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The birth rate of subluminous and overluminous type Ia supernovae

Based on the results of Chen & Li (2009) and Pakmor et al. (2010), we carried out a series of binary population synthesis calculations and considered two treatment of common envelope (CE) evolution, i.e. $α$-formalism and $γ$-algorithm. We found that the evolution of birth rate of these peculiar SNe Ia is heavily dependent on how to treat the CE evolution. The over-luminous 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 sub-luminous SNe Ia from equal-mass DD systems is between 0.1 and 0.3 Gyr for $α$-formalism with $α=3.0$, while 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 sub-luminous SNe Ia from equal-mass DD systems may only account for no more than 1% of all SNe Ia observed. The super-Chandrasekhar mass model of Chen & Li (2009) may account for a part of 2003fg-like supernovae and the equal-mass DD model developed by Pakmor et al. (2010) 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 systems is the progenitor of 1991bg-like SNe Ia.