In the last post, I compared the early universe to a mattress with a number of bowling balls on it, creating divots for matter to fall in and out of. I have to admit that it isn't the best analogy; the behavior I'm trying to describe is relatively universal, however. Imagine a really great vacation spot - initially, people will be attracted to this spot. As more and more people visit it, the pressure of all those people mean that it's no longer an attractive location and they stop coming. Also not a good analogy.
In the end, the point is that local density fluctuations created sources of oscillation. Matter was attracted to regions of high density and fell into the well, before photon pressure became too great and pushed it back out. The final piece of information we need before we can finish this particular section is that regions of high density are hotter than regions of low density. And as we already know, the temperature or energy of a photon is related to its wavelength. Therefore, a photon coming from a region of high density is "hotter" or has a higher frequency than a photon coming from a region of low density. This is how the CMB tells us about the early universe. By looking at the temperature fluctuations of the CMB, we can understand the density fluctuations in the early universe.
To once again plagiarize Wayne Hu's website, he has an expanded version of the movie in the previous post. Here, there are two potential wells with a hill in the middle. When the balls are at the bottom of the well, the temperature is hotter and photons departing at that time are correspondingly hotter. When the balls are not in the well, things are colder and the photons reflect it accordingly (I believe in this movie, hotter is represented by blue and colder by red, since blue light has more energy than red light). By detecting these photons we now know about how uniform the early universe was and we can make conclusions about the distribution of matter and energy. In the next post, I'll start talking about how we decode these photons using Fourier analysis.
Wednesday, August 26, 2009
Saturday, August 8, 2009
Gravitational potential wells (part 2)
In the last post I described gravity as the curvature of space, creating little wells for other masses to fall into. This is the image we want to think about as we imagine the early universe. At that time, the structure we see in the universe today hadn't formed yet - there were no planets, galaxies or clusters of galaxies. Instead, there were small perturbations, small potential wells that contained the seeds of future galaxies. Returning to the image of a bowling ball on a mattress, we can imagine a giant mattress with many small little bowling balls on it. These bowling balls were placed at random, simply because nothing is perfectly smooth. In addition to the bowling balls, there are countless smaller marbles moving at random across the surface of the mattress. None of the bowling balls was very large, but they did create small little divots for the little marbles to fall into or orbit around or bounce in and out.
This isn't the whole picture though. Over a month ago, I described the thermal equilibrium of the early universe, where everything was reacting with everything else, atoms were ionized and electrons were constantly interacting with photons. There was a lot of energy involved in those reactions. In particular, this energy was enough to keep the marbles from settling down in the divots. If too many marbles gathered in a particular place, the pressure caused by all the photons bouncing around tended to push the marbles apart. In this way, a situation very much like the pendulum on the spring was created. The marbles were attracted to the wells created by the bowling balls, but when they tried to reach the center, there was enough energy to push them back out. Once out, they were again attracted to the bottom of the well, and therefore we have an oscillation.
I've taken a nice illustration from University of Chicago Professor Wayne Hu's website. In this movie, the well is caused by the random gravitational fluctuations, or the bowling balls. The marbles are represented by the yellow balls, and the pressure caused by all the photons is represented by the springs, pushing the marbles apart when they get too close to the bottom of the well.
This isn't the whole picture though. Over a month ago, I described the thermal equilibrium of the early universe, where everything was reacting with everything else, atoms were ionized and electrons were constantly interacting with photons. There was a lot of energy involved in those reactions. In particular, this energy was enough to keep the marbles from settling down in the divots. If too many marbles gathered in a particular place, the pressure caused by all the photons bouncing around tended to push the marbles apart. In this way, a situation very much like the pendulum on the spring was created. The marbles were attracted to the wells created by the bowling balls, but when they tried to reach the center, there was enough energy to push them back out. Once out, they were again attracted to the bottom of the well, and therefore we have an oscillation.
I've taken a nice illustration from University of Chicago Professor Wayne Hu's website. In this movie, the well is caused by the random gravitational fluctuations, or the bowling balls. The marbles are represented by the yellow balls, and the pressure caused by all the photons is represented by the springs, pushing the marbles apart when they get too close to the bottom of the well.
