 Let's start the seminar by Mostafa Amin. Oh, Mostafa Amin is visiting us from Rice University. He agreed to give a seminar on dark matter with a short introduction for the students, who are not experts on dark matter. But then later it will be more specific about the vector dark matter model that he's working on. So thanks, Mardad. It's been great. It's been great talking to old friends as well as to the new students that I've met. So I changed my mind a little bit compared to what Mardad just described. So I will talk about this new work on vector dark matter, but instead of giving you an introduction to dark matter generally all the way at the beginning, I'm going to intersperse it along the way, wherever the need arises to give you more details. Okay, so let me start by saying that I'm going to talk about... Most of the talk I'm going to be talking... I'm going to be telling you something about these three papers that are here. And it's mostly about vector dark matter or just dark matter that is made up of particles of higher spin. Not just scalars, but vectors, perhaps even massive tensors. Before I begin, the work is in collaboration with these people up here. On all three papers is Mudad Jain, who's a postdoc at Rice. Hongyi Zhang is on one of them. He's a graduate student and Rohit is an undergraduate and Philip is a research scientist, not at Rice. Okay, let me minimize this. Somebody asked me a question about... What is this going on? It's on my screen. Are you sharing the entire screen? No. I've lost control. Oh, there, it's back. Let me... I think it'll be okay. High video panel, high floating meeting controls. I think this should be fine now. I tried to remove it, so... So, somebody asked me... When I said I was from Rice, I said where Rice is. I thought, given that this is a cosmology school, I would give my cosmic address. And appropriately, so, we are at the center of the universe. At least SDSS claims so. And not quite at the center of the galaxy, but it's okay. For the whole universe, we are the center. It's okay. We are in Houston, Texas, in the US. And if you want to know more about Rice, just ask me afterwards. Okay, so, before I get into the details of the talk, especially given that there is an audience on Zoom as well, and I know how tired I get on Zoom within the first five minutes, so I will give you the punchline first to start with the talk, because I'm sure half the people will not be listening after the first five minutes. Okay, so, the main motivation of this work for me, apart from the fact that it is a lot of fun, is I want to know whether we can tell something about the spin of dark matter from astrophysical observations. I hope that by the end of this talk, you will be able to tell that these two simulations, that look rather similar, have differences between them, and you will be able to tell that one of them is a scalar dark matter simulation and one of them is a vector dark matter simulation. And there will be a test in the end, so pay attention. Okay, in a little bit more detail, I'm going to tell you that if dark matter is made up of a light vector field, then it contains within it a class of solitons, localized over densities that are ground states of the system. This is not different from scalar dark matter. The interesting thing about these is that these solitons are polarized and have macroscopic amount of intrinsic spin angular momentum. And this is a new thing about these guys. I will also tell you that in these models of dark matter, there is a potentially observable signature in the form of interference patterns. This is wave-like dark matter, so there is observable signature in terms of that, which is different from the case when it is made up of scalar dark matter. Somewhat a little bit more speculatively, even the shape of halos is slightly different. Finally, time permitting, I will also tell you a bit more about the formation mechanism of vector dark matter, as well as these objects. As Mehrdad mentioned, interspersed through this sort of research presentation, I will also give you some pedagogical introduction, short bits of it in terms of things about dark matter, things about differences between particle and wave-like dark matter, a little bit about solitons, and a little bit about the power spectrum of density fluctuations as well. All of these things are required in my talk, so I will just do a slightly longer introduction than normal for these concepts as I go along. With that, let me start the main part of the talk. And again, please feel free to interrupt me throughout. My plan is, so Mehrdad told me the talk is about 10 hours, so I will take a break at five in the middle, maybe for five minutes, because I will need to go to use the restroom, but otherwise, you know, I am sure you will all be here. Okay. Well, given the fantastic work that your lecturers have done for me, my life is a lot easier. Asim as well as Marco have done a fantastic job of introducing density fluctuations and also a bit about dark matter along the way. So I will just start by saying, I think we all mostly agree that dark matter exists and we know it exists because of its gravitational interactions. What we don't know is what are the properties of dark matter? What is this charge? What is this spin? What is this mass? What are its interactions within itself and with the stuff that you and I are made up of? Is it an elementary particle or is it some composite thing made up of many elementary particles together? What are its formation mechanisms? What is this cosmic history? Did it form via thermal mechanism? That is, from a thermal plasma? Did it come about via some non-thermal mechanism? All of these things we don't know about dark matter. Let's start with just knocking out a few things out of here. If dark matter had electric charge, that's no good. We've already seen that if dark matter interacted strongly with photons in the early universe, we would not see the structure we see in the cosmic microwave background, which you will learn more about from Blake next week, or the matter power spectrum, which you heard about yesterday from Asim. The fact that dark matter cannot interact very strongly with photons tells you that the charge of dark