Angular scales from the CMB

The physics of the CMB is extremely rich, and I won't do it justice in this series of entries. However, I do want to give one example of how the spectrum of the CMB fluctuations and in particular the location and size of the peaks gives us information about the universe. Let's look at the results from analyzing the CMB maps using the spherical harmonic functions one more time:

As I said last time, this plot tells us about the correlations between different regions of the sky. What exactly does that mean? Well, to the naked eye, the CMB map looks fairly random - some parts are blue, some parts are green, but there's no obvious pattern. What the above graph tells us is that regions of the map separated by 1 degree are actually related to each other. The amplitude of the spherical harmonic with a "frequency" of 1 degree is very high compared to other frequencies, just as the amplitude of the sine curve with the same frequency as the A note was very high when I was decomposing the A chord using Fourier analysis.

The next question to be asked is, "so what?" And the answer relates to stuff I was talking about a while ago in the posts on Gravitational Potential Wells. There, I talked about how early fluctuations in the gravitational potential created oscillations, like balls rolling in and out of a divot. In the early universe, small gravitational fluctuations of all shapes and sizes were created. These fluctuations expanded and contracted, interacting with photons to make hot and cold photons, up until the moment of last scatter when the universe became neutral and no longer interacted with photons at all, creating the CMB. What the large peak at 1 degree in the CMB spectrum tells us is precisely the size of the fluctuation that oscillated one time before the moment of last scatter.

Let me see if I can come up with an analogy for how that works that makes sense. Imagine watching a swimming race where the swimmers do laps in the pool. Let's suppose this race includes swimmers of all abilities, so some are very slow and some are like Michael Phelps. The race starts with all swimmers along the starting line, but as time passes, the swimmers spread out according to their abilities. However, because they have to swim laps, the absolute distance between the swimmers and the starting distance is always less than or equal to the length of the pool - in other words, you can't tell the fast and slow swimmers by how far they are from the start, since there will be times when the fast swimmer is heading back to the start line while the slow swimmer is still at the far end of the pool and vice versa.

From Ironman blogger, Chad Holderbaum

Now let's stop the race and have each swimmer stop exactly where they are. They will be spread out all over the pool. But, a few of them will be near the far end of the pool. And some of them, the ones whose pace was exactly right will be exactly at the far end of the pool. If we measure the maximum fluctuation in the position of the swimmers, we find that all the swimmers who exactly swam the full length of the pool have that maximum fluctuation - they are farthest from the start. The CMB measurement is making a similar type of measurement. It measures the size of the gravitational potential well that was maximally expanded when we stopped the race, or when the CMB decoupled from the universe. Therefore, we can calculate the size of the pool - or the size of the universe at the time of last scatter. The location of the first peak in the CMB is like a ruler for the early universe. And that ruler helps us find all that other information I've been talking about.

As usual, I fear I have not fully done the physics justice in this rather slow developing and superficial treatment of the problem. However, I think I'm done with the CMB for now. If you want to learn more about it, I do recommend Professor Wayne Hu's excellent website at http://background.uchicago.edu/~whu/ which explains the CMB much better than I could hope to.

Understanding the CMB

How do scientists understand the CMB? At this stage, I think we can try to outline the whole process. Over a year ago, I described the CMB as a sea of photons streaming through the universe, not interacting with anything until they reach us on earth. These photons are microwaves and can be picked up by radio antennas; at one point in time, the snow that people saw on their old television sets with bunny-ear antennas contained a component of the CMB. As I described here, the first group to observe the CMB initially interpreted it as an unexplained source of noise in their state of the art radio equipment.

To make the very sensitive measurements necessary to understand this today, we need to measure the CMB in space, where there is less interference from man-made radio backgrounds and the atmosphere. Therefore, in the 1990s, a group of scientists developed the WMAP satellite, which was flown by NASA at the beginning of the 2000s and has been collecting great data ever since.

(Personal aside: the WMAP satellite was originally the MAP satellite. The W was added in honor of Prof. David Wilkinson of Princeton University who passed away in 2002. I had the good fortune of being taught by Dave as a sophomore in college when I didn't know the first thing about experimental physics, and I also worked with him for a summer on the Search For Extraterrestrial Intelligence project [a topic for another time, perhaps]. He was a really great teacher, a wonderful man and one of the reasons I am a physicist today. It's nice that his work has had such a profound influence on physics research today.)

