Matching expansion and light propagation at reduced matter density

We compare two cosmological descriptions with the same prescribed expansion history but different matter densities, and ask under which conditions they can also produce the same effective light-propagation response. Starting from the scalar-averaged cosmological equations, we determine the backreaction and averaged-curvature terms required to preserve the reference expansion. We then use an effective distance equation to derive the additional condition required for the same angular-diameter distance. For a reduced matter density, the missing Ricci-focusing contribution must be compensated by the specified averaged optical-shear term. We further examine whether this compensation can be continued to the observer. Under explicit regularity, normalization, averaging, and common-approach assumptions for the light bundles, the relevant averaged shear contribution vanishes at the reception event. If the matter-density difference remains positive there, the compensation condition therefore cannot hold all the way to the observer. The two distance functions may nevertheless agree in value and first derivative at reception; their local difference arises at higher order. The analysis distinguishes preserved expansion, preserved effective light propagation, and the stronger requirement of a common local realization. It combines established scalar-averaging and optical-propagation results in a single conditional comparison and identifies the precise assumptions under which the comparison succeeds and where its continuation fails.