1 00:00:00,340 --> 00:00:25,240 Some of. I will deviate a bit from from the last two talks in that I will not describe individual gravitational wave events, 2 00:00:25,600 --> 00:00:30,069 but rather what we call the gravitational wave background, 3 00:00:30,070 --> 00:00:39,940 which is some cumulative combination of all the gravitational waves that that that should be here. 4 00:00:41,850 --> 00:00:45,810 And I will be talking a lot about cosmology. 5 00:00:46,260 --> 00:00:52,409 And so this plot is a reminder to all of you that the universe is not static, 6 00:00:52,410 --> 00:01:01,980 but rather the universe is expanding as a function of time and the scale, the size of the universe we call we call it. 7 00:01:02,520 --> 00:01:07,120 Is there a. Oh, that's right. 8 00:01:07,450 --> 00:01:11,070 Yeah. Topper. Yeah. 9 00:01:11,400 --> 00:01:18,450 So the scale of the universe is as a function that we call a a function of time. 10 00:01:19,020 --> 00:01:23,160 And, and we will be using this function a lot. 11 00:01:25,320 --> 00:01:34,050 The rate of expansion. Is is called the Hubble Constant, and that's also a quantity that we need. 12 00:01:34,380 --> 00:01:37,890 So this is something familiar from this morning's talk. 13 00:01:39,060 --> 00:01:53,520 I will be talking about the quantity omega GW, the energy density of gravitational waves, which has been conveniently defined in the first talk. 14 00:01:54,360 --> 00:02:00,810 So just just just to remind you. So there are some factors of PIs and twos which are not important. 15 00:02:01,200 --> 00:02:05,170 There is the power spectral density of gravitational waves. 16 00:02:05,190 --> 00:02:10,680 This is essentially this h squared here inside pointy brackets. 17 00:02:12,000 --> 00:02:17,550 And effect of frequency cubed and then a Hubble constant, 18 00:02:17,850 --> 00:02:28,860 because we are comparing this to the density of the total density of our universe, which is called the critical density. 19 00:02:29,160 --> 00:02:32,219 Now, there is another way to write this down. 20 00:02:32,220 --> 00:02:37,890 So you would you would notice that the Hubble constant has units of one over time. 21 00:02:38,640 --> 00:02:44,070 Therefore this whole coefficient has units of time squared. 22 00:02:44,910 --> 00:02:46,890 Now if you, if you define, 23 00:02:47,250 --> 00:02:57,870 if you define omega GW such that is proportional to this thing h c squared times frequency squared you get that H these dimensionless. 24 00:02:59,590 --> 00:03:06,300 Which means that it is essentially the same thing as your original metric perturbation. 25 00:03:06,310 --> 00:03:08,980 This little h from the from this morning's talk. 26 00:03:10,060 --> 00:03:16,780 So it doesn't have units even though it's it's a four year transform it has units of of just a number. 27 00:03:17,860 --> 00:03:30,249 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, 28 00:03:30,250 --> 00:03:36,910 which is the typical strain that the gravitational wave at frequency F will have. 29 00:03:37,300 --> 00:03:45,700 So here's another familiar plot which summarises the different sources and detectors I'll be describing today. 30 00:03:46,360 --> 00:03:49,570 So here on the on the right at high frequency. 31 00:03:49,580 --> 00:03:57,220 So here here is a frequency axis, a height, relatively high frequencies of 100 times per second. 32 00:03:57,760 --> 00:04:00,700 Let's say you have light doe and Virgo. 33 00:04:02,110 --> 00:04:12,790 This black curve of the sensitivity curves of these detectors, that means roughly that if a signal is above these curve. 34 00:04:12,790 --> 00:04:15,969 So on the y axis, we have the characteristic strain. 35 00:04:15,970 --> 00:04:26,110 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. 36 00:04:27,150 --> 00:04:33,120 Roughly. And if it's below, there is too much noise in the detector to see it. 37 00:04:33,660 --> 00:04:38,280 Now, the noise is approximately because this is in large scale. 38 00:04:38,280 --> 00:04:46,590 It's approximately the area between the two. So here you can see the very first direct competition, a wave detection in in red. 39 00:04:47,460 --> 00:04:51,870 So you see that it's above the light curve. So we did indeed. 40 00:04:52,610 --> 00:04:57,110 The fact that there is another curve, it's called a plus here. 41 00:04:57,120 --> 00:05:03,810 This is what people hope to upgrade light go to in the near future. 42 00:05:04,600 --> 00:05:08,190 You see it's lower, so it's more sensitive. Okay. 43 00:05:08,350 --> 00:05:13,980 At lower frequencies, we have another gravitational wave detector which will now be in space. 44 00:05:13,990 --> 00:05:17,380 This is going. This has been already mentioned by Benza. 45 00:05:18,580 --> 00:05:26,110 It's called Lisa. It's much longer. So it's also an interferometer like logo, but with much longer arm length. 46 00:05:27,850 --> 00:05:32,290 And therefore it will be sensitive to much longer wavelengths because the detector 47 00:05:32,290 --> 00:05:37,930 is essentially sensitive to wavelengths that are similar to the size to its size. 48 00:05:38,740 --> 00:05:42,220 So Lisa will be sensitive to much longer wavelengths. 49 00:05:43,240 --> 00:05:48,940 And these frequencies are roughly ten to the minus two, ten to the minus three hertz. 50 00:05:49,570 --> 00:05:53,320 So these are orders of, let's say, hours, days, maybe. 51 00:05:55,060 --> 00:05:59,530 And sources for Lisa are different. So for Ligo's, 52 00:05:59,530 --> 00:06:05,200 you've had neutron star binaries and black hole binaries which constitute most of the gravitational 53 00:06:05,200 --> 00:06:11,470 wave sources because they they are very close to each other and therefore they orbit very fast. 54 00:06:12,800 --> 00:06:20,770 For Lisa, lower frequencies, longer periods of gravitational waves and also orbits of binaries. 55 00:06:20,770 --> 00:06:25,160 So you would expect to see white dwarves in our own galaxy. 56 00:06:25,670 --> 00:06:28,910 You would also expect to see supermassive black holes. 