Evolution of the number of accreting white dwarfs with shell nuclear burning and the SNe Ia rate

We analyze the time evolution of the number of accreting white dwarfs with surface shell hydrogen burning in semidetached and detached binaries. We consider the case where continuous star formation with a constant rate takes place in a stellar system over 10¹⁰ Gyr and the case of a starburst in which the same mass of stars is formed over 10⁹ Gyr. The evolution of the number of white dwarfs is compared with the evolution of the rate of events that are usually considered as SNe Ia and/or accretion-induced collapses, i.e., the accumulation of a Chandrasekhar mass by white dwarfs or the merger of white dwarf pairs with a total mass greater than or equal to the Chandrasekhar one. In stellar systems with a starburst, the supersoft X-ray sources observed at t = 10¹⁰ yr are most likely not the progenitors of SNe Ia. The same is true for a significant fraction of the sources in systems with a constant star formation rate. In both cases, the merger of white dwarfs is the dominant mechanism of SNe Ia. In symbiotic binaries, accreting CO dwarfs do not accumulate enough mass for an SNe Ia explosion, while ONeMg dwarfs finish their evolution by an accretion-induce collapse with the formation of a neutron star.