And so that question is not something completely new in terms of looking at capture of dark matter by gravitational objects or by compact objects. This again.
Move this until I find something. No. No, I think that I'm okay, I'm gonna do this, because otherwise see this
And then
Had to go back here. Hopefully. Okay. All right. So
So, back in the 80s, people were looking at the possibility that the Earth could capture dark matter. Of course, when the wave came and everyone hoped that there was going to be the, I mean, 100 GV, that's going to be dark matter and interacts quickly with the standard model. Let's see if the Earth can capture
This type of dark matter. So gold and spiral, they in those years they proposed that this could be possible. They wrote a formalism to do it, but they assumed that dark matter would be captured, could be captured after just one collision. So you have
That the Earth is being showered with black matter, and then every now and then these particles will collide with some nuclear nucleus in the Earth, lose the kinetic energy, and then get the gravitationally bound to Earth. And then after a while, you have accumulated some of this. So you have
And that's still something that is considered in the detection experiments, and so it's still quite important. Of course, that was assuming that dark matter would be captured only after one scattering with a target.
More realistically, you would expect that it takes more, especially as you go to heavier and heavier dark matter, you would expect that it takes more than just one collision to get rid of the kinetic energy in the incoming dark matter particle. So in the in recent years, there has been quite a lot of work
on how to calculate the capture of dark matter after multiple collisions.
Now
All these works and most works that study multiple collisions to capture dark matter, they still work with the assumptions that gold has had in the early paper. That is
This is more like a toy model. So you assume that dark matter actually goes through the compact object or through the star or through the planet in a straight path
You assume that the object that you're talking about is homogeneous, so that there is no variations of the density, for example, or the temperature across the across the system. And then that
that the scale velocity, which is important because that is precisely what is going to tell us if the object is gravitationally bound or not. If you're… if the velocity of the particle is already less… after losing some kinetic energy, you have that the velocity is less than the scale velocity of the
The object where you are, then you will get captured. So most of these papers assume that the scale velocity is constant and it's actually that is the value on the radius of the object
Which is already… so this works in terms of estimating the effect, but of course, these are not very realistic assumptions.
So more recently, my collaborators and and a larger group of people, they started developing a formalism where you would have multiple scatterings also with multiple targets with different types of elements. But but in a in a case where you lift all these assumptions, basically where you take the radial dependence
of everything. Of course, the downside of this is that calculations become more computationally heavy, and it becomes more complex, but it's more realistic. So, I mean, if we're going to take seriously this way of probing dark matter through capture
In stellar objects, or in compact objects as well, then we better develop the best formalism that we can. So in that case, in their work, they basically
Enclose the probability that a particle will lose a certain amount of energy that would scatter, and that would
would travel in different… in a curved path, they put all this into something that we call the response function. This response function also considers the gravitational focusing. So it's not straight path. As you start colliding, you are changing your path
And then the fact that the density and the scale velocity are actually functions of the radial position inside the star
So this is the formalism that I'm going to present. I'll do it in a couple of slides, but this is precisely the context in which I started working on this right? So now why Pub 3 stars?
So you can interrupt me at any point. I mean, I know that in our field we do it, so please interrupt me. I'll be happy to comment. Yes? I think you're going to tell me all this. I would have thought all those
Factors that just mentioned were relatively small
Changes in the cancer. Right. So some of them are. So that's precisely what we wanted to check. Yeah. Some of them are, but some of them for us, if it is at least one of the magnitude is already important.
So some of them are less than another magnitude, factor of 2, factor of 3, but some of them are already one order of magnitude.
That's a bigger change than I would have if
Okay. Great to see. All right, so pottery stars. These are very old stats. We believe that they are the first type of stars that formed at redshifts of the order of 20. When they formed, they basically were just hydrogen and a bit of helium
And it is believed by some astronomers that this is precisely how you explain the existence of the rest of the metals in the universe. And they tend to form in smallish halos, so 10 to the 5, 10 to the six solar masses
And then there have been more and more mentioned in the news or in the internet nowadays because with the JWST entering the collection of data, then one of the expectations is that it should be able to observe these type of stats
And indeed, there have been already a few results where people think, well, maybe this thing that the AWST is looking at or that it caught in some region, it is about to restart just recently, two different groups, they reported that
close to GNZ11, that this is one of the most luminous in the UV sector of the spectrum. Galaxies that are very, very early, so
And a ratio of 10.6, I think. So close to this object, they saw something that is emitting a lot of helium or ionized helium, which is a very typical emission that they expect from something that has a lot of
like primitive hydrogen or primogenous hydrogen and and some helium. So some people, there are some things that they still don't convince the community that that this is a poultry style, but they they start seeing more and more objects like this
So that's supposed to be a star or a galaxy of sorts. So this is a galaxy. And then the thing you call this is not they haven't they haven't been able to tell if it is exactly a star, but there is something that emits like
The mission that it has, apparently, is the type of mission that you would expect from a Pop 3 star. So it's localized. Yes, it's localized. Yes.
