 Hello, everybody. Welcome to the Latin American Webinar on Physics. I'm Camilo Garcia-Selli from Desi in Hamburg, and I'll be your host today. This time we have a very interesting talk by Anirvandas, who will tell us about dissipation mechanisms from multilevel dermatoscattering. Anirvandas is a PhD student at the Data Institute for Fundamental Research. And before we start the talk, I would like to encourage you to ask questions. You can do that by typing them in the designated box in YouTube, which you can find on the right-hand side of your screen. At the end of the talk, I will read them loud. Thanks, Anirvand, and please go ahead. Okay. Thanks to all for joining us today, and thanks to Camilo for inviting me today here to give this webinar. So, I think I should share my screen now. Okay. Can you see it? Yes, I can see it. Okay. Yeah. So, today I'm going to talk about dissipation mechanisms from multi-component dark matter scattering. This is based on work, which I did with my supervisor, Barsha Devdashukta. So, I don't think I need to give any formal introduction to dark matter in front of this audience, but still, I'll just touch upon a few basic things. So, we have solid gravitational evidences for dark matter, starting from galactic scales, which is like tens of hundreds of kilo-persec to a cluster scale at the neuro-persec scale, and all the way up to the largest scale in our universe from the CMB observations. And all of these observations can be explained very well by a theory of lambda CDM, which has the total energy density in our universe is dominated by a lambda, a dark energy, and a substantial part comes from the cold collisionless dark matter. So, that's good. But today, I'm telling you about why we need to go beyond lambda CDM, try to motivate for basically self-intracting dark matter system, which can solve some issues with lambda CDM. Then I'll talk briefly about a little bit of physics of multi-component self-intracting dark matter, which is the main topic of this work. And then finally, I'll discuss about the results. So, let's start. Here, this is just to say that in our... Usually, we think to start to begin with, we think that the simplest way to work with dark matter will be just taking a single dark matter species, which produces all of the dark matter abundance that we see today. But that's very asymmetric if you look at the counterpart of dark matter in the visible sector, where we have three families, and each of them has four kinds of particles. So, that's very asymmetric if the dark matter sector has only one kind of species. So, let's be a bit more serious here. There are more things, the more issues with lambda CDM, which exist in the smallest scales in our universe, namely the galactic or subgalactic scale. So, one of them is two-week-to-fail problem, which was first introduced... which was first discussed in this paper by Boil and Kolchin and all, where they had some simulation, but simulations of collisionless cold dark matter, and they observed a milky-wave-sized dark matter halo and found that there are... this simulation predicts way too many massive subhalos around this milky-wave-sized dark matter halo. And they are too big to fail to form visible galaxies in the present universe. So, this is a plot from this paper where there... all these red points are the observed dwarf galaxies around a milky-wave. And this is a plot in the V-infoil, which basically measures the mass of the dark subhalo, and this is the luminosity of the galaxies which are sitting in the subhalo. So, here we see that the simulation predicts a few dark subhalos, which are very massive, but all the observed dwarf galaxies are on the other side of the parameter space. So, there's clearly a discrepancy. There are a few more problems. The second one would be missing satellite problem, which, again, it's a prediction by collisionless cold dark matter simulations, and they predict of the order of few hundreds or even thousands of satellite galaxies around our milky-wave, but you don't see them. We see only... I think the current number is of order of 40 or 50. And there is core Cusp issue in dwarf galaxies, which says that in the dwarf galaxies around our milky-wave, we see that the central density profile of the dwarf galaxies have a core instead of Cusp-1 over R dependence, which is predicted again by n-body simulations with collisionless dark matter. So, these were the problems with lambda CDM, where dark matter is assumed to be an ideal collisionless fluid, cold fluid. But people have shown that these problems can be solved in a scenario where there is self-interaction between dark matter particles. They are not just freely moving. They have some self-interaction. And this kind of interactions arises quite naturally if we have a separate dark sector alongside our visible sector, where this dark sector has more particles than the dark matter species, and they will mediate some kind of interaction between two dark matter particles or between a dark matter and another particle in the dark sector. So, as I'll show later, that this self-interaction can help us ease these small-scale issues. And also, it helps to get the relic abundance of the dark matter in the correct way without violating