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!
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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):
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| These are the tensors |
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| 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
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