily on measuring the anisotropy because it’s the signal from which we learn the most about the universe.
In the late 1960s, the skies were scanned with single room-temperature detectors to look for temperature differences of roughly a thousandth of a kelvin. We have now advanced to instruments with thousands of detectors cooled to just a tenth of a degree above absolute zero that run around the clock and measure CMB temperature differences at the level of a millionth of a kelvin or better. The experimental challenge is to measure these minute differences from position to position on the sky while observing with an instrument in a 300K environment, almost a billion times hotter than the signal. The steady advances in techniques and technologies that enable us to do this have been profound and sustained.
The CMB shines over a broad range of wavelengths, although it is strongest near 0.1 cm as shown in figure 2.1. Between the wavelengths of 30 cm and 0.05 cm it is brighter than anything else in the sky if you look away from the galactic plane and observe from above the Earth’s atmosphere. Especially at wavelengths shorter than 0.3 cm, water vapor in the atmosphere can make the measurements from low-elevation sites difficult if not impossible. This has driven researchers to take their instruments to high and dry locations such as White Mountain in California, the Chilean Andes, and the South Pole, or to fly them on balloons. However, the ultimate platform for measuring the CMB is a satellite.
The first satellite3 dedicated to measuring the CMB and the infrared emission was NASA’s COBE, which we discussed in section 1.1. After that came the Wilkinson Microwave Anisotropy Probe (WMAP), and most recently, Planck. We will focus on these last two, as shown in figure 3.2, as they have given us the best and most complete picture of the CMB anisotropy, and have done so in quite different ways.
Penzias and Wilson measured the CMB at a wavelength of 7.4 cm. This is in the “microwave” band and hence the name cosmic microwave background stuck, even though most of the emission is at shorter wavelengths. Because 7.4 cm is a relatively long wavelength it is not so affected by the atmosphere. Other common devices in the microwave band include TV stations (channels 2-83, with wavelengths 500 cm to 34 cm) and microwave ovens (12.2 cm). Modern TV satellite dishes operate near a wavelength of 1 cm. You can see in figure 3.2 that WMAP looks like it has two back-to-back TV satellite dishes. This is no coincidence. It operates between the wavelengths of 1.3 cm and 0.3 cm in five different bands.
FIGURE 3.2. The WMAP satellite is on the left and the Planck satellite is on the right. For scale, the large reflector or “dish” on WMAP is 140 cm by 160 cm. For Planck it is 150 cm by 190 cm. The satellites are comparable in size with overall heights about 300 cm. For WMAP the sun is in the direction of the bottom of the page, and for Planck the Sun is off to the lower right. The thermal shielding allows WMAP’s primary reflector to cool to 60 K and Planck’s to below 40 K. For both satellites, the detecting elements are just below the primary reflectors. For WMAP, the detectors are passively cooled to 90 K by removing the heat from the detectors with the large radiators. Planck uses an active system to cool the longer wavelength set of detectors to 20 K and the shorter wavelength ones to 0.1 K. Credit: ESA and the Planck Collaboration; NASA/WMAP Science Team.
To get a better sense for how the measurement is done, we can think about an old-fashioned TV. Let’s say you have the type of antenna that attaches directly to your TV. If you then tune to channel 83 and there is no broadcast there, you will just get fuzz or noise on your TV screen. This fuzz comes from two sources. It is a combination of microwaves coming into the TV from the environment through the antenna plus noise from the electronics inside the TV. Consider the antenna component of the fuzz. The incident microwaves move the electrons in the antenna structure. In turn, those electrons tickle the input of the transistors in the TV receiver and the rest of the TV receiver amplifies and packages the signal so you can visualize it. The CMB enters the antenna just like a broadcast signal would, but it looks like noise. To a crude approximation, about 1% of the total noise on the TV screen is from the CMB that enters through the antenna.
