 Can't go through everything. This is 14 pages of it. Okay For those who just came in you might want to come down to get some lecture notes We'll wait a couple more minutes because I think I heard people had a great time last night So most people haven't slept too late So for those who came in late you might not want to sit too far from people because you wouldn't know that the others Actually had lecture notes and of course we'll play games after yeah, okay So today we're gonna go through a little bit more derivations because I think a lot of people were asking me questions Yesterday, and I figure it's not gonna hurt if I go through a little bit of derivations What I'm gonna show you next which is in the slides will be following the beginning of the lecture notes that you just got okay Let's go back to what we left off yesterday so it's about basically something called the Russian space distortions and it's describing a probe that focuses on the velocities Vx Vy Vc and What do you do when you actually have rush shifts in? Velocities while the philosophy is coming from the rush of because we're observing of rush of the actual combination of the position and the Velocities as called the Russian space distortions are described the effect and the derivation is actually in your notes And it will go through a bunch of this so let's just go from what we learned yesterday to what we will learn today We have the correlation function in 1d So this is the one with the little dip on you know the very interesting extra Excess of probability of finding another galaxy a hundred mega parts are over actually right So that's that little dip but you sweep it out in 2d You see a quadrant like this the BAO is in this little quadrant right there that has an extra power Right, but this is in correlation function in real space What you actually observe is something in rush of space So we do not measure in fist we do not measure in physical space We actually measure in rush of space Russia measures a combination of Hubble recession and peculiar velocities So if you look at the equations below is the observed Velocities is a combination between the Hubble flow and the peculiar velocities So your distances observed is a combination of velocity and actual distances Okay, and that's actually coming To the discussions in the lecture notes here So the third dimension of Russia survey is not radio distance is reshift The rate of distance is related to Russia by Hubble expansion But also peculiar velocity feedback that you actually see in your notes on small scales the random motion of the particles Say within the clusters will cause the particle at the same distance to appear to have different reshifts Call the fingers of God no offense to any religion just people name me the first time I'll show you a picture on this one on the largest scales The object force towards something. That's like an overdense region So the object between us and over density will appear to be further away from us I'll show you a picture very soon an object on the other side of the over density will appear to be close to us Because falling to us. So this is what it means here When you have an over density Which is the one that you see in a center at linear flow, which is at the largest scales It will appear to be squashed. So we're observing from below It's falling into this over density Things are moving towards us. We'll make it. You know, we look like it's closer things are moving away from us It looks like his father. So that's actually the second column. So I need to have a laser pointer to realize So this is what you see in your eyes down here. We're observing down here. So this start has moved Look like it's actually further away from us and this thought has moved in so it looks like it's closer to us That's why it's the on the largest scales is squashed and the Lonnie Lonnie a structure such as the clusters You actually see it is moving randomly So the outer boundary looks like it's elongated So of course things that move out, but there's also things that move in because it's random, right? But the one that moving out will actually stood out because the stuff moving in just there's all these other stuff around Okay, so the boundary the boundary itself will elongate it. That's what they call a finger of God So we'll go through a little bit derivation of linear theory of a call Kaiser is some what basically developed by Nick Kaiser 1987 the main assumption the first principle assumed there is that Basically the galaxy numbers are conserved. So let's start. This is actually in your notes You might want to start making notes in that Lecture that you get for those who just came in and don't know why everyone else has this You can come down to get it We denote the Russia space coordinate s and the real space coordinate s are So now we are it's the peculiar velocities in the units of the Hubble constant. So Right here This is the observed Russia space coordinates. This is the real space coordinates Actually, this is the one that is a line of sight velocities. So this is not product with the line of sight are Assuming that originally is zero the velocity and Assuming the object's distance enough that's k times R is very much larger than one. So this is called the distant observer Assumption so we'll relax it in the lecture notes later, but we might not be able to do it today But that's later on in your discussion and then we have Now we have the Jacobian that you between the two different coordinate systems. It's very simple You look at it. It's basically like this. We drop the bunch of the high-order terms. So that's why it looks this simple We're only getting the first order. That's why it's a linear case of theory. Also Hey, have I lost everyone it's early No, okay, good Now the this is the derivation you should be able to do if you just walk not go close your eyes entirely if you can Assume the following the number density conservation