SNR, Second Orders, Special Relativity, and Style

Aloha from Hawaii!

Just kidding. I'm in Flagstaff for the week.

But I acquired a ukulele from Dr. Hartman!! Playing the ukulele while walking to school really makes me feel like I'm in Hawaii, especially if it is warm out. But that is not related to this blog. Moving on!

So last week (or last last week I should say), I showed you some of the error results we got for the cosmic string waveform parameters using the inverse Fisher matrix method. You can see below for the values, but basically, they were very small errors, but not horribly unreasonable. After we got these results, we created two new goals for the week:
1) Calculate the signal to noise ratio (SNR) for cosmic strings. The signal to noise ratio will make our results more meaningful. To be explained.
2) Calculate the second order inverse Fisher matrix for the parameters. No, those results I displayed last week are not finished and this step will make our answers more accurate. Also to be explained.

So let's get explaining.

So my 1st goal was to calculate the signal to noise ratio (SNR) for cosmic strings. What is the signal to noise ratio? I actually mentioned it in an earlier post, but this time I'll go more in detail.

Remember this image?

Clarification Point: LIGO-Hanford is the interferometer located in Hanford, Washington; LIGO-Livingston is the interferometer located in Livingston, Louisiana. There are two instruments located in two different places because it helps confirm that a signal is actually a gravitational wave. If a "signal" is only detected by one instrument, then it probably wasn't a signal, just some strange background noise. 

These are the noise curves for the LIGO and Virgo interferometers. The Advanced LIGO curve can be represented mathematically with the following function: , while Virgo's curve matches this function: . You can try graphing the LIGO curve yourself onto something like Desmos Graphing Calculator online and it'll look quite a bit like the Advanced LIGO curve shown in the graph above. If you look at the curve, there is the least noise between 10^2 and 10^3 Hz. 

The SNR compares the loudness of the signal (in my case, from cosmic strings) to the loudness of the background noise. The higher the SNR, the clearer the signal is. Reasonable SNR values are different for every type of object that emits gravitational waves (this includes neutrinos, black holes, core collapse supernovae, cosmic strings, and more). According to Creighton in his paper "Advanced LIGO: sources and astrophysics," "While the existence of this source is rather speculative, cusp waves in particular could achieve very high amplitudes, depending on the values of the model parameters. For the best reasonable parameters, SNRs of 20 or so are possible on a yearly basis" (Creighton 2003). 

Having an SNR value for cosmic strings will give my error calculations meaning because it'll tell me if my error calculations are truly reasonable. Like I said, the error values I got are very small. This really only makes sense if the SNR is quite large (I'm not sure how large, I'll have to ask), because the higher the SNR, the clearer the signal; the clearer the signal, the less likely we will have trouble measuring the gravitational waves from cosmic strings. The less trouble we have, the more likely are measurements will be accurate and precise, meaning the error will be small. Right now, we have small error measurements. This makes sense if the SNR is large. If the SNR isn't large, we may have screwed up. 

So I've been trying to calculate the SNR for cosmic strings. The SNR equation I'm using is the second one in this picture (Creighton 2003):

|h(f)| is the waveform function I showed you earlier. In the case of cosmic strings, h(f) = B|f|^(−4/3) Ɵ (f_h − f)Ɵ(f − f_l).
S_n(f) is the noise curve. The equation for the Advanced LIGO noise curve is .

So how has my SNR calculating been going? So far, I haven't gotten a reasonable answer. My SNR for when B=6*10^-21 is around 53, more than twice as high as Creighton's estimate. This coming week, I plan to look back over those calculations to find my errors. 

My second goal is to calculate the second order Inverse Fisher Matrix for the parameters. What does that mean? Last week, I showed you the first order Inverse Fisher matrix error results. Those results were pretty good estimates, but not very precise. Although my answers went out to something like 15 significant digits, we're really only sure about the first 2 digits or so. Calculating the second order Inverse Fisher matrix will make our results more precise (and thus more accurate), which means we can trust more of the significant digits. 

If you've taken Calc BC, you kind of know what I'm talking about. When we work with Taylor series, we calculate the answer to so many orders, depending on how close we want our answer to be to the true value. Same deal with the Fisher matrix. The first order Inverse Fisher matrix is a pretty good approximation, but the second order is vastly better. We could go out to the third order and the fourth order, but actually, Dr. Zanolin says we probably don't need to. The second order is close enough to the true value that going to any further order will be not worth the effort. 

