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Some of. I will deviate a bit from from the last two talks in that I will not describe individual gravitational wave events,
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but rather what we call the gravitational wave background,
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which is some cumulative combination of all the gravitational waves that that that should be here.
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And I will be talking a lot about cosmology.
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And so this plot is a reminder to all of you that the universe is not static,
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but rather the universe is expanding as a function of time and the scale, the size of the universe we call we call it.
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Is there a. Oh, that's right.
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Yeah. Topper. Yeah.
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So the scale of the universe is as a function that we call a a function of time.
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And, and we will be using this function a lot.
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The rate of expansion. Is is called the Hubble Constant, and that's also a quantity that we need.
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So this is something familiar from this morning's talk.
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I will be talking about the quantity omega GW, the energy density of gravitational waves, which has been conveniently defined in the first talk.
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So just just just to remind you. So there are some factors of PIs and twos which are not important.
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There is the power spectral density of gravitational waves.
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This is essentially this h squared here inside pointy brackets.
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And effect of frequency cubed and then a Hubble constant,
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because we are comparing this to the density of the total density of our universe, which is called the critical density.
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Now, there is another way to write this down.
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So you would you would notice that the Hubble constant has units of one over time.
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Therefore this whole coefficient has units of time squared.
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Now if you, if you define,
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if you define omega GW such that is proportional to this thing h c squared times frequency squared you get that H these dimensionless.
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Which means that it is essentially the same thing as your original metric perturbation.
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This little h from the from this morning's talk.
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So it doesn't have units even though it's it's a four year transform it has units of of just a number.
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So we will be using both omega three W as as a definition of energy, of density of gravitational waves at a frequency F and also eight C,
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which is the typical strain that the gravitational wave at frequency F will have.
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So here's another familiar plot which summarises the different sources and detectors I'll be describing today.
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So here on the on the right at high frequency.
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So here here is a frequency axis, a height, relatively high frequencies of 100 times per second.
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Let's say you have light doe and Virgo.
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This black curve of the sensitivity curves of these detectors, that means roughly that if a signal is above these curve.
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So on the y axis, we have the characteristic strain.
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So the typical amplitude that the gravitational wave would have, if the signal is above the curve, it's going to be resolved and detected.
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Roughly. And if it's below, there is too much noise in the detector to see it.
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Now, the noise is approximately because this is in large scale.
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It's approximately the area between the two. So here you can see the very first direct competition, a wave detection in in red.
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So you see that it's above the light curve. So we did indeed.
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The fact that there is another curve, it's called a plus here.
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This is what people hope to upgrade light go to in the near future.
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You see it's lower, so it's more sensitive. Okay.
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At lower frequencies, we have another gravitational wave detector which will now be in space.
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This is going. This has been already mentioned by Benza.
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It's called Lisa. It's much longer. So it's also an interferometer like logo, but with much longer arm length.
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And therefore it will be sensitive to much longer wavelengths because the detector
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is essentially sensitive to wavelengths that are similar to the size to its size.
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So Lisa will be sensitive to much longer wavelengths.
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And these frequencies are roughly ten to the minus two, ten to the minus three hertz.
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So these are orders of, let's say, hours, days, maybe.
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And sources for Lisa are different. So for Ligo's,
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you've had neutron star binaries and black hole binaries which constitute most of the gravitational
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wave sources because they they are very close to each other and therefore they orbit very fast.
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For Lisa, lower frequencies, longer periods of gravitational waves and also orbits of binaries.
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So you would expect to see white dwarves in our own galaxy.
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You would also expect to see supermassive black holes.
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These are the black holes that that also contribute to this detection by pulsar timing injuries,
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which I'll describe towards the end of my talk when when they are on the very last moments of merger.
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And also some very interesting early universe potential gravitational waves.
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I'll get to that later. Okay. And lastly, I will be mentioning, even if at even lower lower frequency.
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So these are a few years period, let's say that these are the pulsar timing arrays that have already produced a nano graph result.
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These these measure gravitational waves by looking at correlations between pulsar clocks.
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Okay, here's a picture of Legault. Okay, so there are two ligo's and you see the interferometer arms.
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There's a picture. So here's the drawing of Lisa.
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There will be three satellites here, and these will be two ends of the interferometer.
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It's going to be in space, hopefully in the next decade and then ten years from now, hopefully.
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And this is another radio telescope which looks at pulsars.
