 Well, thank you very much. Good morning, everybody. Yes, I'm Andrea Fevara and so we will have three mornings together and so the organizer the director of the school has asked me to talk about cosmic realization and and this is a this is a very broad topic that involves a number of physical processes and and also cosmological Milestones that we try to cover this broadly in these three lectures so cosmic realization is a is a key process through cosmological evolution and sometimes it's called as a is It's called a phase transition the last phase transition of the universe even though I don't like very much this definition because after all It's not a sudden process. So this is the first thing that we have to learn that realization has been actually relatively smooth and gradual process that has turned most of the of the Cosmic gas, which is as you know essentially hydrogen and helium has turned it from a neutral state into an ionized state as I've already said Cosmic realization it's a process that in that involves hydrogen, but also helium and so this to realization in principle core Slightly different times because of the different ionization potential into species. We will go through these details but We when we when we talk about cosmic realization usually people means hydrogen realization even though Elim realization is also important as we will see for several reasons Now cosmic realization as I said, it's such a process. It's very important. So why do we care about cosmic realization? We care about realization because essentially the impact of this process on the structure formation Galaxy formation intergalactic medium and many other many other properties observable properties of the universe have been Tremendous, so there is a very strong in the universe would be would look very different from what we we observe today if realization would not have occurred and So it makes sense to study the interplay between structure formation and and cosmic realization there is obvious a link between these two these two type of Processes and so the next the second line of my of my title It's how do we study whether we as we will see there are several way to study? cosmic realization from the experimental point of view and Maybe the most important one is just to use The fact that we are looking at hydrogen hydrogen. It's a meeting radiation In the 21 centimeter line and therefore this appears to be a perfect tracer to do the step-by-step study of realization Through cosmic times by looking at the evolution and of the intensity of this line sometimes We talk also intensity mapping in the line So this is something that will become clear later on But so the idea is just to use this line There is a meeting by hydrogen to study how hydrogen is actually disappearing due to ionization and is turned into some proton and electron that are not Combined anymore. So What is my lecture plan? These are the three lectures that we will have today tomorrow and on Wednesday So I've divided the the lecture in three parts that That we need to go through in order to have a coherent framework of realization so the as the as Realization affects the diffuse gas that is called the intergalactic medium or sometimes also the circum galactic medium So we will need to discuss Key properties of what we know about the intergalactic medium before we go into into the actual topic of Cosmic realization and so cosmic realization is essentially divided in the in two parts What is the theory? How do we study theoretically the process by which? the universe gets Ionized and the second part in which we also need to Discuss what are the sources of realization? These are as we will see. This is a problem that it's very Open we have some ideas, but we don't have a firm conclusion on what sources actually have driven realization through from the beginning to to the organization epoch that occur about One one billion year after the Big Bang as we will see and finally we will as I was mentioned already We will look at how we can study The EOR the epoch of realization. This is the acronym that is often used EOR is the epoch of realization that is meant to define the epoch from which realization started to the time at which it was completed and So I will also introduce the physics of the 21 centimeter line Intensity mapping and the way in which we can use it. I I like to stress that the 21 centimeter mapping is not only Is a tool that can be used to study realization, but not only Probably you will hear or maybe you have already heard about the use of 21 centimeter also to study other types of Large-scale problems in cosmology. So it's a I guess it's an important future technique that we will use For all these problems. So before I actually start also, I need to give you some references there is a lot of literature in the in the field and At least in the in the last 10 years there's been a lot of activity and so I'm just referring to to the to the Reviews that I have personally used to prepare the lectures and something that I also Certainly are definitely happy to suggest They go from some more historical material that was pre say pre Well, I don't know pre what but anyway there was before 2007 and then the most recent material that is scattered a little bit in in in different Different reviews and you can find others of course, but these are the one that that I Will be mostly using during my during my lectures so Lecture number one so the intergalactic medium So let's let's start to look to define and and see what are the properties of the intergalactic medium Which is definitely? Broadly speaking is anything left after you form galaxy So you put some of your gas some of the cosmic Material which is as you know hydrogen and helium left over from the from the big bang You put some of it into galaxies whatever is left. Let let is call Let's call it for the moment the intergalactic medium Okay, so intergalactic medium is a diffuse component of the universe which is not Condensed or collapsed into bound objects as galaxies. So this is the working definition for now so the the IGM intergalactic medium, so this is the acronym the IGM physics is commonly the birth of this type of Physics is commonly dated back to 1965 so what happened then it happened that an observer Martin Schmidt in California discovered or was able to take the spectrum of them of what was then the most distant object There was a quizer called 3c9. It was located at the redshift, which today appears almost ridiculous, right to Redshift 2.1, which seems you know yesterday for us But it's actually at that time. It was the most distant object. No and so that Observation was very challenging Because you know you use the best possible technology available at the time, but the discovery of this quizer Was very fundamental in understanding Things that they were not clear before in particular when When Schmidt showed in a in a talk at Caltech it showed the the spectrum of this quizer a young postdoc maybe like you was sitting in the in the in the audience his name was Jim Gunn and Immediately noticed something that there was light there was transmission of photons Even beyond the energies that are larger than the ionization potential of the hydrogen Of the hydrogen atom, which is 13.6 electron volts or 1912 angstrom so the the the spectrum of the quizer was showing there was some transmission of light even in ionizing photons of photons that should have been absorbed by intervening hydrogen along the line of sight, okay, so that was very surprising and so gum Noted that and Immediately said oh, this is very important for cosmology Why because the fact that we were the Schmidt was able to observe this this light from High-energy photons meant that there were two possibilities. So either Galaxy formation as was very efficient and therefore all the cosmic gas was in galaxies And so the intergalactic space was completely empty. So the photos were not absorbed That was the hypothesis number one hypothesis number two was that The gas was still there, but was in a state that was unable to absorb the ionizing photons coming from the quaza towards us That means that the those atoms should have been totally ionized. Okay, so They work out together with another Smart guy a Bruce Peterson. So gun and Peterson Just work on this problem in one week They turn out with a paper a fundamental paper that we will study better later on That was claiming that Quizzars could be used to study the state of the intergalactic gas and before we could understand How much of this gas was in galaxies and also what was the state of the of the intergalactic medium and they concluded that the intergalactic medium had to be Completely almost completely ionized and to explain the observation. So that was the first hint of the physics or the properties of the intergalactic medium and was also the birth of Of the intergalactic medium science and also maybe of realization Because we for the first time we firmly Saw that the the IgM was ionized of this by right you too. Okay So in finally