57 00:06:29,660 --> 00:06:38,200 These are the black holes that that also contribute to this detection by pulsar timing injuries, 58 00:06:38,210 --> 00:06:46,360 which I'll describe towards the end of my talk when when they are on the very last moments of merger. 59 00:06:48,220 --> 00:06:53,200 And also some very interesting early universe potential gravitational waves. 60 00:06:53,230 --> 00:07:02,290 I'll get to that later. Okay. And lastly, I will be mentioning, even if at even lower lower frequency. 61 00:07:02,320 --> 00:07:13,360 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. 62 00:07:14,020 --> 00:07:21,550 These these measure gravitational waves by looking at correlations between pulsar clocks. 63 00:07:22,370 --> 00:07:30,040 Okay, here's a picture of Legault. Okay, so there are two ligo's and you see the interferometer arms. 64 00:07:30,640 --> 00:07:34,810 There's a picture. So here's the drawing of Lisa. 65 00:07:35,230 --> 00:07:41,650 There will be three satellites here, and these will be two ends of the interferometer. 66 00:07:44,000 --> 00:07:51,710 It's going to be in space, hopefully in the next decade and then ten years from now, hopefully. 67 00:07:52,990 --> 00:07:59,290 And this is another radio telescope which looks at pulsars. 68 00:07:59,560 --> 00:08:05,790 It's called the Square Kilometre Array. There are two actually one in South Africa, one in Australia. 69 00:08:05,800 --> 00:08:12,910 It's under construction now and I will just mention that Oxford is heavily involved in the design and development 70 00:08:12,910 --> 00:08:22,330 of S.K. and I put the link here to the relevant website on the sorry web page on the department's website. 71 00:08:24,310 --> 00:08:32,650 And this will will look at pulsar pulsars at a much more precise way. 72 00:08:33,220 --> 00:08:43,420 Okay, so let's, let's talk first about higher frequencies and then go down in frequency as my talk progresses. 73 00:08:43,990 --> 00:08:47,710 So again, I'm talking about the background of gravitational waves. 74 00:08:47,740 --> 00:08:52,000 Now, this background is the sum of all the gravitational waves. 75 00:08:53,040 --> 00:09:03,870 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. 76 00:09:04,170 --> 00:09:11,190 And I should I should emphasise that gravitational waves are weak in their very nature. 77 00:09:11,190 --> 00:09:17,820 Okay is a small constant and gravitational waves therefore are very weak. 78 00:09:18,780 --> 00:09:24,540 So for light go you would see something which goes light, which is ten to the -20 or 30 miles, 21. 79 00:09:26,280 --> 00:09:30,420 So it's zero, followed by 19 other zeros. 80 00:09:30,600 --> 00:09:36,049 And then the one. This is the effect. So gravitational waves are very weak. 81 00:09:36,050 --> 00:09:42,379 So it makes sense that most of these gravitational waves are so weak that we can't see them, 82 00:09:42,380 --> 00:09:46,400 but they do add up to a background of gravitational waves. 83 00:09:47,210 --> 00:09:54,110 So you can you can think about the noisy restaurant, as Stephen mentioned earlier. 84 00:09:54,320 --> 00:10:00,320 So if you sit in a noisy restaurant, you might hear some conversations from the tables around you. 85 00:10:00,350 --> 00:10:06,530 You might pick up a few sentences, but most of the most of the sound waves, 86 00:10:06,560 --> 00:10:12,080 most of the noise is actually in the form of some blurry, constant background. 87 00:10:12,080 --> 00:10:15,860 Hum. Right. So I'm talking about this at the moment. 88 00:10:16,580 --> 00:10:22,160 And and I want to describe to you some of these properties. 89 00:10:23,060 --> 00:10:29,750 Okay. So gravitational waves are waves and therefore they have an amplitude and a phase. 90 00:10:30,020 --> 00:10:39,709 Okay. So the phase is what goes. It goes into the cosine or whatever sign that you had in your undergraduate degree. 91 00:10:39,710 --> 00:10:46,740 And the amplitude is some number. And therefore, you can define a complex number which has this amplitude and phase. 92 00:10:46,760 --> 00:10:53,090 So for each gravitational wave, for each gravitational wave source, I can define its own complex number. 93 00:10:53,390 --> 00:10:57,200 And the background is just the sum of all these gravitational waves. 94 00:10:57,620 --> 00:11:03,589 So we just add all the complex numbers. So we start with the first one, then the second one, the third one. 95 00:11:03,590 --> 00:11:12,120 And they are completely random, right? Because each black hole binary, neutron star binary, each one of them has a different orbital phase. 96 00:11:12,120 --> 00:11:16,159 You know, it's in a different position along along its orbit. 97 00:11:16,160 --> 00:11:24,800 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. 98 00:11:25,280 --> 00:11:28,400 And then the background is just this red arrow, right? 99 00:11:28,820 --> 00:11:34,100 So this is the final total gravitational wave, right? 100 00:11:34,100 --> 00:11:40,489 So if you think about it, each wave is like a step in a random walk in a plane, right? 101 00:11:40,490 --> 00:11:46,610 Because the directions are completely random and the amplitudes are also random. 102 00:11:46,610 --> 00:11:53,750 They are drawn from some distribution of binary separation and masses all around the universe. 103 00:11:54,650 --> 00:11:57,080 So it's just like a random walk in the plane. 104 00:11:57,080 --> 00:12:04,100 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. 105 00:12:05,580 --> 00:12:10,590 Okay, let's introduce the toy model for gravitational wave. 106 00:12:11,520 --> 00:12:19,110 So the gravitational wave measured by, let's say, Lego is equal to some amplitude, 107 00:12:19,110 --> 00:12:26,250 which I will choose to be a constant divided by the distance from the source to us. 108 00:12:27,860 --> 00:12:37,849 Find some oscillation. Okay, so cosine omega dwell omega is also a constant and phi is some initial phase for the black hole binary, 109 00:12:37,850 --> 00:12:45,190 which is a random phase between zero and two pi. Okay, so this would be a toy model I'm ignoring for the moment. 