any direct detection constraints, which are getting tighter and tighter using the dark freezer where dark matter freezes out within the dark sector only. And they don't need to directly couple to the standard model particles. Okay, so let's briefly discuss how self-interaction, self-interaction in dark matter solves this. Here, I have stolen these two plots from these papers where they have done in body simulations with various degrees of self-interaction among the dark matter particles. So here you see in the left plot, this black solid line is a simulation without any self-interaction and all these others are with self-interaction. This is with the strongest self-interaction. So clearly, in this density profile, a core kind of structure is formed instead of CASP1 over R in a develop profile and it can very easily solve the core CASP issue. About the 2-bit-2-file problem, this solves this problem in that way that when the central density of these sub-hellows is reduced due to self-interaction, during structure formation, they will act with less amount of baryonic matter and less amount of baryonic means less stock formation rate and there will be almost barely any visible galaxy to be seen. So these dark sub-hellows will be just sitting out there without any visible emission, so we don't see them. So to solve this missing satellite problem, we need to have dark matter interacting with some other radiation-like species. So that this radiation-like species, it's called the dark radiation. So when dark matter interacts with dark radiation, the longer free streaming length of the dark radiation particles will erase out some structure in the very small scale. So here we see a simulation with only collision-less dark matter, CDM, and these two are simulations with dark matter-dark radiation interaction where we see less number of satellites. So that's how it solves the missing satellite problem. But of course there are some issues, so one should check out these papers and many more where they discuss the missing satellite problem is not really a problem because there are many issues like the visibility of our surveys and the region of observation and all other things which says that in the future we'll discover more and more satellite galaxies and this problem will be resolved in a trivial way. But I don't think this is totally settled though this last paper says boldly that there is no missing satellite problem, rather there is too many satellites problem right now. But I don't know. Okay, let's move on to multi-component CDM. So we know multi-component dark matter system is very much theoretically motivated. It shows up in many kinds of dark matter models from supersymmetric dark matter to atomic dark matter and composite dark matter, et cetera. In supersymmetric dark matter we have many... dark matter comes from a multi-plate in that case theory and there will be many particles in the multi-plate and their masses will be split by their mass renormalization, very small mass gaps will be there. And in atomic dark matter the dark matter is composed of dark atoms and usually they will have some excited states and they will form a multi... and it will form a multi-component dark matter. In the composite dark matter, for example, in dark global theory, in the dark gas theory we can have blue balls as dark matter particles and the lowest state of this blue ball will be as dark matter but there will be, of course, many more higher excited states with different quantum numbers. But to discuss my result, I'll just take the simplest case where the dark matter has only two components, these two particles, chi-1 and chi-2, with these interactions. These chi-1, chi-1 and chi-2, chi-2, they interact with each other through the interaction mediated by scalar row 1 and the cross-optagonal interactions are mediated by particle row 2. But for simplicity, I'll assume the coupling strength to be the same and the mass of the scalar is also the same, which I'll take to be m-row. So here... Okay, so I'll also assume that the dark matter mass, so I'll take chi-1 to be the lightest particle, so it has mass m and chi-2 is a heavier state with a mass m plus delta, where delta is much, much smaller than the dark matter mass, which is generically true in all these multi-component dark matter models. So typically the mass between the dark matter states is much smaller than the bigger scale of the dark matter mass. And okay, so with this Lagrangian, we see that two kinds of interactions possible, two kinds of scattering possible at the late time. These two chi-1 particles can go to, can scatter elastically going to chi-1, chi-1, and also they can scatter elastically going to chi-2, chi-2. So here we should note that we don't need to bother about other processes like chi-2, chi-2 going to chi-1, chi-1, or any other processes involving chi-2s in the initial state because in this kind of model, in the present time, there will be no chi-2 left in our universe because chi-2 decays through this interaction, chi-2 decays to chi-1, and we'll have only the total dark matter population will be formed by chi-1 only. So these two processes are enough to consider. Okay, so now we have to solve the scattering amplitudes for these two