Yeah, that's but it's all sitting on some like incredibly sitting on some cusp of yes, yes, yeah, because yeah, yes.
So okay, so this has brought more and more of our attention to to pot three stars. One thing that we realized once we started working with this, as I will explain in the next slide, is that these stars actually when we think about them as
Once you have the star, just with hydrogen and helium, it doesn't stay like that. It actually evolves quickly. And it actually creates metals. So the fact that they can evolve quickly to actually go from metal free to having some
a trivial metallicity, then give some some avenue for astronomers to try to connect metal free and high metallicity stars and explain how that's the regular stars get
They become as metallic as we observe them. And then
And then as whenever there is an interesting phenomenon like that, then we can start asking, well, and then we probe dark matter since this is so early, can we probe dark matter with this? And there have been some works in similar lines, although with a different approach
So if we want to study poetry stars in, and this is one of the main reasons why I wanted to do this, because I have never even heard about Mesa until I started hearing about this. So I wanted to start to learn about Mesa. So we decided, so Sandra actually is the expert on this
So we started, I started learning about how to simulate the evolution of a star in Mesa. And then so what we did was to take this module that basically astrophysics used to simulate stellar evolution, and we simulated
Single stars with 3 different specific masses with a boundary conditions that basically they form at z equals 20.
I'm sorry, at z equals 10, and then that the metallicity is 0. So that the initial protest star basically as it enters the main sequence has pretty much no metals, just hydrogen and helium. And then we evolve them until they have reached the point where it depletes
Completely the helium, or it enters the red giant phase.
And then so. So starts this semi don't collapse.
If the black holes would cost, they're made out of helium? I mean, as long as they haven't reached the red giant phase, they still have enough pressure to be stable. Yes, yes. Yeah, after the… after the red giant phase, yeah, depending on the mass, yeah, they will
Collapse into something
All right, so these are some screenshots in time of what we obtained. So these are the three different masses, and here I'm plotting the number density of the elements versus this is the profile, the radius of the star. And so we… I just chose 3 different times. This is the latest, this is the earliest, and these are sometimes in between
Just to show some general features of the evolution. So indeed, when it enters the main sequence, this is LOH main sequence, then indeed you see that we have basically blue is hydrogen. And what is that purple is helium
And because, I mean, it's… the geometry is really just like a sphere with all these elements mixed in, right? As the star evolves, at some point after only 7.9 mega years
Then the star actually has some metals in it. So you see here, for example, for this case, we have some carbon, we have some oxygen and neon. But not only that, you will start seeing that the geometry also changes. Now here we have something that has a core. You see how the profile
of the metals, basically ends say 10 to the 5 kilometers for this example. And then there is an envelope, some sort of atmosphere with hydrogen and helium.
So this for us was very interesting because then we started thinking, okay, how is this going to show up in capture? Because now we have not only more than just two elements, but also we have a different distribution of these elements in the object that we
that we want to capture. So I mean, if anything, we knew that the code was going to be more cumbersome. But but we were curious about how this would show up in our process. Okay, sorry. No, no, sorry.
The oxygen and the heavier elements are on the core. Right, so yeah, the heavy, so this is the core where basically you have here, for example, oxygen, carbon, and nil
And then in the envelope, you have the lighter ones, helium. Having never actually done this, I'm surprised that those ratios are that high.
All right, I didn't have an expectation. All right, so now the game that we want to play is basically at different times of the evolution. We want to compute the capture
And see how the capture evolves as the study evolves. Now, at this point, and for the rest of our work as of now, now we're working on the follow up. But at this point, we will not include so this is the evolution, and we are going to assume that that is the evolution. We are not
Like, we don't have any feedback from the capture of dark matter that's being captured into into this evolution in terms of injecting any new heat or anything else, or even changing the gravitational potential that we will look at in a different way
But but yeah, but so this is the evolution of the star. We are going to assume that this is not modified for now. That is not modified by the dark matter that's being captured. Yes, I'm just the actual density is continuous
Exactly. Yes, yeah. That just moves out over time
Yes, yes, yeah
So this all happens continuously without any explosions. Exactly. Yes. Yeah. At least until here is where we stop because this is where most of the helium has been depleted. That's where methane stops
And then after that is that something can happen, something drastic can happen, equalization like stars or something.
Yeah, I was actually Yeah, so yeah, so like payment stability, for example, that's something. But for us to study that, we have to use the feedback of the data into MESA, which we haven't done, but we are working
Junet doesn't have, like, a mesa well yes but it is okay yeah I'll say things about that. Okay, okay, okay
Of course, yeah, there are ways that you can claim that you can do that. Okay, so all right, so of course notice that we're going to have to use two different regimes. The capture in the early time is going to be very different to the capture in the late time just because
Clearly, this is a very different star than that one. So we're going to split that into parts
Okay? Now.
