 Okay. So we're going to start the third lecture on dark matter. So take your seats, please. Can you hear me? So in the first lecture, we had a brief review of why we believe there is some abundance of dark matter in the universe today in our galaxy and even at larger scales. Two days ago in the second lecture, in the second lecture, we went back in time and we looked at possible ways, mostly one way. We looked at one way to produce dark matter in the early universe that was thermal freeze out. And we derived a relation between the abundance and the cross section and we saw that the relic abundance requirement in some sense give us an idea of how much dark matter should interact with us. So what we did the last time was exactly going back in time when the universe was very, very young and we studied the production of dark matter. What we do today, we come back to the present and we have an overview of experimental strategies to actually search for these particles. So we will talk about three main ways that you may have heard of before of detecting dark matter. There are direct detection, indirect detection and collider. And it's good to start this topic by drawing a diagram that you probably have seen before, but just... So this is just a more visual way of visualizing the annihilation process that we were discussing on Tuesday. You have two dark matter particles arriving, finding each other. They interact and they give rise to a final state made of some other particles. So I didn't emphasize that but in the previous lecture I was always assuming that the hour of the time was going from left to right in this diagram. So this is my initial state and this is my final state. And if this is the case, so if time goes from left to right this diagram describes an annihilation. And we saw on Tuesday that this is the process that sets the relative density. So the cross section for this process tells us how much dark matter should be around today. What we will see today, the same process that happened in the universe and the process that set the number density of dark matter can still happen today. For example, at the center of the Milky Way, our galaxy or in dwarf galaxies around us. And this process is how we do indirect detection. So for now I'm just giving you the name but then we will review experimentally how you look for these annihilation processes. Now the time can actually go in two other ways in this diagram. So if you consider time going from the bottom of the blackboard or the top, then you have an elastic collision. So you have a dark matter particle that collides with a thermal particle and in the final state you have the same two states. So it's just a collision where the initial state and the final states are the same. So this is the process that is probing direct detection experiments. So it's an elastic collision between dark matter and us, standard model. And I take advantage of this opportunity to mention something that was more appropriate to cover on Tuesday but there was no time. But this is also the process that set kinetic decoupling in the early universe. So we saw on Tuesday that when gamma for annihilation is equal to Hubble you get chemical fees out. So number changing interactions are not effective anymore and the number density is frozen. But these elastic processes can still take place and this is what allows dark matter particles and the thermal particles to keep the same temperature until lower scales, lower times when the universe is older. And this kinetic decoupling process is very important because it sets the scale of the smallest object we can have in the universe. So you can put a bound on this cross-section by looking at structure similar to what I was discussing when I said that dark matter has to be cold. So if you're interested to hear more about decoupling come talk to me later. But since this lecture is about searches today so we will focus on direct detection and the last option is to read this diagram from right to left so this describes a collision of two standard model particles producing a pair of dark matter particles so from right to left. So this is what happens, for example, what could happen at the LAC. You collide two protons, they collide and you end up producing two dark matter particles in the final state. So this is, here I wrote LAC but it fits within a broader category which is dark matter searches at colliders. So the plan for today is to describe without giving of course too many details because I'm interested more on giving you the idea and how the experiment works and what is the output of the experiment so when you look at the plot if you're sitting in a dark matter talk you understand where the plot comes from and what it means. And I will follow this order direct detection first, indirect and collider. There is no reason to prefer one with respect to the other it's just a choice. OK, so direct detection. So direct detection, the idea of direct detection was suggested in the 80s by Goodman and Whitten and the idea is just to look for events of this kind so the idea is to look for elastic collisions between a dark matter particle and some target. Here I call the target N so this will belong to the experiment and it is typically a nucleus. We will see later that if it's large it's good so typical targets are large nuclei like Xenon or Xenon and Germanium are probably the most diffuse in these experiments. OK, so before we get to the actual experiment let's get an idea why we expect this event to happen in the first place. OK, so what we can do is to take a nucleus of course we don't take a single one, we take many but we put the nucleus in the lab here we wait and then if we see that these things move it means that it was kicked by something by something invisible that we cannot see and so by... Oh, let me close. So if you sit and wait and you see some recoil