processes, elastic and inelastic. How do we solve them? We solve them using Schrodinger equations because we know that the present time dark matter is very much non-literistic and their fields will obey Schrodinger equations or non-literistic equation where the wave functions are basically the transition amplitudes. So here I have a 2 cross 2 system. So the matrix psi has a 2 cross 2 structure and the elements of psi are given by these amplitudes. So it basically, the amplitude of step ii going to jj. So that's the ijth component of the metric psi. And this equation is for a particular partial wave l and we also have a potential, the V matrix, which has this structure where all the elements are given by this Eucar potential which arises due to this kind of exchange of many rho particles among the dark matter states. And this is typically true for any kind of interaction. In any scalar it could be the mediator could be vector particle also but in all cases we'll have a Eucar potential. And we have a simple system where all the potentials, all the elements are the same. That would not be the case if I have different coupling strengths for the rho1 and rho2 and also different masses of rho1 and rho2 but so in some cases, in some limiting cases I can have only diagonal or only optimal systems but I'm keeping them to be equal for simplicity. Okay, so for the inelastic scattering we note that there is a threshold effect here because here too there is a mass gap between this chi 1, chi 1 state and chi 2, chi 2 state and chi 2 delta. So the kinetic energy of the initial state particles have to be at least larger than this mass gap to delta. And what can happen after that is that this chi 2 particles, they can decay to the lighter state and the scalar particles and we have calculated the decay rate which is very fast and decay happens almost instantaneously. So this rate of this whole process will be given by the upscattering rate. So this is an important thing because this will give rise to the underlying mechanism behind all the dissipation rates from inelastic scattering. And if the initial state kinetic energy is not large enough we have only the elastic case where ground state goes to ground state. So that is trivial. Okay, so I'll not describe here how I solved the equations because that's a bit technical and I mean I can discuss if you want up to the top but here I'll just directly show you the results. So in the left plot I have plotted the elastic cross section, the total elastic cross sections with dark matter velocity. So this vertical dashed line is the border between above threshold and below threshold. And we see that this is very much similar to what we have in a single component dark matter scattering. The cross section saturates to a constant value at low velocity and decreases at high velocity. So one can easily satisfy the constants coming from dwarf galaxies, the constants coming from both dwarf galaxies and galaxy clusters simultaneously. And here in this plot I'm showing the three different kinds of cross sections, the transverse elastic cross section, the viscosity elastic cross sections and the inelastic cross section. There's a little difference between these two kinds of cross sections, transverse and viscosity which I am not discussing here because of time constraint but I can if I ask I can show you later. But the thing is that they, so yeah, so for the, and this is a variation with the mediator mass. So for a very large mediator mass, the interaction is more like a hard stress scattering without any directionality. So we start from this bond limit. So the right hand part of this plot is basically the bond limit. And as we keep decreasing the mediator mass, we go into the resonant part of the parameter space where because of this attractive potential, these two dark matter particles can go into some bound states and we see the presence of those virtual bound states in our spectrum. And even for a smaller value of the mediator mass, we'll get the trivial coulomb scattering cross section. And the position of these resonances can be predicted by this simple formula which can be derived from the simple quantum mechanical calculation of the system. Okay, so I'll now discuss about the results of our work. So yeah, so I told you that this inelastic cross section is the main reason behind the dissipation mechanisms that I'll discuss now. So here we have said that these two QI1 particles are upscattering to these QI2 states and these QI2, upscattered QI2s are again taking back to the ground states. And if the scattering rate of this rho2 particles with the dark matter particles is small enough, then this emittered rho2 particles will leave the dark matter halo. So suppose think about the situation when this upscattering is happening in a particular dark matter halo inside a dark matter halo, then they upscatter, then they decay and these rho2 particles are emitted from the halo and they escape the halo directly. And which is true, which can happen because the scattering rate between the lighter state and the dark matter state is quite small. So then what is happening as a result is the dark matter halo as a whole is losing some amount of energy. And from the previous plot we have seen that