How to Hack the Universe into Giving Up Its Secrets and Why This Is Difficult: A Story about Human Error and Math

Good morning humans!

So like usual, I have a lot to talk about, but in this blog post, I'm specifically going to talk about the methods I've been using and how my project has been progressing. Here we go!

:) Look at me posting this ON TIME!!! WOOO!!!

Here is an example of how cosmic strings can emit gravitational waves.
The string (blue line) formed a cusp, giving the string a large Lorentz
boost and a large mass per unit length at that point. These two factors
increase the energy of the string at that point, and this energy is
dissipated in the form of gravitational waves.

Brief recap, remember that my projects goal is to help scientists at LIGO detect the gravitational waves emitted from hypothetical cosmic strings by calculating how precisely the LIGO interferometers can measure the parameters of the gravitational waves--amplitude, lower frequency, and higher frequency. Knowing how precisely LIGO can measure these parameters will tell us what the instruments are capable of and will help us better interpret the data we receive from the interferometers.

I'd like to mention that I'm doing this project in collaboration with a student from BASIS Prescott, Teerth Gill. He's a sophomore and his sister goes to Embry Riddle and does her own research on gravitational waves. I'll talk more about everyone later, but if I ever say, "we did blahblahblah," it's because I'm working with Teerth on the project.

So like I said briefly, we are using two methods to calculate the error on the parameters: the inverse Fisher matrix method and Dr. Zanolin's asymptotic expansion method.

The inverse Fisher matrix method is the easier method, but also less accurate method. Now remember earlier I was talking about the cosmic strings' gravitational waves' mathematical waveform? Here is is:
h(f) = B|f|−4/3 Ɵ (fh − f)Ɵ(f − fl).
B=amplitude
fh=high frequency cutoff
fl=low frequency cutoff
You probably don't want to hear every detail, but basically for the inverse Fisher matrix method, I take the derivative of this waveform with respect to each of the parameters, multiply these derivations together in all possible combinations (there are 6 combinations), divide these combinations by the power spectral density function, ((l/215)^(-4.14) - 5 (l/215)^(-2) + 111 (1 - (l/215)^2 + ((l/215)^4)/2)/(1 + ((l/215)^2)/2)), take the integral of each of these combinations, put these values into a matrix, calculate the inverse matrix, then take the diagonal values of the inverse matrix and square root these values. These square-rooted values are the errors on the parameters. Crazy, right? To be honest, I don't completely understand why this works yet. I'm trying to learn as I go.

The second method, the asymptotic expansion method, involves this very long, very complicated equation that Dr. Zanolin developed himself. I'm still working on the inverse Fisher matrix part of the project, so I can't really explain this part very well. So far, I've learned that I have to calculate more derivative combinations, taking that waveform function to the second and even third derivative. We're primarily going to use a program for this part. A cool note about Dr. Zanolin's long equation: so we're using it to calculate the error on the parameters of the cosmic string's gravitational waveform, but you can actually use this equation to calculate the error on ANYTHING. Originally, the equation was developed to improve underwater acoustics, I think. But really, you could use this equation for baseball statistics, or ecology, or...who knows. Anything where you're gathering data and you want to measure how precise you're values are, you can use the asymptotic expansion. So basically, Dr. Zanolin's the real MVP.

So how has this been going in practice?

It's hard. The very first thing I had to do was go through some studies and find the waveform function. Not too bad right? Well, reading scientific studies isn't exactly the easiest thing in the world, especially if you're not familiar with the terminology. Plus, I wasn't really sure what a waveform function would look like. But I've realized that when it comes to physics research at least, everything is hard when you first start. You won't really know what you're doing in the beginning, but you have to trust that you'll learn eventually and that you are capable of learning.

Calculations I've been doing. This is (hopefully) correct. See if you can find a mistake!

