 Vespoj, da sem tukaj na 2. več. Tako, da sem vse zelo v fronu. To je pravda, da je Philipo Vernizzi v Parisi, in je zelo počke, da je zelo počke. Tako, da sem počke, da sem počke, da je zelo počke. Tako, da se vse... ...zdarso, da sem tako načinja... ...zdebeznači... ...me... ...se postavijo to spavno, ker sem bovalo izstavnja, ...zaprej da je zvonil preko več vzivnač in... ...zaj, da sem bovalo izstavnja vzivnja, ...zaprej da sem bovalo izstavnja. ...saj bovalo po 15 noh, ...v 2006 do 2008. Zato sem zelo vseh taj spas, da je to počkaj izgleda. V 1998, da sem počkaj počkaj počkaj, sem počkaj vseh vseh v kosmologiji, to je to moj prvi vseh vseh vseh, in v mojj delaj je to, da sem počkaj vseh vseh. Zato me zelo jazem vzelo na vrste. V nekaj deli sem je vzelo v počke. Vzelo je to vzelo. Zato je to vzelo, da sem tukaj. Zelo sem na Blackboard. Vzelo sem, da sem vzelo, da sem vzelo. Zato sem vzelo, da sem vzelo. Kaj ki pas sem stajila mesej objetne viti, There will be a couple of exercises. Maybe I-we'll skip some calculations, but you can do them later, okay? So, today well this morning in the first lecture I'm going to discuss about in očneč, da je zelo, da se univeri oprečen, našli začneči. Prvno, da sem postošljala notacij, ovo je, da je, ovo je, tako, da sem pričal, da je frezman Lemaetra vs. Walker matrič, ki se vseče vseče vstaj, zato vseče vseče in z njega. Zelo sem zelo vseče in zelo vseče. Zelo je toga faktor, a in zelo sem zelo vseče. Tako, imamo zelo komuženje angolarne vrstečnje in danes dve angole theta in dfi. Tako, to je spesijalne metriče v t-kostanje seksu in funkciju f of k depenja na kurvacu univiru kajče in ekipalik signusje kajče, kajče, kajče in ekipalik signusje kajče, kajče, kajče, kajče, kajče, kajče, kajče, kajče, kajče. Njega imamo, da še uvrata, nekako imamo, je zelo vse, da je vse, da je vse. Now you cannot observe directly the scale factor, but you can observe... You know that the scale factor disguises basically the average expansion between galaxies. If you neglect pecular velocity, the hub of law is described by the scale factor. So you cannot observe the scale factor directly, but you can observe distances. And in cosmologies there are... two typical ways of describing distances one is to us standard Speed and the other one is standard laders Nakaz, what are standard candles? Sviško. Ah, are categories of- obž forests in stands of Under Boulder, in generally, under the계 of the lord, ob-уч poulager. We are accepted. o sefejice, kaj je izgledaj... Oh, nekaj, da? Ja, vse, da. Ja, da. Zato je, da je izgledaj izgledaj izgledaj izgledaj izgledaj izgledaj izgledaj izgledaj izgledaj izgledaj izgledaj. Zato, kaj sem tudi ... ... sefejice, nekaj sem tudi, če ima, da nekaj se izgledaj? Če so bril nam Tando Chi? Vese SC? Zato ne kaj smo ni izgledali na Wikipedia, in zdajjo tudi nekaj, nekaj, ki sem odprivil. Selo, da sem odprivila, nekaj nekaj nekaj, nekaj govoril, je Tando Chi v starosti, Soana. And you must tell me how you pronounce this name. Louis. How do you pronounce it in English? Love it. Erieta. Ah, sorry. Thanks. I don't know if you know the story behind. So basically we are at the beginning of last century, around 1910. And at the time, so Erieta was working at the observatory of Harvard, I think, and at the time women were not allowed to operate on telescopes. And they were often used to do calculations as a computer. So they were not allowed to use telescopes, but they were allowed to look at photographic plates with images of stars, for instance. And they were doing basically machine learning, although they were the machine, because they had to recognize all the stars and make catalogs on them and recognize the magnitude, the brightness of the stars, et cetera. And what she discovered was that there was a relation between the magnitude, the luminosity of the stars, and the period of surface. So sci-fi star pulsating stars, the luminosity pulsates, and also the diameter of the star pulsates. And she realized that there was this regularity, so a relation between the period and the luminosity that could be used to know the intrinsic luminosity of the star. And at the same time, so at that time, distances were measured only with parallax techniques. So basically you look at the sky, you look at how stars, well, the line of sight of the star moves when the earth move around the sun. So this was a technique that you could use for very close bi-objects, but when you go to very far away objects, the position of the star doesn't move much, so the uncertainty on the angle that you can take can measure, it's very large with the path to the angle. So somebody at around the same time managed to measure the distance to close bi surface with these parallax techniques, and that's all, because once you know the distance of one or a few of the surface, then you know how to calibrate the luminosity of the