Aims. We investigate the contribution of different formation scenarios for type Ia supernovae in elliptical galaxies. The single-degenerate scenario (a white dwarf accreting from a late main-sequence or red giant companion) is tested against the double-degenerate scenario (the spiral-in and merging of two white dwarfs through the emission of gravitational wave radiation). Methods. We use a population-number synthesis code that incorporates the latest physical results in binary evolution and allows us to differentiate between certain physical scenarios (e.g. description of common-envelope evolution) and evolutionary parameters (e.g. mass-transfer efficiency during Roche-lobe overflow). The obtained theoretical distributions of the delay times of type Ia supernovae are compared, both in morphological shape and in absolute number of events, to those which are observed. The critical dependency of these distributions on certain parameters is used to constrain the values of the latter. Results. We find that the single-degenerate scenario alone cannot explain the morphological shape of the observational delay-time distribution, while the double-degenerate scenario (or a combination of both) can. Most of these double-degenerate type Ia supernovae are created through a normal quasi-conservative Roche-lobe overflow followed by a common-envelope phase, not through two successive common-envelope phases. This may cast doubt on the use in other studies of analytical formalisms to determine delay times. In terms of absolute number, theoretical supernova Ia rates in old elliptical galaxies lie a factor of at least three below the observed ones. We propose a solution involving the effect of rotation on the evolution of intermediate mass binaries.