So this is what we want to study, we're going to study using multi-scattering with multiple targets. As you saw in the first profile, we have two different targets, helium and hydrogen. So this story of multi-scattering has to also be found not only multiple scatterings with dark matter with a single type of element
But also with another type of element. And you have 2 options. Either the dark matter is captured until interacting, say, only with hydrogen or after interacting only with the helium. But very likely actually will interact some with some helium and some hydrogen
And we need to include all that in our calculation. So we consider individually helium and hydrogen in the sense of
What if dark matter is captured after interacting mostly with helium and some hydrogen, or mostly with hydrogen and some helium? And then the sum of that, we're going to call it the total capture.
So for a single element or for a single target element, the capture rate, and this is probably something that you have seen before, looks like something like this, where basically there are some factors that have to do with the amount of dark matter that is going
through the object, through the star. So it depends on basically the vicinity, how fast the dark matter is moving, the distribution function for those velocities, and then the size of the star. So you will going to integrate over the radius, so that's what we're going to integrate here. And then there is this other factor that should go here as well
But this is a constant that I'm going to pull out that is basically what is the number density of the dark matter population in the vicinity of the star. So that basically will provide the dark matter to capture. Now, for each of those particles of dark matter, then you have to account for
What is the probability that one black matter particle gets captured after interacting with the targets? And this is where basically we have to take into account some sort of opacity or optical length. And then what is the probability
That after interacting, then you lose enough genetic energy through multiple collisions in order to get captured. So I'm going to to open up each of these
Each of these terms. And this may be
that would affect the increase in velocity of the dark matter as it pulls it
An increase in the velocity of the dark matter as it falls in.
Yeah, that part is included. And because there are real dependencies. So whenever we call this is really, I mean, later when we doing the integral over this, we actually going to take into account this is the velocity
I think affinity and then we are the part exactly
What do you take from lambda?
Okay, so yeah, so in all the analysis that I will show here, I'm going to assume that dark matter interacts with the nucleons through a scalar coupling. And this actually, this is just a coupling, right? So
So we basically, at the end of the day, the analysis, we don't do it by running Lambda, but just running the… we parametrize everything with the proton dark matter scattering cross section. That implies that these lambdas will
will go to very high, but even for but we stop at 10 to the 13, 10 to the 15 GB for the mass of the dark matter, where basically this Lambda would be too big, and then we will have problems with unitarity and and those things. But basically
Rather than going into the details of what's going on with these couplings, we just parametize everything with the scattering cross-section between the dark matter particle and the proton
Okay, so is it weak scale much weaker than we scale? We will go from wheat scale to much stronger than we scale depending on the mass
Yeah, because we scale too much stronger than we scale
Well, actually, we… no, actually, no, we will go from very weak because we will scan the scattering cross-section from very, very small scattering cross-section we will see later in the plane
To above the weak scale, but also things that are constrained by now by LZ with delay detection.
But as we do all this analysis, we don't think about if we are getting or not into things that are ruled out by delay detection, but then later we will superimpose the delay detection in that
All right. Okay, any other questions?
Before I move on
All right, so let's talk about the probability that dark matter can be captured.
Okay, so we can… so of course we start with what is the probability that the dark matter loses a certain amount of energy after just one collision. And that is not, I mean, that is very easy to model. We just model it as a probability density here. We integrate with the recalling energy
And then… so this gives us the probability that the particle will lose certain energy after one collision. And then we can use some convolution to get an expression for the probability that a dark matter will actually lose certain amount of energy after n collisions. And then we get something that looks like this.
Here we are parameterizing with deltas the energy that the particle is losing, the kinetic energy with respect to some typical energy of the target. So this energy, this is the characteristic energy that will come in the form factor for the different targets later. So this is usually of the order of Mevs
And most of these elements is not the other of a few MEDs. So sorry. So what is the master? Like, I know you haven't said that, but it's massive detriment is comparable to the nuclei. No, it's not going to start at a TV, and we're going to go to 10 to the 15 GB. Okay
So it's always much heavier. Okay. Yes
So like they run the walk, this convolution by coalition is
Similar to random walk or not because there are some that is very surprised that we get this type of factors here because I mean, in some ways
is similar to a random walk, but the particle walk where basically you are always losing some
Yeah, and then for example, with the approach of random work, a lot of the works for multiple collisions before, rather than taking into account that you might lose different amount of energy in every collision, they just assume that you you lose some average amount of energy
Here, actually, you are integrating over the energy and later when we integrate, we will integrate over delta, basically. So we will take into account the fact that you can lose energy at different values at every collision.