we also learned that we can use Quasars as Lighthouses to pierce line of sight through the intergalactic medium along which we can study the physics of the intergalactic medium But this is the first thing. So let's Let's become a little bit more quantitative and so this is a This is a kind of budget of the of the gas of the material and the of the burials actually that are in the universe that was put together by Foucault Gita and colleagues a few years ago and that that shows Few things so we have here a number of components that are Evaluate whose abundance is evaluated a redshift zero So we had three columns the central value and maximum minimum So just look at the central value these numbers are given in units of the critical density So they show the density parameter of that particular component, right? So first of all you may expect if I ask you so now a redshift zero Where are all the burials? What is the component in the universe that that contains most of the bears probably naively? You would reply. Okay. Well, maybe stars and galaxies Wrong wrong answer because in fact if you look at this consider the sphere Okay, the total of this is the the total that we have that we observe for example from the C and B So this is 0.021 which is the omega barrier Okay, so the value of the the density parameter of burials are actually zero So this is the total that we need to Explain so how is it distributed among different components? so if you look at stars and by by far the spheroids like you know elliptical galaxies or large spheroidal systems Contained most of the stars, but as you see they make it up barely to something like 10% So only 10% of the burials today are in stars and are in mostly in elliptical galaxies So this is the first lesson But so where is the rest of the 90% it's in the intergalactic medium So even now a redshift zero the intergalactic medium. It's important because it contains 90% of the burials that we know have been created by by nucleosynthesis. So This is a very important thing to appreciate and so and if we further ask so and within the IGM Where are they? Where are they distributed now? I'm introducing yet another acronym, which is ICM, which is the intra cluster medium so there is also Historically there's been a little bit of there's a little bit of ambiguity between the two terms so intergalactic versus intra cluster but you know sometimes they are used in an almost interchangeable way, but in principle They are different as we will see so if we ask now How is the gas that is not in galaxy distributed among different components? Well, we see that the most important The most important by far the most important Component is plasma in groups, okay So this contains almost 60 60 percent or 70 percent of the burials In the intergalactic that are not in galaxies are in Plasma that is Surrounding small groups where maybe 10 or 20 galaxies So this gas is floating around these galaxies the one that Contains most of the material, but there is also Warm plasma in clusters or cool plasmas around around in truly in the intergalactic space But so all these four diffuse components make up about 70 percent of the barrier So the intergalactic means there's a lot of mass in the intergalactic medium. This is the takeaway message And so can we have a little bit more quantitative? understanding of why Why of another important property? So this gas the the IGM and the ICM is Relatively warm or if not hot, okay, so why is that these high temperatures? So, you know for example the mean temperature of the gas in a galaxy like our own is of the other a 8,000 K, so it's relatively cool Okay, so most of the gas in the Milky Way and in other galaxies below 10 to the 4k However, if we go into the IGM the gas is always hotter than that It's good come could get to 10 to the 5 10 to the 6 and even 10 to the 7. Okay, so why is that? well one way to understand why the intergalactic medium is warm hot is Because it's very simple. So the idea is that as structure are forming and the gas collapse into into the Filaments as we will see later collapse in filaments and structures. So it's it's heated up by Shocks, so the accretion shocks that are that are forming as a result of the collapse of this of this otherwise diffused medium into filaments and sheets Then the shocks convert the gravitational energy of the collapse into thermal energy So various to the very zero order you can say that the the velocity at which That is related to the collapse is essentially the wavelength of the linear perturbation That is becoming nonlinear over time which over the time that it takes to collapse and this kinetic energy has to be converted into into thermal energy that is Express in terms of the sound speed of the gas the sound speed I recall you that is proportional to the square root of T. So T is equal to see a square And so if we say the time is the inverse is the able time and so the inverse of H So we get that the temperature by simple simple Dimensional analysis we find that the temperature by redshift zero has to be of the order of Ten to the seven K. Okay, this is the essentially the maximum you can reach that there is dissipation but so that explains why the kinetic energy of the collapse of structure turns the gas into through shocks Turns the gas into a hot component. So the IgM is usually warm hot Okay, so most of the gas it's it's heated up by a structure formation shocks So shocks related to structure formation now In order to see this in a in a more Graphical way Let's let's look now at these different components. So there is gas a different a different temperature because of course the gas that is related to larger larger Larger fluctuation that are connected with clusters. It's hotter than the groups and it's hotter than single galaxies So there's a hierarchy of temperatures and so we see here the the gas is divided in terms of different So we are looking at the volume feeling fact volume fraction Filled by a given component. So you have galaxies and then you have very hot gas Intermediate and warm gas around ten to the five. So you see as a function of redshift You find that by far the the volume is filled by gas the cosmic volume is filled by gas with Temperature of the order ten to the five and ten percent is instead of even higher temperatures There is essentially related to cluster and very large groups. Okay, so this this As as expected from the previous argument as you go from redshift three to zero the volume feeling fire the volume fraction filled by hot gas is increasing and this is because the Larger larger structure are collapsing and therefore the temperature that we derived before are getting higher. So there's more structure formation of Large structures that can heat the gas up to the ten to the seven K We can do the same and now and look at the instead of the volume feeling factor We can look at the mass fraction in the different components and here. We see something Different but essentially reflects what I said before so the cold this cold gas now It's in terms of mass is decreasing from 90 More percent a wretched tree. So a wretched tree. Remember that more than 90 more than 90 percent of the gas It's relatively cold. So it's temperature less than ten to the five, but that Cold gas is disappearing in favor of this warm gas produced by structure formation shocks So by wretched zero you see that the situation is now reversed We have Most of the gas about 60 percent is it's warm and then we have some cold gas 30 percent Hot gas 20 percent and galaxies the 10 percent that I told you before So 90 percent if you sum up cold warm and hot This is all the intergalactic medium that it's outside galaxies. So Star formation in galaxies can only rely on 10 percent of the available gas in order to Form stars and the rest cannot be or has not yet been Put into into bound systems. So this is the first first overview of the intergalactic medium, but we can also learn more about the About the processes that are that are and the distribution of these gas in the cosmos. So for that We have not been able yet to Directly image the distribution of the gas because of course it's very diffused and the absorption like Absorption like techniques that I was mentioning before can only allow us to study the one dimensional Distribution of the properties of the intergalactic medium But we cannot have real maps yet of the intergalactic medium. Maybe we will in the future But as we will discover later, but so far we have to Sorry a bad call today To rely on on numerical simulation like this one, which are showing the This is also this is essentially Factors a factor in outer scale is true for any redshift. So this is a typical distribution of the of the variants In the in the end up matter actually in the in the given field This this one is a 10 by 10 megaparsec inverse H Volume so you see there are a lot of filaments here Then these are exactly the filaments that I was discussing before where the gas it's is squeezed into these into these filaments and because of that it