To measure the anisotropy you would point the TV antenna in a specific direction and record the amount of fuzz, say, by taking a picture of it or recording its hiss. Without changing anything on the TV, you’d then point the antenna to a new location and again record the amount of fuzz. The difference in the amount of fuzz directly corresponds to the difference in temperature of the radiation coming into the antenna.
It is easy to imagine how to improve the measurement. You’d definitely want to get a less noisy TV receiver so that more of the noise came from the CMB. It is somewhat counterintuitive but it is not necessary to make your instrument colder than the CMB to measure it. The key is that the electrons in the detecting elements are free to respond to the CMB. If the transistors are cooled to, say, 100 K, the electrons are freer and the electrical noise from the transistors is reduced. You could increase the signal by listening in multiple channels at once. You’d want to make sure that your antenna only accepted TV waves from directions close to where you were pointing it. You’d want to make a TV that worked at shorter wavelengths, etc. In effect WMAP does all of this, although with very fancy transistors and a lot of attention to detail.
WMAP’s other key feature is that it accepts radiation from two different directions simultaneously and compares them. This is why there are back-to-back dishes. The instrument is not capable of measuring the absolute temperature, only a large collection of temperature differences. Then, a computer program takes all those difference measurements and combines them to produce a map of just the spatial variations as shown in plate 7.
The Planck satellite, shown in figure 3.2, takes a different approach. It has just one primary receiving dish. The idea is to spin the satellite, find the average temperature over the spin, and look at temperature differences around that. In total there are 72 independent detection channels measuring the sky at any time, as opposed to WMAP’s 20, but in both there is quite a bit of redundancy. Planck has two instruments that in combination measure between the wavelengths of 1 cm and 0.035 cm in nine different bands. The lower three bands are similar to WMAP’s, but the upper six are different and use a different technology called bolometry.
The bolometer, a fantastically sensitive device, is a lot like a thermometer. It simply measures the amount of thermal energy dumped onto it. Unlike transistors, bolometers do need to be fairly cold to detect the CMB. The key to using them is to isolate them so well that only the radiation you intend gets to them. The ones on Planck are cooled to 0.1 K above absolute zero. In one second they can measure a temperature difference smaller than one ten thousandth of a kelvin.
Both satellites observed from “the second Lagrange point” or L2. This is a location in the solar system that is about a million miles away from the Earth in the direction opposite the Sun. In 1772, Joseph-Louis Lagrange found there were five places in the solar system where the gravitational pull of the Earth and Sun balanced in just the right way to make an orbit possible. The L2 position is unstable, so miniature jets on the satellite are fired ocasionally to keep it from wandering too far away. Unlike most satellites, WMAP and Planck orbit the Sun and not the Earth.
From L2, the dishes of both satellites generally look away from the Sun, Earth, and Moon. This is important because these bodies are hot compared to the tiny temperature we need to measure. Another important feature of L2 is that it is thermally stable. There are no day/night cycles. This stability is important for being able to measure the sky over and over and then averaging the data together. WMAP observed for nine years; Planck for four years.
Although the function of the satellites is straightforward—they simply measure the radiation temperature of the sky—getting them to work to the limits they have achieved has required unprecedented control of systematic sources of error. The devil is in the details. For example, you have to be sure that a measurement made on one day can be directly compared to a measurement made two years later at a level set by the fundamental noise characteristics of the instrument. By far the most computationally intensive part of the data analysis is checking that you understand the instrument and the environmental impacts on it.
One of the recurring questions when measuring the CMB is: “how do you know that you’re really looking to the farthest reaches of light and not just at something in the Milky Way or in the Local Group?” The primary way to determine this is by measuring the anisotropy at different wavelengths. Just as Planck’s equation precisely describes the amount of power per wavelength band for a blackbody, it prescribes a similar relation for the fluctuations. Emission from the galaxy has a much different pattern of emitted power versus wavelength so it can be distinguished from the CMB. Both Planck and WMAP have multiple wavelength bands that make it possible to clearly separate the “foreground emission” from the cosmic emission.