All right, and then… so this is… given that you have that particle that has collided N times, what is the probability that you lose certain amount of energy? Now, of course, you have to include the fact that you have to include the probability that the particle does go
Through N collisions. And that one is easy, because, I mean, that one is more like the… so this is a Poisson distribution. Okay, so now if you multiply these two, then for a given value of n, you will have the probability that the particle gets captured after that. Okay? And then what you do is that
You basically add over all possibilities. You can get capture after one collision or after many, many, and then we can we can just consider the sum to the infinity, which actually is very convenient because this gives us a closed form for the sum. It's not modified vessel function
A lot of other works, they actually have to put here some cut but then there is some non-trivial dependence on the cutoff. But we were very surprised when we realized that actually, no, you can get a closed form for this. So this basically gives us the response function
That will tell us what is the probability that a particle undergoes, through collisions through a certain optical depth
So this is basically the opacity, the optical desert opacity through the star, and that you lose certain amount of energy.
Okay, now
Notice that if you were to say that the particle has lost most of its kinetic energy at a given point
Well, there are several ways, there are two ways that you have an elliptical orbit, and that's why I have on purpose left this here. You can get to that place if you use conservation of angular momentum 3 different ways, either through the green or through the purple path. So basically.
At all times, whenever you are thinking about capture at a given radius of the star, you have to consider the different paths. So you will have to have what we do is that we average over the response functions that consider the different
optical depth. So that's why later you will see some some other tilde around. Okay, question. Yes. In the diagram, then, are you just showcasing a scenario where it's captured through single? No, I mean, this is really a cartoon. This is basically… no, here, I must… no, I'm assuming that you have lost
Energy going through certain path
But
with any number of collisions between 1 and infinity. So basically, I'm assuming that you have that basically the probability that you get captured at that given radius is given by this response function. So you take into account all the
You have already… this function will tell you what the probability, assuming that you have collected as… I mean, you know the dark matter has collided as many times as needed
So in this formula, the information about what the cross section was is all in tau
Yes. Yes, sir. Yes.
All right, so okay. So there is that this is assuming that you have just one type of target. Okay, notice that I haven't so of course you have to do this for several targets. So now
What if you have then we have tomorrow, what is the probability that you get captured after as many collisions as you need, but interacting mainly with one target, but then some during some of that optical depth with another target. In that case, we have to model them. So that's what we label this
This 2 means that you are interacting with two targets. The targets are I and J. So you have the probability that so this basically is how we can model the probability that you lose certain amount of energy
set a fraction of this energy by interacting with J, and then the rest of it by interacting with
Okay? And then so this is now the probability that you interact, you get captured after interacting with 2 targets, and then we can add the 2 of them. So now I can say, well, then the total probability to get captured in the presence of two different targets where I mainly interact with target I is going to be given by
The probability that I lose certain amount of energy after interacting with I, but not with J, that is basically this is the probability that I don't interact with J at all.
plus the probability that I interact with both of them. Okay? And then this is the response function that I need, for example, for the early star where I only have 2 targets, and then I can just compute my for each of the targets. I compute my my capture rate, and I add them
This twiddle here means that I am averaging over the different optical depths, and then I have my required cross capture rate. Why is only one species dominate
Well, I actually am assuming that the decisions that both of them will dominate, I mean, that
This is the capture rate, assuming that species I dominates
But then I have to add the capture rate, assuming that the species J will dominate because right. And then at some point, I mean, the chances that they dominate or democratic is going to be very small
All right, so, okay, so now with this basically is a description of the probability to capture it. Now, remember that I said that there are these factors that have to do with how much dark matter is. So this is another thing that for me was very interesting because
Whenever we talked about direct detection, because we are talking about the Earth, we always assume that the dark matter speed, the dark matter velocity distribution function is Maxwell Boltzmann
And I said to them, okay, it's Maxwell Boseman. And actually it's a pretty good assumption for the Earth. I mean, I would say most, if not all, of this capture of dark matter works, they assume that
that anywhere in the galaxy, or anywhere in the halo that you are considering the the distribution of the velocities for the dark matter is going to follow up Maxwell-Boltzmann form like this, of course, you have to change the velocity of the star depending on where you are
But that's pretty much the only thing that… that is the only radial dependent quantity as you change your locations, although you'll be surprised. I've seen some of these works that study starts close to Sagittarius A in the vicinity of the center of the galaxy, and they scale assume
for example, which I mean, I would say that that is not accurate. All right, so
Now, Maxwell Boltzmann assumes that the stars that and I know that most of you work with simulation, so I don't have to tell you a lot of this, but basically you assume that this is an isothermal sphere with particles that are in thermal equilibrium
Okay, there is not a lot of structure to that you need to include to get maximum Boltzmann. And but then if you actually take NFW halo, for example, and then you go to very small radii
Then, actually, the Maxwell-Boltzmann distribution function will lead to an overestimation of the low tail, the small velocity population in the distribution. And this was pointed out before
And but of course, if we are closer to the center of the galaxy, we are interested, or more interested in the small velocity dark matter particles, because those are easier to capture
So if we overestimate those, then we are going to overestimate a lot of things. So because of that, then we actually used what people call the Edington inversion method proposed by Edington
110 years ago, where basically you get the distribution function starting from the potential, the gravitational potential of that contains all the matter contents of the of your halo plus galaxy
and everything this way, basically, because you are integrating over when you're first using things that are integration constants like we did when we were in classical mechanics. And then you are integrated over all possibilities rather than just assuming that everything is an isothermal
a sphere with particles going on circles only. But here you are actually taking into account all the components or all the things that are contributing to your potential. And then so this is the this is the velocity distribution function that we will
They include in or that we included in our work, and that I will… everything that I show from now will be assuming that we use the Erdington distribution function just as a comparison. If we were to have one of those stars that I showed at the beginning
And five kiloparsecs and then you were to compute what is the Maxwell-Boltzmann distribution function of the velocities and what is the one that you wrote in with Eddington, this is what you would get
So as you can see, there is a serious overestimation for low velocities, but also an underestimation for
for high velocities.
