Halo model for cosmic shear predictions

The thesis we present is a numerical study of the lensing properties of the large scale structures. The current theory of structure formation assumes that it forms via gravitational instability from initial perturbations in the density field. As the Universe cooled, clumps of dark matter began to condense. Gas and dark matter were gravitationally attracted to the higher density areas and formed the seeds for the primordial celestial objects. This scenario is called top-bottom or hierarchical scenario, because the smaller objects are the first to be assembled, then they merge to give origin to the large scale structure, (Coles and Lucchin, 2002). At present time the LSS shows up in a cosmic web, in which galaxy clusters occupy the nodes and are connected by filaments of matter, and in between there are the voids, (Einasto, 2012; Coil, 2013). Being the most bound and latest forming structures, clusters can be used as an ideal laboratory to test the prediction of the LambdaCDM model, (Davis et al., 1985; Hung et al., 2016). In particular, since clusters are dominated by dark matter, we can use them to investigate the properties of dark matter too (Kunz et al., 2016). Because the dark matter interacts only with gravity, it is not possible to trace it directly, as we can do when we observe the hot gas bremsstrahlung emission in X-ray band (Bahcall and Sarazin, 1977; Sarazin, 1986) and the light coming from galaxies in the optical band (Zwicky, 1937; Munari et al., 2013). To detect dark matter we need to recognize its gravitational effects on the images of background galaxies, (Bacon et al., 2000; Refregier, 2003). In fact the gravitational field produced by massive objects extends far into the space-time and, deforming it, it is able to bend light rays passing close to the objects and to refocus them somewhere else. This produces a pattern of distortions in size and shape of the images. The effect of deflection of light by gravity is called gravitational lensing and the objects responsible of it are called gravitational lenses, (Narayan and Bartelmann, 1996). In the near future several new large scale surveys, like the Large Synoptic Survey Telescope (LSST, LSST Science Collaboration et al., 2009) and Euclid (Refregier et al., 2010, Laureijs et al., 2011) are including among their goals the measurements of weak lensing. The main purpose of this work is to determine the effects of cosmic shear due to the dark matter distribution in the LSS produced on the background galaxies, by understanding how and how much its convergence field changes, changing the model used to describe the density field. In order to represent the large scale structure, to estimate its lensing properties and be able to compare the observations with the numerical results, we have two possible alternatives. The first one is to simulate the evolution of a piece of universe by reproducing mock light cones with cosmological simulations, but, for making a statistical analysis, a large number of realizations is needed while the N-body simulations are very expensive in terms of computational time and space, or we can treat the LSS as an ensemble of dark matter clumps, with a given density profile and a cosmologically consistent distribution, compatibly with the halo-model idea, (Cooray and Sheth, 2002). We proceeded by choosing the second approach. Our work is mainly based on the developing of Weak Lensing-Matter density distributiOn Kode for grAvitational lenses (WL-MOKA), a semi-analytical tool based on MOKA (Giocoli et al., 2012), which is able, given a three-dimensional distribution of dark matter halos, to create the surface density distribution of each spherical halo and then to calculate the convergence fields. This code respect to N-body simulations is very fast, so to test its limits and fine tune it, we took the available dark matter halo catalogues extracted from the COupled Dark Energy Cosmological Simulations (CoDECS) project, in particular the LambdaCDM simulation, implemented with the parallel TreePMSPH N-body code GADGET (Springel, 2005) by Baldi (2012), in which, they added the physical effect due to the interaction between the cold dark matter fluid and the dark energy scalar field. Starting from the outputs of simulation, we created mock light cones with a field of view of 25 deg2 and filled with halos with a given density profile and we calculated the convergence maps. At this step we considered the extension and the density profile outside the virial radius of the halos as free parameters. Then, we proceeded into a one-point and a two-point statistical analysis by deriving the probability distribution function and the power spectrum of the convergence, and we compared our results with the same quantities extracted from the simulation. Once the best match has been determined, we continued our study by analysing the entire a set of 25 light cones of the LambdaCDM simulation and we calculated the covariance maps and the cross-correlation coefficient matrices of the power spectrum. At least we compared them from those derived directly from the simulation. We also considered another cosmological simulation, LambdaCDM-HS8, with the same cosmological parameters of the LambdaCDM simulation and a different value for sigma8, and we applied the previous recipe. All these comparisons allow us both to fix the parameters which better reproduce the simulations results and to point out to the pros and cons of our method. The thesis is divided into four chapters. The first one gives an essential cosmological scenario. Its aim is to provide the theoretical concepts necessary to understand the cosmological framework in which the research takes place. For this reason, after a brief introduction to the cosmological principle and the equations that lead the Universe’s evolution, there is a focus on cosmological models and a dealing with the growth of perturbations in linear and non-linear regime. In the second chapter we depict the large scale structure, under an observational and a numerical point of view. We point out the clusters and their physical properties, in particular their connection with dark matter. The third chapter is dedicated to the gravitational lensing theory. Starting from simple assumptions, we can derive all the quantities we need to understand and give a global description of lensing, with the aim to introduce the concepts used in the research. In the last chapter we described in detail our work, the analysis and the results coming from the comparison with the simulations. In the conclusions we summarize all the results we achieved, the emerged critical points and the future perspectives.