Speaking of effort, calculating the second order will be much harder calculating than the first order. While with the first order I could do some parts by hand, we'll have to do most of the calculations for the second order using a computer program. Luckily, one of Dr. Zanolin's former students who studies at Cal Tech now (and who is also from Flagstaff ;)) has already wrote a code for calculating the second order, so all we have to do is adjust the code to fit with our cosmic strings project and learn how to run it. 

This has proven difficult for me, considering I have no computer programming background. But that's ok. Right now, I'm trying to learn the basics of Maple, the programming language Ron used. In order to use this language, you need to have a license, so we're getting Maple installed onto a desktop in Dr. Zanolin's room. Until Maple is installed, the most I can do is study Ron's code and learn the basics of Maple. 

So that's all the updates I have for last last week. Now for the broader things I've learned.

So in tandem with doing my project, I've also been trying to learn special relativity (and eventually general relativity) because it is a good thing to understand when studying gravitation. And it is very interesting! It really makes you think about perspective a lot, and makes you question what time is, and what space is, and what it means to move through time and space. Not just physically, but philosophically. For example, technically, from my perspective, I am not moving, the world is moving around me. Special relativity challenges you to think is a different way. 

And learning about special relativity has shown me a few things about math as well. There are many different ways to prove something mathematically. To prove the invariance of the interval, Schutz (the author of the book I'm using) uses simple scenarios with easy math to prove all scenarios with complicated math also follow the same rules (it's complicated.) However, Einstein used a completely different method to prove the invariance of the interval (I don't know what this method is, but apparently, it's more complicated than Schutz's way). I'm starting to think that one of the most important parts of physics and math is being able to make proofs, and in order to do that, you have to use some creative problem solving methods to form answers where others may not be able to find them. I've officially decided that creative problem solving skills are very important to have if you want to be a mathematician or a physicist. 

Dr. Zanolin also pointed out to me that many of the greatest discoveries (Nobel Prize worthy) come from finding cracks in other people's work, and exploring those cracks. By cracks, he means flaws in the logic or little areas that have not been heavily explored. If you can find one of these cracks, you might find a whole crystal cave with huge caverns and an underground river underneath. Or it might just be a crack. But basically, what I took away from that is that it is important to look closely and question everything, even the seemingly insignificant things. 

Also, reading the studies has become easier. I understand some of the jargon now, and am more used to scientific language. has that happened for any of you other SRPers? I'd be interested to know.

Well, that's really it. Except for one funny physicist story. By request, here you go:

As you may have heard before, physicists apparently have horrible fashion sense. Of course, this is a stereotype, but stereotypes do have some truth to them. Dr. Zanolin admits he is guilty of fitting the stereotype. He's dreamed of wearing a Dark Vader suit to school ("if it weren't so darn expensive to rent one!"), and sometimes he walks around campus without shoes. The dean is used to seeing Dr. Zanolin without shoes, but one time, because he ran out of clean laundry, Dr. Zanolin had nothing to wear except for a full suit, so while he was walking around campus with his full suit, he ran into the dean, who always wears a suit, and the dean was so shocked, he thought there must have been some huge event going on, but really, Dr. Zanolin just neglected to do his laundry. But Dr. Zanolin is not the only one. There's this Nobel Laureate physicist, Frank Wilczek, who is super famous, but everyday wears a suit jacket and pants with a T-shirt underneath, mismatches his sneakers, and purposefully does his hair so it sticks up on one side and lays flat on the other. Here's a picture:

Physicists are so fly. 

But anyhow, that's all for now. 

Toodleloo!

Carissa

Bibliography:

LIGO noise curve pic: http://www.ligo.org/science/Publication-S6BurstAllSky/Images/S6_VSR23_LHV_sensitivity_logx.png

Advanced LIGO and Virgo noise curve equations: http://relativity.livingreviews.org/open?pubNo=lrr-2009-2&page=articlesu19.html

Creighton 2003: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.194.4908&rep=rep1&type=pdf

Frank Wilczek pic: http://blogs.lse.ac.uk/maths/2015/08/03/5-minutes-with-frank-wilczek/