Chapter 10

Spots in the Cosmic Microwave Background

So far, our study of 3D topology of galaxy clustering has extrapolated backward to deduce something about the random-phase nature of the initial conditions. But what if we could see the initial conditions directly? The cosmic microwave background, which comes to us from an epoch just 380,000 years after the Big Bang, allows us to do just this. When we observe the cosmic microwave background, we are seeing out to a spherical shell (where we hit the fog of the early universe), which has a radius of about 13.8 billion light-years. We are seeing directly the initial conditions of the universe, just 380,000 years after the Big Bang. It’s like looking directly at Einstein’s baby picture, as opposed to doing a computer age regression of a picture of him as an adult, to figure out what he might have looked like as a newborn baby. Because of the finite velocity of light, when we look out in space to the fog bank about 13.8 billion light-years away, we are also looking back in time, to the epoch of recombination about 13.8 billion years ago, just 380,000 years after the Big Bang. At that time, on the largest scales density fluctuations were about 1 part in 100,000, creating temperature fluctuations of about 1 part in 100,000 in the cosmic microwave background as one looks from place to place in the sky. These density fluctuations grow into clusters of galaxies by the action of gravity over the course of the next 13.8 billion years. Standard inflation predicts Gaussian randomphase initial density fluctuations and, therefore, Gaussian randomphase temperature fluctuations for the cosmic microwave background. These are waves, with amplitudes picked from a Gaussian distribution (a bell-shaped curve), going in all possible random directions in the sky with random phases (random locations of their peaks and troughs). Just as a Gaussian random-phase distribution in three dimensions implies a 3D spongelike topology, a Gaussian random-phase distribution in two dimensions (on a map of the sky) implies a particular 2D topology.

Color Plate 13 is a temperature map over the whole sky of the cosmic microwave background made by Wes Colley and me, along with our colleagues, using data from the WMAP satellite. Looking out in all directions, the sky looks like a sphere. The night sky looks like a hemispherical dome overhead, but Earth blocks out the other half of the sky. If Earth were to disappear suddenly from under your feet, you would find yourself floating in space, and if you looked down you would see another hemisphere of stars below you. This completes the celestial sphere. The WMAP satellite scans the entire sky, covering all the cosmic microwave background. This map shows the inside surface of the celestial sphere. In order to plot this on a piece of paper, one has to project the spherical sky onto a flat map. One of the standard map projections used for mapping the spherical Earth onto a flat map can be adapted to plot the celestial sphere. The WMAP team chose the Mollweide equal-area projection to do this. Perhaps you have seen such a map of Earth. A Mollweide projection of Earth shows its entire surface as an elliptical map, with the North Pole at the top of the ellipse, the South Pole at the bottom, and Earth’s equator as a horizontal line stretching across the middle of the map. In the same way, a Mollweide projection of the celestial sphere plots the entire sky as an elliptical map, with the North Galactic Pole at the top, the South Galactic Pole at the bottom, and the galactic equator as a horizontal line spanning the center of the map. If we show the visible sky on such a projection, you will see stars sprinkled over the entire map, with the faint band of the Milky Way as a horizontal band crossing the center of the map from left to right. But the cosmic microwave background map is a radio map, a map showing radiation only from the radio portion of the spectrum; furthermore, radio radiation from the Milky Way has already been carefully subtracted out. Radio radiation from the galaxy has a different spectrum from the thermal spectrum of the microwave background; the WMAP team observed at different frequencies so that the galactic radio radiation could be estimated at each location and subtracted out. Thus the WMAP team produced a map of the cosmic microwave background alone. The Doppler effect due to the motion of Earth relative to the cosmic microwave background has also been subtracted out. Temperature fluctuations in the cosmic microwave background of order 1 part in 100,000 can then be seen.

