 The Dark Energy in the title is Structure Formation and Observational Tests. Okay, can you hear me okay? Okay, so this is a lecture, so I changed gear. So in the last two lectures I talked about these theoretical models and why it's important to think about dark energy and how difficult it is to construct theoretical models. So today, so I will look at dark energy from a slightly different angle so try to understand how we test dark energy models using observations. So in order to do that, I want to specify my model. So basically I want to have a gravitational equation. So this is the Einstein equations. This T mu nu m is the energy momentum tensor for the normal matter including variance and called dark matter. But then I introduced this new tensor, mu nu. This can be either dark energy or the modification of gravity. Okay, so this mu nu describes the modification of gravity in the left-hand side or the new matter in the right-hand side. But I said that you can combine the two and we want to know how we can distinguish between these two possibilities. The only requirement for this tensor is that due to the Bianchi identity the total of the energy momentum tensor of the matter and this new tensor should be conserved. So this is the only condition. And in the background, due to this homogeneity and isotropy I can write down this immune tensor as a diagonal metric and you have energy density and pressure. So if you consider dark energy, this is the energy density of dark energy and these are the pressures of the dark energy. And as I said, the only thing you have to know is the equation of state. So this is the ratio between pressure and density. But this may come from modified gravity but you can still define an effective tensor. Maybe this is not energy but this is related to the modification of gravity but what you have to know is just an equation of state. So now using that just background equation you can only prove the equation of state. So we want to go beyond that. To do that we want to consider structure formation. To describe structure formation I will consider small deviations from Friedman metric described by these two functions psi and phi. So now this depends on time and space. And I will assume these two functions are small. So you can basically expand Einstein equations up to first order and try to find the solutions for these small perturbations. So I will use conformal time eta. So this is the conformal Hubble parameter. So this prime is the conformal time derivative. So in center of cosmic time I will use conformal time. So technical term I will consider linear scalar perturbations with respect to this three dimensional space and I assume three space is flat for the moment. And maybe you hear about this cosmological perturbation theory so in order to deal with these small perturbations you have to understand this cosmological perturbation theory and there are very nice reviews and I recommend to look at these reviews. But just say that the only important thing in my lecture is that remember that I will use the Fourier transformation so I will expand all the scalar function in terms of exponential i k x. So k is a wave number. So k dash means that you are considering very small scales and small k means that you are considering large scales. And we want to construct the index and I try this. Okay so then we want to include the index si so this is a vector but I only want to use scalar so this vector is defined as the derivative. So this is basically the spatial derivative of the scalar function. I can construct also three dimensional matrix and I decompose this into trace path proportional to delta ij and the trace less path sij so trace of sij is 0 and this trace is three dimensional trace so i is x, y and z. So I can decompose any three dimensional tensor into two scalars trace path and trace less path. And then the fourth point of this cosmological perturbation theory is that in GR, GR is the theory where you can change coordinate and does not change your physics. So this means that for linear perturbations there is a symmetry so you change your coordinate and you shift the coordinate the theory is invariant and you should care about this and I do not discuss this in details. Basically having this form of the metric I already chose a gauge so I already fixed my coordinate. So this is I can do always for any theory of gravity. That's right person gauge. You can use any gauge Newtonian gauge is the easiest gauge to use to understand the structure formation but as I said I can use any gauge it's just a convenience. Yeah so this is a linear transformation so this is a small transformation and this cosine mu contains all the information so I will decompose this cosine mu into scalar vector tensor and I only consider scalar transformations. Okay so by doing that so everything now expanded by Fourier transformation I can decompose vector and tensor so let's look at the energy momentum tensor so this part is the same as background you have density and you have density perturbations you have pressure and pressure perturbations so that's the same. So now you have two additional quantities V so this is the velocity as I said I will define this SI this is just a derivative of scalar function so this scalar function V is velocity so this is new quantities new quantity appears in the perturbations then you can have this traceless part of pressure so in the background by symmetry you can have only the part proportional to delta ij but now you can have this traceless part so this is the anisotropic stress so for perturbations you have two new quantities velocity and anisotropic stress so now for the moment I will assume that energy momentum tensor is conserved for matter so I will assume for the moment there is a conservation of energy momentum tensor so I will come back to this because remember that what we need is the sum of this T mu nu and the new tensor mu nu is conserved but for the moment I will assume that the matter energy momentum tensor is conserved so using this conservation equation you get the equations for the density perturbations so this is the energy conservation equation and so this is the Euler equation so this is the equation for velocity so remember in the background you only have one equation but for perturbations now you have two equations so the continuity equation and the Euler equation and you have anisotropic stress and pressure perturbations so these are the equations that determines the evolution of matter quantities as before you have to specify the equation of states so for the background there is a matter has no pressure so the equation of state is zero and in