Gravitational potential wells
I've changed my mind on how I want to proceed with the CMB. I had a post starting to talk about general relativity, but I've decided that it is too much for this particular sequence. I'd want to really talk about special relativity and general relativity to really do it justice, therefore I've decided to skip it for now. However, that still leaves us needing to understand just what is the information encoded in the Cosmic Microwave Background, so I'll try to do a slightly different description.
Imagine a pendulum - like this one!
The pendulum oscillates back and forth, and as it does so, it traces out a well. The pendulum wants to rest at the bott
om of the well, but it has too much energy, and so it continuously overshoots the bottom. The well looks like the line drawn in the still picture to the right. In physics, something like this is known as a potential well - the force of gravity is pulling the weight downward, towards the bottom of the well, but because of the string, the pendulum just bobs up and down in the well.
There are a surprising number of situations like this, and most gravitational interactions can be described in terms of potential wells. For example, the motion of the Earth around the Sun is an orbit that follows the same path as a pendulum in two dimensions. The Earth wants to go straight to the center of the Sun, just like the ball wants to rest at the bottom of the well; instead, the Earth goes around the Sun forever, unable to reach the middle (thankfully).
General relativity is a theory of gravity. Why does gravity create these potential wells? The answer can be thought of in terms of curvature. Large masses tend to curve the space around them, so that other masses will fall in towards the large one. In this framework, one can imagine the Sun as a giant bowling ball on a very smooth mattress. The mattress dips because of the mass of the Sun, and so the space around the Sun curves. Now, one can imagine rolling a bunch of marbles around the divot left by the Sun; if there were no friction, those marbles could roll around the Sun forever in an orbit, just like the planets.
In this sense, then, mass will curve the space around it to attract other masses. But those masses won't necessarily just fall straight in (although that can happen), but can oscillate, much as the Earth oscillates around the Sun, or as the pendulum above keeps going back and forth.
Imagine a pendulum - like this one!
The pendulum oscillates back and forth, and as it does so, it traces out a well. The pendulum wants to rest at the bott
om of the well, but it has too much energy, and so it continuously overshoots the bottom. The well looks like the line drawn in the still picture to the right. In physics, something like this is known as a potential well - the force of gravity is pulling the weight downward, towards the bottom of the well, but because of the string, the pendulum just bobs up and down in the well.There are a surprising number of situations like this, and most gravitational interactions can be described in terms of potential wells. For example, the motion of the Earth around the Sun is an orbit that follows the same path as a pendulum in two dimensions. The Earth wants to go straight to the center of the Sun, just like the ball wants to rest at the bottom of the well; instead, the Earth goes around the Sun forever, unable to reach the middle (thankfully).
General relativity is a theory of gravity. Why does gravity create these potential wells? The answer can be thought of in terms of curvature. Large masses tend to curve the space around them, so that other masses will fall in towards the large one. In this framework, one can imagine the Sun as a giant bowling ball on a very smooth mattress. The mattress dips because of the mass of the Sun, and so the space around the Sun curves. Now, one can imagine rolling a bunch of marbles around the divot left by the Sun; if there were no friction, those marbles could roll around the Sun forever in an orbit, just like the planets.
In this sense, then, mass will curve the space around it to attract other masses. But those masses won't necessarily just fall straight in (although that can happen), but can oscillate, much as the Earth oscillates around the Sun, or as the pendulum above keeps going back and forth.
Sunday, July 12, 2009
CMB Anisotropies (part 2)
The DipoleThe 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.
Sunday, July 5, 2009
CMB Anisotropies (part 1: tricks with figures)
Now that we've had a week since the last post for us all to calm down about how exciting we found the giant map of pink representing the CMB and the implications that single color had for our understanding of the universe, I want to start talking about "anisotropy." Last week, I defined isotropy as meaning that everything looks the same in all directions. My mother, being a woman of letters, will immediately recognize that anisotropy must be the opposite - everything is not the same in all directions. In the last twenty years, it's been the anisotropy of the CMB that has really changed the physics world.