The WMAP satellite detects the CMB as it streams in from all directions, and the data can be used to produce the lovely CMB map that I keep showing. But what is this map? Essentially, it's sort of the inverse of a world map. As we know, the earth is a sphere and flat world maps are projections of that sphere onto a flat surface, as in this nice illustration taken from www.nationalatlas.gov:

The CMB map is very similar. If you look out into the sky the same distance in every direction, you would map out the inside of a spherical surface. Then you could project what you saw onto a flat surface just like the globe projects onto a flat world map. The result is the CMB map.


Ok, now what? We have a map of all the little temperature fluctuations in the CMB photons coming from all directions of the sky. Well, the CMB scientists use a version of Fourier analysis to find correlations in these temperature fluctuations. For those who want more mathematical detail, in the series on Fourier analysis, I stated that any function could be obtained by summing sine functions of different frequencies. Well, there are a class of functions similar to the sine function called Spherical Harmonics that can in most cases recreate any two dimensional function, and the spherical harmonics have many of the same properties as the sine function when it comes to integration. Therefore, one can multiply the two-dimensional signal by a spherical harmonic of a given "frequency" and integrate just as one would in Fourier analysis to find the amount of the signal described by that particular frequency. And the result is something that looks like the following:

Courtesy WMAP/NASA Science Team

This is analogous to the breakdown of the A chord into frequencies, with the difference that "l" or the "multipole moment" refers to the way frequencies are understood in spherical harmonics. Just as the Fourier transform shows us how much of a signal is contained in different frequencies, this plot shows us how much of the CMB are correlated over different angular scales (the lower x-axis in the plot). For example, much of the CMB signal is contained around an angular scale of between 2 and 0.5 degrees. What does that mean? It means that the map is not just a random collection of fluctuations, but that regions separated by about 1 degree are related to each other.

This is a fairly dense post, so I'll leave it at that for now and come back later if I get questions. Next, we'll talk about how the angular correlations tell us about the universe.

Back to the CMB

Over a year ago now (I have been really delinquent), I started talking about the CMB. If you recall, the CMB was like a picture of the universe as it was very early on after the Big Bang. And in this post, I said the following: "using the [noise in the] CMB, we can understand the age of the universe (13 and a half billion years), the geometry of the universe (flat), the amount of energy and density in the universe (the pie charts in the first post of this blog, including the 23% accounted for by dark matter [there is a connection between this and what I have been talking about until now, after all]), the rate of expansion of the universe, and other things."



Now that we've been through the whole sequence on Fourier analysis, we can start to understand how we extract that kind of information from a map that looks completely random to the eye. The key is that by applying a variation of Fourier analysis to the map in the picture, we can look for correlations between the fluctuations in the noise. As I explained it, Fourier analysis was able to extract how much of an apparently noisy and random signal was contained in different frequencies - for example, it could pull the individual notes out of the idealized A chord.

An artificially noisy A-chord.


The Fourier transform of the A-chord, with constituent frequencies easily visible.

In that example, the Fourier analysis effectively looks for correlations in time. Because the signal was made up of discrete frequencies, different parts of the signal were related to each other. For example, the A note has a root frequency of 440 Hz, or 440 cycles per second. What that means, although it can be hard to see by eye, is that two parts of the signal separated by 1/440 seconds are related in the way they appear, and the Fourier transform picks up on that. The premise of the CMB analysis is that two areas on the CMB map are also related in the way they appear (just not by eye). Instead of looking for time correlations, they look for spatial correlations in the map using a similar algorithm to the Fourier transform described above. In the next post, I'll show how they decompose the map shown above into its underlying angular or spatial frequencies.

CMB Anisotropies (part 2)

The Dipole
The above picture is an image of the temperature variation in the CMB with the contrast turned up to 1 part in 1000. Therefore, there is about 0.1% difference between the left side and the right side. This particular pattern appears fairly often in physics and is known as a dipole (there are two "poles" where the temperature is hotter or colder and the rest of the distribution stems from those two centers). Why is there such a distinct pattern in the temperature distribution?

The answer lies in the Doppler effect, which we've talked about at length before. In fact, we've talked about everything we need to explain this pattern. I've mentioned that the temperature is similar to the energy, so that we're effectively showing the energy of the CMB photons as a function of where they are coming from. And we know that the energy of a photon is related to its frequency. Therefore, the above picture shows the change in frequency of photons coming from one direction or another. We know that galaxies rotate, including our own. And finally, we know from the Doppler effect that the relative velocities of a source and an observer can change the observed frequency of light.

