 Valery, are you ready? Yeah, I'm good to go. All right, so we can now start with the last lecture of the day with the second lecture by Valery Domke on gravity waves, please. OK, thank you. So maybe to start, let me briefly recap what we discussed in the previous lecture. So we talked about the linearized Einstein equation. So we started from the general relativity equations of motion, Einstein's equation of motion. We expanded the metric as a kind of a background component, a flat background component, and a small perturbation. And then we expanded all equations to first order in this perturbation. And what we found here was this equation here, which is the equation of motion for this linear perturbation H-menu, which is essentially on the left-hand side, just a usual wave equation with a box operator. And on the right-hand side, we have a source term, which is related to the energy momentum tensor of any matter in the universe. And we also just spent quite a time discussing gauge symmetries, the symmetries of GR, and identifying what are actually the independent degrees of freedom. And starting from initially 10 degrees of freedom in the metric tensor, we understood that there's actually only two degrees of freedom, which are propagating in the end, so which are really physical degrees of freedom. And this corresponds to the two polarization of this spin-2 perturbation, which we call a gravitational wave. And we saw that this is actually really something that oscillates. It really behaves like a wave. It propagates at the speed of light. And we saw, in particular, that we could make this explicit in the transfer stressless gauge where we have these gauge conditions, which then really make it very explicit that we only have two independent degrees of freedom. Now, what I want to talk about today is the stochastic gravitational wave background. So this is, in some sense, the analog on what is the cosmic microwave background for photons. So this is some background radiation from the universe in terms of gravitational waves. And the hope is that by analyzing it, by first detecting it and then analyzing it, we can learn about astrophysics, about particle physics, and about the history of our universe. So let's talk a bit about what it actually is. So this object that we'll be talking about today is stochastic gravitational wave background. So the idea is that you are an observer here. And now we have gravitational waves essentially coming from all directions of space with different frequencies in different directions. And they could have completely different origins. So at the detector level, what you observe is a superposition of gravitational waves with different k-vectors, meaning different frequency, different direction, and really, in some sense, this is very, very similar to what we observe from the CMB. So in the case of the CMB, what we essentially observe is some last scattering surface in cosmic history corresponding to the coupling of photons and then the photons from that last scattering surface want to really call it a sphere. From that sphere, they propagate to us, and we observe them here on Earth. And here it's very, very similar. But essentially, this sphere from where these gravitational waves propagate is not necessarily one particular time in cosmic history, but we can really have gravitational waves coming from all sorts of distances and also from all sorts of times in cosmic history. And we can detect them here. And these sources that we are talking about, which shows these gravitational waves, they can be both astrophysical or cosmological. So astrophysical means that, OK, if we are the observer here, if we have, for example, a black hole, a black hole, and merge it very close to us, then we will really see this as a black hole-black hole merger. We will detect it in 1m5 sigma, and we will know what it is. However, in some sense, our detector has a volume. So the volume essentially means in which, how far can these black holes be away from us? And so the signal when it arrives here is really strong enough that we can actually be certain about what we're seeing. If the black hole merger happens further away, which also means we're going a bit back in cosmic history, then we will not be able to resolve this event individually. So if, for example, this circle here now represents the effective detector volume of the AC LIGO, and then if you have black hole merges happening here on the outside, you will not be able to resolve them individually, but you will essentially see some confusion noise arising from these sources. Or alternatively, you can think of these fields that I'm drawing here really in terms of time. So very similar to the CMB, that there is some event in cosmic time which cause gravitational waves. And then simply due to the finite travel time of the gravitational waves, because they travel with the speed of light, we will perceive this cosmic event as a sphere around us, very, very similar to the CMB. And from the detector point of view, these two things look very, very similar. It just means we get gravitational waves, which are coming at us kind of isotropically, different frequencies in different directions. So we can have astrophysical and cosmological contributions. And we really expect to have both. And typically, not always, but kind of typically, this is what the leading order expectation is. We expect this background to be isotropic. So both for cosmological and astrophysical sources, at least for the astrophysical sources if they are extragalactic sources. So many coming really uniformly from all directions. We expect this background to be unpolarized, meaning that we saw the gravitational waves coming to polarizations, which we call plus and cross. And we expect, now you pay that this background should now prefer either of the two. So we should get the same amount of plus and of cross gravitational waves. And typically, we expect this thing to be Gaussian in its statistics, which for astrophysical sources essentially is because it's a superposition of many, many, many independent sources. And then the central limit theorem tells it should be Gaussian. And for most cosmological sources, a similar argument can be made, not always. And so what we're then actually hoping to measure is the amplitude of this background and its spectral shape. And the spectral shape we've parameterized is this capital omega, which is the derivative of the total energy of the gravitational waves per logarithmic frequency interval normalized to the critical energy density. So this essentially tells us how much per logarithmic frequency interval how much energy is stored in gravitational waves. And so that will give us then you will have omega as a function of k of f. And we will get some spectrum, right? And this is then kind of the thing we want to observe with gravitational wave detectors, which have been sensitive to different frequencies. But then when you actually think about how you want to detect it, in some sense, what you are looking for here at the detector level is additional noise. Let me put it in quotation marks. So if you know the story about the discovery of the CMB, they built this fantastic antenna. And the antenna was simply not performing as well as they thought it should perform. And that then at the end was identified as actually discovery of the cosmic microwave background. And in some sense, here the goal is similar. So the goal is to build a very, very sensitive instrument and then to realize that it's not actually that sensitive. Because there's some kind of additional noise contribution, which you cannot account for with any of the instrument components. And that essentially then comes from this cosmological astrophysical background of gravitational waves. That's, of course, a very tricky measurement, right? Because you have to be very sure that the noise is not coming from somewhere else. And in particular, you cannot turn off gravitational waves. So you cannot, in some sense, turn off the source and see if the signal is still there. Because it's really, it's static, it's isotropic. There's one important trick that you can play, and that is you can cross-correlate different detectors. And essentially, if you have some maybe noise source related to your instrument, if you then cross-correlate, for example, between the two LIGO detectors, you would expect the cross-correlation in most noise sources to drop out, whereas the cross-correlation in the gravitational wave contribution would actually stay. So that is essentially the key strategy of actually detecting this guy. So in some sense, I call this, this really has the potential of becoming a cosmological history book. Because you could imagine in some very distant future, we measure some spectrum like this, right? And then you will know, aha, here, what is this? This is, I don't know. We see some phase transition here. Maybe we see whatever, I don't know, super symmetry, right? And we can really, we can really