 Two remarks to start, first I appreciate those of you who spoke this morning being willing to exchange times with me so I could take my wife to the airport. As a consequence, I inherit a sleepy group having had a heavy lunch. The second remark is that this again is going to be a lecture about problems rather than results, problems that we face in trying to measure the BMOs rather than results. The results will be tomorrow morning. I'll present those. In the first lecture I explained probably in over great detail the experimental problems that you face in looking for the BMOs. Problems with the equipment, the atmosphere and so on that have to be dealt with before you can detect polarization in the CMB. And the aim today is to talk instead about astrophysical foregrounds that we can do nothing about and how to assess them and eventually remove them from the data. After all, think for a moment, we're looking at the CMB out at a red shift of around 1100. We see it through the screen of our own galaxy and through the screen of distributed galaxies as for instance was described a couple of days ago by Professor Jane. All of those intervening pieces of matter can in principle produce a mission that interfere with the detection of polarized signals in the CMB. And what I want to do today is to describe the various mechanisms that produce such outlines and then towards the end talk about how to get rid of them. So it's a two-step process, first which foregrounds matter and it turns out to be primarily galactic foregrounds. And then I'll talk in a little bit of detail about one particular galactic foreground and that is the emission by dust, warm dust in our galaxy which turns out has an important bearing on some of the recent claims of the detection of BMOs. And then towards the end, as I did the other day, I will talk about means we can use to control, to measure, to subtract, to remove these foregrounds in order to leave the underlying CMB signal. So let me start here with a list of galactic foregrounds that we need to take into account. There's a surprisingly or distressingly large number of foregrounds that must be considered before you can talk about the CMB. The first is synchrotron emission. Free free emission is also present and so on. And with apologies to those of you who are astronomers who know all of this material, I'll go over some of these emission processes briefly for those of you who are less familiar with them and then we'll use some of the properties that I talk about as means to eliminate these foregrounds. Synchrotron, relativistic electrons in the cosmic ray field interact with the galactic magnetic field to produce synchrotron radiation. It is intrinsically highly polarized and that of course is bad news if you're looking for weak polarized signals in the CMB. In addition, the spectrum of synchrotron emission depends on the spectrum of the relativistic electrons, the strength of the magnetic field and so on, so it's not completely predictable and we know it varies from place to place in the sky. It's frequent to talk about the spectral index of a continuum emission such as synchrotron, it's not a line emission, it's a continuum process. And we define a temperature spectral index which I'll be using throughout this lecture as the temperature, the measured temperature of an emission proportional to the frequency to the power beta. For those of you who are familiar with radio astronomy, this means the flux density goes up as new to the beta plus 2. In the case of synchrotron emission, as you go to higher and higher frequency, the emission drops off very rapidly, typically with a value of the spectral index minus 2 or something of that sort. And that's shown in cartoon form here, temperature and you'll notice the units here are in Kelvin as a function of frequency and I'll remind you that for observational reasons we like to work in this range of frequency. Here's the synchrotron emission. It's drawn as a band because we don't know its amplitude and it varies from place to place in the sky. But you can see that as you go to higher frequency it falls off very rapidly as suggested by this strong spectral index here. The next component that we need to take into account is free free or Brems-Strahling emission emitted by ionized plasma, primarily ionized hydrogen. The radiation emitted when electrons pass by a proton have their path deflected. The acceleration of a charge produces radiation. Free free or Brems-Strahling emission has a huge advantage from the point of view of microwave background observations that its spectrum is fixed. That is the value of beta is fixed very closely to minus 2.1. So we may not know its amplitude at a particular point in the sky but we do know its spectrum. Please keep that in mind. In addition as in most cases the free free emission is not dominant. As you can imagine since synchrotron is falling off rapidly and free free emission has a flatter spectrum it does cause problems if at all at the higher frequencies. Yes. What is? Oh yes it's just what I've described. You can think of it in two ways. First I'll describe Brems-Strahling emission. You have a plasma. The protons are moving slower than the electrons so we will pretend for a moment that the protons are fixed in place and we'll ignore helium and all the heavier elements. As the electrons move through the plasma they feel a coulomb attraction and their paths are deflected. That means of course that you're having a charge that is accelerated and the acceleration of a charge produces electromagnetic radiation. That's what we see. Alternatively you can think of it as free free emission, its other name. You can think of the proton and an electron as an unbound hydrogen atom. So there are non-quantized transitions possible from one energy level to another. A free electron can transition to another free electron state. In both cases you get electromagnetic emission. And in both cases for reasonable temperatures the spectrum is very tightly constrained. Number three is something that we did not expect until 20 or 30 years ago. And that is dipole electromagnetic emission from rapidly rotating small dust grains in the interstellar medium. They are small enough so that the absorption of a single optical or UV photon delivering let's say a few electron volts worth of energy sets them spinning at frequencies of the order of tens of Geyer Hertz. That is of order 10 to the 10th to 10 to the 11th Hertz. A little surprising. As I say we didn't expect this. But observed to be the case. It's only the smallest dust grains that produce this kind of spinning dust emission. But it's wildly variable intensity and in the peak of the spectrum. The one thing we can say, I'll just draw a cartoon of it using the laser printer here, is that it pops up quite rapidly and falls off quite rapidly. So it's a sort of bump like this at frequencies that are typically below those used for most B mode searches. So it's there but it's not a dominant source. So we have synchrotron which is a big effect. We have free free which we know a fair amount about. We have dipole radiation