I've had a few requests to do an explanation of my project in Plain English, so I figured I'd turn my lack of things to talk about to my advantage, and do some explaining for the rest of my project. I'll start with a short (hopefully) easy to understand/non-technical explanation of gravitational waves and their detection, and then from there go into a little depth on what I'm doing specifically. This will probably be kind of lengthy so maybe grab a comfy seat and a snack.
Gravitational Waves:
Part 1
An Introduction to Waves:
When you think of "waves" you probably think of the motion of the ocean, or of the friendly greeting hand gesture, but in the physical/mathematical sense waves are a little different. A "wave" is probably better described as a repeated 'oscillation' in some 'medium'. An oscillation is some kind of small disturbance in something, which we refer to as the "medium". In the case of the waves you see on the beach the oscillations take the form of pressure, and the medium is the water itself; pressure builds up and pushes the water up onto the shore, then the pressure decreases and the water recedes. The timing of this is [roughly] constant, and is referred to as the "frequency". Waves can happen in anything; sound, for example, is waves of pressure moving through the medium of sound. The frequency of the pressure waves determines the pitch of the sound. Light (and all other forms of electromagnetic radiation like radio waves, x-rays, ultraviolet rays, etc.) is another type of wave, though it's a little harder to explain in an easy, physically intuitive way. The easiest way to think about it is to imagine an invisible "sphere" around some tiny point. This sphere is called an "electromagnetic field" and is basically a force that works like magnets. It moves charged particles in one way or another. Imagine some charged particle at a far distance from our little point, and it will be pulled in some direct, let's say "left". If you change the direction of the field suddenly, the particle will move "right". If you do this fast enough, the particle will vibrate back and forth, sort of like how the end of a jump rope moves around when you shake the other end. By changing the field back and forth like this, you're creating "electromagnetic waves", the frequency of which determines the characteristics of the radiation. Do it slowly, and only big things will be effected (will vibrate) with the field, radio waves are like this, it's why you need big antennas to pick them up. The higher the frequency, the smaller the receiver size needs to be, ultraviolet rays and higher frequencies are dangerous because they vibrate things on the scale of our cells and DNA, and if they vibrate them too fast they will be damaged (which can lead to things like sunburn or even cancer). Visible light is visible because it vibrates the cells in our eyes which detect light.
An Introduction to Gravity:
Gravity is a force that exists between all things with mass. You have gravity, I have gravity, the Earth (obviously) has gravity. Gravity is "attractive", aside from being easy on the eyes, this means that gravity pulls things together. You and your cat aren't dragged together from across the room because gravity is actually pretty weak: you need a lot of mass to have any appreciable effect on other objects.
The other key fact about gravity is that it depends on distance. More importantly, it depends on distance to the center of an object. Two objects whose centers are separated by 10 meters will be pulled together more strongly than if they were separated by 50 meters (this is technically referred to as an "inverse squared proportionality", as gravity actually weakens by the distance squared, objects separated by 1 meter are actually pulled together 4 times harder than objects separated by 2 meters, 2 squared equals 4 [if this bit is hard to follow ignore it]). However if both objects "swell up", but their mass stays the same, and their centers don't move, then the gravity between them stays constant; changing the density of objects does have some ramifications about how their gravitational fields act, but in this case we will not need to discuss them, though it should be noted that by decreasing an objects volume enough, ie. increasing its density, you can create a black hole. (Furthermore, note that this does not change the strength of the gravity at some fixed distance, it merely 'allows stronger gravity to exist', as you can now get closer to the center of the object than you could before, and at the certain closeness the gravity becomes strong enough that even light cannot escape, this is called the "event horizon". If you didn't follow any of that just ignore it, it's of no real importance to what we'll be discussing.)
Gravitational Waves:
I'm now going to throw some simplified lies at you, that while not really being technically correct, will give you an intuitive understanding of the some of the key properties of gravitational waves. For a more accurate explanation I recommend checking out the Wikipedia page https://en.wikipedia.org/wiki/Gravitational_wave, and for those interested in an even more rigorous introductory explanation let me know at pshaff2@gmail.com, and I'll email you some reference .pdfs that I have on the subject.
With that out of the way, let's talk about binary systems. Binaries are astronomical entities composed of two star-like objects that orbit each other. While something like the Earth orbiting the Sun could technically be described as some kind of binary, it's usually not because of the huge mass inequality. Typically binaries are composed of two stars, or a star and a black hole or something. A special class of binaries are called "compact binaries", which means that the objects orbiting each other are "compact", these are things that are very massive, but relatively "small". Neutron stars and black holes "compact".
So why am I talking about compact binaries? Let's go back to gravity now. Imagine that you are an astronaut floating outside some big star, far enough away that you're not being cooked, but close enough that you can "feel" the star's gravity (let's furthermore assume that you are magically anchored where you are, so that you're not being pulled in to the star, but that you have a Gravity-o-Meter, which let's you know how strong the gravity you feel is). Unless you are the mass of a sun, your Gravity-o-Meter is going to give off a reading that is pretty much only proportional to the mass of our star. Let's furthermore say that our meter reads out gravity in "G's", that is to say, in multiples of Earth gravity. So, anchored in space, your meter is telling you that you are feeling 100G, that is to say, the gravity from the star is 100 times stronger than Earth gravity at the distance we are anchored.