decode this frequency axis into a time axis. And we can kind of read off all sorts of different things. Now, in practice, same as kind of with regular archaeology, the trick with this history book is that it has a whole lot of information, but it's very, very hard to decipher. So it can be very, in practice, extremely tricky to actually get this, obtain this information, which is, in theory, however, encoded in this object. And maybe another thing to mention at this point is that there is actually, so far, we have no confirmed detection of this stochastic background yet. However, we do have maybe a hint by what is, I'll maybe get back to that at the end if we have time or tomorrow of these Pulsar timing arrays, in particular the nanograph collaboration. They have reported that they have seen precisely some kind of excess noise, which is compatible with being such a gravitational wave background. It can also still be compatible with other things. So it's a bit too early to say, but within the next months or a few years at the very most, we will know if maybe we already have seen the first glimpse of the stochastic background. OK, so so much for the introduction. So now we see that the size of object here that we're actually talking about is this energy density here. So let's think a bit more about what actually the energy density stored in gravitational waves. And for this, I invite you to go to this polling link again. But you should now see the first question. So you can either scan the code or you can enter this little link here. And so the question which I just wanted you to think about for a moment and we resolve it and then as we continue is if the gravitational waves, the energy density by gravitational wave, if that actually impacts the coverage of the universe, right? So if you remember yesterday, we said that Einstein equation relates kind of the coverage of the universe on one side to the energy density on the right hand side. And we also said that, OK, this rotation wave is somehow a perturbation of the metric. But if we're now talking about energy density, then you know the question is, should this thing be on the left hand side or the right hand side of Einstein equation, can kind of the metric be curving itself? I mean, it seems a bit circular maybe, right? Which let people even ask the question if the gravitational waves can't even carry energy. And Valerij, there is a question for you. Yes. My manual. Oh, thank you for taking my question. What is harder to detect the cosmic neutrino background or the gravitational background? So I would say, OK, so it depends on the amplitude, right? So for the cosmic neutrino background, we have a very clear prediction. And we know it's going to be extremely difficult. For this gravitational wave background, we have come a bit to different sources, right? But for the astrophysical sources, we have pretty clear predictions. And they are really within reach. So for LIGO, it may be water line, but pulsar timing rays will see this. And also the LISA instrument, so space based interferometer, which will launch in 10 years or so, that will also see it. If they don't see it, then something is really, really wrong with our understanding of GR or astrophysics. So in that sense, I think this is easier. Now, however, if we then actually think about doing what I was indicating here, like actually decoding all these cosmological events and doing particle physics, then the game is open, right? Then it's difficult. And then I don't know what will happen first. Because we have predictions, but the predictions depend on model parameters. We'll talk about that tomorrow. So we may be lucky. Nature may be kind. And we may see something very soon. Or nature may be not so nice. And then it's going to be tough. Here's the link again. Maybe somebody can type it in the chat. I cannot copy paste here. I have another question by Max. Yes, hello. So I'm still trying to understand executive in the difference between the CMB and this gravitational wave, so he has the background. So in the case of the CMB, why are we then not also considering that we have not only cosmological sources, but also the astrophysical ones? Well, we do. I mean, the CMB is simply photons, right? So we do, of course, have sources for photons in the universe, right? Just in that case, we call them foregrounds. So I mean, there are, of course, whatever, there's galaxies and hydrogen clouds and whatnot, which are emitting photons. And we do have to subtract all of those before we can actually detect the cosmological CMB. And ideally, we would hope to do the same here. So as a cosmologist, I call the astrophysical stuff foregrounds, right, and say, please subtract them. The astrophysicists call our cosmological stuff foregrounds and say, please subtract them. But yeah, in principle, yes, yes. It can, however, I mean, the thing is that potentially the astrophysical and the cosmological signal are very, very similar. So it's also trivial how you actually subtract. But there's definitely ideas on how to do that. This is what I'm saying. The difference is that in the case of gravitational waves, it might be more than difficult to subtract it than in the CMB case. Yeah, yeah. But there are definitely ideas on how to subtract it. So one maybe a bit crazy idea is that if we build something like the Einstein Telescope, so next generation of ground-based detectors, they will be so sensitive that they will be able to resolve every single black hole, black hole merger in the entire observable universe. So in our entire Hubble patch. And then you can reconstruct the waveforms of the gravitational waves, extrapolate that back to low frequencies, and then subtract that out of the Lisa band. In principle, you can do that. In practice, there are, of course, many, many difficulties in the process. OK, thank you. But then if I look at what people do with LHC, you also think it shouldn't be possible. And then they manage. So OK, so what is the right? OK, so the question here is, essentially, what is even this is we touched a bit about on this yesterday. The question is, what is even the definition of a gravitational wave in curved spacetime? Because yesterday, or if I write this. So yesterday, we simply separated the metric into a flat part, which did not depend on time or space. And the perturbation and the perturbation did depend on time and space. So it was clear how to separate it, too. You just separate out the constant part. But if we have some curved background like this, and then we have some gravitational wave like this, I mean, how do we even decide what is the gravitational wave and what is the background? And in fact, it is only possible if we have a separation of scales. So if there's a typical length scale associated with the background, let's call it LB, and there's a typical length scale associated with the gravitational waves, called the lambda gravitational wave. And only if these are really very different, can we sense if we talk about the gravitational waves? Because then we can have a prescription where we say, OK, let's average over some distance d, such that d should be much, much bigger than the length scale of the gravitational wave. But it should be much, much smaller than the background. And then if we average over d, we will essentially average out the gravitational wave and recover the background. And then if we subtract, take the full thing, subtract this now found background, we can identify what is the gravitational wave. So it really requires this separation of scales. So we need lambda here to be much smaller than d, to be much smaller than l. Or alternatively, so this is now in kind of the length domain. You can also go into the time domain. Then you would need that the frequency of the gravitational wave is much faster than kind of the frequency of the background, so the time scale over which the background changes. And then you can do a similar game in the time domain instead of in the spatial domain. And this is really a necessary requirement. Otherwise, we cannot even sensibly define what is the gravitational wave. And so now with this definition, we can try and get to this question of what is the energy momentum tensor. And to do so, I want to expand the g mu nu, so the left-hand side of the Einstein tensor, in powers of h. So yesterday, we did it at the first order, but of course, we can continue this game. So the full g mu nu is the g mu nu of the background plus the g mu nu, which is first order in h, plus the g mu nu, which is the second order in h, plus and so on and so forth. And now this background part with this definition that we have above, so this has no powers of h in sight by definition. And this means this only has long length scales, meaning small momentum scales, if we go to Fourier space. So this only involves modes with small k. And the typical length scale is kind of this lb scale. Here we have only thing which have one power in h, meaning these are our short length