The temperature fluctuations we see in the cosmic microwave background arise from several effects:

1.  A region that is hotter than average at recombination will appear hotter to us. This is straightforward. Regions that are cooler than average will appear cooler. This is important for regions smaller than 1° in angular scale.

2.  A region that has higher-than-average density at recombination will create a gravitational well; photons must climb out of it. These photons will lose energy as they climb out, and get an extra redshift, making that region look cooler. Underdense regions, by comparison, will look hotter. This is called the Sachs–Wolfe effect, and it is important for regions larger than 1° in angular scale.

3.  Additional Doppler redshifts and blueshifts occur because of the peculiar motions of matter as it is drawn into denser regions and as it oscillates due to sound waves in the early universe (baryon acoustic oscillations). These Doppler effects are important on angular scales smaller than 1°.

4.  Some extra redshifting and blueshifting occurs as photons climb in and out of growing gravitational wells as they pass through the growing superclusters on their path to reach us. This is called the integrated Sachs-Wolfe effect because it integrates what happens to the photons on their way to us. This is important on scales larger than 1°.

5.  When a cluster of galaxies lies in the line of sight, microwave background photons can be scattered by electrons in the hot gas in the cluster and kicked up to higher frequency (the Sunyaev-Zeldovich effect). This depletes the number of microwave background photons seen at low frequency in the direction of the cluster, creating a tiny (approximately 0.03° wide) cold spot in the microwave background. Foreground clusters can be detected in low-frequency (less than 218 Gigahertz), very high-resolution maps as rare cold spots and in highradio-frequency (greater than 218 Gigahertz), very high-resolution maps as rare hot spots. This effect due to hot cluster gas has been observed and has been used to detect massive clusters but is not important on 1° scales. If one observes at a radio frequency of 218 Gigahertz, this thermal scattering should not create a hot or a cold spot, but galaxy clusters can still produce hot or cold spots by this scattering effect if they have peculiar velocities toward or away from us along the line of sight. This is called the kinematic Sunyaev-Zeldovich effect and is about twenty times smaller in amplitude than the regular Sunyaev-Zeldovich effect. Nick Hand and colleagues (2012) have detected this effect statistically by comparing nearby pairs of galaxy clusters, which have—on average—peculiar velocities along our line of sight of opposite signs as gravity pulls them together—also not important on 1° scales.

6.  Gravitational deflection of light along the line of sight (weak gravitational lensing) can move hot and cold spots somewhat on the sky and slightly distort them in shape, but it does not change the topology of the spots when seen at the WMAP satellite angular resolution. This lensing effect creates small but characteristic shape squeezing of the spots, which has been detected statistically.

7.  Finally, there are additional redshifting and blueshifting effects from the squeezing and stretching of spacetime due to primordial gravity waves inherited from the inflationary epoch. This is a smaller effect than the others, not detected yet, and would play a role only on angular scales larger than 1.5°.

All these effects can be calculated theoretically. Effects 1, 2, and 3 give us a direct look at the initial conditions at recombination—accounting for most of the temperature structure we see on small angular scales. We will ultimately see all these effects compared with the observations when the power in the fluctuations at different angular scales is analyzed later in this chapter.

WMAP’s predecessor satellite, the Cosmic Microwave Background Explorer (COBE), had been the first to observe these fluctuations. The satellite mission earned Nobel prizes for George Smoot and John Mather. But the WMAP satellite offers a higher-resolution view. Many people from the original COBE team rejoined to build the WMAP satellite.

