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The birth rate of supernovae from double-degenerate and core-degenerate systems

We make a series of Monte Carlo simulations by a rapid binary evolution code and take two treatments of CE evolution, i.e. $α$-formalism and $γ$-algorithm. We find that only for the $α$-formalism with a high CE ejection efficiency, the shape of DTD from DD systems may be consistent with that derived observationally, i.e. a power low of $\sim t^{\rm -1}$, while the value of the birth rate of SNe Ia marginally matches with observations. For the $α$-formalism with a low CE ejection efficiency and the $γ$-algorithm, neither the shape of DTD nor the value of the birth rate can be compared with that from observations. Metallicity may not significantly change the shape of DTD, but a low metallicity may slightly increase the birth rate of SNe Ia by a factor of 2, especially for SNe Ia with long delay times. If the results from the single degenerate (SD) channel is incorporated into the DTD, both the shape of DTD and its value may be well consistent with observations for SNe Ia of younger than 2.5 Gyr, and SD and DD channels show a comparable contribution to total SNe Ia, while for SNe Ia with delay time longer than 2.5 Gyr, DD is the dominant channel and the birth rate is lower than that derived from observation by a factor up to $\sim4$. In addition, we calculate the evolution of various integral parameters of DD systems, and do not find any one suitable to explain the correlation between the brightness of SNe Ia and its delay time. Moreover, there are three channels producing core-degenerate(CD) systems which may contribute a few SNe Ia, but the contribution from CD systems to total SNe Ia is not more than 1%.