 Okay before starting some some of you asked me about the weekend. There is nothing organized by the school and the main reason is that you are really too many to bring you around. So if you ever if you want some ideas unfortunately the weather will not be very nice but if you want to have some ideas of what you can do in the city or in the nearby there is a map and which is at the info point and there are a bunch of things that you can do in Trieste in the city or in outdoor in the surroundings. Okay this afternoon we'll have the last lecture on dark energy and then we'll have a sort of special seminar which will be about experimental search for gravitational waves. Okay we start with the fourth lecture of Kazuya. Yes thank you can you hear me? Okay so I continue with the observational test and I try to give you some introduction to nonlinear structural formation. I said that for dark energy if dark energy is complicated you have to specify pressure perturbations and analysis of stress but alternative way is to just parameterize your modified Einstein equations instead of introducing these additional quantities and in the end you get is the modification of the relation between Newton potential and your dark matter density and the relation between phi and psi and you can have any function of space and time so this is a wave number and this is time and you can have two functions of time and space to describe the modification of Einstein equations and of course you can combine this to get a different parameterization so this is another set of parameterizations. As I said that the ranging is determined by phi plus psi half so it is more useful sometimes to define the relation between ranging potential and density perturbations and sigma is just a combination of mu and eta so in this case mu and the sigma are the two functions describing the modification of gravity. I say modification of gravity but in fact what we are looking at is the deviations from smooth stock energy because for smooth stock energy you have no corrections to these equations so mu and sigma stays one so this is looking for derivation deviations from smooth stock energy okay and then again emphasizing again that combining two different ways to look at structure formation is very important so we glending measures phi plus psi half, pecular velocities are less if distortions measures the peculiar velocity of matter and this is determined by psi and so psi is determined by this function mu and density and looking at the evolution equation for density this mu appears here so this is evolution equation for density so this means that the peculiar velocity sigma which determines less if distortions only cares about this mu function. If you just use less if distortions so this is a constraint on mu and sigma I assume a very simple time evolution so this mu one and the sigma one are constant. If just use RSD less if distortions and this is a constraint from SDSS and you get just a constraint on this mu so there's no constraint on sigma but if you use weak lensing so weak lensing is determined by phi plus psi so this is sensitive to this sigma function but at the same time this is also determined by delta so delta is determined by mu so there is a degeneracy between mu and sigma so lensing is sensitive to both mu and sigma. So this means that if you use just one proof like the weak lensing or RSD you get a huge degeneracy and you cannot really say anything about these functions but combining these two now you get this constraint. So this is using a real data from SDSS and weak lensing from CFHTLS at the moment the constraint is very weak you see this is three so this is really order one constraint but you can see that lambda CDM is of course consistent and you may be also try to do the model independent test I discussed for equation of state so you have two functions depending on wave number and time so less shift so you make beams in terms of less shift and wave number so this is a two dimensional functions and treat mu and the sigma in each beam as three parameters but then you can use the same technique the principal component analysis by combining these two parameters not two the many parameters to form and correlated parameters and we try to see the deviations from one so this parameter is constructed so that it becomes zero if you have smooth stark energy and if this is non-zero this means that there is something beyond smooth stark energy so this is one example we did in 2010 so we use just two beams in Z and two beams in K so basically you have eight parameters describing deviations from smooth stark energy combining these eight parameters in a suitable way create this eight uncorrelated parameters and these error bars are uncorrelated and if you find the deviation from zero this indicates that there is something going on beyond the smooth stark energy and in 2010 we find there are three parameters you can constrain at like five to ten percent level and we find one mode deviating from zero at two sigma level this means that there is something going on and at that time so we use the weak cleansing data from CFHTLS and later we find out that there is a systematic in this weak cleansing measurement and this is driven by this systematic so unfortunately this is not the evidence of the deviation from laminar CDM but this indicates that using this kind of method you can really find our deviations from a simple smooth stark energy model but at the same time this indicates that you have to be very careful about your observational systematic because observational systematic can fool you to have some deviations again it's good to talk about the latest constraint the game plank paper 2015 gave a constraint on this mu and sigma so they use a slightly different time dependence using the dark energy density so they fix some time evolution and constrain this mu bar and sigma bar and reconstructed mu and sigma today okay so this is constant but remember that they assumed some very special time dependence so again they use plank and the background observations like supernovae weak cleansing BAO RST and combined everything so in their notation mu 0 minus 1 0 is lambda CDM so sigma 1 sigma minus 1 0 is the lambda CDM so this is lambda CDM and you notice that the contour somehow tries to escape from lambda CDM and looking at this if you combine everything there is a three sigma hint of deviation from lambda CDM so this is the current status yeah now they try the different parameterizations but the fact that there is two three sigma deviation it remains the same they didn't probably do that so let's look into a little bit why this deviation come from so there are two things one is the a lensing the cmb lensing I said that is the measuring this 5 plus psi and there is a known tension in lambda CDM that the lensing measured from power spectrum of cmb you need a little bit larger amplitude compared