Scrumdidlyumptious SNRs And Intense Tensors. And the Grand Canyon.

It's 12:12 am!! I always write these posts at the weirdest times.

Anyhow,
Good morning wonderful humans!

Saturday, I hiked into the Grand Canyon with my physicist buddies, Dr. Zanolin, Marek, Wenhui, Dr. Jones, Dr. Rachmanov, and my dad. It was so much fun!! Physicists are so chill. :)

Yeah, I fell asleep, so now it is 7:49pm of the next day. Oh well.

So like I said last week, the goals for this week were the following:
1) Calculate the signal to noise ratio (SNR) for cosmic strings.
2) Calculate the second order inverse Fisher matrix for the parameters.

I made good progress on both those goals, so I have much to tell.

First, regarding the signal to noise ratio calculations, I went over them with Marek, a Polish graduate student who works with Dr. Zanolin, and he said that my calculations might actually be correct for the range of amplitudes I was using.

Yup, I'm bringing up this graph again, but this time a different pic because I was using the old LIGO curve before. Whoops!

Alright, we need to go over the misconceptions I had that I cleared up with Marek before continuing:
1) The Advanced LIGO curve is actually the orange one above. SO, Advanced LIGO actually experiences the least noise between 50-1000 Hz at an AMPLITUDE Spectral Density (ASD) around 6*10^-23 Hz^(-1/2).
2) The y-axis is not the Noise Spectral Density (Power Spectral Density (PSD) but the Amplitude Spectral Density (ASD). They are closely related. ASD=PSD^(1/2). Basically ASD tells you about the amplitude of the signal in a given frequency range, while PSD tells you about the energy of the signal in a given frequency range. If you've taken physics, you can look at the relationship between PSD and ASD in terms of springs. If you graph the amplitude of a spring with respect to time, you get a sine curve, so springs, like cosmic strings, produce their own waveforms in a sense. We know that the potential energy PE of a spring =1/2kx^2, k=spring constant, x=amplitude. Here we see that the amplitude x is proportional to PE^(1/2), illustrating the relationship between PSD and ASD.

Alright, so why is my SNR correct for the range of amplitudes I'm using? The amplitude values I was using, 10^-17~6*10^-21 s^(-1/3), result in an ASD so much higher than the noise curve that the signal would drown out the noise, resulting in a large SNR.

I graphed my waveform function and the noise curve on the same plot, so that you can see how much higher the cosmic string waveform is at certain amplitudes:

The blue curve is the Advanced LIGO curve, the green is the cosmic string waveform when B=10^-20, and the orange is the waveform when B=6*10^-21. I couldn't even graph the waveform for when B=10^-17 on this plot because it basically pushed the noise curve off the picture. The SNR is the integral of your waveform function divided by the noise curve with respect to frequency, so the more area between the waveform and the noise curve, the higher the SNR.  Because there is a lot of area between the waveform when B= 6*10^-21 and the noise curve from f=50-300Hz, a large SNR would make sense.

So I went to Dr. Zanolin with this news, and he told me to calculate an amplitude that would give me an SNR of 20 or so, which was the SNR that Creighton predicted for cosmic strings in his 2003 study. At SNR=20, I ended up getting 1.08577*10^-21 s^-1/3 for the amplitude, six times smaller than the smallest value in the amplitude range I was using earlier, 10^-17~6*10^-21 s^(-1/3). What gives? Well, an SNR of 20 is more likely to be detected than an amplitude between 10^-17~6*10^-21s^(-1/3). Most cosmic strings probably exist very far away from earth, and because the amplitude is dependent on the distance between the cusp and the point of observation, we would probably see smaller amplitudes than the range I was using. Here is a snippet from Robinet et al.'s study showing the amplitude equation:

So what are we going to do with this new amplitude value of 1.08577*10^-21 s^-1/3? We may go back through our inverse Fisher matrix calculations and replace our current amplitude value, 6*10^-21 s^-1/3, with this new value and see how the error on the parameters is affected. But that is for a later time, after we finish the second order inverse Fisher matrix error calculations.

Speaking of those, how has that been going?

First off, I never stated that the "second order Inverse Fisher Matrix" is the same thing as using "asymptotic expansions of the covariance matrix of maximum likelihood estimators." To be honest,
I wasn't sure if they were the same thing until I asked. But they are! So we are on the final step of the project.

