 Hello, everybody, and welcome, again, to this low physics webinar. So after the pause of August, we are going to re-assume with the season 5 that we are doing now. So before to start, let's remind you that if you are following this in YouTube, all the questions for the speaker here in the YouTube chat, you can see it, of course, when you go in YouTube. And if you are watching this video in the future, I mean later, you can leave some comments to the video. So let's talk about the speaker. The speaker is Andrea Vittino. He did a PhD in the University of Turino, and the University of Paris did a lot. And after that, he is now a postdoc in the Technician University of Munich. Basically, the title of the talk is a signature of anisotropic cosmic ray transport in the gamma ray sky. So Andrea, if you are there, you can start whenever you want. And welcome to this low physics. OK, thank you very much, Roberto, for the introduction, for inviting me. So let me just share my screen so you can see my slides. OK, so I hope that everybody can see my slides. Yeah, we can see it. Yeah, so as I said, thank you very much for the invitation. I'm very happy to take part in this series of webinars. Today, I will talk about a signature, as you said, of anisotropic cosmic ray transport in the gamma ray sky. So this talk is based on a paper, on a very recent paper, that I've written in collaboration with Silvio Czerli, Daniela Gaggiero, Carmelo Evoli, and Dario Grasso. And obviously, you can find the paper on the archive. Before I start to dig into, let's say, more complicated things, let me just start with a very basic introduction on the topic of cosmic ray transport. And in order to talk about cosmic ray transport, one has to talk about cosmic rays. So here you see the cosmic ray spectrum for different cosmic ray species, as measured by several experiments. So one thing that you can see is that the energy spectrum of cosmic rays actually covers a wide range of energy, more than 12 orders of magnitude, actually. And these energies are even higher than the energies that can be reached at colliders. So this makes cosmic rays a very good place to look for new physics. You see that antimatter is present in a cosmic ray flux in the form of positrons and anti protons, but also possibly in the form of light anti nuclei. Antimatter is obviously a very good place to look for possible dark matter signatures. The fact that the cosmic ray energy range extends up to very high energy makes us think that some of the highest energy cosmic rays might be of extragalactic origin. So the argument here is very simple, basically, at some high energy. For example, in the case of protons above 10 to the 8GV, the normal radius of cosmic rays exceeds the size of the galactic disk. And therefore, this cosmic rays might come from or has to come from sources that are located outside of the galaxy. You can also see here that what is probably the most striking feature of cosmic rays. So cosmic rays have a very simple spectrum. So basically, the spectrum is a series of power loads with a shape, with a slope, changing at the knee. In particular, below the knee, most of the cosmic ray species show what is usually called a universal spectrum. So they all share these E to the minus 2.7 or 2.8 spectrum. This is at least to a first approximation. Obviously, there are deviations from these behavior. Probably the most famous one is the ardening in the hydrogen and helium spectrum. But nevertheless, to a first approximation, a universal spectrum is observed. The presence of this universal spectrum below the knee suggests the possibility that galactic cosmic rays are accelerated by a homogeneous class of sources with a rigidity-dependent mechanism. Now, this homogeneous class of sources in the standard picture is represented by supernova remnants. So the idea is actually quite simple. The standard scenario is that cosmic rays are produced by stars, so are among the end product of stellar nucleosynthesis, and are then injected in the stellar medium by a supernova explosion. But before being injected in the stellar medium, they are accelerated by diffusive shock acceleration mechanism that take place in the blast wave of supernova remnants. There are two kinds of arguments that can be made in favor of this scenario. One has to do with the energy. So the fact that supernova explosion can power the total luminosity of cosmic ray seems to be plausible. Actually, this idea was introduced in the very old days by Bade and Zvik in 1934. And even today that we can make more precise calculation, we know that if we assume a rate of supernova explosion between 1% and 3% and if we assume that more or less the 10% of the energy released in the explosion is channeled into the acceleration of cosmic rays, we know that if these numbers that are realized in nature, then we know that supernova explosion can power the total luminosity of cosmic rays. Another very important argument in favor of the so-called supernova-revenant paradigm is that if cosmic rays are accelerated by diffusive shock acceleration mechanism within supernova-revenant, then they acquire universal spectrum. And this is because diffusive shock acceleration mechanism are a specific kind of first-order Fermi acceleration mechanism. And within this mechanism, the particles that are accelerated acquire a spectrum that is a power law with an index that depends only on the properties of the shock. So this makes this kind of mechanism very appealing, obviously. So as you have seen, this scenario is quite successful. So it seems that the supernova paradigm is actually a good paradigm. It seems very plausible. But nevertheless, one