First, let's talk about the pink map one more time. What is actually being shown in that map is the temperature of the phot
ons coming from that particular region of the sky (the map is elliptical because we are projecting a spherical surface [the sky around the earth] onto a flat space, much like flat maps of the globe are elliptical). The temperature is in this case a proxy for energy, and recall that the energy of a photon is related to its wavelength. Therefore, we can think of the pink map as showing the wavelengths of photons coming from different parts of the sky, and they all have about the same wavelength or temperature (about -270 degrees Celsius if you're interested).
Now, there's a subtlety here regarding contrast, because I never told you what the color actually represents in terms of temperature. If pink means any temperature between 0 and 4000 C, then no wonder the universe looks the same everywhere! To illustrate what I mean, I'm going to once again draw some of my own really high quality images. I have a gas stove in my apartment with 4 burners. When I turn those burners on, there are four hot spots on my stove. Let's assume the main part of the stove always stays at room temperature (70 degrees Fahrenheit or 21 C). Let's further assume that the temperature in the flame of my burners is 3500 F or 2500 C. I can represent this graphically in two different ways:
In the plot to the left, I've used a reasonable contrast, and we can clearly see the white that represents the room temperature part of the stove and the red that represents the hot part. But in the plot to the right, I've used such a big scale (or a small contrast), that the stove looks the same color, just like the map of the CMB.
Hopefully, you're now all asking the question, "so just how isotropic is the CMB?" since I can apparently make a plot that looks uniform just by changing the scale. The answer is that it is very isotropic, but not perfectly. The pink map is accurate up to 1 part in 1000. Basically, all the photons have the same temperature to within 0.1%. Which is pretty uniform. But, suppose we turned up the contrast, so that colors varied with that 0.1% (this would be analogous to switching from the right plot to the left). Now the CMB looks like this:
What about if we went even further, to a contrast of 1 part in 100,000 (this would be like looking for the difference between adding or subtracting a penny from 1,000 dollars)? Here is where the excitement really enters, but I'll talk about that in the next post (CMB plots courtesy of the WMAP homepage, as usual).
First, let's talk about the pink map one more time. What is actually being shown in that map is the temperature of the phot
ons coming from that particular region of the sky (the map is elliptical because we are projecting a spherical surface [the sky around the earth] onto a flat space, much like flat maps of the globe are elliptical). The temperature is in this case a proxy for energy, and recall that the energy of a photon is related to its wavelength. Therefore, we can think of the pink map as showing the wavelengths of photons coming from different parts of the sky, and they all have about the same wavelength or temperature (about -270 degrees Celsius if you're interested).Now, there's a subtlety here regarding contrast, because I never told you what the color actually represents in terms of temperature. If pink means any temperature between 0 and 4000 C, then no wonder the universe looks the same everywhere! To illustrate what I mean, I'm going to once again draw some of my own really high quality images. I have a gas stove in my apartment with 4 burners. When I turn those burners on, there are four hot spots on my stove. Let's assume the main part of the stove always stays at room temperature (70 degrees Fahrenheit or 21 C). Let's further assume that the temperature in the flame of my burners is 3500 F or 2500 C. I can represent this graphically in two different ways:
In the plot to the left, I've used a reasonable contrast, and we can clearly see the white that represents the room temperature part of the stove and the red that represents the hot part. But in the plot to the right, I've used such a big scale (or a small contrast), that the stove looks the same color, just like the map of the CMB.
Hopefully, you're now all asking the question, "so just how isotropic is the CMB?" since I can apparently make a plot that looks uniform just by changing the scale. The answer is that it is very isotropic, but not perfectly. The pink map is accurate up to 1 part in 1000. Basically, all the photons have the same temperature to within 0.1%. Which is pretty uniform. But, suppose we turned up the contrast, so that colors varied with that 0.1% (this would be analogous to switching from the right plot to the left). Now the CMB looks like this:
What about if we went even further, to a contrast of 1 part in 100,000 (this would be like looking for the difference between adding or subtracting a penny from 1,000 dollars)? Here is where the excitement really enters, but I'll talk about that in the next post (CMB plots courtesy of the WMAP homepage, as usual).