requires a plane with perturbation delta Assuming H is equals to one right now, which is fairly simple What is just the easy assumption to make because you can end to put it back on later? And then you'll have this equation which comes from the Poisson equation for those who might have done the Prodivation theory is the simple prohibition theory for overdensable in how much in this universe You probably recognize it right away. Delta is the over density dot is the derivative over time the velocity k k is the for a component The UDR is equals to mu d BDR This one is just coming from from a very early equation. Do you guys see why that's the case? Yes, maybe maybe not But this is too early for that derivation but basically if you work through and plug in for example Mu is the cosine theta between line of sight and where the pair particle is or the coordinate is and then you Put in ikv, which is right here to delta dot and delta dot is coming back down to the growth factor times Well, this is a change of growth times delta times the over density you plug it back in here Then you actually would derive this very important equation You can actually go through all the equation on your own But I think it's very beneficial if you can make the first calculation on your own Back then to this particular question which relates the over density in rush of space to over density in real space With basically this particular parameter and you're like, okay. What is this right f is defined by the change of Delta as a function of time Mu is the angle in between the line of sight and whatever is moving right this stuff We're just looking at and if you assume the galaxy fluctuation is a linear bias trace of underlying dark matter density You basically take this Correlate with itself, which is the auto correlation Multiplied by bias square, which is basically relating by galaxy over density to dark matter density Then you'll get this While beta is basically a rewrite of f is f over b Okay, I'm gonna take a big pause and see how everyone is doing here too much too fast Okay, okay, so So this is the first equation of linear kaiser theory for rush of space distortion And that's what people usually plots when they plot what rush of space distortions actually does to The real space correlation function or power spectrum So it enhances the power so you can see usually without the rest of the distortions you don't have these terms But it's bias square times dark matter or power spectrum, right these terms enhances it. I have few heads nodding So this will go back to let you know, sorry quickly actually So we have this equation and then what people do is that you now can expand out this dependence on the angles in Lajana polynomials People remember Lajana polynomials. It's something you learn in undergrads and you might not remember at this point Yes, no Some science and coscience. Yep, okay So people expand it out and you can actually see that you can plot the differences between the two Well not plot to write the differences between the two the ratios as a function of these Zero and second and the fourth moment of Lajana polynomials May I ask who knows why we only consider the even moments of Lajana polynomials here, okay? This is time to start having some game Okay, think about why you only have zero two and four Lajana moments and you can talk to your friends Like if that equation does not make sense So let's group together and talk to each other. Okay, so you know how this, you know group one through three four works I'll call on random person to explain this later. Okay Five minutes So since you guys are discussing I want to pose a second question which comes up very quickly For a very long time It is believed that the ratio of the quadrupole to monopole is the best way to measure beta But we discover it's not Do you have a guess why it's the case? So the third very very long time people just try to measure the quadrupole moments of the clustering of the field of the galaxy to the monopole And you get the growth factor which is beta, but if you look at the equations, you know It kind of makes sense. You just kind of this five one by another It should get you beta, but then why that's not good So that's a second question since you're discussing might as well All right parody question for the question about the odd moments think Symmetry and parody and something physical So I have some good answers for the first question Do you guys have volunteers to tell us? Well quiet no volunteers. Yes Yeah, I'm trying the most physical argument we could come up with I think is the fact that a plane with a plane wave with K and a plane wave with minus K That could be the idea of going from mu to minus mu. I mean we have like a fixed observation direction and If I like the mode with K and the mode with minus K would look pretty much the same I mean if you're if it was like a time propagation thing Then it would be like in opposite directions, but if you're just creating a plane wave One with K and one with minus K. They eventually give you the same pattern So it should be the same if you have like mu or minus mu So that's a very mathematical way to look at this equation So that's one way to look at it. Is there someone who think of a physical reason why that's the case? Anyone want to help out? Know someone knows the answer No, I suppose The universe shouldn't have any Good it starts with the universe No, I'm serious I don't feel like the universe should have any preference between these K and minus K mode I mean they feel I mean in every Intuitive way they should look the same Because it's isotropic and there is no preference for the direction But it's exactly so we have a physical and a mathematical argument here I think you guys are going to the same idea But it's good to have one way and the other way looking at it And also, I mean we still have a slight symmetry breaking because you know we have the radial direction