has to take into account that no total cosmic rays are producing stars and then accelerating in supernova remnants because we have the so-called secondary cosmic rays. So to understand the distinction between primary and secondary cosmic rays, one has to look at the relative abundances of elements in cosmic rays and in the galaxy. And here we take the solar system as a proxy for the abundances of elements in the galaxy. So you can see that there are elements that are as abundant in cosmic rays as they are in the solar system. And these elements are, for example, the hydrogen, the helium, or the carbon, et cetera. So these cosmic rays are the ones that are called primary cosmic rays. These are the cosmic rays that fit into the supernova remnant paradigm that I was describing in the previous slide. So these are those cosmic rays that are synthesized in stars and then accelerated by supernova remnants. But on the other hand, there are also secondary cosmic rays. So secondary cosmic rays are the ones for which the abundance is, as you can see, is much higher in the cosmic ray than in the solar system. These elements are, for example, the lithium, the beryllium, and the boron. And the idea is that these cosmic ray species are produced in palatial reactions that involve heavier cosmic ray species that collide against particles of the interstellar medium. And it is important to point out that these cosmic rays are usually not assumed to be accelerated by supernova remnants unless they are produced directly inside the shock region. OK, so this was, let's say, a very basic overview on the topic of cosmic ray and on the topic of cosmic ray origin. The main topic of this talk will be cosmic ray propagation. So the idea is very simple, actually. So cosmic ray, after they have been produced in a source, so for example, a supernova element, as you see here, have to travel across the galaxy before being detected, a third or anywhere else in the galaxy. The cosmic ray propagation is actually a quite complicated process. Under a mathematical point of view, this process is modeled by means of a transport equation as the one that you can see here. So this transport equation contains a lot of term. In particular, well, first of all, one has to say that this equation is in terms of n, where the variable n represents the cosmic ray momentum density. Then it is important to point out that transport equation is time dependent. Usually, as it would be for this talk, one assumes a steady state scenario. So one study cosmic ray in a situation in which n is constant in time, but this is not necessary. Going to the right-hand side of the equation, the first term that we meet is the source term that is basically the number of particles injected per unit of time, volume, and energy. And this term can model both the primary and the secondary emission. Then we have the spatial diffusion that is the result of the interaction of cosmic rays with the turbulent component of the galactic magnetic field. I will talk much more about the spatial diffusion later on in the talk. Basically, this will be the main topic of this talk. Then we have advection. So we have the possibility that cosmic rays are advected away from the galactic disk by a wind. And it is a word to stress that if this wind increases with a vertical height above the galactic disk, then advection also causes energy losses, as you see here. So we have a debatic energy loss. We have reacceleration. So reacceleration is basically a diffusion in momentum space, once again due to the interaction of cosmic rays with the turbulent component of the galactic magnetic field. We can think of reacceleration as being the counterpart in momentum space of spatial diffusion. Finally, we have energy losses. Energy losses are different if we are talking about cosmic ray leptons. Because in this case, we may have synchrotron emission, inverse contour, or brainstorm, which are the dominant processes. In the case of nucleons, we have coulomb losses and ionization. So this was a very basic introduction to the topic of cosmic ray propagation. Now let me just briefly explain how one study cosmic ray propagation. So actually, in order to put all the ingredients inside the equation that I showed you in the previous slide, one has to compare theoretical predictions, which are always based on the formulas that I have described. So one has to compare this prediction with experimental observation. And experimental observation are basically of two different kinds. So we have local observables, which are also direct observables in the sense that what is measured is the cosmic ray particle itself that is measured by detectors that are on satellite experiments or balloon-borne experiments or even on spacecraft. So to make some example of local observable, here you see, for example, the primary flaxes, the secondary to primary ratios, the interstellar flaxes. Here by interstellar flaxes, I mean cosmic ray flaxes measured outside of the area of influence of the solar magnetic field. So this measurement actually was made possible by the Voyager 1 spacecraft that reached the boundary of the heliopause in recent years. So actually, this is actually a measurement that is not so local in the sense that it's pretty far away from Earth. Then we have the measurement of antimatter flaxes, which, as I said at the beginning of the talk, are very important if one is looking for that matter. Together with these local observables, we also have non-local observables, which are indirect, in the sense that what one measures is not the cosmic ray particle itself, but it's the radiation emitted by the cosmic ray particle in