110 00:12:45,190 --> 00:12:48,520 The universe is expansion. I'm ignoring everything else. 111 00:12:48,730 --> 00:12:58,360 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? 112 00:12:58,690 --> 00:13:11,649 So you can. You can run a simulation and generate random gravitational waves according to your favourite formation channel for gravitational waves. 113 00:13:11,650 --> 00:13:16,720 And then you can, you can ask yourself what is the total gravitational wave background going to be? 114 00:13:17,290 --> 00:13:21,820 So here's what's plotted here. This is a simulation by the light going Virgo collaboration. 115 00:13:22,180 --> 00:13:32,170 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. 116 00:13:32,590 --> 00:13:38,610 It fluctuates. Now, this assumes that there is no noise for the detector, okay? 117 00:13:38,980 --> 00:13:49,750 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. 118 00:13:49,750 --> 00:13:52,850 I'm not going to discuss any of that. Okay. 119 00:13:53,080 --> 00:13:59,160 But assuming there was no noise. This is what you would see fluctuations. 120 00:13:59,180 --> 00:14:05,450 You know, you would notice that this is symmetric about zero. This is because the cosine is symmetric about zero. 121 00:14:05,450 --> 00:14:10,910 And that's a physical property is not just because I chose I chose a nice toy model. 122 00:14:11,310 --> 00:14:20,570 Okay. And and the question I want to answer today is how do you model the amplitudes of these fluctuations? 123 00:14:20,650 --> 00:14:28,190 Okay, So, you know, if you if you ask me what is actually going to be a T equals 3000 seconds, 124 00:14:28,670 --> 00:14:31,550 I would say, oh, well, it's just going to be a random quantity. 125 00:14:31,760 --> 00:14:38,270 But I can calculate the probability distribution of the value at 3000 seconds or any other time. 126 00:14:38,360 --> 00:14:44,310 To be to be honest. Right. So that's what I want to show you how you can do that. 127 00:14:44,810 --> 00:14:50,480 And of course, to do it completely, you need to do a complicated integral. 128 00:14:50,490 --> 00:14:55,490 But there are things that you can do analytically in which we could do right now. 129 00:14:57,370 --> 00:15:01,060 Okay, so this is what we're interested in. 130 00:15:01,300 --> 00:15:06,740 The probability that the sum of all the ways from all the sources is equal to some number H. 131 00:15:07,210 --> 00:15:13,040 This is the probability density function. Okay, So let's talk about two limits. 132 00:15:13,520 --> 00:15:18,800 One, limit small values of H. What's what's the probability that this happens? 133 00:15:19,070 --> 00:15:26,570 Well, how can this happen? It can either happen if all of the sources are just not active at the moment. 134 00:15:27,080 --> 00:15:34,129 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. 135 00:15:34,130 --> 00:15:39,010 So it doesn't contribute anything. All this is unlikely, right? 136 00:15:39,010 --> 00:15:45,850 Because you're looking at all the universe, all you could have destructive interference between a lot of them. 137 00:15:45,940 --> 00:15:53,350 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. 138 00:15:53,740 --> 00:15:57,219 It's highly likely that they will destructively interfere. 139 00:15:57,220 --> 00:16:02,020 And if they destructively interfere, you will get that the sum is is small. 140 00:16:02,800 --> 00:16:11,170 Okay. Also, bear in mind that this probability has to be an even function because positive and negative values are equally likely. 141 00:16:11,680 --> 00:16:18,820 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. 142 00:16:19,060 --> 00:16:28,180 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. 143 00:16:28,660 --> 00:16:32,470 Okay. What about very large gravitational wave amplitudes? 144 00:16:33,590 --> 00:16:42,110 There are two possibilities here either. So again, there could be a lot of constructive interference, but there are lots of sources. 145 00:16:42,350 --> 00:16:48,800 This is very unlikely that they will all be in phase right to generate a very large gravitational wave amplitude. 146 00:16:49,700 --> 00:17:01,969 So it's exponentially small, actually. And the other possibility is that you would have one source which is much stronger than the others. 147 00:17:01,970 --> 00:17:09,340 So it dominates. Much stronger, but not strong enough to be resolved as an individual gravitational wave event. 148 00:17:09,350 --> 00:17:12,890 So it's right there below the threshold. 149 00:17:12,920 --> 00:17:22,160 Now, what's the probability for that? So the probability that the strain lives in some interval, the H because of our toy model, 150 00:17:22,310 --> 00:17:26,629 I told you the amplitude was constant, the frequency was constant, the phase was random. 151 00:17:26,630 --> 00:17:31,040 I'm talking about the amplitude here. Right. Because that's what what interests me. 152 00:17:31,550 --> 00:17:42,200 So that means that the probability that h lies in this interval is just the probability that the distance is what is like, 153 00:17:42,410 --> 00:17:48,110 you know, goes like one over. H Okay, so there's this constant in front OC times the. 154 00:17:50,500 --> 00:17:54,160 Radial increment. Now, this is like go. 155 00:17:54,190 --> 00:17:57,070 We are seeing black holes from all over the universe. 156 00:17:59,000 --> 00:18:09,680 So the universe, as maybe some of you remember from your cosmology courses, is homogeneous and isotropic. 157 00:18:09,920 --> 00:18:14,450 That means that matter on large scales is distributed roughly evenly. 158 00:18:15,110 --> 00:18:23,540 And that means that the probability of finding an R inside of shell, of size of the R is proportional to. 159 00:18:25,390 --> 00:18:32,770 The area of the show times the increment the R so it proportional to our squared times the OC. 160 00:18:35,150 --> 00:18:42,320 And R-squared is one over eight squared. Therefore, the gravitational wave probability is just eight to the minus two, 161 00:18:42,320 --> 00:18:47,000 which is this h of the minus two times the R by the H because I divided by the H. 162 00:18:47,390 --> 00:18:54,980 So it goes like to the minus four case we have a constant at low HS and the power low minus fourth largest. 