scale objects, meaning they have large k in Fourier space. And the typical length scale here is this lambda gw. Now this guy is by definition second order in h. And now something interesting happens because now you can really have both small and large k modes. And the reason is that you can always add two large k modes if you choose them parallel with what opposite sign so that they give a small k mode in the end. So here, because we have h squared, so essentially break the, this is now beyond the linearized theory we discussed yesterday. So now you can really have small and large k modes. And because precisely you can have one plus another and then this can be actually close to zero, even if these two individual k's are large, if this k1 is roughly minus k2, right? So two large k's which correspond to the small length scale can conspire to give you a small k's or a large length scale in the end. So now we can take this and plug this back into Einstein equations because Einstein equation in the end tells us how to relate the curvature of spacetime with the energy momentum tensor of the universe. And it suffices now for this argument to only look at the small k part of the Einstein equation. So the idea is we have the full Einstein equation and kind of we do a Fourier decomposition of the entire term on the left side of the entire time in terms of the right kicking into all linear effects, all non-linear effects. And then we're kind of gonna separate out the small k part and the large k part. So now for the purpose of this discussion, we will focus on the small k part. The large k part is also interesting. The large k part would tell you how gravitational waves propagate in curved spacetime. But that's not the question I'm interested in now. So if we look at the small k part, and then to the small k part, we have this thing contributes, right? And part of this guy contributes. So then Einstein equation becomes this guy, then the second order part I'm gonna put on the other side of the equality sign. So there's a contribution from the second order, but not the entire second order term, but only the small k part of the second order term. And then we have the energy momentum tensor. And also the energy momentum tensor, of course, we have to take only the small k part. And what does it mean in practice to take the small k part means that I perform an averaging procedure precisely over this distance D. And this kicks out all the high momentum parts, kicks out all the small length scales and I'm already left with the long length scales. And I do the same here. And here if I do this averaging procedure, this guy is now essentially the energy momentum tensor of the background. And so we're relating now the demunew of the background, which is essentially the curvature, the curved metric of the background with the energy momentum tens of the background, but we have found an additional term, right? And this additional term precisely comes from combining two gravitational waves with high frequency such that when I add it to momentum, I get something an object with low frequency. So this is a new contribution, which will give the suggestive name of eight pi G over C to the fourth, so same prefactors here and they call it small T-menu. And this small T-menu is now indeed the energy density carried by gravitational waves. So here that we now see gravitational waves carry energy density, this curves the background, right? Because it's an additional source on the right hand side. So it impacts the background contribution on the left hand side and this is really the definition of the energy momentum tensor of gravitational waves. So this is a logic which is nearly circular. It feels a bit circular, but it's not quite circular. So we've actually in this way, we've really derived how kind of the energies of gravitational waves impact the background. So if we go back to the question of Paul, right? The correct answer is indeed that second order, because here there's a two, right? So at second order in this age perturbation, gravitational waves do curve the background. And we also now have a prescription of how we compute this energy momentum tensor. And the prescription is to take G-menu, do this perturbation metric plus like background metric plus more fluctuation, expand the entire thing to second order. So one order more than what we did yesterday and then perform an averaging procedure. And that should give you the energy density in gravitational waves. May I order any questions so far? Yes, one. Yes, please. Yui. You can talk. Yui, we don't hear you. Microphone is open. Can you hear me now? Yes. Ah, sorry. So my question was that what does it mean to have a negative energy density for the gravitational waves? Ah, so we would see that actually it will be, so the, I mean, here we have all the components, right? We'll see that when we compute the T-00, which is actually the energy density, we'll see that it's actually positive. Because now, like there's a minus sign here, this is true, right? But we also still need to evaluate this thing, right? And we don't know yet what is the sign of this entire object. Okay, okay, okay. Okay, thank you so much. No. Just one more, Julian. Yes, so it's still a bit unclear to me why averaging the stress energy tensor over D is basically the background energy tensor. Maybe you elaborate on that a little bit further. You mean here on the right-hand side? Yes. Yeah, okay. Right, so I mean, this team you knew was really, so we started from the full Einstein's equation, right? Where we have G-minu, right? Is this pre-factor here, right? T-minu. That's the full story. And so if we do, and now essentially what we're doing, and so this right-hand thing, essentially, okay, so this right-hand thing is really kind of the the mantel content of the universe, right? This a priori, the way it's written here, does not know anything about gravitational waves. This is just the entire energy density of all the matter in the universe. And on the left-hand side, we have something which depends on G-minu and on small G-minu, right? So the metric, small G-minu, enters into this large G-minu. So if we now include gravitational waves, we perform a perturbation here on the left-hand side, but the right-hand side is never changed, right? So in some sense, in some sense, it's maybe not even necessary to do this averaging procedure here, right? I mean, it's just kind of to make things fully consistent. So the trick is that we have to, on the left-hand side, we have to separate the two contributions. On the right-hand side, essentially, which stays unchanged. And then we do this trick of kind of moving this contribution from the left to the right, so that we see that it acts as an energy density. So next stage is Suryush. Yeah, hello. Yeah. Can you hear me? Yeah. Yeah, so I was just thinking that so now here you're using the condition that the length of the background scale is much greater than this lambda. So as I know that during inflation, that some of the wavelengths are straight outside the horizon. So then does this condition will hold there? Yeah, no, it doesn't. So this is, no, no, then it's like you would, because the thing is we want to, yeah. So in fact, when we will talk about that tomorrow, right? So in fact, when the gravitational waves are outside the horizon, they don't contribute to energy in this way. So when we talk about the energy density carried by gravitational waves, we're always talking about sub-horizon processes. So we're talking about kind of something we can see in a detector or something which is maybe measured by ineffective, so the degrees of relativistic, relativistic degrees of freedom. But I would, this logic does not apply for once they are super-horizon, yeah. Okay, thank you. Thank you. Okay, so let's... There is one more question by Maria. Yeah. So I would like to ask why did you make this qualitative difference of separating the tensor that corresponds to the background and the other one that appears at second order? Why did you put one in the left-hand side and the other one in the left-hand side? This corresponds to a qualitative difference between the one corresponding maybe to the energy content and the other to the curvature. Yeah, so this is just kind of a trick to make, kind of to make it a bit more intuitive, right? Of course, the equation will be equally correct if I put the thing on the left-hand side, right? So then I would maybe, I mean, in the end, what I have is something which relates the background, the curvature with the background energy density and somewhere in this equation is an additional term. Now we find it convenient to think about this term as essentially an additional energy density, but of course you could put them in, the equation will be equally correct if you put it on the right-hand side. But this is a kind of, this is maybe a quick and somewhat dirty way of deriving this expression. But of course the equation will be equally correct if you put it