surface. This was truly revolutionary, because at that time, you could measure distance to objects as far as 100 light years. But with these techniques, one could go to 10 millions of light years, so one could really go to a few megaparsecs. And this technique allowed Hubble, for instance, to measure the fact that Andromeda, so the closest galaxies was out of our galaxy. It was not belonging to our galaxies, and there was a big debate at the time about this, and also allowed Hubble to measure the expansion of the universe. So very often, one says that the cosmology starts with Hubble, but probably we should say that it started with heritast only with that allowed at least to the beginning of really of cosmology. So this was very important discovery. So just to show you the relation between the period and the luminosity. So this is the period, and this is the luminosity of surface, and you observe something like that. OK, of course, in cosmology, well, if you use only sapphates, you can go up to a few megaparsecs. But this is not enough. If you want to serve the cosmic acceleration, you have to go to farther distances. So we have to use different standard candles. And as you know, these standard candles are supernovae nowadays. And moreover, also the concept of distance have to be redefined, because the space time is expanding, so we have to find. So the luminosity distance is the way of measuring distances if you use standard candles. So what is the luminosity of an object? It's the radiated electromagnetic energy in time. So it's DE over DT, basically. So it's measured in watts, its power. Then we can define the flux, which is what telescopes measure, which is what you observe. And the flux is given by the luminosity divided by the air. So the idea is that you have a star with a given luminosity, you surround it by a sphere, with the radius, the distance that you are. So here you have an observer, and here you have the star. And this area is the area of the sphere. So it will be given by 4 pi the distance squared. So basically, the flux is the amount of energy that reaches the surface, the unit surface, for instance. OK. Now, in an expanding universe, sorry, the universe is expanding, this area is not simply 4 pi the distance. We have to use the metric that we had before. This time we have to integrate over the surface elements, the two-dimensional surface elements. And if you remember, this was, so I had to make a double integral over a squared t f of k chi, and then the angular, the spherical surface element. And this will give you 4 pi a squared f k chi squared. OK, simple. Now, the definition that we use in cosmology is exactly the definition of luminosity distance that we use in cosmology is exactly the same as the one that we use in flight spacetime. So we will say that the flux observed is equal to the luminosity observed divided by the area. And the area depends on the time of observation and on the angular co-moving distance. And now, we want to relate the observed luminosity to the intrinsic luminosity of the object, knowing that this luminosity will be rejshifted with expansion, OK? The observed luminosity is the emitted energy observed divided by the emitted time. And we can write this as dE observed divided by dE emitted. dT emitted divided by dT observed times dE emitted divided by dT emitted. There are no colored chalks, OK? So this is the luminosity, the intrinsic luminosity of the object. And this will give you, so this will give you a 1 over z, where z is the rejshift factor. And this will give you another 1 over z rejshift factor, OK? You cannot see it. This will give you 1 over z, and this will give you 1 over z. This is just the rejshift of the energy. And this is just due to the rejshift of the time, of the period, if you want. So at the end, I will find that the flux observed is equal to the luminosity of the emitted object divided by 1 plus z squared a0 squared, which comes from here, is this k factor evaluated today. And then f of k chi squared, where chi is the co-moving angular distance from the star to today. And so we define the luminosity distance, right, the luminosity distance, as 1 plus z. Exactly, this is defined as dl squared a0. OK. OK, now we need to compute chi, the angular diameter distance. And to compute it, we can just look at the light geodesics. Ah, thanks. Wow, OK. So we just solve this equation. So chi is equal to the integral between the time of emission to the time of observation of dt over a of t. And then I change the variable. I go from the time of observation to the value of the scale factor, sorry, from the time of emission to the value of the scale factor at observation, da over a dt over da. Then you have defined probably the Hubble rate as a dot over a. So I use this here to rewrite the same, but in terms of the Hubble rate. And finally, I convert the scale factor in