becomes hotter And so there is a lot of structure and then you see these small dots here that Actually are the are the galaxies the form exactly where the where the Igm it's There are crossings between the the filaments So in at the at the knots of the of the filament you are the crossings You form these tiny dots that are the galaxies now you see immediately from here one of the key key properties of the intergalactic medium that is that the intergalactic medium is Fundamental in driving gas Guiding the gas onto onto galaxies. So it's so all all this accretion of gas onto galaxy feeds The star formation activity into a galaxy. So the Igm is funneled into into Collapser or Virialized system like the galaxy and this gas is the is the fuel to form stars Stars in galaxy but as we will see later and actually we see even better in the next slide Galaxies react back on the on the on the intergalactic medium So there is a cross talk between the intergalactic medium and galaxy. So how do galaxy feedback? Well, this is It happens that the the gas that is funneling to into the galaxy Very rapidly goes and form stars for example and the stars start to inject energy and in particular they through radiation and also Supernova explosions and so this gas it's the the stars try to stop the info Okay, so there is a battle between the intergalactic medium that tried to to pierce into the galaxy and the galaxy that reacts and You can see clearly the the feedback this feedback effect as is called So the galaxy feeds back on the intergalactic medium From the the same map, but now we look at the temperature map So it's the same simulation as before, but now you see that we are showing You're looking at the temperature Temperature map and you see that around this yellow regions here Where the these are the the galaxies where start formation are starting to push out winds Okay, so the winds are coming from from the galaxy and they span into the intergalactic medium And they work against the info of gas that is trying to come down into onto the galaxy through the through the filaments So there is a strong interplay The strong interplay between the intergalactic medium and the galaxies and that tells you immediately that If you change the properties of the intergalactic medium for example through cosmic realization then The properties of the galaxy that you would get in a in an university which Organization is drastically change will be also drastically change because it changed this interplay between the intergalactic medium and and the galaxies We can also Have a better look of the to very important diagram in the intergalactic that the Characterizes the intergalactic medium, which is called the phase diagram or the equation of state That relates the temperature so this is taken again from the simulations, but This is in very good agreement with with data as we will see later but from from the same simulation we can look at the Equation of state or phase diagram that relates the temperature of the gas with respect to its over density now the in IGM Physics we often Rather than talking about densities in absolute density. We talk about over densities with respect to the cosmic mean So this is denoted by Delta so log Delta equals zero means is the mean density of the of the in of the of the Gas in the of the variance in in the universe And then if you go to negative value you are living in another density if you have positive value We are living in the regions that are more dense than the average so the Of course as you go to very large over density you enter in the realm of of galaxies For example, we are already at these points here have overdensities are that large are largely non-linear So the intergalactic medium is usually Considered I mean, it's an arbitrary definition But the the the realm of the intergalactic medium ends over density of roughly 100 Where the perturbations start to become non-linear and therefore they start start to form Bound objects now if we concentrate then on this part we see that that denotes the the true IGM You see that the the temperatures range from Roughly 10 to the 10 to the well few thousand K's Up to a million K and this is this one is a ratchet for okay So this is a ready for just so that that the question of state changes very much, but as a function of redshift, but after organization remains more or less stable this With this shape that Essentially tell us a little bit about the physics that we are going to learn in the next in the next few slides so the idea is that the the IGM is divide the the temperature of the IGM is is Not only Determined by by the shocks as I was Discussing before that are able to create million degree gas But you see that there is also a lot of gas that it's in the in the cold phase Remember that I told you a ratchet tree 90% of the gas is in this cold phase With temperature of the order of 10 to the 4 or 10 between 10 to the 4 and then 30,000 K So this is this gas is it's not Shock-eated, but it's or not yet shock-eated, but it's photo ionized. It will be photo ionized by UV radiation that we will discuss in a second So the the phase diagram is the combination of gas that is photo ionized Which is by the way the majority if you look at the this is a mass weighted distribution so You see that the green here means that most of the mass is here in the cold component In agreement with what we said before, but it's also a considerable amount of gas that it's heated by by the shocks By structure formation, but also by supernovae as we have seen before okay, so So now We have this the something like that and so how do we how do we studied it? How can we make a progress? So this is essentially we are still stuck to the idea by by Gunn and Peterson so what we do is just we pierce line of sight through the intergalactic medium. So suppose We have a quasar which is a very luminous source It does also also galaxies can also be used or gamma ray burst can also be used anything Which is bright enough that that is a spectrum that is simple enough that you can easily understand and quasar certainly do Can be used as a background source at the background lamp through which you can shine Light through the intergalactic medium. And so what what you receive is the filtered Quasar spectrum. So you see the spectrum of the quasar, but now it is filtered through the the filaments That I show you before so these are the filaments that I'm showing you in in the previous light So like this one so suppose that now we are piercing this box through with the line of sight and What what we are passing through is a number of? Filaments that contain some neutral hydrogen so the light of the quasar photos that have energies larger than than the limon alpha line are essentially scattered by this by this gas and They are pushed out of the line of sight that it's equivalent to us As to an effective absorption absorption process because we are losing that photo from the line of sight And so what we observe? It's something like this. So this is a Typical spectrum of a quasar that it's filtered through the It's filtered through the through the intergalactic medium and the media to you see a number of things first of all You notice that the most prominent line is the limon alpha The limon alpha line that corresponds to the transition from level two to level one of The hydrogen atom that is emitted by the quasar itself but as you see that what happens here is that as you go to Shorter wavelengths therefore you go to higher energy of the photons then you start to see this this Incredible number of absorption features. So these are all absorption features that are as some structure That that is reminiscent of a forest and that's why it's called the limon forest limon alpha forest Okay, so it's a forest of absorptions that Are not as nothing to do with the with the quasar itself But they are imprinted by the intergalactic medium so as at any time in which you pass through one of these filaments for example here or here or here then That structure in space leaves an absorption or better scattering Imprint into into the quasar spectrum and therefore you can see directly reflected into into the spectrum of the quasar so So each line here corresponds to a discrete absorber that contains sufficient Neutral hydrogen so that the The quasar spectrum is is absorbed the photons from the from the quasar are absorbed and we see them as absorption features Now it's interesting to know that The inter intergalactic medium. There is a little redshift 3.5 There are not only there's not only hydrogen and helium as we would have expected from pure Big Bang Conditions, but we also already see a number of features for example look at this one This is a very prominent one. There's two lines. This is the doublet of What's called a Is an absorption feature related to the carbon atom is carbon in particular ionization state This is three times ionized carbon And you see that this feature is very prominent. That means that in the intergalactic medium by redshift 3 There is not only