Late-time integrated Sachs-Wolfe effect in mimetic Horndeski gravity

In this thesis we perform the analytical calculation of the integrated Sachs-Wolf (ISW) effect in the framework of mimetic Horndeski gravity. We start with a review of the standard model of cosmology and then, introducing also the cosmological perturbation theory and studying the Boltzmann equation for photons, we obtain the equations for the free streaming of photons after recombination. Here we define the ISW effect, a spectral distortion of the cosmic microwave background (CMB) radiation originated by the variation in time of the gravitational potentials. Afterwards we introduce the mimetic framework following its first formulation (mimetic dark matter) and then we generalize the discussion to mimetic Horndeski gravity, a very general scalar-tensor theory of gravity. We then obtain the time evolution equations of the scalar perturbations both in general relativity and in mimetic Horndeski gravity, imposing a Lambda-CDM or a perfect fluid dark energy background and finding the explicit solutions only in the special case of mimetic cubic Horndeski. Finally we perform the analytical calculation of the late-time ISW effect in general relativity and in mimetic cubic Horndeski, showing that they are identical since the time evolution of the gravitational potentials is the same in the two models. The case of general mimetic Horndeski gravity could give us a different result for some reasons that are explained, but numerical calculations would be needed to investigate it.