What caused the transition, and how to we know that this did not warp our calculations for the age of the early universe ?
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What caused the transition,[...]
As the universe expands, the number of particles per unit volume goes down. But in addition to this, photons suffer a cosmological red-shift. So the density of mass-energy due to nonrelativistic particles goes down, and the density due to photons goes down, the latter goes down faster. By extrapolation, we predict that photons dominated in the early universe.
and how to we know that this did not warp our calculations for the age of the early universe ?
Because the models have the correct physics built in. The transition from radiation-dominated to matter-dominated is a prediction of the models, not an assumption. We also have good reason to believe that this aspect of the models is correct, since we can empirically verify predictions of the models, such as big-bang nucleosynthesis, that are sensitive to the early-universe part.