the inelastic cross section is very much comparable with the inelastic cross section. So they are of the same order. So roughly at around 0.1 to 1 centimeter square per gram. So if the inelastic cross section is so large, then we can have the dark matter halo will lose a substantial amount of energy. That's a prediction. So and for this to happen, the dark matter particles need to satisfy a certain hierarchy in the scale where the kinetic energy of the particles, it has to be larger than the mass gap. And also for this, so that this decay phase space is available, the mass of this rho2 has to be smaller than the mass gap. When this is satisfied, then we can have some amount of dissipation from the dark matter halo. And because of this, this will be mostly satisfied. We expect that it's mostly satisfied in the plasticized or bigger halos where the dark matter particles have a typical velocity of the order of thousands of kilometer per second. And not only that, inside a halo, we know that the circular velocity of the particles, they have a typical shape. The most energetic, the hotter part of the halo is at the outer radius. So at the very outer shell of the halo, the dark matter particles are very energetic. And they will have the most probability to scatter with to upscatter and then downscatter and then decay back to the ground state. And we expect that this dissipation, if it is possible, then if it happens, then it will be observed from this part of the dark matter halo, at the outer part of the dark matter halo. So here I have written out the formula for the cooling rate from where I have convoluted this inelastic cross-section with a velocity distribution from the dark matter particles, which can be taken to be the Maxwell Boseman velocity distribution. And here is a plot of this cooling rate where I have assumed this inelastic cross-section to be 1 centimeter square per gram. And that I have assumed in a double profile with a scaling radius of 560 kilopers. So it's a cluster size halo. So basically this parameter corresponds to the halo of the Van Gogh cluster. So we see that the, so this is in the order of 10 to the power 41 arcs per second. So and the cooling rate is maximum at around 1000 kilopers, which is at like the near the radial radius of the dark matter halo. So the outer part has the most dissipation happening. So what does this say? It says that in a virialized halo, the dark matter, this hotter outer part of this halo will start losing energy at a certain rate. And just after this decay, upscattering and decay, they will fall towards the center. And as a result, the dark matter halo can shrink in size and gravitational collapse can happen. So it says that this upscattering and decay can accelerate the rate of gravitational collapse. So to get an estimate of this upscattering rate, here I have written out the time scale associated with this upscattering, which is roughly with this parameter values of 10 to the power 5 solar mass per kilopers per cube, which is typical for a cluster size halo. And with one centimeter square per gram cross section and 1000 kilometer per second velocity, we have roughly 10 to the power 11 years. And that is very close to the age of the universe, so slightly larger than that. So roughly one can say that this upscattering will happen in one age of the universe, it will happen only once per dark matter particle. So each of the dark matter particle will undergo such a process only once. So maybe we are safe here. The collapse rate will not be as fast as to be a very catastrophic process. But we assume in the same way that the self-interaction, the elastic scattering leaves an imprint on the halo profile to a near the center. In the same, in the similar way, we also expect that this inelastic projection through this dissipation mechanism will also leave some imprint on the halo profile and shape. But this is true for a single isolated halo. But in a physical situation, in a cosmological simulation, things are more complicated where there will be many more late-time mergers between dark matter halos. So they will inject energy into the dark matter halos and they might delay these grapho thermal collapse. So we are not sure. It's not very easy to understand the result of this analytically. So the only thing that can settle this is in body simulation with this dissipation mechanism, where we have this kind of dissipation mechanism in the dark sector where there is a possibility that going to dark matter particles collide, they can dissipate some energy and the final particles will have some slightly smaller velocity. So only such in body simulations can say the final words about this thing. So the second result that we found is about the effect on the collision of two halo dark matter halos. So here I have taken this cartoon picture from this paper where they have shown the situation just after two dark matter halos collide. So I'll just explain the components here. The black dots here are the galaxies. The red patches are the gas, the visible sector gas, and these blue patches are the dark matter. So we see that the galaxies are the most collisionless part of the system. So they just pass through each other. And the gas is very viscous and there is a