Sunday, June 28, 2009
The Cosmological Principle
The definition of cosmology is the study of the structure and evolution of the universe. In modern physics, cosmology begins with the application of Einstein's theory of gravity, or General Relativity (recall this post), to the universe. This is a difficult task and would probably not be possible without a basic assumption about the universe - that it is spatially homogeneous and isotropic on large scales. Isotropy is a statement that the universe is the same in all directions (the universe looks the same whether you are looking directly outward from the North Pole or the South Pole). Homogeneity contends that the universe is the same at all points. These two hypotheses are together known as the "cosmological principle," without which much of our presumed understanding of the workings of the universe would be invalid.
Over short scales, this is obviously not true. Looking at the Milky Way is clearly different from looking at other parts of the sky. This makes it hard to test the hypotheses, as we need to go to larger and larger length scales to really see this principle in action, by averaging large volumes (using painting again as an example, imagine a canvas entirely of one color. Up close, you can see individual brush strokes with a great variation from place to place. From far away, though, one section of the canvas looks much like any other section, as they are all one color. Our universe is like that, if you believe the cosmological principle) .
Viewed in that context (I originally wrote "viewed in that light" but didn't want anyone to think I was making a pun), this rather boring picture of the CMB (taken from the COBE satellite in the early 1990s) becomes much more exciting - as already discussed, the CMB photons are coming from all corners of the universe. And they all look exactly alike (to 1 part in 100,000)! The first measurement of the very smooth spectrum of the CMB provided strong supporting evidence to the foundational hypothesis of cosmology, as the universe truly does look the same in all directions (it's slightly harder to convince yourself of homogeneity, that the universe looks the same at every point, but Copernicus can help here - if we proceed under the conservative assumption [although perhaps contentious from a religious point of view] that we do not live in a particularly special place in the universe [the "Copernican principle"], we can conclude that since the universe is isotropic around us, it should be isotropic everywhere. This implies homogeneity).
The Horizon Problem
Of course, that is not the entire story. I will briefly discuss the "horizon problem" here, before talking about the "anisotropies" in the CMB in later posts (these are the 1 part in 100,000 fluctuations that you can't see in the above picture because they are too small). We've decided the universe looks the same in all directions (the left side of the picture is the same color pink as the right side of the picture). But is the entire universe in causal contact?
My mom might ask, "what does causal contact mean?" If two events in space and time can be caused by the same preceding event, they are in causal contact. Here on earth, this is generally understood in terms of time. If something happens after something else (say, for example, I get a book out of the library because my mother recommended it), there can be a causal relationship (I got the book because my mom recommended it). On the other hand, if things are happening at the same time, they can't be causal (if my mom's recommendation comes at the exact moment I'm getting the book [or after I do so], she clearly can't be the cause of my literary enjoyment).
On universal scales, things are slightly complicated by the finite speed of light which adds a dimension of distance to the picture. We all know that the speed of light is constant, but for most of us, this doesn't really mean anything. We turn on a light switch, and the light turns on immediately. That is because the speed of light is so fast that we don't notice the time it took for the information to travel down the wire to the light bulb and back to our eyes. In space, however, this is not the case. For example, it takes about 8 minutes for light from the Sun to reach us. That means that an event in the Sun can only cause a response on Earth 8 minutes later. Suppose there were explosion in the Sun followed by an explosion on Earth 4 minutes later. The Sun's explosion cannot be the cause of the one on Earth, because any information from the Sun cannot reach us in less than 8 minutes (of course, both explosions could have been caused by some event happening in between, but hopefully the idea is clear).
This gives rise to the horizon problem. We know roughly how old the universe is and we know the speed of light. That means we know how far light can have traveled since the "epoch of last scattering." The problem is that the far right side of the pink ellipse is too far away from the far left side of the pink ellipse to have been in causal contact. Imagine running time backwards and following a photon emitted from both edges directed towards the center. At the time of last scattering, those photons would not have reached the center yet. In other words, what is happening on the left side and what is happening on the right side could not possibly have been caused by the same thing. Yet, they clearly look the same. How is this possible, when they could not have been influenced by the same initial conditions? This is the horizon problem, because the two extremes are outside of each other's causality horizon.