Now imagine that the star begins to swell up, until it has doubled in size, though it's mass stays the same. Even though you are now technically "closer" to the surface of the star, your meter is still reading 100G. This is because of what we discussed earlier, ie. that changing the size of an object while keeping it's mass constant has no effect on the gravity at a fixed distance. Now imagine that the star starts spinning on some axis. As you can probably more easily intuit, this produces no change in our meter, it still reads 100G.
Now imagine that, all of a sudden, the star shoots away from you some large distance. Now our meter reads a change in the gravity. Like I said above: gravity weakens at large distances, and strengthens at short ones. If the star begins to come closer to us again the gravity we read will once again increase. If the star does this over and over again, we will read repeated change in the strength of gravity, that is to say, we will observe a wave! The medium is the "gravitational field", and the oscillation is the increase and decrease in the strength of the gravity. Like I said above, this is not exactly true, the actual medium is spacetime, and the oscillations are actually strain caused by a change in the spacetime metric, but my explanation holds a lot of the same 'intuitive' properties as the true explanation, and is much easier to understand and explain, so we'll stick with how I've described it.
Binary stars are great candidates for giving off gravitational waves because the stars in them exhibit this kind of motion: they go away from us, then swing back, then go away, then swing back, all in a regular orbit. There are other ways for gravitational waves to be produced (imagine a spinning dumbbell, anything like that will give off gravitational waves): spinning objects with unevenly distributed mass, large clumps of matter moving irregularly, etc., but compact binaries are expected to produce very "clean" waves, that is to say, they are relatively easy to model, so we have a good idea of what kind of gravitational waves they will give off.
Part 2
Detection:
As of yet we have not "directly" observed gravitational waves. This is because they are very, very small and hard to detect, and there is a lot of "noise". The current detectors we use rely on something called "laser interferometry", which is a whole other topic, and is maybe a little too much for one mouthful, so we're going to tastefully gloss over it. The general idea is that gravity bends light (that's a pretty big truth bomb to fire off so cavalier, but just pretend that it makes sense. If you imagine light moving in a straight line, then in a gravitational field that straight line will curve, that's the easiest way to visualize it), so by firing off really, really long lasers and looking at how they bend we can detect gravitational waves from far away. But there's a lot of things that can mess us up. "Seismic noise", the vibration of the earth because of plate tectonics, plus people driving big trucks and stuff nearby can cause the laser to jitter around and give false readings. Also something called "thermal noise", which is a product of the heat produced by the laser can cause problems. There are many, many other sources of noise, but those are the big ones, and they make detecting GW (gravitational waves) really hard. At the present there are two big, international projects out there trying to make a detection, LIGO and VIRGO. LIGO is an international group with two detectors in the US that includes countries like Germany, Korea, and the US, while VIRGO has detectors in Italy, and is a collaboration between Italy and France.
When LIGO was constructed, in the grant proposal it was freely admitted that it would be very unlikely that an actual detection would be made, the technology just wasn't advanced enough. However what would be gained by constructing it was a chance to experiment with it. By trying different noise reducing technologies we could improve detector sensitivity so that the next detector would have a better chance of actually observing GW. [Incidentally we do in fact know that GW exist, as indirect observation can be made, originally and most famously in the the Hulse-Taylor binary, the orbit of which we observed to slow down in pretty exact accordance with predictions made by the existence of GW.] The next generation detector from LIGO, called Advanced LIGO, or aLIGO, is scheduled to go online in the next year or so, and is predicted to have a detection rate between around 4 and 200 detection events a year. There are other projects in the pipeline right now as well, KAGRA in Japan, and INDIGO in India are both associated with the LIGO group, plans for the incredibly advanced Einstein Detector are being fought for, though that would likely begin construction no earlier than 2020, as well as a space-based detector called LISA are all being discussed or begun.
So let's talk about how detection works. Ignoring the physics of laser interferometry, let's shift focus to a math/computer frame of mind. When we begin to make detections, we're going to be able to want to use them for something. One thing we can do with them is something called "parameter estimation", which is basically looking at the wave, and based on the physics that we know we can make informed estimates of the "parameters" of their source. That is to say, we can figure out the masses of the binary components, their orbital frequency, etc. So obviously we're going to want a good system to do this, and here's where I start to come in.
Matched Filtering and Parameter Estimation:
As I said earlier, we have a pretty good idea of what the waves coming from compact binaries should look like. However the 'waveforms' (basically a graph of the detected waves) that we pull out of the detector will not look like that. Noise from terrestrial and heavenly sources will have "corrupted" or signal and added their own imprint to it. So what can we do? We can perform "matched filtering", that's what! The idea behind matched filtering relies on our theoretical models of GWs coming from binaries. By running computer simulations of all different kinds of binaries with different masses and orbital speeds and such we can generate 'template' waveforms, ie. we can figure out what the GWs should look like for a whole bucket-load of different sources. Then, using some math, we can compare these templates to what we're reading on our GW detectors, if they look reasonably similar we can consider this to be a detection. Even better though, we can logically assumed that because the two waveforms look similar, that the source of the detected waveform has similar parameters to the simulation we used to generate that template. So if we run a simulation with 2 neutron stars with masses 10 times the mass of our sun, and we find that the waves generated by these are reasonably close to what we're seeing on our detector, then we know that the source of the detected waves is probably 2 neutron stars with masses around 10 times the mass of our sun. Kind of.
