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We have seen that gravitational waves are produced by a sudden motion of a bulk mass. Now, that motion could be due to an explosion. And we’ll see later that that explosion has to be asymmetric, and it could be an explosion like a supernova or gamma ray burst, like in this picture. The motion could also be due to a binary system. In a binary system, two stars are rotating against one another through gravitational attraction. And so this induces motion, and so this produces gravitational waves. It must be said that in most cases these binary systems are compact objects like black holes or neutron stars– and we’ll come back to that type of binary systems later.

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The wavelength of a gravitational wave is directly related to the size of the site where there has been a sudden motion of mass. In order to understand this, let me take again the example of a stone thrown in a pond. In that case, you might realise that the distance between the ripples that propagate in the pond is exactly of a size of the stone you have been throwing into the pond. So Let me explain that in more detail.

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So you see the ripples in the water. The distance between two adjacent ripples is what is known as the wavelength.

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And so you can imagine that this distance is precisely of the order of the stone you have been throwing into the water. Similarly, in the case of gravitational waves, you have the wavelength of the gravitational wave, which is, again, defined as the distance between two adjacent maxima of motion. And this distance is of the order of the source, so that means of the order of the size of the cosmic site where there was a sudden motion of mass that created the wave.

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I have mentioned earlier the frequency of gravitational waves. And it turns out that frequency is directly related to the wavelength, and it’s easy to see on this picture.

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The wave is propagating in this direction. And so that means that when I receive the wave, I will receive, for example, the different peaks at a frequency which is lower if the wavelength is larger. So that means the distance between the two peaks. So that means the lower the frequency, the larger the wavelength.

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And, indeed, there is a relation between the two. The wavelength is equal to the velocity of the wave divided by the frequency. Now, this has important consequences for the detectors themselves.

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If I come back to the site where the mass is suddenly moving, if a site is of the order of a few kilometres size, that means the wavelength is of the order of a few kilometres. That also means that the detector has to be of this size. And in this case, that corresponds to a frequency which is of the order of a few Hertz. On the other hand, if I’m looking at a site where the mass is moving rapidly, which is of the order of a few million kilometres, a cosmic site of very large size, then the wavelength is of the order of a few million kilometres.

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In order to measure it, I need a detector of a few million kilometres that corresponds to frequencies of the order of 10 to the minus 4 to 10 to the minus 3 Hertz. And so for this type of gravitational waves, I need a huge detector, and we have to go into space in order to have this kind of detectors.

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I said earlier that cosmic explosions have to be aspherical in order to produce gravitational waves. So let me review the argument. Well, actually, the argument goes in two steps. We first saw that a point mass, the point source, is not moving under the passage of the gravitational wave. Just imagine that we have a gravitational wave passing, then it will modify the texture of space and time that you would need a second point source in order to see that the distance is changing because of the passing gravitational waves. In the absence of such a reference, this point source is not moving. And, inversely, a point source is not emitting gravitational waves.

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Now, let me turn to a spherical astrophysical source, spherical distribution of mass. We have seen in the first week that for an outside observer, everything, from the point of view of gravitation, is as if the mass of this spherical distribution was localised at its centre. And so you see that from the point of this outside observer, this spherical distribution of mass has properties which are identical to a point source. And so, correspondingly, because we have just seen that a point source is not emitting gravitational waves in the same way, a spherical distribution is not emitting gravitational waves for an outside observer.

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The other type of gravitational wave source is a binary system of stars, mostly compact stars such as black holes or neutron stars. So you have such a system here. And you might think that this is a rare system. Well, it turns out that most stars are actually in binary systems, so we have many examples of such binary systems. Now, you see that the stars are rotating around one another. And so this is motion of mass, and so this produces gravitational waves. And it turns out that the frequency of the gravitational waves is directly related to the rotational frequency of the system.

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And from the point of view of an outside observer, when the stars have just rotated by 180 degrees to get into this situation, you see that if the two stars have the same mass, the observer is looking at the same mass distribution. And so there is just a factor of two between the rotational frequency and the frequency of the gravitational wave, the frequency of the distribution of mass, if you prefer.