matter, the electric charge has to be less than about 10 to the minus 7 of the electron charge. That's what it tells you. This is weakly charged. What about dark matter's interaction with itself or with regular matter, say protons or neutrons? The interaction of dark matter with itself can be constrained, for example, with the help of this beautiful observation of merging of two clusters called the bullet cluster. The bullet cluster is a thing that looks like a bullet. This observation tells us that the self-interaction or the scattering cross-section of dark matter per mass has to be less than 10 to the minus 24 centimeters squared per GeV. This is about a nuclear cross-section. That's roughly what the number is. It's a very stringent limit when you think about its interaction with protons. This is from xenon, and as you can see, this number is going down to about 10 to the minus 40-ish. For neutrons, it's even slightly different. So that's charge knocked off. That's a little bit about the self-interactions or interactions with the sun and model particles knocked off a bit. What about the mass? Here, we reveal that we really know nothing. The mass of dark matter particle is highly unconstrained. It can be as light as 10 to the minus 21 EV-ish. This is really light. I don't actually remember the exact number, but I'm not even sure if the constraint on the mass of the photon is that good. So we are allowed to have massive cold dark matter. That's this light. It doesn't contradict observations. You are also allowed to have masses very high, up to a long scale, or if it's composite, even higher. So there's a huge range of dark matter mass that's available, and we don't know what the mass of this constituent is. Very important for our purposes in this talk, we also don't know much about the spin of dark matter. We don't know whether it's a spin-a-half integer spin, such as whether that's a fermion or it's a boson. But we don't know some things regarding spin. If dark matter happens to be a fermion, its mass better be heavier than 100 EV, which corresponds to saying that the de Broglie scale, assuming usual typical velocities in our galaxy, has to be smaller than 10 to the minus 6 meters. Where does this come from? This comes from the simple fact that fermions don't like to sit on top of each other. They already knew about COVID-A before we did. So they were just socially distancing all the way from the beginning of the universe. And that tells us, that maybe even tells us why they have to be heavier, but they cannot be lighter because you cannot pack so many of them in a given volume. We already know what the mass density of dark matter needs to be in a local neighborhood, about 0.3G V per centimeter cubed. Because of this restriction on how many fermions you can pack into a phase space volume, that in turn gives a restriction on what the mass can be. If it's too light, you cannot make up the mass density in a given region. So that restriction tells us that these fermions have, if dark matter is made up of fermions, it has to be heavier than 100 electron volts. On the other hand, if dark matter is bosonic, it has no such restrictions. You can pack as many bosons together as you want in a given volume. So you can make up the local mass density by just packing on many, many, very light bosons together. To give you an example of how many of these there can be in a typical volume, de Broglie volume, look at the numbers. If I make the mass of the dark matter particle about 10 to the minus 5 EV, this is like a QCDA axion-like particle. You have 10 to the 23 of them. This is humongous. If I make it ultra light, like 10 to the minus 20 EV, this is going to be 10 to the 83. So there are many, many particles. So how should I think about these? Particles stack one on top of the other, all sharing a roughly similar de Broglie wavelength, just stack one on top of the other. It doesn't make much sense to talk about these objects individually. They're just sitting on top of each other. Indeed, the de Broglie scale corresponding to these masses that I've just said is macroscopic. So for an axion-like thing, the de Broglie scale, for a QCD axion-like thing, the de Broglie scale is about tens of meters perhaps. So that's macroscopic. For 10 to the minus 20 EV, it can be one parsec in size. So these are astrophysical scales for these particle sizes. And they're all packed on top of each other. So it seems to make very... So it perhaps motivates us that we shouldn't be thinking of them as individual particles. We should just be thinking of them as all sitting on top of each other, a collection of them together, and we should be perhaps thinking of them as a field, the classical field of this stuff. If you want, you can think of this as a condensate, like a BEC. Yes, they would have too much speed. We want them to be cold as well. You would want them to be cold as well, right? Can you repeat the question? Sorry, the question was, why can't we fill up higher levels? I believe the argument would be that they would not remain cold. They would move too fast if you get to very high states. Yes, I guess... Repeat the question. Sorry, I keep on forgetting. The question was, what are promising candidates for fermionic dark matter? I think you could pick something from Susie if that was something, if your flavor is appealing to you. But any fermion that I can pick that has reasonably weak interactions with the standard model will work. There are some questions in the chat. I also have a question. I don't understand the second equation because the second and third term don't seem to be equal. What do you mean? I mean, I'm just saying that the de Broglie scale for this mass particle for different masses is different. Okay, I got it. Let me ask questions from you. Can we say dark matter has both bosons and fermions since neutrinos are also alike? You could certainly have an admixture of fermions and bosons. Nothing is preventing that. In fact, you could have a whole dark sector that's present to somewhat similar to the standard model. Perhaps it's a prejudice of ours to think that dark matter is simple and made up of just one particle. It could be two particles. It could be an entire dark standard model sitting