And
Okay? And so so for us is more solid to work with the Errington method, and that's why we're going to use for now.
Any questions about this?
Now, I'm not saying that doing… just before I move on, I'm not saying that doing this for the
For, say, for example, in theory detection is wrong, because actually it happens to be that if you do… if you did the internal inversion method for for the sun, for example, the position of 8 kiloparsecs, you would get that they are pretty much the
So there are no differences between using the Maxwell-Boltzmann or the area into distribution functions in the solar vicinity. So that's why it actually works very well for us. So we are outside enough of the galaxy so that they actually the isothermal
A approximation is good enough. But if you go inside, once you have a disk, or once you have a profile that is very cuspy, like the NFW, then it becomes not the best approximation.
Alright. Okay, so now we apply all that to the early start where we only have two targets. And then this is what we obtain. Basically, we have here the capture rate in units of inverse seconds versus the scattering procession between the proton
And the dark matter particle here as this is a specific example for 10 to the 9 Geb. I will show more in a second. But here, I just want to show some features. I have 3 different cases
Our work, basically, the main result of our work is the purple line, where I'm showing the capture rate using the Eddington distribution function, and then we are using… using the same formalism for the capture probabilities with the response function and everything, but using the Maxwell-Boltzmann
And then, just for comparison, I'm using the previous formalism that other people have proposed before, where they use the Maxwell-Boltzmann, but also where they don't take into account these radial dependencies. So this goes to a little bit towards Matt's question at the beginning
So you see that first the previous formalism gives you already an order more than one order of magnitude of an overestimation in the capture rate. But now, if you if you take away those assumptions and you apply the formalism that we had
But the difference between using the 2D distribution functions, actually not that drastic. That is really a factor of 2.
Right, but nevertheless, we're going to stick to the Arrington formalism. Now, here we see some features. We see that for very weak scattering cross-sections, then you get some capture rate, and then as you start increasing the scattering cross-section
The particle will actually or the capture rate will actually saturate to some value. Well, this we call it the geometric capture or the geometric capture limit, where basically it's the limit where you have interaction so strong
That anything that goes to the star will get captured
And that is, of course, limited by the area of the star. So that way, basically here there is no scattering cost, there is no cross section. The cross section is really just the area of the number of dark matter particles captured per second. Yes, yeah, exactly
And I guess
losing energy, but you're
Dallas and Connecticut, I think, yes. Yes, but that is, and so you have to get up to start. Well, I mean, what happens with the capture, we'll talk about that, but so for now, I'm interested in how many particles per second you are capturing, okay, given some scattering cross-section.
Between the
between the dark matter and the nucleus.
Okay, all right. And now, just to show how this depends for all the masses. So if I start increasing the ratio of the capture rate to the geometric limit, so that I can actually show something together. Otherwise, it would
very different. And then so you see that for different masses there is a different scaling of the capture rate, and that difference difference scaling starts at a given mass. In this case, for this specific example
starts at 10 to the 9 Gb. For anything that is less than 10 to the 9 Geb. Actually, the
This ratio of the geometric or the capture rate to the geometric limit is the same for different whole different masses. But that is because if you are in this particular example, under 10 to the 9 Geb. The one collision is enough to to capture the dark matter
And then as you start increasing the mass, of course, it's going to be more difficult. You will need more collisions. You enter the multiple scattering regime after you pass 10 to the 9 GB. That's why the works that focus on wings, for example.
They all have to look at single scattering regime because they are way below the transition mass
Okay.
And I understand the community path that you just cross. Well, yeah, this has to do with the fact that when you integrate here, notice that this integral doesn't go to infinity. This integral goes to the scale velocity of the particle from the galaxy, from the whole halo at a given position, right? Otherwise, because you cannot integrate anything because of course you want to capture something that
infinite velocity. But then, so because you have this, that means that for a given mass, there is a largest amount of kinetic energy that you will have to lose, right? That's going to be given by the mass times this, divided by this squared divided by
Right? And then you can ask, okay, for what is that value compared to the standard like to this in order that I have before, to this characteristic energy of the nucleus that will be basically that will give me the characteristic recoil energy in a single collision
So there will be a mass for which that amount of kinetic energy is going to be less than that typical recoiling energy. So just one collision will be able to take that kinetic energy away. And then if you start increasing the mass, then, of course, you will
make a transition to to needing more collisions, and then that's why you get to the multiple scattering.