the neutrino, radio, or gamma ray sky. And these observables are obviously no-local in the sense that one can measure these emission basically everywhere in the sky. In this talk, I will focus on the gamma ray measurement. So here you see a map of the gamma ray sky as seen by the Fermilat experiment. So basically, if we are looking for gamma rays produced by cosmic rays, we may have three different processes. So we may have, in the case of cosmic ray protons, they can impact against hydrogen atoms in the interstellar medium. These lead to the creation of a neutral pion that then decay into a couple of photons. In the case of cosmic relaptences, we can have either inverse-compton scattering, as you see here on the interstellar radiation field, or we can have a brainstorming. In this talk, I will be mainly focusing as you will see on the gamma ray emission across the galactic plane. And the dominant process that leads to the production of gamma ray in this region of the sky is actually the pi Mbk. This is because of the abundance of protons in cosmic rays together with the abundance of hydrogen in interstellar gaps. What is very important, and actually is one of the key things to understand to get into the message of this talk, is that by mapping the emission, this gamma ray emission across the galactic plane, one is able to map the cosmic ray proton distribution across the galactic plane. So this is obviously a fundamental feature. And actually, the starting point for my talk. So in this talk, I will discuss what the map of the proton distribution across the galactic plane might be telling us about cosmic ray transport. So my starting point will be, as I said, the proton distribution inferred from gamma ray observation. And I will try to see what this can be telling us about cosmic ray transport. I will focus on a particular aspect of cosmic ray transport, that is the spatial diffusion, which is a dominant process is if we are studying the transport of galactic protons with energies that are not low. In particular, in this talk, I will try to show that these gamma ray observations might be calling for a more complex modeling of spatial diffusion with respect to the standard picture. So I will try to justify a sort of change of paradigm in the description of the spatial diffusion. So before doing all that, let me just describe what is the standard picture of cosmic ray spatial diffusion. The standard picture is actually very simple. And to understand it, we have to go back once again to the transportation, which you see here. So if you remember some slides ago, I told you that spatial diffusion is modeled by this term here. And in particular, it's modeled by means of a diffusion coefficient that is spatially independent. And it's usually assumed to be a power low in rigidity. So it goes like momentum to this power of delta. And delta is assumed to have no spatial dependence whatsoever. So over the years, this kind of approach has been very successful. So this standard picture of cosmic ray diffusion has proven to be very successful in, for example, in reproducing all the local observables, such as, for example, primary fluxes or secondary to primary ratios. And so given the fact that this is the standard picture and given the success of this kind of interpretation of spatial diffusion in describing experimental observation, one may ask, what is the shape of the proton spectrum that is predicted by such a framework? The answer is, once again, very simple. The sense that if we consider a purely diffusive transport, so if we remove from the transport equation all the terms that are different from the spatial diffusion, so we are left with the source term and the diffusion term. And if we assume that the source term goes like momentum raised to the power of minus alpha injection, so this is basically the index of the spectrum that is injected by sub-normal lambda. So if the shape of the source term is this one, then we have that the solution to the transport equation, to the purely diffusive transport equation, is a power law in momentum with an index that is delta plus alpha injection. So to sum up, in the standard picture of cosmic ray spatial diffusion, the proton spectrum is assumed to be the same everywhere in the galaxy. So it's a power law with an index that does not depend on the spatial coordinates. One can check if this prediction is actually what is seen in nature. And this was, for example, done by the Fermilac collaboration that measured the proton spectral index across the galactic plane at different distances from the galactic center. The result of this measurement is shown here in this plot. So you see here that the proton spectral index is plotted as a function of the galatocentric radius. The prediction from a spatial independent diffusion, as I shown you in the previous slide, is an horizontal line because the spectrum is expected to be the same everywhere across the galactic plane. But what is seen is rather different. So what is seen is an hardening. So there is an int for disarming. And my point is that this hardening might represent a possible challenge to this standard picture of cosmic ray spatial diffusion. Obviously, one here has to consider things with a sort of quotient in the sense that, as you can see, aero bars are quite large. So we can think of an int. It's not really an evidence for an hardening. And also, as you can see, there is this point at 1 kiloparsec that doesn't really fit into this picture. So all these things have to be kept in mind for the reminder of the talk, that's it. One thing that is worth to point out is that this hardening seems to be fairly well compatible