 The number of things, the number of parameters that describe, for example, the production rate of photons, their escape into intergalactic medium and star formation efficiency, so there are free parameters in any realisational model that do not depend on your ability in da je pograbila tradskehώςih posebnih detajov. Da, nekako na mostičnju sezativanju. Teh mora resultočnih, je tudi na parametri. I nabig nald, z tem, da je ino preizülen za nasna crashed. Zato jer觀elim, da je jaz vsoječno vsične, vso parametri, da jih se vših košali odinivno, če sem si je vzetelja, da je svešal za realizacij. Zanim, da smo njiha, vse tudi se pravamo, da se pravamo, na najbolj stran, tako, da smo nača vzeteljene vreštno, tako, da je tudi izgleda, da so lahko tudi trvi jaz, da se ZDM počeče, je zelo pošljenja, ki je dobrovaja učinjavce. Zelo smo se videli, ki je to vse občine, ki je prijevna kod prvih vsevnjamočnih nisotropijov z vsevnjih NB za tudi, je to več zelo še več smal. Tudi vsevnjičnih nisotropijov modulji je obzoril v zelo vsevnjičnih zelo vsevnjičnjih z vsevnjičnih nisotropijov. zelo tega neče zelo začnjenje, vse občasno je neče odstavnja. Nača smo prišli povedanje, da je tudi začnil iz njo izvastovati, da je produkčenja na maloči vsekondarijanja izotropij. Vsekondarijanja izotropij je prvno spetrem, kako se je bila vsekondarijanja izotropij vsekondarijenja izotropij Does, as a result of a modulation of the ionization field with the velocity field and the CMB. Today, just, like before we go into the, we dive into the 21 centimeter physics, I like to discuss few more points on the CMB. In partic 옆uh the large-scale signal, remember that we have defined this electro scattering optical depth, this is something that we covered yesterday, so I don't NE, že ni veliko dolog težne, zato bomo zvonati in neče težne ob saat, po pozdravu elektronu, kdo bomo se pogra, z qualities Cmb-smine, ki ima nekaj zepet, in nam je tudi tomson intensitve interaktio, ki obstartuje in angularne presne, czyli to, da se, are somewhat proportional to the amplitude of this value. So this is a very important number, and this is something that can be in fact derived from CMB observation. We have been lucky that in the last few years we had two fantastic experiments like WMAP and Planck. They have been observing the CMB with exquisite precision, and in particular they have been able to also to obtain polarization maps, because as we will see in a second, the realization affects the polarization spectrum of the CMB. So these maps are different frequencies, and these are the clean maps after we remove all artifacts, the galaxies, and foregrounds, and so on and so forth. And so these are the clean maps. And basically this is the power spectrum that we can get. This is the standard CMB temperature power spectrum, but this is the polarization power spectrum, and this is the cross power spectrum between E-mode and T-mode of the power spectrum. So as you see immediately, the effect of realization is to move from the black curve to the red curve here, and the data that are measured, these are WMAP data, by the way, allow us to set a constraint on tau, so because from the amplitude of this deviation it's possible to derive what is the value of the electron scattering optical depth, and for WMAP the value that was obtained, or the Planck and WMAP, first release of Planck in combined with WMAP, was given this value, which is about 9%, which corresponds in the instantaneous realization model that we discussed yesterday, is often used as a first guide by CMB people that if you want to transform this tau e into a realization redshift by using the previous relation that we have here, so you have tau, so you can get a realization redshift if you assume that, for example, you prescribe that the realization evolution is a step function, then you get something that it's of the order of realization would have occurred at redshift 10, so let me, 10.8 plus or minus 3, roughly, so now with Planck we have improved this data considerably, we have increased the precision, so this is a measurement of the EE power spectrum by Planck, fortunately they don't go yet to the very low L mode, which are the most affected by realization, so we are still using here, we are combining this data at the moment with the WMAP data at low L, but of course the errors here that are so small that can help us improve that limit, and in fact that value has changed now, so this is the latest value that we have available from the latest Planck analysis published this year, so this is one month ago or maybe two months ago, so tau it's now of the order of 6.6% plus or minus 1%, 1.6%, so you see that the error are decreasing, but also the value of tau has been decreasing since in the last 10 years, so each measurement has been contributing to decrease the value of tau, and that means that it's bringing realization closer to us, to a lower redshift, in fact now the redshift range is still certainly above 6, but now it's constrained to be between 7.2 and 10.5, I reiterate that this is a redshift that you should take with some grain of salt because clearly it's based on the assumption that realization is a step function, when you do something a little bit more complicated you see that it's possible to have different values, but nevertheless the fact that tau is certainly decreasing depends on a better treatment of foreground, essentially the tau measurement is affected by the presence of dust in the galaxy that is producing noise, and then the foreground that you need to be able to subtract carefully and Planck has done a very good job on that, and so we think that we are now more or less stabilizing around this value of tau, so in this case, I was saying that in addition to the CMB though, we have a number of other experimental constraints that we can use in order to build models that are fully consistent with data, because as I said before, there are many free parameters that we cannot control, and so we need data to constrain things that we don't know, and among these there are things that should obviously sound familiar to you after these three lectures, so the first is the Gump Peters on tau that we have discussed in detail, and together with the Lyman alpha that you can also do, devise the same type of quantity like the Gump Peters opacity for the Lyman beta transition, the electron scattering optical that we have just discussed, the UV background intensity, I touch upon that in the first lecture, so this is another constraint that is given, and then there is the redshift evolution of Lyman limit systems, so what are Lyman limit systems? Remember when I showed you the simulation of the organization, there were these islands of neutral gas, which are something between the intergalactic medium and the galaxy itself, so they are moderately overdensed regions that retain some neutral hydrogen sufficient to absorb considerably the UV photons, and so from their number, as the number evolves with time, we can also have constraints on the organization history, the temperature of the intergalactic medium discussed, then there are others that are probably less important, just to give you an idea that there are many constraints that the organization is tightly linked with the number of processes it has to do, of course that has to do with the galaxy formation and the intergalactic medium evolution, so now what can we do, one approach is just to run simulation, but these simulations are very expensive, and you cannot vary your parameters to explore the parameter space as thoroughly as you would like to, so another approach is just a little bit like in the spirit of CMB physics, in which we measure a power