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It's called the Square Kilometre Array. There are two actually one in South Africa, one in Australia.
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It's under construction now and I will just mention that Oxford is heavily involved in the design and development
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of S.K. and I put the link here to the relevant website on the sorry web page on the department's website.
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And this will will look at pulsar pulsars at a much more precise way.
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Okay, so let's, let's talk first about higher frequencies and then go down in frequency as my talk progresses.
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So again, I'm talking about the background of gravitational waves.
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Now, this background is the sum of all the gravitational waves.
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That are arriving at us right now. Most of them, I've told are so weak that we can't detect them with Lego or indeed any other detector.
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And I should I should emphasise that gravitational waves are weak in their very nature.
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Okay is a small constant and gravitational waves therefore are very weak.
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So for light go you would see something which goes light, which is ten to the -20 or 30 miles, 21.
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So it's zero, followed by 19 other zeros.
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And then the one. This is the effect. So gravitational waves are very weak.
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So it makes sense that most of these gravitational waves are so weak that we can't see them,
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but they do add up to a background of gravitational waves.
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So you can you can think about the noisy restaurant, as Stephen mentioned earlier.
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So if you sit in a noisy restaurant, you might hear some conversations from the tables around you.
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You might pick up a few sentences, but most of the most of the sound waves,
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most of the noise is actually in the form of some blurry, constant background.
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Hum. Right. So I'm talking about this at the moment.
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And and I want to describe to you some of these properties.
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Okay. So gravitational waves are waves and therefore they have an amplitude and a phase.
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Okay. So the phase is what goes. It goes into the cosine or whatever sign that you had in your undergraduate degree.
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And the amplitude is some number. And therefore, you can define a complex number which has this amplitude and phase.
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So for each gravitational wave, for each gravitational wave source, I can define its own complex number.
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And the background is just the sum of all these gravitational waves.
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So we just add all the complex numbers. So we start with the first one, then the second one, the third one.
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And they are completely random, right? Because each black hole binary, neutron star binary, each one of them has a different orbital phase.
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You know, it's in a different position along along its orbit.
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So the phases are completely random. We're just selector, you know, and we add then we add all of them until we've had the last one.
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And then the background is just this red arrow, right?
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So this is the final total gravitational wave, right?
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So if you think about it, each wave is like a step in a random walk in a plane, right?
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Because the directions are completely random and the amplitudes are also random.
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They are drawn from some distribution of binary separation and masses all around the universe.
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So it's just like a random walk in the plane.
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So that's how you would think about if you want to model the gravitational wave background, you just have to solve a random walk.
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Okay, let's introduce the toy model for gravitational wave.
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So the gravitational wave measured by, let's say, Lego is equal to some amplitude,
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which I will choose to be a constant divided by the distance from the source to us.
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Find some oscillation. Okay, so cosine omega dwell omega is also a constant and phi is some initial phase for the black hole binary,
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which is a random phase between zero and two pi. Okay, so this would be a toy model I'm ignoring for the moment.
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The universe is expansion. I'm ignoring everything else.
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This is just the pure spherical way. Okay, now, this is for one source, and you get something like that from every other source, right?
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So you can. You can run a simulation and generate random gravitational waves according to your favourite formation channel for gravitational waves.
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And then you can, you can ask yourself what is the total gravitational wave background going to be?
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So here's what's plotted here. This is a simulation by the light going Virgo collaboration.
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You have time on the x axis, you have the gravitational wave amplitude on the y axis and you see that it just fluctuates a bit.
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It fluctuates. Now, this assumes that there is no noise for the detector, okay?
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In reality, this is going to be subsumed by by a lot of noise from Earth and from the instruments and thermal noise and so on.
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I'm not going to discuss any of that. Okay.
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But assuming there was no noise. This is what you would see fluctuations.
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You know, you would notice that this is symmetric about zero. This is because the cosine is symmetric about zero.
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And that's a physical property is not just because I chose I chose a nice toy model.
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Okay. And and the question I want to answer today is how do you model the amplitudes of these fluctuations?
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Okay, So, you know, if you if you ask me what is actually going to be a T equals 3000 seconds,
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I would say, oh, well, it's just going to be a random quantity.
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But I can calculate the probability distribution of the value at 3000 seconds or any other time.
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To be to be honest. Right. So that's what I want to show you how you can do that.
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And of course, to do it completely, you need to do a complicated integral.
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But there are things that you can do analytically in which we could do right now.