163 00:18:55,550 --> 00:19:00,560 What happens if you do a correct calculation are proper exact calculation. 164 00:19:00,590 --> 00:19:07,880 Well, you see on the x axis, H on the y axis, the probability h normalised by something, whatever. 165 00:19:08,450 --> 00:19:11,629 Okay. It's just eight divided by something which is very small. 166 00:19:11,630 --> 00:19:15,530 So I get thinks of all the unity here and here. 167 00:19:16,100 --> 00:19:20,210 Okay. So at low, as you see, look at the blue curve. 168 00:19:20,960 --> 00:19:28,760 It goes to a constant and a large h it goes to the asymptotic, which I promise you is an H minus four asymptotic. 169 00:19:29,030 --> 00:19:32,390 And in fact, you can actually calculate exactly the coefficient. 170 00:19:32,720 --> 00:19:39,650 Okay. And so you can, you can verify that this nice physical picture is indeed correct. 171 00:19:40,940 --> 00:19:44,630 Okay. The last curve here is the normal distribution. 172 00:19:44,870 --> 00:19:51,079 I should just say that this is somewhat surprising if you remember the law of large numbers. 173 00:19:51,080 --> 00:19:59,660 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, 174 00:19:59,990 --> 00:20:02,990 should this should be distributed like normal distribution. 175 00:20:03,200 --> 00:20:08,300 Well, a normal distribution is what you see here. It's not distributed like a normal distribution. 176 00:20:08,930 --> 00:20:15,770 So it's a homework problem to figure out why this does not violate the law of large numbers. 177 00:20:16,580 --> 00:20:19,520 Okay, you can do the same thing. 178 00:20:19,520 --> 00:20:30,950 So we've calculated the we've calculated the probability distribution of the gravitational wave amplitude fluctuations. 179 00:20:31,610 --> 00:20:36,490 Right. So what is omega GW, the energy density. 180 00:20:36,500 --> 00:20:40,940 Well, it's h squared, right. So it's like the variance of the gravitational wave. 181 00:20:41,960 --> 00:20:49,190 Right. So once you have the probability distribution, you can calculate this, right? 182 00:20:49,190 --> 00:20:52,850 And so you can do it. And if you do it, you get this plot. 183 00:20:53,090 --> 00:20:57,590 So on the x axis, you have frequency. On the y axis, you have omega. 184 00:20:57,610 --> 00:21:01,940 GW So you look at the blue curve, it looks roughly like that. 185 00:21:02,480 --> 00:21:12,650 Okay, it has this F to the frequency, to the two thirds power law at small frequencies, and it goes down to high frequencies. 186 00:21:13,250 --> 00:21:16,820 Now in purple here is a very interesting thing. 187 00:21:17,450 --> 00:21:21,979 So if you look at like go observations, they haven't seen any gravitational wave background. 188 00:21:21,980 --> 00:21:27,709 They've only seen resolved individual gravitational wave coalescence is so they were 189 00:21:27,710 --> 00:21:32,930 able to put an upper limit on the amplitude of the gravitational wave background. 190 00:21:33,290 --> 00:21:41,420 And this upper limit is the purple line. And you will see that my calculation falls nicely below the excluded range. 191 00:21:41,930 --> 00:21:51,590 But I also put the sensitivity curve of the future lie detector, and you would see that this calculation is above that, 192 00:21:51,920 --> 00:22:00,920 which tells you that there is hope that the gravitational wave background will be detected once Legault reaches this A-plus sensitivity. 193 00:22:01,340 --> 00:22:13,890 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. 194 00:22:14,250 --> 00:22:22,650 I would now like to switch detectors and move to laser and describe two types of gravitational wave backgrounds for this detector. 195 00:22:22,860 --> 00:22:27,450 Now, black holes and neutron stars are things that we know that exist. 196 00:22:28,230 --> 00:22:33,210 But nonetheless, measuring this gravitational wave background will tell us a lot about the history, 197 00:22:33,510 --> 00:22:42,210 how they form, you know, about cosmic structure, formation and so on, all the things that go into this. 198 00:22:43,310 --> 00:22:47,810 Precise modelling of the probability distribution that we've just calculated. 199 00:22:48,950 --> 00:22:52,760 And once we have observations, we could understand more about the history. 200 00:22:53,930 --> 00:23:01,280 So I haven't talked about any of these because it's complicated to do the calculations with them. 201 00:23:01,280 --> 00:23:09,200 So I told you just about the features that that would be the whatever formation history you choose. 202 00:23:09,920 --> 00:23:19,050 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. 203 00:23:19,070 --> 00:23:24,260 This is true for all the other types of backgrounds as well. Okay, So let's go back to Lisa. 204 00:23:24,500 --> 00:23:32,240 Remember, it's a space based experiment, much longer arms length, much lower frequencies, much longer wavelengths, and much longer periods. 205 00:23:33,170 --> 00:23:39,350 And again, just to remind you, the the equation between energy density of gravitational waves and the characteristic strength. 206 00:23:40,390 --> 00:23:45,820 Okay, so in Lisa, you would see white dwarf binaries from our own galaxy. 207 00:23:46,150 --> 00:23:47,820 These are much lower frequencies. 208 00:23:47,830 --> 00:23:57,580 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. 209 00:23:58,880 --> 00:24:07,250 So in the physics of the physics, it's the same. So everything I said beforehand applies, except the frequencies are lower. 210 00:24:09,020 --> 00:24:13,010 And most of the sources are going to come from our own galaxy. 211 00:24:13,820 --> 00:24:19,070 That means that they are no longer uniform, they are no longer homogeneous, 212 00:24:19,070 --> 00:24:23,390 and they are not isotropic because our galaxy is not homogeneous and it's not isotropic. 