Update on my job

I'm still trying to decide what my next topic should be, but I figured I'd tell everyone a little about what I've been doing in my new job. As we might recall, I started a postdoc in July at Fermilab outside of Chicago. In my new experiment, COUPP, I'm still looking for dark matter, but the way we go about it is very different. It's probably worth reiterating how the whole dark matter search works, or you can also read over some of the summary posts on the side bar. To recap a portion of those entries, we are pretty sure that dark matter of some kind exists, and we know this gravitationally; I've described in detail rotation curves and the CMB in prior posts, and there are other observations that support the conclusion that there is some form of matter out there that we can't see.

There are certain classes of theories that predict dark matter might be a new type of particle, one that only interacts weakly. These particles would theoretically be all around us, just like neutrinos produced in the atmosphere and the sun (the villains in the ridiculous, apocalyptic movie 2012 from two years ago from whence comes the picture to the right, that I was quite happy to find my father had rented when I went home for a weekend. A friend of mine at Fermilab, Dave Schmitz, blogged about the physics behind the premise to 2012 back when it came out) that constantly stream through. Every once in a while, we expect these hypothetical particles to hit something on earth, and so we build detectors to catch those hits. At this point, we know that dark matter can't interact more than a handful of times per year in the most sensitive detectors we know how to build. Unfortunately, these detectors are also sensitive to any other radiation that's flying around - say at a rate of about 100 times per second. Or 4 billion times per year. And we are looking for a handful of events.

This is the problem with backgrounds that I went into in detail in those early posts. I said there that the majority of those backgrounds are "electronic recoils," where the radiation has hit an electron in the detector. My new experiment is something called a bubble chamber, which was used a great deal in the heady days of high energy particle physics in the 60s and 70s. They aren't used for high energy work anymore, but we've repurposed the technology for our experiment. In our bubble chambers (and I think I will have to talk about the physics a little at some point), we can set it up so that electronic recoils don't do anything in the detector - we are effectively blind to the main background to dark matter searches!

That's probably a bit more setup than I wanted. Because what I really wanted to say (in one paragraph only) was that I spent all day Friday in a jumpsuit (I do love the jumpsuits) with a full face mask respirator (requiring me to shave the beard I had grown while writing my thesis) trying to shove a 1/2" steel pipe down the neck of an incredibly expensive, delicate, and very clean (for radioactivity purposes) glass bell jar surrounded by a stainless steel vessel (making it rather difficult). And failing. Physics isn't all math and equations and brilliance. In fact, for me, most of the time, it's working down in a tunnel doing plumbing, rebooting computers ("wait, it's not working? Have you tried rebooting it yet?"), or trying to carefully put a steel tube in a pressurized glass jar filled with red fluid that might or might not contain HF acid and a poison gas used in WW1. So all in all, pretty fun, even when it doesn't work.


My experiment

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.

A little more about me and the blog

I've talked to a few people in the last few weeks who asked me to write a bit more of an introduction to both me and what I'm trying to do with this blog, so here goes:

My parents are both extremely intelligent people. Before he retired, my father was in academic publishing as an editor for several years before running the Princeton University Press for the last 20 years of his career. My mother is a writer, author of several books on a rather wide-ranging list of subjects including gardening, architecture, biographies of several people and two novels. I grew up surrounded by books on history, literature, politics, etc. Despite this rather literary background, however, all through school I somehow found myself doing my best work in math and science. I graduated college with a physics degree, but I did not then know what I wanted to do (nor did I have a job). Fortunately, my thesis advisor in college, Dan McKinsey, was hired as a professor at Yale University that summer and asked me to come work for him while I figured things out. Seven years later, I left Yale with a PhD in physics and I think it's pretty safe to say that I am now trying to make a career as a practicing physicist.

Although my family has gotten used to this idea during the (many) years spent lost in graduate school without a real job, I think it's still a bit of a mystery to them how I ended up as a scientist. And perhaps more importantly, I've often felt that they (and in particular, my mom) really don't know what I do nor how I do it. While the question of how I ended up here is sometimes a mystery to me (as far as I can tell, I like doing the physics I do and figure I should keep at it as long as people let me), the second mystery is something I should be able to do something about, especially if I want to be a good physicist. Hence this blog.