with the one that was invoked in a series of paper by Gadget Retal from 2014 on. So basically, in this paper, what was proposed is a sort of phenomenological model in which the diffusion coefficient has this shape here. So you see that the rigidity scaling of the diffusion coefficient has a spatial dependence with respect to the data-centric radius. And the spatial dependence was tuned with respect to some, let's say, some old Fermilac data. So they are not related to this very recent analysis, but still the hardening that is predicted within this model is compatible with the hardening that is now observed by this very recent analysis. Okay, so the message that I want to convey in this talk is that this hardening, so this hardening that seems to be observed by this very recent Fermilac analysis might be qualitatively similar to a hardening that can be obtained by assuming a spatial diffusion to be anisotropic. So this is the, that's the main message of the talk. And the first thing that obviously we have to address now is what is actually anisotropic diffusion. And yeah, let's just start from the basics. So if one consider the gravity magnetic field to be the sum of a regular and a turbulent component as it's usually done. So you see here, we have a regular component is this V zero and delta V that is the turbulent component. Obviously, as I said at the beginning of the talk, it is the interaction with this turbulent component that generates the cosmic ray spatial diffusion. So if this is the situation, then we have obviously the diffusion can be either parallel to the regular component of the magnetic field or perpendicular to this vector. And we talk about anisotropic diffusion if these two diffusion coefficient, the parallel and the perpendicular one are different. So obviously in the standard picture of spatial diffusion, these does not happen. So the two are equal and they are equal to the diffusion coefficient that is basically a number. In anisotropic diffusion, these two are different. Okay, so this was just very basic definition. Let me just dig a bit deeper into anisotropic diffusion and let me present you some theoretical motivation for anisotropic diffusion. So in order to introduce some theoretical motivation, I would like to start from the quasi-linear theory of each angle scattering in a random magnetic field. This is like the basic framework when dealing with cosmic ray diffusion. And a key prediction of these theoretical framework is that the ratio between the perpendicular and the parallel transport goes like the ratio between the turbulent fluctuation of the magnetic field and the regular component of magnetic field squared. So this quantity, this delta V over V zero is expected to be one at the scale at which the turbulence is injected. So for example, if the turbulence is assumed to be injected by supernova explosion, then delta V over V zero is equal to one at a hundred percent. For all the scales that are smaller, that is a hundred percent. So for example, for all the scales that are relevant for the propagation of protons with energy between the GV and the TV, this delta V over V zero is expected to be much less than one. So in this domain, which is also the domain of applicability of quasi-linear theory, these, so since delta V over V zero is much smaller than one, then the ratio between the perpendicular and the parallel transport is also much smaller than one, which means that the quasi-linear theory predicts a highly anisotropic diffusion. Now, quasi-linear theory is not the final word when it comes to spatial diffusion in the sense that there are effects that are not included in the quasi-linear theory treatment that can complicate the picture. This kind of effect are usually studied in the framework of numerical simulations. And as you can see from the results of this numerical simulation that I show you in this slide, so these results are taken from two different papers. So you see that even if this effect might complicate the theoretical framework, nevertheless the bottom line is not altered. So the need for anisotropic diffusion is still there. You can see that the parallel and the perpendicular transport, the parallel and the perpendicular diffusion coefficient have a different normalization and a different rigidity scaling. Okay, so this was a sort of basic introduction and a motivation for anisotropic diffusion. So given this motivation, we implemented a model of anisotropic diffusion and we did that in the framework of the Dragon 2 code, which is the new version of Dragon. If you are interested in the details of the code, I address you to this reference where the code was actually presented and all the details were given. Let me just stress that anisotropic diffusion will actually be one of the key features of Dragon 2. So to describe a bit more detail our setup, we consider, as I said, the transport equation where only the diffusive term is present together with the source term. We restrict ourselves to the two-dimensional case. This was done basically for a technical reason in the sense that working in two-dimensions is obviously much easier than working in three-dimension. We took the source term from the, let's say the standard model of Lorentz-Merital. So basically here Q represents the source term, the proton source term from supernova remnant. And we used a diffusion tensor, that is this DIJ, that can account for possible anisotropy. And what does this mean? Well, it means that we define the diffusion tensor in this way. So you see that it is basically given by the sum of the perpendicular and parallel diffusion coefficient. Each one of these two diffusion