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