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Okay, so this is what we're interested in.
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The probability that the sum of all the ways from all the sources is equal to some number H.
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This is the probability density function. Okay, So let's talk about two limits.
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One, limit small values of H. What's what's the probability that this happens?
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Well, how can this happen? It can either happen if all of the sources are just not active at the moment.
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Right. So that means that the black holes at the moment to widen them are meeting at the frequency which like cannot observe at all.
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So it doesn't contribute anything. All this is unlikely, right?
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Because you're looking at all the universe, all you could have destructive interference between a lot of them.
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Right? This is very probable because if you take a lot of black holes, a lot of sources, each one has a random initial phase.
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It's highly likely that they will destructively interfere.
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And if they destructively interfere, you will get that the sum is is small.
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Okay. Also, bear in mind that this probability has to be an even function because positive and negative values are equally likely.
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Okay, so what's the conclusion of that? The conclusion of that is that the probability should go to some constant as H goes to zero.
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It's not going to be zero at zero strength. It's going to be some constant at zero strength strength as, i.e., gravitational wave amplitudes.
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Okay. What about very large gravitational wave amplitudes?
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There are two possibilities here either. So again, there could be a lot of constructive interference, but there are lots of sources.
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This is very unlikely that they will all be in phase right to generate a very large gravitational wave amplitude.
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So it's exponentially small, actually. And the other possibility is that you would have one source which is much stronger than the others.
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So it dominates. Much stronger, but not strong enough to be resolved as an individual gravitational wave event.
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So it's right there below the threshold.
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Now, what's the probability for that? So the probability that the strain lives in some interval, the H because of our toy model,
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I told you the amplitude was constant, the frequency was constant, the phase was random.
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I'm talking about the amplitude here. Right. Because that's what what interests me.
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So that means that the probability that h lies in this interval is just the probability that the distance is what is like,
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you know, goes like one over. H Okay, so there's this constant in front OC times the.
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Radial increment. Now, this is like go.
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We are seeing black holes from all over the universe.
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So the universe, as maybe some of you remember from your cosmology courses, is homogeneous and isotropic.
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That means that matter on large scales is distributed roughly evenly.
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And that means that the probability of finding an R inside of shell, of size of the R is proportional to.
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The area of the show times the increment the R so it proportional to our squared times the OC.
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And R-squared is one over eight squared. Therefore, the gravitational wave probability is just eight to the minus two,
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which is this h of the minus two times the R by the H because I divided by the H.
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So it goes like to the minus four case we have a constant at low HS and the power low minus fourth largest.
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What happens if you do a correct calculation are proper exact calculation.
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Well, you see on the x axis, H on the y axis, the probability h normalised by something, whatever.
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Okay. It's just eight divided by something which is very small.
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So I get thinks of all the unity here and here.
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Okay. So at low, as you see, look at the blue curve.
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It goes to a constant and a large h it goes to the asymptotic, which I promise you is an H minus four asymptotic.
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And in fact, you can actually calculate exactly the coefficient.
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Okay. And so you can, you can verify that this nice physical picture is indeed correct.
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Okay. The last curve here is the normal distribution.
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I should just say that this is somewhat surprising if you remember the law of large numbers.
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So for the people who remember that the law of large numbers says that a lot of independent quantum variables when you add all of them,
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should this should be distributed like normal distribution.
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Well, a normal distribution is what you see here. It's not distributed like a normal distribution.
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So it's a homework problem to figure out why this does not violate the law of large numbers.
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Okay, you can do the same thing.
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So we've calculated the we've calculated the probability distribution of the gravitational wave amplitude fluctuations.
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Right. So what is omega GW, the energy density.
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Well, it's h squared, right. So it's like the variance of the gravitational wave.
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Right. So once you have the probability distribution, you can calculate this, right?
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And so you can do it. And if you do it, you get this plot.
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So on the x axis, you have frequency. On the y axis, you have omega.
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GW So you look at the blue curve, it looks roughly like that.
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Okay, it has this F to the frequency, to the two thirds power law at small frequencies, and it goes down to high frequencies.
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Now in purple here is a very interesting thing.
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So if you look at like go observations, they haven't seen any gravitational wave background.
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They've only seen resolved individual gravitational wave coalescence is so they were
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able to put an upper limit on the amplitude of the gravitational wave background.
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And this upper limit is the purple line. And you will see that my calculation falls nicely below the excluded range.
