The cosmic star formation rate (CSFR), is an important clue to investigate the history of the assembly and evolution of galaxies. Here, we develop a method to study the CSFR from a purely theoretical point of view. Starting from detailed models of chemical evolution, we obtain the histories of star formation of galaxies of different morphological types. These histories are then used to determine the luminosity functions of the same galaxies by means of a spectro-photometric code. We obtain the CSFR under different hypothesis. First, we study the hypothesis of a pure luminosity evolution scenario, in which all galaxies are supposed to form at the same redshift and then evolve only in luminosity. Then we consider scenarios in which the number density or the slope of the LFs are assumed to vary with redshift. After comparison with available data we conclude that a pure luminosity evolution does not provide a good fit to the data, especially at very high redshift, although many uncertainties are still present in the data. On the other hand, a variation in the number density of ellipticals and spirals as a function of redshift can provide a better fit to the observed CSFR. We also explore cases of variable slope of the LFs with redshift and variations of number density and slope at the same time. We cannot find any of those cases which can improve the fit to the data respect to the solely number density variation. Finally, we compute the evolution of the average cosmic metallicity in galaxies with redshift.
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