A straightforward way of separating galactic emission from the CMB, sufficient for our purposes, is to simply “mask out” the galaxy. This means that you remove, or mask, that part of the map from any analysis. In plate 7 you can do that in your mind’s eye by excluding regions between latitudes plus and minus 20° as indicated by the dashed lines on the top map. Northward and southward of this region, the maps show a seemingly random array of hot and cold regions of various irregular shapes and sizes. This is the CMB anisotropy signal we are after.
When looking at the Hubble Ultra Deep Field in plate 4, you might wonder why CMB experiments don’t see all of those galaxies. There are three reasons for this: those galaxies emit radiation at different wavelengths, there is a lot of space between the galaxies (most of the image is black), and the galaxies are small in angular extent. When we measure the sky with CMB telescopes with high angular resolution, we can see galaxies and clusters of galaxies and thereby determine that their contribution to the maps in plate 7 is practically negligible.
We show two different maps in plate 7 to make an important point. The anisotropy is measured with high precision and is the same as determined with two completely independent satellites with different detectors, observing strategies, and scientists. The data were processed by independent and competing groups, yet, ultimately, the two maps show the same thing. The results are confirmed.
Measuring the CMB is now “big science.” Up until the 1990s, groups of two or three researchers could make a breakthrough measurement with inexpensive equipment. Now there are thousands of people in the field and the instruments cost many millions of dollars, or hundreds of millions for a satellite. The satellite maps will remain the most accurate full-sky measurements, but in select regions of sky and at certain angular scales you can improve on them significantly. A network of ground based telescopes is already in the process of mapping the CMB over half the sky with even greater precision than Planck or WMAP.
3.2 The CMB Anisotropy
Let’s say we have a map from either WMAP or Planck that has been masked or cleaned such that we’re confident the remaining signal is the CMB anisotropy. It’s just a collection of temperatures, a heat map. How do we learn cosmology from it? First, recall that these maps provide a picture of the edge of the observable universe from 400,000 years after the Big Bang. Although the universe went through decoupling throughout its entire volume at this time, we can think of this radiation as coming to us from a surrounding surface because that is the region from which the CMB we now measure originated. The region is sometimes called the “decoupling surface” to remind us that the light we detect decoupled from the primordial plasma there. In plate 5 the decoupling surface is the outermost shell.
What do the hot and cold regions show us? The connection we want to develop is that they trace out a map of the strength of gravity in the universe just 400,000 years after the Big Bang. It will take a few steps, but the connection is important because it will allow us to relate the spatial distribution of matter to the temperature anisotropy of the CMB.
Earlier we gave a simple one-dimensional picture for how mass clumps to form structure. In the universe, that of course happens in three dimensions. When the mass clumps, the strength of gravity in that region of space is stronger than in others. If, say, the Earth were the same size but more massive, we would weigh more because the force of gravity would be stronger. Similarly, the more mass that clumps into a fixed volume, the greater the strength of gravity. We call this variation in the strength of gravity throughout space the “gravitational landscape.” In turn, the gravitational landscape produces the CMB anisotropy, as we describe next.
It is sometimes easier to think of the clumping on a two-dimensional slice through space. We can imagine a two-dimensional section of land with hills and valleys of all sorts of different widths and heights. The different heights of the land represent the different strengths of gravity. Gravity is stronger in the valleys than on the hilltops. As the universe evolves prior to decoupling, the dark matter clumps, and the valleys get deeper. The plasma of photons, electrons, and nuclei tries to fall into the valleys, but it is so energetic that it doesn’t clump.
There aren’t really terrestrial analogues for the process. Loosely one can think of the plasma like rather agitated water trying to settle into an array of plastic egg crates. The crates represent the hills and valleys. Unlike water, though, the plasma is compressible. When it falls into a valley it compresses, heats up, and then bounces back.