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But I also put the sensitivity curve of the future lie detector, and you would see that this calculation is above that,
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which tells you that there is hope that the gravitational wave background will be detected once Legault reaches this A-plus sensitivity.
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Hopefully, at least. Okay, so we've discussed black holes and neutron stars, and I told you a bit about how this background due to them looks like.
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I would now like to switch detectors and move to laser and describe two types of gravitational wave backgrounds for this detector.
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Now, black holes and neutron stars are things that we know that exist.
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But nonetheless, measuring this gravitational wave background will tell us a lot about the history,
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how they form, you know, about cosmic structure, formation and so on, all the things that go into this.
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Precise modelling of the probability distribution that we've just calculated.
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And once we have observations, we could understand more about the history.
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So I haven't talked about any of these because it's complicated to do the calculations with them.
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So I told you just about the features that that would be the whatever formation history you choose.
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But a lot of physics go in, goes into and therefore once you detect the background, you would, you would know more about the history of the universe.
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This is true for all the other types of backgrounds as well. Okay, So let's go back to Lisa.
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Remember, it's a space based experiment, much longer arms length, much lower frequencies, much longer wavelengths, and much longer periods.
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And again, just to remind you, the the equation between energy density of gravitational waves and the characteristic strength.
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Okay, so in Lisa, you would see white dwarf binaries from our own galaxy.
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These are much lower frequencies.
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These are the same physical system as a black hole binary, because this is just two massive particles that go around each other in a capella in orbit.
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So in the physics of the physics, it's the same. So everything I said beforehand applies, except the frequencies are lower.
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And most of the sources are going to come from our own galaxy.
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That means that they are no longer uniform, they are no longer homogeneous,
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and they are not isotropic because our galaxy is not homogeneous and it's not isotropic.
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And of course there are white dwarf binaries, white dwarf binaries in the whole universe,
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but from other galaxies, because these galaxies are so far away, it's basically impossible to see them.
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So the domain, even the gravitational wave background, will be dominated by our own galaxy.
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Okay, so it's not homogeneous and isotropic.
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So if you go back to the equations that I've shown you previously,
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you just have to modify the spatial distribution and you will get a calculation
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of the probability distribution and also the gravitational wave energy density.
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I should also say that there are less sources that are active at the given moment in time.
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Active means that they are at a.
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And an orbital period which corresponds to a frequency which falls inside the detector's frequency range.
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Okay, let's then not go back on sources.
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Okay. So I'm not going to talk a lot about those because it's the same physical system as as for like, except for these.
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Now, let's let's move to something more exotic.
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And these are not gravitational, you know, not binary systems that emits gravitational waves.
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And so, you know that any. So.
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So the gravitational waves comes from the Einstein field equations.
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And on the left hand side of these equations, you have the gravitational field,
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the metric tensor on the right hand side of these equations, you have the energy density.
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So the matter if you have. Things that, you know, violent changes of the energy momentum of matter,
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you will get changes in the gravitational field, which are gravitational waves.
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Okay. So one way to produce that is that you have two black holes which are orbiting each other.
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They change the energy density and therefore they create the gravitational wave.
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Another way is by other physical mechanisms.
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So now I would like to go back to the time when the universe was much, much smaller than it is today in the very early universe.
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And describe some non-exhaustive list of possible sources for gravitational wave background from events that might have taken place at these times.
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So it's non-exhaustive because there are many more that I didn't put there, but these are some main ones.
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I would say that these come from physics beyond the standard model of particle physics.
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So just to remind you, the standard model of particle physics is the best,
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well confirmed theory we have of electromagnetism, the weak and the strong forces.
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But we know that it is not complete because, for example, we don't have gravity is not included.
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And also in the standard model, neutrinos don't have a mass, whereas we have measured them to have a mass.
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Okay, So we know it's not complete and therefore it makes sense to think about things that happen in theories of physics,
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which are beyond the standard model of particle physics.
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Okay, So one, one example is a phase transition.
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For those of you remember, it's it's got to be a first order faith tradition that happens in the early universe, what I'm talking about.
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So think about boiling water. Okay, so boiling water changes from one phase, water to another phase, which is gas.
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Right. And during the phase transition, you generate bubbles of gas.
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Inside the water bubbles expand and more and more bubbles form and they expand until bubbles combine.
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Right. So they collide and then they combine. Right.