momentum and recoil energy without seeing anything else so that means that there was something invisible because the momentum is conserved so something invisible kicked it and the hope is to detect an event where something invisible was this dark matter particle. OK, so how common are these events? What is the typical rate for these events? Let's see, in order to give an estimate of the rate we need to know how much flux we have here on Earth, OK? So if I... Let's see, if I consider one centimeter square, OK? One centimeter square and I consider one second so one centimeter square is this big, OK? Here on my front how many dark matter particles passes my head one centimeter square of my head in one second? Do you have any idea? Ten, five, thousand So it depends, it depends because what we know is the mass density not the number density, OK? So the flux is number density times the velocity, OK? And the velocity we know that around the solar system and in general in our galaxy is ten to the minus three times the speed of light, OK? So they are very relativistic today, this particle in our galaxy, but the number density of course is the ratio between the mass density and the mass of a single particle that's how many particles we have. So this is the number I give you, OK? So we know everything because we know how fast dark matter particles move around our galaxy today, yes? So the velocity is just hydrostatic equilibrium so there is gravitational there is gravity that wants dark matter to collapse at the center of the galaxy and then there is velocity dispersion that acts like a pressure even though it's not correct to speak about pressure but so it's a competition between the dispersional velocity and gravity and once you impose the static equilibrium you see that you need this velocity in order for the system to be in equilibrium so the halo around our galaxy for example, OK? So we know the velocity we know a row, so for each mass we can estimate the flux and know how many particles hit my head in one second. So let's put one number which is 100 GV this 100 GV is as we saw on Tuesday it's a very interesting mass range because there are motivated dark matter candidates from the hierarchy problem they get produced with the right density so this is a good target and we estimate the flux for this case and we get 10 to the 5 per unit surface per unit time, OK? So it's a big number it's a big number it's 100,000 dark matter particles hitting one centimeter on Earth per second it is really big now, it is big or not? Looks big, right? Now for comparison, think about the neutrinos coming from the Sun, OK? Neutrinos produce through nuclear reactions in the Sun we get the flux of those neutrinos which is if I remember the number correctly 65 billion, OK? So it's 10 to the 10 and detecting those neutrinos was a huge pain it was really, really difficult so we were not, probably none of us was born around some of us but so this is a success that we finally managed to so I wasn't around those days but I'm sure it was extremely difficult to detect these neutrinos from the Sun because it's true that the flux is a big number but the rate is flux times cross section and if something is weakly coupled you need really large flux to detect that, OK? So these numbers are just to say that direct detection is an extremely tricky business because we are really looking for something with a very small probability to happen, OK? Because the full rate is flux time cross section, interaction strength and it's just a very small number, OK? So that's why all the direct detection experiments they are never in this room or at the level of the Earth's surface they go deep underground like there is a lab here in Italy at Gran Sasso and there are many others in the world you have to go really deep inside the mountains where you reduce the background because of course if I put this bottle here and I look for this bottle to move there are not only dark matter particles making that move there are other things like cosmic rays and so on so we want to reduce the background as much as possible and that's why we need to go into some environment where there is no much dark matter around OK very good so this is one reason why it's difficult it's because it's something very rare there is another reason why it's difficult so it's difficult twice not only because the events are very rare but because each single event is very hard to detect, OK? The reason why the event is hard to detect is that of course we don't see the dark matter particle we don't see the effects of this dark matter we see these things, this bottle or this target moving so what we measure is the recoil energy of this object and so let's try to make an estimate of what the typical recoil energy is for this process, OK? so the kinematics is this so before with a given momentum which is the mass times the velocity I can use the non relativistic expression because V is small so P is just MV, OK? and you have a nuclear target here which is at rest, OK? so if this collision happens then you see the vector after the collision both chi and the nucleus are moving, OK? and of course we don't see chi moving chi is dark matter is invisible, it just escapes the detector what we see is this nucleon moving and so we can measure through some very sophisticated experimental setup it's not an easy thing to measure we can measure the recoil energy, OK? so direct detection experiments measure the recoil energy of the of the target nucleus so we can try to make an estimate of how big the typical recoil energy is for these processes and see that it's actually very small so not only you have very few events but each event is extremely difficult to to detect so let me use this so I will leave this, I will not erase this blackboard so we will always keep in mind that this is the topic for today but let's go through the kinematics, OK? so