The whole universe at this time is full of a compressing, rarefying, bouncing, oscillating plasma that is trying to collect in the valleys but can’t settle down. Then, in a relatively short time, the universe cools enough for atoms to form, the CMB is set free, and the plasma state ends. This is the decoupling at 400,000 years. The CMB records the state of the universe at this time. The plasma that has fallen into a valley is compressed and heats up, somewhat like gas compressed in a piston. To a good approximation, the hot regions, the red in the maps, show the locations of the gravitational valleys where the plasma is hotter, and the blue regions show us the locations of the hills. The CMB provides a snapshot of the primordial gravitational landscape. The freed up atoms then go on and respond to this landscape to gravitationally collapse further into the cosmic structure discussed earlier.
Typically the level of contrast in the maps, that is, the difference in temperature in different regions, is 100 millionths of a kelvin or 100 microkelvin. We often use the term “fluctuations” as shorthand for “variations in space” and say that the temperature fluctuations are roughly a few parts in 100,000 of the total, 3 K. This is the same fraction as for the clumping in the matter that we discussed in the last chapter. If your mass represents the amount of clumping at decoupling, one region might have a copy of you and another might have a copy of you plus or minus the mass in the tip of your little finger. The clumping of matter and the level of the anisotropy are intimately related—indeed, the source of the fluctuations is the same: the primordial seeds we mentioned earlier.
Plate 8a shows a close-up picture of the region in the small gray box in the top map of plate 7. It is about eight full-moon diameters on a side. Although the hot and cold regions are splotchy and irregularly shaped, they do have a characteristic size. The image clearly does not look like a pointillist painting made up of thousands of tiny patches of color. Nor are the color patches so large that they take up half the image. The characteristic patch size looks like roughly twice the diameter of the full moon.
Why is there a characteristic splotch size? Let’s go back to the gravitational landscape. The primordial plasma will be hottest when it is compressed the most. This happens when the plasma “flows” down the valley walls from all sides just once in the time between when it starts flowing, roughly 50,000 years after the Big Bang, and decoupling. The flow speed is more properly thought of as the speed of a disturbance in the plasma, similar to how sound travels in air. The flow speed of the plasma is fixed by fundamental physics. The time over which it can flow is set by the expansion of the universe because after 400,000 years there is no longer a plasma and the decoupled CMB is free to travel unimpeded. The flow speed multiplied by a time is a distance. Therefore, a special size of valley exists that is particularly effective at creating hot splotches. Similarly, the same special size of hill makes the cold splotches. The speed of the plasma and the optimal valley size may be computed theoretically to high accuracy using conventional physics. To be sure there are valleys of all sizes and depths, but the CMB highlights a special size and this size may be computed in units of light-years.
We can make a simple approximation of the special size. The time over which the plasma can flow is 400, 000 − 50, 000 = 350, 000 years. The speed of the plasma is more difficult to compute. Because the plasma is predominantly composed of photons, the speed of a disturbance in the plasma is very fast, about half the speed of light. Multiplying these numbers together we get around 200,000 light-years, which corresponds to the distance between the bottom of the valley and the edge. This isn’t quite correct because during this time the universe expands by a factor of three (see appendix A.3). A more detailed calculation reveals that the size is closer to 450,000 light-years. This is called the characteristic acoustic scale at decoupling because the plasma flow is like a sound wave. Our special size, corresponding to the diameter of, say, a hot or cold spot, is twice this, or about nine times the diameter of the Milky Way as measured today.
We can now understand what the universe was like back then relative to today. It was a thousand times hotter and much more uniform. If we divided space into volumes of 900,000 light-years on a side, some would have more mass than the average by a few parts in 100,000 and others would have less by the same fraction. This tiny mass difference, traced out by the CMB anisotropy, grows through gravitational instability to form cosmic structure while the universe is expanding.