there. So we don't know. Can gravity overcome the degeneracy pressure due to Pauli's exclusion principle for fermions? Yes, but hopefully not in our vicinity. That happens, of course, when you form black holes. That is happening when you overcome gravity in neutron stars. That's exactly what is happening. You add more mass and you're overcoming the degeneracy pressure to go from a neutron star to a black hole. But not for dark matter, typically. For any fermions, why do we see them? The interactions will be very weak, right? You have to produce them. If the interaction is too weak with the standard model, how are you going to produce them? Is dark matter, its interactions are going to be very weak? It has to be very weak, otherwise it would not be dark matter. No, but suppose I have a particle that is completely non-interacting. How would you produce it? It only interacts via gravity. Can you produce a particle? You can't, right? If it doesn't interact, to produce something, it has to interact with the stuff you're colliding or it has via something. Okay, maybe I'll repeat the question and I'll try to answer the way I understand it. So why don't we see heavy fermions, heavy dark matter fermions at the LHC? I think it would be great if we could see them, but if the fermions don't interact much with the standard model, which is kind of expected for dark matter, it would be very hard to produce them there. Does it have to be that way? No, perhaps we would have gotten lucky if we could produce them at the LHC. We haven't, but that doesn't mean that dark matter cannot be a fermion. It's just that its interactions would have been weak. I don't have a favorite. So I wish... I don't have a good candidate in mind, yes. Why would... They don't have to. Photons don't... This is of course a slightly contrived example, but photons are bosons. You can think of many bosons that are non-rat... What would make them collapse, you think? But to collapse, it has to be somehow that there should be nothing preventing... We'll get to this a little bit in some of the examples. You have to have... Gravity just pulling you inwards, and you have to have enough mass for it to collapse. Typically, there is something preventing that from happening. Usually in the case of a field, I'll interpret the question, whether you're asking that or not, is if you have a field, usually there are gradients in the field. Gradients want to spread out, so they push outwards. And that prevents these things from just coming back. There is, of course, virialization. Sorry? Virialization. That's true, but I think if they were completely cold... I thought the question was if they were completely cold. Right. What would happen? Right, let's move on. This will be 10 hours, I think. Sorry, can I ask another related... Can we assume that dark matter field is excitation or dark matter? Let me read it. Can we assume a dark matter field or a potential and a state that dark matter particles are essentially excitations of this field instead of being fermions or both? I think if we... As we all should, if you believe quantum field theory, every particle is an excitation of a field. So I think in that sense, yes, these are excitations of a field. Okay. Let me now say, given all the good discussion we have had and given what I've said so far, I hope I've convinced you to some extent that when we think of light bosonic dark matter, it's not crazy to think about this as a field, a classical field just with particles sitting on top of each other everywhere. Instead of thinking about them as individual particles, and once you start thinking about them as a classical field, wave dynamics comes into play. You can think about linear and nonlinear wave dynamics. You can think about all the wonderful phenomena that fluid dynamicism and other people have figured out, such as interference effects, solitons, et cetera. These kinds of effects, of course, have been seen and have been explored for a long time in the context of condensed matter systems, such as bosonic condensates, as well as in the early universe. So as I said, now coming back to a little bit away from the review part and getting back to my topic. So I want to talk about the spin of dark matter and whether we can distinguish them. Let me tell you that in the context of scalar dark matter, which is spin zero, this has been explored quite a bit. We also heard a talk yesterday about this. People have been talking about QCD axions, fuzzy dark matter, many different names, and people have done simulations of these already on cosmological scales. And this is well explored by now. What we want to talk about is what are the differences if I think that dark matter is a vector field? Why does it have to be a vector? No. Does it have to be a scalar? No. We don't know. Why am I doing it? It's fun. I have no better reason. That's why I do physics. I wish I could tell you this observational exact motivation for this, but there isn't. But there are implications of this, which can be tested observationally. So you could potentially tell what is the nature. So what is the fundamental underlying model that we are going to look at in case of vector dark matter? The underlying model is the following. It's very simple. It's just electromagnetism with a mass added to it. That's it. So electromagnetic field strength tensor, that's G mu nu, capital G mu nu. W is the vector field. I've added a mass term to it. That's what the second term in the action is. And then there is gravity. This is very similar to a scalar field action that some of you might be familiar with. That's the Klein-Gordon action. If you're worried about loss of gauging variance, et cetera, you could imagine that there is a, this is an Abelian-Higgs model. I've integrated out the Higgs, if you want. But this is the model. It's just a massive vector field. And for the moment, I'm only going to consider gravitational interactions, no others. But I want to do dark matter. One of the things we know about dark matter that in the today's universe is pretty non- relativistic. It doesn't move very fast. That means the spatial and temporal scales are slow. There is nothing relativistic about dark matter. As a result, it would be stupid to take the entire