 Good afternoon everyone, thanks the organization for the opportunity to talk about this work on impact of dark energy perturbations on the growth index. So the outline is like this, I'll talk a bit, like a brief review about red-ships based distortions, the growth rate and the growth index, then how this can be used as a tool to study cosmic acceleration and finally show the impact of clustering dark energy on the growth index. So since 1929 we know about cosmological red-ships, so you have galaxies receding from us, you can associate that, some red-ships, but of course the universe is not homogeneous. So if you take red-ships near some glomerations, for instance some objects here, some galaxies here do not follow the law, so this is due to the peculiar velocities these galaxies have around this massive object, so there is peculiar red-ships and this might affect large scale structure and how we observe this, and in fact we can do linear theory to explain this kind of effects, so we find the red-ships based distortions, which is pretty easy, so we get Newtonian equations, fluctuations for matter, velocity and potential, so we plug continuity equation, Euler equation, Poisson equation, this admits quite simple solutions, so this D is the growth factor, which we can say is dependent from A, then if we take the derivative, the logarithmic derivative of the growth, we call this the growth rate, then after we can associate this with the continuity equation, but galaxies may have some bias in relation to matter, so we should include some bias factor here, and then from this we can define the beta, which will actually appear in the Keiser formula for the distorted power spectrum in red-ship space, so these peculiar velocities are related to the growth rate and also the bias, it also enters here, we also have to take care of that, and then of course if we get information from that and a theory for that, we can measure this beta and then the evolution, how matter evolves in the universe. Some examples of growth index, if we study the matter-dominated universe, so I see the CETA model, we get F1, the function D goes as A, so we get 1, if we study the lambda CDM model, we can parameterize F as omega m to the power gamma, so this gamma is actually the growth index, it's quite a good approximation for F, lambda CDM is gamma 0.55, actually many times this value is regarded as the GR value, what you may find in many models with gravity is described by general relativity, but you can move away a bit from lambda CDM and also study the smooth, dark energy, dynamical dark energy, but it's smooth, and then you have some correction to that, where here W1 is the value of the equation of state at red-shift one, we have this correction for the growth index, and then so this is for GR models, so you can also study modified gravity to try to explain cosmic acceleration, and for this particular Stado-Binsky model, it was computed in this paper, how the growth index changes, so there is some scale dependence here, but it's quite mild, but here is the lambda CDM line is basically constant, and what you get is something about 0.42, something like that, and decreasing it with red-shift, red-shift here is from 0 to 0.5, and all these studies about the growth index are important because in the future we expect to measure the growth rate F, and of course we can relate F to gamma, for example this Euclid forecast, they assume a traditional model here, the lambda CDM1, the green line, they expect to measure F with 1% accuracy, so having this kind of measurements, it would be easy to distinguish between some model of modified gravity, F of R, this is some coupled dark energy, dark matter model, and this is some DGP model, so since we start getting good observations on that, it can be very useful to distinguish between the various models of cosmic acceleration, okay, but up to now I just talked about the smooth dark energy models, which can fit in that linder approximation for gamma, and some guys have studied that, there are also the papers, but this one compares, this one by those at INISHAC physical review for 2003, they study some modified gravity models and some what they may call extreme case of dark energy models, and they find there is little dispersion of the growth index when you actually study these extreme models of dark energy, and in these models they also study the case where dark energy can cluster, this is what they find for the growth index, this is the dispersion, here is the lambda CDM value, here they vary the equation of state, actually this is the value at Z1, and they consider a constant equation of state, the red dots, and also variable equation of state, the blue dots, and the thing here is that they consider CPL parameterization for this variable equation of state, and CPL parameterization is good for many purposes, but it's usually this very smooth variation with redshift, so you might lose some freedom to analyze dark energy perturbations if you stick with CPL parameterization, but they say these differences are small, but now let's move to something more general than CPL, so this is actually the work, so if you take like a corassoniti parameterization, it has four parameters, the value now, the value deep in the matter dominated epoch, basically the time of transition and the width of this transition, so here's four models using this parameterization, and the thing that I hope to show