Recent observations of Type Ia supernovae (SNe Ia) suggest that some of the progenitor white dwarfs (WDs) had masses up to 2.4-2.8 M ⊙, highly exceeding the Chandrasekhar mass limit. We present a new single degenerate model for SN Ia progenitors, in which the WD mass possibly reaches 2.3-2.7 M ⊙. Three binary evolution processes are incorporated: optically thick winds from mass-accreting WDs, mass stripping from the binary companion star by the WD winds, and WDs being supported by differential rotation. The WD mass can increase by accretion up to 2.3 (2.7) M ⊙ from the initial value of 1.1 (1.2) M ⊙, consistent with high-luminosity SNe Ia, such as SN 2003fg, SN 2006gz, SN 2007if, and SN 2009dc. There are three characteristic mass ranges of exploding WDs. In the extreme massive case, differentially rotating WDs explode as an SN Ia soon after the WD mass exceeds 2.4 M ⊙ because of a secular instability at T/|W| ∼0.14. For the mid-mass range of M WD = 1.5-2.4 M ⊙, it takes some time (spinning-down time) until carbon is ignited to induce an SN Ia explosion after the WD mass has reached maximum, because it needs a loss or redistribution of angular momentum. For the lower mass case of rigidly rotating WDs, M WD = 1.38-1.5 M ⊙, the spinning-down time depends on the timescale of angular momentum loss from the WD. The difference in the spinning-down time may produce the "prompt" and "tardy" components. We also suggest that the very bright super-Chandrasekhar mass SNe Ia are born in a low-metallicity environment.
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