 Okay, can you hear me? Sounds good. Well, it's a pleasure to be here. Thank you all for coming. I'm going to use largely blackboard, but some PowerPoint slides. I'll start you off with a little bit of eye candy of fly through the large scale structure of the Sloan Digital Sky Survey. So this is using the Sloan Digital Sky Surveys three-dimensional map of the distribution of galaxies and then the actual images of those galaxies from the Sloan Survey, but blown up by about a factor of 100 so that you can see them. I thought some about what I should take as my title for these lectures, and in the end I settled on precision cosmology with large scale structure. I think that's what I settled on, yes. So I'll be talking about clustering of galaxies, the Lyman Alpha Forest, a little bit about the cosmic microwave background, but particularly with a view to applications to high precision cosmological measurements of the parameters of the universe and trying to test ideas about fundamental cosmology with rather less emphasis on aspects of galaxy formation physics, though galaxy formation will necessarily come in when we're talking about using galaxies as tracers of structure. And so broadly speaking in this field, I think of there being two general classes of questions, so fundamental questions about the matter and energy contents of the universe and about the initial conditions that seeded an isotropy in the microwave background and the formation of structure. And astrophysical questions about particularly the physics of galaxy formation and the intergalactic medium, IGM, and clusters of galaxies and so forth, yes. It's a good fair question, and I think I will mostly mean the regime characterized by linear theory, by linear perturbation theory for the growth of fluctuations. So for the most part, I will assume that these are coming out of inflation or something like inflation, and I won't attempt to address why inflation occurred or alternatives to inflation and so forth. So Justin Corey is going to address that topic in his lectures, and from my point of view, I'd say we're going to use large-scale structure to try to tell us about what came out of that phase. So great question. And I'm about equally interested in both of these classes of questions, and I worked on both of them through my career, but for the most part, these lectures will focus on this second category, sorry, on the first category, and they used to be very closely intertwined, and that's because large-scale galaxy clustering used to be the main thing that we had available to actually try to address these fundamental questions before any anisotropies in the cosmic microwave background had been detected, and in trying to understand the physics of galaxy formation, we had very open questions about the nature of dark matter, about what the initial conditions might be like, about how much dark matter there was in the universe, and so we were really trying to go both directions on a two-way street to simultaneously figure out the underlying cosmology and to understand the physics of galaxy formation that translated that cosmology into our main observables. They're considerably more decoupled now than they used to be because we now understand, we know enough about the cosmological model that the remaining uncertainties, for the most part, aren't uncertainties in how galaxies form. There's still some dependence of galaxy formation physics on things we don't know about the cosmological model, but mostly we've now pinned down what we need to know for purposes of modeling galaxy formation. That doesn't mean we understand everything about galaxy formation, but most of our open questions here have to do with things like star formation and cooling and feedback and not with what the initial conditions were like. And on the other side, we now have many different routes to testing cosmological models that I will come to over the course of this lecture. So a bit of historical introduction. So when you give talks in lectures like these, you think back to when you were a graduate student or a postdoc, and say the first event like this that I attended was a summer school on galaxy formation in Eriche in, I think, 1987. So it was my second year of graduate school just as I was starting my PhD thesis. And also when I was in graduate school, my wife and I made this board game called Galaxy Formation. And so my wife did all of the artwork, including you had the option of playing various famous cosmologists. And I wrote the very complicated rules. And this was not a game that was really designed to actually be played, although it was attempted a few different times. And so in this game, you would play a theory of galaxy formation. And the goal of the game was to have everyone else in the game agree that your theory of galaxy formation was the true and correct one. And so you did this by going and writing papers in these various different areas. You had to roll the dice to get your paper refereed. You had to spin the spinner to get funding to carry out your research. And now that I've done that enough, I realize that I gave far too small odds to getting no funding, which happens a lot more. You could spin from the Hubble constant and the values of the Hubble constant that were allowed were anywhere between 30 and 100, which was sort of the range of debate in the late 1980s. And there were observational developments that could come out one way or another, cosmic microwave backgrounds fluctuations were discovered or not discovered, or cosmic strings were discovered or not discovered. And these would affect the plausibility of various different theories. Now, from the point of view of actually learning anything, what's interesting about this game is to look back at what you could play, at the playing pieces. Because I said you could be a person, but really you're trying to win your theory. And so the options in the middle, lower right, we had cold dark matter. To the left we had massive neutrinos. So this I think was 1987 when we made this. So hot dark matter in which the dominant form of dark matter was neutrinos. Jim Peebles was at the time still advocating fairly strongly a model in which there were only baryonic dark matter, no non-baryonic dark matter. And then cosmic strings were still quite popular, so you could play cosmic strings over here, or superconducting cosmic strings, which would basically blow bubbles, electromagnetic bubbles that would press stuff up. So that's superconducting cosmic strings up there. Explosions in which the supernova explosions from the early generations of galaxies would create expanding bubbles on which further galaxies formed. And any of these were topics, thank you, that you could, yeah, basically anything here, these were things that were in polite discussion and you could get papers published on in the Astrophysical Journal and so forth. This is primordial magnetic fields, which never really was a fully developed theory of galaxy formation, but which people nonetheless talked about all the time. And you could also play the skeptic, you couldn't actually win the game, but you could go try and take away points from everybody else. And so what's interesting is to realize how completely things have collapsed. And now we really just talk about variants on this. And we no longer really even talk about competing theories of galaxy formation. We talk about the parameters of the cosmological model. And so beginning particularly, I think, with the first detections of microwave background fluctuations in the early 1990s, the discussion really