terms of the redshift. So I use that a of z is equal to a0. So the scale factor today divided by 1 plus z, this here. And I can integrate between the redshift at 0 today. Sorry, there is a factor 1 over a0 here. And the integral goes from 0 today to the redshift of the source, the z. And what I left with is just h of z, because I use that da is equal to minus dz 1 plus z squared a0. And this canceled this a squared here, the denominator. OK. So I can finally write an expression that we can use for the luminosity distance. But before doing that, let me define two things. Well, one is, I think, yes, I seem to find this in this course. But anyway, sometimes we will use this quantity. We will use this quantity, which is the ratio of the Hubble rate as a function of the redshift divided by, sorry, by the Hubble rate today. OK. And another definition is omega curvature today, which is defined as k, the curvature, divided by a0 squared h0 squared. And this is defined with a minus, because we want that the sum over all omegas plus omega k is equal to 1. And I will use the same notation as a seam in adding a0 when I am evaluating the omegas today. OK. So omega is the usual critical density, energy density divided by the critical density. So 8 pi g rho i divided by 3h squared for even species i. Now a0, by inverting this relation, can be written as h0 minus 1 over the square root of omega k, evaluated today. OK. And so now I'm going to rewrite this expression by using the expression of the commoving radial distance. The commoving radius. So I will have, and I'm using this passion for a0 from here. So we have a0 minus 1 divided by square root of omega k0. And then f of k, square root of omega k0, the integral between 0 and z, where z is the redshift of a mission, any questions? Why there is a? Well, just because I'm integrating between 0 and z, so I had to change the variable, I'm just adding a tilde. Z is the redshift of a mission. OK. Yeah. So this is infinite, but if you expand this function, so this function is either, so remember, f of k over chi is sinus of chi, chi, or hyperbolic sinus of chi. And for a small argument, for a small chi, all these functions start linear in chi, and then there will be a third chi squared, or plus a third chi squared, or it's just chi. So the zero curvature here cancels with this one. And you have a finite result, of course. And in fact, you see immediately that the dependence on the curvature appears only at cubic order in chi and in z in the redshift. So I can, in principle, I can expand in the redshift. And this is what I'm going to do now. So we can, so supernovae are measured at low redshift. So let's say, around smaller than one. So first of all, if I expand a cleaner order, if I expand this expression at cleaner order in z, I can consider e to be constant. And I find that, so linear order in z. I find that dLz is equal to h0 minus 1, this one, this I can neglect, times z, which is the Hubble law. But of course, I want to go higher in a second order to see the effect of the cosmic acceleration. And so to do the calculation, which I will leave you, I will leave it to do it as an exercise, I can expand this function for a small argument. And as I said, the first term is linear. And the second term will be cubic. So I can neglect the second term in this function. And I will just use the linear term. And then I expand this, I will expand this in Taylor series. So for exercise, you can compute this, of course, will give you, this term will give you h0 minus 1. And it is less as an exercise to compute this term here, the term second order in z. And you will find that dLz to derivative is equal to h0 minus 1, 1 minus a dot dot h over a, evaluated today. So now we see the acceleration appears here in this function of the luminosity distance as in terms of the redshift. And it is traditional to define the deceleration parameter in this way to describe the acceleration of the universe. And the reason why there is a minus sign here, as you know, is historical. Because before discovering the accelerated expansion, people thought that the universe was decelerating. Nobody was, well, nobody. Most people were expecting to see a decelerated universe. And of course, sometimes you use q0, which is all this evaluated today. So we can write down the Taylor expansion of the luminosity distance as h0 minus 1, z, which is the linear term. And then we have 1 plus 1 alpha, 1 minus q0, which comes from this term here, the 1 alpha, just from the Taylor expansion, z plus order z cube questions. OK, very good. So now we have an expression that relates the luminosity distance to the deceleration parameter. So we are going to talk about supernovae as standard candles to measure the luminosity distance as a function of the redshift and infer this q0 parameter. OK. So what our supernovae type 1 is? Well, first of all, as we