coefficients is weighted upon the, let's say the geometry of the regular component of the galactic magnetic field. For the two diffusion coefficients, the parallel and the perpendicular, we assume a very simple form. So they are two power loads. But since we want to include a possible anisotropy in our framework, we use for these two power loads a different rigidity scaling. So in particular for the rigidity scaling of the parallel diffusion coefficient, we assume delta equal to 0.3. While in the case of perpendicular diffusion, we consider values from 0.5 to 0.7. These are not just random numbers, but they are based on a low energy extrapolation of the result of numerical simulation. And also in the case of the normalization, we consider two different normalization for these diffusion coefficients. But actually we are kind of agnostic in the sense that this epsilon b, that is the ratio between the perpendicular and the parallel, we consider it to be between 0.01, which is relatively strong anisotropy and one where one is almost the isotropic case. Okay, so as one can see the key point now that will determine the shape of the diffusion tensor is actually the magnetic field. So the regular component of the magnetic field. We model such magnetic field as being given by the sum of three components. So we have an halo field, a disc field and a poloidal field. For the halo plus disc field, we assume the model by Cherkov et al. So basically our field is purely as a muthal. But in the case of the poloidal field, we take the model from Johnson and Farar. So this poloidal field has this X shape. So basically it's perfectly vertical at the graphic center and then it has an inclination in the RC plane that grows as we move outside of the graphic center. Here you see the total magnetic field, which is the sum of these three components. So you see some of the field lines. So basically just to sum up the basic feature, this field is predominantly directed in the vertical direction when we are close to the graphic center and it becomes increasingly as a muthal as we move outside of the graphic center. Okay, so this was the setup. So I think I've given a complete description of the different ingredients of our setup. Let me show you now some results. So first of all, I will talk about the profile of the spectral index, the proton spectral index. So here you see that I plot the proton spectral index for protons with energy between 10 GB and 100 GB as a function of the galatocentric radius. So these blue lines correspond to the results of our model. This is for a ratio between the normalization of 0.01, delta parallel is fixed to 0.3 and the two different lines correspond to delta perpendicular equal to 0.5 in the case of the blue dashed line and 0.7 in the case of the blue solid line. You will see also in this green dashed line, the results of the phenomenological model by Gacier et al. So you can see that we actually get unardening. So the spectral index becomes smaller. If we are at the galatic center, the entity of this ardent is between 0.27 and 0.58. So these numbers are just the difference between the local spectral index and the spectral index of the galatic center. The reason why we have this ardent is actually, it's pretty simple. So the idea is that at the galatic center, the domian process is the parallel cosmic ray diffusion along the vertical direction. So in this region, the spectral index goes like alpha injection plus delta parallel. On the other end, when we move outside from the galatic center, the domian process is the diffusion in the azimuthal direction. And therefore, if we are working into dimension, this means that we have a perpendicular diffusion both along R and along Z. And this means that the spectral index goes like alpha injection plus delta perpendicular. So given this, given also the fact that delta parallel is smaller than delta perpendicular, one can immediately understand why there is an arvelling. The situation is qualitatively the same, let's say for the case in which the ratio between the normalization of the diffusion coefficients is set to 0.1. The entity of the ardent is different. Actually, we have a smaller ardent, but let's say the agreement with data is still qualitatively good. Things become a bit more tricky if we look at the normalization actually of the flux. So this is the integral of the proton flux. So the proton density, basically. So you see that these colored lines correspond to the result of our anisotropic model. The color code is the same as in the previous plots. So the red lines are for ratio between normalization that is 0.1, while in the case of blue lines, this is for epsilon D equal to 0.01. Again, different lines are for different values of delta perpendicular while delta parallel is still equal to 0.3. We also show the results obtained with the isotropic model. And these are obtained by setting delta to 0.3. So since this is an isotropic model, we have delta equal everywhere. So it's everywhere equal to 0.3, both for parallel and perpendicular diffusion, of course. So you can see that actually it's very difficult to reproduce the data that are observed by Fermilat. So you see that close to the galactic center since our domain process is the cosmic ray escape along the vertical direction, we have a very efficient depletion of cosmic ray from galaxy. This means that the cosmic ray density actually decreases quite a lot, even too much in the case of the strong anisotropy, so in the case in which epsilon is 0.01. But this behavior seems to be at least qualitatively in agreement with the first data point of the Fermilat analysis. So you see that even data seems