which is not particularly effective or dangerous at the frequencies we use. And finally and the big one here is submillimeter reemission from warm dust. The dust involves tend to be large, the dust grains involved tend to be larger. They're heated by the optical or ultraviolet radiation field between the stars and our galaxy. They're heated to temperatures of order 20 Kelvin and they emit. If they were big they'd emit its perfect black bodies at a temperature of 20 Kelvin. But because they're small the emission efficiency at long wavelengths much longer than the size of the particle is very small. So instead of getting a pure black body spectrum you get a black body spectrum that falls off with increasing wavelength. So if I can draw a quick cartoon here log of intensity versus log of frequency that's a typical black body spectrum. And for dust grain at 20 Kelvin this peak would be at around 100 microns. But the dust spectrum itself does not look like this instead it falls off more rapidly because of the reduced efficiency at long wavelengths at low frequencies. And typically the spectrum here goes as indicated here, beta something like 2, parenthetically if you did see perfect black body emission from the dust beta would be zero, temperature would be independent of frequency for a black body, right? It has a single temperature. But instead because of this fall off as indicated here what one sees is a spectral index of order 2. And the final thing we have to take into account for some experiments but not all experiments is the emission of spectral lines primarily from carbon monoxide in the galaxy. Now I've been at pains to go through all this detail which will be very familiar to some and probably not of much interest to others. I hope there's somebody in between. Because all of these effects get in the way or bother us when we try to see the CMB. And that's shown in this cartoon which you've seen before. Here's the CMB with fluctuations, in this case temperature fluctuations. And here are all these foregrounds that need to be stripped away one by one in order to get at the underlying signal. How do we do that? In outline we can use two different properties of the foregrounds to measure them and once they're measured to get rid of them to peel them away. The overall process is called component separation. And that's the name component separation carries an important message. What we want to do is to find each of the components, synchrotron, dust emission, free free, find it, model it and get rid of it. There are a couple of properties that can be used to do that which I'll be talking about for the remainder next 80 minutes or so. First we can use spectral properties. If the foreground has a specific spectrum it can be modeled. But as a particular spectrum the only thing that varies from place to place in the sky is its amplitude. The spectrum doesn't vary. However if you're trying to use spectral information you pretty clearly need to have observations at several different frequencies in order to determine the difference between the thermal spectrum of the CMB, let's say the different spectrum of free free emission and so on. So if you're going to use spectral information you're required to have multi-frequency observations. The clearest example of this is free free emission. We know its spectrum exactly. And we also know that the spectrum of free free emission is different from the thermal spectrum of the CMB. Free free emission has beta minus 2.1, the CMB has beta equals zero. So in principle observations at two or three frequencies can be used to separate those two components, component separation on the basis of spectral information. Another method involves using external data, not your own measurements of the CMB but external data of some sort which gives you information about the distribution of the foreground across the sky which I've indicated here by intensity as a function of the two angular variables of the sky. If you've got that information and if you can make a reasonable extrapolation from the frequency of the observations to the frequency you're using then you can model the emission and subtract it. A couple of examples here. First, remember that I pointed out that synchrotron emission grows weaker with increasing frequency. We observe typically at high frequencies but you can also make them map of the sky at low frequencies. At low frequencies synchrotron is completely dominant. So what you do is you make a map of the sky at low frequencies. That map essentially is telling you where the synchrotron emission is coming from and then you scale that map down to the lower emission at the frequencies at which you're trying to observe the CMB. So that's one example. All sky maps of synchrotron emission, the one that is normally used is made at a frequency of .4 GHz, in other words about a factor of 250 lower frequency. So the temperature is roughly speaking 250 squared greater in this map than in CMB maps. Another example which I'll come back to is to use maps of the emission of H-alpha, the Bomber Alpha line of hydrogen as a proxy for free free emission. Remember free free emission is a proton and an unbound electron. Bomber Alpha is a bound bound transition. It involves the same two elements of proton and electron. It just happens to be that they're bound together in a hydrogen atom. So it's not unreasonable to expect that H-alpha, the Bomber line of hydrogen, is a reasonable proxy for free free emission. And I'll show you that it is a little bit later. So these are the two basic techniques. You use the spectrum or you use external data sets which are in a sense maps of a particular component that you want to get rid of. And here's an example. Instead of doing the entire sky, let's take a small chunk of the sky, a small solid angle that covers, in this case, the small Magellanic Cloud. First let's focus on the observations. They're the points scattered along here. Some of them are Planck data. Some of them are from the earlier WMAP satellites. Some of them are archival data. But these are the observations, okay? In this case, it's flux density, not temperature, as a function of frequency. Intensity versus frequency. Component separation consists basically of saying which components make up this signal. For instance, the yellow curve is a synchrotron component. The free free is the black dash line. The dust emission from spinning grains is the green, which you can barely see here. And you can also see why we generally ignore it, it's not a big effect. And the last is dust emission. Finally, the other thing that is in the field of view of this small Magellanic Cloud, in addition to all these galactic foregrounds, is the microwave background itself. And that's shown by the blue curve. And the idea is that you have all these observations, one, two, three of order 20 observable points of different frequencies. And you use that information to decompose or to make component separation into these one, two, three, four, five possibilities. In this particular case, I think there's six variables fitted. And fortunately, we have more than