The WMAP team presented their map with a rainbow color scheme, with red being the hottest, followed by orange, yellow, green, and, finally, blue, as the coldest. Their picture, which displayed the sky as both intricate and beautiful, has become world famous, appearing in many books and articles. (This WMAP map is shown as an inset at the top of Color Plate 8.) But it was relatively hard to find the average temperature contour (it was somewhere in the middle of the green band), and it was hard to compare hot and cold spots. What shade of yellow on the hot side, for example, corresponded to what shade of blue on the cold side? For our purposes of testing the topology of the microwave background, Wes and I devised a different color scheme. We colored the map in red, white, and blue. The average temperature contour is white. Regions that are hotter than average are colored in varying shades of red: those just slightly above the average temperature are colored light pink, with progressively hotter regions appearing in more and more saturated shades of red. The hottest regions are colored pure red. Regions colder than average are colored in varying shades of blue. Slightly below average temperature is light blue, with colder temperatures depicted in deeper blues. One can then directly compare hot and cold spots of equal magnitude. The scale is linear in ink. If a region is twice as far from the mean in either the hot or cold direction, it will, accordingly, have twice as much red or blue ink. Regions at the mean temperature (which is also the median temperature) are pure white. This color scheme for the WMAP data appears in Color Plate 13.

Measuring 2D Topology

To study the topology of these temperature fluctuations on the 2D surface of the microwave background sphere, we would first have to learn how to measure 2D topology. One day Adrian Melott called me to say excitedly that he had figured out how we could measure topology in two-dimensional slices in the universe. For the cosmic microwave background, one starts by drawing temperature contour lines in the microwave background map. Such contour lines on a 2D geologist’s map usually represent lines of constant altitude. If there are mountains of high altitude, then contour lines will surround them. If there are depressions of low altitude, then contour lines will surround them also. Lakes have shorelines, for example, that are contour lines of constant altitude, and the ground beneath the lake represents a depression that the shoreline encloses. But we are measuring temperature, not altitude. Hightemperature regions will be surrounded by high-temperature contour lines that encircle them; likewise, low-temperature regions will be surrounded by low-temperature contours that encircle them. We often see such temperature contours on 2D weather maps: isotherms (curving contour lines) where the temperature is 70°, 80°, 90°, and so on. We would be making such a “weather map” of the cosmic microwave background.

Adrian had figured out how to use temperature contours to measure the topology of these hot and cold regions in the 2D map of the cosmic microwave background sky. In 2D, we needed a new definition of the genus. Temperature contour lines in a 2D map have a curvature. If you were to drive a truck around a circular temperature contour line on a map, you would have to turn your truck by 360° (or 2π) as you went around and returned to where you started.

Now, imagine that you have a general contour curve that encloses a high-temperature (deep red) region. Imagine this occurs on a map with pixels, like the city-block grid in a city. Drive your truck around a closed contour—say, one that encloses the Empire State Building. As you circle the contour, keep the Empire State Building on your left. Circle the block, and you must make 4 left turns to get back to where you began. A generalized contour could have a complicated shape but must follow the grid lines of the rectangular street plan. (Ignore any diagonal roads like Broadway for this exercise.) You will make many left and right turns as you work your way around a complicated contour that encloses the Empire State Building, but you will find as you complete the circuit that the total number of left turns (of 90° each) will always outnumber the total number of right turns by 4. Try it on the 7%-high-temperature contour enclosing the highest temperature 7% of the pixels shown in Figure 10.1. Put the high-temperature side on your left and drive around on the path indicated by the arrows. You will make 6 left turns and 2 right turns, and 6 − 2 = 4. This would be true of any closed high-temperature contour: (left turns − right turns = 4). You must complete a net excess of 4 left turns of 90° to make a full counterclockwise rotation of 360° as you circle the isolated hot region. By making the contour circuit follow a grid, you are simply ensuring that all the turning occurs at discrete points.

Figure 10.1. A map with examples of 2D temperature contours. The 7%-high-temperature contour encloses the 7% of the area with the highest temperature. The 7%-low-temperature contour encloses the 7% of the area with the lowest temperature. The median temperature contour divides the map in half. Drive around the 7%-high-temperature contour with higher temperature on your left; you circle counterclockwise, making 4 more left turns than right turns. Drive around the 7%-low-temperature contour with higher temperature on your left; you circle clockwise, making 4 more right turns than left turns. The median contour has an equal number of left and right turns. Counting the number of left turns minus the number of right turns required to navigate a contour enables us to calculate its genus. (Credit: J. Richard Gott)

Figure 10.1 also shows the 7%-low-temperature contour, which encloses the coldest 7% of the pixels in the plane. Finally, there is the median density contour, which divides the plane into two equal-area pieces, with the higher-temperature pixels on one side and the lowertemperature pixels on the other.