What does this step entail? Like I said earlier, we are doing most of the work using a program one of our buddies from CalTech created. But first, we must calculate all the pieces that we are going to plug into the program: tensors. Tensors are like vectors, but they have an extra dimension, are more complicated, and are used all over physics. That's not a real definition, that's my definition, don't quote me. And let me just say right now, holy guacamole, there are a lot of tensors. I'm still calculating them! Here, let me show you the tensor list and the second order Inverse Fisher matrix equation(Zanolin, et al, 2011):

These are the tensors

Second order inverse Fisher matrix equation. Pretty gnarly.

I'd have to do a lot of explaining to explain the notation and such, but basically, I have to calculate all the possible first, second, and third derivative combinations of the waveform with respect to each of the parameters and then subtract or multiply them with each other. That results in a ton of combinations. Lots of derivating. Lot's of Mathematica. Praise the Lord for copy and paste though. For that third tensor set, v_abc, I got 9!+8!+7!+6!+5!+4!+3!+2!+1! different tensors. Would that be (9!!)? But yeah. The program does the equation part, which multiplies all these tensor combinations together, creating more combinations, and I don't even know. But for sure, this tensor calculating step exemplifies the tedium that one often has to do in scientific research to produce results. This does not just apply to physics, but biology and chemistry as well. Most research projects require you bust your butt on a really boring, tedious step, but once you're done with that step, you have cool results, so I guess it's worth it.

That's all for last week. I have updates for this week, mostly bad news, but I'll write that later.

Ciao!

Carissa

Bib:

SNR picture with the actual Advanced LIGO curve: https://inspirehep.net/record/963331/files/GWIC_Roadmap+ETD+LIGO3b.png

Robinet et al http://arxiv.org/pdf/0904.4718v2.pdf

Zanolin (the big error equation): http://arxiv.org/pdf/0912.0065v4.pdf

Randomness Because I Was in Flagstaff

Hello!
Another post! Oh my!

So like I said in the last post, I was in Flagstaff this past week, so I didn't make much progress on the project. But I did get to compare school life with research life. Being a school girl is easier than being a researcher. But this research project has helped me think a little more about math. As a researcher, when it comes to math, you have to figure out everything on your own (there's no answer book to the universe), so you get used to checking your answer and trying to figure things out on your own. Now, this comes much more naturally to me in math class when I'm doing my homework. I did this before the research project, too, but now, I just do it.

Some advice from Dr. Zanolin: it is better to understand things deeply rather than quickly. If you have the choice to speed through a class and learn all the required material or take your time and learn all the required material plus some extra background and depth, go for the latter choice. This may be applicable to some of us who have tons of AP credits and can skip tons of classes. Maybe instead of finishing undergrad in three years, spend your extra free time (a years worth of free time) focusing on a research project or some other cause you care about. That is much more rewarding and probably looks better to graduate schools (if you're into that academia life) than skipping a year.

Ok, that was my random blurb for this week.

Here's a puzzle Dr. Zanolin emailed the group today! I still need to think about it. See if you can figure it out!
Can anybody  tell me why, in a party, the number of people
who happen to try an odd number of dancing partners, is always even?

Alright, time to get out of bed. Ciao!
Carissa

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/

Math Hacks and Potato Sacks: Day in the Life of a Physicist

After Years
Today, from a distance, I saw you
walking away, and without a sound
the glittering face of a glacier
slid into the sea. An ancient oak
fell in the Cumberlands, holding only
a handful of leaves, and an old woman
scattering corn to her chickens looked up
for an instant. At the other side
of the galaxy, a star thirty-five times
the size of our own sun exploded
and vanished, leaving a small green spot
on the astronomer's retina
as he stood on the great open dome
of my heart with no one to tell.
—Ted Kooser

One of my favorite poems from many years ago. :) I recently remembered it after seeing an email The Writer's Almanac sent me saying that they'd interviewed Ted Kooser in celebration of National Poetry Month. Check out Ted Kooser's interview here: http://writersalmanac.org/bookshelf/ted-kooser/?utm_campaign=APM%20TWA%2020160401%20NPM%20interview%201%20Ted%20Kooser%20Email&utm_medium=email&utm_source=Eloqua&utm_content=An%20interview%20with%20Ted%20Kooser&elqTrackId=cf8aceae637f40ef83a45f543d53987d&elq=b13c7058852a4232829a4e21c10175f6&elqaid=21346&elqat=1&elqCampaignId=18492

But anyhow, let's swing back onto the SRP train. Choochoo!


So we finally got some results that make sense!