six data points so that we can determine the fit uniquely. And the component separation reveals the synchrotron component, the CMB component, the emitting dust component and so on. This is component separation. And then what we would do to get at the underlying CMB signal is to subtract the dust, subtract the free free, subtract the synchrotron and see what's left. Now what I want to do is to show not a portion of the sky, but the entire sky. So I'm going to show a series of maps of component separation results. These are all from the Plunk papers which are referred to in various places. Here we go, down here. So the first is, how do we separate out the synchrotron emission? Can we make a map of it? And if we can make an accurate map of it, we can then subtract it from the observed map to get at the underlying CMB. So one of the components that separated out a synchrotron and there is a map of the synchrotron. Again, we see here the entire sky projected so that the galactic plane lies along the horizontal midpoint here, north galactic pole, south galactic pole. And there are two features I want to point out. First, not surprisingly, synchrotron emission, which has to do with radio waves, microwaves emitted by relativistic electrons. The synchrotron emission tends to peak up along the galactic plane, where the sources of the relativistic electrons are. But it's not just along the galactic plane. They're features that lie well off the plane. This one is known to radio astronomers as the so-called north galactic spur. And you'll see that popping up time after time after time in these various pictures. And it's a warning that galactic processes, synchrotron in this case, can cause trouble not just in the plane of the galaxy, but also at higher galactic latitudes. We have to be careful everywhere we look in the sky. It's a real nuisance, but that's how nature works. Now, it's reasonable to ask if this is a component that's been separated out, does it have the properties of synchrotron emission? Otherwise, we've done a bad job. And in this particular case, what we see is that at the frequencies we're interested in, 50 to 200 gigahertz, the spectral index does indeed come out reasonably close to what might be expected, about minus 3 or so, with some modest latitude dependence. It's close enough, so there's no gross error here. And here is a plot of spectral index versus frequency over a small range here, and the various colors represent various latitudes. So it doesn't have a universal spectral property. Is this spectral index? Yes, it is. So in the frequency band we're interested, down around here, it's fairly flat. But there's some latitude variation. Yes, I'll go back. Yes, bear with me just a second. I was afraid that question would arise. Again, this is a very, very wide range of frequency. And for CMB work, we typically work between 50 and 200, so we're working just down here. I should have drawn a little circle around this. What's going on here? It's interesting. Synchrotron emission falls off with frequency, right? I presented that. But that is true when the material is optically thin. Now imagine for a moment you've got a synchrotron emitting material that is optically thick. If it's optically thick, it ought to have something like a black body spectrum. So something very different than minus 3. In fact, something more like plus 2. So what you're seeing here is a transition from optically thick to optically thin synchrotron radiation. Are you referring to this difference up here? That's latitude dependence. You can see here that we're making measurements from zero galactic latitude right in the galactic plane where you would expect the emission to be optically thick. And then as we go to higher latitudes, it's less likely to be optically thick. So while we're on this point, from an observational point of view, what I should do is throw away all the diagram except right around the frequencies that we're interested in. And I should also throw away everything that is at low galactic latitude. Then it looks better. Good question. Two more components, which I've lumped together in a single slide. And that is the free free and the emission from these rapidly spinning small dust grains. Again, you see the galactic plane highlighted, not surprisingly. Most of the ionized hydrogen lies roughly along the galactic plane. Down here, you would expect these little spinning grains to be excited by starlight and most of the stars are in the galactic plane. So the concentration is not surprising here. It is interesting, though, if you look at the free free emission, that it spills well out of the galactic plane. So although free free emission is not a strong component compared to synchrotron or dust, it does have a worse latitude dependence. It's more likely to get in the way at high galactic latitudes. So you have to be careful of free free emission. As we'll see in just a moment when I start talking about polarized emission from these various sources, these two sources don't produce polarized emission, so generally we can treat them as close to negligible. Yes, can you shout so that everyone can hear? Good. That's a good observational point. Why be silly and make your CMB observations in the plane of the galaxy? Make them at high latitude. And I'll give you an example. Exactly that line of reasoning was followed by the bicep team. They consciously looked at a piece of the sky that was well above the galactic plane. But not all of the backgrounds are entirely restricted to the galactic plane. So we'll see that there was a little trouble there. But it's a good point. Don't be foolish and make your observations in the galactic plane work at high galactic latitude. Okay, I've skipped over dusk as I'll come back to that. Finally, the question of CO lines. This is a problem for some observational programs, but not others. This diagram is a little hard to follow. But what you see here in these jagged lines are the frequency response of several of the detectors in the Planck satellite. For instance, the one which we refer to as the 100 gigahertz receiver is this one here, shown I guess it's in black. Then superimposed on this are dashed lines, getting closer and closer together. These are the various emission lines of carbon monoxide. You'll see that for Planck, one of the carbon monoxide lines falls in the band pass to which it's sensitive. Another one right in the middle here and so on. So you have to be careful of the contribution of carbon monoxide emitting and getting into your detectors. And here's the picture. Again, one of these components separations, this is the component contributed by CO lines. Again, strongly concentrated along the galactic plane, but considerable amount of emission well off the galactic plane. That has to be taken into account. And here I want to do a little detour and fit together Tuesday's lecture with today's lecture. Here's an instance of an interaction between a foreground and a systematic effect, just to show you the kind of subtlety that can occur. What I've done here in this cartoon is to draw the responses, the frequency