Look at all the vertices that lie on the 7%-high-temperature contour. Each of these vertices is surrounded by 4 pixels. Some of them must be hotter than the 7%-high-contour value, and some of them must be colder. If two are hotter and two are colder, you are going straight and no turn is being made. If one is hotter and three are colder, you are making a left turn at that vertex, and if one is colder and three are hotter, you are making a right turn. For this particular contour level, you can find all the vertices that fall on that contour line and count how many are surrounded by one hotter pixel and three colder pixels (left turns) and how many have one colder pixel and three hotter pixels (right turns). By obtaining the net result for left turns minus right turns and dividing it by 4, you can figure out how many isolated high-temperature regions you have. With this technique, you can count the number of isolated hot spots lying above a particular contour threshold.

Similar arguments apply to the low 7% contour. Here, if you put the high-temperature side on your left, as before, you must circle the contour clockwise, and your number of left turns minus right turns equals −4.

For a general contour, if we calculate the value (number of left turns − number of right turns)/4 required to draw the entire contour with the high-temperature side on your left, it will equal the number of isolated hot spots minus the number of isolated cold spots. This is Melott’s definition of the genus for two-dimensional topology maps:¹

2D genus = number of isolated hot spots − number of isolated cold spots.

The genus of the 7% high contour is +1, because it encloses 1 isolated hot spot and no isolated cold spots. The genus of the 7% low contour is −1, because it encloses 1 isolated cold spot and does not encircle any hot spots. The median density contour has 1 left turn and 1 right turn, making its genus equal to 0; it just wanders around, enclosing neither hot nor cold spots.

Hot Spots Minus Cold Spots

What do we expect the 2D genus to look like for a random slice through our sponge? (After all, the cosmic microwave background is a spherical shell that makes a 2D slice through the 3D sponge.) If the temperature fluctuations have a Gaussian random-phase distribution, as we expect from inflation, the 2D genus formula adapted from Adler (1981) is

A is the amplitude of the genus curve, and exp(x) = e^x, where e = 2.718281828 … is the base of the natural logarithms. The variable ν is a measure of the area enclosed by the temperature contour, just as it was in our 3D contours: the 7% high contour is ν = 1.5, the 7% low contour is ν = −1.5, and the median contour is ν = 0, as before.

The genus curve is shown in Figure 10.2.

Figure 10.2. The 2D genus curve from WMAP compared with theory. Genus (defined as number of hot spots minus number of cold spots) is plotted as a function of temperature threshold. The theoretical curve for Gaussian random-phase initial conditions is a solid line. The WMAP genus data appear as dots, with 68% and 95% confidence error bars. The median temperature contour is ν = 0. For temperature contours above the median, we have ν > 0, and we see many hot spots; the genus is positive. For temperature contours below the median ν < 0, we see many cold spots; the genus is negative. The map shows about 2,200 hot spots and 2,200 cold spots. (Credit: W. N. Colley and J. Richard Gott, Monthly Notices of the Royal Astronomical Society, 344: 686, 2003)

The solid curve is the theoretical formula just given and the data points with the error bars are from our analysis of the WMAP data (Colley and Gott 2003). The error bars (68% and 95% confidence limits) are calculated by looking at the observed variation in the hot-and-cold-spot counts in 12 separate regions of the sky. For a high-temperature contour (ν = 1.5), which is in the red, the genus curve is positive—consistent with the large number of isolated hot spots (red spots) in Color Plate 13. For a low-temperature contour (ν = −1.5), which is blue, we see many isolated blue spots, as expected. One can see at a glance that there are the same number of hot (red) spots as cold (blue) spots. Thus, the genus curve shows that the distribution of red and blue ink in Color Plate 13 is symmetric, as expected. If these fluctuations are due to random quantum fluctuations, as predicted by inflation, then red and blue (positive and negative) fluctuations must be equivalent.