fact matter does not have any pressure and anisotropic stress so the cold dark matter has no pressure and anisotropic stress so these are zero but this nu tensor you have to specify equation of state but in addition you have to specify pressure perturbations and anisotropic stress so these are new parameters you have if you consider linear perturbations okay so now you want to use the Einstein equations the Einstein equations give two equations so phi and psi remember these are the functions in the metric so this is geometry it's determined by matter so I define a new density so the combination of density perturbation and velocity perturbation but basically this is the density perturbations so you have a contribution from matter and you have a contribution from this new tensor either dark energy or modified gravity so this is the person's equation so this determines the three-dimensional curvature in terms of density perturbations so in addition you have this new equation so this is the relation between psi and phi remember psi is the time-time component of the metric phi is the space-space component of the metric this is related by the anisotropic stress so if there is no anisotropic stress there is a very important relation between these two functions so if anisotropic stress is 0 you have the relation that phi equals psi so this is the prediction of gr if there is no anisotropic stress but dark energy or modified gravity you introduce anisotropic stress and you change the relation so these are the two equations you can use so the combination of these two equations with conservation of energy momentum tensor gives the evolution of all these quantities phi, psi, density perturbations and velocities so let's look at the conservation of energy momentum tensor for matter and I consider a small scale that means that the k is larger than h and the first this conformal h is related to the usual Hubble parameter as the scale factor times Hubble parameter so this ratio between k and h is k over a h and k over a is the physical wavelength inverse so you are comparing the wavelength of perturbations against the Hubble horizon so this condition means that you are considering the perturbations whose wavelengths are smaller than horizon scales so we are interested in small scales so using these conditions the matter evolution equations are very simple so this is the continuity equation and this is the velocity but I defined a new velocity it's just the k over h times v so this is known as the velocity divergence and this is in fact related to if you go back to the original definition so this is the spatial derivative of the velocity so that's the reason why it is called divergence but basically this is the velocity and then you get the equations for velocity and this is the Newtonian potential so remember psi is the time-time component so this is the Newtonian potential and then combining these two equations you get a very important equation so this determines the evolution of this small inhomogeneity in dark matter it's determined by the Newton potential so the question is how we can separate tensor perturbations from the scalar perturbations so again coming back to the cosmological perturbation theory there is a theorem that says that if you decompose perturbations into scalar vector tensor they do not mix at linear order but if you go to nonlinear order they will mix but I consider a small perturbations so at linear order they will not mix it doesn't mean that you can ignore tensor you have to consider but it does not change the scalar so this means that in fact the evolution of dark matter density is very simple you just need to know the Newton potential so I didn't use any gravitational theory so far the only assumption is this conservation of energy momentum tensor and the small scales so all of what we need to know is the potential and then you have to use the Einstein equations so how about this new component E so you have to have the equations for this new component so you have density perturbations and again you have for Euler equations to determine the velocity but this new component had four new quantities so density perturbations, pressure perturbations anastopic stress and velocity but you have only two equations so this is similar to the background you have to specify equation of state in the same way you have to specify pressure perturbation and anastopic stress so this you cannot get from the equations so this conservation of energy momentum tensor so you have to specify this so this is basically the same as equation of state in the background eta is the conhomal time so eta is conhomal time h is the conhomal hapl so for pressure perturbations maybe you heard about sound speed we often use sound speed to define pressure perturbations the sound speed is defined as the ratio between pressure perturbations and density perturbations when velocity is zero so in the frame where you are commuting with fluid if you take the ratio between pressure perturbation and density perturbation we call this sound speed and this sound speed in the general frame where the fluid is moving is written by this and this Ca is the adiabatic sound speed so this is very similar but constructed using the background quantity so this P pressure is the background pressure low is the background density and prime is the conhomal time derivative so you can calculate this and in fact this is written just in terms of the equation of state but this sound speed is not determined by adiabatic sound speed and in fact you can have any sound speed depending on what kind of fluid you have so you have to specify this sound speed so in this case this is WE so this is a general formula which can be applied to any component so if you consider E component it must be E okay? yeah oh yeah so I do the Fourier transformation so this was originally a function of time and space but spacetime I do the Fourier transformation that's the reason why I have k so this is a function of k and eta however for each k this is the usual derivative okay? okay so let's look at the several cases I will classify the evolution of this dark matter density according to the different models the simplest case of course is lambda shidem in this case this E menu tensor is just minus lambda times G menu then the Einstein equations are very simple so this is the Poisson equation and as I said you have this condition between phi and psi so this means that you can change psi to phi so remember I had psi here and so you want to find psi so psi is the same as phi and phi can be related to delta and you get these equations for delta so this is just a equation for delta so density