I truly do want to try to explain my work so that my family, and my mom in particular, understands. As I said before, my mom is a very intelligent woman, but she tends to be a touch skittish around mathematical ideas. And if I can explain things like dark matter in an understandable way to her, it will mean that I myself understand what I'm doing. More generally, I've found over the years that people I meet really have a lot of interest in physics, but they always say, “I was no good at it” or even worse, “my teacher was terrible.” In the blog, therefore, I'm trying to write about what I'm doing and why it's interesting in a way that does not require any background while simultaneously not underestimating the audience.

Once I started writing, I quickly discovered the particular format I wanted the blog to take, which I outlined at the bottom of this post: Update and future plans. So that's the summary.

Also, here's a picture of me at my sister's wedding. I'm trying to be funny here.

Update and future plans

To all three regular readers of this blog,
I apologize for not having posted in several months. By way of explanation, I will say that since April 2, I finished and defended my dissertation, spent a month out of the country, moved to a new city, started a new job and finished it off by watching my sister get married in a beautiful ceremony in Maine. That said, I'm now back and I plan to post more regularly for the foreseeable future.

A giant sun dial in Jaipur, India

To give a little more detail about my new job, I have moved to Chicago to work as a postdoctoral associate at Fermilab, which is the location of the second largest particle accelerator in the world now that the new LHC has turned on at CERN in Switzerland. To a large degree, Fermilab has been the focus of high energy particle physics over the last twenty years, and I'm really excited to be here. Fortunately for all of us, I still plan to work on a dark matter experiment, so I won't have to start on a completely new thread in the blog but instead can pick up where I left off in April.

Let me restate the way I imagine this blog looking - to me, the study of physics builds upon the huge amount of effort and thought that humanity has put into the subject for several hundreds of years (with emphasis on the 20th century). Nothing that we do is a completely new idea, but instead we must draw on all the experiments, theories and results that have gone before. My goal in this blog was to illustrate that idea by starting with a very modern, exciting topic of research like dark matter and showing how each argument that leads us to believe both in its existence and that we might be able to detect it depends on other, more established observations. And then I hoped to explain all of those observations in a more or less simple to understand fashion.

"Standing on the shoulders of giants (taken from mushon)."

As an example, I started with the argument that galaxy rotation curves prove that we are missing something, which of necessity led me to the Doppler effect, light as a wave, Newtonian gravity and back to dark matter. I'm currently trying to explain the Cosmic Microwave Background, which led me to a discussion of thermal equilibrium and then Fourier Analysis, and I'm not quite finished yet.

That is the image I have for this blog, but I'm open to suggestions if I'm failing somewhere or otherwise losing your interest, so please do not hesitate to let me know what you think. And in the next post, I'll get back to physics.

Hugh

Fourier analysis - Sines and integrals (part 3)

We're almost there. Let's talk about two more features of integrals and sine curves. First, (and mom, remember the symbol for "integral" is an s-shaped thing):

$\int sin2(x)dx\; >\; 0$

The integral of the square of the sine function is always positive. This makes sense sort of by definition, because anything squared is always positive, so the negative parts of the sine curve become positive when squared. To illustrate this graphically, I'll show the integral of the sine function:

followed by the integral of the sine squared:

So the integral of the sine times itself is always greater than 0.

Now, the key: the integral of the sine times a sine with a different frequency over an entire period is always equal to 0. Let's slow down and read that one more time, since it's hard to say in a small number of words. The integral of two sine functions with different periods is always 0. To take a specific example, let the second sine function be sin(3x), or one with 3 times the frequency like the green curve here:


The statement I'm making can be expressed symbolically,

$\int sin(x)*sin(3x)dx\; =\; 0$ (integrated over a full period)

How about graphically? Well, here is a plot of sin(x)*sin(3x), and if you look, the green regions are equal in area to the yellow regions, for a total integral of 0.


How about if I show another example, with a sine of four times the frequency.

$\int sin(x)*sin(4x)dx\; =\; 0$ (integrated over a full period)



Again, the areas of the green and yellow regions are equal, and the total integral is 0. Now I haven't proven this is true for all frequencies, but it can be done rigorously (or rigourously); I suppose you might have to take my word on it, but it clearly works for the two examples given.