huge lag between the galactic part and this gaseous part. And the dark matter is somewhere in between where since they have SIDM, so they have some kind of drag force from these self interactions. And there are two things to observe here. There is offset from the, if you look at the centroid of the galaxy distribution and the distribution of the dark matter, there is offset between the centroid to centroid. And there is a tail of the dark matter distribution. So in this paper they have discussed what happens in what cases, I mean, they have discussed the evolution of these tails and offsets of these dark matter distributions when two dark matter halos collide. And in a previous paper, it was derived that this drag force between the two dark matter halos is given by this formula, where sigma was the elastic cross section, m is the dark matter mass, rho 2 is the density of the halos, and v0 is the relative velocity between the colliding halos. So the drag force per unit dark matter mass is given by this. So this was already there in this paper. But we now predict, we now expect that this new upscattering and decay process, this gives us to an additional energy loss. So that in turn can give an additional drag force between the dark matter halos. So let's see how that goes. So here this is the loss rate of this upscattering and decay. So this is the average energy loss per such event. This is the dark matter density, number density. This is the elastic cross section. This is the velocity. So if I just simply divide this energy loss rate by the momentum of the particles, then I get the decay drag force, which is this proportional to the decay energy loss and also to the inelastic cross section. Now if we add this term with this previous term here, we get this total drag force here, where this is the old term, but which now contains this inelastic cross section here because that also contributes to the ordinary dynamic friction. And this is a new term where 2 delta is decay coming from the upscattering and decay and sigma inelastic in the cross section. So let's look at how big is this new term compared to the old term. If I just take the ratio between these two terms, I get this, where it is proportional to v0 squared dark matter velocity squared and to delta over m. So if you remember, I have said at the very beginning that the mass gap between the dark matter particles is very small compared to the dark matter mass. So typically delta over m is of the order of 10 to the power 3 at most and the dark matter velocity even in a cluster size halo, it's not in the unit of speed of light, it's not much greater than 10 to the power minus 3 or minus 2. So we see that it's a very small number. This new term, it's a bit small compared to the old term. But nevertheless, there is another observation which is that the old term is velocity dependent. It's a velocity squared sitting in the denominator, but the new term has no velocity dependent. So there might be some way to identify the impact of this new term from using this velocity independence by looking at the mergers of dark matter halo in a simulation of different sizes with different velocities. So again, one needs to do some sophisticated simulation to verify these observations. Okay, so I think with this I have to conclude my talk. So we have said that the long range of diagonal interaction, of diagonal self interaction between the dark matter particles, it leads to large inelastic scattering. And also this inelastic scattering, this large inelastic scattering can give a way to dissipate energy through upscattering and decay. And this dissipation mechanism predicts faster halo cooling, which can be observed in a proper dark matter in body simulation with this extra feature in the dark sector. And we have also seen the same process of upscattering and decay contributes to an extra drag force between two dark matter halos when they collide. So the simulations only dark matter simulations need to be done with this proper extra feature so that we can identify whether these are true or not. So I'll stop here. Thank you all for your attention. Anirvan, thank you very much for your nice talk. And now I will read the questions from the people in YouTube. So we have a question from Lucas Semmelrock, and he's asking the following question. So I will read, as far as I understood, you calculate chi 1, chi 1 to chi 2, chi 2, separately from chi 2 to chi 1, rho. You don't need to evaluate the diagrams, chi 1, chi 1 to chi 1 to chi 1, rho, rho with chi 2 as an intermediate state as a whole. How is this approximation justified? Does it somehow depend on the mass splitting? Sorry, I couldn't quite follow what is the process that he's talking about. Exactly. Okay, I will read it again. So as far as, you can also read it in the shot, I think. Right? So, okay, I read it again. So as far as I understood, you calculate chi 1, chi 1 to chi 2, chi 2, separately from chi 2 to chi 1, rho. Right? So chi 2 decays into chi 1, rho after it's produced from the ox-cuttering process. Yeah. And then the question is, I mean, you don't have to evaluate chi 1, chi 1 to chi 1, chi 1, rho, rho. Right? Okay. Okay, yeah. So the question is, how is this approximation justified? Okay, so