 Let me now track this intrinsic spin. So the colors now are that of the spin density and not of the mass density. And the vectors that I'm showing you in the middle are that of spin density. So you notice that central object at the end of the simulation, at least in this particular one, seems to have the spin density that is aligned in the middle. This is an indication that there is a soliton there. Because solitons should have all the spin density vectors pointing in the same direction. So you see that these solitons form in the middle, and the spin density is what you expect that of a soliton. We've done lots of statistics on this as well, but I will not bore you with the details. Let me now move to just answering something very quickly. There were questions about interactions at some point. Suppose instead of gravitational interactions alone, I also introduce non-gravitational interactions. That is, this dark matter can talk with its neighboring particles. It's a point interaction. In that case, you can still find, and even without gravity whatsoever, you can still find these solitonic objects. But now there is a key difference. The key difference is that the two polarization states, the basis polarization states, no longer have the same energy. The circularly polarized polar attractive interaction, the circularly polarized one will have a slightly higher energy than the linearly polarized one. The somewhat intuitively might at least seems to make sense. And if you look at the level of the Lagrangian massaged a bit, you will see that there is, you can write down that there is a spin-spin interaction that appears in the effective description. So you can see why these two states are different from each other in terms of energy. Again, there's a lot of work to be done here, and I will move on for the moment. You can also include both gravity and interactions together. And these solutions also exist with repulsive and attractive gravity, with repulsive interactions and gravity, or attractive interactions plus gravity. In the repulsive case, the circularly polarized one becomes lower energy than the linearly polarized one. In the attractive interactions one, the opposite takes place. So all of these exist, and there's a lot of work, again, that needs to be done in this context that hasn't yet been done. What about the click? Can I ask some questions on that? If it's quick, or if not, we will try to do it later. Yes. One quick question. So the stability of these solid tons, is it still the competition between gradient pressure and gravity, and it's just the energies and other things that have been modified by the self-interactions or the reason for stability has also changed to self-interactions? It depends on what regime you are in. It can be different terms can balance each other. If there's a lot of self-interaction, that can be the thing that's balancing against gradients or gravity. But I think this is a longer conversation about stability that can be had. But yes, the different terms can compete with each other. OK. Newtonian gravity, well, spin came along for the ride. It's something I can calculate, but doesn't play a role in the dynamics directly. It matters that I have three fields instead of one. So if you want the take-over messages, three is not equal to one. That's all it is. There's nothing else that I've said so far. But spin doesn't intrinsically play a role in Newtonian dynamics. Newtonian gravity is blind to spin. It doesn't know about it. But if you go to GR, you will see that spin now plays a role. These are post-Newtonian corrections. And you'll see terms like spin-orbit coupling, spin-spin coupling. So now, suddenly, these effects would become important. I'm not saying this is easily observable, but if I can make these objects compact, then it becomes interesting. We are thinking about doing full GR simulations of this, but it's still work in progress. Coupling it to photons. This has been done for axions. You can do something similar for these vector fields or even higher spin fields. Again, I'm not saying these as a way of all of these things have been done, but these are examples of things that can be done. Also, I've not even touched upon the fact that when I talk about vector dark matter, there's a whole slew of literature trying to detect these things in the laboratory. And this paper, this is a snow mass paper. You can take a look at all the constraints that people have come up with without thinking about these nonlinear dynamics. What was new in what I presented was not that there's a vector dark matter candidate. That's not the husband and people have been thinking about this for a while. What's new is the nonlinear dynamics and what happens, especially when this field is light. That's the new bit. This is when we were supposed to take a break and stretch out, but we took it earlier. That's OK. We will again stretch out a bit. OK. I wish I could do this, though. Like, this seems a better stretch. Evolution has not been kind to us. OK. So there were questions about formation, which are absolutely critical and important. Do how does vector dark matter form? How do solitons in these in vector dark matter form? Usually, I should be giving a talk with this first and then later, but less work has been done here, especially by me. So I started with it from the other direction. Here I'm going to shift again for five minutes to the pedagogical mode and repeat something already discussed very nicely yesterday by Asim about the matter power spectrum. There were already quite a few questions here, but I'm going to repeat that again because I need to do that for you to understand what is different for scalars and vectors compared to the usual power spectrum that you get. So the plan is to sort of do a very quick review and then I will just say, as a homework exercise, you can derive it for scalars and vectors. That's what we are going to do. OK, you're now familiar with this. This