said, they are ideal to, as standard, to go to high ratio, because they are extremely luminous. They are very bright, so they can be seen at much farther distances than sapphates. So the idea is the following. In fact, I have a, so just to. So you see that the supernova can be seen as luminous, basically, as a galaxy. And let's explain why. So the idea is the following. Let's suppose that you have a white dwarf in spiraling, in binary, with another star, with a companion. So you have something like that. And the companion is, well, the white dwarf is creating material from the companion, something like that. OK. Here you have a nice artistic view of this. So white dwarfs are the remnants of standard stars. And in which the, so as you know, in stars, there is always a competition between gravitational force and pressure, in order to sustain the star. And while, you know, the pressure is due to thermal energy, in a white dwarf is due to the electron degeneracy pressure. And typically white dwarf stars have a mass, which is one of a limit, which is called chandaseka limit, which is about 1.4 solar masses. And above this limit, the degeneracy pressure of the electrons is no longer enough to sustain the gravitational force. So something happens when the mass reaches this limit. So typically, when you have this binary system, the mass of the white dwarf, and here we have a companion, the mass of the white dwarf starts increasing, and hit the point where it grows higher than the chandaseka limit. And the internal temperature of the star increases. And there are processes that were not allowed before that start taking place. In particular, the most violent process is what is called the carbon fusion. So this process here, so typically white dwarfs are made of carbon and oxygen, which are the remnants of the standard stars. And at this temperature, at this value of the mass, the carbon can fuse, it can form heavy elements. And this process is extremely violent. And in fact, it ends with a explosion, which can be seen very far away. This is sometimes called the carbon detonation. So it's a sort of a runaway reaction that basically ends up in an explosion. So because this happens always at this critical mass, the luminosity emitted by the supernova is sort of standard. It's sort of the same for supernova type 1A. In fact, we can show. There is a request to explain the carbon detonation process once more. OK. So, well, just that, as I said before, when the mass of the white dwarf starts becoming higher than the Chandraseka limit, the star starts collapsing because the electron, the generacij pressure, is not enough to counteract the gravitational force. When it starts collapsing, temperature increases. Before the temperature was not even defined in the sense, the pressure was due to the electron, the generacij. But now we have an increase in temperature. And this process that before could not take place now starts taking place. So this is fusion of nuclei, of carbon, to produce heavier elements. And one can show the characteristic luminosity versus time light curve. Here is luminosity. And here is the time from the peak of the luminosity. I'm going to explain what it is in days. So the luminosity of the supernova from explosion peaks. And by definition, we put this peak at day zero. And then the typical time scale is weeks. Is of the order of weeks. And one observes, so by the way, this peak is due to the production of nickel, which when it's produced is extremely luminous, is extremely bright. There is a lot of electromagnetic radiation associated with it. And one can see that those supernova that because of a different composition, for instance, because of less carbon or less oxygen in the supernova, have a smaller peak. Their luminosity time is also short. But however, there is a way of correcting from this effect and the standardized supernova. There is a relation, in fact, a simple relation that has been complexified later on. But there is a simple relation between this luminosity course. And at the end, you can basically find a way of standardize the supernova into standard candles. In fact, this is what. So there is a way of correcting the fact that different supernovs release have a different luminosity. But because we know the relation between a luminosity and a time-like curve, we can standardize them. And we can have a calibrate them. And we can have a standard candle. So the typical relation that is used to standardize supernova is called Philips relation. Yeah, because white dwarfs have different composition. So the amount of carbon, oxygen, et cetera, is different in different white dwarfs. So although they all start exploding at the same mass, more or less, their luminosity is different because of that. So the amount of nickel, for instance, that is produced, or other heavy elements that is produced, is different. OK. So