It's special. So that's why you know, it's not entirely isotropic But it's the general idea is that I thought to be cool the second question anyone. Thank you second question The one that we have the quadrupole divided by the monopoles Why is it a bad way to measure beta beta which is related to grow factor is F divided by bias Why is that bad? We're not good enough Because there are distortions in the radial direction with K, but The cosine of the angle so that's perpendicular and you wouldn't get any ridges based distortions from that So there's a mathematical way to look at it. There's a simpler reasons a simpler one much simpler I think you're a first year grasser who going in and measuring this What could go wrong? Think about signal to noise of the measurements. That's where I will start I'm gonna try to raise the hands. Are you trying to raise your hands? Maybe we don't know the normalization of the power spectrum and so it's impossible to have a We have a constant that we don't know So this is one very good guess. We actually really don't know the bias So that's one problem with Russia's face distortions, but it's not because of that. We don't do, you know Quadruple divided by monopole, but that's a really good reason for why Russia's face distortions when combined with gravitational lensing It's way more powerful because the gravitational lensing tells you about the bias Which is amplitude which is basically modulating the amplitudes up and down, which is what you're pointing out Good very very good point, but different reason. Okay. I'm gonna give you this is a very simple reason The quadruple it just really really really noisy So for most of the measurements the quadruple is a very very noisy term So you're basically dividing a very noisy term by another noisy term So these are two very like the monopole itself was quite noisy back in the days So for very long time you're taking very two very noisy measurements You're trying to compute a ratio out of it And it was just a very hard way to do it now people measure and then Actually try to model both of them together and don't forget the next reason why we don't want to do that Is this is linear theory? So any when you're doing this ratio you're mixing a lot of modes and scales together And you don't want to do that So when people what they do nowadays actually some of the students who know this Is to actually model both monopole and quadruple Together with whatever theory you're doing usually it's not linear theory Usually it's something higher order than linear theory and you want to use something a little bit better than just Kaiser theory these days So Kaiser theory is the basic of freshers-faced distortions There are many other things that actually in your lecture notes I think about page 10 or so you start doing Prohibition theory that's what people usually do now So if you're interested to see what people actually do nowadays, that's also in your lecture notes about page 10 or so quickly take a look Cool. Okay. Have I scared everyone and put everyone to sleep yet? No, yes question. Oh There's other moments too, so I only show you know The first three because even the fourth and the fourth one is also really really the hexadecabot is also really hard So right now we haven't seen a very good measurement of even hexadecabot, which is the fourth So that's why I don't show all the other even moments because they completely noisy Yeah, yeah, but then there is sound. They're just super noisy. We're not measuring Yeah, for the non-linear thing you could they go forever Exactly cool. Okay numerical simulations. All right the next thing. So I want to show you one more thing Back to this slide So now do you see this plot and you say, okay, I'm only showing one squashing effect in the largest scales That's the one that's squashing That is why the BAO peak you can see that it's usually like a circle around it a shell around it But then you actually have a Squashing comes down a little bit. Well, this size went out a little bit So this is why the squashing effect is and can you reconcile with your mathematics that you have seen in the equations just now? a little bit So the physical effect is a squashing in the middle term here. I mean in the middle column and I'm only showing you the quadrant That makes sense Now going back to something physical. I see a few heads nodding. How's everyone? Okay? This is hard to questions in the morning Well, someone asked me to have harder lecture yesterday so blame it on them. Okay Okay, so the next part So the correlation function so some of you have seen this before so I think in the inflation lecture You've seen a correlation function. You've seen the power spectrum, but that's usually they do it for the early universe here I'm showing you basically what you do for the lower shifts. I mean, it's the same math But you might not have seen it in this way before so So it's straightforward to derive the correlation function corresponding to the Kaiser power spectrum So you have the Lagrange polynomials So I use L in my light at my power point It's actually P here for the Legendre polynomials the usual way and then you have the correlation functions on the right-hand side and Then you bring it down here. This is this CFR The Rayleigh expression is boy down here where you have the best of functions Okay If you use the correct recursive currents relationship, you can actually write it down in this particular way How's everyone any questions what those are so the next part that I'm not gonna be able to go too much It's the beyond the plane parallel approximation So there's a approximation you made at the first derivation There's a plane plane parallel approximation and if you relax that which is not necessary You will want to do this following expansion so there's a best of function and the spherical harmonics basically you want to do those two And you do it in 3d Of course Why don't you guys ask