is the matter power spectrum. The meaning of this, I think this was already described by Marco and Asim, is that this is p of k. It is some measure of the level of fluctuations that you have in a field at a given length scale. More concretely, k cubed times p of k, where k cubed is k is a wave number, square root of k cubed times p of k is the amount of density fluctuations you have typically on a length scale of k inverse. k is a wave number, so k inverse is a length scale. So that's what this is. Now, this you know, so what is it that I'm going to tell you? Let's try to derive this shape very quickly in five minutes, and then as an exercise, we will do the rest. What do we need to derive this? You need to know that the density in dark matter, this fractional density, this is just for CDM, but can be usual CDM, just evolves based on the continuity equation and the Euler equation. And Asim already discussed this in a sub-horizon and matter dominated universe, sub-horizon scales matter dominated universe, this fractional over density just grows as a scale factor. On super horizon scales, it doesn't evolve. This is a gauge dependent statement, but ignore it. But it's okay, it's constant, okay, for our purposes. Marco in his lecture already described that inflation makes a prediction for this. Inflation makes a prediction that this k cubed p of k is scale invariant. So at initial conditions, this power spectrum is a constant across all wave numbers. So this is a simple, approximately constant across all wave numbers. And it's on super horizon scales, it doesn't evolve. There are two scales here which are important, wave number today that corresponds to the horizon scales and the wave number at matter radiation equality that corresponds to the horizon scale at that time. You evolve the power spectrum from inflation all the way up to matter radiation equality. At the approximation we care about, nothing happens. There is a logarithmic growth, blah blah, on sub-horizon scales, but let's ignore that. It's logarithmic, don't care. So the power spectrum remains the same. Now we enter into the fun era, the matter dominated era. In the matter dominated era, everything that is sub-horizon will grow as a scale factor. So everything before matter radiation that is on scales here grows as a scale factor and as long as it's sub-horizon, it also grows but there is a slope there. So this is, I'm showing you the picture of everything that's happening for every k-mode as a function of time. A little bit later, it grows even further until it gets to this shape. So it's just growing as a scale factor. So why, so what gives that slope that's over there? The reason it has a slope is because modes only get to grow once they enter the horizon. The longest scales mode enter the horizon the latest so they get to grow less than the modes that entered earlier. And you can calculate this show quite nicely. You can look at the notes. So this is the shape of the power spectrum. How do you get this bump? Well, this is k cubed p of k or square root of that. You take the square and divide by it and you immediately get the shape that we saw in the first slide. So the ingredients are scale invariant spectrum, growth as a scale factor and you only grow once you enter the horizon period. So this is a way of seeing this. So why was I trying to do this? The reason I want to show you this is because we are going to use this understanding to understand what the initial power spectrum of vector dark matter and scalar dark matter should be. For a scalar case, that same spectrum looks something like this. You see a difference, right? So you see a difference here for that it's somewhat suppressed. For the vector case, for the CDM case, you just have this dashed line that was there. Now this is suppressed. Why is it suppressed? It's suppressed because remember in the, there is a gradient pressure that's when we're treating things as fields, there's a gradient pressure associated with them that prevents structure from forming on small scales. Very simply, the equation that you know for the growth of structure gets a correction. It looks like a sound speed, a non-zero sound speed, but it's a sound speed that's dependent on the wave number. So when that wave number is large compared to this characteristic scale which is often called the quantum or gradient gene scale, then you don't get growth of structure, otherwise you do. And this is the reason why you get suppression of scales. You can again start with the initial power spectrum which is scale invariant. Nothing happens up to matter relation equality. After matter relation inequality, same behavior on long wavelengths, but on short wavelengths you don't get to grow. So you notice that beyond the gene scale there is no growth of structure, okay? Again, you square this, take the k cubed out and you will get the shape of the spectrum that you saw. So you will again derive nicely and you can derive all the power laws here. Look at that funky power law of k to the minus 11. But you should be able to derive this just without doing any calculations. Well, by scaling calculations. And this matches nicely with what you do with a full solving of differential equations. What about the vector case? The vector case is different because at matter radiation equality it's not almost scale invariant. It's scale invariant on long wavelengths, but on short wavelengths close to the gene scale there is already a bump in the spectrum. This is very different from the scalar case and this is very different from CDM case. This bump is present because if you produce vectors the two formation mechanisms I know of how to produce vector dot matter. Both have the, let me be more careful, the gravitational production mechanism, the way you produce particles during inflation usually, that production mechanism naturally has a k squared tilt in terms of the amplitude of the field fluctuations and then the fact that they're