of course, you measure supernova very far away, but then how do you remember before, we talked about sapphates. And we said that at the same time, as strongly with discovery, there were people who measured the distance of sapphates using parallax. Now you also have to measure the distance of some supernova to infer the absolute brightness, so the intrinsic luminosity of the supernova. So how do we do that? Can you imagine? Very good. Exactly. So yes, so very far away supernova are calibrated using close by supernova. This close by supernova are calibrated using sapphates. And these sapphates, which are far away, are calibrated using close by sapphates, which are calibrated using parallax. OK. So it's a complicated process, but it's what it is. I mean, this is what we can do. So at the end, we always use parallax. Of course, now we have much better instruments than before to measure the position of stars, et cetera. But there is what is called the cosmic distance ladder. So it's sort of a ladder. And I think I have here a plot. So you see, first we use parallax to calibrate nearby sapphates. And then we calibrate farther away sapphates that are used to calibrate supernovae, which are used to calibrate farther away supernovae. So we expect that there could be also some possible redshift dependence in this model. But let's first look at what we can obtain with supernovae. OK, the discovery of q0. OK, this is so the discovery that the universe is accelerating, yes? So if we wanted to show a supernova, which is farther away on that same plot, how would it look like? Would it just be a fainter one, or the shape would be different? Well, on this plot, I'm showing the intrinsic luminosity of the supernova. So it will be the same, whether it's farther away. If there are no evolutionary effects, it will be the same. So this is the intrinsic luminosity in this plot. But somehow you want to break the degeneracy with distance. In a sense that you are saying there is this variability in luminosity, even at the same distance. Yeah, at the same distance there is the variability in luminosity. But you can break it because the time like curve is shorter for dimmer supernovae at the same distance. So the time like curve lasts much longer for brighter supernovae than for dimmer supernovae. OK, so if I see something that is farther away, it would be faint but broader. Exactly, so yeah. So you can standardize them with a simpler relation. Well, the Phillips relation does not depend on call, or et cetera, but then you can improve the simple relation. But there is a, yeah, if that is one parameter relation. Yeah, but this you can take in into. The question was, yeah, so, good. The question was, there is also a time delay for distance supernovae. Yes, there will be, and you can take it into account. However, when you calibrate the supernovae, you do it rather close. So there the time delay is totally negligible. There are also lensing effects. There are many effects that people take into account. So, but the calibration is done at much lower rate than what we use to explore, to measure the acceleration. Yes, so, yeah, the question was, can you explain again the calibration? So, I'm not going to show you the relation. It's very, in fact, it's very simple. It's a very simple relation. But the idea is the following. You observe a pattern. So the fact that brighter luminosity, so when the peak associated to nickel is higher, have a longer time of, so the time light curve, this one, is much longer. You can see it also here. Is much longer for the supernovae. Then those that have a lower brightness and a lower peak. Here the time is much shorter. And there is a relation with one parameter that you can make. Basically, to, yes, to relate or to find the same curve, the same luminosity. Yeah. Ah, the relation between, OK, OK. So the question was, how you calibrate them? You calibrate the absolute magnitude of this supernovae. Yes, OK. So this is shown here. So you have a far away supernovae. Sometimes, ah, no, sorry, it's my fault. So you have far away supernovae, OK. Well, let's start here. First of all, you calibrate the sapphates. So you know the luminosity of the sapphate. And therefore, you know the distance, OK. Of all sapphates, also of a far away sapphates. Then if you have a sapphate in a galaxy, which also contains a supernova, then you can calibrate that supernova, OK. So you observe a supernova, for instance. One of those is very bright and has a longer time like curve. Then you calibrate that one. So you know it's intrinsic luminosity. And you can infer the intrinsic luminosity of all the others if you trust this relation, OK. Well, you observe, how can you know that they are nearby? Is that OK? Well, I think that you see that they are, well, first of all, around there is nothing. And then, probably, you can also see that they are bound. They