collapsed around variants of this model and all these other ones basically went away. So a somewhat more serious version of the same point is to go to, I dug this out of a drawer of my overhead transparencies. So back in the day when we give talks with overhead transparencies. And I would usually begin my talks with stating the big questions of the field I was working in. And those big questions, I had variations of this phrasing, but is the gravitational instability picture basically correct in structure formed by gravitational growth of small primordial fluctuations? If so, what were the properties of those primordial fluctuations? And where did they come from? So inflation was around as hypothesis for where they came from, made quantitative predictions, and we wanted to test those predictions. What is the dark matter? So at that point it was very much open whether dark matter was massive neutrinos, was some other kind of exotic particle, or was baryonic matter that had been packaged into non-luminous form, black holes, white dwarfs, or Jupiters. And what is omega? And at the time, you could just write omega without subscripts. And the... So when I wrote this, I'm sure I was thinking of omega as being representing the density of matter in the universe, and it wasn't really thinking about that possibly being different from the total energy density. And then the questions below the line, that's the one where my phrasing changed a lot depending on what I was actually attempting to... What the focus of my talk was, but... So sometimes that would be questions about how do galaxies form, what determines the luminosities and masses and sizes and morphologies of galaxies. But from the point of view of constraining cosmology with large-scale structure, the big question is what's the relation between the distribution of galaxies and the underlying distribution of mass, and particularly important in the late 1980s, early 1990s was the question of when you look at typical galaxies, things with luminosities similar to the Milky Way, is their clustering similar to the clustering of underlying dark matter, in which case the density of matter that you infer from the clustering of galaxies should be similar to the total density of mass in the universe, or are the kind of typical galaxies strongly biased, more clustered than the dark matter by a factor of two or so, in which case you could have a critical density of dark matter and galaxies just tracing the highest density region. So this was largely a question about galaxy bias. And what's striking to look back at these questions is they are basically answered. Or at least we have answers that are, we've greatly narrowed down the range of possible answers. So one, answer is yes. We really know structure formed by gravitational instability, and we know that the properties of primordial fluctuations are very close to the predictions of simple inflation models, Gaussian, nearly scale invariant, adiabatic, I'll have more to say about that later on. We still don't know what dark matter is, but we're virtually certain it's not baryonic, and that it's a weakly interacting particle, and that it is cold or pretty cold. And the answer to the question about omega turned out to be the most interesting of all, that I would say when I would give this talk in the late 80s, the question was is omega 0.2 or is omega 1? And the answer turns out to be both, that the matter density is about three-tenths of the critical density, but there is an additional energy component that makes the total energy density of the universe approximately critical. And the relation between distribution of galaxies, we now know that typical galaxies like the Milky Way are in an RMS sense about as strongly clustered as dark matter, but more luminous or redder galaxies are more strongly clustered, and there's lots of details to this last question, which will be a focus of my second lecture. And this is the kind of, the principle sort of data that we were using in the 1980s were maps of the distribution of galaxies. The lower left was one slice from the first CFA Regis survey, which totaled about 3,000 galaxies. And this big picture in the middle, which is still kind of iconic, though becoming less so, but this appeared my first year of graduate school, and this kind of visual impression of structure over scales of hundreds of mega, tens of megaparsecs, had a big impression on people and was playing a big role in these discussions about structure formation. So, if I were to write my list of cosmological questions today, and again focusing on this fundamental category, I would pick why is the universe accelerating? Still, what is dark matter? We know a lot about what it isn't, but we still do not know what it is. We know that neutrinos have non-zero mass, and cosmology is perhaps the best tool we have for figuring out what their masses are. And more generally, we could ask about other properties of neutrinos, how many neutrino species are there, and are there sterile neutrino species that contribute to the energy density of the universe, but don't show up in most of our particle physics experiments? And then, on this question of initial conditions, I'll phrase this in two sort of different ways. What are the departures of the initial conditions from scale invariant, Gaussian, adiabatic scalar? And by scalar, I mean fluctuations just in the density, so departures would mean gravity waves, tensor fluctuations. And we've detected that there are departures from scale invariance. Everything seems consistent so far with fluctuations that are Gaussian, adiabatic, and scalar. So we're now interested in the small departures from the most generic predictions. And another way to phrase this question would be to focus more. This is sort of observational, but the underlying physics we're interested in is what is the physics of inflation or is inflation correct at all? Is that really where these primordial fluctuations come from? And so this is, I think, how I think of the main questions we're after today. And alternatively, you can phrase what you're doing in terms of questions you want to answer or you can phrase it in terms of, well, I've got a standard model and I'm just going to try to break it. I'm going to measure things with increasing precision and see if something breaks down and hope that if something breaks that that teaches me some new physics. So I'll go on and say more about that standard cosmological model. Let me just pause for a moment in case people want to ask questions so far. So the question is, what was the discussion of the cosmological constant in the 1980s and early 90s? And there was a fair amount of discussion of it and particularly... So obviously it was a possibility. I've been around as a possibility ever since Einstein 1917. And it was still, I would say the main reason that people were interested in it was to reconcile the inflationary prediction of omega equal to one with observational evidence for a low matter density. Nonetheless, I think that if you... For the most part, when people thought about a low-density universe, with a matter density that was 0.2 or 0.3, of the critical density, people would typically think of that as corresponding to an open universe with curvature, but then say, or alternatively, there could be a cosmological constant and it could be flat. And then some people would say that the cosmological constant was the better... That was their preferred choice in that alternative. So both of those things were under discussion and certainly by the time you got to the... If you