213 00:24:23,690 --> 00:24:28,610 And of course there are white dwarf binaries, white dwarf binaries in the whole universe, 214 00:24:29,300 --> 00:24:36,520 but from other galaxies, because these galaxies are so far away, it's basically impossible to see them. 215 00:24:36,530 --> 00:24:41,300 So the domain, even the gravitational wave background, will be dominated by our own galaxy. 216 00:24:42,110 --> 00:24:45,260 Okay, so it's not homogeneous and isotropic. 217 00:24:45,560 --> 00:24:48,560 So if you go back to the equations that I've shown you previously, 218 00:24:48,560 --> 00:24:53,600 you just have to modify the spatial distribution and you will get a calculation 219 00:24:53,600 --> 00:24:58,850 of the probability distribution and also the gravitational wave energy density. 220 00:24:59,990 --> 00:25:04,870 I should also say that there are less sources that are active at the given moment in time. 221 00:25:04,880 --> 00:25:08,210 Active means that they are at a. 222 00:25:09,690 --> 00:25:17,010 And an orbital period which corresponds to a frequency which falls inside the detector's frequency range. 223 00:25:17,730 --> 00:25:21,000 Okay, let's then not go back on sources. 224 00:25:22,570 --> 00:25:30,880 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. 225 00:25:31,420 --> 00:25:36,670 Now, let's let's move to something more exotic. 226 00:25:38,580 --> 00:25:44,400 And these are not gravitational, you know, not binary systems that emits gravitational waves. 227 00:25:45,110 --> 00:25:50,040 And so, you know that any. So. 228 00:25:50,230 --> 00:25:54,320 So the gravitational waves comes from the Einstein field equations. 229 00:25:54,740 --> 00:26:00,320 And on the left hand side of these equations, you have the gravitational field, 230 00:26:00,610 --> 00:26:06,170 the metric tensor on the right hand side of these equations, you have the energy density. 231 00:26:06,590 --> 00:26:17,070 So the matter if you have. Things that, you know, violent changes of the energy momentum of matter, 232 00:26:17,370 --> 00:26:22,320 you will get changes in the gravitational field, which are gravitational waves. 233 00:26:22,700 --> 00:26:28,010 Okay. So one way to produce that is that you have two black holes which are orbiting each other. 234 00:26:28,020 --> 00:26:32,100 They change the energy density and therefore they create the gravitational wave. 235 00:26:33,120 --> 00:26:37,979 Another way is by other physical mechanisms. 236 00:26:37,980 --> 00:26:47,160 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. 237 00:26:47,160 --> 00:27:00,600 And describe some non-exhaustive list of possible sources for gravitational wave background from events that might have taken place at these times. 238 00:27:01,260 --> 00:27:09,510 So it's non-exhaustive because there are many more that I didn't put there, but these are some main ones. 239 00:27:10,770 --> 00:27:16,020 I would say that these come from physics beyond the standard model of particle physics. 240 00:27:16,020 --> 00:27:20,280 So just to remind you, the standard model of particle physics is the best, 241 00:27:21,930 --> 00:27:28,410 well confirmed theory we have of electromagnetism, the weak and the strong forces. 242 00:27:29,370 --> 00:27:34,710 But we know that it is not complete because, for example, we don't have gravity is not included. 243 00:27:35,070 --> 00:27:43,410 And also in the standard model, neutrinos don't have a mass, whereas we have measured them to have a mass. 244 00:27:43,890 --> 00:27:51,000 Okay, So we know it's not complete and therefore it makes sense to think about things that happen in theories of physics, 245 00:27:51,000 --> 00:27:55,320 which are beyond the standard model of particle physics. 246 00:27:55,590 --> 00:27:59,610 Okay, So one, one example is a phase transition. 247 00:28:02,530 --> 00:28:09,430 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. 248 00:28:09,610 --> 00:28:16,330 So think about boiling water. Okay, so boiling water changes from one phase, water to another phase, which is gas. 249 00:28:16,720 --> 00:28:21,610 Right. And during the phase transition, you generate bubbles of gas. 250 00:28:21,610 --> 00:28:29,720 Inside the water bubbles expand and more and more bubbles form and they expand until bubbles combine. 251 00:28:29,740 --> 00:28:33,340 Right. So they collide and then they combine. Right. 252 00:28:33,550 --> 00:28:43,840 These bubble collisions are very violent events, and if we had enough energy in the water, they would generate gravitational waves. 253 00:28:44,530 --> 00:28:46,680 Now, think about the early universe. 254 00:28:46,690 --> 00:28:53,860 Okay, so the universe was in some phase of matter, doesn't matter what phase, and it transitions to another phase. 255 00:28:54,100 --> 00:28:58,360 And during this phase transition, you get bubbles, expanding bubbles. 256 00:28:58,930 --> 00:29:03,549 And if these bubbles are energetic enough, they collide. 257 00:29:03,550 --> 00:29:09,220 And once they collide, you have these violent changes of the energy momentum tensor and therefore you get gravitational waves. 258 00:29:09,670 --> 00:29:19,570 Okay, so this is one picture of a phase transition at the very early universe which could generate gravitational waves. 259 00:29:20,020 --> 00:29:26,139 Now, I put here a picture, a plot, rather, of what that would look like. 260 00:29:26,140 --> 00:29:31,030 So again, we have my favourite plot frequency, gravitational wave, energy density. 261 00:29:31,360 --> 00:29:38,770 We have the laser sensitivity curve in blue, and we have one prediction of a theory for such a phase transition in black. 262 00:29:39,550 --> 00:29:43,720 There is this website b d plot, which is what I use to create this plot. 263 00:29:44,050 --> 00:29:49,090 You just put the parameters of your favourite beyond the Standard Model phase transition. 264 00:29:49,510 --> 00:29:56,020 Okay, into the website you click submit and it produces a prediction for the gravitational wave background. 265 00:29:56,650 --> 00:30:06,310 Okay. I choose some parameters. Chose some parameters that are physically motivated by some physical, some theory. 