yourself a following question? Why do we use the spherical harmonics and the best of function here? For this particular expansion Talk to each other for like five minutes. I mean you can Okay, I heard many good answers already. I suspect multiple group knows the answer Do you someone want to volunteer and tell us? Okay, do you guys want to pass the mic? Where's the microphone? It was so original press the red button. It's not on Yeah, so The plane way, okay, so this is going beyond the plane wave approximation, but then again So whatever wave you have which is coming at a distance r for example You can you have a sphere and on the sphere the eigenfunctions. We know are the spherical harmonics So that is the that part of the decomposition and then we know the wave has to decay when it comes to us So decaying part is taken care of by using the spherical whistle functions. So yeah, that's that's what I think Any other supplemental suggestions? Well, so this is a general idea basically that's the parts about the radio way radio parts is described But the best of functions and the part of the sphere is a spherical harmonic. So that's great Everyone can understand this and what I'm showing right here is the Why we actually care about versus my distortions. So I'm showing the quadruple and the monopole. This is an animation So hopefully it's not so bad after all the equations you've been staring at You'll have a key test of dark energy versus modified gravity is the growth rate of structures and Fixed expansion the geo actually predicts a very specific growth rate So for different growth rate, which is f here f is a logarithmic growth rate of the structure You will see that the monopole and quadruple actually looks very different So as f changes you change the model and quarter mostly the quadruple more dramatically than the monopole And if you can tell which one is the real f so a lot of the measurements these days They will measure something called f sigma 8 by modeling those two curves Then you can actually tell someone that may be modified gravity is correct or incorrect. Maybe gr is deviated I mean something's deviating from the gr So a quick question for some of don't don't have to start a discussion might know the answer right away Is how can I prove that a model of modified gravity is correct? Given a measurement, so you have a bunch of galaxies and you can calculate the correlation function You get the monopole and the quadruple And can I immediately say I think f of our gravity is correct? Is there some steps? I'm missing here questions Maybe not right away. So discuss for like two minutes not too long But you should know you should start thinking about what do we need to do before we can claim geo is incorrect And the second thing is that what do we do if want to claim a specific of about gravity model or a specific modified gravity model is correct? Can you speak loudly sorry? Right, so so he's really good about saying that you need to have a hypothesis and you have to test a model So say if your model is general activity We can test its deviation from general activity if you don't want to prove that a modified gravity model is correct You need to say to start your hypothesis in a modified gravity model part And that's actually hard because most of the modified gravity models don't have Nonlinear evolution that we understand very easily So you need to actually run the simulation even the simulations are not done most of the time because it's actually very hard to Code in some of those into a normal and body simulations So most of these tests are done in a way to test the deviation from gravity of GR general activity gravity Because that's something we have we have the animal simulations of GR type stuff Okay, so that's how people do it now. Let me just go to the next slide This is to give you a sense what the Lockworth growth rate f this f parameter that you've seen Many times in your equations actually how does it change as a function of scale and time scale and rush shifts? So distance on this axis is same as matches and rush shifts. This is in Einstein's theory of gravity Hey, very different From f of our gravity model So we can measure f as a function of a scale and rush shifts You have a pretty good shot of telling people whether Chameleon gravity model is correct f of our gravity model is correct or Einstein's theory of gravity and Notice that for Einstein's theory of gravity the Lockworth growth rates this f parameters is scale independence So that's something very easy if you see something scale Dependent for sure, then it's something already telling you that maybe it's different from GR question Wait, this is this is not sure. It's just the angle is it's flat. It's just the way it's shown. It's a little weird Yeah, I know it's even for me too. It's it's the it's an illusion because of the color Okay, so how do we do this so now going back to how we actually do this right you create a clean sample We did that yesterday We want to test the theory model the theory model I show you here is something called conflux of Lagrangian conflux Lagrangian perturbation theory It's the LPT it's by started by Actually started by white modern whites and Jordan Carson and company and this is the most recent version that shows you here is the radio distance versus transfer assistance and The dash line here. I believe is the simulations The solid line is theory And you can see that they match pretty well in small scales in large scales is easy because they actually need theory So the kinds of stuff that you just did is mostly found in fairly large scales But for the smaller scales like 30 megaparsec and below maybe 40 even sometimes That you will need to have a description of the theoretical model of the larger structure How does it look like under Russia space in correlation function? And that's what you get a squashed that's something you actually have seen before so you expected that and