perturbations you can solve it so this just means a very simple equation this is the gravity so the gravity of dark matter basically makes the density growth and this is the friction term which we already saw in the scalar field example and this friction is coming from the expansion of the universe and a simple case is the matter-dominated theta so forget about lambda in very early universe you can forget about lambda so the scale factor in terms of conformal time scales like eta squared and you can calculate this density in terms of the conformal Hubble parameter and this you can calculate from scale factor and substituting all these you can solve this equation and you find that this equation has a solution where the solution grows like scale factor so d plus so because this is a second-order equation you have two solutions one growing, one decaying but we are interested in the late-time solution so I will choose growing mode of solutions d plus so in the matter-dominated error the matter density grows as scale factor so that's the important result of this equation so where is the effect of lambda? so effect of lambda indeed happens only here because the conformal Hubble parameter is related to the density of matter and lambda at late times lambda becomes important and the expansion of the universe becomes faster and faster and the friction term wins over gravity and what happens is that if you normalize this growth function by scale factor in matter-dominated universe it always won but due to this lambda expansion becomes fast and the structure grows basically slows down so in terms of d plus over a this is lambda-schedem prediction so you see the suppression of growth so this makes sense right so if you have lambda it's faster and faster and the structure does not grow as fast as you imagine so this suppression is the evidence of lambda so let's consider a general dark energy model so now we want to extend this to the general models so let's consider a dark energy model in the background you can have any equation of state however in this case I only consider a very smooth dark energy so dark energy does not have any perturbations so this means that the density and anisopheic stress sorry this is a capital Pi is 0 so there is no perturbation of dark energy so in this case in fact you get exactly the same equations for delta because if there is no density perturbations from dark energy and there is no anisopheic stress basically Einstein equations are the same so it doesn't affect any gravitational equations so you get exactly the same equations so the effect of lambda dark energy is included in this again the Harper expansion and this is a very good exercise so you can write down this equation and the background equation using this N so N is the logarithmic of a scale factor so this dot is the derivative with respect to N and you can derive this equation so this takes the growing solution and you can derive this equation from this and this is coming from just the background equation conformal time derivative and I change conformal time derivative to N derivative so dot is N derivative oh so lambda has no perturbations so there is no perturbations by definition for dark energy you have to consider perturbations here I neglect the perturbations okay so then look at this D plus over A as I said if there is no lambda you should be just one this is lambda CDM now you can change equation of state so this is one parameterization so W naught is constant W A is constant but this part is describing the time dependent of W and so W 0 can be different from minus 1 and you can solve these equations and you can derive this growth rate and I have three cases so this blue line is W naught is minus 0.8 and there is no time dependent so this is constant equation of state and this is W naught minus 0.8 but W A is minus 0.3 this is W naught minus 0.8 W A is plus 0.3 yeah that's very good question yeah is it consistent to neglect perturbations I will come back to this so assumption is that because we are considering very small scales small scales you can ignore perturbations that's the assumption but I will come back to this okay so then you notice that so this is the lambda CDM and if W is larger than minus 1 you get more suppression and this depends on normalization in this figure I fixed the present state dark energy energy density so if equation of state is larger than minus 1 dark energy density changes and in fact dark energy density is larger in the past compared with lambda CDM that's the reason why you get more suppression however you see that due to these parameterizations if W naught is minus 0.8 and W A is minus 0.3 effectively it's very similar to minus 1 so there is a degeneracy between lambda CDM and dark energy okay so this will show up later if we consider observational constraint and another interesting quantity you often see is the gross light so the definition is the logarithmic derivative of this density logarithmic of scale factor and this is basically the derivative of scale factor in terms of A and you multiply A over D and this basically measures how fast the delta is changing and again this is a very good exercise you can also derive the equations for this F gross light because this is already fast derivative you get the first derivative equations and again you can solve this very easily dot is again derivative with respect to log A and again if you specify W you can solve this and find F and there is a very interesting prediction for this gross light in dark energy model with no perturbation of dark energy this gamma is defined like F is proportional to omega matter to gamma if you define F in this way gamma is 0.545 for lambda CDM and there is a very little dependence on equation of state so if W is very close to minus 1 gamma is always this number so this is a very interesting prediction of this smooth dark energy but remember that this doesn't mean that F itself does not depend on equation of state so there are a lot of examples with different parameters and clearly you see that F itself depends on equation of state that's just because omega M depends on equation of state but if you can measure gamma this is a very robust prediction of lambda CDM and smooth dark energy so that's the reason why you often see this gamma for example in the Euclid mission statement because this gamma is an indication for the non-lambda CDM physics okay so let's move on to more complicated scenarios so there was a question so can we really neglect the perturbation of dark energy that's the question so let's consider the model where I have the density of dark energy but still let's assume there is no pressure