This process was first spelled out in the 1970s by Jim Peebles and Jer Yu and in a related paper by Rashid Sunyaev and Yakob Zel’dovich. The model has been refined and augmented over the decades, but the basic picture we have today is the same. It is a testament to the universality of physics that predictions can be made for what should happen in the early universe based on measurements made on Earth, and that those predictions can be tested. The physics behind the hot and cold regions is more involved than we described above, but the process we presented is the dominant one that gives rise to the features in the CMB maps.
3.3 Quantifying the CMB
To compare with theoretical models, we need to quantify the maps. In other words, we need to reduce the randomly spaced set of hot and cold regions of different temperatures and irregular sizes to a set of numbers. Mathematically, the anisotropy maps are two-dimensional sets of random numbers on a sphere. Over the decades a few methods have been developed to characterize maps like these. We will look at two.
The first method is simple and powerful. We simply go through the map and everywhere there is a hot spot we extract a 4° × 4° section of map centered on the hot spot. We have to be careful that we do not double count but we can experiment and devise an algorithm for that. On the left of plate 8a, we see that there are about a dozen hot spots, so in the area around this region we’d make a dozen 4° × 4° patches. Over the region of the sky well away from the galactic plane, north and south of the dashed lines in the top of plate 7, there are about 10,000 hot spots. We then take all those 4° × 4° maps and average them together. Through this process, the average hot spot emerges while the features in the maps that are not common average away. Although we focused on the hot regions, the same could be done to the cold regions.
The right side of plate 8a shows the average hot spot map for Planck. This is an amazing picture. It is showing us that special valley size in enormous detail. To the eye it looks to be roughly two full moons across. A more detailed analysis gives the angular diameter as 1.193°, which we will round to 1.2°. Along with the CMB temperature, this is one of the most precisely measured numbers in cosmology. It has far-reaching consequences, to which we will return later.
The second method for making sense of the maps is more involved but it displays the details of the spot more clearly. This results in a plot called a “power spectrum” as shown in figure 3.3. In essence the plot tells us the magnitude of the fluctuations of the temperature in the map for different angular sizes. We already know from the above that the largest fluctuations in temperature will be for regions around a degree in size, which corresponds to the maximum in figure 3.3 near 1°.
FIGURE 3.3. The power spectrum of the CMB anisotropy from WMAP and Planck. The variance or magnitude of the fluctuations is on the y-axis, and the angular size is on the x-axis. The angular scale markers are not evenly spaced and decrease by half with each tick. The maximum is near 1° roughly corresponding to the diameter of the hot spot on the right side of plate 8a. The gray line shows the best fit model based on the six parameters that describe our universe. The measurement uncertainties are indicated by the vertical black lines at each point.
One way to think about this plot is as a graphic equalizer, or simply “equalizer,” for a fancy audio system. This is a device that lets you amplify or soften different frequencies of sound. For instance, you might want to emphasize the bass notes over the treble notes in a piece of music. A car radio might do this with one “tone” knob, but an equalizer lets you do it with fine control. A typical equalizer has five to ten columns of little lights. The number of lights lit up in the left-most column indicates the loudness of the bass notes; the lit-up ones on the right tell you about the treble. In our analogy, the map of the CMB corresponds to the music. The bass notes are on the left side of the plot and the treble notes on the right, arranged similar to a piano keyboard. The y-axis then corresponds to the loudness of each audio frequency or pitch. If the peak near 1° corresponds to middle C at 261 Hz, the second peak would be at 635 Hz or just below E in the next octave higher, and the third peak would be at 963 Hz in this same octave but just below B. In the map, these second and third peaks are not easily discernible to the eye, but they are there. Somewhat prosaically, the plot thus shows us the music of the cosmos. Or, more accurately, it shows us the harmonic content of the cosmos.
