Initial state fluctuations and anisotropic flows of the Quark-Gluon Plasma created in Ultra-relativistic Heavy Ion collisions

The fundamental theory of strong interactions is the so called Quantum Chromo Dynamics (QCD) that is a quantum field theory with an extremely rich dynamical content. The main features of this theory are the asymptotic freedom and confinement. The study of QCD, under extreme conditions of temperature and density has been one of the most difficult problem in physics during the last decades, capturing increasing experimental and theoretical attention also in connection with its relation to the Early Universe physics.\\ In this work of thesis it is extensively discussed the effect of the primordial QCD phase transition during the first part of the evolution of our Universe: the Big Bang nucleosynthesis.\\ On the other hand, the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) programs have been used to probe the properties of nuclear matter under such extreme condition. In the light of the experimental results accumulated in these years in these ultra relativistic heavy ion collisions, the main purpose of this thesis is to study the dynamical evolution of the Quark Gluon Plasma (QGP) in the framework of kinetic theory.\\ In particular, recent experimental data show that the momentum anisotropy of the emitted particles is an observable that encodes information about the transport properties of the matter created in these HIC and also that it is an observable sensitive to the shear viscosity to entropy density ratio $\eta/s$. Hence, in this work we have investigated within an event-by-event transport approach at fixed viscosity this elliptic flow $v_{2}$ and high order harmonics $v_{n}$.\\ The principal results presented in this thesis concern the different sensitivity to the $\eta/s(T)$ at different energies (RHIC and LHC) for both ultra-central and mid-peripheral collisions, especially in the cross over region of the transition. Moreover we highlighted the effect of the inclusion in our simulation code of a realistic kinetic freeze out. Finally, we discussed the correlation between the initial spatial anisotropies $\epsilon_{n}$ and flow coefficients $v_{n}$