relativistic before and non relativistic later, sub horizon, super horizon evolution, all of those things combined to give you this bump. So you can ask me details about how to derive this or you can look at that nice paper up there. Also you can look at the papers mentioned at the bottom. But you can take the spectrum initially, you can evolve it using the same tricks that I told you before. Sub horizon evolution is just scale factor, super horizon, nothing happens and then suppression below a certain scale. Same physics that we did for seed before. You can apply it, you can derive all these power laws and you can derive the shape of the power spectrum. This is just a linear power spectrum. The key thing I want you to notice is there is a bump in the power spectrum on short length scales or on large wave numbers. This bump is very interesting because it will lead to extra structure formation on small length scales. It will give a boost to structure formation on small length scales. This bump is actually quite large in the sense that it's already dealt a row of order unity at the time of matter radiation equality. So it's already non-linear then. So you would have a lot of substructure on small length scales in these models. This also gives a scale, for example, in this mechanism for the formation of these solitonic objects. So people are asking me questions about how do you form them. If you start with these initial conditions with already a large amount of power on the scales, it's on this scale that those solitons will form because they are the lowest energy states and they are issued form. Don't take my word for it. There were really nice simulations done recently by Gorghetto et al where they started with this power spectrum and they saw formation of the solitons that I was talking about. So from these Gaussian, not Gaussian, but these initial conditions with actually, I think they're assuming Gaussian. These initial conditions there, you start with this power spectrum and very quickly you form the solitons that I was talking about. One caveat here is the mechanism that they use, this gravitational production, restricts the mass of the dark matter particle to be greater than 10 to the minus 5 EV. So it cannot be fuzzy dark matter in this regime. We are currently looking into trying to do the same thing with a different mechanism which allows for ultra light masses. This is again recent work by some of the authors listed here. You can get a similar early power spectrum but now you can allow for length scale for fuzzy masses which means the length scales are much more accessible to cosmology. I will skip this part. Everything I've done for vectors you could do for a general spin S field. There was a nice talk about massive tensor fields yesterday as well but I'll skip this, you can ask me questions about this later. And of course there is a lot more to do. This is new to some of the possibilities like a laundry list of things that can be done with this. And now for the final, very important part, the test. I hope you were all paying attention. So there are three panels here. They all have the same density to begin with. There will be one which is a vector, one which is a scalar and one which is a massive tensor field. You might say, ah, this was not on the syllabus, tensors. But I'm not a good, but I hope you can extrapolate from what I've said. If I've been a good teacher, this should work. All right, so they begin to merge. Anybody venturing a guess? Excellent, see it was an easy question. Yes, it's a scale of vector sensors. Hopefully you can tell just by the level of interference that you see for this exact thing. On the screen it's a little bit harder, because you can't see them. And there's also a logical reason that the middle one had to be a vector. It couldn't have been anything else. So it's okay. But thank you very much, I appreciate it. So is there an epoch in the linear regime where the peak, or the bump that you see is not nonlinear, where it is in fact? So why is that happening? Because there is, so remember it's nonlinear in the sense of dark matter, not in the sense of the total density in the universe. So during radiation, right. I understand, yes. So the reason is that, so the production mechanism, so let me give the two production mechanisms that I know of. One is you transfer energy from a scalar field on which it is stored, and you transfer it to the vector. The production basically stops because, for example, let's say it stops because of back reaction. So the fluctuations in the dark matter because it's stopping because of back reaction is already order unity, okay. Because it's already fluctuating when it's back reacting. So you start with a delta rho over rho of order unity because of that. The other reason, the gravitational production, for scalars, it would be scale invariant. A vector, at least a longitudinal mode of the vector, you can think of it as a gradient of a scalar. So if a scalar is scale invariant, a vector will be k squared, okay. So the k squared creates already a UV-dominated thing. The evolution will create a peak in there because, and that peak because if you think about it, the total dark matter density, which is just an integral over this peak, will be roughly the same as what the value of the peak is. So delta rho over rho will again be order unity. So it's natural to expect, and I was discussing with Meridot, thanks to him for clarifying this, it's natural to expect that delta rho over rho in these mechanisms will be order unity. At the time of production. Essentially at the time of production, yes. The peak is always delta rho over rho in dark matter as unity. But it doesn't evolve much, right? It just stays until maturation equality, and then you start nonlinear structure formation. What happened if instead of non-Abelian gauge boson? Excellent question. You should talk to Mudejane, who's a post-doc at RISE. He's written a paper on recently