here, saying that the chi 2 particles are produced in the, in as off shell, as off shell states so that the scattering goes through. Okay. So yeah, that is possible, but we didn't consider it here. We have assumed this chi 2 particles are in the, are on shell or always on shell in the final state. But I think, but I think that even if you consider, even if you include that process, the result will not change much because the masses between the chi 1 and chi 2 are very small. I mean, yeah, the mass gap between the two states are very small. So, I don't know. Yeah, but I agree that one should consider that process also. Yeah, but we did that. Yeah. Okay. So I have a question myself. You mentioned at the beginning of your talk and during the motivation, the missing satellites problem. Right? Yeah. So, in this scenario that you're considering of multi, multi particle scattering, you don't really have a solution for the missing satellite problem, right? It's just that to be able to fail and the core versus cost problem. Did I get that right? No, actually, we can, we can, I think solve this missing satellite problem also. So this missing satellite problem requires to have the dark matter interact with some other radiation like species. This is some dark radiation. And in this model, in this simple model, we can treat the row 1 and row 2 as those dark radiation species. And we have seen that with this Lagrangian, there is chi 1 can scatter with row 1 and chi 2 can scatter with row 1 and all of these processes are possible. So, I think one can, we have not, we were not bothered about that solving all the problems here in this paper. But in principle, one can do the, calculate the scattering cross sections between the lighter particles and the dark matter particle. And I think it's possible to solve the missing satellite also in this model. Okay. It's my guess. I have not done the calculation. Okay. I mean, my question was more like, I mean, if you calculate what you just said, in the presence of the extra particle, the results will probably change compared to what people have done in the case of only one particle species. That's right. Yeah, but here we should also think that at late time, we don't have any chi 2. And the dark matter hellos, they mostly have chi 1 particles, the ground state particles, because as soon as the chi 2 is produced, it will decay back to the ground state. So, we need to compute the scattering between chi 1 and row 1, row 2, they are stable. We have to assume that. Yeah, if they are stable, then they will scatter with the row particles and hopefully can solve the missing satellite problem also. Okay. Thank you. Yeah. Okay. May I ask one question? Hi, Yung. Yes. Yeah. Okay. Yeah. I guess in your calculation of cooling rate, you always take a constant velocity distribution, sorry, constant velocity dispersion with the Maxwell-Bortzman distribution, right? Yeah. I wonder in that case, I mean, because scattering changes the dark matter velocities and so I wonder if you should also take into account the relaxation timescale of the halo and compare it with cooling rate. Do you know the relaxation timescale of the halo? Yeah. So, yeah, one should compare these two timescales. But here in my slide, I have compared the dark matter scattering rate with the age of the universe. So, and we have seen that the scattering rate is slightly larger than the age of the universe. And of course, the halo relaxation timescale will be smaller than that, right? Okay. Okay. So, of course, the scattering rate will be larger. The timescale for the scattering rate is larger than the halo relaxation rate. Yeah. So that's another point which I have not mentioned in the talk, but it's there in the paper, which is that we expect this upscattering and decay, this energy loss thing. It will not be episodic. I mean, it will not be the other usual energy loss, the energy loss processes which are there in the baryonic sector. We see the supernova feedback and active galactically all other things which are episodic, which happens very quick and then it goes away. But if this upscattering decay thing, this is not like that. It happens on a very larger timescale. So it is very continuous. Okay. Thank you. Maybe just something out of curiosity. I mean, in your calculation, the potential matrix is, I mean, you have four same elements, right? Sorry. I mean, the matrix of your potential in calculating the scattering cross-section, you have four same elements. I wonder if you're assuming all those off-diagonal elements to be zero, would it change the elastic scattering cross-section a lot or not really? Yeah. That's a very good question. So I think I should show you a slide we have had in my backup. So here. Okay. Can you see this slide? Yes. So this is a plot with where I have switched off all the diagonal potentials. So the potential matrix now reads zero, v, v, zero. So with this potential matrix, I have plotted, calculated the cross-sections and plotting the ratio of the inelastic cross-section to the elastic cross-section. So you see these two cross-sections are comparable in some region of the parameter space and in some areas just nearby the resonances, the ratio can go below. But the important thing is this for the