this is a very simple problem in classical mechanics it's a binary collision of two non relativistic objects it's something that you can use by imposing conservation of momentum and energy so let's without putting all the factor of two in the right place, let's just get a feeling of what is the typical order of magnitude so the momentum transfer which is just the final momentum of chi minus the initial momentum of chi is defined, I define it as q and the size of q, the typical size of q is let me make sure I yes, so it's this is just a straightforward kinematics so as you may remember there is the reduced mass of the two body system that enters the equation and the typical momentum transfer is the mass times the relative velocity, the relative velocity is V because the target is at rest at the beginning, OK? so the typical recoil energy for the nucleus is, well, it's M sorry, it's the momentum over twice the mass why? because the initial momentum of the nucleus was zero so all the momentum transfer went to the nucleus just by conservation and I know that in the non relativistic regime the energy is momentum square over twice the mass OK, so I substitute the first equation into the second equation so this is the typical size of the recoil energy, now keep in mind that V chi is ten to the minus three V chi square is ten to the minus six OK, so typically matter particle and the target have mass in between ten 100 gV so this factor within the parenthesis of the order of ten gV, ten 100 gV you multiply that by ten to the minus six so the typical recoil energy it depends of course on M chi and on M n but you can convince yourself that it's in this range here OK? and so these are very very small energies to detect and that's the second reason why direct detection is difficult so two reasons why it is difficult and when we see the experimental results we will be that it's amazing how much progress they made over the last 30 years and how much regional parameter space they were able to exclude so before I do that let me just give you an expression OK, so there is another important quantity that plays a role in the experimental setup which is the threshold recoil energy for each experiment so each experiment can measure a recoil energy up to some value, below that value it will be just too small, OK, so you lose the ability of reconstructing that recoil energy so I call this ET OK? T stands for threshold, it's a threshold below which my experiment doesn't work anymore so now you see that let me rewrite this very important equation so there is a minimum velocity that the dark matter particle needs to have in order to observe an event OK, if V gets too small then ER becomes too small and ER gets below this threshold value OK, so I say this because I gave you this 10 to the minus 3 number but actually dark matter particles in our galaxy they have a distribution, OK and this distribution is a Gaussian some actual, well to a very good approximation but there may be corrections but for now let's assume it's a Gaussian this V0 is precisely 10 to the minus 3 the speed of light OK, so once I compute the actual rate I have to take an average over all the possible incoming velocity of the dark matter and this distribution holds as long as the velocity is not big enough that the particle can escape from the galaxy, so the escape velocity is something you can estimate the same way you estimate the one for the earth OK, is there any question? Yeah, yes, yes this one this is 0, the first one is for non-activistic particle, yes, but I mean you should just think that this is not an equilibrium, not in the sense that you reach thermal equilibrium, just that you get a distribution in the velocity that it's a Gaussian distribution with a typical value V0, OK, other questions? OK, so I wanted to say that because let's compute the minimum velocity that you need in order to get a rate OK, so the minimum velocity I just solve this equation by imposing that the recoil energy is equal to the threshold energy OK, so it's 2 mn over mu square e threshold to the one-half OK, and now I want to write down this equation in two, for two different cases because here there is the reduced mass so if you have a two-body system where the mass are very different from each other the reduced mass is basically the mass of the lightest object OK, so let's take the case where the mass of chi is much, much less than the mass of the nucleon then this equation becomes 2 mn m chi square Et, sorry, and here instead you get 2 Et over mn OK, so square root, thank you, yes, yes, thank you, thank you OK, so let's just go through the logic again we see that if a dark matter particle arrives with a given V chi there is a typical recoil energy that is produced the smaller V chi the smaller ER will be and there is a point where we eat the threshold which is typical of the experiment, if V chi is too small then we don't produce much now the reason why I emphasize these two cases is you see that V min gets large for small m chi OK because you see that from the first equation when m chi is small the reduced mass is basically the same as m chi and then you pay a huge price and this is bad because the escape velocity for our galaxy is more or less 600 kilometers per second but we made that the same way you do for the escape velocity if I throw this shock on the air to just escape the gravitational field of the earth so this means that our experiment will be effective to test dark matter particle up to some mass below which we lose sensitivity because the velocity the minimum velocity I need for that specific mass is going to be either bigger on the escape or you start to pay this exponential suppression so you lose sensitivity very quickly so for small mx we see that we don't do very well what about large mx m chi so I call it chi, not x for large m chi so it's very useful to write an