266 00:30:06,550 --> 00:30:07,060 Anyway, 267 00:30:07,570 --> 00:30:16,160 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. 268 00:30:16,180 --> 00:30:23,010 What what I would like to do is describe the shape. So it turns out that all of these have the same shape. 269 00:30:23,020 --> 00:30:26,650 They go up, they reach the maximum and they go down. 270 00:30:27,070 --> 00:30:30,640 Okay. And this increase goes like frequency cubed. 271 00:30:31,270 --> 00:30:32,680 Okay. And then it goes down. 272 00:30:32,890 --> 00:30:42,160 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. 273 00:30:43,630 --> 00:30:52,930 Okay. Another source of gravitational wave from the very early universe is called is from a collision of objects which have not been detected. 274 00:30:53,290 --> 00:30:56,740 Called Cosmic Strings. Cosmic strings are. 275 00:30:59,050 --> 00:31:03,550 I know. Sorry. Cosmic strings. 276 00:31:04,930 --> 00:31:08,370 Let's say energy concentrations, which are very long. 277 00:31:08,380 --> 00:31:13,660 So they basically have one dimension. They look like a string. They could form loops, for example. 278 00:31:14,800 --> 00:31:19,959 And. And when two strings collide, they combine into one string. 279 00:31:19,960 --> 00:31:24,100 So imagine, you know, those two strings and they collide, right? 280 00:31:24,100 --> 00:31:29,950 So at the point when they collide, you could create those things that go like this. 281 00:31:30,790 --> 00:31:37,030 Right. And. And this collision will generate gravitational waves. 282 00:31:39,020 --> 00:31:46,600 Just like phase transition. So again, these violent changes of of of of of the energy momentum tensor, 283 00:31:47,210 --> 00:31:54,530 I'm the third thing that I want to mention which actually doesn't look like that and doesn't have these frequencies. 284 00:31:54,530 --> 00:31:59,300 But I should also mention it because it's a gravitational wave emission at the very early universe. 285 00:31:59,540 --> 00:32:04,190 And this is actually something that we do expect and we have good reasons to expect. 286 00:32:05,510 --> 00:32:10,070 And some may be experimental evidence that that this should be detected. 287 00:32:10,760 --> 00:32:14,809 It's gravitational channel waves emitted during the period of inflation. 288 00:32:14,810 --> 00:32:19,850 So we have evidence that the universe expanded very rapidly. 289 00:32:23,420 --> 00:32:27,140 Very rapidly after the Big Bang. Right. 290 00:32:27,620 --> 00:32:28,459 Very fast. 291 00:32:28,460 --> 00:32:41,090 And and and and quantum effects during this period of expansion create created the distribute the distribution of matter or gave rise to it. 292 00:32:41,450 --> 00:32:46,600 Right. And these same effects are supposed to create also gravitational waves. 293 00:32:46,610 --> 00:32:51,140 We haven't seen them. But if we believe in inflation, they should be there. 294 00:32:52,130 --> 00:32:56,570 Right. So let's describe this frequency you. 295 00:32:57,040 --> 00:33:00,070 I should say this is a bit technical and mathematical. 296 00:33:00,670 --> 00:33:05,650 So for those of you who don't remember all the mathematical details, it doesn't matter. 297 00:33:07,930 --> 00:33:10,990 But if there are students here, then it does matter for you. 298 00:33:13,480 --> 00:33:18,490 Right. So we've talked about the metric for an expanding universe. 299 00:33:18,500 --> 00:33:19,780 The metric looks like this. 300 00:33:19,790 --> 00:33:29,770 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. 301 00:33:30,370 --> 00:33:34,300 We can change coordinates. So we define this at a conformal time. 302 00:33:34,630 --> 00:33:42,310 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. 303 00:33:43,240 --> 00:33:49,930 If we do that and we write down the wave equation from Stephen's talk, we get a very simple equation. 304 00:33:50,560 --> 00:33:57,430 So again, we take the growing tension of a wave and we take out a fact of a from it to make things simpler. 305 00:33:58,300 --> 00:34:06,700 And we define K. So K double prime is the second derivative with respect to time E plus. 306 00:34:08,300 --> 00:34:14,690 Wave number vector squared times K So this is a harmonic oscillator, equal source. 307 00:34:16,930 --> 00:34:21,520 So the solar sigma is whatever energy density you had. 308 00:34:21,610 --> 00:34:26,140 Right? So we have an armonica oscillator once we Fourier transform in space. 309 00:34:27,250 --> 00:34:34,180 So we have a harmonic oscillator with a source, so a driven harmonic oscillator at each wavelength. 310 00:34:34,480 --> 00:34:39,040 We can solve this. Everybody learns how to solve a harmonic oscillator. 311 00:34:39,040 --> 00:34:48,970 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. 312 00:34:49,310 --> 00:34:55,130 Okay, That's what matters. Okay, good. 313 00:34:55,970 --> 00:35:00,130 So I'm talking about low frequencies at low frequencies. 314 00:35:00,140 --> 00:35:06,230 What do I mean by low frequencies? Well, I mean wavelengths that are much longer than the typical size of the source. 315 00:35:06,230 --> 00:35:13,250 What is the typical size of the source? So, again, I'm going go back to a boiling water metaphor. 316 00:35:13,580 --> 00:35:17,450 This is the typical size of a bubble of a water bubble. 317 00:35:18,350 --> 00:35:26,930 Right. So if you're talking about wavelengths, wavelengths that are much longer than the size of a typical bubble, 318 00:35:27,230 --> 00:35:33,560 then waves from, you know, energy density is gravitational waves squared. 319 00:35:33,890 --> 00:35:42,650 Okay. So I take 11h from one side of the saucepan and add another H from another side of the saucepan. 320 00:35:43,070 --> 00:35:54,220 And if. If if the wavelength is very big, much larger than a typical bubble size, then these ages cannot know about this other. 321 00:35:54,230 --> 00:36:03,980 So. So the sigma that created them must be uncorrelated between one point and another point. 