have a velocity that corresponds to the galaxy. I must say, I'm not an expert of that. But yeah, I think this is probably quite standard to know. Yes, I'm sure that this is done. That the spectra, first of all, we use the spectra to measure the redshift. Of course, we haven't talked about the redshift. But because this is well known, you can measure the redshift by looking at the spectra and by looking at the redshift of the spectra. But yes, the luminosity and the color of the supernova is used. We know that we can study the boundaries. We can try to use also the boundaries of the elements of the white drafts to improve these relations. Is the redshift related to? Well, this, I don't think. So the question was whether this parameter here was related to the abundance of the elements. Well, I think that all these models are very empirical. And that we don't have a full understanding of the supernova explosion, like that we can really simulate it on exactly as a function of the bond, et cetera, on a computer. Most of these relations that are used to standardize the supernova are really empirical. So there is some uncertainty also there. OK. How much time do we have? 50 minutes. OK. OK. So, well, now we want to measure this acceleration. And so what the luminosity distance. So let's plot something which has no dimension. So it's the luminosity distance as a function of redshift times h0. You remember the formula that I had written before. This was given by z1 plus 1 alpha, 1 minus q0 z. OK. So we are plotting this. Let's put some 1.5 here, 1 here, 0.5 here. And the redshift here. And let's go up to 1 more or less. OK. And we can, so the curve starts at 0. And let's draw three different curves. 1, well, it's 1 for q0 equal to 0, a universe that is not accelerating. 1 for q0 smaller than 0. So here we have acceleration. And 1 for q0 larger than 0. So here the universe is decelerating. OK. And what we measure, in fact, is this. So if you, let's say that at fixed redshift, if you consider a supernova at fixed redshift, the supernova that we measured look dimmer. OK. So they look more far away, farther away. So they look dimmer than what they would be if there was no accelerated expansion or if the universe was decelerating. And at fixed distance, they look farther away. Or they look, sorry, at fixed distance. No, they look closer. So they look that they are shifting slower. OK. Yeah, so this makes sense, because the expansion of the universe increases in time. So they should have started slower. OK. They should be closer. And they should have started with a slower redshift, because later on, this expansion has increased. OK. So it makes sense that we find something like that. And I think I have a plot here that I took from a paper of 2000 where you can see the difference between, well, several universes, for instance, a universe where there is no cosmogical constant and omega matter, the amount of the matter, or normal electricity component, is one. And you can clearly see that supernova are showing us that we are not clearing in such a universe. I don't remember what I have. Oh, yes, here have a compilation of supernova as for today. This was from 2001. It's the pantheon compilation. So it gives you an idea of the fact that we can go to higher redshift. And we have many more supernova in the plot before. OK, so let me. So I would like to discuss a different way to measure the luminosity distance, which is not using light, but gravitational waves. So you know that if, well, in 2015, we observed binaries of black holes emitting gravitational waves. So we observed the gravitational waves emitted from these binaries. And the strain, so the amplitude of the gravitational wave emitted is inversely proportional to the luminosity in the distance of these objects. And then is proportional to some power, 5 over 3, of a quantity, which is called the shear per mass, which is, well, you can define it. It's not very important here the definition, but it's given by n1 times n2, so the product of the two binaries to the 3-th, divided by the sum of the masses of the two binaries to the 1-th. And this combination comes out really from Kepler laws. And then the amplitude of gravitational waves emitted depends also on the frequency of the gravitational waves to the power of 2-th. So if you know, so you can observe the frequency, is the one that you see in your instrument, in your interferometer. And if you knew the shear per mass of these objects, you could infer the luminosity distance. And in fact, you can infer the shear mass because by using the quadruple formula, you can basically compute the change in frequency of the gravitational waves, which is due to the fact that these two objects are, since they release gravitational waves, are getting closer and closer. So the frequency increases in time. And it increases in time due to the release