Okay.
Alright. So this is for early capture. Now, what about the late start? So this is where things became more interesting, because now you have a geometry that is non-trivial. You have some core with metals, and you have some envelope or atmosphere with helium and hydrogen
So now you and then you have many more targets. Of course, some of them are very subliterating. For example, in this case, for the last stage of the start. In this case, you see that helium in the core is of leading even
Carbon is definitely less than neon and oxygen also the atomic number is different. So you have a factor of the atomic number squared in the scattering cross section
to the 4
So… so then… then in that case, this is even sublidden. So we can… we can assume that we have, let's say, 3 different targets for a single particle going through the star.
And then so we basically have to model our response function. Now we have to have targets i, j, and k, we have the probability that you get if I dominates, then that you interact only with target I or only with target I and J, or i and k, or the ij and k
So this is where mathematica crashes
Because now you have tomorrow
The response function, when you lose so many with one target and then some energy with the other target, and then the rest of it with the third target and so on. But then you write your total probability or response function where you have, I mean, if i dominates
You have the probability that you don't interact with J and K only with i or only with i and j, or only etc. So so this because this became a pain. But we managed to do it. We actually did it for 4 targets, but then we realized that the 4 target wasn't really
Not very much. So it was pointless to use a lot of time in the computer if it wasn't going to give us anything different. This is just to compare. So this is the result for a given mass.
And here we want to compare the difference between considering for this profile the collisions with three targets or naively with two only.
And then, so let's start with three targets. So we see the same form that we had before, except that now this is where the core actually shows up. We see that, okay, so for very weak scattering cross-sections, you have some capture
As you start increasing the cross section, the capture starts increasing. At some point, you have some momentary saturation. And this is because you have reached basically the core geometric limit
That means that now this interaction is strong enough so that anything that goes to the core will get captured. So you get some some mini saturation there. But of course, as you start increasing the scattering cross section, now you start getting captured really even in the entry to the atmosphere, and then at some point you will saturate the whole
Geometric limit
And then you can ask, well, but if you're going to get to this point and saturate the whole star geometric limit, what's the point of using three targets rather than two? Because if you were here at this type of… I mean, notice that this is very strong interactions
But if you were on this regime, notice that if we were to consider only two targets, it would be underestimating the capture, because we're missing a lot of collisions since there is another element there. So you will have that particle that if you don't take into account that, you would think that the particle actually went through and then got out, but maybe actually
interacted with something in the atmosphere or something else in the nucleus. And then so that this is what we wanted to point out that it's important when you have this non-trivial compositions or stars to take into account the fact that you have more than two targets
And this is just to show the similarly to the early star. Then you have here just for one of the cases, and then as you go to mass is heavier than 10 to the 9
You also have the same scaling less than 10 to the 9. You have only one scattering as necessary. And then that case, the features are very similar
Okay, so now that you have captured the dark matter, what do you do with that? Okay, so then this is where we started asking, well, maybe then the particles, if they are interacting with the elements of the star, at some point they will reach some equilibrium
Right? And then the question is, well, so for that, basically, we solve this Boltzmann looking differential equation that basically tells me this is an equation for the energy that is transferred from the dark matter to the nucleus and as they interact
And then for different masses, this is what we obtain. We are carrying it here. This is time, and this is the closed section. We are carrying it here because this is the time when the star for this particular star, when the star enters the giant phase. So all the helium has been depleted. So we we basically
Call that the end of the study, because at that point is where some explosion might happen, or where something drastic. I mean, basically the start gets out of the of the main sequence, but then you see that for different masses, of course, it takes different time, different values of the time
to thermalize, the heavier the star, well, it's going to take longer, and then so even for some scattering cross-sections, where a given value of the mass that is heavy enough, the star won't probably live enough to see all the dark matter that is captured
But then if we are in that case where the dark matter actually thermalizes, then in that case, the dark matter will basically occupy some sphere that we call the isothermal sphere and is described by basically just a matter
Kinematics tool to show that this is the model is the region. And then we can ask and then we can write an equation for the particles in that isothermal sphere. What is the what is the evolution
of these particles where we have to take into account the fact that you are capturing particles, but also
And you have the possibility that these particles might manipulate
We know that you have this particular equilibrium, then you can aniculate into standard model, and then you are going to deplete the number. So basically, this, again, a Boltzmann-looking differential equation that we can solve. And this is where our optimism sort of starts diminishing because
If you allow the dark matter to maniculate, unfortunately, there is not a lot of exciting things that happen in that case because very quickly the amiculation rate will sort of match the capture rate and then the number of particles inside the star, it basically remains almost constant
And it remains constant at a value here, I'm just showing two different values of the cross section. It remains constant at a value where it does nothing to the start, because this is basically about 20 orders of magnet. In this case, it is about 22 orders of magnitude
less than the number of particles in the in the star. So so it wouldn't do much.