spectrum, and then we have a cosmological parameter with many three parameters, like omega m, omega b, all the cosmological parameters, so what we do, just we do some type of statistical analysis based on, for example, Monte Carlo Markov change, just to adjust the parameters to their best fit to the data, so you can take the same approach for realization, just having a relatively simple, but physically motivated model, and then you can try to do the same type of analysis in order to fit all the available data that you can find in the literature, so this is an example of such an approach in which you have a semi-analytical but physically complete model that depends on the parameters that we don't know, as the ones that we mentioned already, and by doing principle component analysis or Monte Carlo Markov change type of study, we can derive the possible realization history, so you are not finding a single realization history that fits all the data, but you find fiducial models and give some idea of the confidence level that the model has with respect to the data, so these are six predictions, for example, for one of these models that are tuned to fit the data of different types, so what do we have here? The first is the famous parameter that the number of ionizing photon per barrier that you put in stars that go into the IGM, the second here, this is the photonization rate, remember we defined the photonization rate as a function of redshift, and so these are the data points that are measured up to redshift six, so you have a model that fits that and that this is the prediction, or this is the number of limelimid systems, the one that I just described, so the number of neutral islands that are left over by realization as a function of redshift, so you predict this, and this is the feeling factor that we have also defined yesterday, so this is how the feeling factor grows as a function of time, there is a spread, of course, but you see that you can get models that essentially complete realization between six and ten, so which are certainly within even the latest WMAP determination, and so this is probably the key function that is the evolution of neutral and neutral hydrogen fractions, so the universe starts neutral and then the neutral hydrogen fraction drops almost precipitously down to very low values that we measure redshift four, five, and six, so these are the gambiters or limits, and also we have the, we can exploit the CMB data not only in the, for what concerns tau, which is a single number, but there is more information that can extract from the CMB that has to do with the full power spectra that you have, so you have many data points here, so maybe it does not make sense to fit only one number, which is tau, when you have all this wealth of data, so if you fit all these models together, then, sorry, you fit all these data at the same time, then you get an idea on constraints and bounds on how realization may have proceeded, so any, this is something that any theoretical model of realization should satisfy, because we have this data and you cannot invent any possible realization history without fitting those data. Now, what do we, something that we have learned, once you have this model, you can then do other things, so you can ask questions, we have fixed the model, so you have a fidučal model with some possible, with some confidence level prediction, and so you can ask questions now to that model, so for example, if you ask me now, what are the, what are the galaxies that mostly contributed to realization, so you can ask the model the question and here is the answer, so basically what we have here, this is the fraction of ionizing photons that have been pumped into the into the IgM produced by galaxies residing in halos with mass larger than m, where m is written here, okay, so these curves show at any different red shift ranging from 6 to 10, so this is 6 and this is 10, so this is the fraction of ionizing power that has been produced by a given type of galaxies, and you see here that, for example, by red shift 7, which is close to the end of realization, you find that more than 80% of the ionizing power has come from galaxies that reside in halos with masses less than 10 to the 9 solar masses. Now, these are very, if that doesn't tell you much, I can tell you that, for example, for comparison, the halo of the Milky Way, our own galaxy has a mass of 10 to the 12 solar masses, so that means that the galaxies that contributed most to realization are galaxies that are very small, these are dwarf, tiny galaxies, so even smaller than the Magellani clouds, which are our satellites, so these are one of the key points, there are many other things that you can extract from these models, but one clear answer that comes out of that is that if you want to do realization, you have to do it with a large number of small objects that are very well diffused, that can efficiently realize the gas. So another thing, and this is the last point I want to make on this realization, is that once you have your fidušal model, you can even try to explore and extract more information from the CMB on the degeneracy between realization model and cosmological models, and again so this is the figure I showed before, these are the data from the CMB for the three power spectra, so what you can do is just okay, let's now, instead of keeping the cosmological parameters fixed, like we know them and we only vary the astrophysical unknown parameters to fit the data, now once you have your fidušal model, which gives you the evolution of the neutral hydrogen, the electron fraction or neutral hydrogen fraction, which is the same, this is the fidušal the fidušal model that you have derived from the previous analysis, then you can explore variations around that and you can expand in principle components the actual evolution of the electron fraction in terms of some eigen functions of the fissure matrix that describe the relation of the dependence of the polarization on the electron fraction, so these are some eigen function, and so you can compute the amplitude, the principal component amplitude, it's a little bit technical, but just to give you an idea that you can do an analysis that at the end of the day allows you also to explore the dependence of the link between the realization and the cosmological parameters, so from this analysis, this is the, for example, this is the W map values that you would get, for example, for omega b and the primordial power spectrum index, without, with the step function this is what is usually done by CMB people when they do their analysis, now if you look at the third column here where we also add the fidušal model for realization derived or fixed by satisfying all the astrophysical constraints, the gan Peterson and the thermal history, blah, blah, blah, then you see that there is some variation which may or may not be significant, but certainly this is probably a more complete analysis rather than taking a single step function, so that means that the realization is also a cross link with cosmology, so the cosmological parameters that we derive are also affected by the way in which realization proceeded because the two things are definitely a couple, so now it's time to go to the final part of my lecture that I have to do with the use of 21 centimeter mapping to study realization and not only realization, of course 21 centimeter is an entire new area of knowledge of the universe because it allows us to push our observation, our experiments into a realm of the very young universe where there is no other way to study the processes that occur at that time, so essentially within the first half billionaire, the