In addition, I make tons of mistakes. When doing the calculations, I've dropped negatives and multiplied incorrectly and made faulty assumptions. In fact, we've had to calculate the inverse Fisher matrix a few times because the first time, we assumed multiplying two dirac delta functions together isn't much different from having one, which is wrong. In the end, we had to develop our own math hack involving cosine curves, which made the math much more complicated but doable. So yeah, mistakes are common. We use several methods including dimensional analysis (or unit analysis) to check our answers, and sometimes, you just have to sift through your work until you find that one forgotten negative sign, or notice you dropped a square. Dr. Zanolin said something like 20% of physics research is doing the actual research; the other 80% is checking your work for mistakes.

It's cool though, because the answers to these math problems will literally help reveal the secrets of the universe. Knowing about cosmic strings will give us insight into how the universe formed, and early phase transitions, and so much stuff. But you have to do the math first.

So that's how it's been going so far.

Have a nice day! :)

-Carissa

Bibliography:

cosmic string cusp pic: http://aether.lbl.gov/eunhwa_webpage_2/gw.jpg

cosmic string waveform equation: http://arxiv.org/pdf/0904.4718v2.pdf

Awesome Videos related to Gravitational Waves!!!

Guys, physics is awesome. Here are some great videos I found that are very relevant to my project, and will blow your mind.

I watched these NOVA documentaries as a kid, and look! He mentions gravitational waves! It's all coming together!  https://www.youtube.com/watch?v=6MJ0lTEhoL8

Animation that illustrates Newton's Gravity vs Einstein's General Relativity: https://www.youtube.com/watch?v=DdC0QN6f3G4

Physics Girl: https://www.youtube.com/watch?v=GHCc9b2phn0

There's a lot of stuff out there on general relativity, so if you're interested, the internet is your friend!

Carissa

Citations for March 11th Post

Whoops, I should probably include some citations for all that information on cosmic strings. Here are some of the sites I used, primarily the first and second ones:

http://www.ligo.org/science/Publication-S5S6CosmicStrings/flyer.pdf

http://www.damtp.cam.ac.uk/research/gr/public/cs_home.html

http://www.andersoninstitute.com/cosmic-strings.html

http://www.pbs.org/wgbh/aso/databank/entries/dp76st.html

http://www.space.com/9315-cracks-universe-physicists-search-cosmic-strings.html

And here are some of the studies I've been using:

X. Siemens T(the study we are going to compare our results to): http://arxiv.org/pdf/gr-qc/0603115.pdf

Zanolin (the study with the more computationally intensive error equation): http://arxiv.org/pdf/0912.0065v4.pdf

LIGO (lots of authors) (this paper gives the waveform equation and some of the equations for the parameters-- aplitude, lower frequecy and higher frequency): http://arxiv.org/pdf/0904.4718v2.pdf

Weeks 1-3 at Embry Riddle: A Lot Happened

Hello Readers!
About time I put something on here again.
So first I'm going to talk about my project, what exactly I'm doing, how I'm doing it, etc. Some parts I will go further in depth with later. Then, I'm going to talk about my general experience actually doing the project. Here we go!

At the beginning of my project, I knew little about how physicists go about discovering “the secrets of the universe.” They can’t exactly just observe black holes in their natural habitats like you would a troupe of chimpanzees. Instead, physicists often have to observe the universe as if the chimpanzees are invisible and all they can see is how the chimps affect the space around them. Like if some physicists saw a floating banana slowly peel itself and disappear at a rate within the accepted range for chimpanzee consumption, they might think, “Hmm, that there might be a chimp.”

