The nature of the Type Ia supernovae (SNIa) progenitors remains still uncertain. This is a major issue for galaxy evolution models since both chemical and energetic feedback play a major role in the gas dynamics, star formation and therefore in the overall stellar evolution. The progenitor models for the SNIa available in the literature propose different distributions for regulating the explosion times of these events. These functions are known as the Delay Time Distributions (DTDs). This work is the first one in a series of papers aiming at studying five different DTDs for SNIa. Here, we implement and analyse the Single Degenerate scenario (SD) in galaxies dominated by a rapid quenching of the star formation, displaying the majority of the stars concentrated in the bulge component. We find a good fit to both the present observed SNIa rates in spheroidal dominated galaxies, and to the [O/Fe] ratios shown by the bulge of the Milky Way. Additionally, the SD scenario is found to reproduce a correlation between the specific SNIa rate and the specific star formation rate, which closely resembles the observational trend, at variance with previous works. Our results suggest that SNIa observations in galaxies with very low and very high specific star formation rates can help to impose more stringent constraints on the DTDs and therefore on SNIa progenitors.
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