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One example of such binary systems was studied in detail by the astrophysicists Hulse and Taylor and provided the first evidence of gravitational waves. In the system studied by Hulse and Taylor, the distance between the two stars is something like 700,000 kilometres. One of the stars has been identified, and it is what is known as a pulsar. So that’s a neutron star that is emitting a beam of electromagnetic waves in the radio frequency, and that beam reaches the earth every 59 milliseconds. So every 59 milliseconds, we get a light beam coming from this pulsar. And so that is acting almost like a clock.

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Now, these two stars are moving around one another, and so there is a slight Doppler shift because of the relative motion of this star with respect to the Earth. And so we were able to deduce, or at least Hulse and Taylor were able to deduce the exact rotation period of this binary system. It’s 7 hours, 45 minutes, and 7 seconds. Now, this system is presumably emitting gravitational waves. So it is losing energy. And so, little by little, these two stars will get closer. It will go faster, and the period of the system, the period of rotation, should be decreasing. And this is exactly what Hulse and Taylor discovered.

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They measured that indeed there is a decrease in the period of rotation of this system. And that decrease is exactly what is predicted by Einstein’s theory following the release of energy due to gravitational waves. And so you see that such a system provides an indirect detection of gravitational waves, and this is why Hulse and Taylor received the Nobel Prize in 1993. The first phases of the evolution of the universe could also have produced gravitational waves. The best example that we know is inflation. At the time of inflation, we have seen that there are fluctuations of the vacuum, and these fluctuations induce the fluctuations in the cosmic microwave background. Now, these fluctuations also induce fluctuations of curvature.

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These fluctuations of curvature grow because of the expansion and turn into gravitational waves. So those are gravitational waves of a primordial origin. Now, these gravitational waves at the time of recombination of the hydrogen when the cosmic microwave background was produced, these waves are imprinted in the light of the cosmic microwave background because they polarise this light. Now, what is the polarisation of light? Well, a classical example is light reflected on a surface. It could be the surface of the sea. It could be the surface of a puddle on the road. And you know that in this case, for example, if you want to prevent this reflected light, you might use sunglasses. Or you might use the windshield, which is polarised.

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Well, this is precisely because the light reflected on the surface is polarised. Now, the same effect comes from these primordial gravitational waves. They polarise the light. And so one is looking for this polarisation of the cosmic microwave background light. This is exactly what the BICEP2 experiment believed to have detected in 2014, this primordial polarisation of the cosmic microwave background light. Now, there are other sources that could polarise this cosmic microwave background light, and one source is the dust which is in the interstellar medium. In the interstellar, intergalactic medium, there is some dust. And the light reflected on the dust becomes polarised. And so that’s another source of polarisation.

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Since the result of the BICEP2 experiment, the Planck mission has studied in detail the presence of dust in the region of the sky where the measurement was made and has identified that there was more dust than previously anticipated. And so most probably, and unfortunately, the result of the BICEP2 experiment is not coming from polarisation due to primordial gravitational waves, but polarisation due to the reflection on the dust in the intergalactic medium. But it remains that the ultimate goal of cosmology is to identify this primordial polarisation because that would tell us information not just about 380,000 years after the Big Bang.

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It would give us information coming from the gravitational waves produced at inflation 10 to the minus 38 seconds after the Big Bang. And so it would lead to the first instance of the evolution of the universe.

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Let us summarise what we have learned so far about gravitational waves. These waves are a deformation of space-time, or more precisely of curvature, which are propagating through the universe. The sources of gravitational waves could be either violent cosmic phenomena, which generate large displacement of mass, or they could also be binary systems of masses which rotate one around the other. The gravitational waves have been indirectly discovered thanks to the study of pulsars, the famous Hulse and Taylor pulsar. And we have seen that the detection of gravitational waves requires to make extremely precise measurements of distances. And so we’ll see next time how one can do this type of very precise measurements.