with lambda CDM so they parameterized this a lens so this is one in lambda CDM but they introduced some additional parameter to enhance the lensing and they find that they need slightly larger lensing for cmb and this modified gravity or the deviation from smooth stock energy parameters you can do that using this sigma so remember that sigma is describing the lensing so if you make sigma larger than one you can enhance your lensing so you can explain this so basically in order to explain this lensing anomaly you need to have sigma larger than one so that's the reason why contour is shifted toward this way so sigma must be larger than one but this is not the only reason and there is another probably more interesting tension with lambda CDM in prank data so if you measure the amplitude of fluctuations using cmb you can predict what is the amplitude today and you can compare this amplitude with the amplitude you measure from weak lensing so they use CFHTLS again and comparing this amplitude they notice that the amplitude measured by weak lensing is smaller than the amplitude you expect in lambda CDM so in this case the amplitude is smaller but in order to explain this lensing anomaly you wanted to have larger lensing signal so you cannot use sigma but you can use this mu parameter because weak lensing as I said is sensitive to both sigma and mu so what you can do is to suppress your growth by making a mu smaller than one okay mu smaller than one means that the gravity is weaker once the gravity is weaker the density grows slowly this suppresses the lensing so that means that you can explain this by making a mu smaller than one so that's the reason why the contour becomes like this so you want to have larger sigma to explain cmb lensing you want to have smaller mu to explain weak lensing measurement so that's the current status and we still need to see whether this is due to against systematics in the weak lensing measurement just mentioned that there is another way to measure cmb lensing using a tricepectrum so this is the four-point functions instead of par-spectrum and in fact we do not see this anomaly so it seems there is an internal tension within plank data so we still need to see but it's interesting that we are now getting some evidence that there is something going on beyond the simple smooth stock energy model I just probably skip the details so the plank paper also tried theoretical based parameterizations so I will not go through this but so I'm using a very phenomenological two parameters but there is a way to parameterize directly the action but then you get like six free functions of time instead of two functions of space and time and they did the constraint on just one parameter assuming very simple time dependence but theoretically this is a very interesting way and I encourage you to looking at the recent papers because now there is a modified Boltzmann code is available so you can play with these parameters and you can predict everything using this code but I have to say that it is a very challenging task to constrain six free functions however the future is bright so this is try to see how well we can constrain this parameter so let's focus on mu for example so this is the principal component number meaning that the how many parameters in this new function you can constrain so this is a function of time and space so you can parameterize this in many ways so there are basically number of parameters you can constrain so this is the errors and this is the constraint from the current observations so this is a bit old using double map but the prank doesn't change much so this means that there's only one parameter which can be constrained and the constraint is like 40-50% so that's the current status however now the dark energy survey is ongoing so this is a weak lensing measurement so imaging survey so they take image of galaxies and measure the shape so this is a weak lensing experiment ebos is looking at the position of the galaxies so you can measure RST and you see that using this dark energy survey the constraint is improved a lot so you now have four five parameters constraint at 20% level but now as I said combining weak lensing and RST even by 2018 you may get one to three parameters less than what 10% accuracy so within three years there will be a transition in our knowledge of these parameters and we will get a very good constraint on these parameters so deviations from simple lambda CDM model at 5 to 10% level and in fact in 2020 Euclid satellite will be launched so this will do both weak lensing and RST and in fact this is the future forecast you will get like 10 parameters constraint at 1% level after 2020 so these parameters are precisely measured so that's the reason why people want to do these experiments so we are talking about dark energy survey and ebos survey today but then after 2020 we have next stage experiment so I talked about Euclid so this blue one is imaging so this will do weak lensing so this measures 5 plus psi, red one will do the less shift distortions this will measure the peculiar velocities so measure the Newton potential so it's very good to have both of them as Euclid will do the both we have very large weak lensing survey RST we have a very big spectroscopic survey Daisy so having all these we expect that within 10 years 15 years we will know much about these parameters parameterizing C deviation from simple lambda CDM so I hope I convinced you that that's the reason why people are interested in these huge surveys and the reason why they want to do both of these surveys so this is what we want to do so in the first two lectures I talked about theoretical ideas of dark energy and in fact we don't know what it is so that's the question so we need to know what is the dark energy theory and then we want to parameterize the structure formation I introduced these two parameters mu and sigma in the end just to describe structure formations you just need to know these two functions in terms of space and time then using observations you can reconstruct or constrain these parameters and then hopefully you will fill the gap between the two and this kind of thing can be done in the next 10 years but there are a lot of problems and there are a lot of reasons that we fail for example as I said if you have a with a rational systematics this can fool you to have some modifications the problem is that there are so many models so how we parameterize mu and sigma at that moment for example this theoretical approach have too many functions so even for the future surveys it's not very clear we can really constrain these many functions so this is open questions so there are a lot of things you can welcome okay so any questions so far okay so let's move on so far it looks like we have a very good framework to test generalized dark energy model however there is a very important things hidden in the