this is for a Specific type of halo mask. So that's what that 13.8 so it's 10 to 13 to 10 to 14 or so for the halo mask right there so people use this model because It's very similar to the theory the theories can predict the simulations pretty well Okay, but what we have done so far is only for general Einstein's theory of gravity Nothing about modified gravity, right? So I cannot prove as far as I can tell that certain modified gravity models correct I can only test deviations from GR and Then you also want to test one thing is whether you can recover all your pipeline is correct You want to test F? Which is what you want to measure this growth rate of structure the logarithmic one as a function of the minimum Scale you can use in your data analysis and the input is the dash line the dash black line and The cyan distribution is basically the one-sigma variance of 600 marks here So if you run your 600 simulations Through your pipeline and they achieve your recovered F You can see there's a this a distribution of F what you actually can get From all your pipeline and they're not always the truth There's a distribution to it So you want to know how wide I is and at what point it's really bad So if you start deviating if the means does deviating from the truth that you want to really not use it Okay, so that's how people actually go about doing it to test whether GR is correct Okay, so next part have another question for you so what people do here is that they want to do more than just Constraining F sigma 8 this measurement of the growth rate times sigma 8. Do you know what sigma 8 is probably learned this? I think from the previous lectures. Yes. Okay. See multiple has nodding You want to combine with multiple surveys Because you test multiple rush of ranges You remember we talked about the growth rate as a function rush shift and scale will help you tell whether GR is correct So you want to have multiple rush of because mostly surveys only give you one or two rush of bins and So What can go wrong when we combine multiple surveys to get at the best constraint of gravity Suggestions if someone has to answer raise your hands now because I don't have to go into discussion mode I give yourself two minutes two minutes. So think about this. Let me just pull this up a little bit Right now you have a bunch of these Measurements of all the different surveys All the different surveys and you want to combine them to get the best constraint on gravity Because you know that F as a function of rush shift and scale tells you whether it is gravity or maybe it's chameleon gravity something not GR What can go wrong when you combine them my question is are these surveys Independent of each other. I mean there's no Interdependence between them. That's definitely one very good thing to ask are these surveys covering different parts of sky Right or different chunks of the sky some of them are some of them are not So that's something you need to take care you look at the covariance matrix amount of different surveys So we can't take care of that. That's very good point any other suggestion. You want to pass them my back Maybe the completeness of the series are different Yeah, complete a survey is fairer than every one of them is different And so you need to take care of that, but that's assuming all the service did the job, right? So they all provide you you know the completest functions there So what do you do? But it's very good point Any other suggestion you raise your hands? I'm not sure about f6 my 8 but some measurements depend on the calibration and for example ground-based Instruments and space-based instruments have trouble like relating amplitudes because the atmospheric effects are really really Complicated sometimes. I don't know. It's a really good point So this is actually a question people have been starting to deal with for future generation of surveys like we have something called a LST you probably heard of before and then you have something like Euclid or WFIRST as space-based Missions and if we want to analyze all the data together, you might actually need to Say look at the shapes of the galaxies together with all the different data sets So that's basically what you're hinting at. There's atmosphere on the ground. There's no atmosphere in the space Which one should I be using? How do I come complement our data sets? Very good point So you guys are raising very very interesting points. The one I'm looking for is much easier than this I'll show you what they are Well different measurements and made assuming different cosmological models say if the survey was done when WMF 9 was you know The cosmology it will be assuming WMF 9 cosmology or assuming Plan 13 cosmology or assuming Plan 15 cosmology So when you do that, you want to make sure that you calibrate everything the same cosmology So that's something kind of easy, but you have to not forget that And it's not trivial because the growth is scale dependence in modified gravity models So while most measurements also assuming scale independent growth So that's the second part that we need to be careful about most people most experiments actually assuming scale the independence models when they're measuring this Even though we're testing gravity Okay, so these are results that you can get you can get very good constraints of multiple modified gravity models These are some examples that you can see this is about an f of our gravity This is the first constraint on general scale scale with Henson theory I actually do not do theory of modified gravity. So don't ask me exactly what these models do because I actually don't know improve the architecture constraint because if you know the growth of structure And also you can fit the whole shape of model point quadruple. You also get the distances Like the angle of diameter distances and the Hubble parameter that you were just asking me about earlier from the BAO from the full-shape Measurements