action and try to simulate this and try to get structure formation. Because I'll get information that I don't need. I will get, so for example, this has a mass. The mass, it's a field. It's a real field. It's just going to oscillate with that frequency of the mass. And that frequency is going to be on the mass scale, which is much higher than the Hubble scale. There is no way I can track that many oscillations numerically or otherwise for an entire Hubble time. So I shouldn't be tracking that. A better thing to do would be to write down the action for the slow degrees of freedom. What I do in detail is you take the real-valued vector field. You split it into a slow part. That's the field psi. And a fast part, which is e to the i m t. I've written down c's and h bars here. I will not refer to them because we don't refer to such things. But I have written them just because talking to different audiences sometimes is useful to have them in there. OK, so the fast part. Can I ask a question? So when we couple a massive vector to other fields, unless we couple it contracting it with a current, we usually introduce a cutoff to the theory. I was wondering when we couple it to gravity, which is going to be important, do we also... In the theory that you wrote down, can I think of the cutoff as in Planck or do I... the fact that I have a massive vector, does that mean I have a lower cutoff? I don't know if for this field there is a lower cutoff. This is just a massive vector field. I guess one way of thinking about a cutoff here is also that I've integrated out the Higgs here. So that will introduce a scale. That's the ultraviolet theory that we began with. But otherwise, I don't think there is... but maybe I haven't thought enough about this. Yeah, I just mean, so if I couple this w... if I wrote down, say, w squared times phi squared, where phi is a real scalar, it means there's a cutoff, and secretly there's a cutoff in the theory. That's true, that's true. But I don't know... I've never seen it... when you couple it to gravity in the minimal way, I was wondering if you knew if there's a cutoff. I can tell you a little bit about it, but maybe if you don't mind, it will take us too far away from the talk. Let's talk a little bit later about this. Thank you. Okay, so I integrate out the fast modes and I write down an action for the slow modes. Is this new? No. This has been done for scalars for a while now. In the case of scalars, this psi is just a complex number at each point in space and time. In the case of a vector, this is just supposed to be three complex numbers. The field is actually real. I'm just decomposing it into this part, and then I'll take the real part eventually if I need to. Surprisingly, this four vectors hadn't been done. I don't know why until recently, but it's straightforward enough. You can also do this for higher spin fields. Phi, by the way, here is the Newtonian potential. What are the equations of motion? The equations of motion for this field are the Schrodinger and the Poisson equation. We've already been introduced to the Poisson equation before. That's the Laplacian of phi equals, you're familiar with it by saying it's 4 pi g rho. The rho here is m times psi dagger psi. Psi squared is your number density. The Schrodinger equation, I'm sure all of you are familiar with, but we have to be careful. It is a Schrodinger equation, but psi does not represent any probability density. There is no probabilistic interpretation here. This is not quantum mechanics. It's the same equation, but there is no probability interpretation here. Psi squared is just a number density. There's no normalization of the wave function here. It's not a wave function in that sense. People sometimes call this a wave function, but I think it's a bad nomenclature to say that. Compared to the scalar case, this is a Schrodinger equation which is multi-component. It has three components here, but otherwise it's the same. You can repeat the same exercise for a spin S field, which has two S plus one components, and the result is the same. This is a very general construction. You can start with the action, but you will end up with a Schrodinger Poisson system if you only have gravity. You could worry about whether in the ultraviolet a higher spin field is well-defined or not, but that's not our concern today. At this level, this is nothing more than just a collection of two S plus one scalars. For a vector, it's just three scalars with the same mass. For those who are interested, here is a fluid version of those equations as well. You can translate a Schrodinger Poisson system into an equation that's perhaps familiar from the lectures on structure formation. You can translate it into continuity and Euler equations and things like that. As always, once you want to understand something about the theory, you should write down the conserved quantities in the theory, because as I'll show you, I do numerical simulations as well, and this is very useful to know these kinds of things are conserved. They're respected by your simulations, and they're respected by the theory itself. Because it's a non-relativistic theory, the particle number is conserved, that's defined in the first line, the energy is conserved, the total angular momentum is, of course, conserved, rotational invariance, but importantly, in this non-relativistic limit, the spin angular momentum and the orbital angular momentum are separately conserved. It's not just the sum which has to be conserved, but they're individually conserved as well, which is very useful for us. Okay, so now let me tell you a little bit about some fun effects about this. My plan here is to do it for pedagogical purposes. I'll first present you with the analytic results. Then I will show you that those analytic results are okay by showing you detailed results of simulations as well. And I will also, for pedagogical reasons, explain the answers to what you would expect from the scalar case. So that's the plan. The first result I want to present is embarrassingly simple, which is great. And this is a result that we should know from freshman or from the first course in waves, which is waves have interference. You take two waves and add them together. Take two