We look at galaxies that are billions of light years distant and draw conclusions based on the red shift of the light. How is the age of the light accounted for. We see the reality of a galaxy perhaps a billion years ago. Why not conclude that redshift is related to time instead of distance?

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    $\begingroup$ With respect for its venerable age, of course. $\endgroup$ – Lucian Jun 24 '19 at 11:22

The redshift you're referring to is a cosmological redshift, which is fundamentally different from a Doppler redshift. At the heart of it, it has nothing to do with neither time, nor distance. It is a consequence of light traveling through an expanding space, no more and no less.

Because space expands, the emitting source is carried away to a certain distance from us, so we can relate the redshift to a distance. And because it takes time for the light to reach us, we can relate the redshift to a time. But this requires a model for how the Universe has expanded through time, and how light travels.

The model that best describes our reality is the so-called standard cosmological model, or ΛCDM ("Lambda-CDM") model, which is a mathematically based, physically motivated, and observationally constrained model. It uses the so-called Friedmann–Lemaître–Robertson–Walker metric, which is a description of space itself, the Friedmann equations, which describes how space expands, and an equation of state, which describes the relation between pressure and density. The various parameters that enter the equations are then observed and constrained in a number of different ways.

With this model, we can relate the distance to an object, and its age, to the redshift we observe.

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