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Galactic and Cosmic Type Ia SN rates: is it possible to impose constraints on SNIa progenitors?

We compute the Type Ia supernova rates in typical elliptical galaxies by varying the progenitor models for Type Ia supernovae. To do that a formalism which takes into account the delay distribution function (DTD) of the explosion times and a given star formation history is adopted. Then the chemical evolution for ellipticals with baryonic initial masses $10^{10}$, $10^{11}$ and $10^{12} M_{\odot}$ is computed, and the mass of Fe produced by each galaxy is precisely estimated. We also compute the expected Fe mass ejected by ellipticals in typical galaxy clusters (e.g. Coma and Virgo), under different assumptions about Type Ia SN progenitors. As a last step, we compute the cosmic Type Ia SN rate in an unitary volume of the Universe by adopting several cosmic star formation rates and compare it with the available and recent observational data. Unfortunately, no firm conclusions can be derived only from the cosmic SNIa rate, neither on SNIa progenitors nor on the cosmic star formation rate. Finally, by analysing all our results together, and by taking into account previous chemical evolution results, we try to constrain the best Type Ia progenitor model. We conclude that the best progenitor models for Type Ia SNe are still the single degenerate model, the double degenerate wide model, and the empirical bimodal model. All these models require the existence of prompt Type Ia supernovae, exploding in the first 100 Myr since the beginning of star formation, although their fraction should not exceed 15-20% in order to fit chemical abundances in galaxies.