| Abstract
| - Context. Based on the single degenerate (SD) scenario, a super-Chandrasekhar mass model derived from the rapid rotation of a progenitor star may account for the overluminous type Ia supernovae (SNe Ia) like SN 2003fg. Previous authors calculated a series of binary evolution and showed the parameter spaces for the super-Chandrasekhar mass model. Another team developed an equal-mass double degenerate (DD) model to explain subluminous SNe Ia like SN 1991bg. But they did not show the evolution of the birth rate of these peculiar SNe Ia or compared them with absolute birth rates from observations. Aims. We aim to show the evolution of the birth rates of these peculiar SNe Ia based on the results of these other works, and compare the birth rates with observations to check whether these model may account for all the peculiar SNe Ia. Methods. We carried out a series of binary population synthesis calculations and considered two methods of common envelope (CE) evolution, i.e. α-formalism and γ-algorithm. Results. We found that the evolution of the birth rate of these peculiar SNe Ia heavily dependen on how the CE evolution is treated. The overluminous SNe Ia may only occur for α-formalism with low CE ejection efficiency, and the delay time of the SNe Ia is between 0.4 and 0.8 Gyr. The upper limit of the contribution rate of the supernovae to all SN Ia is less than 0.3%. The delay time of subluminous SNe Ia from equal-mass DD systems is between 0.1 and 0.3 Gyr for α-formalism with α = 3.0, but longer than 9 Gyr for α = 1.0. The range of the delay time for γ-algorithm is very wide, i.e. longer than 0.22 Gyr, even as long as 15 Gyr. The subluminous SNe Ia from equal-mass DD systems may only account for no more than 1% of all SNe Ia observed. Conclusions. The super-Chandrasekhar mass model may account for a part of the 2003fg-like supernovae and the equal-mass DD model may explain some 1991bg-like events, too. In addition, based on the comparison between theories and observations, including the birth rate and delay time of the 1991bg-like events, we found that the γ-algorithm is more likely to be an appropriate prescription of the CE evolution of DD systems than the α-formalism if the equal-mass DD system is the progenitor of 1991bg-like SNe Ia.
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