on these objects in non-Abelian fields. It's, you still get these solitons, but now you have spin and isospin. He included a Higgs in it. The non-Abelian, the Yang-Mills interaction gives you a repulsive interaction. The Higgs part gives you an attractive interaction. And so you have a much broader class of objects. So you can have objects with spin, true spin, and you can have objects with isospin as well. But these objects do exist in those theories as well. It seems. And can it be dark matter? Can what, these objects? No, this non-Abelian, massive gauge boson. Could they also constitute total dark matter? I think so, but I don't, I forget the details. There are some worries about formation of this as dark matter, but I'm not too comfortable with, because I didn't work on it, so I'm not too comfortable knowing with giving a clean answer to that. But I think in the paper by Moudetrain, he discusses the formation mechanism in detail. Formation not just of the solitons, but of dark matter. So there are different types of solitons. How do you sketch yours amongst them? Have you done a, for example, a sketch of the type of soliton that you have in your solutions? And also, how can you incorporate it in other fermionic theories? Are there what theories? Fermionic, like a field, sort of, if you had. Fermion is tough, right? Like how are you going to make a soliton with the occupation? I mean, it's, you know, it's the poly exclusion will prevent you from putting them on top of each other. So I don't know personally taking that into account how I build a soliton. Of course I can build a soliton in the sense that I can build a neutron star, right? I can stack fermions on top of each other and just build a neutron star. That's my, that's a collection of objects that is held together by gravity against the generacy pressure. But I don't know of how to think of it coherently. These are just a bunch of things we're adding together. In the class of solitons, oops, that kicked me out because I was going too fast. Okay, maybe I'll just draw it. So I was drawing this axes. Okay, so this is, let's say gravity, this is self-interactions. You can have solitons that are held together by gravity, solitons that are held together by self-interactions. Think of an axis here, we'll call it spin. The new solitons include all of this region, but now they're ones with macroscopic spin. So that's how it fits in. Another axis that I would have drawn if we were living in four dimensions would be that four spatial dimensions would be that you could also think of the solitons in real-valued fields versus complex-valued fields. Real-valued fields, in the non-rativacy climate, have a conserved particle number, so they're almost like complex fields, which obviously have a U1. So you can think of them like cue balls, for example, that have been discussed in the literature, are analogs of this, but in a complex-valued field. The other context in which I can put this is that these are all non-topological. These don't have topology protecting them. A classic example of a topological soliton is a kink. It goes from one vacuum to another, and it's protected because it's pinned at two ends. So that's a topological soliton, village-deboundary conditions. These are not related to that. There is a very, it's a very interesting thing to think about about why these objects are so stable. Why don't they just disperse away? Why don't they annihilate into their own quanta and just go away? But it's a whole other talk. The essential point is that the wave numbers at which they can radiate are very short compared to the power they have in Fourier space at that wave number. So they're very bad antennas. They can't do it very well. This is a very glib answer because there's a much more detailed calculation that you can do that it just naive estimates would lead you astray. You would have to do it carefully to see that this is really a configuration-dependent statement. You can't see it in the Lagrangian. You have to do it based on the configuration. So thinking of it in a form of a wave propagation, yours seems like a bulb, or does it also have an overlap kind of structure to it? What, sorry? Thinking how it kind of went, for example, for the water wave example, it goes and it's like a bulb, right? And then the bulb goes, right? But there are many other solitones that are overlapping on top of each other. And the solution is basically considered as a circle, but the circle has a bulb that goes around it. So that, yeah. You're thinking of sort of vortices kind of, is that what you're thinking? Yeah, so this is not quite, I mean you can have, in the simplest, I didn't discuss them today, but there are vortices are possible in some contexts in these models. Some of them are not really solitons. They're just chance superpositions. But certainly you could have those solutions that are available, string-like solutions, or vortices that are possible in these theories as well. When I make them real-valued, you can't have them being stable, okay? The other question about, you saw the water waves were moving, of course these objects can also move. They're perfectly stable. For the mathematicians amongst you, when mathematicians say the word soliton, they mean something very specific. It's a field configuration that when you collide them together, they also emerge without any distortions. All that happens is the phase change. They would call what we are describing here, solitary waves. They are stable by themselves, but when you crash them together, there is nothing that really is conserved. Meaning they can get destroyed. So that's another context in which I can put these solitons. So they're, mathematicians would call them solitary waves, not solitons. Okay, so I'm gonna use my power and stop the session because we have another long session after. Let's leave the rest of questions for after a while and thank Mustafa for the very beautiful talk. Thank you.