were writing papers in this subject in the early 1990s, you would typically... You would often have SCDM, OCDM, LCDM, TCDM, tau CDM. These were all variations on cold-dark matter. S meant standard and usually referred to omega-matter equal to 1. OCDM was open with a low matter density in an open universe. LCDM was with a cosmological constant. And then there were other variants in which you played around with the initial power spectrum or something else. And part of the reason that the supernova evidence for the... So particularly once we had the cosmic microwave background fluctuations, and you started to be able to compare the amplitude of fluctuations in the early universe to what we saw today, that's when I think the notion of a cosmological constant was taken much more seriously because it was clear that if you put that in, then a lot of things fit better. And so at that point, the decision was kind of clear that LCDM was the empirically preferred model. And the question was just, well, how outlandish did you think it was to have a cosmological constant? But this is the reason that the supernova evidence for acceleration was accepted so quickly by the community, was partly that there were two different groups that had found the same answer despite being in competition. And so that kind of ruled out a lot of the easiest ways to have screwed things up. But mostly, you know, there was already a lot of evidence that this was actually the model at best fit observations of large-scale structure in the cosmic microwave background. And once you had really direct evidence for acceleration, then instead of thinking, oh, well, this is a ridiculous thing to put into a cosmological model, everybody thought, oh, I guess that's why this is the best fit because it's actually there. So that would be my take on the history. I don't know, Ravi, you lived through a lot of the same stuff. Do you basically agree? Yeah. Okay. So let's talk about this standard cosmological model, usually abbreviated lambda CDM for cold dark matter with a cosmological constant. Yes. So good question. So under here, I did not write this question, what is dark energy? And the reason I didn't write it is what is dark energy is I think that the answer to why the universe is accelerating could quite well be because of alternatives to... because of modifications of gravity. And so really, I think, as soon as you break this question down, you break it into, is the universe accelerating because GR is incorrect, or is it accelerating because of dark energy within GR? Now, you could also ask, all right, can you explain the observations that imply dark matter with alternative gravity? And people still do investigate that. My view at this point is that the evidence that the phenomenology of dark matter really is because of dark matter rather than modified gravity is very strong. So I used to pay a lot of attention to that question and I no longer do. But when we come to acceleration, I think it could very well be whether it could very well be alternatives, and you'll hear more on that from people who know the subject better than me. Good question. So this means cold dark matter with a cosmological constant and although it's not written into the title here, it really means with inflationary initial conditions or at least initial conditions that have the same kinds of properties that inflation predicts. And so in vanilla lambda CDM, the assumptions are nearly scale invariant, Gaussian adiabatic primordial fluctuations. And so by nearly scale invariant, we mean that, we call this p primordial, is approximately a power law, some amplitude times k to the ns, and that ns, the scalar spectral index, is approximately equal to one so that we have equal fluctuations in the gravitational potential as a function of scale. And adiabatic means the fluctuations are equally present in matter and radiation in the early universe. Dark matter is weakly interacting. It could be just gravitationally interacting. Really, we mean it has weak interactions with itself. It has no electromagnetic or strong interactions with itself or with baryons. And that it is too cold to affect galaxy formation. So that could be because the particles are massive and they're formed in the early universe with low thermal velocities or they're actually formed with low velocities because they arise from some physics that doesn't put them in thermal equilibrium. And from a cosmologist's point of view, cold means that it didn't affect the formation of even the smallest galaxies. Dark energy, that is constant in space and time. So that's the lambda. A flat universe and omega total equal one. And then we can discuss a variety of alternatives to that. So there are a number of parameters to this model. And if we take this simplest form, then there are various ways to book keep those parameters. But one way, a sort of cosmic microwave background oriented list of those parameters would be omega C H squared. So C standing for the cold dark matter here. H being the Hubble constant divided by 100. So this is the physical density. This being the physical density of baryons. Omega lambda would be the energy density of dark energy. Something is written in various ways, but I'll write as As is the amplitude of the primordial density fluctuations. Really, I mean the amplitude of the power spectrum of those fluctuations. So it's that thing there. Let's see. Does this start to get blocked if I write down here? Can everybody see down here? It's good? Okay. Most likely to be blocked, so I'll write lower. So Ns would be the index of the primordial power spectrum. And then the last one is a little bit different in character from these. It's the optical depth for scattering of CMB photons after re-ionization. So it's different in that, you know, this is really an astrophysical nuisance parameter. It's a physically interesting one. We're interested in knowing about when the universe re-ionized and over what epoch. But it's not sort of something characterizing the matter density of the universe or the initial conditions. But we need it because when we consider the amplitude of the cosmic microwave background power spectrum here, what that's actually proportional to is, so the CMB power spectrum is proportional to this amplitude of the actual matter fluctuations times e to the minus 2 tau because when those, if the CMB photons are scattered at late times, that washes out the fluctuations of the factor of e to the minus tau and then the power spectrum goes like the square so we get e to the minus 2 tau. So if we're going to use microwave background fluctuations then we have to consider this as a parameter. And at low redshift, when we're talking about the amplitude of fluctuations we often refer not to AS but to sigma 8, which is the RMS matter fluctuation amplitude megaparsec spheres as computed by linear theory. And the 8 megaparsecs is sort of historical but it's there because this is roughly the scale on which the amplitude of matter fluctuations is about 1 today. So sometimes I'll talk about amplitude of, we're talking about the microwave background, AS will mostly be, I'll mostly use AS but later on when we're talking about the large scale structure will frequently refer to sigma 8. So questions, yeah. Between tau and, so there's relations in the sense of the, they both affect what you observe. There's also, I mean there's relations in the sense of you change the initial conditions, you'll change when galaxies form and you'll change when reionization occurs. But in terms of, I think that, and maybe what you're getting