There are some theories on how to solve the horizon problem (with the leading candidate being "inflation") but they are probably beyond the scope of this blog (an argument can be made that the CMB is beyond the scope of this blog, but I hope my loyal reader(s) ignores that argument).
Over short scales, this is obviously not true. Looking at the Milky Way is clearly different from looking at other parts of the sky. This makes it hard to test the hypotheses, as we need to go to larger and larger length scales to really see this principle in action, by averaging large volumes (using painting again as an example, imagine a canvas entirely of one color. Up close, you can see individual brush strokes with a great variation from place to place. From far away, though, one section of the canvas looks much like any other section, as they are all one color. Our universe is like that, if you believe the cosmological principle) .
Viewed in that context (I originally wrote "viewed in that light" but didn't want anyone to think I was making a pun), this rather boring picture of the CMB (taken from the COBE satellite in the early 1990s) becomes much more exciting - as already discussed, the CMB photons are coming from all corners of the universe. And they all look exactly alike (to 1 part in 100,000)! The first measurement of the very smooth spectrum of the CMB provided strong supporting evidence to the foundational hypothesis of cosmology, as the universe truly does look the same in all directions (it's slightly harder to convince yourself of homogeneity, that the universe looks the same at every point, but Copernicus can help here - if we proceed under the conservative assumption [although perhaps contentious from a religious point of view] that we do not live in a particularly special place in the universe [the "Copernican principle"], we can conclude that since the universe is isotropic around us, it should be isotropic everywhere. This implies homogeneity).
The Horizon Problem
Of course, that is not the entire story. I will briefly discuss the "horizon problem" here, before talking about the "anisotropies" in the CMB in later posts (these are the 1 part in 100,000 fluctuations that you can't see in the above picture because they are too small). We've decided the universe looks the same in all directions (the left side of the picture is the same color pink as the right side of the picture). But is the entire universe in causal contact?
My mom might ask, "what does causal contact mean?" If two events in space and time can be caused by the same preceding event, they are in causal contact. Here on earth, this is generally understood in terms of time. If something happens after something else (say, for example, I get a book out of the library because my mother recommended it), there can be a causal relationship (I got the book because my mom recommended it). On the other hand, if things are happening at the same time, they can't be causal (if my mom's recommendation comes at the exact moment I'm getting the book [or after I do so], she clearly can't be the cause of my literary enjoyment).
On universal scales, things are slightly complicated by the finite speed of light which adds a dimension of distance to the picture. We all know that the speed of light is constant, but for most of us, this doesn't really mean anything. We turn on a light switch, and the light turns on immediately. That is because the speed of light is so fast that we don't notice the time it took for the information to travel down the wire to the light bulb and back to our eyes. In space, however, this is not the case. For example, it takes about 8 minutes for light from the Sun to reach us. That means that an event in the Sun can only cause a response on Earth 8 minutes later. Suppose there were explosion in the Sun followed by an explosion on Earth 4 minutes later. The Sun's explosion cannot be the cause of the one on Earth, because any information from the Sun cannot reach us in less than 8 minutes (of course, both explosions could have been caused by some event happening in between, but hopefully the idea is clear).
This gives rise to the horizon problem. We know roughly how old the universe is and we know the speed of light. That means we know how far light can have traveled since the "epoch of last scattering." The problem is that the far right side of the pink ellipse is too far away from the far left side of the pink ellipse to have been in causal contact. Imagine running time backwards and following a photon emitted from both edges directed towards the center. At the time of last scattering, those photons would not have reached the center yet. In other words, what is happening on the left side and what is happening on the right side could not possibly have been caused by the same thing. Yet, they clearly look the same. How is this possible, when they could not have been influenced by the same initial conditions? This is the horizon problem, because the two extremes are outside of each other's causality horizon.
There are some theories on how to solve the horizon problem (with the leading candidate being "inflation") but they are probably beyond the scope of this blog (an argument can be made that the CMB is beyond the scope of this blog, but I hope my loyal reader(s) ignores that argument).
Tuesday, June 16, 2009
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.
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.
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