Hydrogen and helium, but they are already heavy elements. They've been Injected by galaxies remember we were discussing about the winds before and these winds are not only carrying Energy and momentum into the intergalactic medium, but they also carry a lot of a lot of heavy elements And these heavy elements are although they are very relatively rare There are for example, they are typically 1,000 times more rare than in our galaxy In spite of that we can see them Imprinted in in in several transitions carbon, silicon and oxygen and we see many many species So the intergalactic media has already been subject to injection of energy and and also polluted with with heavy elements by Early on so how early we will see later okay, so this is the the General overview of what we do expect in general terms So let's do a little bit more a little bit more of physics now. So I told Yeah, say it again. Do I see the gum Peterson effect in in that plot? You mean here This one. Yeah, I mean the I have to explain a few things more. We will see the next few slides, but essentially you can Compute the tau Gumby. I've not introduced the gum Peter. So yes, so if you can wait for a second But yes, we do a certain we do a tau we observe we can compute a Optical depth the gum Peter son of the optical depth and in this case It turns out I don't know the precise value, but I'll show you later a plot that show how it evolved with red Sheep so we do yes the answer is yes, but sorry It's a little bit I had to introduce a few more things because before I can go in more details so I Was I was told telling you that actually Probably for the true intergalactic medium at the high redshift because it says the structure formation shocks has not yet been Able to eat it up to very large temperatures. So most of the gas is cold We sold 90 percent as a number as a guideline 90 percent of the of the gas It's it's called a redshift above three or four And so what is heating up this gas to the temperature 10 to the fourth that we are observing? well, this is mostly due to it's a photo ionization process and I'll tell you I'll tell you later. What is producing the Where the energy comes from so this is a very simple slide that however shows the basic The basic physics that you need to include if you want to compute for example, how much neutral hydrogen it's present in the intergalactic medium and what is the temperature that you would expect so There are two things that you need to compute the first is the photo ionization rate and the second is the photo heating rate so the photo ionization rate is the Tells you how many ionizations there means how many times an hydrogen atom? It's divided by its own electron. So it's ionized per unit time or per second. So This is the it's called gamma gamma h1 and is nothing else that the that the integral from the Minimum energy that you need to give to an hydrogen atom in order to strip away the electron Which is one Rindberg or 13.6 electron volts to infinity so you can use all photos that have energies larger than that And so this is the energy density of the radiation field that the atom sees and this is a of new which is the Which is the cross section for photo ionization, which typically depends to the Frequency to power minus three so as soon as you go to larger larger energies the ionizing power decreases okay, so at the same at the same in the same way you can Every time you Every time you you divide the electron from the you give energy to electron to to be stripped away from the Hydrogen atom that the excess energy that that you have given to the atom goes into Kinetic energy of the electron so this electron then Thermalizes by collisions with with the rest of the gas and that corresponds to an eating rate So almost inevitably every time you produce an ionization You also heat up the gas because you give More energy that you need usually and then this extra energy goes into kinetic energy of the electron and the electron Distributes this energy to the other particles. Okay by thermalization process So there is a heating rate that is related to the photo station rate So given what I said is not surprising to see that the expressions are between the photo station rate and the photo eating rate as very similar And and the only difference that here we have an extra factor that Gives the difference between the actual energy of the pinging photon with respect to the threshold energy For the ionization of that space now this has been done for Hydrogen, but of course you can do it for any species any atom in practice There is also photo ionization not only of hydrogen But also velum and all the other if you have other species like carbon, silicon, every elements that would work the same Of course hydrogen is the most abundant by far atom and therefore we are caring about that So with these two numbers the photo ionization rate and the photo eating rate you can learn a lot about the Properties of the IGN so if you're able to compute those numbers then many things can be derived up to then so the first thing is that You can write a simple equation that tells you how what is the? The evolution the time evolution of the fraction of the gas which is in neutral form And this is x h1, which is the ratio between the density of hydrogen neutral hydrogen with respect to density of hydrogen atoms, so this is simply this is a detailed balance equation that simply is it's a Determined by two terms The first is the fact that the neutral hydrogen fraction decreases as you Ionize the gas and this is the ionization rate there So you see that this is a negative term that tends to decrease the the abundance of neutral hydrogen and the second one is the Balance is there is a process that balances the ionization And this is due to essentially Recombination, so if you have an electrons a bath of electrons and protons at a given temperature There's a certain probability that they Recombine and form a neutral hydrogen and so that depends on temperature and is called the recombination coefficient And and there's also a proportion to the number of Electro free electrons that you have the density of free electrons that you have available to do this recombination So this is something that also is a similar equation, although Not exactly the same with respect to what that you use when you study the cosmic recombination Epoch right so when you study cosmic recombination at all that also give rise to the To the physics of the cosmic microwave background, so the process is exactly the same even though With some some differences due to the different conditions so this is the first equation and then the second is just a such a conservation of the number of I don't add hydrogen atoms that say that the the sum of neutral atoms and other ionized atoms has to be constant So if you solve that equation you find that that equilibrium, so you say you forget about the time evolution if you look for timescale that are short enough for the that are sufficient for the ionization to reach a steady-state equilibrium Then you find this equation that tells you that actually the neutral hydrogen fraction of equilibrium It's as a very simple expression. There is essentially proportionate proportional to the recombination rates and inversely proportional to the Photonization rate, of course the largest of what is this your rate the lower is the fraction of Neutral hydrogen that you have available at the end So you can do the same for so this is a fundamental equation that very simple, but it gives you an idea of Given the condition of the intergalactic medium. What is the fraction of atoms that are in the? neutral state and you can also Write the the energy equation, so these are called the ionization equations These are they call the energy equation that deal with the balance of energy that That that they are providing because as I was saying before so for each ionization You produce also some eating and so they are photo eating the gas But the gas is also losing energy because it emits radiation We will see that more clearly in the next slide and so this this photo eating term has to be Diminished by the amount of radiation or cooling losses that the gas is is suffering So I'll I'll specify later what this function is but for now it's just a function of temperature that Despresses the cooling of the gas so This this term is the the usual expression for the entropy and the entropy is a function of time Evolves like like this. This is the energy equation that written in a specific way And so you can also get the temperature from By solving that you can get the temperature the interesting thing is that if you are likely if you want to know how the Temperature of the intergalactic medium evolves well We see that this is proportional to the ratio between the photo eating rate and the photo ionization rate So as a function of time, this will evolve in this way. So there's a proportionality Now I was telling you about this function of the so you're eating the gas through some form of radiation Photo ionizing