response of two mutually orthogonal polarized detectors of the same experiment. Ideally, the band passes, these patterns would be exactly the same, but they rarely are. Now what happens if CO lines happen to fall in these band passes? What you can see is that the response of the detectors to a continuum source with a smooth spectrum going across here is going to be different than their response to CO. So if you calibrate using some sort of smooth spectrum source, like the crab nebula, which I mentioned on Tuesday, CO won't match that and will produce a false polarization. So it's a combination of a foreground at a particular frequency and imperfections in the design or the way the equipment works can produce a false polarization. So, systematics interact with foregrounds in unpleasant ways. Now I want to switch gears. We're talking about the B modes. Let's now talk about what these foregrounds do, CO, free-free, synchrotron and so on, in terms of polarization. Not total intensity now, but polarization. How do they matter? So we'll start with synchrotron, which as I already said is intrinsically highly polarized, if the synchrotron emission is coming from a uniform magnetic field. Let's contrast a uniform magnetic field like this, large-scale structure. There's a directionality in that, and that tends to produce high degrees of polarization. Of course, if my magnetic field is completely jumbled, there may be directionality associated with each of these little vectors, but it's going to average away. But intrinsically, at least, synchrotron emission could be highly polarized. I've already pointed out that the synchrotron emission we're concerned with is from optically thin material, and its spectral index is around minus three. For optically thin synchrotron emission, the polarization percentage, or fraction, is given by this formula here, where gamma is the index of the electron energy distribution. For values of gamma that range typically around two or three, so that the energy is falling off as e to the minus two, e to the minus three, this value of p, the polarization fraction, can be as high as 70 or 75 percent. So if the magnetic field and the galaxy were uniform, and if the material is optically thin, then we expect synchrotron emission to be highly polarized. And that, of course, is a problem when we start looking for polarized CMB signals. However, if the magnetic field is tangled, as I've indicated over here, the fractional polarization average is down along a line of sight. But it remains true that for many of the frequencies used in CMB searches, the polarization foreground is dominated by synchrotron emission. These two are fairly quick. Free-free is intrinsically unpolarized. There's no preferred direction in the passage of the electrons around the protons. And the AME, this rapidly spinning dust, is observed to be not heavily polarized, to be weakly polarized. And that's an important result. You might expect if you had little tiny spinning grains that they might align along the magnetic fields. And if they're aligned along the magnetic fields, their spinning would produce some sort of polarized emission. So I'm going to leave it as a homework problem. Why is it that these very small grains, which produce the spinning dust emission, don't align along the magnetic fields? There's a reason, and it's fairly straightforward physically, but think about it. Are you going to answer? No. Please don't, because I want people to work on it. Okay? It is a homework assignment. We can talk at the end of my lecture, okay? However, you don't want to trust your intuition. You also want to go out and physically observe that this emission is weakly polarized, and that's discussed in the paper referred to here. Now, when I talked about reemission from dust, I was talking about larger grains, which can be aligned, and do tend, therefore, to emit in a polarized way. And we know that the grains are aligned because we can look at the scattering produced by those grains of background starlight in the optical. That is clearly polarized, so there's at least a substantial potential that the emission from these larger grains will also be polarized. So we have to expect that dust is going to be polarized. And that was shown to be the case in a sort of precursor of the Planck satellite, a very important experiment, Archaeops, and here is the paper reference. So to summarize, synchrotron we know is polarized and is likely to be a problem, particularly where it's dominant at low frequencies. Dust emission is likely to be polarized and is expected to be dominant at high frequencies, where it's dominant. And here's some comparisons. This, again, is just a small chunk of the sky. In this case, I think, yes, this is a large Magellanic cloud. What we have here are Planck images of this small galaxy. This is the synchrotron component separation. So this is a map of the synchrotron emission of this source, and this is a map of its polarization. You'll notice that the scales here, expressed in micro-Kelvin, are not all that different. So in this bright spot here, the polarization is substantial, above 10%, roughly speaking. And here is the same object, but this time seen in the emission from warm dust, thermal dust, emission, polarization, emission, polarization. But notice the difference in scale. Emission is thousands of micro-Kelvin. Polarization is tens of micro-Kelvin. So this is brighter in thermal dust than it is in synchrotron, but the polarization amplitudes are about the same. So the polarization fraction of the thermal dust is a lot less. And those numbers, I've done it roughly here. For synchrotron, the polarization would be as high as 50%. For dust, we have to do some extrapolation. But the polarization percentage is below 10%. So the dust is polarized, but not as strongly polarized. Finally, here's a Planck map, I believe at 100 GHz, of the polarization itself. Polarization intensity expressed in temperature units. It can be directly compared to the 2.7 Kelvin temperature of the CMB. It's a logarithmic scale because there's quite a lot of variety here, but what you can see is that it runs from about, roughly speaking, a micro-Kelvin up to a small fraction of a micro-Kelvin. I think this is actually Kelvin. There's a substantial range here, but it's considerable compared to the 2.7 Kelvin temperature of the CMB. Also notice the following. This is a map of polarization intensity. Not surprisingly, the polarization intensity is strongest along the galactic plane. But there are also a few bright point objects. We'll come back to those in just a moment. At least some of those point objects are extra galactic sources. There's nothing to do with our own galaxy, but radio galaxies at much larger distances. They tend to be highly polarized. The other thing is that you can see here this fan or fountain of high polarization intensity. That's the North Galactic Spur. We think that is a region where the magnetic field is fairly highly correlated. We've got the conditions of a fairly uniform magnetic field, which we know can produce high polarization in synchrotron