anisotopic stress so I assume there is no anisotopic stress so I only consider the density perturbations so how we get this clustering of dark energy so in order to understand that let's consider a very simple example where you have some pressure perturbations so pressure perturbation of this fluid is proportional to the density perturbation and this is the sound speed and assume that this fluid dominates the universe and you repeat the same calculation as before and you get the equations for this density so this is the same equation you saw for dark matter if there is no pressure but if there is a pressure you have this additional term determined by the sound speed for dark matter we say that there is no pressure perturbations so there is no term like this so then you get the previous equation but if there is a pressure then you get this equation so what this term does so this is basically coming from pressure and this is gravity so basically you are looking at the situation that you have some inhomogeneity and if you have gravity basically gravity won't collapse but there is a pressure then this pressure won't support this system so this is a comparison between pressure and gravity and there is a wavelength determined by this combination determined by the sound speed and if we consider very large k basically pressure wins over gravity and density does not grow so I assume this must be positive if it is negative it becomes unstable you can see from this if this is negative you get an exponential function and it grows so this means that in order to have clustering of dark energy the sound speed of dark energy must be small that's the condition so let's look at the quintessence model so we looked at the scalar field example of dark energy so how does this fit in so if you compute the sound speed of the scalar field so this is the definition so in the left frame of the scalar field the velocity is zero sound speed is one so this means that the scalar field is very different from dark matter it has a pressure so sound speed is one and you can also compute the equation motion for the scalar field and you find that a similar gradient term so this k-square term this works so in the scalar field equation you have this term and this is one for scalar field if the scalar field is described by this Lagrangian so this is the quintessence model so this means that the scalar field with this Lagrangian has a sound speed very large larger than zero so this means that the scalar field does not cluster on the small scales so that's the reason why quintessence can be regarded as a smooth stark energy on small scales so you can ignore the perturbation of the scalar field so this is an example of the smooth stark energy however there was a very nice question it is indeed consistent to ignore scalar field perturbations and the answer is no and this is a very good example so this is a CMB power spectrum so this is a CMB power spectrum, temperature power spectrum this is a multiple so this is small scales, this is large scales so you have two cases equation state is minus one third so it's a very extreme case and this is the case when you ignore the scalar field perturbations by hand and then you get this huge effect but then if you include the scalar field perturbations this is your answer so this means that you overestimate the difference from lambda CDM if you ignore the perturbation of the scalar field by hand so this means that you now I think so this is an effect inconsistent so if you start from Lagrangian you have to include perturbation of the scalar field but by hand you can set this perturbation zero it's not consistent but you can do that you get the very wrong answer however you get the same answer on small scales this is what I'm saying if you go to small scales you can ignore the scalar field perturbations so on small scales you can think this is a smooth dark energy but remember that this is not always the case on horizon scales, near horizon scales you have to worry about perturbations so how we get this small sound speed if you want to have clustering of dark energy so there is an example again using the scalar field the action is like this so instead of having this kinetic term you can have any function of this kinetic term and then you compute the sound speed and sound speed depends on this function k and if you k is a linear function this is a standard quintessence then there is no second derivative so this comma x is a derivative with respect to x so this is zero and sound speed is one but if there is a second derivative you see that you can have small sound speed so this is an example where you can have a small sound speed for scalar field and if you have this kind of model scalar field can cluster even you can make the situation that the sound speed of this scalar field is very close to zero so it behaves like dark matter even if it looks like dark energy in the background so this is the example of the clustering dark energy so let's look at the effect on the growth on this from this clustering of dark energy sorry can you say that so this has no ghost because there is a condition but if you choose k properly there is no ghost in this background so remember I choose the correct sign here so I'm not changing the sign so this must be so in the natural unit there is no dimension any other questions yeah sorry yeah it is just a scalar field theory and you can write on any you know Lagrangian you say this is just a function of kinetic term you can do so where this comes from that's a different question but as a Lagrangian for scalar field you can write down this so going back to the effect on the growth so now the Einstein equations become this so this is the passing equation so now you have the clustering of dark energy so you have a contribution from dark energy clustering but I assume that there is no anisopic stress so phi is still the same as psi so this is the condition for this class of model so this means that indeed the clustering of dark energy is like modified gravity because for dark matter this is basically effective Newton constant so you are changing gravity for dark matter because there is some unknown matter dark energy which clusters and they make gravity so this changes the gravity for dark matter so it looks like modified gravity but this can be modified gravity or this can be just half a dark energy for each cluster so this is the point so the clustering of dark energy looks like modification of gravity but the one assumption I used is the anisopic stress so remember that this relation between phi and psi so this relation is proportional to the anisopic stress I assume that this dark