on the other side. Okay, one more question by Arushi. Hi, so I wanted to ask about this averaging over D. So I understand that the scale has to be between like the length scale of the gravitational waves and the length scale of the background, but it can be anything I choose, right? And if there's some dependence on D, how do I get rid of it? Or how do I make sure that there's no dependence on D on the right-hand side? Yeah, there shouldn't be. So if you have the separation of scales, then it becomes independent of what value of D you choose. So that's a good way to check that you actually have the separation of scales. I'll say, right? Because essentially from the point of view of this very long fluctuation, this scale is so small that you don't even know that you're averaging. And from the point of view of this small fluctuation, the length scale is so big that you've completely averaged it to zero. But it really does not depend on what exactly is D. Right, so I mean, maybe I wanna ask like what happens to the K-modes which have wavelength of order D is there some? Because they are like neither too small nor too big. Yeah, no, so then you can do this procedure. So you can really only do this procedure if your entire, your full metric, let's say, right, has things with very slowly and things with very fast and nothing in between. If, cause otherwise you really cannot do this procedure. And in fact, if you think about, say concretely LIGO, right? So LIGO has the problem that the gravitation waves we want to measure have a length scale of kilometers roughly, right, or hundreds of kilometers actually. But now the gravitational potential of the earth and kind of maybe the gravitational potential of buildings and of different components of the earth that actually also has kind of a length scale of a couple of hundred of kilometers. So there you have for LIGO, we have no separation of scales in the length domain. And the only reason that LIGO works is because there's this separation of scales in the time domain. So the gravitational waves that LIGO detects are oscillating very fast, right, in the Hertz regime. And however, the background, the gravitational background of the earth is pretty static. And hence they can do the separation. But if you had a situation where you have a background which oscillates, which is very important in space, on a similar distance as you want to measure your gravitational wave, then you just, then you're lost. You cannot kind of even define properly what you're looking for. I see. So if you have like a gravitational wave background, which in the frequency domain or in the lambda domain, it's just has a flat or like very slowly dropping spectrum, then it would be very difficult to kind of do this separation of scales on this because it has energies at every time. It's not the spectrum that matters. It's really the frequency of the gravitational wave. But yeah, yeah. Okay, thank you. Okay, so now the remaining task, which I want to do before the break is to explicitly now compute this T mu nu. So what we need to do is an explicit computation of G mu nu at a second order in TT gauge. So if at some point you have a weekend with nothing to do now, like a workday with nothing to do, right? Then this is an interesting computation, but lengthy computation. So we did it yesterday to first order, right? And even at first order, I didn't do it fully explicitly. I skipped a couple of steps. So a second order you can imagine the computation really explodes. And if you lose one minus sign on the way you are completely lost. But so in principle what you need to do is you need to compute our mu nu as we did yesterday. It's a second order, which is a long, long, long computation. But in the end what you get is you get this term. So which is the derivative two powers of H of course and derivative acting on both powers of H. And then you have 12 additional terms, which I'm not going to all write down. But then you can go, you can make your life easier by going to the TT gauge that we discussed yesterday. And in particular, and because that kills, that will kill a lot of these terms, right? Because as soon as you have kind of this index contracting with this index, it's a zero, right? As soon as one of these two indices is zero, it's zero. So that kills a lot of terms. And also you need to do integration by parts. And in fact, this computation you can find it in Majora's textbook, the full computation, or at least the relevant steps of the full computation. And then you will find that the second order tensor is given by minus one four, the new H alpha beta TT, essentially the expression that I wrote above. So essentially all of these 12 order terms get killed as soon as we go to the, we choose a clever gauge and we do this integration by parts. And of course we are interested here in the average quantity. So also here we have to take, oh, there's an H missing here, the average quantity. So you will often, yeah. And then of course, now you have the Ritchie tensor, right? Then you also need the Ritchie scalar to second order in H menu. And this turns out to be zero as does this R1 guy. So this is essentially the full, so then this is essentially once you have R menu, this means you have directly G menu, right? So then with this condition, this R menu is G menu. And then so you can get that this T menu that we were looking for, the energy momentum tensor, is just C to the 432 pi G, right? This was just a pre fact that we introduced a bit by hand up here, including this factor four, del nu H alpha beta TT. There are new H alpha beta TT. And now what we really want to know is the energy in gravitational waves. So this is the T00 component. And this is just a 00 component of this guy. So here, the only thing that survives is time derivative in both cases, right? So we have alpha beta TT H dot. And I can also, because I'm in transverse spatial sketch, actually I don't have the whole alpha beta indices, I just have the spatial index. So that makes it another bit easier. TT. Okay, so the prescription is, once I, if I have my gravitational wave, I need to take the time derivative, and then I need to take the two point function of two gravitational waves. And that is in the end, my energy density of gravitational waves. And that is the thing which essentially enters, right? This is essentially the stochastic background that we will be looking for. So this is the energy density of gravitational waves. Okay, any more, then that's as much as I wanted to do before the break. I don't know if there's maybe some more quick questions, otherwise we can take a break now. I don't see any questions. So I guess we can take now five minutes and we'll see you at 48 minutes. Okay, perfect. All right, I guess whenever you ready we can resume. Yeah, okay, so let's continue. And we're going to essentially, yeah. Now address one of the questions that was asked, and that is, well, to some sense, address one of the questions that was asked us, what is about kind of rotation waves in an expanding universe? What about kind of sub-horizon and sub-horizon gravitational waves? So the second part here is a gravitational waves in an expanding Friedman-Robertsen-Woffe universe. And so now essentially what we're doing is we're saying that we're in an expanding universe. So our background metric is not actually just a flat Mankowski metric, but at the very least we have the factor in that we have an expansion of space. So the metric that we're now using is minus DT squared. And then the spatial components are multiplied by the scale factor, which is a function of time and in particular grows over the history of the universe. And here we just have the usual spatial components. And then it can sometimes be useful to introduce conformal times. And from conformal time, we take the scale factor all the way to the front, and then we introduce conformal time tau, such that kind of everything inside here is just described by the flat Mankowski metric and we have the scale factor on the outside. And I'll be also now taking C for one in this part now. So now that we have an expanding universe, I have another quiz question for you. And that is how does the amplitude of the gravitational wave actually redshift? So we know that for photons, we know that kind of there's an in an expanding universe, the energy density decreases and the amplitude of the photon and the amplitude of the photons redshift as one over the scale factor. A is the scale factor. But the question is now, what would rotation wave? So they redshift in the same way as photons or do they redshift in a different way? Yeah, and the link is in the chat if you missed it earlier. Okay, so we'll see how this evolves. It seems to be a close race this time. Nobody likes my A cubed. So, okay, let's try and answer the question. Ah, somebody does look like my A cubed, okay. So the equation is over the linearized, with the right linearized Einstein equation yesterday. Now, if we take into account