Is the metallicity of their hosts a good measure of the metallicity of
  Type Ia supernovae?
  The efficient use of Type Ia supernovae (SNIa) for cosmological studies
requires knowledge of any parameter that can affect their luminosity in either
systematic or statistical ways. Observational samples of SNIa commonly use the
metallicity of the host galaxy, Z_host, as an estimator of the supernova
progenitor metallicity, Z_Ia, that is one of the primary factors affecting SNIa
magnitude. Here, we present a theoretical study of the relationship between
Z_Ia and Z_host. We follow the chemical evolution of homogeneous galaxy models
together with the evolution of the supernova rates in order to evaluate the
metallicity distribution function, MDF(Delta Z), i.e. the probability that the
logarithm of the metallicity of a SNIa exploding now differs in less than Delta
Z from that of its host. We analyse several model galaxies aimed to represent
from active to passive galaxies, including dwarf galaxies prone to experience
supernova driven outflows. We analyse the sensitivity of the MDF to uncertain
ingredients: IMF, star-formation law, stellar lifetime, stellar yields, and
SNIa delay-time distribution. There is a remarkable degree of agreement between
the mean Z_Ia in a galaxy and its Z_host when they both are measured as the CNO
abundance, especially if the DTD peaks at small time delays, while the average
Fe abundance of host and SNIa may differ up to 0.4-0.6 dex in passive galaxies.
The dispersion of Z_Ia in active galaxy models is quite small, meaning that
Z_host is a quite good estimator of the supernova metallicity. Passive galaxies
present a larger dispersion, which is more pronounced in low mass galaxies. We
discuss the use of different metallicity indicators: Fe vs. O, and gas-phase
metallicity vs. stellar metallicity. The results of the application of our
formalism to a galactic catalogue (VESPA) are roughly consistent with our
theoretical estimates. (abridged)