assumption so I used a linear perturbation theory and this mu and sigma parameters can be used only for linear theory however the weak lensing and let's see if distortions both you need to understand nonlinear clustering so nonlinearity you can see this in the power spectrum so you compute the power spectrum of density perturbation of dark matter and then you get this power spectrum as a function of wave number and this basically measures the amplitude of this density perturbations so this is the plot of the power spectrum so starting at less if 5.5 this is 2.5 1 and 0 so this growth is described by linear growth function I derived okay so you can describe this changing amplitude with less shift on very large scales again small k means large scales using linear theory very well so these dotted line from simulations blue lines are linear theory so linear theory is doing very well at small k but then you notice that simulations curve deviate from linear theory at high k and this deviations happen smaller and smaller k at late times okay this is a nonlinearity so the density becomes large because density grows like scale factor even if dark energy suppresses the growth density becomes nonlinear at some point then you cannot trust linear theory so this is the ratio between this full power spectrum compared with linear power spectrum removing the value on acoustic oscillations so that you can see these oscillations very well so linear prediction is always here the always amplitude is constant but this is a full power spectrum and you see that if less shift to become a smaller and smaller you see the enhancement of your power spectrum this is coming from the nonlinearity and if you do the observations at less shift lower less shift then you have to worry about all these nonlinearity so this smooth is a linear power spectrum but I removed this value on oscillations so by removing this you see these oscillations but because we use the linear power spectrum the amplitude is always one so this one is nonlinear power it's smooth means that I removed this value on acoustic oscillations by hand that's right so this is a nonlinear yeah that I will talk about later so why this is important so I said that I want to use weak lending and less shift distortions to do the test so let's look at the impact of this nonlinear effect on the convergence power spectrum so as I said that the convergence determines the measurement of your shapes of galaxies so this is what you want to measure from the observations so in this case you measure this from simulations so this dotted line is the simulations so this is the multiple so this is similar to K so for large L you are looking at the small scales small L you are looking at large scales so this is a prediction from linear theory so this is a simulation result so all this enhancement comes from nonlinearity so if you use linear theory you get a completely wrong answer here why this is important if you look at the predictions or with different dark energy model so this is a for our gravity model for example this is the constant dark energy equation of state model all these lines are different and you want to distinguish between these lines but then you see that on large scales we have cosmic variance because you can observe only a finite mode so the error bars are large but you see that the linear theory you can trust up to only 100 here so the difference between different models are hidden inside the cosmic variance and the interesting difference happens on nonlinear scales where you cannot use linear theory so you really need to understand the nonlinear clustering this is already a program in lambda CDM and you need computer simulations so you had a nice lectures on that by including our complicated dark energy model you have the same problem but the problem becomes more complicated so this is the same for less if distortions measurement so less if distortions because of the peculiar velocity of galaxies along the line of sight your clustering is enhanced so the path spectrum becomes anisropic this depends on your line of sight and then this mu is the angle between your line of sight and the wave number and you can expand this in terms of the general function so this is a general function and you can look at the monopole and the quadruple and you can compute this using linear theory I showed so this is a simple case of lambda CDM so you have the clustering of density in the less if space this is depending on the growth rate f and we want to measure this how we do that we measure the spectrum in less if the space to the this decomposition looking at the monopole and the quadruple taking the ratio and this is a function of f so you can measure the growth rate so now we try to predict this monopole so this is a monopole p0 again I normalize this using linear smooth power spectrum so linear theory prediction is always constant so this dotted line is a linear prediction it's always constant at any less if so this is a linear prediction again the data is from simulations so this is what you will measure and you see that at the high less if linear theory is doing fine but at lower less if you see that simulation result are completely different from linear theory so then if you use linear prediction and try to measure f just using linear theory you get a completely wrong answer and this is more serious for the quadruple so this is a quadruple p2 even at z equals 1 this is a linear theory so this is what you measure so it's completely different so this means that you really understand the nonlinear physics yeah so the question is why the deviation is larger for quadruple so quadruple is mainly coming from velocity velocity per spectrum and it is known that the nonlinearity is more important for velocity so what happens is that on larger scales you have a coherent this motion and this is the origin of velocity divergence spectrum but on smaller scales you have a lot of velocity dispersions this delays this coherent motion and you see this dumping so this means that the amplitude of this coherent motion is destroyed so maybe you will hear more about this next week in the large scale structure lectures but the fact is that basically this is determined by the per spectrum of velocities and the velocities have larger nonlinearity so you see this dumping because of this nonlinearity yeah monopole is coming from delta so density density per spectrum so this is basically density density per spectrum so this is density velocity and velocity velocity per spectrum so this means that we have to understand the nonlinear structure formation and I don't think I have time to of course explain all the nonlinear structure formation but again the idea is that I don't want to use gravitational theory to describe nonlinear structure formation because in some cases people start from gr and then discuss nonlinear structure formation but we want to