at is that the, so really my statement that the fluctuations are suppressed by e to the minus 2 tau. That's true over most scales, that's true over sort of all of these scales. And out of these largest scales, things are not suppressed because the scale of the fluctuations is large compared to the size of the horizon at reionization. And so there is also some influence of tau on NS because basically this NS you're trying to get by sort of comparing the overall tilt here, comparing these fluctuations to these fluctuations. Taking into account all of the astrophysics that's causing these oscillations and so forth. And so the principal effect of tau is that it moves this whole thing up and down. But then a secondary effect is that it moves this stuff up and down a little bit relative to that. Other questions? In space and time. Let's see, let me find where I write. Sorry, this is the lambda. So for a cosmological constant, by definition, it means that the dark energy is constant throughout space and constant in time. Now this is the standard model and then we're going to think about alternatives to it which would include variations in that. So now let me run through just about three slides on the empirical basis of this standard model. And maybe that will answer your question if not ask it again. So one important part of the empirical basis is the cosmic microwave background. Here we see the fluctuations in temperature, power spectrum of the fluctuations in temperature of the fluctuations in polarization and the cross correlation between polarization and temperature. And there is remarkably good agreement with the red curve and the red curve is the predictions of the standard cosmological model where there are these six free parameters that are adjusted to give a fit, but many more than six data points. So this is a big piece, this spectacular agreement with this high precision data is a big piece. And I'll say a little bit, I'll say more in a few minutes about what the different pieces of the CMB constraint. A second important pillar of the standard cosmological model is just direct measurements of the expansion history of the universe and in particular cosmological constant really became part of the standard cosmological model because of measurements of supernovae in the late 1990s. So using the luminosity of Type 1A supernovae as a standard candle and thereby inferring the distances to things of high redshift and finding that the supernovae were fainter than they would be even in a universe that was just freely expanding that was what implied that there had to be acceleration. So in the first papers from 97, 98, 99 that was a kind of two-sigma effect in a couple of different experiments but today there's pretty spectacular versions of the supernovae evidence and so here these are plots from four different surveys principally the Sloan Survey and the Supernova Legacy Survey from CFHD with some high rates of supernovae from Hubble Space Telescope and local calibrators and the line going through this is again the prediction of the standard cosmological model treating these supernovae as constant standard candles to infer distance throughout and if you vary for instance you go to an open universe then you'll get substantial departures from this at the high end and then we also have measurements of the expansion history from Baryon acoustic oscillations so measuring the clustering of galaxies to pick out the feature that represents the same physics that produces those fluctuations in the cosmic microwave background imprints this calculable scale in the clustering of galaxies that's showing up here is this peak in the correlation function I'll have quite a bit more to say about Baryon acoustic oscillations over later today and in the upcoming lectures but the main thing is that we're able to measure the location of this peak with very high precision at several different redshifts so that's an alternative to supernovae for measuring the expansion history of the universe and gives a consistent result again favoring lambda CDM model with the same cosmological parameters so those are expansion history measurements and then more generally we have measurements of the shape of the power spectrum and so really we'd like to measure the shape of the matter power spectrum but we can't really do that we're getting to the point where we can do that with weak gravitational lensing but it's pretty limited but at present but we can measure the shape of the power spectrum of galaxies or of the power spectrum of the Lyman Alpha Forest and so here are measurements of the power spectrum of galaxies again compared to the lambda CDM prediction and we see these oscillations here the same things that are showing up as this peak in the correlation function and the overall shape is clearly also in good agreement with the theoretical predictions although compared to the cosmic microwave background we don't have as much precision and we don't have as much interesting structure nonetheless we can measure the clustering of matter at a much lower redshift and get a consistent answer and then finally there's the amplitude of matter clustering how strongly is dark matter clustered and again the challenge is that we don't see the dark matter directly so we have to use various techniques weak gravitational lensing, clusters of galaxies redshift space distortions being shown in the lower right this is the clustering correlation function of galaxies as a function of separation perpendicular to the line of sight and along the line of sight and the fact that these contours are not spherical but they're squashed is a sign of the peculiar velocities of galaxies and by measuring that squashing we can infer the characteristic peculiar velocities and ask how strongly the matter has to be clustered to produce that and at the sort of 10% level that agrees with the model predictions so let me create some more space so empirical basis say is C and B fluctuations expansion history from supernovae and from baryon acoustic oscillations to the shape of the matter power spectrum galaxies galaxy clustering the Lyman alpha forest which I haven't shown you here but I'll have more to say about today and tomorrow the amplitude of dark matter clustering so basically sigma 8 that's something that agrees at the 10% level with the predictions I think I'll skip that slide but that's showing that you've got lots of constraints on different combinations of parameters and typically this is from combinations of cosmic microwave background and baryon acoustic oscillations but the standard model is always near the center of those error ellipses but on this particular point of measuring the amplitude of matter clustering good question so on these the things that are going in are cousin microwave background and just the constraints on expansion history from baryon acoustic oscillations and the bias of galaxies affects the height of this peak but it appears to have very little effect on the location of the peak and the scale and it's that location it's the physical scale that's being used not the height so I'll say more probably on Wednesday about making that case that the galaxy bias doesn't really affect the baryon acoustic oscillations but that is the reason that b isn't an additional parameter in here is because the only thing for this particular set of plots that's being used is the is that BAO location so in terms of the this test about the amplitude of dark matter clustering we're really taking the level of fluctuations observed in the cosmic microwave background which is