it, but you're also losing energy because the gas is cooling. So what what is produced in this cooling? This is a very important Function not only in for the intergalactic medium or you know for galaxy formation But also for interstellar medium it enters essentially everywhere So this is a function called the cooling function that expresses the ability of a gas that is sitting at a given Temperature that's been heated up a temperature T to lose its thermal energy By carrying away with photos with radiation Okay, so you're you're you're losing the thermal energy of the gas and so the way in which a gas Does that depends essentially on what? Processes at a given temperature are able to create photons that carry away the thermal energy So you convert thermal energy into radiative energy, which is lost from the system so It makes a lot of difference if If it makes a lot of difference if your gas as a primordial composition like hydrogen and helium only or if they're also pollutants like Heavy elements or more more heavy species like carbon oxygen and silicon do the one we discussed before and And this is clear because the more atoms you put the more species you put the more Available process you have more lines you have available in order to Transform kinetic energy of the gas or into into radiation so for the primordial in the simplest possible case, which is by the way for the for the Integrating medium is it's an extremely good approximation because we see that most of the gas is just hydrogen and helium with traces of Heavy elements so the primordial gas to a first approximate a primordial composition to a first approximation is a very good Gives us a very good idea For to perform actual calculation in fact even in the most sophisticated numerical Simulation is still we are using this cooling function because the the contribution from every species is essentially close to zero now The this is the cooling function so that means this is the the energy lost by parcel of gas a temperature T Per unit time per unit volume now you see that the the solid line is is the the total of the of the Of the of the curve, so is the actual curve, but there are also What I'm showing here also the different processes that are contributing to to build up this curve And as you see there are the most prominent one that gives you these two peaks are the Is the is the cooling related to the liman alpha emissions? So for example, what is plot is telling us if you have a gas with temperature of roughly 20,000 30,000 K It will lose a lot of energy Because this gas is hot enough so that the electrons can excite the neutral atoms Excited to level 2 they decay into a liman alpha photon that liman alpha photo is gone and that energy has been lost by the system Okay, so this is this is why this You have to speak and you have the same for for helium. Okay, so helium is slightly higher energy So these are the two most important processes of liman alpha cooling both from hydrogen and also from helium is The key cooling species for gases for for a gas of primordial composition around 10 to the fire 10 to the fire Kelvin But there are also continuum processes like free free recombination And and others that are not even listed here So but most of the most of the most of the the peak of for what it's concerning us it's essentially liman alpha excitation and also free free when you go to very High high energy. So this is the cooling function one thing that you may note here is that because this the The cooling function is dropping to zero Below 10 to the 4k. So that means that in the primordial Composition gas with temperature Sorry with the primordial composition gas. It's impossible to cool that gas below Essentially 10 to the 4k. So there's a sharp drop and there's no way to cool that gas below That to the for by radiative processes, of course, you can always do it by adiabatic expansion. Okay, that's a different thing but purely from from Radiative losses, it's impossible if you have a gas which is not expanding that is bound For example, if you put that gas into a galaxy and and you don't have any other way to cool the gas The gas will remain at 10 to the 4k. So it's only when you start to pollute that gas with a lot of Heavy elements that a new branch will appear here and that what happens in inside galaxy But we are not discussing Galaxy formation here. We just if we restrain to the IgM. It's so the temperature Cannot go below the the key point is that the temperature cannot go below 10 to the 4 by Radiative processes can only go below that if you expand the gas through adiabatic expansion. So this is a very important point now Remember that one let's step back one second one second. So before I was I was When I was discussing the photonization rate and the photo eating rate. I was Mentioning that there's a UV Which is the energy density of radiation? Okay, both in the photonization rate But also in the photo eating rate. I did not specify after this time What is the what is the source of this radiation? So where is what is what is you? I mean or at least what is the relevant? energy density of radiation that heats and and ionizes the IgM Well, this is called for the IgM This is the this comes from a background of radiation. It's called the ultraviolet background or UV background Okay, so this UV background is produced by all the sources that emit light there could be Quasars and galaxies or could be also other sources as maybe we'll see later But essentially the first approximation is quasar and galaxies so Obviously the energy density you is a function of redshift and Also, it has a spectrum that depends on what type of sources are emitting that radiation so What do we have an idea of of this UV background? Well, this is something that we are we have We have ideas and we have good models and we have some tests There we are not 100% sure exactly if for example, there's a long-standing debate If the UV background is mostly produced by quasars of galaxies and the the relative balance between these two type of sources It's actually very debate is an open problem in this type of studies, but at least theoretically with some assumption we can compute the the evolution and spectrum of the of the intergalactic You ultraviolet background. Okay, so the this is the mean background that a nitrogen atom sees so in this figure I'll show you exactly the spectrum of the of the UV background that is shining on top of this All the intergalactic medium and we are seeing as it evolves from redshift 7 or so Down to redshift 0. So these are units of KV So and I just put the lining and just to show you where the ionization potential of the hydrogen is so remember that Only photons with energy larger than that are entering in that integral in the photonization rate because you see that We need to these photons that are below the threshold of 13.6 cV They just do not ionize the the hydrogen. So One thing that you may notice is that starting from high redshift to low redshift there is a Substantial change in the in the shape of the of the inter of the UV background and the spectrum is determined by a number of Radiative transfer process because this radiation of the UV background is filtered through the IGM. So if you want to measure it as a function of redshift for example, we need to Clarify how the evolution occurs, but you can see for example that There's a there's a in the in This is a well as this is a lot of variation in in in terms of the spectral shape One thing to appreciate is that The UV background strictly speaking can only be Established after the ionization and the reason is that when I say UV background when people refers to background means some Radiation field that whose intensity is independent on location Okay, so if I measure the intensity here is the same idea. So it's homogeneous, okay And and this is what is called a background usually a background is shows very little fluctuations very little spatial fluctuations however The this is clear that before Realization when there was a lot of neutral hydrogen still floating around A true background could not be established because the fluctuation must be very large For example, if you are in a ionized region the intensity is high if you are in a neutral region The intensity is low because it's absorbed by the hydrogen itself. So the Strictly speaking, we should only talk about a UV background after the completion of realization But with this caveat in mind, we also extend it to larger larger red ship So this is the the final the red ship zero background and so we see that these are the The theoretical expectations and these are measurements that we have available Today, so you see that there's a very good agreement. So we think we have a good idea of the Of the background at least a lower red ship. So we we are confident that we can extend the extend the Prediction of the model to to very high red ship. Of course, as I said before as you enter the realization epoch things become a little bit more Difficult but at least as a first guideline, this