first billionaire of the universe, so 21 centimeter as I'll show you is, of course we are coming from the perspective of cosmic realization, it's very important for the study of cosmic realization, but it's also important for many other applications that I'll try also to give some example of those, so why do we care about, why do we want to look, why do we want to use 21 centimeter to study realization, well or in general the irachiv universe, the thing is that as we have seen already, the gum Peterson opacity increases very rapidly as you go beyond ratio of 6, so the universe essentially becomes completely opaque to UV radiation and so this is bad news because a lot of energy produced by galaxies and quasers comes in that region, so the universe is becoming opaque, on the other hand the C and B as we have just discussed through the electron scattering optical debt provides only integrated measurements, so we have this electron scattering layer and we can only have measurement that depend on the integrated depth of this layer, so we are not sensitive to specific redshift, so if I want to ask what happened exactly redshift 7.85, I can't answer that with the C and B, but as we will see I can do that with the 21 centimeter experience, so what is that 21 centimeter line, it's the line that corresponds to spin flip transition of the neutral hydrogen, you know that hydrogen is proton and as an electron and there are two states in that determine the hyperfine structure of the atom in which the two spin can either be parallel or anti-parallel, so and that is the coupling between electron and proton magnetic interactions, so this is a line that it's very important and has been discovered back, it has been used in astrophysics back in the fifties when van der Hulst, a Dutch astronomer had the first idea to use this line to study the neutral hydrogen distribution in the universe, of course that makes sense because as we know hydrogen is by far the most abundant element in the universe, so if you find something that can powerfully trace the most important component of the baryonic fluid, then you have a lot of advantage in understanding the mass distribution in the universe, so as I said this is the fundamental thing, so we have the line that is emitted when the there is a spin flip between the of the electron, so from parallel to anti-parallel and so the line that has been emitted as a specific frequency of 1420 megahertz that corresponds to essentially 21.1061 centimeters, so this is the line that is emitted by neutral hydrogen atoms, the atom has to have an electron and a proton, so ionized gas would not emit that. So again we have to deal a little bit, if you want to understand the intensity and the transfer of this energy that is emitted, we have to solve a radiative transfer equation which physically is similar to the one for the UV ionizing photo that we studied already, but in this case you are dealing with the line, so this is line transfer, line radiation transfer, so the line has a specific line profile which called phi of nu, so it depends, the intensity of the line depends on frequency and we have the energy of the photon and then we have two terms here, again we have a source term so clearly the line is powered by transitions from the upper level in which the two spins are parallel to the ground level in which they are anti-parallel, so for now on I call it 1 and 0 respectively, so the intensity is proportional to the number density of atoms that are sitting in the excited level times the spontaneous decay coefficient and then we have an absorption term which is again the number of, so we have the intensity field and this can be absorbed by atoms that are sitting in the ground level that are excited to the level 1 and this is the relative coefficient minus the stimulated emission, so these are essentially it's a stimulated emission that occurs when the photos from the, of the radiation field interact with an atom in the upper level and make it decay, so this is a stimulated emission, so a and b of course are the coefficient, so this is the equation that give us the intensity of the line and so as we have these two numbers there that are appearing in the radiative transfer so the number density of atoms in the excited level versus the number density of atoms in the ground state so we have to the degeneracy factor, so remember that the upper level as the degeneracy 3 while the ground state as the degeneracy 1, so the ratio is 3 and then we have an exponential of the contents of ratio of two temperatures t star which is the energy difference between the two levels, so the energy difference corresponding to the 21 centimeter line corresponds to a temperature of 68 millikelvin, so it's a very tiny, it's a very tiny energy difference between the two levels that in terms of energy is roughly 6 micro electron volts, so it's a very tiny energy but yet it's what allows us to detect this gas and so and then we have another temperature which is fundamental for the 21 centimeter which is the spin temperature, so the spin temperature actually you can read this formula as a definition of the spin temperature, so the spin temperature expresses in other words the ratio between the excited and ground level of the hyperfine transition so this is by all means this is a definition of the spin temperature which is, keep in mind, is related to the level population ratio of the two levels. Now usually because t star is so small, it's only 68 millikelvin, in all essentially all astrophysical conditions t star is much smaller than the spin temperature and t gamma which t gamma I indicate the C and B temperature at that redshift, so from now on remember that t gamma is the C and B temperature that evolves like t zero which is 2.73 Kelvin times 1 plus z, so because of that because t is small then usually n1 is close to 3 times n zero so it's just the general factor that means that the stimulated emission is in general important because remember that in the previous slide that was showing you that this is the stimulated term so n1 is larger than is almost 3 times n not and therefore stimulated emission is something that we cannot neglect when we try to predict the 21 centimeter intensity. Now in this type of studies usually you define another quantity which is important and it's called the brightness temperature and this is the way in which essentially you measure the intensity of the light, so instead of referring to an intensity we refer to something which is probably more familiar to us, it's a temperature and in particular it's the brightest temperature, so what is the brightest temperature? The brightest temperature is the effective temperature at which you can put, at which you should put black body radiator in order to have the intensity that you measure, so instead of parametrizing the intensity of the radiation field by some number i you refer to black essentially, the brightest temperature that is the temperature at which a black body wouldn't be the same amount radiation at that frequency. Now because we are working at radio frequency, we measure 21 centimeter, it's a frequency that it's part of the radio regime, we are working on the Rayleigh genes tale of the black body function, the Planck function and so we can work in this approximation that in which Tb can be written simply like that because you are just working the Rayleigh genes regime where you can neglect the exponential drop-off of the V in tail, so this is the brightest temperature, it's simply related, it's almost proportional apart from the frequency to I nu and the radiative transfer equation that I wrote before for I can now be written in terms