to point towards this depletion. And this is not observed in the isotropic model. On the other hand, within our anisotropic setup, it's basically impossible to reproduce this peak at three kiloparsec. So it's always too smeared. So it's important to stress that this peak is already present in the source function. So cosmic rays are injected with this peak and in our setup, this peak is too smeared. This is still actually an open issue which we are investigating. We are considering two different possibilities at the moment. So for example, this disagreement with data might be alleviated if one consider a different source term. This is actually kind of obvious. And also, this makes sense if we consider that the source term is not that well known. On the other hand, one may have also a better agreement with data by working in a full three-dimensional framework because at this distance from the galactic center, so around three kiloparsec, there is one of the spiral arms of the galaxy. So one may assume that cosmic ray diffusion might be very efficient along this azimuthal direction and this might reduce the impact of the vertical escape. But these are, as I said, all hypothesis which we are still investigating. So it's time to get my conclusions since I'm already over time. So in this talk, I discussed anisotropic diffusion and I've shown you some possible signatures of anisotropic diffusion in the gamma-ray diffusion. In particular, I've shown you that some of these features might be similar to some features that are hinted by this very recent Fermilat analysis. Nevertheless, I've shown you that picture is still not totally clear. Things are not so easy to model and therefore it's very important to, let's say to investigate also more complex full three-dimensional setup because in this case, we will be able to explore an even wider phenomenology and this will make us able to perform much more precise comparison between theory and the data. So that's it. Thank you very much for your attention. Thank you very much, Andrea. It was very interesting talk indeed. So before going to the question round, let's just remind to the people that is following this webinar that you can do all the questions to Andrea via the YouTube chat. And yeah, now that we are open to questions, so please, everybody in the room that can unmute themselves and ask questions on there. Please. Okay, I have a question. Can you guys hear me? Yeah, yes. Okay, Andrea, so at the very beginning, you talk about this acceleration of cosmic rays and you talk about this supernova random hypothesis. Yeah. Actually, is there a way for testing that? I don't think so, but I'm just wondering if it's possible to do it. You mean to test the supernova hypothesis? Yes, I mean, yeah. I mean, cosmic rays are really accurate, it is there. But I think that, well, a basic test is whatever I already briefly mentioned that is the energy balance of the galaxy. So if you consider what is the total luminosity of cosmic ray, you will actually realize that there are not many sources that can provide such a high luminosity. So I think that this is already quite a test for these hypotheses. Now, I'm really not expert in this field, but I mean, I think that there are no other galactic source that can provide that much energy. For example, you can consider the case of pulsars, but I think that the maximum energy that they can provide is at least one or the magnitude too smaller to power the total amount of galactic cosmic ray. Okay, so basically the only option? Yeah, I think yes, but as I said, I'm not really expert in cosmic ray origin or in cosmic ray production, so maybe there are some other options that I don't know. Okay, thanks. So more questions, I have a couple of questions, but maybe Nicolas, you want to make one? I have another one, but you can go with one. Yeah, I was just wondering, Andrea, for the case of the hardening, I mean, you invoke the isotropic diffusion, but is it also possible to achieve the same effect for the reason with compaction, changing the standard picture of vertical galactic wing to something that can also be especially dependent? Yeah, actually it's possible, but there is a difference between this scenario and the one in which the hardening is related to action, and the difference is that if you explain the hardening in terms of advection, you have an effect that it's present only at low energies. Because if you compare the effect of advection with the one of diffusion, then that's some energy you have that diffusion will start to dominate. So if you want to explain the hardening in terms of advection, you will have an hardening that is efficient at low energy, but it's not efficient at high energy. And it seems that, I mean, both from the Fermilac analysis, but also from the, for example, the measurement of the multi TV gamma ray emission that was performed by Milagros some years ago. It was, what the measure was the gamma ray emission around 20 TV at some point in the galactic plane. I don't really remember the region of the sky, but so if you consider this measurement, there are some ink that the hardening is present also in the multi TV region of the proton spectrum. And it's very, I think it's very difficult to have an hardening that extends up to this high energy by means of advection. This would be my answer. Okay, so it's not possible in this. Yeah, because of the, no, I was wondering because since the hardening is just with the data of Fermi at lower than 10 GB proton energy. So actually it's at 10 GB, I mean, it's, I saw a red opposite the less than or greater than. So, and the other question that I was also wondering