Testing the initial-final mass relationship of white dwarfs

White dwarfs are the final remnants of low- and intermediate-mass stars. About 95% of main- sequence stars will end their evolutionary pathways as white dwarfs and, hence, the study of the white dwarf population provides details about the late stages of the life of the vast majority of stars.Since white dwarfs are long-lived objects, they also constitute useful objects to study the structure and evolution of our Galaxy. For instance, the initial-final mass relationship, which connects the final mass of a white dwarf with the initial mass of its main-sequence progenitor, is of paramount importance for different aspects in modern astrophysics. This function is used for determining the ages of globular clusters and their distances, for studying the chemical evolution of galaxies, and also to understand the properties of the Galactic population of white dwarfs. Despite its relevance, this relationship is still relatively poorly constrained.The main aim of this thesis is the study of the initial-final mass relationship. For such purpose we have used two different approaches. From an observational perspective, the statistical significance of the current initial final mass relationship can be improved by performing spectroscopic observations of white dwarfs for which some important parameters are available. Since this approach involves the use of theoretical stellar evolutionary tracks the resulting initial-final mass relationship is, in fact, semi-empirical. In this thesis we present a promising method which consists in using common proper motion pairs comprised of a white dwarf and a FGK star. It is sound to assume that the members of the system were born simultaneously and with the same chemical composition. Moreover, these stars are well separated and it can be considered that they have evolved as isolated stars, since mass exchange between them is unlikely. Thus, a careful analysis of the observational data of both members of each pair allows us to derive the initial and final masses of the white dwarf components, something which is totally impossible when white dwarfs are isolated. Considering the new data that we have obtained with this work and the observational data currently used to define the initial-final mass relationship we have carried out a revision of this relationship, giving some clues on its dependence on different parameters, especially on metallicity.The second approach to improve the initial-final mass relationship involves an indirect measurement, which has been carried out by studying its influence on one of the powerful tools related to the white dwarf population, the white dwarf luminosity function. We have computed a series of luminosity functions using different theoretical initial-final mass relationships, and also, considering the semi-empirical relation derived in this thesis. We have compared these computations with the available observational data in order to evaluate the validity of each of these relations.In order to increase the statistical significance of the white dwarf luminosity function and to improve the initial-final mass relationship it is necessary to extend the amount of accurate and reliable observational data. For this reason part of the thesis is devoted to the Alhambra Survey, which is a good example of the new deep surveys currently under development. These observational projects will detect thousands of new white dwarfs, some of them belonging to common proper motion pairs, which could be eventually used to extend our analysis. Thus, we have performed an exhaustive study to optimize the identification procedure of the white dwarf candidates which will be eventually detected by the Alhambra survey