is fine So again, this is a different same thing by the with a different view of the of the spectrum that allow us to appreciate a few features So this is a again the intensity of the UV background Which is usually measured in these units of erics per square centimeter per second per ounce per steradian From red ship to to red ship five five point five and you see that the That there's an evolution you see features here that are prominent and I don't know if you can recognize them but these are essentially the the lima now falline and the analog for the Ilium Atoms so these are the lines that are these are emission lines that essentially are due to They are produced anytime that there are Recombinations between electrons and protons so you have these lines which are more prominent at high red shift And then they as you see they they decrease and almost disappear as you go to very low red shift And this is because the lower red shift the session is no very few Neutral hydrogen atoms left to to produce a lima now falline So the lima alpha line is disappearing very rapidly. So this is our understanding of of the of the UV background and so we can we can use these to perform calculations of various types and This these calculations that are typically done through numerical simulations can then be compared with actual data Okay, so in order to make this comparison We need to learn a little bit more in detail how we a few a few things about how we measure What type of measurements we are making on the intergalactic medium? So so As I was saying before most of the studies so far But we'll see that things are changing with a 21 centimeter there. I will discuss later on So far the intergalactic media has been studied mostly Through this quiz absorption line. So the gum Peterson idea, okay? and Historically all these studies started when the the the the the resolution the spectral resolution of the instrument and of the instruments were relatively small people were looking at the quantity that is It's called the flux decrement of a of a spectrum. This is called like the sub a and It's it's nothing else that the average over all the Over all the absorption feature that we saw before in the liman alpha forest So we have it over all these discrete absorbers of this quantity one minus the observed flux divided the Continuum flux. So in other words, let me show you again this figure that we're looking before right so for So you so what that that formula says is that for each of these absorption features here you compute the ratio of the absorption fly the flux that is Absorbed with respect to the one that is the continuum of the quasar, of course You need to know where the continuum here is so this is also a tricky point But you may imagine to draw Connect the points here with a line and that will be the continuum the quasar which is now all Featured by by this absorption features, so if you do you can construct the quantity which is the dis a bay the flux decrement by averaging over all these discrete Absorption features this quantity and you can therefore determine an effective optical depth So I come back to your question before so f obs is the observed or residual flux What do you see at that specific frequency and then the estimated flux of the unabsorbed continuum and so This is you get the tau which is the line optical depth as a function of wavelength or range which is the same so Of course as I was mentioning you need to know exactly what the continuum level that the of the quasar is Over over large regions, but so you usually what you do You're extrapolated from regions that are red work of the lima-alpha emission line where the lima-alpha forest is not present so you try to to fit a power low to the spectrum and so you get the the optical the optical depth as a function of of red ship now This is how you get it from your spectra, but then you want also to To understand how this tau is physically determined so in actually your Effective this effective tau that you have just measured the way that I described Is the result actually of of an integration over three quantities? That is the the number of absorbers you have along the line of sight their Doppler parameter that I decided that I described in a second and the Red ship interval okay that you have so this is the defective tau Then it's the one minus the tau of each individual individual absorber which is determined by n which is the the column density of of gas that is the the density integrated along the line of sight Times and also by the Doppler parameter. What is the Doppler parameter? Well the Doppler parameter is gives you gives you the the essentially the The width of the of the full width of maximum of the line is related to the to the width of the line that you are the of the lima-alpha line that you are that is scattering the the photos and It's a function of the temperature. So this is the temperature of the gas over the mass of the hydrogen atom plus and some extra Broadening of the line which is not related to a thermal effect but is also due to bulk motions of the gas So the Doppler parameter is the sum of the square root of of the sum of the of these two broadening Okay, so if you now Assume that the the column densities and the and the B the Doppler parameter are independent on the redshift and You assume what is usually done that you can write the this quantity this function as a product of the redshift evolution times the Properties of individual observers then you can write the effective optical depth in this way and where where you define also the Rest frame the equivalent width of the line, which is nothing else that the amount of the amount of light that you are Absorbing within the line with okay, so this is the equivalent width so this is the formal expression for for the optical depth and so Having having made the connection between theory and observation. You can also try to obtain something from the spectrum So what do you do? Well, so this is a much Expanded view of the quasi-absorption spectra that I show you before so you see the spectrum is the is the blue lines here So what you what you do you try to? Indeed to to fit to each absorption feature that you observe in the spectrum a number of components that come from individual individual observers, so So you try to model each of the absorbers that the The light ray finds along its path into into different component each one characterized by Doppler parameter and a neutral hydrogen column density and each one I recall that the column density is the integral of the number density of hydrogen Integrated along the line of sight so as a unit of one over an area so one over centimeter square and Okay, so you you find you do This was called the void profile the composition so you fit void profiles. What is a void profile? Well, you know that the lines Atomic lines as a as a they have a natural line width, which is described by Lorentzian and then they have this Doppler Extra Doppler Broadening that is described by a Gaussian so avoid profile. It's an empirical form that for the profile of the for these profiles that you are trying to fit to the lines that it's It's a convolution between the Lorentzian Lorentzian natural broadening and the and the Doppler broadening Okay, so you fit this void profiles and for each observer you can find these two quantities so you can find our width our large is Wide is the line and also what is the column density of that of that specific absorber And so what you get from from this type of exercise is for a number of information about the individual absorbers so the first is the the distribution of of the column densities of different absorbers and it turns out to be So these are data the the points are data. So we have the number or the yeah the frequency of of the of Absorbers with a given column density shown here in in h1. So these are the the data and these are Results from the numerical simulation. So see that they fit very well and it turns out that this Function is essentially a power law of of the column density to power minus 1.5 so now we know that this This Power law extends to even larger column density actually goes down up to 10 to the 20 even 21 So there is a as a very simple distribution of the column densities of the absorbers as a power law that Actually can be understood is in agreement with what? Numerical simulation predicts so that means that numerical simulation are able to catch the physics of structure of structure and filament formation coupled with a good treatment of the UV background and the Photonization rate of the gas so we are we are we think we understand The the the physics of the liman alpha forest to to a very good extent and so know that also this Functional form is almost independent of wretches so from wretches 2 to wretches 6 There is no much change in the in the in this functional form So this is one thing we have learned the second We can also learn about the temperature from the data We can learn about the temperature of of the gas. This is a little bit