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