of the brightest temperature and so you write the brightness temperature that in the rest frame of the source here, you're still working the prime in the case that we are still in the in the rest frame of the source because there is also red shift in radiation of course, so the brightness temperature is written now by introducing the spin temperature that that I introduced before, it's written like the spin temperature times 1 minus e to the minus tau nu, where tau nu is again the optical depth of the line that I'll come back in a second, plus the background radiation that you have along the along the along the line of so the this background usually is the CMB but because we are interested in cosmological application of the 21 centimeter but if you're doing if you're using 21 centimeter observation locally you could be you know some radiation from gas behind your cloud or whatever so in general this is whatever it's coming from behind the emitting region in the 21 centimeter then keep in mind that when we talk about the observed brightness temperature there is an extra factor 1 plus z if you move from the cloud framework to the observed framework. Now tau that enters in the the enters in the in the in the previous equation in the radiative transfer equation it's simply as we are used to is the is the integral along the line of site of your absorption coefficient per unit length. So tau is a non-dimensional quantity alpha nu is the unit of one over length so multiplied by a length is along the path you integrate and you get your tau. The absorption coefficient is exactly what we saw in the in the first slide. Remember that I told you there is a source and a sink. So this is the sink that enters in this equation and essentially this is the expression for the for the absorption coefficient that depends on again on the population of the levels and so on so forth. So now once you if you want to to well it's very it's very useful to make a parallel with the computation of the gum Peterson optical depth that we that we did so the it's exactly the same the same thing in that case we are considered the optical depth due to limon alpha scattering in this case we are we're looking at the 21 centimeter line but the physics is the same so it's a little bit it takes a little bit of math but it's relatively simple you have to do the integration and substitute for the Einstein coefficients and and also choose a profile that usually is taken like one of the width of the line so it's like a top hat with a given width so there are some some you know technical things here but at the end of the day this is very easy and you compute tau of the one zero transition that depends on Einstein coefficient the spin temperature frequency so these are all things that are more or less constant and then we have know that we have transformed the number density of atoms into total number density and this is a fraction of neutral hydrogen that you have at that time know that this is exactly what organization models predict so organization model predict how the neutral hydrogen fraction evolves with redshift and finally because it's a line and you have a profile also the peculiar velocities along the line of sight may change may enter into the computation of tau because the line is a shape and therefore velocities redshift of blue shift the the absorption sorry the redshift of blue shift the photos that they that the atom sees if the atom is moving along along the line of sight so then you can recast there's nothing other than some definitions here that you can find in test books probably so you enter so the the elbow constant that enters in the you can write in terms of the of the of the elbow constant because the density can be written in terms of the critical density so on so forth so there are some passages but at the end of the day you find simple expression for tau that depends on the over density of the gas remember the delta is the fluctuation amplitude with respect to the mean depends on redshift so tau increases as you go to i redshift as that reflects the fact that the universe becomes more and more dense as you go to i redshift neutral hydrogen fraction spin temperature and some cosmological factors now actually the final step before we can can make actual prediction is the as to consider the fact that what we observe it's the difference between the emission from our neutral patch of the universe and the c and b okay if this so what we actually measure is the contrast between the that that is the difference between the brightest temperature of the emitting neutral patch and the background the temperature of the of the background radiation which as i said in this case is the c and b okay so so in order to to get that we use our modified equation for the radiative transfer remember that this is the radiative transfer equation and we use it we use this thing in the in the small tau limit because we have seen that tau is very small so in this case in the in the small tau limit that equation simply reduces to simple expression which is the difference between the spin temperature and the c and b temperature at the redshift times tau and then there's one positive factor because this is the absurd one so given the expression that we had before for tau so now that reduces to a simple expression for the brightness temperature of the of the 21 centimeter line which again it's proportional to neutral neutral hydrogen so the first thing we note is that from a fully after realization if x h1 goes to zero the signal goes to zero because there is no h1 to emit the the intensity will be stronger from more dense regions of the universe where delta is larger will be stronger at higher redshift and there is this factor here which is very critical now what happens if ts the spin temperature is larger than the c and b temperature then this factor will be positive and tb will be positive so we see the signal in emission okay against the c and b so the cloud will be brighter than than the background however if ts is less than is smaller than than than the c and b temperature at that redshift then the signal will appear in absorption because this term becomes negative so we can have either a emission or absorption depending on the relation between the spin temperature and the c and b temperature note that in emission the signal saturates because at best this term can become equal to one but if in absorption this factor can be arbitrarily large so you can have while while you can you're saturated in terms of the maximum amount of emission that you can see in 21 centimeter if you are looking it in absorption the signal could be as strong in principle could be infinitely large okay so this is an important thing so that that tell us immediately that absorption would be probably easier to be detected than emission okay so we've seen that in this so basically this formula is telling us that all the the actual physio what happens is determined by this ratio okay apart from the amplitude but this ratio is telling us if the signal is emission or in absorption and how big is the signal so ts is the quantity that we need to compute and so far we have said nothing about that but so we need to start to investigate the how to determine the spin temperature of the 21 centimeter so the 21 centimeter remember the spin temperature remember is the essentially a way to express the ratio of the population at the excited level with respect to the ground level so there are three physical processes that govern the the spin temperature one is the interaction of c and b photos of c and b photos try to try to force the atom to be in thermal equilibrium with the radiations who