apply this to general dark energy or even modified gravity so I try not to use any gravitational theory in the derivations and try to understand what you have to do if you want to understand the dark energy model so one way to do the nonlinear structure formation is to use a fluid approximation so let's treat dark matter particles a collection of these dark matter particles as a pressure less fluid and use the Newtonian theory to describe the dynamics of this fluid and if you remember the fluid dynamics in the Newtonian theory we have two equations one is the continuity equation so this describes the conservation of energy of the fluid this is the Euler equation so mu is the velocity of the fluid this is free nonlinear in terms of the variables of fluid so density can be nonlinear and velocity can be nonlinear the only assumption is that this is driven by Newtonian force Newtonian potential so in cosmology we want to separate the expanding universe so what we do is to use the commuting coordinate this is scaled by the scale factor and I remove the expansion contribution to the velocity so this just comes from the fact that the universe is expanding and we are interested in this velocity perturbations but I said perturbations but this v can be as large as possible here I just separate background and homogeneous part I do the same for density perturbations but again this density perturbations can be as large as possible so by doing that the previous equations can be written in this form so this is the continuity equation this is the Euler equation and maybe this looks familiar to the equations I used for linear perturbations and in fact as I will see you will see that this includes that linear equations to see that we need to change variable so again I use the velocity divergence so theta instead of v changing v to theta and using conformal time we get these two equations and if you remember what you compared with the lecture note last time so you have a nonlinear term like delta times theta and v times delta but if you forget about all this nonlinear term so this is delta plan equals minus h theta this is what we had before and for the Euler equations we have this term so this is exactly the same as linear theory and we have these contributions from Newton potential so then you get nonlinear collections so this is a very natural extension of linear equations we used that's good because we didn't use any assumptions about gravity the only information of gravity or dark energy is indeed encoded in this potential so I can use these equations for any dark energy model assuming that there is a conservation of this energy momentum tensor for that matter so this is a way to extend linear theory to nonlinear theory so in order to understand what happens nonlinearly the problem is that that equation is free nonlinear so it's very difficult to solve but there is one special situation where you can solve this nonlinear equation exactly in an analytical way so I will explain a little bit about this and how we extend this to general dark energy model so this is to assume the spherical symmetry I'm sorry for the equations but I need to change variables again first using this log A so this is just changing the variables so from conformal time to this variable and then I use a new derivative known as the convective derivative so this is the total derivative so this is using the fact that the density at the position of the fluid is the function of time but remember that the fluid is moving so this is position of the fluid also depends on time so if you want to take the derivative with respect to eta you have to first take the derivative with respect to time but then you have to take derivative with respect to the space coordinate so this is the part dealing with time dependence here the only reason I'm doing this is that the then the equation is simplified so this is the same equation using this new derivative and you are live at these equations so then looking at this in order to find the solutions we want to write on everything in terms of theta but we still have this part and this part cannot be written by theta because theta is the divergence so theta is the divergence of the velocity and looking at this tensor structure you cannot write this down in terms of theta but good thing about spherical symmetry is that the velocity is isotropic so this means that this term vanishes so this term disappears so this means that all equations can be written by theta and you can derive the second-order differential equations for delta so forget about all the derivations you get this second-order equations again good thing is that neglecting all the nonlinear terms like this and this delta times number of square psi this is the equation we used for linear theory so this is the extension of linear theory but now you can include nonlinear terms and this is now the ordinary differential equations so you can solve it and this equation is indeed coming from a very simple picture of spherical collapse so let's look at the spherical object with radius initially a scale factor times r0 and you have some cosmological background and this region just expands due to the expansion of the universe but now you consider the situation that you have the same mass but within the smaller radius r so you put your mass inside this outside sphere into this small sphere so then see what happens so this spherical shell first expands due to the expansion of the universe but then you have the density perturbations because you put more mass into this sphere so then due to gravity this sphere will collapse this is described by this simple Newtonian equation the acceleration of this object is determined by Newtonian force so this is a very simple physics and then I need to separate the Newtonian potential coming from the background and coming from this inhomogeneous density inside the sphere and then the density perturbations is determined by the ratio of these radius because we put more mass inside this smaller radius so if delta is 0 this means that r is the same as a times r0 so there is no density so you can compute density perturbations in this way and this is a very good exercise so by doing this definition you can derive these equations try this one if you are interested in this so basically this means that you are solving a very simple physics it's just a Newtonian physics so let's look at the simplest case with lambda CDM and even forget about lambda so the scale factor is given by time to the 2 3rd so this is the expand the expansion without lambda and then this equation becomes very familiar Newtonian physics so the acceleration of this shell is determined by GM over r squared Newton force so maybe you can easily solve this by integrating what once so this M is the mass within this radius then you can do the integration once and then you can find the solution for time and r in terms