But then, of course, and given that then you ask for the case, I will give you more of them is what if we have asymmetric dark matter that cannot annihilate? In that case, of course, we are getting rid of that, this differential equation becomes trivial, and then you actually see that
that you have a non trivia population of these stars, and then now that you have this dark matter particles. Now, at that point, then you can ask, well, what do I do with that? Then, of course, you remember that at some point there will be enough of this dark matter particles. Yes
Oh, sorry, you said that the annihilating dark matter wouldn't
Affect the star in any way with these numbers because this is 10 to the 35. And then in the sun you have something like 10 to the 55 particles. So it really is not adding a lot of
of interesting stuff to the start. Now, that in terms of annicilating, because later I will talk about what happens if you actually include this into MESA. But yeah, in terms of affecting the star by saying
Providing some luminosity or anything, it won't do much. Sorry, Mark.
I mean, each one of these particles tend to 7 Geb. And I guess the I I'm sure that it's going to be way right? So then you are here. Sorry. Yes.
But still less than if I did it in terms of the energy output of the core, it's still I'm going to guess it's still many orders of magnitude lower
Yes, that would be even more interesting figure, but it was really 10 to the 15th GDP
Yes. So the number that I mean, at some point you're saturated by just the amount of dark matter passing right? It doesn't matter right like you can convert all of it into energy.
But that
Yeah, I'm sure it's way, way below the energy output of the core. Yes, that part
Especially because we are not assuming, and I actually don't have it here. I forgot to say the hello that we are using, we got it from simulations about primitive halos and everything. We are not assuming anything atypical, like there is a spike of dark matter, or that there is some crazy 10 to the 14 GeV per centimeter cubed density
We're assuming so all these calculations are at syncope parsecs from the
Sorry, I don't know why… just because today, Finco de Mayo. I don't know why I have the cinco in my mind. It's fine. At 5 parsecs from the center of the star
The density is something like 5 GB per centimeter cubed, which is only another magnitude than the simulated energy. It's fundamentally limited by the fact that not that much dark matter ever passes through a star. All right, okay.
All right, but then at some point you have so much black matter stored
I mean, that, of course, they have mass. So at some point they will enter some regime that we call self-gravitating regime, where basically they are contributing as much gravity as whatever is in that same sphere from the baryonic point of view. And that basically happens when you saturate this inequality
Okay, and then if you have now self-gravitation, then we are talking about fermions, then you can ask, at what point you are basically winning over the Fermi pressure, and you have a collapse of these particles
And again, so you have this undersecret limit. You have a number that you need to reach to have to have the collapse. And then at this point, you form a black hole
And then the question is
Turn this black hole is going to be small anyways, but it's going to start accreting the material around. We are going to assume for days in this case, it's still valid the assumption that you have Bondiacretion. So basically, everything is very
It's very called ansymmetric, and you are not breaking the speed of sound as you would do, for example, in a neutron star. So you can compute you can calculate the time that it would require for this black hole to eat the whole star
Okay? And then for the 3 different stars, the answers are here. So I'm going to show. So here we have the three different four different shades. Basically, blue means if you are sitting in one of these regions.
The dark matter will… you won't have enough time to thermalize the dark matter that is getting captured. If you are in the purple pink region, then you actually can… you have enough time to achieve the self-gravity regime
Okay, now in some of those regions, you might actually, that might be actually worth putting some region where you actually collapse. But in some other regions, you will collapse into a black hole
Okay, but the question that we wanted to ask is, okay, for in which cases you have enough time to eat the black bone? And that is what the collapse region should be collapsed and accretion or accretion just. This is the yellow regime
And then the last shade is the gray that shows the limits that LZ have, that the collaboration has published, assuming that the scattering cross-section for these values of the mass scales like 8 at the end of atomic number to the 4
These are the exclusion limits. So notice that only in the case of 10 of a thousand solar masses, just because they live very short, you don't have time to to do anything. You are very quite well above the about this in the detection limits. But for this too, I mean, there is a little bit of
of the space that can be actually… you should be able to probe with
with a with this all right. So having this like, are we going to Little Rocks little dots or not?
You're forming, like, black holes that maybe are
getting public more. So like can you like this is too small
You'll get 1,000 solar mass black hole out of it, and from there, you might be able to create it from a solar mass. No, no, the star blows up before it collapses. Right, so here, this is the only, yeah, here you also form the black holes, but you don't have enough time. I mean, the star, yeah, the star basically
It's the redundant phase before the black hole has any chance to do anything. Yeah, of course, the black hole that you form is not, yeah, it's much, much lighter than the center, yes, yeah.