have the correct distribution relative distribution of exciters versus ground state electrons that is relative to the temperature of the c and b so that drives ts to t gamma on the other end the hydrogen atom also have collisions with other particles could be other hydrogen atoms or electrons if there are free electrons and so these try to scramble or mix up the level population and therefore by doing so they drive ts away from the c and b temperature and finally they are also scattering with uv photos a very important process actually that again mix this hyperfine level structure and so they move away the population from the thermal equilibrium population and so this again drives ts away from t gamma so how do you put the oldest how do you compute the the the relative importance of these effects what you have to do is write down a detailed balance equation that it's expressing the the population of two levels n1 and n0 as a result of the combination of these various processes so here for example the level electrons of level 1 can go to level 0 make a transition a de excitation due to collisions or uv photos can also do stimulated emission that drives the electron to ground level this is a spontaneous emission and this is stimulated by the c and b and the inverse processes are also the ones that produce a transition from the ground level to the upper level so this essentially if you solve this equation that gives you the ratio between n1 and n0 and with the definition of ts you can get ts you get the spin temperature is determined by that so this is a essentially the way in which you compute the temperature the spin temperature now this this equation if you now introduce the temperatures the the definitions of the brightness temperature and the and the spin temperature that we had before so the previous equation in the religion's regime can be written in terms of the temperatures remember that we have we have that freedom that because we are working in the radio regime so the previous equation can be written in more sensible way like that where now we have number of temperatures that are appearing so this is the spin temperature this is the c and b temperature this is the kinetic temperature which by definition is defined by the ratio of the excitation and de excitation coefficient by collisions which is given by the standard Boltzmann equilibrium and so you define the kinetic temperature in the usual terms then we have another temperature that refers to the to the uv radiation field now if there is some uv radiation and then we'll see how important is that so that is defined that defines a color temperature in the same way formally is defined as the ratio of the excitation coefficient and the excitation coefficient due to the uv irradiation of the atoms and finally you have this xc and x alpha which are the the coupling and coupling coefficients now this color temperature it's essentially tells us about the properties of the radiation field around the liman alpha the liman alpha transition which is the one as we will see in a second that it's important now most often we can neglect the the the fact that tc is slightly different than tk because there is an interplay between hydrogen atoms and liman alpha photos there are recoils that that tend to equalize these two temperatures so these are the complications that we can forget for a second and so most often tc of this color temperature it's we can take it as equal to the kinetic temperature so in this case that equation can be written in a very simple way which tell us that the the different remember that this is the factor that is telling us if radiation comes this is this factor right so the factor that tells us if the 21 centimeter piece in emission or in absorption and what is its amplitude so it's a key factor so that key factor it's proportional it's to a constant which are these coupling coefficients times the ratio between the gamma the c and b temperature and the kinetic temperature now you notice imediately that for example in order to have this factor different from zero then this coefficient have to be different from zero so the collisional collisional coupling and the liman alpha or uv coupling have to be different from zero also you don't get anything also if the kinetic temperature is equal to the c and b temperature okay so there are conditions for which this factor is different from zero because if it is zero you see nothing okay so it's this factor has to be different from zero in order to see it either in emission or in absorption and that depends on this combination so let's concentrate for a second about the these two coupling coefficients so the collisional coupling and the liman alpha coupling the collisional coupling as a you know simple expression that depends on on some on the coefficient from the the absorption coefficient on the excitation from level one to zero and the ratio of t star the energy different between the levels and the c and b temperature and i collisions can be produced by either gen atoms or electrons now the interesting point about this xc is that there is a critical over density delta for which this xc this coupling coefficient becomes equal one for hh collisions and so this is the expression on which xc becomes equal equal to one remember that i told you before that if xc suppose for a second that we have no uv radiation okay so we are only left with xc the collision excitation so if the collision excitation becomes very small then that number goes to zero and we see nothing so the this expression is telling us when under what condition xc is becoming very small and therefore the signal disappears now a redshift 70 or one plus z equal 70 we take the temperature of the gas that corresponds to the temperature of an adiabatic expansion adiabatically expanding gas a redshift 70 which is 88 k and so we find that a redshift smaller than 70 xc becomes less than one and and very small and therefore ts goes to t gamma okay the spin temperature becomes equal to the c and b temperature and by redshift 30 the igm would become invisible okay so if you have no radiation no uv radiation in the universe you would not see the signal at redshift larger sorry redshift below 30 okay and barely up to 70 so because collision are not able to scramble the the level distribution very far from the c and b one so you see no contrast between the background and the 21 centimeters so in order to see the signal basically practically you need to have some sort of uv radiation field which must be created but whatever you want to whatever you want to think about but so it could be stars could be quzards or it could be even dark matter as we will see later now so this uv radiation produces a new type of of coupling which is is so important that i also deserve a name it's called the 4000 field effect from the name of two scientists in dachuan 4000 and george field was american astrophysicist that realize that the scrambling of the hyperfine levels which are these two ones so this is the one where the parallel spin this is antiparallel so what you can do if you want to instead of playing only with these two with these two hyperfine structural levels what you can do if you have uv photons you can do the following for example you can take an atom which is in the antiparallel state if you make it absorb a liman alpha photon that brings it up to level with n equal to and then from there you decay into the into the excited hyperfine state or you can also do other combinations okay so so these transitions that start from one level and end up in another one can mix up the the population of the hyperfine level away from the distribution that the c