Same deal with the universe. In fact, the discovery of gravitational waves was so exciting mostly because now we can use gravitational waves to observe other aspects of the universe. That’s actually what I’m doing with my project. Physicists think there are these things in the universe called cosmic strings. They are hypothetical objects made of mass-energy that formed in the early universe during phase transitions. In the beginning, these strings were smaller than quarks, but as the universe expanded, the strings expanded. They’re a lot like the lines you see in ice when it cools from liquid, except cosmic strings can stretch, interact, oscillate, and disintegrate. These mass-energy strings are extremely dense—just a 1.6 km long string would be as massive as earth. They are also one-dimensional—no height or width. If cosmic strings existed, they could give us valuable insights into the early universe, so scientists have been trying to detect them. One of the most promising ways to find cosmic strings would be to detect the gravitational waves they emit when they decay. As explained by the LIGO Scientific Collaboration, “Gravitational-wave emission is the main mechanism for cosmic strings to dissipate energy. When a string in a cosmic string network crosses itself, a loop separates from the string. Once formed, a loop is doomed. It oscillates, radiates gravitationally, shrinks and eventually evaporates. Strong gravitational emission occurs at the pinch-off points of the loop, the cusps, which move with a velocity [expected to be] the speed of light. Powerful bursts of gravitational waves are expected to be produced by cosmic string cusps. The amplitude of the signal depends on the tension in the string and the size of the loop. These signals could be detected by the ground-based laser interferometers, LIGO and Virgo.”

            But before we start sifting through data collected from LIGO and Virgo, we need to know what the interferometer can measure (not all the features of these cosmic strings may be equally easy to measure). The gravitational waves emitted from cosmic strings have a mathematical form with depends on physical parameters, such as amplitude, lower frequency, and higher frequency.

It is also important to understand that the noise in LIGO depends on the frequency. We want signals to have as little noise as possible because with less noise the actual signal is much more prominent. In other words, we want a high signal to noise ratio (SNR). See the graph below:

Graph explanation: As you can see, between approximately 10^2 and 10^2.5 Hz, LIGO and Virgo experience the lowest noise spectral densities (approximately between 10^-22 to 10^-23 1/(Hz)^1/2). The lower noise spectral density, the lower the noise, and thus, signals that are between 10^2 and 10^2.5 Hz give LIGO and Virgo scientists the most precise data. 

 The goal of my project is to understand how precisely the laser interferometers can measure these physical parameters. These errors in turn will tell us which physical conclusions LIGO will be able to make and which are beyond the current reach of the instrument. Just to give an analogy, when people tested the prediction of general relativity that light from distant galaxies get bent by the sun (gravitational lensing), it was necessary to have sufficient precision in measuring the trajectories to distinguish between general relativity and Newton's gravitational theory.

The work will involve first estimating the error with a technique called the Inverse Fisher Matrix approach. Then we will use a more accurate technique (which is also computationally much more intensive) developed by Dr. Zanolin, which uses asymptotic expansions of the covariance matrix of maximum likelihood estimators. The idea is to compare the prediction of these analytical studies with previous studies that were performed with computer simulations done by X. Siemens et al.

So that is what my project is all about. So how has it been going in practice? First, I've had to learn SO much stuff. I've had to learn new math and some programming (although my work buddy does most of the programming) and of course, lots about cosmology and gravitation. Here are some of the topics I've had to learn to do my calculations:

general info about LIGO and gravitational waves

cosmic strings: what are they and how do they work?

inverse matrices

estimation theory, maximum likelihood error, and cramer rao bound

dirac delta functions

fourier transforms

gamma functions and incomplete gamma functions

gaussian distributions

noise spectral density

how to use Mathematica

I don't even completely understand all these topics yet. Some I'm still learning as I go, but basically, my brain has been exploding. But it's pretty great. Also, it's cool to see how the math we learn in school is actually applied to learn things about the universe. Yes, calculus is actually used in the real world. And I didn't realize there is so much math I haven't learned yet. I remember thinking in 5th grade that once you've learned calculus, you've basically learned everything. Ha was I wrong! There is a lot more math out there I haven't even heard of, but now, I'm having to use a tiny fraction of this more advanced math, which really makes me realize how much I haven't learned. 

Well, that's enough for now. In my next post, I'm going to talk about the people I've been working with because they're pretty awesome! But anyhow, ciao!

Carissa