proportional to AS e to the minus 2 tau and then using either linear theory and we'll come to this equation in a few minutes but that's an equation for how fluctuations grow in linear perturbation theory or using embody simulations to advance the structure from redshift of a thousand to some low redshift when we've got observations and then we're using some kind of measurements of low redshift structure to try to infer the clustering of the dark matter and of course the difficulty is that this is a map of the galaxies so to infer the clustering of dark matter we need to use things like the abundance of galaxy clusters redshift space distortions in galaxy clustering or weak gravitational lensing to try to measure the dark matter clustering directly and I'll be saying more about each of these but those are basically the techniques that people use to try to measure sigma 8 the amplitude of fluctuations today and things almost fit but they don't quite fit so having I guess I'll add to this in terms of empirical basis that the standard model gives a plausible setting for galaxy formation and particularly that's important to the cold dark matter part of it that when you try to follow the formation of galaxies within this standard model you get reasonable results at least putting in somewhat reasonable assumptions about cooling of gas formation of stars and influence of energy input from those stars so that you can take this model and construct a reasonable theory of galaxy formation that's in pretty good agreement with observations of galaxies in the local universe and with the evolution of the galaxy population out to high redshift so the caveats to this empirical basis are first of all the lowest multiples in the cosmic microwave background let's go back you can see in particular that this quadrupole is surprisingly low and there's also some disagreements around here and it's generally thought I think that these are probably statistical flukes possibly statistical possibly systematic effects in the observations although those are better ruled out than they used to be but it's possible that somehow that particularly this very low fluctuation amplitude on the largest scales that we can measure is telling us something interesting about the structure of the universe so that's one of the caveats that people worry about a second is that the Lyman-Alpha forest BAO measurements don't quite fit what I'm going to say about that probably on Wednesday another is direct distance ladder measurements of the Hubble constant so when you really go try to measure H0 from distances to Cepheids to calibrate distances to supernovae and then supernovae to calibrate distances to galaxies these tend to give 70 to 75 kilometers per second per megaparsec versus the Lambda CDM expectation of about 67 kilometers per second per megaparsec and then the low redshift measurements of matter clustering typically 5 to 10 percent below the prediction so let's let me address one point that I forgot here which is the Hubble constant itself this H looks like it should be another parameter in the model and the reason it's not is that the assumption of a flat universe if we have omega matter sorry, omega cold dark matter plus omega barion plus omega radiation plus omega Lambda equal to 1 for a flat universe that determines the value of H so if you know this and this and this and the radiation density we know from measuring the cosmic microwave background adding those up and requiring the equal one determines the Hubble constant so that's not a separate parameter however as soon as we allow departures from this model if we allow curvature or if we allow extra energy components then the Hubble constant is no longer a prediction but in the standard case it is the let me come back to that in a minute so right so there are various various ways we can play with this in order to loosen that prediction well no okay I'll address it now so so I think that I actually forget how I think it is difficult to resolve this with omega k because that's sufficiently well constrained by the cosmic microwave background fluctuations alone that even when you open up that degree of freedom I'm actually not certain of this but I think it remains difficult to get the Hubble constant correct however you can go away from constant dark energy or you can go away from you can add extra neutrino species so you change omega radiation in ways that will resolve this difference and in fact I think it is most likely that these measurements are wrong rather than new physics but possibly this indicates new physics and I'll tell you more later about why I think this is why I think it's most likely that this is an international problem but the nonetheless this is certainly one of the discrepancies that people worry about and all of these about two sigma fluctuations in terms of two sigma deviations in terms of statistical significance and so the and in particular here where we talk about the low range of measurements of galaxy clustering this particular plot from paper by the boss collaboration Oburgadol is comparing the predicted combination sigma 8 times omega matter to the point 4 and it turns out that most of these measurements galaxy clusters, weak lensing red chip space distortions they constrain a combination of the amplitude of matter fluctuations and the mean matter density because you can get a stronger weak lensing signal either by having more stronger fluctuations or by having more matter that's there fluctuating and this is roughly speaking the combination that's well constrained by the observations and this is the prediction for the lambda CDM model and that's taking the parameter constraints from the Planck CMB measurements these are actually the 2013 measurements not the 2015 measurements but they haven't changed much so that corresponds to an omega matter that's a little bit bigger than 0.3 and a sigma 8 that's about 0.81 or so and so that's what's predicted and then the subsequent the black points as you go down here show what happens if you allow partures from if you allow curvature if you allow dark energy not to be constant so we allow increasing degrees of complication and there's more freedom in this prediction but the central value doesn't really change very much except when you add extra relativistic species the predicted value actually goes up but so this predicted value is fairly stable although the more parameters you have in the model the larger the range of predictions and then the red points here are observational estimates and the vertical position of the red points has no significance it's not that this corresponds to that they're just sort of lined up these two are from weak lensing this is from cosmic shear by hymens et al from the CFHT this is galaxy-galaxy lensing by mandel-bombe-dell in the Sloan survey these three are different measurements of galaxy cluster abundances this is using x-ray or weak lensing or synyaev-zeldovich data and this is a more recent measurement of from galaxy clusters using weak lensing mass calibration and this one agrees with the predictions but most of these are low this is at redshift fairly low these are measurements of using the redshift space distortions the motions of galaxies are 0.6 and here the statistical significance of the discrepancy is low particularly because these are all using the same data set they're different analyses of the same data but they still come in on the low side this is a measurement from the Lyman alpha forest at redshift 2.5 and this one actually comes the measurement is above the predicted amplitude