Thanks to a stellar evolution code that is able to compute through the C flash we link the binary population synthesis of single degenerate progenitors of Type la supernovae (SNe la) to their physical condition at the time of ignition. We show that there is a large range of possible ignition densities and we detail how their probability distribution depends on the accretion properties. The low-density peak of this distribution qualitatively reminds of the clustering of the luminosities of Branch-normal SNe Ia. We tighten the possible range of initial physical conditions for explosion models: they form a one-parameter family, independent of the metallicity. We discuss how these results may be modified if we were to relax our hypothesis of a permanent Hachisu wind or if we were to include electron captures.