energy does not have any anisopic stress and this is usually the case for example scalar fields does not have any anisopic stress so for quintessence or kessence you do not have the anisopic stress so this is 0 for the scalar field psi is the same as phi and for the normal matter like radiation so radiation has anisopic stress however the anisopic stress usually is smaller than density perturbations and of course it's very uneasy to talk about normal matter for dark energy because we don't know what is dark energy and dark energy can have large anisopic stress but all the known matter has a very small anisopic stress small compares to density so there is a reason to assume that anisopic stress is smaller than density but at the moment this is the assumption ok so finally let's look at the case when the anisopic stress is non-negrisible and this is the case of modified gravity so let's consider the branch-stick gravity I showed and in fact a simplest example is the f of r gravity this is equivalent to this theory so there is no kinetic term but the scalar field couples to the curvature and in this case this is a very special case of branch-stick gravity f of branch-stick parameter is 0 and then you can consider a small perturbation of scalar field so at the leading order this is just a gravitational constant but you can consider small perturbations and in the last lecture I said that this small perturbations of scalar field couples to matter and changes gravity so in the Poisson equations you get the contribution from this perturbation of scalar field and this mediates the additional force ok so this looks like just a clustering dark energy if you compare this clustering dark energy case you can identify that the clustering of dark energy is coming from this additional scalar field sorry can you revisit again yeah so we consider linear perturbations at the moment and in fact if of our gravity this parameter is always 0 and the solar system constraint is evaded by the potential I didn't write it down but for linear perturbations you can linearize the potential ok so but the new thing is this anisopic stress the equations from this Lagrangian you get a contribution from this scalar field perturbations to this equation so now the relation between phi and psi is modified and this is determined by this scalar field perturbations oh vector field as I said that I do the scalar vector tensor decomposition and for this linear order vector is decoupled from scalar you can just think about scalar of course there is a vector perturbations but you can think this separately so what is the question ah sorry thank you yeah in the Einstein frame in fact anisopic stress disappears but the effect appears in a different way so I will show this is the next class ok so let's look at the examples as I said that if of our gravity is the basically gravity with a branch parameter zero but the difference is that you have potential in addition to this and if you linearize it you have this mass term so this is the equation for phi so these are the Einstein equations and you need the equations for phi and in this theory equations for phi is given by this and so phi is sourced by delta sorry this is this delta so in addition there is a mass term depending on this function so now you solve this equation publish it into this and then you can find the evolution of dark matter density perturbations and this is all determined by these equations for Newton potential so it's a bit complicated so let's look at it the solutions for the Newton potential in this theory looks like this so this part 4 pi g times this one looks the same as laminar CDM but then due to this mass term you get these corrections so there are two cases so again this mass term as I said inverse of this mass is the Compton wavelength so if we consider large scales larger than Compton wavelength so large scales means that the k is small so this term becomes large this term becomes large so then taking the ratio you see that this ratio is 1 so you go back to GR so on large scales in this theory you go back to GR this is because the scalar field has a mass so it doesn't propagate beyond this Compton wavelength so this scalar disappears and you go back to GR but if you go to small scales k becomes large so this term becomes small this term becomes small and you notice that the gravitational constant becomes 4 third times g so gravity is stronger in this theory so in fact this is a picture so inside this Compton wavelength gravity is stronger so this is due to this additional force you have in this theory so this is the program I mentioned last time so if you modify gravity you have this additional force and this mass is indeed of the order Hubble scales so this scale is a cosmological scale and inside of which you have a heath force and you get a stronger gravity so having these passing equations the evolution of this growth function becomes very complicated now it depends on k so it depends on the scale you are looking at so this is the case of Ramda CDM so Ramda CDM has this behavior so due to the Ramda the structure growth is suppressed so this is a two-dimensional figure of showing this so this is less shift so this is today so this is today and you start from constant and then it decays but this doesn't depend on the k so wave number but in this gravity model you get exactly the same behavior on large scales for small k so you look at this looks like Ramda CDM and in fact for small k the growth rate behaves like Ramda CDM but if you go to small scales you have an enhancement of gravity so that you get the enhancement of the growth in this model the structure is more enhanced on small scales and you get a huge increase of your structure so if you look at this k-mode so for small k this is larger than one so this means that in fact structure is more structure growth is more pronounced compared with even the matter-dominated universe so this is due to the stronger gravity but you see that once you have this complicated modified gravity then even the growth rate becomes very complicated and you notice that then you have to have screening mechanism for small scales your gravity is completely different from GR so you have to have screening mechanism to recover solar system control so finally this is the final class I said that the only condition I had is that the combination of these two tensors is conserved but you can have the situation that in fact you can have energy exchange between dark matter and this new field and this is known as the interacting dark energy model so if you say e comes from dark energy m is dark matter so dark matter and dark energy is interacting