Mom, can you now guess why this pattern looks the way it does (I'm not sure how I feel about directly addressing anyone in this blog, since there's clearly no possibility of a direct response, but I'll leave it for now)? If you guessed that the Earth's motion through the galaxy resulted in a Doppler shift of the CMB photons depending on whether they are coming from in front of us or behind us, you would be exactly right. In effect, the Earth (and the Sun and the entire solar system) is moving towards one of those poles and away from the other, and thus we see the Doppler shifted dipole pattern shown above.

That is pretty interesting, but not revolutionary. We understand the Doppler effect and we know our galaxy is rotating, so if that were the only thing in the CMB anisotropy, it wouldn't be that big a deal. The real excitement (I keep pushing it forward, don't I?) arrives when we subtract the dipole effect (it's fully understood, so we can do that), leaving the smaller part in one hundred thousand variations.



Tiny variations
Finally (finally!), I will talk about what the CMB is showing us. The above is a map with the contrast turned up to that part in 100,000. And now there's no obvious pattern, which is good, because the universe is supposed to look the same in all directions. Basically, these little fluctuations are the imprint of noise in the very early universe (remember, at one point I described the CMB as a snapshot of the universe at 400,000 years old). And by studying the distribution of this noise, we can infer things about the properties of the universe.

I plan on going into this in more detail (with a detour through something called Fourier analysis), but using the CMB, we can understand the age of the universe (13 and a half billion years), the geometry of the universe (flat), the amount of energy and density in the universe (the pie charts in the first post of this blog, including the 23% accounted for by dark matter [there is a connection between this and what I have been talking about until now, after all]), the rate of expansion of the universe, and other things. I think (and I hope you agree with me) that this is really impressive - this one measurement has answered several deeply fundamental cosmological questions about how the universe works all in one go, just by carefully studying the snow picked up by the rabbit ears on my mom's now useless analog television set.

Some history

In the 1940s and 50s, a few scientists (George Gamow, Ralph Alpher and Robert Herman among others) predicted the continued existence of the photons that last scattered in the very early universe. Theoretically, those photons had continued to travel through the universe, cooling as the universe expanded. The early theorists tried to predict what the temperature of these photons would now be (with varying degrees of success). These photons should be all over the place and hence providing a constant "background" to any antenna on earth. In addition, they should have cooled enough that now their wavelength would be in the microwave range. Thus, these photons came to be called the cosmic microwave background.

In the mid 1960s, a group at Princeton led by Robert Dicke began building a radiometer to detect the CMB. At the same time, Arno Penzias and Robert Wilson at Bell Labs observed some noise in a sensitive antenna they were planning to use for radio observation. After careful work, they decided that this noise had to be external and coming from all directions in the sky. Eventually they made contact with the Princeton group, and this background noise was interpreted as being the CMB (after first talking to Penzias and Wilson, Dicke supposedly got off the phone and told his collaborators, "Boys, we've been scooped"). The two groups published companion papers on the observation and the interpretation, and in 1978 Penzias and Wilson received the Nobel Prize.

Although important, that first observation is not on its face all that exciting. The CMB is remarkably smooth or isotropic (meaning it looks the same in all directions). The picture below shows what Penzias and Wilson might have seen if they'd been able to observe the CMB in all directions (courtesy http://map.gsfc.nasa.gov/), and it's hard to see what all the fuss is about. But I'll leave that for the next post.

Thermal equilibrium recap

The last post is rather long and involved, so I will try and recap in briefer terms. The early universe was very hot, so that everything was in thermal equilibrium. In particular, because reactions were constantly taking place, the universe was strongly "ionized" or charged. Therefore, photons were constantly scattering off the charged particles.

Eventually the universe began expanding and cooling.* As it did so, the ions and free electrons "recombined" (during the time romantically referred to as the era or epoch of recombination) to form neutral atoms, after which photons no longer scattered (romantically referred to as the "surface of last scattering," a phrase that always puts me in mind [for whatever reason] of the "Last Homely House" in the Lord of the Rings [yes, I am a physicist and I love Tolkein and I write a blog for my mom]). Those photons remain unmolested since that time.