Figure 3.3 is one of the most important plots in cosmology. It is the culmination of more than five decades of work by scientists from around the world. At the start of the quest to make the plot, no one knew what we would find or even how much we could learn once the measurement was made. We now interpret every little bump and wiggle in detail. Later we’ll interpret the full plot, but to give you an appreciation of what it has to offer, we can determine the composition of the universe from the positions and amplitudes of the peaks.
The plot is so important that we’ll consider it from another angle. While the equalizer analogy interprets the plot in musical terms, the plot is really telling us about fluctuations in a two-dimensional map of the CMB. Let’s think about the spatial aspect. Imagine that you are far from the beach looking out on the ocean. Think of what you would see if you immediately froze the pattern on the surface. In this frozen seascape you’d see large swells, medium-size waves, and small ripples on top of them. The frigid expanse is like the map of the anisotropy with the heights of the frozen water representing the temperature fluctuations in the CMB. The average depth of the ocean could represent the average CMB temperature of 2.725 K. In that frozen ocean, the swells have the longest wavelengths and the largest heights (if you are way out in the ocean and not in a storm), the waves have shorter wavelengths and medium-size heights, and the ripples have the shortest wavelengths and the smallest heights. Now let’s say we got in a plane and looked at our frozen ocean from above. The angular separation of the peaks of the swells would be larger than the angular separation of the peaks of the waves and would be larger still than the angular separation of the peaks of the ripples. On a power spectrum plot, the swells would be on the left side of the x-axis at large angles and in our example they’d have a high value on the y-axis. The waves would be in the middle of the x-axis with a medium value on the y-axis, and the ripples would be on the right with the smallest value on the y-axis. You can see the effectiveness of this type of plot. In a compact way it tells you the characteristics of all the different fluctuations—swells, waves, ripples, etc.— on a frozen ocean. Conceptually, finding the power spectrum of the CMB is not too different from finding it for the ocean surface. The peak in the CMB plot near 1° would correspond to unusually high frozen waves of this angular size as viewed from our plane. We shouldn’t push the analogy too far, though, because the CMB fluctuations are random, whereas the ocean’s fluctuations, that is the waves, are not.
Last, let’s think about the plot in terms of how we might make it. The actual procedure involves specialized algorithms and, like the measurements, has evolved and matured over the years. However, it is not too difficult to get an operational sense for how the algorithms work. The following details are not important for other sections but ideally will give more insight into figure 3.3. To start, take a map, cut out the region contaminated by the Milky Way, and then cut the remainder of the map into disks, say, 8° across (16 full moons). For each of those 8° diameter disks, compute the average temperature. Of course, in an 8° disk there will be lots of smaller hot and cold regions, but they will average out. You’ll end up with a set of average temperatures for all the 8° disks. Some will be hotter than zero, others colder. We don’t so much care about the average disk temperatures; we just want to know how much they scatter around zero. The common way to determine this is to subtract the average of all the disk temperatures from each individual disk, square the remaining part of each disk’s temperature, because this makes them all positive, and then average those. This is called a “variance” and is why the y-axis of the plot has units of (μK)2. Now, repeat the process for a list of, say, 100 disk sizes ranging from 16° in diameter all the way down to 1/8° in diameter. Then, because a smaller disk will have all the variance of the next larger disk and then some, you have to go through and subtract entry 99 from entry 100, subtract entry 98 from entry 99, and so on. You’ll end up with a new list that has the variance associated just with each disk’s angular size. Finally, go through the list and multiply each number by the disk’s angular size. You then plot the entry in the new list on the y-axis and the disk diameter on the x-axis. In broad outline, the resulting figure will resemble figure 3.3 but without all the details.