There's one more theorem that I need to state before finally explaining Fourier analysis, although I hope it won't be too difficult to understand. This is an associative statement, that the integral of the sum of two functions (any functions, let's just call them f(x) and g(x); for example, they could be sin(x) and sin(3x)) is equal to the sum of the integrals done separately:

$\int \; (f(x)\; +\; g(x)\; )dx\; =\; \int \; f(x)dx\; +\; \int \; g(x)dx$

Let me know if that is not clear, because I'm so excited about the punch line, I'm inclined to skip past some of this stuff.

Finally...

To recap, so far we know the following things:

1. Any periodic shape can be expressed as the sum of sine functions with different frequencies.

2. The integral of a curve is the area under the curve.

3. The integral of a sine times itself is greater than 0.

4. The integral of a sine times a sine with a different frequency is equal to 0.

5. The integral of a sum of functions is equal to the sum of the integrals done independently.

Who can guess what the next step is?

Suppose I have an unknown function (like the A chord from the November post).

By Rule 1, I know that this function can be expressed as the sum of many sine curves of different frequencies. Now, suppose I want to understand what the signal actually is - I want to break it down into the frequencies that went into its construction.

What if I multiplied the unknown function by a sine curve of a given frequency that I know and integrated the result over an entire period? From Rule 4 above, if the frequency I control does not match one of the frequencies that make up the unknown function, the integral will be 0. But if I do find a match, all of a sudden, the integral is positive (by Rule 3) and I've identified one of the component frequencies in my unknown function!

Now, I scan my known frequency over all frequencies, and at the end of the scan, I've found all of the elements that went into making the unknown signal, producing a plot like this:

In this graph, I'm basically plotting the value I get when I integrate the product of the A chord function times a sine with a frequency given by the value on the x-axis. In most situations, I get 0, but when I find a match, the integral (or "power") is positive and I see a spike!

Isn't this exciting? And I'm being completely serious here, none of the vaguely self-mocking tone you might find elsewhere in this blog - I find Fourier analysis completely awesome and elegant and beautiful. Simply using mathematical formalism, we can completely deconstruct a complicated and unknown signal into its individual constituents and understand exactly what is going on. It's stuff like this that makes me love physics and math. If I didn't quite manage to get the beauty and simplicity across in the last few posts, let me know and I'll do what I can to fix it.

Fourier analysis - Sines and Integrals (part 2)

In the last post, I attempted to remind the reader of the definition of a sine curve. I particularly wanted to highlight that the sine function is the mathematical representation of a wave, and since waves are representations of musical notes, a sine curve is also a mathematical representation of a musical note. Those who are familiar with music (or perhaps with my post on the guitar back in November) will be aware that different musical notes are simply waves with different frequencies. That is easily related to the sine curve by multiplying the variable by some number. For example, I've been showing plots of just the basic sine function, y = sin(x) or y = sin(θ). If, however, I decided to plot y = sin(3x), then all of a sudden the frequency of the wave would be tripled, as in this figure:

It is pretty clear that during the time it takes the standard sine function (the red curve) to undergo a full oscillation, the higher frequency curve (the green curve) represented by y = sin(3x) has undergone three full oscillations. Thus, I can represent any of the musical notes using the sine function, simply by changing the multiplication factor. This is essentially what I did in the graphical representations from the November post, by just adding different functions together to produce the sounds I wanted.

We are now ready to talk about the two fundamental keys to Fourier analysis. The first is that any periodic signal can be obtained simply by adding sine curves of different frequencies. To illustrate this, I'm going to draw on everyone's friend, Wikipedia, which has a great entry on Fourier analysis and Fourier series (which does raise the question, "why am I bothering to do this when so many other people have already done it before?" but then this is my blog and I can do it again if I want to. In general, for those who are interested, Wikipedia is really good at mathematical concepts, and I use it as a reference all the time). In the Fourier series article, the unnamed Wikipedia author is exactly illustrating the point I'm making here, that any periodic function can be expressed as the sum of sine curves with different frequencies.

The first example is the square wave. This is a wave that alternates between two values, for example either 1 or -1, so that it looks like a box. The sine function is very smooth, so it may seem hard to believe that you can get a square wave from sines. But, as in the following picture, it doesn't take very many iterations before the sines do a pretty good job at imitating the square wave:


A second example, featuring our favorite gimmick animation, is the the sawtooth wave. Assuming I get this to work right, the animation should show a sawtooth wave along with the sum of sines as each additional sine is added to the total. As with the square wave, the approximation gets pretty good without too many steps:


From here, it should be easy to imagine creating the shapes from my guitar post simply by adding different notes together. The second key will be the subject of the next post.