of this tau variable so this is the analytic solution you can even compute the density analytically in terms of this tau and I fixed this constant assuming that delta is 0 initially a question so far okay so the good thing about this is that everything is analytic so now you can understand what happens so this is the radius of this sphere so sphere expand due to the expansion of the universe but at some point due to the homogeneity this shell collapses so there's a turn around radius so then once this shell expand at some point shell becomes collapsing and eventually it goes to 0 so this is a spherical collapse okay and then you compute the density so this is the density and you compute the density it's not easy to see but this line is the nonlinear density and you can also compute the linear density so you linearize everything so this is basically the linear density proportional to scale factor so this is a linear density this is a nonlinear density so you see that once the density becomes nonlinear they behave very different way and eventually if the shell collapses you get the infinite density and often used number is that you can basically measure when this collapse of the object happens and you can estimate this using this linear density so you can estimate this time and you extrapolate this linear density up to this point of course the real density is diverging but you can extrapolate your linear density and find that this happens when linear density is 1.6 H6 so if you just use linear theory of course linear theory is not correct but you can extrapolate your theory then you find that this collapse happens when delta is 1.6 H6 and we want to know the effect of dark energy on this number for example but of course in reality we don't have singularities so if this spherical shell collapses of course density is infinity so you mean that you have a singularity but this does not happen in reality in reality what happens is the biliarization so according to the biliar theorem there is a stable point when the potential energy and the kinetic energy satisfies this relation so this is the biliar theorem and assuming that the potential is inverse function of r so this is assumption so this is valid for lambda CDM this condition is given by this equation so the potential energy plus two times kinetic energy is zero then the system is stabilized you don't see this in the spherical collapse because you assume that everything is spherical but in reality you have a velocity divergent and dispersions inside this sphere and once this condition is satisfied in fact you don't get this collapse as it is a spherical shell becomes stable and you can estimate when this happens using total energy conservation so the total energy is potential energy plus kinetic energy at this turn around radius there is no kinetic energy because the velocity is zero so the total energy just comes from potential energy so now you want to know when this biliarization happens you use the total energy at biliarized time is potential energy plus kinetic kinetic energy but then you know that to have a biliarization's kinetic energy must be minus potential energy half so using that this is one half of the potential energy at biliar radius so remember that I assume that the potential energy scales like one over r so this means that biliar radius must be half of the turn around radius this is a very simple argument and then you see that so this is a turn around radius because i normalized this by turn around radius then it the radius becomes one half of this then it realizes means that it becomes a stable object so this is what we call dark matter heroes okay so this is the formation of dark matter heroes and you can estimate the density non-linear density at this time so this is the number 178 in the matter dominated universe so the beauty of spherical collapse is that everything can be done analytically and you can quite easily understand the physics so it's quite good exercise to extend this to dark energy model and see what happens for example you can extend this analysis to the smooth dark energy very easily because there is no funny clustering of dark energy your physics is the same right so you consider spherical shell this spherical shell is controlled by the newton potential this is coming from the inhomogeneity inside the shell and expansion of the universe and a smooth dark energy just changing the expansion of the universe and then basically this term gives the additional contribution from dark energy density in the background so this is the only modification so then you can solve everything in this case numerically but you can easily find for example the effect of equation of state for this density at the collapse time so this is the linear density i said this is 1.68 in the matter dominated universe but changing the omega matter this number changes so this is the effect of lambda and if you change equation of state this will change the this density but you can compute this in an analytic way using this spherical collapse model you can also characterize the bdli density it was 178 in the matter dominated universe so if less shift is 2 universe is dominated by dark matter so lambda cdm the bdli over density is 178 but then lambda becomes important bdli density becomes larger and you can also compute the effect of the equation of state so this is a very good way to understand the effect of dark energy equation of state on the non-linear structures so the application of this technique to the more complicated theory is in fact very difficult but it can be done for some cases like clustering dark energy so in the clustering dark energy case dark energy clusters so not only dark matter the dark energy is clustering and in fact if sound speed is very small dark energy clusters and basically you cannot distinguish between dark energy and dark matter and in this case you can compute the effect on this clustering dark energy so this is again the collapsed density this is 1.686 for matter dominated universe this is lambda cdm so this is the dark energy model with equation state minus 0.7 now you can change the sound speed and then predict what happens so this is one way to understand non-linear collapse and in fact this was done by power so if you have any questions you have to ask power and but modified gravity situation is more complicated because for this simple equation I assume that the dynamics of this spherical shell is determined by just the mass inside this spherical shell and this is known as the Birkoff's theorem so in gr there is a nice theorem saying that if we consider a spherical shell you don't care about the environment okay you just need to know what happens inside this sphere but in modified gravity basically this does not fold and in fact you have to worry about what happens outside this spherical shell so this kind of calculations become a bit difficult