you is that if you have something like this, some sharp, some small width decay at low redshifts and dark energy perturbations can be important, so this magenta line actually is a good guide for us, the equation of state is 0.95 today, but it was like minus 0.2 in the past and has a rapid transition at low redshift, these models are not totally crazy, these parameterizations don't produce a huge difference in Hubble parameter, for instance, this is the difference against lambda CDM, so this magenta line is not for low redshifts around here, it's not too much difference, it's less than 80%. That's just to say that this is not something totally crazy that might affect the background too much, so there's no point to analyze. Okay, so if you want to describe dark energy perturbations, now we need to evolve two perturbations, matter and dark energy perturbations, so these are the equations in the Newtonian gauge, we have to pay some attention to the pressure perturbation, and here's the CF, or just the sound speed, the effective sound speed, which enters in these equations here for dark energy, and then when I evolve these equations, then I compute the growth index from this expression, so let's try to clarify something about dark energy perturbations, so the key ingredient is this sound speed, effective sound speed computed at the rest frame, it cannot be negative, the squared one, otherwise you get divergent perturbations. Okay, you may think of no negligible sound speed, in this case dark energy basically follows the potential, there is this prefector here, which is important, but since matters go with k squared phi, on small scales this is much more important, so usually people neglect dark energy perturbations, but if you have negligible sound speed for dark energy, then dark energy follows the dark matter, also there's this prefector here, this prefector is important, because if you go to lambda CDM, if you go to the lambda limit, this goes to zero, so you erase dark energy perturbations. Okay, so in this case you might have comparable amounts of dark energy and dark matter perturbations, so this is based on fluids, but you can also find the Scholarfield models that might have quite small sound speed, or even you can use Lagrange multipliers and get exactly no sound speed. Okay, so you can come up with some Scholarfield model that has this great behavior. All right, so the results are here, so for those models I showed you, and this k, I compute CF1 and CF0, so the case of CF0 which you might think as a smooth dark energy has some impact, some difference, this is the lambda CDM value, and here are the different models, they are a bit above, but not too far, there's not a huge difference here, okay, but if you consider no sound speed, things may change a lot. For instance, this magenta line which get around zero redshift, you get minus 0.95, so it would be for low redshift quite similar to lambda CDM in the expansion history, might have a very different growth index. Of course, these differences all diminishes at low redshift, but if you remember what we saw from the modified gravity example, we got values around 0.4, that plot was for 0 to 0.5, so in this range is still a bit above, but if you go a bit further in redshift, you also find this value, so this is already showing us that when you allow dark energy to cluster, you might get confused if it's a modified gravity model or some dark energy model in GR. Professor Jain told us something about this in this morning, this is a particular example where you can find this difficulty about distinguishing from GR with some strange kind of matter in a modified gravity. So there's something else, which is about the parameterizations people use to the growth rate, so this is the usual form. You may consider this as constant or some redshift dependent function, so if I feed the numerical values with a constant gamma, I get several percent errors in the description of F. If I use this parameterization which was proposed by the authors of Dosset and Nishak, things get better, but still I can have several percent, about two percent, one percent error at low redshift. So even if you try to parameterize these effects of dark energy clustering, it's difficult to do in the usual way. You get some important error by doing that. Okay, so here are my conclusions. As I told you, clustering of dark energy can strongly impact the growth rate. This is due to this factor here, so as I showed you, the models have like minus 0.2 at high redshifts and then decay. So this value here is large at high redshifts and then only at very low redshifts, you have this thing approaching minus one and then you start to raise dark energy perturbations. But if they can live for a long time, in the past, they can impact the evolution of matter today. So this is why you need some feature like this. So this is how dark energy impacts. Distinguishing between modified gravity and clustering of dark energy is not straightforward. There are other examples of this in the literature. This is another one where the growth index may be more or less the same for both. So if you have some uncertainty in the measurements, it's not guaranteed. Okay, I got gamma 0.4, so this is modified gravity. It's not straightforward to do that. And of course usual as I just showed, usual parameterizations for the growth rate are not very accurate for this kind of models. Okay, that's it. Thanks.