emission. The next picture shows something quite different, and I want to make sure that the contrast is clear. What's shown here is not polarization intensity, but polarization fraction. The polarization intensity divided by the total intensity. You see something rather interesting. The galactic plane is not particularly highly polarized, and the reason for that is this jumbling of magnetic fields along the line of sight. Where you tend to see a fairly high degree of polarization, not a lot of polarized light, but a high fraction of polarization, regions like our friend the North Galactic Spur, regions down here. I'm going to call your attention to this little blob here. High degree of polarization well off the galactic plane, and it will turn out by the end of today's lecture that we'll be paying a lot of attention to this region, because that's where the bicep observations were made. Well off the galactic plane to avoid the obvious problem of emission from the galaxy, but still in a region with substantial polarized emission. Now for the next few slides, what I want to do is to talk in much more detail about polarized emission from dust. The first hint that we would have trouble from the polarized emission from dust in addition to synchrotron, came from the Archaeops experiment that I've already referred to. And I'll summarize it by saying that about 1% of the sky is heavily contaminated by highly polarized emission, and that 1% of the sky is not necessarily confined to the galactic plane. It can be anywhere, like the North Galactic Spur. So a lot of the sky is ruined, in a sense, by highly polarized emission. Just to give you a comparative figure, typical polarizations involved in the EE signal are of order of 5%. That is, if you take the polarized flux and compare it to the temperature flux of 2.7 Kelvin, you get about a 5% polarization, roughly. BB lies at least in order of magnitude below that. So we're talking about maybe half a percent in terms of polarization. And here we discover that a substantial chunk of the sky has intrinsic polarization from foregrounds of 10% or more. So we have to be careful. This is the approach that we've taken with the Planck satellite to the problem of polarization primarily by dust. The aim is to find general useful properties of the dust emission and then to make the assumption that the dust behaves in a statistically reproducible way to make models of the dust emission where it's weaker. So you measure the properties where dust emission is strong, including its polarization properties, so that you've got a good measurement of it. And then you assume that those statistical properties work where it's weaker. So the notion is, find the general properties, generate a power spectrum of the dust emission, and then extrapolate it to higher galactic latitude where the dust emission is overall weaker. In addition, we already know that the plane of the galaxy is strongly emitting in thermal dust. So it's prudent, as someone has already pointed out, not only not to look in the plane of the galaxy, but also to mask it from the maps, get rid of it. And I'll show you the results of these two steps. First the modeling and then the masking in the next few slides. This is a complicated slide, but deserves some attention. Okay. First, the curve that you'll be most familiar with from the earlier lecture is the predicted power spectrum of the emode polarization over a range of multiples from about 50 to 500. This characteristic shape here. Notice that the amplitude of the signal is of the order of 10 micro-kelvin squared, something like that, okay? And you'll also remember from the lecture I gave on Tuesday that this curve is exactly predicted by the cosmology that's derived from the temperature. So this is a fixed curve. Down here in the black and gray are the residual effects of instrumental systematics of various sorts. All the stuff that I was talking about on Tuesday. Beam mismatch, band mismatch, ground pickup, whatever it is, in the case of the Planck satellite, represented by these points here. And what you can see very quickly is that for a good stretch of angular scale or multiple, the systematic effects are well below the size of the emode signal. In other words, Planck should be able to pick up the emode signal with no particular problem. However, these points up here are the noise in emodes of the various polarized foregrounds. In other words, you make a map of the polarized foregrounds and then you decompose it in the same way as you would a map of the CMB to look for the characteristic shape of the emodes of radial signals. And this is what you get. Signals here, which are substantially bigger by a couple of orders of magnitude than the emode signal you're looking for. 353 gigahertz, this particular one. So if you're clever, you work at a different frequency where the effect is smaller, but this is an example. You see here very large signals. You notice also that the noise contributed by polarized foregrounds tends to be pretty flat in L, unlike the emode signal. So here where the emode signal is fairly strong, we have a reasonable chance of detecting it, even with noise present in the maps. Here there's no hope. Furthermore, if you look over here, you'll see that various fractions of the sky are included. As we go from the yellow to the orange to the red point, we're including more of the sky, which means we're masking less of the galaxy. So there's more galaxy here than there is here than there is here. So the contribution to the emode signal and the noise introduced by the foregrounds does depend strongly on the masking that you use. It's not surprising. We know that a lot of effects, including polarized emission, tend to peak up along the galactic plane. How much of the galactic plane you cut out does affect things. If we were not able to subtract polarized foregrounds, we would not be able to detect the emodes. Well, maybe marginally here, but you certainly couldn't make any claim about the detection of emodes if you could not subtract this polarized signal. So foreground subtraction in polarized light is absolutely required. You've got to do a good job of component separation and subtract all of this stuff out to get down to this level. Okay, with that? The next slide shows you how much worse it gets when we switch to the B-modes. Here's the B-mode signal, the solid line again. Okay? I've forgotten for which value of R it's drawn, but my guess is it's R of 0.1, which is sort of a reasonable value. In the case of the B-modes, the instrumental effects are the same, essentially, as they are for emodes, same level, but you can see that they dominate the B-modes. So in order to see the B-modes, we have to correct for instrumental effects. We have to be a lot more careful about instrumental effects than was the case for the E-modes. And up here are the predicted foreground contribution to the B-mode signal. Again, you take a map of the polarized sky and you decompose it looking for the characteristic B-mode type signals at various points on various multiple or angular scales, and this is what you