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In this paper, we want to check whether or not the carbon abundance can beaffected by initial metallicity. We calculated a series of stellar evolution.We found that when $Z\leq0.02$, the carbon abundance is almost independent ofmetallicity if it is plotted against the initial WD mass. However, when$Z>0.02$, the carbon abundance is not only a function of the initial WD mass,but also metallicity, i.e. for a given initial WD mass, the higher themetallicity, the lower the carbon abundance. Based on some previous studies,i.e. both a high metallicity and a low carbon abundance lead to a lowerproduction of $^{\rm 56}$Ni formed during SN Ia explosion, the effects of thecarbon abundance and the metallicity on the amount of $^{\rm 56}$Ni areenhanced by each other, which may account for the variation of maximumluminosity of SNe Ia, at least qualitatively. Considering that the centraldensity of WD before supernova explosion may also play a role on the productionof $^{\rm 56}$Ni and the carbon abundance, the metallicity and the centraldensity are all determined by the initial parameters of progenitor system, i.e.the initial WD mass, metallicity, orbital period and secondary mass, the amountof $^{\rm 56}$Ni might be a function of the initial parameters. Then, ourresults might construct a bridge linking the progenitor model and the explosionmodel of SNe Ia.

The correlation between C/O ratio, metallicity and the initial WD mass for SNe Ia