Like I said, we are calculating the error on the parameters using the inverse Fisher matrix method using a program called Mathematica to do all our calculations. We set B=6*10^-21 s^-1/3, f_h=75Hz, and f-l=50Hz and calculated the terms. However, when we were trying to find the inverse matrix, Mathematica kept producing an error message. Mathematica was having trouble calculating the inverse matrix because our matrix terms were very different in scale. For example, our first row first column term was 1.9258`*^43, our first row second column term was 4.6191*10^21, and our second row second column term was -.94581. Here's a screenshot of our matrix and inverse matrix calculations.

Woohoo, bleeding off the blog. I don't care.

Let me explain the matrix a little. So the diagonal values (1st row, 1st column; 2nd row, 2nd column; 3rd row, 3rd column) of the inverse matrix are the values that will give us the error on our parameters: amplitude (B), high frequency cutoff (f_h), and low frequency cutoff (f_l). The square root on the 1st diagonal term=the error on B, the square root of the 2nd diagonal=error on f_h, and the square root of the 3rd diagonal=error on f_l.

But anyhow, so after we got this error message, we had to get creative. How could we get our terms to be closer in scale to each other? So Dr. Zanolin created this math hack where we set B=e^c, c=-46.5625274836 when B=6*10^-21. This way, when we take the derivative with respect to B, nothing happens(!) because the derivative of e^c is just e^c.

When Dr. Zanolin showed us this, I thought it was kind of cheating. You can't just manipulate the equations like that. But time and time again, he shows us that you can just look at the math in a different way, do some manipulating, and get an answer where before you couldn't. That's one thing about physicists and mathematicians. They just create new math sometimes. I also noticed this in Physics C when we were learning how to use Gaussian surfaces to calculate the electric field of a 3-D object. It's such an out-of-the-box way to do it, you wonder how Gauss even thought it up. So, in order to be a good physicist/mathematician, you have to be a very creative problem solver. And, you know, understand math very well.

However, I feel like we don't really learn how to be creative problem solvers in math class. We only learn how to be problem solvers. And you don't really need to understand math super well to solve the problems, you just need to recognize the patterns and practice practice practice, and if you practice enough times, you don't really have to know what you're doing, you just replicate what you did before. Does that count as problem solving even? Well, you can probably develop creative problem solving skills just like you develop other skills, so maybe practice does help. We'll see.

Here is the new matrix and inverse matrix calculated using the e^c method.

Notice how close in value the second diagonal (2nd row, 2nd column) and third diagonal (3rd row, 3rd column) terms are in the e^c inverse matrix to their corresponding terms in the original inverse matrix. Crazy huh? But what about the 1st diagonal (1st row, 1st column)? Because we set B=e^c, the first diagonal=the error on c, not B.

So once we calculated the error on B based on the error on c, we got our results! Here they are:

Remember that B=6*10^-21, f_h=75, and f_l=50, so these values for error are reasonable. We still need to do checks to make sure these are correct, but many mistakes and math hacks later, we have results that make sense.

I think I'm done. I kind of want to talk about how quirky physicists are, but I'm tired. I'll tell one story.

Dr. Zanolin was telling me one day about how weird physicists are, and he gave me a bunch of examples from his own experiences. I think he was at MIT, and all the physicists decided to have a potato sack race. Who knows why. But there they all were, a bunch of old white guys lined up on the field, standing in their potato sacks. The race began. You wouldn't think these old white guys with their heads in the stars would be very energetic, but when it comes to potato sack races, physicists can get competitive. They may have Nobel Prizes and PhDs, but none of that matters in the sack. Some guy in the back of the pack started elbowing his coworkers to the ground to get in front. Someone gave the Director of the Physics Department a bloody nose. And all of them were fierce sack hoppers.

I have other funny stories, especially about how physicists dress, so if you enjoyed that one, I'll post more.

But I'm done for now.

Goodnight (or good morning or good afternoon, for whenever you read this).

With utmost sincerity,

Carissa

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

My Blog Title Definitely Wasn't Wrong-- Thoughts on the Detection of Gravitational Waves

Sooo.... gravitational waves have officially been detected.

It was announced Thursday. I'm sorry I didn't update this then, I really didn't feel like just explaining the whole deal. Because, man, that would be a lot of explaining to do, in my own words, when there's already so much written about it already. But then about five minutes ago, as I was about to fall asleep, I thought, why not just do a reflection and reaction? I mean, I do have some things to say. So that's what I'm gonna do.