get. And you'll notice here that we're many orders of magnitude above the B-mode signal. So the takeaway message from this slide is you cannot claim to be detecting B-mode signals unless you do a very, very careful job of subtracting out foreground emission. And you also have to be quite careful with your instrumental effects. So when you read papers that claim to detect or set up or limits on B-modes, please keep this figure in mind. It's a little bit of a nightmare. This is what you're looking for. This is what you're fighting or trying to get rid of. Okay, so how do we go about getting rid of those polarized signals? We can't leave them in if we have any hope of detecting the B-modes. And the trick is to look at general properties of the polarized dust emission as a sort of model. So it appears to be the case, irrespective of sky coverage, that C of L varies as a power alpha of spatial frequency or multiple number. You determine this where the dust emission is strong and the polarization of the dust emission is strong. And what you discover is, as you go from one scale of L to another, there's a sort of power law dependence. We then make the assumption that that is a universal property of the dust. The sort of statistical spatial dependence of the dust doesn't change. All it changes is how much dust there is or the amplitude of the signal, but not its statistical properties. So that's the assumption of the model we use. You then ask the next question, okay, if we know how the dust behaves spatially, how does its amplitude of, let's say, the E component or the B component contamination, how does that scale with the overall emission, the strength of the emission? And that's plotted here. As we go to stronger and stronger measurements, megajanskis per steradian of emission at 353, how does the dust behave? The polarization behaves, sorry, the polarization. And here is this curve. It doesn't reproduce very well, but it's in the Planck paper that I've been referring to. And what we see is that there's this sort of universal behavior that seems to be the same for EE and BB. So the amount of noise kicked in in terms of amplitude can be predicted on the basis of how bright the overall emission is. So we can get the amplitude and we can get the spatial properties, at least on a statistical basis. Those allow us to make a model of the dust emission, not where it's strong, but where it's weak, where we make the actual CMB observations. Again, this is a polarization amplitude from Planck data. It's a horrendous mess. Again, you'll notice that it's non-zero down here. There's the North Galactic spur again. This is a polarization amplitude, but because of the curve I just showed, we know how the polarization percentage varies with amplitude, so we can simply take each pixel here of amplitude, multiply it by the appropriate scaling to get the polarization at any given point in the sky. And that is shown in this slide. We've divided the sky up into great big pixels. This is the North Galactic hemisphere, the South Galactic hemisphere, so essentially the whole sky is shown here, not quite. Big pixels. In each pixel we know what the amplitude of the dust emission is, and then we can scale to that the polarization of the dust emission, and that's what's done here. The closer you get to the Galactic plane, which in this projection runs around here and around here, the bigger the polarized signal is. As you move towards the Galactic poles, North and South, the emission tends to get less. Okay? So this is a map of the polarization primarily of dust. And it's done in a particular way. What's plotted here is the polarized contribution to, or contamination of, the B-mode signal. And we've consciously expressed it here in units of R, which you remember is the sort of measure of the amplitude of the B-mode signals, or measure of the gravitational wave contribution of the tensor modes from inflation. So, along here is the log of the dust, the equivalent dust contribution to R, which ranges from minus two to one. Okay? So regions that are red in this diagram have a contribution from dust that's equivalent to a value of R of about ten. And what we're interested in, if you'll remember the earlier lecture, is values of R that are closer to one-tenth. So if we're going to measure values of R down around one-tenth, we have to be using chunks of the sky that are down here in this color scale. Okay? There are such chunks. You're seeing the equivalent noise contributed by dust emission of R equal 0.01. Here, R is going to be around 0.1 contributed by noise. Okay? Yes, two questions. First, back there. I can't hear you. So do we have a microphone? Wait for the microphone, then I can hear you and other people can hear you as well. And then there's another question quite nearby. You are using this 150 GHz that I think is biceps. And the other one in the planks is in 353 Hz. Yes. And you are translating the same noise to the other frequency. I don't know why you can make this assumption. We're not doing that. The work we did to characterize the properties of the dust was all done at 353 GHz. And if you'll allow me just a moment, I'll explain why. Let's go back to this spectrum, okay? 353 GHz is here where the dust emission is strong. We're interested in making observations where it's much weaker here at 150. So what we do is to model the dust where it's strong and then we extrapolate knowing this spectrum reasonably well to the properties of the dust at a lower frequency where bicep worked. So this plot here is extrapolated to 150 GHz. If I plotted it at 350 GHz, it would be totally red. It would be a disaster, okay? Is this plot saying that the heart is less than 0.01? It can never be detected from a measurement because your log heart is of minus 0.2. Okay. The scale stops at minus 0.2. What this is saying is that even in the best parts of the sky, here and here, there is residual noise from this contaminating dust emission at the level you're trying to reach. So you've got to figure out some way of modeling the dust better to subtract that out if you're going to go below R.01. Yes. Now, just while I have this up here, this is the bicep field shown here. It's sort of an intermediate region down around here. Interestingly, down around here, the midpoint of this logarithmic scale is R equal the geometric mean of 10 and 0.01. The geometric mean of those two numbers is 0.3, right? And the claim of the bicep 2 team was 0.2, okay? So just keep that in mind. So I'm not saying that it's impossible to measure the b-molds. I'm saying you've got to do a better job with the dust than Planck was able to do if you're going to go to 0.01. Or you have to be very lucky and observe just in the right spots. Before I go on to other things, there's a crucial check here. We've made a model of the dust. Does it behave the way astrophysical dust does? Does the spectrum of this inferred dust emission make sense? And this is a useful sort of pullback and check the results to make sure that they are sensible. There's another plot where the dash curve here represents a dust spectrum of T proportional to