and you actually break some degeneracies because you know the growth rates of the universe So same thing how we glancing actually measure the growth of structure of the universe that will also help break some degeneracies on Basically, W naught w a plane So here we showing Plunk. I think Plunk 13 measurements for W naught w a with Plunk only I believe there might be some priors and then there's easy mass which is basically just See mass, which is a one particular example a high ratio sample of boss galaxy with Russia space distortions and very acoustic oscillation measurements and then the third one was just a blue one actually shows all the different constraints from This one with all the different Russia space measurements from all the surveys That's the blue one and it really shrinks the measurements really shrinks the constraining power Increase the constraining power. All right, so we have Russia space distortions I think I gave you a tour of Russia space distortions with some of the theory in your hands Hopefully you get to study them a little bit more We did not relax a lot at approximation when we did the first derivation So go back and relax all the approximation if you go beyond linear theory what happens And then I show you a little bit about how you actually do the work with Russia space distortions And next thing I want to show you is something new because one group actually send me a cool Answer yesterday about what is the new probe you can create by combining seem be a larger structure group one here But people might have moved so don't know yesterday's group one Has sent me an answer and they talked about Sunil Sadovich effect I say other people has done the homework and want to tell the class about it other than group one. Go ahead Where's the mic? Do you want to pass the mic down someone? It takes a little time to react lens by large-scale structure So Planck has produced lens map of lensing potentials So that map is unbiased lesser of integrated mass from current epoch to the last scattering surface So this can be cross-correlated with the other tracers of large-scale structure like galaxy positions So this will give us more information about the cosmology Any other suggestions? Yes, do you want to pass the mic down don't don't turn the mic off here right here Extending on from that you could use the lensing maps to I think we talked about another lecture course to Find a cluster counts, which is also another way you can probe Find the clusters Yes, so the cluster mass function could be a very good way to do cosmology. Very good. You can also do SC It's the thermal Sunil Sadovich effect. You can also do the clusters any other suggestions Maybe you get the one I'm gonna tell you about then I don't have to tell you anything You know pass the mic back The lecture hall doesn't propel the sound. Yeah, so Planck has also produced these Compton scattering y-parameter maps so those can also be cross-correlated with the last course of time What does it do? Do you want to tell people a little bit about the Y map? Yeah, so so that is that generally traces the gas that is in the clusters So that map is a tracer of gas in the cluster so which causes the gas in the cluster causes Compton scattering of CMB photons Okay, I feel like there's more. Okay. You want to pass the mic back Last row. Thank you Well, the galaxy clustering is proportional to the galaxy bias squared Well, the CMB is proportional to just the galaxy bias so you can break the genesis with This parameter very good So this is basically saying that the galaxy lens in cross larger structure give you one factor of bias Well, if you do a galaxy clustering By itself you have bias square so you can help break some degeneracies right there and it's very useful Way to look at parameters to boil down from the galaxy-gassy correlation to dark matter correlations So very good. Very good point. Okay, so I'm going to show you quickly something what we're doing That's new not one of these and I'm just gonna show you quickly because we have maybe five minutes left Okay, perfect. Okay, so what happened when we combine the both so I'm going to show you one thing There's something called the EG that our group has been doing something a little newer It's a combination between lensing Graphitational lensing the velocity field and the clustering of the density field so a lot of things all together So if you look at equations proportional to CL which is the angular power spectrum of the kappa kappa being the Gravitational lensing a potential gravitational lensing kappa and then galaxy so those two will pull out one factor of bias as we just talked about and then you have beta which is coming from F of a B Right, so the growth rates divided by bias and then you have COg which is the angle of power spectrum of galaxies When you combine all of that it's kind of interesting because you've proved both the metric potential This is the metric. We're looking at here Non-relativistic particles feel the gravitational potential so motions of these particles probe the dynamical mass Well, the relativistic particles like photons in gravitational lensing are deflected by a spatial curvature So it probes basically different metric potentials and it's very interesting Well, okay one of them probe both metric potentials the other one probes only one of them So it's really interesting combine them And why is the case that this particular form is good is because it's independent of the galaxy bias when you combine it correctly So as the one in the back just talked about the galaxy lensing is on top Velocity probe by the bottom and the galaxy clustering there if you combine it properly You have the bias one fast term on top The others become also exactly cancel out all the biases all cancelled out EG itself ranges from point about point two eight to six with a variety of Gravity models and it is scale independent in GR, but it's not so in other gravity