of gravitational waves. And there is a relation between this change of frequency and the shear mass. This is the following. So you can infer the shear mass. Even the shear mass, you can measure the strain and you can infer the luminosity distance. So it's a way of measuring the luminosity distance of an object. The problem is that, yes, this is, in principle, is the redshifted shear mass. And this is exactly what I was coming to. The problem is that you don't know the redshift. And there is, so, in principle, this mass is the redshifted mass, which means that, for instance, the same signal from a supernova with shear mass m at redshift 0 is given by a supernova with shear mass measured, shear mass m over 1 plus z at some redshift z. So there is a degeneracy with the redshift, which is called mass redshift degeneracy, which does not allow you to infer directly the redshift from the supernova, from the binary. So the nice thing of the binary is that it's self-calibrated, concerning the luminosity distance. We can know the luminosity distance. But just looking at it, we don't need the sapphates. We don't need to calibrate it without their objects. But the problem is that we are missing the redshift. So nowadays, there are several ways that people are exploring to get the redshift. So I'm just going to list them here. Well, yeah, I will show something later. OK. So, well, the first of all, is to have a direct counterpart. Example, in 2017, we measured the merging of two neutral stars. And we could detect, at the same time, the gravitational wave and the gamma rays emitted. And we could really observe the redshift. Yes? This is coming from the fact that all these quantities here have to be redshifted at the moment of. So you observe this, but this law holds, for instance, the frequency that I put here is the frequency at that time of emission, which is redshifted with respect to mine. But also the real definition of a redshift mass that I put here is a redshifted chip mass. So all the quantities that, because to infer this equation, I'm using, so for instance, the quadruple formula, I'm using quadruple formula over there, so I'm using frequencies over there, et cetera. But then I have to redshifted them here. And OK, one is to detect, one is that the binary release electromagnetic energy. But this is very rare. For the moment, we have seen, well, we have seen other emerging on neutrostars or neutrostars in the black hole, but we haven't seen a electromagnetic light associated with them. Another one is to see galaxies around, localized at the same position as the meters of the gravitational waves. So to look at catalogs and try to correlate the emission with the presence of galaxies. Then another way could be knowing the source, so the mass distribution of binaries. Imagine that we know that for central reasons we have a lot of binaries with certain masses, or we have a mass gap. For the moment, it's not clear whether, so this mass distribution is not totally clear for the events that we have in LIGO. But maybe one day we will manage to measure the distribution of the number of binaries as a function of their mass, and there will be some peak mass gap or something like that. Well, in this case, we have something that tells you where these masses are, and you can infer the true masses. And finally, tidal deformations. So what I said here is restricted to the point-particle approximation, so it is restricted to treating the two objects as a point-particle without an internal structure. But in the case of the neutral stars, for instance, tidal deformations are important. They determine the internal structure of the star, and this breaks this redshift mass degeneracies. And, in principle, you could infer something about the redshift. So, in principle, these are potentially interesting for the future. Let's see. Well, I think I'm going to stop here. Any question? Any further question? OK, I don't see any. Is there a question over there? Sorry, maybe I missed it, but when you say galaxies, you mean like determine the host galaxy of the binary? So these binaries will be in galaxies. So ideally, you would like to infer which galaxy the binary belongs to. This is not always possible, but you may try to do it statistically. So you have a lot of binaries emitted. You have a galaxy catalog, and you can try to fit parameters using trying to associate. So these binaries will come from a place where there are galaxies. And you know these galaxies here because of galaxy catalogs. And you try to associate this binary to a galaxy. But if you cannot associate to this single galaxy, you can try to do it in a statistical way. In a sense, you will have some ratio parameters that you will fit. And yeah, you try to fit in a statistical way. Maybe we can postpone a third of questions to the discussion session. So now we take a break. Let's thank Filippo.