Bibliography: pages 112-113.This thesis discusses the Sachs-Wolfe effect, which is the variation in the observed temperature of radiation emitted at the last scattering surface which occurs at the place where matter and radiation decouple at about 4000 degrees Kelvin. The work is in two parts, with the first part dealing with extensions made by George Ellis, Chongming Xu, Bill Stoeger and myself to the paper by Miroslaw Panek [13] where the gauge invariant formalism of cosmological density perturbations by James Bardeen [1] has been used to find the SW effect in the case of a perturbed Friedman-Lemaitre-Robertson-Walker (FLRW) universe with a barotropic equation of state describing the matter in the unperturbed case. In our work we extend the example given by Panek for a flat universe (K = 0) filled with dust where the density perturbations are adiabatic, to the case of non-fl.at universes (K = -1, 0 + 1) filled with a mixture of N types of matter where the density perturbations are nonadiabatic. The second part shows the agreement between the formalisms of Sachs and Wolfe's pioneering paper and the recent work of George Ellis and Marco Bruni which presents the study of cosmological perturbations in a gauge invariant and covariant way. After the overview of the work covered in this thesis, the gauge invariant formulation of Bardeen is discussed where we follow the description by Panek of a universe whose energy content is described by a mixture of N ideal fluids coupled only by gravity. From the Einstein equations we get Bardeen's evolution equation for the gauge invariant energy density perturbation which is now given for the N different matter fluids as it appears in Panek. We then checked Panek's equations where he finds an expression for the placing of the perturbed last scattering surface, after which he derives an equation for the fractional temperature variation and writes it in terms of the perturbation variables. The equation found by SW for their particular choice of K = O, pressure free dust, where the last scattering surface is placed at its unperturbed position, is verified in terms of the Bardeen formalism. Now we extend this simple case to nonadiabatic perturbations in the same scenario and find the SW effect for a mixture of two fluids: dust and radiation, with nonadiabatic perturbations in a not necessarily flat universe. We then generalise to the case of a mixture or baryons and radiation and N types of matter. This section then ends with a calculation of the difference between temperatures taken from two different directions in the sky and is written in terms of the fractional temperature perturbation defined by Panek. The second part puts forward the formulation of the gauge problem by Ellis and Bruni (EB), and then writes out their gauge invariant quantities in terms of the SW variables. Their evolution equations are verified in this form, and the shear and vorticity determined as well. Now all of the EB cosmological quantities are listed for the special gauge that SW use and then we explore the relation between the SW metric and that of Bardeen before ending off by verifying that the form for the redshift in the EB approach is in agreement with that given by Panek