Fourier analysis - Sines and integrals

In case anyone is still reading this, now that it is being updated so sporadically, I'm finally managing another post on Fourier analysis. In this post, I'll try to set up a little bit of the math behind the theory. To do so, I'm going to first remind everyone about the sine function, which I wrote about when talking about the Double Slit Experiment. In that post, I said that the sine function was a mathematical representation of a wave. Here is a plot of y = sin(θ):
Now, that looks an awful lot like the sound waves I was looking at with my guitar back in November. Because they are the same. In fact, when I wanted to depict the sound waves graphically, I used the sine and its partner, the cosine to do it. Going back to the post on the double slit experiment, I believed I compared these functions to a part of speech; by using them, I can now describe a whole host of different phenomena that were previously inaccessible. Including sound waves.

Next, I want to talk about integrals. My mother never took calculus and says she has no idea what an integral is, which means I'm going to try to give a brief introduction (without going into details, alas). The first thing I was taught about integrals is that they represent the "area under the curve," and I think that's really all we need to know about them. If I draw a curve on a coordinate system, for example, like the sine curve above, then the integral is the area between the curve and the x-axis. Therefore, we need to know one other thing to define it, and that is the range of the integral. For example, I am going to zoom in slightly on that sine curve, and then I'll take the integral from x = 0.5 to x = 2.5, which is just the area below the curve between those limits, or the region shaded green.



Now, things can get a bit trickier conceptually when the curve crosses the x-axis and becomes negative-valued. In this case, the integral is still the area under the curve, except that it is now negative. This is represented by the yellow shading.



Finally, if you look carefully, you'll notice that the sine function appears to be symmetric. This will be really important for Fourier analysis. If you integrate the sign function over an entire period, the positive part and the negative part cancel each other out, and we're left with a total integral of 0.



I want to make two final comments about integrals. The key to calculus is finding out that you can generally solve for these areas if you know the functional form of the curve (in this case, for example, we know the curve is a sine curve, so I could write down the function representing the area from calculus). And because I know this is the kind of thing that might interest my mom, in math, we represent an integral with a symbol that looks a little bit like an "S". For example, the integral of sin(x) is written like this:

$\int sin(x)dx$

The "dx" is there partly to let the reader know that the integral is being performed over the x variable.

Carl Wieman and learning science

This will be the second non-Fourier post I will write, and again I apologize. Who knew that writing a thesis and applying for jobs was so demanding? The subject of this post is learning and teaching science. This week, we had Nobel Laureate Carl Wieman visiting Yale, and he gave two great talks on research people have done on how students actually learn science. Professor Wieman has been applying scientific methods to scientific learning for some time now, and among other things, he writes a blog about it.

One of the more interesting conclusions is that the standard lecture format of undergraduate courses is poorly matched to the way people actually learn and retain scientific understanding - in fact, often students come out of these classes thinking more like a "novice" scientist than when they started. By novice, I mean the following: there are certain ways that an expert in a scientific field thinks about that field that are very different from the way a novice thinks about that field. For example, a novice believes that scientific content consists of isolated pieces of information that have been handed down by some authority and require memorization. An expert believes that scientific content consists of a coherent structure of concepts that build on each other, being accurate descriptions of nature and established by experiment. Sad to say, but students coming out of intro science classes are even more likely to believe that science is bits of memorization based on nothing more than faith, as opposed to a coherent argument based on reality.

These results resonated with me, because in this blog, I've tried to emphasize how one builds to a conclusion (like "dark matter exists") from a variety of physical observations and theories (like the 20 posts that followed my original three). I'm sure that sometimes (often?) I fail in communicating this key point about the way I look at physics, but that is ultimately the goal of this blog. And when I start writing it again regularly, I'll try not to forget that.

Finally, Wieman and his group have developed a series of simulations for students to play with that really demonstrate key concepts of physics. One example that caught my eye is something that I tried to explain in a post a few months ago, the photoelectric effect. If a reader really wants to understand what I was trying to say in that post, I highly recommend trying out Wieman's simulation, located here. Especially you, mom (although she's currently in India right now, and therefore not reading this blog at all. By the time she gets back, I'll be writing more regularly...)