and b would is trying to impose okay so the the the coupling coefficient then can be it's obviously apart from constants is proportional to this quantity pa which is not in else that the integrate the the intensity of the uv flux integrated over over the liman alpha line because you remember that this transition corresponds to the liman alpha line emission or absorption so we have a rate this is a rate and this is the essentially the coupling coefficient that you would get from the uv so you need some uv in order to do this job and but the intensity doesn't have to be very large so it's very the the intensity is relatively small and this is obvious because also the the difference between the two levels is also very small and so you don't need a very large intensity to that but you need some some uv photons okay now the last step before we can do we can see what what are the implications of that is that we need also to to define the power spectrum of the of the brightness temperature so the or as we said more in short the power spectrum of the 21 centimeter so what you do you define the fractional perturbation to the brightness temperature delta 21 at the given position x in space which is equal to the different between the brightness temperature at that position minus the brightness the mean of the brightness temperature over your volume divided the mean so this is a zero mean field and what you do is you perform the the Fourier transform of that quantity to go in k space and so the 21 centimeter power spectrum is then define as the ensemble average over the of the convolution of the of the Fourier transform of the fractional perturbation two different case and and then you define p21 as that quantity with the delta direct k1 minus k2 okay so this is standard definition of a power spectrum and you can also what what we use usually is the delta delta square k which is p of k multiplied by k cube which give us an idea of the variance of the field of this quantity is proportional to the to the variance of the fluctuation of the of the brightness temperature field so let's go to now with all this theory now what can we what can we learn good so from from these calculations we the first thing that we can that we can obtain is the global history of the of the temperature so how the different temperature that enter in the 21 centimeter business evolved with time and recall you that we have the three temperature the c and b temperature the kinetic temperature and the and the speed temperature okay so the first line that you see here is the the dash line is the evolution of the c and b temperature not in special about that it just an evolution as 1 plus z that we are drawing here and then we have the the dotted line which is the kinetic temperature of the gas as as as you know the the gas is evolving adiabatically and therefore the temperature evolution of the gas drops like after recombination drops like 1 plus z square so it drops faster than the than the c and b so the spin temperature as leaves in between these two curves why is that because at high redshift the the collisions the collision or coupling is sufficient to keep the spin temperature bound to the kinetic temperature okay because the collision or coupling of this collisions between hydrogen atoms make sure that the population levels is is the one set by the kinetic temperature but as i told you before around redshift 70 the collision the collision the coupling of the xc factor the collision or coupling becomes inefficient and becomes very small and becomes so small that also becomes zero and you see that the spin temperature by redshift 30 goes to the goes to the c and b temperature so from as i was mentioning before anticipating before from here down you would not see anything so you see an absorption feature here because you know that the spin temperature is below the the c and b temperature but here you see nothing and so this is the fractional delta the variation of the brightness temperature okay so the brightness temperature that you will compute you see here goes to zero but here you have an absorption feature of the order of 40 mK which is substantial and that would occur around redshift 1900 okay so that is the that would be what you would see if nothing has happened in the universe after recombination so this is simply the simplest case in which are not considering so realization is not entering here now let's add realization and see what happens so now this graph is the same the same as before so we have a temperature as a function of redshift so this is the this is the c and b temperature and the thing line is the is the kinetic temperature and the thick line is the spin temperature but now so you see that initially again as before tk and ts are coupled and they are below the c and b temperature but then if you now add realization look at this curve first you see that realization produces a lot of heating so the because he's eating the gas remember yesterday we were looking at the temperature distribution within ionized region that was 10 to the 4k so it's much hotter than the c and b which is here at the order of 70 or 80 k and so so the what you see depends on realization model so these are two different realization model with different prescription and this is the evolution that has been assumed for different course it's not important now to see the details but clearly realization changes the the evolution with respect to the to the previous case you see that here the the spin temperature was going close and matching the the c and b temperature now we don't see that anymore there is a there's a growth so the after realization when realization starts the 21 centimeter starts to become to be to appear in emission rather than in absorption and you can see that also from the differential brightness temperature in millikelvin see we still have this peak around no 50 100 depending on on details in absorption and then we go in emission near of of the order of 30 30 millikelvin this is the signal that you would expect from from realization okay now we can implement all this in in in numerical simulations these are very large boxes of much larger boxes that that I show you yesterday so these are rough this is roughly one gigaparsec so it's a light cone of one gigaparsec size here and we're going for redshift 6 to redshift 178 here okay so what am I showing here is the brightness temperature differential brightness temperature in millikelvin now the blue is where the signal is in emission red and yellow is where is in absorption so this is essentially a 2d view of what we have seen before but now it's coming from simulation rather than from analytical models but the physics obviously has to be the same so at very high redshift you see that the signal is is in absorption this is the depth the very the deep that we saw in in in the previous slide so this one would correspond to something yes to this deep here so we are going along this this we are following along this path in redshift but now in 2d so so there is a in the first in the first part here which are called the the so-called dark edges okay because everything is dark there is nothing else the cmb photons and this c of neutral hydrogen and helium gas so the igm is called it in the cmb and but the the coupling the collision coupling becomes weaker and weaker and it disappears you see that goes to black here where the reason where the the collision coupling becomes very weak and therefore the the signal disappears but then as the the first light appears and we can now use the uv pumping the 2000 field effect then again the igm is still called the the cmb but there is a coupling the 2000 