Ok, but first, I feel like if you haven't read up, I'm not making much sense. Here, PLEASE read these articles!! They explain everything really well, I don't think I could do better on this blog

https://www.theguardian.com/science/2016/feb/11/gravitational-waves-discovery-hailed-as-breakthrough-of-the-century

http://www.nytimes.com/2016/02/12/science/ligo-gravitational-waves-black-holes-einstein.html?_r=1

http://www.nytimes.com/2016/02/14/opinion/sunday/finding-beauty-in-the-darkness.html?action=click&contentCollection=Science&module=RelatedCoverage&region=Marginalia&pgtype=article

That last one is a NYT opinion article, which I just read and which actually just inspired me to do the reflection.


For me, the detection is mind blowing on so many different levels. For one, it's crazy that I've been actually reading up on gravitational waves and the graviton for a while now, crazy that I'm doing my SRP project on THIS TOPIC, and then BAM, HUGE HUGE HUGE SCIENTIFIC DISCOVERY PERTAINING TO THE TOPIC I'VE BEEN EXPLORING HAPPENS!!! LIKE WHAT!?! Now gravitational waves are all over the national news, international news, Facebook... This finding was going to be revolutionary regardless of when it was found, but the fact that it was found NOW??? That's crazy. Because in order for the waves to be detected, you have to have a huge redistribution of mass happening somewhere in the universe, which is hard to find, but we found one and registered it and... only a little after LIGO was updated, too, just weeks after the updates. It's actually pretty lucky the gravitational waves (GW) were detected this soon. So props to the universe for being timely. But more craziness about timeliness, the black holes collided (did you read those articles??? 'Cause I ain't explaining!!) about 1.2 BILLION years ago. And even MORE craziness about time, the signal only lasted for 20 thousandths of a second. A wave 20 thousandths of a second long from 1.2 billion years ago happened to ripple through earth just in time for LIGO to detect it and just in time for me to be doing my SRP on this subject. Seems pretty amazingly perfect to me.

But speaking of perfection, holy guacamole, those instruments are SENSITIVE. They can detect a difference in the size of the tunnels (due to the warping from the gravitational waves) of less than one ten-thousandth the size of a single proton. That's right, proton.  As Lawrence Krauss from the New York Times noted in his opinion article "Finding Beauty in the Darkness," "This difference is so small that even the minuscule motion in the position of each mirror at the end of each tunnel because of quantum mechanical vibrations of the atoms in the mirror could have overwhelmed the signal. But scientists were able to resort to the most modern techniques in quantum optics to overcome this." You've got to be just astonished at the technological advances we have reached thus far that allow us to detect changes one-thousandth the size of a proton. And also to detect a change this tiny without interference from the outside world. If I were typing this blog post inside the LIGO tunnels while they were receiving the signal, I would've ruined everything with my super disruptive clickclickclicking on the keys. I also would be dead since it's a vacuum in there. But anyhow, everything about LIGO is crazily amazing and crazily advanced. The mirrors are suspended by glass thread and insulated in layers of metal and concrete by the tunnels. Can we just take a moment to appreciate the people who designed and built, calibrated and calculated everything pertaining to LIGO? Because of our technology, detecting gravitational waves is now possible.

There are so many more amazing numbers and facts pertaining to GW and the black holes (one 36X the size our sun, the other 29X) but if you've read the articles you saw them. They're not just numbers. If you really think about them, they're insane, seemingly impossible ( like two colossal black holes spinning around each other at 30 times per second??? ), but they are our reality.

In that NYT opinion article, Krauss opens up with the lines, "WITH presidential primaries in full steam, with the country wrapped up in concern about the economy, immigration and terrorism, one might wonder why we should care about the news of a minuscule jiggle produced by an event in a far corner of the universe." But for me as a child, the question was always the other way around. Do our day to day struggles and the stories on the radio have any significance in relation to our great grand universe? Over the years, I've developed a greater awareness and appreciation for the close-to-home issues, but I still always try to keep things in perspective. The black hole collision unleashed a peak power 50 times the power output of all the stars in the universe combined. If that doesn't make you feel small, I don't know what will. But at the same time, once the waves reached LIGO, they registered as only quiet chirps, something we would've never noticed if we weren't paying attention. For something that large to get dispersed into something that small really says something about the vastness of our universe. But also, it shows that even the greatest, most colossal of events eventually fade into almost nothing. In the grandest of grand schemes of things, everything becomes insignificant with time, from black hole collisions to you and me. While matter and energy are neither created nor destroyed, entropy will eventually break down all objects and events into one meaningless everything with no defined edges, nothing to distinguish one point in space-time from another.

