The luminosity of distant galaxies

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Abstract
The usual expression for the apparent luminosity of a distance source is valid only if most of the matter content of the universe is intergalactic. If not, two corrections are needed: one, which leads to a decrease in luminosity, is due to the missing matter on the light path; the second, which brightens the source, is due to the gravitational focusing by galaxies lying near the line of sight. A general formula which takes into account these corrections is derived and applied to the relations between observables (apparent magnitude, apparent diameter, red shift and number count) with a power expansion in terms of the red shift z. It turns out that the first term, linear in z, is not affected; but the second contains as a parameter the ratio f of the intergalactic matter density to the total density. Precise measurements of the first two coefficients would lead, in principle, independently of the particular cosmological model chosen, to an independent determination of f. The focusing effect due to local inhomogeneities is-on average-of third order in the red shift.