scalar waves and add them together. What do I mean by that? The square of the sum will not be equal to the sum of the squares. There'll be a cross term. For a scalar field, this has to happen. You can't avoid this. Now think of a vector field. A vector field has three components. Each component necessarily interferes with the same component from another wave. And if I put two waves into orthogonal components, there'll be no interference. So from this simple argument, you realize that given if you fix a total amount of mass density or particle density in a given region or some average quantity in a given region, then the amount of interference will be lowered in vectors compared to scalars. Does that make sense? You can analyze to a spin-ass field. The more components you have in the field, the less interference there can be because you can distribute the same amount of stuff into many different components, which are orthogonal to each other. I'm showing you a cartoon version below where I have superpose a large number of plane waves. The orange is vector dark matter. The blue is scalar dark matter. The average density is the same in both. You notice that the peaks are higher and the valleys are deeper than the blue compared to vectors because that's what you should expect. Interference is what causes the large excursions. You can do this a little bit more formally, but there's nothing more to it. The amount of interference because you have three components will be down by a square root of three for vectors compared to scalars. You can generalize this as a very difficult homework problem to the fact that this will be square root of 2s plus 1 if you have a spin-ass field. So that's the result there. Next I'm going to tell you about another implication which I really love solitons, so I can talk ad nauseam about these, but stop me if I go off on tangents. Okay. Solitons are these fascinating things that exist in non-linear field theories. Okay. They were first discovered by the gentleman on the horse there with a top hat. He was riding along the canal somewhere near Edinburgh and he saw that a boat came to a stop and created a pulse in the channel. And the pulse, he claimed, he followed it on horseback, did not dissipate. It just maintained its shape and kept on going and he followed it for two and a half miles or something. He didn't use those units and then it disappeared into the woods. And that was a discovery of a soliton in water waves. This is the type of soliton. The theory was developed later. So on the top you're seeing the recreation of that same event done more recently in the same canal. Let me see up there. By the way, solitons, this is not just a curiosity, they're definitely seen in fluid dynamics. They've been seen in optics, hydrodynamics, both are in certain condensates. The green things are places where they've seen them. We would love to see them in high energy physics and cosmology. There are many ideas, many theories proposed that they should be there. We haven't seen, we don't have a detection of them in any form yet. It would be great to have them. Okay. We're going to be talking about a particular class of solitons which are called non-topological, which means they are not, so the solitons generally, why do they exist? They exist because some two opposing things are preventing them from dispersing or decaying away. Something holds them together. Sometimes it's topology. Other times it's nothing to do with topology. It's to do with the configuration itself. When it's got to do with the configuration or there's some conserved charge in them, even approximate, then they're called non-topological solitons. For our purposes there are two types, those that are held together by gravity and those that are held together by self-interactions. There's a long history of this. I won't bore you with the details. But these things, if you look at the real field inside them for each component, it's just oscillating up and down. This is very weird. You would think that they would just disperse away. But some sort of attractive interaction is holding them in place and preventing the gradient pressure that would try to push them apart. So that's what's holding them together. We keep on doing this for a very long time in some cases for cosmologically long time. The scalar solitons here have been known in the literature for a while. These are sometimes called boson stars. They're basically a collection of bosonic particles, a coherent collection of bosonic particles. The real valued field is oscillating inside. The density profile just looks roughly like a Gaussian. You can describe the solution, the non-relativistic part of the solution with the help of this profile times e to the i mu t mu, you should think of as a binding energy per particle, whereas m is the mass of the boson itself. I think something that's important is that the total mass of this object is m-plunk squared over m. That's the coefficient that appears. M-plunk is a huge number. m, which is the mass of the boson, is a very small number. That means this mass is a huge number compared to just the mass of the particle. So it's a macroscopic object. It's not at the level of a quantum. It's a macroscopic object. The same is true for the radius. It's a macroscopically sized object. The thing that's determining why it's macroscopic in terms of the radius is this ratio m over mu. m is the rest mass. Mu is the binding energy for a non-relativistic particle. The binding energy will be very small compared to the rest mass, so the radius will be very big. Even though that ratio is going to be small, it doesn't affect this m-plunk squared over m very much. Good. What types of solitons do you have now that I've told you about a scalar case? You said mu is a free parameter. Sorry? Mu is a free parameter. Mu is a free parameter. You have a one-parameter family of solutions. For any mu, you have a solution. So there are many masses and there are many radii, meaning for a given mass, there's a given radii, but there's a whole family of solutions. Thanks, Bertha. Yes? I mean, it's numerically known, but it's at the level of one ordinary differential equation in the sense that it's just if you have a shooting solution. And if you