and in fact this is nothing but the Einstein frame version of the same theory I talked about in the Branstich theory can be written in this way so now this becomes Einstein gravity so it's the same as Einstein theory but now you have interaction between this scalar field and dark matter and if this scalar field is the dark energy you got the interactions and this you have to specify this energy exchange and in this example energy exchange is determined by the scalar field and this alpha so the coupling strings so how this changes the structure formation you don't change Einstein equations because gravity is Einstein but now you change the evolution equations for matter because now you change this equation in this example you don't change continuity equation but you change the Euler equation and if you look at the collections you see that this perturbation of the scalar field appears here and in fact this is similar to modified gravity because now you are changing the Newtonian potential but this is coming from the coupling and in addition you have a different friction term coming from the coupling so in this case remember that the matter is coupled to this scalar field so again you have to worry about the local constraint and in this case as I said you can debate this constraint by assuming that this coupling happens only to dark matter not to variance so this is the summary of all the models and so we start from lambda CDM smooth dark energy so this model is characterized only by the equation of state you can have a k-sense type model where you have a clustering of dark energy but no anisopic stress modified gravity is an example where you have clustering of dark energy and anisopic stress and we can have an interacting dark energy so in this case you have to specify these interactions all these models have a different impact on the structure formation so the question is how to distinguish between these models so I think I started from okay so any questions so far? okay so let's look at the so far kind of observations we can use so you will have very nice lectures on large-scale structure and CMB next week so it is not my intention to review all the observations my aim here is to again make the assumption very clear so for kind of assumption you are using to measure these things and sometimes if you look at the formula sometimes GR is used in a hidden way so I don't want to do that so I try to make sure what is the assumption to do these observations so the background I don't need to repeat so we have supernovae and baryonacostic constellations and what this is doing is to measure this background expansion history so I do not repeat so you can do this background observations so for the structure formation what we can do so one option is to measure the weak lensing so the lensing is the phenomena that you have a mass the trajectory of the photon is bent due to this mass distribution so here the assumption is I can use this metric so small perturbation from freedom universe but I do not use any gravitational theory so this is just a geometrical assumption the assumption is that photons follow geodesics of this metric and then you can calculate so called convergence convergence which basically measures how the light is bent by the structure between the source and the observer and this is a bit complicated formula so let's look at it separately so this first term so chi is the commuting distance okay so chi s is the distance between us to the source so this is the distance between us to the source chi is the distance to the lens so this is the dark matter distribution at chi and chi is the distance between this lens to us okay so this is a geometrical factor which comes from the formula for the lensing and first determines lensing is this combination phi plus psi over half so basically if you have this lensing potential this is phi plus psi half here this lensing potential changes your trajectory of photons and this changes your image at your observer position and this is a two-dimensional derivative along this plane but important thing to remember is that this is measuring this combination phi plus psi and then this convergence is related to the share and this share creates the difference of the shapes of galaxies that we can measure so of course there are long stories from here to here but from a theoretical point of view important thing is that lensing is determined by this combination phi plus psi does not use any gravitational so CMB is the background observations but there are two things you can measure which is sensitive to the structure formation so one is known as the integrated saxophone effect so this is again related to this lensing potential so if you have a lensing potential and this changes with time so eta is a conformal time due to this change of the potential CMB photon temperature is changed and this is determined by the time derivative of this lensing potential so this is another way to measure this combination but in this case this is determined by the time derivative also the CMB photon is lensed in the same way so you get exactly the same formula but of course the source is CMB so LSS is the last scattering surface so CMB photon is emitted but due to the dark matter distributions between last scattering surface and us CMB photon is lensed so the position is shifted and this shift is also determined by phi plus psi so we have all these three measurements measuring phi plus psi and one final very important observation is the less shift distortions so if you observe galaxies and the galaxies have peculiar velocities they are moving and of course you measure the clustering of galaxies not in the real space but using less shift and using less shift you cannot distinguish between the expansion of the universe and the velocities of galaxies so let's imagine you have a one dimensional distribution in the real space but the galaxies are moving which means that in the real space this is the distribution in the same line but if you measure this clustering in the less shift space the position of the galaxies appears to be here so this enhances the clustering so this is known as the less shift distortions and this is determined by the velocity of the galaxies so this density of the less shift space that we measure in the galaxy survey is determined not only by density but also the velocity again there is no assumption about gravity here the only assumption is that the velocity of galaxies is the same as velocity of dark matter that's the assumption so I will use theta to theta m so I assume galaxy velocity is the same as dark matter this may be not the case if there is an interaction between