But the thing is, if you go about living your life in despair because we all inevitably, as said by Shakespeare's Hamlet, "returneth to dust," you're missing the point. The universe and life and everything is not meant to be mourned, but marveled. While, yes, from one perspective, all of space-time is tiny and insignificant, you must remember that you have a perspective, too, that is just as valid, if not more valid than the universal perspective. We as humans cannot perceive everything, no. We cannot perceive every birth, every death, every cataclysmic event, every star, every grain of dust, every moment in time from the beginning to end. But we can perceive what's important. From our very human perspectives, the universe is meaningful. People around the world have cried and gawked and celebrated these "insignificant" chirps emitted from two black holes 1.2 billion years ago.  We can feel emotional about gravitational waves and graduations, caucuses and coffee, because while we can't perceive everything, what we do perceive, we make meaningful.

But let us not forget, the story of LIGO is a story about how we strive to expand our perspective. As Szabolcs Marka said, we can now hear the universe, when before we could only see. “Finally, astronomy grew ears. We never had ears before.” With LIGO, we will be able to learn more about black holes and the very beginnings of the universe. We will be able to perceive more. And that is a noble goal. For as long as humans have existed, we've tried to unravel the universe's mysteries. It is true that we are always striving to get closer to the universal perspective. But will we ever get there? With each new mystery solved, new ones arise. With the discovery of the gravitational wave, we now only have more questions about the big bang and black holes, not less. But that's what makes science so fun. Why do we strive to reach the universal perspective, even if we'll never get there? To make our lives more meaningful, I guess. For me, the more I know, the more I have to appreciate.

But ahh, now I feel like I just contradicted that last paragraph. I'm still figuring this out. For now, I'm gonna go to school!

Have a nice day! :)

Carissa

Defying Gravity!

Dear Readers,

Hello! You've entered the Lair of the one and only Carissa Cirelli.
Welcome.

Thank you for stopping by.

So you're probably wondering what I'm researching and where I'm going and who I am and all that. Well, then, let's get down to business.

So I'm Carissa Cirelli, a senior at BASIS Flagstaff and these coming months, I'm going to be researching THE GRAVITON with Dr. Darrel Smith and Dr. Michele Zanolin down at Embry Riddle in Prescott. What's the graviton, you ask?

Basically, it's not a real word, according to this blog's spell check. And technically, it's not a real particle either, because it hasn't yet been discovered. The graviton is a hypothetical particle that mediates the gravitational force. What does that mean?

Well, by mediates, I mean "carries" or "transmits." Part of the reason physicists think the graviton exists is because all the other fundamental forces (the strong force, the weak force, and the electromagnetic force) are mediated by particles. The strong force is mediated by gluons, the weak force by W and Z bosons, and the electromagnetic force by photons. If you've taken a high school physics course, you're probably most familiar with photons and the electromagnetic force. The electromagnetic force occurs in electric fields, magnetic fields, and in light . You've probably heard how light is both a wave and a particle? A photon is the particle form of light, and it is what light is made of. More on all this in the coming blog posts.

So the graviton is analogous to a photon. It is a particle that transmits gravity. It is what gravity is made of (I'm simplifying, but basically).

Taking this analogy further, gravity is also thought to be a wave, just like how light is both a wave and a particle. In addition to being in the form of the graviton, physicists think gravity can be in the form of a wave, the gravitational wave. These gravitational waves ripple out from objects with mass (stars, planets, even us) and distort the space-time fabric.

So that's a little intro on the graviton and the gravitational wave.

So what exactly is Dr. Zanolin researching? Well, Dr. Zanolin is one of the many physicists working on detecting the graviton. He's working with the LIGO (Laser Interferometer Gravitational-Wave Observatory) Scientific Collaboration, a group of more than 1000 scientists worldwide whose aim is to directly detect gravitational waves using the detectors LIGO and GEO600. More info here
http://www.ligo.org/
and here
https://ligo.caltech.edu/page/about

I'll go more into the specifics of Dr. Zanolin's focus over time, but for now, here's his site:
http://mercury.pr.erau.edu/~zanolinm/

And cool article posted today! As you can see, this is a pretty hot topic. I'm super excited!
http://www.sciencemag.org/news/2016/02/woohoo-email-stokes-rumor-gravitational-waves-have-been-spotted

That's all for today, folks. I've got lots yet to explain, from the four fundamental forces to how the LIGO instruments work so I'll be posting all this information over time.

Have a great day! :)

Bibliography:
http://www.pbs.org/wgbh/nova/blogs/physics/2014/05/what-are-gravitons/
http://www.bbc.com/news/science-environment-34815668
http://www.fnal.gov/pub/science/inquiring/questions/graviton.html