wanted it, there are very good analytic approximations to that, but there's no, it cannot be written down in terms of simple functions. No trig functions or something like that. You can't write it down. The question was about the form of size solution of the UNX. Sorry. I will repeat from now on. Otherwise, Merda, you can throw chalks at me. Okay. So what do you do for vectors? Now it's simple. You already know about the scalar case. You have to do it for a vector. You just take the scalar solution and get it by a polarization vector. Done. That's your vector soliton. You know from electromagnetism, you can think of polarization in terms of circular polarization and linear polarization. Those are your bases. So let's think of the epsilon that I've written, multiplying the profile or multiplying the full solution as one of those. So linearly polarized one is the first one that's written up here and I'm showing you what the real-valued field would do inside. It would just oscillate up and down. The vector would just oscillate up and down everywhere. So it's pointing this way and just going up and down. For a circularly polarized one, at each point the vector field will be moving in a circle. So these are two possible polarized solitons which are only possible because of the vector nature of the field. For a scalar, there is no such quantity. There is no polarization to talk about. Yes, please. Is the polarization not a dynamical variable? No. I mean, dynamical in what sense? Sorry? Chalk, man. Chalk. So the question was is polarization a dynamical variable? Maybe I should answer more carefully. Here, epsilon being a constant provides a solution. And all solitons that we know of come with epsilon being a constant. All solitons in this, they don't depend on time. There could be higher energy states or some other configurations where this epsilon would also vary with time. So general solutions could of course vary. But for solitonic configurations that I'm talking about it's a constant in space. And time. Okay. Can I ask a question? Yes, from the nebulous internet. Yes, sorry. I'll turn on my camera. I don't know if you can see me. So these objects are stationary and the other hand I guess we want to identify its constituent particles with some kind of very realized velocity. My question is can I still interpret different polarization vectors as transverse versus longitudinal? I would not do it at the level of individual particles. I would do it at the level of the field itself. It's not, I think we've already taken the non relativistic limit where I can't, I've integrated in some sense I've dropped all scales relevant for the individual particles at the Compton scale. The other reason I wouldn't think of this as transverse and this is transverse with respect to what? This is not a massless particle. There's no sort of fundamental direction I can talk about. So here when I say transverse is longitudinal polarization and transverse polarization in that sense but with respect to what? With respect to some axis that I pick. It's an arbitrary axis that I picked. It's not something inherent to the particle that I can pick. Hopefully that answers the question. Sorry but so if a particle has a small velocity at the level of the particle if it has a small velocity I think it still makes sense to think about transverse as longitudinal. But this is a massive particle right? What is the unique how are you going to split it uniquely? You can talk about it if you want. So the longitudinal polarization would be the one which is in parallel to the three momentum and the transverse would be short. I think if you want to do it at the level of the particles that's perfectly, I'm not saying it's not impossible but I don't see a big benefit to doing that at least in what I'm going to say. I'm not saying it's not possible. It's also I don't think about it this way. It's just because you're when so many particles are sitting on top of each other forming this field I find it almost becomes confusing to think of each one of them having an independent identity. The other question you asked which I did not answer which was about realization and so on the velocity dispersion for this is roughly the inverse of the length scale of the object here. Okay. There is also a question in the Zoom asking how the scalar and vector dark matter are getting related to the scalar and vector soliton. So scalar dark matter is being organized itself into scalar solitons. Vector dark matter can organize itself into vector solitons. So they are not, it's not that solitons are all of dark matter. It's just these are configurations possible within specific classes of dark matter. Okay. Let me move on. One of the important things that comes out of these solitons, vector solitons is that they can have spin angular momentum. You can calculate the spin by taking a cross product of the polarization vectors and for the linearly polarized one it has zero spin for a circularly polarized one it has a spin equal to h bar times a particle number. Again particle number you can just get by integrating the mass over the mass of the soliton getting the total mass of the soliton if you want. So this can be because the mass of the soliton is big and mass of the boson is small this ratio can be macroscopically large or really big. It's just a number of particles. So that's why these things I got really fascinated by this, that this has an object with a gigantic macroscopic spin. Not big enough to make it elliptical. Not big enough to make it elliptical because in the non-relativistic limit gravity knows nothing about spin. It's just a collection of three scalars so it doesn't know that there is an angular momentum that gravity cares about. So it doesn't in a full GR case it will care. In the non-relativistic limit no. Yes. By the way these are two examples of these solitons. You could take because both of these end up having the same total energy for fixed particle number that means I can take linear super positions of these guys and create a fractionally polarized object. So I can take I can put this basis solitons together in appropriate coefficients and create a fractionally polarized object which will have a spin between zero and this maximum