more Tricky because if you plot for example look at these plots we are plotting the the Doppler parameter Which remembers proportion to the square root of the temperature as a function of the? Neutral hydrogen column density and see that there is a Scatter there, but you see there is also a there seems to be a minimum line now Why is that important? Well, this is important because if you want to Measure the temperature of the intergalactic medium, of course the the Doppler parameter is telling us something But remember that the Doppler parameter contains also this term Which is the turbulent? Broadening and so when we measure something we measure the total be while we would like to know only temperature But there's also these extra terms that that Confuses us a little bit and in fact this is This is reflected into the fact that the be parameters Many given column density have a spread which could be produced by the by this double the Turbulent broadening but this line here is Sets like a minimum of these points and we can think that this is what we good We would get if the turbulent motion would be equal to zero So it's not straightforward to obtain the temperature directly from B, but we know we can have at least Lower limit to the temperature set by the lower envelope of the of the distribution now With that technique we can also reconstruct the evolution of the temperature of the intergalactic Me of the photo ionized part of the intergalactic medium Which again at high redshift is essentially the almost the totality of the gas So the temperature as a function of redshift You see it's it's initially decreases. So if you're coming from my redshift Decreases down to redshift 3 and then apparently shows a sudden Jump here, and then it decreases again now. How do we understand that? Well the first part it's almost obvious because we as I was mentioning before the IGM is cooling down as the Because of expansion so it's is adiabatically expanding and so the temperature drops but There's this intermediate phase here of of apparent reheating of the of the IGM it goes from a few times said to well 10,000 K to maybe 50,000 K that it's commonly associated with the extra heating due to Ilium Realization so while as we will see later Hydrogen realization was completed around redshift 6 or so the Ilium The Ilium realization that means from Ilium singly ionized doing Ilium doubly ionized was only completed later on and so that process May have left an imprint in the temperature Due to the extra eating the photo eating produced by the full realization of Ilium Now there is a lot of debate on this the data are difficult exactly because of the fact that it's difficult to interpret the B as purely due to temperature So this is another open problem. So if Ilium realization is truly a core at redshift 3 And if the eating connected to that has been Responsible for this apparent jump in the in the temperature that we see now Here we come to the Where we started from that is the most important Way to characterize the realization one of the key quantities that Often used to characterize realization is the so-called Gump Peterson optical depth. Okay, so this is a paper by Gump Peterson 1965 And this is the original formula that they used they were not talking about the optical depth but something similar in non-dimensional units which is P So there's the probability of a photon to be Scattered as it travels to towards us. So this probability is the integral between zero and the given redshift So this is n which is the number of hydrogen atom along the line of sight So this is the frequency Sorry, this is a cross section computed at the frequency Appropriate to the redshift. So this is the Lyman alpha frequency Which the the photon can can be absorbed by the Can be scattered by the neutral hydrogen and this is the mean free path Along the radial direction Express in the in Commoving coordinates and as a function of redshift. So the cross section sigma For the Lyman alpha is very simple. So it's a product of Fundamental constant time the oscillator strength and this is the line profile. So if you do the the integral Assuming a density which is Evolving like one one plus e to the cube as a function of redshift then you get the standard formula for the Gump Peterson This is the solution to that equation and in the more modern way, which you can usually find it written it's with the coefficients that are The cosmological parameters that are extracted that put in evidence So you see that the gun Peterson is a for Tao. So this is the optical depth of the gun Peterson As a function of redshift depends on first of all it depends on the cosmology so it depends on the cosmological parameters like the this is the matter density parameter depends on the Hubble constant and the the density parameter of variance and it depends on redshift as Powered three ups. So immediately you see that as you go to higher redshift the you would expect from simple basic principles that The optical depth should increase and it increases because this this term is driven by the increase of the density of the mean density of the universe you go to higher redshift. So as you go to sample higher and higher Sources then you would see towards them an optical that which is increasing but also it depends on the on the relative abundance of Hydrogen atoms with respect to the total number of atoms Total hydrogen atoms. So this is what we call before the x h1. So the neutral hydrogen fraction You see that the important thing is that this number is very large The coefficient here in front of that is very large It's Roughly five times ten to the five that means that even if you if you put here in a number In a neutral hydrogen fraction as small as ten to the minus four then you still You're that still gives you an optical that which is much larger than one and so you have a complete Gun pit is an absorption. So the the Actually that depends on the fact that the cross section of the Lyman alpha is very large And so as soon as you have a little bit of neutral hydrogen the Absorption capacity of that little amount of neutral either is very large So that would obscure very distant very distant Quasars and so this is the original paper of gun and Peterson that That I strongly recommend you to read because a very nice reading is also an example of how you can write a very important Scientific result in a little bit more than two pages. So it is very short and sharp Clear, I mean, it's beautiful. So go ahead read it. And so I think I'll just finish for today by showing you what we what we know about the The data so this is the gun Peterson effect. I just show just Illustrated you see that we have quiz as a spec different spectra from quizzes located at redshift 5.7 up to Redshift 6.37 and you see immediately that apart from the fact that the the Lyman alpha feature Which is the peak here that I show you also before is shifting towards larger wavelength This is expected because we are going to higher redshift. So it's just a red shifting of the of the photos But you see that while the Lyman alpha forest that was very prominent Remember the the original plot that I show you of the quasar with redshift 3 So there was a lot of Lyman alpha forest and a lot of features here See that these features here are disappearing and more so as you go to our to go towards high redshift And this is because each individual because the gun Peterson optical depth becomes Larger than one as you go to a redshift and therefore the all the radiation that is Short words of the Lyman alpha gets completely Blocked it's interesting to know that even if on average the you see that the the the the flux goes to zero Here there are some regions of of transmission. So that means that even a redshift 6.2 That there are also some there are still some some Regions where the gas is locally a little bit ionized. So it allows a little bit of transmission of these photos so that means that indicates that the Realization is the first in that the realization. It's a little bit patchy So some regions have been ionized some others are not yet completely and we will see this later on when we study the realization so the gun Peterson Evolution of tau is a functional redshift that you can get by analyzing all these Quasar samples shows a steady increase With respect to redshift and then there's a very sharp Rise year of redshift 6 that that has a power even larger than 10. So this is this is very Important and that is a maybe is a signal that We are entering into the realization epoch in the epoch of realization Around redshift 6 where the gun Peters of the gas starts to become considerably Neutral you can also transform this tau into using the formula that I showed before in in the volume-feeling fraction of Of h1 again shows the same trend So as you go to very high red she looks like that they run between redshift 6 and 6.5 something dramatic happens and instead of smooth increase in the gun Peters on that you would expect from simply from the Increase of the density of the