variance and dark energy and CDM but this is the assumption so if galaxies velocity is the same as dark matter so you are measuring the dark matter velocity and then you can light down the clustering in the less shift in this way so mu is the angle between your line of sight and k because this happens to be along the line of sight that's right I assume that galaxies are moving because dark matter is moving and I assume that the motion of galaxies are determined by the peculiar velocity of dark matter that's the assumption here this applies only for large scales so on large scales everything follows the dark matter peculiar velocities on small scales it's not the case so you have to worry about body things like that so the velocities of clusters for example is not necessarily the same as the velocity of dark matter so this is really a large scale so we can ignore all the things so I do not assume anything about so I assume variance and dark matter and galaxies velocities are all the same that's the assumption and this may not be the case if you have interacting dark energy so the point is that you can change this theta to delta using this continuity equation so that's the assumption if you have this equation this is related to this gross function I mentioned so the safety distortion is a way to measure gross rate but remember that I am using this equation so this is the assumption that continuity equation follows okay so the rest of my lecture let's look at the observational test what you can do let's start from background the simplest thing is to parameterize so the dark energy equation of state the popular one is to theta expand around A equals 1 so this is the parameterization I already showed so W0 and WA so WA is describing the changing of equation of state so now we have a very good paper by Planck last year summarizing the current status so I try to show the current status of the observational constraint so this figure shows how good this parameterization is so there are many lines but just look at these two examples so this is the two examples of quintessence model with some specific potential and you can reproduce the equation of state dynamics the time dependence using this parameterization very well the constraint from Planck so BSH means that this is the background of observations like supernovae so using that you get this constraint so WA, WA 0 is consistent with Ramadashidem but you see that the errors are quite large so in fact the errors on WA is huge so this is the current status and also you see this degeneracy I talked about because fat matters is basically the sum so there is always a degeneracy along this line and you cannot break and other observations you can also use weak lensing because weak lensing contains some geometrical factor which is affected by W and also you can use RAS RISD, RIS-50 distortions and again error by the large and they are not that consistent but by combining this and you see that the Ramadashidem is still consistent but remember that the constraint are not that strong at the moment so let's see okay let's do that so one thing we may want to do is to try to constrain this function in a model independent way so the problem of parameterization is that the result you get will depend on how you parameterize so in order to avoid this what you can do is to basically make bins in Z so let's see and treat W in each bin as an independent parameter so you want to deconstruct some function of W of Z so what you can do is to make many bins and then you say W is like this and try to constrain W in each recipe bins however this is not a very good idea because if you compute the errors on this parameter so this W bar is the best fit parameter in each bin and you compute the errors and you find that the errors are in this bin so you have errors and errors here and errors here are very correlated so this means that it's very difficult to assess how significant the deviations from lambda CDM is just looking at each bin so this is expressed by this error matrix called covalent matrix and this covalent matrix is highly non-diagonal meaning that errors here and errors here are highly correlated so what you can do is just a linear algebra so you just recombine your WI in a clever way so that the new parameters errors are correlated so this is known as the principal component analysis so basically you want to diagonalize this error matrix and if you find the eigen mode you expand this function not this step function you just expand using this new basis and alpha is just a linear combination of all these parameters so this is just a clever trick to rearrange your parameters so that you can have a new parameter and errors on this new parameters are uncorrelated so this is oh yeah that's a good question so the question is whether this depends on the size of bin and if you have enough eigen mode it will not depend on the size of the bin that's a good point I mean at this point everything depends on the size of the bin but once you go to this basis you can remove this size dependence to do that of course you need quite a huge number of well defined well measured eigen mode but eventually it will be independent of the bin size so just look at the example so using this technique basically what you can do is to you can reconstruct double of z so this is one way not using the parameter so this is the eigen mode so this is the combination of w in each bin so how you combine different dependence of this w however there is one question so I said this is independent of the parameterizations but in fact you cannot escape from theory the reason is that from observations you want to measure this function however you cannot measure all of these functions the errors becomes larger and larger if we increase this number for example this is the example this is basically how well you can measure and how many this function you can measure so you can measure first three this function very well so large number here means that the errors are small so you can measure these three so one to three mode very well but then the errors becomes huge and the fact you might want to do is to say that you cannot measure all these mode and set the amplitude of all these mode zero but this is a very bad idea because I think that the amplitude of all these modes are zero means that you measured all these mode in a very precise way the amplitude of zero with no errors so this introduce a bias so what you have to do is to put some theoretical expectations how these not well measured mode behave from theory which means do not tell us about these mode so you have to put theoretical prior so just a caution if you see