value. And all of these configurations have the same total energy. Yes. If you are only using Newtonian gravity none. Okay. If Newtonian gravity is your only probe far away you cannot tell the difference. It's the same. It's exactly the same as it is for scalars and it's the same for each polarization. So the question was that from the point of view of a distant observer is do different spins have some implications. Thank you. See if I was taking an exam I would fail. Many times over. Okay. Yes. Yeah. You can take think of it that way. Projection. Projection along the Z. I mean here I've taken them to be along the Z axis. Okay. One thing I want to make sure I mentioned right is that this so these solitons are new. The ones I've talked about essentially this we found them in the last year you would think this is such a simple thing why hasn't been this has been just in literature these polarized solitons with macroscopic spin. I suspect and I don't know for sure of course is that we were misled perhaps by symmetry. So the simplest configuration you could think of perhaps at the vector field which would be spherically symmetric not just in the mass density but also in the field configuration. You would think that the field should also point radially outwards or inwards. That is a simplest symmetric situation you could think of. But if you think about it a little bit if it's a vector field and it's either pointing in or out then it has to go to zero at the origin. How will it choose which direction to go in? It cannot be multi-valued. That means that the profile of the soliton if it was a vector that was pointing radially outwards has to have a node at the origin. You perhaps know from quantum mechanics that solutions with nodes usually cost more energy than solutions without nodes. So this gives you some intuition for why even though this is a perfectly good solution it might be a higher energy state than the ones I've been discussing whose profile doesn't go to zero at the origin. It's a nice Gaussian profile. So these the hedgehog ones the ones that are all pointing outwards porcupine one they've been known in the literature. They're often called pro-castiles. Those are fine. They seem to be higher energy than the ones we are discussing here. In simulations the ones we have done we have only seen these objects forming the lower energy states not the hedgehogs very easily. It doesn't mean they don't exist. It's harder. I should also mention a caveat that everything I'm talking about is a non-relativistic limit. If I do a full GR case where I'm getting close to compactness of neutron stars and so on we haven't done that yet. So there are some questions in the chat. Do you take them now? One is that why do you assume or do you assume that the spins are aligned in this solid or is there a reason for them to be aligned? The other one is why doesn't this boson star collapse because of gravitation and form a black hole? Let me answer both of them. They are very relevant questions. The reason it doesn't collapse is because there is a gradient pressure. The field wants to disperse off because of gradients. You can think of this as gradient energy or gradient pressure. Sometimes people call it quantum pressure. I don't like the word, but it's just gradients wanting to move outwards. Gravity is trying to pull you in. There is a balance between them if you want. It's a static solution between the two. This is the usual case of why stars exist. Something is pushing out. Gravity is usually pulling in. The same reason stars exist. These boson stars also exist. It's a pressure associated with gradients. The other question was about what do I assume the spins are aligned? No, I don't assume they are aligned in the sense that for a soliton solution what I mean actually this polarization vector that you see on the top for a minimum energy solution the polarization vector is a constant in space which means the cross product of it will also be a constant in space at every point in space. In that sense the spin density at each point is pointing in the same direction. It doesn't matter whether it's linearly polarized, circularly polarized or a combination of them and you will see that there's nothing deep about there being a constant spin density. You take a collection of three fields put them together in a configuration with having the same profile unless they're perfectly in phase you will end up generating a spin density. Let me just tell you what we have said so far. I've told you that there are solitons in scalars. There could be solitons with higher spin so it's extending a new axis in this plot. How much time have I taken? I have already taken it? Wow, okay. I will speed up a little bit so and we'll take a break maybe after 10 minutes from here so that we we recover a bit and then we'll start over for the next 15 minutes if that's okay. So let me give you some phenomenology for these. Sorry, the numerical phenomenology that will be connected to what I've just said. This is I think perhaps the most fun part. So notice that I've got two panels on here. The first part is the first row is a simulation of vector dark matter. The bottom row is a simulation of scalar dark matter. The colors are the mass density, projected mass density. I start both simulations with the same initial conditions in terms of their density. There are some idealized halos, in this case they're solitons themselves. And I let this go. There's a full 3 plus 1 dimensional simulation. You and some of the first ones that have been done on this so you start this you move this forward and you notice at the end of it the time is running towards me at the end of the simulation you see some very similar structure. There's some dense region in the center with some fuzzy stuff around. I don't know if the screen is allowing it but hopefully you can see some differences between the two. Slight differences. Anybody want to take a guess what the differences are? Let's not look at the boundaries too much. Let's look doesn't have to be in the center. Yes, go ahead. Exactly, interference patterns. Here we see more interference in the scalar one. Look, there are more dark and bright spots here compared to the one on top. This is what we should