In figure 3.3 we see there is more going on than just the degree-scale fluctuations. The other ups and downs come from different ways the plasma oscillates and interacts with the gravitational landscape. For each data point in figure 3.3 there is an uncertainty represented by the vertical “error bar.” The smooth line that goes through the data is the standard model of cosmology. You can now see why measurements of the CMB anisotropy are so powerful. They are extremely accurate and highly constraining. Any potential theoretical model of the universe has to fit these data. If the model doesn’t fit, it is ruled out. If a model cannot make a prediction for this plot, it is not a contender. You can also see why cosmologists are confident that we have the basic picture correct even though we don’t know all the elements of the model in depth. In the next section, we discuss how the model relates to the gray curve.
Before moving on, let’s do a quick thought experiment to put our map of the CMB anisotropy in a broader perspective. Let’s imagine that we could live 13.8 billion years and witness all of cosmic history. You might wonder, where did the Big Bang happen? It happened everywhere at the same time. In particular, it happened right were we are. Of course the universe was much more dense then but still, as far as we are concerned, infinite. If we started our stopwatches just after the Big Bang, we would experience the formation of the light element nuclei at 3 minutes, decoupling at 400,000 years, the formation of first stars at 200 million years, and so forth. After decoupling took place, the CMB photons, free at last, had 13.8 billion years to travel to the edge of the observable universe. Around us here on Earth, there was a valley in the gravitational landscape, so that the Local Group (figure 1.2), including the Milky Way, could form. The same physical processes took place everywhere in space at the same time, though some locations were in the bottoms of valleys, others at the tops of hills, and most others somewhere in between.
Now imagine that you were instantaneously transported today to the edge of the observable universe and looked back toward Earth. What would you see? The galactic environment around you would be similar to the environment we see around us now. Recall that at any fixed age, the universe looks the same everywhere. Your environment would be different in terms of the particulars—that is, you would see galaxies that we cannot see from Earth—but it would look the same on average. As you looked back towards Earth and our Local Group you might see a CMB hot region because, as we know, matter had to clump in our vicinity to make the Local Group. We say “might” because the Local Group would appear small in angular extent compared to a typical CMB fluctuation. But you would not see any galaxies in the Local Group because the light from them would not have reached you yet.
Now that we have a sense of the bigger picture, and how to think about the observational measurements of the cosmos in a physically intuitive way, it is time to change gears and introduce the major theoretical elements of the standard cosmological model. This will require more advanced concepts from physics, and you may have to take a bit more on faith. However, the reward is that we will reach a description of the six cosmological parameters that characterize our universe and all the measurements made so far of its large-scale properties. We begin the next chapter by considering the geometry of the universe.
1 The central horizontal red swaths in the panels in plate 7 correspond to the line marked as the galactic plane, or GP, in plate 2 and to the central horizontal red swath in plate 3.
2 Firsthand accounts of the discovery and interpretation of the CMB, including contributions from Penzias and Wilson, may be found in Finding the Big Bang by Peebles, Page, and Partridge, Cambridge University Press (2009). With near misses and false leads, the story shows the all-too-human path of establishing scientific fact.
3 Physicists in the Soviet Union mounted a CMB radiometer on the Relikt satellite that came close to detecting the CMB anisotropy.
PLATE 1. This is a map of the variations in temperature of the remnant light from the birth of the universe as measured across the full sky. The temperature difference between the bluest and reddest regions corresponds to 400 millionths of a degree Celsius. The goal of this book is to explain this image and what it tells us about the universe. (Credit: NASA/WMAP Science Team)
PLATE 2. The Milky Way, running diagonally upward at roughly 45° from the lower left. This picture, by Giulio Ercolani and Alessandro Schillaci, was taken from just outside of San Pedro de Atacama, Chile. Cerro Licancabur is in the lower left. The bright dots away from the galactic plane are stars in our galaxy. The galactic plane lies between the arrows on the periphery labeled “GP.” The intersection of the galactic plane and the line indicated by “GC” marks the galactic center. The dark regions in the galactic plane are “dust lanes.” The dust blocks visible light but emits thermal radiation, as shown in plate 3.