322 00:36:04,340 --> 00:36:15,230 So at very large wavelength that the sigma so the source fluctuations have to be what is technically known as white noise. 323 00:36:16,150 --> 00:36:23,990 Fact means that the variance of the amplitude of these source fluctuations at very large wavelengths, so very small. 324 00:36:23,990 --> 00:36:31,490 KS And very small frequencies, because K goes like frequency for gravitational waves has to be a constant. 325 00:36:33,300 --> 00:36:40,470 And indeed, if you look at the phase transition, though, this is from one one model of a phase transition, 326 00:36:40,830 --> 00:36:48,150 you see that the gravitational wave source fluctuation goes like a constant and then it reaches some. 327 00:36:49,730 --> 00:36:58,770 Value. And then it goes down and say, here the wave factor is measured in units of the typical source length scale. 328 00:36:58,790 --> 00:37:09,260 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. 329 00:37:10,160 --> 00:37:16,810 So one over the typical source wavelength, a length scale, it's a constant, much larger. 330 00:37:16,820 --> 00:37:20,210 It's it goes down very quickly to zero. 331 00:37:21,270 --> 00:37:30,060 Okay. But for us at low frequencies, it's a constant. And from my solution, from our solution to the Einstein field equations, 332 00:37:30,330 --> 00:37:36,450 we know that H goes like sigma over K, So h squared goes like sigma squared of a case squared. 333 00:37:37,810 --> 00:37:43,960 Right. But we want but we know that omega three W is squared times characteristic strain. 334 00:37:44,170 --> 00:37:48,370 So we have two reactions form back. This introduces a factor of cubed. 335 00:37:49,620 --> 00:37:56,370 Frequency cubed and therefore are gravitational wave. 336 00:37:56,670 --> 00:38:01,110 Energy density goes like frequency cubed, right? 337 00:38:01,110 --> 00:38:05,580 So there's an F squared here, times that are cubed, which is F cubed. 338 00:38:05,790 --> 00:38:11,099 So it's after the five and then there is a K to the minus two, which is after the minus two. 339 00:38:11,100 --> 00:38:14,970 This is a constant. So 5 minutes till we get the execute. 340 00:38:15,930 --> 00:38:20,610 This is true for any causal source of gravitational waves. 341 00:38:20,850 --> 00:38:24,420 Cosmic string collisions are also causal. What do I mean by causal? 342 00:38:24,600 --> 00:38:32,200 I mean that that that this emission of gravitational wave event happened in a way that 343 00:38:32,230 --> 00:38:37,559 that information propagates at the speed of light or lower than the speed of light, 344 00:38:37,560 --> 00:38:40,920 but not faster than the speed of light. Okay. 345 00:38:40,980 --> 00:38:45,290 So. Right, so we've explained this rise like a few. 346 00:38:45,310 --> 00:38:50,610 There's a peak. The peak depends on the specific physics of the theory. 347 00:38:50,940 --> 00:38:54,660 And, and, and it changes. 348 00:38:55,080 --> 00:39:04,200 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. 349 00:39:04,470 --> 00:39:08,370 But the fact that there should be a decline does not. 350 00:39:08,610 --> 00:39:13,230 And it's because the sigma squared variance cannot be. 351 00:39:13,560 --> 00:39:20,320 It has to go down to zero as you go to scales which are much smaller than the typical source length scale. 352 00:39:22,190 --> 00:39:30,890 Okay. So I explained to you, I mean, what what these omega three WS look like this gravitational wave energy densities look like for. 353 00:39:32,420 --> 00:39:34,810 Four phase transitions and very early universe. 354 00:39:34,820 --> 00:39:40,970 Of course, if we find something that's great because we find it, find evidence for physics beyond the standard model. 355 00:39:42,350 --> 00:39:54,860 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. 356 00:39:55,610 --> 00:40:02,360 So we've measured this. So the nano graph collaboration is measured this for paper from July. 357 00:40:02,930 --> 00:40:06,050 Right. So it's very recent. This is the Helens and Downs curve. 358 00:40:06,470 --> 00:40:10,760 You see that it's not zero, right? So there must be a signal here. 359 00:40:11,030 --> 00:40:21,350 And the the the idea is that this signal comes from the background of a supermassive black hole coalescence. 360 00:40:21,860 --> 00:40:26,210 So, again, these are binaries of masses. 361 00:40:26,510 --> 00:40:33,800 So the theory for the ligo's sources also applies here, except that frequencies are much lower decades. 362 00:40:34,640 --> 00:40:44,390 And I should also mention that individual events, which are bright enough, will be seen by Lisa when it is operational. 363 00:40:45,440 --> 00:40:52,310 Okay. Where does this background come from? The background comes from coalescence of supermassive black holes. 364 00:40:52,640 --> 00:40:57,890 We know if if, if our understanding of physics is correct. 365 00:40:59,090 --> 00:41:04,160 That every galaxy has a supermassive black hole at its centre. 366 00:41:04,550 --> 00:41:12,200 These are like a million times the mass of the sun, or even a billion times the mass of the sun, even heavier sometimes. 367 00:41:13,070 --> 00:41:17,330 And each galaxy has one of them. So how do you get them together to merge? 368 00:41:17,480 --> 00:41:24,440 You have to merge the galaxies. And indeed, that happens. But before that, let's see how you described that. 369 00:41:24,470 --> 00:41:31,610 So this is something. Very nice because it's also related to Oxford. 370 00:41:32,240 --> 00:41:36,560 It's called the Mongolian Relation, and that's named after John McGauran, who is a physicist. 371 00:41:38,030 --> 00:41:43,219 And and this relation says that if you look at the mass of the stars in a galaxy 372 00:41:43,220 --> 00:41:46,820 and you look at the mass of a black hole in the at the centre of the galaxy, 373 00:41:47,180 --> 00:41:52,460 the two are correlated and the Coalition says that one you, you can, 374 00:41:52,490 --> 00:41:57,470 if you know the mass of the stars, you can predict roughly the mass of the supermassive black hole. 375 00:41:58,050 --> 00:42:04,100 Okay. So if you have a theory that tells you how massive each galaxy should be, 376 00:42:04,550 --> 00:42:10,070 you would know from the Mongolian relation how to calculate the mass of the black holes. 