field effect or also some x-ray preheating if you have if you have x-rays there could create a very strong absorption signal okay this is around redshift 20 or so that it's a very strong signature of of the formation of the first star so the first luminous sources that is reflected into this feature and then eventually as these sources continue to pump uv radiation and energy into the igm the signatories in emission and then we are entering the the real epoch of realization in which the igm is hotter than the cmb and there is also strong spin temperature kinetic temperature coupling due to the 2000 field effect so at some point then the neutralizer will disappear completely and the signal drops away okay this is the the brightness temperature evolution that explains the explains the the global evolution of the 21 centimeter we can also look at the at the power spectrum i have a small animation from the same simulation now you can compute in i'll show you an animation of that you can compute the the power spectrum that that we have defined before as a function of the of the wave number in inverse mega parsec so as a year you have the redshift we have the mean neutral hydrogen fraction one means it's all neutral and this is the value of the delta tb the brightness temperature that there we see so let let's see if i can run if i can run an animation that i have for that okay i can't see it i see it on my screen but i can see it there i don't know what to do sorry i can see it for me but doesn't have that much i don't know why it's not showing up on the screen but okay forget about that we will we'll live without that yeah so anyway it was showing how the this line here was going going up and down and changing shape from the as the as the realization as through time right from redshift 120 to redshift 6 so it's the same simulation that we have seen before in terms of the global signal in this case it's just the the power spectrum that is that is changing and so by measuring this power spectrum we can really learn a lot about the the topology and the evolution of cosmic of cosmic realization now together we have not said much about the possible eating processes that are occurring as the first sources appear and that would make the signal make a transition from absorption to emission okay so what are these eating sources the eating sources can be of different nature very important is the x-ray heating from astrophysical sources as we have seen also yesterday all galaxies emit all to any star formation rating galaxies there is an associated x-ray emission and x-ray emissions are important because they they warm up the the the i g m on very large scales because the x-ray have a very long mean free path so they can eat up the i g m very even in a uniform manner so x-rays are certainly important liman alpha heating is it's also of some importance not as important as the x-rays so that means that when you when you essentially there is a recoil effect associated with the absorption of of liman alpha photos that that is then transferred into kinetic energy of the atom so is a small effect but is there we can have shock heating by for example if you have supernova explosion then that or also structure formation shocks that could eat the gas but also an interesting interesting possibility and this is the the last example i like to do before we as a final applications of the 21 centimeter connected to the realization but also connected to other important important problems like dark matter in cosmological context so another application of 21 centimeter would be to study the heating produced by the annihilation or decay by dark matter particles which is certainly very exciting at least to me perspective in the use of iridesh 21 centimeter so what is the so let me concentrate on that in the last 10 minutes or so and let me show you what we can do what we can learn on dark matter by using the 21 centimeter observation that we will do anyway for realization okay so what happens is that once you have an episode of annihilation of two dark matter particles then there is a the the injector particle whatever it is we can have we can have several channels of annihilation and decay so but in general terms you have some high energy injector particle with typically is of starts with 100 gv or one one tv depending on the mass of your favorite candidate dark matter particle that creates a cascade in which the energy is eventually thermalized so you can have all sort of particles or electrons or photons and and eventually you can have processes like photoelectric absorption, comptons cutting all I mean this is a very complicated cascade but we can model nevertheless that eventually produce we can have three final outcomes once is the thermalization of part of the energy is thermalized and goes into eating of the gas some of the energy is lost to ionize the gas and in particular hydrogen and also some other some other energy goes into limon photons which are also useful for us in terms of exciting the 21 centimeter line through the votuism field effect so this seems a very nice situation for the it's a perfect situation I would say for for the 21 centimeter because as I just said we have limon photos to excite the to to to make the limon the votuism field effect we have eating that again makes a clear signature of the presence of energy input from from something else which is not the the the global evolution of the igm and finally have also a little bit of ionization so fortunately ionization by this high energy particle is not very efficient so it leaves most of is a little bit like x-ray so it leaves is not it does not produce a full ionization of the igm and that would be bad because otherwise the 21 centimeter line would not be observed but they produce a little bit of ionization that that is also produces electrons and collisional coupling so can we model that in a little bit more detail yes so if you want to understand what are the effects of dark matter decays on to on to this we can write down the two equations the ionization equations and the energy equation that is telling us how the gas is ionized as a result of the dark matter annihilation decays so we are studying the case in which there are no stars yet okay so we are working at very high redshift so we are working in the in the conditions in which there is nothing else that adiabatic expansion and dark matter energy injection so the simplest situation possible which is the one in which actually we are interested in because this is the a clear signature of the effects of dark matter because there is only dark matter other than the adiabatic expansion of the gas so this is the ionization equation so how the ion the electron fraction evolves as a function of redshift which is produced by the ionization rate due to dark matter annihilation decays and the standard recombination so this is seen exactly the same as the granization equation that we wrote yesterday with the only exception that now gamma is produced by dark matter rather than stars and here again this is the evolution of the kinetic temperature that enters into the 21 centimeter business and again there are several terms but the important and important terms here is this energy deposition which could come from Compton eating if you have astrophysical access sources or in this case dark matter annihilation so this equation gives you gives you how the ionization fraction and the temperature evolve as a function of as a function of redshift so the eating rate that was the last term that show you in case of dark matter can be essentially written as the as is proportional to the to the annihilation cross-section