nu to the plus 1.9. We expect something close to nu squared, nu to the second power. So what we see here is that for both EE and BB, we get reasonably good behavior. In other words, the dust polarization has the proper spectral expected properties. Our model is consistent with what we understand from the astrophysics of dust emission. Here is a nagging concern that I want to raise. This is the first time I'm criticizing our work in Planck. Okay? I'm going to make the following statement and this is another homework assignment for you to think about. If you have a bunch of totally random polarized points and you just plop them down anywhere, the noise when you decompose into E-modes and the noise when you decompose into B-modes ought to be equal. In fact, there's a very nice little paper. If someone could make a nice little theoretical argument that makes that point concisely, it would be a nice contribution. We know it's true. We've done some modeling, but surely some bright student out there can figure out a way to prove it. A bunch of random polarized rods decompose into E, decompose into B. You ought to get the same amplitude and Planck doesn't for the galactic contamination. The E-modes signal is about a factor of two bigger than the B-mode signal. I think most of my colleagues in Planck are too worried about that, something about the configuration of the magnetic field or something or other, but it worries me. You would expect the numbers, these observed points here, to be up here where the ratio is one to one and they're not. They're down here. And it's true whether or not you mask a lot of the sky or a little bit of the sky. So in addition to writing this beautiful little paper on why E should be equal to B, if you can figure out why it isn't for the galaxy, that would be a nice contribution. So that's a slight concern that we have, or at least I have. Okay, I want much more briefly to mention the problem with extragalactic foregrounds. Remember when we look at the CMB we're not only looking through our galaxy, we're also looking through a forest of point sources. The galaxies that Bouvne's Jane mentioned, radio sources and so on. How do we deal with them? In the case of the Planck satellite, because its beam on the sky is fairly large, many arc minutes across, its sensitivity to point sources is fairly limited, so we only see the very brightest ones. Unfortunately, the very brightest ones tend to be the most strongly polarized. They're typically radio sources in which synchrotron emission dominates. But how do we handle the emission of these radio sources? The first point I'd make is that there's no preferred direction to the polarization of radio sources. This one can be pointing this way, this one can be pointing that way, and so on. So it's an entirely random placement of little polarization rods. So I'm predicting that the E-mode noise ought to be the same as the B-mode noise for these guys. Furthermore, if there are lots of radio sources in your field of view with polarizations running all over the place, the signal should tend to average out. So it's not a dominant signal, but it is there. There's also continuous backgrounds, the so-called cosmic infrared background, the light of all the dusty emitting galaxies in the universe. How do these affect our results? I've already mentioned that for random polarization, the noise in B-modes does or should equal the noise in E-modes. The other, and another, this is yet another homework assignment. If you've got a bunch of randomly distributed, Poisson-distributed point sources in the sky, they will contribute to the power spectrum as spatial frequency squared. Prove that. And I'll show you some curves that demonstrated in just a moment. This is a quick observational result from the South Pole Telescope. Our colleagues working at the South Pole. What this shows at various frequencies, 94 gigahertz, 150 gigahertz, this is where Bicep operates, 220 gigahertz. The total emission, or counts rather, counts of sources, the little green squares are radio sources, the little blue crosses are dusty galaxies. So rather to our surprise, at essentially any of the sensitivities reached by ground-based or-plunk observations, the sources that dominate the counts are ordinary radio galaxies, quasars, blazars, and so on. And those are synchrotron emitters. So we have to take into account the possibility that their light is going to be polarized. I'm going to skip over some things here and go now to a model of the polarized contribution to E and B modes. First, we'll look at the predicted signals. The dashed line here is the familiar and exactly known plot of the E-mode signal, drawn in this case at 143 gigahertz, very close to the bicep observing frequency. Down here are the corresponding values for b-modes. This one is R equal 0.1, I think this is probably 0.01 and so on. So we don't know the amplitude of the b-modes, but we do know that they're smaller than the E-modes. And here, rising rapidly with L, in fact proportional to L squared, is the contribution of extragalactic sources. So what you can see here is that for low values of L, even for tiny values of R, the extragalactic sources are not going to be a particularly big problem with the b-modes, which is comforting. I'll explain in just a moment what this range of predictions is. Apparently not right away. So let me go back for a moment. Why are there so many predictions here in this model by Tucci and Tofolati? And the answer is nothing to do with assumptions about the sources themselves, but instead an assumption about how many of the sources you simply mask out. Since Planck, SBT, bicep only see very bright sources, there are a finite number of them, and you can simply remove those chunks of your CMB map. Let's just throw that part of the data away. The more you mask, the fainter you go in masking, the smaller the contribution of the point sources becomes. So you've got some control here. And in any case, they are not the problem. The cosmic infrared background, this is a smooth distribution of the light from all of the star-forming galaxies in the universe. Incidentally, roughly speaking, equivalent in terms of energy density to the optical visible light of all galaxies, but it's smooth. It's largely isotropic. It's intrinsically not strongly polarized. Remember, dust emission is less intrinsically polarized than synchrotron. And in addition, there's no particular preferred direction. So the polarization tends to average out very, very quickly. So the CIB is not a problem for the B-modes. It's even less of a problem than point sources, which are less of a problem than galactic emission. It's all a galaxy that's a problem. Now, in the remaining 15 minutes, how do we get rid of these effects? How can we fight our way down so that we can make confident claims about R, let's say R less than one or R less than a half or what have you? First, the zeroth order, as already pointed out by one of you, is to work at high galactic latitude. Don't be senseless and work in the galactic plane where polarization can be strong. Work at high galactic latitude. And