PLATE 3. The glowing dust in the Milky Way as observed by the DIRBE instrument aboard the COBE satellite. This image shows far-infrared radiation at a wavelength of 100 microns or about 200 times longer than that for plate 2. While in plate 2 the dust obscures starlight, at this wavelength you see the dust glowing. The center of the Milky Way is in the center of the image. The blob above the center is the Ophiuchus Complex, a large dust cloud. The feature on the far left is the Cygnus region, and the bright spot to the lower right is the Large Magellanic Cloud, a nearby dwarf galaxy. (In plate 2, the Large Magellanic Cloud was off to the right and below the horizon when that picture was taken.) This picture covers a quarter of the sky. If you were out in the desert and could see the Milky Way at these wavelengths, the image width would span from horizon to horizon. (Credit: NASA/COBE Science Team)
PLATE 4. The Hubble Ultra Deep Field. The vast majority of the objects in this image are galaxies. The light from the nearer ones has been traveling to us for a billion years; the light from the farthest ones for about 13 billion years. This picture was taken while observing in the direction of the constellation Fornax. (Credit: NASA, ESA, and S. Beckwith (STScI) and the HUDF Team)
PLATE 5. Telescopes are like time machines. As we look out in space we look back in time. With the Hubble Ultra Deep Field image we look back to when the galaxies began to form. Light from the first stars was emitted when the universe was roughly 200 million years old and has been traveling to us since then. We can think of it as coming from a shell out near the edge of the observable universe. The CMB comes to us from a shell essentially at the edge of the observable universe. In this picture the CMB is the outer yellow ring. The label “Big Bang” marks the beginning of the time line. (Credit: NASA)
PLATE 6. A composite image of the Bullet Cluster made with data from the Chandra X-ray Telescope, the Magellan Telescope, and the Hubble Space Telescope. The image is about one-sixth of the full Moon across. The white and yellowish objects are mostly galaxies. The pink areas show the normal matter, which is primarily in the form of a hot X-ray emitting gas. The blue areas show the location of the dark matter as revealed by gravitational lensing. Note the concentration of galaxies in the blue regions. (Credit: X-ray—NASA/CXC/CfA/M. Markevitch et al.; Optical—NASA/STScI; Magellan/U. Arizona/D. Clowe et al.; Lensing Map—NASA/STScI; ESO WFI; Magellan/U. Arizona/D. Clowe et al.)
PLATE 7. Maps of the full sky showing the CMB anisotropy and galactic emission in a Mollweide projection. Top: The Planck map at a wavelength of 0.2 cm. Radiation from the relatively nearby Milky Way is primarily between the dashed lines. Most of the signal above and below these lines is the CMB anisotropy, although in a few places the Milky Way emission pokes through. The little square box on the left just above the top dashed line is centered on the North Star and shown in plate 8a. Bottom: WMAP map. The features in the two maps are the same once you get away from the Milky Way. The temperature color scale runs from −300 millionths of a degree to +300 millionths of a degree. The “μ” sign means “millionth.” (Credit: ESA and the Planck Collaboration; NASA/WMAP Science Team)
PLATE 8A. Left: A close-up of a 4° × 4° section of the Planck map in plate 7 centered on the north celestial pole. For scale, the white circle shows the size of the full moon. Of course, the full moon is not near the North Star. Right: The average of more than 10,000 hot (or red) spots in the Planck map also shown as a 4° × 4° image. The irregularities of individual spots average out. On this plot, blue is near the average of all hot and cold spots, 2.725 K, whereas red, the average hot spot temperature, is 45 μK greater than that. (Credit: ESA and the Planck Collaboration)
PLATE 8B. A depiction of measuring the size of the hot and cold spots in the CMB. This average hot spot size is shown in plate 8a and corresponds to the peak of the power spectrum in figure 3.3. By combining the measured angle and the computed size of the spots with knowledge of the Hubble constant, we can determine the geometry of the universe. (Credit: ESA and the Planck Collaboration)