Some Issues on the Foundations of Modern Cosmology, Gravitation and Quantum Physics

After the first clear evidence of the recession-at very high speeds-of the spiral nebulae was announced by V.M. Slipher in 1914, as a result of his work started in 1912, it still took several decades to properly understand the phenomenon in terms of an expansion of the Universe. Some historical issues around that crucial discovery and the contemporary attempts at determining the scale of the visible Universe will be discussed. Presently, very important questions to answer are: What is the precise value of the expansion rate? What drives the acceleration of the Universe's expansion? The latter is called dark energy, but what is it actually? The possibility that this could be the result of a sort of Casimir effect at the cosmological level has not been discarded, yet. One of the main technical problems in tackling this issue is constituted by the regularization and corresponding renormalization procedures. Beautiful but rather non-trivial mathematics, involving the zeta function of pseudodifferential operators (associated with physical quantities), are key in this respect. A discussion of those items is provided here.