expect. This is exactly what we see. There's also detailed information about the soliton but you can't see it on the screen. So let me tell you about the detail difference in the central regions. So the scalers here I'm looking at the final result of this merger process of the polarizations. By the way, I should tell you that you might have a question about what are the what do I choose for the polarization of the vectors and all of these things because vectors have so much freedom. We do lots of things meaning we choose random polarization vectors, we choose unpolarized cases, we do basically do 100 simulations with all different initial conditions. And we will compare when I present you the results it's either an average of them or all of the simulations together. And then I plot the average, yes. These are two simulations for the same mass of the particle itself. But if I change the mass I would imagine that the interference pattern would be affected because of the variety of elements. Does that make sense? If I give me a few moments we'll exactly get to this point. I'm just worried that when you say that this is vector the scalar and the scalar are more interference. Could it just be that I could see a different scalar with a different mass with less interference and mistake it for a vector? I will try to argue that there is a degeneracy that can be broken. But certainly it's a very valid question. But we are just getting in a couple of sites we'll get to this. I did not repeat the question. The question was whether there is a degeneracy between scalar and vector interference patterns depending upon if I am allowed to choose different masses. Okay. So here I'm showing this for the same mass of the particle same total initial mass. Everything is the same. As much as I can make it. And you notice that in this simulation the profile, the density profile this is angularly averaged is different for the two cases. The vector case is in red. The scalar case is in blue. The vector case is as you can see is shallower in the center compared to the scalar case. The scalar is allowed to get more dense in the center. Another important thing hopefully you notice is that far away from the core they start looking similar. It doesn't really matter whether it's a vector or scalar. You might be worried that I won't know what the initial mass of this is. Here I it's very idealized simulations where I know the initial conditions perfectly. What if I didn't know the central core? Well if you normalize by the central core density done over many different masses you notice that there is a difference in shape as well. The difference in shape is that the transition from the inner region to the outer region for vectors is smoother than that for scalars. I cannot give you a full analytic argument for this but it's very it's basically because of interference again. There are three components that don't allow you to have a very distinct region between the two. Just some confirmation about interference here is a density pdf or a histogram of densities in my box you notice that the scalar case which is in can go all the way to zero densities but the vector case doesn't it can't have that much interference. Similarly the excursion at the high density regions is more for scalars compared to vectors this is again coming from coming from interference or it's basically dominated by a central soliton. By the way I should have mentioned that the central if you can fit the shape of these profiles very well in the center to help by taking a soliton profile that's why I'm saying that there's a soliton in the middle. So there's a central soliton profile that fits very well and then the outer core. Sorry to butt in from the sky why do you have solitons as initial conditions? Very good question because it was easy to do I will talk about because I know the solution it's easy to start with I will talk about formation mechanisms and I'll tell you what other people have done where they don't sort with solitons as well the whole idea here for us was to do a control simulation where we could tell the difference very exactly when we start with identical conditions and this was the easiest thing we could do but we don't have So ideally would you want to start with some Gaussian distribution? I'll get to it I'm not sure if I will but I might not get to it but I will at least you can look at the slides and I talk a lot about formation in the second half of the talk which will not happen Okay here is a picture of the correlation length of these interference patterns so just a two point correlation function in density and what you find is basically normalized by the central density you find that it's the two point correlation function sits exactly on top of each other which means that they have the same de Broglie length whether it's scalars or vectors so this Asim is going towards answering a bit of your question they have the same de Broglie scale here why? because this de Broglie scale is just a gravitational thing it's 1 over mv for the same mass they will have the same de Broglie scale it doesn't matter whether it's a vector or a scalar it's just a smooth potential that determines the scale however because of interference the amplitude of the interference pattern or the amplitude of the fluctuations is going to be lower for vectors compared to scalars the same de Broglie scale but different amplitude for scalars versus vectors that's how you can tell the difference so one example that I really like which is very recent and we'll change the mass I think the delta rho over rho has to be the same regardless of the delta rho over rho it typically is of order unity this is just an interference pattern on top of a smooth smooth thing it's just a difference between vectors and scalars will be 1 over square root of 3 that's to say it's not a statement about the nature of the field or anything right it's just okay I will skip this in view of time if you don't mind let's take a I'll take a 5 minute break because we've been going on for quite some time and I cannot imagine anybody's attention given that I'm giving the talk and I'm bored I'm sure you are as well so let's take a 5 minute break and then we can continue if that's okay