the expansion of the universe and essentially the left hand side, the g-menu, if we evaluate the g-menu, the expression changes a bit with respect to what we had yesterday. But the change is actually pretty minimal. So if you do the same exercise again, what you will get is that box H-menu of X and tau, minus, and this is the new term, two A prime, but the prime denotes the rotor with respect to conformal time, two A prime over A. H prime menu, so first the rotor here of X and tau is minus 16 PgT menu, okay. So the new term is this guy. And if you have a constant scale factor, right, so in particular if you have flat space, then this term will just disappear and you will produce the results of yesterday. So this disguise zero in a static universe, but now it is non-zero. So now we can do a Fourier transform of this equation. And we will introduce just for notation H tilde, which is just A times the original H. And the two porosations I will denote by lambda. So lambda is plus and cross. So then the Fourier transform of this equation, and again, recall that the reason I can simply do a Fourier transform here is because we're dealing with linearized equations, right? So all the k-modes decouples will create my life easier. So the equation you get is H double prime. So this comes from the box operator. Then again from the box operator, you get a k-squared term. Here you get A double prime over A, H tilde lambda k and tau. And here on the right-hand side, I get 16 pg. And now let me call it t lambda. So this is the energy momentum tensor, but projected into the plane of the polarization. So there's some projection operator, which we're not gonna discuss in detail, which takes the menu indices and projected on the two helicities that you have. And now you can distinguish two cases. So one case is that we have very small modes. So in particular, k is much bigger than A times H. So you can actually, so H is the Hobbit parameter, right? So this is A dot over A, where dot is proper time, or cosmic time. And this is essentially what is here, right? So this thing you can rewrite and this is approximately A H squared. So now the kind of question is, is this term or this term dominant in this equation? And now if the first term is dominant, it means, okay, you can essentially forget again about the scale factor, right? So everything is just exactly the way it was beforehand, right? So now this is what happens for sub-horizon scales, for small scales. If we're doing physics here in the lab, we don't care about the expansion of universe very much at least. So then we get that H till the lambda is just k squared H. Lambda is zero, so this is the usual wave equation, which is just solved by something like A, whatever, some cosine k tau plus some phase maybe. But actually we were not interested in H tilde, but the original equation was H, right? So if we put H here, then we have to divide by one power of the scale factor. So here the amplitude for sub-horizon scales, the amplitude shifts as one power of the scale factor. This is exactly the same as it is for photons. But now if we go to super-horizon scales, we go to the opposite limit, we can essentially in this expression neglect the k and only keep A H, so this is the super-horizon limit. So now we're thinking of gravitational waves, which in some sense are besides, which is bigger than the horizon of the universe. In this case, something more strange happens. So then this case, you can rewrite this equation up here. There's two A prime H lambda prime plus A H lambda double prime equals zero. And then you can look this up, you can plug this into your favorite computer program and ask what is the solution of this differential equation and the solution is some constant plus another constant, right? There's two constants of integration because this is a second-order differential equation. And then here there's some integral, zero to tau d tau prime A squared of tau prime. And the observation is now that this guy goes to zero in any expanding universe. Okay, so this is true for all the phases of the universe that we know this is true during inflation, this is true during matter domination, this is true during radiation domination, this is true during CC domination. So this thing just goes like after some time, you can forget about this term and all that you are left with is this term, which was just integration constant. And this integration constant is in particular constant. So the solution is that your gravitational wave is not actually a gravitational wave. I mean, it's not a wave, right? It's just some constant amplitude enough oscillating, so really you wouldn't normally call it a wave at all. And it's also not redshifting, right? So essentially what we say is that the gravitational waves are somewhat frozen out outside the Hubble horizon. So they just kind of stored there at constant amplitude and they only do anything interesting once they are inside the Hubble horizon. We say, gravitation waves frozen outside Hubble horizon. So there was two correct answers to the quiz. Both of both, it does not redshift is correct. This is correct, but a super horizon case and one of the A is correct for the sub-horizon case. Any questions on this? Okay, so now we can use this information to essentially write down a useful parameterization for gravitation waves. This is just to make our lives a bit easier. So we already understood that Fourier decomposition is very useful because we're working in the linearized theory. So we can write a bit more clever Fourier transform. So the object is the gravitation wave and transfer stressless gauge is a function of X and tau. And now I'm going to write that as a sum over the two polarizations, lambda, D3K integral, so we're doing a Fourier transform. And now there's essentially an amplitude h lambda from K, but I'm going to divide my amplitude into two factors. I'll explain in a moment. So it was a T of K, sorry. I'm going to divide here my amplitude essentially into a K dependent part and a tau dependent part. And I'll explain in a second why I can actually do that. Then I will have a polarization tensor because I need to kind of go from these A, J indices to the plus minus indices. I know that I only have these two degrees of freedom. And then I have the exponential function of the Fourier transform. So K tau minus K X plus Hermitian conjugate. So this guy was the polarization tensor. This guy is what we refer to as the transfer function. And essentially what I want to do is I want to separate this one, the primordial amplitude or the initial amplitude A from kind of the trivial scaling of this one over small a. I want to factor out the expansion of the universe. So this transfer function does exactly that. It takes care of the expansion of the universe. So this is what we call transfer function. And it is essentially the scale factor at some initial time divided by the scale factor that appears at the time that appears here on the left-hand side to the scale factor which we described the gravitational wave. And this initial time could be either at the time when this gravitational wave is formed or if you have formed it in some super horizon process then this time would be the point in time when the gravitational waves enters inside the horizon. So in both cases, this time T star is when the gravitational wave begins actually behaving like a wave. So oscillating begins redshifting as one of A. And then consequently, the thing that is left here is the Fourier coefficient at this time here. So this is the, or something we call it the primordial Fourier coefficient. So in general, of course, if you have a completely general Fourier transform you cannot just separate the k-dependent part and the tau-dependent part kind of in this product. But here we can do it because we understand the physics and we understand that the entire evolution of universe is just kind of encoded in this factor here. So now we can use this parametrization and go back into our expression for the energy density of the gravitational wave. So we now want to take this expression and plug it essentially into this guy here. And we're gonna use, make use of one additional observation and that is we're gonna make use of homogeneity and isotropy in the universe. Because if you recall the energy momentum tensor, right? This is the energy density that we computed. It contained a two point function of two gravitation waves. So it has some h, with h dot, h dot, right? But here now let's first look at h lambda of k h lambda prime of k, right? So these are the type of objects we need to evaluate to compute the energy density. And the cosmological concept of homogeneity and isotropy tells us that this two point function will be given by, okay, so a the two polarizations are orthogonal. Then homogeneity tells us that we have a shift symmetry and we can evaluate the two point functions at different points in space and you should always get the same. And so this translates to this delta