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These bubble collisions are very violent events, and if we had enough energy in the water, they would generate gravitational waves.
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Now, think about the early universe.
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Okay, so the universe was in some phase of matter, doesn't matter what phase, and it transitions to another phase.
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And during this phase transition, you get bubbles, expanding bubbles.
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And if these bubbles are energetic enough, they collide.
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And once they collide, you have these violent changes of the energy momentum tensor and therefore you get gravitational waves.
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Okay, so this is one picture of a phase transition at the very early universe which could generate gravitational waves.
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Now, I put here a picture, a plot, rather, of what that would look like.
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So again, we have my favourite plot frequency, gravitational wave, energy density.
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We have the laser sensitivity curve in blue, and we have one prediction of a theory for such a phase transition in black.
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There is this website b d plot, which is what I use to create this plot.
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You just put the parameters of your favourite beyond the Standard Model phase transition.
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Okay, into the website you click submit and it produces a prediction for the gravitational wave background.
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Okay. I choose some parameters. Chose some parameters that are physically motivated by some physical, some theory.
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Anyway,
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what I want to describe today is obviously not the specifics of these phase transitions because it's too complicated and it would take too much time.
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What what I would like to do is describe the shape. So it turns out that all of these have the same shape.
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They go up, they reach the maximum and they go down.
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Okay. And this increase goes like frequency cubed.
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Okay. And then it goes down.
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So I'd like to explain why it goes like frequency cubed And and all of the physics really is in the position of the peak and amplitude of the peak.
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Okay. Another source of gravitational wave from the very early universe is called is from a collision of objects which have not been detected.
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Called Cosmic Strings. Cosmic strings are.
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I know. Sorry. Cosmic strings.
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Let's say energy concentrations, which are very long.
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So they basically have one dimension. They look like a string. They could form loops, for example.
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And. And when two strings collide, they combine into one string.
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So imagine, you know, those two strings and they collide, right?
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So at the point when they collide, you could create those things that go like this.
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Right. And. And this collision will generate gravitational waves.
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Just like phase transition. So again, these violent changes of of of of of the energy momentum tensor,
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I'm the third thing that I want to mention which actually doesn't look like that and doesn't have these frequencies.
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But I should also mention it because it's a gravitational wave emission at the very early universe.
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And this is actually something that we do expect and we have good reasons to expect.
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And some may be experimental evidence that that this should be detected.
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It's gravitational channel waves emitted during the period of inflation.
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So we have evidence that the universe expanded very rapidly.
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Very rapidly after the Big Bang. Right.
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Very fast.
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And and and and quantum effects during this period of expansion create created the distribute the distribution of matter or gave rise to it.
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Right. And these same effects are supposed to create also gravitational waves.
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We haven't seen them. But if we believe in inflation, they should be there.
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Right. So let's describe this frequency you.
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I should say this is a bit technical and mathematical.
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So for those of you who don't remember all the mathematical details, it doesn't matter.
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But if there are students here, then it does matter for you.
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Right. So we've talked about the metric for an expanding universe.
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The metric looks like this.
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So it has the same time piece and the distances in space increase, as you know, in a way that's proportional to a scale factor.
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We can change coordinates. So we define this at a conformal time.
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So at a data is equal to the T, and then we have a metric that looks like main health T multiplied by the scale factor.
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If we do that and we write down the wave equation from Stephen's talk, we get a very simple equation.
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So again, we take the growing tension of a wave and we take out a fact of a from it to make things simpler.
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And we define K. So K double prime is the second derivative with respect to time E plus.
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Wave number vector squared times K So this is a harmonic oscillator, equal source.
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So the solar sigma is whatever energy density you had.
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Right? So we have an armonica oscillator once we Fourier transform in space.
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So we have a harmonic oscillator with a source, so a driven harmonic oscillator at each wavelength.
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We can solve this. Everybody learns how to solve a harmonic oscillator.
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And this is the solution. What I care about is that the solution goes like one over K and it's linear in the source.
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Okay, That's what matters. Okay, good.
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So I'm talking about low frequencies at low frequencies.
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What do I mean by low frequencies? Well, I mean wavelengths that are much longer than the typical size of the source.
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What is the typical size of the source? So, again, I'm going go back to a boiling water metaphor.
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This is the typical size of a bubble of a water bubble.
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Right. So if you're talking about wavelengths, wavelengths that are much longer than the size of a typical bubble,
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then waves from, you know, energy density is gravitational waves squared.