so this statement down here the low redshift measurements are typically low basically comes to the fact that most of these red points are below the black points and this is again something that may simply go away with better observations or could be statistical flukes or could be systematic errors in the measurements I consider this one the most interesting of the current tensions okay questions yes in this particular case the so this sigma 8 is by definition this the matter fluctuation amplitude in linear theory however in terms of predicting the observables so for instance if you want to actually go from the observations of galaxy clusters to a constraint on sigma 8 then you need and body simulations or something else so I would say that I'm referring to a linear theory prediction but there is nonlinear calculations going into actually deriving the empirical constraints but that's one potential source of the uncertainty and that'll be sort of we'll talk more about that tomorrow there was another question here yes the H0 is not very sensitive to the optical depth for reasons I'll say in a couple of minutes I wouldn't say it's completely insensitive but it's not very sensitive to it right so when none of these are using just the amplitude of galaxy clustering to use to compute the dark matter clustering however so the only one that's actually using galaxy clustering in the middle panel are the measurements of this red shoe space distortions and the and so the thing that you can get out from those measurements you can basically get out this combination of the amplitude of fluctuations times the growth rate of fluctuations independent of the galaxy bias but yes we can't simply measure this clustering and infer this because we would need some separate constraint on the galaxy bias okay let me go on and the answer is to some of these may become more evident and I guess the last of these caveats let's see actually I don't want to erase that yet so let me go to here would be that there are long standing challenges in the inner sort of kiloparsec scale regions of galaxies that the shapes of galaxy rotation curves do not agree with the predictions of the structure of dark matter halos and also the number of observed satellites of the Milky Way and other galaxies is small compared to the number of dark matter substructures that are predicted to be around predicted to be in the halos of galaxies like the Milky Way so this is sometimes the first of those is sometimes referred to as the cusp problem of the second one is the missing satellites problem the too big to fail problem is kind of a combination of those two things I actually I'm happy to I know quite a lot about that subject and I'm happy to answer questions about it but I'm not going to focus on it for these lectures I will say my view is that most likely the resolutions to these things lie in the baryonic physics and the influence of the baryons on the dark matter and the ability of low mass halos to actually condense their baryons and make galaxies but it's possible that those things are telling us something interesting about the properties of the dark matter and that remains an open question yeah right so the I would say there are simulations of galaxy formation that get down to resolution of scales of tens of parsecs even and the so I think that it's certainly been a longstanding and one of the early questions was whether these discrepancies reflected numerical errors and I think that's pretty clear that it doesn't reflect errors in the numerical predictions but they are sensitive to the assumptions about the underlying physics about the feedback from star formation and that's where most of the debate about these things lies at the moment so it's different assumptions for each red point the so because that goes into how you went from the observational data to a constraint on these parameters so I will let me not go into that now but for each of these I'll have somewhat more to say about each of them or at least about several of them over particularly in tomorrow's lecture okay yeah so I think that the relative to when this was the missing satellites were first highlighted as a problem on the one hand we've discovered many more satellites first with the Sloan survey and then with the Dark Energy survey but that it turns out it's not enough to sort of resolve what all of those satellites are pushing fainter and therefore getting to lower halo masses which are still more numerous so actually I would say in simple form it's even though we found a lot more dwarf galaxies by searching only small areas of the sky in itself that hasn't resolved the issue so we can now say more about alternatives that we might consider how might we vary this standard model I'll just leave that up there but so there are a number of these underlying questions here that we can vary and the there are lots of very natural things or very crazy things that we could think about but which aren't interesting because they're just ruled out so certainly a natural thing to consider would be something where the dark matter is only baryons but all those models fail and there are so when I make my list of alternatives to lambda CDM I'm going to only consider models that at least can come roughly as close to existing observations as the standard model because lots of physical interesting alternatives are just empirically ruled out so one thing is non-zero neutrino mass and this is really something that's guaranteed because we know from oscillation experiments that neutrinos have a non-zero mass and actually we know that that minimum value of the sum of the neutrino mass is sum over the three species is about 0.06 electron volts and at this level that already is enough to change the growth of structure enough that it should eventually be detectable so current constraints from cosmology are at the level of something like 0.2 electron volts and as we further improve precision we expect to eventually get to the point where even at the minimum neutrino mass we ought to be able to measure it so that I would say it's not really that's not violating any fundamental assumption but it is introducing a new free parameter to the model which is omega nu h squared the energy density of the neutrinos and then there is tensor contribution to cosmic microwave background fluctuations so gravity waves and this is considered at least a perfectly natural extension of the model many inflation physics models predict that there should be tensor contributions so I wouldn't say this one's guaranteed but this one is plausible and here your new parameters are usually written r, the ratio of tensor fluctuations to scalar fluctuations and nt which is the spectral index of those tensor fluctuations analogous to the ns for the scalar fluctuations and then we get to things that are not guaranteed but could be there so dynamical dark energy meaning that the equation of state of dark energy so a cosmological constant corresponds to an equation of state pressure over rho c squared equal to minus one and if w is not minus one then the energy density of dark energy evolves over time and w itself can be a function of time and we can have extra relativistic species which would be the number of neutrinos you would think it means the number of neutrinos not equal to three but in fact the way these calculations are done usually three refers to three species of neutrinos that decouple entirely before electron-positron annihilation and because of the injection of energy from electron-positron annihilation it turns out that the standard