because, okay, this is the analysis based on proton plus kind of pre-stralum and Compton losses that the emission of gamma rays and bioproduction and so on and so forth. So is it possible that also with electrons and inverse Compton, you can have an extra information about your anisotropic diffusion but to high Z, for instance, or high vertical because since electron, they're gonna meet inverse Compton by the interaction with the C and B for instance, you are gonna be able to map what is the dependence of this anisotropic diffusion to higher latitude. It's just a question, I don't know if travel is already included this type of effect for electrons but I think it would be a possibility to test furthermore this hypothesis. Yeah, I think it would be interesting actually. So, yeah, I agree that electrons can give much, can give a complimentary information for regions away from the graphic disk. Under a technical point of view, this is rather difficult to model because if you are dealing with leptons, actually you must work in a three-dimensional framework. So to include, for example, the emission in the, I mean, the spiral pattern in the source term, for example, and just by a technical point of view at the moment is actually very difficult to work in this three-dimensional framework. But at some point we will try to do it. We are trying to do it. Yeah, for instance, this type of anisotropic diffusion is not included in GALPROP at least. I think, I mean, in the current version of GALPROP, the one that for me is a fairly useless for analyzing data or not. As far as I know, I mean, this formalism, I mean, this fully anisotropic diffusion at the moment should be only present in GALPROP. So it's not present in GALPROP. So in some sense you are one step ahead of GALPROP? Yeah, but the three dimensions are not simple because you have a, I mean, just to give you some technical detail, when you are modeling anisotropic diffusion, you have all the mixed derivative terms in the diffusion equation. And the discretization of these terms is actually very difficult. So you cannot use the cranking code some method. You have to use a fully explicit scheme. And because of this, you have to use a time step that is very small in the evolution of the numerical solution. And therefore it's actually very slow to get to the solution. No, yeah, one more dimension is crazy because it's already, let's say, two dimensions, your code, I mean, the spherical, cylindrical, symmetrical is two dimension for the spatial plus energy. Yeah. Let's say at three, and also if you want that this stuff is stabilizing time, you have to add a time also. So actually to give you an idea, these results are based on some runs that lasted for some days. So if you add a dimension, you add at least a factor of, I don't know, 40 or 50 to this time. So the moment is not unfeasible, but it's rather, I mean, it's not simple. So I don't know, I guess there are other questions from people from the hangout. I have a second one. Please. No, it's about the galactic magnetic field. So you show this model with three components. So I was wondering if you're taking to account some uncertainties on that. I don't know if I understood, completed the question. So you say if we consider uncertainty on the magnetic field model. Right. Actually, no, in the sense that for these results, we only consider one magnetic field moment, one magnetic field model. And this was done basically before, because of the technical difficulties that I have described earlier. So since the runs were actually kind of long, it's very difficult to actually take into account all the uncertainty. But if you look at our paper, we also consider a toy model for the magnetic field. So before presenting these results obtained with a realistic model, we also investigated a toy model in which we introduced in this type model some parameters and we studied the effect of the variation of these parameters. So yeah, the answer is that we did not really take into account all the uncertainties, but in the paper we investigated what might be the impact of some ingredients of the magnetic field modeling in the result. Okay, thanks. So other questions, because now that you were saying this stuff with the hardening of the spectral index, then when you play with the ratio between the zero and the parallel, I mean the rate of between all the magnetic fields, the turbulent of the regular or the other, at the end you can also invert the type of the hardware, it's complete. In the solution you can have both the scenario in the sense that it's harder in the galactic center and less harder outside, and then also inverted because of this ratio. If you instead to use 0.1, you use 10, for instance, just to play with the... Yeah, I mean under a mathematical point of view, yes, for sure. Yeah, yeah, of course. I mean just physically realistic because it's kind of obvious that the diffusion along the magnetic field is easier to realize than the diffusion in a perpendicular direction. So what expect the parallel to be bigger than the perpendicular? Yeah. Oh no, yeah, that was just when, because you were mentioning that you were working on a paper in which you were exploring more the variation with the parameter. So since this ratio was essentially a parameter in all the description, so just to know the other type of example that can be covered. Yeah. So I don't know, anybody has question? I'm gonna go just to remember the people that were still in the question round. So if you have question for Andrea, please write it in the comment. Also, all the questions are possible. For instance, now I was remembering that in the case of if you go to