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Okay. So I take 11h from one side of the saucepan and add another H from another side of the saucepan.
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And if. If if the wavelength is very big, much larger than a typical bubble size, then these ages cannot know about this other.
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So. So the sigma that created them must be uncorrelated between one point and another point.
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So at very large wavelength that the sigma so the source fluctuations have to be what is technically known as white noise.
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Fact means that the variance of the amplitude of these source fluctuations at very large wavelengths, so very small.
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KS And very small frequencies, because K goes like frequency for gravitational waves has to be a constant.
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And indeed, if you look at the phase transition, though, this is from one one model of a phase transition,
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you see that the gravitational wave source fluctuation goes like a constant and then it reaches some.
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Value. And then it goes down and say, here the wave factor is measured in units of the typical source length scale.
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Okay. So that's why this transition happens around one. So what do you what do you see from this plot at K, which is much lot lower than the typical.
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So one over the typical source wavelength, a length scale, it's a constant, much larger.
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It's it goes down very quickly to zero.
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Okay. But for us at low frequencies, it's a constant. And from my solution, from our solution to the Einstein field equations,
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we know that H goes like sigma over K, So h squared goes like sigma squared of a case squared.
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Right. But we want but we know that omega three W is squared times characteristic strain.
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So we have two reactions form back. This introduces a factor of cubed.
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Frequency cubed and therefore are gravitational wave.
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Energy density goes like frequency cubed, right?
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So there's an F squared here, times that are cubed, which is F cubed.
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So it's after the five and then there is a K to the minus two, which is after the minus two.
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This is a constant. So 5 minutes till we get the execute.
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This is true for any causal source of gravitational waves.
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Cosmic string collisions are also causal. What do I mean by causal?
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I mean that that that this emission of gravitational wave event happened in a way that
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that information propagates at the speed of light or lower than the speed of light,
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but not faster than the speed of light. Okay.
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So. Right, so we've explained this rise like a few.
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There's a peak. The peak depends on the specific physics of the theory.
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And, and, and it changes.
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And then there is a decline. And that the slope of the decline also depends on the specific physics of what created the gravitational waves.
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But the fact that there should be a decline does not.
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And it's because the sigma squared variance cannot be.
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It has to go down to zero as you go to scales which are much smaller than the typical source length scale.
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Okay. So I explained to you, I mean, what what these omega three WS look like this gravitational wave energy densities look like for.
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Four phase transitions and very early universe.
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Of course, if we find something that's great because we find it, find evidence for physics beyond the standard model.
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Okay, So towards the end of my talk, I want to move down even further in frequency this and describe let's say, something that we've measured.
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So we've measured this. So the nano graph collaboration is measured this for paper from July.
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Right. So it's very recent. This is the Helens and Downs curve.
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You see that it's not zero, right? So there must be a signal here.
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And the the the idea is that this signal comes from the background of a supermassive black hole coalescence.
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So, again, these are binaries of masses.
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So the theory for the ligo's sources also applies here, except that frequencies are much lower decades.
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And I should also mention that individual events, which are bright enough, will be seen by Lisa when it is operational.
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Okay. Where does this background come from? The background comes from coalescence of supermassive black holes.
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We know if if, if our understanding of physics is correct.
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That every galaxy has a supermassive black hole at its centre.
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These are like a million times the mass of the sun, or even a billion times the mass of the sun, even heavier sometimes.
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And each galaxy has one of them. So how do you get them together to merge?
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You have to merge the galaxies. And indeed, that happens. But before that, let's see how you described that.
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So this is something. Very nice because it's also related to Oxford.
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It's called the Mongolian Relation, and that's named after John McGauran, who is a physicist.
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And and this relation says that if you look at the mass of the stars in a galaxy
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and you look at the mass of a black hole in the at the centre of the galaxy,
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the two are correlated and the Coalition says that one you, you can,
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if you know the mass of the stars, you can predict roughly the mass of the supermassive black hole.
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Okay. So if you have a theory that tells you how massive each galaxy should be,
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you would know from the Mongolian relation how to calculate the mass of the black holes.
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And of course, these are important because they influence the amplitude of the gravitational wave emitted by these black holes.
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Okay. So do we have a theory for the masses of galaxies?
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The answer is yes. It's called the Halo model, started by President Chester and many more people.
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It basically gives an equation for the number of galaxies with a certain mass and certain mass range.
