It is thought that type Ia supernovae (SNe Ia) are explosions of carbon-oxygen white dwarfs (CO WDs). Two main evolutionary channels are proposed for the WD to reach the critical density required for a thermonuclear explosion: the single degenerate scenario (SD), in which a CO WD accretes from a non-degenerate companion, and the double degenerate scenario (DD), in which two CO WDs merge. However, it remains difficult to reproduce the observed SN Ia rate with these two scenarios. With a binary population synthesis code we study the main evolutionary channels that lead to SNe Ia and we calculate the SN Ia rates and the associated delay time distributions. We find that the DD channel is the dominant formation channel for the longest delay times. The SD channel with helium-rich donors is the dominant channel at the shortest delay times. Our standard model rate is a factor five lower than the observed rate in galaxy clusters. We investigate the influence of ill-constrained aspects of single- and binary-star evolution and uncertain initial binary distributions on the rate of type Ia SNe. These distributions, as well as uncertainties in both helium star evolution and common envelope evolution, have the greatest influence on our calculated rates. Inefficient common envelope evolution increases the relative number of SD explosions such that for $alpha_{rm ce} = 0.2$ they dominate the SN Ia rate. Our highest rate is a factor three less than the galaxy-cluster SN Ia rate, but compatible with the rate determined in a field-galaxy dominated sample. If we assume unlimited accretion onto WDs, to maximize the number of SD explosions, our rate is compatible with the observed galaxy-cluster rate.
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