function here, this is homogeneity. And then there will be some power spectrum, p lambda and the power spectrum can only depend on the absolute value of k. And that is the concept of isotropy, which says that the direction of k cannot matter. So with what we're gonna do now is we wanna compute the energy density. We're gonna use this expansion of the gravitation waves, we're gonna use this relation. And then taking all this together, we get for the energy density in the gravitational wave at the time when we observed them, so t zero. Now we have the pre-factor 32 high g. Then from taking the time derivatives, so recall that this row that we're computing, right? The row was up to pre-factor something like h dot h dot, right? So this is what we're computing. So you have to perform these time derivatives on this expression here. So this gives you, will give you a pi squared and an eight squared of tau naught. And then you have the d3k integral, you will have from this two point function, you will have the power spectrum and then you will have the transfer function squared. So a squared of tau star divided by a square of tau naught. Okay, so this is essentially our final expression for the energy density in gravitational waves in an expanding universe. And essentially we see again that there's two parts. So here we have the transfer function which encodes the cosmologic history, right? So this tells us how much has the universe expanded since the point in time when the gravitational wave was emitted. So here we have all the information on the cosmologic history. And here this is the two point function of the primordial Fourier coefficients, right? So these are really like these are the two point function of these guys here. So this contains the information about the source, right? Or whatever kind of happens to generate the gravitational waves in the first place. So this is the primordial power spectrum. And one observation is now that, say you managed to observe this thing, you're always observing a convolution of whatever is the primordial thing and whatever is the cosmologic history. So typically you have to assume that you know one of the two, so typically we assume that we understand cosmological history and then we can kind of learn something by measuring this, we can learn something about the primordial power spectrum. And one should also always keep in mind the possibility that maybe we don't actually entirely know our cosmologic history. And so you could be kind of misinferring things. But you could also kind of hope to whatever, at some point have a good understanding of the primordial spectrum, right? And then you could use this relation to infer the cosmologic history. But at the end of the day, we are convoluting two unknowns here. So there's a kind of a non-driven exercise to actually reliably extract physics here. And now just for notation, we can rewrite this energy density as kind of the critical energy density times integral dL and k, one over the critical density, dRawGw dL and k, right? So this is just a complicated way of writing the left-hand side. And the reason I'm writing it like this is that this object here is precisely this capital omega gw as our function of k evaluated today. And this is essentially the thing that typically you can predict in a given model and then you try to observe in gravitational wave detectives, right? So this is what we usually call the gravitational wave spectrum after stuff has a background. And what it really is kind of at the fundamental level, this is, we see it here, right? It's the two-point function of gravitational waves. So that also tells you that how do you want to observe this? The way to observe it is really to look for this energy is to really look at two-point functions of gravitational waves. Okay. Are there any questions on this? Yes. Yeah, so usually, can you hear me? Yes. Yeah. So usually in cosmology, we define that the omega parameters has a density of example radiation or matter over critical density. So what is there a derivative here? Yeah, yeah, this is confusing notation. I fully agree, right? Because also normally, yeah, because normally you would call it omega, you would just call kind of the total fraction, right? With respect to the physical energy density, but here it's not, right? Here it's a spectrum. And if I integrate over omega, so I need to, if I integrate over omega, then I get back what you would normally expect to be called omega. Yeah, yeah. So the reason we do it in this way is the one can debate if the name is clever, right? But it's what people do. But the reason you want to do it in this way is that kind of the spectrum information is actually important, right? Because you really kind of want to, at the end of the day, you really kind of want to see the spectrum of gravitational waves. And the contribution of different frequencies, like we do it for photons, right? Like if we think about kind of, I don't know, astrophysical observations, right? We can, we look at whatever gamma rays, we look at radio waves, right? And we kind of check how much energy do we have in each different energy bin. And here we want it to be the same, right? And hence the spectrum is important, but I fully agree with you that calling it omega is confusing, but this is kind of what the standard thing has become. Okay, and also why is it logarithmic derivative? Well, okay, I mean, the motive, okay. So I guess the reason is that typically, kind of if you have cosmological processes, but also if you have astrophysical processes, I mean, typically this thing looks sensible if you plot it on a log-log plot. So kind of taking the logarithmic derivative kind of tells you kind of how much do I have per decade or something of energy. And also if you do it this way, it doesn't matter so much here, right? If you have K or if you have F, right? That because both is correct and the two pi factors drop out. So maybe that's another motivation. Oh, okay, thank you. Okay, so then let's look at this in a, so tomorrow we'll talk more about sources, but let's just have a first glimpse and look at essentially what we expect here for a single field slow roll inflation. So inflation is a phase of exponential expansion in a very, very early universe invented to explain why the universe is so homogeneous and so is a tropic. And essentially the reason it leads to gravitational waves is that kind of you have this phase of an expanding universe. And in this phase, during this time we have some point of fluctuations of any particles that we have around in particular quantum fluctuations of the inflaton which is a particle driving inflation and also quantum fluctuations of the metric. And in this expanding universe, these quantum fluctuations get stretched to very large length scales. In fact, they get stretched to these super horizon scales. And then they just stay frozen there not red shifting at all. And then they can stay there for very long time until at some point later in the universe they can reenter into the horizon and then they can evolve. And this is why via this mechanism one can really hope to observe and we think actually we have observed for the scalars we can observe today really the quantum fluctuations that were tiny, tiny perturbations at this extremely early time in the history of our universe. So essentially the prediction of inflation is that the power spectrum of these gravitational waves or the two point function of these gravitational waves that are produced in this way as often called delta T squared which is related to kind of the some sense related to the usual power spectrum by a k-cube factor. I'm just putting this here because you might see it written in a different way in different literature. And the prediction of the simplest model of inflation is that the power spectrum here is just given by the scale of inflation divided by the Planck mass because we're producing metric fluctuations so gravity is somewhat important so this is where the Planck mass enters and essentially energy scale of inflation tells us how easy it is to produce these tensor perturbations. And the slow roll inflation essentially means that it's a hover parameter that this scale of inflation is pretty much constant which means that these perturbations that you're producing are pretty much constant constant meaning frequency independent. So when you're generating these perturbations during inflation, they do not depend on frequency you kind of just get a flat spectrum primordially. So now a question for you in the poll I'm telling you that kind of the spectrum is scale invariant when I generate it primordially. So essentially this thing here is the source by a scale invariant spectrum. What will be the spectral shape of this omega GW that we're actually detecting in the end? So you don't have to know it, right? Obviously just take a guess and then we'll see kind of how the true answer relates to your intuition. Yeah, so if you don't know just take a wide guess