very high energy, this scheme also is break down or, I mean, in a sense, can it possible to try to explain this transition between the so-called galactic extragalactic break in the spectral index or? Well, I don't think you can really explain that. Actually, at some point, it will break down in the sense that since the rigidity scaling of the perpendicular diffusion is with a higher index, at some point, at some energy, they will become equal. I mean, the diffusion will become isotropic. And yeah, so therefore, there will not be the ardent anymore, but this is, I think, a very high energy. Yeah, more than the energy that are... For sure, more than the TV scale that we are considering here. Yeah. So that could be a kind of prediction of this description, to expect that it's hardly going to disappear after certain energy. Yes, I mean, if you have data up to very... No, of course. No, yeah, yeah. So, okay, let's... If there are no more questions here, and the question that they were in the YouTube chat, I already did. Let me ask one question here. All right, good. So I was wondering if neutrino data could be useful here, like the neutrinos coming measured in ice cube. Actually, I think yes, in the sense, I mean, if there is an arcing in the proton spectrum, so you expect on one side to have more gamma rays at high energy, but also more neutrinos. So the galactic component of the neutrino emission will be larger. So in principle, yes, neutrino can provide further confirmation of this scenario, provided that one has sufficient statistics. And this is probably the weak point. Yeah, but you wouldn't expect the neutrino flux to be affected by the magnetic field, right? So it could provide some sort of normalization of the flux, maybe? But you don't expect the magnetic field to impact the neutrino, but you have... I mean, within our model, you have that the proton flux is harder towards the galactic center, so you have a much larger number of high-energy protons, and therefore you have a higher neutrino emission from these protons. So what you expect is that from the inner galaxy, you should see more neutrinos. Okay, okay. Thank you. This is, of course, only the galactic component. So one has to see if this is visible in the total neutrino flux. So more questions. Now that it seems a kind of a regular question in all the webinars, what about the role of the possible source related with that matter in this scenario? But do you expect also to be able to explain, for instance, the galactic center emission without breaking down other type of observables related with cosmic rays? Quentin, how is affected by Oversea also? I mean, to make everything compatible in this scenario? Well, it's... I guess there are two questions, let's just focus on the dark matter, I guess. Not in the bioversea that I just mentioned. Well, I mean, so you have to assume that these gamma rays that Fermi is seeing are produced by a dark matter component. Maybe this could be plausible. I don't know, it depends a lot on the dark matter model that you are considering. I, well, I would say probably it's not excluded. I don't know how plausible it is, but... Yeah, I mean, also there is a direct emission from gamma rays, but I was thinking in the case for instance, anti protons and stuff like this. What in the... So you are saying that, well, that's kind of, I think it's difficult to have a dark matter article that, so it will... I mean, because if you wanted to explain, for example, this Ardening that I was talking about, well, I mean, you can always, I think you can think about a model in which you produce no anti protons, but you produce gamma rays. So this will work at least with respect to the anti proton constraint. I don't know if you can evade all the gamma ray constraints. Yeah. Yeah, yeah, I was thinking this because since there is very strong constraint on the anti proton over proton ratio, especially for the case of dark matter. So I don't know if the anisotropic diffusion could also affect this type of ratios. You know that this is a local measurement. It's not a measurement at the galactic center or you cannot infer this ratio at different levels of distance with the galactic center. One thing that one has to point out is actually kind of interesting is that within our scenario, the anti proton constraints to dark matter will be a bit weaker because if you assume that cosmic ray time to escape from the galactic center along the vertical direction, then since dark matter is mostly concentrated in the galactic center, then you will weaken the constraint because the anti proton produced by dark matter will just escape along the vertical direction. It will not reach us. And so in the local observable, you will see a weaker anti proton flux from dark matter. Now this probably will not be a huge effect in the sense that if you consider, for example, an NFW profile for dark matter, and I think that the emission that comes from the inner kiloparsecs of the galaxy should be no more than 40% or something like that. So you can lower the bounds on dark matter and elation cross-section by this quantity at most. If you assume that all the anti protons within the inner parsecs simply go away. I mean they go away in the vertical direction. Yeah, okay. Ah, okay, yeah, interesting. So, okay, let's check if there are questions. There are no questions in the YouTube chat. I don't know if somebody else has questions for Andrea. So if not, I guess it's time to finish this webinar. First of all, I want to thank Andrea for this very interesting talk. I mean, I like it a lot, especially because of the topic. 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