equation for the rate so now I will explain what I mean with this equation but let me just write that first so we saw yesterday that an interaction rate is typically n s v how many interactions per second so this is n, this is s and then I take an average over all the possible v with the distribution f that is here so this is just an average of n s v to give us the rate per unit recoil energy and of course I have to plug the cross section per unit energy now let me rewrite this equation a different way as we already said we don't know n but we know rho we know rho very well around the solar system which is where we perform the experiment so I replace n with rho over m and once I do that I see immediately that if m gets large then I get less events in other words I'm paying a price because a bigger mass corresponds to a smaller number density around the galaxy and so if the mass gets very large eventually I get no events so we see that we have two bounds on the mass below which the experiment doesn't have enough sensitivity because we require a minimum velocity that's too large and we also lose sensitivity at high masses because high masses correspond to low number density so we don't have particles coming here at any rate that would work so now let's go and see what is the typical output of a data detection experiment put that here, skipping many steps and a lot of experimental good work if you have a dark weather model where the relevant parameter space is the mass which I put on the horizontal axis and the cross section to scatter of some nucleon let me put also some scale here so these are the typical scales that these experiments probe for the mass so above the GV and by the time you get to 10 TV you see the limit gets very weak so let's stop there and actually I have a real plot here that was public only a few weeks ago and this is the current best limit data detection 45 ok so these are all small numbers and I emphasize that the dimension that they used to plot are cm2 so when you read this plot and you say 10 to the minus 46 that's 10 to the minus 46 cm2 for the cross section and if you look at the experimental bound that comes from an experiment that is called Xenon 1 ton so it's easy to remember because Xenon is the target material they use and 1 ton is how big the target is and this is an experimental Gran Sasso and if you look at the bound you see that something like this ok so this is the experimental bound whenever you see a line on a plot like this it means that you are not allowed to live on one side of the line and of course in this case means that whatever is here is excluded and that makes sense because you exclude cross section bigger than some given value you want the cross section to be small enough because they have observed no events that look like dark matter ok so few comments you see that the rise here is very sharp and you lose sensitivity very quickly because of this effect I was saying here so once we mean gets very very small then your experiment is not going to perform well for that mass range so when the mass gets around 1gV the details depend on the experiment the overall scale is always around 1gV ok so a 1gV you lose sensitivity and then your bound it goes almost along a vertical line ok below that mass you cannot say much so here it's also easy to understand why the bound gets worse because this corresponds to if you measure the slope of this line it's like 1 over m chi and this is the flux factor that I mentioned before here that you pay because if you consider larger particles masses then you have less number density so you have less incoming flux and your limit gets weaker ok ok yes yes so it is model dependent of course it is model dependent so it doesn't matter because it's a non relativistic cross section so Fermin or Boson it doesn't matter in this case but no but that's a good comment because of course if you have a model when you know the theory and you can compute cross sections as a theory you can actually give you can make a plot of this quantity ok you say ok I know the rho I know v more or less I compute the sigma dr with my Lagrangian and the tools I have and then I compare that with data now for this plot there is an assumption and the assumption is that sigma is the so-called spin independent ok so there are it means two things first the cross section is taken in the limit when q goes to 0 the momentum transfer and that is a constant that approach so it doesn't vanish in that limit and then there is the assumption that the dark matter particle interacts coherently with the nucleus and not just with a single nucleon and so the cross section scales so the cross section to scatter off the full nucleus is the one for a nucleon times the number of nucleon that are in the target squared because you have to square the amplitude so there are some assumptions but this plot is valid for this class of theories spin giving spin independent cross section there are other cases like spin dependent where the dark matter particle in that case is a fermion and interacts with the spin of the nucleon and that's another expression and the plot looks different so there are several cases but this is if you want the case where the cross section predicted is the largest possible one and so these are the best limits available so just some questions yes in that region I would say something at the end if there is time there are new ideas using different targets not nuclei but the experiments dark detection experiments the old way the ones suggested in the 80s they cannot say much now below the GV there has been some improvement there is this experiment called super CDMS that they can really get to a very small threshold pre-coil energy so their bound goes like this it gets worse but not too much but I think it's fair to say that it's a territory that is going to be explored in the next 5-10 years