The agreement between the WMAP data and the Gaussian random phase theory is extraordinary. Earlier, Changbom Park and I had helped the COBE satellite team study the 2D topology results from their lowerresolution map of the cosmic microwave background. It also fit the theoretical random-phase curve well. The higher-resolution WMAP data just show more detail and more hot and cold spots, making the error bars smaller. Departures from a Gaussian random-phase distribution can be quantified by a parameter f_NL invented by Komatsu, Spergel, and Wandelt (2005). A perfect Gaussian random-phase distribution would have f_NL = 0. As observations have gotten better and better, the limits on f_NL have gotten closer and closer to zero, honing in on the near-zero value predicted by standard inflation.² The cosmic microwave background looks just like what a 2D section through a Gaussian randomphase 3D cosmic sponge should look like.

1D Topology

We’ve covered 3D topology and 2D topology; is there any way to treat 1D topology? Barbara Ryden from our group discovered a way to do topology in a 1D rodlike survey. Here, a density contour would divide the rod into line segments that were either higher or lower than the contour. Going from left to right, one would find regions that were higher, then lower, then higher, then lower than the contour. Just as in a road trip, you would be crossing above and below a certain altitude contour. Barbara showed that the 1D topology was measured by the number of up or down contour crossings. We could define this as the 1D genus. The formula to relate this to the density enhancement of the density contour as measured by ν had already been derived in 1945 by the mathematician S. O. Rice:

A is the amplitude and exp is the exponential function. The genus curve is a bell-shaped (Gaussian) function that is high in the center (at ν = 0) and low on the sides as ν becomes either large and positive or large and negative. That means that the median density contour has the most up or down crossings as one looks down the rod survey. If the rod is divided into equal high- and low-density parts, as one goes along the rod survey from one end to the other, one will often be either going above or crossing below the median density in a random-phase Gaussian field. Suppose one has a rodlike survey of galaxies. If one sets a high threshold, such as ν = 1.5, which encloses the 7% high regions, one finds only a few isolated clusters and, therefore, relatively few up or down crossings are encountered. How many fewer? The answer is exp(−1.5²/2), or about ⅓ as many up or down crossings as the median density contour.

Eventually, the 1D topology formula was checked by David Weinberg using long, rodlike samples containing neutral (nonionized) hydrogen clouds along the lines of sight to distant quasars. Each hydrogen cloud along the line of sight to a distant quasar would absorb some of the light from the distant quasar at a particular wavelength, which would be red-shifted, depending on its distance from us, in accordance with Hubble’s law. A series of hydrogen clouds along the line of sight to a distant quasar would absorb a series of narrow lines from the quasar’s spectrum. If the hydrogen clouds were equally spaced along the line of sight, the absorption lines would be equally spaced in the spectrum. If the hydrogen clouds were clustered, then the absorption lines would also be clustered. By smoothing the spectrum, we identify higher- and lower-density regions, corresponding to dense and sparse regions of hydrogen clouds. The results for threshold crossings closely followed Ryden’s Gaussian random-phase bell-shaped curve, showing that the distribution of hydrogen clouds along the lines of sight to quasars followed the 1D topology expected for a Gaussian random-phase distribution.