but still you can get some guess what can happen at non-linear scales okay so any questions about spherical collapse model that's right but on the background it's dark energy so equation of state is not zero it's close to minus one but perturbations behave like dark matter so another way this is just one slide to show that there is another analytic why which is useful especially for less if distortions is to use perturbation theory so our task is to solve this horrible non-linear equations and one way I showed is to assume symmetry to make these ordinary differential equations another way to solve these equations perturbatively assuming that delta is small but want to include the non-linearity in a perturbative way clearly this cannot be applied to free non-linear system but you can use this technique to include non-linear effect to your linear theory predictions so idea is very simple just expand delta and theta order by order assuming delta and theta are small at the linear level you get exactly the same equations I showed in the last lecture you get the linear theory but then in the next order for example this delta times theta you get the non-linear collections and you try to solve this non-linear collections order by order again as I said this these equations are very general so I do not assume any gravity theory the only assumption is that the matter energy momentum tensor is conserved the only problem is to predict this newton potential and this in this approach basically what you have to do is to find psi perturbatively in terms of delta so everything is encoded in this relation you have clustering dark energy you have a non-linear relation between psi and delta then you have to expand this relation remember that in gr laminar CDM you have a linear person equation but this can be non-linear in this complicated model but in the perturbation theory approach you expand everything and then you compute the power spectrum order by order so the first order is just using this linear perturbations but then the next order you can have a second order times second order contribution you can have first order times third order contribution so this is called p13 this is p2 2 so this is called one loop collections and this is a non-linear collections to the linear perspective so by including this you can improve your predictions including some non-linear effect and this approach is very useful if you want to apply this to the less safety distortions okay so again i forgot when i started so i started 15 okay okay so so far i discussed how to include non-linearities using fluid but in fact even the fluid approximation breakthrough okay so i treated the collection of dark matter particles as a fluid but this is not the case on very small scales and the fluid approximation breakdown and the fact that you have to do is to solve this Boltzmann equation so basically you have a distribution of this dark matter particles and this Boltzmann equation describing the evolution of distribution function in the phase space so f depends on spacetime time and space but also the momentum so momentum of this particle so you have a distribution of this function a definition is that if you integrate this function over momentum you get density and the evolution of this distribution function is described by very simple principle that our distribution function is conserved so this is conserved against the change of time change of space and change of momentum the only complication comes from this dependence of momentum with respect to time so here you have to use geodesic equations and then you can define velocity in this way so if you multiply the momentum divided by mass and integrate over distribution function you get velocity and if you have two momenta PIPJ divided by mass squared you get density times velocity times velocity but then there is an additional contribution known as velocity divergence so this is a new quantities you have then applying this to this equation you get these equations so you can do this order by order and then the first equation you get is the continuity equation so this is the same equation we had before the second equation you get the equation for velocity so this is the same equation we had so this part is the fluid equation so you can solve using a spherical collapse model of alteration theory the problem with is this velocity dispersions so to get velocity dispersion you have to go to the next order you get the evolution equation for velocity dispersions but this equation contains another new parameters so then you have to continue to all the orders so this means that there's no way to truncate this equation and the fluid approximation is assuming that these velocity dispersions are small you ignore this then you just solve these two equations but this is not the case on small scales so at some point fluid approximations break down and you have to worry about these velocity dispersions so how we do that the only answer is n body simulations so you already had excellent lectures on n body simulations so I don't repeat anything just say that the n body simulations are solving this collisionless Boltzmann equation using many particles that's one way to do that so you can calculate these velocity dispersions for example using this method but just to emphasize what we need is the geodesics of dark matter particles and you need to know the Newtonian potential in the lambda c dm Newton potential is determined by density and density is just a collection of mass of dark matter particles and a good thing about lambda c dm is that this equation is linear meaning that you know the solution so this is just one over r so this is the reason why you can do the n body simulations using this formula and then there are a lot of ways to speed up these simulations however in the clustering dark energy or modified gravity this person equation becomes very complicated for example in the modified gravity models this is one example so you have a person equation but then you have this additional scalar field and this additional scalar field can satisfy a very complicated nonlinear equations okay so this is the problem of this clustering dark energy and modified gravity models so this part is the same but the calculation of the force is very difficult because we cannot use this one over r formula you have to solve this complicated scalar field equations and then you compute this additional force and then you have to add this additional force here but luckily there are now n body code that can do that and in fact there are a variety of n body code proposed and basically what they are doing is similar so you have a distribution of dark matter particles and you want to compute the force here in lambda c dm you just compute the potential and then you can compute the force but in the modified gravity models you have additional scalar field you have to solve the scalar