Theoretical tools for the physics of CMB anisotropies

This review presents, in a self-consistent manner, those analytical tools that are relevant for the analysis of the physics of CMB anisotropies generated in different theoretical models of the early Universe. After introducing the physical foundations of the Sachs-Wolfe effect, the origin and evolution of the scalar, tensor and vector modes of the geometry is treated in both gauge-invariant and gauge-dependent descriptions. Some of the recent progresses in the theory of cosmological perturbations are scrutinized with particular attention to their implications for the adiabatic and isocurvature paradigms, whose description is reviewed both within conventional fluid approaches and within the the Einstein-Boltzmann treatment. Open problems and theoretical challenges for a unified theory of the early Universe are outlined in light of their implications for the generation of large-scale anisotropies in the CMB sky and in light of the generation of stochastic backgrounds of relic gravitons between few Hz and the GHz.