377 00:42:10,430 --> 00:42:18,950 And of course, these are important because they influence the amplitude of the gravitational wave emitted by these black holes. 378 00:42:19,430 --> 00:42:23,809 Okay. So do we have a theory for the masses of galaxies? 379 00:42:23,810 --> 00:42:31,940 The answer is yes. It's called the Halo model, started by President Chester and many more people. 380 00:42:32,060 --> 00:42:39,020 It basically gives an equation for the number of galaxies with a certain mass and certain mass range. 381 00:42:39,030 --> 00:42:43,240 And the important thing to know. So here you have a log of mass. 382 00:42:43,250 --> 00:42:50,569 Here you have log of, uh, relative frequency that at high mass as it goes down. 383 00:42:50,570 --> 00:42:59,540 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. 384 00:42:59,720 --> 00:43:03,680 Hmm. These are simulation results. 385 00:43:03,890 --> 00:43:11,690 Okay. But there are certain. But. But the black lines are equations that that you can give in some closed form. 386 00:43:12,590 --> 00:43:16,280 They have parameters that are determined by simulation. 387 00:43:16,790 --> 00:43:22,910 Okay. But anyway, we have equations that tell us how many galaxies there are of each mass. 388 00:43:23,540 --> 00:43:27,499 And once we know that masses of galaxies and black holes are correlated, 389 00:43:27,500 --> 00:43:38,120 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. 390 00:43:38,360 --> 00:43:42,560 So it turns out that galaxies merge as the as the universe evolves. 391 00:43:43,820 --> 00:43:49,370 And it is possible using cosmological simulations to calculate how many mergers 392 00:43:49,370 --> 00:43:54,949 you should have to give in moment in time and as a function of the sorry, 393 00:43:54,950 --> 00:43:58,100 as a function of the masses of the components. 394 00:43:59,030 --> 00:44:07,609 So the total rate of gravitational wave events from supermassive black hole combination coalescence as is the number of galaxies with mass and 395 00:44:07,610 --> 00:44:16,310 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. 396 00:44:16,490 --> 00:44:22,600 Now, I have assumed here that once the two galaxies merge, the black holes at the centres. 397 00:44:22,610 --> 00:44:28,580 Also note this is a known, known non-trivial assumption. 398 00:44:28,790 --> 00:44:35,030 But let's let's keep assuming that anyway. And the gravitational wave energy density. 399 00:44:35,420 --> 00:44:40,640 Well, it's just. The energy emitted. 400 00:44:41,090 --> 00:44:44,060 So you remember, this is energy emitted per frequency. 401 00:44:44,810 --> 00:44:50,960 So it's the energy emitted well, per frequency, but I've multiplied by T in bottle on both sides. 402 00:44:51,230 --> 00:44:58,970 So it's the power emitted by the gravitational wave times this rate of change of frequency as a function of time, 403 00:44:59,690 --> 00:45:05,060 inverse times the rate that I've described integrated over the parameters. 404 00:45:05,360 --> 00:45:12,590 And it turns out that this depends on frequency, like after the two thirds, after the two thirds is the same, 405 00:45:12,890 --> 00:45:19,430 after the two thirds that you get, if you look at black holes that have the mass of a few solar masses, 406 00:45:19,430 --> 00:45:24,620 a few times the mass of the sun, which I've showed showing you before, and actually if you have an F two, 407 00:45:24,650 --> 00:45:31,940 two thirds, it's a hint that your background comes from binaries of massive particles that are merging. 408 00:45:33,330 --> 00:45:36,930 Now here is another plot from the Nano Graph collaboration. 409 00:45:38,430 --> 00:45:50,270 These grey green vertical lines are essentially the data and if you assume so. 410 00:45:50,300 --> 00:45:52,710 So here again you have the omega. 411 00:45:52,920 --> 00:46:00,090 That look that goes should go like after the two thirds, it's it's equal to F squared times the characteristic strain. 412 00:46:00,450 --> 00:46:04,350 That means that the characteristic strain should go down. 413 00:46:07,340 --> 00:46:14,240 Like the minus. So. So the eight C squared should go like F to the minus four over three. 414 00:46:14,480 --> 00:46:18,350 And therefore H.S. should go like after the minus two of three. 415 00:46:18,740 --> 00:46:20,450 Okay. And if you look at that, 416 00:46:20,450 --> 00:46:29,240 you see this is a slope of two over three and it does really agree with the signal that goes like to f to the to the minus two over three. 417 00:46:30,230 --> 00:46:36,140 If you assume that the signal comes from a supermassive black hole binary background, you can, 418 00:46:36,770 --> 00:46:43,760 you can tune your parameters and go into this calculation of the right phi and get the blue line. 419 00:46:44,540 --> 00:46:51,110 Okay. So it means that once you measure, once you measure the slings and downs curve, 420 00:46:51,290 --> 00:46:55,040 you can learn something about the history of galaxy mergers in the universe, 421 00:46:55,040 --> 00:47:00,950 the history of black hole, supermassive black hole, mergers in the universe as a function of time. 422 00:47:01,730 --> 00:47:06,220 Okay. So that's the third type of fourth type of black. 423 00:47:06,470 --> 00:47:10,040 I'll be talking about all of these backgrounds today. 424 00:47:12,660 --> 00:47:20,850 Each one of them is different, but each one of them, if detected, the one detected, will tell us a lot about new physics, 425 00:47:20,850 --> 00:47:28,470 about physics that we don't know and will provide us with a way to measure a phenomenon that we don't know about yet. 426 00:47:31,170 --> 00:47:41,760 And and that does not require not does not always require sorry it does not always require resolving individual gravitational wave events. 427 00:47:42,930 --> 00:47:49,050 Thank you very much and I'll be happy to speak more about any of this in the break or afterwards.