and there are many ideas to do that yeah okay so there are it's very easy if you play with these things for a while then you can always find way to hide so the most famous example is actually the a Majorana fermion interacting with the standard model Z boson so the process you have is this you have chi, chi then you have Z and then you have the nucleus here and if this is a Majorana fermion this vertex can only be the axial vector current because you apply the self-conjugation property of Majorana fields and you see that the vector current is too vanish and this cross section is spin dependent not independent so you don't gain the da2 factor here so you all interact with one nucleon not with the full nucleus and then the bound for that type of particle is quite weak there are other cases where sigma is proportional to V squared and we know that V is small so that's even more suppressed so it is a model dependent so there is a broad class of models where you this plot is valid but there are models where this is not okay wow it's already 240 okay so let's conclude this direction with this plot with two comments I'm sure you may have heard there is a neutrino floor which is something like this so there is a irreducible background to these experiments that I don't even know if it's correct to call background because it looks like a signal okay neutrinos coming from the sun, atmospheric neutrinos and other neutrinos they give an experimental signal identical to the one of a dark matter particle so by the time the bound will get here then we will see something but we don't have something it's standard model physics so when you see this plot there is also a shaded region here which is the neutrino background region but we are still ah they didn't put that here okay we are still few orders of mind to the way so there is reason to push this bound here and that's something that you will happen with conventional well conventional with the same techniques just increasing the amount of material in the target and increasing the exposure time and so this bound will be improved here when the same bound improved it means either we improve the bound or we see something and we discover something as was mentioned before there is all this region here GV below the GV and this is there is some bound there but there are more ideas than bound which is good because there are ideas for new experiments to be built that were put forward in the last five years so it's a very active field also with a substantial contribution from theories too not only experimentalist and there were a lot of ideas again something very recent and some of these experiments will be built and we will also explore this window with the same caveat in mind that in the sub MEV region it's tricky because we know BBN right so having something below the MEV scale that is interacting with us strongly enough to give us a signal to be in thermal equilibrium in the early years so we will spoil BBN and so let's say that I see two interesting region MEV to GV where there is really nothing to worry about there can be particles living there with a good cross section we see that sub MEV we need to be more creative with the models because there are these BBN bounds but there can still be models that will be tested so yes this is not the sigma it's just the bound on sigma so it's not a result from field theory that I do the calculation and I find that it goes like 1 over M I'm just saying that if you look at the plot and you want to find an equation for this line it's sigma is a constant over M because you lose the incoming flux by increasing the mass so I will what is the boss? I have to finish at 315 so let's do indirect detection then if there is time I'll say something about collider otherwise I'll just do that so before I start indirect detection if there is more question about direct detection think about that while I clean the blackboard ok so if there is no other question let's move let's look now at this diagram from left to right ok so we want to detect something like this kai kai finding each other in the universe today annihilating and giving rise to standard model final states so whatever I say in the last half an hour will be true for annihilations kai kai going to SMSM but it's actually true also for decay so if you have as we discussed I think already before dark matter particles do not have to be absolutely stable they can be metal stable they can be stable until today but with a long lifetime long enough to be around but also to give us some of these events here so what I say here is valid for both ok and how do we do that well we need to point our telescopes that can be either telescope on earth or on satellites spinning around the earth towards regions where there is a high density of dark matter ok because we want this process to happen so we need to look at places where there is dark matter and moreover this place this environment where there is a lot of matter they should not be too far away from us because if they are far away from us then we lose in terms of the flux because if they emit if this process happens and they emit isotropically around the environment then we lose one over our square one over the distance ok so they have to be rich of dark matter particles and not too far away so there are two main targets one is the one is the center of the Milky Way so it's very it's very close because we are in the Milky Way so we don't pay a huge price in terms of distance and we know there is dark matter there ok so that's a good one and the other one is to look at dwarf galaxies that are not too far away from our galaxy and these are good too because they are small but they are rich of dark matter and they don't have too much background as I will discuss in a second ok so these are the places where these reactions can happen but then what we detect is here on earth ok