All these genus statistics for Gaussian random-phase distributions are related: they all start with a bell-shaped Gaussian function, Aexp(−ν²/2), which is then multiplied by a polynomial that depends on the dimension of the space: (1 − ν²) for the 3D genus, ν for the 2D genus, and 1 for the 1D genus. These are related to the Minkowski functionals in N dimensions long known to mathematicians, as previously noted. We have found agreement with the Gaussian random-phase results using the 3D topology of galaxy clustering in the cosmic web, independently using the 2D structure of the hot and cold spots on the microwave background, and, finally, using the 1D topology of the distribution of hydrogen clouds along the lines of sight to distant quasars. All these tests support the standard picture of inflation, whereby the structures we see today originated as random quantum fluctuations in the first 10⁻³⁵ seconds of the universe.

Power at Different Scales

Inflation also predicts the size distribution of the spots in the microwave background with remarkable precision. We can measure the power in the fluctuations we see (the square of the amplitude of the waves) as a function of angular scale. This is called the power spectrum of the fluctuations: it displays the power in the waves as a function of angular scale.

In Color Plate 14 we show the power spectrum of the fluctuations observed by the Planck satellite. The curve has many bumps in it. The main bump at a scale of about 1° is the characteristic size of the hot and cold spots we see in the picture. The other bumps in the curve are essentially overtones—like those you get when you ring a bell. These are baryon acoustic oscillations—due to sound waves in the early universe. The green band indicates the predictions of inflation with cold dark matter. The fit is astonishing. The difference in amplitude of the even and odd overtones in the sequence allows us to measure the amounts of normal matter and cold dark matter in the universe. The green band (prediction of the theory) broadens as we get to large angular scales, because only a few modes occur at these large angular scales, and sampling them at random from a Gaussian random-phase distribution leads to a larger random cosmic variance. The measured tilt in the primordial power spectrum relative to the Harrison–Zeldovich constant amplitude hypothesis is −0.032 ± 0.006 (68% confidence; Planck Collaboration 2015a). This compares amazingly well with the value predicted by inflation—for a simple model slowly rolling down a hill in the landscape: −0.0333.

Inflation in the early universe produces causal horizons that are only a tiny distance (10⁻³⁸ light-seconds = 3 × 10⁻²⁸ centimeters) away. Due to the uncertainty principle, this produces uncertainties in the geometry of spacetime: ripples, which according to Einstein’s equations propagate at the speed of light—gravity waves. These gravity waves should produce characteristic swirls in the polarization of the cosmic microwave background radiation. So far, their detection has remained elusive. The BICEP2/Keck and Planck Collaborations (2015) has now set upper limits on gravity waves comparable to, but slightly below, the gravity wave amplitudes predicted from Linde’s simplest version of chaotic inflation from 1983. Interestingly, a model of inflation that fits extremely well with the new Planck + Keck + BICEP2 data (cf. Planck Collaboration 2015a) is the older Starobinsky (1980) version of inflation and generalizations of it (Kallosh and Linde 2013; and Kallosh, Linde, and Roest 2013). The amplitude of gravity waves produced depends on the contour of the hill in the landscape you are rolling down. Linde assumed a valley with walls having a parabolic shape. This was based on the physics of a scalar field associated with a massive particle. Starobinsky’s idea was based on one-loop quantum corrections to the vacuum state and his model has a valley with contours like an inverted bell carved out of a plateau. It produces gravity waves with an amplitude that is about a factor of 6 less than Linde’s model, significantly and safely below the current observational upper limits. In the Starobinsky model, the doubling time for inflation in our universe after its formation would be a factor of 6 longer than in the Linde model; namely, the universe would be doubling in size every 3 × 10⁻³⁸ seconds at the end of the inflationary epoch rather than doubling every 5 × 10⁻³⁹ seconds, as in the Linde model. This sixfold-less-violent expansion would produce gravity waves 6 times smaller in amplitude. Follow-up experiments (from the South Pole Telescope, from the BICEP3/Keck experiments, and from the SPIDER highaltitude balloon experiment—all in Antarctica) as well as searches using the Planck satellite itself are in progress now. Cosmologists are watching with interest to see if these studies can open a new window on the very early universe.