of the dark matter over the mass of the particle and then there are cosmological factors here the critical the critical density and a boost factor that has to do with the fact that the annihilation not only come from the dark matter at the mean density but as we know there are enomogenities and non-linear structure and where the density is higher therefore the also the annihilation rate goes up because it depends on the square of the density and so this boost factor must be estimated from for example by feats to numerical simulation that follow in detail the growth of structure and therefore you can estimate or get simple forms for the for this structure formation boost and so you have a fit that that describes the average dark matter density enhancement from collapsed structure and these three parameters there are bh, zh and delta here must be determined from numerical simulations but they depend a little bit on the what is the cut of the dark matter power spectrum that you are assuming for example if you cut your dark matter power spectrum different masses like 10 to the minus 3 of the solar mass or 10 to the minus 9 of the solar mass these parameters change so there is a dependence on the spectrum of the of the primordial spectrum of the of the of the dark matter and so what is the minimum scale on which dark matter particle can cluster okay so this is the that number and depending on that you have some dependence on this b factor and therefore in the heating rate now to make a specific example showing you what we can do for example with the interesting dark matter candidate which is a light wimp with the with the mass of 10 gv that annihilates into two muons a pair of muons with the annihilation cross section which is given by this number so why do we choose that well above these different candidates is in the light wimp particle has been advocated to explain for example the signals that we have from the galactic center the galactic haze synchroton emission or also low energy signal from from direct detections like dama or crest even though you know that this we don't know exactly what what the what the particle mass is but this is a particle that is interesting because it's more or less consistent with all the constraints that we have so far but of course you can you can play around with any favorite dark matter candidate you may have so the the constraints that that you can get from from this particle by by fitting the by by looking at the at the for example of the power spectrum of the c and b so you can constrain the mass of the of the cross section versus the mass of the particle for three different channels of different annihilation channels in tau mu or in electrons and you can find constraints that are allowed by the by these experiments on on the mass of this particle so depending on on what type of channel you are assuming you can have different constraints but so what what is the result for the 21 centimeter then well 21 centimeter in this case considering just as I said before the channel in which annihilation course in the in the morning channel so this is the brightness temperature that we have seen many times so that the black line would be the the the case in which there is no dark matter decays but if you allow for dark matter decays which are the purple red and yellow curves then that correspond to different cuts in the in the spectrum in the power spectrum that I mentioned before so 10 to the minus 9 10 to the minus 6 10 to the minus 3 of solar masses in in the clustering mass scale so you see that there are strong deviations from from from this peak right from this peak that of course are 20 which is the absorption peak corresponding to to the the cosmic dawn just before the cosmic dawn at the end of the dark ages so if you measure this peak and this is something that we should know because it's simply due to the adiabatic special of the of the universe which we think we understand precisely so any deviation that you can measure from this black curve would be signaling the presence of dark matter annihilation and so depending also on how big the deviation is you may also be able to say something not only on the particles but also on the power spectrum of the of the of the dark matter as well so you see that for example if you have dark matter the clusters of very tiny scales one billion of the solar mass then you the eating rate is so large that you can also turn from absorption to an emission signal even before well before the formation of the star so this is very intriguing and also what you can do is just to look at the at the analogous power spectrum at the fixed scale so you fix the scale the power spectrum which in this case is 0.1 inverse mega parsec and you look at the evolution of the of the power spectrum on that scale as a function of so the plot is very busy so let me drive it through it very slowly so the black line here is the the standard model so the one without dark matter effects okay and so the solid line turns into in a dash line when the signal goes from absorption to emission okay so before I go into the other details let me show you that there are experimental detection limits here that these are instrument like lofar and w8 that are already available right now there is an upper limit by another instrument by paper here but in the near future we will have available two other instruments that go to interferometers here and SKA that will will be able to take these measurements in a in a in a few hundred hours actually with 1000 hours you would get down here so it would be a perfect representation of all the spectrum so anyway the black curve is the no dark matter fiducial standard case and then we have the the three cases in the dimension so 10 to the minus three 10 to the minus six 10 to the minus nine cut so the first thing you see that for example in 10 to the minus three and 10 to the minus six this peak that corresponds to the cosmic dome it's it's it's depressed okay so it's depressed because the the gas is becoming heated up by the dark matter so the power spectrum is decreased and also if you if you go to the stream case of 10 to the minus nine you've given as we were discussing before you go to a null point here where the signal turns from being absorption to being emission a very high regent okay so by looking all this all these features here we can be able to measure the presence and the nature of dark matter annihilations in in the very high regent universe before the the formation of the star so it's a very clean measurement that we should be able to do uh well SKA supposed to be online in 2020 so in 2020 we should be able if not anything else worked before to set very strong constraints on on dark matter and the and also on the mass of the particle so i just conclude with the with the summary of this of the effects of the dark matter because i went a little bit fast perhaps so the effects of the dark matter is that they depress the the the peak at the second peak of the power spectrum and interestingly the second peak of course by the signal is already the mission so this is a very clear prediction this feature cannot be produced by by any astrophysical source and in particular if dark matter dominates the eating considerably then the brightest temperature goes to zero before any x-ray source appears so a new detection of the power spectrum at very high red she would indicate that there is no matter and this is preheating the gas by annihilations and that in turn would lead us to strong constraints on the mass of the particle and it's a different way it's an indirect way to study the dark matter in the in the universe and i think i'll stop here