so what do we detect so what is SM in this case it cannot be something like a muon if you know what it is so a muon is a particle like the electron just heavier 200 times heavier than the electron with the same properties but it's unstable so even if the dark matter goes to mu plus mu minus then the muon eventually decays and it decays before you make it to the earth where we detect that so SM here really means stable standard matter particles so I gave the example of the muon but it's true for any other unstable somewhere so if you go to a couple of X bosons for example in this annihilation and then again the X boson is unstable so you get the decay products here ok so stable some other particles and there is not many of them ok there is not many why so we can make a list of the most popular targets so let's see neutral so we can look for something that carries electric charge like electrons positrons protons antiprotons we can look and detect these objects here but then we can also look at something neutral like photons or neutrinos ok and there is a why well it depends so there are advantages and disadvantages about both ok so if you look for something charged for example there is the advantage that E plus is antimatter the antiproton is of course antimatter and we know that there is there are no many well known mechanisms to produce antimatter in the universe at the very high energy so if you observe something which is positrons or antiprotons at very high energy it's something we don't know and it can be very likely to come from annihilation like this where the energy scale is set by the mass of these particles or by the decay ok so this is what is good about charged stuff the bad thing is that charged stuff not propagate straight because they get deflected in magnetic field of the galaxy so once we detect them here we don't know where they came from ok if you have an positron being created somewhere in the galaxy it moves as a consequence of the magnetic field and it hits our detector with a direction that in principle has nothing to do with the incoming direction and the other thing is that they lose energy so once they get here we don't know the energy they were produced with ok so there are good and bad things so today since I had to make a choice because the amount of time I have is finite I will discuss one specific final state which are photons ok so let's discuss photons in more details but I want you to emphasize that when people speak about indirect detection it doesn't mean only photons even though I'm going to discuss only photons today it means positrons it means antiprotons it means neutrinos and so if you want to know more about the others you can ask me of course ok photons so first of all dark matter we say that it doesn't interact too much with photons that's something we discussed in the first lecture and also in the second also in the Q&A session it was brought up this issue so how do we produce photons from dark matter annihilations ok there are several ways so let's make a distinction between primary and secondary photons so primary photons are photons produced by processes like this so the dark matter particle can go to two photons ok it may be small the rate but there could be a rate for example if you we know that the dark matter doesn't interact too strongly with the photon but the dark matter can interact with something that is charged and then you go to the next next to the leading order perturbation theory for example you compute one loop diagram if you know what it is you get two photons in the final state ok so this is definitely possible even if the dark matter doesn't have an electric charge that is big ok just perturbation theory you go to the next order then there is also the possibility of producing photons in this way you have I don't know what can we do W ok so this is one example the W the photon is the mediator of one of the mediators of weak interactions and so the dark matter can talk with the W with no problem it doesn't mean it's charged but then these W's are unstable they evolved they decay and you can get photons from final state radiation because it's true that the dark matter is not charged but if dark matter annihilation produce something charged then these final state particles they can radiate photons this is just Bremstra-Lungov of the W's that are produced with the very high energy ok moreover we know that the W also has Hadron decay modes so it can decay to stuff a nice color charge and in this chain you can produce neutral pions pi zeros and we know that the main decay mode of the pi zero is to gamma gamma ok so you see that there are multiple ways that from a process well you either go to two photons straight and that's very easy or even though you don't go to a photon pair you go to W plus W minus then you follow the evolution of these two and you get photons in the final state these are called primary photons because they are produced basically at the same location of dark matter annihilation ok now there are also what are called secondary photons so why secondary because there is nothing to do in principle with the production at the annihilation site ok but imagine that you have kai-kai annihilating to something and this something gives you electron or positrons ok then what happens is that these electron and positrons they travel through the galaxy and they can excite photons through inverse compton these can be CMB photons, photons from stars but if you produce electrons these are very energetic because the energy here is it could be hundreds of gv so these could be photons with a very big kinetic energy and they may hit CMB photons, photons from stars and then you get photons that are at the energy much higher than the corresponding energy before like CMB photons are 10 to the minus 4 electron volts ok and also there is synchrotron radiation because we