field equations so what we have to do is to prepare a mesh inside n body simulations and using this mesh you for example solve the scalar field equations you compute the scalar force and you also compute the gravity and add together and then move particles and for this we have to use a usually adaptive mesh technique I guess you heard about this so you define your mesh around these very dense regions and you can solve scalar field using this adaptive mesh and then you can find this gravitational potential and all these code are based on the lambda c dm code probably you heard about lamussets so the first one third one is based on lamussets the second one by name it is based on gadget so now we have these extensions of n body code so that we can study non-linear clustering in these complicated models so in the last few minutes I just yeah you mean the caspy problem of dark matter hello I think modified gravity does not solve that problem because you enhance probably the gravity usually and you'll get the more clustering and in fact it's the opposite way so it doesn't solve that problem but you can look at this kind of problem using simulations my answer is that probably it doesn't solve but yeah so the I will show the example from the second order Eulerian perturbations but let's see if to zero I guess k equals 0.1 is the kind of the limit of perturbation theory so beyond the k equals 0.1 you need n body simulations so let's look at the example of f of r gravity and so we use a very simple model you have lambda c dm plus some collections and this collection scales like 1 over r and this collections are determined by this parameter fr0 and if this is 0 it's lambda c dm so this is describing the deviation from lambda c dm I already looked at the linear perturbations for this theory so this theory you are getting the enhanced force on small scales so we can do the full simulations solving the full scalar field equations so the scalar field is now this delta fr so this is a scalar field this is sourced by density but there is the nonlinear potential so this is complicated but this is the nonlinear functions of delta fr so this is the origin of the screening mechanism I talked about so due to this nonlinear potential as I will see this suppresses the scalar force on small scales but you have to solve these nonlinear equations to compare what is the effect of having this nonlinearity you can also consider the model where you linearize this potential term so you get the mass term so this is the linear theory I used to describe linear theory so instead of these full equations I can also solve this linear equation for scalar field and comparing the two you can understand the effect of screening so the nonlinear suppression of the scalar force so this is the example of the simulations in this model so this is a distribution of density with different parameters so as I said if this parameter is small you go back to gr so this is a distribution of density and this is the distribution of the scalar field and this is the distribution of Newton potential so now you want to compare the scalar field and the Newton potential so this scalar field is giving the additional force okay so this enhances gravity and for this parameter for 10 to minus 4 you see that there is no suppression of scalar field compared with potential in fact there is a relation between phi and the scalar field so this means that there is no suppression of the heath's force so gravity is enhanced but making this parameter small so this is a different model with 10 to minus 6 comparing the two you see that the scalar field has much smaller amplitude compared with Newton potential so this is coming from the chameleon screening I talked about so in this theory there is a screening mechanism once you get this nonlinear structure you suppress the additional force the point is that in order to see this you really need to do simulations so it's very difficult to predict this using a narrative method so antibody simulations are probably the only why it understand this nonlinear physics and this is very important because without this mechanism you cannot satisfy solar system constraint for comparison if you remove this nonlinear interaction of the scalar field you see that there is no suppression at all so you still have the enhanced heath's force okay so this means that this chameleon mechanism is working very fine and you can also look at the time evolution of this chameleon effect and so at early times the chameleon is very working so the comparing scalar field and Newton potential you see there is no scalar force so chameleon is very active at early times and you cannot see the difference between gr and this model because there is no scalar force but at late times for this choice of parameter the chameleon mechanism is not strong enough so you are getting a scalar force at some point chameleon is to exist and then at late times for this parameter you get this enhanced force but again this time dependence you can see only by solving this nonlinear scalar field yeah so for this parameter yes so now you can look at the past spectrum and and compare it with linear theory and you can understand how bad the linear theory is so this is looking at the difference between the past spectrum in lambda c dm and f of r gravity so delta p so i compute the past spectrum in this modified gravity model and subtract the lambda c dm past spectrum and divide it by lambda c dm past spectrum so if this is zero this means that it's the same as lambda c dm okay and for 10 to minus four this line is a linear prediction so this is what i talked about in the last lecture so the black data is coming from full simulations and the green data is coming from simulations without chameleon mechanism so for 10 to minus four the chameleon is not very important today so this is the result today so this means that there is no difference between chameleon and the full chameleon and the linear simulations but you see that there is a huge difference between linear theory and nonlinear theory so this happens around like k equals one but then if you make this parameter small now you see that chameleon is working even today so you see the difference between non-chameleon simulation and chameleon simulations for example for 10 to minus six this is the linear prediction this is the simulation without chameleons and this is the full simulations so you see that in the full case the deviation from lambda c dm is very small because of this screening mechanism but if you do not use nonlinear simulations you just use this linear predictions you get a very wrong answer so this means that you really need to understand this nonlinear physics if you want to change gravity okay and just say that then you say that can you trust nabadi simulations that's another question and recently we have done the comparison between different code based on