larger mediator mass. Well, this ratio goes very high. It becomes very larger than one. So what is happening here, the inelastic scattering rate is much, much larger than the elastic scattering rate. And this is happening because in this kind of model, at the tree level, there is no process which will lead to elastic scattering. For elastic scattering, you will need at least one loop. But the inelastic scattering will still happen at the tree level. That's why this ratio is going up. And if you consider the results of this, one can see that if our true dark matter model sits somewhere here, then we'll see more inelastic scattering than the elastic scattering. And each of these inelastic scattering will also have this decay, this energy loss. So in that case, maybe this energy dissipation mechanism that I talked about will be most magnified. So it will be most easy to see. In that case, it means it's not in the non-perpetitive region anymore. So you could calculate the tree level and loop level. Yeah, so in that case, one don't need to do the calculation. One can just directly state the... That's it. Thank you. Okay. Are there more questions? Yes, I have a question. Can you guys hear me? Yes. So for the inelastic process, and even you studied the 1-1 into 2-2. But what about 1-1 into 1-2? You can always have it if you have mixing between row 1 and row 2. Yeah, that's another extra feature that you can add in the model. But in our simple case, we didn't consider any other mixing or anything between the... So for example, one may consider the same scalar particle instead of considering 2 different row 1 and row 2. So in that case, one can have 1-1 going to 1-2. So that's also possible. But yeah, so I did not consider that. So I'm just trying to think what would happen if one considers that model. Yeah, I think in that case, it will be easier to have inelastic... Easier to have the energy dissipation from the dapf and adapter scattering because you have only... There's one more channel is opening up. So the energy scale hierarchy that I showed, the kinetic energy has to be greater than 2 delta. It becomes kinetic energy has to be larger than 1 delta, larger than delta. I think it will ease the situation here. Okay, thanks. Great. Is there any other question? Yeah, I have a couple of questions. Okay. First of all, thank you for the talk. It was very interesting. It was very... For me, for the area... I mean when this self-interactive matter is... I'm not so expert, but yeah, it was very nice to learn about it. So I have two questions. One is regarding this mediator, the row, row one or row two. You assume that they're very light, but have you include, for instance, constraint that could come from the effect of the constraint on the relativistic of... I mean of the number of relativistic species in the early universe because these particles are going to be either produced by the dark matter before the result. If it is a kind of wind, the capon also could be produced by standard model particles in case of... Dependent of the Lagrangian, of course. Yeah, so... Yeah, exactly. So wait, I did not talk about the... Or any EUH completion of this model. We just started from this interaction Lagrangian, but one may try to build some model which has this kind of... Which has this kind of Lagrangian in the IR. But yes, if I have these lighter particles, row one and row two, which are typically much lighter than the dark matter particles, then they will contribute to the ineffective in the early universe. So before CMB or before BBN, they will act like a radiation species and it will contribute to the... We'll see some imprints on the Delta ineffective measured from these two observations from both BBN and CMB. Yeah, and actually one may look these papers. I think one of Camillo's papers with Alejandro, where they have considered a similar kind of model where there are two... There is a Dirac fermion. At UV, there is a Dirac fermion and a complex killer, which... And symmetry breaking happens in the dark sector, which gives rise to two Myronov particles. The Dirac fermion splits into two Myronov particles and the complex killer splits into a massive mode and a lighter ghost on mode. So in that case, also we have a similar kind of... Similar kind of IR Lagrangian. So yeah, one may... There are other examples also, but one may do various kinds of model building with this thing. We did not consider... We did not consent on building a model with this. We just took this simple Lagrangian and studied the phenomenology in the scattering part only. So just regarding to this UV completion or your simple model. In that case, you assume that Roy is just a kind of real scaler or complex scaler. But what happens if there is cutting like pseudo-scalar effect? Do you expect some modification in your plastic scattering because of the nature if it is scalar or pseudo-scalar? Well, as far as I know, the potential... So these are... I assume them to be real scalers. So the potential which arises from the mediation of these scalers is simply attractive Ucava. But if you calculate the potential from pseudo-scalar, mediation of pseudo-scalers, you get potential which has some spin-dependent terms. So it depends on the spins of the