know that there is a magnetic field in our galaxy so if the electron propagates in the magnetic field it radiates photons and if you plug the number both for the electron energy and the typical size of the magnetic field in our galaxy which I think is microgauss more or less you expect photons in the radio band of the electromagnetic spectrum ok so why I made this distinction I made this distinction because from a single particle kai let's say kai 100 gv we have ways to look at the sky and add photons in different range of the electromagnetic spectrum because primary photons are produced at energy very close to the mass but these secondary photons like either inverse, counter or synchrotron they correspond to lower energy ok and so we look at the sky with different experiments we look for photons and we can also try to correlate if we see a signal or not ok so it's clear the idea more or less ok so now let's do something analogous to what we did with direct detection let's see how we estimate the flux how we estimate the number of photons we expect to see on Earth and how we derive bounds the same way a plot similar to the one I drew before ok so how many photons do I expect so let's make the estimate for one specific case so we can give names to all the particles and we know what to call what ok so I have this example here so we have kai kai that goes to tau plus tau minus so tau is a lepton of the third family so it's a heavier causing of the electron it's 9gb it's heavy and other than the mass it's identical to the electron and our goal is to estimate this number so I have a model where my dark matter particle annihilates two tau's I want to estimate how many photons I produce for example if I look at some dwarf galaxy I look at the dwarf galaxy I want to know how many photons are supposed to come to Earth because of this interaction here the way it works is the following so we need to specify the cross-section ok so this process will have an associated cross-section which will tell us how big the interaction strength is giving rise to this annihilation and then I write down the equation then I will explain what I mean ok so let's go through this equation and this is a two so numerical factors are not important I just want to explain the different things ok so the flux of photons is given by and remember as I said before that charged stuff gets deflected but photons do not get deflected so when we see a photon coming from there we know it was produced there it was the guys produced and it traveled straight to us so we look at this dwarf galaxy and we integrate over the fraction of solid angle that we observe the galaxy and this is important because of course if this is me here ok and this is a dwarf galaxy so when I look at the dwarf galaxy with my eye I'm only observing a given fraction of the solid angle but photons get produced in all the directions in the dwarf galaxy so it's unavoidable that I pay a price which is the whole solid angle over 4 pi so that's easy to understand it just means that the galaxy emits photons isotropically we only detect the ones that are produced toward us ok this is also easy to understand because it's the integral along the line of sight this is the line of sight so I look toward the galaxy of what? n squared times sigma v why n squared? because they need two chi's to have an annihilation I need a chi here, a chi here and then they collide with probability given by the cross section so that's why there is this combination n chi squared sigma v and chi is the density, the number density as a comparison if instead of the annihilation I'm considering a decay so if I'm considering this process here so my dark matter is metastable and then decays to tau plus time minus then here instead of n chi sigma v the product of the first power of n chi times the decay width of chi gamma why? because I don't need two particles anymore to produce tau, I just need one and wait that it decays, that's gamma that means how many I have in a single volume and finally this is a product that tells us how many photons with the given energy I produce per single annihilation so imagine you have one single annihilation chi chi going to tau plus time minus and you want to know for that single given annihilation how many photons you produce with respect to the energy so differential in the energy and this is something by the way that you can compute by using well established techniques in particle physics because this is something you can test at colliders and all you have to do is to produce two tau's with an energy in the center of mass as given by twice the dark matter mass and follow the evolution so there is a very nice way to rewrite this equation I will rewrite in that way and then we will get to the experimental bounds ok so I didn't do much, all I did I took this equation and instead of n I replace n as I did all the time in the last week rho over m, number density is mass density over the mass. Once I do this then I collect this j factor that is defined here and I write the rate as a product of multiple factors this is a nice decomposition because let me write it here ok so 1 over 8 pi is just a number and there is not much to say ok, j, the j factor so the j factor is something that depends on the target only ok because you see by definition you are just integrating over the line of side on the solid angle of the square of the dark matter density so this quantity knows nothing about your dark matter model if it goes to tau or if it goes to W's it's just something that it depends on the target only if you look at the dwarf galaxy and you must know that there is an associated j factor with that target yes, the cross section is no it's not in the integral the cross section is, I'm taking it as a constant here yeah no, it's outside, it's outside so this is, you are just integrating over matter density along the