just to then see how the true result will contrast your expectation. Well, this is a really close race. So if we look, I can tell you that when we look for these yeah, when we look for these objects in the in the CMB it's also the CMB can look for gravitational waves. And if we look for them in the CMB the expected signal is in fact a peak which is related to the essentially to the energy scale of the or to the length scale of the CMB. But here it's a may or may not be the same. Okay, so let's do the computation. So what we want to compute is this omega today. So index zero. And so we take the formula from above and we plug in this expression here for the power spectrum. And so what we'll get is delta T squared should be a small t just an index. This, this looks like time, there's no time here. Delta T squared divided by 12. K squared A zero squared H zero squared A of tau star of K divided by A of tau zero. Okay, so this really just comes by taking this expression here plugging it in here for the power spectrum. Then this K integral simplifies because we have such a simple spectrum to start with you can actually do the K integral. And then you just have to kind of do this operation. Well, or you don't even have to do the K integral, right? Because K integral is here, K integral is here. So you just compare the two integrants take into account all these three factors and you get this expression for omega. Right here there's of course a square, there was a square of both. And now this thing here, this tau star of K, right? This indicates that the time that I, this indicates that the time of horizon entry of a given gravitational wave is K dependent. So for each given K, I need to actually use a different tau star. So in particular, I mean the definition for something being super or super horizon what we saw above is that A star, so A of tau star times H of tau star equal K, right? This is the definition of horizon crossing. So right, this is, let me write this expression is H of tau star, H of tau star like this. And so this for any given K, assuming a cosmological history, this defines the tau and this is the tau which goes in here. So we can now write this slightly differently as delta T squared divided by 12. Here I'm going to now write this as K divided by A star H star squared. So this is now by definition here, definition of the star, this is one. And now I have to of course, now I've essentially expanded with this parameter here, right? So now I have to compensate for that. So I'll get on the right hand side and I have two powers of A star from here, two powers from here. So if A star to the four have to compensate for this, so there's an H star squared, then I have A zero to the four, right? Two powers here, two powers here, and then H zero squared from here. And so this thing here is a constant. This thing here is just one. So the entire dependence just comes from this term. This can be the only term which gives us anything interesting. But this term is just the evolution of the scale factor and the Hubble parameter, right? So we call it the Hubble parameter is just A dot divided by A. So if I know in my cosmological history, I know A of tau. I know kind of how the universe is expanding. And as soon as I know that, I can evaluate this term here. So this we can now just evaluate in standard cosmological model, so in lambda CDM. So we see that here really the two unknowns we factor very neatly. We just have one overall amplitude which kind of comes from the details of inflation. And we have here this term here which comes from the cosmology. And now evaluating this now, however, depends if you are in matter already, like if this A star is in matter domination or radiation domination or in CC domination because the essentially the time dependence of this of destruction here will depend on that. So if you evaluate that carefully, what you will get is that omega GW as a function of a frequency has this type of a shape. So it goes, I would say it's a, more or less it's really good at broken power law. Where essentially here at the very left, that is a small frequency. So this corresponds to an equation of state parameter of zero. So this is matter domination, late time matter domination. These are the gravitational waves which entered the horizon at late times during matter domination. Here this plateau corresponds to gravitational waves entering the horizon during radiation domination. And here you want to also put a question of here we understand the cosmological history pretty well. Here, this is kind of a reheating phase which I'm saying again at the reheating phase as an equation of state of zero. So matter dominated reheating phase. This depends on your model, right? So this could also actually be quite different. But here on the left hand side, we understand the cosmology pretty well. So here we have something that scales as F to the minus two. Here we have something that is constant. Here we have a bound actually somewhere which comes from the CMB observations of tensor perturbations which is essentially a constraint on this delta T squared which is at the end of today a constraint here on the energy scale of inflation. But this only really applies here at very, very small frequencies. And yeah, overall like the time axis here. So the time of horizon goes from at the time of horizon entry that goes from right to left T of horizon entries. It is what I was explaining with a different equation of state parameters. So in a given phase of the cosmological history, you can evaluate in this term here and then you can find the frequency dependence of the signal. So if hypothetically, you could measure this guy, you could one, you could kind of measure this pre-factor. This would be the overall normalization of the spectrum but you could also kind of see any changes in the equation of state parameter of the universe. You could also see any changes in the number of degrees of freedom because here I'm being a bit sloppy, right? But it actually like say you have supersymmetry or something, right? And you may kind of see some little bump here, right? Associated, well probably will be at high energy scale but you could see a little bump, right? But the degrees of freedom in the universe change. So you really have kind of the full information of the cosmological history and of the primordial spectrum. The downside is that in this very simple model as we'll see tomorrow, in the simple model, the bound coming from the CMB kind of gives us a maximum amplitude for this plateau. And this maximum amplitude is such, I think it's off the, what is this? This max of the amplitude here is roughly, if I remember correctly, of the order of 10 to the minus 16 or so, even smaller, which, okay, which means, I mean, the relative thing to compare this to is the sensitivity of experiments. And this kind of very simple model, cosmological model, this amplitude is way too small to be seen with LIGO or even with Lisa. So this is an extremely difficult signal to find, something because it's so very, very small. But I find that it illustrates in a nice way kind of what types of information you can extract here. So just to add the labels here. So essentially here, this line corresponds to matter radiation quality, right? This line here corresponds to reheating, which separates the different equation states of the universe. Okay, and do we have any more questions? Do you have time? Yeah, so I have a question regarding the W star equals 7 by 3 and W star equals 0 phase. So when we are sort of building a model to explain and generate these parameters, so of course, W star equals 0, the A-land is model-dependent, right? So will our model also have to ensure that the other two quantities stay as such or is there some leeway in it as well? Well, I'm not sure I fully got your question. So this W equals 0 phase on the left, right? So this is just matter domination. So this is just kind of to late-time matter-dominated phase, right? So dark matter domination. So this I think we understand pretty well, right? So this we understand pretty well, this we understand pretty well. This depends on the details of your equation model. Okay, and almost any model we design will automatically follow the other two. You don't have to specifically, but have boundaries. Yeah, so no, the other two are simply observed. Yeah, the other two are simply the usual lambda CDM, right? So that's simply the observation that we've measured the amount of matter in the universe. We've measured the amount of radiation in the universe. We've measured the expansion rate today. And from that, we can kind of infer, you know, taking into account that matter redshift as a scale factor to the minus three and radiations goes a scale factor to minus four. So we can then kind of compute back in time just extrapolating things and kind of see that kind of now we're in a phase of cosmological constant domination then it's a fairly recently on cosmological time scales, right? In our