Unfortunately it's not that exact. We know that the source system is probably 2 neutron stars with masses around 10 times the mass of our sun, but what if one of them has mass 9.8 times the mass of our sun? With all the noise in the detected wave, then our template might still look like detected waveform. What we need to do is use some statistics. Using our template and the detected wave, we can actually develop something referred to as a "posterior", which is basically an equation which tells us how probable certain source parameters are. That is to say, we may not be able to say with certainty that the compact binary we are detecting has 2 neutron stars with 10 solar mass each (masses 10 times greater than the sun), but we can say that we are, say, 30% sure that they have 10 solar mass, and 15% sure that they have either 9.8 solar mass, or 10.2 solar mass. Because probabilities are nice, we can actually just add this 30%, and these two 15% to say that we are 60% sure that the masses of the neutron stars are between 9.8 and 10.2. If you've ever heard the phrase "confidence interval" in the news (usually around election season) this is what they are referring to (though election statistics are terrible, most of the time they have no idea what they are talking about and rarely draw the correct conclusions from their data, physicists are much, much better at it. This is where the idea that you can "lie" with statistics comes from). Here's a picture:
Ignoring the numbers, this is called the "normal" or "Gaussian" distribution. It's a really important tool in statistics, because it just so happens that many, many different types of measurements give this shape. There's a reason why, but it's complicated and I'm not going to talk about it. As always, if you want more in-depth reasoning I'd advise you check out Wikipedia. The general idea is this: the horizontal axis represents some value of the parameters the describe the source, let's say the mass of one of the binary components. The vertical axis represents how probably that value is. So the highest point is the most probable value of the mass of the neutron star. Note that there are several graphs overlayed here, each color represents a different one. See how some are "sharper" while others are "flatter", and that the blue is the "flattest"? Flat is bad, because that means that a larger range of values are more probable. Think about it this way, a huge, thin, sharp peak, means that there's one value that's really likely to be our best estimate for the true mass of the neutron star. If the graph is flat, it means that there are a lot of values that are all roughly equally likely to be a good estimate. We want one value, not a bunch, so we like to see sharp graphs.
Recall how I said above that we get these graphs from equations called "posteriors", which rely on what our detector picks up, but also on our template? As it turns out, the "better" our template, the sharper the graph. This makes some intuitive sense, as a better template means that it's closer to the real value of the mass of the neutron star, while a worse template means a worse guess, so more values are likely to be true. So we can tell how good our template is by sharp the graph it produces is. Unfortunately, this takes a lot of time. It's not so bad for one measurement, and for one parameters, like mass, but there are a lot of parameters (between 9 and 15 depending on which model you use) and there is a lot of data to process, so calculation these graphs can become very time consuming. Fortunately, there is a better way! This called the "Fisher Information Matrix".
Fisher Information Matrix:
Fisher Information Matrices (here shortened to FIM) are hard to explain easily, as they're not very intuitive without a math background (seem my previous presentation for a slightly more math-y, but still fairly informal treatment). But the general idea is this: FIM give us numbers which correspond to how sharp or flat the graph is going to be, but all you need to calculate them is the template you want to use. That is to say, we can use FIM to figure out how good our measurement will be without ever actually having to make a measurement. As a lot of detectors and detection techniques are still being developed and honed, this is obviously incredibly useful. Now, instead of having to go through all the time, pain, and computation resources of generating these graphs, we can just compute FIM and use them to compare quality of templates. This allows us to add or remove certain facets and features, and really optimize our "detection pipeline" (how we process the incoming data to extract information). But they do have drawbacks: FIM do not account for bad noise, and certain parameters are not well suited to them either. Because of this, we need to generate "posteriors" and use them to calculate graphs for a bunch of parameters at different noise levels, and compare them to different types of FIM to see how well they work at predicting the sharpness or flatness of the graphs. This is basically what my project is. I am working with some called an "Effective Fisher Matrix" (effFM), which is a further adaptation of FIM for this usage, and is a technique that is more "robust" to noise. It allows us to get a much better idea of the quality of our measurements of certain parameters (hopefully). I am currently exploring the usefulness of effFM over two parameters called the "chirp mass" and the "symmetric mass ratio" of compact binaries made of one neutron star and one black hole (NS-BH Binaries).
So far I have computed the effFM for different values of these parameters, and now I need to do the more time-costly part and generate the posteriors, then compare them to what I see from the effFM. With any luck, they will agree.
So that's about the best I can do for simplifying my project. If you're still curious about it and/or hate your brain, I have plenty of .pdfs for you which go through all of this more rigorously. Feel free to ask questions in the comments as well, or shoot me an email at pshaff2@gmail.com.
Adios!
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