line of side so the cross section is outside now the second piece is this and this is really where I have to specify what dark matter model I am considering so if I am writing a paper on a given dark matter model and I want to see what is the flux of photons in that model, well j I look for a reference and these things are known they've been observed and it's just something that doesn't know anything about my model but once I write down my model then sigma v and m chi these are parameters of my model I can compute sigma v, I can choose a mass so this is a model dependent quantity finally, so this is something that it doesn't depend on the target and it doesn't depend on the model this is something that is I would say this is standard model physics so it's something we know very well not only how to describe theoretically but we know it works because we tested this at collider for many many years and this is just a quantity that describes given the single annihilation into tau and given a dark matter mass which here enters just as the energy the ambient mass of the tau plus tau minus pair what is the spectrum of photons that I get after I let tau plus time minus evolve so that's why it's standard model physics because this is something that can happen in the vacuum it can happen at the LAC, it can happen I don't need the galaxy to describe this so this is standard model physics so this is a nice way to factorize the rate into three contributions very different nature and if you are an observational astronomer you care about the J factor and this is a very important job because there are still uncertainties on the J factor of dwarf galaxies and they can be improved if you are a dark matter model builder you specify sigma v and m chi and this is something that I think by now we can take from Monte Carlo codes that generate evolution of tau plus and minus pairs at the given center of mass energy and we can just use that with a lot of confidence ok ok so now let's look at the limits so now I have a plot from the where is the echo yes ok so this is a plot from the Fermi collaboration so the Fermi satellite is satellite that was launched I don't know more or less ten years ago so this result is the outcome of the observation of a set of dwarf galaxies for which the J factor was known they specified this annihilation channel that I mentioned before so this plot is valid for tau plus and minus and these were the two parameters that they use to bound the system because they say ok we predict how many photons we expect to see from this dwarf galaxies using this equation we count the number of photons we impose that the number we see cannot be accounted for so this is an upper limit on the cross section otherwise you would see more you would predict more than what you actually saw and then you put bounds on this parameter space so let's go through the details this is sigma v here this is m chi now sigma v they use these units centimeter cube over second so sigma is a cross section v is a velocity so the dimension looks right but if you remember yes two days ago the wind miracle let me call it thermal there was this cross section that was the magic number to get the rally density I express this in natural units but you can also explain that with these other units so the number we have to face once we look at the bound is this magic number this is the number of the wind miracle I just translated that from natural units to units where I restore the speed of light okay so NH bar so let's see the plot again what are the ranges of the mass so this is 10g let's just say mass in gv 10 100 so again we are exploring a range of mass very similar to the one from that actual experiment that we just discussed and on the vertical axis here we have what are the typical numbers the typical numbers are 10 to the minus 27 10 to the minus 26 25 24 so we can immediately identify on this plot the magic so this is the one so in some sense and I remind you this result was very independent on the dark matter mass we saw on Tuesday that regardless on the mass up to some very tiny logarithm dependence if the annihilation cross-section is equal to this value then you get a successful production in the early universe so this is an interesting benchmark for experimental observations because in some sense you expect this number to be the one of your model as I say there are caveats to this number as I discussed there is an assumption about knowing the history of the universe all the way to very high temperature so that doesn't have to be the case but it's an interesting benchmark so now let's go to the result and then I will stop 27 ok so this is the result from the Fermi collaboration ok and again as I said before this region is excluded so we see already something interesting here that if the dark matter mass is below more or less 100 gV thermal production seems to be excluded for this channel by the way Fermi collaboration in their paper released this plot for many other final states not just out plus time ions and they all look very similar ok there is not really a sharp difference there are there is some difference but more or less the messages is the same ok so it is very interesting that we managed to test the thermal up to masses of 100 gV of course this region is still viable and the fact that this region is excluded doesn't mean that these models cannot exist with a successful production rate because as I say this line is a big caveat it was derived under the assumption that Hubble is the one we extrapolate from the thing so this ends the review on the protection what I couldn't cover today was the collider researchers so you can ask me in private if you want I would be more than happy to say a few things about that and tomorrow in the last lecture we will discuss the QCD action that is a dark matter candidate that we never discussed until now thank you