three-nose species in the way these things are usually parameterized corresponds to n nu of 3.046 so if there's an extra neutrinos species or any other form of energy density then that's an alteration of the standard model there can be curvature or features in the primordial power spectrum so instead of this being a power law it might curve or it might have jumps so that's often referred to in terms of running of the spectral index that would mean this NS is changing as a function of scale there's non-gaussianity so some models of inflation predict departures from Gaussian fluctuations so you'll hear more about that so I'm probably from Justin Kuri and now I need more space it's always here so a non-flat universe omega k not equal to zero and that would have important implications for the sort of global structure of the universe because the basic geometry is different if we have curvature and even an omega curvature of 10 to the minus 3 would be quite hard to understand in typical inflation models and would give us some interesting clue to what's going on and departures from GR so modified gravity as an explanation for cosmic acceleration dark matter there are various ways in which dark matter properties could be different so we could have dark matter that is self-interacting or has some low thermal velocities that would affect these issues about the challenges on subgalactic scales or if dark matter decays over the history of the universe that would affect the history of energy density and alter some of these other things so yes, other variations of dark matter properties and other variations of the primordial fluctuations so in particular if we have not adiabatic fluctuations but isocurvature fluctuations so by fooling with the early universe physics we can get fluctuations that are not present equally in all the different energy density species and those produce distinctive signatures in the cosmic microwave background so all of these are things which we could imagine changing about the standard cosmological model each one of them brings in some new free parameters and all of them are fairly tightly constrained we actually have already from existing observations pretty good constraints on any of these on the parameters of any of these alternatives but this is where a lot of the action is in precision cosmology today is trying to look for any of these departures or constrain those parameter variations okay, questions yeah so I think that the tensor contribution is mostly that's really mostly in the realm of cosmic microwave background so you might get some I would say there a whole of large scale structure is that it helps you constrain some of the other parameters of the cosmological model and therefore you can better use your cosmic microwave background on isotropes to focus on tensor contributions so for instance you know the constraints on tensor contributions from the CMB are much stronger if you include baryon acoustic oscillation measurements of the expansion history so there's not really a direct signature that we're looking at in large scale structure it's more in the CMB other questions yeah, that's I think a reasonable good so I think let me write down what I think you're talking about and then you could add more if I missed it but yeah something I didn't put on here would be some coupling between dark energy and that could take many different forms you know one of these species decaying and producing the other or direct interactions by which one affects the other so in the standard model these are just two entirely different things but we could have interactions of the dark matter with itself interactions of the dark matter with the baryons or between the dark energy and the dark matter so yes, this is also a big focus of some current work is that what you were after I have I've got quite a lot on one hand the I've got quite a lot more in my notes that I'm going to get through in the next 10 minutes the last of these topics about forecasting experimental performance kind of naturally slides into the last lecture so I'll probably put that at the beginning of Wednesday but the intermediate stuff is about the different probes we have of of these kinds the different observational probes that we have and what their information content is and so what I think I will do is I'll try to cover the cosmic microwave background specifically and then baryon acoustic oscillations and weak lensing and the Lyman Alpha forest I will basically cover in tomorrow's lecture and some of the other things I'll either skip or go through faster somewhere along the way so I and I apologize I wasn't able to post all my notes long enough in advance for you to print them out and bring them with you but there is quite a lot more words in the notes than I attempt to write on the blackboard and I will get the notes for tomorrow's lecture sometime this afternoon so that you can look for those so with that let's go specifically to the cosmic microwave background and there are other people lecturing in this school who are more expert on this subject than me however no one seemed to you're going to hear about the search for B modes in the cosmic microwave background as a probe of gravity waves particularly from Bruce Partridge but just in terms of what we generically learn from the cosmic microwave background that didn't it may be familiar to many of you most of you but it's pretty basic to everything that comes further so I wanted to make sure we covered the basic ground so what's great about the cosmic microwave background is that the observations are hard but not impossible and so over decades of effort people have gotten to the point where we pretty much measured it free of observational systematics the observational systematics are under control over most scales there are some challenges when you get to polarization and small scales but mostly that works and the physics is straightforward because the fluctuations are small so it's complicated in linear perturbation theory and that fluctuation spectrum is very complicated it's got a lot going into it so it's responding to many different physical effects so it actually has influence has signatures of all these different parameters of the standard model but the summary of what we get from it at least my version of it would be that the heights of the peaks constrain the dark matter density and the barion density so this you're seeing in the top two plots no sorry the bottom two plots and the a great way to see this is if you go to MaxTagmarkCMBmovies website so just google MaxTagmarkCMBmovies and you can change any one of these parameters and it will run through things and show you how the predicted CMB spectrum changes but particularly you can see that as you change the matter density or the barion density what mostly changes the heights of these peaks and their relative height so both of these things influence the heights of the peaks but not in the same way and so kind of largely independent of other stuff the relative heights of the peaks give you the matter density and the barion density and from that you can compute the sound horizon which is the integral from 0 to T star the time of recombination of the sound speed divided by the expansion factor and the sound speed is C over the square root of 3 times 1 plus 3 row barion over 4 row photon to the minus a half so this is the speed at which sound waves propagate through the photon barion fluid before recombination the photons are coupled to the electrons the electrons are coupled to the protons and at early times the sound waves