model independent reconstruction of W there is a theory always hidden so you have to always check but this is better than parameterization because to some extent you can remove the dependence on the parameter so I will come back to this probably next lecture so another way is to use theory so if you remember the equation of state can be parameterized in the scalar field theory in this way this is a slow roll parameter this is the density parameter so why not use this form so this was done by prank paper but they don't want to basically look at huge number of potentials what they did is to define two parameters so one is the parameterization of this is from phi so this is from s this parameter when dark matter density and dark energy density are the same so this is just a definition to parameterize this and you have to parameterize this omega phi the another parameter is this combination at early times so this is epsilon s this is epsilon infinity and this is lambda so 0 and if you have a potential you can predict these parameters for given potentials and you can put the observational constraint and this parameter epsilon infinity basically measures the importance of dark energy at early times so this is the limit where A goes to 0 at early times so if dark energy is very important even at early times omega f is large so epsilon phi is large so this is basically the freezing model so at early times dark energy is important like inverse of phi appears here so dark energy density is large at early times and this is already excluded and then there are a bunch of the scalar field potentials so finally let's look at the structure formation so how we use the structure formation so we are talking about just the distance so this is the moving distance and in the smooth dark energy if you change equation of states so this is a simple example of changing constant equation of states so this is lambda c dm this is say w is minus 0.7 and as I said the nature of lambda c dm and smooth dark energy is that once you fix background everything is fixed so this is the gross rate divided by a so this is 1 for no lambda you have lambda c dm and this is smooth dark energy with w minus 0.7 the point is that there is a one to one correspondence so if you measure this you predict structure formation so this is a prediction of lambda c dm and smooth dark energy and this changes if you have a clustering so this is a predictions for clustering dark energy or modified gravity so the same let's say you have the background with equation of states minus 0.7 you have two models smooth dark energy and clustering dark energy or modified gravity you cannot distinguish between the two just using background it's the same however if you look at structure formation you have additional contribution from clustering of dark energy this changes how structure grows so these two models you cannot distinguish using just background but looking at structure you now see the difference so this is smooth dark energy now you can break the degeneracy so that's the reason why we want to combine the measurement from background and the measurement from structure formation so in this way you can distinguish between the smooth dark energy and the c dm from this more complicated clustering dark energy modified gravity this is a very good example of how this shows up so let's imagine that our universe is described by clustering dark energy or modified gravity you are living in a universe where dark energy clusters of course you don't want to trust this so you want to try to fit your data using this simple parameterizations so now you use background observations supernovae and cmb and you get constrained so now you use a combination of supernovae and weak lensing so weak lensing measures the structures you get a very different sorry this is using cmb but the point is that having weak lensing you get a very different constraint why? that's because of this difference right so your universe is described by this blue line and you try to fit this using smooth dark energy so you can do that in the background but if you want to try to fit this for weak lensing you have to have a different equation of state so there is an inconsistency between background and structure formation so if you see this kind of constraint this indicates that dark energy is not as simple as you think so dark energy should have some kind of clustering to see this inconsistency so this is the thing you can do so you can combine structure formation and the background you can test the nature of dark energy so this is the final slide so how then you distinguish between clustering dark energy and modified gravity so what I said is that using structure formation you can distinguish between smooth dark energy and clustering dark energy but clustering dark energy and modified gravity looks the same so let's remember that the difference between clustering dark energy and modified gravity in general is that you have a large anaesthetic stress so remember that large anaesthetic stress means that phi is not the same as psi so remember using weak lensing you can measure phi plus psi and in GR this is the same as psi because phi is the same as psi but if you have anaesthetic stress phi plus psi is not the same as psi so if you remember we have two kind of measurement one is to use the lensing potential like weak lensing, CMB lensing ISW all of these are sensitive to phi plus psi but the peculiar velocities are determined by only by psi so this means that so this is the case so peculiar velocities are determined by psi so if phi plus psi is different from psi this means that there is a difference between weak lensing and peculiar velocities so this is another step within the structure formation combining different way to measure structure formation you can distinguish between classing dark energy and motor gravity for example you can construct an estimator comparing phi plus psi to theta so you can measure this from weak lensing you can measure this from peculiar velocities and if this relation is not false then you can see some inconsistence and this was applied to the real data so this must be the prediction from lambda CDM for example for VAR predict on this as you see the data points are quite scattered the error bars are very large so at the moment we cannot say anything about this test but in the next lecture I will show that in the future this kind of test becomes realistic okay so I think it's time so I finish my lecture here so the important thing is that it's very important to look at many different proofs of dark energy but you have to remember the assumptions you make to make these observations okay