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And the important thing to know. So here you have a log of mass.
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Here you have log of, uh, relative frequency that at high mass as it goes down.
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So you don't have a lot of galaxies with too high a mass at low, and it has some peak at low mass as it goes to some power low.
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Hmm. These are simulation results.
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Okay. But there are certain. But. But the black lines are equations that that you can give in some closed form.
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They have parameters that are determined by simulation.
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Okay. But anyway, we have equations that tell us how many galaxies there are of each mass.
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And once we know that masses of galaxies and black holes are correlated,
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we can calculate how many supermassive black holes we would have of each month's last stage is how many mergers you have of galaxies.
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So it turns out that galaxies merge as the as the universe evolves.
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And it is possible using cosmological simulations to calculate how many mergers
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you should have to give in moment in time and as a function of the sorry,
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as a function of the masses of the components.
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So the total rate of gravitational wave events from supermassive black hole combination coalescence as is the number of galaxies with mass and
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one times the number of galaxies with mass and two times the rate at which galaxies with mass and one merge with galaxies with one and two.
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Now, I have assumed here that once the two galaxies merge, the black holes at the centres.
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Also note this is a known, known non-trivial assumption.
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But let's let's keep assuming that anyway. And the gravitational wave energy density.
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Well, it's just. The energy emitted.
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So you remember, this is energy emitted per frequency.
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So it's the energy emitted well, per frequency, but I've multiplied by T in bottle on both sides.
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So it's the power emitted by the gravitational wave times this rate of change of frequency as a function of time,
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inverse times the rate that I've described integrated over the parameters.
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And it turns out that this depends on frequency, like after the two thirds, after the two thirds is the same,
405
00:45:12,890 --> 00:45:19,430
after the two thirds that you get, if you look at black holes that have the mass of a few solar masses,
406
00:45:19,430 --> 00:45:24,620
a few times the mass of the sun, which I've showed showing you before, and actually if you have an F two,
407
00:45:24,650 --> 00:45:31,940
two thirds, it's a hint that your background comes from binaries of massive particles that are merging.
408
00:45:33,330 --> 00:45:36,930
Now here is another plot from the Nano Graph collaboration.
409
00:45:38,430 --> 00:45:50,270
These grey green vertical lines are essentially the data and if you assume so.
410
00:45:50,300 --> 00:45:52,710
So here again you have the omega.
411
00:45:52,920 --> 00:46:00,090
That look that goes should go like after the two thirds, it's it's equal to F squared times the characteristic strain.
412
00:46:00,450 --> 00:46:04,350
That means that the characteristic strain should go down.
413
00:46:07,340 --> 00:46:14,240
Like the minus. So. So the eight C squared should go like F to the minus four over three.
414
00:46:14,480 --> 00:46:18,350
And therefore H.S. should go like after the minus two of three.
415
00:46:18,740 --> 00:46:20,450
Okay. And if you look at that,
416
00:46:20,450 --> 00:46:29,240
you see this is a slope of two over three and it does really agree with the signal that goes like to f to the to the minus two over three.
417
00:46:30,230 --> 00:46:36,140
If you assume that the signal comes from a supermassive black hole binary background, you can,
418
00:46:36,770 --> 00:46:43,760
you can tune your parameters and go into this calculation of the right phi and get the blue line.
419
00:46:44,540 --> 00:46:51,110
Okay. So it means that once you measure, once you measure the slings and downs curve,
420
00:46:51,290 --> 00:46:55,040
you can learn something about the history of galaxy mergers in the universe,
421
00:46:55,040 --> 00:47:00,950
the history of black hole, supermassive black hole, mergers in the universe as a function of time.
422
00:47:01,730 --> 00:47:06,220
Okay. So that's the third type of fourth type of black.
423
00:47:06,470 --> 00:47:10,040
I'll be talking about all of these backgrounds today.
424
00:47:12,660 --> 00:47:20,850
Each one of them is different, but each one of them, if detected, the one detected, will tell us a lot about new physics,
425
00:47:20,850 --> 00:47:28,470
about physics that we don't know and will provide us with a way to measure a phenomenon that we don't know about yet.
426
00:47:31,170 --> 00:47:41,760
And and that does not require not does not always require sorry it does not always require resolving individual gravitational wave events.
427
00:47:42,930 --> 00:47:49,050
Thank you very much and I'll be happy to speak more about any of this in the break or afterwards.