recent history, the universe was dominated by matter. And before that, the universe was kind of just a plasma of relativistic particles. And all of this is nothing to do with inflation, right? All of this is simply the observation that we live in an expanding universe and that different components variation matter, the CC that they redshift at different rates. So this is nothing to do with inflation. Where inflation comes in only kind of in telling you here the amplitude of these primordial fluctuations, right? And in fact, that they are fluctuations at all. And then kind of how the transition from the inflation phase into the usual standard model of cosmology happens that depends on the details of the inflation model. So that depends how you couple your inflation sector to the standard model part of physics. And so this is why this phase here on the right is model dependent. Okay. And so what about, how does it look like if you're looking at things like post inflation in terms of like beta genesis or something else like? Right, so this is the spectrum the way we really observe it today, right? So this is the spectrum that we observe today taking into account in higher cosmological history, right? But looking only at the gravitation wave sourced during inflation, right? Then there are other sources of gravitational waves, right? And we'll talk about those tomorrow. There's other cosmological processes and of course, actual physical processes which also source gravitational waves, right? So then you have maybe a different source of gravitational waves, right? Which can give you completely different spectrum, right? Maybe you have a phase transition or something, right? And you could get maybe an additional contribution like this, right? Okay. All right, thank you. So while you're thinking of the more questions I encourage you to do the last poll in the last poll on the webpage. Yeah, but this is all I wanted to cover today, right? So the rest now is just questions. Okay, well, then thank you. Any more questions? Yeah, so this picture really illustrates a bit kind of this image that I was explaining of the cosmological history book, right? Because essentially here you really, in some sense you can associate the frequency axis with the time axis. I mean, here is the time of horizon entry of the mode. And you can really kind of this curve here essentially in some sense attracts the evolution of the scale factor over time. And so hypothetically, if you could kind of measure this thing over this entire time, right? Which is an arbitrary difficult process for many, many reasons. Then you could really, in principle you could map the evolution of the universe or you could map the scale factor to arbitrary high times, right? Way beyond the energy scales that we can ever reach with colliders. And you could in principle also probe like all the details of the model, right? Of the inflation model. Though of course, since the two are convoluted, right? That can be difficult in practice. But the bigger difficulty arises that from the fact that your experiments have limited sensitivity and that you have other gravitational wave contributions, in particular the ones from astrophysics. So kind of residuals of black hole, black hole mergers and these things I was explaining in the beginning. And these will typically be much larger than the signal that I'm showing here. So actually measuring this thing is really, really a very difficult process. Go Tom. All right, that one, Tom? I'm sorry, this might just be not understanding things. So in terms of the experiments and stuff we've done so far. So for the tail end of the spectrum which is model independently, are there any bounds that you've generated or is it just only for the other parts? Right, so if I would draw the experiments here then so typically this depends a bit, right? So this thing we know or what frequency this happens. If I had right the number from the top of my head it's probably wrong, right? But this is like no frequencies, right? This is something like nanohertz or so. This line here, this depends on the reheating temperature. So again, it is model dependent. But typically for a lot of models that put it that way you'd have kind of LIGO maybe sitting here. And so in order to really detect this tail you would need something again, something which is sensitive at higher frequencies unless you have a very low reheating temperature. So a very low maximal temperature of the universe at earlier time. But then you could move this to the left. So in fact, there are some constraints on kind of this guy, right? So if you say, okay, this thing was kind of growing then you have the, actually I think that the most so there is a constraint by LIGO. There's also a constraint from simply saying that we know how much and we'll talk about that all the tomorrow but we know how much energy density there is in radiation in the universe because if we had like extra radiation even if it's kind of dark radiation, right? And even if we can't see it directly but it would impact the expansion history of the universe. So we have an idea of how much radiation there is totally in the universe. And we know of course, what is the prediction for the standard models, right? Because we have the photon, we have the neutrinos, that's it. So you cannot put an arbitrary amount of energy into gravitational waves even at these very high frequencies where we have no direct detector. And so from that you can kind of put a bound on this part of the spectrum where it cannot kind of go up arbitrarily high. So you can constrain some models and some models have already been constrained but it tends to be more the slightly baroque models that you can constrain in this way for the moment. But that will change kind of with better sensitivity. Right, and I mean, you mentioned this and asking about like the wave generator from phase transitions and then it provides peak into the spectrum unless you're going to talk about this tomorrow, could you just elaborate a little bit more? Yeah, let's postpone that to tomorrow because tomorrow I'll actually explain why it's a peak and what are the parameters of the peak and so on, yeah. All right, thank you. All right, more questions. I see one by Martin. Martin. Cici. Oh, hello. Go ahead, please, Mick. Hello. So thanks for taking my question. I was just wondering if you could briefly comment about this a few pages back when we did this averaging of H lambda and H lambda prime and we used the homogeneity and isotropy. So this averaging, it's still on this length scale D, right? But this is much smaller than the kind of length scale. Yeah, so yeah, this is a very good question, right? Because yeah, here the notation is a bit sloppy. So here, this statement is true. This homogeneity and isotropy statement is true. For any, essentially for any two-point function of any cosmological perturbation, right? So this is essentially a statement about the statistics of these perturbations. So it's true that we still have, we still have the averaging going on in somewhere, right? Because we need the averaging to even like define really properly what is a gravitational wave, right? But once we have kind of separated out what is a gravitational wave, and then this is kind of a property about the statistics of the two-point functions of these objects. Okay, so it's more like referring to how they are sourced. That's of the homogeneity and isotropy, I guess. Yeah, yeah, yeah, yeah. Yeah, thanks. It's also not something that really comes from any, it's not derived from any fundamental principle, right? But we simply, in cosmology, we tend to say that we kind of conjecture that homogeneity and isotropy are kind of underlying principles of cosmology, right? We kind of think Earth is not a particularly special place to live, and from these kind of general thoughts is kind of where you get these types of relations from. All right, more questions. It seems there's an internet connection, problem connection in the CDP, so that's why Giovanni left Madrid. And also the ACTS support for my CDP also is connected, so. Okay. Okay, I'm connected from home, so it should be okay. I hope the Zoom session is connected somehow. Let's continue in the new way. In case I went down for some problem, you are a co-host, by the way, so I think you also have rights to unmute people that ask questions. Okay, yeah, so any more questions? I mean, otherwise, I guess it's been a very long day, could you guys? Yeah, lots of Zoom for everyone. Yeah, yeah, indeed, indeed. But not bad at the still, about 60 brave people staying here in front. Yeah, indeed, indeed. Okay, but if there's no more, I mean, if there's no more questions, then let's let people go. And yeah, so tomorrow we'll talk about sources and also a little bit about kind of detectors and kind of what are the prospects of actually finding these things. So tomorrow it gets a bit more speculative. Sounds good. Thanks for the lecture and see you tomorrow. Ciao.