gadget based on lamb says and up to k equals one we get one percent agreement does it mean we can trust the code probably not because everyone can be wrong right but we are using a different algorithm we are using a different baseline nabadi simulations so gadget is mg gadget is using gadget so this isis a bit bad name but eco smoke is using the lamb says so so it's good that we get the same answer so why this is important so it says that we want to use lessive distortions so again this is the power spectrum in lessive space so this is the monopole and this is the quadrupole and this is from our simulations so this is the predictions at z equals one this is wave number so this is monopole and quadrupole and again I divide by linear prediction so linear prediction is here and these are the simulation result so again the difference is the nonlinearity so you have to model this nonlinearity properly if you want to test this model a good thing about lessive distortion is that you can use part of a deep approach I talked about and the fact these lines blue line and red lines are coming from prediction from perturbation theory so this is not from nabadi simulations you can use perturbation theory to predict but you can see that this is valid only up to k equals 0.15 so validity regime is quite limited compared with simulations but thing is that you can compute this line using perturbation theory probably 1000 times faster than nabadi simulation and you need this because you cannot do simulations for all the cosmological parameters and all the modified gravity parameters so perturbation theory is working very fine that's the reason why this can be applied to real data so we apply this to stss data and we get the constraint from this parameter so it's like 10 to minus 3 yeah so these are so this dotted line is the usual one-loop standard perturbation theory so this solid line is the regularized perturbation theory so there's a way to improve perturbation theory so this is the one using some technique to improve the perturbation theory now so the on very large scales anyway these techniques will fail two loops so the question is so this is one loop so this is what I showed so two loops in fact you have to do the six dimensional integrations and this takes time and sometimes maybe takes more time to do simulations so you need to be careful there is a quick way to do that so you need that technique to improve that okay so the next step is to look at dark matter hellos so I talked about this dark matter hellos in the spherical collapse model and then you can predict the number density of this dark matter hellos so again this is the difference between gr and f of r depending on the parameter so this is again today so this is the mass of the dark matter hellos so for 10 to minus 4 there is no camera your mechanism and gravity is enhanced meaning that you get huge numbers of massive hellos so this is 10 to 15 solar mass dark matter hellos compared with gr you have 100 percent more massive dark matter hellos but then if you make this parameter small so this blue line is the full simulation result you notice that the number of massive hellos in this f of r gravity model is the same as gr this is because chameleons so chameleon mechanism suppresses the modification of gravity for massive hellos so this means that you don't see any modification if you ignore this chameleon effect you still have enhancement so this is very important to be included so again spherical collapse model I talked about can predict this line so this shaded line is the prediction of spherical collapse as I said it's very difficult to predict precisely but still it can give some indications what you expect and this kind of analytic method is very important and then you can see that you can constrain this parameter very well just looking at the abundance of clusters because we don't see this huge abundance of massive hellos and you can get this constraint so the constraint is now like less than 10 to minus 5 so this is the constraint on this f of r gravity parameter f r 0 so you can combine many observations to constrain this parameter I talked about isw cmb lensing rst all these gives a constraint like 10 to minus 3 10 to minus 4 but using cluster abundance you get the much better constraint so in this type of theory in fact nonlinear structures are the best way to constrain models and to do that you really need simulations you need a very good analytical understanding of nonlinearities okay I think and this is the end of my fourth lecture so any questions about the fourth lecture question is about finger of god so what would you ascribe as the causes for the peculiar velocities at that region 0 to 20 for megaparcy coverage which is where we see the finger of god effect yeah so the finger of god effect is coming from this velocity divergence inside very small nonlinear structures right so sorry what was the question yeah what would you ascribe as a causes for the velocities which cause the finger of god effect yeah so the finger of god effect is the random motions right so this random motions is not caused by this velocity divergence so it cannot be described by linear disco field and motion so that's the reason why we get huge suppression of velocity divergence per spectrum one thing which I had tried seeing is if you remove that region off and then try doing an analysis let's say correlation function analysis okay then it messes up all the parameters that includes the alcoke-paksinsky parameter and things like that would you have any comment on why that happens I don't know about that particular problem so maybe we have to talk about but just say that when I showed the prediction from perturbation theory I didn't say that but I have a free parameters for finger of god effect so we have a velocity divergent dispersion parameter I tuned it so in this sense we have the understanding of this truly nonlinear finger of god effect is very difficult so somehow we treat this as a phenomenological way giving free parameter why if we can remove it from your observations of course that's better but maybe we can talk later thank you sorry clustering dark energy or smooth dark energy yeah which of these scenarios is are in better agreement with observation observations at the moment I think it's very difficult to distinguish so at the linear scales as I said this is described by mu and sigma parameter so at the moment we see some deviation from lambda CDM smooth dark energy but in fact deviation says that gravity is weaker and this is very difficult to do either in the clustering dark energy or modified gravity because you have to suppress gravity and if you have a clustering dark energy this enhances gravity so at the moment we have no idea how to do any other question I don't see any so let's thank Kazuya for his set of lectures notice that we are going to have a small break so we just started