mean density of the gas now something very dramatic increase that that pushes the optical debt to very large values and to That signals that we are entering in the epoch of realization So I think I'll stop here for today. So you have five more minutes if you have questions I will be happy to take that actually Given that I have a microphone I can start so in the Ratio between hydrogen and helium in the IGM is is a good racer of the primordial one In what tracer in what sense is the same is is the one that you calculate from nucleosynthesis or it's Position you mean the the ratio between hydrogen and helium. Ah, okay Yes, they so far. There is no there's no indication from from the IGM that there is a that is In terms of the abundance there is a difference with respect to the primordial value I have to say that in order to do this you need to Correct for the neutral fractions right if you want to compute the true Nucleosynthetic ratio between alia and helium and nitrogen need to do do it for the total amount So total number of helium total number of Of hydrogen, but what you see is essentially is only the neutral fractures. You need to correct for that So it's you know, you can do it by the very uncertain measure There is no evidence that no no no within the errors though Actually, I also have a second question Can you show the data for the UV background? Yes, but I didn't get 30 yes, how do you how do you get these measurements? Oh, these are essentially see they are in the x-rays see these are if you look at there around 1 kv So you can from the x-ray background you can measure these are essentially Chandra and XMM data From from the background says you can measure the the spectral the spectral evolution of the of the background In the average over that be it is average over what? So you run average of her Next lady be no the next light Okay, so you show an average of her be Sorry, which be be the Doppler parameter you mean No, I'm saying that I Was saying that when you measure be In this case you you you're the be that you measure includes not only the thermal motion of the gas that Related to the given temperature both of the bulk motion for example, if you have You don't have turbulence, but you're something that similar to turbulence in which for example You have gas is accreting onto the filaments So you don't know exactly if there are other other motions that are not related to Random motion Related to temperature. So but when you measure you measure both. So you it's up to you. That's why There is a spread. I mean there's a large spread Yeah, there is larger than what you would expect from the temperature alone. So there must be some other Component due to bulk motions of the gas and but the fact that That you have it seems to be a lower envelope that means that this is the minimum amount of Broadening that is related only to the temperature and the rest is added by by bulk motions What I'm asking is is this feature you've seen it all the directions like you yes, they see yeah This is each point here. Well Each point is a different system in different quasar spectrum So you do it for several quasars and for each quasar you have several systems, right? So all these points refer to I guess these are 19 quasars Well, I'm not sure. Well, these are this one. So three six nine nine quasars and each quasar different systems So you put them all together In one of the slide yes, you have showed the average of that optical depth term Yeah, so what is that average taken over? Yeah, this is the average over over all the observers that that the all the Absorption system that you have in the quasar spectrum So each one for each one you can measure how much Flux is absorbed by that by the system with respect to what you would expect from the continuum and then you measure You do it for each observing here for each absorption system in your spectrum, and then you have a job at that So he's an ensemble average. Okay Thanks. Any other question Running exercise Well This is the this is a budget very on that it's a It's a complicated study because there are many uncertainties, but certainly what we know for example what you do first you start by by checking the The stars right so the first thing you do is check stars. How do you do that? You have luminosity functions of galaxies. So you take Luminosity function is a function that explains that tells you how many How many galaxies are per unit luminosity per luminosity been okay, so by integrated over this luminosity function You get all the light produced by those galaxies, but then you have to convert and this is another uncertainty Light into mass, okay And you do that by knowing what type of stellar population these galaxies made from and so on so forth So you get the mass of the stars so that is one one So you get the stars and this is at least, you know for sure that all these stars here can only make 10% you do it for different type of galaxy this exercise for ellipticals Irregular spirals, so you have the gas then you're left with the rest of the gas. So the hot gas is measured essentially through Two ways the hot gas you do it through x-ray observations So again you for example look the particularly the hot gas in in clusters you measure through x-rays So you know the x-rays you have to convert again from light to mass and for cool cool gas you do it with Neutral hydrogen so a 21 centimeter observations, so you measure all the amount of cold gas There is this intermediate regime Which is that by the way that the one that is the most difficult to determine with temperatures between 10 to the 5 and 10 to the 6 which are too cool to produce x-rays and are too hot to produce 21 centimeter to have to be neutral So these are essentially left. This is the most uncertain part, but this is what essentially you get it by subtraction once you have Well, you have the you have to comply with the total as to be the omega-barium right that you have from different Constraint for a C and B or or any other so the total you know the total and And you have you you count all the component that you can measure It is only one that is very hard to measure but that you can get it from subtraction. That is what we can do for the density fraction density of h1 to what It was measured to what to what redshift in the picture when you put the the gun Peterson No, no, not the gun Peterson's The distribution The the density of h1 The densities density coding this one. No, no density distribution No, no the density and function of redshift Density as a function of redshift Density of new riders and fractures. So the one of the last ones. Yes. Okay. Is this one? No, no, it's just This one yes to what Redshift because we know that at higher redshift the wavelengths get stretched and We don't get any absorption. So how can we be accurate on Well This has been obtained as I was mentioned before remember the way in which this is obtained Okay, so you take you decompose your your observed spectrum into into components and For each one, you know the column density the amount of gas that it has and the be parameter And then you simply take that value and you You do the distribution now You can you can obtain this function for different redshift And so typically these are these got this measure been obtained from a wretched two to five Okay, then if you go beyond five what happens is the spectra become like This let me show you. Yeah So the spectra they don't show the liman alpha forest ever. So it is nothing you can measure, right? So you don't you don't have features anymore. You see it's blank. Yeah dark So we can only do it up to redshift. You're ready at five point seven You still see a lot. I mean still see a little bit of the forest But there's nothing comparable to the case of redshift treated I was showing before in which the forest is obviously very evident at lower redshift, right? You see this one Here you have a lot of features as you go to our redshift is become Black zero, right? So you can't do the experiment anymore. Okay. Yes well, you know that strictly speaking the the full homogeneous UV becker is only established once the The mean free path of all the photons becomes longer than the elbow radius. That's the form of the fish So so that each point in the universe is the same Intensity okay, and that is the the the mean actually the we will see in the next lecture, but The this condition is achieved essentially at realization or slightly earlier. I should say but And that it's around redshift six. So If it depends on where you think realization has really ended But so by the end of realization The mean free path of photon is comparable to the elbow radius and therefore you establish a truly homogeneous Position independent UV background intensity before that there are still large fluctuations And the more you go to higher redshift fluctuation become even larger But the condition is that the mean free path of photon has to become Equal to the elbow radius in order to have a background. I Think it's time for coffee. Let's thank Andrea