All right, so this is the end of my talk. I'm going to show some conclusions, basically saying that the answer is maybe. But before that, so if I had one next to me, I had a slide. Yes, just to mention some
So naive point of view that that actually was the thing that motivated me to do this this way rather than propagate the literature. And it's that so a lot of effects that people look for in this capture business
They have to do with assuming that the dark matter then it leads into standard model particles. And it's, I mean, it's not difficult to convince yourself that the luminosity that you get from that annihilation is going to be of the same order of magnitude
As the… as the capture rate. Okay, so it's very easy. So you can. You can compute the luminosity that the star is emitting due to dark matter annicculation that is the the dark stars business. And then and then and then you can. You can say, I'm going to set a bound again how
Well, funded that bound S is again for astrophysicists that is apparently not a serious thing, but let's suppose that we take seriously that a star cannot shine anything at a higher luminosity than the Erington luminosity
Right? So so then that would tell you that you can set a bound on this
And then you can set constraints on this plane, right? If you wanted to be to do that and you wanted to be competitive with direct detection, then you would have to require that you dark matter
In the local vicinity of the Daguire density is of the order of 10 to the 14, or even 10 to the 9 Geb per centimeter Q, which, of course, those are huge numbers.
Now, this type of analysis, I did it that way to show how nice this is. This type of analysis also have a problem. That was the problem that motivated us to do actually the work that we did without getting rid of any R-dependence or anything. Remember that I said
The Maxwell-Boltzmann distribution function overestimates things when you don't take into account the contents. Well, imagine if I told you that in the center of the star, you have a spike with this density. It would be naive not to think that this is going to affect the gravitational potential, and therefore it's going to really mess the distribution function of the velocities of the star
So you have to redo basically everything every time that you change this value, this is not just a constant in front of the capture rate. As I would say 90% of the literature papers do that this is not a constant because if you change that, then you have to change also
The distribution over which you are integrating. But anyways, and that's as far as I'm gonna… and that includes the work about the parent stability. So that is as far as I will go with that comment. Anyways, so the answer to this question is maybe
Right now, we are looking, we are working with Fabio Yoko and Sandra in some… in doing some similar work to this, but now with regular massive stars using the model of the Milky Ways, the models that we have available now with some simulations and from some
Observations, including guide staff, and then we are figuring out how to code into Mesa the gravitational effect of capturing this dark matter. But that's going to take me a while
Thank you.
Why don't do the I mean, for the very massive steroids like so this is
this future thing. I mean, that's you're trying to look at the stars that you can see today. Right, yeah, yeah, yeah. These massive stars are 20 solar masses now, yeah. Yes, yes, that's not a thousand
And for the black hole, you're saying if you had a black hole in the center, it's going to change the equation. Well, we want to see if we feed now the new gravitational effect and the presence of a black hole, we're trying to figure out how to do that in Mesa
I mean, at what point, at what point we introduce what you have now a collapsed object.
I mean, my understanding is that the matter to energy conversion in the accretion disk of a black hole was like an order of magnitude more effective
Then nuclear reflgin. Yes, I see. Coming from just from the black hole will actually produce like a significant amount of energy that couples back into the because it's sitting in the middle of quite a lot of water. Yeah, yep yes no that is a physics that is the physics that I don't know that we're trying to figure out also because
A lot of the works that you see about a Christian, like AGA and all this, it's a very different environment. So yeah, you have no idea how… I mean, even even so I come from back in the beginning with Willie, I was in more formal stuff, and I did a lot of calculations with Hawking radiation
But the standard Hawking, I mean, when people talk about evaporation of black holes in centers of neutron stars, I haven't seen and this is already out of the talk, so out of the record. So this is just expressing my confusions. I see how
A lot of people assume, like, the standard rate of evaporation of a black hole that they get from Hawking papers or etc. But again, if you go to the calculations, the calculations are only assume there is backyard surrounding the black hole, but now you have a neutron star, which
Very different from vacuum. So I… I don't know how… I mean, I don't know how much the calculation is going to be affected by all this thermal and strong effects that… but people use those rates. But it would be great to understand that. But the similar notes, then also understand how accretion inside a very dense
environment works also when you have fusion going on and everything. But but
Can I get some intuition? Why? Like, also like you can just do this with like
So like, I know that smaller, maybe there are no more heavy. So you get in there, but you have less radio, so you lose there. But also you have more time.
So, yeah, so like yeah that's why we want to do it with no primitive stuff. Yeah, okay, because then you get gain for every time. Yeah, but on the other hand, again, you have more regular
densities of that. So the comment on that you are limited by the amount of dark matter that you can pass to this stuff.
That's why people usually feel forced to introduce spikes or into deny
Because of course this the more dark matter you have, the more effective it is. But again, it's those are assumptions are very strong. Also, they want to do a better job for planets than the other people
Yeah, that's how okay
You take no more questions, let's thank again the speaker.
And let me know if you want to go to dinner, but lately the dinner have become a negotiation harder than the negotiating things. So yeah, but yeah, but like we're gonna find a place to go and yeah.