propagate just at the speed of light over the square root of 3 and then as barions become important then that also affects the the growth of this sound horizon this plays an important role in barion acoustic oscillation measurements but it's also the physical scale that you're seeing in those fluctuations and so once you can measure omega b8 squared you know this we can calculate that from basic physics and so we can calculate this sound horizon and then the angular scale of peaks determines rs over the angular diameter distance to the redshift of recombination and that is particularly sensitive to curvature and that you see in the upper left that as you change the curvature of the universe away from omega total equal 1 would be a flat universe as you go to an open universe those peaks shift to subtly smaller angular scales because you've got the same sound horizon but this angular diameter distance increases and then there are other things that affect this angular diameter distance there's the Hubble constant there's the dark energy equation of state and other things and so you can also see that dark energy produces small shifts in the angular location of the peaks in the upper right so there's some constraint on omega lambda and on w in the state of dark energy and it's less sensitive to that but the measurements are very high precision the overall shape constrains ns so if we go from that's not shown here anymore but as you just compare what's happening at large scales to small scales that's constraining this overall tilt of the spectrum as well as other approaches from scale and variance like curvature or steps the amplitude the overall heights of these curves constrains the actual amplitude of matter fluctuations as times this e to the minus 2 tau the polarization on large scales is what constrains tau itself so we use that polarization signal to figure out how much late time re-scattering as there's been because that scattering induces polarization so then once you can constrain that you can back as out but uncertainties in that are a big source of the still a big source of the remaining uncertainty in as this tail down to small scales like the so-called damping tail where the fluctuations are damped constrains the detailed history of recombination and so anything you do that screws around with recombination by for instance changing physics of atoms at redshift of a thousand compared to today or by introducing extra energy species or something will affect that damping tail and be ruled out by that overall shape to small scales almost done low multiples constrain tensor to scalar and the reason for that is that the tensor fluctuations contribute on large scales but they're redshifted away on small scales so that overall amplitude on the largest scales compared to smaller scales is a signature of the contribution of gravity waves something that's just recently become powerful is that lensing of the CMB constrains matter clustering and so there's a distinctive signature in the cosmic microwave background maps that you can look at to measure how much the microwave background fluctuations have been lensed on the way to us and that's most sensitive to the redshift range of about 2 to 4 that's where maybe I should say 1 to 4 so it's the matter clustering in that redshift range that produces most of the detectable CMB lensing and you could also now look for CMB lensing behind clusters and other things but for the first time we're actually getting a measurement of low redshift matter clustering from the cosmic microwave background high multiple so high L small angular scales is sensitive to the Sunayev-Zeldovich effect from hot electrons in collapsed structures and this in turn depends on low redshift matter clustering sorry say again yeah I'll say here L less than 40 but the lower L you go the more sensitive you are and high L I would say greater than 2000 so at L greater than 2000 the primary anisotropies are basically gone and what you're left with is the Sunayev-Zeldovich I think the answer here is basically L it seems to be in principle it should be something like L less than 100 because at that point you're looking at scales that are larger than the horizon reionization but mostly people at least seem to focus on L less than 10 for this and then the last thing is that the large angle and here again I think this is sort of L less than 100 but you'll get better answers on this from Bruce Partridge B mode polarization sorry B mode polarization is a direct measure of the gravity wave contribution and thus tensor to scalar so there's all of these different I think the things to take away from this list are one is that there's a lot of different pieces of information and hence lots of different you're able to constrain many different pieces but the second is that that information it's at least somewhat separable at some level you put in all of these things and you make a prediction for that whole curve but really what's telling you about the matter density and telling you about the angular distance to last scattering telling you about NS these observations and that's why we're sometimes able to get good constraints on one thing even if we can't get good constraints on another let me take one or two final questions and then we'll break for coffee yeah so on large on large angular scales the integrated sax wolf effect influences the fluctuations and particularly what you're seeing here this big boost out here and this boost here are both due to the changes in the gravitational potential because of what's called the integrated sax wolf effect so one can break that out as a separate thing affecting the large scale the lowest multiples and I think the unfortunate thing about the integrated sax wolf effect is that because it's confined to these largest scales where where you just can't get very high precision because there's only one universe to look at it's actually at this point it's quite difficult to come up with any model that can fail measurements of the ISW effect but it's still giving you some information about dark energy and I'd say in addition to looking at the CMB alone and maybe this is what you're getting at you can try to isolate the ISW contribution by looking for correlations with cross correlations with galaxy maps for instance the lensing good question so one way of distinguishing the lensing is through polarization and particularly lensing will produce B mode polarization on small scales which you don't get from the primary fluctuations but I think one way to think about the lensing signal is you've got some patch of the microwave background and if you have an over density of matter in front of it then you'll squeeze that you'll focus that patch make it smaller in size if you've got an under dense region you'll stretch it and so really that's sort of taking this power spectrum and it's shifting its angular scale up or down up or down in different patches on the sky and so the thing that isolates the CMB lensing signal is actually not the power spectrum but four point clustering measure but as I the thing that makes the most intuitive sense to me is that you can think of this four point signal as being you're really asking how much does the CMB power spectrum fluctuate from one region of the sky to another so you've got variance of the power spectrum that's being produced by convergence and divergence and a variance of a variance is a four point measure I am happy to answer more questions during the break during the lunch breaks etc but let's take our coffee break and I think we restart at 11.30