models So it's also a good way to look at deviation from general activity So, let me just show you what EG looks like in various gravity models This is in Einstein's theory of gravity as a function of scale and Russia. You've seen something similar before but for the growth factor F of our gravity model Chameleon model so fairly different. So now if you see anything different from the original expected GR Measurement of EG then you can say, okay At least there's a deviation from GR and if you know what it is as a function scale in Russia You might be a pin down what gravity model it should be So this was implemented the first time using galaxy lensing and clustering from Sloan to the sky survey not our work Someone else's work actually right about Reyes, and I mentioned about over subject 2010 it ruled out actually modified gravity model teffas Back in the days when teffas was still okay model So you see this data points as a function of distance EG as a function of distance right here Teffas model was actually ruled out by quite a number of sigma because of them So it's actually interesting and then so but there's no other measurement since 2010 2015 why why do you think that's the case? Maybe you already know the answer. So if you do raise your hand anytime Otherwise have a quick maybe not five minutes two minutes discussion So we have this amazing measurement like it just wrote out teffas model 2010 Why is then not repeated till 2015 your friends so it combines gravitational lensing galaxy clustering and Velocity measurements, which is from Russia's face distortions Think what you need to do that measurement Because you can assume this plus some kind of a hint, but that's how I was playing it very soon Any suggestions yet nobody come on Okay, someone actually have the answers can I have the mic down here wherever the mic is It's on it's on it's on don't touch it. It's on People have exhausted all the data. It was that they are present at that time. So we'll need to go for high-rate ship galaxies high-rate ship lenses Which will be lens by in between? Matter so we don't have that sample of galaxies at high-rate ship Beyond this point six or so. So what I was looking for is that you basically need a combination of many things, right? so you need lenses and sources photograph galaxy lensing and That means you have a very good imaging survey that's able to do galaxy lensing Somehow at high-rate shift. So that's what his point is So we already measure it once at this rush of at certain amount of sky So we want something with huge imaging Nice spectroscopic measurements. We can do galaxy clustering and You need to have us well galaxy cluster to get retrospective distortions Remember, we need the philosophy field and then you need the galaxy galaxy clustering that one is a little easier Okay, so you actually need to be able to measure the retrospective distortion effect of the lens population and you require good imaging to do graph is your lensing and The galaxy clustering so you need large volume and medium high and density of spectroscopy over the same area as the sky As the imaging so that's actually hard. There's not that many out there So we start thinking can we do this a little better? Can you guys come up with ideas that you can just change this around and make it possible? People have mentioned it earlier in class about 15 minutes ago. That's the hints. We will have just talked about it They still do e.g. But slightly differently That does not require good ground in ground-based imaging and or space-based imaging in large volume of spectroscopy You have the microphone up there We have a surface CMB, which is lensed and we have that available I did not tell him earlier, but he just figured out but he actually told us about there's a gravitational lensing Done because there is a CMB lensing that's possible So yes, so what if we place the galaxy lensing with CMB lensing? Very simple replacement, but that changes many things And I will show you quickly So it dramatically increases the range of number of traces we can use because anything in front of the CMB is lensing the CMB Right. So instead of we have the lens and the source playing like this The source is the galaxy and the lensing of the galaxy right by this by the stuff in between the sources are being lensed by this lens population You actually have the source 1100 rush shift because what the CMB is and you have everything in front of his lensing it so that helps and CMB is a lot cleaner than galaxy lensing at least for me No astrophysical system acts such as things like intrinsic alignments for people might not have I know you don't have a gravity's gravitational lensing lecture, but that's something that is a major astrophysical System acts right now is called something called intrinsic alignments in Galaxy lensing so something we have not figured out how to deal with at all while in CMB is quite Gaussian before the lensing But the galaxies before the lensing is already aligned to some tidal fields And if they already have alignments that could mean that you get the wrong amounts of matter Lensing because the lensing I should just increase the alignment of galaxies in a certain way So that's very important CMB does not have that and we have a very well-known source playing rush shift because we know where CMB is much better than Trying to get the photometric rush shift of all the galaxy sources because you actually need to do that if you do Galaxy lensing you need to get the rush of distribution of the sources at the minimum So we don't have to do that in CMB lensing because we already have to CMB rush shift Okay, so I'm not going to show you everything here because we have to go to the next lecture and the break I Will show you in the next lecture how people actually do this and I think this is great People have come up with many good ideas and if you have questions on the lecture notes, please let me know Okay. Thank you