interactant dark matter particles. And it depends on the inner products of the spin vectors. So and in a realized system, in an isotropic system, that potential will be zero on average. So my guess is that those kinds of potentials will not contribute anything in the nature in the dark matter scattering because they have spin-dependent term. I'd like to add a little thing about on this. So here I considered only scalar particles, but one can build some models with vector particles. So then in that case, the sign of the potential will depend on the nature of the dark matter particles. So it can be either particle particle scattering or particle end particle scattering. So it could be plus or minus in that case. So as a result, the scattering rate will be enhanced or suppressed. So in the sense that also just in your case, also there is a factor related to the type of dark matter. If it is fermion scalar or factor. Yeah, I assume the rules to be only scalars. One can take them to be vector, but then one has to worry about the signs of the potentials. It could be both positive or negative. For if they mediate some fermion particle among them, I don't know what would be the potential in that case. So I'm not sure about potential. Okay, I guess there was a question from Federico. Hello, no, yes. Yeah, yeah. Yes, thinking about these scalars is wrong. Do you have to hide them in colliders? I mean, or would they appear in some cascades for instance? Okay, so these are like the further aspects of this model. So one can go on and let the scalars mix with our standard model Higgs particle. So and if they mix with Higgs, then there will be all kinds of signatures of the, let's say the invisible decay of the Higgs. So it will contribute to the invisible decay width of the Higgs. And so I'm not an expert on collider physics. So that's the first thing that came in my mind, but there might be some other signatures in the collider. I'm not very sure about that. Okay. This is a trivial extension. Yeah, sure. But I thought when you have this scattering, k1, k1 to k2 decaying again to k1, k1 and the sigma and the rows. What happens with the rows after that? Okay, so if they are stable, then they just stay there in the universe. And if they are not stable, then one has to couple the scalars with the standard model particle. So they can either mix with Higgs and decay to some lighter fermions or photons. Or yeah, I think one can build some kinds of, one can just have to couple the scalars with the standard model particles. So which is of course model dependent. Thanks. Okay, yeah, but this can be a very slow decay. So it doesn't make me avoid it. Thanks. Yeah, but I think they finally have to decay because otherwise my guess is that the masses of these particles, row one and row two, they might over close the universe because they are not totally massless. So there might be some issues with the cosmology, but yeah, I don't want to guess. Okay. Are there more questions? Yes, maybe one. Okay. If the elastic process you don't have some kind of velocity dependence, these affect in something or not? I don't think, if I understood your question properly. Diego, can you repeat the question, please? A bit louder. Yes, if the elastic procession have some kind of velocity dependence, this can have some effect. Yes, if this can have some effect in the analysis. Okay, so yes, the elastic cross section, both the elastic and the elastic cross sections, they have a strong velocity dependence, especially towards the larger end of the velocity spectrum. But that's not directly relevant to the dissipation mechanisms, which I was talking about. If you say the velocity, here the velocity dependence is important to have to satisfy all kinds of cross sections, all kinds of constraints which come on the dark matter cross sections. So like we have the constraints coming from the dwarf galaxies to solve, we have to solve the co-cause problem in the dwarf galaxies, and we also have to solve the too big to fail problem. So for that we need some minimum amount of self-interaction among the dark matter particles. So typically this kind of observations gives lower bound on the dark matter scattering. And on the other hand, and we must remember that the typical velocity dispersion of the dark matter particles in these kinds of objects is very low, roughly 10 km per second or a few tens of km per second. But in galaxy clusters, let's say from blue clusters and other colliding galaxy clusters, we see that these dark matter distributions are just moving through them without much effect, without much observable effect. So that puts an upper bound on the dark matter scattering. So we have to satisfy these two things together. So this kind of velocity profile, we have velocity dependence where it's constant for low velocity and decreases at the larger velocity. It automatically helps us to solve, to satisfy both these constraints. So we can have a larger cross section for dwarf galaxy, but a smaller cross section for clusters. Okay. Are there any other questions? Okay. I would like to thank again Anirvan for his very nice talk and for answering all our questions. And yeah, thank you very much and see you next time on the Latin America webinars. Thank you.