*Aside: my mom asks in a comment "why did the universe cool?" The short answer to that is because it expanded. Temperature is in some sense a measure of how many collisions occur in a space [recall my analogy about money in the last post] - at high temperature, there are lots of collisions. Suppose we expanded the space, but kept the number of particles the same. All of a sudden, the number of collisions would go down, because the particles wouldn't be able to find each other to collide. Therefore the temperature drops. Many [if not all] refrigerators operate this way, by allowing a compressed gas to expand rapidly and thereby drop in temperature. A follow-up question is then "why did the universe expand?" and I have a less satisfactory answer to that. My best explanation is that there was a lot of energy released in the big bang, and it was that energy that drove the expansion. We may have more to say on this subject at later times).

Thermal equilibrium

Last week was graduation at Yale, and a few of my closest friends here were getting their degrees. As such, some celebrating ensued. One of my friends is now doing post-doctoral work at UCLA, while another is working for a financial firm outside of New York. One night we spent some time in the early morning hours discussing the economy and the stock market. In that discussion, I came up with a somewhat stilted metaphor that I'm now going to invert to describe the concept of thermal equilibrium, which is where I want to begin the series on the CMB. In physics, temperature plays a similar role to that of money (or liquidity) in the markets.

First, I'm going to define "ionization" by referring briefly to the Bohr model I described here. Ionization is the process by which an atom loses (or gains) an electron and becomes charged. In the old post, I compared an atom to a building with an elevator which could transfer people (or electrons) between discrete levels. Using that image, ionization would occur if the elevator dropped you off on the roof, at which point you could leave the building entirely. As long as you were within the building, you remained trapped, just as an electron remains trapped by the electric field of the protons at the center of the atom (or as the Earth is trapped by the gravitational field of the Sun). On the roof, however, you have gained enough energy that you can leave the building; if an electron gains enough energy, it can escape from the electric field and be free, leaving the atom positively charged. This positively (or negatively, if it picks up an electron) charged nucleus is referred to as an ion.

One more thing that we should keep in mind about charged particles is that they interact rather strongly with light (or photons, as faithful readers will remember that light is a particle called a photon). A photon traveling through a cloud of charged particles will scatter many times, so that the photon that appears on the other side of the cloud will have very little to do with the one that entered it.

I'll now switch gears completely to describe the relationship between temperature and money. Suppose my mother in her younger days was living in a rather small apartment in London. My mom is a rather accomplished amateur interior decorator, and we'll assume she had those skills in her flat in London. I'm going to go one step further and ascribe a fickle nature to my mother which I would like to emphasize for posterity that she does not in actuality possess; in my hypothetical situation, this invented nature of hers combined with her penchant for interior design led her to continually change her mind on how she wanted to decorate her small house.

Ok, now we'll add money. If my mom had a lot of money, she could indulge her ever-changing whims. One week she could go for ultra modern and the next for antiques. Basically, the furniture would be coming and going, styles would be in and out, her little flat would be in a constant state of flux. Suppose, however, that she suddenly lost all her money; my mother would be forced to pick the cheapest option with which to decorate her house and stick with it. While she may still desire a change, she would have to settle for the most practical option.

In the physics of chemical reactions, temperature is like money. If my mom has money, she can change her flat at will - she can bring in new stuff, get rid of the old stuff easily whenever she wants. If the temperature is very high, a chemical reaction can occur easily and can go in both directions. Specifically for the purposes of the CMB, at high temperatures atoms can easily lose electrons and become ionized, before quickly finding other electrons freed from other atoms to become neutral again. In the early universe, the temperature was very hot and this was happening all the time; the universe was a soup of charged particles and photons bouncing off each other constantly. In particular, the photons never went very far before hitting another charged particle.

However, when my mom no longer had any money, she was forced to pick the cheapest option and stick with it. Similarly, after the big bang the universe began expanding and cooling. As the temperature dropped, it was no longer so easy to ionize atoms. Eventually, the universe cooled enough that it dropped out of thermal equilibrium. That meant that all the atoms had to neutralize, because a neutral atom requires less energy than an ionized atom and free electron, and nature prefers to minimize the amount of energy in any system (just as my mom had to settle for the cheapest decor). Once the atoms were all neutral, any photons that were bouncing around no longer had to travel through a soup of charged particles. In effect, the photons that were produced just as the universe become neutral did not scatter again. These photons are still traveling through the universe and we can detect them now; they are the CMB. They still contain information from the last time they interacted with matter, which was 13 billion years ago, right when the universe became neutral.