that's what the BISEP team did. This is the South Galactic Pole. Because of their location at the South Pole, they could not easily look directly at the South Galactic Pole. So they picked a region consciously to be fairly free of dust emission. In a sense, you can argue in retrospect, they just missed. They should have been here, but they were here. But this is the BISEP field, well away from the galactic plane. The main defense against foregrounds is to model them and subtract them. It's that process I showed very early on. You simply peel away the synchrotron. You peel away the free-free. You peel away the dust emission. And leave the underlying stuff. The process of doing that, at least as done by Planck, is using the commander code. Here's a reference for it. And here are the papers that discuss the results in the case of Planck. So for instance, just to remind you, one way of subtracting things is to use templates. The synchrotron template, that's what synchrotron emission should look like on the sky. We model it and subtract it. Here is the Bomber Alpha H Alpha proxy for free-free emission. We model it and subtract it. And a final remark here. This is harder to do for polarization because the polarization signals are weaker and there's not a guaranteed one-to-one connection between emission and polarization except for the dust. So it's a little trickier with polarization, but that's your main defense against foregrounds. Model them, subtract them. Earlier I said that the H Alpha line was a proxy for free-free emission. How good a proxy? If it were an exact proxy, all the observed points here would simply fall along this line. The free-free scaling exactly with H Alpha brightness. It's not bad. So it's a reasonable proxy to use. The map of the distribution of H Alpha emission is the same modulo amplitude as the map of free-free. I can't ask you to remember it, but the map we got for free-free emission is an almost exact reproduction of the map for H Alpha. So this trick of using a proxy works pretty well. Second level of defense. We've got the zero level of working at a sensible latitude. We model the foregrounds and subtract them. And then we also mask out the plane of the galaxy to get rid of the galactic emission and we also mask out all these little dots up here are the bright sources. Get rid of the bright sources too. And then do your analysis on what is left of the sky. When it comes to polarization, it's a bigger problem because we're interested in much weaker signals, so we have to use a more aggressive mask. Again, we wipe out the galactic plane, but we wipe out more of it. You don't have to worry about the sources. You could, but you don't have to worry about them and we don't. But you do have to worry about the North Galactic Spur and there it is because it's a region of high percentage polarization. That's why this mask has a very funny shape it does. And again, let me point out that the bicep region is down in here. So it's a region that Planck thinks should be at least partially masked. Okay? As I've already said, you can mask the point sources. The more you mask the point sources, the smaller the contribution they make. It's always proportional to L squared if it's a Poisson process, but the amplitude is reduced by more and more and more masking. If, for instance, you cut out all the sources that are brighter than a Janski and at Planck frequencies, that's of the order of several hundred, these are the resulting curves. If you go down to a tenth of a Janski, in other words, 10 times weaker, which means incidentally about 30 times more sources, the contribution becomes much less. The third defense, the third level of defense against foregrounds, goes back to something I said on Tuesday. That is, there's certain signals in the sky that from symmetry arguments ought to be null. The E cross B signal and the T cross B signal. Symmetry tells you that unless there's some peculiar and interesting variant on standard cosmology, these signals ought to be null. Now, after you've done all your foregrounds of traction and fixed up your instrumental effects, is it the case that if you plot TB and EB, you get zero. If you don't, something is wrong. It might be your instrument, it might be your foregrounds of traction, but it's a flag or a signal that something is wrong. And here are the bicep results, not Planck bicep in this case, of precisely that test. And it's done in two ways. You'll see there are two TB plots here and two EB plots here. This is the real data. And this is simulated data. Run through the same analysis pipeline. In both cases, there's some scatter. Of course, nothing is perfect, but the simulated data where you put in zero signal produces zero signal with some scatter. And the real data does too. So there's not an instrumental or dominant foreground effect in the bicep data. So keep that in mind for tomorrow's lecture. And I'll end with this slide. On Tuesday, I gave you theorists some guidance on how to read B-mode papers. Watch out for the instrumental effects, check the polarization calibration, and so on. Now I want to add some that summarize some of the warnings that I've talked about in today's lecture. First, if there's any claim that component separation is using the spectral properties of the foreground, you must look for multi-frequency observations. It's really tough to make observations at one frequency and start subtracting off foregrounds on that basis. A point I'll come back to. Second, I know this is painful, but read the papers or the sections of the papers on foreground component separation. Read the stuff that describes the steps I've been talking about for the last 85 minutes. It's painful, but you can't trust the result unless you know those huge foregrounds have been correctly subtracted. A slightly subtler point. Look carefully at the masks that are used and look to see whether changing the masks changes the results. If, for instance, as you mask more and more of the galaxy, your derived value of R, or your derived value of the Baryon density or any other quantity, shows a systematic variation, there's a problem. It means looking at different parts of the sky with more or less of the galaxy is affecting the result, the cosmological result, the value for R, what have you. So be careful about that. And finally, as I've done a couple of times, look at the models that are used for the various parts of the foreground, the synchrotron model, the dust emission model, and so on. And ask yourself, do those models make astrophysical sense? If, for instance, the model for free-free emission, just to use an example, the model for free-free emission shows a spectrum that differs from minus 2.1, it can't be right. Nature tells you that the spectral index has to be minus 2.1. If you get something else from your model, the model's wrong. So those are tests to use. I will end there, but once again, as I did on Tuesday, I'll stand down here for a while before I go and get a much-needed coffee to answer any questions that you may have, but I'm also willing to answer questions from the group as a whole.