Detailed stellar evolution calculations have been performed to quantify the influence of the main-sequence mass MMS and the metallicity Z of the progenitor on the structure of the exploding white dwarf (WD), which are thought to be the progenitors of Type Ia supernovae (SNe Ia). In particular, we study the effects of progenitors on the brightness-decline relation M(ΔM15), which is a cornerstone for the use of SNe Ia as cosmological yardsticks. Both the typical MMS and Z can be expected to change as we go back in time. We consider the entire range of potential progenitors with 1.5-7 M☉ and metallicities between Z = 0.02 and 1 × 10-10. Our study is based on the delayed detonation scenario with specific parameters that give a good account of typical light curves and spectra. Based on the structures for the WD, detailed model calculations have been performed for the hydrodynamical explosion, nucleosynthesis, and light curves. The main-sequence mass has been identified as the decisive factor to change the energetics of the explosion and, consequently, dominates the variations in the rise-time-decline relation of light curves. MMS has little effect on the color index B-V. For similar decline rates ΔM15, the flux at maximum brightness relative to the flux on the radioactive tail decreases systematically with MMS by about 0.2m. This change goes along with a reduction of the photospheric expansion velocity vph by about 2000 km s-1. A change in the central density of the exploding WD has similar effects but produces the opposite dependency between the brightness-to-tail ratio and vph and therefore can be separated. The metallicity alters the isotopic composition of the outer layers of the ejecta. Selective line blanketing at short wavelengths decreases with Z and changes systematically the intrinsic color index B-V by up to -0.06m, and it alters the fluxes in the U band and the UV. The change in B-V is critical if extinction corrections are applied. The offset in the calibration of M(ΔM15) is not monotonic in Z and, in general, remains ≤0.07m. We use our results and recent observations to constrain the progenitors and to discuss evolutionary effects of SNe Ia with redshift. The narrow spread in the fiducial rise-time-decline relation in local SNe Ia restricts the range of main-sequence masses to a factor of 2. The upper limit of 1 day for the difference between the local and distance sample supports the need for a positive cosmological constant. The size of evolutionary effects is small (ΔM ≈ 0.2m) but is absolutely critical for the reconstruction of the cosmological equation of state.
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